ECEU692
Subsurface Imaging
Course Notes
Part 12: Imaging with Light (4):
Diffusive Optical Tomography
Profs. Brooks and DiMarzio
Northeastern University
Spring 2004
February 2004
Chuck DiMarzio, Northeastern University
10471-12-1
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
10471-12-2
The Matrix Elements
P
P
t
DC
AC Amplitude
AC Phase
t
February 2004
Chuck DiMarzio, Northeastern University
10471-12-3
Radiometric Quantities
February 2004
Chuck DiMarzio, Northeastern University
10471-12-4
Radiometry and Photometry
Notes: Spectral
F, Flux
x=dx/dn or dx/dl:
Add subscript n or w,
divide units by Hz or
mm.
1 W is 683 L
at 555 nm.
E, Flux/Area Rcd.
Radiant Flux M, Flux/Proj. Area
Radiant Exitance
Watts
Watts/m2
Luminous Flux
Luminous Exitance
Lumens
Lumens/m2=Lux
 / A
I, Flux/W Radiant
Intensity
Irradiance
Watts/sr
Watts/m2
Luminous
Illuminance
Intensity
Lumens/m2
Lumens/sr
=Lux
 R2
1 Ft Candle=1L/ft2
1 Candela=1cd=1L/sr
February 2004
Chuck DiMarzio, Northeastern University
 / W
L,Flux/AW Radiance2
Watts/m /sr
Luminance
Lumens/m2/sr
1 Lambert=
(1L/cm2/sr)/p
1 ftLambert= (1L/ft2/sr)/p
1mLambert= (1L/m2/sr)/p
10471-12-5
What Is Radiative Transport?
L
dW
dW L+dL
ds
• The Radiative Transport Equation
Lnˆ 
m
n Lnˆ 
  mLnˆ    Lnˆ ' pnˆ, nˆ 'dW' 

s
4p
c t
February 2004
Chuck DiMarzio, Northeastern University
10471-12-6
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
10471-12-7
Resolution Limits (M-C)
– Monte-Carlo
– Reciprocity
– Fourier Transform
• Parameters
– Depth 1 cm.
– Thickness 2 cm.
• Transillumination
MTF
• Approach
Tissue Parameters
ma = 0.03 /cm
ms = 200 /cm
g = 0.95
d = 1 cm
125 150
200-ps Gate
Spatial Frequency, /cm
Dunn, 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.
February 2004
Chuck DiMarzio, Northeastern University
10471-12-8
Photon Diffusion Approximation
• The Radiative Transport Equation
Lnˆ 
m
n Lnˆ 
  mLnˆ    Lnˆ ' pnˆ, nˆ 'dW' 

s
4p
c t
• Taylor Series:  is Fluence Rate, J is Flux
L(nˆ)  /p  (J/p )  nˆ
• Result
J
 ma c
· J  ( 1/c)

 q
t
n
c
J 
  0
3nm s 1  g   ma 
February 2004
Chuck DiMarzio, Northeastern University
n̂
10471-12-9
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=ma
– Used to Determine  in Monte-Carlo
February 2004
Chuck DiMarzio, Northeastern University
10471-12-10
Dispersion Equation
• The Diffusion Equation
F
D
   F 
 aF  0
t
c
D
3nm s 1  g   m
• Wave Solution
  0e

i ( k r wt )
 cma iw
k 

nD
D
k2

Im
k
w=0
2
February 2004
a
mac
a
n
Chuck DiMarzio, Northeastern University
Re
10471-12-11
Dispersion Results
February 2004
Chuck DiMarzio, Northeastern University
10471-12-12
Spherical Waves
February 2004
Chuck DiMarzio, Northeastern University
10471-12-13
Different Types of Waves
8
10
Light
(Real)
6
-1
k/(2p), Wavenumber,
m
10
1mm
4
10
Sound
2
10
0
DPDW
(Imag)
1mm
(Imag)
1m
(Real)
10
-2
10
1km
10059_1
-4
10 0
10
February 2004
5
10
10
10
f, Frequency, Hz.
15
10
Chuck DiMarzio, Northeastern University
20
10
10471-12-14
Physical Reason for Dispersion
200 MHz.
500 MHz.
10
0.5
20
0
30
Sample
Sample
10
-0.5
40
0
30
-0.5
Imaginary part
of k increases
with frequency
50
0
5
10
0
5
10
Easy to
understand in
terms of
multiple paths.
50
Signal
50
Signal
20
40
50
0
-50
0.5
0
5
Time, ns
February 2004
10
0
-50
m100574a.m
0
5
Time, ns
10
Chuck DiMarzio, Northeastern University
10471-12-15
Watch the Photons Migrate!
• 20 Photon Tracks
• 20,000 Photon Tracks
– Pabs=0.1
– Pext=0.3
• Received Photons
90
Photons in Box
80
70
60
50
40
30
20
10
0
0
20
40
60
Time Step
February 2004
80
100
Chuck DiMarzio, Northeastern University
10471-12-16
How Diffuisve Waves
Begin?
Tissue
Extrapolated
Boundary
• Generation
– From Light Wave
Detector
• Wave Behavior
–
–
–
–
–
–
Absorption
Reflection
Refraction
Diffraction
Interference
Scattering
February 2004
Image
Source
Image
Source
Effective
Source
Input
Chuck DiMarzio, Northeastern University
10471-12-17
Noise Issues
1.2
Noise proportional
to square root of
DC signal.
1
Signal
0.8
0.6
0.4
0.2
m100574a.m
0
0
1
2
February 2004
3
4
5
Time, ns
6
7
8
9
10
Chuck DiMarzio, Northeastern University
10471-12-18
DOT Instrumentation at MGH
Imaging Center
TECHNOLOGY
•Near-infrared light
•Fiber optics
•Computed Tomography
ADVANTAGES
•Optical contrast
•Portable - bedside, ambulance
•Continuous
•Inexpensive
•DISADVANTAGES
•Resolution
From David A. Boas - MGH NMR Center
•Depth penetration
Chuck DiMarzio, Northeastern University
February 2004
10471-12-19
Functional Imaging of a Neonate
4 cm
6 cm Mid-line
At Rest
Detectors
Sources
Passive movement of Passive movement of
right arm
right arm
Data Set I - 98-05-14
From David A. Boas - MGH NMR Center
February 2004
Chuck DiMarzio, Northeastern University
10471-12-20
Keeping the Matrix Rank Up
Source
Detector
Z axis
0
-1
-2
-3
-4
-5
Object
6 5
4 3
2 1
0
Y axis
56
34
2
0 1 X axis
0.05
0
y=4
0.02
-4
-5
0.01
0
1
2
3
4
5
6
0
0.12
0.1
0.08
-3
0.06
-4
0.04
x
0
1
2
3
4
5
0.02
6
0.15
0
-1
0.1
-2
-3
-3
0.14
-2
Reconstruction with
Reflection only
(Top Sources)
0.03
-2
z
-1
-5
0.04
-1
0
Reflection and
Transmission
(All Sources)
0.05
-4
-5
0
2
4
6
0
DiMarzio, et. al., Presented at Photonics West, Jan 1999
February 2004
Chuck DiMarzio, Northeastern University
10471-12-21
API Virtual Source
Optical
Source
Optical
Source
Optical
Receiver
Optical
Source
Optical
Receiver
Ultrasound
Beam
Optical
Receiver
Ultrasound
Focal Point
All Light from
Source Fiber
February 2004
Light from
Source to
Receiver
Chuck DiMarzio, Northeastern University
Light from Source to
Receiver through
Ultrasound Focus
10471-12-22
Solving the Wave Equation (1)
February 2004
Chuck DiMarzio, Northeastern University
10471-12-23
Solving the Wave Equation (2)
February 2004
Chuck DiMarzio, Northeastern University
10471-12-24
The First Born Approximation
February 2004
Chuck DiMarzio, Northeastern University
10471-12-25
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.
February 2004
Chuck DiMarzio, Northeastern University
10471-12-26