Controlled drug delivery is a rapidly advancing technology of high economic importance to the pharmaceutical and health industries [1]. Sustainable drug delivery systems (DDSs) are designed so that a desired drug release rate of the active pharmaceutical ingredient (API) can be realized over a specified time period. In general, DDSs possess several distinct advantages over the conventional drug dosing formulations including: (i) sustainable administration of the therapeutic API at the recommended therapeutic level; (ii) decrease of the amount of administered API, thus limiting potential toxicity side effects; (iii) reduced number of dosages; and (iv) improved patient compliance due to infrequent administration [2], [3], [4], [5].

Polymers are commonly used as drug carriers for they can enhance drug safety and efficacy and impart a controlled and sustainable drug release rate at the specified site of action. Note that polymeric drug carriers can also minimize unwanted interactions of unprotected API with other molecules (e.g., enzymes) that could cause changes in the chemical structure, activity and efficacy of the administered drug. In general, polymeric drug delivery systems can be classified according to the governing drug release mechanism into three main types, namely, (i) diffusion-controlled; (ii) swelling-controlled; and (iii) erosion-controlled systems [5], [6], [7], [8].

Poly(lactic-co-glycolic acid) (PLGA) is a commonly used polymer in drug delivery systems. It is a biocompatible and biodegradable polymer that has been approved by the Food and Drug Administration (FDA) and the European Medicine Agency for use in drug delivery applications. It can provide a sustained drug release rate profile by proper selection of polymer molecular and morphological properties (e.g., molecular weight distribution, copolymer composition, particle size distribution, particle porosity, etc.). Additionally, PLGA carriers can be loaded with different types of drugs (e.g., hydrophilic and hydrophobic small molecules or/and biomolecules). It is important to note that the polymer-drug carriers can be properly engineered by tuning the material properties in order to achieve a desired drug release rate over a specified release period [5], [9], [10].

In general, the drug release rate is the net result of a combination of physical and chemical processes including drug dissolution, water absorption, polymer swelling and dissolution, polymer hydrolysis and degradation, drug diffusion, etc. The molecular weight distribution (MWD) , degree of crystallinity and degree of hydrophilicity of the polymer as well as the chemical nature and molecular weight of API, drug-polymer interactions (acidic or basic), particle size distribution, pore size distribution, drug loading distribution, drug solubility are some of the key drug-carrier properties that can affect the drug release rate from a polymer-drug formulation. Yet, the underlying drug release mechanism is not always clear due to the complexity of the drug delivery phenomena and the fact that the rate-determining process step is often obscured [11], [12], [13]. In Figure 1, typical drug release profiles from biodegradable polymer carriers are illustrated [11].Fig 2.Fig 3.

Note that case 2 in Figure 1 represents a characteristic tri-phasic drug release profile exhibiting three drug release phases: (i) A burst-like drug release rate(Phase I), during which drug molecules entrapped at the carrier’s external surface are rapidly released when the drug-loaded carriers come into contact with the release medium. Additionally, particle disintegration can contribute to a burst drug release behavior. (ii) A diffusion-controlled drug release rate (Phase II), during which the API molecules do slowly diffuse through the relatively dense polymer matrix and primarily via the water-filled pores of the drug-carrier. During this phase polymer degradation can take place, which further enhances the drug diffusion rate. (iii) Phase III is associated with a second burst in the drug release rate, mainly attributed to the onset of polymer bulk erosion [11], [13], [14].

In the open literature, there is a great number of publications dealing with mathematical modeling of drug release from drug-loaded polymer carriers and matrices. In the review works of Sackett and Narasimhan [15], Lao et al. [16], Siepmann and Siepmann [17], Ford Versypt et al. [6], Xu et al. [13], Parmar and Sharma [18], the most important contributions to the subject are cited. Ford et al. [19] developed a polymer degradation-drug diffusion model by considering an autocatalytic kinetic mechanism for PLGA degradation and a time-varying drug diffusion coefficient in terms of particle’s porous network. The model revealed the dependency of the drug release rate on particle size and drug diffusion coefficient. Casalini et al. [20] proposed a kinetic model accounting for the autocatalytic degradation of PLGA chains and transport of generated oligomers out of the polymer matrix. The postulated polymer degradation model was coupled with a drug diffusion model that accounted for both drug dissolution as well as diffusion of the dissolved drug through the polymeric matrix. An effective diffusion coefficient expressed in terms of the average molecular weight of the PLGA was introduced to account for the increase of drug diffusion coefficient due to polymer degradation. Busatto et al. [21] investigated the effect of polymer degradation on drug release rate from biodegradable polymer carriers. Busatto et al. [14] combined together the hydrolytic degradation PLGA model of Busatto et al. [21] with a new drug diffusion and dissolution model to analyze the effects of particle size, polymer molecular weight and drug loading on drug release rate.

In the present study, a model-based optimization approach is detailed to calculate the optimal particle size and drug-loading distributions of a population of biodegradable PLGA drug carriers so that a desired drug release rate, over a specified period of time, can be realized. In particular, in Section 2, a comprehensive kinetic mechanism is postulated to describe the hydrolytic degradation of PLGA chains in terms of the absorbed water concentration. The decrease in the polymer molecular weight is quantitatively described through the spatial-temporal variation of the leading moments of the molecular weight distribution in terms of the spatial-temporal water concentration in the polymer particles. In Section 3, a polymer degradation-diffusion model is formulated to predict the drug release rate from a size-distributed population of drug-loaded biodegradable particles. The model takes into account the spatial-temporal variation of the drug and water diffusion coefficients due to polymer degradation that results in a decrease of the molecular weight of the polymer. In section 4, the unknown model parameters are estimated by fitting model predictions to the experimental data of Berchane et al. [22] and Raman et al. [23]. In Section 5, a multi-parametric optimization problem is formulated to calculate the optimal particle size and drug loading distributions of drug-loaded polymer particles to realize a desired drug release profile of a therapeutic drug dose over a drug administration period of several weeks. In the final section of the paper, the key findings of the present work are summarized and the significance of the proposed model-based approach for optimal design of DDSs is underlined.