Biomedical Engineering Reference
In-Depth Information
r R l ð t Þ
R c ð t Þ R l ð t Þ ;
x ¼
for R l r R c ;
ð 45 Þ
r R o
R c ð t Þ R o ;
x ¼
for R c r R o ;
ð 46 Þ
were used to map the moving intervals R l ð t Þ r R c ð t Þ and R c ð t Þ r R o to the
single fixed interval 0 x 1 : This made the equations amenable to the NAG
routine d03ph which solves two-point boundary value problems comprising
systems of parabolic-elliptic partial differential equations coupled to systems of
ODEs.
The substitution ( 45 ) transforms the non-dimensional version of Eqs.
( 4 ) and
( 40 ), appropriate for the variables defined within the submucosa, into
uv þ J r R l þ x ð R c R l Þ u
o u
ot ¼ 1
o
ox
r ð x ; t Þ
R c R l
r ð x ; t Þ
ð 47 Þ
þ s ð x ; t Þ u
R c R l
þ 2
h I u ð r ð x ; t Þ; h = 2 ; t Þþ f ð u ; r ð x ; t Þ; t Þ;
¼ H A vc d col ð 1 þ C tgf
K ct
;
1
r ð x ; t Þ
o
ox
r ð x ; t Þ v
R c R l
C col Þ N fib
ð 48 Þ
where,
J r ¼ D u
R c R l
ou
ox ;
ð 49 Þ
r ð x ; t Þ¼ R l þð R c R l Þ x ;
ð 50 Þ
s ð x ; t Þ¼ R l ð R c R l Þð R c R l Þð R l þ 2 ð R c R l Þ x Þ;
ð 51 Þ
and the dots over the variables R c and R l represents the time derivative. Similar
equations hold for the variables defined within the cartilage (which are omitted
here for brevity). Continuity of the values of u and the flux of u were applied at
x ¼ 1 (the boundary corresponding to r ¼ R c ) to the variables defined in the
submucosa and cartilage for the same species.
The transformed equations corresponding to Eqs. ( 30 ) and ( 32 - 39 ) and the
ODEs ( 31 ) and ( 41 ) were solved numerically using the NAG routine d03ph .A
spatial domain with 40 grid points was used because it was found there was no
significant change in the solution values when a finer grid was employed. At each
time step v was approximated by solving Eq. ( 48 ), with the condition v ð R o Þ¼ 0 ;
as a matrix equation obtained using a first-order approximation for the derivative.
For the results of simulations shown in Fig. 4 b the average chondrocyte density
and average GAG concentration were calculated using the formula
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