Erosion controlled release

5.3.3.1. EMPIRICAL MODELS

Hopfenberger supposed that the rate of drug release from an erodible polymer is proportional to the surface area of the device and derived the following empirical model:

n

a C

t k M

Mt ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ −

−

∞ = ₀

1 0

1

where k0 is a rate constant, a is the radius of a cylinder or sphere, or the half thickness of a slab. According to the geometry of the device the shape factor n becomes n = 3 for spheres, n = 2 for cylinders or n = 1 for slab geometry. Thus, according to the Hopfenberger model a zero-order release can be achieved with slab-shaped systems [114, 207].

5.3.3.2. MECHANISTIC MODELS

Mechanistic models for degradable matrices either treat polymer erosion as a combined diffusion and chemical reaction process, or consider erosion as a random event by applying the Monte Carlo simulation [8, 207].

A model belonging to the first subclass was proposed by Heller and Baker. Here, based on the Higuchi equation, a mathematical solution for bulk eroding polymers was developed by considering that the permeability of the polymer matrix increases due to polymer cleavage. Relating the change of permeability to a function of the numbers of remaining bonds and assuming first order kinetics for the polymer degradation the following equation was established [207]:

t C kt P

A dt

dMt 2 ₀exp( ) ₀

= 2

where P is initial permeability, A the surface area, and k the first order rate constant.

Due to the increase in diffusion pathways the release rates first decrease with time.

After a certain time point this is compensated by the exponential term, which represents the permeability of the progressively eroding polymer. Hence, the release rates increase when erosion takes place [207].

Several other models were developed that apply the equations of diffusion with a time-dependent increase in the diffusion coefficients. The diffusion coefficients are thereby calculated based on the molecular weight loss of the polymer matrix [8, 114,

207]. However, these approaches neglect the acceleration of the drug release due to the evolving pore network concomitant to the erosion process [8].

Especially in our case of macromolecule delivery, the microporous structure of the matrix is of outstanding importance since the release will be restricted to the diffusion through water filled pores.

In order to study the release of a model macromolecule (FITC-dextran) from porous PLGA microspheres Ehtezazi and Washington introduced an alternative attempt to obtain the effective diffusivity within the matrix. Here, based on the percolation theory (see Chapter I.5.3.1) the diffusivity within the polymeric matrix was assumed. This data allowed the subsequent calculation of the released fraction with a derivation of Crank´s solution for diffusion in spheres [63].

Langer and co-workers developed another model for the release of proteins from PLA and PLGA microspheres that took into account alteration of the microstructure [12]. The structural change of the matrix was thereby related to the mass and molecular weight loss of the polymer during erosion. The following assumptions were made to describe the release process: after microsphere preparation with the double emulsion technique the protein is localised in occlusions that are entirely embedded within the polymer matrix. Consequently, the observed initial burst release was ascribed only to the desorption of protein from the exterior microsphere surface or from the surface of pre-existing mesopores. The concomitant hydration gives rise to polymer erosion. As a result, micropores grow up and coalesce to mesopores. Since the large size of the protein restricted the release until a sufficient number of mesopores was created, there was an “induction phase” with low or no protein release, followed by a second burst.

While the developed model predicted well the release of a glycoprotein, it failed for tetanus toxoid. The release of the latter did not show biphasic release behaviour. The authors assumed that in this case, the desorption was either very slow or the hydration was very fast and thus both phenomena were overlapped [12].

A subclass of mechanistic models simulates the polymer degradation as a random event using the Monte Carlo technique. This attempt was introduced in the late 1980’s by Zygourakis to describe the drug release from surface eroding polymers.

The complexity of simultaneous polymer and drug dissolution, which in turn changes continuously the morphology of the erosion front, is solved by generating a two-dimensional grid that simulates the microstructure within the device (Figure 10). Each

cell/pixel represents one of the device elements: drug, solvent, polymer, or pore void, respectively. In order to model the drug or polymer dissolution for each cell a lifetime is defined. According to this lifetime expectance the cellular grid is updated after distinct time-intervals. For example, if a drug cell is adjacent to one or more solvent cells, then its lifetime will end and the cell may change to a solvent cell at the next time interval [264]. However, the diffusional resistance to the transport of the drug to the bulk fluid is neglected and the drug release rate is thought to be determined only by the detachment of the molecule from the matrix surface. Furthermore, the model does not consider other mass transport processes, such as diffusion of water or polymer degradation products [207].

Figure 10: Zygourakis´ model for the simulation of drug release from surface eroding polymers.

The pictures depict the configuration of the cellular array modelling the cross section of the controlled release device. The two-dimensional grid is shown at four different stages of release: (a) initial situation, (b) 25 % released, (c) 50 % released, (d) 75 % released drug. Four types of pixels are distinguished: polymer, drug, filler, and solvent [264].

More sophisticated models were developed by Göpferich and co-workers by taking into account Monte Carlo simulation based polymer degradation and diffusional mass transport phenomena. These models are not limited to surface eroding polymers and can also be applied to bulk eroding matrices. Due to the fact that polymer degradation is affected by porosity, device geometry, and crystallinity these parameters were also taken into account [207].

While these models were developed for cylindrical implants, Siepmann et al.

proposed a model based on Monte Carlo simulation for the release kinetics from PLGA microspheres. According to the symmetry conditions of an idealised sphere the mathematical analysis could be reduced to one quarter of the sphere (Figure 11).

That way the computation time for numerical analysis was minimised. Importantly,

the proposed model covers well complex triphasic drug release kinetics with an initial burst, followed by zero-order release, and a second burst. Due to the continuous update of the grid it was possible to take into account the time dependent changes of the microsphere morphology. As outlined above especially in the second release period a superposition of drug dissolution, diffusion, and polymer erosion occurs.

Since the matrix porosity increases, the drug diffusivity within the matrix is enhanced.

Consequently, the time- and direction-dependent porosities within the microparticle were calculated for any grid point. Based on this information the time-, position-, and direction-dependent diffusivities were calculated. Furthermore, the schematic representation of the microsphere by a two dimensional pixel grid allows elucidating the influence of drug dissolution on the resulting release kinetics. At each time interval the concentration of drug and water of each grid are calculated and only the amount of soluble drug is considered to be available for diffusion [211].

Figure 11: Monte Carlo-based attempt to simulate polymer degradation and diffusional drug release from PLGA microspheres.

The schematic structure of the system is illustrated before exposure to the release medium (a) and during release (b).

Recently, a Monte-Carlo scheme was developed to simulate the diffusion and the release of proteins from cross-linked dextran microspheres. Importantly, with the obtained scaling relations it was possible to predict the release curves of various proteins [239].

The apparent advantage of Monte Carlo-based models is that they provide a very realistic comprehension of the drug release mechanism. In contrast to simple analytic solution, these models take into account the complexity of several overlapping physical and chemical phenomena which influence the drug release kinetics.

However, no explicit mathematic model is derived, hence it is difficult to get an idea of the effects of certain device design parameters on the drug release [8, 209].