Biomedical Engineering Reference
In-Depth Information
concentrations in the liquid (
c
0
) and solid (
c
e
) phases considered separately. By using
a microstructure argument [
17
], the governing equations are:
∂
c
e
∂
t
=−
ʲ
0
c
e
+
ʴ
0
c
0
in
(
−
l
0
,
0
)
(1)
2
c
0
∂
∂
c
0
∂
D
0
∂
t
=
+
ʲ
0
c
e
−
ʴ
0
c
0
in
(
−
l
0
,
0
)
(2)
x
2
where
D
0
(m
2
s
−
1
) is the diffusion coefficient of the unbound solute,
0(s
−
1
)
ʲ
0
≥
0(s
−
1
) are the unbinding (
dissociation
) and binding (
re-association
)
rate constants in the coating, respectively [
10
]. It should be noted that
and
ʴ
0
≥
ʲ
0
depends
on the porosity of the coating
ʵ
0
[
17
]. The binding rate constant is defined as the
inverse of the characteristic solid-liquid transfer time scale,
t
−
0
. The ratio of
the unbinding and binding rate constants is the equilibrium dissociation constant
K
0
ʴ
0
=
ʴ
0
ʲ
0
=
1
−
ʵ
0
ʵ
0
=
.
t
0
and
K
0
are quantities that can typically be determined
experimentally.
The associated initial conditions are:
c
e
(
x
,
0
)
=
C
e
c
0
(
x
,
0
)
=
0
(3)
expressing that, at initial time, the entire drug exists in the solid phase at a maximum
constant concentration, and it is subsequently released into the liquid phase. Since the
metallic strut is impermeable to the drug, no mass flux passes through the boundary
surface
x
=−
l
0
; hence, we impose a no-flux condition:
D
0
∂
c
0
∂
x
=
0
at
x
=−
l
0
(4)
2.2 The Two-Phase Wall Model
Drug that enters the arterial wall is transported by convection and diffusion through
the tortuous paths of the extracellular matrix surrounding the wall's cells. Similarly
to the coating, a phase change can occur in the wall from the free state (
c
1
)ofthe
drug to the bound state (
c
b
) and vice versa at the surface of the cells within the wall.
The free drug (
c
1
) binds to specific receptors on the surface of SMCs to form a bound
complex (
c
b
). The maximum density of specific receptors available to the drug is
denoted as
c
ma
b
. The formed drug-receptor complex is not permanent and can be
dissolved after a typical time scale
t
1
. These two processes (binding and unbinding)
are modeled by a second order reversible saturating binding equations [
1
]:
