Biomedical Engineering Reference
In-Depth Information
1.2 The CPM Algorithm
The CPM core principle consists of an iterative stochastic minimization of
a system free energy, described by the Hamiltonian functional H that will
be defined in detail below. Simulated objects, in fact, rearrange and evolve
to gradually reduce such a pattern energy looking for a global minimum,
rather than toward a configuration in which multiple local minima coexist.
Since the energy gradient is not completely smooth, on its way to global
minimization, the system has to move through transient states characterized
by higher energies than the previous configurations. This energy minimization
philosophy is implemented by adopting a modified version of the classical
Metropolis algorithm for Monte Carlo{Boltzmann thermodynamics [169, 263].
It evolves in time using repeated probabilistic updates of the site identi-
fication spins. Procedurally, at each simulation time step, t, a lattice site x,
belonging to an object , is randomly selected (source voxel ), and proposed
to copy its spin (x) into an arbitrary unlike neighbor x 0 2 (target voxel ).
The proposed change in the lattice configuration (also called spin flip) is ac-
cepted with a classical Boltzmann transitional probability, which is a relic of
the CPM descent from statistical physics [315]:
8
<
e Hj ( x )!( x 0 ) =T
Hj ( x )!( x 0 ) > 0 ;
P((x) ! (x 0 ))(t) =
:
1
Hj ( x )!( x 0 ) 0 ;
(1.2)
where
Hj ( x )!( x 0 ) = H after spin flip H before spin flip ;
(1.3)
is the net variation in the total energy of the system as a consequence of the
spin update and T 2R + is a Boltzmann temperature measured in units of
energy. T does not reflect any conventional thermal temperature, but in basic
CPMs broadly correlates to an overall system motility. The nomenclature of
T originates from the fact that membrane agitation rates in biological individ-
uals play a role analogous to real temperatures in ordinary thermodynamics
[268]. From a statistical point of view, T represents the likelihood of the ener-
getic unfavorable changes in lattice configurations, since it determines the rate
of their acceptance. For very small values, the system evolution is almost de-
terministic, and it can be trapped in local minima. For very large T all moves
are accepted, and the simulated objects are characterized by a biased random
walk in the absence of potential barriers. Their motility, in fact, overcomes
the constraints set by the local environment, since Hj ( x )!( x 0 ) =T ! 0 for
all the proposed displacements of their sites. It is useful to underline that in
[162] the motility T is allowed to depend on the object type (T = T ( ) );
however, it is also possible to relate it to each single unit, T = T . This
aspect is described in more detail in Sections and 4.3 and 4.4.
 
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