Biomedical Engineering Reference
In-Depth Information
After the discrete object
has evolved through a spin flip, both equations
that describe the variation of continuous fields, and the attributes of all the
objects are rederived on the basis of new lattice configuration.
The basic step of the Metropolis algorithm is then iterated until the end
of the simulation time or until the whole system reaches an energetic global
minimum, if it exists. The unit of time of all CPM approaches is the Monte
Carlo Step (MCS). An MCS corresponds to a fixed number of trial lattice
updates, which is usually a multiple of the total number of sites of (i.e., 1
MCS = k Volume
) and which has to be translated into the actual unit of
time (i.e., seconds, hours, days). A direct correspondence between the model
and the actual time scale may therefore not be straightforward, giving rise to
one of the main criticisms of the method. However, a realistic correspondence
is usually set by fitting a posteriori the temporal dynamics of the simulated
phenomenon with the relative experimental counterparts.
1.3 The Hamiltonian
The discrete effective energy of the system, given by the Hamiltonian H, may
contain a variable number of terms, that can be grouped as:
H(t) = H
adhesion
(t) + H
constraint
(t) + H
force
(t):
(1.4)
H
adhesion
describes the adhesive/repulsive interfacial energy between all
the pairs of discrete objects that interact across their common membrane.
H
adhesion
is based on Steinberg's Dierential Adhesion Hypothesis (DAH)
[169, 377, 378]. The DAH proposes that individuals in the same aggregate
adhere to each other with different strengths, according to their type. Such
a hierarchy of contact forces is one of the main driving mechanisms behind
the evolution of biological systems, whose final organization maximizes the
overall strength of interface interactions (or, in other words, minimizes the
overall adhesion energy). However, the DAH says nothing about the dynam-
ics of moving objects: differential adhesion itself, in fact, only helps to select
the most favorable configuration among the different possibilities that have
been explored. Evidence supporting DAH has been observed in a wide array
of biological systems, especially in the embryonic stage of life. For example,
it successfully explains how cellular adhesive properties can operate to de-
termine tissue reorganization during cell sorting [16, 164, 169]. The typical
formulation of DAH-derived H
adhesion
is
X
H
adhesion
(t) =
J
(
(
x
)
);(
0
(
x
0
)
)
(t);
(1.5)
x
;
x
0
2
0
x
:
(@
x
2@
)\(@
x
0
2@
0
)6=;
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