Mathematical Modelling of Regeneration of a Tissue-Engineered Trachea - Computational Modeling in Tissue Engineering - page 404

when they are inside the cartilage ring they differentiate into chondrocytes at a rate
that is stimulated by TGF-b1 : The equation governing the density of MSCs is thus

o N msc
ot

¼ 1
r

o
or

rvN msc þ rD msc o N msc
or

N msc Þ k cm f ð r Þ C tgf N msc ; ð 9 Þ

C tgf
K mt

þ C tnf
K mn

þ b msc N msc ð N msc

for R l r R o : The function f ð r Þ in the third term on the right-hand side of Eq. ( 9 )
is equal to 1 inside the cartilage and equal to zero inside the submucosa,

f ð r Þ¼ 0 fR l r R c ð t Þ;
1 fR c ð t Þ \r R o :

ð 10 Þ

The MSCs enter the trachea through the outer surface, at r ¼ R o ; by being applied
during implantation surgery. The boundary conditions on the MSCs are therefore,
from Eq. ( 3 ),

o N msc
or

D msc o N msc
or

ð R l ; t Þ¼ 0 ; ð R o ; t Þ¼ I msc g ð t Þ; ð 11 Þ

where g ð t Þ specifies the time course of the infiltration, which is taken to be con-
stant for a duration T inf :

g ð t Þ¼ 1if 0 t T inf ;
0

ð 12 Þ

if t [ T inf :

Fibroblasts enter the submucosa axially, migrate randomly within it and prolif-
erate according to a purely logistic growth law, hence Eq. ( 4 ) is for the fibroblasts

þ 2

o N fib
ot

¼ 1
r

o
or

rvN fib þ rD fib o N fib
or h I fib þ b fib N fib ð N fib N fib Þ

ð 13 Þ

for R l r R c : The numbers of fibroblasts is assumed not to be influenced by
macrophages or growth factors; fibrotic activity is attributed only to the quantity of
collagen they secrete (see 'Fibroblasts' below). Fibroblasts do not enter or leave
the submucosa radially hence the boundary conditions are

o N fib
or

ð R l ; t Þ¼ o N fib
or

ð R c ; t Þ¼ 0 : ð 14 Þ

Chondrocytes are produced by the differentiation of MSCs. They are assumed to
proliferate up to a certain density and undergo apoptosis at a rate that is propor-
tional to the concentration of TNF-a : The equation governing the density of
chondrocytes, N chn ; is thus