Biomedical Engineering Reference
In-Depth Information
(a)
(b)
Figure 12.1
Anaerobic metabolism (a) and aerobic metabolism (b) of PEG.
The mathematical model (12.1) was originally proposed for the PE biodegrada-
tion. However, it can be viewed as a general biodegradation model for exogenous
depolymerization processes, which covers not only the PE biodegradation but also
other polymers such as PEG. A PEG molecule is fi rst oxidized at its terminal, and
then an ether bond is separated (Figure 12.1) [6, 7]. This process corresponds to β
-oxidation for PE, and we call it oxidation because oxidation is involved throughout
the depolymerization process [6, 7]. Note that
L
=
44 (CH
2
CH
2
O) in the exogenous
depolymerization of PEG, whereas
L
28 (CH
2
CH
2
) in the β - oxidation of PE.
Equation (12.1) forms an initial value problem together with the initial
condition
=
wM f M
(, )
0
=
( )
(12.2)
where
f
() represents the initial weight distribution. Given the total consumption
rate
α
(
M
and the oxidation rate
β
(
M
, the solution of the initial value problem is
a function
wtM
(, ) that satisfi es Eq. (12.1) and the initial condition (12.2). Given
the initial condition (12.2) and an additional fi nal condition at
tT
=>
0
wT M
(, )
=
gM
( )
(12.3)
Equation (12.1) forms an inverse problem together with the conditions (12.2) and
(12.3). It is a problem to determine the degradation rates
α
(
M
and
β
(
M
for which
the solution
wtM
(, ) of the initial value problems (12.1) and (12.2) also satisfi es the
fi nal condition (12.3). It has been shown that the following condition is a suffi cient
condition for a unique positive total degradation rate
α
(
M
to exist, given the β
- oxidation rate
β
(
ML
+
)
and the weight distribution
wM
(
+
L
)
[4, 5]:
MM L
ML
β
(
+
)
T
∫
0
<
gM
()
<
f M
()
+
wsM
(,
+
L s
)
d
(12.4)
+
0
