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Magnetic Induction Tomography for
Imaging Cerebral Stroke
Massoud Zolgharni, BSc, MPhil
A thesis submitted to Swansea University in fulfilment of the
requirements for the Degree of
Doctor of Philosophy
Swansea University
School of Medicine
June 2010
ii
Summary
___________________________________
Magnetic induction tomography (MIT) is a relatively new contact-less soft-field
imaging modality which is used for imaging the internal distribution of the passive
electromagnetic properties of an object. The main purpose of this study is to
numerically investigate the feasibility of using MIT for imaging cerebral stroke in the
human brain. Stroke is a major cause of mortality and morbidity in Western society, and
conventional brain imaging techniques (e.g. MRI and CT) are expensive and,
sometimes, inaccessible for diagnosing stroke. It has been conjectured that MIT may
eventually be an attractive and low-cost alternative for rapid diagnostic imaging.
An anatomically realistic, multi-layer head geometry is used for the MIT simulations,
which is the first application of its kind in MIT. The finite element and finite difference
techniques are used for solving the forward problem and predicting the MIT signals. In
order to make the simulations more realistic, noise in the measuring hardware and two
examples of systematic errors are added to the simulated data. 3D images from the
conductivity distribution within the head are reconstructed by solving the ill-posed
inverse problem. To this end, a Tikhonov regularized minimization technique is used.
It is concluded that provided the noise in the hardware and the systematic errors in the
measuring protocol remain within a certain level, images of peripherally located strokes
may be reconstructed. However, apart from the unrealistic noise-free cases, imaging the
small volume strokes, located deep within the brain, is unfeasible. This problem may be
overcome by de-noising the MIT signals and employing other types of the inverse
solution techniques in the future. It is also shown that hemispherical MIT arrays
perform better than the conventional cylindrical arrays for imaging brain lesions.
iii
Declaration
This work has not previously been accepted in substance for any degree and is not being
concurrently submitted in candidature for any degree.
Signed .............................................................. (Massoud Zolgharni)
Date ................................................................
Statement 1
This thesis is the result of my own investigations, except where otherwise stated. Where
correction services have been used, the extent and nature of the correction is clearly
marked in a footnote(s).
Other sources are acknowledged by footnotes giving explicit references. A bibliography
is appended.
Signed .............................................................. (Massoud Zolgharni)
Date ................................................................
Statement 2
I hereby give consent for my thesis to be available for photocopying and for inter-
library loan, and for the title and summary to be made available to outside
organisations.
Signed .............................................................. (Massoud Zolgharni)
Date ................................................................
iv
To my parents
for their love and sacrifices...
v
Acknowledgements
___________________________________
First and foremost, I am deeply indebted to my experienced supervisors, Professor Huw
Griffiths from the School of Medicine and Dr. Paul D Ledger from the School of
Engineering, for their patience, guidance, advice, criticism, and insight throughout this
research.
I wish to thank all my colleagues at the South-Wales MIT group and the University of
Manchester with whom I had many fruitful discussions. I would also like to thank
Professor David S Holder for providing the UCL head mesh.
Critical reading and helpful suggestions by the PhD examiners, Professor Bill Lionheart
from the University of Manchester and Professor Kenneth Morgan from Swansea
University, is appreciated.
The financial support from the EPSRC and the ABM University Health Board is
gratefully acknowledged.
Special thanks to my friends with whom I shared my moments when there was no
forward or inverse problem to be solved.
My love to my family who have always encouraged and supported my efforts.
Finally, my deepest gratitude to the one, for the provision of joys, challenges, and grace
for growth.
vi
Contents
___________________________________
Part I: Preliminaries
Chapter 1. Introduction .......................................................................................... 1
1.1. MIT ......................................................................................................................... 1
1.2. MIT applications ..................................................................................................... 3
1.3. Motivation and objectives ...................................................................................... 3
1.4. Outline of the thesis ............................................................................................... 5
1.5. Contributions to knowledge ................................................................................... 6
1.6. LCOMIT ................................................................................................................... 8
Chapter 2. Principles of MIT .................................................................................... 9
2.1. Introduction ............................................................................................................ 9
2.2. Physical principles................................................................................................... 9
2.2.1. MIT problem ..................................................................................................................11
2.2.2. Eddy-current problem ....................................................................................................12
2.2.3. Solving the (forward) problem........................................................................................14
2.3. Working principles .................................................................................................14
2.3.1. The MIT signal ................................................................................................................16
2.3.2. SNR and operational frequency ......................................................................................17
2.4. MIT studies for brain imaging ................................................................................18
2.5. MK1 and MK2a systems .........................................................................................22
Chapter 3. Anatomy of Head and Stroke Pathology ............................................. 23
3.1. Introduction ...........................................................................................................23
3.2. Head anatomy .......................................................................................................23
3.2.1. Scalp ..............................................................................................................................24
3.2.2. Muscles .........................................................................................................................24
3.2.3. Skull ...............................................................................................................................25
3.2.4. Meninges .......................................................................................................................25
3.2.5. Brain ..............................................................................................................................27
3.2.6. Ventricles and cerebrospinal fluid ..................................................................................30
3.2.7. Spinal cord .....................................................................................................................30
3.2.8. Eyes and optic nerves .....................................................................................................31
3.2.9. Cavities ..........................................................................................................................32
3.2.10. Other tissues .............................................................................................................32
3.3. Stroke ....................................................................................................................33
vii
3.3.1. Classification of stroke ...................................................................................................33
3.3.2. Diagnosis and treatment ................................................................................................35
Chapter 4. Dielectric Properties of Head Tissues and Pathophysiology ................ 39
4.1. Introduction ...........................................................................................................39
4.2. Dielectric properties of head tissues ......................................................................40
4.2.1. Literature before 1996 ...................................................................................................41
4.2.2. The 1996 database .........................................................................................................41
4.2.3. Literature after 1996 ......................................................................................................43
4.3. Pathophysiology and conductivity change .............................................................59
4.4. Can MIT be applied for stroke imaging? ................................................................62
4.5. Adopted values for simulations and discussion .....................................................64
Part II: Discretized Forward and Inverse Problems
Chapter 5. Discrete Forward Problem ................................................................... 68
5.1. Introduction ...........................................................................................................68
5.1.1. BVP simplification ..........................................................................................................69
5.2. Finite element method ..........................................................................................71
5.2.1. FE discretization .............................................................................................................72
5.2.2. Comsol Multiphysics ......................................................................................................73
5.2.3. Model construction ........................................................................................................74
5.2.4. Solution technique .........................................................................................................76
5.3. Finite difference method .......................................................................................77
5.3.1. BVP ................................................................................................................................77
5.3.2. FD discretization ............................................................................................................79
5.3.3. Solution technique .........................................................................................................80
5.3.4. FD solver ........................................................................................................................81
5.4. Benchmark test ......................................................................................................82
5.4.1. FE vs. FD vs. Analytical....................................................................................................85
5.5. Screen implementation in FD model ......................................................................88
5.5.1. Screen for MK1 and MK2a systems.................................................................................89
5.5.2. Simulated vs. measured data .........................................................................................93
5.6. Discussion ..............................................................................................................96
Chapter 6. Discrete Inverse Problem ..................................................................... 99
6.1. Introduction ...........................................................................................................99
6.2. Time-difference imaging ......................................................................................101
6.2.1. Non-linearity ................................................................................................................ 101
6.2.2. Underdetermined ........................................................................................................ 103
6.2.3. Ill-posedness ................................................................................................................ 105
6.3. Sensitivity analysis ...............................................................................................106
viii
6.3.1. Sensitivity computation................................................................................................ 106
6.3.2. Ill-conditioning ............................................................................................................. 112
6.4. Truncated singular value decomposition .............................................................115
6.5. Tikhonov regularization .......................................................................................119
6.5.1. Standard Tikhonov ....................................................................................................... 119
6.5.2. Generalized Tikhonov................................................................................................... 122
6.5.3. Neighbouring matrix .................................................................................................... 123
6.6. Error estimation and regularization parameter ...................................................126
6.6.1. GCV ............................................................................................................................. 127
6.6.2. L-curve ......................................................................................................................... 129
6.7. Frequency-difference imaging .............................................................................131
6.8. Absolute imaging .................................................................................................134
6.8.1. Iterative non-linear solution ......................................................................................... 134
6.8.2. Newton-type methods ................................................................................................. 137
6.8.3. Absolute imaging using phase changes ......................................................................... 138
6.8.4. Three-layer geometry .................................................................................................. 147
6.8.5. Edge-preserving ........................................................................................................... 148
6.9. Imaging centrally located features.......................................................................150
6.9.1. Differential imaging...................................................................................................... 150
6.9.2. Absolute imaging ......................................................................................................... 152
6.10. Other iterative methods ......................................................................................153
6.11. Inverse crime .......................................................................................................154
6.12. Discussion ............................................................................................................155
Part III: Stroke Imaging
Chapter 7. Simulated Stroke Imaging Using the MK1 System ............................ 158
7.1. Introduction .........................................................................................................158
7.2. Head model .........................................................................................................158
7.2.1. UCL head mesh ............................................................................................................ 158
7.2.2. Stroke domains ............................................................................................................ 160
7.2.3. FE mesh construction ................................................................................................... 160
7.2.4. FD mesh construction .................................................................................................. 163
7.2.5. Dielectric properties..................................................................................................... 163
7.3. Forward problem .................................................................................................164
7.3.1. FE mesh convergence................................................................................................... 164
7.3.2. MIT signals (FE results) ................................................................................................. 167
7.3.3. FD vs. FE ...................................................................................................................... 169
7.4. Time-difference imaging ......................................................................................172
7.5. Frequency-difference imaging .............................................................................182
7.6. Specific absorption rate .......................................................................................184
ix
7.7. Discussion ............................................................................................................187
Chapter 8. Customized Coil Arrays, Systematic Errors, and the MK2b System.... 189
8.1. Introduction .........................................................................................................189
8.2. Forward and inverse problems ............................................................................191
8.3. Hemispherical and annular arrays .......................................................................192
8.3.1. Hemispherical (Helmet) array ....................................................................................... 192
8.3.2. Annular array ............................................................................................................... 193
8.3.3. Forward problem ......................................................................................................... 193
8.3.4. Sensitivity analysis ....................................................................................................... 194
8.3.5. Image display ............................................................................................................... 196
8.3.6. Image reconstruction ................................................................................................... 197
8.4. Simulated errors ..................................................................................................203
8.4.1. Phase noise .................................................................................................................. 203
8.4.2. Systematic errors ......................................................................................................... 205
8.5. MK2b system .......................................................................................................210
8.5.1. Concept of MK2b array ................................................................................................ 210
8.5.2. Dimensions .................................................................................................................. 211
8.5.3. Number of coils............................................................................................................ 212
8.5.4. Other coil arrangements and quality measurement ...................................................... 214
8.5.5. MK2b construction....................................................................................................... 219
8.6. Discussion ............................................................................................................220
Part IV: Experiments and Conclusions
Chapter 9. Experiments ...................................................................................... 224
9.1. Introduction .........................................................................................................224
9.2. Experimental setup ..............................................................................................224
9.3. Simulations ..........................................................................................................227
9.4. Signal analysis ......................................................................................................227
9.5. Image reconstruction ...........................................................................................230
9.5.1. Differential imaging...................................................................................................... 231
9.5.2. Absolute imaging ......................................................................................................... 234
9.6. Further discussion ................................................................................................235
9.6.1. Discretization error ...................................................................................................... 237
9.6.2. Phase noise .................................................................................................................. 238
9.6.3. Conductivity value error ............................................................................................... 238
9.6.4. Systematic errors ......................................................................................................... 240
Chapter 10. Conclusions and Future Work ............................................................ 243
10.1. Introduction .........................................................................................................243
10.2. Remarks and conclusions .....................................................................................243
x
10.2.1. Chapter 1 ................................................................................................................ 243
10.2.2. Chapter 2 ................................................................................................................ 243
10.2.3. Chapter 3 ................................................................................................................ 244
10.2.4. Chapter 4 ................................................................................................................ 244
10.2.5. Chapter 5 ................................................................................................................ 244
10.2.6. Chapter 6 ................................................................................................................ 244
10.2.7. Chapter 7 ................................................................................................................ 245
10.2.8. Chapter 8 ................................................................................................................ 246
10.2.9. Chapter 9 ................................................................................................................ 247
10.3. Discussion and future work .................................................................................247
10.3.1. Immediate follow up research ................................................................................. 247
10.3.2. Forward and inverse problems ................................................................................ 249
10.3.3. Practical measurements and systematic errors ........................................................ 250
10.3.4. Computational vs. Measurement Accuracy .............................................................. 251
10.3.5. Sensor array design ................................................................................................. 253
10.3.6. Image resolution ..................................................................................................... 254
Part V: Appendices and Bibliography
Appendix A. System Description ............................................................................... 257
A.1. MK1 system ..............................................................................................................257
A.2. MK2a system.............................................................................................................258
Appendix B. Dielectric Theory and Terminology ....................................................... 260
Appendix C. Supplements to chapter 6 ..................................................................... 265
C.1. GCV method ..............................................................................................................265
C.2. Steepest descent .......................................................................................................267
C.3. Landweber method ...................................................................................................267
C.4. Conjugate gradient method.......................................................................................268
C.5. ART and SIRT .............................................................................................................268
C.6. Noise .........................................................................................................................269
References 270
xi
List of tables
___________________________________
Table 3.1. Performance of CT and MRI for diagnosing stroke; values in percentage [83]. .......................36
Table 4.1. Weighted average dielectric properties assigned to heterogeneous or minor organs in the
head (frequency of 10 MHz).........................................................................................................65
Table 6.1. Conductivities (in S m-1) of white matter and blood at 1 and 10 MHz, and conductivity change
between two frequencies. ......................................................................................................... 133
Table 7.1. Conductivity (S m-1) and relative permittivity values assigned to each tissue type at the
frequencies of 1 and 10 MHz. The mass density of tissues (kg m-3), adopted from [205], are to be
used for the power absorption calculations later in section 7.6. a filled with air. ......................... 164
Table 7.2. Series of meshes for the convergence tests; NS: not solved due to memory restrictions. ..... 165
Table 7.3. Series of FD meshes for the convergence tests. ................................................................... 170
xii
List of illustrations
___________________________________
Figure 2.1. Schematic view of boundary value problem in MIT; C: conductive domain, N: non-
conducting region (free space). ....................................................................................................12
Figure 2.2. Block diagram of a typical MIT system (after Peyton et al [30]). ............................................14
Figure 2.3. Phasor diagram representing primary (Vp) and secondary ( V) voltages. ..............................16
Figure 3.1. Skeletal muscles: (a) structure of a skeletal muscle fibre (after Saladin [66]), (b) muscles of
head and neck (after Mader [65]). ...............................................................................................24
Figure 3.2. (a) simplified side view of human skull (after Villarreal [67]). (b) illustration of a typical long
bone showing the location of cortical and cancellous bones (picture adapted from [68]); skull
bones have a similar internal structure. .......................................................................................25
Figure 3.3. Meninges; (a) meningeal membranes surrounding brain, (b) frontal section of head to show
the meninges (after Seely [71]). ...................................................................................................26
Figure 3.4. Human brain; cerebrum, seen here in longitudinal section, is the largest part of brain in
humans (after Mader [65])...........................................................................................................27
Figure 3.5. Cerebrum: (a) lateral view of right hemisphere with insula exposed; colours in this figure
distinguish the lobes of cerebral hemispheres (after Shier et al [72]), (b) coronal section of brain
showing cortex, white matter, and corpus callosum (after Scanlon and Sanders [73]). ..................28
Figure 3.6. Illustration of the brainstem, diencephalon and cerebellum (after Shier et al [72]). ..............29
Figure 3.7. Spinal cord: (a) illustration of entire length, (b) drawing of a segment (after Seely [71])........31
Figure 3.8. Illustration of eye balls and optic nerves (after Mader [65]). .................................................31
Figure 3.9. Air-filled cavities in head; (a) nasal cavity, (b) ear canal (after Mader [65]). ...........................32
Figure 3.10. Stroke; ischaemia due to embolism (a) and thrombosis (b), haemorrhage due to rupture of a
blood vessel (c) (after Broderick and Santilli [77]). ........................................................................34
Figure 3.11. Diagram showing different types of haemorrhage based on its location within head. .........34
Figure 3.12. Examples of intra- and extra-axial haemorrhages (after Caplan [80]). .................................35
Figure 3.13. Stroke images: (a) CT scan showing a large cerebral infarct, (b) MRI scan showing a left
cerebral infarct, (c) CT scan showing a haemorrhage in deep portion of brain, (d) MRI scan showing
a brain haemorrhage similar to that in (c) (after Caplan [79]). ......................................................37
Figure 3.14. CT scan of a haemorrhage worsened by thrombolytic therapy (after Harold and Adams [85]).
....................................................................................................................................................38
Figure 4.1. Dielectric properties of muscle against frequency; longitudinal and transverse directions. ...52
Figure 4.2. Dielectric properties of grey matter against frequency. ........................................................53
Figure 4.3. Dielectric properties of white matter against frequency. ......................................................54
Figure 4.4. Dielectric properties of cancellous bone against frequency. .................................................55
Figure 4.5. Dielectric properties of cortical bone against frequency. ......................................................56
Figure 4.6. Dielectric properties of CSF against frequency. .....................................................................57
Figure 4.7. Dielectric properties of blood against frequency. .................................................................58
Figure 4.8. Current flow through cells at both low and high frequencies. Due to ischaemia and
subsequent oedema, brain cells swell resulting in impedance increase at low frequencies. ECS:
extra-cellular space (low-frequency and high-frequency diagrams are redrawn from Holder [116]
and Holder [118], respectively). ...................................................................................................59
Figure 4.9. Current flow through cells. Due to haemorrhage, increase in conductive blood volume results
in impedance decrease. BV: blood volume (diagram redrawn from Holder [118]). ........................61
xiii
Figure 4.10. CT scan showing haemorrhage growth in a patient with basal ganglion intra-cerebral
haemorrhage (after Adeoye and Broderick [122]). .......................................................................62
Figure 4.11. Dielectric properties of head tissues used for simulations. .................................................66
Figure 5.1. A 2D representation of the grid showing the electrical connections between a node and its 4
neighbours in the grid (6 neighbours in 3D). The admittances Y and vector voltage generator
components e are known [153]. ...................................................................................................79
Figure 5.2. Exciter coil and conductive sphere. ......................................................................................82
Figure 5.3. Typical FE meshes generated for benchmark test; a coarse (a), and a fine (b) mesh. Note that
the presence of coil, sphere, and layer of infinite elements (separate subdomains) may result in
generation of non-uniform meshes, e.g. in (a). .............................................................................83
Figure 5.4. Error in the numerically computed eddy current inside the conducting sphere versus the
number of degrees of freedom; results for first- and second-order elements plotted in broken red
and solid blue lines, respectively. .................................................................................................84
Figure 5.5. Cross section of sphere showing r. .......................................................................................85
Figure 5.6. Plots of current density |J| along r for = 1 S m-1 (b) and = 100 S m-1 (c). ..........................86
Figure 5.7. FE results: real, imaginary, and norm of current density along r for = 1 S m-1 (left) and =
100 S m
-1 (right). ..........................................................................................................................87
Figure 5.8. Maps of induced current densities in A m
-2 on a cross-section of sphere for FE results when
= 1 and 100 S m-1. ........................................................................................................................87
Figure 5.9. FE models for computing A-field for MK1 and MK2a systems. ..............................................90
Figure 5.10. Plots of |A| for 4 meshes along a test line. .........................................................................91
Figure 5.11. Induced secondary voltage at 16 sensor coils for Ex1 activated (FE mesh comprised 108,811
tetrahedral elements). .................................................................................................................92
Figure 5.12. Plan view of the MIT array showing the position of the sample and coils within the
electromagnetic screen. Red (outer) and black (inner) coils are exciters and sensors, respectively.
....................................................................................................................................................93
Figure 5.13. Values of Im( V/Vp) in percentage against conductivity: (a) FE and FD results, (b) measured
data where straight fitted line is also shown (after Griffiths et al [8]). ...........................................94
Figure 5.14. Values of Re( V/Vp) in percentage against conductivity: (a) FE and FD results, (b) measured
data (after Griffiths et al [8]). .......................................................................................................95
Figure 5.15. Total values of conduction and displacement current densities induced throughout the
cylindrical vessel. .........................................................................................................................96
Figure 6.1. One geometrical example of the model (to scale): (a) diagram of the 3D model, (b) x-y and y-z
cross-sectional planes for image representations. ...................................................................... 100
Figure 6.2. Diagram illustrating the conductivity distributions for two states in x-y planes. .................. 101
Figure 6.3. Diagram illustrating the conductivity distribution for two inhomogeneous states. .............. 103
Figure 6.4. Perturbation method for sensitivity calculation; highlighted voxels represent the perturbed
elements (pixels). ...................................................................................................................... 106
Figure 6.5. Comparison of sensitivity matrices computed based on perturbation and E.E methods.
Percentage values correspond to conductivity change in the perturbed voxel with respect to the
background. .............................................................................................................................. 109
Figure 6.6. Comparison of non-linear and linear models; is contrast between added inclusion and
background. .............................................................................................................................. 111
Figure 6.7. Comparison of non-linear and linear models; is contrast between added inclusion and
background. .............................................................................................................................. 111
Figure 6.8. Singular spectrum plotted on a logarithmic scale, normalized to the largest singular value.
Also shown some arbitrary singular vector maps; index = 1, 3, 60, 136, 223. .............................. 114
Figure 6.9. Levels of truncation of singular values. ............................................................................... 116
xiv
Figure 6.10. Reconstructed images using TSVD; (a) columns from left: target and images with different
levels of truncation, rows from top: x-y and y-z cross-sectional planes, colour-map shows
conductivity change in S m-1
, (b) 1D representation of target and reconstructed conductivity
changes along a line in x-y plane. ............................................................................................... 117
Figure 6.11. As figure 6.10, but with 60 dB noise added to simulated data. .......................................... 119
Figure 6.12. Modified singular values (si + /si) for 3 different levels of regularization. ......................... 121
Figure 6.13. Reconstructed images using standard Tikhonov regularization with 60 dB noise added to
simulated data. Columns from left: target and images with different levels of regularization
indicated by value of . Rows from top: x-y and y-z cross-sectional planes. Colour-map shows
conductivity change in S m-1. ...................................................................................................... 122
Figure 6.14. Correlation between neighbouring elements with common face (a), common node (b), and
a combination of faces and edges (c) in a 2D image. Values shown in voxels correspond to a central
finite difference approximation of Laplacian operator. ............................................................... 123
Figure 6.15. As in figure 6.13, but reconstructed using (6.25) with R-face and xref = 0. .......................... 124
Figure 6.16. Comparison of standard and generalized Tikhonov: 1D plots showing reconstructed
conductivity changes ( = 1 10-18) along cross-sectional lines in y-direction (a) and x-direction (b).
.................................................................................................................................................. 125
Figure 6.17. Regularization parameter selection using GCV method for three noise levels. .................. 128
Figure 6.18. Reconstructed images by standard Tikhonov method and using suggested value of by GCV
method. Colour-bar in S m-1. ...................................................................................................... 129
Figure 6.19. Plots of L-curve for selecting the regularization parameter for three noise levels. ............. 130
Figure 6.20. Regularization parameter selected by GCV (a) and L-curve (b) methods for three noise levels
(40, 50, and 60 dB). .................................................................................................................... 131
Figure 6.21. Frequency-difference imaging; (a): target, (b) and (c): reconstructed images using
generalized Tikhonov method (1 m added phase noise, R-face, xref = 0, = 1 10-9), but plotted
with different colour-bars in S m-1. ............................................................................................. 133
Figure 6.22. Flowchart for iterative non-linear MIT image reconstruction. ........................................... 136
Figure 6.23. Simulated target; coil array is placed at three levels along height of cylindrical tank. Colour-
bar in S m-1................................................................................................................................. 138
Figure 6.24. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1
, l = 0.5 and 1 = 1 10-8
. First row:
plots of ||rk||2 and ||ek||2 for iteration procedure. Second row: same plots as in first row but
magnified at iteration 52. Also shown reconstructed images at iterations 52, 53, and 54. Upper
limit of colour-bars (S m-1) is set to maximum conductivity value for each image. ....................... 140
Figure 6.25. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1, l = 0.5 and 1 = 1.8 10-20. Errors
||rk||2 and ||ek||
2 are plotted against iterations. Reconstructed images correspond to iterations
1, 2, and 8. Colour-bar (S m-1) limits are set to maximum conductivity for each image. ............... 142
Figure 6.26. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1, l = 0.1 and 1 = 1 10-8. Errors ||rk||2
and ||ek||2 are plotted against iterations. Reconstructed images correspond to iterations 10, 18,
and 19. Colour-bar (S m-1) limits are set to maximum conductivity value in each image. ............. 143
Figure 6.27. Plots of ||rk||2 and ||ek||2 for absolute imaging using (6.39) with R-face, 0 = 1 S m-1, l = 0.9
and 1 = 1 10-8. ......................................................................................................................... 144
Figure 6.28. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1, l = 0.5 and 1 = 1 10-8. with 1 m
additive phase noise. Errors ||rk||2 and ||ek||2 are plotted against iterations. Reconstructed
images correspond to iterations 7, 10, and 14. Colour-bar (S m-1) limits are set to maximum
conductivity value in each image................................................................................................ 145
Figure 6.29. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1
, l = 0.5 and 1 = 1 10-8
. with 10 m
additive noise. Errors ||rk||2 and ||ek||2 are plotted against iterations. Reconstructed images
correspond to iterations 3, 6, and 7. Colour-bar (S m
-1) limits are set to maximum conductivity
value in each image. .................................................................................................................. 146
xv
Figure 6.30. Absolute imaging of a 3-layer model including CSF, white matter, and blood. Colour-bar in S
m-1............................................................................................................................................. 147
Figure 6.31. Surface plots of x-y plane in images for noise free (upper row) and 5 m noise (lower row).
Left column is for images reconstructed using R-face, and right column is for edge-preserving
regularization matrix.................................................................................................................. 149
Figure 6.32. Time-difference imaging. Left: target, Middle and Right: reconstructed images using
generalized Tikhonov method (R-face, xref = 0) and suggested by L-curve and GCV methods,
respectively. Colour-bars in S m
-1. .............................................................................................. 151
Figure 6.33. Frequency-difference imaging. Left: target, Middle and Right: reconstructed images using
generalized Tikhonov method (R-face, xref
=
0) and different values. Colour-bars in S
m
-1. ......... 151
Figure 6.34. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1
, l = 0.5 and 1 = 1 10-8
, using noise-
free data and with 1 m additive noise. Colour-bar (S m
-1) limits are set to maximum conductivity
value in each image. .................................................................................................................. 152
Figure 7.1. UCL head mesh, triangular surface elements of skin and internal compartments. ............... 159
Figure 7.2. Transverse planes showing internal tissues separated by different colours. ........................ 159
Figure 7.3. Stroke regions in the right hemisphere of brain; side view represents the vertical level of the
LP which was similar for all three strokes. .................................................................................. 160
Figure 7.4. FE mesh construction; details given in text. ........................................................................ 162
Figure 7.5. Level of coil array relative to head and LP stroke (red = exciter, blue = sensor). .................. 162
Figure 7.6. FD mesh with two cuts illustrating the internal structure, including the LP stroke. .............. 163
Figure 7.7. Convergence of the induced voltage, for activated Ex1, in sensors Se6, Se9, and Se13 for first-
and second-order elements plotted in broken red and solid blue lines, respectively. .................. 166
Figure 7.8. Typical results. (a) Projection on the transverse plane of magnetic field lines with Ex13
activated with a current of 1 A. The colour of a field line indicates the magnitude of the magnetic
field at that point; the colour-bar indicates the field strength in A m−1. (b) Map of induced current
density (A m−2) in a transverse slice through the normal brain, 25 mm above the centre of the coils.
(c) as (b) but with the LP stroke present. .................................................................................... 166
Figure 7.9. Profiles of for three strokes, simulated at frequency of 10 MHz and with Ex1 activated.
Also shown is the phase angle for normal head relative to empty space (values are divided by 20
for convenience of display). ....................................................................................................... 167
Figure 7.10. The 256 values of magnitude of sorted in descending order for each of three strokes,
simulated at frequency of 10 MHz. ............................................................................................ 168
Figure 7.11. The 256 values of magnitude of sorted in descending order for LP stroke with different
blood percentages. .................................................................................................................... 169
Figure 7.12. Secondary voltage in 16 sensors for Ex1 activated, in normal head at frequency of 10 MHz.
Numbers 7.5, 6, 5, 4, 3, and 2.5 indicate grid sizes used in FD model. Converged FE results are
plotted in broken line for comparison. ....................................................................................... 171
Figure 7.13. Discrepancy between FE and FD (2.5 mm grid size) results in figure 7.12. ......................... 171
Figure 7.14. Regularization parameter selection using GCV method for three phase noise levels (0, 1 m ,
17 m ) added to phase changes for LP stroke. ............................................................................ 174
Figure 7.15. Columns left to right: simulated LP target, and reconstructed images with 0, 1 m , 17 m
additive noise. Rows from top to bottom: transverse, coronal, and sagittal cross-section planes.
Colour-bars in S m
-1. is selected by GCV method. ..................................................................... 174
Figure 7.16. Regularization parameter selection using L-curve method for three phase noise levels (0, 1
m , 17 m ) added to phase changes for LP stroke....................................................................... 176
Figure 7.17. As figure 7.15, but selected by L-curve method. ............................................................ 176
Figure 7.18. Reconstructed images of LP stroke with 1 m phase noise added to phase changes, and for
different values of . Upper limit in each colour-bar (S m-1) is set to maximum value in the
corresponding image. ................................................................................................................ 177
xvi
Figure 7.19. As figure 7.15, but selected empirically. ........................................................................ 179
Figure 7.20. As figure 7.19, but for SP stroke. ...................................................................................... 180
Figure 7.21. As figure 7.19, but for SD stroke. ...................................................................................... 180
Figure 7.22. (a): profiles of noise-free for SD stroke (solid blue line) and those with 1 m added phase
noise (broken red line). (b): difference between two plots in (a). ................................................ 181
Figure 7.23. Regularization parameter selection using GCV and L-curve method for noise-free frequency-
difference data for LP stroke. ..................................................................................................... 182
Figure 7.24. Columns left to right: simulated LP target, and reconstructed images from noise-free data
with selected using GCV and L-curve methods. Rows from top to bottom: transverse, coronal,
and sagittal cross-section planes. Colour-bars in S m
-1. ............................................................... 183
Figure 7.25. SAR distribution in normal head for Ex1 activated at 10 MHz; (a) skin surface, (b) slice at
through centres of coils. Colour-bar indicates logarithm to the base 10 of value in W kg-1........... 185
Figure 8.1. Helmet MIT arrays: (a) side view of hemispheres H100 and H120 in position over head, (b)
coil positions on hemisphere H100, magnified in relation to (a). Angles of elevation of coil centres
in each ring are indicated with respect to lowest ring. Each of first three rows from bottom
includes 16 coils, and the upper row includes 8 coils. Exciters and sensors are shown as red and
black coils, respectively. Origin of coordinate system is defined as centre of helmet. .................. 192
Figure 8.2. Annular MIT array. Three levels of array for each plane of measurement are indicated as Z1,
Z2, and Z3 which are level with first lower rows of H100 (see figure 8.1b). Two pairs of coils for
each plane of measurement are shown. Exciters and sensors are shown as red and black coils,
respectively. Coordinate system is that in Helmet arrays. ........................................................... 193
Figure 8.3. Singular spectrum for three MIT arrays, normalised to the largest singular value and plotted
on a logarithmic scale. ............................................................................................................... 194
Figure 8.4. Total sensitivity plots for three MIT arrays. Rows from top to bottom: transverse, coronal,
and sagittal cross-sections. Colour-bar in radian.m.S-1. ............................................................... 195
Figure 8.5. A reconstructed image of LP stroke, shown with different colour-bars in S m-1. The values
given above each image are upper limits in the colour-bars. ...................................................... 196
Figure 8.6. Integration volumes, shown as blue regions, for computing Ehead, Ebrain, and Eupper. ............. 197
Figure 8.7. Error Ehead in reconstructed images plotted in percentage as a function of , for three MIT
arrays. ....................................................................................................................................... 198
Figure 8.8. Columns left to right: simulated LP target, and reconstructed images from phase changes
with 1 m added noise for annular (three planes), H100, H120, and annular (with one plane) MIT
arrays; was optimised based on Ehead. Maximum conductivity change occurred in each images is
given as ( )max. Rows from top to bottom: transverse, coronal, and sagittal planes. Colour-bar in
S m
-1. ......................................................................................................................................... 199
Figure 8.9. As figure 8.7, but for Eupper (a) and Ehead (b). ........................................................................ 200
Figure 8.10. As figure 8.8, but reconstructed using values of minimising Eupper. Also fifth column from
left shows image reconstructed for H100 and using a high level of regularization. ...................... 201
Figure 8.11. Columns left to right: simulated SP target, reconstructed SP image, simulated SD target,
reconstructed SD image, simulated normal brain, reconstructed normal image. Rows from top to
bottom: transverse, coronal, and sagittal cross-sections. Images reconstructed using phase
changes with 1 m added noise to H100 data, and optimised for Ebrain. Colour-bar in S m
-1. ...... 202
Figure 8.12. Value of error, Enoise, in reconstructed images using H100 (λ = 2.5x10-9) for different levels of
added Gaussian phase noise (standard deviation). Three simulations (different colours) for each
noise level are shown along with the fitted straight line. ............................................................ 204
Figure 8.13. Columns left to right: simulated LP target, and reconstructed images for H100 with different
levels of added phase noise ( = 2.5 10-9). Rows from top to bottom: transverse, coronal, and
sagittal cross-sections. Colour-bar in S m
-1. ................................................................................. 204
xvii
Figure 8.14. Exaggerated geometrical errors: light pink and dark blue colours show the 'true' and
'erroneous' boundaries, respectively. (a) displacement in y-direction. (b) contraction. ............... 206
Figure 8.15. Columns left to right: simulated LP target, and reconstructed images with different
boundary errors in x-direction. Rows from top to bottom: transverse, coronal, and sagittal planes.
.................................................................................................................................................. 207
Figure 8.16. As figure 8.15, but for y- and z-direction displacements.................................................... 207
Figure 8.17. As figure 8.15, but for contraction and expansion scaling errors. ...................................... 208
Figure 8.18. Image error, Eboundary, for different displacements of head or percentage scaling. ............. 209
Figure 8.19. Conceptual diagram of MK2b system (not to scale). ......................................................... 211
Figure 8.20. Standard measurement indexes for head (after ARTUR [218]). ......................................... 211
Figure 8.21. Top view of three coil arrays; red and black coils are exciters and sensor coils, respectively.
.................................................................................................................................................. 212
Figure 8.22. Columns left to right: simulated LP target, and reconstructed images for 30-coil, 46-coil, 60-
coil, and 46-coil+screen arrays; reconstructed with 1 m noise, and optimised for Ehead. Rows
from top to bottom: transverse, coronal, and sagittal cross-sections. Colour-bar in Sm-1
. ........... 213
Figure 8.23. Examples of separated coil arrangements favoured from the point of view of the electronics
layout; red coils are exciters....................................................................................................... 215
Figure 8.24. Columns left to right: simulated LP target, and images reconstructed from noise-free data
and using different values of for asymmetric arrangement shown in figure 8.23 (left array). Rows
from top to bottom: transverse, coronal, and sagittal cross-sections. Colour-bar in Sm-1. ........... 215
Figure 8.25. Six selected coil arrangements for the 46 coil configuration; red coils are exciters. D6 is the
arrangement used previously in figure 8.21 (middle array). ........................................................ 216
Figure 8.26. Quality measures for 6 designs, D1-D6, normalized to Qm = 100128. ................................ 217
Figure 8.27. Columns left to right: reconstructed images for D1 to D6 designs, respectively, where was
optimised for Ehead. Rows from top to bottom: transverse, coronal, and sagittal cross-sections.
Colour-bar in S m-1 with upper limit of 0.23. ............................................................................... 218
Figure 8.28. As figure 8.27, but with 1 m added noise. ....................................................................... 219
Figure 8.29. Photographs of MK2b system at early stages of construction. Left: screen and coil formers
before mounting. Right: array (screen, formers, plastic shell) prior to adding the electronics and
coil windings. ............................................................................................................................. 220
Figure 9.1. Conceptual diagram of targets in case-A and case-B. Upper row shows x-y cross-sections, and
lower row shows vertical cross-sections along broken lines shown in upper row. Distances show
displacement of inclusion relative to centre of tank. .................................................................. 225
Figure 9.2. Photographs of experimental setup showing inclusion within tank (a) and tank within MK2a
array (b). ................................................................................................................................... 226
Figure 9.3. Planes of measurement: (a) levels of 9 planes (P1-P9) relative to tank, (b) & (c) highest and
lowers positions of tank within array, respectively. .................................................................... 226
Figure 9.4. Phase changes in degrees for uniform tank: (a) simulated data for tank of 19.2 cm diameter,
(b) measured data for tank of 19.5 cm diameter, (c) difference between (a) and (b). .................. 228
Figure 9.5. Phase differences caused by addition of inclusion for case-A, relative to uniform tank: (a)
simulated, (b) measured. ........................................................................................................... 229
Figure 9.6. As figure 9.5, but for case-B. .............................................................................................. 230
Figure 9.7. Transverse slices cutting through the tank used for image representations. ....................... 231
Figure 9.8. Case-A: Columns from left: level of cross-section along tank, target on 1 cm grid, differential
images from simulated, measured data. .................................................................................... 232
Figure 9.9. As figure 9.8, but for case-B. .............................................................................................. 233
Figure 9.10. Absolute imaging of uniform tank of 0.205 S m-1 conductivity. Transverse slides cutting
halfway up the tank. Rows from top to bottom: images reconstructed from noiseless simulated
xviii
data, simulated data with 3 m additive noise, and measured data, respectively. Colour-bar in S m-
1. ............................................................................................................................................... 235
Figure 9.11. Comparison of measured and simulated phase changes for uniform tank of 0.205 S m-1
conductivity. Blue, red, and green colours represent Ex1-Se8, Ex2-Se9, and Ex12-Se6 coil pairs,
respectively. For each coil pair (each colour), 9 points are shown which correspond to 9 planes of
measurement for that particular coil pair. The line of identity is also drawn. .............................. 236
Figure 9.12. Discrepancy between the FD and FE phase changes for the uniform tank and middle plane
of measurement. ....................................................................................................................... 237
Figure 9.13. Discrepancy between the noise-free and noisy simulated phase changes for the uniform
tank. .......................................................................................................................................... 238
Figure 9.14. Values of r.m.s. for the phase error as a function of conductivity; phase error is computed as
the discrepancy between the measured and scaled simulated phase changes. ........................... 239
Figure 9.15. Discrepancy between the measured phase changes for the uniform tank and the simulated
ones after the conductivity adjustment. ..................................................................................... 240
Figure 9.16. Discrepancy between the simulated phase changes for the uniform with adjusted
conductivity when placed at the centre of sensor array and when displaced for 2.8 mm. ........... 241
Figure A.1. An image of frond-end of the 16-channel MK1 system [33]. ............................................... 257
Figure A.2. Diagram of the MK1 configuration; dimensions in mm: (a) the coil array inside the
electromagnetic screen; labels represent the coil numbering, (b) detail of an individual coil
module; the coil windings are shown in black (sensor) and red (exciter). .................................... 258
Figure A.3. An image of frond-end of the 14-channel MK2a system. .................................................... 259
Figure A.4. Geometrical specifications of the system; dimensions in mm: (a) exciter and sensor coils sit
on red and blue broken lines, respectively, (b) details of an individual module. .......................... 259
Figure B.1. Normalized permittivity ( ' - )/( s - ), loss factor ''/( s - ), and conductivity 0 ''/( s -
) for a single time constant relaxation plotted against f/fr (after Barnes and Greenebaum [91]).
.................................................................................................................................................. 262
Figure B.2. Dispersion regions , , and (after Grimnes and Martinsen [113]). .................................. 264
xix
Abbreviations
___________________________________
A&E Accident and emergency
ABC Absorbing boundary conditions
ART Algebraic reconstruction technique
BVP Boundary value problem
CG Conjugate gradient
CNS Central nervous system
CSF Cerebrospinal fluid
CT Computed tomography
CVA Cerebro-vascular accident
ECT Electrical capacitance tomography
EIT Electrical impedance tomography
FD Finite difference
FDM Finite difference method
FDTD Finite difference time domain
FE Finite element
FEM Finite element method
FGMRES Flexible generalised minimum residual
GCV Generalized cross validation
GSVD Generalized singular value decomposition
xx
IBC Impedance boundary conditions
LS Least squares
MIT Magnetic induction tomography
MRI Magnetic resonance imaging
PML Perfectly matched layer
RF Radio frequency
RMS Root mean square
SAR Specific absorption rate
SIRT Simultaneous iterative reconstruction technique
SNR Signal to noise ratio
SOR Successive over relation
SSOR Symmetric successive over relation
SVD Singular value decomposition
TLM Transmission line matrix
TSVD Truncated singular value decomposition
Part I
Preliminaries
Chapter 1.
Introduction
1.1. MIT
Magnetic Induction Tomography (MIT) is a relatively new non-contact imaging
modality which is used for imaging the internal distribution of the passive
electromagnetic properties (conductivity, permittivity, and permeability) of an object by
measuring the mutual inductances between pairs of coils placed around its periphery
[1]. MIT relies on coupling an alternating magnetic field, the primary field, to the
region under investigation to induce interrogating eddy currents, which depend on the
internal electrical property distribution of the region. These generate a secondary
magnetic field. The secondary field is then measured with a sensor coil.
Note that the primary and secondary fields are established simultaneously, but here are
defined separately for the convenience of explanation. By having an array of exciter
coils to be activated in turn for generating the primary field, and an array of sensor coils,
a set of projections can be acquired to form an MIT data set from which the internal
distribution of the electrical properties may be inferred, i.e. reconstruct an image.
MIT is one member of the larger group of electrical tomographic techniques which are
used for non-invasive imaging of dielectric properties of materials. Related techniques
include:
Electrical Impedance Tomography (EIT): estimating the conductivity or
permittivity distribution of the subject from surface electrical measurements.
2 Introduction
Typically, conducting electrodes are attached to the skin (surface) of the subject
and small alternating currents are applied to some or all of the electrodes. The
resulting electrical potentials are measured to form the EIT data set (e.g. [2]).
Electrical Resistance Tomography (ERT): a geophysical technique for imaging
the conductivity of sub-surface structures from electrical measurements made at
the surface or by the electrodes in one or more boreholes. It is closely related to
the EIT. In contrast, however, ERT is essentially a direct current method. For
instance, in imaging the content of process vessels and pipelines, the electrodes
make electrical contact with the fluid inside the vessel (e.g. [3]).
Electrical Capacitance Tomography (ECT): determination of the dielectric
permittivity distribution in the interior of an object from external capacitance
measurements. The measurement electrodes must be sufficiently large to give a
measureable change in capacitance. ECT has been applied to, for instance,
imaging of flow systems such as oil-gas and oil-water flows (e.g. [4]).
Induced Polarization Tomography (IPT): a geophysical technique which, similar
to ECT, is used for the determination of spatial distribution of the capacitance
characteristics of the soil (e.g. [5]).
Metal detectors: a kind of manually scanned MIT which are used for the
detection of foreign bodies in food, land mines, weapons, and treasure (e.g. [6]).
Electromagnetic geophysics tomography: used for the reconstruction of the
conductivity and permittivity profiles of the geophysical medium under
investigation (e.g. [7]).
Compared to EIT and ERT, MIT has the advantage that it does not require direct contact
with the object but operates through an air gap. The contact-less nature of MIT means
that errors due to the electrode contact impedances and the associated image artefacts
can be avoided completely. Furthermore, the magnetic field easily penetrates the
barriers posed by the electrically resistive tissues such as skull bones and fat in medical
imaging. Unlike ECT, MIT is sensitive to all three passive electromagnetic properties;
MIT signals contain information that can be used to reconstruct conductivity,
permittivity, and permeability [8,9].
3 Introduction
1.2. MIT applications
There has been considerable interest shown over the past decade in the MIT imaging
technique and its application to various problems. If there is a contrast or change in
dielectric properties of the object of interest, MIT may potentially be used as an imaging
tool [10]. MIT has possible applications in fields such as:
Medical imaging: in medical applications, the non-invasive measurement of the
impedance of biological tissues can potentially yield data of diagnostic relevance
(e.g. [1]).
Industrial application: using conductivity images in monitoring of industrial
processes such as molten steel flow visualization in continuous casting (e.g.
[11]).
Non-destructive testing: eddy-current testing is very well known in non-
destructive testing which, for instance, is used for crack detection in metallic
objects (e.g. [12]).
1.3. Motivation and objectives
The main aim of this study was to numerically investigate the feasibility and sensitivity
of MIT imaging for the detection of cerebral stroke. Cerebral stroke is a major cause of
mortality and morbidity in Western society that is likely to rise with the ageing
population. The cost of aftercare places a significant burden on the UK‟s health and
social care budget. Conventional brain imaging techniques (e.g. MRI and CT) are
expensive and sometimes inaccessible, and MIT may eventually be an attractive and
low-cost (< £10k) candidate for rapid diagnostic imaging.
The major challenge for using MIT imaging in medical applications, however, is the
fact that the induced MIT signals are small. Biological tissues have very low
conductivities ( < 2 S m
-1) and their interaction with the magnetic field is weak, leading
to extremely small MIT signals. Consequently, the noise contaminating the MIT data
due to the hardware measurement errors can limit the accuracy of MIT medical
imaging. Therefore, a realistic feasibility study can be of crucial importance for each
MIT application and, to this end, the ability to accurately model the MIT signals first is
vital to the process of hardware design.
4 Introduction
Prior to commencing this research, all the numerical and experimental studies on brain
imaging used models or setups, consisting of simple geometries, which were unlikely to
accurately estimate the MIT signals in in-vivo imaging. An overview of the history of
MIT employment for medical imaging is provided in the next chapter. Therefore, in
order to assess and optimize the sensitivity of the MIT systems for clinical imaging,
more realistic simulations are required. The key questions relating to the use of MIT
technology for imaging the stroke were:
What is magnitude of the MIT signals produced by the head and stroke, and can
they be measured using the existing MIT systems?
Can an image of the stroke be reconstructed?
What is the design for the MIT coil array in order to obtain the most sensitivity
for stroke imaging?
The image reconstruction process in MIT is highly sensitive to the error in the data; a
low level of noise in the signal can cause large errors in the calculated conductivity
distribution. Consequently, contamination of the measured MIT signal with the inherent
noise in the hardware can introduce considerable artefacts in the reconstructed images.
Hence, efficient image reconstruction methods that can reduce the effect of the noise on
the images are required for the MIT technique. Therefore, the main objectives of this
study were to:
Develop a realistic head model of a typical human head in terms of geometry
and dielectric properties of different head tissues.
Simulate the MIT problem using an approximate numerical method in order to
estimate the MIT signals due to the occurrence of the stroke in the brain.
Investigate and adopt the image reconstruction algorithms for the intended
application.
Investigate different MIT coil arrays in order to improve the sensitivity for
stroke imaging.
5 Introduction
1.4. Outline of the thesis
This thesis comprises ten chapters and three appendices. In the first section of every
chapter, there is an introduction on the subject of that chapter, followed by the main
body:
In chapter 1, the MIT imaging technique is introduced and a brief account of the
research study and this thesis is provided.
In chapter 2, the working principle of a typical MIT system is explained and
physical concepts are discussed. The eddy-current problem in MIT is described.
An overview of MIT studies for medical imaging reported by different research
groups is also provided.
Chapter 3 briefly describes the anatomy of the head and its tissues and
introduces the different types of stroke.
A literature survey of dielectric properties of the head tissues at different
frequencies is carried out in chapter 4, and the effect of stroke on the dielectric
properties is explained.
The discretized MIT problem is dealt with in chapter 5. Two approaches for the
numerical approximation of the MIT problem (i.e. finite element and finite
difference) are explained. A benchmark test is conducted to assess the accuracy
of the employed numerical techniques.
Chapter 6 is devoted to the image reconstruction process. It discusses different
single-step and iterative image reconstruction algorithms. Images of
geometrically simplified models are reconstructed as evaluation tests prior to
stroke imaging using the anatomically realistic head model.
MIT simulations for a 16-channel cylindrical system are presented in chapter 7.
The MIT problem is solved for strokes with different sizes and sites in the brain
and 3D images are reconstructed from the simulated data.
In chapter 8, the performance of customised MIT systems for stroke imaging is
investigated. Different numbers and sizes of the exciter/sensor coils are
evaluated in order to enhance the sensitivity of the system. Details of the
adopted design are also provided. Chapter 8 also describes examples of
systematic errors which are likely to occur during in-vivo imaging, and evaluates
their effect on the images.
6 Introduction
Preliminary imaging studies based on the measured MIT data is provided in
chapter 9.
Chapter 10 concludes the study while some suggestions for further development
are also provided.
A bibliography is placed at the end of this thesis, and is preceded by the appendices.
Appendix A provides a brief description of two existing MIT measurement systems,
constructed by the South-Wales MIT group. Appendix B explains the basic concepts of
dielectric theory and terminology. And finally, Appendix C provides some insight into
some of the image reconstruction techniques.
1.5. Contributions to knowledge
Considering the novel elements of the research undertaken, the following contributions
to knowledge are claimed:
An up to date review of the dielectric properties values for the human head
tissues published since 1996.
Extension of an existing EIT head geometry for MIT applications.
Comparison of the finite element and finite difference models for the MIT
problem.
Numerical simulation of MIT signals due to the presence of stroke in a realistic
head model.
Reconstruction of 3D images of stroke using the realistic numerical model.
Improvement of the geometrical design of MIT system for stroke imaging.
Investigation of the effect of systematic errors on the stroke images.
Preliminary MIT imaging experiments using practical measured data for simple
geometries.
During the course of this PhD study, outcomes of the research have been published in
three journal articles, and presented at several relevant national and international
conferences:
7 Introduction
Journal articles:
1. M. Zolgharni, P.D. Ledger, D.W. Armitage, D.S. Holder, H. Griffiths, Imaging
cerebral haemorrhage with magnetic induction tomography: numerical
modelling, Physiol. Meas. (2009) 30:187-200
2. M. Zolgharni, P.D. Ledger, H. Griffiths, Forward modelling of magnetic
induction tomography: a sensitivity study for detecting haemorrhagic cerebral
stroke, Med. Biol. Eng. Comput. (2009) 47:1301-1313
3. M. Zolgharni, H. Griffiths, P.D. Ledger, Frequency-difference MIT imaging of
cerebral haemorrhage with a hemispherical coil array: numerical modelling,
Physiol. Meas. (2010) in press
Conference proceedings:
4. M. Zolgharni, P. D. Ledger, H. Griffiths, D. S. Holder, Numerical Modelling of
the Cerebral Stroke in Human Brain, 16th UK Conf. on Computational
Mechanics (ACME), Newcastle upon Tyne 2008, UK , pp.139-142.
5. M. Zolgharni, P.D. Ledger, D.W. Armitage, H. Griffiths, D.S. Holder,
Detection of haemorrhagic cerebral stroke by magnetic induction tomography:
FE and TLM numerical modelling, EIT 2008, Dartmouth NH, USA, June 16-18.
6. Y. Maimaitijiang, S. Watson, M.A. Roula, M. Zolgharni, H. Griffiths, R.J.
Williams, An Iterative Absolute Image Reconstruction Algorithm for Magnetic
Induction Tomography, EIT 2008, Dartmouth NH, USA, June 16-18.
7. M. Zolgharni, H. Griffiths, D.S. Holder, Imaging haemorrhagic cerebral stroke
by frequency-difference magnetic induction tomography: numerical modelling,
2008 IFMBE Proc. 22 2464-7 ISBN 978-3-540-89207-6.
8. B. Dekdouk, M.H. Pham, D.W. Armitage, C. Ktistis, M. Zolgharni, A.J.
Peyton, A feasibility study on the delectability of edema using magnetic
induction tomography using an analytical model, 2008 IFMBE Proc. 22:736-9
ISBN 978-3-540-89207-6.
9. M. Zolgharni, P.D. Ledger, H. Griffiths, Imaging cerebral haemorrhage with
MIT: frequency-difference imaging with a customised coil array, EIT 2009,
Manchester, UK, June 16-19.
8 Introduction
10. M. Zolgharni, P.D. Ledger, H. Griffiths, High-Contrast Frequency-Difference
Imaging for Magnetic Induction Tomography, EIT 2009, Manchester, UK, June
16-19.
11. S. Watson, H.C. Wee, R. Patz, R.J. Williams, M. Zolgharni, H. Griffiths,
Detection of peripheral haemorrhagic cerebral stroke by magnetic induction
tomography: phantom measurements, EIT 2009, Manchester, UK, June 16-19.
12. M. Zolgharni, P.D. Ledger, H. Griffiths, Difference imaging of cerebral stroke
in the human brain using edge finite element simulation of magnetic induction
tomography, CMBE 2009, pp.288-91, Swansea, UK, ISBN 978-0-9562914-0-0
13. H. Griffiths, M. Zolgharni, P.D. Ledger, S. Watson, The Cardiff Mk2b MIT
head array: optimising the coil configuration, EIT 2010, Gainesville FL, USA,
April 4-8
1.6. LCOMIT
This study forms part of a larger 3.5-year EPSRC1-funded collaborative project,
LCOMIT2
, for imaging low-conductivity objects involving academic partners at
Swansea University [13], University of Manchester [14], University of Glamorgan [15],
University College London, ABM University Health Board, and industrial/NHS
partners at Philips research laboratories in Aachen, Germany [16]. The work at Swansea
University is devoted to the application of MIT for stroke imaging, while the work by
other partners involves the development of hardware and software aspects of MIT in
general and for two other specific industrial applications, involving a three further
industrial partners.
End of chapter 1.
1 Engineering and Physical Sciences Research Council 2 LCOMIT: Low conductivity magnetic induction tomography
Chapter 2.
Principles of MIT
2.1. Introduction
The purpose of this chapter is to outline the MIT imaging technique. First the physical
concepts and governing equations of the forward and inverse boundary value problems
are explained. Next, the working principles and the constituent parts of a typical MIT
system are discussed. An overview of the history of employing MIT for brain imaging
applications is also provided.
2.2. Physical principles
MIT imaging, and electrical tomography techniques in general, consist of solving
boundary value problems that are governed by Maxwell's equations. The corresponding
'forward' and 'inverse' problems are defined as:
The purpose of solving the forward boundary value problem is to compute the
electromagnetic fields for a given geometry, distribution of dielectric properties,
operating frequency, and source current, subjected to known boundary
conditions.
In contrast, the purpose of solving the inverse boundary value problem is to
recover the distribution of dielectric properties from the measured fields on
(parts of) an enclosed surface surrounding the region of interest.
10 Principles of MIT
For time-harmonic fields varying with an angular frequency of (s-1
), the governing
Maxwell's equations are expressed by:
(2.1)
(2.2)
(2.3)
(2.4)
In the above equations, the field quantities E (V.m-1
) and H (A.m-1
) represent the
electric field and magnetic fields, respectively, D (C.m-2
) and B (T) represent the
electric and magnetic displacements, and j2 = -1. The term s (C
m
-3) is the charge
density. The current density J (A.m-2
) has two components:
(2.5)
where Jc represents the conduction current density induced by the presence of the
electric field, Js is a source current density that induces the electromagnetic fields (here
assumed divergence free throughout), and (S.m-1
) is the conductivity of the
constituting medium.
The auxiliary constitutive relationships between the field quantities and electric and
magnetic displacements provide the additional constraints needed to complete the
equations (2.3) and (2.4). These equations characterize a given material on a
macroscopic level in terms of the two quantities, permittivity (F.m-1
), and permeability
(H.m-1
) as:
(2.6)
(2.7)
where 0 = 4 ×10-7
H m
-1 and 0 = 8.85×10
-12 F
m
-1 are the permeability and
permittivity of free space, respectively. Also, r and r are the relative permeability and
permittivity of the constituting medium, respectively. For isotropic linear materials,
which are of interest in this study, the electrical properties ( , , ) are scalar quantities
(material properties of the tissues in the head are discussed in chapter 4).
11 Principles of MIT
Forward problem:
For a bounded domain , the forward problem is: Find E and H such that:
(2.8)
where 3, = D N with D N = , GD and GN are general Dirichlet and
Neumann boundary conditions, vector n is a unique outward normal at each boundary
or interface, Js is a given current source such that Js = 0 in , ( , , ) are given
distributions of the medium parameters, [...] represents the jump in the quantity, and
finally i j denotes the interface between the regions of different materials [17].
Inverse problem:
Find the distributions of ( , , ) such that (2.8) holds for given E, H.
2.2.1. MIT problem
Figure 2.1 shows a typical boundary value problem for the MIT application. The
domain of the problem, , consists of a conducting region C, ( C) > 0, and a non-
conducting region N, ( N) = 0, so that = C N.
12 Principles of MIT
Figure 2.1. Schematic view of boundary value problem in MIT; C: conductive domain, N: non-
conducting region (free space).
In reality, N should be replaced by N which is of infinite extent. In this case,
would become unbounded and the boundary value problem described by (2.8) would
need to be approximated by a radiation condition [18]:
(2.9)
However, to ensure the boundedness of , N is truncated at a finite distance from the
conducting object, and the artificial domain N, and consequently the artificial
boundary , are created. By relaxing the constraint that D N = , and providing
the exact values of both n E and n H on all of , no approximation is made. The
choice of appropriate approximate boundary conditions for is discussed in chapter 5.
The conducting region will, in general, consist of a number of materials, and the
appropriate interface conditions described in (2.8) should be applied.
The exciter/sensor coils lie in N: E represents an exciter coil on which the current
source Js is applied, and S is a sensor coil on which the voltage is computed as:
(2.10)
where dl is the length element of the coil.
2.2.2. Eddy-current problem
For certain low-frequency application (e.g. non-destructive testing [19]), the problem
described by (2.8) may be simplified to the so-called eddy-current model. The eddy-
E
C
σ, ,
N : σ =0, = 0, = 0
S
13 Principles of MIT
current model is a magnetostatic approximation of the full Maxwell system, and
involves neglecting the displacement currents.
A number of justifications for this exist in the literature. These include those based on
the heuristic arguments that the eddy-current approximation is meaningful if [20]:
the dimension of the conducting object is small compared to the wavelength
the conductivities are high ( << )
More recently, these arguments have been found to be insufficient for certain
geometries, and have been refined to include the topology of the conducting object
[18,21]. Neglecting the displacement current leads to a problem which reduces to
solving a magnetostatic problem in the non-conducting region N (and consequently
N), and also allows the radiation condition to be replaced by the simpler requirement
that [18]:
(2.11)
For the medical MIT applications, however, the material properties are such that the
ratio / can be as large as 0.6 at 10 MHz (electrical properties of head tissues are
provided in chapter 4). Therefore, on the grounds of the earlier heuristics, the eddy-
current approximation must be rejected and the displacement current should be retained,
despite the fact that the frequency of operation is relatively low ( 10 MHz). Retaining
the displacement current means that wave propagation effects are included in the
problem. As the frequency increases, these effects become more important.
It is remarked that MIT systems used for imaging metallic conductors do employ the
eddy-current approximation of (2.8); e.g. [22]. Nevertheless, eddy currents (Je = E)
will form an important part of the discussion in later chapters, as will the skin depth.
The skin depth is defined as the distance measured from the surface of the medium over
which the magnitudes of the fields are reduced to 1/e, or approximately 37%, of those at
the surface of the medium. For the case of a semi-finite conductor with uniform
conductivity and permeability , it can be calculated as = (2/ 0 r )1/2
[23].
14 Principles of MIT
2.2.3. Solving the (forward) problem
Analytical solutions for the forward problem are only available for a small number of
relatively simple geometries, e.g. a conducting sphere [24,25,26,27,28] or disk [29], and
when the materials are isotropic. For complex geometries such as a realistic human head
model, or in case of dealing with anisotropic materials, numerical techniques must be
employed where the solution to the problem is approximated by discretizing the model.
The discrete forward and inverse problems are discussed in chapters 5 and 6,
respectively.
2.3. Working principles
The main elements of a complete MIT system are shown in figure 2.2. According to
Peyton et al [30], the system typically comprises three main sub-systems: a sensor
array, the electronics components, and a host computer. The sensor array consists of:
Exciter coils array
Sensor coils array
An electromagnetic confinement shield
Electronic circuitry for exciter and sensor coils
Figure 2.2. Block diagram of a typical MIT system (after Peyton et al [30]).
The outer electromagnetic shield (screen) performs several functions. In particular, it
confines the interrogating field within the imaging volume, thereby removing the
potential for interference by external conductive or magnetic objects in the vicinity. The
Screen
15 Principles of MIT
screen reduces the susceptibility to external magnetic field interference by providing a
shunt path around the object space [31]. It may also influence the sensitivity of the
system. In addition, the screen can act as a ground plane that helps to reduce undesirable
electric-field (capacitive) coupling between the coils [32]. Details of the electronic
components and circuitry are outside the scope of this study; useful information for MIT
electronic developments can be found in [33,34,35,36]. The image reconstruction,
which requires the solution of the inverse problem, is performed on the host computer.
By solving the MIT inverse problem, the temporal-spectral conductivity differences or
the static (absolute) conductivity values may be reconstructed from the measurements.
The MIT imaging can, therefore, be categorized into two groups; difference and
absolute imaging:
Difference imaging: differential or difference imaging is to estimate a change in
conductivity between two states from the MIT signals measured either at
different times (time-difference imaging), or with different frequencies
(frequency-difference imaging). The differential technique has been used for
imaging temporal-spectral phenomena in medical applications, such as lung
imaging reported by Gursoy and Scharfetter [37].
Absolute imaging: a special case of difference imaging is absolute imaging,
when one of the states is free space; an image is produced using a single set
(frame) of MIT data. Absolute imaging, thus, attempts to calculate the actual
values of the conductivity in S m
-1. The use of absolute imaging is more difficult
as the accuracy of both the measured and predicted voltages is crucial (further
discussed in chapters 6 and 9).
Depending on the application, the appropriate imaging type must be adopted. The
overall MIT procedure, excluding the hardware preparation, can be summarized as
follows:
I. Input: measure the MIT signals induced by the object to be imaged.
II. Define problem: define the MIT setup including the coil arrays and screen,
operating frequency, geometry (outer boundary) of the object. In some cases, the
geometry data is not available.
III. Forward problem: develop the physical model that represents the boundary
value problem.
16 Principles of MIT
IV. Sensitivity analysis: analyse the sensitivity of the output (MIT signals) to the
input parameters (conductivity distribution).
V. Inverse problem: apply the inverse algorithm using the sensitivity data and the
measured signals.
VI. Output: extract the conductivity distribution.
In some cases, the inverse problem is solved iteratively, in which case the steps (III) to
(VI) are repeated until convergence is obtained.
2.3.1. The MIT signal
As shown in figure 2.3, the change in the voltage due to the presence of an object
(target) has real and imaginary components. The total measured signal, i.e. primary plus
secondary, lags the primary signal by an angle .
Figure 2.3. Phasor diagram representing primary (Vp) and secondary ( V) voltages.
Provided that the skin depth of the electromagnetic field in the target is large compared
with the dimension of the target, which will often be so for a segment of the human
body [33], Griffiths et al [8] have shown that the induced voltage change, V,
normalized to the primary voltage, Vp, can be expressed as a phase change:
(2.12)
where C1 and C2 are the geometrical constants, and V << Vp.
The conduction currents induced in the target give rise to a component of V which is
proportional to frequency and conductivity. It is also imaginary and negative (it lags the
primary signal by 90 ). Displacement currents cause a real (in-phase) component
Vp Re( V)
Im( V)
Vp + V
17 Principles of MIT
proportional to the square of the frequency and permittivity. This component is
independent of the conductivity. A non-unity relative permeability also gives rise to a
real component, but with a value independent of frequency. However, since r is very
close to unity for biological tissues (| r - 1| < 10-5
), the second term on right hand side of
(2.12) can be dropped for most purposes.
In general, the real and imaginary parts of the phase change = tan-1
( V/V) may be
used for imaging the permeability-permittivity and conductivity of the target,
respectively.
For larger conductivities (e.g. in metals), the skin depth may become comparable with
the target's dimension. In this case, Re( V/V) will be affected by the conductivity and
equation (2.12) fails to predict the correct values for this component (further discussed
in chapter 5). As described by Griffiths et al [8], in such cases the target acts as a screen
between the exciter and sensor coils, and cause a negative Re( V/V).
In a practical system, an additional (undesired) coupling mechanism exists via the
capacitance between the exciter and sensor coils, and the target. This can cause large
errors in the MIT signal, and efforts by the system designers are made to reduce the
capacitive coupling. A detailed discussion of the capacitive coupling is outside the
scope of this study. Further details can be found in Goss et al [38].
2.3.2. SNR and operational frequency
Since the MIT signals are measured by a practical data-collection system, they are
inevitably contaminated by the noise in the hardware. The noise in the MIT data is
defined by the signal-to-noise ratio (SNR), which is the ratio of the signal (meaningful
information) to the noise corrupting the signal:
(2.13)
where P is the average power and A is the amplitude of the voltage. Employing higher
frequencies for MIT operation can be beneficial. High-frequency (3–30 MHz) operation
of MIT systems offers advantages in terms of the larger induced signal amplitudes
compared to systems operating in the low- or medium-frequency ranges [39].
18 Principles of MIT
While the noise amplitude in the hardware is independent of the frequency, large
induced signal amplitudes results in larger SNR. However, a higher frequency range not
only can make the simulations more complicated, it can also present difficulties in terms
of the phase stability of the distributed signals, and of achieving the required isolation
between channels in a multi-channel system [39]. Furthermore, as will be explained in
chapter 4, frequency-dependence of the dielectric properties of biological tissues may
suggest particular frequency ranges for biomedical MIT applications. Therefore,
multiple factors need to be taken into account in designing the MIT system for a
particular application.
If the MIT signals are normalized to the primary signal as in (2.12), the noise is defined
in m rather than dB, and is often referred to as phase noise.
2.4. MIT studies for brain imaging
Griffiths provided two comprehensive reviews in 2001 [40] and 2005 [1], looking at
different MIT systems for industrial and medical applications. The main focus of this
section, however, is on the numerical/experimental MIT studies for brain imaging
applications.
Tarjan and McFee in 1968 [41] measured an average value for brain conductivity
inductively. Later in 1993, Netz et al [42] suggested that brain oedema might be
detected more rapidly from conductivity changes than is possible from CT or MRI
imaging. A single-channel MIT system for in-vivo imaging was reported by Al-Zeibak
et al in 1993 [43]. However, only imaging of metallic and ferromagnetic samples at 10
MHz were reported, which were not a suitable representation of the head tissues.
Later in 1999, construction of a single-channel MIT system operating at 10 MHz was
reported by Griffiths et al [29] from the South-Wales MIT group. The MIT signals were
measured for saline solutions with conductivities ranging from 0.001 to 6 S
m
-1
encompassing the range of conductivities of biological tissues. Since then, there have
been a number of numerical and experimental studies on the feasibility of using MIT for
imaging brain lesions.
19 Principles of MIT
Merwa and Schaefetter [44] from the Graz MIT group, reported on the sensitivity study
of a single-channel MIT system to an oedematous region using a semi-realistic model of
the human brain. Their model comprised the cerebrospinal fluid (CSF) around the brain,
grey matter, white matter, ventricles, and the oedema (a spherical inclusion). The finite
element method (FEM) was used to study the effect of different sizes and positions of
the oedema on the MIT signals. It was shown that a centrally located oedema with a
conductivity contrast of 2 with respect to the background and radius of 20 mm could be
detected at 100 kHz. The same authors later reported the reconstruction of time-
differential images from the simulated oedema using a multi-channel MIT system
comprising 16 exciter and 32 sensor coils [45].
The spectroscopic distribution of the MIT signal as a function of the relative volume of
oedema and haematoma in the brain was investigated at the Berkeley MIT group by
Gonzalez and Rubinsky [46]. They considered a simple circular disk to represent the
brain and used the analytical solution to compute the signals. The total volume of the
disk was considered as the oedema/haematoma region. It was shown that the MIT signal
was sensitive to the relative volume of the oedema/haematoma at frequencies higher
than 10 MHz, and increasing the volume of tissue had the effect of lowering the
required frequency at which the signal became sensitive to the volume of the
oedema/haematoma. However, considerable simplifications were used in their model.
Later they reported the experimental studies for their simulated model [47,48]. In other
reports [49,50], they numerically compared the MIT signals due to the haematoma in a
circular sensor coil array with those in a magnetron sensor coil. They suggested that a
system with a magnetron sensor is a preferential configuration with greater sensitivity
for the centrally located lesions.
The same group recently reported on the detection of brain ischaemia in rats [51]
(ischaemia is one of the stroke types and is introduced in chapter 3). Their system
consisted of two electromagnetic coils placed around the head. Ischaemia was induced
in the rat's brain by occlusion of the right cerebral and carotid arteries. Experimental
subjects were monitored for 24 hours and measurements were made at five frequencies
in the range of 0.1-50 MHz. A mathematical head model comprising the brain, bone and
skin was also employed to simulate the MIT signals, for comparison with the
experimental data. They concluded that the experimental results were roughly in
qualitative agreement with those of the numerical model. Both theoretical and
20 Principles of MIT
experimental results suggested that the inductive signals increase as a function of
frequency and with the level of the simulated ischaemic damage, i.e. volume of the
lesion.
Researchers from China used the finite difference time domain (FDTD) method to
investigate the effect of oedema in a single channel MIT system operating at frequency
of 10 MHz [52]. Simplified spherical models were used to represent a three-layered
head geometry comprising skull, CSF and brain, and also the inclusion (oedema). By
assessing the MIT signals, it was concluded that the MIT imaging of oedema is feasible.
However, no images were reconstructed. Later they reported on the sensitivity of the
system to the site, size and conductivity of the oedema [53].
Construction of a 15-channel helmet shaped MIT device for brain imaging has been
recently reported by another group from China [54]. A hemispherical glass bowl filled
with the salt solution was used as the brain, and agar blocks of different conductivities
were placed at different peripheral locations in the solution to represent the
haemorrhage. Measurements were carried out at different frequencies raging 30 to 120
kHz and it was concluded that the MIT signal values increased linearly with the
frequency. They used interpolation algorithms to obtain 2D plots; this method suffered
from poor spatial resolution for brain imaging applications. Reconstruction of 3D
images using this MIT system has not been reported as yet.
Korjenevsky [55] from the Moscow MIT group, has obtained the first in-vivo image
using a 16-coil MIT system operating at frequency of 20 MHz. An absolute image,
reconstructed using neural network, appears to show a cross-section of the human head,
with two bright features interpreted as the lateral ventricles of the brain filled with CSF.
Vauhkonen et al [56] from the Philips MIT research group in Germany (one of the
partners in LCOMIT project), reported a numerical investigation of imaging stroke
using a 16-channel MIT system. They reconstructed images of geometrically simplified
stroke lesions with different volumes and locations in the head. Cylindrical tanks and
objects were used to simulate the head, neck, shoulder, and the stroke. It was concluded
that centrally located small inclusions (i.e. strokes) are extremely difficult to detect
since they affect the measurements only slightly. In a recent report [57], they compared
the cylindrical and the hemispherical coil setups of two MIT systems using sensitivity
analysis by numerical simulations. Different parameters for the size and the number of
21 Principles of MIT
measurement and excitation coils were tested. Their findings indicated that the
hemispherical MIT system with a smaller distance between the layer of coils and the
object shows a clearly higher sensitivity compared to the cylindrical MIT system. They
concluded that coils should be placed as close as possible to the object which, for the
human head, suggests a hemispherical coil arrangement.
The South-Wales MIT group used a saline bath to represent the brain, with an immersed
block of agar simulating stroke [58]. The conductivity contrast between the agar and the
saline was a factor of 3.3, being similar to the contrast between blood and brain at
frequency of 10 MHz. The images were reconstructed based on the measured signals
using a 16-channel MIT system, and it was concluded that stroke could be identified in
the images.
Dekdouk et al [24] from the Manchester MIT group (one of the partners in LCOMIT
project), reported a feasibility study on the delectability of oedema using a single MIT
channel. They used an analytical multilayer spherical model to represent the head. The
delectability of the oedema with regard to noise limitations of the MIT systems was
analyzed, but no images were reconstructed. They recently used the anatomically-
realistic head geometry, also used in this thesis, to solve the forward problem for strokes
occurring at different locations within the head. The simulated MIT configuration was a
16-channel cylindrical system, operating at frequency of 10 MHz. They concluded that
the MIT signal decays when the stroke approaches the centre as opposed to the
peripheral region. No images were reconstructed.
As mentioned in chapter 1, the results of this thesis have been published in literature.
Zolgharni et al [59] from the South-Wales MIT group, reported on time-differential
imaging of stroke using a simulated 16-channel MIT system, operating at frequency of
10 MHz. A realistic model was employed for solving the forward and inverse problems,
and 3D images were reconstructed [60]. In another report [61], frequency-differential
imaging (1-10 MHz) using the same simulated MIT configuration was examined.
Frequency-differential imaging of stroke using hemispherical coil arrays was recently
reported [62]. Two types of systematic errors were added to the simulated MIT signals,
and 3D images were reconstructed. Griffiths et al [63] have reported on the construction
of a helmet shaped MIT array, for which the geometrical design was based on the
findings of this thesis. To the author's knowledge, at the time of writing, the use of an
22 Principles of MIT
anatomically realistic model to estimate the MIT signals due to a stroke in the head,
reconstruction of differential stroke images, and addition of systematic errors is the first
application of its kind.
2.5. MK1 and MK2a systems
Most of the studies carried out in this thesis, have simulated one of the two existing
annular (cylindrical) measurement systems in the South-Wales MIT group (MK1 and
MK2a) as the MIT configuration. A brief description of these systems is provided in
Appendix A. From here on in this thesis, for the discussion of results and when
explaining the simulation methods, the simulated MIT configuration is simply cited as
the MK1 or MK2a system.
End of chapter 2.
Chapter 3.
Anatomy of Head and Stroke Pathology
3.1. Introduction
In order to be able to predict the MIT signals produced by the head, a realistic model is
required for the simulations. The model should consist of: (i) an accurate geometry
which represents the complex shape of the human head, and (ii) the most accurate
dielectric properties, available in literature, assigned to each head tissue. The former
needs a knowledge of gross anatomy of the head, and the latter requires a knowledge of
tissue electrical properties. The purpose of this chapter is to review the internal structure
of a healthy human head. It also describes the pathologies of interest (i.e. stroke). The
electrical properties of the tissues are discussed in the following chapter, while details of
the simulations with a realistic model are provided in chapter 7.
3.2. Head anatomy
The human head comprises various organs with particular functionalities and a detailed
review is required to explain all different structures in the head. However in this study,
regardless of the physiology, the main tissues, which are expected to mostly contribute
to the MIT signals, are discussed. Their contribution to the MIT signals will be
influenced by: (i) their volume, (ii) their location relative to the surface of the head and
consequently exciter-sensor coils, and (iii) their electrical properties. These tissues
include bones, muscles, cerebrospinal fluid, brain with its sub-tissues, and the air-filled
24 Anatomy of Head and Stroke Pathology
cavities. In the following, the tissues are introduced from outside in, and the cavities in
the head are described at the end.
3.2.1. Scalp
The scalp is the anatomical area bordered by the face anteriorly and the neck to the sides
and posteriorly [64]. The scalp electrical impedance plays a major role in head EIT
imaging, as it is engaged with currents of the highest density coming from the
electrodes and, therefore, is the first stage in which distribution of current into the rest
of the layers is determined. However, it is of less importance in MIT imaging, which is
a contact-less technique, and yet the scalp is the closest tissue to the coils.
3.2.2. Muscles
Skeletal muscles are contractile tissues which are made up of long and cylindrical
individual components known as muscle fibres (figure 3.1a). According to Mader [65],
the term muscle refers to multiple bundles of muscle fibres held together by connective
tissue. Muscles show anisotropic properties; electrical conductivity along the length of
the fibre is higher than that in the transverse direction. However, as will be discussed in
chapter 4, the anisotropy becomes less important as the frequency increases. Figure 3.1b
illustrates the head, neck, and facial muscles. The muscles of facial expressions are
located on the scalp and the face. Long strap-like neck muscles extend from base of the
human skull to the neck.
Figure 3.1. Skeletal muscles: (a) structure of a skeletal muscle fibre (after Saladin [66]), (b) muscles
of head and neck (after Mader [65]).
(a) (b)
temporalis
frontalis
masseter
sternocleidomastoid
trapezius
zygomaticus
buccinator
myofibril:
bundle of
fibres
sarcolemma
25 Anatomy of Head and Stroke Pathology
Figure 3.2. (a) simplified side view of human skull (after Villarreal [67]). (b) illustration of a typical
long bone showing the location of cortical and cancellous bones (picture adapted from [68]); skull
bones have a similar internal structure.
3.2.3. Skull
The human skull consists of 22 bones, 8 cranial and 14 facial, and contains several
cavities that house the brain and sensory organs. McCann et al [69] describe the skull as
a geometrically complex structure which can be divided into two major parts: the
cranium and the mandible. The mandible (jawbone) is the only moveable portion of the
skull. A skull that is missing a mandible is a cranium. The largest cavity in the skull,
cranial cavity, accommodates the brain and has a capacity of 1,300-1,350 cc [70].
Figure 3.2a shows a side view of the simplified human skull. The rigid part of the bone
is called osseous tissue or bone tissue which is not homogeneous. There are two types
of osseous tissue; compact and spongy (figure 3.2b). Compact (also cortical or dense)
bone forms the extremely hard exterior. Spongy (also cancellous or trabecular) bone
fills the hollow interior and has a higher surface area but is less dense, softer, weaker,
and less stiff. The tissues are biologically identical; the difference is in how the
microstructure is arranged. A realistic model of the head may consider heterogeneous
bones with their different electrical properties.
3.2.4. Meninges
The brain is protected by three membranous connective tissue coverings called the
meninges. According to Seely [71], the meninges have three layers: dura mater,
(a) (b)
26 Anatomy of Head and Stroke Pathology
arachnoid mater, and pia mater (figure 3.3a). The most superficial and thickest
membrane is the dura mater, which is in contact with the skull. The next meningeal
membrane, arachnoid, is the middle of the three meninges. The thin pia mater is tightly
bound to the convolutions of the brain and the irregular contours of the spinal cord.
The space between the arachnoid membrane and the dura mater is the subdural space
and contains only a very small amount of serous fluid. Between the arachnoid mater and
the pia mater is the subarachnoid space (figure 3.3b). The meninges are one of the
regions at which stroke may occur.
Figure 3.3. Meninges; (a) meningeal membranes surrounding brain, (b) frontal section of head to
show the meninges (after Seely [71]).
27 Anatomy of Head and Stroke Pathology
Figure 3.4. Human brain; cerebrum, seen here in longitudinal section, is the largest part of brain in
humans (after Mader [65]).
3.2.5. Brain
As Mader [65] describes, the brain is that part of the central nervous system (CNS)
contained within the cranial cavity. It is the control centre for many of the body‟s
functions and consists of the brainstem, the cerebellum, the diencephalon, and the
cerebrum (figure 3.4).
Cerebrum: the cerebrum is the largest and most obvious portion of the brain.
According to Shier et al [72], it accounts for about 80% of the mass of brain and
consists of two halves, the right and left hemispheres, which are incompletely separated
by a deep groove called the longitudinal fissure. The falx cerebri, which is a strong and
arched fold of dura mater, descends vertically in the longitudinal fissure between the
cerebral hemispheres (see figure 3.3b). Many ridges called convolutions, or gyri,
separated by shallow grooves, mark the cerebrum‟s surface. Each cerebral hemisphere
is subdivided into five lobes which are named according to the overlying cranial bones;
frontal lobe, parietal lobe, temporal lobe, occipital lobe, and insula (figure 3.5a).
28 Anatomy of Head and Stroke Pathology
Figure 3.5. Cerebrum: (a) lateral view of right hemisphere with insula exposed; colours in this
figure distinguish the lobes of cerebral hemispheres (after Shier et al [72]), (b) coronal section of
brain showing cortex, white matter, and corpus callosum (after Scanlon and Sanders [73]).
A thin layer of grey matter, 2 to 5 mm thick, called the cerebral cortex constitutes the
outermost portion of the cerebrum (figure 3.5b). Just beneath the cerebral cortex is a
mass of white matter that makes up the bulk of the cerebrum. Portions of the two
hemispheres are connected internally by the corpus callosum, a large tract of white
matter (figure 3.5b). While the bulk of the cerebrum is composed of white matter tracts,
there are masses of grey matter (so-called basal nuclei or basal ganglia) located
bilaterally deep within the white matter [73]. Since the white and grey matters have
different dielectric properties, the distinction between these tissue types is important in
modelling.
(a)
(b)
29 Anatomy of Head and Stroke Pathology
Brainstem: the brainstem connects the brain to the spinal cord. It consists of the
midbrain, pons, and medulla (see figure 3.4). These structures include many tracts of
nerve fibres and masses of grey matter called nuclei. The midbrain, the smallest region
of the brainstem, is a short section of the brain stem between the diencephalon and the
pons. According to Shier et al [72], the pons appears as a rounded bulge on the
underside of the brainstem where it separates the midbrain from the medulla (figure
3.4). The medulla is the most inferior part of the brainstem.
Diencephalon: the hypothalamus and thalamus are in the diencephalon which is the
part of brain between the brainstem and cerebrum, and is almost completely surrounded
by the cerebral hemispheres (figure 3.6). Saladin [66] describes the thalamus as a large
oval mass of grey matter which constitutes nearly four-fifths of the diencephalon. The
hypothalamus, named for its position below the thalamus, contains several masses of
nuclei that are interconnected with other parts of the nervous system (see figure 3.5b).
Cerebellum: the cerebellum is a large mass of tissue located inferior to the occipital
lobes of the cerebrum and attached to the brainstem posterior to the pons and medulla
(figure 3.4 and figure 3.6). It consists of two lateral hemispheres partially separated by a
layer of dura mater. Like the cerebrum, the cerebellum is primarily composed of white
matter with an outer thin layer of grey matter [65].
Figure 3.6. Illustration of the brainstem, diencephalon and cerebellum (after Shier et al [72]).
30 Anatomy of Head and Stroke Pathology
3.2.6. Ventricles and cerebrospinal fluid
Each cerebral hemisphere contains a relatively large cavity, the lateral ventricle. The
first lateral ventricle in the left cerebral hemisphere, and the second one in the right
cerebral hemisphere. The lateral ventricles are separated from each other by a thin
channel which lies in the midline just inferior to the corpus callosum, and according to
Graaff [70], are usually fused with each other (see figure 3.4 and figure 3.5b). A smaller
cavity, the third ventricle, is located in the centre of the diencephalon between the two
halves of the thalamus. The fourth ventricle is in the superior region of the medulla at
the base of the cerebellum. It is continuous with the central canal of the spinal cord, and
the subarachnoid space of the meninges.
Cerebrospinal fluid (CSF) is a fluid which bathes the brain and the spinal cord and
provides a protective cushion around the CNS from mechanical injury. The brain
weighs about 1,500 grams but, suspended in CSF, its buoyed weight is about 50 grams.
CSF fills the ventricles, the subarachnoid space of the brain and spinal cord, and the
central canal of the spinal cord. According to Seely [71], approximately 23 mL of fluid
fills the ventricles, and 117 mL fills the subarachnoid space.
The CSF flowing in the subarachnoid space is of some importance in head MIT
imaging. Due to its vicinity to the surface of the head, and thus to the exciter/sensor
coils, and its relatively higher conductivity (the highest of all tissues as explained in the
next chapter), CSF can have a dominant contribution to the induced MIT signals. This
can make it more difficult to reconstruct an image of a less conductive feature located
deep in the brain (an example is presented in chapter 6).
3.2.7. Spinal cord
The spinal cord is a cylinder of nervous tissue that begins at the base of the brain and
extends through a large opening in the skull (figure 3.7a). It consists of centrally located
grey matter, and peripherally located white matter. The relative size and shape of the
grey and white matter varies throughout the spinal cord [71]. The core of grey matter
roughly resembles the letter H (figure 3.7b). Within the grey commissure is the central
canal. It is continuous with the ventricles of the brain and is filled with CSF. In this
study, only the neck portion of the spinal cord, which is expected to contribute to the
MIT signals due to its vicinity to the coils, is of interest.
31 Anatomy of Head and Stroke Pathology
Figure 3.7. Spinal cord: (a) illustration of entire length, (b) drawing of a segment (after Seely [71]).
Figure 3.8. Illustration of eye balls and optic nerves (after Mader [65]).
3.2.8. Eyes and optic nerves
The visual system includes the eye balls and the optic nerves. The eye ball is made up
of three layers, enclosing three transparent structures. The outermost layer is composed
of the cornea and sclera. The middle layer consists of the choroid, ciliary body, and iris.
The innermost is the retina. Within these layers are the aqueous humor, the vitreous
humor, and the flexible lens [65]. The optic nerves transmit visual information from the
retina to the brain and are considered to be part of the CNS (figure 3.8).
Despite their relatively small dimensions, optic nerves are particularly important in head
EIT imaging. They can be used as a pathway for current injection or as electric potential
(a) (b)
optic nerve
32 Anatomy of Head and Stroke Pathology
measurements site for EIT data acquisition [74]. The reason why eyes could be used for
this purpose arises from the topology of the brain and skull. The optic nerve connects
the eye with the related occipital cortex without being interrupted by the skull and the
CSF shunting effect. It is therefore a relatively conductive pathway for the current to
flow through. In MIT imaging, however, the magnetic field can easily penetrate the
electrically-resistive skull and, therefore, the conductive pathway provided by the optic
nerves will not be as important as in EIT.
3.2.9. Cavities
As described by Mader [65], there are three large air-filled cavities in the head; nasal
cavity and two ear canals. The nasal cavity or nasal fossa is a large air-filled space
above and behind the nose in the middle of the face (see figure 3.9a and figure 3.2a).
The nose contains two nasal cavities, separated from one another by a septum composed
of bone and cartilage. The ear canal (also external auditory meatus), is a tube running
from the outer ear to the middle ear (figure 3.9b).
3.2.10. Other tissues
There are other tissues and organs such as lips, teeth, skin, and hair which are assumed
to have negligible contribution to the induced MIT signals; due to their low conductivity
and/or their small volumes. Furthermore, the brain comprises some small sub-tissues
which, in terms of the structure, are similar to white and grey matter tissues. In this
study, for simplicity in the head model, they were not accounted for as separate tissues,
but were represented as white or grey matter. More details are provided in table 4.1 in
chapter 4.
Figure 3.9. Air-filled cavities in head; (a) nasal cavity, (b) ear canal (after Mader [65]).
(a) (b)
auditory canal
olfactory
bulb
nasal
cavity
33 Anatomy of Head and Stroke Pathology
3.3. Stroke
A cerebro-vascular accident (CVA) or stroke is the rapidly developing loss of brain
function due to disturbance in the blood supply to the brain which deprives the brain
tissue of oxygen. This can be due to ischaemia (lack of blood supply) or due to a
haemorrhage (bleeding). As a result, the affected area of the brain is unable to function.
A stroke is a medical emergency and can cause permanent neurological damage,
complications, and death. It affects between 174 to 216 people per 100,000 population
in the UK each year [75], and accounts for 11% of all deaths in England and Wales
[76]. It is the major cause of serious disability and the third leading cause of death in the
United States [77,78].
3.3.1. Classification of stroke
Strokes can be classified into two major categories: ischaemic and haemorrhagic.
Ischaemia is due to the interruption of the blood supply. Haemorrhage is due to the
rupture of a blood vessel or an abnormal vascular structure. Therefore, haemorrhage and
ischaemia are opposites; haemorrhage is characterized by too much blood inside the
skull, while in ischaemia there is not enough blood supply to allow continued normal
functioning of the effected brain tissue. Brain ischaemia is much more common than
haemorrhage. Approximately 69-85% of the strokes are due to ischaemia, and the
remainder are due to haemorrhage [76,77,79].
In ischaemia, the interruption of the blood supply may be caused by embolism or
thrombosis. An embolism occurs when an object (the embolus) migrates from one part
of the body (through circulation) and causes a blockage (occlusion) of a blood vessel in
another part of the body (figure 3.10a). Thrombosis is the formation of a blood clot
(thrombus) inside a blood vessel, obstructing the flow of blood through the circulatory
system (figure 3.10b).
Bleeding (figure 3.10c) can have many different causes such as head injury or
hypertension. Caplan [79] provides a comprehensive list of common causes of
haemorrhage. There are several different subtypes of haemorrhage, named for their sites
inside of the skull. An intra-cranial haemorrhage is a bleeding within the skull which
includes two major types; intra-axial and extra-axial (see figure 3.11).
34 Anatomy of Head and Stroke Pathology
Figure 3.10. Stroke; ischaemia due to embolism (a) and thrombosis (b), haemorrhage due to
rupture of a blood vessel (c) (after Broderick and Santilli [77]).
Figure 3.11. Diagram showing different types of haemorrhage based on its location within head.
Intra-axial or cerebral
haemorrhage
(bleeding within brain)
Extra-axial
(bleeding within skull but
outside brain)
Intra-parenchymal
(bleeding within brain
tissue)
Intra-ventricular
(bleeding within
ventricles)
Epidural
Subdural
Subarachnoid
Intra-cranial haemorrhage
35 Anatomy of Head and Stroke Pathology
Figure 3.12. Examples of intra- and extra-axial haemorrhages (after Caplan [80]).
Intra-axial haemorrhage is bleeding within the brain itself, or cerebral haemorrhage.
This category includes intra-parenchymal haemorrhage, or bleeding within the brain
tissue forming a gradually enlarging haematoma, and intra-ventricular haemorrhage,
bleeding within any of the brain's ventricles (lateral, third or fourth ventricles).
Extra-axial haemorrhage, bleeding that occurs within the skull but outside of the brain
tissue, falls into three subtypes (see figure 3.12). Epidural haemorrhage, which occurs
between the dura mater (the outermost meninges) and the skull, is caused by trauma. It
is the least common type of meningeal bleeding and is seen in 1% to 3% cases of head
injury. Subdural haemorrhage results from tearing of the bridging veins in the subdural
space between the dura and arachnoid mater. Subarachnoid haemorrhage occurs
between the arachnoid and pia meningeal layers.
3.3.2. Diagnosis and treatment
If stroke is suspected, a prompt and accurate diagnosis and treatment is necessary to
minimise the brain tissue damage. An ischaemic stroke may be dealt with using
thrombolytic drugs. Thrombolytic drugs are used in medicine to dissolve blood clots in
a procedure termed thrombolysis and can limit the damage caused by the blockage of
the blood vessel. However, according to the guidelines [76,81], thrombolytic treatment
can be helpful provided that:
36 Anatomy of Head and Stroke Pathology
it is administered within three hours of onset of stroke symptoms
possibility of a haemorrhage has been definitively excluded
It should be noted that thrombolytic therapy in haemorrhagic strokes is contraindicated,
as its use in that situation would prolong bleeding into the intracranial space and cause
further damage. Therefore, a quick classification of the stroke, i.e. distinguishing
between ischaemia and haemorrhage, is of crucial importance.
Apart from the physical examination, stroke can be diagnosed through several imaging
techniques [82]: computed tomography (CT), magnetic resonance imaging (MRI),
doppler ultrasound, and arteriography. Among these, CT and MRI scans are the most
conventional and reliable techniques, and their sensitivity and specificity have been the
subject of various studies.
For instance, Chalela et al [83] compared the performance of CT and MRI in a
consecutive series of patients referred for emergency assessment of suspected stroke.
They concluded that MRI is more effective than CT for detection of ischaemia; MRI
detected stroke (ischaemic or haemorrhagic) more frequently than did CT (see table
3.1). On the other hand, CT is less affected by patient motion than conventional MRI
and also allows full access to a patient during scanning, making it of use in restless or
critically ill patients [84].
CT and MRI images allow doctors to separate brain haemorrhages from infarctions1.
They can also show whether bleeding has occurred within the skull, but outside of the
brain in the subarachnoid, subdural, or epidural spaces. Haemorrhages and infarcts
appear with different CT numbers2 on CT scans, making it quite easy to separate the
two stroke types.
Table 3.1. Performance of CT and MRI for diagnosing stroke; values in percentage [83].
MRI CT
Sensitivity Specificity Sensitivity Specificity
Haemorrhage 81 100 89 100
Ischaemia 83 98 16 96
1 Tissue death due to ischaemia is called infarction. 2 A normalised value of the calculated X-ray absorption coefficient of a pixel (picture element) in a
computed tomogram, expressed in Hounsfield units, where the CT number of air is -1000 and that of
water is zero.
37 Anatomy of Head and Stroke Pathology
Figure 3.13a is a CT scan showing a large brain infarct involving the right cerebral
hemisphere. The yellow arrow points to the infarct, which appears darker than the
surrounding brain tissue. Figure 3.13b is an MRI scan showing a small brain infarct
within the left cerebral hemisphere. The infarct appears much whiter than the
surrounding normal brain.
In contrast, the CT scan in figure 3.13c shows a brain haemorrhage. The region of
bleeding appears whiter than the surrounding tissue. The arrow points to the area of
bleeding. Finally, an MRI image from the same haemorrhage is shown in figure 3.13d.
Figure 3.13. Stroke images: (a) CT scan showing a large cerebral infarct, (b) MRI scan showing a
left cerebral infarct, (c) CT scan showing a haemorrhage in deep portion of brain, (d) MRI scan showing a brain haemorrhage similar to that in (c) (after Caplan [79]).
(a) (b)
(c) (d)
CT MRI
Hae
mo
rrhag
e
Infr
act
(Isc
hae
mia
)
38 Anatomy of Head and Stroke Pathology
Figure 3.14. CT scan of a haemorrhage worsened by thrombolytic therapy (after Harold and
Adams [85]).
Figure 3.14 shows a CT scan of a brain haemorrhage in a patient who had received
inappropriate treatment with thrombolytic therapy, resulting in a gross worsening of the
haemorrhage in the left hemisphere of the brain. This illustrates the importance of
accurate early diagnosis.
As can be seen from figure 3.13 and figure 3.14, MRI and CT scans provide excellent
spatial resolution and tissue contrast. However, both modalities involve large,
immobile, and expensive scanners, which are sometimes inaccessible within the 3-hour
window1.
Although MIT will probably never achieve the spatial resolution of MRI and CT, our
research goal is to develop it into a low-cost and portable system, that could possibly
even be carried in ambulances to allow a rapid initial diagnosis to be made.
End of chapter 3.
1 Note that the 3-hour time window is the time when drug must be administrated, and the imaging must be
conducted at a shorter time followed by the diagnosis and assessment.
Chapter 4.
Dielectric Properties of Head Tissues and
Pathophysiology
4.1. Introduction
The purpose of this chapter is to review the dielectric properties of the head tissues
introduced in the previous chapter. Each biological tissue has a distinct frequency
dependence in its passive dielectric properties, which results in a distinct spectroscopic
signature. Therefore, it should be possible to determine the tissue type by this signature.
Furthermore, various pathophysiological mechanisms can alter the biochemical
composition and structure of the cells and their environment and, therefore, result in a
change in the spectroscopic behaviour. As a result of these changes, differentiation
between various tissues‟ conditions by means of bioimpedance measurements may be
possible.
A knowledge of these mechanisms and their impact on the dielectric properties of the
head tissues is important for development of the head MIT imaging technique. For
modelling purposes, this information enables assignment of appropriate dielectric
properties to the tissues involved in the lesion, thereby estimating the expected change
in MIT signals due to the local changes introduced by the pathology. These model-
based predictions can assist in studying the feasibility of MIT applications, such as
distinguishing between ischaemic and haemorrhagic strokes.
40 Dielectric Properties of Head Tissues and Pathophysiology
A brief outline of the basic concepts of dielectric theory and terminology, and various
relaxation mechanisms (dispersions) is provided in Appendix B. In this chapter, a
literature review of the measured or estimated dielectric values of different head tissues
is carried out. The associated dielectric property changes resulting from the abnormal
conditions due to occurrence of the stroke are explained. Finally, the corresponding
values adopted in this study are presented in the end.
4.2. Dielectric properties of head tissues
Research into the dielectric properties of biological materials and their variation with
frequency has been ongoing for most of the past century. In the last few decades, the
research has been driven by the need to establish a credible database of dielectric
properties of all body tissues for use in electromagnetic dosimetry studies, where the
object is to quantify the exposure of people to external electromagnetic fields from the
knowledge of effective internal fields and currents induced in them. In these studies,
tissues are characterized by their measured dielectric properties.
One of the main problems one encounters using bioimpedance measurements is the
reliability of the results. The scatter of the data for the electrical parameters of tissues
indicates the problem of measurement reproducibility. Some of the scatter may result
from problems in measurement techniques, but some may simply be caused by the
individual variability among samples. At present, the relative importance of these two
factors in determining the overall scatter is not clear [86]. This scatter makes it difficult
to establish criteria of normality or reference value for particular measurement results.
However, in spite of these differences between reported data on dielectric properties,
some estimation of the dielectric values at frequencies of interest can still be reached.
Large differences exist in electric properties between different biological tissues. These
differences are determined, to a large extent, by the fluid content of the tissue. For
example, blood and brain conduct electric current relatively well. Skin, fat, and bone are
relatively poor conductors. Moreover, some tissues, such as bone and skeletal muscle,
are distinctly anisotropic. Therefore, when referring to published conductivity and
permittivity values, care must be taken in the orientation of the measuring electrodes
41 Dielectric Properties of Head Tissues and Pathophysiology
relative to the major axis of the tissue (e.g., longitudinal, transverse, or a combination of
both).
In the last decade, most dosimetric studies drew on data published in the scientific
literature in 1996 and made widely available on the Internet thereafter. In this chapter,
the published data is reviewed in three subdivisions; literature before 1996, the 1996
database, and literature after 1996. When available, details of tissue sample,
measurement techniques, and measuring environment are mentioned. This is because
the measurements are known to be affected by factors such as temperature, mammalian
species, subject's age, the particular electrode arrangement used, and whether the
measurement was performed in-vivo, in-situ after death, or in-vitro.
4.2.1. Literature before 1996
In 1996, Gabriel et al [87] (hereinafter as „Gabriel-I‟) conducted a comprehensive
review, extracting the dielectric properties of tissues from the literature of the past five
decades (from 1950 to 1996), and presented the results in a graphical format. The data
reported were those that corresponded most closely to living human tissues. Therefore,
human tissue and in-vivo measurements were selected in preference to animal tissue and
in-vitro. For in-vitro measurements, data obtained at temperatures closest to that of the
body and nearest to the time after death were used. Later, they made the data available
in tabular format [88].
In this study, regardless of source of the data and measurement details, the tabulated
values from Gabriel-I are plotted as „Pre-1996‟ data in figure 4.1 to figure 4.7. Each
figure, corresponding to one tissue type, will be dealt with separately in section 4.2.3.
The 'Pre-1996' data is re-visited in this study, for comparison with those published
thereafter.
4.2.2. The 1996 database
The backbone of the 1996 database is a large experimental study, conducted by Gabriel
at al [89] themselves (hereinafter as „Gabriel-II‟), providing data pertaining to human
and animal tissues at 37 C. A comparison was made between the dielectric properties of
human tissue (autopsy material) and its equivalent in one or more animal species
42 Dielectric Properties of Head Tissues and Pathophysiology
(mostly ovine). All animal tissues used were as fresh as possible, mostly within 2 hours
of the animal‟s death. Human material was obtained 24 to 48 hours after death.
Comparisons were also made between measurements carried out in-vivo on accessible
parts of the body and in-vitro on freshly excised tissue. For most tissues, the
characterization was over a wide frequency range, 10 Hz to 20
GHz, using three
previously established experimental setups with overlapping frequency ranges (10 Hz to
10 MHz, 300 kHz to 3 GHz, 130 MHz to 20 GHz). Each frequency range was covered
by a different measurement setup and it was observed that the agreement among
measurements on the three machines was particularly good at the overlapping frequency
ranges. The data obtained in Gabriel-II were compared to those presented in the context
of a review in Gabriel-I. By and large, the experimental data were well within the
confines of corresponding values from the literature.
In a follow up study by Gabriel et al [90] (hereinafter as „Gabriel-III‟), the dielectric
spectra were parameterised using a multi-dispersion model consisting of four Cole–Cole
terms and one ionic conductivity term (equation (B.8) in Appendix B, n =
4). This was
done based on the data from their two previous studies. For each tissue, the parameters
of the model enable the reconstruction of its spectrum; a procedure that can be
incorporated in the numerical studies to provide dielectric data that are broadly in line
with the vast body of literature on the subject.
While useful, the 1996 database has several limitations, as pointed out by its authors, of
which two are particularly relevant at frequencies below 1 MHz:
Most measurements were carried out on excised tissue, while data pertaining to
live tissue would have been more relevant in bio-electromagnetics studies.
For most tissues, the predictions of the model can be used with confidence for
frequencies 1 MHz because of the availability of supporting data in the
literature.
At lower frequencies, where the literature values are scarce and have larger than
average uncertainties, the model should be used with caution in the knowledge
that it provides a „„best estimate‟‟ based on the available knowledge. This is
particularly important for tissues where there are no data to support its
predictions.
43 Dielectric Properties of Head Tissues and Pathophysiology
Electrode polarization, an inevitable source of error at low-frequencies, was not
totally accounted for. It affects the data at frequencies below 100 Hz.
Because of the geometry of the sampling probe, the electric field was not
oriented in appropriate directions, making it difficult to demonstrate the
anisotropy of the dielectric properties.
The values of measured (Gabriel-II) and modelled (Gabriel-III) dielectric properties are
plotted in figure 4.1 to figure 4.7. Each figure, corresponding to one tissue type, will be
dealt with separately in section 4.2.3.
4.2.3. Literature after 1996
Although comparisons were made between the parameterised data and those from
practical measurements in 1996 and before, in order to include the most up to date
values available, a review covering the reported data since 1996 is now provided. This
serves to extend the very comprehensive work of Gabriel-I and Gabriel-II. The review is
carried out for five main tissues in the human head; i.e. muscle, brain, skull, CSF, and
blood.
Most of the literature data were in graphical rather than tabular form, and in a
logarithmic rather than linear format; such data were retrieved from the graphs. The data
were translated from the various authors‟ preferred set of parameters and units to
relative permittivity and conductivity expressed in S m
−1. In some cases, data from
recent studies are compared with the model in the 1996 database. The data are presented
in a graphical format, in order to highlight the information with respect to the frequency
coverage, and the scatter in the data. The data of the reports giving only one or two
values are not displayed in the figures, but mentioned in the review. In each figure,
details of the tissue, measurement temperature, and the reference are included in the
legend, identified by the first author's name and date of report.
In this thesis, MIT imaging in a particular range of frequency (1-10 MHz) is of interest,
being the operational frequency range of the MK1 and Mk2a systems. However,
literature pertaining to a wider frequency range is covered. The reason for that is
twofold: (i) for some tissues, the reported data in recent studies is scarce in the
mentioned frequency range and, therefore, looking at other frequencies can help to
scrutinise the 1996 database as a whole, (ii) MIT researchers are designing systems
44 Dielectric Properties of Head Tissues and Pathophysiology
operating at lower frequencies (down to 100 kHz) and, consequently, the dielectric
properties for a wider range of frequencies can be employed.
Muscle: the following 'Post-1996' data for muscle, presented in this section, is
displayed in figure 4.1 along with the 'Pre-1996', Gabriel-II, and Gabriel-III data.
Muscle tissue can be skeletal, myocardial, lingual, or other. However, we only consider
skeletal muscles which encompass the skull. Muscles are composed of fibres that are
very large individual cells and are aligned in the direction of muscle contraction (see
figure 3.1a). Electrical conduction along the length of the fibre is thus considerably
easier than conduction between the fibres in the extra-cellular matrix because the extra-
cellular matrix is less conductive than the cell. The -dispersion is more prominent and
the -dispersion is less defined in the longitudinal direction. This is in accordance with
the predictions of effective permittivity modelling of elongated structures [91].
Therefore, muscle tissue manifests typical anisotropic electric properties.
The longitudinal conductivity is higher than the transverse conductivity, even when path
differences in the charge transport are taken into account, especially in the low
frequency range. Moreover, tissue anisotropy is frequency-dependent [86]. If the
frequency of the current is high enough, the anisotropic properties disappear. For
muscle tissue this happens in the MHz frequency range. At higher frequencies, charge
movement takes place over shorter distances thus large-scale structures become less
important and capacitive coupling across membranes becomes more important.
Among different muscle types, data on skeletal muscle is the most abundant in the
literature. As a result of the anisotropy of this tissue, the data are usually presented
separately for the transverse and longitudinal directions, although some results with
random orientation have been reported. In particular, low-frequency data are very
scarce, and there are differences in what is available in the literature, partly due to the
fact that many authors do not specify the measurement orientation. The situation is not
helped by the fact that the apparent anisotropy depends on the measurement procedure,
in particular the inter-electrode distance in relation to the size of the muscle fibre [91].
Hart et al in 1999 [92] undertook 2-terminal resistivity in-vivo measurements of the
bullfrog muscle in the frequency range of 1 kHz to 1 MHz. In addition to the analytical
45 Dielectric Properties of Head Tissues and Pathophysiology
anisotropic models, they also approximately calculated the contribution of each
direction. Based on a finite difference model, they calculated the electrical resistance
between two electrodes, inserted longitudinally or transversely into the muscle, in terms
of longitudinal and transverse muscle conductivities. With their method, they confirmed
that the longitudinal conductivity is consistently higher than that of the transverse
direction. However, the difference in conductivity between these directions appeared
smaller than in previous publications, especially for the lower frequencies considered.
Gabriel et al in 2009 [93] carried out measurements on pig tissue, in-vivo, at eight
frequencies in the range between 10 Hz to 1 MHz. Conductivity data was obtained for
skeletal muscle from average values from measurement on samples from at least three
pigs. They did not observe the expected anisotropy in electrical conductivity and values
were lower than the 1996 database. They suggested that one possible explanation was
the lack of rigorous correspondence between the geometry of the measurement probe
and the structure of the tissue.
In a study by Peyman et al in 2001 [94], the dielectric properties of ten rat tissues at six
different ages (0-70 days) were measured at 37 C in the frequency range of 130 MHz
to 10 GHz. All measurements were performed within 2-4 hours of the animal's death.
Their results showed a general decrease of the dielectric properties with age which was
attributed to the changes in the water content and the organic composition of tissues.
Values shown in figure 4.1 are the average values taken from all ages.
Brain tissues: the following 'Post-1996' data for grey and white matter, presented in
this section, is displayed in figure 4.2 and figure 4.3 along with the 'Pre-1996', Gabriel-
II, and Gabriel-III data.
Many studies have modelled the entire brain as a single homogeneous domain, for the
sake of geometric simplicity (e.g. [95]), and several others attempted to provide an
estimation for the impedance of the whole brain, as the ratio between the impedance of
scalp, skull and brain (e.g. [96]). However, the fidelity of such approximations is
doubtful, particularly for the inner regions of the head. Considering the whole brain as
one homogeneous compartment does not account for the fact that white matter is more
46 Dielectric Properties of Head Tissues and Pathophysiology
resistive than the grey matter. Furthermore, grey and white matter may be affected by
brain pathologies in different ways1.
Latikka et al in 2001 [97] measured the resistivity of brain tissues in-vivo in 9 patients
with brain tumours during surgery with a monopolar needle at a single frequency of 50
kHz. White and grey matter had mean values of 3.91 and 3.51 m (conductivities of
0.25 and 0.28 S m
-1), respectively. Their measurements were subject to several sources
of error; the bleeding, the saline used for washing, and the leakage of CSF decreased the
resistivity of the measured brain tissues. On the other hand, the increase in the
intracranial pressure caused by the tumour increased the resistivity values. The authors
noted that the measured values differ from those given by Gabriel-I (figure 4.2 and
figure 4.3). They ascribed the differences to the fact that their measurements were done
in-vivo, thus more reliable.
Data by Peters et al in 2001 [98] were obtained by modelling the brain cortex as a
suspension of randomly distributed elongated cells suspended in CSF. Assuming the
extra-cellular volume fraction to be between 0.18 (for a rat in-vivo) and 0.28 (for a pig
in-vitro) and the conductivity of the CSF to range between 1.5 and 1.8 S m
−1, the
effective conductivity of the mixture fell to within the range 0.1-0.27 S m
−1.
In a human study by Schmid et al in 2003 [99], the dielectric properties of grey matter
in the frequency range of 800-2450 MHz were measured on 20 human brains
immediately after excision, less than 10 hours after death. The tissue temperature at the
measurement sites ranged between 18 and 25 C, and 160 different measurements were
performed on each brain. The measurements yielded a mean value (standard deviation)
of grey matter equivalent conductivity of 1.13 and 2.09 S m
-1 at 800 and 2450 MHz,
respectively. The mean value of measured relative permittivity was 58.2 and 54.7 at 800
and 2450 MHz, respectively.
Peyman et al [94] reported variation in the dielectric properties of rat brain tissue as a
function of age, at microwave frequencies. Their data pertained to the whole brain. The
observed variation was ascribed, at least partially, to the change in the ratio of grey to
white matter, which is known to occur throughout the developmental stage.
1 Blood flow in white matter is lower than in grey matter, and white matter ischaemia is typically severe,
because there is little collateral blood supply in deep white matter. Moreover, cells within the grey and
white matter have different susceptibilities to ischaemic injury [231].
47 Dielectric Properties of Head Tissues and Pathophysiology
In another study on porcine tissues [100], they were able to investigate grey and white
matter separately. In this case, no variations were observed in the dielectric spectrum of
grey matter, while statistically significant variations were observed in the dielectric
spectrum of white matter. They suggested that the observed variations were probably
related to the process of myelination1, which begins at birth and lasts to maturation. In
another study in 2007 [101], the dielectric properties of pig cerebrospinal tissues were
measured in-vivo, in the frequency range of 50 MHz - 20 GHz. Comparisons were made
for the measured values of white and grey matter with the 1996 database, and both
conductivity and permittivity values were found to be in good agreement.
In a recent study in 2009 [102], dielectric properties of ageing porcine tissues were
measured, in-vitro, in the frequency range of 50 MHz - 20 GHz. Their results showed
considerable reduction with age in both permittivity and conductivity of white matter,
where dielectric properties of grey matter remained unchanged. In figure 4.2 and figure
4.3, measured values are shown for 50 kg pigs.
Bao et al in 1997 [103] presented in-vitro complex dielectric measurements of grey and
white matter of rat brains in the frequency range between 45 MHz and 26.5 GHz at
body and room temperatures (24 and 37 C). The measurement data exhibited two
separated dispersions and they suggested an empirical model containing two Cole-Cole
functions to describe the experimental data. In figure 4.2, the measured data at 37 C is
shown.
Bone: the following 'Post-1996' data for bone, presented in this section, is displayed in
figure 4.4 and figure 4.5 along with the 'Pre-1996', Gabriel-II, and Gabriel-III data.
Bone, cortical or cancellous, has an anisotropic structure and, in consequence,
anisotropic dielectric properties. For cortical bone, the ratio of the conductivity in the
axial and radial dimensions is about 3.2 [93]. Cancellous bone is less anisotropic; the
conductivity is higher in the longitudinal direction compared to the lateral and anterior–
posterior directions. While the skull impedance is not affected by brain pathology or
functional activity, the resistivity of the skull affects the EIT measurements as it acts as
1 The change or maturation of certain nerve cells whereby a layer of myelin forms around the axons
which allows the nerve impulses to travel faster.
48 Dielectric Properties of Head Tissues and Pathophysiology
a current barrier. However in MIT imaging, conductivity of the skull is of less
importance as the magnetic field can penetrate the electrically-resistive bones.
Oostendorp et al [104] performed both in-vivo and in-vitro 4-terminal skull
measurements in the frequency range 100 Hz to 10 kHz. The in-vitro measurement was
performed on a sample of fresh skull placed within a saline environment in which the
temperature was maintained at 37 C. For the in-vivo measurement, a small current was
passed through the head by means of two electrodes placed on the scalp. The potential
distribution thus generated on the scalp was measured in two subjects for two locations
of the current injecting electrodes. Both methods revealed a skull conductivity of about
0.015 0.003 S m
-1 for the mentioned frequency range.
The dependence of the conductivity on the position and bone type was investigated by
Akhtari et al in 2002 [105], during in-vivo measurements using 4-terminal technique at
10-90 Hz on skull flaps, excised from 4 patients during surgery, with a 15-65 minutes
delay between excision and measurement. Holes were drilled in the samples to allow the
resistivity to be measured at the different layers in the bone sample. The results
indicated that the conductivities of the spongiform and compact layers of the skull have
significantly different and inhomogeneous conductivities.
A year later in 2003, the frequency dependence of the conductivity of different layers
and the bulk was measured and modelled in the same frequency range [106]. The
samples were cut from temporal, temporal-parietal and inferior occipital regions of the
skull (these regions are shown in figure 3.2).
Further conductivity measurements were made in 5 temporal bone samples temporarily
removed during epilepsy surgery by Hoekema and Wieneke in 2003 [107]. The
conductivity values were about 10 times higher than any previously published data. This
difference could be due to the fact that the measurements were made a very short time
after excision, while other samples have been dried and then re-soaked. The high
conductivity could also be due to a thin layer of saline on the surface of skull.
Sierpowska et al in 2003 [108] demonstrated the dependency of volumetric trabecular
bone mineral density and its relative permittivity. The dielectric and mechanical
properties of femoral bovine trabecular bone samples from various regions were
measured. The bones were frozen and defrosted prior to the measurement procedure.
49 Dielectric Properties of Head Tissues and Pathophysiology
The frequencies were between 50 Hz and 5 MHz, and the conductivity values ranged
from 0.02 to 0.06 S m−1
depending on the site. This compares with 0.07 S m−1
for ovine
cancellous bone in the 1996 database.
Later in 2005 [109], they measured the impedance of cadaver human trabecular knee
bone in the same frequency range. Samples of tibia and femoral trabecular bone were
defrosted to room temperature (22 C) after being kept frozen for 12 months. While
there were no significant differences between the tibia and the femoral bone throughout
the entire frequency range, human trabecular bone was found to be twice as conductive
as bovine. The authors have recently investigated the relationships between electrical
properties and microstructure of human trabecular bone [110,111].
Peyman et al in 2001 [94] observed variation in the dielectric properties of rat skull
bone as a function of developmental stage from neonate to 70 days old; no reference
was given to the type of the bone. In figure 4.4 and figure 4.5, the results are shown for
two frequencies (900 MHz and 1.8 GHz) as the average values of those with different
ages. Similar results were recently reported for porcine skull [100].
Gabriel et al in 2009 [93] carried out in-vivo measurements on the mid-sectional part of
the skull of 50 kg pigs: this part of the skull has flat areas which allow good contact
with the electrodes. The bone exposed after stripping the periostium membrane
appeared pitted and filled with red bone marrow. The measured conductivities, at eight
frequencies in the range of 10 Hz to 1 MHz, were higher than previously reported data
for bone. However, a recent in-vitro study, conducted by Peyman et al in 2009 [102],
showed a significant reduction with age in both permittivity and conductivity at 37 C.
The skull bone of 250 kg pigs (mature animals) had much lower conductivities. At the
frequency of 450 MHz, the skull conductivity values for 10, 50, and 250 kg pigs were
0.62, 0.52, and 0.19 S m
-1, respectively.
CSF: the following 'Post-1996' data for CSF, presented in this section, is displayed in
figure 4.6 along with Gabriel-II and Gabriel-III data (there was no 'Pre-1996' data
reported for CSF).
As an ionic fluid containing no cell membranes, CSF is expected to have a high
conductivity. Baumann et al in 1997 [112] measured the properties of human CSF using
50 Dielectric Properties of Head Tissues and Pathophysiology
the 4-terminal technique in a Perspex measuring cell at 10 Hz - 10 kHz. The samples
were frozen and then defrosted. They compared the measured values at room
temperature (25
C) with those obtained at body temperature (37
C). The measured
conductivity values at body temperature (1.45 S m
-1) were in better agreement with
previous data than those at room temperature (1.8 S m
-1).
Latikka et al in 2001 [97] measured the CSF resistivity in-vivo in 2 patients with brain
tumours during surgery with a monopolar needle at a single frequency of 50 kHz; it had
a mean value of 0.80 m (conductivity of 1.25 S m
-1).
Peyman et al in 2007 [101] measured the dielectric properties of pig CSF, in-vitro, in
the frequency range of 50 MHz - 6 GHz. Gabriel et al in 2009 [93] carried out
conductivity measurements on pig tissue, in-vitro, at different frequencies in the range
of 10 Hz to 1 MHz. This study was designed to obtain data at frequencies below 1 MHz
to supplement the 1996 database. The results of both studies were in good agreement
with Gabriel-III, albeit the conductivity values were slightly smaller (see figure 4.6).
Blood: the following 'Post-1996' data for blood, presented in this section, is displayed in
figure 4.7 along with the 'Pre-1996', Gabriel-II, and Gabriel-III data.
Blood is a circulating tissue composed of fluid plasma and cells. The plasma, which
comprises 55% of blood fluid, is mostly water (90% by volume). Apart from the
plasma, the blood consists of several kinds of corpuscles; about 96% of them are red
blood cells (erythrocytes), 3% are white blood cells (leukocytes), and the remainder are
platelets (thrombocytes). In haemolysed blood, erythrocytes are disrupted and their
intra-cellular material is discharged into the liquid. Therefore, the electrical properties
of whole blood and haemolysed blood are naturally very different.
Whole blood exhibits - and -dispersion, but no -dispersion [113]. The -dispersion
has a dielectric increment of about 2000, and an accompanying rise in conductivity by
0.3 S m-1
, centred around 3 MHz (haematocrit1 40%). Erythrocytes in suspension have
a frequency independent membrane capacitance with very low losses. The impedance of
lysed erythrocytes in suspension shows two clearly separated single relaxation
1 Haematocrit is the volume proportion of blood occupied by red blood cells.
51 Dielectric Properties of Head Tissues and Pathophysiology
frequencies (Debye dispersions as described in Appendix B). As will be used in later
chapters, the mentioned increment in dielectric properties of blood, centred around 3
MHz, can be exploited in frequency-differential MIT.
The -dispersion is in the lower kHz range, and the -dispersion is in the lower MHz
range. Erythrocytes are poor conductors of electric current at low frequency compared
with the extra-cellular medium or plasma which surrounds them. The conductivity of
the blood has almost linear dependence on haematocrit, so the concentration of
erythrocytes critically determines the blood‟s resistivity.
Gabriel et al in 2009 [93] carried out blood conductivity measurements on pigs
weighing around 50 kg, at different frequencies in the range of 10 Hz to 1 MHz, and
temperature of 37 C. Blood was sampled in-vitro immediately after extraction by
immersing the probe in the blood. The measured conductivities were in good agreement
with previously reported data.
Jaspard et al in 2003 [114] studied the influence of haematocrit on the dielectric
properties of animal blood, cow and sheep, in the frequency range of 1 MHz to 1 GHz
and temperature of 37 C. They found that an increase in haematocrit level induced an
increase in the number of cell membranes with a decrease in the volume of plasma.
Similarly, when haematocrit increased, a drop of conductivity and an increase of
relative permittivity were observed. Figure 4.7 shows the measured dielectric values for
41 and 42% haematocrit in cow and sheep blood, respectively.
In another study [115], they investigated the temperature dependence of the electrical
parameters of human and animal blood in the same frequency range. The results
indicated a weak sensibility and a change of sign of the temperature coefficient of the
relative permittivity (about 0.3% C−1
at 1 MHz and -0.3% C−1
at 1 GHz). The
conductivity presented a more significant variation (of the order of 1% C−1
) over the
whole operating frequency range.
52 Dielectric Properties of Head Tissues and Pathophysiology
100
102
104
106
108
1010
1012
100
102
104
106
108
Perm
ittivity
Muscle
100
102
104
106
108
1010
1012
10-2
10-1
100
101
102
Frequency (Hz)
Conductivity
Figure 4.1. Dielectric properties
of muscle against frequency;
longitudinal and transverse
directions.
Conduct
ivit
y (
S m
-1)
Rel
ativ
e p
erm
itti
vit
y
53 Dielectric Properties of Head Tissues and Pathophysiology
100
102
104
106
108
1010
1012
100
102
104
106
108
Perm
ittivity
Grey Matter
100
102
104
106
108
1010
1012
10-2
10-1
100
101
102
Frequency (Hz)
Conductivity
Figure 4.2. Dielectric properties of
grey matter against frequency.
Conduct
ivit
y (
S m
-1)
Rel
ativ
e p
erm
itti
vit
y
54 Dielectric Properties of Head Tissues and Pathophysiology
100
102
104
106
108
1010
1012
100
102
104
106
108
Perm
ittivity
White Matter
100
102
104
106
108
1010
1012
10-2
10-1
100
101
102
Frequency (Hz)
Conductivity
Figure 4.3. Dielectric properties of
white matter against frequency.
Conduct
ivit
y (
S m
-1)
Rel
ativ
e p
erm
itti
vit
y
55 Dielectric Properties of Head Tissues and Pathophysiology
100
102
104
106
108
1010
1012
100
102
104
106
108
Perm
ittivity
Cancellous bone
100
102
104
106
108
1010
1012
10-2
10-1
100
101
102
Frequency (Hz)
Conductivity
Figure 4.4. Dielectric properties of
cancellous bone against frequency.
Conduct
ivit
y (
S m
-1)
Rel
ativ
e p
erm
itti
vit
y
56 Dielectric Properties of Head Tissues and Pathophysiology
100
102
104
106
108
1010
1012
100
101
102
103
104
105
Perm
ittivity
Cortical bone
100
102
104
106
108
1010
1012
10-3
10-2
10-1
100
101
Frequency (Hz)
Conductivity
Figure 4.5. Dielectric properties of
cortical bone against frequency.
Conduct
ivit
y (
S m
-1)
Rel
ativ
e p
erm
itti
vit
y
57 Dielectric Properties of Head Tissues and Pathophysiology
100
102
104
106
108
1010
1012
100
101
102
103
Perm
ittivity
CSF
100
102
104
106
108
1010
1012
100
101
102
Frequency (Hz)
Conductivity
Figure 4.6. Dielectric properties of
CSF against frequency.
Conduct
ivit
y (
S m
-1)
Rel
ativ
e p
erm
itti
vit
y
58 Dielectric Properties of Head Tissues and Pathophysiology
100
102
104
106
108
1010
1012
100
101
102
103
104
Blood
Perm
ittivity
100
102
104
106
108
1010
1012
10-1
100
101
102
Frequency (Hz)
Conductivity
Figure 4.7. Dielectric properties of
blood against frequency.
Conduct
ivit
y (
S m
-1)
Rel
ativ
e p
erm
itti
vit
y
59 Dielectric Properties of Head Tissues and Pathophysiology
4.3. Pathophysiology and conductivity change
The brain cells can be modelled as a group of three compartments: extra-cellular space,
intra-cellular space, and the cell membrane. Both the extra- and intra-cellular spaces are
highly conductive, because they contain salt ions [116]. The lipid membrane of cells is
an insulator, which prevents currents at low frequencies from entering the cells; see
figure 4.8 for normal condition at low frequencies. Therefore, almost all the current
flows through the extra-cellular space only. Due to the cell membrane permeability at
high frequencies, the current can cross the capacitance of the cell membrane, and so
enter the intra-cellular space as well.
Ischaemia: as discussed in chapter 3, ischaemic stroke occurs due to inadequate blood
supply to part of the brain, initiating the ischaemic cascade. Brain tissue ceases to
function if deprived of oxygen for more than 60 to 90 seconds and after a few hours will
suffer irreversible injury possibly leading to death of the tissue, i.e. infarction. The
ischaemic cascade is a series of biochemical reactions that take place in the brain and is
typically secondary to stroke. It results in cerebral oedema, i.e. swelling of the brain
cells as shown in figure 4.8. This is due to the leakage of large molecules from blood
vessels [117]. These large molecules pull water into the brain tissue after them by
osmosis. Oedema causes compression and damage to the brain tissue.
Figure 4.8. Current flow through cells at both low and high frequencies. Due to ischaemia and
subsequent oedema, brain cells swell resulting in impedance increase at low frequencies. ECS:
extra-cellular space (low-frequency and high-frequency diagrams are redrawn from Holder [116]
and Holder [118], respectively).
Normal (20% ECS) Ischaemia ( 15% ECS)
Current
Cells
normal & swollen
low
frequencies
high
frequencies
60 Dielectric Properties of Head Tissues and Pathophysiology
The extra-cellular space of grey matter is about 20%, and even a small change in its
volume results in a considerable impedance increase; average reduction of extra-cellular
space from the normal can be 4-10% [119]. This effect is most pronounced at low
frequencies as the current flows cannot penetrate the cell membrane.
In summary, the dielectric behaviour of ischaemic tissue can be explained as follows:
During ischaemia there is a shift in fluid volume from the extra-cellular space to
intra-cellular space. This leads to increases in impedance at the low frequencies
due to the capacitive nature of the cell membranes. Thus, there is an increase in
impedance in the - and -dispersion regions due to ischaemia.
The impedance at high frequencies (in or around the -dispersion region) is
relatively unaffected by ischaemia, tissue excision, and tissue death. This is
largely due to the fact that the existence of cellular membranes does not affect
the high frequency measurements.
The -dispersion breaks down much more quickly than the -dispersion
(typically within a few hours after the onset of ischaemia).
Based on the above evaluations of the three different dispersion regions, it may be
concluded that the - and/or -region would be the most suitable for investigating the
electrical properties of ischaemic tissue. A number of studies demonstrated the increase
in tissue resistivity after the onset of ischaemia. For instance, Van-Harreveld and Ochs
[120] measured cerebral impedance of rabbit brain using a 2-terminal method, during
circulatory arrest. The measurements were performed at frequency of 1 kHz. Five
minutes after circulatory arrest, there was a sudden large drop in conductivity, assigned
to the development of a cortical negativity. After this initial rapid decrease, the
conductivity continued to drop at a decreasing rate. After 4-5 hours, the conductivity
became only 20-30% of the original value. The reactance change was smaller than that
of the resistance in amplitude though similar in terms of percentage change.
Holder [121] measured in-vivo, in-situ cerebral impedance changes in rats during
ischaemia. Cortical and scalp electrode arrays were positioned in the frontal occipital
plane. Four-terminal measurements were performed at frequency of 51 kHz. Cortical
measurement showed an impedance increase of 50-200%, whilst the scalp
measurements resulted in 10% of that increase (i.e. 5-20%).
61 Dielectric Properties of Head Tissues and Pathophysiology
Figure 4.9. Current flow through cells. Due to haemorrhage, increase in conductive blood volume
results in impedance decrease. BV: blood volume (diagram redrawn from Holder [118]).
Haemorrhage: haemorrhagic strokes result in tissue injury by causing compression of
tissue from an expanding haematoma. Two pathophysiological mechanisms may occur
during haemorrhagic stroke:
a ruptured vessel causes a focal or diffused blood volume expansion, which
results in a small local impedance decrease due to the high conductivity of the
blood
a rapid ischaemia of the surrounding brain region due to a loss of cerebral blood
flow and increased intracranial pressure
Therefore, a haemorrhage may appear as an inner bleed, surrounded by the ischaemic
tissue. No measurement studies appear to have been published to directly quantify the
effect of haemorrhage on the dielectric properties of the brain tissues. The physiology
suggests that the inner part of the lesion will have a mixed impedance value of the
original brain tissue and blood according to their volume proportions [74]. An increase
in the conductive blood volume will allow more current to flow, resulting in a decrease
in the bulk impedance of the region affected by haemorrhage (see figure 4.9).
The extent of blood saturation depends on the severity of stroke. For moderate cases,
the whole extra-cellular fluid will be displaced by blood, which can account for 20%
volume in grey matter. In severe cases, the original tissue could be deformed out by the
blood and, therefore, the affected region will contain blood only; a number of MIT
modelling studies have replaced the whole brain tissue with blood (e.g. [46]). While the
inner part of the region will be saturated with blood to some extent, the surrounding
layer will comprise ischaemic matter. Further studies are required in order to accurately
quantify the dielectric changes expected in this pathology.
Normal (10% BV) Haemorrhage (>10% BV)
Blood Cells
Current
62 Dielectric Properties of Head Tissues and Pathophysiology
4.4. Can MIT be applied for stroke imaging?
For haemorrhagic stroke, two types of differential imaging in clinical MIT may be of
interest: time-difference and frequency-difference. The rationale for using time-
difference MIT is based on the fact that the conductivity of blood is higher than that of
normal brain tissues, except CSF. For instance, according to the data from section 4.2.3,
blood is between 3.6 and 6.9 times more conductive than grey and white matter at a
frequency of 10 MHz. Therefore, the larger induced eddy currents in the haemorrhagic
region and the resulting MIT signals shall be distinguishable from those of a normal
brain. In other words, time-difference imaging should be able to image the conductivity
rise in the brain from before to after the onset of a haemorrhage. This could in principle
be useful for long-term monitoring at the bedside for patients at risk of recurrence or
progression of a haemorrhage; see the example of haemorrhage growth within about 4.5
hours in figure 4.10.
Figure 4.10. CT scan showing haemorrhage growth in a patient with basal ganglion intra-cerebral
haemorrhage (after Adeoye and Broderick [122]).
According to Joseph [117], multiple factors contribute to the importance of monitoring
for such patients:
Many strokes occur while patients are sleeping (also known as "wake-up"
stroke) and are not discovered until the patient wakes.
Stroke can leave some patients too incapacitated to call for help.
Occasionally, a stroke goes unrecognized by the patient or their carers.
63 Dielectric Properties of Head Tissues and Pathophysiology
An MIT device employed for stroke monitoring could set off an alarm when some
abnormal signals are detected.
For an initial diagnosis, however, frequency-difference imaging will be necessary since
a „before-stroke‟ data frame will not be available. Multi-frequency imaging relies on
being able to distinguish the „frequency signature‟ of the blood from that of other
tissues, all of which will have frequency-dependent conductivities.
The conductivity increase in white and grey matter between 1 to 10 MHz is
approximately 0.06 S m
-1 (figure 4.3) and 0.13 S
m
-1 (figure 4.2), respectively, while
blood has a conductivity change of 0.3 S m
-1 (figure 4.7). By obtaining MIT
measurements at two frequencies, it should be possible to reconstruct an image of the
conductivity difference between the two frequencies.
It would be desirable for a clinical MIT system also to be able to detect ischaemic
stroke. This will be more difficult as the existing MIT systems are operated in the
frequency range of 1-10 MHz to increase the signal to noise ratio (see section 2.3.2). In
this frequency range, -dispersion remains unaffected by the ischaemia and, therefore,
no significant impedance change occurs. Consequently, normal and ischaemic brain
tissues are likely to produce similar MIT signals.
Extension of the range of biomedical MIT systems down to 100 kHz is a realistic goal
(e.g. system reported by Scharfetter et al [123], operating at 50 kHz - 1.5 MHz). This
may allow the impedance change in -dispersion region of the brain tissue to be
exploited for MIT imaging of an ischaemic region. Even so, an MIT system capable of
detecting just haemorrhagic stroke could still be of clinical value for excluding
haemorrhage prior to administration of drugs. In this study, therefore, the intention was
to investigate MIT imaging of haemorrhagic stroke.
Absolute imaging at a single frequency may also be feasible. However, as will be
shown in chapters 6 and 9, it will be more difficult due to the fact that the errors in the
data have a larger effect on the images, and the computational time for the solution of
the inverse problem can be considerably longer.
64 Dielectric Properties of Head Tissues and Pathophysiology
4.5. Adopted values for simulations and discussion
Considering the reported dielectric properties for three main tissues in the head (brain,
CSF, and blood), there appears to be a good agreement between the suggested values by
Gabriel-III and those from the measurement studies (pre-1996, Gabriel-II, and post-
1996); evident in figure 4.1 to figure 4.7. However, there is considerable scatter of the
data for other tissues which show degrees of anisotropy, namely bones and muscle. A
critical review which investigates the reasons of this scatter is outside the scope of this
study; this can be due to the measuring techniques, sample tissues, sample age,
temperature, etc. However, by looking at different studies, the fidelity of the 1996
database can be assessed.
For skeletal muscle, there are considerable differences between the longitudinal and
transverse values from each study. Furthermore, various studies reported different
values for dielectric properties, in particular, conductivity (figure 4.1). However, both
discrepancies diminished as the frequency increased. If the frequency of current is high
enough (MHz range as in the present MIT systems), the anisotropic properties and
scatter of the data are much less important.
The orientation of the measuring electrodes relative to the axis of the bones (i.e.
longitudinal and transversal) was not specified in the reported studies and, therefore, the
level of anisotropy cannot be estimated for the bones. Noticeable differences exist
between the conductivities of the cancellous and cortical bones, with cancellous being
relatively more conductive, and the conductivities reported by various studies for each
type of bone. However, similar to muscle, the discrepancies appear to become
significantly smaller at higher frequencies (figure 4.4 and figure 4.5).
Therefore, based on the data for five important tissues in the head, it was decided that
the values of the dielectric properties suggested by Gabriel-III can be established as the
main reference for the frequency range of interest in this study, i.e. 1-10 MHz. For
inhomogeneous tissues such as bone, there are no anatomical maps to accurately model
the inhomogeneity, and here a homogeneous domain was considered to represent the
tissue. The values of the properties were calculated as weighted average of those of the
sub-tissues; in this case 50% cortical and 50% cancellous bone.
65 Dielectric Properties of Head Tissues and Pathophysiology
Similarly for those organs, e.g. eye balls, for which there are no reported properties as a
whole organ, an estimation was made based on the weighted average values of sub-
tissues' properties. Table 4.1 provides a list of such tissues/organs together with the
details of sub-tissues, and estimated values at a frequency of 10 MHz. The dielectric
properties of the domains existing in the head model are plotted in figure 4.11 for the
frequency range of 100 kHz-100 MHz. The assignment of appropriate dielectric
properties to the tissues in the head model can allow a more realistic estimation of the
expected MIT signals in the simulations. Further description of the head model is
provided in chapter 7.
Table 4.1. Weighted average dielectric properties assigned to heterogeneous or minor organs in the
head (frequency of 10 MHz).
Organ/tissue Constituent Conductivity
(S/m)
Relative
permittivity
Skull 50% cortical, 50% cancellous 0.0828 53.8
Spinal cord 33.3% white matter, 33.3% grey
matter, 33.3% nerve cells 0.225 248
Eye ball 60% vitreous humour, 10% retina,
10% sclera, 10% cornea, 10% lens 1.20 127
Ventricles 100% CSF 2.00 109
Thalamus &
hypothalamus 100% white matter 0.159 176
Basal ganglia 100% white matter 0.159 176
Cavities 100% air 0 1
Olfactory organ 100% air 0 1
Cerebral haemorrhagic lesions were also assumed to be isotropic and homogeneous
regions with blood and brain tissues as its constituents. Different percentages of blood
volume in the haemorrhage were considered in order to calculate the weighted average
of the lesion properties; this ranged 25-100 % representing moderate to severe strokes.
As previously discussed, a haemorrhage can appear as a bleeding region surrounded by
the ischaemic tissues. Any possible ischaemia surrounding the lesion was ignored as the
tissue properties are usually considered to undergo negligible conductivity changes at
MHz frequencies.
66 Dielectric Properties of Head Tissues and Pathophysiology
End of chapter 4.
105
106
107
108
101
102
103
104
105
106
107
108
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Figure 4.11. Dielectric properties of head
tissues used for simulations.
Conduct
ivit
y (
S m
-1)
Rel
ativ
e p
erm
itti
vit
y
Frequency (Hz)
Part II
Discretized Forward and Inverse
Problems
Chapter 5.
Discrete Forward Problem
5.1. Introduction
As stated in section 2.2, the purpose of solving the MIT forward problem is to predict
the measured signals for a given geometry, distribution of dielectric properties,
operating frequency, and coil excitation current. There are two different ways of solving
such problems: analytical and numerical methods. Although theoretically exact,
analytical methods have the major drawback of the lack of generality. Analytical
solutions may handle two-dimensional problems, and in certain cases three-
dimensional, as long as the geometry and the boundary conditions are simple, and the
material properties are isotropic.
When exact analytical solutions are not available, approximate methods are sought.
Numerical solutions may handle any geometrical configurations and both isotropic and
anisotropic materials. When the problem is solved numerically, the model of the
physical system must be discretized, requiring an approximate formulation of the
problem. There are several numerical techniques that can be employed. However, any
numerical field computation involves the following steps:
Definition of the boundary value problem (BVP) that includes describing the
region over which the governing equations are to be solved and the boundary
conditions which should be applied.
69 Discrete Forward Problem
Formulation of the discrete problem by sub-dividing the domain into a series of
points or elements, and introducing an approximate variation of the field over
the elements.
Derivation of a system of equations whose approximate solution gives the
numerically computed approximate field.
The purpose of this chapter is to introduce two of the most established numerical
methods, finite element (FE) and finite difference (FD), which are employed for the
simulations in this study. The basis of each method is briefly explained and its
implementation for the MIT simulation is discussed.
In order to assess the performance of these methods in solving the forward problem, a
benchmark test is carried out, and the numerical results are compared against each other
and with that from the analytical solution. The starting point for these numerical
methods is the BVP in (2.8), introduced in section 2.2.
5.1.1. BVP simplification
Before embarking on the numerical approximation of (2.8), it is transformed into an
alternative form and a number of approximations are made. There are a number of
alternative approaches to solving (2.8), which express the problem either in terms of
primal physical field variations (e.g. E and H), or in terms of non-physical potentials.
One common approach to solve the problem is by introducing the vector potential A as
B = ×A. Considering that E = -j B, the electric field then becomes:
(5.1)
where is the electric scalar potential. Now realising that H = -1
A and by
substituting it into (2.4) we have:
(5.2)
Also, taking the divergence of (5.2) gives:
(5.3)
The equations (5.2) and (5.3) are known as the retarded potential formulation.
70 Discrete Forward Problem
Any A and can be chosen such that they give rise to the physical field B. This is
known as the gauge. To completely define the potentials and obtain a unique solution,
an appropriate gauge is required. Possibilities for gauging include the Coulomb gauge,
.A = 0, or the temporal gauge, = 0 (for further details see [124]).
By Adopting the temporal gauge, the resulting vector wave equation, known as the A-
formulation, can be derived from (5.2) as:
(5.4)
Ledger and Zaglmayr [124] have used this formulation to discuss the eddy-current
problems on multiply connected domains.
For alternative approaches, which express the forward problem in terms of a vector
wave equation for electric field E, see e.g. [34,125,126]. It must be noted that, at this
point, solving either (5.4) or a vector wave equation is equivalent to solving the first
order Maxwell's equations.
Explicitly, the forward BVP for MIT, expressed in terms of A, can be written as:
Find A such that
(5.5)
where [...] denotes the jump, and vector n is a unique outward normal at each boundary
or interface. The domain of the problem, , consists of a conducting region C, and a
non-conducting region N, so that = C N (see figure 2.1 in section 2.2.1).
Assuming that the wave propagation is negligible in N, by choosing N to be large
enough so that n A = 0 can be applied on , the approximate BVP for the forward
problem then becomes:
71 Discrete Forward Problem
Find A such that
(5.6)
By solving this problem, the A-field can be estimated everywhere in , and the induced
signals in the sensor coils can be computed by:
(5.7)
where dl is the length element of the coil.
5.2. Finite element method
The finite element method (FEM) is a numerical technique that is used to solve a BVP
governed by a differential equation and a set of boundary conditions
[127,128,129,130,131,20]. The FEM originated from the need for solving complex
elasticity and structural analysis problems in civil and aeronautical engineering.
However, it has now become widely adopted as a means of the numerical
approximation in many problems in science and engineering.
The main idea behind the method is the representation of the domain with smaller
subdomains called the finite elements (FE). The distribution of the primary unknown
quantity inside an element is interpolated based on the values at the nodes, provided that
nodal elements are used, or the values at the edges, in case that vector elements are
used; vector elements are a special kind of FE and are introduced later in this section.
The interpolation or shape functions must be a complete set of polynomials. The
accuracy of the solution depends not only on the size and distribution of the elements,
but also on the order of these polynomials, which may be linear (first-order), quadratic
(second-order), or higher order.
72 Discrete Forward Problem
The numerical solution corresponds to the values of the primary unknown quantity at
the nodes or the edges of the discretized domain. The solution is obtained after solving a
system of linear equations. To form such a linear system of equations, the governing
differential equation and associated boundary conditions must first be converted to an
integro-differential formulation by, for example, minimizing a weighted-residual
method such as the Galerkin approach [132,133]. This integro-differential formulation
is applied to a single element and with the use of proper weight and interpolation
functions the respective element equations are obtained. The assembly of all elements
results in a global matrix system which represents the entire domain of the BVP [130].
5.2.1. FE discretization
The weak form of the BVP in (5.6) can be shown to be:
Find A H0(curl) such that
for all W H0(curl), where
(5.8)
where W is a test function [17]. One may attempt to approximate each vector
component of A separately using the common nodal shape functions. However, such an
approach leads to nonphysical solutions, referred to as spurious modes [129,131]. This
can be avoided by using a special kind of FE called edge elements, also known as vector
or Nédélec elements [129,134,135]. These elements are very well suited for
approximating electromagnetic fields. The edge elements associate the degrees of
freedom to the edges of the mesh rather than the nodes.
The shape functions for edge elements are constructed such that their tangential
components are continuous across the elements' borders, whereas their normal
components are allowed to be discontinuous. In the case of the lowest order edge
elements, the vector potential A can be expressed as:
73 Discrete Forward Problem
(5.9)
where M is the number of the edges in the mesh. Note that the shape functions Ni are
vectors, and coefficients Ai are scalars that associated with the tangential components of
A on the edges of elements. Constructing the approximation (5.9) in a piecewise manner
over the domain, and using the method of Galerkin weighted residuals where Wi = Ni,
the weak form in (5.8) leads to a linear system of equations as:
(5.10)
where E is the number of elements in the mesh, e is the volume of each element, and
à = (A1,A2,...AM)T are the unknown coefficients associated with the edge element vector
shape functions.
5.2.2. Comsol Multiphysics
A commercial FE package, Comsol Multiphysics [136], was adopted to solve the MIT
forward problem. Comsol has been previously used for simulations in MIT (e.g. [50]).
The „AC/DC‟ module of Comsol is suitable for solving (5.8) and employs the FE
discretization described in 5.2.1. The AC/DC module of Comsol offers the possibility to
use elements of different order applied uniformly throughout the mesh. In the case of
higher order edge elements, the elements have additional degrees of freedom associated
with the edges and faces of the elements.
Unstructured tetrahedral FE elements, which are available in Comsol, are particularly
suitable for representing the complex anatomy of the head as automatic mesh generation
procedures are available for generating a series of elements. Hexahedral and or prism
elements are also available in Comsol, but for these elements, the construction of the
mesh is more difficult.
74 Discrete Forward Problem
Adopting Comsol allowed a faster development of the model to estimate the signals for
feasibility tests. However, the use of commercial packages such as Comsol has several
drawbacks which are discussed later in this chapter.
5.2.3. Model construction
The applied current density Js is introduced at the exciter coils. Rather than modelling
the coils as toroids, they were modelled as collection of edges forming a closed loop.
This approach, which approximates coils as 1D entities embedded in 3D space, does not
require the meshing of a separate region for the coils. The source currents were applied
as boundary conditions; the exciter coils are assumed to carry an impressed current with
a predetermined tangential component. Therefore, Js in (5.8) may be set to zero. When
employed as a sensor, the induced e.m.f. around the coil was computed using (5.7).
There are a number of ways in which the screen could be approximately modelled. Let
us first estimate the skin depth in the aluminium screen. Given = 3.77 107 S
m
-1 and
r = 1, the skin depth for the screen at the frequency of 10 MHz is 27 m which is very
small compared to the thickness of the screen (2 mm).
One possibility is to discretize the screen as an additional material, with its individual
3D domain, using tetrahedral elements. However, given the skin depth of the screen,
this would require extremely small elements to be generated in this region to capture the
variation of the electromagnetic field. Beyond the screen, the region of free space would
then extend to the truncating boundary and, in a similar manner to the other material
interfaces, the interface conditions would be naturally handled by the edge element
formulation.
In order to avoid the generation of such small elements within the screen, an alternative
approach can be employed. This consists of truncating the overall domain at the inner
surface of the screen so that the outer boundary becomes = S , and imposing a
suitable boundary condition on S. If the screen is modelled as a perfect electrical
conductor, which corresponds to setting n A = 0 on S, the skin effects in the screen
will not be taken into account. A more physically realistic approach, adopted in this
study, is to model a short distance penetration of the fields inside the screen. This can be
realised by applying the impedance boundary conditions (IBC) as:
75 Discrete Forward Problem
(5.11)
on S [137,138,139,23]. In (5.11), the dielectric properties are those for the screen. This
boundary condition is implemented by adding a boundary integral of the form:
(5.12)
to the left-hand side of (5.8), and substituting (5.11) into the integral. Since the
boundary condition involves unknown values of the electric field, this boundary integral
form will contribute to the FE system matrix. Comsol offers the possibility of using the
IBC in the model.
At the open ends of the screen (upper and lower as, for example, in the MK1 system),
the truncated infinite region N extends to the artificial boundary . The boundary
conditions that the AC/DC module of Comsol offers for imposing on are suitable for
modelling the decay of static fields in infinite regions. These boundary conditions are
appropriate if the displacement current is neglected as in the eddy-current problem
which is a magnetostatic problem in the non-conducting region. In such cases, theory
predicts that far away from the source, A should be zero and decay at the rate of r-2
(as
described by (2.11)), with r being the distance from the source. Therefore, placing the
artificial truncation boundary sufficiently far from the target and coils, and imposing
either n A = 0 or n A = 0 allows a reasonable approximation to the decay of the
static fields to be obtained.
Following this approach often leads to the requirement for excessively large
computational domains, which can make the simulations prohibitively expensive.
However, an accurate description of the decay of a static field on a much smaller
computational domain may be achieved if infinite elements are instead employed on the
upper and lower open ends [140,141]. The reason for this is that the mapped infinite
elements lead to the correct decay rate for A as r . This consequently results in a
greater accuracy compared to using either above mentioned boundary conditions, when
is placed close to the target. For the MK1 and MK2a simulations, infinite elements
were introduced at the open ends of the screen in order to capture the far-field.
76 Discrete Forward Problem
On a continuous level, the correct treatment of wave propagation effects would require
replacing the boundary condition n A = 0 on by a radiation condition (see (2.9)).
In a computational context, this would require employing some absorbing boundary
conditions (ABC) which can serve to absorb outgoing waves without reflection. A
popular set of such ABCs is the perfectly matched layer (PML) [142,143]. The PML is
a layer of artificial material surrounding the computational region, and designed to
damp waves propagating in the normal direction. The region may then be terminated by
a perfect electric conductor. The PML is not available in the AC/DC module of Comsol.
5.2.4. Solution technique
For a mesh with a small to moderate number of first-order tetrahedral elements (e.g.
150,000 elements corresponding to 180,000 degrees of freedom), the dimension of
the linear system makes it amenable to direct solution techniques. However, for a mesh
with a large number of elements, and in particular for discretizations employing second-
order elements, the increased size of the linear system makes it too expensive to be
solved by direct solvers, either in terms of the memory space requirements or in CPU
time. Therefore, an iterative strategy must be employed.
Comsol offers a variety of different iterative solution techniques. In this study, the
flexible generalised minimum residual (FGMRES) iterative solver, which is suitable for
the complex symmetric matrix obtained in this model, was employed [144,145]. The
speed of convergence was accelerated by using the symmetric successive over-
relaxation (SSOR) vector preconditioner [144].
Only first- and second-order edge elements were considered since it is well known that
when the shape functions for higher-order edge elements are not properly designed, they
lead to a rapid growth in condition number for the FE system matrices and consequently
require a large number of iterations for the iterative solver. The shape functions
employed for the higher-order edge elements in Comsol do not lend themselves to fast
iterative solution as they lead to a rapid increase in condition number and, even after the
application of the available preconditioners, their convergence remains slow. There are
alternative higher-order basis functions which lead to only a moderate growth in
condition number; e.g. see Ledger and Morgan [146,147] and references therein.
Discussion of such shape functions, however, is outside the scope of this thesis.
77 Discrete Forward Problem
5.3. Finite difference method
One of the earliest attempts to solve the partial differential equations is the finite
difference method (FDM) [148,129,149]. In the FDM, the partial derivatives are
replaced by difference equations. Space is divided into FD cells, and the equations are
written for the unknown potential of the field values at nodes. The partial differential
equations are replaced by a set of simultaneous equations. The unknown potentials are
computed on a finite set of points or nodes. The values at other locations may be
obtained by interpolation.
5.3.1. BVP
In the conducting region C, Js = 0 and the right hand side in (5.3) becomes zero. Using
the Coulomb gauge .A = 0, then (5.3) may be re-written as:
(5.13)
In arriving at this equation, the Coulomb gauge has been used, but not imposed. The
BVP before the FD discretization is therefore:
(5.14)
where n is the unit outward normal to . The boundary condition of the BVP is
obtained by equating the normal components of the total current on C to zero [150].
The BVP in (5.14) is the coupled partial differential equation which has to handle the
solution of and A together. However, by making certain assumptions, the problem
may be simplified:
wave propagation delays are negligible
the effect of the dielectric volume on A is negligible (contribution of the induced
currents to the magnetic vector potential is insignificant, and A is entirely
produced by the current source outside the dielectric volume)
skin depth is large compared with the dimensions of the dielectric volume
78 Discrete Forward Problem
Therefore, A in the BVP (5.14) is replaced by the approximation of the magnetic vector
potential created by the current source, Â, computed by:
(5.15)
where Js is the current source, and R is the distance between the source and the field
points [151,150]. In this case, the filamentary current source is applied along the exciter
coils. Therefore, Â in (5.15) can be expressed in cylindrical coordinates (r, , z) by:
(5.16)
where I is the current in the coil, a is the radius of the coil, K(k) and E(k) are elliptic
integrals of the first and second kinds [152]. This results in decoupling of the potential
equations, and the magnetic vector potential in the absence of the dielectric volume is
used directly as a source term for the computation of the electric potential. The BVP
reduces to solving (5.14) for the scalar electric potential within the dielectric volume.
This way, the displacement currents flowing outside the dielectric volume are ignored,
and the problem reduces to a magnetostatic problem in the non-conductive region N.
After solving the BVP, is known, and the total induced current density, JI, is the sum
of conductive and displacement components as:
(5.17)
This current creates a secondary magnetic vector potential, AI, which can be computed
by the Biot-Savart law:
(5.18)
where rI is the distance between the induced current and the field points. The voltage
due to the induced currents at the sensor coils may then by computed using (5.7), where
A is replaced by AI.
79 Discrete Forward Problem
Figure 5.1. A 2D representation of the grid showing the electrical connections between a node and
its 4 neighbours in the grid (6 neighbours in 3D). The admittances Y and vector voltage generator components e are known [153].
5.3.2. FD discretization
The dielectric volume is discretized using regular cubic voxels (size x), with the voxel
node located at the centre of the voxel. A constant conductivity and permittivity value is
assigned to each voxel. The complex conductivity within each voxel is considered as
= + j . The local admittance between the nodes (i,j,k) and (i-1,j,k) in figure 5.1 is
therefore computed by [154]:
(5.19)
The magnetically induced electric field is modelled as voltage generator e as:
(5.20)
where  is evaluated at midway between the two nodes of interest [152]. The current
between the nodes (i,j,k) and (i-1,j,k) is therefore given by:
(5.21)
(i,j,k) (i-1,j,k) (i+1,j,k)
(i,j+1,k)
(i,j-1,k)
Y(i+1,j,k) e(i+1,j,k)
Y(i,j-1,k)
e(i,j-1,k)
e(i-1,j,k) Y(i-1,j,k)
e(i,j+1,k)
Y(i,j+1,k)
x
y
80 Discrete Forward Problem
Similarly the current through the other five boundaries of node (i,j,k) can be computed,
and based on Kirchhoff's current law we have:
(5.22)
In this way, the scalar potential at a node can be calculated using the six nearest
neighbour values.
5.3.3. Solution technique
The BVP is solved by the successive-over-relaxation (SOR) method, which effectively
applies Kirchhoff's current law repeatedly throughout the grid as:
(5.23)
where m
(i,j,k) and m-1
(i,j,k) are the updated and existing values of the scalar potential at
node (i,j,k), and m
(i,j,k) is the change in the potential between two iterations obtained
by:
(5.24)
81 Discrete Forward Problem
In equation (5.23), f is the acceleration or relaxation factor. A value of 1.9 was deemed
to be a good choice for f and is used throughout this study. For further details on the
SOR method and its convergence properties, see e.g. Quarteroni [155] or Ledger [131].
The residual is calculated as the sum of squares of the increments in scalar potential for
each iteration over the whole grid:
(5.25)
where N1, N2, and N3 are the maximum voxel numbers is three directions. The residual
reduces continuously with each iteration, and is used as a measure of the convergence
rate and as a stopping criterion. The iteration is stopped when the residual is fallen by a
factor of 1/
m 10
20. Once is known, the induced currents within each voxel in the
dielectric volume are computed by (5.17). The resulting secondary magnetic vector
potential can be obtained by approximating the Biot-Savart law at the sensor coils as:
(5.26)
where JIn is the induced current density vector within the n-th voxel, rIn is the distance
between the node of the n-th voxel and the coil, AIn is the contribution of n-th voxel to
the secondary magnetic vector potential, and N is the number of voxels in the grid.
Subsequently, the e.m.f. in the sensor coil is computed by (5.7); note that rIn in (5.26) is
the distance to one element dl of the coil.
5.3.4. FD solver
A previously developed code [153], available in the South-Wales MIT group, was
employed for the FD simulations in this study. The FD code, hereinafter as 'FD solver',
had been developed based on the FDM described in this section, and was written in
Fortran1. The existing code was further improved to implement the electromagnetic
screen into the model (explained later in this chapter).
1 Fortran is a programming language that is suited to numeric computation and scientific computing.
82 Discrete Forward Problem
Figure 5.2. Exciter coil and conductive sphere.
5.4. Benchmark test
In order to validate the numerical techniques employed to solve the MIT forward
problem, a benchmark test was first attempted before solving the realistic head model
for haemorrhage simulations. The setup comprised a coil carrying an alternating current
as the exciter, placed coaxially with a spherical conductive object with material
properties , , and (see figure 5.2).
An analytical solution for the electric field inside a multilayered conducting sphere in
spherical coordinates (r, , ), has been derived by the LCOMIT partners based on
modified Bessel functions, In+1/2 and In-1/2, and associated Legendre polynomials, Pn
[24]. For simplicity, a homogeneous sphere is considered here for which the general
analytical solution reduces to:
(5.27)
where J0 is the current in exciter coil, 0 is shown in figure 5.2, and (r, ) are coordinates
of the point of evaluation.
83 Discrete Forward Problem
The validity of the analytical solution is restricted to cases when / << 1 , so that the
displacement current can be neglected. Following the calculation of E, the induced
eddy-current density inside the sphere can be computed from J = E. Values of r0, rC
and a were 143.7, 25 and 60 mm, respectively (see figure 5.2). The properties of the
sphere were = 0 and = 1 S m-1
, the frequency was 10 MHz, and the current in the
coil was J0 = 1 A. The ratio / was then 5.6 10-4
in the conducting region.
The benchmark problem was modelled using Comsol in which an artificial boundary
was introduced at a radius of 20 cm. The region beyond the boundary was modelled
using infinite elements. A series of meshes comprising 2,342 (figure 5.3a) to 1,135,307
(figure 5.3b) tetrahedral elements was generated in Comsol. The FE model was then
solved using first- and second-order basis functions, and the numerical results were
compared against the analytical solution.
The relative L2 error measure for the numerically calculated induced eddy current in the
sphere was approximated using an 11-point Gauss Quadrature rule:
error = (5.28)
where EL is the total number of FEs, and wp is the Gauss coefficient in a reference
tetrahedron multiplied by the determinant of the Jacobian. The term ep = |Jnu - Jan|2 is
the magnitude of the error at each integration point inside the tetrahedron, where Jnu and
Jan are the numerical and analytical current densities, respectively.
Figure 5.3. Typical FE meshes generated for benchmark test; a coarse (a), and a fine (b) mesh. Note
that the presence of coil, sphere, and layer of infinite elements (separate subdomains) may result in
generation of non-uniform meshes, e.g. in (a).
(a) (b)
84 Discrete Forward Problem
Figure 5.4. Error in the numerically computed eddy current inside the conducting sphere versus
the number of degrees of freedom; results for first- and second-order elements plotted in broken
red and solid blue lines, respectively.
As expected, the error when using second-order elements was smaller than that with
first-order elements for the same number of degrees of freedom (figure 5.4). As the
mesh was refined, the numerical results tended to the true (analytical) solution.
Examining the gradient of the error curves revealed that the second-order elements
converged approximately twice as fast as the first-order elements. The error for second-
order elements ranged from 3.76 to 0.25% which indicates the numerical results are in
good agreement with the analytical solution.
In order to attain a global error of 3%, approximately 1,135,000 and 3,800 elements of
first- and second-order were required, respectively, which stresses the superiority of
second-order edge elements in terms of the accuracy.
Comsol uses Nédélec‟s edge elements of the first type, so that a first-order element has
6 degrees of freedom, whereas a second-order element has 20 [17,134]. It is expected
that with further refinement of the mesh (i.e. increasing the number of elements), the
error would decease further, but the time taken to solve the model would become
prohibitively long.
103
104
105
106
107
10-1
100
101
102
Degrees of freedom
err
or
%
85 Discrete Forward Problem
5.4.1. FE vs. FD vs. Analytical
In order to assess the performance of the FDM, the sphere was modelled using a cubic
grid of 1 mm voxel size. The model was solved for two conductivities:
= 1 S m
-1 where the skin depth was = 160 mm
= 100 S m
-1 where the skin depth was = 16 mm
The FE model was also solved for = 100 S m
-1 using the finest mesh generated in the
previous section; for = 1 S m
-1 the FE solution was already available. The values of
the radial current density along r (see figure 5.5) were plotted for the analytical, FE, and
FD solutions.
Figure 5.5. Cross section of sphere showing r.
For = 1 S m
-1, the skin depth is larger than the sphere diameter. In this case, there is a
good agreement between the three models (figure 5.6a). Close to the surface of the
sphere (r 60 mm), the FD model slightly departs from the analytical and FE solution.
This is due to the discretization error on the boundary, whereas a tetrahedral mesh in the
FE model can properly represent the curved geometry.
When the skin depth is smaller than the sphere diameter, i.e. as in the case when =
100 S m
-1, the FDM fails to predict the correct values of the current density (figure
5.6b). However, there is a good agreement between the analytical and FE solutions.
Note that the FD solution for either of the conductivities can be obtained simply by
multiplying the values in the other one by the corresponding ratio of the conductivities.
exciter coil
86 Discrete Forward Problem
Figure 5.6. Plots of current density |J| along r for = 1 S m-1
(b) and = 100 S m
-1 (c).
Figure 5.7 shows the real and imaginary components of the current density together
with its norm along r, from the FE results computed for two conductivities. As can be
seen, for the large skin depth at = 1 S m
-1, the induced current is predominately
imaginary. However, for the smaller skin depth at = 100 S m
-1, the real component of
the current density is considerably larger.
0 0.01 0.02 0.03 0.04 0.05 0.060
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
r (m)
|J|
Analytical
FE
FD
0 0.01 0.02 0.03 0.04 0.05 0.060
2
4
6
8
10
12
14
16
18
20
r (m)
|J|
Analytical
FE
FD
(a)
(b)
87 Discrete Forward Problem
Figure 5.7. FE results: real, imaginary, and norm of current density along r for = 1 S m-1
(left)
and = 100 S m
-1 (right).
Figure 5.8 illustrates these quantities as the current maps on a cross-sectional plane
cutting through the sphere. The high magnitude current densities on the surface are
evident for = 100 S m
-1.
Figure 5.8. Maps of induced current densities in A m
-2 on a cross-section of sphere for FE results
when = 1 and 100 S m-1
.
Real Imaginary Absolute value
r (m) r (m)
Curr
ent
den
sity
(A
m-2
) |J|
|Im(J)|
|Re(J)|
|J| |Im(J)|
|Re(J)|
= 1 = 100
= 1
= 100
88 Discrete Forward Problem
5.5. Screen implementation in FD model
As stated in section 2.3, the MIT sensor array usually includes an electromagnetic
screen which can have a significant effect on the MIT signals. Therefore, implementing
the screen into the simulations is an essential aspect of the MIT models. The method by
which the screen is modelled in Comsol was explained earlier in section 5.2.3.
However, in the FD solver the region outside the conductive domain is not discretized,
so the screen could not be included in the FD models in a straightforward way. As
described by Ktistis et al [156], in order to add the screen to the FD solver, the forward
problem can be solved using the exciter-sensor mutual impedance changes, Z, for a
given target. Based on the Lorentz reciprocity theorem, Z for a pair of exciter-sensor
coils (a and b) can be derived in terms of the real inner product of the magnetic and
electric fields as:
(5.29)
where
Za is the mutual impedance between the coils when the domain properties are μa,
εa, σa and an alternating current I, of angular frequency ω, is applied to coil a to
produce the magnetic field Ha and the electric field Ea.
Zb is the mutual impedance for domain properties μb, εb, σb and Hb, Eb fields
produced by coil b when excited by the same current and frequency.
The volume of integration, , should include the regions where domain properties are
changed. Equation (5.29) is not exact as it ignores the error term ((σb-σa , εb-εa , μb-μa)2)
which is second-order with respect to the dielectric properties changes (for further
details see [157]). Assuming the permeability change to be negligible, which is usually
valid for biological tissues, (5.29) can be simplified as:
(5.30)
If the impedance changes are measured with respect to empty space, the effect of the
permittivity εa is negligible and σa = 0, and therefore:
89 Discrete Forward Problem
(5.31)
In empty space (air), the electric field is solely produced by the magnetic vector
potential A as:
(5.32)
The current density inside the target may can be computed by Jb = ( b + j b) Eb. Using
(5.31) to obtain the voltage changes in the sensor coils as V = I Z, then gives:
(5.33)
which can be used to compute the voltage change in the sensor coil (coil a) with respect
to the empty space. The vectors Aa and Jb are computed for the volume of integration,
i.e. conductive volume C. Note that Aa (hereinafter as 'A-field') is calculated for empty
space and is not affected by the target; it is related only to the geometry of the sensor
array and its operating parameters.
Therefore, if the A-field is available for an MIT sensor array, computed in presence of
the screen, it can be fed to the FD solver for computing V for any target of interest.
The A-field can be computed by the FE model in Comsol which includes the screen:
the A-field calculated for the exciter coils replaces (5.16) for solving the eddy-
current problem and computing Jb within the conductive volume.
the A-field calculated for the sensor coils is used for computing the voltage in
the sensors using (5.33), instead of (5.7) and (5.26).
5.5.1. Screen for MK1 and MK2a systems
In order to the assess validity of (5.33), FE models were constructed in Comsol for the
MK1 and MK2a systems (figure 5.9). Instead of truncating the domain at the internal
surface of the screen and adding the infinite elements at the above-bottom open ends,
the overall domain was considered to be a sphere of 30 cm radius. The screen domain
was added to the model as a vacuum, with IBC on its boundaries; material of the screen
was not discretized. A layer of infinite elements was then added to capture the far-field.
90 Discrete Forward Problem
Figure 5.9. FE models for computing A-field for MK1 and MK2a systems.
Although this approach requires the generation of more elements in the non-conductive
region (extra elements beyond the screen are required), the construction time of a one-
piece spherical layer of infinite elements is considerably shorter in Comsol than that for
the two-piece upper-lower positioned layers. However, the solution time will be longer.
A series of increasingly fine meshes of tetrahedral elements was generated, and the FE
model was solved using the second-order elements:
Mesh-1: 16,968 elements, 101,816 degrees of freedom
Mesh-2: 30,708 elements, 188,860 degrees of freedom
Mesh-3: 53,923 elements, 333,636 degrees of freedom
Mesh-4: 108,811 elements, 683,740 degrees of freedom
In order to assure the accuracy of the results, the magnitude of the vector potential, |A|,
was monitored along an arbitrary test line passing through the imaging domain within
the sensor array; line specified by (-0.1,-0.1,-0.1)-(0.1,0.1,0.1) in a Cartesian
coordinates whose origin was the centre of the spherical domain, and z-axis was
directed along the axis of the cylinder. Figure 5.10 shows the convergence results for
the MK1 system, with Ex1 (exciter coil 1) activated (for coils numbering in the MK1
system, see Appendix A). Mesh-4 deemed to have converged and further refinement of
the mesh did not significantly alter the results. For other activated exciter coils, similar
behaviour was observed. Therefore, the computed A-field from this mesh was employed
for the FD simulations.
MK1 MK2a
screen layer of infinite
elements
imaging region
91 Discrete Forward Problem
Figure 5.10. Plots of |A| for 4 meshes along a test line.
The A-field was extracted for a centrally located cylinder, 12 cm radius and 50 cm
height; this is the imaging region where the target is placed (see figure 5.9). The A-field
was extracted on regular cubic grids of 5 and 10 mm. The FD solver was amended to
implement the extracted A-field as the source. The discrete form of (5.33) can be
expressed by:
(5.34)
where x is the grid size, N is the number of voxels in the grid, Aan and Jbn are the fields
within the n-th voxel, and Jbn is computed by (5.17). Equation (5.34) gives the voltage
for one exciter-sensor coil combination, and it must be applied to each pair of coils.
In the following step, a conductive cylinder of 0.1 S m
-1, 10 cm radius and 20 cm height,
was centrally located within the sensor array. For this cylinder, the skin depth ( = 50.3
cm when r = 1 and f = 10 MHz) was larger than its dimensions. The FE model was
solved using the same mesh selected for the A-field computations. The solution time for
one activated exciter was 15 min1.
1 All simulations in this thesis were conducted on a Dell Precision 390 workstation, which has a CoreTM2
6700 CPU, with an internal clock frequency of 2.66 GHz and 8 GB of memory.
92 Discrete Forward Problem
The FD model, with the screen incorporated, was also solved using the grid sizes of 5
and 10 mm. The solution time for one activated exciter was approximately 6 and 0.4s,
for 5 and 10 mm gird sizes, respectively. The induced secondary voltages obtained from
FE and FD simulations are plotted in figure 5.11, for Ex = 1 and Se = 1,2,..,16 (as
described in Appendix A, throughout this thesis, exciter and sensor coils are labelled as
Ex and Se, respectively).
As it can be observed, there was a good agreement between the FE and FD results. The
FD results with 10 mm grid size slightly departed from the FE results at Se1 and Se16,
where the magnitude of the voltage was at its largest (see magnified view for Se16). The
5 mm grid size can represent the curved surface of the conductive cylinder more
accurately. The FD model was also solved without the screen. Large discrepancies
between the results from the model without the screen, and those with the screen
included are evident in the figure.
Figure 5.11. Induced secondary voltage at 16 sensor coils for Ex1 activated (FE mesh comprised
108,811 tetrahedral elements).
A similar procedure was carried out for computing the A-field for the MK2a system.
The converged FE mesh comprised 103,375 tetrahedral elements corresponding to
643,900 degrees of freedom.
0 2 4 6 8 10 12 14 16
-20
-10
0
x 10-5
Se
Secondary
voltage (
V)
FE
FD - 10 mm
FD - 5 mm
FD - 5 mm (without screen)
14.5 15 15.5 16 16.5
-5
-4
-3
-2
-1
0
1x 10
-5
Se
Secondary
voltage (
V)
10-5
-3
-5
93 Discrete Forward Problem
5.5.2. Simulated vs. measured data
With the screen included in the FD model, the FD and FE results can now both be
compared against the practical measured data. In this section, the results from the
experimental study carried out by Griffiths et al [8] are reproduced through simulations.
As the target, they used a cylindrical glass vessel (diameter 120 mm, height 200 mm)
filled with saline solution with the following nominal conductivities: 0.5, 1.0, 1.5, 2.0
and 3.0 S m
−1. Tap water (0.012 S
m
−1) was used for an additional measurement to
provide a sample with a very low conductivity.
In their study, the measurements were undertaken using Ex1-Se9 combination (opposite
coils) of the MK1 system operating at the frequency of 10 MHz (see figure 5.12). First,
the primary voltage Vp was measured (sample absent). Then sample was placed at the
centre of array, and the secondary voltage Vs was acquired.
Here, the same setup was simulated by the FE and FD models. Recalling from section
2.3.1, for a fixed frequency, when the skin depth is larger than the target dimensions,
then we can expect Re( V/Vp) and Im( V/Vp) .
Simulated and measured results for Im( V/Vp) are shown in figure 5.13a and figure
5.13b, respectively. As can be seen, the imaginary component closely obeyed the
expected linear relationship with the conductivity in all three data sets (measured, FE,
and FD). A fitted linear line for each data set is shown in the figure, where the
coefficients of the line equation are also provided.
Figure 5.12. Plan view of the MIT array showing the position of the sample and coils within the
electromagnetic screen. Red (outer) and black (inner) coils are exciters and sensors, respectively.
9
1
Screen
Sample
94 Discrete Forward Problem
Figure 5.13. Values of Im( V/Vp) in percentage against conductivity: (a) FE and FD results, (b) measured data where straight fitted line is also shown (after Griffiths et al [8]).
The real component Re( V/Vp), plotted in figure 5.14, was much smaller in magnitude
than the imaginary and was positive, but only for the two lowest values of conductivity
(0.012 and 0.5 S m
−1). As the conductivity increased, the curve in the measured and FE
results became negative and increased in magnitude. It is likely that the small, positive
value at low conductivities arose from the permittivity of the saline solution, when the
eddy currents were predominantly displacement currents.
As the conductivity increased, and the skin depth became comparable with the diameter
of the sample, the assumption of a weak interaction, on which the FD model is based,
became invalid. Therefore, the FD model failed to predict the correct values for the real
component and, as can be seen in figure 5.14a, the values remain unchanged for a fixed
0 0.5 1 1.5 2 2.5 3-10
-8
-6
-4
-2
0
Conductivity (Sm-1)
Im (
V/V
p)
(%)
(a)
(b)
FE
-3.28 - 0.03
FD
-3.28 - 1.6 10-10
95 Discrete Forward Problem
value of permittivity ( r = 80), regardless of the conductivity. The skin depth would be
equal to the diameter of the sample, 120 mm, at a conductivity of 1.76
S m
−1.
Figure 5.14. Values of Re( V/Vp) in percentage against conductivity: (a) FE and FD results, (b) measured data (after Griffiths et al [8]).
As an extra step and in order to investigate the frequency dependence of the conduction
and displacement currents, the FE model was solved for two fixed conductivity values
of 0.012 and 3 S
m
-1 ( r = 80), where frequency was varied between 10 kHz to 10 MHz.
The measures of the conduction and displacement current densities induced throughout
the cylindrical vessel were computed by integrating the norm of current density over the
cylinder using an 11-point Gauss Quadrature. These values are plotted in figure 5.15
against the frequency.
0 0.5 1 1.5 2 2.5 3-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Conductivity (Sm-1)
Re (
V/V
p)
(%)
(a)
(b)
FD
FE
96 Discrete Forward Problem
The norm of displacement current was linearly proportional to the frequency squared,
and remained the same for a fixed value of relative permittivity and frequency,
irrespective of the conductivity. The norm of conduction current, however, was almost
linearly proportional to the frequency. It was also proportional to the conductivity. The
average ratio between the values for 3 and 0.012 S m-1
was 249.2 where 3/0.012 = 250.
Note that at the frequency of 10 MHz and for = 0.012 S m-1
, the displacement currents
are larger than the conduction currents, resulting in a positive value for Re( V/Vp)
shown in figure 5.14.
Figure 5.15. Total values of conduction and displacement current densities induced throughout the
cylindrical vessel.
5.6. Discussion
Comparison of the FE results with the analytical solution (section 5.4) and practical
measurements (section 5.5.2) ensured the validity of the employed FE techniques. A
very strong advantage of the FEM is its ability to deal with complex geometries by
using unstructured grids. This can be of importance for modelling a realistic human
head geometry, with its complex features. The employment of Comsol for the solution
of the forward problem in this study, allowed a rapid development of the model to
104
105
106
107
10-12
10-10
10-8
10-6
10-4
10-2
Frequency (Hz)
Induced c
urr
ent density thro
ughout cylin
der
(mA
)
Displacement
Conduction
= 3 S m-1
Conduction
= 0.012 S m-1
97 Discrete Forward Problem
estimate the signals for feasibility tests. However, the use of commercial packages, such
as Comsol, has two major drawbacks:
For large 3D models comprised of more than 100,000 elements, most of the
computational effort associated with the FE simulation is devoted to the solution
of a large sparse set of linear equations. For meshes of this size, direct solvers
are inappropriate due to the excessive computational resources that they require.
Therefore, iterative forward solution techniques must be adopted. However,
without suitable pre-conditioners, these can be slow to converge. Although
commercial FE packages offer a range of solution techniques, their performance
for MIT was not found to be satisfactory, and led to excessive run times. This
could jeopardise one major advantage of using MIT which is a rapid diagnosis.
For an optimised implementation of iterative inverse algorithms, it is necessary
to solve multiple forward problems in order to update the Jacobian (sensitivity
distribution matrix); the sensitivity matrix is introduced in the next chapter. To
this end, it is essential to have unrestricted access to the source code in order to
introduce the necessary modifications. Besides the excessive computational time
for solving multiple forward problems, another major drawback is the fact that
access to the source codes is normally impossible in commercial packages.
In contrast, the FD solver offers full flexibility and accessibility, allowing the necessary
amendments to be done. Furthermore, the FD solution time is considerably shorter; for
the particular example presented in section 5.5.1, the FD model performed 150 times
faster than the FE model, with comparable results.
Comparison of the FD results against those from the analytical (section 5.4.1), measured
(section 5.5.2), and FE (section 5.5.1) data indicated that the FD model can be used for
the MIT simulations. However, strictly its applicability is limited to the cases when the
target's dimension are larger than the skin depth. Although by examining the imaginary
part of the signal computed by the FD model, there is surprisingly good agreement with
the FE model and the experimental results when the target is of comparable dimension
to the skin depth (figure 5.13).
Therefore, when appropriate, the FDM has been employed for the simulations in the
remainder of this thesis. Furthermore, when conducting preliminary studies for the
98 Discrete Forward Problem
introduction of various image reconstruction algorithms (chapter 6), and system
geometrical optimisation (chapter 8), FDM was the method of choice.
When using the measured data, it is crucial to simulate the MIT system as accurately as
possible. This includes implementing the screen into the model. Figure 5.11 indicated
that the effect of screen can be significant on the induced voltages.
End of chapter 5.
Chapter 6.
Discrete Inverse Problem
6.1. Introduction
As stated in section 2.2, the aim of solving the MIT inverse problem is to recover the
electrical properties from the measured signals at the sensor coils. The purpose of this
chapter is to explain the general framework of the discretized MIT inverse problem, and
to review the image reconstruction algorithms with details of their implementations.
The discussion starts with time-difference imaging, and describes the difficulties
associated with the MIT inverse problem which are common to all types of MIT
imaging. This is followed by frequency-difference and absolute imaging.
Although time- and frequency-difference imaging techniques are discussed in different
sections, they are essentially similar, and all the inverse algorithms applicable to time-
difference imaging can also be used for frequency-difference imaging. Absolute
imaging requires iterative image reconstruction techniques, and is dealt with in a
different section. When necessary, images are reconstructed after introducing each
individual algorithm and the results are discussed.
In practice, the distribution of dielectric properties cannot assumed to be continuous. In
order to approximate the inverse solution, similar to the forward problem, the inverse
problem is discretized which yields a solution in matrix representation:
The imaging volume is discretized into n elements (also voxels or pixels).
The conductivity value throughout each element is assumed to be constant.
100 Discrete Inverse Problem
n unknown conductivity values are arranged in a vector.
After solving the inverse problem, the obtained conductivity values are then
assigned to the corresponding elements in order to reconstruct the image.
Figure 6.1. One geometrical example of the model (to scale): (a) diagram of the 3D model, (b) x-y
and y-z cross-sectional planes for image representations.
Throughout this chapter, geometrically simplified 3D models are used for the
simulations. The model comprises a homogeneous cylindrical tank (background) of 10
cm radius and 16 cm height (figure 6.1a). One or more cylindrical inclusions, 2 cm
radius and 4 cm height, are placed within the background. The inclusions have various
conductivities different from that of the background. The forward problem (and inverse
problem) is solved using the FD solver and for the MK1 system, with the screen
included. The discretized problem has a voxel size of 10 mm resulting in n = 5056
conductive voxels in 3D. Throughout, the parameter to be estimated is the conductivity.
For displaying the reconstructed images, two cross-sectional planes, passing through the
tank and inclusion, are extracted from the reconstructed volume image in order to show
the conductivity values (figure 6.1b). In some cases, 1D images are also plotted in order
to closely investigate the computed conductivity values along a line passing through the
imaging volume.
It is remarked that by using the same set of voxels for the forward and inverse problems,
and in some cases using the noise-free simulated data, a so-called inverse crime is
committed (inverse crime is discussed at the end of this chapter). In this regard, it must
be noted that the purpose of this chapter is to introduce the concept of the inverse
problem and the difficulties associated with solving it, briefly review some of the
y-z
x-y
inclusionbackground x-y y-z
(a) (b)
101 Discrete Inverse Problem
existing image reconstruction algorithms, and provide a basis for dealing with more
realistic cases later in this thesis. The inverse algorithms and techniques, explained in
the following sections, are well-established methods and have been presented before for
EIT simulations. Here, they are explained in the context of MIT, and MIT related
examples are provided.
6.2. Time-difference imaging
As stated in chapter 2, time-difference imaging is to calculate the conductivity change
between two states, separated in time. Depicted in figure 6.2, is an example of the two
states with different conductivity distributions (hereinafter as „State-I‟ and „State-II‟).
The goal of the inverse problem in this case is to estimate from V, where V is the
voltage change measured in the sensor coils between the two states. There are several
difficulties associated with solving this problem including its being non-linear,
underdetermined, and ill-posed.
Figure 6.2. Diagram illustrating the conductivity distributions for two states in x-y planes.
6.2.1. Non-linearity
In MIT, the field strength, computed by (5.6) or (5.14), varies as a function of position
and distribution of dielectric properties throughout the domain. Therefore, eddy currents
(J = E) and the signals induced by them, are a non-linear function of the distribution
of dielectric parameters of the entire domain. In order to obtain the conductivity
State-I State-II
102 Discrete Inverse Problem
distribution one must solve a system of equations which relate every voxel in the
domain to every measurement.
One approach for solving the discretized inverse problem can be obtained by linearizing
the model. First, the forward problem, when approximated by either (5.6) or (5.14), may
be considered in the form:
(6.1)
where n is a column vector of the (real) conductivity values in each voxel, V m
is
the column vector of voltages induced in the sensor coils resulting from , and F is the
corresponding non-linear function which represents the approximate solution of the
forward problem [158]. If F and are known, the voltage can be calculated. This is the
forward problem that was discussed in chapter 5, and for the presumed boundary
conditions, has a unique solution.
Assuming small conductivity changes between the two states, one can expand (6.1)
using a truncated Taylor series. With the second and higher order terms in the series
sufficiently small, the non-linear problem may be approximated by:
(6.2)
Here, F/ is the Jacobian of the (discrete) forward operator F( ), computed at some
initial conductivity estimate 0, which relates interior conductivity perturbations to
perturbations in the measured signals. In this context, the Jacobian is often called the
sensitivity matrix.
Let us assume that 0 is the conductivity distribution for the homogeneous State-I
(figure 6.2) and is known. Using the linearized problem in (6.2):
(6.3)
The Ax = b is essentially a system of linear equations in which A is the sensitivity
matrix, b is measured data (voltage changes), and x is the vector of conductivity
changes to be determined. Now let us consider the case when State-I is not
homogeneous and its conductivity distribution ( 1) is not known (see figure 6.3).
103 Discrete Inverse Problem
Figure 6.3. Diagram illustrating the conductivity distribution for two inhomogeneous states.
Similarly, the problem can be linearized for the two states around an initial guess 0:
(6.4)
which again yields a system of linear equations. Note that the linearization is valid as
long as there are no large differences between 0 and the conductivity distribution for
State-I. For instance, if State-I has a conductivity distribution with large contrasts, a
homogeneous 0 may result in a sensitivity matrix that does not accurately represent the
variations in the voltage obtained in the forward model for changes in the conductivity
distribution. This is further discussed in sensitivity analysis in section 6.3.1.
6.2.2. Underdetermined
The MIT inverse problem will usually suffer from a shortage of information. The reason
for this is twofold. Firstly, the number of available measurements is typically fairly
small. Secondly, it is a common practice to discretize the domain by a large number of
elements (voxels), inside of which the conductivity is assumed to be constant, in
particular in 3D imaging.
For instance in the MK1 system, with 16 exciter and 16 sensor coils, a total of m =
16 16 = 256 coil combinations is obtained. However, since the exciter and sensor coils
State-I State-II
104 Discrete Inverse Problem
are placed closely, only 136 of these are independent due to the reciprocity theorem
which states that reversing the sensor and exciter coils would give an identical mutual
inductance between the coils [158,159].
One may attempt to overcome this problem by increasing the number of channels in the
system as the MIT is a contact-less technique. In general, a C-channel system, with each
channel acting as both the exciter and sensor, will deliver C (C+1)/2 independent coil
combinations [160]. More information could also be obtained by conducting the data
collection for several planes of measurement; e.g. positioning the MKI sensor array at
different heights with respect to the target, which results in more coil combinations.
However, the number of coils that can be implemented, and the number of planes of
measurement that can be undertaken are limited by the hardware and processing time
restrictions.
Furthermore, one could choose to reduce the number of voxels (unknowns), although
this may result in blocky images. Finally, another possibility would be to use an
alternative basis for representing the unknown conductivity (e.g. a Fourier basis
[161,162,163,164]). In this way, the number of unknowns can be reduced, but still
maintaining the smoothness of the solution. However, in this study, the conductivity is
discretized by constant values inside each voxel, and the number of unknowns will be
larger than the number of measurements. Therefore, the problem will be
underdetermined. A more precise definition of over- and underdetermined problems is
provided in the following.
In the overdetermined case (A m×n
, m >
n), there is at least one solution to Ax = b, if
the number of linearly dependent rows of A (i.e. r = rank(A)) is less or equal to the
number of columns of A. The least squares (LS) solutions are the ones that satisfy:
(6.5)
where ||.|| is the Euclidean norm. In the case of full column rank (r = n), the LS solution
is unique and may be obtained by:
(6.6)
where A+ = (A
TA)
-1A
T is the Moore-Penrose pseudo-inverse [165].
105 Discrete Inverse Problem
In the underdetermined case (A m×n
, m <
n), there are infinitely many solutions or no
solution to Ax = b. In the case of the former and full rank (r = m), the solution with the
smallest Euclidean norm ||x|| can be computed by:
(6.7)
where in this case, A+ = A
T (AA
T)
-1 is the Moore-Penrose pseudo-inverse [165,166]. In
general, the pseudo-inverse for under- and overdetermined problems (with or without
full rank) can be obtained using the singular value decomposition which is explained
later in this chapter. However, even in situations with full row or column rank,
computing x is not trivial as AAT and A
TA are ill-conditioned with respect to inversion,
i.e. the inverse problem is ill-posed. Further numerical difficulties can arise when the
right hand side is not consistent due to the errors in the measurements (i.e. b does not lie
in the range of A).
6.2.3. Ill-posedness
Hadamard [167,168] attempted to describe the generic properties of the inverse
problems which arise from physical and natural phenomena. A mathematical model of a
physical problem is well-posed if the solution of the mathematical model:
is consistent with the observations (in this case, the MIT signals).
is unique.
depends continuously on the data (observations).
The MIT inverse problem is severely ill-posed. Although by imposing some constraints
on the solution, one may ensure the validity of the first two criteria, it certainly fails the
third one. This means even small perturbations in the measurements, b, can cause large
changes in the solution, x. In practice this means for any given measurement precision,
there may be arbitrarily large changes in the obtained conductivity (since b is a
measured quantity, it is subject to various observational errors). The ill-posedness of the
linearized inverse problem is due to the fact that the matrices AAT or A
TA are ill-
conditioned with respect to inversion.
The ill-posedness of the problem does not mean that a meaningful approximate solution
cannot be computed. Rather, it implies that the standard methods in numerical linear
106 Discrete Inverse Problem
algebra for solving (6.3), e.g. pseudo-inverse, cannot be used in a straightforward
manner to compute a solution. Therefore, solving the inverse problem requires special
treatment. A common approach includes the regularization or truncation of a singular
value expansion. But first, computation of the sensitivity matrix and its reliability to
represent the model is discussed.
6.3. Sensitivity analysis
6.3.1. Sensitivity computation
The Jacobian or sensitivity distribution matrix may be computed numerically by altering
the electrical properties of each degree of freedom (i.e. each voxel or element) and
solving the forward problem. This way, the sensitivity of each point in the domain to the
conductivity change can be assessed. For instance, figure 6.4 illustrates a homogeneous
initial guess 0 = 1 S m-1
, where the conductivity of the perturbed element is raised to
twice of that in the domain (2 S m
-1).
This „perturbation method‟ requires the solution of multiple forward problems (once for
every perturbed element), and can be cumbersome, in particular, for 3D image
reconstruction. The sensitivity matrix may be pre-computed for single-step differential
image reconstruction algorithms. However, for the non-linear absolute imaging
algorithms, which requires the sensitivity to be updated after each iteration (section 6.8),
the use of perturbation method for the computation of sensitivity matrix would be
prohibitively long.
Figure 6.4. Perturbation method for sensitivity calculation; highlighted voxels represent the
perturbed elements (pixels).
1st perturbed pixel 2nd perturbed pixel
107 Discrete Inverse Problem
An alternative approach is to use the mutual impedance changes, Z, based on the
Lorentz reciprocity theorem and derive the generalized representation for each pair of
exciter-sensor coils in terms of the inner product of electric fields [169,156]. To this
end, similar to section 5.5 and neglecting the permittivity in (5.30), V for one pair of
coils can be expressed by:
(6.8)
where Ea and Eb are the electric fields produced when either of the coils is activated.
For a discretized domain comprising n homogeneous elements, then we have:
(6.9)
where k is the volume of k-th element. Assuming a current of I =
1
A, the sensitivity
entry Ak of the k-th element which measures the rate of voltage change Vk due to the
small change in the properties of that element is then defined as:
(6.10)
Note that k represents the change in the conductivity of the k-th element between a
and b. Therefore, using the same conductivity for a and b results in Ak being the
gradient of the non-linear model, F, at k.
Strictly speaking, what has been computed in (6.10) is a Fréchet derivative which is
obtained by neglecting higher order terms [125] and, hereinafter, is referred to as the
E.E sensitivity. The employment of this method to compute the sensitivity matrix is,
therefore, significantly faster than the perturbation method, which is a finite difference
approximation of the Fréchet derivative [125]. Compared to the perturbation method,
which requires the solution of n forward problems for each pair of coils (and
consequently each row of the sensitivity matrix), here the same entries using the E.E
sensitivity approach can be derived from the solution of two forward problems; solution
for the activated exciter, and solution for the sensor when employed as the exciters.
108 Discrete Inverse Problem
The entry Ak for one element not only depends on the electrical properties of the
material within that element, but also on the distribution in the whole sensing area since
the surrounding area affects the current flow in that element. Therefore, the initial guess
of conductivity distribution for computing the sensitivity matrix is of some importance.
The initial conductivity distribution can be assumed using some prior knowledge. For
instance, it can be taken as the typical distribution that is expected in the target. For a
low contrast target the nearest guess could be a uniform distribution.
On a regular grid of size x, the entries can be simply computed by:
(6.11)
where (Ea)k and (Eb)k are the electric fields within the k-th voxel, and Ne and Ns are the
total number of the exciter and sensor coils, respectively. The sensitivity matrix of the
entire system, A, has therefore as many columns as there are elements in the discretized
volume (k = 1,2,...,n), and as many rows as there are coil combinations or coil pairs (i =
1,2,...,m) where m = Ne Ns. Each row of A corresponds to the sensitivity of one coil
combination to all elements. Each column corresponds to the sensitivity of all coil
combinations to one element. In order to compute the sensitivity matrix based on the
reciprocity method, the FD solver was amended to implement the required sub-routines.
In order to compare the perturbation and E.E sensitivity matrices, a homogeneous
conductivity distribution of 1 S
m
-1 was considered for the cylinder shown in figure
6.1a. The E.E sensitivity matrix was computed simulating the MK1 system which
provides 256 coil combinations. The entries of an arbitrary column in the sensitivity
matrix, corresponding to the sensitivity of all coil combinations to one voxel in the
discretized cylinder, are plotted in figure 6.5 (thick black line). Note that the plot is
shown only for a limited number of the coil combinations for the convenience of
display.
The perturbation sensitivity was computed by raising the conductivity of the same
voxel, while the conductivity of the remaining voxels remained at 1 S
m
-1. Three levels
of conductivity change were considered for the perturbed voxel: 0.1, 0.5, and 1 S
m
-1
corresponding to 10, 50, and 100% conductivity change, respectively. The computed
sensitivity entries are also shown in figure 6.5.
109 Discrete Inverse Problem
Figure 6.5. Comparison of sensitivity matrices computed based on perturbation and E.E methods.
Percentage values correspond to conductivity change in the perturbed voxel with respect to the
background.
As can be seen, when the conductivity change in the perturbed voxel with respect to the
background is large ( 50%), the entries in the perturbation matrix deviate from those in
the E.E one. This is to be expected as the E.E method assumes small conductivity
changes, i.e. ignores the second-order terms with respect to the conductivity change (see
section 5.5). However, for a small conductivity change ( 10%), there is a good
agreement between the entries of the two sensitivity matrices.
If the MIT signals are normalized to the primary signal to obtain the phase changes, the
entries of the sensitivity matrix in (6.11) should also be divided by the primary signal
for the corresponding coil combination:
(6.12)
Therefore, the equation (6.3) will be in the form:
(6.13)
100 105 110 115 120 125 130 135 140
-6
-4
-2
0
2
4
x 10-8
Coil combination
A
E.E
10%
50%
100%
110 Discrete Inverse Problem
In order to examine the reliability of linearization in equations (6.3) and (6.4), an
investigation was carried out using the setup in figure 6.1 as follows:
i. Ex1 (exciter 1) and Se9 (sensor 9) in the MK1 system were considered to
represent a single channel system with opposite coils (for coil numberings see
Appendix A).
ii. State-I denotes the case of a homogeneous background conductivity distribution
of 0 = 1 S m-1
. The forward problem was solved for this distribution.
iii. The sensitivity matrix and voltage in the sensor coil, V0 = F( 0), were recorded.
Note that for a single channel system, there is only one sensor coil and the
voltage can be expressed as a scalar.
iv. An inclusion was added to the homogeneous background conductivity
distribution (see the diagram in figure 6.6). State-II denotes the conductivity
distribution 1 which includes the inclusion.
v. The conductivity of the inclusion was varied from 0.1 to 1.9 S m-1
in 0.1 steps;
equivalent to 90% perturbation with respect to the background.
vi. For each case, the forward problem was solved and the voltage V1 = F( 1) was
recorded.
vii. The difference between the voltages ( V = V1 – V0) is plotted against the
conductivity change (solid blue curve in figure 6.6). This is the voltage change
computed by the non-linear model, i.e. forward model.
viii. In the next step, using the linearized model V = b = Ax = A , the sensitivity
matrix was used to predict the voltage changes; values of V are shown as red
circles.
As can be observed, there is good agreement between the non-linear and linear models
when the conductivity change is small. However, as the conductivity changes become
larger, the predicted voltage changes by the linear model depart from those computed by
the non-linear model. At 90% conductivity change (inclusion of 0.1 S m-1
), the error in
the linear model was 35%. For large values of | | > 4 S m-1
(i.e. 400%, not shown in
the figure), the change in the voltage tends to saturate the change in the conductivity.
Steps (i) to (viii) were repeated for an inhomogeneous background as the State-I, where
the inclusion of 1.2 S m-1
was placed in the background of 1 S m-1
(see diagram in
figure 6.7). The sensitivity matrix, however, was computed for the homogeneous
111 Discrete Inverse Problem
background as before. Similarly, State-II was simulated for an added inclusion with up
to 90% conductivity change with respect to the background.
The results are plotted in figure 6.7. As expected, a similar behaviour was observed,
indicating that for small enough conductivity changes, the inverse problem can be
treated using the linear model. Examining other exciter-sensor coil combinations in the
MK1 system, revealed the same conclusion.
Figure 6.6. Comparison of non-linear and linear models; is contrast between added inclusion
and background.
Figure 6.7. Comparison of non-linear and linear models; is contrast between added inclusion
and background.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-6
-4
-2
0
2
4
6
8
x 10-7
x =
V
F(1) - F(
0)
A0 (
1 -
0)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-6
-4
-2
0
2
4
6
8
x 10-7
x =
V
F(2) - F(
1)
A0 (
2 -
1)
0
1
1
2
112 Discrete Inverse Problem
6.3.2. Ill-conditioning
One suitable numerical tool that can be used for the analysis of discrete ill-posed
problems is the singular value decomposition (SVD) of the sensitivity matrix A. The
SVD reveals the difficulties associated with the ill-conditioning of A. An idea of how
severely ill-conditioned the sensitivity matrix is, can be obtained by evaluating its
singular values. To this end, the SVD of A is computed as a decomposition of the form:
(6.14)
where Um m = (u1,u2,...,um) and Wn n = (w1,w2,...,wn) are unitary matrices with
orthogonal columns, UTU = Im and W
TW = In. The matrix is the m by n matrix
diag(s1 , s2 , .... ) padded with zeros and s1 s2 .... > 0 [170,171,172]. The numbers si
are the singular values of A while the vectors ui and wi are the left (output) and right
(input) singular vectors of A, respectively. The matrix A is said to be ill-conditioned if
both of the following criteria are satisfied:
Singular values of A decay to zero.
Ratio between the largest and smallest non-zero singular values (s1/sm) is large.
The singular values of the sensitivity matrix, which was computed for the example in
the previous section, are plotted in figure 6.8. The singular values fell abruptly by
several orders of magnitude, indicating that the sensitivity is severely ill-conditioned
and the inverse problem is, therefore, ill-posed (s1/s256 = 9.86×106).
Using (6.14), matrix A can be expanded into the summation of a number of matrices:
(6.15)
As a result, the application of a linear transformation in the form Ax is equivalent to the
application of a sequence of linear transformations which have a contribution of
decreasing magnitude. Notice that coefficients si are ordered in terms of magnitude. The
SVD can be used as a tool for solving over- and underdetermined problems. The LS
solution of Ax = b corresponds to:
113 Discrete Inverse Problem
(6.16)
The solution of this problem is equivalent to [168]:
(6.17)
where A+ is the pseudo-inverse given by:
(6.18)
Since W and U matrices are unitary, their inverse is simply their transpose, and + is the
transpose of with the non-zero singular values si replaced by their reciprocals 1/si.
Equation (6.18) generalises the earlier expressions given in (6.6) and (6.7) for the over-
and underdetermined problems with full row/column rank and, in the case of exact
arithmetic, the results are the same. Note that in the overdetermined case with r = n,
there is only one solution to , and this solution has minimal norm. In
the overdetermined case without full column rank and in the underdetermined case, r <
n and there are an infinite number of solutions to , and the solution
with minimal norm is sought.
Equation (6.17) is referred to as the singular value expansion of the solution from
which, multiple results can be observed:
The solution is now a summation of r components instead of simply A+b.
For each component of the series, the singular value si is in the denominator and
a very small value can potentially have a large impact on the solution.
Since each term uiTb/si is a scalar coefficient, it leaves the solution x as
essentially a linear combination of wi vectors. Notice that x represents the
reconstructed MIT image. Therefore, the right singular vectors wi (i = 1,...,r) are
the orthogonal bases comprising the image.
Among the singular values of the sensitivity, the smallest ones are the most vulnerable
to noise. These are often extremely small, and when contaminated by the numerical
error and then inverted, they grow into drastically magnified errors, causing instability
114 Discrete Inverse Problem
in the solution. They also amplify the measurement noise in data vector b. This can lead
to 'blowing-up' of the solution (artificially large values in the solution), and introducing
artefacts in the reconstructed image. This explains why computing the solution using the
equation (6.7) may lead to poor quality results.
Similar to the work by Lionheart et al for EIT [173], some arbitrary singular vector
maps are shown in figure 6.8, adjacent to their corresponding singular values. Each
singular vector plot was constructed by placing the elements in column vector wi in the
appropriate voxel positions in the grid to reconstruct the 3D volume, and then summing
the absolute values in voxels along the height of cylinder (z-direction) to obtain the 2D
(x-y) maps.
It can be appreciated from the plot that the singular vectors corresponding to the large
singular values contain information mainly related to the periphery and locations close
to the surface of the domain. As the magnitude of the singular values decrease, the
corresponding singular vectors tend to contain information about the centre of the
domain and/or details of spatial resolution. Note that by using absolute values to obtain
2D maps, the changes in sign of the singular vector components are not observed.
Figure 6.8. Singular spectrum plotted on a logarithmic scale, normalized to the largest singular
value. Also shown some arbitrary singular vector maps; index = 1, 3, 60, 136, 223.
0 50 100 150 200 25010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Singular value index
Norm
alis
ed s
ingula
r valu
e
115 Discrete Inverse Problem
In order to eliminate the cause of instability in equation (6.17), filtering out the small
singular values is essential; a process which also improves the conditioning of the
sensitivity matrix. Removing singular values from the sensitivity matrix, however,
causes the sensitivity to drop. Since some of these small singular values may represent
important details and desirable parts of the images, by filtering out the singular values
the information within their singular vector is lost, and the reconstructed image may
suffer from a low spatial resolution. The filtering process is often performed with
numerical tools such as TSVD and Tikhonov regularization.
6.4. Truncated singular value decomposition
As previously discussed, complications arise due to the terms with small singular values
in the series in (6.17). Therefore, the most straightforward way to avoid this situation is
to eliminate these components entirely from the summation in (6.17). This can be
achieved by setting a truncation level for the minimum magnitude of singular values
that are to be retained in the summation. Let 0 < k < r be the number of terms retained
after truncation. The solution then becomes:
(6.19)
which is the truncated singular value decomposition (TSVD) solution corresponding to
subject to [168]. Here, k is the truncation level, and
(6.20)
By truncating the small singular values, all the noise components parallel to uk+1, ...,ur
which would be strongly amplified, are rejected. The equation (6.19) can be rewritten
using a filtering function, f, in the form:
(6.21)
116 Discrete Inverse Problem
In order to examine the effect of truncation, consider the case where State-I corresponds
to a uniform background of 1 S m-1
, and State-II consists of the same background, but
with the addition of two inclusions (0.5 and 1.5 S m-1
). The sensitivity matrix is
computed according to State-I, and the inverse problem is solved with three different
levels of truncation in (6.19): k = 100, 180, 256 corresponding to sk/s1 = 1.4 10-3
,
4.0 10-5
, 1.0 10-7
(see figure 6.9). Note that k = 256 indicates no truncation is done, and
all the singular values are included in the solution.
Figure 6.9. Levels of truncation of singular values.
The first column in figure 6.10a shows the target to be imaged, and the second to fourth
columns are the reconstructed images with different levels of truncation. As can be
observed, with a high level of truncation (k = 100), more than half of the singular values
are dropped (also see figure 6.9); this results in a blurred but relatively smooth image.
Incorporating 180 singular values, details of the inclusions are clearly better visualised.
Both the shape and magnitude of the reconstructed conductivity change are improved;
see figure 6.10b which shows a 1D image of the conductivity values along a line in x-y
cross-sectional plane.
A further improvement is observed when all the available information is used (k = 256).
For this case, the recovered inclusions are slightly better defined, are less displaced
towards the boundaries, and the maximum magnitude of 0.3 and 0.42 S m-1
is obtained
0 50 100 150 200 25010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Singular value index
Norm
alis
ed s
ingula
r valu
e
k = 100
k = 180
k = 256
117 Discrete Inverse Problem
for positive and negative conductivity changes, respectively. The fact that even with
incorporating all 256 singular values, and using the noise-less data, the reconstructed
image is still different from the target clearly indicates the shortage of information (the
inverse problem being underdetermined).
Figure 6.10. Reconstructed images using TSVD; (a) columns from left: target and images with
different levels of truncation, rows from top: x-y and y-z cross-sectional planes, colour-map shows
conductivity change in S m-1
, (b) 1D representation of target and reconstructed conductivity
changes along a line in x-y plane.
Next, the simulated data is contaminated with a relatively small level of error of 1 10-3
0.1%. A randomly generated noise of 60 dB (standard deviation) is added to the MIT
signals (for further details see Appendix C.6).
0 -8 -6 -4 -2 0 2 4 6 8 10
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Length (cm)
Co
nd
uctivity c
ha
ng
e (
S/m
)
True values
k = 100
k = 180
k = 256
5 10 15 20
2
4
6
8
10
12
14
16
18
20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
18
20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
18
20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
18
20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
18
20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Target k = 100 k = 180 k = 256
y-z
x
-y
(a)
(b)
5 10 15 20
2
4
6
8
10
12
14
16
18
20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
118 Discrete Inverse Problem
In this case, the introduction of noise leads to the situation where the right hand side
vector b is no longer consistent. Therefore, as the problem is underdetermined and
nearly rank deficient, no meaningful solution can be found using either the pseudo-
inverse A+ = A
T (AA
T)
-1. In the case of SVD without truncation, we have:
(6.22)
which is large due to the appearance of small singular values and an inconsistent right
hand side. However for the TSVD, the resulting solution has a smaller norm [168]:
(6.23)
and, consequently, the use of TSVD does still allow meaningful solution to Ax = b to be
obtained.
The results for the noisy data are shown in figure 6.11. For the high level of truncation
(k = 100), effect of noise is insignificant as the incorporated singular values are above
the noise level (see figure 6.9). For the intermediate level of truncation, although a
portion of the singular values fall below the noise level, the reconstructed image is
almost identical to its noise-free counterpart. However, a closer scrutiny of different 1D
images reveals small differences between the two images, with the noise-free image
being more accurate.
With no truncation (k = 256), the solution is unstable and large artefacts, both in terms
of magnitude and volume, can easily be observed in 2D representation of the image.
The visualization of the inclusions is unfeasible as the artefacts are dominant. Similarly,
large oscillations can be seen in the 1D image.
While TSVD cuts the data, the generalized Tikhonov regularization can do both
filtering and implementing a-priori information. In the next section, the Tikhonov
regularization method is first explained in the standard form, followed by the
generalized form.
119 Discrete Inverse Problem
Figure 6.11. As figure 6.10, but with 60 dB noise added to simulated data.
6.5. Tikhonov regularization
6.5.1. Standard Tikhonov
The Tikhonov regularization [173], is often presented as a technique for obtaining
approximate LS solutions to overdetermined problems that are ill-posed. It has
advantages over the TSVD approach as the SVD of a matrix does not have to be
computed. The disadvantage is that it requires a regularization parameter that must be
tuned to individual problems and applications. An approximate LS solution to the
0 -8 -6 -4 -2 0 2 4 6 8 10
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Length (cm)
Co
nd
uctivity c
ha
ng
e (
S/m
)
True values
k = 100
k = 180
k = 256
5 10 15 20
2
4
6
8
10
12
14
16
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Target k = 100 k = 180 k = 256
y-z
x
-y
5 10 15 20
2
4
6
8
10
12
14
16
18
20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
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16
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-0.4
-0.3
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0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
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16
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0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
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20
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0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
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20
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-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
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20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
18
20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
5 10 15 20
2
4
6
8
10
12
14
16
18
20
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
(a)
(b)
120 Discrete Inverse Problem
overdetermined problem Ax = b with the standard Tikhonov regularization is
characterized by minimizing the objective function:
(6.24)
where is the scalar regularization parameter and ||x||2 is the penalty term, constraining
the solution. Regularization can be viewed as a tool to trade-off the influence of the
measurements on the solution against its compliance with the prior assumptions, i.e. not
letting ||x||2 get too big. The value of controls this trade-off; as 0, x tends to the LS
solution A+b. The solution of the minimization (6.24) is given by:
(6.25)
Here, In and Im are identity matrices of the appropriate sizes. The positive definite
matrices (AAT+ In)
-1 and (A
TA+ Im)
-1 can be interpreted as filters in the image and data
spaces respectively: the former is back-projection followed by spatial filtering, and the
latter is filtering of the data followed by back-projection [174].
In order to see whether the Tikhonov regularization can also be applied to the solution
of underdetermined systems note that the Tikhonov solution (6.25) is equivalent to the
solution in the SVD form in (6.21) when the following filtering is employed [168]:
(6.26)
Note that for si >> then f (si) 1, and if si << then f
(si) 0. Therefore, the Tikhonov
regularization has a similar effect as the TSVD. We know that the application of the
TSVD leads to a LS solution satisfying subject to .
Therefore, it can be concluded that the Tikhonov regularization can also be applied to
the solution of ill-posed underdetermined systems [175]. Computationally, the second
of the rearrangements shown in (6.25) will be advantageous for underdetermined
problems as it leads to the creation of a smaller matrix. In the following, only the so-
called left inverse form of the Tikhonov based schemes will be presented, although
advantageous right inverse form are also available [168].
By replacing the definition of f (si) given by (6.26) together with (6.21), the modified
singular values of the sensitivty matrix may be computed. These are shown in figure
121 Discrete Inverse Problem
6.12 which allow comparison with the results for the TSVD in figure 6.9. As the
original singular values decrease, the modified singular values grow larger and,
therefore, their effect in the solution (6.19) is inceasingly diminished.
Three different values of are adopted in order to examine how the singular values are
affected by different levels of regularization. Note that = 0 denotes no adjustment,
similar to no truncation for k = 256.
Figure 6.12. Modified singular values (si + /si) for 3 different levels of regularization.
A similar method, called the damped SVD method [176], replaces si2 by si in the filter
factors defined by (6.26). This method has a similar effect, but with a more gradual
transition between including large and rejecting small singular values. The use of the
dynamic (variable) in (6.26) has also been suggested for seismic tomography [177].
Reconstructed images using the standard Tikhonov method are shown in figure 6.13. As
expected, for the high and intermediate levels of regularization ( = 1 10-15
to 1 10-18
),
the inverse problem is stable, but the images are blurred. With no regularization, the
noise added to the data is amplified in the solution which, as before, is dominated by the
noise. The Tikhonov images are almost identical to their counterparts reconstructed by
the TSVD method.
0 50 100 150 200 25010
-12
10-10
10-8
10-6
10-4
10-2
Singular value index
Sin
gula
r valu
e
= 1 10-15
= 1 10-18
= 0
122 Discrete Inverse Problem
Figure 6.13. Reconstructed images using standard Tikhonov regularization with 60 dB noise added
to simulated data. Columns from left: target and images with different levels of regularization
indicated by value of . Rows from top: x-y and y-z cross-sectional planes. Colour-map shows
conductivity change in S m-1
.
6.5.2. Generalized Tikhonov
The penalty term ||x||2 in the standard Tikhonov form prevents extreme values of the
reconstructed conductivity change, but does not enforce any smoothness on the values
[173]. As an alternative, ||Rx||2 can be used for the penalty term where R is called the
regularization matrix, which can introduce additional constraints on the computed
conductivity distribution.
The choice of R depends on the particular application being considered. However,
common choices involve approximate first- or second-order (finite difference)
differential operators. Further refinement of the Tikhonov regularization may be
achieved by penalizing the differences between the solution, x, and some reference
background, xref, which can include some known non-smooth behaviour. Finally, there
is the possibility that we may not wish to fit all measurements to the same accuracy, in
particular as some may have larger errors than others. This can be achieved by
considering some diagonal weighting matrix [173]. The generalized Tikhonov
procedure is then of the form:
(6.27)
5 10 15 20
2
4
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14
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18
20
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Target = 10-15 = 10-18 = 0
y-z
x
-y
5 10 15 20
2
4
6
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123 Discrete Inverse Problem
which reduces to the standard Tikhonov form for the choice of R = I, W = I, and xref =
0. The solution for (6.27) will be [34,178]:
) (6.28)
Throughout this study, the weighting matrix = I has been employed. Notice that xref is
different from the initial conductivity 0 corresponding to State-I. Recalling from
equations (6.2) and (6.3), we have x = = - 0. By applying ||x - xref||2, the
difference between the reconstructed conductivity change, x, and the expected
conductivity change, xref, is penalized. For instance, consider a case when the a-priori
information includes some anatomical features such as the approximate location of
different tissue types in the domain. Then in time-difference imaging of haemorrhage,
the conductivity change for the elements representing any tissue type in which the
bleeding is unlikely to occur (e.g. bones) can be set to zero in the vector xref.
Similar to the standard form, the generalized Tikhonov method can be explained in
terms of the singular values. However, in this case, the generalized singular value
decomposition (GSVD) needs to be employed. For further details on the GSVD see e.g.
[176,168].
6.5.3. Neighbouring matrix
A common choice for the regularization matrix is to use a discrete approximation of the
difference operators on the discretized volume. This corresponds statistically to
assuming a correlation between neighbouring voxel values. Elements sharing a face as
in figure 6.14a (e.g. [125]) or at least a node as in figure 6.14b (e.g. [179]) may be
considered neighbours. Hereinafter, the former is called R-face, and the latter R-node.
Figure 6.14. Correlation between neighbouring elements with common face (a), common node (b),
and a combination of faces and edges (c) in a 2D image. Values shown in voxels correspond to a
central finite difference approximation of Laplacian operator.
-1 -1 -1
-1 8 -1
-1 -1 -1
-1
-1 4 -1
-1
-1 -2 -1
-2 12 -2
-1 -2 -1
(a) (b) (c)
124 Discrete Inverse Problem
For a discrete Laplacian operator in a regular cubic mesh comprising n voxels, each
element in the regularization matrix is defined as:
(6.29)
Here, N is the number of elements neighbouring the central element. Another type of
operator can be realised by taking into account the degree of neighbourhood or
correlation between the elements (figure 6.14c).
Applying the penalty term ||Rx||2 enforces some degree of smoothness on the image by
preventing the neighbouring elements having sharp conductivity contrasts. Those
regularization matrices that can enforce the smoothness while preserving the desired
edges and genuine sharp contrasts, are introduced later for absolute imaging algorithms.
Detailed description of neighbouring regularization matrices can be found in [166].
Images were reconstructed using the generalized Tikhonov method, where R is the
neighbouring matrix (R-face) and xref =
0. For a high level of regularization (
= 10
-15),
the smoothness is significantly enforced on the image which makes the internal
inclusions nearly invisible (see figure 6.15).
Figure 6.15. As in figure 6.13, but reconstructed using (6.28) with R-face and xref = 0.
The magnitude of the obtained conductivity changes for the inclusions are comparable
to that for the image reconstructed using the standard Tikhonov in figure 6.13. For =
10-18
, the inclusions are clearly visible, while the background is smooth. The small
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125 Discrete Inverse Problem
artefacts, which appreaed in the image reconstructed using the standard Tikhonov, are
removed here due to the effect of R. Finally, for no regularization ( = 0), the
reconstructed image is entirely degraded by the presence of artefacts.
In order to compare the results obtained from the standard and generalized Tikhonov
methods, 1D plots of the conductivity change are considered. Figure 6.16a shows the
values along a cross-sectional line cutting through the inclusions. As can be seen, while
the magnitude of the reconstructed values for the inclusions is comparable in both
methods, the transition from negative to positive conductivity changes is smooth in the
results obtained from the generalized Tikhonov with R-face. Also, the positions of the
features, where the maximum magnitude is obtained, are more accurately recovered
using the generalized Tikhonov form.
Figure 6.16. Comparison of standard and generalized Tikhonov: 1D plots showing reconstructed
conductivity changes ( = 1 10-18
) along cross-sectional lines in y-direction (a) and x-direction (b).
-10 -8 -6 -4 -2 0 2 4 6 8 10-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Length (cm)
Co
nd
uctivity c
ha
ng
e (
S/m
)
-10 -8 -6 -4 -2 0 2 4 6 8 10
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0
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Length (cm)
Co
nd
uctivity c
ha
ng
e (
S/m
)
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(a)
(b)
True values
Standard Tikhonov
(Identity)
Generalized Tikhonov
(R-face)
126 Discrete Inverse Problem
Figure 6.16b shows the 1D plots corresponding to a different crossing line. This plot
shows the smoothness of the reconstructed conductivity changes in the background. For
the standard Tikhonov with Identity matrix, some artefacts with magnitudes as large as
0.08 S m
-1 can be observed. The plot for generalized Tikhonov method shows an
smoother conductivity change in the background.
6.6. Error estimation and regularization parameter
Figure 6.11b illustrated the importance of selecting the regularization parameter
properly. For very low levels of regularization, reconstructions will have oscillatory
artefacts due to the noise amplification. In contrast, with too much regularization, the
reconstructed images tend to be smooth or damped. The study of optimal regularization
parameters is an active area in ill-posed problems. The majority of the suggested
methods for this purpose exploit certain types of the error in solution of the inverse
problem, and attempt to minimize it. In order to derive the mathematical representation
of the error, let us express the linearized model in the form:
(6.30)
where xtrue is the sought true distribution in the target, and represents the discrete
noise vector with zero mean (E( ) = 0) and2 variance (standard deviation = ).
Following (6.25), the error in the solution computed for regularization parameter can
be expressed by:
(6.31)
where e is the solution error [180]. Note that in practice, this error is not available as
xtrue is unknown. Therefore, it cannot be used for optimizing the regularization
parameter. However, when comparing the sensitivity of spherical and cylindrical MIT
arrays for stroke imaging in chapter 8, e is employed for selecting ; this helped to
obtain the best feasible images for each system.
It is worth mentioning in passing that while a mathematical method attempts to
minimize the errors in the solution, it may not necessarily result in the best potentially
127 Discrete Inverse Problem
useful images. For instance, if the image artefacts consistently appear in certain regions
of the head, which can be excluded as the bleeding sites by a radiologist, then a noisy
image with a clear stroke may be preferable from a clinical point of view, even if with
larger mathematical errors. Examples of such images are provided in chapter 8.
The predictive error, P , estimates the discrepancy between the predicted voltage by the
linearized model and that obtained from the true solution as:
(6.32)
where C = AB . Again, this error cannot be employed for error minimization as it
involves the usually unknown true solution. For the purpose of optimising , the
regularized residual may be defined as:
(6.33)
Several methods have been suggested for selecting the regularization parameter. Two of
the most commonly used methods are:
Generalized cross validation (GCV) method proposed by Wahba [181].
L-curve method proposed by Hansen [168].
Each method is briefly explained in this section. Detailed studies and comparisons of
these methods can be found in the work of Hansen [168] or Vogel [180].
6.6.1. GCV
The GCV method is based on statistical considerations [181]. It attempts to minimize
the predictive error, P . Since xtrue is not usually available, the GCV method
alternatively aims at minimizing the function:
(6.34)
which is computable (for further details see Appendix C.1).
To evaluate the GCV function, the regularized inverse C = AB = AW++
UT must be
computed for different values of , where ++
is the inverse of if the singular values
128 Discrete Inverse Problem
are replaced by (si + /si). Also, the residual r can be computed using (6.33). For
differential imaging, this can be easily obtained by using the SVD of A.
However, since the sensitivity matrix is updated for absolute imaging (explained later in
this chapter), this can be a cumbersome process when computing the SVD of A must be
repeated. In addition, the GCV function can have a very flat minimum, making it
difficult to isolate a clear minimum [182].
To examine the efficiency of the GCV method, three levels of noise were added to the
simulated data in previous sections. For the standard Tikhonov method and each data
set, a plot of the GCV function for different values of is shown in figure 6.171.
For a relatively low level of 60 dB noise, the optimized is 3.6 10-18
. As the noise level
increases, a higher level of regularization is suggested by the GCV method. For 40 dB
noise, is increased by approximately a factor of 100.
Figure 6.17. Regularization parameter selection using GCV method for three noise levels.
In order to illustrate the importance of using the correct values of for a specified level
of noise, the images presented in figure 6.18 were reconstructed using the computed
1 For the GCV and L-curve methods, a regularization toolbox, which is available as freeware, was
employed. The toolbox is developed in Matlab [233,232]. The command 'fminbnd', which finds the
minimum of a function in a given domain, is used in the toolbox to find the optimum value of .
10-25
10-20
10-15
10-10
10-21
10-20
10-19
10-18
10-17
10-16
10-15
G(
)
3.6050e-018
6.4012e-016
3.4753e-017
40 dB
50 dB
60 dB
129 Discrete Inverse Problem
values of suggested by the GCV method. The fourth column from left, shows the
image with 40 dB added noise, but reconstructed using the optimized for the 60 dB
case, i.e. 3.6 10-18
. Clearly, this level of regularization is low for 40 dB noise as the
image is dominated by the artefacts. To avoid these artefacts, a larger is required, as
suggested by the GCV method (6.4 10-16
). However, the use of a larger value of has
the disadvantage that the image becomes more blurred.
Figure 6.18. Reconstructed images by standard Tikhonov method and using suggested value of by
GCV method. Colour-bar in S m
-1.
6.6.2. L-curve
The L-curve method is a more recent technique for choosing [168]. The general idea
is to display the trade-off between the two quantities of the objective function in (6.24);
i.e. the penalty term versus the corresponding residual error norm for each of a set of
regularisation parameter values. This results in a plot which typically has an L-shaped
curve; thus the L-curve name. Intuitively, the best regularisation parameter should lie on
the corner of the L-curve, since:
for larger values of , the residual increases rapidly without a significant
reduction in the norm of the solution.
for smaller values of , the norm of the solution increases rapidly without much
decrease in the residual.
The sharpness of this corner varies from problem to problem; not all curves have a
pronounced enough corner to allow unambiguous selection of [182].
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0.5Target = 3.6 10-18 = 6.4 10-16 = 3.6 10-18
130 Discrete Inverse Problem
Figure 6.19. Plots of L-curve for selecting the regularization parameter for three noise levels.
Similar to the previous section, the optimized value of was computed for three levels
of noise using the L-curve method. The suggested values are close to those obtained by
the GCV method (see figure 6.19).
When attempting the generalized Tikhonov regularization with R-face, however, both
methods failed in predicting an optimized value for . Figure 6.20a indicates that care
must be taken when examining a range of regularization parameters to minimize the
GCV function. The GCV function has points of inflection, making the automatic
detection of the desired minimum more difficult. For 50 dB noise, the smallest value in
the range ( 9 10-37
) is suggested as the optimized , whereas the desired local
minimum occurs at a larger value ( 10-20
). The optimized values for all three noise
levels were too small resulting in under-regularized solutions. This is in line with the
observations of others [182,183,184].
The L-curve method resulted in the same value of ( 1.6 10-15
), regardless of the noise
level; the corners for all curves were coincident (figure 6.20b). However, the solution
obtained for this choice of would be over-regularized. Notice that other L-corners
occur for each curve which depend on the noise level (shown by green lines), but the
algorithm has found the sharpest. In general, the L-curve criterion does not work well
when the solution has a higher degree of smoothness [185], which is the case when
applying the neighbouring matrices.
10-10
10-9
10-8
10-7
10-6
10-5
10-2
10-1
100
101
102
103
residual norm || A x - b ||2
so
lutio
n n
orm
|| x || 2
2.2183e-018
5.6477e-017
40 dB 50 dB 60 dB
3.9390e-016
131 Discrete Inverse Problem
Figure 6.20. Regularization parameter selected by GCV (a) and L-curve (b) methods for three noise
levels (40, 50, and 60 dB).
6.7. Frequency-difference imaging
Although time-difference MIT imaging, at a fixed frequency, could potentially be useful
for many applications, it cannot be applied for imaging when:
No change in the conductivity distribution of the target occurs.
The data set for one of the two states occurring in time, State-I or State-II, is
unavailable.
The latter, for instance, applies to an initial diagnosis of haemorrhage when no previous
data-set for the patient is available. Frequency-difference imaging may alternatively be
attempted. This relies on being able to distinguish the frequency signature of the
10-40
10-35
10-30
10-25
10-20
10-15
10-21
10-20
10-19
10-18
10-17
10-16
G(
)
4.4128e-022
8.2933e-037
8.5527e-020
40 dB
60 dB
50 dBlocal
minima
10-10
10-9
10-8
10-7
10-6
10-5
10-6
10-4
10-2
100
102
104
106
108
residual norm || A x - b || 2
so
lutio
n s
em
i-n
orm
|| L
x || 2
5.5e-22 1.1e-21 4e-19
1.663e-15
60 dB 50 dB 40 dB
(a)
(b)
local
minimum
132 Discrete Inverse Problem
constituents of the target from each other. Note that in order to be able to apply
frequency-difference imaging, at least one of the constituents must have a frequency-
dependent conductivity.
In order to form the reference data set for State-I, the measurement/simulation is carried
out for one frequency (f1). Subsequently, data for State-II is acquired at a different,
usually higher, frequency (f2). Throughout this thesis, the phase changes are employed
as the MIT data sets for frequency-difference imaging. Of interest is the change in tissue
conductivity, , as the frequency changes from f1 to f2. However, as the equation
(2.12) shows, the change in the signal's phase is proportional to both and f.
A corrected phase change, , is computed so as to represent only the change in
conductivity:
(6.35)
where V1 and Vp1 are the State-I voltages measured at frequency of f1, and V2 and Vp2
are for State-II measured at frequency of f2. The sensitivity matrix must be computed
accordingly at frequency f2 (Scharfetter et al [186] applied a similar correction
procedure to the voltages).
Once the corrected phase difference has been obtained for all coil combinations ( ),
the linearized inverse problem is then treated in the same way as discussed in the
foregoing part of this chapter (see equations (6.12) and (6.13)). As before, the linearized
model assumes that there are no large conductivity changes between the two states. In
other words, the sensitivity matrix computed for one frequency can be used for solving
the inverse problem if the morphology of the sensitivity map does not depend on the
frequency [186].
If f2 > f1 in (6.35), and given the fact that the conductivities of all known biological
materials either increase or remain unchanged as the frequency increases, the estimated
conductivity changes by the inverse problem are expected to be non-negative. This
physical constraint may be applied as a-priori information by truncating the negative
values from the estimated conductivity changes before plotting the image.
133 Discrete Inverse Problem
Table 6.1. Conductivities (in S m
-1) of white matter and blood at 1 and 10 MHz, and conductivity
change between two frequencies.
1 MHz 10 MHz 10 MHz - 1 MHz
Background (white matter) 0.1 0.16 0.06
Inclusion (blood) 0.8 1.1 0.3
The constituents of the cylinder with inclusion (figure 6.21a) were considered to be
white matter (representing brain) and blood (representing haemorrhage). Two
frequencies of 1 and 10 MHz were considered. The conductivity values are given in
table 6.1. The acquired data for State-I and State-II corresponded to 1 and 10 MHz,
respectively.
Frequency-difference images were reconstructed from the simulated phase changes with
1 m added phase noise, and using the generalized Tikhonov method (R-face, xref = 0)
and (6.35). Again, neither of the GCV and L-curve methods worked properly. The
solution was slightly under-regularized for both cases ( GCV = 1.3 10-10
and L-curve =
2.9 10-11
). A better image in terms of the smoothness was obtained for an empirically
adjusted value of = 1 10-9
.
Figure 6.21. Frequency-difference imaging; (a): target, (b) and (c): reconstructed images using
generalized Tikhonov method (1 m added phase noise, R-face, xref = 0, = 1 10-9
), but plotted with
different colour-bars in S m
-1.
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134 Discrete Inverse Problem
As can be seen in figure 6.21b, although the inclusion is reconstructed, it can be barely
seen, and the obtained conductivity change values are under-estimated ( 0.1 S m
-1).
The under-estimation also happened in time-difference images reconstructed in the
previous sections. However, it is more evident here. The inclusion may be better
visualized if the maximum limit in the colour-bar is reduced as in figure 6.21c.
If State-II measurements are carried out for multiple frequencies (multi-frequency
imaging), more information will be available, which may allow the spectroscopy of the
target; e.g. biological tissue characterization [187]. This was not pursued in this thesis.
6.8. Absolute imaging
Absolute imaging is an attempt to quantify the actual conductivity distribution of the
target relative to empty space (0 S m
-1), where the conductivity of the target is supposed
not to change during the data acquisition. In absolute imaging, there is only one MIT
data set available. While single-step linear methods are useful for differential imaging,
absolute imaging is likely to require a non-linear algorithm to be successful. This is due
to the fact that the use of linear approximation is only valid for small deviations from
the reference conductivity (see figure 6.6).
In medical applications, the conductivity contrasts can be considerably large; e.g.
contrast between CSF (2 S
m
-1) and skull (0.08 S
m
-1) at frequency of 10 MHz. A
linearized model may be able to calculate a small change in conductivity between two
states, but reconstructing the absolute conductivity values will require an iterative non-
linear solution.
6.8.1. Iterative non-linear solution
Let us assume the voltage Vm is measured for an unknown conductivity distribution .
Revisiting equation (6.2), this voltage can be expressed by a Taylor expansion around
some initial conductivity 0. In equation (6.3), the voltage V0, which corresponded to
the measured voltage for State-I, is not available in absolute imaging. Instead, it can be
estimated by solving the forward problem for the known initial guess 0 (usually a
homogeneous distribution in lack of any a-priori information).
135 Discrete Inverse Problem
The sensitivity matrix is also computed for the same conductivity distribution. By
setting x = = - 0 and b = V = Vm - V0, the absolute values, , can then be
calculated by, for example, the standard Tikhonov method as in (6.25):
(6.36)
In presence of a large contrast, the estimated absolute values may not be accurate as the
linearized approximation fails to properly model the problem. However, the new
estimated conductivity distribution, , can be employed as a closer guess to solve the
linearized model with an updated sensitivity matrix. This results in an iterative
algorithm as:
(6.37)
where k+1 is the conductivity distribution computed at (k+1)-th iteration, and k is the
conductivity estimated by the previous iteration, which is used as the initial guess for
the current iteration. The terms Vk and Ak are the voltage vector and sensitivity matrix
calculated by solving the forward problem for the conductivity distribution k.
Therefore, as stated by Lionheart et al [173], the essence of a non-linear algorithm is to
repeat the process of calculating the sensitivity and solving a regularized linear model. It
starts by taking advantage of the known voltages due to an initial guess for the
conductivity distribution, and approaches the optimum solution in an iterative manner.
Obviously, the iteration procedure must be conducted in a way to improve the
convergence of the solution. This may be done by adopting the proper choice of
regularization level, and applied constraints (further discussed later). However, as will
be shown, in presence of a high level of noise in the data, the solution may not
converge.
In absolute imaging, similar to frequency-difference imaging, negative values can be
truncated from the solution as a-priori information. This may be applied after each
iteration prior to using the most recent conductivity distribution for the computation of
the sensitivity matrix. A flowchart for the iterative image reconstruction algorithms is
shown in figure 6.22.
136 Discrete Inverse Problem
Figure 6.22. Flowchart for iterative non-linear MIT image reconstruction.
There is no single correct choice for the stopping criterion. One can monitor the
difference between the voltages ||rk||2 = ||Vm – Vk||
2 and the iterations may be stopped
when the error reaches a pre-defined tolerance value which indicates that the assumed
conductivity, k, and the absolute values, k+1, are close enough. For instance, the
Morozov stopping criterion suggests that the tolerance value can be estimated from the
known level of noise in the measured data [168]. Various types of stopping criteria can
be found in [168,180,188,189]. As an alternative approach, || k+1 k||2 may be
monitored.
In order to assess the performance of an algorithm by the numerical experiments, the
error in the conductivity itself can also be monitored; ||ek||2 = || k+1 true||
2 where true
is the known conductivity distribution of the simulated target. Obviously, this method of
monitoring will not be feasible for practical MIT imaging where true is unknown.
Solve forward
problem
Stopping
criteria
Measured
voltages
Start with initial
guess
Compute
sensitivity
Compute
voltages
Solve inverse
problem
stop
Continue
Update
conductivity
Solution verified
137 Discrete Inverse Problem
6.8.2. Newton-type methods
The iterative algorithm presented in the previous section, falls in the category of
Newton-type methods for solving the inverse problem. The key idea of any Newton-
type method consists in repeatedly linearizing the forward operator, V = F( ), around
some approximate solution , and then solving the linearized problem. There exist
various formulas in the literature for the Newton-type iterative algorithms, which are
explained by the LS minimization and using Tikhonov regularization1. The only
difference is the penalty term added to the LS solution.
For instance, the penalty term || ||2 which attempts to constrain the conductivity
change between two iterations results in equation (6.37), referred to as Levenburg-
Marquardt [190]. One may consider constraining || k+1||2 (absolute conductivity values)
or || k+1 - ref||2 (difference between the expected and estimated absolute conductivity
values). The latter results in a formula which is referred to as Gauss-Newton [173].
In general, one may allow the regularization parameter to vary between the iterations
( k), or consider it to be constant ( ). The automatic update of between the iterations
is possible if a reliable method for selecting is available.
With the aim of increasing the computational speed of the iterative process, the
sensitivity matrix can be calculated only once, although the forward solution is
calculated repeatedly as the conductivity is updated. Since the sensitivity matrix
remains the same in all iterations (same as that in the first iteration), it can be pre-
calculated off-line. This semi-linear algorithm is also called the Newton-Kantorovich
method [2,191]. Furthermore, if rather than solving the forward problem, the MIT
voltage is simply calculated by multiplying the sensitivity matrix with the most recent
conductivity distribution (i.e. Vk =
A k), then the method is referred to as the iterative
Tikhonov regularization [190]:
(6.38)
However as figure 6.6 indicated, the sensitivity matrix can only predict the voltages if
the conductivity change is small. Therefore, the applicability of semi-linear methods
may be limited to low-contrast targets.
1 Regularization methods other than Tikhonov regularization for Newton-type algorithms can be found in
Kaltenbacher et al [230].
138 Discrete Inverse Problem
6.8.3. Absolute imaging using phase changes
As discussed in the forgoing sections for differential imaging, it is also often useful to
normalize the MIT signals to the primary voltage for absolute imaging. This may help
to reduce some of the errors in the data. For instance, if there is an error in the gain of
one of the measurement channels in the hardware, it will affect the secondary and
primary voltages in the same way, and the error will be cancelled when forming =
tan-1
( V/Vp). This can be of great importance when attempting absolute imaging for
which the difference between the measured and simulated data is used.
The equivalent to formulation (6.37) for the phase changes in the generalized Tikhonov
form is:
(6.39)
where sensitivity entries are computed by (6.12). As a case study, absolute imaging of
the cylinder with inclusion, was performed. The simulations were conducted at the
frequency of 10 MHz. The conductivity values of the target were, therefore, 0.16 and
1.1 S m
-1 for the background (white matter) and inclusion (blood), respectively (see
figure 6.23). In order to gain more information, the MK1 array was placed at 3 different
levels as shown in the figure, resulting in 256 3 = 768 MIT signals.
The images were reconstructed from the phase changes, i.e. equation (6.39). It was
found that normalization slightly reduces the ill-posedness of the inverse problem in this
case. The condition number of a normalized sensitivity matrix, computed for a uniform
conductivity, was some 5 times less than that of an un-normalized one.
Figure 6.23. Simulated target; coil array is placed at three levels along height of cylindrical tank.
Colour-bar in S m
-1.
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1x-y y-z
4 cm
0
-4 cm
139 Discrete Inverse Problem
As stated previously, both the GCV and L-curve methods require the solution of the
inverse problem for different levels of regularization in order to select . This can be
done by computing the SVD of A, which must be repeated for each iteration when the
sensitivity matrix is updated. Therefore, the automatic update of between iterations
can be prohibitively long. Furthermore, figure 6.20 indicated that these methods, when
employed with the generalized Tikhonov solution, may result in under- or over-
regularized solutions.
Here, an alternative approach is suggested for selecting k. The iterations can be started
with a relatively large value of 1. This assures the stability and robustness of the
algorithm for the first few iterations. The initial guess can, therefore, be chosen with
less caution. In order to accelerate the convergence and achieve a better spatial
resolution, the regularization level is reduced after each iteration as:
(6.40)
where l 1 is a predefined ratio. Note that k is used to compute k+1 which is the
solution computed at (k+1)-th iteration. The residual error ||rk||2 = || m – k||
2 (radians)
and the solution error ||ek||2 = || k true||
2 (S
m
-1) are monitored between the iterations.
The iteration process can be stopped once an increase in the residual error is observed
(||rk+1||2 > ||rk||
2) and the last updated conductivity may be considered as the converged
solution. This is the solution for which the minimum residual is achieved. Here,
however, the process was continued for a few more iterations after the increase in the
residual. This allowed examining the solutions obtained thereafter.
For the mentioned case study (figure 6.23), the iterations were conducted using R-face,
1 = 1
S
m
-1, and 1
=
1 10
-8. As the first attempt, a 50% drop in the regularization level
between the iterations were considered (l = 0.5). Solving the forward problem for each
iteration, including all 3 planes of measurement, took 75 s, while 6
s was the solution
time for the inverse problem. Plots of the monitored quantities are shown in figure 6.24.
The residual error ||rk||2 fell abruptly during the first few iterations and, after a slow
decreasing trend, started increasing from iteration 53; this occurred at 52 = 4.4 10-24
.
The second row in figure 6.24 shows the magnified plots at iterations 51-53. The
solution error ||ek||2 showed a similar trend to that in the residual.
140 Discrete Inverse Problem
Figure 6.24. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1
, l = 0.5 and 1 = 1 10-8
. First
row: plots of ||rk||2 and ||ek||
2 for iteration procedure. Second row: same plots as in first row but
magnified at iteration 52. Also shown reconstructed images at iterations 52, 53, and 54. Upper limit
of colour-bars (S m
-1) is set to maximum conductivity value for each image.
Based on the residual plot, the solution obtained at iteration 52 can be considered as the
converged solution. The reconstructed images obtained from the 52th, 53th, and 54th
0 10 20 30 40 50
-0.05
0
0.05
0.1
0.15
0.2
Iteration No.
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5
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6.5
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8
Iteration No.
50 51 52 53 54
0
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1
1.5
2
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x 10-3
Iteration No.
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5.25
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Iteration No.
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10
52th 53th 54th
y
-z x
-y
||rk||2 ||ek||
2
||ek||2 ||rk||
2
141 Discrete Inverse Problem
iterations are also illustrated in the figure where the colour-bar limits are set to the
maximum conductivity value in each image. In order to enhance the visualization, for
each image, the colour-bar limit is set to the maximum value in that particular image.
The conductivity values of the inclusion (blood) are clearly under-estimated in the
converged solution. A conductivity of 0.47 S
m
-1 is obtained for blood in the 52th
iteration (target is 1.1 S m
-1).
If the iterations are continued after the increase in the residual, the process will be
unstable due to the solution being under-regularized. A large ||ek||2 and high amplitude
artefacts can be observed in the plot and reconstructed image at the 54th iteration.
One may start the iteration process with a different level of regularization (in previous
example 1 = 1 10-8
). Obviously, starting with a larger will require the algorithm to be
carried out for a few more iterations to arrive at the converged solution. However, a
small value of can have a significant effect on the solution. In order to examine this,
let us consider 1 = 1.8 10-20
to start the iterations with. This is equivalent to 40 =
1.8 10-20
in the previous example for which the algorithm was stable (see the plots at
iteration 40 in figure 6.24).
Absolute imaging was performed where all the parameters (except 1) were similar to
the previous example. Results are shown in figure 6.25. The algorithm was unstable and
the residual started increasing from iteration 4. The smallest solution error ||ek||2 was
obtained at the first iteration which corresponds to the initial guess 1. Also the solution
errors were considerably larger than those in figure 6.24. The reconstructed images are
dominated by the artefacts which clearly indicates the inverse solution being under-
regularized.
The reason why the algorithm was unstable in this case can be realised from the
algorithm itself. Revisiting the equation (6.39), we can express the inverse problem as x
= B b where x = k+1 - k, b = m - k, and B = (AkTAk + kRk
T Rk)
-1Ak
T. For the first
few iterations, there are large values in vector b. This is due to the difference between
the initial guess ( 0) and target conductivity distribution.
Therefore, for a small value of k, B can amplify such large values in b, resulting in
high amplitude artefacts in the solution. These artefacts produce more artefacts in the
next iterations; evident in images in figure 6.25.
142 Discrete Inverse Problem
Figure 6.25. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1
, l = 0.5 and 1 = 1.8 10-20
.
Errors ||rk||2 and ||ek||
2 are plotted against iterations. Reconstructed images correspond to iterations
1, 2, and 8. Colour-bar (S m
-1) limits are set to maximum conductivity for each image.
For the example in figure 6.24, however, the solution at iteration 40 is much closer to
the target than the initial guess (evident in the ||ek||2 plot), particularly for the
background which has a large volume. Therefore, the values in b (contributions from
both the background and inclusion) are smaller (evident in the ||rk||2 plot). This prevents
the large oscillations in the solution until grows extremely small, making the
algorithm unstable at iteration 53.
Therefore, unless a close estimate for 1 is available, it is suggested to initially over-
regularize the inverse problem. The only disadvantage would be the need for a few
more iterations to arrive at the solution.
0 2 4 6 8 1050
100
150
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250
300
350
400
450
Iteration No.
0 2 4 6 8 100
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Iteration No.
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10
1st 2nd 8th
y
-z x
-y
||ek||2 ||rk||
2
143 Discrete Inverse Problem
Figure 6.26. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1
, l = 0.1 and 1 = 1 10-8
. Errors
||rk||2 and ||ek||
2 are plotted against iterations. Reconstructed images correspond to iterations 10, 18,
and 19. Colour-bar (S m
-1) limits are set to maximum conductivity value in each image.
Another parameter which influences the algorithm is the ratio used to reduce the
regularization level between the iterations; l = 0.5 was previously employed. In order to
examine the effect of this ratio on the solution, the results in figure 6.26 were obtained
using l = 0.1, where all other parameters were similar to the example in figure 6.24.
As can be seen, the residual increased from iteration 19, corresponding to 18 = 1 10-25
,
which, not surprisingly, is at an earlier stage of the iteration process compared to the
53th iteration in figure 6.24. However, unlike the diverged solution at iteration 53 which
provides a still useful image (see figure 6.24), here the diverged solution at the 19th
iteration is useless. This is due to a quicker drop in the regularization level when l = 0.1
is adopted.
0 5 10 15 20 25
-0.05
0
0.05
0.1
0.15
0.2
Iteration No.
0 5 10 15 20 254.5
5
5.5
6
6.5
7
7.5
8
Iteration No.
5 10 15 20
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4
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20 0
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10th 18th 19th
y
-z x
-y
||rk||2 ||ek||
2
144 Discrete Inverse Problem
Figure 6.27. Plots of ||rk||2 and ||ek||
2 for absolute imaging using (6.39) with R-face, 0 = 1 S m
-1, l =
0.9 and 1 = 1 10-8
.
If l = 0.9 is used, the iterations reach the maximum allowable number defined in the
algorithm (100 iterations) before arriving at the converged solution (see figure 6.27).
Based on the presented examples, l = 0.5 deemed to be a good compromise. This allows
the algorithm to reach the solution in a more gradual and stable trend than that for l =
0.1. On the other hand, fewer iterations are required than that for l = 0.9.
One may reasonably argue that 53 iterations for l = 0.5 is still a large number making
the process long. It must be noted that in all the mentioned examples, noise-free data
have been used. When using a noisy data, it is likely that the divergence in the residual
will occur at the earlier stages of iteration. In order to demonstrate this, two phase noise
levels of 1 and 10 m were added to the simulated phase changes. Absolute imaging was
performed in a similar way ( 1 = 1 10-8
and l = 0.5).
For 1 m noise, the first increase in the residual ||rk||
2 occurred at iteration 8,
corresponding to 7 = 1.5 10-10
(see figure 6.28). However, since the regularization
level is still high, the divergence of the residual is gradual and the following iterations
of the algorithm do not produce a substantial increase in the residual. As can be seen,
the image reconstructed at the 10th iteration is still useful. Thereafter, as grew
smaller, large increases occurred in the residual. The solution error ||ek||2 showed a
similar trend.
Clearly for this example, it is more difficult to decide where to stop the iterations.
According to the Morozov stopping criterion [168], the residual error cannot be
0 20 40 60 80 100
-0.05
0
0.05
0.1
0.15
0.2
Iteration No.
0 20 40 60 80 1004.5
5
5.5
6
6.5
7
7.5
8
Iteration No.
||rk||2 ||ek||
2
145 Discrete Inverse Problem
expected to be smaller than the noise in data, thus stopping the iterations when arriving
at the noise level. Here, the phase noise is 1 m = 1.7 10
-5 radians, and the minimum
residual error at iteration 7 is ||rk||2 = 4 10
-4 radians, which is larger than the noise level.
Therefore, the Morozov stopping criterion cannot be applied.
One possible approach is to continue the iterations for a few more steps after the first
increase in the residual, and then examine the obtained images. In this particular
example, the minimum solution error occurs at the 7th iteration where also the residual
is minimum. However, the image obtained from the 10th iteration shows a closer
conductivity value for the inclusion ( 0.35 S
m
-1), despite being less smooth in the
background (white matter). The solution at the 14th iteration is under-regularized.
Figure 6.28. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1
, l = 0.5 and 1 = 1 10-8
. with 1
m additive phase noise. Errors ||rk||2 and ||ek||
2 are plotted against iterations. Reconstructed images
correspond to iterations 7, 10, and 14. Colour-bar (S m
-1) limits are set to maximum conductivity
value in each image.
0 5 10 15
-0.05
0
0.05
0.1
0.15
0.2
Iteration No.
0 5 10 154.5
5
5.5
6
6.5
7
7.5
8
Iteration No.
5 10 15 20
2
4
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8
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7th 10th 14th
y
-z x
-y
||rk||2 ||ek||
2
146 Discrete Inverse Problem
Figure 6.29. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1
, l = 0.5 and 1 = 1 10-8
. with 10
m additive noise. Errors ||rk||2 and ||ek||
2 are plotted against iterations. Reconstructed images
correspond to iterations 3, 6, and 7. Colour-bar (S m
-1) limits are set to maximum conductivity
value in each image.
For the phase noise level of 10 m , the increase in the residual ||rk||
2 occurred at even an
earlier stage at iteration 5, corresponding to 4 = 1.2 10-9
(see figure 6.29). None of the
reconstructed images showed a clear inclusion within the background, and the artefacts
are dominant in all images. Notice the solution error ||ek||2 is larger than that obtained
from the noise-free data in figure 6.24. This level of noise was simply too high for the
image reconstruction algorithm to succeed.
As stated in Appendix A, a phase noise level of 1 m is achievable in the recently
developed MK2a system in the South-Wales MIT group. Absolute imaging using the
measured data by the MK2a system is presented in chapter 9.
0 5 10 15
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3rd 6th 7th
y
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147 Discrete Inverse Problem
6.8.4. Three-layer geometry
In order to make the geometry more realistic, a layer of peripheral CSF, 1 cm thick, was
added to the geometry. As the a-priori information, conductivity values below zero and
above 2 S m
-1 were not admissible; CSF is the most conductive constituent in the head.
Absolute images were reconstructed using the iteration procedure explained in the
previous section, and the noise-free simulated data.
As can be observed in figure 6.30, although the CSF layer is reconstructed ( 1.6 S m
-1),
the blood (inclusion) is entirely invisible. The likely reason for this is the fact that the
presence of a highly conductive layer on the surface and close to the coils, produces
very large signals. This, in turn, makes it more difficult to visualize internal features
with lower conductivities. The solution started diverging after 3 iterations, despite
initially over-regularizing the inverse problem. Varying different parameters in the
algorithm (e.g. 1 and l) did not improve the images.
Although the meningeal CSF has a relatively smaller thickness in the human head, the
CSF filled ventricles, located within each hemisphere, may also have the same
dominating effect on the signals. Therefore, absolute imaging of haemorrhage may be
difficult using the existing inverse algorithms.
Figure 6.30. Absolute imaging of a 3-layer model including CSF, white matter, and blood. Colour-
bar in S m
-1.
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Target Image
148 Discrete Inverse Problem
6.8.5. Edge-preserving
The quadratic penalty term ||Rx||2 in Tikhonov regularization, when R is a neighbouring
matrix, penalizes the discontinuities resulting in smooth edges and blurred images. In
some applications, the distribution of the electrical properties is approximately
piecewise constant; e.g. in medical imaging where each tissue type has its own electrical
properties. Therefore, the use of algorithms which can preserve the edges in the target is
desirable.
One approach to reconstruct the edges is to substitute the neighbouring matrix in
Tikhonov regularization by a modified difference operator which allows the presence of
the discontinuities:
(6.41)
where (R )i evaluates the conductivity gradient between the element of interest and its
neighbours, n is the total number of elements, and is an auxiliary function that is used
to mark the edges, and then encourage their reconstruction [192].
The process starts with using one of the difference operator types for the first iteration.
Once the new conductivity distribution is computed, the gradient between each two
neighbouring elements is examined. Based on this gradient, the regularization matrix for
the next iteration is adjusted in such a way that its smoothing effect is reduced for the
large gradients; the larger the gradient, the less the penalization. This is based on the
assumption that sharp gradients are genuine edges in the target and, therefore, should
not be smoothed out. After a few iterations, the edges are gradually introduced. The
edge-preserving regularization has been employed for MIT imaging by Casanova et al
[193]. Details on modifying the regularization matrix, and various types of the auxiliary
functions can be found in [193,192,194,195].
The edge-preserving technique was examined for the model in figure 6.23, and the
results were compared with those obtained by the R-face. For noise-free simulated data,
there was a visible improvement in the reconstructed image, both in terms of the
sharpness of the inclusion edges and the conductivity magnitude. This is evident in the
surface plots of the x-y cross-sectional plane in figure 6.31, upper row.
149 Discrete Inverse Problem
Figure 6.31. Surface plots of x-y plane in images for noise free (upper row) and 5 m noise (lower
row). Left column is for images reconstructed using R-face, and right column is for edge-preserving
regularization matrix.
However, when a noise level of 5 m was added to the simulated phase changes, the
edge-preserving matrix did not show any superiority over the ordinary neighbouring
regularization matrix (see figure 6.31, lower row). This was due to the fact that the
presence of noise in the data can produce artefacts and, consequently, large gradients in
the image. Such gradients are considered as desired edges, and are amplified in
subsequent iterations.
The iteration procedure was stopped after only 4 iterations where the residual error
started diverging. Continuation of the iterations would result in clearer edges for the
inclusion, however, with highly magnified artefacts. Therefore, this method of edge-
preserving can be very sensitive to the noise in the data.
An alternative approach to preserve the edges is the total variation (TV) regularization
function which is also appropriate for problems where discontinuous jumps in the
model are expected. The particular advantage of the TV regularization is that the
regularization term does not penalize discontinuous transitions in the model any more
than smooth transitions.
R-face, noise-free Redge, noise-free
R-face, 5 m noise Redge, 5 m noise
150 Discrete Inverse Problem
As described by Aster et al [176], the discretized TV regularised inverse problem can be
formulated as:
(6.42)
where L is a discretization of the gradient operator. The employment of ||L || as the
regularisation penalty term allows the reconstruction of discontinuous profiles. The TV
minimisation belongs to the important classes of problems known as „Minimisation of
sum of norms‟ or „Linear l1 problems‟ [173]. Two good reviews on the TV
regularization are provided by Vogel [180] and Borsic et al [196]. They suggest various
methods for solving (6.42), including the 'lagged diffusivity fixed point' and 'Primal-
Dual' methods, which are outside the scope of this thesis. However, as part of the
LCOMIT project, the TV method was applied for producing 2D images of oil/water in
industrial pipelines.
6.9. Imaging centrally located features
MIT imaging of the centrally located inclusions is examined in this section. This is of
direct relevance for the case of bleeding deep within the brain.
6.9.1. Differential imaging
Time-difference imaging: the model comprised the white matter as the background
(0.16 S m
-1) with and without blood (1.1 S
m
-1) for State-II and State-I, respectively (see
target in figure 6.32). In order to gain more information, similar to absolute imaging, 3
planes of measurement were used (see figure 6.23). A phase noise level of 1 m was
added to the simulated phase changes. The images were reconstructed using the
generalized Tikhonov method (R-face, xref = 0) and (6.35), and using the regularization
levels suggested by the GCV and L-curve methods (see figure 6.32).
Clearly, the L-curve method resulted in an over-regularized solution (middle column).
A better visualization of the inclusion was achieved if the GCV method was used to
select (right column). In both cases, the obtained conductivity change values for the
inclusion were highly under-estimated ( 10-3
S m
-1) where the target value is 0.94 S
m
-1.
151 Discrete Inverse Problem
Figure 6.32. Time-difference imaging. Left: target, Middle and Right: reconstructed images using
generalized Tikhonov method (R-face, xref
=
0) and suggested by L-curve and GCV methods,
respectively. Colour-bars in S m
-1.
Figure 6.33. Frequency-difference imaging. Left: target, Middle and Right: reconstructed images
using generalized Tikhonov method (R-face, xref
=
0) and different values. Colour-bars in S
m
-1.
Frequency-difference imaging: the model comprised the white matter and blood
where the conductivity values for 1 and 10 MHz were adopted from table 6.1 (see target
in figure 6.33). Similarly, 3 planes of measurement were employed and 1 m phase
noise was added to the simulated phase changes. The images were reconstructed using
the generalized Tikhonov method (R-face, xref = 0) and (6.35). Both the GCV and L-
curve methods suggested under-regularized solutions ( GCV= 5 10-11
, L-curve= 9 10-11
).
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Target L-curve = 9 10-5 GCV = 2 10-6
y-z
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152 Discrete Inverse Problem
The middle column in figure 6.33 shows the image reconstructed using L-curve. Since
GCV was similar to L-curve, the image in the right column was instead reconstructed
using a larger value of , in an attempt to reduce the artefacts in the image. This resulted
in a spread inclusion of under-estimated amplitude ( 0.07 S
m
-1), where the target is 0.3
S m
-1. As can be seen, however, the background conductivity change was reconstructed
more accurately ( 0.05 S m
-1), where the target value is 0.06 S
m
-1.
6.9.2. Absolute imaging
Absolute imaging was performed using the model in figure 6.23, but when the inclusion
was placed in the centre of the cylinder (see target in figure 6.34). The same iterative
algorithm explained in section 6.8.3 was employed for image reconstruction ( 1= 1 10
-8
and l = 0.5). The images were obtained from the noise-free simulated phase changes and
those with 1 m additive noise, and are presented in figure 6.34.
An under-estimated ( 0.24 S
m
-1) and spread inclusion was reconstructed from noise-
free data at the 22nd iteration (again, this is the point at which the residual error started
increasing). The obtained background conductivity ( 0.15 S m
-1) was more accurate,
where the target values is 0.16 S m
-1. Continuing the iterations resulted in an oscillatory
behaviour of the residual error, but a better reconstruction of a the inclusion.
Figure 6.34. Absolute imaging using (6.39) with R-face, 0 = 1 S m-1
, l = 0.5 and 1 = 1 10-8
, using
noise-free data and with 1 m additive noise. Colour-bar (S m
-1) limits are set to maximum
conductivity value in each image.
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153 Discrete Inverse Problem
The presence of even a small level of noise (1 m ) made the inclusion invisible (see the
right column in figure 6.34). Despite the fact that the background conductivity (0.16 S
m-1
) is much smaller than the inclusion (1.1 S m
-1), and the lack of a highly conductive
layer such as CSF, the contribution of the inclusion to the induced MIT signals is
relatively small. This is due to its location at the centre and further away from the
exciter-sensor coils, and the fact that the eddy current is zero at the centre, which results
in a zero sensitivity at the very centre. Therefore, such small signals produced by the
inclusion are highly affected by the noise, making it more difficult to reconstruct the
feature. Compare the image obtained at the 9th iteration here with that at the 10th
iteration in figure 6.28 (both are affected by the same noise level and are reconstructed
using the same level of regularization). The peripheral inclusion was clearly visible.
6.10. Other iterative methods
A drawback of Newton-type methods is the need for the application of the discretized
inverse of F( ), in order to build the search direction for minimizing the objective
functions. In general, when dealing with large 3D problems, the SVD-based pseudo-
inverse and Tikhonov regularization solutions can become cumbersome and, in some
cases, impractical.
Alternatively, some iterative methods, which have been introduced to solve linear
algebraic systems, can also be used to regularize the solution of linear ill-posed
problems. The basic feature is that the number of iterations plays the role of a
regularization parameter [197]. These iterative methods do not require the inversion of
sensitivity matrix to find the search direction. Although more iterations may be
required, the cost of each iteration is usually cheaper.
From these iterative algorithms, a few have been employed for electrical tomography
techniques (MIT, EIT, ECT) and for both absolute and differential imaging. These
methods include Steepest descent, Landweber, Conjugate gradient, Algebraic
reconstruction technique (ART), and Simultaneous iterative reconstruction technique
(SIRT). These algorithms are outlined in Appendix C. Among these, the Landweber
method was briefly examined for time-frequency imaging of haemorrhage [60], where a
detailed investigation was not pursued due to the time restrictions.
154 Discrete Inverse Problem
6.11. Inverse crime
As introduced in this chapter, the concept of the inverse problem is to find the
constituents of a material from a set of observations. As was done in this chapter, a
relatively common practice to examine the inverse solvers‟ efficiency is to predict the
observations (e.g. voltages in MIT) through numerical simulations, and then solve the
inverse problem. Since the practical observations are subject to measurement errors,
there will be differences between the predicted and measured data. The errors in
measured data can have several sources:
SNR: noise in the electronics is always present in the MIT signals. This
type of error is measured in terms of SNR, introduced in section 2.3.2.
Geometrical error in the sensor array: any inaccuracies in the geometry of
the MIT sensor array can influence the signals. These include the location
and orientation of the coils, and also the shape and dimensions of the
electromagnetic screen. An accurate model must simulate the exact
geometrical design.
Boundary errors: a common strategy in electrical tomography techniques is
to assume the boundary of the eddy current region as the a-priori, and solve
the forward problem for this specified region. In the case of head imaging,
this would be the skin surface. In practice, however, this a-priori needs to
be defined by some other measuring means, such as optical techniques.
Therefore, error is inevitable. Furthermore, once the boundaries of the
eddy-current region is known, it needs to be registered to the MIT sensor
array; i.e. defining its location within the array. This, in turn, may introduce
additional errors.
The employment of the same solver for predicting the observations and estimating the
material distributions, with no noise added to the predicted data, is known as the inverse
crime. Lionheart [2] and Lionheart et al [173] have suggested several guidelines in
order to avoid the inverse crime. These guidelines and the strategies employed in this
study are as follows:
In the absence of measured data, different models should be employed as
the predictor and estimator. If two models are not available, two different
155 Discrete Inverse Problem
discretizations must be used. This way, when solving the inverse problem,
the boundaries of the inclusions will not be forced to be of that in the
forward problem. The discretization for the estimator should not be just a
subdivision of the mesh in the predictor; e.g. making each cubic voxel into
eight smaller ones. In the next chapter, when using the realistic head model,
the FE and FD models are used as the predictor and estimator, respectively.
In chapter 8, where the imaging performance of different MIT arrays is
examined, the FD solver is employed for all simulations due to its faster
performance. However, different discretizations are used.
Noise in the practical data-collection systems must be added to the
predicted data. For the simulations of our practical MIT data-collection
systems, realistic levels of noise for the MK1 and MK2a are used. When
using random noise generators, multiple runs are examined.
Simulated errors in the boundary of the eddy-current region are added to
the predicted data in chapter 8. These include displacement and scaling
errors.
It is also suggested to use objective regularization parameters, rather than
subjective ones, i.e. using an automatically selected . In this thesis,
automatic and empirically selected parameters are both examined.
6.12. Discussion
Differential and absolute MIT imaging were introduced in this chapter. First, the general
framework of the discretized MIT inverse problem and the difficulties associated with
solving it was reviewed. Next, one of the most commonly used image reconstruction
techniques, the Tikhonov regularized solution, was applied to imaging geometrically
simplified models. Time- and frequency-differential images of peripherally located
inclusions were reconstructed, where the effects of noise and the regularization matrix
on the inverse solution were examined. The concept of error estimation and the
automatic selection of the regularization level was introduced. The application of two
automatic methods (GCV and L-curve) for selecting revealed that while these
methods can suggest a proper level of the regularization in some cases, they may result
in under- or over-regularized inverse solutions for other cases.
156 Discrete Inverse Problem
In theoretical studies, when the simulated target is known, one can empirically adjust
the regularization level to obtain the images that are most similar to the target; as was
done for some examples in this chapter. However, in practice the target is unknown, and
no single optimised can be selected empirically.
As stated previously, the automatic methods require the solution of the inverse problem
for various levels of regularization to provide one value for . Therefore, it is suggested
rather than the automatic dismissal of different levels of the regularization and using
only one value of , a visual inspection of the reconstructed images for the examined
range of values may be useful. Note that since the inverse problem solutions are
already available for different values of , the only extra step would be the graphical
representation of the images.
This approach might be helpful for imaging cerebral haemorrhage, where one may be
able to tell the difference between a genuine haemorrhage and an artefact produced due
to an under-regularized solution, even if the target is unknown. Examples of such cases
are provided in the following chapters. This visual inspection of the images can be used
until a reliable automatic method, applicable to the various cases, is available. Note that
the development of such an automatic method was not the purpose of this study.
For absolute imaging, even if reliable, the automatic methods for selecting can be
computationally expensive when the sensitivity matrix is updated between the
iterations. On the other hand, a visual inspection of the images after each iteration is
simply impractical. Therefore, an alternative approach was suggested in section 6.8.3,
where the inverse problem is initially over-regularized and is gradually decreased as
the iterations proceed. This approach will be applied to absolute imaging using the
measured data in chapter 9.
When the inclusion was centrally located, both differential and absolute imaging proved
to be more difficult. This is due to the fact that the MIT technique is inherently less
sensitive to the central regions, where the field strength and, consequently, the eddy
currents are smaller. Therefore, imaging a haemorrhage deep within the brain may be
much more difficult. This will be examined in the haemorrhagic simulations for a
realistic head model later in this thesis.
End of chapter 6.
Part III
Stroke Imaging
Chapter 7.
Simulated Stroke Imaging Using the
MK1 System
7.1. Introduction
In this chapter, realistic simulations for stroke imaging with the MK1 system are
presented. First, the head model is described. Introduction of the stroke domains into the
model is explained, followed by the finite element (FE) and finite difference (FD) mesh
construction. Next, the results of the forward problem are presented. This includes mesh
convergence tests and discussing the MIT signals.
After extracting the predicted MIT signals due to the stroke, time- and frequency-
difference imaging is performed. Images are reconstructed from the simulated data after
adding random noise of the same level existing in the practical data-collection systems.
Finally, the power absorbed in the head due to its exposure to the electromagnetic fields
is investigated.
7.2. Head model
7.2.1. UCL head mesh
A tetrahedral head mesh, originally developed by UCL for EIT, was used as a basis for
the description of the head geometry in this study. It is an anatomically-realistic, multi-
159 Simulated Stroke Imaging Using the MK1 System
layer mesh which was developed from segmented MRI data and using I-Deas1
[198,199]. This mesh has been employed for a number of EIT imaging studies (e.g.
[199,200,201,202,203,204]). The UCL mesh comprises 12 tissues types: muscle, skull,
CSF, grey matter, white matter, ventricles, spinal cord, optic nerves, eye balls, nasal
cavity, ear canals, and the olfactory organ.
Figure 7.1 shows the triangular surface elements of the UCL FE mesh, and figure 7.2
illustrates 6 transverse cross-sectional planes cutting the head at different levels along
the z-axis. Each tissue type is defined by a separate domain, shown in different colours.
Since this mesh was developed for EIT studies, which involves much lower frequencies
than MIT, the air-filled cavities within the head (e.g. olfactory organ) were not
discretized. The total volume of the head, excluding the cavities, is 3,684 cm3. There
are 53,336 tetrahedral elements in the head.
Figure 7.1. UCL head mesh, triangular surface elements of skin and internal compartments.
Figure 7.2. Transverse planes showing internal tissues separated by different colours.
1 Integrated design engineering analysis software.
x
y
z
160 Simulated Stroke Imaging Using the MK1 System
Figure 7.3. Stroke regions in the right hemisphere of brain; side view represents the vertical level of
the LP which was similar for all three strokes.
7.2.2. Stroke domains
Intra-cerebral haemorrhages were introduced into the head within the right hemisphere
of the brain. This was done by separating some of the existing tetrahedral elements
within the head as the stroke domains. Three lesions of varying sizes and sites were
considered: a large peripheral (LP), a small peripheral (SP), and a small deep (SD) with
volumes of 49.4, 8.2, and 7.7 cm3, respectively (see figure 7.3). These volumes ranged
from 4.9 to 0.8% of the total volume of the brain. These particular examples were
considered in order to assess the sensitivity of the MIT signals to the location and
volume of the lesion.
7.2.3. FE mesh construction
In order to prepare the mesh for the MIT simulations, the missing components such as
the electromagnetic screen and the coils were required to be added to the model.
Furthermore, the air-filled cavities within the head needed to be included in the
discretized domain.
As the first attempt, the geometry of the head was extracted from the UCL mesh. Note
that only a tetrahedral mesh was received from UCL, but not the geometry description.
To this end, the UCL mesh was imported into Comsol and the geometry was extracted
from it. However, the complexity of the geometry prevented its extension to add the
missing components. Comsol could not perform any changes in the geometry, but it
allowed generation of new meshes for the existing geometry.
161 Simulated Stroke Imaging Using the MK1 System
Therefore to overcome this problem, an alternative approach was undertaken. It
included addition of the missing components in the form of tetrahedral elements to the
UCL mesh, and then extracting the geometry from the completed mesh.
The FLITE1 mesh generation software was employed to extend the mesh in a five-step
process as shown in figure 7.4:
I. First, the missing components, external to the head, were inserted into a
cylindrical domain with a centrally located ellipsoidal vacuum which was
slightly larger than the head.
II. A tetrahedral mesh was then generated for this domain using Comsol (II).
III. Next, based on the triangular boundary elements on the surfaces of the
ellipsoid and head, a tetrahedral mesh was grown to fill the space between
them using FLITE (III).
IV. In a similar way, the internal cavities were filled with a tetrahedral mesh; as
an example, nasal cavity shown in (IV).
V. Finally, all separate meshes were merged to construct the complete MIT
mesh (V).
The total volume of the head, including its air-filled cavities, was 3,709 cm3.
The completed mesh was then imported into Comsol, and the geometry was extracted
from this mesh. A series of tetrahedral meshes were generated using the extracted
geometry in Comsol for subsequent simulations. Further details on these meshes will be
provided later in this chapter.
Figure 7.5 shows the location of the coils relative to the head and LP stroke within the
brain, where red and blue coils represent the exciter and sensor coils, respectively.
1 FLITE is a fully automatic mesh generation software that is developed by O. Hassan and K. Morgan of
the Civil and Computational Engineering Research Centre, School of Engineering, Swansea University.
For further details see e.g. [235,234].
162 Simulated Stroke Imaging Using the MK1 System
Figure 7.4. FE mesh construction; details given in text.
Figure 7.5. Level of coil array relative to head and LP stroke (red = exciter, blue = sensor).
y
z
(II)
(III)
(IV)
(V)
163 Simulated Stroke Imaging Using the MK1 System
Figure 7.6. FD mesh with two cuts illustrating the internal structure, including the LP stroke.
7.2.4. FD mesh construction
Since in the FD solver, only the conductive volume (head) is discretized, the UCL head
mesh, with its internal cavities first filled with tetrahedra (figure 7.4IV), was discretized
into cubic voxels for the FD model. Uniform cubic grids of different sizes, ranging from
2 to 7.5 mm were prepared for subsequent simulations. In a grid of 2.5 mm size, for
instance, 243083 cubic voxels are used to represent the head. Figure 7.6 shows the 2.5
mm cubic grid, where the red domain in the right hemisphere is the LP stroke.
7.2.5. Dielectric properties
All dielectric properties were assumed to be isotropic. As explained in section 4.5, the
stroke lesions were assumed to be homogeneous regions with brain tissues and blood as
constituents. The lesion conductivity was set to a weighted average such as:
(7.1)
where 0.25 p 1 shows the proportion of the volume occupied by blood, and can be
thought of as the severity of bleeding. The permittivity was computed in a similar way.
Table 7.1 gives the adopted values for different tissues at frequencies of 1 and 10 MHz,
based on the survey carried out in chapter 4. As an example, for a lesion in white matter
saturated with 75% blood, and at frequency of 10 MHz: lesion = 0.25 0.159 +
0.75 1.097 = 0.863 S m
-1.
x y
z
164 Simulated Stroke Imaging Using the MK1 System
Table 7.1. Conductivity (S m
-1) and relative permittivity values assigned to each tissue type at the
frequencies of 1 and 10 MHz. The mass density of tissues (kg m
-3), adopted from [205], are to be
used for the power absorption calculations later in section 7.6. a filled with air.
1 MHz 10 MHz
Mass density Organ/tissue Conductivity
Relative
permittivity Conductivity
Relative
permittivity
Muscle 0.503 1836 0.617 171 1040
Skull 0.057 197 0.0828 54 1850
CSF 2.000 109 2.002 109 1010
Grey matter 0.163 860 0.292 320 1030
White matter 0.102 480 0.159 176 1030
Ventricles as CSF as CSF as CSF as CSF as CSF
Spinal cord 0.133 670 0.225 248 1040
Optic nerves 0.130 926 0.223 155 1040
Eyeballs 1.127 897 1.20 127 1037
Nasal cavitya 0 1 0 1 -
Ear canalsa 0 1 0 1 -
Olfactory organa 0 1 0 1 -
Blood 0.822 3026 1.097 280 -
7.3. Forward problem
The forward problem was solved using both the FE and FD methods. In this section,
first the FE simulations are discussed, followed by the FD results.
7.3.1. FE mesh convergence
The FE model was solved using the techniques explained in chapter 5. In order to
ensure that the simulations with the head were accurate, a convergence test was first
performed. As described in section 7.2.3, a series of increasingly fine meshes of
tetrahedra, comprising 43538 to 776852 elements, were generated using Comsol (see
table 7.2). The models were solved for the normal head (no stroke present) and a
frequency of 10 MHz. In order to represent the normal head, the dielectric values of the
haemorrhage domain was set to those of the brain tissues, i.e. white or grey matter.
165 Simulated Stroke Imaging Using the MK1 System
Table 7.2. Series of meshes for the convergence tests; NS: not solved due to memory restrictions.
Mesh Number of elements Degrees of freedom
First-order elements
Degrees of freedom
Second-order elements
1 43,538 51,294 277,470
2 74,812 87,940 476,024
3 82,852 98,114 529,318
4 97,144 114,025 617,530
5 155,253 182,305 987,062
6 183,369 216,540 1,169,898
7 229,379 270,249 1,461,356
8 276,787 326,283 1,764,010
9 306,808 361,534 NS
10 33,488 393,344 NS
11 383,093 453,105 NS
12 471,113 556,270 NS
13 587,941 694,427 NS
14 776,852 918,028 NS
The computations were performed for both first- and second-order elements, and for the
activated Ex1 (for coil numbering see figure A.2 in Appendix A). For models with
degrees of freedom larger than 2000000 (cells marked 'NS' in table 7.2), the memory
required to solve the model exceeded the maximum memory available (8 GB).
For the solved models, the induced voltage in the sensor coils was monitored. As
expected, the second-order elements performed better than the first-order in terms of the
accuracy, and achieved convergence of the voltage at a smaller number of elements (see
figure 7.7). From Mesh-2 onwards, second-order elements showed a steady predicted
sensor voltage. However, for a sufficiently fine mesh (Mesh-7), first-order elements
returned comparable results albeit at greater computational cost. Figure 7.7 shows the
typical convergence results for the computed voltage for three arbitrary sensors.
Monitoring the results at all sensors revealed similar behaviour.
Based on the convergence results, all subsequent calculations employed Mesh-3 (3rd
point from left on each curve for second-order elements), with 82,852 second-order
tetrahedral elements of which 54,019 elements represented the head. This mesh was
deemed to be a good comprise between the accuracy of solution and computational run
166 Simulated Stroke Imaging Using the MK1 System
time. Solving the model for one activated exciter for this mesh took approximately 4
hours, and to compute a complete set of data (i.e. 16 exciters) took 64 hours.
A typical result of the magnetic field produced by one exciter coil (Ex13), at frequency
of 10 MHz, is illustrated in figure 7.8a. The corresponding map of the induced eddy-
current density in a transverse slice, for the normal head, reveals a high current density
in the CSF due to its larger conductivity relative to the skull and brain tissue (figure
7.8b). With the LP stroke lesion present, it too contains a higher current density than the
surrounding normal brain tissue because of its blood content and, consequently, higher
conductivity (figure 7.8c).
Figure 7.7. Convergence of the induced voltage, for activated Ex1, in sensors Se6, Se9, and Se13 for
first- and second-order elements plotted in broken red and solid blue lines, respectively.
Figure 7.8. Typical results. (a) Projection on the transverse plane of magnetic field lines with Ex13
activated with a current of 1 A. The colour of a field line indicates the magnitude of the magnetic
field at that point; the colour-bar indicates the field strength in A m
−1. (b) Map of induced current
density (A m
−2) in a transverse slice through the normal brain, 25 mm above the centre of the coils.
(c) as (b) but with the LP stroke present.
105
106
-0.5
0
0.5
1
1.5
2
2.5x 10
-5
Degrees of freedom
Vo
lta
ge
(V
)
13
9
6
Shield
Exciter
Sensor
(a) (b) (c)
167 Simulated Stroke Imaging Using the MK1 System
7.3.2. MIT signals (FE results)
Simulations were first carried out for the 256 coil combinations with the conductivity of
all head tissues set to zero, and their relative permittivities set to unity, so that the
simulated problem corresponded to that obtained if no head were present. The primary
voltage Vp was recorded. Simulations were then carried out for the normal head, and the
in-quadrature voltage, V1, was recorded. Similarly, with a lesion present, the in-
quadrature voltage, V2, was recorded. The phase difference, , was calculated for each
coil combination as:
(7.2)
Profiles of for each of the three strokes saturated with 75% blood, and for Ex1
activated at 10 MHz, are plotted in figure 7.9. Note that the values of due to the
presence of three haemorrhages are in the range of milli-degrees which indicates that
sensitive hardware is required to measure them accurately.
The LP stroke produced the largest signals ( max = 65 m in Se11). This is not
surprising as a larger volume of blood with higher conductivity replaced the brain
tissues.
Figure 7.9. Profiles of for three strokes, simulated at frequency of 10 MHz and with Ex1
activated. Also shown is the phase angle for normal head relative to empty space (values are
divided by 20 for convenience of display).
0 2 4 6 8 10 12 14 16-10
0
10
20
30
40
50
60
70
80
Sensor number
(m
)
LP
SP
SD
/20
168 Simulated Stroke Imaging Using the MK1 System
The values of for the two smaller strokes (SP and SD), reveal that the more centrally
located the bleeding, the smaller the MIT signals are. This is also to be expected as the
sensitivity of the MIT system decays towards the centre of brain as a result of the
reduced inductive coupling of the lesion with the coils. For comparison, the phase
angles relative to the empty detector volume when the head was first introduced, =
tan-1
(V1/Vp), are also shown. They are very much larger, with a maximum of 1560 m .
In figure 7.10, the complete sets of 256 values of are plotted in magnitude, sorted in
descending order, for each of the three stroke lesions. As expected, the largest signals
were produced by the LP stroke (| |max = 71
m ), and the smallest were from the SD
lesion (| |max = 3.4
m ). Given the phase noise level of the MK1 system (17
m ),
approximately 70 of the 256 signals (27%) would be detectable for the LP stroke, but
none for the smaller strokes1.
In order to be able to measure the same percentage of the signals produced by the SD
stroke as detectable in the LP stroke, a noise level of about 1 m would be required. This
level of the phase noise is achievable in the MK2a system [206]. The tolerable noise
level for producing acceptable images will be examined in sections 7.4 and 7.5.
Figure 7.10. The 256 values of magnitude of sorted in descending order for each of three strokes,
simulated at frequency of 10 MHz.
1 A transmission line matrix (TLM) model was also used as an independent check of the FE simulations.
Detailed discussion can be found in a report by Zolgharni et al [60].
0 50 100 150 200 250
0
10
20
30
40
50
60
70
Coil combination
| |
(m
)
LP
SP
SD
169 Simulated Stroke Imaging Using the MK1 System
Figure 7.11. The 256 values of magnitude of sorted in descending order for LP stroke with
different blood percentages.
Finally, the phase changes for the LP stroke, containing different percentages of blood
(25 to 100%), are plotted in figure 7.11. Clearly, as the blood percentage increases, the
phase changes produced grow larger. For only 25% blood in the lesion, | |max = 34
m
is obtained. This indicates that at early stages from the onset of haemorrhage, it would
be more difficult to detect the presence of bleeding. When the blood entirely replaces
the brain tissues, i.e. 100% blood content, then | |max = 94
m . However, in the
remainder of this thesis, the strokes saturated with 75% blood are considered in order to
be compatible with the EIT studies using this head model (e.g. [202]).
7.3.3. FD vs. FE
For the FD model, similar to the FE simulations, the conductivity values were adopted
from table 7.1. Since the FD solver cannot handle holes within the conductive domain,
for the cavities within the head, a small enough conductivity of 10-4
S m
-1 was
considered to represent the air.
The permittivity of all tissues, however, was assumed to be zero. The reason for this
approximation is that it allows a stationary iterative solver (such as SOR explained in
section 5.3.3), to be used for solving the linear system of equations. For the FD
0 50 100 150 200 2500
10
20
30
40
50
60
70
80
90
100
Coil combination
| |
(m
)
100%
75%
50%
25%
170 Simulated Stroke Imaging Using the MK1 System
methods, stationary iterative solvers can be implemented without the need to assemble
the large matrix. Clearly, this leads to considerable computational savings.
If the true permittivity values were to be included in the model, this would change the
system matrix from being symmetric positive definite to being complex symmetric. For
such matrices, the convergence of stationary iterative solvers can no longer be
guaranteed. For the permittivity values listed in table 7.1, convergence was not
obtained.
By adopting an alternative iterative solution technique (or a direct approach), linear
systems with complex matrices can be solved, but this would require the assembly and
storage of a large matrix. Considering the large number of cubic voxels used to
represent the head, this would have substantial negative impact on the computational
footprint of the FD algorithm. As the computational saving was one of the main reasons
for adopting the FD algorithm in this thesis, this approach was not pursued here.
In order to examine the accuracy of the FD results, and the effect of ignoring the tissue
permittivity, the FD results can be compared with those obtained from the FE model
that included the true permittivity values (available from the previous section). To this
end, the normal head (no stroke present) was simulated at frequency of 10 MHz, using
different grid sizes, generated in section 7.2.4 (see table 7.3).
Table 7.3. Series of FD meshes for the convergence tests.
Mesh Grid size (mm) Number of conductive voxels
1 7.5 9 012
2 6 17 583
3 5 30 401
4 4 59 338
5 3 150 865
6 2.5 243 083
Note that the MK1 system includes an aluminium screen. Therefore, the A-field was
extracted from Comsol for each one of these grids (screen implementation in FD solver
was explained in section 5.5), and the FD models were solved.
171 Simulated Stroke Imaging Using the MK1 System
The induced voltages in 16 sensors for Ex1 activated are plotted in figure 7.12, where
the converged FE results are also plotted for comparison (thick broken line). As can be
seen, the FD results showed a consistent trend. As the grid size reduced, the discrepancy
between the FE and FD results grew smaller. For the grid of 2.5 mm size, the FD results
were comparable to those obtained from the FE model.
The discrepancy between the FE results and FD results (2.5 mm grid size) is plotted in
figure 7.13. The largest differences are observed at sensor coils where the magnitude of
the induced voltage is relatively large (e.g. Se1 and Se9).
Figure 7.12. Secondary voltage in 16 sensors for Ex1 activated, in normal head at frequency of 10
MHz. Numbers 7.5, 6, 5, 4, 3, and 2.5 indicate grid sizes used in FD model. Converged FE results
are plotted in broken line for comparison.
Figure 7.13. Discrepancy between FE and FD (2.5 mm grid size) results in figure 7.12.
0 2 4 6 8 10 12 14 16
-4
-3
-2
-1
0
1
x 10-5
Se
Secondary
voltage (
V)
0 5 10 15
-3
-2
-1
0
1
2
3
4x 10
-7
0 5 10 150
5
10
15
20
0 2 4 6 8 10 12 14 16
-4
-3
-2
-1
0
1
x 10-5
Se
Secondary
voltage (
V)
7.5
6
5
4
3
2.5 FE
VFE - VFD (V)
Sensor number
Sensor number
172 Simulated Stroke Imaging Using the MK1 System
The error obtained for other coil combinations showed a similar behaviour, indicating
that there was a good agreement between the FE results and those obtained from the FD
model using a 2.5 mm grid size. This is despite the fact that the permittivity values in
the FD model were set to zero, therefore, neglecting the displacement currents within
the head. The error observed in figure 7.13 can have several reasons such as the
discretization errors.
In order to examine the effect of neglecting displacement currents, while excluding the
other sources of error, the FD model must be solved with and without the true
permittivity values. This would require implementing better iterative solvers in the FD
code. Alternatively, the FE model could be solved were the permittivity values are set to
zero. However, the iterative solvers, available in Comsol, did now allow zero
permittivities. The use of a direct solver was also impractical for such a large model.
The error values shown in figure 7.13 deemed to be in the acceptable range for the
purpose of this chapter which is to use different models as the predictor and estimator,
therefore avoiding the inverse crime (see section 6.11). Solving the FD model for the
grid of 2.5 mm size, activating all 16 exciters, took 65 min, which is considerably
shorter than 64 hours in the FE simulations. In the remainder of this chapter, this grid
will be used for computing the sensitivity matrix in the FD model. As the predicted MIT
measurements, signals computed by the FE model will be employed.
7.4. Time-difference imaging
From the values of (obtained from the FE model), time-difference images of the
conductivity change were reconstructed. Here, the conductivity change corresponded to
that from before (normal head) and after stroke.
First, the sensitivity matrix was computed by solving the FD model for the head with a
uniform conductivity of 1 S
m
−1, activating all 16 exciters and 16 sensors in sequence.
The entries of the sensitivity matrix were computed using (6.12), i.e. normalized to the
primary signal. In solving the inverse problem, it was assumed that the boundary of the
eddy-current region (skin surface) was known as a-priori information. The entire eddy-
current region (head), therefore, remained fixed at 243,083 cubic elements.
173 Simulated Stroke Imaging Using the MK1 System
Since the sensitivity was an excessively large matrix (256 243083), the Tikhonov
regularized solution in (6.25) was used for reconstructing the images, where the
regularization matrix was Identity matrix. The images were reconstructed from the
noise-free and noisy phase changes where two levels of Gaussian phase noise, 1 and 17
m (standard deviation), corresponding to the noise in MK1 and MK2a systems, were
added to the simulated data. Three cross-sections were extracted from the reconstructed
volume image for display. For each image, the normalized solution error was computed
as:
(7.3)
where rec and true are the reconstructed and true conductivity change distributions,
and ||.||2 is the Euclidean norm. Note that the target ( true) and image ( rec) have
different discretizations, simulated by the FE and FD models, respectively. For
computing E, the target was interpolated onto 2.5 mm voxels, similar to that in the
image, as if it had been simulated using this grid.
GCV method: first, the regularization parameter was selected by using the GCV
method (explained in section 6.6.1). Plots of the GCV function for different values of
are shown in figure 7.14, while the images presented in figure 7.15 were reconstructed
using the values of λ which minimized the GCV function. Note that, for convenience of
display, the sagittal plane (third row) is not to scale.
For the noise-free data, the solution is clearly under-regularized, where the magnitude of
the obtained conductivity changes are as large as 40 S m
-1 (2nd column from left). These
correspond to negative and positive values appearing as small volume artefacts evident,
mainly, on the boundary of the transverse plane. A large error of E = 4.4 103% was
obtained for this image.
Also the GCV plot shows that the GCV function is constantly decreasing as grows
smaller until 10-22
. Therefore, for any range of examined values of , the minimum
value within the range would always be considered as the optimised . However, an
inflection point in the curve, suggestive of the development of a minimum shown by the
arrow, occurs at = 1 10-11
. As will be shown in later images, this is where the
optimum value of lies.
174 Simulated Stroke Imaging Using the MK1 System
Figure 7.14. Regularization parameter selection using GCV method for three phase noise levels (0,
1 m , 17 m ) added to phase changes for LP stroke.
Target - LP
0
= 7.3 10-22
E = 4415%
1 m
= 3.3 10-11
E = 93.7%
17 m
= 4.0 10-8
E = 96.7%
Figure 7.15. Columns left to right: simulated LP target, and reconstructed images with 0, 1 m , 17
m additive noise. Rows from top to bottom: transverse, coronal, and sagittal cross-section planes.
Colour-bars in S m
-1. is selected by GCV method.
10-30
10-25
10-20
10-15
10-10
10-5
10-16
10-14
10-12
10-10
10-8
G(
)
4.02e-8
3.29e-11
7.28e-22
17 m
1 m
0
1 10-11
175 Simulated Stroke Imaging Using the MK1 System
For 1 m of phase noise, a clear minimum in the GCV function can be observed at =
3.3 10-11
. A visible stroke feature of under-estimated magnitude ( 0.15 S
m
-1) is
reconstructed in the image (3rd column). However, its visibility is degraded slightly by
the presence of small volume artefacts of higher amplitude ( 3 S
m
-1). The visibility of
the stroke can be improved by lowering the upper limit in the colour-bar, which in the
current image is set to the maximum value obtained in the inverse solution. An error of
E = 93.7% was obtained for this image.
With 17 m noise in the simulated measurements, the stroke is feature is entirely
displaced towards the head surface; evident in the transverse and coronal planes of the
image in 4th column. This is due a high level of regularization for = 4 10-8
that
minimizes the GCV function (see GCV plot). A smaller value of would improve the
localisation of the stroke, but artefacts would have degraded the image. The computed
error (E = 96.7%) is slightly larger than that in the image with 1 m noise.
L-curve method: next, the L-curve method (as described in section 6.6.2) was used for
selecting , and the results are shown in figure 7.16 and figure 7.17. Apart from the
noise-free case, the inverse solutions are comparable with those suggested by the GCV
method, but slightly more regularized.
The optimised regularization parameter for the noise-free data ( = 3.3 10-12
) resulted
in a stroke feature with an under-estimated magnitude ( 0.15 S
m
-1) in the image (2nd
column from left). The smallest error is obtained for the noise-free image (E = 93.2%)
which is considerably smaller than that obtained based on the GCV method (E =
4.4 103%).
For the noisy data, slightly over-regularized solutions were obtained for the values of
suggested by the L-curve method (1.6 10-9
for 1 m and 1.2 10
-7 for 17 m ). The stroke
feature is blurred and spread in the vertical direction, evident in the coronal planes of
the images (3rd and 4th columns). It is also displaced towards the surface. However, the
asymmetry in the images due to the stroke is clearly noticeable.
176 Simulated Stroke Imaging Using the MK1 System
Figure 7.16. Regularization parameter selection using L-curve method for three phase noise levels
(0, 1 m , 17 m ) added to phase changes for LP stroke.
Target - LP
0
= 3.3 10-12
E = 93.2%
1 m
= 1.6 10-9
E = 95.3%
17 m
= 1.2 10-7
E = 97.2%
Figure 7.17. As figure 7.15, but selected by L-curve method.
10-6
10-4
10-2
100
102
104
106
residual norm || A x - b ||2
solu
tion n
orm
|| x || 2
3.9659e-026
1.26e-71.62e-9
3.32e-12
17 m
1 m
0
177 Simulated Stroke Imaging Using the MK1 System
= 1.0 10-13 = 1.0 10-12 = 1.0 10-11 = 1.0 10-10
= 1.0 10-9 = 1.0 10-8 = 1.0 10-7 = 1.0 10-6
Figure 7.18. Reconstructed images of LP stroke with 1 m phase noise added to phase changes, and
for different values of . Upper limit in each colour-bar (S m
-1) is set to maximum value in the
corresponding image.
Now let us adjust the regularization level empirically, assuming that the target is
unknown. Different values of , ranging from 1 10-13
to 1 10-6
, were considered and
the images were reconstructed from the phase changes with 1 m noise (see figure 7.18).
These are the typical images that a radiologist might obtain by varying .
178 Simulated Stroke Imaging Using the MK1 System
For 1 10-12
, the reconstructed images may be dismissed given the magnitude of the
obtained conductivity change (> 2 S m
-1) and their appearance within the images. Note
that while the background (head volume) appears smooth, there are high amplitude
values in the solution which indicates the presence of small volume artefacts, spread
throughout the volume. A few high amplitude voxels can just be seen in the sagittal
plane at the back of head for = 1 10-13
. Therefore, one may decide either the data is
too noisy, or the solution is highly under-regularized.
On the other hand, if the assessment starts with values of 1 10-7
, an evident
asymmetry can be observed in the images, arousing suspicions of some abnormality in
the head. However, as will be shown in the next chapter, this asymmetry could also be
due to some errors in the assumed boundaries for the eddy-current region (head). An
increase in would result in disappearance of the asymmetry, and reconstruction of
smoother images.
Therefore, one may appreciate that the desired level of the regularization should lie in
the range 1 10-11
1 10-8
, for which the best visibility of the stroke is achieved,
while no significant noise artefacts are introduced.
The best visualization of the stroke for 1 m noise in the data is achieved at = 1 10
-10
in figure 7.18. This image is reproduced in figure 7.19 along with those for other noise
levels (0 and 17 m ), all of which were obtained by the empirical adjustment of .
Note that these images are obtained using the values of optimised based on the visual
assessment of the images. Such images may not necessarily result in the smallest values
of E for the definition of the error given by (7.3), and the minimum error may be
achieved using a different value of .
In chapter 8, while comparing the performance of different configurations for the MIT
sensor arrays by reconstructing the images for each array, the regularization level is
optimised based the computed error. In this way, the best feasible image, from an
strictly mathematical point of view, is obtained when the minimum error is achieved for
each MIT system. Obviously, this way of adjusting requires the targets to be known.
179 Simulated Stroke Imaging Using the MK1 System
Target - LP
0
= 1.0 10-11
E = 91.6%
1 m
= 1.0 10-10
E = 92.6%
17 m
= 1.0 10-8
E = 96.7%
Figure 7.19. As figure 7.15, but selected empirically.
Similarly, the images were reconstructed for the smaller strokes, and are shown in
figure 7.20 (SP stroke) and figure 7.21 (SD stroke), where the regularization parameter
was selected empirically. As can be seen, the magnitude of computed conductivity
change for the lesions in all images is considerably under-estimated ( 0.06 S m
-1).
For the SP stroke with 1 m added noise, the stroke is visible, although spread and
blurred. In case of 17 m noise, the stroke is barely visible as an asymmetric feature in
the coronal plane, displaced towards the boundaries.
For the SD stroke, apart from the unrealistic noise-free case where a spread and under-
estimated feature is reconstructed, the stroke is not visible. The likely reason is the fact
that when regularizing the solution to diminish the effect of noise, the genuine phase
changes due to the stroke are also dismissed. This is in line with the observations in
section 6.9, which indicated that MIT imaging of the centrally located features is more
difficult than the peripheral ones.
180 Simulated Stroke Imaging Using the MK1 System
Target - SP
0
= 5.0 10-12
E = 97.8%
1 m
= 1.0 10-9
E = 99.2%
17 m
= 1.0 10-7
E = 99.9%
Figure 7.20. As figure 7.19, but for SP stroke.
Target - SD
0
= 1.0 10-10
E = 99.5%
1 m
= 1.0 10-8
E = 100%
17 m
= 1.0 10-7
E = 100.5%
Figure 7.21. As figure 7.19, but for SD stroke.
181 Simulated Stroke Imaging Using the MK1 System
Figure 7.22. (a): profiles of noise-free for SD stroke (solid blue line) and those with 1 m added
phase noise (broken red line). (b): difference between two plots in (a).
As an example, figure 7.22a shows plots of the phase differences, noise-free and with 1
m noise, for the SD stoke. Notice that the noisy data deviates significantly from the
noise-free one. Figure 7.22b shows the difference between the two data sets, i.e. noise,
on a similar y-axis scale. As can be seen, the noise is comparable to the genuine phase
diferences induced by the stroke. The variance of the noise-free data and the noise was
1.094 and 1.030 m , respectively.
0 50 100 150 200 250-3
-2
-1
0
1
2
3
4
5
6
Coil combination
(m
)
0 50 100 150 200 250-3
-2
-1
0
1
2
3
4
5
6
Coil combination
no
ise
-fre
e
- n
ois
y
(m)
(a)
(b)
182 Simulated Stroke Imaging Using the MK1 System
7.5. Frequency-difference imaging
Frequency-difference imaging was performed using the approach introduced in section
6.7. To this end, all simulations mentioned in section 7.3.2 were repeated, but now at a
frequency of 1 MHz. Note that the convergence test presented in section 7.3.1 was
carried out at frequency of 10 MHz. Here, it was assumed that the convergence results
will also apply at the lower frequencies.
Frequency-difference images were reconstructed from the phase changes computed by
the FE model and corrected using (6.35). The sensitivity matrix was the same as the one
used for time-difference imaging: computed by the FD model at frequency of 10 MHz
using a grid of 2.5 mm size and uniform conductivity of 1
S
m
-1. Its entries were
normalized to the primary signal as in (6.12).
The Tikhonov regularized solution, expressed by the equation (6.25), was used to solve
the inverse problem, and to reconstruct the conductivity changes between 1 and 10
MHz. Any negative values were truncated from the solution as a-priori information
(this physical constraint was discussed in section 6.7).
Figure 7.23 shows the GCV and L-curve plots for optimising for the noise-free data
corresponding to the LP stroke, and figure 7.24 shows the reconstructed images using
these suggested levels of regularization.
Figure 7.23. Regularization parameter selection using GCV and L-curve method for noise-free
frequency-difference data for LP stroke.
10-30
10-20
10-10
10-20
10-15
10-10
G(
)
10-15
10-10
10-5
100
101.2
101.3
101.4
101.5
101.6
residual norm || A x - b ||2
solu
tion n
orm
|| x || 2
GCV L-curve
= 2.88 10-21
= 4.1 10-10
183 Simulated Stroke Imaging Using the MK1 System
Target - LP
GCV
= 2.9 10-21
E = 84.5%
L-curve
= 4.1 10-10
E = 70.9%
Figure 7.24. Columns left to right: simulated LP target, and reconstructed images from noise-free
data with selected using GCV and L-curve methods. Rows from top to bottom: transverse,
coronal, and sagittal cross-section planes. Colour-bars in S m
-1.
While, similar to time-difference imaging, GCV suggested an extremely small value of
= 2.9 10-21
for the noise-free data, the L-curve method returned a reasonable value of
= 4.1 10-10
. However, as can be seen in the reconstructed images, the stroke feature is
not visible in either case. A slight left-right asymmetry can be barely observed in the
transverse and coronal planes of the images corresponding to the L-curve method (3rd
column from left).
Even the empirical adjustment of the regularization parameter did not improve the
images. Furthermore, other attempts for imaging the smaller strokes, or reconstruction
from the noisy data, also failed.
Note that in frequency-difference imaging, the change in the conductivity of blood in
the lesion needs to be visualised against changes in all the other tissues (see target in
figure 7.24). Therefore, the non-zero values in the reconstructed images are not
necessarily artefacts, but can be genuine conductivity changes in other tissues. Such
non-zero values can be seen in the coronal and sagittal planes of the image in 3rd
column from left.
184 Simulated Stroke Imaging Using the MK1 System
However, since the sensitivity is not uniform throughout the head, the conductivity
change is not reconstructed equally well in all tissues. For instance, the central region of
the head in the sagittal plane seems to have no conductivity change ( 0 S m
-1), whereas
this region is filled by frequency-dependent tissues (see target).
In this chapter, MIT measurements were simulated with the coil array in just one
position relative to the head (see figure 7.5). As will be shown in chapter 8, by
conducting the measurements (simulations) for multiple planes and therefore acquiring
more MIT data, meaningful frequency-difference images of the stroke can be
reconstructed. Since, in this chapter, the FE model was employed for predicting the
MIT signals, simulation of more planes of measurement was prohibitively long.
7.6. Specific absorption rate
For reasons of safety, it is important that the power absorbed in the tissue from the
electromagnetic field should be within international safety limits. The absorbed power
can lead to a temperature increase in the tissues due to Joule heating. The Joule heating
is particularly important when dealing with the brain tissues. An increase in temperature
by even 1 C can have profound effects on a single neuron and neuronal network
function [207].
Specific absorption rate (SAR) is a measure of the rate at which the energy is absorbed
by the tissue when, for instance, exposed to a radio frequency (RF) electromagnetic
field. At a given point in the tissue, SAR is defined as the power absorbed per unit mass
of tissue and has units of W kg
-1 as:
(7.4)
where m(x) is the local mass density of the tissue (table 7.1) and |E(x)| is the magnitude
of the electric field at this point [208,209]. While SAR is commonly used to measure
power absorbed from mobile phones (e.g. [210]) and during MRI scans (e.g. [211]),
tissue heating is generally considered to be negligible in MIT. However, no values,
either measured or modelled, appeared to have been published prior to commencing this
study. Here, the FE model was used to compute the SAR throughout the head.
185 Simulated Stroke Imaging Using the MK1 System
Figure 7.25. SAR distribution in normal head for Ex1 activated at 10 MHz; (a) skin surface, (b)
slice at through centres of coils. Colour-bar indicates logarithm to the base 10 of value in W kg
-1.
Eyes are potentially the most critical organs (tissues) regarding the heat generation here.
The reason for this is twofold:
due to the low perfusion in the eye, the temperature may increase even with
relatively low SAR levels [212].
as an external organ, and due to its vicinity to the exciter coils, eye is located in
the areas of high current density where the fields are relatively large, resulting in
higher SAR levels (see figure 7.25).
Therefore, in terms of the heat generation in the eyes, the worst conditions may happen
when the exciter coils 1-3 and 15-16 in the MK1 systems are activated; these are the
closest coils to the eyes (for coil numbering in the MK1 system see Appendix A.1).
Average SAR in one eye was maximum with Ex1 activated, and was 4.1 10-5
W kg
-1.
The computed SAR with Ex1 activated at frequency of 10 MHz was maximum near the
eyes, with a value of 1 10-3
W kg
-1 (see figure 7.25). The overall maximum (not
shown in the figure) was slightly larger, at 3 10-3
W kg
-1, and occurred in the skin and
facial muscle inferior to the eye when Ex3 or Ex15 were activated. This computation
was for an exciter coil of 1 turn carrying a current of 1 A.
Before proceeding to an assessment of the SAR values encountered in practice, let us
consider the effect of changing the excitation current or the number of turns on the
(a) (b)
186 Simulated Stroke Imaging Using the MK1 System
excitation coil, the product of which can be taken as a single variable, the Ampere-turns.
If the Ampere-turns is changed by a factor, , the value of the source current density Js
in the boundary value problem (5.6) also changes by the factor . It follows from the
linearity of equations (5.6) and (5.7) that the vector potential, A, and the signal, V,
measured in the sensor coil will all then change by the same factor .
This will be true without the presence of the conductive domain, C, (see figure 2.1),
and in this case the measured V will be the primary signal, Vp. It will also be true for the
in-quadrature part of V (V1 or V2) measured in the presence of C. Hence, as Vp, V1 and
V2 all change by the same factor , a change in the Ampere-turns has no effect on the
values of produced by a given stroke lesion which is computed by (7.2).
The phase noise in the measurement will, however, be affected by the value of . If the
phase noise arises from a fixed level of voltage noise in the receiver amplifier, as will
normally be so, changing the magnitudes of the signals by a factor , will cause an
inversely proportional change in phase noise. Thus, changing the Ampere-turns of the
exciter coil will cause the phase noise to change by a factor 1/ . Finally, the SAR at any
point in C also depends on the Ampere-turns of the exciter coil. Being proportional to
the square of the electric field, as identified by the equation (7.4), the SAR is
proportional to 2.
For the MK1 system (coils of 2 turns, current of 0.14 A as described in Appendix A.1),
= 2 0.14 = 0.28 relative to the numerical model. Multiplying the modelled values of
SAR by 2 = 0.0784 leads to a maximum SAR of 2.4 10
-4 W
kg
-1. The IEC
requirements for MRI scanners, which cover similar frequencies to those used here,
place a limit for the head of 10 W kg
-1, averaged over 6 min [213]. The estimated SAR
for the MK1 system is thus many orders of magnitude below this limit and will indeed
produce negligible heating.
However, in order to reduce the phase noise and the integrating time in the data-
collection systems, new MIT designs are now tending to plan for much higher
excitation currents, possibly up to at least 10 A, to improve the performance in terms of
the noise and speed. For a coil of 2 turns, a current of 10 A ( = 20) would bring the
maximum SAR to 1.2 W kg
-1 which is still below, but is now approaching the IEC limit.
SAR modelling should therefore now become an integral part of MIT system design.
187 Simulated Stroke Imaging Using the MK1 System
7.7. Discussion
In this chapter, it was intended to make the simulations as realistic as possible by:
using an anatomically correct geometry
avoiding the inverse crime by employing different solvers as the predictor and
estimator
avoiding the inverse crime by adding phase noise to the simulated data; the noise
levels were comparable with those currently achievable in practical data-
collection systems
MIT imaging of the stroke lesions of different volumes and sites in the brain was
examined. Based on the obtained results, time-difference imaging of peripherally
located haemorrhages appears to be feasible, given the phase noise levels as low as 1 m
which is achievable in the MK2a system (see Appendix A.2).
Imaging of a centrally located small haemorrhage, apart from the unrealistic case of
noise-free data, was not successful using the adopted reconstruction algorithms. The
likely reason is the fact that the system is considerably less sensitive to the central
regions and, therefore, smaller signals are produced by the lesions located in such
regions; as was shown in figure 7.9. Regularizing the inverse solution diminishes the
effect of these small signals on the solution.
Automatic selection of the regularization parameter was found to be unsatisfactory in
some cases. The best images, in terms of stroke visibility, were obtained by visual
inspection of the images and empirically adjusting the regularization level. In this
regard, it is remarked that the purpose of this study is not to improve the inverse
solution techniques, but to employ existing methods to examine the feasibility of stroke
imaging using MIT. Therefore, by adjusting the regularization parameter, the most
potentially useful images were obtained. The inverse solution for different values of
can be obtained using the SVD of the sensitivity matrix, as explained in section 6.6.1.
Subsequently, the reconstructed images could be examined by a radiologist to extract
the most meaningful ones.
Frequency-difference imaging proved to be more difficult as nearly all the materials in
the imaging domain have frequency-dependent conductivities. It is possible that the use
188 Simulated Stroke Imaging Using the MK1 System
of geometrically customised systems, with higher sensitivity to the region of interest
(brain), might help to improve the visualisation of the desired feature, i.e. stroke. This is
investigated in chapter 8.
Although absolute imaging is increasingly used in geophysics [214], its application to
stroke imaging currently seems impractical using the available numerical techniques
and computing facilities. This would take more than one day for the FE model, and one
hour for the FD model, to perform each iteration of the non-linear solution for the head
model. This could jeopardise one of the main reasons for employing MIT for stroke
imaging, which is a quick diagnosis. Furthermore, as will be shown in chapter 9,
absolute images are very sensitive to the different types of errors, in particular
systematic errors.
Further improvements in the computational facilities and simulation methods, such as
the development of more time-efficient numerical packages together with parallel
computing1, may allow the feasibility of using iterative absolute image reconstruction
for stroke imaging. However, at the moment, differential imaging of the stroke is
preferable provided that:
'before-stroke' data set is available to perform time-difference imaging
hardware is capable of multi-frequency measurements to perform frequency-
difference imaging
SAR computations for the head revealed that, at present, MIT is probably still a
harmless and safe imaging method regarding heat generation. However, the SAR
computation should become an integrated part of MIT system design in future, as higher
coil currents are being considered by hardware designers.
End of chapter 7.
1 Parallelization approaches used in the FD solver are currently being investigated at the University of
Glamorgan as part of the LCOMIT project (e.g. [236]).
Chapter 8.
Customized Coil Arrays, Systematic
Errors, and the MK2b System
8.1. Introduction
As discussed in chapter 7, a customised MIT coil array may help to improve the
sensitivity of the system to the regions of interest in the target. The induced eddy-
current patterns in MIT and, therefore, the sensitivity distribution of the system not only
depend on the material property distribution of the target, but also on the geometrical
design of the MIT sensor array. Aspects of the geometrical design include:
the screen geometry
the number of channels and the number of coil turns in each channel
the coil dimensions (i.e. radii for circular coils)
the spatial coil configuration (position and orientation)
Peyton et al [32] investigated the effect of screen dimensions on the sensitivity maps.
They concluded that the screen geometry has little influence on the spatial sensitivity
distribution for a particular exciter-sensor pair; entries of the sensitivity matrix are
mainly only scaled by a constant factor. Watson et al [33] have shown that a coil-screen
standoff distance of 60-80% of the coil diameter gives a reasonable suppression of inter-
coil capacitive coupling without excessively damping the inductive signals. Therefore,
once the coil diameter and their spatial location in the MIT sensor array are known, an
appropriate diameter for the screen can be estimated.
190 Customized Coil Arrays, Systematic Errors, and the MK2b System
As discussed in section 7.6 for SAR computations, the coil turns (and the excitation
current as an operating parameter) have no effect on the eddy-current patterns; the
induced currents are scaled by a constant factor. The phase noise in the measurements,
however, will be affected by these factors; more coil turns induce larger signals,
resulting in a larger SNR (see equation (2.13)). The remaining factors are the number of
channels (coil combinations) and their spatial configuration. The coil radii are also
limited by the space available to accommodate the specified number of coils.
In a numerical study, Gursoy and Scharfetter [215] examined the effect of the location
of the sensors on the Tikhonov regularized time-differential images, for some fixed
exciter coils. They observed that by positioning the sensor coils at certain locations, the
resolution of the reconstructed image could be improved for a particular region, albeit at
the cost of other regions' resolution. They referred to such designs as 'regionally
focused' MIT.
In a later study [216], they investigated the effect of orientation of the sensor coils on
the same images, again assuming that the exciter coils were fixed. They concluded that
each coil orientation had its own merits and shortcomings for different imaging regions.
Eichardt et al [57] compared cylindrical (annular) and hemispherical coil arrays using a
sensitivity analysis. Their simulation results indicated that the hemispherical array
showed a higher sensitivity at the top of the brain compared to the cylindrical MIT
system. These findings, in fact, confirmed the results of the current chapter in this
thesis, which had been published prior to their report [62].
The purpose of this chapter is to compare the imaging performance of the customised
annular and hemispherical MIT arrays. To this end, frequency-difference imaging of
stroke lesions in the realistic head model is examined. Furthermore, the effect of
different values of SNR and examples of systematic errors (in terms of inaccuracies in
the boundaries) on the reconstructed images are investigated.
Later in this chapter, the geometrical design study of the latest MIT system in the
South-Wales group (MK2b) is briefly explained. The MK2b system, under construction
at the time of writing this thesis, is optimised for stroke imaging in terms of the 12-
tissue head model.
191 Customized Coil Arrays, Systematic Errors, and the MK2b System
8.2. Forward and inverse problems
In this chapter, the FD solver was used as the predictor and estimator models for solving
the forward and inverse problems, respectively. Employing the much faster FD solver
for all simulations allowed a variety of different arrays to be examined in a shorter time
(recalling from section 7.3.3, the FE and FD models took 64 hours and 65 min to
solve the forward problem for the MK1 system, respectively).
In order to avoid an inverse crime, two different discretizations were used in the FD
model throughout this chapter:
the forward problem was solved using a grid of 2 mm size, comprising 474897
voxels
the inverse problem was solved using a grid of 2.5 mm size, comprising 243083
voxels (sensitivity matrix was computed using this grid)
Based on the comparisons made between the FE and FD results in section 7.3.3, the FD
model with a grid of at least 2.5 mm size provided results of acceptable accuracy.
However, the simulations in that section were performed for the MK1 system. Here, it is
assumed that the same accuracy could be achieved if similar comparisons were to be
made for the MIT arrays under investigation in this chapter. This assumption meant that
the convergence tests were not carried out for every single MIT array, in the interests of
the computational time.
Initially, MIT arrays without an electromagnetic screen were considered, again in the
interests of the computational time. In this way, the simulations could be performed
without the process of introducing the screen into the FD model which requires the FE
model to be solved for free space (see section 5.5).
Therefore, the second assumption made in this chapter is that the effect of screen on the
reconstructed images, if any, would be similar in all the examined MIT arrays. In other
words, if a particular array performs better than the other ones, this superiority would be
the case with and without the screen. Therefore, the comparisons made between
different arrays without the screen are still meaningful.
As discussed in section 2.3, the screen helps to minimize the interference from
surrounding dielectric objects in a practical MIT system, and also the capacitive
192 Customized Coil Arrays, Systematic Errors, and the MK2b System
coupling between coils. However, these issues are irrelevant in the simulations
presented in this chapter. Later in this chapter, when the final design of the MK2b
system is defined, the screen is added to the system in order to obtain practically
realistic images. As will be shown, the images are influenced by the presence of screen.
8.3. Hemispherical and annular arrays
8.3.1. Hemispherical (Helmet) array
The first helmet shaped MIT array considered in this chapter consisted of 28 exciter and
28 sensor coils of 10 mm radius, positioned in 4 rows on a hemisphere (figure 8.1). Two
different helmet radii were considered:
100 mm, referred to as 'H100'
120 mm, referred to as 'H120'
The closest distance between a coil and the surface of the head was 5 mm for H100.
This particular configuration of the coils was adopted as it conformed better to the
region of interest for imaging, i.e. the brain. These arrays provided 28 28 = 784 coil
combinations.
Figure 8.1. Helmet MIT arrays: (a) side view of hemispheres H100 and H120 in position over head,
(b) coil positions on hemisphere H100, magnified in relation to (a). Angles of elevation of coil
centres in each ring are indicated with respect to lowest ring. Each of first three rows from bottom
includes 16 coils, and the upper row includes 8 coils. Exciters and sensors are shown as red and
black coils, respectively. Origin of coordinate system is defined as centre of helmet.
(a) (b)
z
y
67.5
45
22.5
H100 H120
193 Customized Coil Arrays, Systematic Errors, and the MK2b System
Figure 8.2. Annular MIT array. Three levels of array for each plane of measurement are indicated
as Z1, Z2, and Z3 which are level with first lower rows of H100 (see figure 8.1b). Two pairs of coils
for each plane of measurement are shown. Exciters and sensors are shown as red and black coils,
respectively. Coordinate system is that in Helmet arrays.
8.3.2. Annular array
The annular coil array modelled was similar to the MK1 system, consisting of 16 pairs
of exciter/sensor coils arranged in two concentric rings, but here the rings were smaller,
having radii of 100 and 110 mm for exciters and sensors, respectively. The radii of the
coils themselves were also smaller, at 10 mm, to match the coils of the helmet.
The array was positioned at 3 different heights with respect to the simulated head as
shown in figure 8.2. This provided a total of 768 (3×16×16) coil combinations,
comparable to that in the helmet systems (784). The middle plane passed through the
centre of the LP stroke. The spacing of the three planes of measurement are different in
order to match more closely the vertical extent of the helmet configurations; these
planes are level with three lower coil rings in H100.
8.3.3. Forward problem
In order to predict the phase changes due to the stroke, the forward problem was solved
using the grid of 2 mm size. Solving the FD model to compute each of the MIT data
sets, e.g. 784 measurements for H100 array at one frequency, took approximately 6
hours. The forward problem was solved for the frequencies of f1 = 1 MHz and f2 = 10
MHz.
Z3 = 70 mm
Z2 = 38 mm
Z1 = 0
1 cm
194 Customized Coil Arrays, Systematic Errors, and the MK2b System
Figure 8.3. Singular spectrum for three MIT arrays, normalised to the largest singular value and
plotted on a logarithmic scale.
8.3.4. Sensitivity analysis
The sensitivity matrix, A, was computed using the grid of 2.5 mm size, for a uniform
conductivity of 1 S
m
-1 and at the frequency of 10 MHz. All the sensitivity entries were
computed using (6.12), i.e. normalized to the primary signal. For each of the MIT
arrays, this took approximately 2 hours.
The SVD of A was computed as described in section 6.3.2. The singular values,
normalized to the largest singular value s1 in each system, are plotted on a logarithmic
scale in figure 8.3. The ratio between the largest and smallest singular values for the
annular, H100, and H120 sensitivity matrices was 9.1×107, 4.0×10
5, and 1.9×10
6,
respectively. This indicates that, for the particular discretization used, the sensitivity of
the H100 array is 230 times better conditioned than that of the annular system.
Total sensitivity maps for each system are plotted in transverse, coronal, and sagittal
cross-sections in figure 8.4. To this end, the sum of absolute values in each column of
the sensitivity matrix, corresponding to all coil combinations for one voxel in the grid,
were used to compute the total sensitivity map values (entries):
(8.1)
where aij are the entries of the sensitivity matrix A, m and n are the number of rows
(coil combinations) and columns (voxels) in A.
H100
H120
Annular
195 Customized Coil Arrays, Systematic Errors, and the MK2b System
Therefore, Map1 n = (map1, map2,..., mapn) is a row vector from which the 3D map is
constructed by placing each entry mapj in the corresponding voxel of the grid.
Absolutes values of aij were considered for the convenience of display.
Among the three arrays, a maximum total sensitivity value of 18 10-4
radian.m. was
obtained for the annular array in the nose region, evident in the sagittal plane. The upper
limit of the colour-bar in all maps was, therefore, set to this value.
As can be seen, the maps show a strong decay of total sensitivity values from the
periphery close to the coils towards the centre of the head. In the centre, the values are
almost zero; dark regions evident in all cross-sections.
The H100 array has a better sensitivity distribution in the desired brain region (see all
cross-sections in middle column). This is due to the coils' locations and orientations
conforming to the upper part of the head. Since the coils are positioned closer to the
scalp, it reveals higher values of total sensitivity than does the H120 array.
Annular H100 H120
Figure 8.4. Total sensitivity plots for three MIT arrays. Rows from top to bottom: transverse,
coronal, and sagittal cross-sections. Colour-bar in radian.m.S-1
.
10-4
196 Customized Coil Arrays, Systematic Errors, and the MK2b System
Target, ( )max = 0.24 ( )max = 0.27 ( )max = 0.24 ( )max = 0.18
Figure 8.5. A reconstructed image of LP stroke, shown with different colour-bars in S m
-1. The
values given above each image are upper limits in the colour-bars.
8.3.5. Image display
Prior to discussing the image reconstruction and the obtained results, let us examine the
image plots in terms of the colour representations. Figure 8.5 shows an arbitrary image,
reconstructed for the LP stroke. The maximum value of the estimated conductivity
change, occurring in one of the image voxels, is ( )max = 0.27 S m
-1. The second
column from left, shows the plotted image when the upper limit of the colour-bar is set
to 0.27. The visibility of stroke is poor. In the third column, the upper limit is set to the
maximum value in the simulated target, i.e. 0.24. Although still blurred, the stroke
visualization is slightly improved. The last column shows the same image when the
upper limit is further adjusted to 0.18. As can be seen, the visibility of stroke is now
clearly better. An extremely low upper limit would result in a totally uniform image.
Therefore, the presence of even a small volume artefact occurring in one voxel, but with
a high magnitude, may jeopardise the visibility of the desired feature. Hence, the proper
choice of the colour scales can be important for the visual assessment of the images. In
practice, once the image values are computed, the radiologist should be able to easily
change the colour scales in a visualization toolbox, in order to see whether a meaningful
image can be observed.
197 Customized Coil Arrays, Systematic Errors, and the MK2b System
8.3.6. Image reconstruction
A phase noise level of 1 m was added to all the predicted phase changes. The Tikhonov
regularized solution as in (6.25) was used to solve the inverse problem, and reconstruct
the conductivity changes between 1 and 10 MHz. Any negative values were truncated
from the solution as a-priori information (this physical constraint was explained in
section 6.7). The error in the solution obtained using a regularization parameter, , was
computed as:
(8.2)
where ||.||2 is the standard L2 norm. Note that the distribution of the conductivity change
in the target ( true) and image ( ) have different discretizations. For the purpose of
computing E, the target was interpolated onto 2.5 mm voxels, as if it had been simulated
using this grid.
The error was computed for a wide range of values (10-12
- 10-6
), and the optimised
regularization parameter was considered to be the value of that minimized the error.
In this way, the best feasible image, from a mathematical point of view, was obtained
for each array. The reconstructed images using the optimised values of were then
compared. For computing the norms, three volumes were considered for integrations:
entire head, to compute Ehead
only brain region, to compute Ebrain
the volume of the head superior to the base of brain, comprising the brain and
upper parts of the skull-CSF-muscles, to compute Eupper
These volumes are shown in blue in figure 8.6.
Figure 8.6. Integration volumes, shown as blue regions, for computing Ehead, Ebrain, and Eupper.
Ehead Ebrain Eupper
198 Customized Coil Arrays, Systematic Errors, and the MK2b System
Figure 8.7. Error Ehead in reconstructed images plotted in percentage as a function of , for three
MIT arrays.
The optimal λ, which gave the minimum value of Ehead, was 7.5 10-10
, 1.0 10-9
, and
5.0 10-10
for the annular, H100, and H120 arrays, respectively (see figure 8.7). The
corresponding minimum values of the error were 57.5%, 58.5%, and 55.5%.
The reconstructed images using the optimised values of , are presented in figure 8.8,
three middle columns. Since the reconstructed features were under-estimated in terms of
the conductivity values, an upper limit of the colour-bar (0.18), lower than the
maximum value in the simulated target (0.24), was chosen for displaying the
reconstructed images.
The maximum value for each particular image, ( )max, is provided to clarify whether
and to what extend the high magnitude values are being suppressed for the illustration.
Only some of the artefacts in the reconstructed images saturate the colour-bar, and none
of the reconstructed stroke features.
A blurred stroke feature, of amplitude 0.13 S m
-1, can be seen in the image from the
annular array (target is 0.24 S m
-1). The stroke is better visualised with H100 where the
feature has amplitude > 0.16 S m
-1. Increasing the radius of the helmet to 120
mm
(H120) has the advantage of reducing the artefacts of raised image value occurring near
the coils (see sagittal planes), but at the expense of reducing the stroke visibility.
10-12
10-11
10-10
10-9
10-8
10-7
10-6
50
60
70
80
90
100
Eh
ea
d
10-12
10-11
10-10
10-9
10-8
10-7
10-6
50
60
70
80
90
100
Eb
rain
Annular
H100
H120
199 Customized Coil Arrays, Systematic Errors, and the MK2b System
Target - LP
( )max = 0.24
Annular
( )max = 0.35
= 7.5 10-10
H100
( )max = 0.32
= 1.0 10-9
H120
( )max = 0.32
= 5.0 10-10
Annular - 1 plane
( )max = 0.35
= 5.0 10-11
Figure 8.8. Columns left to right: simulated LP target, and reconstructed images from phase
changes with 1 m added noise for annular (three planes), H100, H120, and annular (with one
plane) MIT arrays; was optimised based on Ehead. Maximum conductivity change occurred in
each images is given as ( )max. Rows from top to bottom: transverse, coronal, and sagittal planes.
Colour-bar in S m
-1.
Compared with the images obtained from the MK1 system in the previous chapter
(figure 7.24), conducting the measurements now for three planes, to acquire more data,
clearly improves the performance of the annular array. In order to further examine this,
data obtained from only one plane of measurement in the present annular array (figure
8.2) was used for image reconstruction. This was the lowest coil ring (Z = 0), which is
almost at the same level as the MK1 coils shown in figure 7.5. The reconstructed image
is displayed in the last column from left of figure 8.8, where was optimised in the
same way explained. The visibility of the stroke feature is clearly much degraded.
Figure 8.9 shows the errors Eupper and Ebrain as a function of . The minima exhibited by
the curve for Eupper were broad, indicating that there was no sharp optimisation of λ. The
likely reason for this is the fact that for high levels of regularization (the range shown
by a green arrow), the stroke feature vanishes, but so do the high amplitude peripheral
200 Customized Coil Arrays, Systematic Errors, and the MK2b System
artefacts outside the brain, thus lowering the error. The three middle columns in figure
8.10, show the reconstructed images using the optimised values of for Eupper. All
images are slightly more regularized than those for Ehead.
Figure 8.9. As figure 8.7, but for Eupper (a) and Ehead (b).
The fifth column from left in figure 8.10, shows an over-regularized image for H100 ( o
= 2.5 10-7
). According to the values of Eupper, two images shown for H100 have very
similar errors; 63.1% for optimised , and 64% for the over-regularizing o. This is a
clear example of where from a strictly mathematical point of view and using the
previously defined error measure by equation (8.2), the inverse solutions in both cases
exhibit similar accuracies. However, one is clinically more useful by visualizing the
10-11
10-10
10-9
10-8
10-7
10-6
60
65
70
75
80
85
90
95
100E
up
pe
r
10-12
10-11
10-10
10-9
10-8
10-7
10-6
50
60
70
80
90
100
Eb
rain
10-12
10-11
10-10
10-9
10-8
10-7
10-6
50
60
70
80
90
100
Eb
rain
Annular
H100
H120
10-12
10-11
10-10
10-9
10-8
10-7
10-6
50
60
70
80
90
100
Eb
rain
Annular
H100
H120
(a)
(b)
201 Customized Coil Arrays, Systematic Errors, and the MK2b System
stroke feature much more clearly, despite the presence of high-amplitude artefacts in the
scalp (see sagittal plane in middle column, figure 8.10).
It is therefore suggested that visual inspections should be conducted alongside with the
mathematical methods for assessing the images. In practice, when the target is
unknown, the suggested value of by an automatic method can be employed as an
initial estimation for deciding on the range of values to be examined (for this case
GCV = 8 10-11
, L-curve = 1 10-10
). A series of images using these values can be
reconstructed, and then visually inspected.
The images reconstructed using the optimised values of for Ebrain were slightly more
regularized than those for Ehead (not shown here as they were very similar to figure 8.8).
Because of the better visualisation of the stroke feature in the images obtained by H100
(see figure 8.8 and figure 8.10, middle column), only this array will be considered for
the remainder of this section.
Target - LP
( )max = 0.24
Annular
( )max = 0.29
= 2.5 10-9
H100
( )max = 0.27
= 2.5 10-9
H120
( )max = 0.27
= 7.5 10-10
H100
( )max = 0.22
o = 2.5 10-7
Figure 8.10. As figure 8.8, but reconstructed using values of minimising Eupper. Also fifth column
from left shows image reconstructed for H100 and using a high level of regularization.
202 Customized Coil Arrays, Systematic Errors, and the MK2b System
SP-target
( )max = 0.24
SP-image
( )max = 0.25
= 7.5 10-9
SD-target
( )max = 0.22
SD-image
( )max = 0.26
= 2.5 10-9
Normal-target
( )max = 0.13
Normal-image
( )max = 0.25
= 5.0 10-9
Figure 8.11. Columns left to right: simulated SP target, reconstructed SP image, simulated SD
target, reconstructed SD image, simulated normal brain, reconstructed normal image. Rows from
top to bottom: transverse, coronal, and sagittal cross-sections. Images reconstructed using phase
changes with 1 m added noise to H100 data, and optimised for Ebrain. Colour-bar in S m
-1.
Images reconstructed by minimizing Ebrain, are shown in figure 8.11 for the smaller
strokes and also for the normal head (no stroke present). Notice that, in this figure only,
the upper limit was further reduced to 0.15 S
m
-1. This is because the reconstructed
conductivity changes were further under-estimated compared to those obtained for the
LP stroke.
All images appeared very similar. The SP stroke can be barely recognised as a slight
left-right asymmetry in the transverse and coronal planes (2nd column from left). The
SD stroke cannot be visualised at all (4th column).
The image of the normal head (last column) is not completely symmetrical. The likely
reason is an unsymmetrical head geometry and the presence of 1 m random phase noise
in the data.
203 Customized Coil Arrays, Systematic Errors, and the MK2b System
8.4. Simulated errors
In this section, different types of error are added to the data, and their effect on the
reconstructed images of the LP stroke is investigated for the H100 array.
8.4.1. Phase noise
Although the images in the previous section were all reconstructed from the data
containing 1 m noise, here different levels of Gaussian phase noise (up to 20 m°
standard deviation) were considered in order to test the effect of the noise level on the
images.
First, using noise-free data, the regularization parameter was optimised as described in
the previous section: the value of that minimised Eupper was adopted ( = 2.5 10-9
).
Subsequently, noise of different levels were added to the data and all the noisy images
were reconstructed using the same = 2.5 10-9
. Strictly speaking, each noisy data set
will have its own optimal , which may be different from the adopted one. However, the
optimised for the noise-free data was used throughout this section.
The image error due to the added noise was evaluated as:
(8.3)
where noisy and noise-free are the images reconstructed using the data with and
without noise. The integration was carried out over the upper portion of head; Eupper in
figure 8.6.
The error Enoise thus is a measure of the spurious content in the image due to the noise,
as a percentage of the noise-free image content, and increases roughly in proportion
with the phase noise added to the simulated measurements (figure 8.12). Three runs of
the model, each with its own randomly generated noise for each noise level, were used.
Examples of images are shown in figure 8.13, including two noise levels of particular
interest: 17 and 1
m° relating to the MK1 and MK2a practical data-collection systems
(see Appendix A). Compared with the noise-free image, the image is slightly degraded
204 Customized Coil Arrays, Systematic Errors, and the MK2b System
by a noise level of 1 m°, and the stroke feature remains clearly visible. For 17 m° added
phase noise, however, artefacts dominate the image and the stroke cannot be visualized.
Figure 8.12. Value of error, Enoise, in reconstructed images using H100 (λ = 2.5x10-9
) for different
levels of added Gaussian phase noise (standard deviation). Three simulations (different colours) for
each noise level are shown along with the fitted straight line.
Target - LP
( )max = 0.24
Noise-free
( )max = 0.25
1 m
( )max = 0.26
3 m
( )max = 0.34
17 m
( )max = 1.44
Figure 8.13. Columns left to right: simulated LP target, and reconstructed images for H100 with
different levels of added phase noise ( = 2.5 10-9
). Rows from top to bottom: transverse, coronal,
and sagittal cross-sections. Colour-bar in S m
-1.
0 2 4 6 8 10 12 14 16 18 200
20
40
60
80
100
120
140
Noise level (m )
En
ois
ey = 6.43 x
205 Customized Coil Arrays, Systematic Errors, and the MK2b System
Of course, the noise artefacts will change for every run of the random number generator,
but qualitatively the images will remain fairly similar. The artefacts in the image can be
reduced by increasing λ, but at the disadvantage of blurring or losing the stroke feature.
The purpose here is to define the tolerable noise level which does not degrade the
visibility of stroke.
By inspection of the images for different runs of randomly generated noise, it appeared
that an acceptable noise level would be between 1 and 4 m , and could reasonably be set
at 3 m . This leads to an acceptable value of the error measure, Enoise, of approximately
20% (see figure 8.12).
8.4.2. Systematic errors
It is widely appreciated that MIT image reconstruction will be improved by introducing
as much a-priori information as possible. One example of this is to determine the
boundary of the eddy-current region, in this case the shape of the head, by some
independent means such as by optical shape scanning (see e.g. [217]). This boundary is
of great importance as it marks the large contrast in dielectric properties between the
tissues and air.
The sensitivity matrix will then be computed for this known region. Accurate
calibration of the optical method will be important, and so will accurate registration of
the optically-determined boundary with the known geometry of the MIT array.
Inevitably there will be errors in these processes.
In order to examine the effect of such inaccuracies, two examples of geometrical
systematic errors were simulated. The first simulated a mis-registration equivalent to a
displacement of the head relative to the helmet, separately in x-, y- and z-directions
(figure 8.14a). The second involved a simple scaling error simulated by an expansion or
contraction of the entire head (figure 8.14b).
Since the coils do not form part of the FD discretization, but instead are perfect circles
whose coordinates and orientations are specified separately, it is very easy to model
these geometrical changes. A shift of the head was introduced by altering the
coordinates of all the coils by a fixed amount, whilst leaving the FD mesh unchanged.
206 Customized Coil Arrays, Systematic Errors, and the MK2b System
An expansion or contraction of the head was introduced by altering the FD mesh size
whilst leaving the coils unchanged.
Figure 8.14. Exaggerated geometrical errors: light pink and dark blue colours show the 'true' and
'erroneous' boundaries, respectively. (a) displacement in y-direction. (b) contraction.
Note that the data computed using the changed („erroneous‟) boundary will differ from
the measured data, i.e. data computed using the original („true‟) boundary. This
difference is the systematic error due to the use of wrong geometry that may happen in
practice. For image reconstruction, the sensitivity matrix was computed using the
changed boundary of the head. The simulated MIT measurements at 1 and 10 MHz
were computed using the original head boundary. No random noise was included in any
of these simulations. The image in the 2nd column from left in figure 8.15 (true
boundary) is identical to that in figure 8.13 (noise-free), and is reproduced here for ease
of comparison.
Displacement of the head in the negative x-direction, produced artefacts which appear
as increased values on the surface that faces towards the positive x-direction, and
decreased values on the surface that faces towards the negative x-direction (figure 8.15,
3rd and 4th columns from left).
A displacement in the opposite direction produced the opposite effect (figure 8.15, 5th
and 6th columns). Obviously, larger displacements resulted in more pronounced
artefacts. For instance, compare those for x = +3 and x = +5 mm in columns 5 and 6.
The artefacts for x = +5 mm have degraded the stroke feature. Similarly,
displacements of the head in y- and z-directions produced artefacts near to the surfaces
facing the corresponding directions (see figure 8.16).
(a) (b)
207 Customized Coil Arrays, Systematic Errors, and the MK2b System
Target - LP
( )max = 0.24
True boundary
( )max = 0.25
x = -3 mm
( )max = 0.32
x = -5 mm
( )max = 0.39
x = +3 mm
( )max = 0.31
x = +5 mm
( )max = 0.38
Figure 8.15. Columns left to right: simulated LP target, and reconstructed images with different
boundary errors in x-direction. Rows from top to bottom: transverse, coronal, and sagittal planes.
Target - LP
( )max = 0.24
True boundary
( )max = 0.25
y = -5 mm
( )max = 0.36
y = +5 mm
( )max = 0.42
z = -5 mm
( )max = 0.39
z = +5 mm
( )max = 0.27
Figure 8.16. As figure 8.15, but for y- and z-direction displacements.
y
z
x
y
z
x
208 Customized Coil Arrays, Systematic Errors, and the MK2b System
Contraction of the head (scaling error) produced increased values near the surface of the
head (figure 8.17, 3rd and 4th columns from left). This is to be expected. Note that in
contracting the geometry, a layer of conductive material is in fact removed (ignored)
from the periphery. This would be the pink layer in figure 8.14b. Therefore, when the
sensitivity matrix is computed for the contracted geometry, the signals received from
the absent material is not accounted for in the sensitivity matrix. The contribution to the
signals from this material results in higher values of conductivity at the surface in the
images. Therefore, artefacts of raised conductivity will appear on the boundary.
Target - LP
( )max = 0.24
True boundary
( )max = 0.25
3% Cont.
( )max = 0.36
5% Cont.
( )max = 0.45
3% Exp.
( )max = 0.21
5% Exp.
( )max = 0.20
Figure 8.17. As figure 8.15, but for contraction and expansion scaling errors.
Expansion of the head, however, produced artefacts as reduced image values on the
periphery (figure 8.17, 5th and 6th columns). Note that since haemorrhage is always
going to be a region with positive image values, its visibility may even improve in the
case of an expansion of the geometry.
209 Customized Coil Arrays, Systematic Errors, and the MK2b System
Indeed, in the absence of a sufficiently accurate method of a-priori boundary
determination, it may be worth deliberately using an oversized boundary for the
computation of the sensitivity matrix in order to ensure that all image errors will be
negative, and not confused with a haemorrhage. This possibility deserves further
investigation.
In a similar way to the calculation of Enoise in section 8.4.1, the error due to inaccuracies
in the boundary was computed as:
(8.4)
where erroneous and errorless are the images reconstructed with and without boundary
errors ( errorless is identical to noise-free in equation (8.3)).
This time, Eboundary is a measure of the error in the image content due to the systematic
errors expressed as a percentage of the original image content (image without boundary
error). The values of Eboundary are plotted in percentage in figure 8.18, for displacement
and scaling errors.
Figure 8.18. Image error, Eboundary, for different displacements of head or percentage scaling.
Applying the same acceptance criterion as for the level of random noise, i.e. Eboundary <
20%, indicates that the mis-registration displacement error should be no more than 3-4
mm, and the size scaling error should be no more than about 3%.
-5 -4 -3 -2 -1 0 1 2 3 4 50
5
10
15
20
25
30
35
40
Displacement (mm) - Scaling (%)
Eb
ou
nd
ary
x-direction
y-direction
z-direction
Scaling
210 Customized Coil Arrays, Systematic Errors, and the MK2b System
8.5. MK2b system
As discussed previously, conducting the data acquisition for multiple planes of
measurements in annular arrays can help to improve the spatial resolution of the
reconstructed images, despite the fact that not all the added information is independent
from those obtained from the first plane of measurement. In practice, this may be done
by applying a scanning protocol such as coil or target movements.
However, any type of movement could result in the introduction of further errors in the
data; e.g. boundary errors. Therefore, it is preferable to carry out the measurements in
one simple process, avoiding the movement of the head or sensor array.
Based on the results of previous sections in this chapter, it appears that hemispherical
arrays might be the most suitable coil configurations for stroke imaging. For a simulated
phase noise level of < 3 m , acceptable images of a peripheral stroke were produced for
this configuration. The electronics developed by the South-Wales group for the MK2a
system, can achieve a low level of phase noise (1 m°), but is not compatible with 56
coils as used in the H100 array.
In order to employ the new electronics architecture, the number of exciters must be a
multiple of 16, and the number of sensors a multiple of 7 [63]. By taking into account
the physical and practical constraints to design a feasible system, this section describes
a numerical study for the design of the MK2b system.
8.5.1. Concept of MK2b array
As shown in figure 8.19, the system is envisaged as an array of separate exciter and
sensor coils positioned on a hemisphere (radius RC), inside an hemispherical
electromagnetic screen (outside radius RS). An inner, concentric, plastic hemispherical
shell (inside radius RP) is to be fixed inside the coil array so as to completely enclose
and protect the coils.
For the electromagnetic screen, hemispherical spun aluminium bowls of 3 mm thickness
are commercially available. Plastic shells (3 mm thickness) can be vacuum formed in
our own Radiotherapy Physics Mould Room.
211 Customized Coil Arrays, Systematic Errors, and the MK2b System
Figure 8.19. Conceptual diagram of MK2b system (not to scale).
Figure 8.20. Standard measurement indexes for head (after ARTUR [218]).
8.5.2. Dimensions
The plastic shell needs to be small enough to allow the coils to be placed as close as
possible to the head surface in order to maximise the MIT signals, and at the same time
large enough to accommodate most adult heads. The 1988 U.S. Army Anthropometry
Survey provides head measurements from 1774 males and 2208 females [218]. By
examining the data, the most relevant dimension is the length (anterior–posterior
distance from between the eyebrows to the back of the cranium, as shown in figure
8.20). A hemisphere based on the head length, will certainly be large enough to suit the
breadth and coronal arc for that head.
The measurement data fell within the length of 209.5 mm for 96.5% of the males and
99.95% of the females. A further margin of 10 mm was allowed for cushioning, leading
to an inside shell radius of RP =
115
mm in figure 8.19. The coils are positioned further
out by another 10 mm, on a hemisphere of radius RC = 125 mm. The coil-shell
separation is, therefore, 7 mm. A coil-screen standoff distance of 32 mm is chosen as
this allows the use of a commercially available size of spun aluminium hemisphere with
outside radius of RS = 160 mm.
coils
screen
shell
head
RS
RC
RP
Length Breadth Coronal Arc
212 Customized Coil Arrays, Systematic Errors, and the MK2b System
Figure 8.21. Top view of three coil arrays; red and black coils are exciters and sensor coils,
respectively.
8.5.3. Number of coils
The number of coils in the array determines the maximum coil diameter that will fit
onto the hemisphere. Initially, three arrays were considered as shown in figure 8.21:
30 coils of 45 mm diameter: 16 exciters + 14 sensors
46 coils of 35 mm diameter: 32 exciters + 14 sensors
60 coils of 30 mm diameter: 32 exciters + 28 sensors
The exciter and sensor coils are considered to be distributed in the array (figure 8.21). In
order to compare the performance of these three arrays, frequency-difference imaging of
the LP stroke was performed. The FD solver, with grid sizes of 2 and 2.5 mm, was used
for the forward and inverse problems (see section 8.2). The screen was excluded from
the models at this stage, so that the simulations could be conducted without the need for
time consuming step of solving the FE model (see section 8.2).
The forward problem was solved for the frequencies of 1 and 10 MHz, and 1 m noise
was added to the simulated phase changes. In order to find the optimised , the error
Ehead was minimised (see equation (8.2) and figure 8.6). The minimum computed values
of Ehead were 67, 64.4, and 64.5% for arrays comprising 30, 46, and 60 coils,
respectively.
The reconstructed images are shown in figure 8.22 (three middle columns). All three
arrays produced images which were poor compared to those obtained from the 58-coil
array (H100) in the previous section (e.g. see figure 8.13 for 1 m noise). The likely
reasons are:
30 coils 46 coils 60 coils
213 Customized Coil Arrays, Systematic Errors, and the MK2b System
Target - LP
( )max = 0.24
30-coil
( )max = 0.50
= 7.5 10-11
46-coil
( )max = 0.31
= 2.5 10-10
60-coil
( )max = 0.34
= 2.5 10-10
46-coil + screen
( )max = 0.45
= 2.5 10-10
Figure 8.22. Columns left to right: simulated LP target, and reconstructed images for 30-coil, 46-
coil, 60-coil, and 46-coil+screen arrays; reconstructed with 1 m noise, and optimised for Ehead.
Rows from top to bottom: transverse, coronal, and sagittal cross-sections. Colour-bar in Sm-1
.
larger distance between the coils and head (RC = 125 mm in figure 8.19
compared with 100 mm for H100 array).
use of fewer exciter-sensor coils (for 30-coil and 46-coil arrays)
Nevertheless, the stroke feature can still be recognised in all images.
For the 30-coil array (2nd column from left), the stroke is considerably displaced
towards the face (evident in transverse plane), and the image contains artefacts of high
amplitude (( )max = 0.5 S m
-1). The 46-coil and 60-coil arrays have a fairly similar
performance, both in terms of the error Ehead and the visualization of stroke, which is
better than the performance of the 30-coil array.
Since adding a further 14 sensors to form the 60-coil array would have increased the
cost of the electronic components, the 46-coil array was chosen for the practical
implementation.
214 Customized Coil Arrays, Systematic Errors, and the MK2b System
For the 46-coil array with 35 mm diameter coils, the screen standoff distance (32
mm) is
91% of the coil diameter, which is slightly larger than the range 60-80% used by
Watson et al [33].
Finally, following the method described in section 5.5, the screen was added to the 46-
coil array by solving the FE model. The reconstructed image is shown in the last
column in figure 8.22. Comparing to the image obtained by the same array without the
screen (middle column), adding the screen clearly improved the stroke visibility. It is
less under-estimated in amplitude ( 0.17 S m
-1). However, the minimum error remained
the same (Ehead = 65%). This is due to the presence of artefacts of high amplitude on the
periphery (( )max = 0.45 S m
-1). The remainder of this chapter employs the 46-coil
array with the screen included.
8.5.4. Other coil arrangements and quality measurement
Another investigation in the design of the sensor array was to decide which coils acted
as the exciters, and which ones as the sensors. A distributed arrangement1, with the
exciters and sensors interspersed, was previously considered (see figure 8.21).
According to the Lorentz reciprocity theorem discussed in section 5.5, the mutual
inductance between a pair of exciter-sensor coils is unchanged if they are interchanged.
However, their relationship with other coils in the array will change. Therefore,
different coil arrangements may not perform in the same way.
The number of possible ways to select 32 exciters from 46 coils is 46C32 = 46!/(32! 14!)
= 239,877,544,005 arrangements. Clearly, a number of assumptions should be made to
limit the possible arrangements, since not all of them can be numerically investigated
due to time restrictions.
For practical implementation, it is preferable to separate the exciter and sensor coils, so
to make the circuitry simpler; e.g. reducing cable lengths. Two examples of such
arrangements are shown in figure 8.23, one symmetrical about the x- and y-axes (right
array), and one asymmetric about y-axis (left array).
1 Note that throughout this thesis, the term 'coil configuration' is used to describe the spatial positioning of
the coils (location and orientation) and the number of exciters and sensors. The term 'coil arrangement'
refers to which coils are exciters and which are sensors for a given coil configuration.
215 Customized Coil Arrays, Systematic Errors, and the MK2b System
Figure 8.23. Examples of separated coil arrangements favoured from the point of view of the
electronics layout; red coils are exciters.
Target - LP
( )max = 0.24
= 1 10-9
( )max = 0.28
= 1 10-10
( )max = 0.40
= 1 10-11
( )max = 0.81
Figure 8.24. Columns left to right: simulated LP target, and images reconstructed from noise-free
data and using different values of for asymmetric arrangement shown in figure 8.23 (left array).
Rows from top to bottom: transverse, coronal, and sagittal cross-sections. Colour-bar in Sm-1
.
The use of asymmetric coil arrangements, however, results in artefacts on the
reconstructed images. In order to demonstrate this, the asymmetric arrangement in
figure 8.23 (left array) was simulated. Images of the LP stroke were reconstructed from
noise-free data and with three different levels of regularization: optimised value of =
10-10
, one smaller ( = 10-11
) and one larger ( = 10-9
) value than the optimised value
(see figure 8.24).
y
x
216 Customized Coil Arrays, Systematic Errors, and the MK2b System
As can be seen in the transverse and coronal planes in figure 8.24, artefacts of relatively
large volume are evident in the left hemisphere of brain, on the opposite side to the
stroke location, and also in the central regions. Therefore, symmetric arrangements were
intuitively favoured in this chapter. A detailed investigation is required in order to
realise the possible reasons for this behaviour in asymmetric arrangements.
Even after excluding the asymmetric arrangements, the number of possibilities is still
too large to permit assessment of every individual arrangement. Therefore, six
symmetric arrangements, separated and distributed, were considered for comparison;
labelled as D1 to D6 as shown in figure 8.25.
Figure 8.25. Six selected coil arrangements for the 46 coil configuration; red coils are exciters. D6 is
the arrangement used previously in figure 8.21 (middle array).
A quality measure, which has been employed in geophysics for tomographic problems
[219], was used for a quantitative comparison of the different coil arrangements. Gursoy
and Scharfetter [215] used this measure for comparing the performance of sensor coils
placed in different locations in a MIT array (i.e. different coil configurations). Here, the
coil configuration is fixed, but the arrangement is varied as in figure 8.25.
A measure of the orthogonality of the rows in the sensitivity matrix is given by:
D1 D2 D3
D4 D5 D6
217 Customized Coil Arrays, Systematic Errors, and the MK2b System
(8.5)
where ri and rj are the i-th and j-th rows of the sensitivity matrix. The quality measure is
defined as:
(8.6)
where m is the total number of rows (coil combinations) in the sensitivity matrix. The
quality measure reflects the linear independence of the different exciter-sensor
combinations [219]. The more orthogonal the rows are, the more independent
information they provide, and the less ill-conditioned the sensitivity matrix is. A matrix
with orthogonal rows ( ij = 0) will result in the maximum value of Qm = m (m-1)/2. On
the contrary, a matrix with totally dependent rows ( ij = 1) will have the worst quality
measure of Q = 0. For the 46-coil array, m = 32 14 = 448 and the maximum value of Q
is Qm = 100128.
The sensitivity matrix was computed for arrangements D1-D6, where its entries were
computed using (6.12), i.e. normalized to the primary signal. The quality measure was
calculated for each design using (8.6), and was normalized to Qm (=100128). The
normalized values are shown in figure 8.26 as percentages. As can be seen, these values
spanned only a small range, (79.2-80.3%), indicating that there was no clear best coil
arrangement. No quality measure values in the work by Gursoy and Scharfetter [215]
were reported to be compared against the values obtained here.
Figure 8.26. Quality measures for 6 designs, D1-D6, normalized to Qm = 100128.
D1 D2 D3 D4 D5 D679
79.2
79.4
79.6
79.8
80
80.2
80.4
80.6
Qu
alit
y (
%)
218 Customized Coil Arrays, Systematic Errors, and the MK2b System
According to the quality measurements, none of the considered coil arrangements (D1-
D6) will have a distinctly better performance than the rest; the reconstructed images
confirmed this.
Initially, for comparison, no noise was added to the simulated data so to avoid any
random error affecting the images. In order to find the optimised , the error Ehead was
minimised (see equation (8.2) and figure 8.6). In the presence of screen, the computed
conductivity changes tended to have higher amplitudes. Therefore, the upper limit of
colour-bar was set to the maximum value in the simulated LP target (( )max = 0.24 S
m-1
) for illustrations in figure 8.27.
The stroke feature can be clearly visualized in all images, with some being more under-
estimated in amplitude (D1-D2 & D5), and some being slightly more spread (D3 & D6)
evident in the transverse planes.
D1
( )max = 0.33
= 2.5 10-10
D2
( )max = 0.35
= 2.5 10-10
D3
( )max = 0.58
= 1.0 10-10
D4
( )max = 0.33
= 2.5 10-10
D5
( )max = 0.42
= 2.5 10-10
D6
( )max = 0.42
= 1.0 10-10
Figure 8.27. Columns left to right: reconstructed images for D1 to D6 designs, respectively, where
was optimised for Ehead. Rows from top to bottom: transverse, coronal, and sagittal cross-sections.
Colour-bar in S m
-1 with upper limit of 0.23.
219 Customized Coil Arrays, Systematic Errors, and the MK2b System
D1
( )max = 0.53
D2
( )max = 0.37
D3
( )max = 0.61
D4
( )max = 0.40
D5
( )max = 0.39
D6
( )max = 0.37
Figure 8.28. As figure 8.27, but with 1 m added noise.
Figure 8.28 shows the images with 1 m added noise, where was optimised using the
noisy data. A value of = 2.5 10-10
was obtained for all coil arrangements. In the image
obtained from D6 (last column from left), the stroke feature was slightly better
visualized than in the rest of the noisy images obtained from D1-D5 coil arrangements.
More investigations are being carried out in order to make the final decision on the
arrangement of the coils.
8.5.5. MK2b construction
At the time of writing, the MK2b system is under construction. Figure 8.29 presents two
photographs from the early stages of the construction at the Medical Physics & Clinical
Engineering Department, Singleton Hospital, Swansea. The array is now with the
LCOMIT partners at the University of Glamorgan for the addition of the electronics and
finalizing the sensor array.
220 Customized Coil Arrays, Systematic Errors, and the MK2b System
Figure 8.29. Photographs of MK2b system at early stages of construction. Left: screen and coil
formers before mounting. Right: array (screen, formers, plastic shell) prior to adding the
electronics and coil windings.
8.6. Discussion
Hemispherical arrays conform more closely to the shape of the head than a conventional
annular array, thus increasing the sensitivity of the MIT array to conductivity changes.
By examining the total sensitivity maps, it was shown that for the coils placed
sufficiently close to the head, a better sensitivity distribution can be achieved, in
particular, in the upper parts of the head (map for H100 in figure 8.4).
Hemispherical arrays offer the possibility of using a larger number of the coils, than in
the annular ones. This allows the measurements to be carried out without the need to
move the head or the array, and therefore avoids further systematic errors.
For the chosen discretizations and the inverse solution techniques, no clear optimised
level of regularization could be found for brain imaging. In practice, it may be useful to
reconstruct a series of images with different levels of regularization, and inspect them
visually in order to decide whether a meaningful feature can be recognised in the
images. For instance, just the detection of an asymmetry in the brain may be very
significant clinically. Furthermore, presenting the images with different colour limits
may improve the visualization of the features under-estimated in amplitude.
Frequency-difference MIT imaging of a large intra-cerebral haemorrhage ( 50 ml),
peripherally located, seems to be feasible, provided that the phase noise level in the
hardware is small enough ( 3 m ), and any error in the assumed a-priori boundaries
remains low ( 4 mm for displacement and 3% for scaling).
Screen
Former
221 Customized Coil Arrays, Systematic Errors, and the MK2b System
However, imaging of a small haemorrhage (8 ml), located deep within the brain, proves
to be more difficult, even with noise-free measured data, and no systematic errors in the
assumed boundaries of the head. This is due to the lower sensitivities of MIT to the
central regions of the imaging domain.
In this chapter, only two types of the boundary errors were examined; displacement and
scaling. Mis-registration of the measured boundaries may also occur, for example, in
terms of rotation of the head about all three axes. Further errors can also be introduced
if the patient's head moves during the measurement process, as suggested by Gursoy
and Scharfetter [220], unless the head's position is somehow fixed within to the sensor
array. Further investigations are required in order to examine the effect of such errors on
the images.
Contraction of the head geometry, as a scaling systematic error, produced artefacts of
small volume but high amplitude on the periphery (figure 8.17). Such raised
conductivities close to the scalp could be mistaken for small subarachnoid
haemorrhages. On the contrary, artefacts of negative sign were produced by an
expansion of the head (same figure). However, such artefacts cannot be confused by
haemorrhage which appears as a raised conductivity region. The stroke visibility may
even improve, as shown for the particular example presented in the images.
Therefore, in case of uncertainty about the accuracy of the measured boundaries, it is
recommended that using a head geometry, slightly larger than the available one, might
help to avoid the appearance of artefacts of high amplitude. If the only geometrical data
available is the approximate overall dimensions of the head in three directions, one
might consider computing the sensitivity matrix for a sphere or ellipsoid large enough to
accommodate the head. In the FE mesh construction in figure 7.4(II), such an ellipsoid
was shown in which the head was confined. Obviously, detailed investigations are
required in order to examine the images that one would obtain using these approximate
boundaries for the eddy-current region.
The design process of the MK2b MIT system was described. Different coil
configurations and arrangements were examined, and frequency-differential images of
the LP stroke were reconstructed. The conductivity distribution in the target was
assumed to be known, and was employed for adjusting the regularization level during
image reconstructions. This allowed achievement of the best possible images for each
222 Customized Coil Arrays, Systematic Errors, and the MK2b System
MIT array, using the particular inverse solution technique employed. The best image
was taken as that which produced the minimum error in the estimated conductivity
change values.
End of chapter 8.
Part IV
Experiments and Conclusions
Chapter 9.
Experiments
9.1. Introduction
One of the main purposes in the LCOMIT project was to design a practical MIT
imaging system for biomedical and other low-conductivity applications. This has been
pursued by the LCOMIT partners at the Universities of Glamorgan and Manchester.
The MK2a system, described in Appendix A.2, is a recently developed hardware with a
low level of phase noise (1 m ), which may allow imaging low-conductivity targets.
The purpose of this chapter is to carry out the first practical MIT imaging using the
MK2a system and the image reconstruction algorithms introduced in chapter 6. To this
end, a geometrically simplified model comprising cylindrical domains, similar to those
employed in chapter 6, are used. Absolute and differential imaging of the targets are
performed.
9.2. Experimental setup
All the measurements described in this chapter were carried out using the MK2a system
(see Appendix A) and at frequency of 10 MHz. The experimental setup comprised a
cylindrical tank (19.5 cm diameter) as the background, and a smaller solid cylinder (4
cm diameter, 4 cm height) as the inclusion. The inclusion was placed within the tank at
two different positions as shown in figure 9.1; two setups are referred to as 'case-A' and
'case-B'.
225 Experiments
Figure 9.1. Conceptual diagram of targets in case-A and case-B. Upper row shows x-y cross-
sections, and lower row shows vertical cross-sections along broken lines shown in upper row.
Distances show displacement of inclusion relative to centre of tank.
The tank was filled with aqueous saline solution ( = 0.205
S
m
-1) to the depth of 16 cm,
and the inclusion was a piece of Perspex ( = 0
S
m
-1). The position of the inclusion
within the tank was maintained using a length of nylon fishing line (see photograph in
figure 9.2a), and the tank was placed at the centre of the MK2a array (see figure 9.2b).
The secondary voltage, Vs, was measured for 9 equally-spaced (2 cm) planes of
measurement by moving the tank within the array in z-direction (figure 9.3). This
provided 14 14 9 = 1764 coil combinations. In order to adjust the vertical level of the
tank, wooden spacer disks of 1 cm thickness were placed beneath the tank. A plastic
guide was employed for the horizontal placement of the tank within the array (see figure
9.2b). Three runs of measurements were conducted:
uniform tank with no inclusion
inclusion placed within the tank according to case-A (figure 9.1)
inclusion placed within the tank according to case-B (same figure)
Prior to placing the target within the array for each plane of measurement, the primary
signal for empty space, Vp, was acquired. In theory, the primary signal measured at
different times should remain the same. However, multiple measurements for Vp were
performed so the Vs could be normalized to a Vp measured close in time so as to
minimise the effect of drift in the hardware.
5 cm
5 cm
5 cm
case-A case-B
Tank
Inclusion
y
x
z 16 cm
19.5 cm
226 Experiments
Figure 9.2. Photographs of experimental setup showing inclusion within tank (a) and tank within
MK2a array (b).
Figure 9.3. Planes of measurement: (a) levels of 9 planes (P1-P9) relative to tank, (b) & (c) highest
and lowers positions of tank within array, respectively.
(a)
(b)
Nylon line
Inclusion
Tank
Tank
MK2a
array
2 cm
(a) (b) (c)
Screen
Tank
Ex array
Se array
P1
P1 P9
P9
Guide
P5
227 Experiments
9.3. Simulations
The experiment was simulated using the FD model. Revisiting figure 5.11 in chapter 5,
grid sizes of 1 cm provided results of acceptable accuracy for simulating a cylindrical
domain of 20 cm diameter and height. Here, in order to achieve a better accuracy, the
smallest grid size which the FD solver could handle was considered. This corresponded
to a grid of 0.05 cm size, allowing a better approximation of the curved surfaces in the
geometry using the cubic voxels. The use of smaller grid sizes than 0.05 cm resulted in
excessive numbers of voxels for representing the cylindrical tank, thus preventing the
FD solver to run (the arrays sizes defined in the FD solver grew too large). For the
employed grid, solving the forward model for each run of measurement (9 planes of
measurement corresponding to 1764 coil combinations) took approximately 15 hours.
All three runs for the uniform tank, case-A, and case-B were simulated.
9.4. Signal analysis
Figure 9.4a is a plot of the 1764 simulated phase changes, , due to the uniform tank.
The data from each plane of measurement can be distinguished from those from the
other planes by their maximum values. The maximum magnitude of the phase changes
predicted by the model is 1.83 , which occurs for the middle plane of measurement.
Figure 9.4b shows the same plot, but for the measured data plotted on the same y-axis
scale. The maximum magnitude of the phase changes is 2.27 . As can be seen, despite
the presence of some 'spikes' in the phase changes, the data corresponding to each plane
of measurement can be distinguished from those for other planes.
The difference between the simulated and measured data, plotted in figure 9.4c, shows a
discrepancy as large as 0.44 in magnitude. This phase error is significant compared
with the genuine phase changes produced by the tank itself. The largest discrepancies
are observed for the phase changes of large magnitude, corresponding to the middle
plane of measurement. The root mean square (r.m.s.) phase error is 0.0843 .
The discrepancy between the simulated and measured data can significantly affect the
absolute imaging, which is examined later in this chapter. Also, the possible reasons for
this discrepancy are investigated in the discussion.
228 Experiments
Figure 9.4. Phase changes in degrees for uniform tank: (a) simulated data for tank of 19.2 cm
diameter, (b) measured data for tank of 19.5 cm diameter, (c) difference between (a) and (b).
0 200 400 600 800 1000 1200 1400 1600 1800-2.5
-2
-1.5
-1
-0.5
0
0.5
Coil combination
Phase c
hanges (
)
0 200 400 600 800 1000 1200 1400 1600 1800-2.5
-2
-1.5
-1
-0.5
0
0.5
Coil combination
Phase c
hanges (
)
0 200 400 600 800 1000 1200 1400 1600 1800-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Coil combination
Phase e
rror
()
(a)
(b)
(c)
one plane of
measurement
tank (simulated)
tank (measured)
tank (measured)
- tank (simulated)
229 Experiments
Figure 9.5. Phase differences caused by addition of inclusion for case-A, relative to uniform tank:
(a) simulated, (b) measured.
The phase differences due to the addition of the inclusion, = tank + inclusion - tank, are
plotted in figure 9.5 for case-A. The simulated data predicts a maximum phase
difference of 0.064 magnitude (figure 9.5a). There are a few spikes in the measured
data with much larger magnitudes (| |max = 0.67 ). These spikes are out of y-axis limits
in figure 9.5b which is set to [-0.15,0.15] for convenience of display. By closer
scrutiny of the two plots, the genuine phase differences, which can be used for
differential imaging, can be recognised in the measured data. However, the presence of
random high magnitude spikes is also significant. These can produce artefacts in the
reconstructed images. Similar plots are shown in figure 9.6, for case-B.
200 400 600 800 1000 1200 1400 1600 1800-0.15
-0.1
-0.05
0
0.05
0.1
Coil combination
Phase d
iffe
rence (
)
200 400 600 800 1000 1200 1400 1600 1800-0.15
-0.1
-0.05
0
0.05
0.1
Coil combination
Phase d
iffe
rence (
)
(a)
(b)
tank + inclusion - tank
(simulated)
tank + inclusion - tank
(measured)
230 Experiments
Figure 9.6. As figure 9.5, but for case-B.
9.5. Image reconstruction
In order to solve the forward problem, a grid size of 0.05 cm was employed in section
9.3. However, the same grid cannot be used for solving the inverse problem as it would
result in an excessively large sensitivity matrix. Furthermore, it would be time-
consumingly expensive to perform any iterative absolute imaging which requires
solving multiple forward problems for updating the sensitivity matrix.
Therefore, similar to chapter 6, a grid size of 1 cm was used for solving the inverse
problem. By using this grid size, a tank of 20 cm diameter (0.5
cm larger than the one
200 400 600 800 1000 1200 1400 1600 1800-0.15
-0.1
-0.05
0
0.05
0.1
Coil combination
Phase d
iffe
rence (
)
200 400 600 800 1000 1200 1400 1600 1800-0.15
-0.1
-0.05
0
0.05
0.1
Coil combination
Phase d
iffe
rence (
)
(a)
(b)
tank + inclusion - tank
(simulated)
tank + inclusion - tank
(measured)
231 Experiments
used in the experiments and forward simulations) was simulated for computing the
sensitivity matrix. The tank was 16 cm in height.
9.5.1. Differential imaging
For differential imaging, the sensitivity matrix was computed for a uniform conductivity
of 1 S
m
-1, and its entries were normalized to the primary signals as in (6.12). The
generalized Tikhonov method was used together with the neighbouring regularization
matrix as in (6.28). The regularization level was chosen empirically as none of the
automatic methods suggested an appropriate value for . For image representations, 8
transverse slices cutting the tank at different heights, were selected from the
reconstructed volume image (see figure 9.7).
Figure 9.7. Transverse slices cutting through the tank used for image representations.
The first column from left in figure 9.8, shows the target in case-A, when interpolated
on a 1 cm grid size. The second column represents the reconstructed images from the
noise-free simulated data ( = 1 10-8
).
As can be seen, the inclusion is clearly visible in slices 7 and 9. Two upper and lower
slices (5 and 11) also show the feature, which does not exist in the target. The likely
cause for this is the use of neighbouring matrix which does not allow sharp gradients in
the image, thereby slightly spreading the reconstructed inclusion. Also the reconstructed
inclusion is under-estimated in amplitude where the maximum magnitude of the
obtained conductivity change is 0.15 S m-1
(target is 0.205 S m-1
).
Slice 1
Slice 15
z
232 Experiments
Target
case-A
Image
(simulated data)
Image
(measured data)
1
3
5
7
9
11
13
15
Figure 9.8. Case-A: Columns from left: level of cross-section along tank, target on 1 cm grid,
differential images from simulated, measured data.
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
5 10 15 20
2
4
6
8
10
12
14
16
18
20 -0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
5 10 15 20
2
4
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10
12
14
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18
20 -0.2
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-0.1
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0
0.05
0.1
0.15
5 10 15 20
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5 10 15 20
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5 10 15 20
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233 Experiments
Target
case-B
Image
(simulated data)
Image
(measured data)
1
3
5
7
9
11
13
15
Figure 9.9. As figure 9.8, but for case-B.
5 10 15 20
2
4
6
8
10
12
14
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18
20 -0.2
-0.15
-0.1
-0.05
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234 Experiments
The third column shows the images reconstructed from the measured data. An inclusion
can be observed in the images in the correct location (slices 7 and 9), with an even more
under-estimated magnitude ( 0.06 S
m
-1) in comparison to the second column. This is
because a higher level of regularization level, = 1 10-7
, was used in order to reduce
the artefacts.
The presence of artefacts is evident in all slices in this column; some with lower
conductivity change (slice 1), and some with raised values (slice 15). The likely
reasons for these artefacts are discussed later in this chapter.
The differential images for case-B are shown in figure 9.9.
9.5.2. Absolute imaging
Absolute imaging was performed using the iterative method described in section 6.8.3
for the uniform tank. Initially, the images were reconstructed using noise-free simulated
data (figure 9.10, upper row). The computed conductivity values were slightly higher
( 0.22 S m
-1), with lower conductivities on the periphery. This is likely due to the fact
that the signals were computed using a tank of 19.5 cm diameter, and the sensitivity
matrix was computed for a tank of 20 cm diameter. Recalling from chapter 8 for stroke
imaging, an expansion in the assumed boundaries resulted in lower conductivity values
on the head surface.
On the day of measurements, a phase noise level of 3 m was estimated in the MK2a
system. In order to see how such a noise level may affect the images, this noise was
added to the simulated phase changes, and change in the image was insignificant (see
middle row).
Absolute imaging from the measured data was not successful, and the inverse solution
diverged (the residual started increasing) after a few iterations (lowest row). As can be
seen, the image at iteration 8 is totally degraded by the presence of artefacts which are
produced because of the noisy signals. The application of various levels of
regularization and types of regularization matrix (Identity, neighbouring) did not
improve the images.
235 Experiments
Figure 9.10. Absolute imaging of uniform tank of 0.205 S m
-1 conductivity. Transverse slides cutting
halfway up the tank. Rows from top to bottom: images reconstructed from noiseless simulated data,
simulated data with 3 m additive noise, and measured data, respectively. Colour-bar in S
m
-1.
9.6. Further discussion
The discrepancy between the simulated and measured data in, for example, figure 9.4c
can have several reasons. Apart from the possible discretization error in the simulated
data, the error in the measured data may be due to the phase noise in the MK2a system,
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Simulated +
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Measured
Target
236 Experiments
systematic errors, and the conductivity values. In this section, the phase changes for the
uniform tank are examined more closely. To this end, three arbitrary coil pairs (Ex1-
Se8, Ex2-Se9, Ex12-Se6) are chosen for the purpose of comparison between the
simulated and measured phase changes. Figure 9.11 shows the measured phase changes
plotted against the simulated ones, where each colour represents one pair of coils and 9
points for each colour correspond to the 9 planes of measurement.
As can be seen, for some coil pairs, the measured data is larger than the simulated,
whereas for other coil pairs the simulated data is larger. Furthermore, if one particular
coil pair is considered (e.g. the one represented by the blue colour), the measured data
for some planes of measurement is larger than the simulated one, while for other planes,
the simulated data is larger.
It is a common practice to consider a calibration factor for the measuring devices, where
the measured data are calibrated by, for example, dividing or multiplying them by a
factor. However, as discussed, coil pairs show an inconsistent behaviour for each plane
of measurement. Therefore, a single calibration factor cannot be applied to adjust the
measured signals. In the following, the effect of each potential source of error on the
signals is investigated separately.
Figure 9.11. Comparison of measured and simulated phase changes for uniform tank of 0.205 S m
-1
conductivity. Blue, red, and green colours represent Ex1-Se8, Ex2-Se9, and Ex12-Se6 coil pairs,
respectively. For each coil pair (each colour), 9 points are shown which correspond to 9 planes of
measurement for that particular coil pair. The line of identity is also drawn.
-2.5 -2 -1.5 -1 -0.5 0 0.5-2.5
-2
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-1
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0
0.5
Measured ( )
Sim
ula
ted
()
237 Experiments
9.6.1. Discretization error
Considering the grid size of 0.05 cm used for solving the forward problem, the
simulated signals are expected to have a good accuracy. Solving the FD model for a
smaller grid size was impractical as mentioned in section 9.3.
However, in order to further assure the accuracy of the FD results, the FE model was
solved for the uniform tank. Since the solution time in the FE model is long, the model
was solved for only one plane of measurement (middle plane, P5 in figure 9.3a). For a
fine mesh, comprising 65,000 second-order elements, the solution time was 6 hours.
The phase errors, difference between the FE and FD phase changes, are plotted in figure
9.12, where the y-axis limits is similar to that in figure 9.4c for the ease of comparison.
Note that the coil combinations in x-axis refer to the middle plane of measurement.
Figure 9.12. Discrepancy between the FD and FE phase changes for the uniform tank and middle
plane of measurement.
As can be seen, the phase errors are much smaller than those in figure 9.4c. The r.m.s.
phase error is 0.006 with a maximum magnitude of 0.01 . Therefore, the large
discrepancies between the measured and simulated data cannot be due to the
discretization error in the FD model.
700 750 800 850 900 950 1000 1050-0.5
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0
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Coil combination
Phase e
rror
()
238 Experiments
9.6.2. Phase noise
As mentioned previously, the phase noise in the hardware was estimated to be 3 m
during the measurements. This level of noise had no significant influence on the
reconstructed images in figure 9.10, middle row. In order to assess the effect of this
noise noise on the phase changes, the difference between the noise-free and noisy
simulated phase changes is plotted in figure 9.13. The discrepancy in the data is small
compared to those in figure 9.4c. The r.m.s. phase error (noise) in figure 9.13 is 0.003
with a maximum magnitude of 0.01 . This indicates that the large discrepancies
between the measured and simulated data cannot be due to the phase noise in the
hardware.
Figure 9.13. Discrepancy between the noise-free and noisy simulated phase changes for the uniform
tank.
9.6.3. Conductivity value error
The conductivity of the saline, measured at different times during the experiment, varied
from 0.2 to 0.21 S m-1
. This variation in the observed values may be due to the error in
the sampling equipment for which the accuracy was unknown. Note that a nominal
conductivity of 0.205 S m
-1 was considered and used for the forward simulations.
According to equation (2.12), for a fixed frequency and when the skin depth is larger
than the target dimensions, the phase change is linearly proportional to the conductivity.
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0
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Coil combination
Phase e
rror
()
239 Experiments
Revisiting figure 5.13, the imaginary part of the phase changes is linearly proportional
to the conductivity, regardless of the value of conductivity itself (in the range < 3 S m
-1
shown in the figure). Therefore, a slightly larger or smaller conductivity will affect the
imaginary component of all the phase changes by a constant factor.
In order to take out the possible conductivity errors from the measured data, the
conductivity was adjusted as follows:
A range of conductivity values between 0.19 and 0.25 S m
-1, with a step size of
0.001 S m
-1, was considered.
The phase changes for each conductivity were obtained by scaling the data set
computed for = 0.205 S m
-1.
The phase error or the discrepancy between the measured data and each of the
obtained data sets was computed as in figure 9.4c.
The r.m.s. phase error was calculated and is plotted in figure 9.14 as a function
of conductivity.
Figure 9.14. Values of r.m.s. for the phase error as a function of conductivity; phase error is
computed as the discrepancy between the measured and scaled simulated phase changes.
The adjusted conductivity was considered to be the value for which the minimum r.m.s.
in the phase error was obtained. This corresponded to a = 0.223 S m
-1.
0.19 0.2 0.21 0.22 0.23 0.24 0.250.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
Conductivity (S/m)
R.M
.S.
= 0.205 S m-1
r.m.s. = 0.0843 a = 0.223 S m-1
r.m.s. = 0.0613
240 Experiments
The phase errors, corresponding to a, are plotted in figure 9.15; this corresponded to
the difference between the measured and adjusted simulated data. In comparison to
those shown in figure 9.4c, the phase errors now have relatively smaller magnitudes
with a maximum of 0.28 (this was 0.44 in figure 9.4c). Nevertheless, there still exist
phase errors indicating that the remaining discrepancy between the simulated and
measured data may not be due to the phase noise in the hardware or the conductivity
error, but due to systematic errors.
Figure 9.15. Discrepancy between the measured phase changes for the uniform tank and the
simulated ones after the conductivity adjustment.
9.6.4. Systematic errors
Systematic errors can arise if the boundary (i.e. position and/or shape of the tank) was
not exactly as assumed in the model, or the tank was not exactly centred within the coil
array. Furthermore, the surface of the saline in the tank may have been in error by
perhaps a few millimetres due to inaccurate filling or movement during data acquisition.
This may contribute, for example, in producing the apparently the low conductivity in
slice 1 in figure 9.8 and figure 9.9.
The vertical alignment of the tank within the sensor array for different planes of
measurement was achieved by placing wooden spacer disks beneath the tank. A total
number of 16 disks of 1 cm thickness was used to obtain the required spacing between
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Coil combination
Phase e
rror
()
241 Experiments
the upper and lower planes of measurement. The height of all 16 disks together was
16.2 cm, again resulting in an error of a few millimetres. A further error of about the
same magnitude could occur during the horizontal placement of the tank within the
guide as shown in figure 9.2b.
In order to simulate an error in the tank alignment within the sensor array, the tank was
displaced in x- and y-directions for 2 mm. This corresponded to the tank being 2.8 mm
off-centre within the array. The forward problem was solved for the displaced tank and
the adjusted value of saline conductivity ( a = 0.223 S m
-1).
The phase errors, being the difference between the simulated phase changes for the
displaced tank and the tank at correct position, are plotted in figure 9.16. The r.m.s.
phase error is 0.033 with a maximum magnitude of 0.11 . As can be seen, these phase
errors are comparable with those in figure 9.15 in terms of the magnitude. Therefore,
the dis-positioning of the tank can be one of the likely contributing reasons for the large
discrepancy between the simulated and measured data.
Figure 9.16. Discrepancy between the simulated phase changes for the uniform with adjusted
conductivity when placed at the centre of sensor array and when displaced for 2.8 mm.
Other potential contributing factors may be due to the inaccurate geometry of the sensor
array itself. These may include the size, positioning, and orientation of the coils, and
also the geometry of the screen in the hardware.
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0
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Coil combination
Phase e
rror
()
242 Experiments
Since the data-collection hardware design was stable, with a relatively small phase noise
level of 3 m , the discrepancy between the measured and simulated data deemed to be
mainly due to a combination of the systematic errors.
The measurements reported here were the very first conducted on the MK2a data-
collection system. There is much scope for further investigation of the errors, and this is
suggested for future work in the last chapter.
End of chapter 9.
Chapter 10.
Conclusions and Future Work
10.1. Introduction
The feasibility of MIT imaging of cerebral haemorrhage was investigated through
numerical simulations. This chapter summarises the work carried out in preparation of
this thesis, and discusses the main conclusions drawn from the study. It also outlines
potential future work in a continuation of this study.
10.2. Remarks and conclusions
10.2.1. Chapter 1
The MIT imaging technique was introduced, and potential applications of MIT were
discussed. The motivation and objectives of this research were also provided.
10.2.2. Chapter 2
The physical principles of MIT imaging were described together with the governing
equations. The concept of the forward and inverse problems was introduced. The main
sub-systems of a practical MIT measurement device were outlined, and the working
principles were explained. The history of MIT studies for brain imaging, conducted by
different research groups, was also provided.
244 Conclusions and Future Work
10.2.3. Chapter 3
The anatomy of a healthy human head was briefly reviewed. Different types of stroke
pathology (ischaemic and haemorrhagic) were introduced, and further insight into the
motivation for this study was provided.
10.2.4. Chapter 4
A literature survey of the dielectric properties of the head tissues at different frequencies
was carried out. The most reliable dielectric property values available in the literature
were extracted to be employed for the simulations. The pathophysiology of the stroke
and its influence on the dielectric properties of the tissues were described, and the
applicability of MIT for stroke imaging was discussed. It was concluded that, for the
current range of operating frequencies, MIT imaging of haemorrhage may be feasible.
10.2.5. Chapter 5
Two of the most common numerical techniques, finite element (FE) and finite
difference (FD), were used for solving the discretized forward problem. The boundary
value problem for each technique was described, and its implementation for solving the
forward problem was explained.
The validity of the employed numerical techniques was examined by conducting
benchmark tests. These included performing convergence tests and comparison of the
numerical results against the analytical solutions for a simplified geometry (conducting
sphere). The numerical results were also compared with the measured practical data.
The results revealed that both methods can be reliably used for MIT simulations.
However, the application of the FD model is limited to the cases when the skin depth is
larger than the target dimensions. In terms of the solution time, the FD model performed
better and, for the examples provided, it was up to 150 times faster than the FE model.
10.2.6. Chapter 6
The discretized MIT inverse problem was outlined, and difficulties associated with
solving it were explained. The Tikhonov regularized method was initially examined for
245 Conclusions and Future Work
solving the inverse problem for geometrically simplified 3D models. Differential and
absolute images of an inclusion placed within a tank were reconstructed. Imaging
centrally located inclusions proved to be difficult, indicating that the MIT system is less
sensitive to the central regions.
The effect of the regularization matrices on the images was examined. Furthermore, the
regularization level, controlled by the regularization parameter, was found to have
profound influence on the accuracy of the inverse solution, and therefore quality of the
reconstructed images. Two automatic methods for the selection of the regularization
parameter were tested and it was shown that, for some of the presented examples, they
may result in under- or over-regularized solutions, so can be unreliable.
For non-linear iterative absolute imaging, an alternative scheme for the adjustment of
the regularization level between the iterations was employed. This included an initial
over-regularization of the inverse problem, followed by a gradual decrease in the
regularization parameter. For differential imaging, it was suggested that a subjective
assessment along with the use of automatic methods can be helpful for selecting a
proper level of regularization.
10.2.7. Chapter 7
An anatomically realistic head geometry, employed for the simulations, was described.
Three stroke lesions, of various sizes and sites within the brain, were considered. The
MIT forward problem was solved for a 16-channel annular MIT system (MK1). The
largest signals were produced by a peripherally located large stroke (50 ml), where the
smallest ones were due to a small lesion (7 ml), located deep within the brain. It was
also shown that more severe strokes (containing a higher percentage of blood within the
lesion), result in stronger MIT signals. For all strokes, the produced signals were small
(< 100 m phase difference) which indicated that sensitive and robust hardware is
required to measure them accurately.
In order to the make the simulations more realistic, phase noise at different levels was
added to the simulated data. Time- and frequency-differential imaging of the stroke
lesions was performed. For a phase noise level of 1 m , peripheral strokes could be
visualized in time-difference images. However, apart from the unrealistic noise-free
246 Conclusions and Future Work
cases, images of a deep stroke were degraded by the noise and the lesion was not
visible.
Frequency-difference imaging using the MK1 system was less successful, and only the
large peripheral stroke could be visualized as a left-right asymmetry in the images.
The amount of absorbed energy by the head was examined by computing the specific
abortion rate. It was concluded that, at present, MIT is a harmless and safe imaging
method regarding heat generation.
10.2.8. Chapter 8
Customised MIT arrays, hemispherical and annular, were investigated for stroke
imaging. For the particular arrays examined, the sensitivity matrix of the hemispherical
arrays were found to be slightly less ill-conditioned. Plots of the total sensitivity maps
revealed that for the coils placed sufficiently close to the head, a better sensitivity
distribution can be achieved, in particular, in upper parts of the head. This is because
hemispherical arrays conform more closely to the shape of the head than a conventional
annular array.
The imaging performance of the arrays was investigated by reconstructing frequency-
differential images. The true conductivity distribution of the target was assumed to be
known, and was used for adjusting the regularization level during image reconstruction.
This allowed achievement of the best possible images for each MIT array.
It was shown that making measurements in multiple planes with the annular array can
improve the reconstructed images by providing more independent data. On the other
hand, hemispherical arrays offers the possibility of using a larger number of coils,
allowing the measurements to be carried out without the need to move the head or the
array, and therefore avoiding further systematic errors.
Two types of systematic errors in the assumed a-priori boundaries of the head for
computing the sensitivity matrix were considered: displacement and scaling. It was
shown that displacement and scaling should remain less than 4 mm and 3%,
respectively. Otherwise, the artefacts produced by the errors may degrade the images.
However, the displacement errors were examined separately in the three coordinate
247 Conclusions and Future Work
directions. A simultaneous displacement in two or three directions, may lower the 4 mm
limit.
A scaling error in terms of contraction of the head boundaries produced misleading
artefacts of positive sign on the periphery, which could be mistaken for a genuine
feature due to a haemorrhage. Conversely, artefacts of negative sign were produced by
an expansion of the head which could not be mistaken as a haemorrhage.
The design study of the MK2b data-collection system was discussed. This included
examining different coil configurations and arrangements. A hemispherical array,
comprising 32 exciter coils and 14 sensor coils, was adopted as the final design based
on which the system is currently under construction.
10.2.9. Chapter 9
Practical MIT imaging using the measured data with the MK2a system was performed
for the very first time. While acceptable dime-differential images were reconstructed,
absolute imaging was less successful. The likely reasons for the observed discrepancy
between the simulated and measured MIT data were discussed. These included
discretization error, phase noise, conductivity error, and systematic errors. It was
concluded that systematic errors could be the main contributing reason to the large
discrepancies between the two data sets.
10.3. Discussion and future work
In this section, first the suggested future work that could be carried out as an immediate
follow up to this thesis is discussed. Next, potential research areas which could be
considered as long term investigations in order to improve MIT imaging in general are
described.
10.3.1. Immediate follow up research
a) In chapters 7 and 8, a single generic geometry was employed without variation
of the geometry for different subjects. It may be useful to perform the
248 Conclusions and Future Work
simulations for various subjects with different head shapes and sizes. This can
be done by extracting the head geometry from the MRI scans.
b) It is worth investigating the imaging of various types of stroke such as
subarachnoid or epidural haemorrhages.
c) More realistic simulations may also model the small substructures in the brain,
and take into account the effects of age and temperature.
d) Following the results in chapter 9 which were the very first from the MK2a
system, the electronics must be checked carefully to ensure the stability and
robustness of the hardware components.
e) The cylindrical phantoms and the measurement protocol (correct adjustment of
the phantom within the array and conductivity of the saline solution) should also
be improved in order to minimise the discrepancies between the simulated and
measured data.
f) Measurements can be repeated several times in order to acquire more data, and a
similar examination of the measured MIT signals as in chapter 9 could be
performed following the steps (d) and (e).
g) For stroke imaging, the construction of the MK2b system, designed based on the
findings in chapter 8, should be completed as the first step.
h) Based on the findings in chapter 8, it is of importance to accurately define the
boundaries of the eddy-current region (head surface) for computing the
sensitivity matrix. Therefore, immediate follow up work will include
examination of different shape measuring systems such as optical tools, and
utilizing photogrammetry software packages to determine the boundaries of the
head.
i) An accurate method must be developed for registration of the obtained boundary
in step (h) with the MK2b coil array so as to avoid systematic displacement
errors.
j) In order to validate the numerical feasibility tests, it would be timely to carry out
practical measurements using anatomically realistic phantoms representing the
head and using the MK2b system. The phantoms should be real head-shaped and
its constituents should have dielectric properties similar to those in the head
tissues. It is appreciated that it will be impractical to construct a phantom with
all the characteristics and complexity of a human head. However, it must be
noted that by being realistic, the phantoms should be multi-layer, representing
249 Conclusions and Future Work
the most important tissue types such as brain, CSF, and muscles. They might
also include real animal blood to simulate a haemorrhage. The dielectric
properties of all components of the phantoms must first be checked using
wideband impedance analyser (available at the University of Glamorgan).
k) If images, reconstructed from experiments in step (j), provide the required
confidence, an application for funding for a clinical pilot study on stroke should
be made.
10.3.2. Forward and inverse problems
Most numerical simulations of MIT, including this study, have neglected the effects of
wave propagation. When measuring small signals in biomedical MIT, propagation
delays might be significant and would appear at the sensor coil as a phase lag which
could be confused with the phase lag caused by the eddy-current signal. An analysis of
the error due to neglecting the wave propagation delays warrants a detailed numerical
investigation.
A long simulation time could jeopardise one major advantage of using MIT which is a
rapid diagnosis. On the other hand, the application of too many assumptions to simplify
the simulations could jeopardise the accuracy of the results. For the numerical
techniques and simulation environment, it would be invaluable to develop a dedicated
MIT software package which is:
comprehensive (implementing forward and inverse problems)
flexible (adjustable for particular imaging applications)
robust (employing improved iterative solvers)
time-efficient
The use of such a package would allow conducting the feasibility tests for stroke
imaging or any other application of interest.
Future research in the solution of the inverse problem may include exploring more
efficient ways of image reconstruction and, in particular, developing automatic methods
for selection of the regularization level.
250 Conclusions and Future Work
One possibility to improve the image quality for monitoring purposes, could be the use
of an individualised conductivity distribution of the head before stroke. The internal
structure of the head can be mapped from MRI or CT scans, and then incorporated into
the reconstruction algorithm (e.g. through the reference conductivity vector in
generalized the Tikhonov technique in equation (6.28)). For those tissues outside the
likely regions for bleeding (e.g. bones and muscles), the conductivity change can be
forced to be zero. This can be done by modifying the regularization matrix to highly
penalize the conductivity change for these tissues (opposite to what is done to preserve
the edges in equation (6.41)).
Even if it is impractical to map each individual's head before the application of MIT
monitoring, the use of a generic map obtained from a subject with the same head size
may be useful.
10.3.3. Practical measurements and systematic errors
With an achievable phase noise of 1 m and a low phase drift [206], it seems that the
current generation of practical measuring systems should be accurate and stable enough
for imaging peripherally located haemorrhages. Systematic errors will occur during the
data acquisition process and may include:
inaccuracies in the assumed a-priori boundaries of the eddy-current region (e.g.
size-scaling)
inaccurate registration (misalignment) of the assumed boundaries to the sensor
array (displacement and rotation)
movement of the target during measurements (displacement and rotation)
While the effect of size-scaling and displacement errors on the images was examined in
this study, further research may include investigating the patient's movement, which has
previously been reported only for simplified geometries [220].
Another source of systematic errors is the inaccuracies in geometry of the sensor array
itself; e.g. the axes of the coils in the hemispherical or annular arrays are not exactly
directed towards the centre of array. One may consider this as a fault in the hardware, or
an error in the model which does not simulate the hardware exactly as it is. Either way,
this will result in discrepancy between the predicted and measured signals. One method
251 Conclusions and Future Work
for correcting these error may be to examine the primary signals (no target present),
which depend only on the geometry of sensor array and driving parameters (current and
frequency). This may allow estimation of correct coordinates and orientations of the
coils.
10.3.4. Computational vs. Measurement Accuracy
Ideally, we would have liked to have shown that the simulations are at least as accurate
as the measurement system and data, if not more so. However, the question of accuracy
and how it is measured poses a number of interesting points.
Firstly, the accuracy of a numerical model is not governed solely by the numerical
discretization. The accuracy of a numerical simulation is in fact governed by a number
of different factors including the approximation of the mathematical model to the
physical phenomena, uncertainty in the geometry, boundary conditions and materials,
the fidelity of the numerical discretization, and the magnitude of the tolerance used in
the iterative solvers.
Secondly, an interesting question arises as to how the accuracy should be measured. In
the simulations conducted in this study, various outputs and fields were computed that
vary with position in space. When analytical solutions were available, the accuracy was
measured by computing the norm of the error in a given quantity, as was done in the
case of the convergence study presented in figure 5.4. However, analytical solutions
were not available for the complex geometries such as the head model, making such
studies impossible. On the other hand, experimental data was available for a limited
number of realistic case studies that allowed the comparison with computed quantities
at specific locations in the model, such as the comparison of the computed and
measured voltage in the sensor coils shown in figure 5.13. The problem with such
comparisons is that although the experimental data may show a close agreement with
the computed voltage, this may not necessarily mean that the other fields elsewhere in
the domain have the same level of accuracy.
The derivation of the sensitivity matrix shown in chapter 6 required that the
conductivity changes were small. The comparisons shown in figure 6.5 indicated that
the reciprocity based sensitivity can accurately predict the signal if the conductivity
252 Conclusions and Future Work
changes are small. However, for changes in the background conductivity which
exceeded 10%, the reciprocity based sensitivity became inaccurate.
Thirdly, there is a considerable level of uncertainty in the measured data obtained from
practical measurement system. For example, one known uncertainty is the SNR. In
reality, however, the overall uncertainty could be much worse than 1 m (1.74 10
-5
radians) suggested by the SNR alone due to the existence of various systematic errors
(see chapter 9). Also, the exact positioning and geometry of the coils may in practice be
slightly different to that set out in the system design and, therefore, lead to uncertainties
in the measurement system. Thus, unless the magnitude of these uncertainties is
known, it could lead to the false conclusions that the simulated model was not as
accurate as the measurement system.
Nevertheless, there is strong evidence to support the claim that the numerical
approximations of the MIT forward and inverse problems were accurate. This was
illustrated by the closeness of the agreement between the numerical and analytical
results presented in figure 5.4, and the agreement between the numerical and practical
results illustrated in figure 5.13.
For instance, the least squares fit in figure 5.13 showed that the difference between the
FE and the measured results was 0.052%, the difference between the FD and measured
results was 0.055%, whereas the allowable SNR translates to 1.74 10-3
%. An initial
observation of these results might lead to the conclusion that the results are not as
accurate as the SNR. However, we reject this conclusion on the basis that there are a
large number of uncertainties that are associated with both the simulations and the
measurements, and that for the discretizations chosen. These uncertainties dominate
over the numerical discretization error. That is to say, that for a finer discretization
(small meshing spacing and/or the use of higher order shape functions), we would not
expect the quoted differences to reduce substantially.
The comparisons presented in chapter 5 were used as a basis for choosing the
discretization for the complicated head model in later chapters. Clearly for a new
problem, the discretization required to achieve a specified level of accuracy may be
different to that required by the examples where the analytical solutions and
experimental data were available. However, for cases where no such data is available, a-
253 Conclusions and Future Work
posteriori error estimation could be used to evaluate the error in the FE discretization
and, subsequently, adaptively refine the discretization.
The topic of a-posteriori error estimation is widely established in many fields, and a
large number of estimation techniques have been developed (see e.g. [221]). An
interesting class of error estimation is that of goal orientated error estimation and
adaptivity. In this technique, the numerical discretization is adaptively refined to reduce
the error in a chosen output of interest. Such techniques have been successfully applied
to the reduction of the error associated with the scattering width output of Maxwell‟s
equations [222]. A wide range of other error estimators exist for the Maxwell problem;
e.g. the a-posteriori error estimation technique developed by Schöberl [223]. Other error
estimators have also been applied and developed in the context of the eddy-current
problems (see e.g. [224]).
To the author's knowledge, a-posteriori error estimation and adaptivity have not yet
been applied to the problems in MIT. However, an adaptive meshing approach has been
developed by Molinari [225] for EIT applications. An interesting extension of the
current project would be to incorporate a goal orientated error estimation and adaptivity
procedure for refining the discretization so as to reduce the error in the chosen outputs
of interest. In the context of MIT problem, this could be reduction in the error
associated with the voltage in the forward problem, or reduction of the error in the
sensitivity matrix in the inverse problem. Such error estimator could then be used to
subsequently adapt the discretization of the both forward and inverse problems [175].
A-posteriori error estimation can be seen as one aspect of the uncertainty quantification.
Further techniques of the uncertainty quantification could also be explored for
quantifying the uncertainties that are not associated with the numerical discretization;
e.g. mathematical modelling and measurement errors (for further details see [226]).
10.3.5. Sensor array design
Although it is stated that the MK2b system is optimized for stroke imaging, in the
author's opinion, care must be taken when using the term optimization. An optimized
design for a system is a design that offers the best possible performance regarding all
different aspects of that system. In an MIT device, for instance, this will include
hardware components comprising the electronic circuitry and sensor array, host
254 Conclusions and Future Work
computing unit and its software components comprising the image reconstruction
algorithms, and the data acquisition protocol.
Even optimizing the sensor array alone, will require exploration of all different
possibilities for the geometrical design of its sub-systems. In this study, only annular
and hemispherical arrays were examined. A sensor array comprising coils of different
sizes, with an irregular spatial positioning may generate better eddy-current patterns
throughout the head. Obviously, the possibilities can be limited by the practical
implementations.
Therefore, as a follow up to the work undertaken in this study, it might be worth
investigating the effect of a larger variety of coil configurations (size, location, and
orientation of the coils), targeting specific regions within the imaging domain. For
example, imaging of meningeal haemorrhages (subarachnoid or epidural) may be
realised by employing small sized coils located close to the interrogating region.
10.3.6. Image resolution
MIT inherently suffers from poor spatial resolution. This is due to insufficient
independent information and, more importantly, the fact that the underlying continuum
inverse problem is exponentially ill-posed. One may identify the low resolution of the
MIT imaging technique as a weakness. In this respect, it must be noted that the
motivation for using MIT is somewhat different from that of the conventional imaging
modalities such as MRI. Despite its limited resolution, MIT is employed to provide a
reliable, portable, and low-cost imaging tool. Furthermore, MIT is measuring different
physical parameters from those measured in CT or MRI, namely conductivity and
permittivity, that may even give an earlier indication of disease in some cases. In this
context, the reconstructed MIT images can be considered as a source of information
aiming to answer certain questions; in the case of stroke imaging, the presence of the
bleeding and its approximate location and volume.
However, in order to improve the resolution, a hybrid approach combining MIT with
ultrasound has been proposed by the South-Wales MIT group. This potentially provides
similar electrical conductivity information to MIT but with the spatial resolution of
ultrasound (in principle sub-mm).
255 Conclusions and Future Work
The ultrasound wave causes the charged ions within the tissue to move in presence of a
strong static magnetic field. A moving charge is subject to the Lorentz force. Positive
and negative ions move in opposite directions and this acts as the source of current. The
electric current density generated depends on the velocity of the ions (ultrasound wave),
static magnetic field strength, and density and mobility of the ions (conductivity). If the
induced current can be measured by some means outside of the object, the internal
conductivity distribution may be reconstructed.
One approach is to sense the magnetic field produced by the currents using a sensor
coil. This should provide the advantages of ultrasound, in terms of spatial resolution,
but with the additional conductivity information overlaid on the ultrasound images. The
proposed hybrid approach could be sensitive just to the small region of tissue excited by
the ultrasound pulse and, therefore, may be a much better posed problem than MIT.
The numerical techniques employed in this study could be used as a basis for
preliminary tests. The Lorentz force current multi-poles produced by the ultrasound
pulse and also static magnetic field throughout the domain of interest may be computed
by the addition of extra terms to the boundary value problems presented in this thesis.
Therefore, one interesting follow up study could be to examine the feasibility of
employing this novel approach.
End of chapter 10.
Part V
Appendices and Bibliography
Appendix A. System Description
A.1. MK1 system
The MK1 data-collection system consists of a circular array of 16 coil modules
(channels), switchable between exciter and sensor modes (figure a.1). The object to be
imaged is placed inside the coil array. The array is enclosed within an aluminium
electromagnetic screen which is 350 mm in diameter and 250 mm in height. The coils
are wound on Perspex formers of 50 mm diameter, each of 2 turns, mounted inside the
electromagnetic screen. The standoff distance of the coils from the screen being 30 mm
for the exciters and 40 mm for the sensors (figure A.1). The coils are placed halfway up
the screen, 125 mm height from the base. Sixteen corresponding exciter and sensor
circuit modules are housed in metal enclosures attached to the outside of the screen. The
electronic components have been described in more detail elsewhere [33].
Figure A.1. An image of frond-end of the 16-channel MK1 system [33].
258 Appendix A. System Description
Figure A.2. Diagram of the MK1 configuration; dimensions in mm: (a) the coil array inside the
electromagnetic screen; labels represent the coil numbering, (b) detail of an individual coil module;
the coil windings are shown in black (sensor) and red (exciter).
Activating the exciter coils in turn and measuring the signals in the sensors, provides
256 coil combinations. Numbering of the coils is shown in figure A.2a. Throughout this
thesis, the exciter and sensor coils will be referred to as Ex and Se; e.g. exciter coil 1 is
labelled as Ex1.
The exciter coil current is 141 mA in amplitude (100 mA r.m.s.). The system operates at
a frequency of 10 MHz. The averaged phase noise across all channel combinations is 17
m . For simplicity, the simulations described in this thesis have all assumed coils of just
1 turn and an excitation current of 1 A in amplitude. This simplification has no effect on
the phase changes; the only implication is for the heat generation in the target (head),
which is discussed in chapter 7.
A.2. MK2a system
The MK2a data-collection system consists of two circular arrays of 14 coil modules,
one of exciters and the other of sensors (figure A.3). The arrays are enclosed within an
aluminium electromagnetic screen, which is 350 mm in diameter and 300 mm in height.
The centres of the exciter and sensor coils lie on planes 40 mm above and below the
centre of the screen (figure A.4a). The coils are wound on Perspex formers of 44 mm
diameter, each of 2 turns, mounted inside the electromagnetic screen. The standoff
distance of the coils from the screen is 40 mm for all coils (figure A.4b).
(a) (b)
13 5
9
1
259 Appendix A. System Description
The MK2a system is capable of operating at multiple frequencies in the range of 0.5-14
MHz. The system achieves a measurement phase noise of less than 1 m for a time
constant of 1s for frequencies in the range of 1-14 MHz [206]. Similar to MK1, the
simulation for the MK2a system considered coils of 1 turn. The MK2a system provides
196 coil combinations for each set of the measurements, i.e. one plane of measurement.
Figure A.3. An image of frond-end of the 14-channel MK2a system.
Figure A.4. Geometrical specifications of the system; dimensions in mm: (a) exciter and sensor coils
sit on red and blue broken lines, respectively, (b) details of an individual module.
End of Appendix A.
350
300
190
110
40
44
(b) (a)
Appendix B. Dielectric Theory and
Terminology
In a polarisable material such as biological tissue, the polarization does not respond
instantaneously to an applied electric field as the rates at which polarizations can occur
are limited. Therefore, as the frequency of the applied electric field is increased, some
polarizations will no longer be able to attain their d.c. or low frequency values.
There are a number of different dielectric mechanisms, connected to the way a medium
reacts to the applied field. Each dielectric mechanism is centred around its characteristic
frequency, which is the reciprocal of the characteristic time1 of the process. According
to Pethig [227], the slowest polarization mechanism is often the dipolar orientation of
macromolecules and this is usually the first polarization term to disappear as the
frequency is increased. The fall of the polarizability, with consequent reduction of
permittivity and the occurrence of energy absorption, is referred to dielectric relaxation
or dispersion [227].
The frequency at which this fall in dipole polarization occurs can vary from very low
frequencies of the order 1 Hz for large macromolecules to frequencies up to 1012
Hz for
small molecules. Atomic and electronic polarizations occur at frequency comparable
with the natural frequencies of vibration of the atoms in a molecule ( 1014
Hz) and
corresponding to electronic transitions between different energy levels in the atom
( 1015
Hz), respectively.
1 Characteristic or relaxation time refers to the time needed for a system to relax under external stimuli.
261 Appendix B. Dielectric Theory and Terminology
In order to derive a mathematical model for the relaxation process, the constitutive
relationship (2.6) can be re-written as:
(B.1)
where the polarization P is defined as the induced dipole moment per unit volume. The
electric field and polarization are related by the dielectric susceptibility, , as P = 0 E.
The polarization P is composed of two parts P1 and P2, where P1 arises from atomic and
electronic polarizations and P2 arises from the much slower process of dipolar
reorientation. For the frequency range of interest in MIT applications (< 10
8 Hz), it can
be assumed that P1 responds instantly to the applied field E, and has the constant value
P1 = 1E. The part P2 lags behind E in such a way that at any moment, P2 approaches its
final value 2E at a rate of:
(B.2)
where is the characteristic relaxation time. This shows that the polarization approaches
its final value, in response or relaxation mode, exponentially with the time constant .
By integration of (B.2), the total polarization for a harmonic field can be expressed in
the frequency domain as [227]:
(B.3)
From (B.1) and (B.3), the permittivity can be treated as a complex function of the
frequency of the applied field:
(B.4)
where is the permittivity measured at a sufficiently high frequency for the dipolar
polarization to have been disappeared and s is the limiting low frequency permittivity.
The real and imaginary parts of the permittivity are given by:
262 Appendix B. Dielectric Theory and Terminology
(B.5)
Here, ' is related to the stored energy (a measure of the induced polarization per unit
field) and '' is related to the dissipation of energy within the medium (out-of-phase loss
factor). These equations are commonly known as the Debye dispersion formulas which
describe the relaxation for ideal systems in terms of permittivity as a function of
frequency; it is assumed that there is only a single orientational process and the
equilibrium is attained exponentially with time. It is often useful to define the complex
conductivity as in-phase or lossy ( ') and capacitive ( '') components:
(B.6)
where dc is the static ionic conductivity and ac is the conductivity increase due to the
dispersion. The total conductivity is thus made of two terms corresponding to the
residual static conductivity and polarization losses. Figure B.1 illustrates the idealised
variation in the permittivity, loss factor, and conductivity with frequency for a Debye
single dispersion relaxation, plotted based on the equations (B.5) and (B.6). Note that
the relaxation time corresponds to a relaxation frequency fr = 1/2 . At the relaxation
frequency, the permittivity is halfway between its limiting values and the loss factor at
its highest.
Figure B.1. Normalized permittivity ( ' - )/( s - ), loss factor ''/( s - ), and conductivity
0 ''/( s - ) for a single time constant relaxation plotted against f/fr (after Barnes and Greenebaum [91]).
263 Appendix B. Dielectric Theory and Terminology
Biological tissue is a heterogeneous material containing water, dissolved organic
molecules, macromolecules, ions, and insoluble matter. The presence of ions plays an
important role in the interaction with an electric field, providing means for ionic
conduction and polarization effects. Ionic charge drift creates conduction currents and
also initiates polarization mechanisms through charge accumulation at structural
interfaces, which occur at various organizational levels. Their dielectric properties will
thus reflect contributions to the polarization from both structure and composition.
The complexity of both the structure and the composition of biological material is such
that each dispersion region may be broadened by multiple contributions to it, and
biological materials exhibit a certain asymmetrical distribution of relaxation times. The
broadening of the dispersion can be compensated empirically by introducing a
distribution parameter. This results in an alternative to the Debye equation known as
Cole-Cole equation [91]:
(B.7)
where the distribution parameter, , is a measure of the broadening of the dispersion.
Furthermore, biological materials rarely show a single time constant Debye response.
Therefore, the complex permittivity is usually a complicated function of frequency; a
superimposed description of dispersion phenomena occurring at multiple frequencies.
Dispersion data in a biological material is based on the electrical examination of the
material as a function of frequency, i.e. dielectric spectroscopy. The relaxation
mechanisms can be divided in three groups of -, -, and -dispersions; each relating to
cell membranes, organelles inside cells, double layer counter-ion relaxation,
electrokinetic effects, etc [113]:
The - or low-frequency dispersion (typically below 1 kHz) is characterized by
very high permittivity values and can be ascribed, at least partially, to counter-
ion diffusion effects. Other mechanisms, e.g. interactions in the vicinity of the
cell membrane, may contribute to -dispersion.
The -dispersion occurs at intermediate frequencies (in the hundreds of kilohertz
region) and originates mostly from the polarisation of the cellular membranes
and those of membrane-bound intra-cellular bodies. Cellular membranes act as
barriers to the flow of ions between the intra- and extra-cellular media. Any
264 Appendix B. Dielectric Theory and Terminology
damage to the cell membrane changes the features of the -dispersion. Other
contributions to the -dispersion come from the polarization of protein and other
organic macromolecules.
The -dispersion is due to the dipolar polarization of tissue water at frequencies
in excess of a few hundred MHz, where the response of tissue water is
dominant.
Figure B.2. Dispersion regions , , and (after Grimnes and Martinsen [113]).
Figure B.2 shows that the permittivity in biological materials typically diminishes with
increasing frequency. The dispersion regions are here shown as originating from clearly
separated Cole–Cole dispersions. The dielectric spectrum of a biological tissue may be
expressed as the summation of multiple (n) Cole–Cole dispersions with the addition of
the static conductivity term [91]:
(B.8)
Where adequate experimental data is available, and with a choice of parameters
appropriate to each tissue, equation (B.8) can be used to predict the dielectric behaviour
over the desired frequency range.
End of Appendix B.
Appendix C. Supplements to chapter 6
C.1. GCV method
The GCV method works in the basis of 'leave-one-out' cross validation [176]. It is based
on the philosophy that if an arbitrary element bi of the vector b is left out, then the
corresponding regularized solution should predict this observation well, and the choice
of regularization parameter should be independent of an orthogonal transformation of b.
In the context of MIT, the models are obtained by leaving one of the m data points,
corresponding to one coil combination, from the fitting process in the LS solution.
Consider the modified Tikhonov regularization problem at which a data point bk is
ignored from the measurement vector b:
(C.1)
The solution to this problem is called xk, which would accurately predict the missing
data bk. A regularization parameter, , is sought to minimize the predictive errors for all
k:
(C.2)
Computing G( ) involves solving m problem of this form. In order to simplify this
computation, consider the definition:
266 Appendix C. Supplements to chapter 6
(C.3)
The solution xk
, therefore, solves:
(C.4)
This is known as the 'leave-one-out lemma', which gives:
(C.5)
where is B defined in equation (6.31). It is straightforward to derive:
(C.6)
where C = AB . Subtracting both sides of the equation from 1 gives:
(C.7)
Considering:
(C.8)
the equation (C.7) can then be simplified as:
(C.9)
By substituting (C.9) into (C.2) , the function can be expressed by:
(C.10)
The equation (C.10) can be further simplified by replacing (C )k,k in the sum with the
average value 1/m trace(C ). Therefore, the GCV methods aims at minimizing:
(C.11)
which will be minimised at the same as that in (6.34). For further details see [176].
267 Appendix C. Supplements to chapter 6
C.2. Steepest descent
The steepest descent method is generally used to find the local minimum of a nonlinear
function using its gradient. The search direction used in this method is the negative of
the gradient vector at every iteration point. It requires an initial estimated solution 0. In
the searching process, at the k-th iteration, k is replaced by k+1, which is a better
estimate of the solution.
The name of steepest descent method is earned because it uses the steepest descent
direction, - f( ), as the search direction, when the goal is to minimize f( ) = ||A -
Vm||2. If A is available, then we have:
(C.12)
Therefore, moving in the direction opposite to the gradient of f( ), the iteration
procedure can be written as:
(C.13)
where the positive scalar αk is the optimal step-length [190]. At each iteration, the
search direction and the optimal step-length in the search direction are updated. The
detailed procedure for finding the optimum step-length can be found in [166,170,228].
C.3. Landweber method
If the step-length in the steepest descent method is chosen to be fixed, then one obtains
the Landweber method expressed by [180]:
(C.14)
The Landweber method may require many iteration steps before reaching a minimum
point. It is an easy to implement algorithm, and has been employed for MIT imaging
(e.g. [229]). The modified types of the Landweber method can be found in [230].
268 Appendix C. Supplements to chapter 6
C.4. Conjugate gradient method
A reduction of the iteration steps may be achieved for the steepest descent method, if
instead of (C.13), one uses an iteration procedure, where the search direction is different
from the gradient direction. In the conjugate gradient method, one can obtain a set of
linearly independent conjugate gradient directions at each iteration from an
orthogonalization of the successive gradients [166]. For quadratic functions, the
conjugate gradient method has been proved to converge to the unique global minimum,
by moving along such successive non-interfering directions [228]. Implementation of
the conjugate gradient method has been suggested for EIT technique [173]. However, its
convergence, when applied to non-linear ill-posed problems has not been proved [230].
C.5. ART and SIRT
The algebraic reconstruction technique (ART) and simultaneous iterative reconstruction
technique (SIRT) are the variants of the Kaczmarz‟s algorithm [176]. These algorithms
were originally developed for tomographic applications, and have been applied to MIT
technique (e.g. [35]). ART is a version of Kaczmarz‟s algorithm that has been modified
especially for the tomographic reconstruction problem. It uses only one set of projection
data in each iteration step. The ART iterative formula for solving the system A = Vm
is:
(C.15)
where ak+1 is the (k+1)-th row vector of the sensitivity matrix A, and Vk+1 is the (k+1)-th
element of the vector voltage Vm. Because only one set of measurement data is used in
each iteration step, ART may not converge if the measurement contains a significant
error [190]. SIRT is a variation on ART which overcomes this disadvantage. Assuming
that the sensitivity matrix is composed of m rows, then m vectors can be obtained
according to equation (C.15) using ART. The conductivity is updated using the average
of those m vectors and the iterative formula of the SIRT algorithm is:
(C.16)
269 Appendix C. Supplements to chapter 6
where is the relaxation factor, and diag(AAT) is a vector composed of diagonal
components of AAT where the division means that each numerator is divided by the
corresponding denominator. If necessary, can be replaced by a vector; this means that
the m vectors based on ART are not treated equally when trying to obtain the average
vector [190]. SIRT can give slightly better images than ART, at the expense of being a
slightly slower algorithm [176]. Note that SIRT formula in (C.16) is similar to the
Landweber method in (C.14) with a re-scaled matrix.
C.6. Noise
Throughout this study, Gaussian pseudo-random noise is used for the simulations.
When examining the tolerable noise level for the image reconstruction, several draws of
randomly generated noise, but with the same standard deviation, were employed. The
following is a MATLAB function for generating noise based on equation (2.13):
function Vnoisy = addNoise(V,SNRdB0) % SNRdB0 is the noise level in dB % V is the column vector noise-free voltage % randn(n,1) : returns n values with Standard deviation=1
n = length(V); Vpower = V'*V/n; % or Vpower = mean(V.^2); SNR0 = 10^(SNRdB0/10); Npower0 = Vpower/SNR0; Noise = sqrt(Npower0)*randn(n,1); Vnoisy = V + Noise;
If the data set is in terms of phase changes (i.e. voltage normalized to the primary
signal), the noise is generated with a specified standard deviation in milli-degree:
function PHInoisy = addNoise(PHI,SNR0) % SNR0 is the noise level in milli-degree % PHI is the column vector noise-free phase change % randn(n,1) : returns n values with Standard deviation=1
n = length(PHI);
PHInoisy = PHI + SNR0*(pi/180000)*randn(n,1);
End of Appendix C.
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