Mathematical Modelling of Regeneration of a Tissue-Engineered Trachea - Computational Modeling in Tissue Engineering - page 412

Z 1

2
R o þ R c ð t Þ

u av ð t Þ¼ r ð x ; t Þ u ð r ð x ; t Þ; t Þ dx ; ð 52 Þ

0

which was evaluated at each time point in the simulations using Simpson's rule.

5 Results

The model equations, defined by Eqs. ( 30 - 41 ) with boundary conditions given by
( 42 ) and initial conditions by ( 43 ), were solved numerically using the methods
described in Sect. 4 with the parameter values listed in Sect. 3.6 . To compare
different types of therapies, simulations were performed with or without the intra-
operative application of MSCs, and with or without preseeding of the tracheal
lining with EPCs. Simulations were carried out up to t ¼ 60 days, consistent with
typical post-operative recovery and monitoring time frames [ 26 ]. The main
objective is to compare how these different seeding strategies perform with regard
to minimising the amount of stenosis and regenerating the cartilage of the trachea,
since these are the major determinants of the long term viability of the implant.
Figure 3 shows how the densities of the cells and the concentrations of ECM
and cytokines at various times vary radially through the trachea, for the case of
application of MSCs only and without seeding of the lumenal surface with EPCs.
Spatial profiles are given at t ¼ 0 (curves with triangles), at t ¼ 5 days (curves
with crosses) and at t ¼ 60 days (curves with circles) after implantation of the
trachea. At t ¼ 0 there are no cells present inside the trachea, only collagen and
GAGs being present. By 5 days after implantation macrophages have fully infil-
trated the submucosa invoking inflammation by secreting large amounts of TGF-
b1 and TNF-a through the full radial cross section of the trachea. At this time the
concentration of IL-10 and the densities of MSCs and chondrocytes remain low.
However by t ¼ 60 days, MSCs have infiltrated right the way through the cartilage
ring, regenerating the chondrocytes, and have migrated into the submucosa where
they inhibit macrophage activity by secreting IL-10. This reduces the levels of
TGF-b1 and TNF-a thereby resolving the inflammation. The continued secretion
of small amounts of TGF-b1 and TNF-a by the small population of residual
macrophages supports the presence of a small number of MSCs at large times.
Figure 4 a shows the time courses of the density of EPCs on the lumenal surface
for different seeding strategies. In the case where only MSCs are applied to the
trachea (see also Fig. 3 ) there are initially no EPCs present on the lumenal surface
but EPCs slowly infiltrate axially from the edges of the graft. However, EPC
proliferation is inhibited during the inflammation because of the high concentra-
tion of TGF-b1 : Through the down regulation of TGF-b1 secretion once the MSCs
invade the submucosa, reepithelization of the trachea can occur, as indicated by
the solid curve in Fig. 4 a. Without application of MSCs to the trachea the
epithelial lining fails to regenerate (dashed curve). In the case where EPCs
are preseeded onto the lumenal surface and MSCs are applied to the outer surface