Biomedical Engineering Reference
In-Depth Information
EXAMPLE 10.7
An erodable spherical polymer of radius
R
(0.5 mm) contains a drug nerve growth factor
to treat brain disorder. Experiments were performed in vitro to characterize the release of
the drug assuming that the rate of diffusion is fast relative to the rate of dissolution of the
polymer. At the surface of the polymer (
r
=
R
) the drug concentration is
C
0
(700 ng/mL).
Far from the surface (
r
→
∞
), the drug concentration drops to zero. Determine the steady-
state release rate from the polymer.
Solution:
At steady state with no chemical reactions, (10.3) reduces to
1
d C
r
⎛
⎞
2
i
=
0
⎜
⎟
2
⎝
⎠
r
dr
dr
Integrating twice results in the expression
C
i
=
B
−
A
/
r
rom the boundary condition at
r
→
∞
,
B
=
0
rom the boundary condition at
r
=
R
, we get
A
=
−
RC
0
The resulting concentration profile and flux are
CR
CR
C
0
and
N
D
0
2
=
=
i
ir
AB
r
r
As the distance from the surface increases, the flux decreases, due to the increase
in the cross-sectional area through which transport occurs. The steady state release rate
(mol/s) at the polymer surface is the product of the flux times the surface area.
2
Release rate
N
4
R
4
DC R
=
π
=
π
ir
rR
0
=
10.4.4 Complex Model Systems
The comprehensiveness and complexity of models are increasing due to the neces-
sity of developing more realistic and integrative models for physiological subsys-
tems of the human body. The goal is to base the development of diagnostic and
therapeutic procedures on mathematical models. The questions being asked are
much more demanding and the level of interdependence of the component models
has increased dramatically in order to answer complex physiological questions.
In many cases several time scales and levels of functionality are considered. Fre-
quently varieties of models require interfacing of different models from many dif-
ferent sources with different local design rules. For example, in order to express the
convective component of the mass transport equation, it is necessary to know the
velocity profiles. The fluid dynamics can be described by the equation of continuity
or mass balance and the equation of motion, which describes the momentum bal-
ance (described in Chapter 4). Analytical solutions to these sets of differential equa-
tions are difficult to obtain and there may be multiple solutions. One widely used
