Biomedical Engineering Reference
In-Depth Information
huge population (for example, a full model of organ morphogenesis requires
the simulation of millions of cells), the level of detail for each element can be
significantly decreased. In other words, researchers should always follow this
rough principle:
number of simulated individuals details of each individual = constant.
However, it is not always possible to either simplify the internal structure
of simulated individuals or to reduce their number. For instance, a full ex-
tended CPM of the morphogenesis of a complete organ or of an entire embryo
would require the representation of 10 6 -10 8 compartmentalized cells, with the
relative subcellular dynamics. Given that CPM realizations are usually run
hundreds of times to average out the results determined by the same values
of parameters, the first outcomes might be obtained after months.
Ecient computational techniques that able to significantly increase the
simulation speed would be therefore needed. As commented in [20, 81], one
of the main issues in CPM implementations is that too low acceptance prob-
abilities of spin flips (i.e., 10 4 -10 6 ) often waste much computational time.
In this regard, among the non-Metropolis Monte Carlo algorithms, rejection
free dynamics such as N-fold way and kinetic Monte Carlo are particularly
productive [41, 131, 231]. For each time step, they do not consider a trial index
copy, which may or may not be accepted, but choose only from among the
set of allowed lattice updates (i.e., those that decrease the system energy).
Obviously, the net computational gain will depend on the balance between
the average number of possible spin flips and the average acceptation rate.
The Random-Walker (RW) algorithm instead reduces, but does not elimi-
nate, the rejection rate by selecting as target sites only those belonging to an
object boundary [73]. The automatic rejection of non-boundary sites, charac-
teristic of a normal algorithm, is therefore eliminated.
However, the increment of the computational speed is appreciable only in
the presence of large individuals. The most attractive area of improvement
in CPM computational implementation is represented by the use of distribu-
tive computing, where the overall simulation domain is divided in equivalent
subdomains, which are in turn assigned to different nodes, as addressed again
in [20]. A first parallel version of the original Potts approach has been im-
plemented in [412] on a model of grain growth, where the effective energy
consisted only of local grain boundary interactions, so that each spin flip
changed only the energies of its neighbors.
Moreover, a recent RW distributing implementation of the CPM ran sig-
nificantly faster [174]. However, the proposed parallel scheme required shared
memory with all processors sharing the same subdomains. This issue therefore
limited the total domain extension to the memory size of a single computer.
Indeed, following the useful and detailed explanations provided in [20, 81],
the main diculty in all forms of CPM parallelization is that the effective en-
ergy is nonlocal: when a given object crosses between nodes, any modification
to it requires ecient parameter passing between nodes, so that the overall
 
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