Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 227

2 Mathematical Modelling Approaches
As outlined above, it is clear that a wide variety of factors influences the formation
of tissue, and considerable effort has been invested in elucidating the mechanisms
by which cells experience and respond to these stimuli. The bioreactor system
employed is specific to the tissue engineering application under consideration, and
generates its own unique biochemical and biomechanical environment, tailored to
the growth of a particular tissue. Such variety necessitates a range of versatile
mathematical modelling approaches, reflecting the environment of the cells and
the experimental questions being posed. Such theoretical models can be used to
predict the flow and nutrient transport characteristics within a specific bioreactor
system, and in particular to determine local information about nutrient and shear
stress fields that is not straightforward (or even possible) to obtain experimentally.
The resulting models may be validated against measurable experimental data, such
as perfusion flow rate, outlet nutrient concentration, and then exploited to reveal
details of the mechanical and nutrient fields within the bioreactor system. Once
validated, the model can then be used to predict the outcome of a particular
experimental scenario (limiting the need for numerous and expensive bioreactor
experiments, potentially saving time and resources) and to optimise the bioreactor
operating conditions.
We focus our review on the use of multiphase modelling to describe biological
tissues, and present a brief overview in Sect. 2.1 below. We conclude this section
with a short description of alternative modelling approaches, before describing in
detail models specific for bioreactor systems in Sect. 3 .
2.1 Multiphase Modelling
Biological tissue is a composite material, comprising a wide variety of interacting
constituents, including, for example, a large number of different cell types, their
associated ECM and other deposited materials, and interstitial fluid. Multiphase, or
mixture theory, models provide a natural continuum framework within which to
investigate such interactions. These models are based upon the idea that tissues
may be represented by a mixture of continua, which are able to occupy the same
region of space; interactions between the different 'phases' are specified via mass
and force balance equations, together with appropriate constitutive relations, the
choice of which allows a wide variety of physical systems to be modelled. This
methodology also reflects the idea that, as composite materials, tissue properties
are reflected by the relative volume fractions (and properties) of their constituents
(Trelstad and Silver 1981 ); changes in tissue composition occur via processes such
as mitosis, apoptosis, necrosis, (de-)differentiation and ECM production.
Since the study of (Treusdell and Noll 1960 ), an enormous number of studies
have been dedicated to formulating a rigorous framework of conservation laws for