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ECEU692 Subsurface Imaging Course Notes Part 12: Imaging with Light (4): Diffusive Optical Tomography. Profs. Brooks and DiMarzio Northeastern University Spring 2004. Topic Outline. Goal: “Find the Matrix Elements” A Bit of Radiometry Terminology and Units Radiative Transport - PowerPoint PPT Presentation
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-1
ECEU692Subsurface Imaging
Course NotesPart 12: Imaging with Light (4):Diffusive Optical Tomography
Profs. Brooks and DiMarzio
Northeastern University
Spring 2004
February 2004 Chuck DiMarzio, Northeastern University 10471-12-2
Topic Outline
• Goal: “Find the Matrix Elements”• A Bit of Radiometry
– Terminology and Units– Radiative Transport
• Approximation to Radiative Transport Equation– Diffusion Approximation– Wave Solution– Generating the Diffusive Waves
• Examples• Adding Ultrasound• Solving for the Matrix Elements
February 2004 Chuck DiMarzio, Northeastern University 10471-12-3
The Matrix Elements
t
P
t
P
DCAC AmplitudeAC Phase
February 2004 Chuck DiMarzio, Northeastern University 10471-12-4
Radiometric Quantities
February 2004 Chuck DiMarzio, Northeastern University 10471-12-5
Radiometry and Photometry, Flux M, Flux/Proj. Area
I, Flux/ L,Flux/AE, Flux/Area Rcd.
Radiant FluxWattsLuminous FluxLumens
Radiant ExitanceWatts/m2
Luminous ExitanceLumens/m2=Lux
A /
RadianceWatts/m2/srLuminanceLumens/m2/sr1 Lambert=(1L/cm2/sr)/
1 ftLambert= (1L/ft2/sr)/1mLambert= (1L/m2/sr)/
Radiant IntensityWatts/srLuminous IntensityLumens/sr
1 Candela=1cd=1L/sr
/
IrradianceWatts/m2
IlluminanceLumens/m2=Lux
1 Ft Candle=1L/ft2
Notes: Spectral x=dx/d or dx/d: Add subscript or , divide units by Hz or m.
1 W is 683 L at 555 nm.
2R
February 2004 Chuck DiMarzio, Northeastern University 10471-12-6
t
nL
c
ndnnpnLnL
s
nL ˆ''ˆ,ˆ'ˆ
4ˆ
ˆ
What Is Radiative Transport?
• The Radiative Transport Equation
dsd dL L+dL
February 2004 Chuck DiMarzio, Northeastern University 10471-12-7
Solutions to RTE
• Monte-Carlo
• Low Scattering
• High Scattering – Diffusion Approximation– Usually Valid in Tissue, Except...
• Certain Tissue Types
• Certain Imaging Modalities (eg. Confocal, OCT)
• Close to Source or to Rapid Changes in Parameters
February 2004 Chuck DiMarzio, Northeastern University 10471-12-8
• Approach– Monte-Carlo
– Reciprocity
– Fourier Transform
• Parameters– Depth 1 cm.
– Thickness 2 cm.
• TransilluminationDunn, Andrew, and Charles A. DiMarzio, “Efficient Computation of Time--Resolved Transfer Functions for Imaging in Turbid Media,” Journal of the Optical Society of America A 13, No. 1, January 1996. Pp. 65--70.
Tissue Parametersa = 0.03 /cms = 200 /cmg = 0.95d = 1 cm
125 150
200-ps Gate
Spatial Frequency, /cm
MT
F
Resolution Limits (M-C)
February 2004 Chuck DiMarzio, Northeastern University 10471-12-9
t
nL
c
ndnnpnLnL
s
nL ˆ''ˆ,ˆ'ˆ
4ˆ
ˆ
n )/ ( / ) nL( ˆˆ J
qn
c
t/c)(· a
1J
013
gn
c
as
J
Photon Diffusion Approximation• The Radiative Transport Equation
• Taylor Series: is Fluence Rate, J is Flux
• Result
n̂
J
February 2004 Chuck DiMarzio, Northeastern University 10471-12-10
Fluence Rate?
• Another Radiometric Quantity– Fluence is Energy/Area– Fluence Rate is Energy/Area/Time
• =Power/Area
• Units Like E or M, but Different Meaning
• Relation to Absorbed Power/Volume– A=a
– Used to Determine in Monte-Carlo
February 2004 Chuck DiMarzio, Northeastern University 10471-12-11
t 0
D as gn
cD
13 n
ca
)(0
tie rk
D
i
nD
ck a
2 Re
Imk2 k
Dispersion Equation• The Diffusion Equation
• Wave Solution
=0
February 2004 Chuck DiMarzio, Northeastern University 10471-12-12
Dispersion Results
February 2004 Chuck DiMarzio, Northeastern University 10471-12-13
Spherical Waves
February 2004 Chuck DiMarzio, Northeastern University 10471-12-14
Different Types of Waves
100
105
1010
1015
102010
-4
10-2
100
102
104
106
108
f, Frequency, Hz.
k/(2
), W
aven
umbe
r, m
-1
Sound
(Real)
DPDW
Light(Real)
(Imag)(Imag)
10059_1
1m
1mm
1km
1m
February 2004 Chuck DiMarzio, Northeastern University 10471-12-15
Physical Reason for Dispersion
-0.5
0
0.5
Sam
ple
200 MHz.
0 5 10
10
20
30
40
50
-0.5
0
0.5
Sam
ple
500 MHz.
0 5 10
10
20
30
40
50
0 5 10-50
0
50
Sig
nal
Time, ns0 5 10
-50
0
50
Sig
nal
Time, ns
Imaginary partof k increaseswith frequency
Easy to understand in terms of multiple paths.
m100574a.m
February 2004 Chuck DiMarzio, Northeastern University 10471-12-16
Watch the Photons Migrate!• 20 Photon Tracks • 20,000 Photon Tracks
– Pabs=0.1– Pext=0.3
• Received Photons
0 20 40 60 80 1000
10
20
30
40
50
60
70
80
90
Time Step
Pho
ton
s in
Bo
x
February 2004 Chuck DiMarzio, Northeastern University 10471-12-17
ExtrapolatedBoundary
Tissue
ImageSource
EffectiveSource
Input
Detector
ImageSource
How Diffuisve Waves Begin?• Generation
– From Light Wave
• Wave Behavior– Absorption
– Reflection
– Refraction
– Diffraction
– Interference
– Scattering
February 2004 Chuck DiMarzio, Northeastern University 10471-12-18
Noise Issues
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
Sig
nal
Time, ns
m100574a.m
Noise proportionalto square root ofDC signal.
February 2004 Chuck DiMarzio, Northeastern University 10471-12-19
TECHNOLOGY•Near-infrared light•Fiber optics•Computed Tomography
ADVANTAGES•Optical contrast•Portable - bedside, ambulance•Continuous•Inexpensive
•DISADVANTAGES•Resolution•Depth penetration
From David A. Boas - MGH NMR Center
DOT Instrumentation at MGH Imaging Center
February 2004 Chuck DiMarzio, Northeastern University 10471-12-20
DetectorsSources
6 cm
4 cm
Mid-line
Data Set I - 98-05-14
At RestPassive movement of
right armPassive movement of
right arm
From David A. Boas - MGH NMR Center
Functional Imaging of a Neonate
February 2004 Chuck DiMarzio, Northeastern University 10471-12-21
0 1 2 3 4 5 6
0123456-5-4-3-2-10
X axisY axis
Z a
xis
0 1 2 3 4 5 6-5
-4
-3
-2
-1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 1 2 3 4 5 6-5
-4
-3
-2
-1
0
0
0.01
0.02
0.03
0.04
0.05
0 2 4 6-5
-4
-3
-2
-1
0
0
0.05
0.1
0.15
Keeping the Matrix Rank UpSource
Detector
Object
y=4z
x
Reconstruction with Reflection only(Top Sources)
Reflection and Transmission(All Sources)
DiMarzio, et. al., Presented at Photonics West, Jan 1999
February 2004 Chuck DiMarzio, Northeastern University 10471-12-22
UltrasoundUltrasoundFocal PointFocal Point
UltrasoundUltrasoundBeamBeamOpticalOptical
SourceSource
OpticalOpticalReceiverReceiver
OpticalOpticalSourceSource
OpticalOpticalReceiverReceiver
OpticalOpticalSourceSource
OpticalOpticalReceiverReceiver
All Light fromSource Fiber
Light from Source to Receiver
Light from Source to Receiver through Ultrasound Focus
API Virtual Source
February 2004 Chuck DiMarzio, Northeastern University 10471-12-23
Solving the Wave Equation (1)
February 2004 Chuck DiMarzio, Northeastern University 10471-12-24
Solving the Wave Equation (2)
February 2004 Chuck DiMarzio, Northeastern University 10471-12-25
The First Born Approximation
February 2004 Chuck DiMarzio, Northeastern University 10471-12-26
Why Do We Want a Model?
• Applications– Forward Model
• Will it work?
– Inverse Algorithms• How Much Does k
Change?– ie. Is there a Tumor?
• And Where?
• Understanding– What is k?– See Panel to Right.