Biomedical Engineering Reference
In-Depth Information
D
mac
;
D
msc
and D
fib
;
were all taken to be equal to 1
:
6
10
13
m
2
s
1
which is at the
lower end of the range of values quoted for the neutrophil motility within collagen
gels [
58
]. The degradation rates of the cytokines were estimated from published
data of their half-lives measured in vivo: the half-life of IL-10 is quoted as 2 hrs in
[
63
] giving d
ilk
¼
8
:
3 day
1
;
the half-life of TGF-b1 as 100 min in [
80
] giving
d
tgf
¼
10 day
1
;
and the half-life of TNF-a as 70 min in [
79
] giving d
tnf
¼
14 day
1
:
Applying the formulae listed in (
44
) the non-dimensional parameter
values are:
9
=
D
mac
¼
D
msc
¼
D
fib
¼
0
:
0138
;
D
ilk
¼
12
:
5
;
D
tgf
¼
D
tnf
¼
11
;
d
ilk
¼
8
:
3
;
d
tgf
¼
10
;
d
tnf
¼
14
;
d
mac
¼
1
;
d
chn
¼
10
;
d
col
¼
d
gch
¼
0
:
1
;
d
gag
¼
0
:
01
;
k
cm
¼
1
;
b
epi
¼
b
msc
¼
b
fib
¼
b
chn
¼
b
ile
¼
b
tch
¼
1
;
I
msc
¼
0
=
0
:
01
;
I
epi
¼
I
fib
¼
0
:
01
;
K
mi
¼
0
:
01
;
K
et
¼
0
:
1
;
K
mn
¼
K
mt
¼
K
ti
¼
K
ct
¼
1
;
A
vc
¼
0
:
9
;
A
vg
¼
1
:
5
;
T
inf
¼
0
:
05
;
R
o
¼
10
;
h
¼
4
;
p
¼
1
;
N
epi
;
0
¼
0
=
1
;
R
l0
¼
7
:
02
;
R
c0
¼
7
:
5
:
;
ð
44
Þ
The parameter values that could not be obtained directly from published data were
estimated using a heuristic approach. For example based on typical cell division
rates of the order 1 day
1
the proliferation rate parameters for all cells i.e.
d
mac
;
b
msc
etc. were all taken to be unity. Because ECM has a slower turnover rate
than cells the ECM production/degradation rate parameters, d
col
;
d
gch
and d
gag
;
were taken to be 1-2 orders of magnitude smaller. Further, the rate of turnover of
cells inside the trachea was assumed to be dominated by cell proliferation, initiated
by a relatively small rate of cell infiltration, hence the values of the flux parameter,
I
;
for all cells was set to 0.01. The parameter values K
mi
¼
0
:
01 and K
et
¼
0
:
1in
the equations for the macrophages and EPCs were chosen to give strong bistable
behaviour in the submodel for inflammation arising from the interaction between
these two cell types. Similarly the value d
chn
¼
10 was chosen to strengthen the
deleterious effect of TNF-a on chondrogenesis. The parameter values A
vc
¼
0
:
9
and A
vg
¼
1
:
5 were selected to give realistic amounts of stenosis [
26
] at the end of
the simulations.
In the absence of further available information the values of the other param-
eters were set to unity. It must be emphasised that the parameter values listed in
Eq. (
44
) are tentative and further work is required to identify the values with more
precision.
4 Numerical Methods
The approach for solving Eqs. (
30
-
43
) numerically was to first split each variable
defined over the full radial extent of the trachea i.e. the MSCs, IL-10, TGF-b1 and
TNF-a
;
into two new variables defined exclusively within the submucosa and
cartilage. Then the transformations
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