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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