Biomedical Engineering Reference
In-Depth Information
=
o N mac
or
o N mac
or
o N msc
or
ð R l ; t Þ¼ 0 ;
ð R c ; t Þ¼ 0 ;
ð R l ; t Þ¼ 0 ;
o N msc
or
o N fib
or
oN fib
or
ð R o ; t Þ¼ I msc g ð t Þ= D msc ;
ð R l ; t Þ¼ 0
ð R c ; t Þ¼ 0 ;
oC ilk
or
o C ilk
ot
o C tgf
or
ð 42 Þ
ð R l ; t Þ¼ b ile N epi = D ilk ;
ð R o ; t Þ¼ 0 ;
ð R l ; t Þ¼ 0 ;
;
o C tgf
or
o C tnf
or
o C tnf
or
ð R o ; t Þ¼ 0 ;
ð R l ; t Þ¼ 0 ;
ð R o ; t Þ¼ 0 ;
v ð R o ; t Þ ¼ 0 :
3.5 Initial Conditions
At t ¼ 0 the only cells occupying the trachea will be EPCs on the lumenal surface
(if the trachea has been seeded with EPCs during the operation or ex vivo). Note
that seeding with MSCs is modelled as an influx of cells for t 0 : The trachea is
assumed to initially have uniform distributions of collagen and GAGs represen-
tative of a healthy trachea. These are calculated by solving for the steady-state
concentrations of Eqs. ( 38 ) and ( 39 ) which assuming C tgf ¼ 0 i.e. no inflamma-
tion, and neglecting the small rate of GAG degradation, are both equal to unity.
The initial conditions on the variables are therefore
9
=
;
N mac ð r ; 0 Þ¼ 0 ;
N epi ð 0 Þ¼ N epi ; 0 ;
N msc ð r ; 0 Þ¼ 0 ;
N fib ð r ; 0 Þ¼ 0 ;
N chn ð r ; 0 Þ¼ 0 ;
C ilk ð r ; 0 Þ¼ 0 ;
C tgf ð r ; 0 Þ¼ 0 ;
C tnf ð r ; 0 Þ¼ 0 ;
ð 43 Þ
C col ð r ; 0 Þ¼ 1 ;
C gag ð r ; 0 Þ¼ 1 ;
R l ð 0 Þ¼ R l ; 0 ;
R c ð 0 Þ¼ R c ; 0 ;
where N epi ; 0 ¼ 0 was used to model experiments without seeding with EPCs, and
N epi ; 0 ¼ 1 used where EPCs are applied to the lumenal surface resulting in an
initially-confluent density of cells.
3.6 Parameter Values
A typical adult human trachea is approximately 20 mm in diameter, with the wall
being 3 mm thick consisting of 2.5 mm of cartilage and 0.5 mm of mucosa, and
each ring segment being 4 mm high [ 27 ]. Using L ¼ 1 mm as the characteristic
length scale these dimensions give the non-dimensional parameter values R l ; 0 ¼
7 ; R c ; 0 ¼ 7 : 5 ; R o ¼ 10 and h ¼ 4 :
An appropriate value for s ; commensurate with the time scale over which
recovery of the patient proceeds, is 1 day. Hence to simulate topical application of
MSCs for 1 h during surgery a value of T inf ¼ 0 : 05 was used.
Following [ 9 ] the diffusion coefficients of IL-10, TGF-b1 and TNF-a were
calculated using the formula D ¼ 6 : 7 10 10 ð M = 180 Þ 1 = 3 m 2 s 1 where M is
the molecular weight. The values of the diffusion coefficients of the motile cells,
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