Biomedical Engineering Reference
In-Depth Information
forces (both effective and generalized) that act on the simulated individuals.
All these contributions are described with the same architecture [99]:
H
force
(t) =
X
x
2
X
k
(
x
)
(t) F
k
(t) r
x
;
(1.8)
kforce
where r
x
= (i
x
;j
x
;k
x
)
T
is the position vector of lattice site x, which is the
application point of force F
k
, and
k
is the relative Potts parameter, which
measures the effective strength of the force on object
(i.e., also in this case
it can be common for all the objects of the same type:
k
=
k
(
)
) .
The most diffused example in CPM applications is the action due to the
presence of extracellular chemical substances (which are described as continu-
ous CPM objects) on a population of cells (which are typical discrete objects):
(t) =
X
X
H
chemical
force
chem
(
x
)
(t)c(x;t);
(1.9)
x
2
where c(x;t), dened with Equation (1.1), is the concentration of the chemical
sensed by the cell in x (which can be modeled as the local chemical concen-
tration in site x itself [203, 259], or in its neighborhood [350, 351]), and the
Potts coecient
chem
is, in this case, an effective chemical potential of cell
. Moreover, the net energy difference caused by such a chemical force is
(
x
)!(
x
0
)
=
chem
H
chemical
force
(
x
)
[c(x;t) c(x
0
;t)];
(1.10)
where x 2
and x
0
2
are the two neighboring lattice sites randomly
selected during the trial update at time t [342].
If
chem
is a constant,
has a linear chemical sensitivity. In particular,
chem
> 0 yields to its motion up the gradient of c (which is thus a chemoat-
tractant, and the relative force is called
chemotaxis), while
chem
< 0 yields
to its motion in the opposite direction (and c is a chemorepellent ). Moreover,
if c is a nondiffusive fixed substrate, Equation (1.9) is a representation of a
haptotactic force, as in [246, 395].
The importance of each term in the Hamiltonian (i.e., of each simulated
biological mechanism) is defined by the magnitude of the relative Potts pa-
rameters, which act as a sort of penalty coecient (whose unit of measure is
obviously established by the type of the relative energetic contribution). In
this respect, a crucial role in determining the evolution of the system is there-
fore played by the hierarchy of the Potts coecients, and not by their exact
values. This consideration allows one to overcome one of the main limitations
of the CPM: given its energetic nature, a direct one-to-one correspondence
between Potts parameters and experimental quantities is not straightforward,
it being only possible to infer empirical relationships. This issue will be ap-
proached more in details in Chapter 4. However, it is useful to underline that,
in the following, we will use the terms low and/or high to set the place of a
single parameter in the overall hierarchy of Potts coecients.
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