Transcript Epithelial tissue - Tufts University

Lecture 10
The time-dependent transport
equation

N (r , sˆ, t )

V t dV  V csˆ  N (r , sˆ, t )dV
Photons scattered to direction ŝ'
Spatial photon gradient
Absorbed photons




  c s (r ) N (r , sˆ, t )dV   c a (r ) N (r , sˆ, t )dV
V
V
Photons scattered into direction ŝ from ŝ'
Light source q



  c s (r )  p ( sˆ  sˆ)N (r , sˆ, t )d dV   q (r , sˆ, t )dV
V
4
V
Time-Dependent Transport Equation
• Typically the transport equation is expressed in terms
of the radiance (I(r,ŝ,t) =N(r,ŝ,t)hnc) , and after
dropping the integrals

1 I (r , sˆ, t )


 sˆ  I (r , sˆ, t )  (  s  a ) I (r , sˆ, t ) 
c
t


 s  I (r , sˆ, t ) p( sˆ  sˆ)d   Q(r , sˆ, t )
4
Time-Independent Transport
Equation
• For the steady-state situation, we assume that
radiance is independent of time, and the transport
equation becomes

I (r , sˆ)


sˆ  I (r , sˆ) 
 (  s   a ) I (r , sˆ) 
ds


 s  I (r , sˆ) p( sˆ  sˆ)d   Q(r , sˆ)
4
Approximations
• The transport equation is difficult to solve
analytically.In order to find an analytical solution we
need to simplify the problem.
• Discretization methods


N (r , sˆ, t ) 
– Discrete ordinates method
– Kubelka-Monk theory
– Adding-doubling method
• Expansion methods
– Diffusion theory
• Probabilistic methods
– Monte Carlo simulations
 N (r , sˆ , t )
i
i
Diffusion approximation
• Expand the photon distribution in an
isotropic and a gradient part
1  
3 


ˆ
ˆ
N ( r , s, t ) 
r (r , t )  J (r , t )  s 

4 
c

• Where r(r,t) is the photon density


r (r , t )   N (r , sˆ, t )d
• And J(r,t) is the photon current density
(photon flux)
Fick’s
C
J 
x
st
1
law of diffusion
Movement or flux in response
to a concentration gradient in
a medium with diffusivity 
 
J (r , t )  cDr
 
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
1
1
D

3tr 3(a  (1  g )  s )
Diffusion approximation
Transport equation:

N (r , sˆ, t )

ˆ
dV


c
s


N
(
r
, sˆ, t )dV
V t
V




  c s (r ) N (r , sˆ, t )dV   c a (r ) N (r , sˆ, t )dV
V
V



  c s (r )  p ( sˆ  sˆ)N (r , sˆ, t )d dV   q (r , sˆ, t )dV
V
4
Photon distribution expansion:
V
1  
3 


ˆ
N (r , sˆ, t ) 
r
(
r
,
t
)

J
(
r
,
t
)

s

4 
c
Photon source expansion:
 
1


qo (r , t )  3q1 (r , t )  sˆ
q(r , sˆ, t ) 
4
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)
 1
1 r
1
3
   a r    J  qo  q1  sˆ
c t
c
c
c
• Assume a constant D and use the relation for
the fluence rate


(r , t )  chnr (r , t )
• To get





1 
2
 (r , t )  D  (r , t )   a  (r , t )  S o (r , t )  3D  S1 (r , t )
c t
Types of diffuse reflectance
measurements
intensity
It
I0
2
It
1.5
1
0.5
0
t=0
0
5
10
t (ns)
15
20
tissue
~ns
intensity
I0
frequency domain
(TD)
Time domain (TD)
intensity
Continuous wave (CW)
phase
shift
2
1.5
ac
1
0.5
dc
0
0
5
10
t (ns)
15
20
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
2


r
3 / 2
 (r , t )  c(4Dct)
exp 
  a ct 
 4 Dct

• This is known as the Green’s function
solution and can be used to solve more
complicated problems
Point source solution: frequency
domain
• Harmonic time dependence is given by factor
exp(-it), so that ∂/∂t -i
• Diffusion equation takes the form




 i
2
 (r )  D  (r )   a  (r )  So (r ) 
c


S o (r )
2
2
(  k ) (r )  
,
cD
i  c a
2
where k 
cD
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
intensity
frequency domain
(TD)
2
1.5
 (r ,  ) 
P( ) e
4D r
ikr
k2 
with
1
i  c a
cD
0.5
0
0
5
 r (  a / D )1 / 2
PDC e
 I DC
4D
r
Abs[ (r ,  )]  AC (r ,  )  ac amplit ude
Abs[ (r ,0)]  DC (r ) 
20
15
20
intensity
phase
shift
2
1.5
ln(r * I DC )  r Re[k ]  ln[4D]
t (ns)
15
tissue
Arg[ (r ,  )]  phase
And it follows t hat
ac
1
phase  rIm(k)
2
10
0.5
intercept (s’)
phase
ln(r2*Idc)
0
slope (a, s’)
r
0
slope (a, s’)
intercept = 0
r
5
10
t (ns)
dc
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
– Illuminate tissue at 2 wavelengths
straddling isosbestic point (eg. 650
and 805 nm)
• Isosbestic point: wavelength where
Hb and HbO2 spectra cross.
10
1
0.1
– Detect signal transmitted through finger
0.01
300
400
500
600
700
800
900
1000
1100 1200 1300
• Isolate varying signal due to
2
2
2




ln
10
*

HbO


pulsatile flow (arterial blood)
a
HbO2
2
Hb Hb
1
1
• Assume detected signal is proportional
a1  ln10*  HbO
HbO


2
Hb Hb
2
to absorption coefficient
(Two measurements, two unknowns)
• Calibrate instrument by correlating detected signal to arterial
saturation measurements from blood samples

ArterialO 2 saturation
HbO2 

HbO2  Hb


*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
source
detector
source
detector
source
detector
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
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
absorption coefficient (cm-1)
ultraviolet
410 nm
near infrared
600
770 nm
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)
1300
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?
intensity (a.u.)
Frequency-domain spectroscopy (FD)
2
1.5
1
0.5
0
0
5
10
15
t (ns)
20
intensity (a.u.)
tissue
phase
shift
2
1.5
ac
1
0.5
dc
0
0
5
10
t (ns)
15
20
Diffusion equation: frequency
domain
• Harmonic time dependence is given by factor
exp(-it), so that ∂/∂t -i
• Diffusion equation takes the form




 i
2
 (r )  D  (r )   a  (r )  So (r ) 
c


S o (r )
2
2
(  k ) (r )  
,
cD
i  c a
2
where k 
cD
1
1
D

3tr 3(a  (1  g )  s )
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
intensity
P ( ) e
 (r ,  ) 
4D r
frequency domain
(TD)
2
i  c a
k 
cD
ikr
1.5
2
with
1
0.5
0
 r (  a / D )1 / 2
PDC e
Abs[ (r ,0)]  DC (r ) 
 I DC
4D
r
Abs[ (r ,  )]  AC (r ,  )  ac amplitude
5
10
t (ns)
15
20
15
20
tissue
intensity
Arg[ (r ,  )]  phase
And it follows that
ln(r * I DC )   r Re[k ]  ln[4D ]
0
phase
shift
2
1.5
phase  rIm[k]
ac
1
0.5
0
phase
ln(r*Idc)
intercept (s’)
slope (a, s’)
r
0
slope (a, s’)
intercept = 0
r
5
10
t (ns)
dc
TISSUE OXIMETRY
Time-domain oximetry
Miwa et al., Proc. SPIE 2389, 142 (1995)
Configuration for tissue oximetry
detector optical fiber
detector
RF
electronics
source optical fibers
laser driver
measuring probe
laser
diodes
multiplexing circuit
main box
100
saturation (%)
90
80
70
60
50
40
ischemia
30
20
0
2
4
6
8
10
12
14
16
18
HbO2 and Hb (M)
Frequency-domain oximetry
90
80
70
60
50
40
ischemia
30
20
0
2
4
5
0.24
4.8
830nm
0.22
10
12
750nm
0.2
18
4.4
4.2
0.19
ischemia
0.18
16
750nm
4.6
0.21
14
time (min)
0.25
0.23
8
s’ (1/cm)
a (1/cm)
time (min)
6
4
830nm
ischemia
3.8
0.17
0
2
4
6
8
10
12
time (min)
14
16
18
0
2
4
6
8
10
12
time (min)
14
16
18