Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 230

metabolites is by diffusion, are reviewed in Sect. 3.1 . Finally, in some applications,
detailed information regarding culture medium flow and cell population dynamics
is required; here it is necessary to consider explicitly the cells as a separate phase
(with their own volume fraction, local velocity, and so on), and to consider their
interaction with the surrounding fluid flow. Examples of models using this
approach are given in Sect. 3.3 .
2.1.1 Mathematical Reduction
When adopting a multiphase approach (and the various modelling simplifications)
the resulting continuum models comprise coupled partial differential equations
(PDEs). However, their solution may be computationally intensive and may not
reveal details of the mechanisms underlying observed tissue growth phenomena.
An alternative approach is to exploit the typically disparate length- and time-scales
inherent in these systems: for example, the bioreactor may be long and thin, or the
timescale for cell growth may be long compared to that for diffusion of solutes. In
the former case, the spatial dependence of the problem may be reduced, in the
latter, quasi-steady assumptions may be made. Alternatively, the magnitude of
problem specific parameters (such as the Reynolds number or Peclet number) may
be exploited to simplify the equations. For example, the Reynolds number char-
acterises the ratio of inertia to viscous effects, and the Peclet number the ratio of
advection to diffusion timescales, and by considering these parameters to be either
large or small, the governing equations may be simplified to retain only the key
aspects of the underlying physics of the system under consideration. In all cases,
sophisticated asymptotic methods can be used rationally to simplify the governing
equations, leading to reduced models that are tractable, yet remain physically well-
grounded. The reduced models can then be attacked using analytical techniques
and simpler numerical methods. By allowing maximum analytical progress to be
made (enabling, for instance, the influence of various governing parameters to be
relatively easily investigated), the analysis of reduced models can provide sig-
nificantly more physical insight into the underlying mechanisms than would be
obtained by computational simulations of the full system alone. Such approaches
are reviewed in Sect. 3 , and contrasted with more computational studies.
2.2 Alternative Modelling Approaches
When modelling cell population growth, many authors have employed a discrete,
rather than a continuum, approach and considered individual cells explicitly. Such
models provide a natural framework within which to accommodate, for example,
cell signalling interactions, movement and proliferation, thereby providing com-
prehensive and detailed information about the dynamics of the cell population.
Representative studies include Ouchi et al. ( 2003 ), in which a cellular Potts