<s>
Modeling	O
photon	O
propagation	O
with	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
is	O
a	O
flexible	O
yet	O
rigorous	O
approach	O
to	O
simulate	O
photon	O
transport	O
.	O
</s>
<s>
This	O
is	O
equivalent	O
to	O
modeling	O
photon	O
transport	O
analytically	O
by	O
the	O
radiative	B-Algorithm
transfer	I-Algorithm
equation	I-Algorithm
(	O
RTE	O
)	O
,	O
which	O
describes	O
the	O
motion	O
of	O
photons	O
using	O
a	O
differential	O
equation	O
.	O
</s>
<s>
However	O
,	O
closed-form	O
solutions	O
of	O
the	O
RTE	O
are	O
often	O
not	O
possible	O
;	O
for	O
some	O
geometries	O
,	O
the	O
diffusion	B-Algorithm
approximation	I-Algorithm
can	O
be	O
used	O
to	O
simplify	O
the	O
RTE	O
,	O
although	O
this	O
,	O
in	O
turn	O
,	O
introduces	O
many	O
inaccuracies	O
,	O
especially	O
near	O
sources	O
and	O
boundaries	O
.	O
</s>
<s>
In	O
contrast	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
can	O
be	O
made	O
arbitrarily	O
accurate	O
by	O
increasing	O
the	O
number	O
of	O
photons	O
traced	O
.	O
</s>
<s>
For	O
example	O
,	O
see	O
the	O
movie	O
,	O
where	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
of	O
a	O
pencil	O
beam	O
incident	O
on	O
a	O
semi-infinite	O
medium	O
models	O
both	O
the	O
initial	O
ballistic	O
photon	O
flow	O
and	O
the	O
later	O
diffuse	O
propagation	O
.	O
</s>
<s>
The	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
necessarily	O
statistical	O
and	O
therefore	O
requires	O
significant	O
computation	O
time	O
to	O
achieve	O
precision	O
.	O
</s>
<s>
In	O
addition	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
can	O
keep	O
track	O
of	O
multiple	O
physical	O
quantities	O
simultaneously	O
,	O
with	O
any	O
desired	O
spatial	O
and	O
temporal	O
resolution	O
.	O
</s>
<s>
Thus	O
,	O
while	O
computationally	O
inefficient	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
often	O
considered	O
the	O
standard	O
for	O
simulated	O
measurements	O
of	O
photon	O
transport	O
for	O
many	O
biomedical	O
applications	O
.	O
</s>
<s>
Microscopy	O
techniques	O
(	O
including	O
confocal	O
,	O
two-photon	O
,	O
and	O
optical	B-Application
coherence	I-Application
tomography	I-Application
)	O
have	O
the	O
ability	O
to	O
image	O
these	O
properties	O
with	O
high	O
spatial	O
resolution	O
,	O
but	O
,	O
since	O
they	O
rely	O
on	O
ballistic	O
photons	O
,	O
their	O
depth	O
penetration	O
is	O
limited	O
to	O
a	O
few	O
millimeters	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
provide	O
a	O
flexible	O
framework	O
that	O
has	O
been	O
used	O
by	O
different	O
techniques	O
to	O
reconstruct	O
optical	O
properties	O
deep	O
within	O
tissue	O
.	O
</s>
<s>
Photoacoustic	B-Algorithm
tomography	I-Algorithm
In	O
PAT	O
,	O
diffuse	O
laser	O
light	O
is	O
absorbed	O
which	O
generates	O
a	O
local	O
temperature	O
rise	O
.	O
</s>
<s>
Diffuse	B-Algorithm
optical	I-Algorithm
tomography	I-Algorithm
DOT	O
is	O
an	O
imaging	O
technique	O
that	O
uses	O
an	O
array	O
of	O
near-infrared	O
light	O
sources	O
and	O
detectors	O
to	O
measure	O
optical	O
properties	O
of	O
biological	O
tissues	O
.	O
</s>
<s>
The	O
forward	O
model	O
is	O
often	O
generated	O
using	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Due	O
to	O
the	O
nature	O
of	O
PDT	O
,	O
it	O
is	O
useful	O
to	O
use	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
for	O
modeling	O
scattering	O
and	O
absorption	O
in	O
the	O
tissue	O
in	O
order	O
to	O
ensure	O
appropriate	O
levels	O
of	O
light	O
are	O
delivered	O
to	O
activate	O
chemotherapy	O
agents	O
.	O
</s>
<s>
Presented	O
here	O
is	O
a	O
model	O
of	O
a	O
photon	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
in	O
a	O
homogeneous	O
infinite	O
medium	O
.	O
</s>
<s>
Responses	O
to	O
arbitrary	O
source	O
geometries	O
can	O
be	O
constructed	O
using	O
the	O
method	O
of	O
Green	O
's	O
functions	O
(	O
or	O
convolution	B-Language
,	O
if	O
enough	O
spatial	O
symmetry	O
exists	O
)	O
.	O
</s>
<s>
Responses	O
to	O
sources	O
with	O
arbitrary	O
time	O
profiles	O
can	O
then	O
be	O
modeled	O
through	O
convolution	B-Language
in	O
time	O
.	O
</s>
<s>
Note	O
that	O
as	O
Monte	O
Carlo	O
modeling	O
is	O
a	O
statistical	O
process	O
involving	O
random	O
numbers	O
,	O
we	O
will	O
be	O
using	O
the	O
variable	O
ξ	O
throughout	O
as	O
a	O
pseudo-random	B-Algorithm
number	I-Algorithm
for	O
many	O
calculations	O
.	O
</s>
<s>
Below	O
is	O
a	O
basic	O
form	O
of	O
photon	O
step	O
size	O
selection	O
(	O
derived	O
using	O
the	O
inverse	B-Algorithm
distribution	I-Algorithm
method	I-Algorithm
and	O
the	O
Beer	O
–	O
Lambert	O
law	O
)	O
from	O
which	O
we	O
use	O
for	O
our	O
homogeneous	O
model	O
:	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
of	O
photon	O
migration	O
in	O
turbid	O
media	O
is	O
a	O
highly	O
parallelizable	O
problem	O
,	O
where	O
a	O
large	O
number	O
of	O
photons	O
are	O
propagated	O
independently	O
,	O
but	O
according	O
to	O
identical	O
rules	O
and	O
different	O
random	O
number	O
sequences	O
.	O
</s>
<s>
The	O
parallel	O
nature	O
of	O
this	O
special	O
type	O
of	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
renders	O
it	O
highly	O
suitable	O
for	O
execution	O
on	O
a	O
graphics	O
processing	O
unit	O
(	O
GPU	O
)	O
.	O
</s>
<s>
The	O
release	O
of	O
programmable	O
GPUs	O
started	O
such	O
a	O
development	O
,	O
and	O
since	O
2008	O
there	O
have	O
been	O
a	O
few	O
reports	O
on	O
the	O
use	O
of	O
GPU	O
for	O
high-speed	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
of	O
photon	O
migration	O
.	O
</s>
<s>
One	O
example	O
is	O
the	O
"	O
GPU	O
Cluster	O
MCML	O
,	O
"	O
which	O
can	O
be	O
downloaded	O
from	O
the	O
authors	O
 '	O
website	O
(	O
Monte	B-Algorithm
Carlo	I-Algorithm
Simulation	I-Algorithm
of	O
Light	O
Transport	O
in	O
Multi-layered	O
Turbid	O
Media	O
Based	O
on	O
GPU	O
Clusters	O
)	O
:	O
</s>
