Mathematical Modelling of Regeneration of a Tissue-Engineered Trachea - Computational Modeling in Tissue Engineering - page 418

rather than a more complete systems biology approach that would seek to include
as much information about cells and signalling pathways as possible.
A simple radial geometry was chosen to represent the trachea which simplified
the description of the transport of species axially. This allowed a 1D formulation to
be used thereby reducing the amount of computational effort required to solve the
model equations. If a less simplified geometry were used, for example if the
posterior wall were to be taken into account by solving the equations on a domain
comprising a 2D ring segment (see Fig. 1 b), this would give a more realistic
description of how a real trachea changes shape under stenosis. However because
the model is purely local the mechanisms that give rise to stenosis do not depend
critically on the overall extent of the deformation of the trachea. Hence more
emphasis has been given in the modelling to the interactions between the different
types of cells and cytokines rather than to the details of the tracheal geometry.
For tractability many aspects of the biological mechanisms were given highly
simplified descriptions in the modelling. For example the complex processes of
homing and migration of macrophages, EPCs and fibroblasts into the trachea from
the host tissue was not modelled in detail. Instead their influx was assumed to
occur at a constant rate characterised by the parameters I mac ; I epi and I fib : Also
consideration was not given to the complex mechanisms of interconversion of
TGF-b1 between an inactive form bound to the ECM and an active soluble form
diffusing throughout the domain [ 35 ] and only consideration was given to a small
number of specific effects of TGF-b1 on cells pertinent to its inflammatory role.
The ability to devise a model that is tractable numerically and analytically, but is
also realistic from a biologist's perspective is a key skill of a mathematical
modeller.
Continuum modelling approaches to describe cell migration and proliferation
such as the reaction-diffusion formulation used in this chapter have been much
used in biomedical modelling applications; for other examples see [ 9 , 69 ]. Such
formulations have the advantage over individual-based models of being relatively
economical to solve computationally. In some cases these models are also ana-
lytically tractable and so mathematical techniques can be used for analysing their
solutions. For example because of the mutually antagonistic effects of TGF-b1
(pro inflammatory) and IL-10 (anti inflammatory) in the model, a hysteresis effect
arises whereby steady states corresponding to inflamed tissue and healthy tissue
can occur for the same values of the model parameters but arise from the use of
different seeding conditions. Bifurcation and stability analysis of the equilibrium
solutions of the model could yield precise conditions on parameters and initial
seeding densities that gives rise to inflammation and stenosis, and in principle be
used to guide therapeutic strategies. Relating the bifurcation structure of the
solutions of a model to aberrant and normal states in real tissue is a useful concept
in biomedical modelling.
It is important to note that the continuum modelling approach adopted in this
chapter is essentially a phenomenological description of what is on the spatial
scale of individual cells a very complex process. For example, implicit in the