Biomedical Engineering Reference
In-Depth Information
of information from finer to coarser levels, where the kinetics of molecular sub-
cellular networks strongly deliver the mesoscopic cell properties, described, as
explained, by typical Potts parameters. The resulting modeling framework,
utilizing the advantage of both discrete and continuous techniques, is there-
fore able to span all the different biological spatiotemporal scales involved,
which range from 10 8 m and 10 9 s for the intracellular molecular processes
to 10 6 m and hours for the cell-level phenomenology.
The sample applications given in the following chapters will be useful to
demonstrate the ecacy of the above-defined procedure and to clarify the
complex notation adopted in this section.
4.4 Motility of Individuals
The motility of individuals plays an important role in all biological phenom-
ena, both physiological (such as embryo development and organogenesis, or-
ganism growth and survival, or wound healing) and pathological (such as in-
flammation and atherosclerosis, cancer invasion, or metastatization). An accu-
rate description of the motility of individuals is therefore a fundamental issue
for all computational approaches, and it is one of the most attractive features
of the CPM. The Metropolis algorithm allows, in fact, CPMs to represent nat-
urally the continuous, exploratory behavior of migrating organisms through
biased extensions and retractions of their boundaries. Moreover, by updat-
ing only one spin at a time, the individuals move gradually, rather than in
jumps, as in some other approaches (for instance, multispin dynamics such as
Kawasaky dynamics are also possible, as commented in [165]). The CPM tech-
nique also allows one to differentiate between the isotropic intrinsic motility
of each individual, which is described by its Boltzmann temperature T (which
can be approximately compared to a diffusion coecient from a continuous
point of view), and the directional, force-based component of its motion. Since
a difference in a potential energy might be related to a force, at any given time
t, for any site x of the domain , the local negative gradient of the functional
H can be related to the local applied force.
Indeed, since a difference in a potential energy might be related to the
work done by a force, we can compute the change of energy due to the spin
flips related to an object as
t = = X
x 2
t = X
x 2
H
F x 2 x
F x 2 v x 2 ;
(4.12)
where is the power of all forces F acting on site x of the object and
v x 2 the local velocity.
In CPMs, it is assumed that for any element (as usual, both a whole
 
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