Mathematical Modelling of Regeneration of a Tissue-Engineered Trachea - Computational Modeling in Tissue Engineering - page 400

TNF-α
C tnf

production

macrophages

N mac

MSCs

N msc

IL−10

C ilk

EPCs

N epi

TGF-β 1

production

chondrocytes

C tgf

N chn

collagen GAGs fibroblasts

C gag

N fib C col

production

+ degradation

submucosa cartilage

r

Fig. 2 Diagram showing the network of relationships between the different species in the
mathematical model. Below the name of each species is the corresponding variable name in the
model. Large thick arrows represent inward migration of cells into the trachea, and differentiation
of MSCs into chondrocytes. The thin arrows represent activation (solid) or inhibition (dashed)of
one species by another. Red arrows represent interactions that are relevant to inflammation, blue
represents maintenance of ECM in the submucosa, green represents maintenance of ECM in the
cartilage

Sect. 3.2 ). MSCs are assumed to be able to migrate through both layers of the
trachea, whereas the fibroblasts and macrophages are confined to the submucosal
layer.

3.2 Modelling Framework

The model is constructed as a set of continuum reaction diffusion equations for
each of the species [ 51 ]. However, the need to describe stenosis of the implant
suggests the need for a moving boundary approach, so the reaction diffusion
equations will be considered to be on an evolving domain [ 19 ], whose boundary
must be calculated at the same time as the other unknowns. In such an approach
the derivation of the equation for the amount of a species, u ; is applied in a