Mathematical Modelling of Regeneration of a Tissue-Engineered Trachea - Computational Modeling in Tissue Engineering - page 401

reference frame that moves with the tissue with velocity v (Lagrangian frame of
reference) leading to

o u
ot ¼rð vu þ J u Þþ f ð u ; x ; t Þ:

ð 1 Þ

The quantity f in Eq. ( 1 ) is the local rate of change of mass of the species, which
for cells may result from division or differentiation from another cell type. The
quantity J u is the flux of the species as it pertains to a stationary reference frame;
for a cell species an appropriate form for the flux is

J u ¼ D u r u þ uv u r a : ð 2 Þ

The two terms on the right hand side of Eq. ( 2 ) respectively model diffusion and
chemotaxis in response to a generic chemoattractant with concentration a (see for
example [ 51 , Chap. 9]). If u is a soluble chemical messenger or chemokine only
the first term is used.
The dilation term in ( 1 ) associated with the r v describes the rate of expansion
of the bulk of the tissue to which a species is embedded, and is specified by
consideration of the dynamics of the extra cellular matrix (ECM) as follows.
Tissue in the submucosa comprises cells attached to a collagen (type I) mesh that
is actively secreted and degraded by fibroblasts. As collagen is secreted it rapidly
polymerises and expands to form a fibrous mesh. Because the collagen matrix
serves as a substrate for cell attachment, as the matrix expands it takes cells with it.
The cartilage ring on the other hand consists of collagen (type II) and GAGs
(chondroitin sulfate). The mechanical properties of the cartilage are dominated by
the GAGs which impart structural rigidity and compressive strength to the tissue.
If the GAGs are lost (malacia) there will be a loosening and expansion of the
cartilage which takes embedded cells. Such changes may be contributed to by the
loss of collagen II which imparts cohesiveness to the cartilage however for sim-
plicity only the effects of GAG degradation will be considered. The preceding
considerations motivates making the dilation of the submucosa and cartilage tissue
elements a scalar function of the net rate of secretion of ECM in such a way that
that the tissue domain can expand (stenosis) or contract (involution/atrophy).
Further details of how this is done are given in Sect. 3.3 A similar idea has been
used to model size increase of tissue engineering scaffolds in a bioreactor [ 56 ].
In a 3D domain the velocity, v, has three components and in general additional
assumptions have to be made to close the model. This could be done, for example,
by specifying the spatial distribution in the rate of expansion of the ECM. Since
the velocity of the matrix in the vicinity of a particular point in the domain r ¼ r 0
relative to that point is dv ¼ðr v Þj r 0 dr ; the principal axes of r v could be chosen
to reflect some preferred direction in the tissue. However, care needs to be taken
that such a choice is compatible with physical constraints imposed on the growth.
This issue can be avoided by considering a 1D geometry thereby requiring only
one component of v to be determined. The geometry of the trachea is highly
suggestive of the use of cylindrical polar coordinates (see Fig. 1 d) and neglecting