Mathematical Modelling of Regeneration of a Tissue-Engineered Trachea - Computational Modeling in Tissue Engineering - page 398

submucosa to the host tissue through inter-cartilaginous zones which lie between
the cartilage rings (see Fig. 1 c). Blood vessels cannot penetrate the rings because
of anti-angiogenesis factors present in the cartilage [ 31 ]. Hence the chondrocytes
must rely on the delivery of oxygen by passive diffusion through distances in the
order of mm. However if the hypoxic conditions are prolonged the cartilage can
degenerate due to cell necrosis and the migration of chondrocytes up oxygen
gradients [ 20 ]. The decellularisation process may leave behind angiogenic factors
in the trachea, such as bFGF, that stimulate angiogenesis [ 67 ]. Also ECs can be
preseeded into the tracheal implant to encourage revascularisation [ 92 ]. In the
tissue-engineered trachea this revascularisation occurs rapidly, with new blood
vessels forming inside the trachea 24 h after implantation [P. Macchiarini, private
communication].
The preceding description identifies the major roles for the different types of
cells, ECM components and cytokines that participate in regeneration of the tissue-
engineered trachea in-situ. This is used as the basis for formulating the mathe-
matical model in the following section.
3 Mathematical Model
The previous section illustrates how the tissue engineering of a trachea typifies
many of the challenges that arise in the mathematics of regenerative medicine,
such as describing cell division, signalling and differentiation within mixed cell-
type populations. The first decision to be made is to decide on which scale, e.g.
cell, tissue or organ, the model should operate—here the intermediate (tissue-
scale) approach is adopted.
This section begins with a summary of those mechanisms described in Sect. 2
that are included in the mathematical model of tracheal regeneration, and key
assumptions and simplifications that are made in the model formulation are set out.
This is followed in Sect. 3.2 by a description of the general methodology used to
construct the model. In Sect. 3.3 the mathematical model is set out in detail,
whereby equations appropriate for each of the species are given. The model is then
nondimensionalized in Sect. 3.4 with appropriate initial conditions given in
Sect. 3.5 and a discussion of the choice of parameter values provided in Sect. 3.6
3.1 Summary of Included Mechanisms
The complexity of tracheal regeneration poses significant challenges for modelling
the process in its entirety and a crucial element of a useful mechanistic model is
that it includes sufficient, but not too much, of the relevant details. For simplicity
the mathematical model presented in this chapter provides a simplified description