Biomedical Engineering Reference
In-Depth Information
n-1
d
C
t
•
n
=∑
k
k
=∑
=∑
=∑
Ck
C
-2
2
Ck
-2
CC
∑
∑
CC
=∑
=∑
k
=∑
j
n-j
Ck
n-j
Pn
Pn
CC
Pn
j
C
j
Ck
n-j
Ck
n-j
d
=∑
=
j=1
n-j
Pn
j=1
k
K
k
K
•
P
∑-
P
eQ
[4.47]
+2
C
∑-
∑-
C
(-
C
(-
(-1)
(-
C
W
∑-
W
Cn
W
W
(-
(-
C
(-
(-
(-1)
1)
C
n
n
n
W
j=n+1
eQ
where
C
i
,
C
0
i
and
k
C,i
are the molar concentration, the molar concentration in
the external environment, and the effective mass transport coeffi cient for the
i
ith species, respectively,
k
P
is the polymerisation kinetic constant,
K
eQ
is the
polymerisation equilibrium constant,
S
eXT
is the external coating surface,
V
R
is
the device volume and
V
M
is the volume of the degraded inner zone. In order
to simplify the model, which would result in a large system of differential
equations, the method of moments was applied. as the water and monomer
mass balance equations (equations [4-45]-[4.47]) can be written in terms
of statistical moments (
m
i
), the following system is obtained:
d
C
t
S
k
M
eX
S
eX
eX
T
0
P
[4.48]
=
k
S
eX
eX
T
(
C
-)
0
CC
0
C
-)
-) -2
kC
M
kC
MP
+
k
P
C
(
(
( -
C
M
)
m
+
P
C
C
(
(
(
(
W0
m
m
m
C,
C,
M
CC
M
CC
MP
-)
MP
-) -
MP
-2
M0
M
m
M0
+
(
W
0
(
W0
(
W0
(
W0
(
(
(
(
m
W0
W0
-
d
V
(
W0
(
W0
(
W0
W0
(
(
m
W0
C,
M
V
K
K
V
M
M
eQ
d
C
t
k
C,W
S
eX
k
W
S
eX
V
V
R
eXT
R
0
2
P
[4.49]
k
C,W
(
CC
-
CC
-
0
0
CC
) +
C
(
0
)
m
2
P
C
(
m
m
=
C,W
W
CC
WP
)
WP
) +
WP
+
WP
WP
m
-
W
m
1
m
1
1
1
-
m
0
0
d
K
K
eQ
d
d
S
k
m
m
0
0
eX
S
eX
eX
T
0
2
P
[4.50]
=
k
S
eX
eX
T
(
CC
-
CC
-
0
0
CC
)
C
(
0
)
m
2
+
+
P
C
(
m
m
C,M
C,M
M
CC
MP
)
MP
) -
MP
-
MP
MP
m
W
m
1
m
1
1
m
1
-
m
0
0
V
K
t
C,M
V
K
V
M
M
eQ
d
d
S
m
m
1
t
[4.51]
1
eX
S
eX
eX
T
0
=
k
S
eX
eX
T
(
CC
-
CC
-
0
CC
0
)
C,M
C,M
M
CC
M
V
C,M
V
V
M
M
Ê
ˆ
˜
2
d
d
S
k
PW
k
PW
m
m
2
m
mm
2
m
m
1
m
21
m
21
m
m
0
2
eX
S
eX
eX
T
0
2
+
PW
21
=
k
S
eX
eX
T
(
CC
-
CC
-
0
0
CC
) + 2
2
- 2
Ê
Á
Ê
m
m
m
2
2
+
+
PW
PW
Ê
Ê
Ê
Á
m
+
Á
C,M
C,M
M
CC
MP
)
MP
) +
MP
+ 2
MP
2
MP
m
1
1
m
1
m
m
1
3
Ë
Á
Ë
Ë
V
MP
MP
m
1
K
t
C,M
V
3
K
Ë
Ë
V
M
M
eQ
1
0
[4.52]
Quantities such as average molecular weight and poly-dispersity can be
directly computed from statistical moments.
4.3 Conclusions and future trends
The purpose of this chapter is to underline the importance of transport
phenomena and mathematical modelling in biomedical applications where
a thermodynamic system, not in an equilibrium condition, undergoes a
spontaneous irreversible transformation, as in classic drug delivery systems.
The irreversibility of thermodynamic processes was thoroughly investigated
from a mathematical point of view, using phenomenological and kinetic
approaches. Three biomedical examples were analysed in detail: membranes
Search WWH ::
Custom Search