Biomedical Engineering Reference
In-Depth Information
Basic CPM
1.1 The CPM Domain
All cellular Potts model approaches include a list of objects, a description
of their interactions, and rules for their dynamics. The CPM domains are
d-dimensional lattices R
d
, where d = 1; 2; 3. The term lattice denes a
regular repeated graph, formed by identical d-dimensional closed grid sites
x 2R
d
, and characterized by periodic or fixed boundary conditions in each
direction. The volumetric extension of is equal to the total number of its
sites, that therefore represent the basic unit of length of the system. Each
site x 2 is uniquely identied by its location and is labeled by an integer,
(x) 2N, where can be interpreted as a degenerate spin value coming from
the original Ising approach [85, 199, 315]; see Figure 1.1(A). With an abuse
of notation, x also usually denotes the closed elementary spatial region (e.g.,
the voxel ) centered in x. As classically adopted in CPM models, the border of
a lattice site x is denoted by @x, one of its neighbors by x
0
, while its overall
neighborhood by
0
x
, i.e.,
0
x
= fx
0
2 : x
0
is a neighbor of xg.
Objects in the CPM are either discrete or continuous: this coexistence
provides to the method its hybrid characteristic. Discrete objects
are finite,
spatially extended lattice subdomains of contiguous sites with the same spin
, i.e.,
= fx 2 : (x) = g:
They are therefore undifferentiated functional units that, in basic CPMs, rep-
resent single biological elements with the typical mesoscopic length scale of
a cell. Examples are bacteria, unicellular organisms, single cells, ECM fibers,
or other substrates; see Figure 1.1(B-C). Trivially, a collection of N discrete
individuals
is dened by N integer spin = 1; 2;:::;N. The borders be-
tween sites with different degenerated spins that are thus shared between a
couple of objects define their membranes, i.e.,
[
(@x \@x
0
):
@
=
x
2
;
x
0
2
0
x
:
x
0
2
Each unit
has a set of attributes (both geometrical, such as volume and sur-
face, and biophysical, such as velocity and elasticity), and an associated type
3
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