Extended Cellular Potts Model - Cellular Potts Models: Multiscale Extensions and Biological Applications - page 57

However, as pointed out in [165, 262], little has been done to obtain an
accurate description of the internal state of individuals whose evolution di-
rectly controls their phenomenology. In fact, microscopic mechanisms have
only been modeled to influence the behavior of CPM objects via appropri-
ate extra terms in the Hamiltonian [254], or with changes in cell mitotic rates
[203] or types. In this regard, an intriguing approach has been used in a model
for chick limb-bud development in [75, 76], where a threshold local concen-
tration of activator TGF- drove the dierentiation of responsive cells in the
active zone, eventually varying their properties (i.e., they became fibronectin-
producing and upregulated the intercellular adhesion). In interesting models
of thrombus formation [414, 415, 416], the activation of platelets was instead
controlled by the level of chemical components, which derived from biochem-
ical reactions of coagulation pathways in blood flow and on cell surface. Such
approaches have given qualitatively correct results, and represent a useful
starting point for further improvements of the method, aiming at accurately
linking the cell-level phenomenology of simulated individuals to their micro-
scopic level of organization, i.e., able to give to the CPM environment a nested
characteristic.
Our main assumption is that the internal state of a biological individual
(i.e., the microscopic level) regulates its biophysical properties (described by
mesoscopic Potts coecients) and not directly its dynamics (described by the
terms in the Hamiltonian). An analogous idea was introduced in [186] for a
specific case, but was barely developed. On the basis of this hypothesis, the
comprehensive and general procedure to nest microscopic models for individ-
ual internal states within the mesoscopic CPM reads as follows.
Let denote a certain discrete object (which, as seen, can now represent
a whole individual or one compartment): we define its internal state vector
s 2R n . The length n of s is defined by the number of internal factors (i.e.,
nutrients, proteins, growth factors, . . . , all described by continuous objects)
considered in the microscopic model, and represents a sort of internal degree
of freedom of . Each component s ;l , where l = 1;:::;n, can be local (i.e.,
per site) and/or time-dependent (i.e., linked to a specific regulatory pathway,
which needs to be modeled, as it will be explained hereafter). Hence, in general,
s = s (x;t), where x 2 . The spatial localization of s is mandatory to
accurately represent internal inhomogeneities of , while its time dependence
is necessary to reproduce its microscopic evolution.
For any , let us consider a generic Potts coecient
2f i ;T ; k ;:::g:
We now dene s 2R m , where m n, the subvector of s whose compo-
nents influence the biophysical property of described by . Therefore, the
spatiotemporal evolution of can be expressed as

(x 2 ;t) = (s (x;t)) = f (s (x;t));

(4.9)

where f :R m 7!Ris a continuous function, which obviously needs to be