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.