Biomedical Engineering Reference
In-Depth Information
3.3 The Detailed Formulation
The modelling framework described in
Sect. 3.2
is used below to set out the model
equations in full, whereby Eqs. (
4
) and (
5
) are used to formulate equations for each
of the cellular and chemical species together with appropriate boundary condi-
tions. For the sake of brevity only the salient features of the equation for each
species is described. A detailed guide to the techniques for modelling biochemical
reactions and tissue growth using reaction-diffusion equations can be found in
[
51
].
In what follows a variable of the form N
is used to denote a cell density while
C
denotes a concentration. Besides the parameters introduced in
Sect. 3.2
,a
parameter of the form b
appearing in an equation refers to a rate of cell division
or a rate of production of a chemical species whereas d
refers to a rate of cell
death or apoptosis, or a rate of chemical degradation. Parameters of the form K
characterise the strength of the effect of one type of species on another.
Macrophages are assumed to be excluded from the cartilage ring, but enter and
migrate randomly within the submucosal region in their classically-activated form
while undergoing apoptosis at a rate which increases with increasing IL-10 con-
centration. The equation governing their density is, applying Eq. (
4
),
þ
2
N
mac
;
1
þ
C
ilk
K
mi
o
N
mac
ot
¼
1
r
o
or
rvN
mac
þ
rD
mac
o
N
mac
or
h
I
mac
d
mac
ð
6
Þ
for R
l
r
R
c
:
Macrophages cannot enter or leave the tracheal segment radially
hence the boundary conditions are, from Eq. (
3
),
o
N
mac
or
ð
R
l
;
t
Þ¼
o
N
mac
or
ð
R
c
;
t
Þ¼
0
:
ð
7
Þ
Epithelial cells are assumed to remain attached to the basement membrane ECM.
The radial symmetry assumption implies that the surface density of EPCs on the
lumenal surface, N
epi
;
and the speed v
¼
v
ð
R
l
;
t
Þ
depend only on t and, because
cells cannot migrate radially, J
r
¼
0
:
Hence Eq. (
4
) for the EPCs reduces to
N
epi
Þ:
dN
epi
dt
¼
v
ð
R
l
;
t
Þ
N
epi
þ
2
1
C
tgf
ð
R
l
;
t
Þ
K
et
h
I
epi
þ
b
epi
N
epi
ð
N
epi
ð
8
Þ
R
l
The second term in the right-hand side of Eq. (
8
) models infiltration of EPCs that
migrate into the implant from where it is grafted to the patients own trachea. The
third term in the right-hand side of Eq. (
8
) models epithelial cell proliferation,
where N
epi
is the surface density of EPCs at confluence i.e. the carrying capacity.
The effect of TGF-b1 is to suppress the proliferation of EPCs, thereby reducing
their maximum surface density.
MSCs. Consistent with their regenerative potential, MSCs can occupy both the
cartilage ring and the submucosal layer, and diffuse passively through both
regions. Their proliferation rate is increased by both TGF-b1 and TNF-a
;
and
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