Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 326

equilibrium shape and size. The parameter a [ 0 is a measure of the stiffness of
the cell. The position of the point i, which depends on time, is denoted by x i ð t Þ:
The velocity of the point is denoted by v i : Further, the position of the cell center is
given by x c ð t Þ and the initial positions of the boundary nodes minus the initial
position of the cell center are denoted by x i : The above equation guarantees that
the velocity is directed towards the largest increase of the concentration, and that
its magnitude depends on the magnitude of the concentration gradient. Note that in
the case of repulsion, for instance due to a poison, the sign of the b-term should be
reversed.
The movement of the cell boundary makes the cell deform and change its
position. Furthermore, the cell area or volume changes as well. Since the cell
consists of both fluids and solid polymeric matter, the cell is classically modeled as
a visco-elastic medium. This means that the volume of the cell is not necessarily
conserved. It is possible to inhibit volumetric changes by enlarging the a-param-
eter if the volume of the cell increasingly differs from the initial cell volume. The
model is described in more detail in Vermolen and Gefen [ 17 ]. An example of a
three-dimensional computation of the model is shown in Fig. 3 . The input-data
were the same as in Vermolen and Gefen [ 17 ], see Table 1 .
In this figure it can be seen how a cell deforms and migrates to engulf the
bacteria. Once the bacteria have been neutralized, the cell deforms back to its
equilibrium shape. In [ 17 ], the model is extended to multiple cells where each cell
secretes an agent that attracts the other cells. The model is based on the assumption
that a cell registers the difference between the present concentration profile and the
concentration profile from its own secretion. A repulsive force between gridnodes
on different cells is introduced to prevent the cells from overlapping. The phe-
nomenological relation of the repulsive force is inspired by the Lennard-Jones
potential from electromagnetics. Since the medium through in which the cells
deform is nonhomogeneous and anisotropic, a stochastic component is added to
the equation via a Wiener process. This makes Eq. ( 26 ) stochastic:

dx i ¼ b r c ð t ; x i Þ dt þ a x c ð t Þþ x i x i ð t Þ ð Þ dt þ rdW ð t Þ; for i 2f 1 ; ; N g; ð 26 Þ

where W ¼ð W x ; W y ; W z Þ is a vector with Wiener processes W x ; W y and W z and r
is a measure for the uncertainty (standard deviation) induced by nonhomogeneities
of the medium. The first two terms are deterministic and hence represent classical
drift. Some computed results with a stochastic contribution can be found in
Vermolen and Gefen [ 17 ]. In this manuscript we only show a deterministic run in
Fig. 3 . In Fig. 4 , we plot the times of engulfing a bacterium versus the cell stiffness
and mobility. It can be seen that an increase of cell stiffness and/or a decrease of
cell mobility delay the engulfment of bacteria. This computation can be used to
quantify the influence of cell stiffening and motility decrease due to certain dis-
eases. This simulation models the effectiveness of the immune response as a
function of the properties of the immunity cells like white blood cells.