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```									Lecture 10

The time-dependent transport equation


V

 ˆ N (r , s, t ) t

 ˆ ˆ dV    c s   N ( r , s , t ) dV
V

Photons scattered to direction ŝ'

Absorbed photons

    ˆ ˆ   c  s ( r ) N ( r , s , t ) dV   c  a ( r ) N ( r , s , t ) dV
V

Photons scattered into direction ŝ from ŝ'

V

Light source q

  ˆ ˆ ˆ   c  s ( r )  p ( s   s ) N ( r , s , t ) d  dV 
V 4

  q ( r , sˆ , t ) dV
V

Time-Dependent Transport Equation
• Typically the transport equation is expressed in terms of the radiance (I(r,ŝ,t) =N(r,ŝ,t)hc) , and after dropping the integrals

 ˆ 1 I (r , s, t ) c t

  ˆ ˆ ˆ  s   I (r , s, t )  ( s   a ) I (r , s, t ) 

  ˆ ˆ ˆ ˆ  s  I ( r , s , t ) p ( s   s )d    Q ( r , s , t )
4

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

 ˆ ˆ s   I (r , s) 

 ˆ I (r , s ) ds

 ˆ  ( s   a ) I (r , s) 

s

4



  ˆ ˆ ˆ ˆ I ( r , s ) p ( s   s )d    Q ( r , s )

Approximations
• The transport equation is difficult to solve analytically.In order to find an analytical solution we need to simplify the problem. • Discretization methods  
ˆ N – Discrete ordinates method ( r , s , t )  – Kubelka-Monk theory – Adding-doubling method
– Diffusion theory

ˆ  N (r , s , t )
i i

• Expansion methods

• Probabilistic methods
– Monte Carlo simulations

Diffusion approximation
• Expand the photon distribution in an isotropic and a gradient part
1   3     ˆ, t )  ˆ N (r , s  r (r , t )  c J (r , t )  s  4  

• Where r(r,t) is the photon density   ˆ r ( r , t )   N (r , s , t ) d  • And J(r,t) is the photon current density (photon flux)

Fick’s
J   C x

st 1

law of diffusion
  J ( r , t )   cD  r

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

  J ( r , t )   cD  r
Photon flux (J cm-2 s-1) in response to a photon density gradient, characterized by the diffusion coefficient D, defined as

D 

1 3  tr



1 3 (  a  (1  g )  s )

Diffusion approximation
Transport equation:


V

 ˆ N (r , s, t ) t

 ˆ ˆ dV    c s   N ( r , s , t ) dV
V



    ˆ ˆ c  s ( r ) N ( r , s , t ) dV   c  a ( r ) N ( r , s , t ) dV 
V V



  ˆ ˆ ˆ c  s ( r )  p ( s   s ) N ( r , s , t ) d  dV  
V 4

 ˆ q ( r , s , t ) dV 
V

Photon distribution expansion:

1   3     ˆ ˆ N (r , s, t )  r (r , t )  J (r , t )  s  4  c  
Photon source expansion:

Diffuse intensity is greater in the direction of net flux flow

  1   ˆ ˆ q o ( r , t )  3 q 1 ( r , t )  s  q (r , s, t )  4

Diffusion approximation
• Plug in, integrate over , and assume only isotropic sources (refer to supplementary material for full derivation)
1 r  1 3  ˆ    a r    J  q o  q1  s c t c c c 1

• Assume a constant D and use the relation for the fluence rate
   ( r , t )  ch r ( r , t )

• To get
1  c t      2  ( r , t )  D   ( r , t )   a  ( r , t )  S o ( r , t )  3 D   S1 ( r , t )

Types of diffuse reflectance measurements
Continuous wave (CW)
I0
It

Time domain (TD)
intensity

frequency domain (TD) intensity
2 1.5 1

I0 It

0.5 0 0 5

t (ns)
tissue

10

15

20

t=0

~ns

intensity

2

1.5 1

phase shift

0.5 0 0 5

ac

dc
15 20

t (ns)

10

Point source solution: timedomain
• The solution to the diffusion equation for an infinite homogeneous slab with a short pulse isotropic point source S(r,t)=d(0,0) is
 ( r , t )  c ( 4  Dct )
3 / 2 2  r exp    4 Dct   a ct 

   

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

• Harmonic time dependence is given by factor exp(-it), so that ∂/∂t -i • Diffusion equation takes the form
 i c
2

Point source solution: frequency domain

    2  (r )  D   (r )   a  (r )  S o (r ) 
2

 (  k )  ( r )   where k
2

 S o (r ) cD

,



i  c  a cD

Point source solution: frequency domain
intensity

• Green’s function for homogeneous, infinite medium containing a harmonically modulated point source of power P() at r=0 is
 (r , )  P ( ) e 4 D
ikr

frequency domain (TD)
2 1.5 1

with r

k

2



i  c  a cD
1/ 2

0.5 0 0 5

Abs [  ( r , 0 )]   DC ( r ) 

PDC e 4 D

r(a / D )

t (ns)
tissue

10

15

20

r

 I DC

Abs [  ( r ,  )]   AC ( r ,  )  ac amplitude

And it follows
2

that phase  rIm(k)

intensity

Arg [  ( r ,  )]  phase ln( r * I DC )   r Re[ k ]  ln[ 4  D ]

2

1.5 1

phase shift

0.5 0 0 5

ac

phase

ln(r2*Idc)

intercept (s’) slope (a, s’)

t (ns)

10

15

dc

20

slope (a, s’) intercept = 0

r

r

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

• How:

The pulse oximeter
100 10 1 0.1 0.01 300 400 500 600 700 800 900 1000 1100 1200 1300

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

• Isolate varying signal due to     a  ln 10 *  HbO  HbO 2    Hb  Hb  pulsatile flow (arterial blood)  • Assume detected signal is proportional ln 10 *  1  HbO    1  Hb   a1 HbO 2 2 Hb to absorption coefficient (Two measurements, two unknowns) • Calibrate instrument by correlating detected signal to arterial saturation measurements from blood samples
2 2 2 2

– Detect signal transmitted through finger

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

Arterial O 2 saturation



HbO 2  Hb 

HbO 2 

* 100 %

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

Near-infrared spectroscopy and imaging of tissue
Sergio Fantini
Department of Biomedical Engineering Tufts University, Medford, MA

volume probed by near-infrared photons

outline
Near-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

source detector

source

detector source detector

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

Why near-infrared spectroscopy and imaging of tissues?
     Non-invasive Non-ionizing Real-time monitoring Portable systems Cost effective

Dominant tissue chromophores in the near infrared
ultraviolet
absorption coefficient (cm-1)
410 nm 600

near infrared
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)

Diffusion of near-infrared light inside tissues

low power laser

optical fiber

optical detector
biological tissue

high scattering problem
is there a car in front of me?

is there a cookie in the milk?

Frequency-domain spectroscopy (FD)
intensity (a.u.)
2 1.5 1 0.5 0 0 5

t (ns)

10

15

20

tissue
intensity (a.u.)
phase shift ac
0 5

2 1.5 1 0.5 0

dc
15 20

t (ns)

10

Diffusion equation: frequency domain
• Harmonic time dependence is given by factor exp(-it), so that ∂/∂t -i • Diffusion equation takes the form
 i c
2

    2  (r )  D   (r )   a  (r )  S o (r ) 
2

 (  k )  ( r )   where k 
2

 S o (r ) cD

,

i  c  a cD
1

D 

1 3  tr



3 (  a  (1  g )  s )

Point source solution: frequency domain
intensity

• Green’s function for homogeneous, infinite medium containing a harmonically modulated point source of power P() at r=0 is
 (r , )  P ( ) e 4 D
ikr

frequency domain (TD)
2 1.5 1

with r

k 
2

i  c  a cD
1/ 2

0.5 0 0 5

Abs [  ( r , 0 )]   DC ( r ) 

PDC e 4 D

r(a / D )

r

 I DC

t (ns)
tissue

10

15

20

Abs [  ( r ,  )]   AC ( r ,  )  ac amplitude Arg [  ( r ,  )]  phase And it follows that phase  rIm[k]

intensity

2

1.5 1

phase shift

ln( r * I DC )   r Re[ k ]  ln[ 4  D ]

0.5 0 0 5

ac

phase

intercept (s’)
ln(r*Idc)

t (ns)

10

15

dc

20

slope (a, s’)

slope (a, s’) intercept = 0

r

r

TISSUE OXIMETRY

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

Configuration for tissue oximetry

detector optical fiber source optical fibers

detector

RF electronics

laser driver

measuring probe

laser diodes

main box

multiplexing circuit

Frequency-domain oximetry
90

HbO2 and Hb (M)

100

90 80 70 60 50 40 30 20 0 2 4 6 8 10 12 14 16 18

saturation (%)

80 70 60 50 40 30 20 0 2 4

ischemia time (min)
6 8 10 12 14 16 18

ischemia time (min)

0.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18 0.17 0 2 4

5

4.8

a (1/cm)

750nm

s’ (1/cm)

830nm

750nm

4.6

4.4

4.2

ischemia time (min)
6 8 10 12 14 16 18

4

ischemia
0 2 4 6

830nm
12 14 16 18

3.8

time (min)

8

10

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