<s>
Photon	O
transport	O
in	O
biological	O
tissue	O
can	O
be	O
equivalently	O
modeled	O
numerically	O
with	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
or	O
analytically	O
by	O
the	O
radiative	O
transfer	O
equation	O
(	O
RTE	O
)	O
.	O
</s>
<s>
Overall	O
,	O
solutions	O
to	O
the	O
diffusion	O
equation	O
for	O
photon	O
transport	O
are	O
more	O
computationally	O
efficient	O
,	O
but	O
less	O
accurate	O
than	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
.	O
</s>
<s>
The	O
flow	O
of	O
radiation	O
energy	O
through	O
a	O
small	O
area	O
element	O
in	O
the	O
radiation	O
field	O
can	O
be	O
characterized	O
by	O
radiance	B-Application
.	O
</s>
<s>
Radiance	B-Application
is	O
defined	O
as	O
energy	O
flow	O
per	O
unit	O
normal	O
area	O
per	O
unit	O
solid	O
angle	O
per	O
unit	O
time	O
.	O
</s>
<s>
Several	O
other	O
important	O
physical	O
quantities	O
are	O
based	O
on	O
the	O
definition	O
of	O
radiance	B-Application
:	O
</s>
<s>
Current	O
density	O
(	O
energy	O
flux	B-General_Concept
)	O
.	O
</s>
<s>
The	O
RTE	O
is	O
a	O
differential	O
equation	O
describing	O
radiance	B-Application
.	O
</s>
<s>
In	O
the	O
RTE	O
,	O
six	O
different	O
independent	O
variables	O
define	O
the	O
radiance	B-Application
at	O
any	O
spatial	O
and	O
temporal	O
point	O
(	O
,	O
,	O
and	O
from	O
,	O
polar	O
angle	O
and	O
azimuthal	O
angle	O
from	O
,	O
and	O
)	O
.	O
</s>
<s>
These	O
assumptions	O
lead	O
to	O
the	O
diffusion	B-Algorithm
theory	I-Algorithm
(	O
and	O
diffusion	O
equation	O
)	O
for	O
photon	O
transport	O
.	O
</s>
<s>
Two	O
assumptions	O
permit	O
the	O
application	O
of	O
diffusion	B-Algorithm
theory	I-Algorithm
to	O
the	O
RTE	O
:	O
</s>
<s>
Likewise	O
,	O
after	O
numerous	O
scattering	O
events	O
,	O
few	O
absorption	O
events	O
will	O
occur	O
and	O
the	O
radiance	B-Application
will	O
become	O
nearly	O
isotropic	O
.	O
</s>
<s>
In	O
a	O
primarily	O
scattering	O
medium	O
,	O
the	O
time	O
for	O
substantial	O
current	O
density	O
change	O
is	O
much	O
longer	O
than	O
the	O
time	O
to	O
traverse	O
one	O
transport	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
over	O
one	O
transport	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
,	O
the	O
fractional	O
change	O
in	O
current	O
density	O
is	O
much	O
less	O
than	O
unity	O
.	O
</s>
<s>
Radiance	B-Application
can	O
be	O
expanded	O
on	O
a	O
basis	O
set	O
of	O
spherical	O
harmonics	O
n	O
,	O
m	O
.	O
In	O
diffusion	B-Algorithm
theory	I-Algorithm
,	O
radiance	B-Application
is	O
taken	O
to	O
be	O
largely	O
isotropic	O
,	O
so	O
only	O
the	O
isotropic	O
and	O
first-order	O
anisotropic	O
terms	O
are	O
used	O
:	O
</s>
<s>
Radiance	B-Application
is	O
expressed	O
with	O
4	O
terms	O
;	O
one	O
for	O
n	O
=	O
0	O
(	O
the	O
isotropic	O
term	O
)	O
and	O
3	O
terms	O
for	O
n	O
=	O
1	O
(	O
the	O
anisotropic	O
terms	O
)	O
.	O
</s>
<s>
Substituting	O
the	O
above	O
expression	O
for	O
radiance	B-Application
,	O
the	O
RTE	O
can	O
be	O
respectively	O
rewritten	O
in	O
scalar	O
and	O
vector	O
forms	O
as	O
follows	O
(	O
The	O
scattering	O
term	O
of	O
the	O
RTE	O
is	O
integrated	O
over	O
the	O
complete	O
solid	O
angle	O
.	O
</s>
<s>
The	O
diffusion	O
approximation	O
is	O
limited	O
to	O
systems	O
where	O
reduced	O
scattering	O
coefficients	O
are	O
much	O
larger	O
than	O
their	O
absorption	O
coefficients	O
and	O
having	O
a	O
minimum	O
layer	O
thickness	O
of	O
the	O
order	O
of	O
a	O
few	O
transport	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
.	O
</s>
<s>
Using	O
the	O
second	O
assumption	O
of	O
diffusion	B-Algorithm
theory	I-Algorithm
,	O
we	O
note	O
that	O
the	O
fractional	O
change	O
in	O
current	O
density	O
over	O
one	O
transport	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
is	O
negligible	O
.	O
</s>
<s>
The	O
vector	O
representation	O
of	O
the	O
diffusion	B-Algorithm
theory	I-Algorithm
RTE	O
reduces	O
to	O
Fick	O
's	O
law	O
,	O
which	O
defines	O
current	O
density	O
in	O
terms	O
of	O
the	O
gradient	O
of	O
fluence	O
rate	O
.	O
</s>
<s>
The	O
direction-integrated	O
radiance	B-Application
at	O
the	O
boundary	O
and	O
directed	O
into	O
the	O
medium	O
is	O
equal	O
to	O
the	O
direction-integrated	O
radiance	B-Application
at	O
the	O
boundary	O
and	O
directed	O
out	O
of	O
the	O
medium	O
multiplied	O
by	O
reflectance	O
:	O
</s>
<s>
The	O
diffusion	O
approximation	O
gives	O
an	O
expression	O
for	O
radiance	B-Application
in	O
terms	O
of	O
fluence	O
rate	O
and	O
current	O
density	O
.	O
</s>
<s>
Convert	O
the	O
pencil	O
beam	O
into	O
an	O
isotropic	O
point	O
source	O
at	O
a	O
depth	O
of	O
one	O
transport	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
'	O
below	O
the	O
surface	O
and	O
power	O
=	O
'	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
of	O
photon	O
transport	O
,	O
though	O
time	O
consuming	O
,	O
will	O
accurately	O
predict	O
photon	O
behavior	O
in	O
a	O
scattering	O
medium	O
.	O
</s>
<s>
For	O
a	O
photon	O
beam	O
incident	O
on	O
a	O
medium	O
of	O
limited	O
depth	O
,	O
error	O
due	O
to	O
the	O
diffusion	O
approximation	O
is	O
most	O
prominent	O
within	O
one	O
transport	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
of	O
the	O
location	O
of	O
photon	O
incidence	O
(	O
where	O
radiance	B-Application
is	O
not	O
yet	O
isotropic	O
)	O
(	O
Figure	O
3	O
)	O
.	O
</s>
