Biomedical Engineering Reference
In-Depth Information
Solution
xwtM
(, ) of (12.6) is associated with the initial condition (12.2). Given
the degradation rate
=
β
(,
tM
, Eq. (12.6) and the initial condition (12.2) form an
)
initial value problem.
Time factors of the degradation rate such as microbial population, dissolved
oxygen, or temperature affect molecules regardless of their sizes. The dependence
of degradation rate on those factors is uniform over all molecules, and the degra-
dation rate should be a product of a time-dependent part
(
t
and a molecular
σ
dependent part
λ
()
M
β
(,
tM
)
=
σ
() (
t
λ
M
)
(12.7)
Note that
(
M
represent the magnitude and the molecular dependence
of degradability, respectively.
In order to simplify the model, let
σ
(
t
and
λ
t
∫
τ
=
σ
(
ss
d
(12.8)
0
and
WM wtM XWM YWML
(, )
τ
=
(, )
,
=
(, )
τ
,
=
(,
τ
+
)
Then
d
d
Xx
t
d
d
d
d
t
1
()
d
d
x
t
=
=
τ
τ
σ
t
and the exogenous depolymerization model (12.6) is converted into the equation
dX
d
M
ML
Y
=−
()
++
+
λ
Mx
λ
(
M L
)
(12.9)
τ
This equation governs the transition of weight distribution
wM
(, )
τ
under the time-
independent or time-averaged degradation rate
(
M
. Given the initial weight
distribution
f
(
, Eq. (12.9) forms an initial value problem together with the
initial condition
λ
WM f M
(, )
0
=
( )
(12.10)
Given an additional condition at
, Eq. (12.9) forms an inverse problem
together with the initial condition (12.10) and the fi nal condition (12.11), for which
the solution of the initial value problems (12.9) and (12.10) also satisfi es the fi nal
condition
τ =Τ
WM gM
(, )
Τ=
( )
.
(12.11)
