# Mathematical Modeling of Desorption

```Mathematical Modeling of Desorption-Diffusion
Controlled Drug Release from Polymer Matrices
Rami Tzafriri
Institute of Computer Science
and
The Department of Neurobiology,
The Hebrew University, Jerusalem, Israel
This research was supported by a grant from the Isreali MOS
The model of Singh et al. (1994)
The binding of charged polypeptides onto a hydrogel matrix is
modeled as Langmuir adsorption and the release of free drug is
assumed to be governed by simple diffusion from a 1D slab.
(1)
M  k C   M max  M   k M
a


d


t
(2)
C  D  2C  P M
t
t
x 2
(3)
(4)
C(0)  C0 , M (0) 
C  0 x  0, 2L
0  x  2L
KC0
1 KC0
, K
ka
kd
Qualitative Analysis
This problem involves 2 typical time scales:
2
L
D

ffid
and
 des  1
kd
Accordingly, two limiting cases are of interest:
Instantaneous desorption
Fast diffusion
 diff L2k
d 1

 des
D
L2 kd
1
D
Instantaneous desorption and linear binding
In this limit
M
KC  KC
1 KC
and
C  D  2C
eff
t
x 2
with
Deff 
D
1 KM max P
This can be solved analytically and has been used to model the
release of polylysine and gentamicine from collagen matrices
Analytical solution of the linear binding case
For small L (films, microspheres) we are more likely to
2
L
kd
encounter the case of fast diffusion:
1
D
for which no analytical solution is available in the literature
and the experimental results can only be analyzed by solving
the equations numerically, which is rather cumbersome.
However, it turns out that Eqs. (1)-(4) can be solved
analytically in the case linear binding:
KC  1  kaC  kd
In this case:
(1`)
M   k M
d
t
Integrating this yields:










M  M (0) exp  k t  K
d

M max C0 exp  k

d
t





and
(2`)
C  D  2C  k P M max C exp  k t 
a
0
2
d 

t
x
The system (2`), (3)-(4) can be solved analytically for slabs,
spheres, long cylinders and rectangular parallelepipeds, and
predicts a bi-phasic release profile.
Example: lysozyme release from a gelatin film
Kuijpers et al. (1998) measured the release of lysozyme from
cross-linked gelatin films into a PBS solution under perfect
sink conditions. They considered the case:
L  0.03cm, P  250 mg/ml, C0  0.5 mg/ml
and estimated:
Dwater 1.110 6 cm2 / s ,
K M max  0.020 mg/ml
onto the gelatin matrix.
Experimental estimate:
Fitted values:
D  0.39 10 6 cm2 / s ,
K M max  0.016 mg/ml
kd  0.0633 h-1

2
L kd
 0.163 1
D
(mg/mg)
0.25
0.2
0.15
0.1
0.05
0
-0.05 0
-5
5
10
15
free drug (mg/ml)
Figure 2: experiment vs. fit
relative lysozyme
release
K M max  0.020 mg/ml
150
100
50
0
0
50
100
150
time (h)
According to this fit 19.6% of the initial drug load is released
in a short diffusive burst at a rate of 3.85/h, whereas 80.4% is
adsorbed and released at a rate of 0.06/h.
```