Perturbation Method for Magnetic Field Calculations ofNonconductive Objects

Mark Jenkinson,* James L. Wilson, and Peter Jezzard

Inhomogeneous magnetic fields produce artifacts in MR im-ages including signal dropout and spatial distortion. A novelperturbative method for calculating the magnetic field to firstorder (error is second order) within and around nonconductingobjects is presented. The perturbation parameter is the suscep-tibility difference between the object and its surroundings (forexample, �10 ppm in the case of brain tissue and air). Thismethod is advantageous as it is sufficiently accurate for mostpurposes, can be implemented as a simple convolution with avoxel-based object model, and is linear. Furthermore, themethod is simple to use and can quickly calculate the field forany orientation of an object using a set of precalculated basisimages. Magn Reson Med 52:471–477, 2004. © 2004 Wiley-Liss, Inc.

Key words: susceptibility; distortion; simulation; field calcula-tion

Theoretical calculation of the magnetic B field, given adistribution of tissue, allows modeling of various phenom-ena, such as MRI signal dropout, geometric distortion,interaction of B field and motion effects, manipulation ofthe B field using active and weakly magnetic passiveshims, and respiration effects. Existing methods for calcu-lating the B field use full finite element calculations (1,2)or approximate solutions to Maxwell’s equations giveneither surface models of matter interfaces (3–5), voxel-based elements (6), spherical elements (7,8), or Fourierrepresentations (9). By using a perturbation approach tosolving Maxwell’s equations (10), a linear first-order solu-tion can be found which is fast and appropriate for mostMR imaging applications. In addition, the perturbationmethod allows the magnitude of the errors to be calcu-lated, and hence the accuracy and appropriateness of themethod to be estimated for various applications.

THEORY

Assuming the object is nonconductive (so J � 0), the rel-evant Maxwell’s equations (ƒ � H � 0 and ƒ � B � 0) canbe reduced to a single equation by using the magneticscalar potential (11): H � ƒ�. This gives:

�0ƒ � ��1 � ��ƒ�� � 0, [1]

where � is the tissue dependent susceptibility.Let the susceptibility, �, be expanded as:

� � �0 � �1, [2]

where �0 is the susceptibility of air (4 � 107) and is aconstant, equal to the difference in the susceptibility of thetissue under consideration and air (e.g., 9.5 � 106 in thecase of brain tissue and air). Note that �1 is a “scaled”version of the susceptibility difference and can take oncontinuous values when using a complex tissue model,but will take values of 0 or 1 for a simple two-tissue model.

Similarly, expand � in a series:

� � �0 � �1 � 2�2 � . . . [3]

This perturbation expansion in can be substituted backinto Eq. 1 to give:

�0�1 � �0�ƒ2�0 � 0 [4]

�1 � �0�ƒ2�1 � ƒ � ��1ƒ�0� � 0, [5]

for the zeroth and first-order terms in .The first-order equation is a 3D Poisson equation:

ƒ2�1 �1

1 � �0�ƒ � ��1ƒ�0�� [6]

with solution:

�1�x� � ���G�x � x��f�x��dx�, [7]

where G(x) � (4�r)1 is the Green’s function, r � �x�

� �x2 � y2 � z2 and f �1

1 � �0�ƒ � ��1ƒ�0��.

This convolution can also be more concisely written as�1 � G � f.

The z-component of the B field is given by:

Bz � �Hz � � �

z

� �0�1 � �0� �0

z

� �0 ��1

�0

z� �1 � �0�

�1

z � � O�2�. [8]

Oxford Centre for Functional Magnetic Resonance Imaging of the Brain(FMRIB), University of Oxford, Department of Clinical Neurology, John Rad-cliffe Hospital, Headington, Oxford OX3 9DU, UK.Grant sponsors: UK MRC, GlaxoSmithKline, UK EPSRC.*Correspondence To: M. Jenkinson, Oxford Centre for Functional MagneticResonance Imaging of the Brain (FMRIB), University of Oxford, Department ofClinical Neurology, John Radcliffe Hospital, Headington, Oxford OX3 9DU,UK. E-mail: mark@fmrib.ox.ac.ukReceived 3 November 2003; revised 17 March 2004; accepted 3 April 2004DOI 10.1002/mrm.20194Published online in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 52:471–477 (2004)

© 2004 Wiley-Liss, Inc. 471

The zeroth-order term is Bz(0) � �0(1 � �0) �0/ z, and the

first-order term is:

Bz�1� �

�1

1 � �0Bz

�0� � �0�1 � �0� �1

z. [9]

Using the fact that:

x�G � f� � G �

f x

� G x

� f

holds for any G and f, together with Eqs. 4, 5, 7 and 9,gives:

Bz�1� �

�1

1 � �0Bz

�0� �1

1 � �0�� 2G

x z� � ��1Bx�0��

� � 2G y z� � ��1By

�0�� � � 2G z2� � ��1Bz

�0��� . [10]

Lorentz Correction

The solution derived above is valid for a continuous me-dium but not for a medium composed of discrete particles.For MR calculations, however, it is the field that is appliedexternally to the discrete nuclei that is of interest. Thisfield can be calculated from the continuous media solutionusing the Lorentz Correction (11,12) (or local field repre-sentation). The corrected field is given by BLC � B

�23

�0M, where M � �H is the magnetization of the

material. Therefore, the Lorentz Corrected field can beexpressed in terms of the scalar potential, � (the continu-ous media solution) as BLC � �0(1 � �/3)ƒ�. Comparingthis with the uncorrected (continuous media) solution,B � �0(1 � �)ƒ�, shows that the Lorentz Corrected solu-tion can be found simply by replacing all instances of �with �/3 in equations involving only � and � terms.

Consequently, the zeroth and first-order corrected fieldscan be written as:

BLC�0� � �0�1 �

13

�0�ƒ�0 [11]

BLC�1� �

13

�0�1ƒ�0 � �0�1 �13

�0�ƒ�1 [12]

which, together with Eq. 10 gives:

BLC,z�1� �

�1

3 � �0BLC,z

�0� �1

1 � �0�� 2G

x z� � ��1BLC,x�0� �

� � 2G y z� � ��1BLC,y

�0� � � � 2G z2� � ��1BLC,z

�0� �� . [13]

For the remainder of this article only the Lorentz Cor-rected fields will be used and the LC subscript will bedropped.

Single Voxel Solution

Equation 13 allows the first-order Bz(1) field to be calcu-

lated if the zeroth-order field B(0) and the susceptibilitydistribution �1 are known. The zeroth-order field repre-sents the field which would be present if there were noobject in the scanner. For example, with a constant field inthe z direction, Bx

(0) � By(0) � 0 and Bz

(0) � B0, so that Eq.13 simplifies considerably.

The susceptibility distribution represents the object inthe scanner and needs to be specified at each point inspace. For complicated functions, though, the requiredconvolutions are difficult, if not impossible, to do analyt-ically. For most purposes, however, it is sufficient to ap-proximate the object using small rectangular volume ele-ments (voxels). The advantage of this is that the convolu-tion can be done analytically for a single voxel.

Consider a single voxel of dimensions (a, b, c) and,without loss of generality, let it be centered at the origin (0,0, 0) with a susceptibility of �1 � 1 within the voxel and�1 � 0 outside the voxel. Given this, the required convo-lutions in Eq. 13 can be written as:

� 2G v z� � ��1Bv

�0��

� ����1�x � x��Bv�0��x � x��

2G v z

�x��dx�

� �xa/2

x�a/2

dx��yb/2

y�b/2

dy��zc/2

z�c/2

dz� 2G v z

�x��Bv�0�, [14]

where v stands for either x, y, or z and it is assumed thatBv

(0) is not spatially varying.The last integral can be easily calculated from the indef-

inite integral, which we will denote here as F (x). That is:

F�x� ���� 2G v z

�x�Bv�0�dx [15]

giving the single voxel solution as:

Hv,z�x� � � 2G v z� � ��1Bv

�0�� � F�x �a2, y �

b2, z �

c2� � F�x

�a2, y �

b2, z �

c2� � F�x �

a2, y �

b2, z �

c2� � F�x �

a2

, y

�b2, z �

c2� � F�x �

a2, y �

b2, z �

c2� � F�x �

a2, y �

b2

, z

�c2� � F�x �

a2, y �

b2, z �

c2� � F�x �

a2, y �

b2

, z �c2�,

where the following specific solutions can be substituted:For a constant (normalized) field along the z-axis:

B(0)(x) � (0, 0, 1):

472 Jenkinson et al.

F�x� ���� 2G z2 dxdydz

�1

4�atan �xy

zr� . [16]

For a constant (normalized) field along the x-axis: B(0)(x) �(1, 0, 0):

F�x� ���� 2G x z

dxdydz

�14�

sinh1� y

�x2 � z2� . [17]

For a constant (normalized) field along the y-axis: B(0)(x) �(0, 1, 0):

F(x) is the same as for the x-axis case except all x and yterms are swapped.

Combining Voxels

Due to the linearity of Eq. 13 the single voxel solutions canbe added together to give the total field:

Bz�1��x� � �

x�

�1�x��H�x � x��, [18]

where x� are the source points (locations of the voxelcenters) and x is the field point where the field is evalu-ated. This equation takes the form of a discrete convolu-tion of the voxel-based susceptibility map, �1, with thesingle voxel solution, H, if both field points and sourcepoints lie on the same rectangular grid. Therefore, thecalculation can be efficiently implemented by using the 3DFast Fourier Transform (FFT) as:

Bz�1� � �1���Z��1����H��, [19]

where �(. . .) is the FFT and Z(. . .) is a zero-padding func-tion, used to ensure that there is no periodic wrap-aroundin the convolution. Note that in previous sections theequations have been general enough so that any field pointcould be chosen. However, in order to take advantage ofthe efficiency of the discrete convolution the generality ofthe field point locations must be restricted to a grid (withspacing equal to the source voxel size and with the pointslocated at the source voxel centers).

Gradient of the Perturbed Field

The exact analytical gradient of the perturbed field Bz(1)

can be calculated by simply replacing the convolutionkernel, H, with its gradient:

Bz�1�

q� �

x�

�1�x�� H q�

�xx��

[20]

where q stands for x, y or z.

Changing Orientation

The field produced by the object in another orientation canbe calculated by rotating the applied B(0) field. Note thatthis calculation is in the reference frame of the object, notthe scanner, and so calculating the scanner defined z com-ponent of the perturbed field requires projection of the fullB(1) field vector onto the scanner frame z-axis unit vector.Hence the desire to express Eqs. 10 and 13 in a general

FIG. 1. Spherical object model (a), analytical field (b) and perturba-tion field calculation (c) for the case of a sphere of radius R0 � 8voxels. The displayed scale of the field maps is �0.4 ppm.

Perturbative B0 Calculation 473

form containing Bx(0) and By

(0) as well as Bz(0). Furthermore,

since B(1) is a linear function of B(0), a rotation can bereconstructed using a basis set of B(1) images resultingfrom B(0) fields in the x, y and z directions:

Bz�1� � �0 0 1�R1�Bx

�1��1,0,0� Bx�1��0,1,0� Bx

�1��0,0,1�By

�1��1,0,0� By�1��0,1,0� By

�1��0,0,1�Bz

�1��1,0,0� Bz�1��0,1,0� Bz

�1��0,0,1�R�0

01,

[21]

where R is a 3 � 3 matrix (13) that represents the rotationfrom the scanner coordinates, xsc, to the object coordi-nates, xob, so that xob � Rxsc, and Bp

(1) (q) is the fieldcalculated in the object coordinate system with the p di-rection (x, y, or z) specifying the component of the fieldresulting from an applied field B(0) � q, where q is a unitaxis vector—either x � (1, 0, 0), y � (0, 1, 0) or z � (0, 0,1). For example, p � y and q � x gives By

(1) (x), whichrepresents the y component of the field due to a unitapplied field along the x direction (all in the object coor-dinate system). Note that R. [0 0 1]T represents the scannerunit z-axis vector (the direction of the applied field) asrepresented in the object coordinate system. This vectorspecifies the direction of the applied B0 field in the objectcoordinate system.

The matrix of perturb fields, [Bp(1)(q)], represents a set of

nine basis images, which can be precalculated and thencombined as specified above to give the desired field at anyorientation. This does not involve further approximation;it is precisely the same perturbed field that would becalculated for the object in the new orientation. In addi-tion, although the perturbed field is linear in the basisimages, it is not linear in the rotation angles, since theelements of R are nonlinear functions of these angles.

A similar calculation can be done for the gradients of thefield, ƒB(1), although this requires 27 basis images.

VALIDATION AND RESULTS

Analytical Sphere

The analytical magnetic field (including Lorentz Correc-tion) produced by a spherical object of radius R0 andsusceptibility �i inside a medium of susceptibility �e isgiven analytically by (12) as:

B � B0z ��i � �e

3 � �e�R0

r �3

B0�3 cos���r � z� r � R0

B0z r � R0,[22]

where the angles � and � are defined by r � (cos�sin�,sin�sin�, cos�) and z � (0, 0, 1). This expression gives allthree components of the field: B � (Bx, By, Bz).

Figure 1 shows, qualitatively, the z component of thefield distribution (less B0) for a sphere, comparing theanalytical solution given in Eq. 22 with the solution cal-culated with the perturbation method for the case of �9.5 � 106. Figure 2 shows plots of this field taken alongthe z-axis through the centers of these spheres for a rangeof different radii, R0 � 8, 16, 32 voxels, where in each case

FIG. 2. Plots of the Bz distribution along the z-axis for sphericalobjects of radius R0 � 8, 16, 32 voxels. Results are shown for theanalytical calculation (theory), the perturbation calculation, and thedifference between the two.

474 Jenkinson et al.

the voxel size is 1 � 1 � 1 mm. This illustrates therelationship between spatial extent and size of the error.

To investigate the dependence of the error on voxel sizemore quantitatively, the absolute difference between theanalytical and perturbation-based calculated fields weregenerated for a larger range of radii: R0 � 4, 8, 16, 32, 64voxels. In each case, several measurements of the distri-bution of this absolute difference (error) were made—mean, median, maximum and 90th, 95th, 99th and 99.9thpercentiles—which are shown in Table 1. Note that themeasurements include all field points in the 3D volume,not just along an axis, as in Fig. 2.

In addition, to test the ability of the method to calculateother components of the field the same absolute differencecalculations for the x component of the field, Bx, weregenerated. The results from this test are shown in Table 2.Note that the x-component of the field was calculatedusing a form of Eq. 13 derived for Bx rather than Bz (wherethe first term becomes zero and the last term interchangesx and z).

The tables indicate that the errors are very small overalland that as the resolution of the object improves (i.e., morevoxels or, equivalently, increased radius) the averaged er-rors (mean, median, percentiles) all decrease, such that forR0 � 32 voxels, 95% of the voxels, or field points, have anerror less than 1% of the maximum field, and 99% of thevoxels have an error less than 10% of the maximum field.However, some larger errors still persist at any resolution,indicated by the maximum error values. These errors oc-cur in a very small number of voxels which are typicallylocated near the surface of the object, where the suscepti-bility changes sharply.

These results are very similar to those shown in Refs. 1,2, 5, 6 despite the range of different object models (e.g.,boundary element methods vs. voxels) and approxima-tions used. However, superior performance is shown by

Ref. 3, where the object is modeled with piecewise planarpatches and a boundary-element style calculation is used.In addition, the method presented in Refs. 7, 8, whichmodeled the object as a collection of spheres, shows veryaccurate performance, although a direct comparison is dif-ficult from the available results. However, these methodsare more numerically intensive and require significantlygreater amounts of computational time.

In Vivo Human Head

An experimentally acquired field map of an in vivo humanhead was used to validate the method in practice. The MRfield map sequence used a symmetric-asymmetric spin-echo pair (14,15) (2.5 ms asymmetry time; 128 � 256 � 20voxels of size 1.5 � 1.0 � 6.0 mm). The 3D theoretical fieldmap was calculated using a 3D object susceptibility map112 � 164 � 156 (1.0 � 1.0 � 1.0 mm), that was created bysegmenting a whole-head CT image into bone, tissue, andair and then assigning a value of 1.0 (for �1) to the voxelsclassified as bone or tissue and a value of 0.0 to thoseclassified as air. This susceptibility map was then regis-tered to the magnitude component of the MR image corre-sponding to the acquired field map in order for the theo-retically calculated field map to be registered to the exper-imentally acquired map. A full 3D theoretical field mapwas calculated where the source and field points weretaken as the voxel centers from the input susceptibilitymap (1 mm spacing).

Figure 3 shows 2D slices from the 3D CT image used todefine the object susceptibility map, plus slices from boththe 3D experimentally acquired field map and the 3D fieldmap calculated using the voxel-based perturbation methoddescribed above (execution time was 9 min on a 1.8 GHzAthlon, 2 GB memory running Linux). Note that both fieldmaps have been masked so that only brain tissue is in-

Table 1Quantitative Error Measurements of Bz for Spherical Objects

R0 Mean Median P90 P95 P99 P99.9 Max

4 0.158 0.0248 0.498 0.733 1.96 3.93 3.938 0.085 0.0094 0.221 0.485 1.25 2.03 4.63

16 0.0388 0.0027 0.0505 0.181 0.865 1.77 5.1132 0.0199 0.0007 0.0111 0.0504 0.549 1.60 5.4864 0.0096 0.0001 0.0031 0.0108 0.246 1.29 5.72

Quantitative error measurements of the absolute difference in Bz (in units of ppm) between the analytical result and the perturbationcalculations for a spherical object of radius R0 voxels, with � 9.5 � 106. The maximum field value (less B0) for the analytical solutionis 6.33 ppm. P90 represents the 90th percentile, etc.

Table 2Quantitative error measurements of Bx for spherical objects

R0 Mean Median P90 P95 P99 P99.9 Max

4 0.141 0.0274 0.428 1.02 1.39 1.60 1.608 0.0763 0.0107 0.167 0.413 1.19 1.92 2.20

16 0.0364 0.0031 0.0432 0.146 0.856 1.79 2.0432 0.0180 0.0009 0.0098 0.0405 0.475 1.62 2.5364 0.0087 0.0001 0.0025 0.0091 0.200 1.30 2.41

Quantitative error measurements of the absolute difference in Bx (in units of ppm) between the analytical result and the perturbationcalculations for a spherical object of radius R0 voxels, with � 9.5 � 106. The maximum field value for the analytical solution is 4.75 ppm.P90 represents the 90th percentile, etc.

Perturbative B0 Calculation 475

cluded (although the simulation included all tissuespresent, with �brain � �bone � 1.0) and have had the firstand second-order spherical harmonics removed in order tofactor out the effect of the shims on the field maps.

Qualitatively it can be seen that the match is good.Quantitatively (see Table 3) the mean absolute differencebetween the field maps is 0.0465 ppm, while the typicalrange of the field values (used for the display range in Fig.3) is �0.4 ppm. Furthermore, � 90% of the voxels were inerror by less than 0.1 ppm.

The calculated error can be compared with the neglectedsecond-order terms in the perturbation expansion. Thesesecond-order terms have an approximate magnitude of2 � 1010 � 0.0001 ppm, which is two orders of magni-tude less than the observed errors. Therefore, it is likelythat the observed errors are due to inaccuracies in thevoxel-based modeling, as investigated in the previous sec-tion.

DISCUSSION

In this article we present a perturbation method for calcu-lating the magnetic B field for an object with varyingspatial susceptibility. A fast, first-order calculation is pre-sented for voxel-based objects, using the analytical voxelsolution. The accuracy of this method was tested using theanalytical solution for a sphere as well as by empiricalcomparison with a human head dataset. These resultsindicate that highly localized errors of less than 1 ppm areachieved generally, which is very similar to other calcula-tion methods, and sufficient for most MR imaging pur-poses. This implementation is available as a free download

from www.fmrib.ox.ac.uk/�mark/b0calc and will be dis-tributed with future versions of FSL (www.fmrib.ox.ac.uk/fsl).

There are several main contributions of this work. Thefirst is the use of a principled, perturbation method forarriving at the field approximation. This is useful in that itallows the magnitude of the error terms (second-order andhigher) to be estimated, which then permits the relevantapplicability of the method to be assessed. For instance,the method cannot be used for metallic objects where J �0 or objects where the susceptibility difference, , is large,but it can be used for some substances with slightly highersusceptibility than biological tissues, such as graphite (15).It is also possible, although potentially analytically intrac-table, to extend the approximation to higher orders toincrease the accuracy. In addition, the formulation of theperturbation equations is separate from the object modelspecification and could be used with other object models,such as boundary element methods.

Another significant contribution of this work is the abil-ity to calculate more than just the z component of the field.In particular, the x and y components can be calculatedjust as easily (although separately) as well as the gradientsof the fields (evaluated at the voxel centers), and formula-tions are provided for all these cases. More interestingly, itis possible to calculate the field for different object orien-tations by linearly combining precalculated “basis” im-ages. This allows the field to be determined, without fur-ther approximation, at any orientation and in a very effi-cient manner. Such calculations will allow the interactionbetween susceptibility fields and motion artifacts to beexplored more easily, a current research interest of theauthors.

In the field calculations used here there are two mainsources of approximation beyond the requirement for zeroconductivity: 1) neglecting all perturbation terms beyondfirst-order, and 2) representing the object by a voxel-basedmodel. The first approximation limits the range of objectsfor which this method could be applied, as discussedabove. The second approximation is potentially more lim-iting, as the use of a voxel-based model for the object willcause errors that are not as easily estimated as the pertur-

FIG. 3. Corresponding axial 2D slices from 3D images of an in vivo human head, showing the CT image used to derive the susceptibilitymap (a), the experimentally acquired field map (b), and the field map from the voxel-based perturbation calculation (c). The displayed scaleof the field maps is �0.4 ppm.

Table 3Quantitative error measurements on in vivo human brain

Mean Median P90 P95 P99 P99.9 Max

0.0465 0.0270 0.105 0.156 0.316 0.589 3.27

Quantitative error measurements of the absolute difference in Bz (inunits of ppm) between the experimentally measured field map andthe perturbation calculations for the in vivo human brain after re-moval of first- and second-order spherical harmonics. P90 repre-sents the 90th percentile, etc.

476 Jenkinson et al.

bation approximation errors and appear to dominate, asdemonstrated in the results on the human head.

Quantitative investigation of the object modeling errorswas conducted on a range of spherical objects and theresults are shown in Fig. 2 and Tables 1 and 2. Theseindicate that the spatial extent of larger errors is limited toa few voxels, which lie near the surface of the sphere (Fig. 1).This is to be expected, as this is closest to the area where thevoxel-based object model deviates from the real, continuousobject. It also suggests that using higher-resolution imagesleads to both smaller and more spatially localized errorsoverall, although at the cost of extra computational effort (thenumber of computations for N voxels is proportional to N logN). For models of the human head with air-filled cavities, thesignificant errors are therefore only likely to occur within afew voxels of the air–tissue boundaries.

Alternative models, such as boundary element methods(1–5,7,8), are likely to be physically accurate in capturingthe object shape, but have two main disadvantages. One isthat boundary meshes are more difficult to instantiate fromtypical voxel-based images (although one possible solu-tion to this problem is given in Ref. 3) and the second isthat they require more computation for the field calcula-tion as each element (triangle of the mesh) is normallyunique and requires separate calculations. In contrast,voxel-based models (6,10) are easy to instantiate and veryefficient to calculate (using Fast Fourier Transforms). Fur-thermore, the numerical results on the spherical objectindicate that similar errors are obtained by many of theproposed methods. Finally, all of these object models havean advantage over finite Fourier representations (9) sincethey can ensure that the object has finite spatial extent,which is not possible with the Fourier method.

ACKNOWLEDGMENT

We thank Dr. Bob Cox for suggesting that a perturbationcalculation could be useful for magnetic field calculationsin notes from a workshop presentation on motion in FMRI.

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