Mathematical Modelling of Regeneration of a Tissue-Engineered Trachea - Computational Modeling in Tissue Engineering - page 419

diffusion term that models migration of MSCs within the cartilage [in the first term
of the right-hand side of Eq. ( 9 )] is a process which involves proteolysis of GAGs
by the MSCs so as to clear a path for migration through the ECM [ 91 ]. The fact
that such mechanisms are not modelled in detail does not necessarily invalidate the
use of a continuum modelling approach, though the determination of parameter
values should come from in vivo measurements taken at the tissue scale e.g. the
thickness of the trachea wall, rather than at the scale of individual cells e.g.
motility coefficients derived from in vitro cell migration assays. Much current
research in mathematical biology concerns trying to develop continuum models
that sensibly link microscopic cell behaviour to macroscopic tissue growth char-
acteristics; however, at present this requires the use of unrealistically simple
models of individual cells.
The alternative is to model cells as discrete entities and perform simulations
involving large numbers of such cells, such as in the model used by our group to
study the growth of blood vessels into porous biomaterials [ 42 ]. The complexity of
cell migration within the fully 3D tracheal tissue geometry, especially when
consideration is also given to angiogenesis, suggests that the modelling of tracheal
regeneration would benefit from applying individual-based modelling techniques
to complement the PDE methods adopted here. Indeed the current trend in bio-
medical modelling, boosted by the increasing speed and power of computers, is
towards making progressively more detailed and sophisticated multiscale models
that can link sub-cellular processes to the outcomes at the level of a whole
organism [ 33 ]. The trend is in part motivated by the desire to create mathematical
and computational models that can represent real tissue faithfully enough to be
effective for the development of new drugs and therapies. This in silico revolution
[ 21 ] is spurred on by the establishment of modelling projects involving large
multi-disciplinary and cross-institutional collaborations between specialist
researchers [ 77 ]. Such projects make extensive use of systems biology approaches,
which allows models to be obtained formally from existing information contained
in databases. Mathematical and computational modelling for tracheal tissue
engineering could benefit from the use of such techniques and collaborations in the
future.
Although individual-based modelling approaches are appealing, continuum
models also have an important role to play in biomedical modelling because they
can allow a deeper understanding of the underlying dynamics of a system than is
possible by performing numerical simulations alone. The study of continuum
models also forms an important interdisciplinary link between traditional areas of
mathematics and biology, thereby stimulating research in both areas, and helps
scientists to relate similar phenomena in disparate fields. These motivations form a
core philosophy of mathematical biology. Alan Turing's model for diffusion-dri-
ven instability [ 74 ] is compelling because it can explain biological pattern for-
mation in diverse areas including embryology and ecology. His genius established
general principles of what gives rise to patterning in tissues; identifying the precise
mechanisms in specific cases came later [ 48 ].