Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 334

Fig. 7 A snapshot of the
solution to the Fisher-
Kolmogorov equation as an
elementary model for wound
healing

one needs to formulate a boundary condition, such as setting the cell density equal
to the equilibrium undamaged cell density or by imposing a no-flux condition. The
Fisher-Kolmogorov equation admits traveling wave solutions, see [ 30 ] for
instance. Furthermore, in a bounded domain, one often solves this PDE by the use
of discretization techniques, such as the finite-element method. The wound
boundary, which moves in time, is classically tracked as a level-curve of the cell
density. The choice of the value for the level-curve is somewhat arbitrary, how-
ever, it gives a good qualitative picture of the kinetics of wound healing predicted
by this simplified version of the PDE-continuum based models. Using the prin-
ciples that were outlined Sect. 2.1.1 , one can also regard the cell density as a
measure for the likelihood to encounter a cell at a certain position and time. If cells
are considered as point masses, then the principles outlined in Sect. 2.1.1 are
helpful. However, if cells get compressed, then one should incorporate the cell
volumes or areas. In the case of 'supersaturated' cell colonies, the cells are
compressed and thereby their sizes are small. On the contrary, the 'subsaturated'
cell colonies, in general contain elongated cells and thereby the sizes are larger. In
both cases the same portion of the computational domain can be considered and
hence the likelihood to find a cell at a certain position and time could thereby be
equal. Hence the probability measures nonlinearly with the cell density in case of
relatively large cells which cannot be treated by the use of point sources. There-
fore, it was argued by Vermolen et al. [ 21 ] that the likelihood proceeds nonlinearly
and that the curve of the likelihood versus the cell density exhibits a concave-
downward relationship. An alternative point of view is to use the solution u to
represent the quality of the tissue at a certain position, where u ¼ u 0 represents the
undamaged quality which could be equal to unity. The quality of the overall tissue
could be quantified by

Q ð X Þ¼ R X udX

u 0 Area ð X Þ :

ð 43 Þ