Computational models can be useful tools for increasing the understanding of polymer degradation and drug release processes from controlled-release microspheres. Although our main focus is the development of general methods, for specificity this study considers the biocompatible polymer poly(lactic-co-glycolic acid) (PLGA). Models have been published which take into account many of the factors influencing PLGA degradation and subsequent drug release such as drug diffusivity and dissolution, pore structure development, polymer composition, and device geometry (Arifin et al. 2006, Batycky et al. 1997). However, few models have fully considered the autocatalytic PLGA hydrolysis kinetic mechanism which is believed to be a key factor leading to particle size-dependent heterogeneous polymer degradation (Grizzi et al. 1995, Richards Grayson et al. 2005). Hydrolysis is catalyzed by acids including the carboxylic acid end groups of the polymer chains. Hydrolysis eventually leads to erosion when sufficiently small water-soluble oligomer fragments from polymer degradation are transported out of the device. With autocatalysis, increasing the diffusion distances as in larger particles leads to slow diffusion transport and a build up of acidic fragments which further catalyze hydrolysis. Experimentalists have reported evidence of local pH drop due to accumulation of the acidic byproducts of the polymer hydrolysis and degradation rates which increase with polymer particle size (Richards Grayson et al. 2005, Fu et al. 2000).
Despite the size dependence of autocatalytic PLGA degradation, we are unaware of any previous model which tracks hydrogen ion concentration as a function of space and time in addition to modeling degradation kinetics, molecular weight distribution, and drug transport with varying diffusivity. Often polymer degradation is assumed to follow well-mixed pseudo-first-order kinetics in models that aim to include autocatalytic effects (Batycky et al. 1997, Siepmann et al. 2005, Raman et al. 2005). Researchers (Siparsky et al. 1998) have shown that for poly(lactic acid), a polymer very similar to PLGA, pseudo-first-order kinetics are a good approximation for hydrolysis catalyzed by an external strong acid but are insufficient for modeling autocatalysis. They proposed two alternative kinetic expressions. The first considers quadratic autocatalysis. This model had limitations because it did not capture the effects of partial dissociation of the carboxylic acid end groups. The second kinetic expression did consider partial dissociation effects and had half-order dependence on carboxylic acid. This model fits the data very well except near the extrema of the data set. It is believed that this model can be improved upon by removing the assumptions imposed by Siparsky et al. Deriving the model results in a kinetic expression for autocatalysis which is dependent on the concentrations of water, external acid source, and dissociated carboxylic acid end groups.
The model being developed in this work uses the autocatalytic kinetic expression described above to determine the rate of hydrogen ion generation and accumulation and the molecular weight distribution of polymer within the microspheres. The effective diffusivities of protons and encapsulated drug molecules are dependent on the polymer molecular weight and pore size distributions, with the effective diffusivities monotonically increasing as the polymer degrades and erodes. The partial differential equations for Fickian diffusion (and reaction, in the case of hydrogen ion) are solved numerically using the finite volume method for a variety of initial and boundary conditions. This model allows for variations in microsphere size, intraparticle pH, and polymer molecular weight which all influence transport of drug and polymer fragments through the polymer.
References
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