Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 231

(Graner-Glazier-Hogeweg) model is employed and formal rules for cellular
interactions are postulated; such a modelling approach has been widely used,
capturing cell sorting phenomena as well as a range of morphogenetic processes
in, for example, the development of the slime mould Dictyostelium discoideum.To
accommodate the physics of cell-cell or cell-environment interactions, Meineke
et al. ( 2001 ) incorporate linear over-damped springs connecting cell centres within
the modelling framework. This approach was included in a multiscale computa-
tional model developed to investigate the role of Wnt signalling in regulating
tissue renewal in the intestinal crypt by Van Leeuwen et al. ( 2009 ). The preceding
models employ a deterministic approach to model cell behaviour; however, at the
single-cell scale, biological processes are inherently stochastic. Biological systems
are profoundly affected by such stochastic noise; for example, genetic selection
may be influenced by the stochastic nature of mRNA transcription (see Wilkinson
2009 and references therein). The importance of such stochasticity to a variety of
tissue growth processes has been widely investigated. Common approaches
include the use of master and Langevin equations (Othmer et al. 1988 ; Hadeler
et al. 2004 ).
The analysis of discrete models typically necessitates a computational
approach, since realistic simulations demand large numbers of cells, for which
analytic results may be difficult or impossible to obtain. In an attempt to cir-
cumvent this problem, multiscale (homogenisation) techniques have been
employed to derive continuum models directly from underlying discrete systems,
enabling some of these discrete effects to be incorporated into tissue-scale models
in a mathematically precise way. Representative examples include Turner et al.
( 2004 ); Murray et al. ( 2009 ) and Fozard et al. ( 2010 ), who employ such tech-
niques to represent collective movement of adherent cells within a continuum
framework, while (O'Dea and King 2011a , b ) analyse pattern formation due to cell
signalling processes. Shipley et al. ( 2009 ) determine the macroscale flow and
transport properties of a specific type of tissue engineering scaffold, in order to
specify criteria for effective tissue growth. In all cases, the resulting continuum
models are formulated as small systems of PDEs which may be amenable to
analysis and/or numerical investigation. For these reasons, much research has
concentrated on continuum representations of tissue growth, and it is models of
this type for tissue engineering bioreactors that we now consider in detail below.
3 Mathematical Modelling of Bioreactor Systems
The biochemical and biomechanical cues which lead to optimum growth are
specific to the tissue under consideration; correspondingly, the mathematical
modelling approach required is determined by both the bioreactor system and the
particular biological processes under investigation. In the following we review
theoretical studies of such systems, concentrating on the following broad themes:
(i) models of static culture systems which focus on biochemical effects such as