<s>
In	O
computational	O
and	O
mathematical	O
biology	O
,	O
a	O
biological	O
lattice-gas	O
cellular	O
automaton	O
(	O
BIO-LGCA	B-Algorithm
)	O
is	O
a	O
discrete	O
model	O
for	O
moving	O
and	O
interacting	O
biological	O
agents	O
,	O
a	O
type	O
of	O
cellular	O
automaton	O
.	O
</s>
<s>
The	O
BIO-LGCA	B-Algorithm
is	O
based	O
on	O
the	O
lattice-gas	O
cellular	O
automaton	O
(	O
LGCA	O
)	O
model	O
used	O
in	O
fluid	O
dynamics	O
.	O
</s>
<s>
A	O
BIO-LGCA	B-Algorithm
model	O
describes	O
cells	O
and	O
other	O
motile	O
biological	O
agents	O
as	O
point	O
particles	O
moving	O
on	O
a	O
discrete	O
lattice	O
,	O
thereby	O
interacting	O
with	O
nearby	O
particles	O
.	O
</s>
<s>
Contrary	O
to	O
classic	O
cellular	O
automaton	O
models	O
,	O
particles	O
in	O
BIO-LGCA	B-Algorithm
are	O
defined	O
by	O
their	O
position	O
and	O
velocity	O
.	O
</s>
<s>
BIO-LGCA	B-Algorithm
applications	O
include	O
cancer	O
invasion	O
and	O
cancer	O
progression	O
.	O
</s>
<s>
As	O
are	O
all	O
cellular	O
automaton	O
models	O
,	O
a	O
BIO-LGCA	B-Algorithm
model	O
is	O
defined	O
by	O
a	O
lattice	O
,	O
a	O
state	O
space	O
,	O
a	O
neighborhood	O
,	O
and	O
a	O
rule	O
.	O
</s>
<s>
Particles	O
are	O
restricted	O
to	O
occupy	O
only	O
certain	O
positions	O
,	O
typically	O
resulting	O
from	O
a	O
regular	O
and	O
periodic	O
tesselation	B-Algorithm
of	O
space	O
.	O
</s>
<s>
In	O
BIO-LGCA	B-Algorithm
,	O
multiple	O
particles	O
with	O
different	O
velocities	O
may	O
occupy	O
a	O
single	O
lattice	O
site	O
,	O
as	O
opposed	O
to	O
classic	O
cellular	O
automaton	O
models	O
,	O
where	O
typically	O
only	O
a	O
single	O
cell	O
can	O
reside	O
in	O
every	O
lattice	O
node	O
simultaneously	O
.	O
</s>
<s>
As	O
every	O
cellular	O
automaton	O
,	O
BIO-LGCA	B-Algorithm
evolves	O
in	O
discrete	O
time	O
steps	O
.	O
</s>
<s>
In	O
BIO-LGCA	B-Algorithm
,	O
the	O
rule	O
is	O
divided	O
into	O
two	O
steps	O
,	O
a	O
probabilistic	O
interaction	O
step	O
followed	O
by	O
a	O
deterministic	O
transport	O
step	O
.	O
</s>
<s>
Typically	O
,	O
particles	O
are	O
assumed	O
to	O
obey	O
an	O
exclusion	B-General_Concept
principle	I-General_Concept
,	O
such	O
that	O
no	O
more	O
than	O
one	O
particle	O
may	O
occupy	O
a	O
single	O
velocity	O
channel	O
at	O
a	O
lattice	O
site	O
simultaneously	O
.	O
</s>
<s>
Since	O
an	O
exact	O
treatment	O
of	O
a	O
stochastic	O
agent-based	O
model	O
quickly	O
becomes	O
unfeasible	O
due	O
to	O
high-order	O
correlations	O
between	O
all	O
agents	O
,	O
the	O
general	O
method	O
of	O
analyzing	O
a	O
BIO-LGCA	B-Algorithm
model	O
is	O
to	O
cast	O
it	O
into	O
an	O
approximate	O
,	O
deterministic	O
finite	O
difference	O
equation	O
(	O
FDE	O
)	O
describing	O
the	O
mean	O
dynamics	O
of	O
the	O
population	O
,	O
then	O
performing	O
the	O
mathematical	O
analysis	O
of	O
this	O
approximate	O
model	O
,	O
and	O
comparing	O
the	O
results	O
to	O
the	O
original	O
BIO-LGCA	B-Algorithm
model	O
.	O
</s>
<s>
However	O
,	O
the	O
term	O
on	O
the	O
right	O
,	O
is	O
highly	O
nonlinear	O
on	O
the	O
occupation	O
numbers	O
of	O
both	O
the	O
lattice	O
site	O
and	O
the	O
lattice	O
sites	O
within	O
the	O
interaction	O
neighborhood	O
,	O
due	O
to	O
the	O
form	O
of	O
the	O
transition	O
probability	O
and	O
the	O
statistics	O
of	O
particle	O
placement	O
within	O
velocity	O
channels	O
(	O
for	O
example	O
,	O
arising	O
from	O
an	O
exclusion	B-General_Concept
principle	I-General_Concept
imposed	O
on	O
channel	O
occupations	O
)	O
.	O
</s>
<s>
In	O
order	O
to	O
cast	O
it	O
into	O
a	O
more	O
familiar	O
finite	O
difference	O
equation	O
with	O
temporal	O
increments	O
only	O
,	O
a	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
can	O
be	O
applied	O
on	O
both	O
sides	O
of	O
the	O
equation	O
.	O
</s>
<s>
After	O
applying	O
the	O
shift	O
theorem	O
and	O
isolating	O
the	O
term	O
with	O
a	O
temporal	O
increment	O
on	O
the	O
left	O
,	O
one	O
obtains	O
the	O
lattice-Boltzmann	O
equation	O
where	O
is	O
the	O
imaginary	O
unit	O
,	O
is	O
the	O
size	O
of	O
the	O
lattice	O
along	O
one	O
dimension	O
,	O
is	O
the	O
Fourier	O
wave	O
number	O
,	O
and	O
denotes	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Constructing	O
a	O
BIO-LGCA	B-Algorithm
for	O
the	O
study	O
of	O
biological	O
phenomena	O
mainly	O
involves	O
defining	O
appropriate	O
transition	O
probabilities	O
for	O
the	O
interaction	O
operator	O
,	O
though	O
precise	O
definitions	O
of	O
the	O
state	O
space	O
(	O
to	O
consider	O
several	O
cellular	O
phenotypes	O
,	O
for	O
example	O
)	O
,	O
boundary	O
conditions	O
(	O
for	O
modeling	O
phenomena	O
in	O
confined	O
conditions	O
)	O
,	O
neighborhood	O
(	O
to	O
match	O
experimental	O
interaction	O
ranges	O
quantitatively	O
)	O
,	O
and	O
carrying	O
capacity	O
(	O
to	O
simulate	O
crowding	O
effects	O
for	O
given	O
cell	O
sizes	O
)	O
may	O
be	O
important	O
for	O
specific	O
applications	O
.	O
</s>
<s>
BIO-LGCA	B-Algorithm
models	O
have	O
been	O
used	O
to	O
study	O
several	O
cellular	O
,	O
biophysical	O
and	O
medical	O
phenomena	O
.	O
</s>
<s>
Angiogenesis	O
:	O
an	O
in	O
vitro	O
experiment	O
with	O
endothelial	O
cells	O
and	O
BIO-LGCA	B-Algorithm
simulation	O
observables	O
were	O
compared	O
to	O
determine	O
the	O
processes	O
involved	O
during	O
angiogenesis	O
and	O
their	O
weight	O
.	O
</s>
<s>
Active	O
fluids	O
:	O
the	O
macroscopic	O
physical	O
properties	O
of	O
a	O
population	O
of	O
particles	O
interacting	O
through	O
polar	O
alignment	O
interactions	O
were	O
investigated	O
using	O
a	O
BIO-LGCA	B-Algorithm
model	O
.	O
</s>
<s>
Epidemiology	O
:	O
a	O
spatial	O
SIR	O
BIO-LGCA	B-Algorithm
model	O
was	O
used	O
to	O
study	O
the	O
effect	O
of	O
different	O
vaccination	O
strategies	O
,	O
and	O
the	O
effect	O
of	O
approximating	O
a	O
spatial	O
epidemic	O
with	O
a	O
non-spatial	O
model	O
.	O
</s>
<s>
Cell	O
jamming	O
:	O
in	O
vitro	O
and	O
Bio-LGCA	B-Algorithm
models	O
were	O
used	O
for	O
studying	O
metastatic	O
behavior	O
in	O
breast	O
cancer	O
.	O
</s>
<s>
The	O
BIO-LGCA	B-Algorithm
model	O
revealed	O
that	O
metastasis	O
may	O
exhibit	O
different	O
behaviors	O
,	O
such	O
as	O
random	O
gas-like	O
,	O
jammed	O
solid-like	O
,	O
and	O
correlated	O
fluid-like	O
states	O
,	O
depending	O
on	O
the	O
adhesivity	O
level	O
among	O
cells	O
,	O
ECM	O
density	O
,	O
and	O
cell-ECM	O
interactions	O
.	O
</s>
