Biomedical Engineering Reference
In-Depth Information
(
) (e.g., endothelial cell, ECM fiber, fibroblast, . . . ). Mesoscopic, cell-level
objects rearrange their boundaries to realistically reproduce shape changes
and motion. Moreover, they can grow, die, duplicate (and the daughter ob-
jects typically inherit their parent's properties [186, 270, 314]), and carry a set
of possible rules for transitions between types (a Cell Type Map, CTM [84]).
The notation
0
is usually used to identify a discrete object neighbor of
.
Continuous objects, or fields, represent the spatiotemporal evolution of
microscopic entities, that may reside within the discrete objects (as DNA,
RNA, cytosolic ions, and proteins), or in the external environment (as nutri-
ents, growth factors, matrix proteins, matrix metalloproteinases). They are
described as variable concentrations with standard reaction-diffusion (RD)
equations, whose general form is
@c
@t
(x;t) = r
[D
c
(x
;t
)rc(x;t)
]
+
F(c)
|{z}
reaction term
;
(1.1)
|
{z
}
diffusion
where c(x;t) denotes the local concentration (i.e., at site x) of the chemical
substance, D
c
its diffusion coecient, and F :R
+
!Ris the reaction term.
In classical CPM applications, D
c
is homogeneous in space and constant in
time; however, it is possible to drop these restrictions. Equations of type (1.1)
may apply to the entire domain or to selected subregions, with xed or
moving boundaries (as in the case of intracellular chemicals). Indeed, as we
will explain more in detail in Appendix A, these continuous equations are
numerically solved using finite difference schemes on grids that exactly match
the CPM domain, and that are discretized at the same resolution.
The specific interactions between discrete cell-level objects and continuous
molecular-level objects can be characterized either by the reaction term in
Equation (1.1), as in the case of cell absorption and secretion of chemical
diffusants [25, 183, 254], or by constitutive laws relating phenotypic behavior
of discrete individuals to the evolution of specific microscopic variables, as we
will see in Chapter 4. The coupling between the dynamics of microscopic and
mesoscopic objects provides to the CPM its hybrid characteristic. Obviously,
each application of the method needs the specification of the initial condition
of the lattice, i.e of both the initial condition of the continuous fields and the
initial spatial configuration of the discrete objects.
Search WWH ::
Custom Search