Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 323

are based on the solution of partial differential equations, and often the level-set
method is used to compute the position of the cell boundary, see for instance [ 26 ].
The level-set method requires the solution of a set of partial differential equations,
such as the level-set function itself, whose zero level curve typically coincides
with the cell boundary, the extension of the boundary velocity and a tedious re-
initialization procedure which can be done by the fast-marching method based on
the shortest-path optimization procedure, or via the solution of another nonlinear
partial differential equation. Despite the enormous flexibility of the level-set
method in terms of the ability to track interfaces also in cases where topological
changes take place, the method is very expensive. Therefore, we choose to present
a simpler method, which has been published only very recently in Vermolen and
Gefen [ 17 ]. This model is based on the sensitivity of cells to a chemical and can
therefore be applied to simulate cell migration and deformation as a result of
chemotaxis. To this extent, the cell boundary, either in 2D or in 3D, is divided into
gridnodes, which have the ability to move according to the gradient of the con-
centration of a certain chemical. This chemical could be a source of nutrition,
oxygen, a growth factor or a poisonous chemical. Further, these points are con-
nected to their neighbors and to the nucleus via springs, see Fig. 2 for a schematic
representation. In this way, surface tension of the cell membrane and the con-
nection between the membrane and nucleus via the ligaments in the cytoplasm are
dealt with. First, we consider the modeling of the chemical sources and subse-
quently we consider the equations of motion of the points on the cell boundary.
To approximate the concentration of the chemical that gives raise to chemo-
taxis, we will use an approach based on Fundamental solutions of the diffusion
equations in unbounded domains such as Eq. ( 5 ). For the release of the chemical
Fig. 2 A schematic of the
distribution of springs that
forms the backbone of the
cell skeleton in the model