Biomedical Engineering Reference
In-Depth Information
forces it can exert in response to the local micro-environment. Knowing this for
each cell, the movement of each single cell, and hence of one collection of cells
can be estimated through the force equilibrium equation. Normally, the effect of
inertia is ignored in this equation and two kind of forces are considered: interaction
forces and viscous drag forces. Two main interaction forces are considered,
coming from cell-matrix and cell-cell interactions. The viscous drag force is
generated from the cell movement through a viscous environment and is dependent
on the cell velocity of each cell. Therefore, establishing the force equilibrium the
velocity of each cell can be evaluated.
Using this approach, a mathematical model for cell movement in multicellular
systems has been developed by [
93
] to simulate in 3D cell movement during
aggregation and slug stage of Dictyostelium discoideum, embryogenesis, limb
formation and wound healing. The movement of individual cells in 3D considering
a more realistic cell-matrix interaction under the consideration of the dependency
on the receptor-ligand adhesivity has been simulated by Zaman et al. [
127
].
More recently, a cell force-based modelling framework for bone tissue engi-
neering applications has been presented [
45
]. In this study, cell-cell and cell-
environment interactions such as adhesion, repulsion and drag forces have been
considered to model cell aggregate behavior, although not angiogenesis.
Although in 2D, a more realistic mechanical behavior of the cell body has been
considered to model collective cell behaviour incorporating sub-cellular effects
with multiple viscoelastic elements [
60
].
Nevertheless, these approaches are really focused on modelling collective cell
migration and not angiogenesis. As far as we know, there are not previous works
that have simulated angiogenesis using this approach.
4 Examples of Application
Next, discrete as well as continuum approaches are presented which are analyzed
from a computational perspective and applied to three distinct phenomena: wound
healing, distraction osteogenesis and individual cell migration in 3D. The results
shown in these examples of application are preliminar results.
4.1 Continuum Models: Wound Healing
As it has been previously mentioned, a number of mathematical and numerical
angiogenesis models have been developed in the last few years. Angiogenesis is
present in the second stage of the wound healing process, the proliferative phase,
so it is very important to understand how every factor present in angiogenesis
affects the evolution of the wound.
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