1
PHYS 6715 - Lecture VIII
1
Lecture VIII
The Diffusion Model
1.
The diffusion approximation
2.
A boundary-value problem
3.
The Green function method
4.
Similarity principle
5.
Boundary condition
6.
Solving the BVP
PHYS 6715 - Lecture VIII
2
Different components of light
•
The light radiance L in a turbid medium can be separated into two
major components:
one with unscattered light
Æ
the primary component L
p
one with scattered light
Æ
L
s
•
The distribution of the primary component L
p
is correlated with the
incident light on the turbid medium but reduces exponentially as
exp{-
μ
t
d} with d as the pathlength
Æ
for
μ
t
=10(mm
-1
), only about 10% photons from an incident light
beam or internal light source belong to the primary component after
traveling over a distance of d
c
=0.23mm
VIII–1
Diffusion Approximation
PHYS 6715 - Lecture VIII
3
Different components of light
•
The scattered component has a much more complicated distribution so
we further divide it into two sub-components:
those being scattering only a few times
Æ
“snake” sub-component
those being scattered many times
Æ
diffuse sub-component
•
The snake sub-component still has a strong correlation with the
incident light and can be detected using the coherence technique
Æ
Optical coherence tomography (OCT)
•
The diffuse sub-component essentially lost its correlation with the
incident light and can be well described by a diffusion approximation to
the radiative transfer equation
Æ
Diffuse Optical Tomography (DOT)
VIII–1
Diffusion Approximation
PHYS 6715 - Lecture VIII
4
VIII–1
Diffusion Approximation
Different components of light
Diffusion approximation of radiative transfer equation – neglect the
portion of light scattered only a few time or less.
Diffuse component
Snake
component
absorbed
component
incident
light
tissue
primary component
5
VIII–1
Diffusion Approximation
Separation of RTE
Let L(
r
,
s
)=L
p
(
r
,
s
) + L
s
(
r
,
s
) and denote
as
4
( ,
)
(
) ( ,
)
( ,
') ( ,
')
'
( ,
)
s
a
s
e
L
L
p
L
d
s
π
μ
μ
μ
ε
∂
= −
+
+
Ω +
∂
∫
r s
r s
s s
r s
r s
( , )
( , )
p
t
p
L
L
s
μ
∂
= −
∂
r s
r s
4
( , )
( , )
( , ')
( , ')
'
( , )
( , )
s
t
s
s
s
p
e
L
L
p
L
d
s
π
μ
μ
ε
ε
∂
= −
+
Ω +
+
∂
∫
r s
r s
s s
r s
r s
r s
( ,
)
L
s
∂
∂
r s
⋅∇
s
A. Ishimaru,
Wave propagation and scattering in random media
, vol.1 (1978)
4
( , )
( , ')
( , ')
'
p
s
p
p
L
d
π
ε
μ
=
Ω
∫
r s
s s
r s
PHYS 6715 - Lecture VIII
6
VIII–1
Diffusion Approximation
The scattered component
•
The 2
nd
equation containing both of the primary and scattered
components is much harder to solve
•
The way out is to replace the scattered component with its diffuse
sub-component or ignore the snake sub-component
L
p
(
r
,
s
)
Φ
s
(
r
)

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