the mathematical and probabilistic equations that describe the drug release kinetics from bio-erodible polymers

1. Drug release from erodible
matrices
Presented by : Hadeia Mashaqbeh

2. Contents
• Polymer degradation and erosion
• Factors affect polymer erosion
• Classes of polymer erosion
• Mechanisms involved in the
release from erodible polymer
• Drug release models

3. Degradation vs Erosion
Degradation is most important process in erosion
Erosion is more complex than degradation
4

4. Factors influence
the degradation
kinetics
5
type of polymer
bond and
composition
Molecular mass
Polymer crystallinity
Porosity

5. Factors influence
the degradation
kinetics
6
 Nature of drugs
incorporated, initial
drug loading,
 pH and ions
 Physical size of the
matrix

6. Surface
and bulk
erosion
10

7. Chemical and physical processes occur in
erodible drug delivery systems
water imbibition into the system
drug dissolution
polymer chain cleavage
diffusion of the drug and of polymer degradation products out of the
device
the creation of water-filled pores
micro environnemental pH changes Inside polymer matrix pores by
degradation products
autocatalytic effects during polymer degradation
osmotic effects
the breakdown of the polymeric structure.
11

8. Models for
Erodible system
Mechanistic
models
Mathematical
Models
Probabilistic
models
Empirical
models
12

9. Hopfenberg
model
General equation, valid for:
spheres, cylinders and slabs:
• spherical (n=3),
• cylindrical (n=2)
• slab geometry (n=1).
• Mt and M∞ are the
cumulative amounts of drug t
released at time t and at
infinite time, respectively;
• c0 denotes the uniform initial
drug concentration within
representing
14
surface erosion
Empirical model

10. Mechanistic methods
Mathematical models
16

11. Heller and Baker assumed the following
relationship between the drug permeability at
time t, P t and the initial drug permeability, Po :
where N is the number of initial bonds; and Z
is the number of cleavages during the time
interval [0; t]
assume simple first order kinetics where the
rate of bond cleavage dZ/dt is proportional to
the number of cleavable bonds present,
:
Then
Ρ = Ρ°𝑒 𝑘𝑡
Heller and Baker
model

12. Heller and Baker
model
18
Mt is the cumulative absolute amount of drug
released at time t; A is the surface area of both
sides of the film; P is the permeability of the
drug within the polymer matrix; and c0 is the
initial drug concentration within the system.
BULK erosion
A > Cs
Assumptions:

13. Lee 1980 model
scheme of the drug
concentration profile within
the system according to Lee
(1980).

14. Lee 1980
model
21
B is the surface erosion rate
constant having the dimension of a
velocity;
• ᵟ is the relative separation
between the diffusion and
erosion fronts defined as [δ=
(S−R)/a];
where S is the time dependent

15. Lee 1980
model
assumptio
ns
surface eroding
A > Cs
the erosion front
moves at a constant
velocity
Uniform drug
distribution
film geometry
edge effects are
negligible
perfect sink conditions
Mass transfer
resistance and swelling
neglected
24

16. Multi-phasic drug release pattern
Generally, bi- or triphasic drug-release behavior is observed from bioerodible matrices
models combine diffusion and other modes of transport mechanisms

17. Chemical reaction-diffusion release mechanism
Scheme of the chemical reactions taken into account
Model developed by Thombre, Joshi and Himmelstein quantifying drug release
and polymer degradation in poly(ortho ester)-based delivery systems containing
acid-producing species

18. Chemical reaction-diffusion
• chemical reactions were then coupled with
diffusion controlled mass transfer processes as
follows:
27
Ci and Di are the concentration and diffusion coefficient of species i,
ʋ is the net sum of synthesis and degradation rate of species i, and
x is the space variable.
A:water
B:acid generator
(for e.g. acid
anhydride)
C:acid
E : drug

19. Chemical reaction -diffusion
• To account for the effect of degradation, the diffusion coefficient of all species is related to the local
extent of polymer hydrolysis and is given by :
• Di,0 is the diffusion coefficient of species i when the polymer is not hydrolyzed, CD,0 and CD are the
concentrations of species D (ester linkages in the polymer) at time zero and time t, respectively, and
µ is a constant.
28

20. Lao and
coworkers
model
(2008)
developed a novel model that postulated that the total
fraction of drug release from bulk-degrading
polymer is a summation of three mechanisms/steps
that occur in sequence:
1) burst release, Solvent (water) penetration into the
matrix accompanied by rapid drug dissolution
2) A degradation-dependent ‘‘relaxation of the
network’’ that allows for more release/dissolution of
hydrophobic drug
3) diffusional release. (Drug removal by a diffusional
process to the bulk aqueous phase)

21. 31
The proposed model is given by a summation of these three steps.
-poly(DL-lactide-co-glycolide),
P(DL)LGA 53/47
-films were prepared through
solution casting method

22. 32
For hydrophilic
drug фr=zero
The proposed model is given by a summation of these three steps.
the fraction of drug released through initial
burst, b due to immediate desorption of drug
particles located at or near the surface of a
film. Its kinetics follows an exponential
relationship
kb denotes the rate of drug desorption
while the end of burst release is given by tb
r is the coefficient of relaxation-induced release, kr is the
degradative relaxation constant and tr is the end of
relaxation-induced release.
describes the relaxation-induced drug
dissolution release.
describes the diffusional release,
adapted from the exact solution by
Crank (1975) for Fick’s second law of
one-dimensional diffusion
D is diffusion coefficient, 2l =thickness of films

23. Comparison of Lao et al. model and experimental data release from bulk-degrading
PLGA 53/47 films. (from Lao et al., 2009)
hydrophilic (metoclopramide) hydrophobic (paclitaxel)
34

24. Lao and
coworkers
model
(2008)
Similar observations were reported by Faisant et
al. as follows: When the release medium was
changed, drug solubility decreased and partially
caused release patterns to change from
monophasic to bi- and tri-phasic.
Assumptions:
for thin films
perfect sink conditions
(C0 < Cs, monolithic solutions).
uniform drug distribution
MATLAB, a programming software, was used to
fit Eq.to the experimental data of in vitro release
35

25. Probabilistic models
37

26. Zygourakis
Monte Carlo-based models
surface eroding matrices.
simulate drug or polymer ‘dissolution’ a ‘life
expectancy”
The lifetime of a specific solid can be constant
for all pixels of this type, or distributed according
to some distribution (e.g., Poisson distributions).
only the upper side of the system is ‘exposed’ to
the release medium
the model does not take into account mass
transport processes, such as the diffusion of
water, drug or polymer degradation products.
38

27. Zygourakis’ model
39
Zygourakis’ model : Four types of pixels are distinguished: drug, polymer, filler and pore void.
Four stages of the device are illustrated, corresponding to the time points when (a) 0%; (b) 25%; (c) 50%;
and (d) 75% of the drug is released

28. Modeling
monomer
release from
bioerodible
polymers
• Both, Monte Carlo Simulation-
based polymer degradation and
diffusional mass transport
processes are taken into account in
the model.
• Besides erosion other phenomena
such as the diffusion of monomers,
crystallization of polymer
degradation products, and
microclimate pH effects were taken
into account.
40
GÖpferich and Langer1995

29. The description of porosity, using a
Monte Carlo Model:
45

30. Siepmann and coworkers (2002)
47
Schematic of a single bio-erodible microparticle for mathematical analysis: (a) three-dimensional
geometry; (b) two-dimensional cross-section with two-dimensional pixel grid used for numerical
analysis.

31. 49
Principle of the Monte Carlo-based approach to simulate polymer degradation and diffusional drug
release; schematic structure of the system: (a) at time t=0 (before exposure to the release
medium); and (b) during drug release. Gray, dotted, and white pixels represent nondegraded
polymer, drug, and pores, respectively.

32. Structural modeling of drug release from
biodegradable porous matrices based on a combined
diffusion/erosion process
V. Lemaire, J. Bélair, P. Hildgen
International Journal of Pharmaceutics 258 (2003) 95–107
53

33. Hypothesis
and
representation
for simplification purpose,
each element idealized as
a cylinder of constant
radius.
each element will release
its drug molecules outside
the sphere independently
of the other elements
55

34. Each element is idealized as a cylinder of length L and radius R with a pore
embedded coaxially in the center with radius r (r < R) and length L
56
Domain 1

35. Modeling
equations
Assumption
-the drug is not chemically bond to the
polymer.
-no solid aggregate of drug
-Uniform drug distribution
The growth of the mean pore radius due to polymer
erosion
r(t) = kt + r0
k is a velocity of erosion and r0 is the initial pore radius.
The symmetry about the midpoint z = L/2
57

36. Modeling
equations
the equation describing the evolution of the
concentration C(,z,t) under Fickian diffusion
is given by
σ1 = σ/R
where R is called the retardation factor
σ2 = Kr σ1
where Kr is called the restriction factor
58
where p and z are the radial and axial axes,
respectively. D = σ1 in domain (1), and D = σ2
in domain (2).

37. 60

38. 61
Release curves generated by the model when k varies over a range of values,
namely, 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.36,

39. 62
Release curves generated by the model for different values of σ2, namely, 4150,
5000, 6500, 8600, 11,300, 15,000, 25,700, 170,000. The other parameters are kept
fixed

40. 63
Cumulative release of an antihypertensive drug from PLA microspheres with the
indicated ratios (in percentage) of high/low
molecular weight PLA.
the burst, are not accounted for by
the model. (between 1 and 5%).

41. Conclusion
• This model confirms the major role that the relative dominance
between diffusion and erosion in the release kinetics.
• the velocity of erosion, the effective diffusion coefficient of the
drug molecule in the wetted polymer, the average pore length,
and the initial pore diameter are sensitive parameters
• the porosity and the effective diffusion coefficient of the drug in
the solvent-filled pores is seen to have little influence
• by using release data from biodegradable microspheres with
different ratios of low and high molecular weight PLA…..by
varying two parameters for all types of experimental kinetics:
from the typical square root of time profile to zero-order
kinetics to concave release curves.
66

42. Summery and Conclusions
• various mathematical and probabilistic models
developed to describe the release from eroding
systems.
• Proper characterizations of the systems should be
studied to provide input values to the model
parameters
• Models based solely on diffusion with
time/degradation dependent diffusion coefficient
(diffusional-based models) are generally simpler
and easier to use. their applications may be
limited.( monophasic release )
67

43. Summery and Conclusions
• Multi-phasic release, (more difficult to use and
almost require the aid of computer/programming
languages).
• Probabilistic model is another interesting approach
that describes polymer degradation
• As all these models have own advantages and
limitations, researcher should carefully select the
appropriate model that can represent the systems
under study.
68

44. Cited
works
Faisant, N., Akiki, J., Siepmann, F., Benoit, J.P., Siepmann, J.,
2006. Effects of the type of release medium on drug release
from PLGA-based microparticles: experiment and theory. Int.
J. Pharm. 314, 189–197.
Crank, J., 1975. The Mathematics of Diffusion , second ed.
Clarendon Press, Oxford.
Gِ pferich, A., Langer, R., 1995. Modeling monomer release
from bioerodible polymers. J. Control. Release 33, 55–69.
Gِ pferich, A., Langer, R., 1993. Modeling of polymer
erosion. Macromolecules 26, 4105–4112
Heller, J., Baker, R.W., 1980. Theory and practice of
controlled drug delivery from bioerodible polymers. In:
Baker, R.W. (Ed.), Controlled Release of Bioactive Materials.
Academic Press, New York, pp. 1–18.
Higuchi, T., 1961. Rate of release of medicaments from
ointment bases containing drug in suspension. J. Pharm. Sci.
50, 874–875.

45. Cited
works
Joshi, A., Himmelstein, K.J., 1991. Dynamics of controlled
release from bioerodible matrices. J. Control. Release 15,
95–104.
Lao, L.L., Venkatraman, S.S., Peppas, N.A., 2008. Modeling of
drug release from biodegradable polymer blends. Eur. J.
Pharm. Biopharm. 70, 796–803.
Lao, L.L., Venkatraman, S.S., Peppas, N.A., 2009. A novel
model and experimental analysis of hydrophilic and
hydrophobic agent release from biodegradable polymers. J.
Biomed. Mater. Res. Part A 90, 1054–1065.
Lee, P.I., 1980. Diffusional release of a solute from a
polymeric matrix – approximate analytical solutions. J.
Membr. Sci. 7, 255–276.
Siepmann, J., Faisant, N., Benoit, J.P., 2002. A new
mathematical model quantifying drug release from
bioerodible microparticles using Monte Carlo simulations.
Pharm. Res. 19, 1885–1893.
Lemaire, V., Belair, J., Hildgen, P., 2003. Structural modeling
of drug release from biodegradable porous matrices based
on a combined diffusion/erosion process. Int. J. Pharm. 258,
95–107.

46. Cited
works
Joshi, A., Himmelstein, K.J., 1991. Dynamics of controlled release from
bioerodible matrices. J. Control. Release 15, 95–104.
Lao, L.L., Venkatraman, S.S., Peppas, N.A., 2008. Modeling of drug release
from biodegradable polymer blends. Eur. J. Pharm. Biopharm. 70, 796–803.
Lao, L.L., Venkatraman, S.S., Peppas, N.A., 2009. A novel model and
experimental analysis of hydrophilic and hydrophobic agent release from
biodegradable polymers. J. Biomed. Mater. Res. Part A 90, 1054–1065.
Lee, P.I., 1980. Diffusional release of a solute from a polymeric matrix –
approximate analytical solutions. J. Membr. Sci. 7, 255–276.
Siepmann, J., Faisant, N., Benoit, J.P., 2002. A new mathematical model
quantifying drug release from bioerodible microparticles using Monte Carlo
simulations. Pharm. Res. 19, 1885–1893
Thombre, A.G., Himmelstein, K.J., 1985. A simultaneous transport–reaction
model
for controlled drug delivery from catalyzed bioerodible polymer matrices.
AIChE J. 31, 759–766.
Zygourakis, K., 1989. Discrete simulations and bioerodible controlled release
systems. Polym. Prep. ACS 30, 456–457.
Zygourakis, K., Markenscoff, P.A., 1996. Computer-aided design of
bioerodible devices with optimal release characteristics: a cellular automata
approach. Biomaterials 17, 125–135.