Biomedical Engineering Reference
In-Depth Information
Collagen (type I) is secreted and degraded by fibroblasts within the submucosa
and, since it comprises large polymeric molecules, is assumed to have negligible
diffusivity. The rate of collagen secretion is assumed to be increased by TGF-b1 :
The equation governing the density of collagen is therefore
d col C col Þ N fib ;
o C col
ot ¼ 1
o
or
1 þ C tgf
K ct
ð
rvC col
Þ ð b col
ð 22 Þ
r
for R l r R c : Although collagen (type II) is also secreted within the cartilage
ring by chondrocytes, this is not included in the model because thickening of the
cartilage region is attributed to the loss of GAGs [ 26 ] (see 'ECM remodelling'
below).
GAGs are also assumed to have negligible diffusivity, and are produced and
degraded by chondrocytes. Also the GAGs are assumed to undergo proteolysis at a
constant rate independent of chondrocytes. The equation governing the concen-
tration of the GAGs is thus
þð b gch d gch C gag Þ N chn d gag C gag ;
o C gag
ot
¼ 1
r
o
or
rvC gag
ð 23 Þ
for R l r R c :
ECM remodelling. Motivated by the discussion in Sect. 3.2 the dilation, r v ;
was formulated so as to be positive when there is a net increase in the collagen
concentration in the submucosa and negative when the there is a net decrease in
GAG concentration in the cartilage ring. This dependence is formulated in
cylindrical polar coordinates as
d col C col Þ N fib
1
r
o
or
1 þ C tgf
K ct
r ðÞ¼ H A vc ð b col
A vg ð b gch d gch C gag Þ N chn d gag C gag !
ð 24 Þ
þ H
;
for R l r R o : The function H ð h Þ; which relates the net rate of ECM production
to the dilation, was chosen to be
H ð h Þ¼ h if h 0 ;
0if h\0 :
ð 25 Þ
This choice was made to ensure that the dilation does not become negative,
thereby making stenosis irreversible. Because collagen and GAGs do not occupy
the same points in space in the model, the first term in the right-hand side of
Eq. ( 24 ) will be zero in the cartilage and the second term will be zero in the
submucosa. The boundary condition on Eq. ( 24 ) is that the outer surface of the
trachea remains fixed:
v ð R o ; t Þ¼ 0 :
ð 26 Þ
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