Lecture 10

The time-dependent transport equation

VV

s

V

a

V

s

VV

dVtsrqdVdtsrNssprc

dVtsrNrcdVtsrNrc

dVtsrNscdVt

tsrN

),ˆ,(),ˆ,()ˆˆ()(

),ˆ,()(),ˆ,()(

),ˆ,(ˆ),ˆ,(

4

Spatial photon gradient

Photons scattered to direction ŝ' Absorbed photons

Photons scattered into direction ŝ from ŝ' Light source q

Time-Dependent Transport Equation

• Typically the transport equation is expressed in terms of the radiance ( I(r,ŝ,t) =N(r,ŝ,t)hc) , and after dropping the integrals

),ˆ,()ˆˆ(),ˆ,(

),ˆ,()(),ˆ,(ˆ),ˆ,(1

4

tsrQdssptsrI

tsrItsrIst

tsrI

c

s

as

Time-Independent Transport Equation

• For the steady-state situation, we assume that radiance is independent of time, and the transport equation becomes

)ˆ,()ˆˆ()ˆ,(

)ˆ,()()ˆ,(

)ˆ,(ˆ

4

srQdsspsrI

srIds

srIsrIs

s

as

Approximations• The transport equation is difficult to solve

analytically.In order to find an analytical solution we need to simplify the problem.

• Discretization methods– Discrete ordinates method– Kubelka-Monk theory– Adding-doubling method

• Expansion methods– Diffusion theory

• Probabilistic methods– Monte Carlo simulations

i

i tsrNtsrN ),ˆ,(),ˆ,(

Diffusion approximation

• Expand the photon distribution in an isotropic and a gradient part

• Where (r,t) is the photon density

• And J(r,t) is the photon current density (photon flux)

dtsrNtr ),ˆ,(),(

strJ

ctrtsrN ˆ),(

3),(

4

1),ˆ,(

Fick’s 1st law of diffusion

x

CJ

cDtrJ ),(

))1((3

1

3

1

satr gD

Movement or flux in response to a concentration gradient in a medium with diffusivity

Photon flux (J cm-2 s-1) in response to a photon density gradient, characterized by the diffusion coefficient D, defined as

cDtrJ ),(

Diffusion approximation

VV

s

V

a

V

s

VV

dVtsrqdVdtsrNssprc

dVtsrNrcdVtsrNrc

dVtsrNscdVt

tsrN

),ˆ,(),ˆ,()ˆˆ()(

),ˆ,()(),ˆ,()(

),ˆ,(ˆ),ˆ,(

4

strJ

ctrtsrN ˆ),(

3),(

4

1),ˆ,(

Transport equation:

Photon distribution expansion:

Photon source expansion:

strqtrqtsrq o ˆ),(3),(4

1),ˆ,( 1

Diffuse intensity is greater in the direction of net flux flow

Diffusion approximation• Plug in, integrate over , and assume only

isotropic sources (refer to supplementary material for full derivation)

• Assume a constant D and use the relation for the fluence rate

• To get

sqc

qc

Jctc oa ˆ

31111

),(),( trchtr

),(3),(),(),(),(1

12 trSDtrStrtrDtr

tc oa

Types of diffuse reflectance measurements

Continuous wave (CW) Time domain (TD)

I0 It

I0

It

t=0

~ns

inte

nsi

ty

frequency domain (TD)

0

0.5

1

1.5

2

0 5 10 15 20

0

0.5

1

1.5

2

0 5 10 15 20

phase shift

tissue

t (ns)

t (ns)in

ten

sit

yin

ten

sit

y dc ac

Point source solution: time-domain

• The solution to the diffusion equation for an infinite homogeneous slab with a short pulse isotropic point source S(r,t)=(0,0) is

• This is known as the Green’s function solution and can be used to solve more complicated problems

ct

Dct

rDctctr a

4exp)4(),(

22/3

Point source solution: frequency domain

• Harmonic time dependence is given by factor exp(-it), so that ∂/∂t -i

• Diffusion equation takes the form

cD

cik

cD

rSrk

rSrrDrc

i

a

o

oa

2

22

2

where

,)(

)()(

)()()()(

Point source solution: frequency domain

• Green’s function for homogeneous, infinite medium containing a harmonically modulated point source of power P() at r=0 is

rIm(k)phase ]4ln[]Re[)*ln(

thatfollowsit And

)],([

amplitude ac ),()],([

I4

)()]0,([

4

)(),(

2

)/(

2

2/1

DkrIr

phaserArg

rrAbsr

e

D

PrrAbs

cD

cikwith

r

e

D

Pr

DC

AC

DC

DrDC

DC

aikr

a

frequency domain (TD)

0

0.5

1

1.5

2

0 5 10 15 20

0

0.5

1

1.5

2

0 5 10 15 20

phase shift

tissue

t (ns)

t (ns)

inte

nsit

yin

ten

sit

y

dc

ac

ln(r

2*I

dc)

r

intercept (s’)

slope (a,s’)

phase

rintercept = 0

slope (a,s’)

Frequency domain measurements

• The slope of r*IDC as a function of r and the slope of the phase as a function of r depend on a and s'.

• Find the slopes and extract the optical properties

Medical applications of reflectance spectroscopy

Pulse Oximetry

Frequency domain NIR spectroscopy and imaging

Steady-state diffuse reflectance spectroscopy

The Pulse oximeter

• Function: Measure arterial blood saturation

• Advantages:– Non-invasive– Highly portable– Continuous monitoring– Cheap– Reliable

The pulse oximeter• How:

– Illuminate tissue at 2 wavelengths straddling isosbestic point (eg. 650 and 805 nm)

• Isosbestic point: wavelength where Hb and HbO2 spectra cross.

– Detect signal transmitted through finger• Isolate varying signal due to

pulsatile flow (arterial blood)• Assume detected signal is proportional

to absorption coefficient (Two measurements, two unknowns)

• Calibrate instrument by correlating detected signal to arterial saturation measurements from blood samples

0.01

0.1

1

10

100

300 400 500 600 700 800 900 1000 1100 1200 1300

HbHbO HbHbOa11

2

12*10ln

HbHbO HbHbOa22

2

22*10ln

%100*saturation O Arterial

2

22 HbHbO

HbO

The pulse oximeter

• Limitations:– Reliable when O2 saturation above 70%

– Not very reliable when flow slows down– Can be affected by motion artifacts and

room light variations– Doesn’t provide tissue oxygenation

levels

Department of Biomedical Engineering

Tufts University, Medford, MA

Near-infrared spectroscopyand imaging of tissue

Sergio Fantini

outlineNear-infrared spectroscopy and

imaging of tissues

applications to skeletal muscles hemoglobin oxygenation (absolute) hemoglobin concentration (absolute) blood flow and oxygen consumption

applications to the human breast detection of breast cancer spectral characterization of tumors

applications to the human brain optical monitoring of cortical activation intrinsic optical signals from the brain

volume probed by near-infrared

photons

source

source

detector

source

detector

detector

Why near-infrared spectroscopyand imaging of tissues?

Non-invasive Non-ionizing Real-time monitoring Portable systems Cost effective

Dominant tissue chromophoresin the near infrared

ultraviolet near infrared

410 nm 600 770 nm 1300

wavelength (nm)

Hb, HbO2 from: Cheong et al., IEEE J. Quantum Electron. 26, 2166 (1990)

H2O from: Hale and Querry, Appl. Opt. 12, 555 (1973)

ab

sorp

tion

coeffi

cien

t (c

m-1)

Diffusion of near-infrared light inside

tissues

low power laser

biological tissue

optical detector

optical fiber

high scattering problemis there a

car in front of me?

is there a cookiein the milk?

Frequency-domain spectroscopy (FD)

0

0.5

1

1.5

2

0 5 10 15 20

0

0.5

1

1.5

2

0 5 10 15 20

phase

shift

tissue

t (ns)

t (ns)

inte

nsit

y

(a.u

.)in

ten

sit

y

(a.u

.)

dc

ac

Diffusion equation: frequency domain

• Harmonic time dependence is given by factor exp(-it), so that ∂/∂t -i

• Diffusion equation takes the form

cD

cik

cD

rSrk

rSrrDrc

i

a

o

oa

2

22

2

where

,)(

)()(

)()()()(

))1((3

1

3

1

satr gD

Point source solution: frequency domain

• Green’s function for homogeneous, infinite medium containing a harmonically modulated point source of power P() at r=0 is

rIm[k]phase ]4ln[]Re[)*ln(

thatfollowsit And

)],([

amplitude ac ),()],([

I4

)()]0,([

4

)(),(

2/1)/(

2

DkrIr

phaserArg

rrAbsr

e

D

PrrAbs

cD

cikwith

r

e

D

Pr

DC

AC

DC

DrDC

DC

aikr

a

frequency domain (TD)

0

0.5

1

1.5

2

0 5 10 15 20

0

0.5

1

1.5

2

0 5 10 15 20

phase shift

tissue

t (ns)

t (ns)

inte

nsit

yin

ten

sit

y

dc

ac

ln(r

*Id

c)

r

intercept (s’)

slope (a,s’)

phase

rintercept = 0

slope (a,s’)

TISSUE OXIMETRY

Time-domain oximetryMiwa et al., Proc. SPIE 2389, 142 (1995)

Configuration for tissue oximetry

0 5 10 20 25 30 H

emog

lobi

n Sa

t ura

tion

(%

)Time (min)

measuring probe main

box

2.0 cm laser diodes

source optical fibers

detector optical fiber

laser driver

detectorRF

electronics

multiplexing circuit

Hb

O2 a

nd

Hb

(M

)

20

30

40

50

60

70

80

90

0 2 4 6 8 10 12 14 16 18

Frequency-domain oximetry

750nm

time (min)

a (

1/c

m)

830nm

750nm

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

0 2 4 6 8 10 12 14 16 18

830nm

3.8

4

4.2

4.4

4.6

4.8

5

0 2 4 6 8 10 12 14 16 18

ischemia

time (min)

ischemia

s’

(1/c

m)

time (min)

satu

rati

on (

%)

time (min)

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10 12 14 16 18

ischemia ischemia