Hydrodynamics in Porous Media with Applications to Tissue Engineering - Porous Media: Applications in Biological Systems and Biotechnology - page 95

3.5 Theoretical Models of Active Porous Media
The term “model” can have different meanings according to the scientific
domain. For instance, in biology, an animal model is used, in a study of pathol-
ogy on a given animal species, to understand the same type of pathology in a
human being. In physics or mathematics, a model is a simplified representa-
tion of a process that can be treated, owing to mathematical expressions. In
this chapter, the term model is used in its mathematical acceptation.
In general, a model can be very useful to understand experimental phe-
nomena and promote new ideas in experimental investigations. In tissue engi-
neering, many different physical parameters participate in tissue growth. The
modeling tool can be useful for independently studying the principal phys-
ical mechanisms and thus identifying those that mainly govern the culture
processes.
3.5.1 Length and Time Scales of the Different
Physicochemical Phenomena
Various physicochemical phenomena are associated to the biological steps of
the implant development, and they interact together at very different time
and length scales (see Figure 3.6).
Physicochemical phenomena occurring during the tissue development are
(this list being nonexhaustive) transport and consumption of nutrient and
oxygen, generation of waste by cells, mechanical loading of tissue and/or cells,
osmosis phenomena (transport trough the cell membrane), electrochemical
phenomena (Debye length), electromechanical phenomena (piezoelectricity),
or chemomechanical phenomena (swelling).
The time and length scales related to these physicochemical phenomena are
showing important variability in their extent: the characteristic time for cell
adhesion is about few hours while cells synthesize ECM over several weeks. As
regards the length scales, few hundreds of nanometers have to be considered
by speaking about focal adhesion points of cells while few centimeters are
the characteristic length scale of the substrate and bioreactor. As a result,
because of the difference in magnitude of both time and length scales as well
as the plurality of the implied phenomena, a good understanding of the in
vitro three-dimensional cell culture is still very complex.
3.5.2 Convection-Diffusion-Reaction Phenomena: Basic
Equations and Characteristic Nondimensional
Parameters
Porous medium numerical simulations have been designed to study phys-
ical and biochemical factors that control the functional development of