Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 250

of sophisticated numerical schemes. To this end, Osborne and Whiteley ( 2010 )
consider a numerical technique specifically for the solution of the multiphase flow
equations. They demonstrate that these equations can be written as a mixed system of
PDEs, consisting of first-order hyperbolic equations for the volume fraction of each
phase, generalised Stokes equations for the velocity of each phase, and elliptic PDEs
for the concentration of nutrients and messengers. This complex system of coupled
nonlinear PDEs is solved via the development of finite element techniques.
As the mathematical and computational models being used to simulate tissue
growth become more detailed and complex, and as they are extended by different
researchers, it becomes increasingly important that the underlying software is
robust, reliable and fully tested. Several groups are developing software to tackle
such problems. For example, the software Cancer Heart and Soft Tissue Envi-
ronment (CHASTE) has been developed and is continually being advanced to
solve multiscale and multiphase problems in areas of physiology that encompass
cardiac, cancer and tissue engineering applications (Pitt-Francis et al. 2000 ).
• Modelling challenges
Within the context of multiphase modelling, a key challenge is the specification
of appropriate constitutive laws for the material properties of the constituent
phases, and their interactions via interphase mass and momentum transfer. Current
modelling approaches, such as O'Dea et al. ( 2010 ), have proposed simple candi-
date constitutive laws and have shown how the characteristic tissue morphologies
that arise depend on the mechanical stimuli: by comparing model predictions with
experimental data, it should be possible to determine the dominant regulatory
stimuli for a given cell line. However, in order to account accurately for the
material properties of the tissue construct in question (highlighted as a key con-
sideration in in vitro tissue engineering in Sect. 1.1.2 ), simplified constitutive
assumptions such as those employed by O'Dea et al. ( 2010 ) and Osborne et al.
( 2010 ) are inappropriate. Detailed modelling of growth-induced stresses and tissue
construct deformation is required. Such approaches necessitate a dramatic increase
in model complexity (see e.g. Loret and Simões 2005 ; Ricken and Bluhm 2010 )
and are therefore heavily reliant on numerical simulation, or model simplification
in order to make analytic progress.
Once appropriate models have been developed for the bioreactor culture sys-
tem, the models must be parameterised for the particular tissue type under con-
sideration e.g. bone, cartilage or cardiac tissue. Such model parameterisation
requires ongoing and close collaboration between modellers and experimentalists,
in order that the parameters of interest can be identified and successfully measured.
A significant barrier to the development of biologically realistic and powerful
models is the generation of suitable experimental data against which they can be
validated. Since the engineering of a tissue construct is a dynamic process, and it is
often technically difficult (or infeasible) to track a single experiment, the collection
of reliable quantitative data is an extremely difficult problem. An important output
of theoretical models is detailed information regarding the spatio-temporal dis-
tributions of (e.g.) stress, cell density, or nutrient levels within evolving tissue