Biomedical Engineering Reference
In-Depth Information
(such as BioSpice
4
or Systems Biology Workbench
5
, see [20]). Our choice was
made to avoid possible compatibility problems that usually arise due to the
interoperability between different software programs.
However, in the freeware spirit of CC3D and of the other programs that
implement the CPM, any interested readers can contact the authors of this
topic to obtain further information on the program modications and, if nec-
essary, get either the entire software or parts of it.
Procedurally, the grid for the numerical solution of each PDE is matched
with the CPM lattice and, at every time step, each computational module
is used as the initial condition for the others. Indeed, the main lines of the
computational algorithm are as follows:
The discrete CPM evolves through an MCS, following the standard
Metropolis rule.
The local quantities of the chemicals are computed, based on the new cell
configuration. In particular, after the spin flip, the target site, x
target
,
is assigned the same concentrations of the microscopic variables as the
moving source site, x
source
.
The continuous equations of the fields are rederived, according to the
new distribution of the continuous variables themselves and to the new
boundaries of the simulated discrete objects, and solved using a finite
element scheme, characterized by 10 diffusion time steps per MCS (this
temporal step is suciently small to guarantee numerical stability).
The biophysical properties of each discrete individual (given by its Potts
coecients) are updated, given its new intracellular state and the new
configuration of the lattice.
The Hamiltonian functional of the system is updated, and the system
is ready to evolve again.
Finally, it is evident that the improvements in the CPM presented so far re-
quire increasing computational power, and therefore cause a significant slow-
down of the speed of the relative simulations. For example, a realization of
the multilevel wound healing model presented in Chapter 5 takes almost 4
h when using a normal notebook, while a simulation of the analogous phe-
nomenon approached with the basic CPM as in Chapter 2 lasts less than 30
min. To mitigate this issue, we strongly advise researchers to make a prelimi-
nary study of the problem of interest and to decide what is the level of detail
really needed.
Typically, if the biological phenomenon requires an accurate description
of the microscopic scale of individuals, it should also allows one to take into
account a small number of individuals, while if it requires the simulation of a
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