Biomedical Engineering Reference
In-Depth Information
(
)
number
N
of elements in the system may be subject to a stochastic dynamics,
driven itself by the mentioned underlying fields. Vice versa these fields are usually
strongly coupled with the dynamics of individuals in the population. The strong
coupling of the kinetic parameters of the relevant stochastic branching-and-growth
of the capillary network, with the family of interacting underlying fields, is a major
source of complexity from a mathematical and computational point of view. This
is the reason why in literature we may find a large variety of mathematical models
addressing some of the features of the angiogenic process, and still the problem of
integration of all relevant features of the process is open [
1
,
9
,
33
,
34
,
36
,
40
,
41
].
Thus our main goal is not to provide additional models for the angiogenic phe-
nomenon, but to address the mathematical problem of reduction of the complexity of
such systems by taking advantage of its intrinsic multiscale structure; the dynamics
of cells will be described at their natural scale (the microscale), while the dynamics
of the underlying fields will be described at a larger scale (the macroscale).
In our specific cases, in a Lagrangian approach, endothelial cells are described by
a family of stochastic processes
t
X
k
Y
k
{
(
(
)
,
(
))
}
t
∈
R
+
,for
k
∈{
,...,
(
)
}
t
t
1
N
t
,where
X
k
(
)
(
)
(
)
t
denotes the position of the
k
th cell out of
N
t
,where
N
t
counts the random
∈
R
+
.
Y
k
(
)
number of cells at time
t
will represent either the velocity or the type
of the cell. The branching mechanism of blood vessels is modelled as a stochastic
marked counting process describing the birth of endothelial cells, while capillary
extensions are described by a system of a random number of stochastic differential
equations; the whole network of vessels is thus obtained as the union of their
individual trajectories. As a matter of mathematical treatability, for the capillary
extension we have adopted a system of Ito type stochastic differential equations with
additive noise, modelled by a family of independent Wiener processes; in this way
we may take advantage of the results of the Ito calculus. As anticipated above, the
kinetic parameters of the branching and extension stochastic processes depend upon
a family of underlying fields, whose evolution will be discussed later. Alternately,
one may adopt an Eulerian description of the system of cells according to which we
consider their collective behavior via the corresponding empirical measure
t
N
(
t
)
k
=
1
ε
(
X
k
(
t
)
,
Y
k
(
t
))
,
1
Q
N
(
t
)=
(
)
N
t
where
denotes the usual Dirac measure.
As anticipated above, the underlying fields are modelled at a larger scale; they
are taken as spatially continuous fields admitting spatial densities with respect
to the usual Lebesgue measure; their evolution is then modelled in terms of a
system of partial differential equations for such spatial densities, with reaction terms
depending upon
ε
Q
N
(
the empirical spatial distribution of the cell population,
hence on the evolving capillary network.
Here it comes the complexity of the whole system; since the capillary network
carries a significant randomness, the evolution equations of the underlying fields
are a set of random partial differential equations, leading to random kinetic
t
)
,
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