Mechanistic computational modeling of implantable, bioresorbable, drug release systems

Abstract

Implantable, bioresorbable drug delivery systems offer an alternative to current drug administration techniques; allowing for patient-tailored drug dosage, while also increasing patient compliance. Mechanistic mathematical modeling allows for the acceleration of the design of the release systems, and for prediction of physical anomalies that are not intuitive and might otherwise elude discovery. This study investigates short-term drug release as a function of water-mediated polymer phase inversion into a solid depot within hours to days, as well as long-term hydrolysis-mediated degradation and erosion of the implant over the next few weeks. Finite difference methods were used to model spatial and temporal changes in polymer phase inversion, solidification, and hydrolysis. Modeling revealed the impact of non-uniform drug distribution, production and transport of H+ ions, and localized polymer degradation on the diffusion of water, drug, and hydrolyzed polymer byproducts. Compared to experimental data, the computational model accurately predicted the drug release during the solidification of implants over days and drug release profiles over weeks from microspheres and implants. This work offers new insight into the impact of various parameters on drug release profiles, and is a new tool to accelerate the design process for release systems to meet a patient specific clinical need.

Graphical Abstract:

Mathematical treatment of polymer solidification, degradation, and diffusivity accurately predicts drug release profiles from a range of size and molecular weight. This mechanistic model offers new insight into the impact of various parameters on polymer degradation and drug release profiles, and is a new tool to accelerate the design process for tailored release systems.

1. Introduction

Approximately 50% of the US population takes at least one prescription drug, and the number of patients is anticipated to increase as the population ages. Furthermore, as many as 50% of patients do not self-administer their medication as prescribed, which ultimately costs billions of dollars per year in avoidable healthcare costs [1, 2]. Reliance on repetitive doses of one-size-fits-all pills not only results in low patient compliance, but even when taken as prescribed, results in fluctuations of drug concentration around a target zone, resulting in poor clinical outcomes for the patients [3–5]. Controlled release drug delivery systems provide a platform to maintain blood plasma levels of drugs without the need for repeating dosing [6].

Polymer-based controlled release systems have already entered the market and are being used to treat a wide range of diseases, from cancers to infections, with a controlled systemic or local drug dosage [7–15]. One such category of release systems, referred to as in situ forming implants (ISFIs), have been used to treat prostate cancer by increasing the body’s systemic concentration of leuprolide acetate, which reduces the production of testosterone to treat cancer [7, 8]. Other ISFIs have been designed to release doxycycline to treat gum disease (periodontitis) [8, 9], and for the release of antiretroviral drugs for the treatment or prophylaxis of HIV [11]. Microspheres have also been developed for controlled release of leuprolide acetate to treat prostate cancer. Additionally, they have been used for transarterial chemoebolization for treatment of hepatic tumors to cut off the blood supply and release doxorubicin locally into the tumor [10].

These two controlled release platforms (ISFI and microspheres) can be designed to encapsulate a wide range of drugs and can be placed nearly anywhere in the body via a single minimally invasive injection. Typically, these release vehicles consist of a biodegradable, biocompatible polymer. In the case of ISFI, the polymer is dissolved in a biocompatible and water miscible solvent, and the solution is injected directly into the body where counter exchange of water from the tissue and solvent from the implant solution drives phase inversion of the polymer, resulting in the formation of a solid drug-eluting depot [13]. Implantable microspheres are first formed via phase inversion in a collection tank before the solid depot is implanted into the body [16, 17]. Water simultaneously acts as a stimulus for phase inversion and a catalyst for the degradation of the bioresorbable polymer through hydrolysis of ester bond linkages [18]. These release systems give a characteristic burst of drug followed by a period of diffusion limited release, then as the polymer degrades, a period of degradation enhanced release. The release profile dictates the drug concentration in a patient’s blood plasma over the course of the drug therapy [19].

To better meet patient needs, drug release profiles could be tuned by varying parameters such as the polymer composition, hydrophobicity of the solvent, and polymer chain length [19]. Developing these technologies into personalized healthcare requires that the rate of drug release is predictable and tunable. However, experimentally optimizing the parameters that influence the drug release is time consuming and expensive [15]. Development of a mathematical model in which experimentally controllable parameters can be varied to accurately predict drug release profiles would allow for efficient testing of parameter regimes that produce reliable control of desired drug release profiles and would facilitate the rational design of implant formulations.

Building upon decades of research on the modeling of swelling and degradation of polymer implants, the current study progresses previous models in four aspects: (a) consideration of the impact of phase inversion on the drug release of both microspheres and ISFIs; (b) investigation of observable phenomena such as complex implant geometries, nonuniform drug distribution, and swelling of larger implants; (c) explicit modeling of the production and transport of H⁺ ions, which catalyze the degradation and diffusion of acid as separate processes; (d) inclusion of a full probability model for the distribution of oligomers produced from random chain scission as proposed by Flory [20]. The ultimate goal of this work is to create a mechanistic model of these implants with as few estimated parameters as possible such that the model is able to predict a wide range of drug release profiles from multiple systems while also providing insight into the processes that most impact drug release profiles in different time regimes.

2. Materials and methods

2.1. Materials

Poly(DL-lactide-co-glycolide) (PLGA 50:50, acid endcap, 4A, MW 53 kDa, inherent viscosity 0.38 dl/g) was obtained from Evonik Birmingham Laboratories (Birmingham, AL). Poly(DL-lactide-co-glycolide) (PLGA 50:50, acid endcap, MW 15 kDa) was obtained from PolySciTech (West Lafayette, IN). N-methyl-2-pyrrolidone (NMP) was obtained from Fisher Scientific and sodium fluorescein (MW 376.28) was obtained from Acros Organics. All supplies were used as received.

2.2. Preparation of polymer solutions

Polymer solutions were prepared by combining poly(lactic-co-glycolic) acid (PLGA), N-methyl-2-pyrrolidone (NMP), and fluorescein in a 39:60:1 mass ratio. First, fluorescein was dissolved in NMP and then PLGA was added. The solution was stirred overnight to ensure complete dissolution. The polymer was stored at room temperature for less than a week before use.

2.3. Drug release studies

Implants were formed by injecting 60 μL of polymer solution into 10 mL of phosphate buffered saline (PBS). Implants were kept at 37°C on a shaker at 100 rpm for the duration of the study. Samples were taken from the bath side solution at 0.25, 0.5, 1, 2, 4, 6, 24, 48, 72, 96, 120, 144, 168, 240, 336, 408, and 504 hours (h) post-exposure to aqueous conditions (post-exposure). The bath solution was completely replaced at these time points to maintain sink conditions. Residual drug mass after 21 days (d) was determined by degrading the implants in 2 M NaOH. The fluorescence of all samples was quantified using a SpectraMax M5 microplate reader using an excitation of 485 nm and emission of 525 nm, and results were compared to a standard curve to obtain the cumulative mass of fluorescein released.

2.4. Scanning Electron Microscope (SEM) imaging

To prepare implants for SEM imaging, implants at selected time points were first freeze-fractured on dry ice and then lyophilized for 4 days. After lyophilization, implants were mounted on aluminum stubs and sputter-coated with platinum for 60 s using a Cressington 208 HR sputter coater. Imaging was done using a NovaNanoSEM with a spot size of 3 and a voltage of 5.00 kV.

2.5. Diffusion-Weighted Magnetic Resonance Imaging (DWI)

To perform DWI, implants were formed as described above. At each time point (0.25, 1, 2, 4, 6, 24, 48, 72, 96, 120, 144, 168, 240, 336, 408, 504 h post-exposure), implants were removed from solution and placed into a 3D-printed insert centered in a water-filled phantom bottle. DWI was conducted using a Bruker BioSpec 70/30 USR 7T Preclinical MRI system and Bruker rat head/mouse body RF RES 300 ¹H 075/040 QSN TR volume coil. A standard diffusion-weighted spin echo protocol was utilized (TE=17.5 ms, TR=2500 ms, FOV=35×35 mm², slice thickness=0.80 mm, b=0,1000 s mm⁻²). The raw diffusion data was used to create apparent diffusion coefficient (ADC) maps of the implant as previously described [21].

2.6. Fluorescence Recovery After Photobleaching (FRAP)

Three different types of samples were prepared for FRAP analysis. The first sample was a solution of 2 μg mL⁻¹ fluorescein in phosphate buffered saline (PBS) placed under a coverslip on a microscope slide. The second was approximately 60 μL of polymer solution, prepared as described in section 2.2 above, placed under a coverslip on a microscope slide. The third sample was the same polymer solution placed under a coverslip on a microscope slide with the slide then placed in a PBS bath for 24 h to allow for polymer solidification via phase inversion.

FRAP was performed on an upright Zeiss LSM T-PMT confocal microscope (Carl Zeiss, Thornwood, NY) with a 40x water immersion objective lens. Experiments were performed at room temperature. Photobleaching was achieved by focusing the 405 nm laser line at 90% power (27mW) on circular sample areas 37.4 μm in radius. Fluorescence recovery was then recorded at 2% laser power (0.6mW). Measurements were taken in mid-plane of the cover and bottom glass slides (as verified by reflectance imaging).

2.7. Mathematical Modeling and Numerical Analysis

The numerical solution of the system of PDEs was evaluated using a finite difference method solver in Python 3.7 with the development environment Spyder 3.3.6. To ensure numerical accuracy while maintaining unconditional stability of the solution, the system of PDEs modeling the reaction-diffusion equations was solved with a conserved 2^nd order central (CN-2) and backwards differential formula (BDF-2) schemes with adaptive time stepping. The nonlinear equations were linearized through decoupling and solving for intermediate steps. The system of ODEs modeling the acid dissociation was solved with the 4^th order Runge Kutta on a much finer time scale. Simulations in which large parameter sweeps were performed utilizing the Brown Community Cluster on the Purdue campus [22]. The code for the model is available at the following DOI on the Purdue University Research Repository (PURR) DOI#.https://doi.org/10.4231/ASPM-SY16 directly on the Kinzer-Ursem Lab GitHub repository: https://github.itap.purdue.edu/TamaraKinzerursemGroup/DrugReleaseSystemModel.git

2.8. Modeling Parameterization and Verification

Model parameter values, when available, were taken from previous literature (Table S5). However, model development produced a set of parameters whose values were unknown, or that varied over a large range. These included diffusion coefficients for drug, oligomers, monomers, and solvent through precipitated polymer as well as four polymer degradation rates for end or random, and noncatalytic or auto-catalytic chain scission. To find values for these parameters, diffusivity coefficients values were derived from experimental data and polymer degradation rates were approximated by fitting a series of degradation profiles (see section 3.5.2). Simulations were used to predict a host of outputs that were compared to independent empirical measurements. Given that there was uncertainty in some of the unknown model parameter values, global sensitivity analysis was utilized to access the sensitivity of model output (drug release profile) to a range of unknown parameter values. Latin Hypercube Sampling (LHS) was used to efficiently sample input parameter space and partial rank correlation coefficient (PRCC) analysis was used to assess model output sensitivity [23–26]. LHS/PRCC studies were conducted to evaluate model sensitivity to parameters involved in diffusion separately from parameters involved in degradation rates.

3. Results

3.1. Model Development

In this work, our modeling efforts center around the presence of water in the release systems (Figure 1). As water diffuses into the controlled release systems, the polymer precipitates into a solid state thereby decreasing the diffusivity of drug out of the depot. In the case of ISFIs, the solidification of the implant occurs over a time scale of days resulting in an initial burst of drug out of the implant. The introduction of water also catalyzes the hydrolysis of the ester bonds of immobile long polymer chains to produce short diffusible polymer chains [18]. As the small polymer chains are transported out of the implant, the porosity of the polymer increases over time resulting in increased diffusion of drug out of the ISFI. To capture the various aspects of these water-dependent processes we take a comprehensive approach to modeling time and space varying polymer solidification (Section 3.2) and water-dependent hydrolysis of the ISFI polymer (Section 3.3). These processes are then combined and solved through a sophisticated numerical analysis scheme (Section 3.4) that allows for 3D simulation of temporally and spatially dependent drug release profiles.

Figure 1. Scheme of solidification and degradation.

(Bottom) The drug release from the implant is controlled by the solidification of the ISFI, and at later time points by degradation and erosion of the solid polymer matrix (white circles indicate pore formation). (Top) The influence of these two mechanisms on the drug release profile.

3.2. Modeling Polymer Solidification

The impact of varying the size of the phase-sensitive controlled release vehicles was considered by evaluating experimental data taken from 10 μm and 50 μm microspheres and 5 mm ISFIs [16, 27]. Each of these systems developed a unique geometry resulting from the phase inversion process. Rather than relying on the full geometries derived from imaging, simplified model geometries were designed to capture the most salient features of the release vehicles (i.e., shell thickness, pore size/distribution, interior matrix thickness, and implant size). One major benefit of reducing the complexity of the geometry with regard to space is that the geometric parameters can be easily modified to sweep parameter regimes for optimal conditions to produce desired drug release profiles.

Confocal images from literature were used to establish the geometric parameters of microspheres. The confocal images taken by Pack and colleagues depict completely solid 10 μm microspheres, however, the 50 μm microspheres had pores distributed throughout the volume of the release vehicle [16, 28]. To account for these random pores, the average distance of each pore from the center and the size of the pore was evaluated (17.99, 3.80 μm), along with standard deviations (6.08, 1.48 μm), and skew (−0.65, 1.48), respectively. A skewed gaussian function was then used to randomly select the position and size of the pores throughout the microsphere for the stochastic simulations (Figure 4h–i).

Figure 4. Numerical Verification.

(A) COMSOL simulation of normalized drug concentration throughout a sphere in three spatial dimensions. (B) Python simulation of normalized drug concentration out of a sphere in a single spatial dimension. (C) Overlay of drug release predictions from A and B. (D) Python simulation showing drug diffusion out of simple 2D slice geometry of an ISFI, 2 days after formation. (E) Python simulation showing drug diffusion out of a complex 2D slice geometry derived from SEM images of an ISFI, 2 days after formation. (F) Overlaying the entire 30 day time course of drug released from the 2D geometries shown in D and E. (G) COMSOL simulation for drug diffusion out of 50 μm microsphere with a 2D slice geometry and heterogeneous initial drug distribution derived from confocal images, 6 hours after formation [16]. (H) The 1D radial geometry produced from a single run of the Python stochastic simulation for the diffusion of drug out of the 50 μm microspheres. (I) Comparison of the drug released from 2D COMSOL simulation with heterogeneous initial drug distribution (solid grey line), 2D COMSOL simulation with homogeneous initial drug distribution (dashed grey line), the average of 50 simulated slices (Figure 4h) of 1D Python simulations (solid blue line), and experimental data for 67 kDa 50 μm microsphere [16].

To design the geometry for the 5 mm ISFI, a SEM image (Figure 2a) from a 2 day old implant, well into the solidification phase, was converted to a binary image in Python (Figure 2b) and reduced to a simplified geometry (Figure 2c). Previous data on the swelling of implants of varying molecular weight (15, 29, and 53 kDa) was used to update the size of the implant at discrete time points [6]. The implants swelled at different rates but reached a maximum cross-sectional area of 2.40, 2.20, and 1.73 fold increase from their initial area for the 15, 29, and 53 kDa implants, respectively [27].

Figure 2. Model development of polymer solidification.

(A) SEM image of a cryosectioned 5mm implant after 2 days post-exposure. (B) Binary geometry built from section of SEM image (red box in A). (C) Simplified geometry of the implant built in Python. (D) Plot of mask as a function n of radial distance at a cross section of the simplified geometry (red line in B). (E) Volume fractions of water $({\phi }_w)$ and solvent $({\phi }_s)$ after 2 days in the coagulation bath as a function of radial distance; implant boundary is denoted by the dashed black line. (F) Final Mask function after ${\tau }_{ter}$ (derived from the volume fractions of polymer, water, and solvent in the ternary phase diagram) is applied to mask. (G) Empirically derived ternary phase diagram for PLGA 50:50, data from Exner et al. [29]. Blue circle represents the initial volume fractions of the ISFIs. (H) Sample of the empirical data utilized to develop the ternary phase diagrams [29].

Each of the geometries for the ISFIs and microspheres were built as an array of 1s and 0s representing polymer rich and lean regions, respectively. Figure 2d shows one such array, which we refer to as a mask, that is then applied to the mathematical model to distinguish the spatial state of the precipitated polymer. This reduced geometry was verified to introduce negligible error, see section 3.4.

However, mask is only useful for a completely precipitated implant and does not take into consideration the time-varying nature of the solidification process, especially at early time points when water is still diffusing into the system.

To model the time dependent nature of the polymer solidification, we solved for the diffusion of water in and solvent out of the implant or microsphere utilizing theory for multicomponent diffusion developed initially by Maxwell and Stefan [30, 31]. The flux of polymer molecules is gradual, and the momentum imparted on the solvent and water molecules is negligible. The solidification rate, as well as the diffusivity of analyte through the solidified and dissolved polymer were considered, allowing the phase inversion to be modeled by the Maxwell-Stefan equation for an ideal, binary mixture:

$$\sum _{k=1}^2\frac{x_i{x}_k}{{\mathrm{Đ}}_{ik}}(u_i-u_k)=-\frac{x_i}{RT}({\nabla }_{T,P}{\mu }_i),\mathrm{i}=\mathrm{1,2}$$ (1a)

where $x_i=c_i/c_t,c_i$ is the molar concentration of component $\mathrm{i},c_k$ is the molar concentration of component $\mathrm{k},c_t$ is the total molar concentration, $u_i$ is the molar velocity of component $\mathrm{i},u_k$ is the molar velocity of component $\mathrm{k},{\mathrm{Đ}}_{ik}=RT/{\mathrm{\varsigma }}_{ik}$, where ${\mathrm{\varsigma }}_{ik}$ is the frictional coefficient between components i and k, R is the ideal gas constant, T is the temperature of the system, and ${\mu }_i$ is the chemical potential. Introducing component molar fluxes ${\mathrm{N}}_i$, driving force ${\mathrm{d}}_i$, and flux ${\mathrm{J}}_i$ gave equation 1b:

$$\sum _{k=1}^2\frac{x_k{J_i-x_{i}J}_k}{c_t{\mathrm{Đ}}_{ik}}=-d_i,\mathrm{i}=\mathrm{1,2}$$ (1b)

which was modeled as

$${\mathit{c}}_{\mathit{t}}\left(\begin{array}{c}{\mathit{d}}_1\\ {\mathit{d}}_2\end{array}\right)=\left[\begin{array}{cc}{\mathit{B}}_{1,1}& {\mathit{B}}_{1,2}\\ {\mathit{B}}_{2,1}& {\mathit{B}}_{2,2}\end{array}\right]\left(\begin{array}{c}{\mathit{J}}_1\\ {\mathit{J}}_2\end{array}\right)$$ (1c)

$$B_{ii}=\frac{x_i}{{\mathrm{Đ}}_{i1}}+\sum _{j\ne i}^2\frac{x_j}{{\mathrm{Đ}}_{ij}},B_{ij}=\frac{{-x}_i}{{\mathrm{Đ}}_{ij}}+\frac{x_i}{{\mathrm{Đ}}_{i1}}$$ (1d)

[32–35]. The friction coefficients were found by using the following relationships

$$D_1=\frac{RT}{\frac{{\rho }_1{\mathrm{\varsigma }}_{11}}{{Mw}_1}+\frac{{\rho }_2{\mathrm{\varsigma }}_{12}}{{Mw}_2}},D_2=\frac{RT}{\frac{{\rho }_2{\mathrm{\varsigma }}_{22}}{{Mw}_2}+\frac{{\rho }_1{\mathrm{\varsigma }}_{21}}{{Mw}_1}},{\mathrm{\varsigma }}_{12}={({\mathrm{\varsigma }}_{11}{\mathrm{\varsigma }}_{22})}^{1/2},{\mathrm{\varsigma }}_{12}={\mathrm{\varsigma }}_{21}$$ (1e-h)

determined by Vrentas et al. and Bearman et al. [36, 37], where ${\rho }_i$ is the density of component $\mathrm{i}$ and ${Mw}_i$ is the molecular weight of component $\mathrm{i}$.

Using equations 1c–h, the mass transport of solvent and water was evaluated, while monitoring the volume fractions of each component, ${\phi }_i$. The volume fractions are defined as

$${\phi }_i=\frac{{Mw}_i}{{\rho }_i}{c}_i$$ (2a)

$$\sum _{i=1}^3{\phi }_i=1,i=\mathrm{1,2},3$$ (2b)

where ${\phi }_i$ is the volume fraction of component $i$.

The solidification of the polymer was determined by overlaying the volume fractions with a ternary phase diagram (Figure 2g). The ternary phase diagrams for vary polymer Mw (Figure 2g) were developed by fitting empirical data [29] with a Hill equation (Figure 2h). The solidification was represented by the variable ${\tau }_{ter}$ and applied to spatially dependent variable mask to give a spatially and temporally dependent variable Mask. The function Mask is defined as

$$Mask={\tau }_{ter}\mathrm{*}mask$$ (2c)

where ${\tau }_{ter}$ is derived from a ternary phase diagram for PLGA, (NMP or DCM), and water (Figure 2g). The variable ${\tau }_{ter}$ is a scalar field whose value varies from 0 to 1, representing completely dissolved or solidified polymer, respectively. Ternary diagrams produced in Figure 2g agree with ternary phase diagrams published previously [19, 38–40].

The rate of solidification has the impact of controlling the encapsulation efficiency of microspheres [41–46], and the initial burst of drug from ISFIs [47–49], as well as the final polymer geometry formed [47].

3.3. Mathematical Model of Polymer Degradation

Polymer degradation occurs through the hydrolysis of linking ester bonds, which was modeled by a system of reaction-diffusion equations tracking the change in concentration of chain scission, ${\mathrm{R}}_{\mathrm{s}}$. Depending on where the chain scission occurs along the polymer chain either an oligomer, ${\mathrm{C}}_{\mathrm{o}\mathrm{l}}$, or monomer, ${\mathrm{C}}_{\mathrm{m}}$, is produced. These small polymer chains are able to diffuse out of the release system, which increases the porosity and consequently the diffusivity of the polymer. Another important characteristic of ${\mathrm{C}}_{\mathrm{o}\mathrm{l}}$ and ${\mathrm{C}}_{\mathrm{m}}$ are that these small polymer chains terminate in a carboxylic acid end group, which further catalyzes the cleavage of ester bonds [50–52]. This produces two feedback loops: the production of ${\mathrm{C}}_{\mathrm{o}\mathrm{l}}$ and ${\mathrm{C}}_{\mathrm{m}}$ further catalyzes the polymer degradation and increases ${\mathrm{R}}_{\mathrm{s}}$; and the diffusion of ${\mathrm{C}}_{\mathrm{o}\mathrm{l}}$ and ${\mathrm{C}}_{\mathrm{m}}$ out of the release system increases the diffusivity of the polymer (Figure 3).

Figure 3. Scheme of degradation/erosion mathematical model.

Water $\left({\mathrm{C}}_{\mathrm{H}2\mathrm{O}}\right)$ diffuses into the release system initiating chain scission (${\mathrm{R}}_{\mathrm{s}}$). End chain scission (${\mathrm{R}}_{\mathrm{e}\mathrm{s}}$) occurs when the terminal ester bonds (${\mathrm{C}}_{\mathrm{e}\mathrm{n}\mathrm{d}}$) are hydrolyzed producing monomers $\left({\mathrm{C}}_{\mathrm{m}}\right)$, while random chain scission $\left({\mathrm{R}}_{\mathrm{r}\mathrm{s}}\right)$ occurs when the interior ester bonds (${\mathrm{C}}_{\mathrm{e}}$) are hydrolyzed producing oligomers (${\mathrm{C}}_{\mathrm{o}\mathrm{l}}$). The total production of oligomers $\left({\mathrm{R}}_{\mathrm{o}\mathrm{l}}\right)$ and monomers $\left({\mathrm{R}}_{\mathrm{m}}\right)$ are monitored to evaluated the porosity $\left({\mathrm{V}}_{\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{e}}\right)$ of the polymer which directly impacts the diffusivity of the polymer for oligomers $\left({\mathrm{D}}_{\mathrm{o}\mathrm{l}}\right)$, monomers $\left({\mathrm{D}}_{\mathrm{m}}\right)$, and the loaded drug $\left({\mathrm{D}}_{\mathrm{d}\mathrm{r}\mathrm{u}\mathrm{g}}\right)$. The dissociation of H⁺ ions (C_H+) from the carboxylic acid terminals of the polymer are evaluated due to their catalytic impact on the degradation of the polymer. The swelling of the polymer (V(t)) also impacts the diffusivity of the polymer. The increase in diffusivity increases the diffusion rate of oligomers, monomers, and drug out of the release system.

3.3.1. Rate of Chain Scission

In developing the mathematical model for polymer degradation, we began in a similar fashion as the work of Wang et al. [50–52] with an equation to describe the rate of chain scissions per unit volume. The equation for chain scission is a function of non-catalytic and autocatalytic rate of degradation, given by:

$$\frac{d{R}_s}{dt}=k_1{C}_{ester}{C}_{H_{2}O}+k_2{C}_{ester}{C}_{H+}{C}_{H_{2}O}$$ (3)

where ${\mathrm{R}}_{\mathrm{s}}$ is the concentration of chain scissions, $C_{ester}$ is the concentration of ester bonds available for degradation, $C_{H+}$ is the concentration of acid produced, $C_{H_{2}O}$ is the concentration of water, $k_1$ is the non-catalytic degradation rate constant, and $k_2$ is the auto-catalytic degradation rate constant.

Previous work assumed that water saturation of the polymer is on a much shorter time scale than degradation and erosion, and therefore not considered [50, 52–57]. However, we accounted for the time and spatial saturation of water by applying the weighting function Mask (section 3.2), Eq 4.

$$\frac{d{R}_s}{dt}=\left[k_1{C}_{ester}+k_2{C}_{ester}{C}_{H+}\right]Mask$$ (4)

Flory observed that polymer scission occurs through both random and end chain scission at different rates that have a varied impact on the polymer [20]. Random chain scission has the effect of dramatically reducing the average molecular weight of the polymer, while end chain scission more efficiently produces acidic byproducts. Experimental data has shown that both are necessary in modeling the degradation of polymer [53, 58–62], therefore we expand Eq 4 to include both end and random chain scission, which is defined by the equations:

$$\frac{d{R}_{es}}{dt}=\left[k_{e1}{C}_{end}+k_{e2}{C}_{end}{C}_{H+}\right]Mask$$ (5a)

$$\frac{d{R}_{rs}}{dt}=\left[k_{r1}{C}_e+k_{r2}{C}_e{C}_{H+}\right]Mask$$ (5b)

$$\frac{d{R}_s}{dt}=\frac{d{R}_{es}}{dt}+\frac{d{R}_{rs}}{dt}$$ (5c)

where $R_{es}$ is end chain scission, $k_{e1}$ is the non-catalytic end scission rate constant, $C_{end}$ is the concentration of terminal ester bonds available for end chain scission, $k_{e2}$ is the auto-catalytic end scission rate constant, $R_{rs}$ is random chain scission, $k_{r1}$ is the non-catalytic random scission rate constant, $C_e$ is the concentration of interior ester bonds available for random chain scission, and $k_{r2}$ is the auto-catalytic random scission rate constant.

3.3.2. Crystallinity

To account for the observed change in crystallinity that occurs during polymer degradation, we followed the work by Han et al., which considered the formation of crystallites, ${\chi }_c$, due to the raveling of polymer chains into parallel lattice structures [54]. During the solidification of the release vehicle, the dissolved polymer has a greater degree of freedom resulting in initial crystallinity, ${\chi }_{c0}$. During a chain scission, the end of the polymer has an increased flexibility and has a probability, $p_c$, of forming a crystallite. Work by Tsuji et al. has shown that once formed the crystallites take months to degrade, which is outside of the simulations timescale and therefore degradation of crystallites was not considered [55, 63]. The equation for unrestrained growth of crystallites, ${\chi }_{ext}$, is given by

$${\chi }_{ext}=p_c{\eta }_A{R}_s{V}_c$$ (6)

where ${\eta }_A$ is Avogadro’s constant and $V_c$ is the volume of an average crystallite. To account for growth restrictions ${\chi }_{cmax}$ is introduced as a limiting term for the maximum crystallinity and $\lambda$ is an empirically derived constant set to 1 [55].

$$\frac{d{\chi }_{ext}}{{d\chi }_c}={\left[{\chi }_{cmax}-{\chi }_c\right]}^{\lambda }$$ (7)

$${\chi }_c={\chi }_{c0}-\frac{{({\chi }_{cmax}-{\chi }_c)}^{\lambda }}{\lambda +1}$$ (8)

Integrating Eq 7 results in Eq 8 and allows for modeling the production of crystallites during chain scission with a volume restraint [55].

3.3.3. Production of Acidic Byproducts

Work by Flory et al. used probability distributions to evaluate the average size of the polymer chains resulting from linear condensation and hinted that a similar approach would be useful in modeling random chain scissions [20]. The probability of a polymer chain of x units being produced by a chain scission is governed by the binomial probability distribution:

$$N_x=Np{\left(1-p\right)}^{x-1}=N_0{p}^2{\left(1-p\right)}^{x-1}$$ (9a)

where $N_x$ is the probability of having a polymer chain of $\mathrm{x}$ units, $N$ is the total number of polymer chains, $\mathrm{p}$ is the probability of an event, and $N_0$ is the total number of units. Eq 9a is then converted to a weight fraction:

$$w_x=\frac{{xN}_x}{N_0}$$ (9b)

$$w_x=x{p}^2{\left(1-p\right)}^{x-1}$$ (9c)

where $w_x$ is the weight fraction of polymer chain of $\mathrm{x}$ units. Eq 9c is then generalized for a distribution of oligomers:

$$w_{ol}=\sum _{x=2}^{L}x{\left(p\right)}^2{\left(1-p\right)}^{x-1}$$ (9d)

where $w_{ol}$ is the weight fraction of all oligomers, and $\mathrm{L}$ is the maximum length of mobile oligomers. Therefore, the probability function used for predicting the weight fraction of oligomers produced during random chain scission is given by:

$$\frac{R_{ol}}{C_{ester0}}=\sum _{l=2}^{L}l{\left(\frac{R_s}{C_{ester0}}\right)}^2{\left(1-\frac{R_s}{C_{ester0}}\right)}^{l-1}$$ (9e)

$$\frac{R_{ol}}{C_{e0}}=\sum _{l=1}^{L}l{\left(\frac{R_s}{C_{e0}}\right)}^2{\left(1-\frac{R_s}{C_{e0}}\right)}^{l-1}$$ (9f)

where $R_{ol}$ is the total concentration of short polymer chains produced over reaction time, ${\mathrm{C}}_{\mathrm{e}0}$ is the initial concentration of interior ester bonds and $l$ is the length in ester bonds of the oligomer chain [62, 64]. In this work we continue by applying equation 9f to evaluate the production of oligomers, while Pan et al. continues by generalizing eq 9f to allow for empirical tuning [65].

The distinguishing characteristic between $C_{ol}$ and $R_{ol}$ is that $C_{ol}$ is reduced when the oligomers diffuse out of the polymer matrix, while $R_{ol}$ accounts for all the ester bonds of oligomers that have been produced. Unlike random chain scission, end scission produces exactly one monomer during each scission event. Therefore $R_m=R_{es}$, where $R_m$ is the total concentration of monomers produced over reaction time.

The concentration of ester bonds available for degradation is a function of both degradation and formation of crystallites. $C_e$ is equivalent to the initial ester bond concentration reduced by the concentration of ester bonds of all oligomers and monomers formed, and the concentration of crystalline esters bonds, Eq 10a. Eq 9 was substituted into Eq 10a to producing, Eq 10b, the functional equation for the concentration of ester bonds available for random chain scission:

$${{C_e=C}_{e0}-(R}_{ol}+R_m)-\omega {\chi }_c$$ (10a)

$$C_e=C_{e0}\left[1-\sum _{l=2}^L{l\left(\frac{R_s}{C_{e0}}\right)}^2{\left(1-\frac{R_s}{C_{e0}}\right)}^{l-1}-R_{es}\right]-\omega {\chi }_c$$ (10b)

where $\omega$ is the molar concentration of ester bonds in the crystalline phase. The concentration of terminal ester bonds available for end chain scission is determined by:

$$C_{end}=2N_{chain0}+2(R_{rs}-\left(\frac{R_{ol}}{m}\right))=2N_{chain0}+2(R_{rs}-\left(\frac{1}{m}\sum _{l=2}^L{l\left(\frac{R_s}{C_{e0}}\right)}^2{\left(1-\frac{R_s}{C_{e0}}\right)}^{l-1}\right))$$ (11)

where $N_{chain0}$ is the initial number of chains, and $m$ is the average chain length of oligomers formed during degradation.

3.3.4. Acid Dissociation

Water hydrolyzes the ester bonds of the polymer to produce two new terminal groups, a hydroxyl and a carboxylic acid group. The carboxylic acid functional group has a high degree of reversible acid dissociation, thereby promoting further degradation of the polymer. The total concentration of $H^{+}$ ions distributed throughout the release system is regulated by both the acidity of the drug and the production of carboxylic acid end groups through the chain cleavage. Therefore, the concentration of $H^{+}$ ions was modeled by:

$$C_{H+}=C_{H+,polymer}+C_{H+,drug}$$ (12)

where the contribution of $H^{+}$ ions by the polymer, including long polymer chains, oligomers, and monomers, are defined as $C_{H^{+},polymer}$, and the contribution $H^{+}$ ions by the drug are defined by $C_{H^{+},drug}$.

Due to the rapid fluctuations of diffusivity of the polymer release system for these acidic byproducts during solidification and degradation, it is probable that diffusion can be rate limiting. It then becomes necessary to solve for acid dissociation as a time dependent event, rather than rely on an equilibrium expression. It has also been previously noted that when the diffusivity of polymer is too low, the equilibrium concentration of reactants is not maintained, and the reaction may not reach equilibrium [20]. Therefore, acid dissociation from the carboxylic acid was modeled using mass action kinetics:

$$C_{COOH}\begin{array}{c}{k}_{on}\\ \rightleftharpoons \\ k_{off}\end{array}{C}_{H+}+C_{COO-}$$ (13a)

where $C_{COOH}$ is the concentration of acid, $C_{H+}$ is the concentration of dissociated ions, $C_{COO-}$ is the concentration of conjugate base formed, $k_{on}$ is the rate of acid association, and $k_{off}$ is the rate of acid dissociation. Eq. 13a was then rewritten in the form of the ordinary differential equation:

$$\frac{d{C}_{COOH}}{dt}=k_{on}{C}_{H^{+}}{C}_{{COO}^{-}}-k_{off}{C}_{COOH}$$ (13b)

which was then expanded to account for long polymer chains (${\mathrm{C}}_{\mathrm{p}\mathrm{o}\mathrm{l}}$), oligomers (${\mathrm{C}}_{\mathrm{o}\mathrm{l}}$), monomers (${\mathrm{C}}_{\mathrm{m}}$), acidic drug ${\mathrm{C}}_{\mathrm{D}\mathrm{r}\mathrm{u}\mathrm{g}}$, and water/buffer (${\mathrm{C}}_{\mathrm{H}2\mathrm{O}}$). The protonation of the solution from these five proton donors is represented by:

$$\sum _{k=1}^5{C}_{H+,k}$$ (14)

where $C_{pol},C_{ol},C_m,C_{drug}$, and $C_{H_{2}O}$ are represented by $\mathrm{k}=\mathrm{1,2},\mathrm{3,4}$, and 5, respectively.

Substituting Eqns 10b, 11, 12, and 14 into 5a and 5b generated the final expression for the degradation rates:

$$\frac{d{R}_{rs}}{dt}=C_{e0}\left[1-\sum _{l=2}^L{l\left(\frac{R_s}{C_{e0}}\right)}^2{\left(1-\frac{R_s}{C_{e0}}\right)}^{l-1}-R_{es}-\omega {\chi }_c\right]\left\{k_{r1}+k_{r2}\left[\sum _{k=1}^5{C}_{H+,k}\right]\right\}Mask$$ (16a)

$$\frac{d{R}_{es}}{dt}=\left[2N_{chain0}+2(R_{rs}-\left(\frac{1}{m}\sum _{l=2}^L{l\left(\frac{R_s}{C_{e0}}\right)}^2{\left(1-\frac{R_s}{C_{e0}}\right)}^{l-1}\right))\right]\left\{k_{e1}+k_{e2}\left[\sum _{k=1}^5{C}_{H+,k}\right]\right\}Mask$$ (16b)

The analytical solution for the rate of production of the short chain polymers, $\frac{d{R}_{ol}}{dt}$ was determined as:

$$\frac{d{R}_{ol}}{dt}=\frac{d}{dt}\left(\sum _{l=2}^L{l\left(\frac{R_s}{C_{e0}}\right)}^2{\left(1-\frac{R_s}{C_{e0}}\right)}^{l-1}\right)=\sum _{l=1}^L\frac{-C_{e0}{\left(\frac{R_{rs}}{C_{e0}}\right)\left(1-\frac{R_{rs}}{C_{e0}}\right)}^{l-2}\left(l\left(\frac{R_{rs}}{C_{e0}}\right)+\frac{R_{rs}}{C_{e0}}-2\right)\left(\frac{{dR}_{rs}}{dt}\right)}{{\left(\left(\frac{R_{rs}}{C_{e0}}\right)-1\right)}^2}$$ (17a)

Every end chain scission produces a monomer so the concentration of monomers produced, $R_m$, is modeled by equation 17b.

$$\frac{d{R}_m}{dt}=\frac{d{R}_{es}}{dt}$$ (17b)

3.3.5. Polymer and Drug Transport

Assuming the polymer degradation products are well-mixed and soluble, the erosion of $C_m$ and $C_{ol}$ was modeled by the conserved form of Fick’s second law of diffusion to account for the non-homogeneous material. The diffusion-reaction equations for $C_{ol}$ and $C_m$ become:

$$\frac{d{C}_{ol}}{dt}=\frac{d{R}_{ol}}{dt}+\sum _{i=1}^3\frac{\partial }{\partial x_i}\left(D_{Col}\left(x_i,t\right)\frac{\partial C_{ol}}{\partial x_i}\right)$$ (18a)

$$\frac{d{C}_m}{dt}=\frac{d{R}_{es}}{dt}+\sum _{i=1}^3\frac{\partial }{\partial x_i}\left(D_{Cm}\left(x_i,t\right)\frac{\partial C_m}{\partial x_i}\right)$$ (18b)

Diffusion of drug, ${\mathrm{H}}^{+}$ (which is transported as hydronium), and the conjugate bases out of the system were modeled in a similar manner with a conserved form of Fick’s second law of diffusion, represented by $\psi$. The diffusion terms are then rewritten for a spherical coordinate system:

$$\sum _{i=1}^3\frac{\partial }{\partial x_i}\left(D\frac{\partial C}{\partial x_i}\right)=\frac{1}{r^2}\frac{\partial }{\partial r}\left[D{r}^2\frac{\partial \psi }{\partial r}\right]+\frac{1}{r^{2}sin\theta }\frac{\partial }{\partial \theta }\left[Dsin\theta \frac{\partial \psi }{\partial \theta }\right]+\frac{1}{r^2{\mathrm{s}\mathrm{i}\mathrm{n}\theta }^2}\frac{\partial }{\partial \varphi }\left[\frac{\partial \psi }{\partial \varphi }\right]$$ (19)

and reduced to a one dimensional equation, which was applied to all of the simulations in this study, with the exception of the geometric reduction studies in section 3.4:

$$\sum _{i=1}^1\frac{\partial }{\partial x_i}\left(D\frac{\partial C}{\partial x_i}\right)=\frac{1}{r^2}\frac{\partial }{\partial r}\left[D{r}^2\frac{\partial \psi }{\partial r}\right]$$ (20)

3.3.6. Porosity/Diffusivity of Polymer

Following the work by Pan et al. [56], the porosity of the polymer matrix was defined by the loss of drug and short polymer chains from the release vehicle, evaluated by the equations:

$$V_{pore,ol,m}=\frac{R_{ol}+R_m}{C_{e0}}-\frac{C_m+C_{ol}-C_{ol0}-C_{m0}}{C_{e0}}$$ (21a)

$$V_{pore,drug}=1-\frac{C_{drug}}{C_{drug,0}}$$ (21b)

The contribution of each is proportional to $f_{Drug}$, which is the volume ratio of drug to polymer. Therefore, the final equation for porosity was given by Eq. 22 [56].

$$V_{pore}=V_{pore,ol}\left(1-f_{drug}\right)+V_{pore,drug}{f}_{drug}$$ (22)

The diffusivity of each component through the polymer was modeled as:

$$D_i=D_{i,polymer}+\left(1.3V_{pore}^2-0.3V_{pore}^3\right)\left(D_{i,pore}-D_{i,polymer}\right)$$ (23)

where $D_{i,polymer}$ is the diffusivity of the ith component (i.e., drug, water, polymer degradation products, etc.) through pure polymer, and $D_{i,pore}$ is the diffusivity of the ith component (i.e., drug, water, polymer degradation products, etc.) through dissolved polymer [62].

For the ISFI, significant swelling was observed, and to account for the increased diffusivity of each of components through the polymer during the swelling of the implant we modified Eq 23 to include for free-volume theory. The relationship between component diffusivity through the polymer and the implant volume expansion is based on the generalized free-volume theory proposed by Fujita et al. [66–68] based on work by Turnbull and Cohen [24], and further developed by Peppas et al. [69–75]. The function for the polymer diffusivity was

$$D(t)=D_0\mathrm{*}{e}^{-{\beta }_{sw}\left(1-\frac{V(t)}{V_0}\right)}$$ (24a)

where $\mathrm{D}(\mathrm{t})$ is the diffusivity as a function of time, ${\mathrm{D}}_0$ is the equilibrium diffusivity, $\mathrm{V}(\mathrm{t})$ is the volume as a function of time, ${\mathrm{V}}_0$ is the initial volume, and an empirical term (${\beta }_{sw}$) [70–73].

$$D_{i,polymer}=D_{i,polymer,0}\mathrm{*}{e}^{-{\beta }_{sw}\left(1-\frac{V(t)}{V_0}\right)}$$ (24b)

where $D_{i,polymer,0}$ is the equilibrium diffusivity of component $i$ through the polymer matrix. Substituting the expression for the diffusivity of each component through polymer matrix, Eq 24b, into the general expression for diffusivity of each component, Eq 23, yields

$$D_i=(1-Mask)\mathrm{*}{D}_{i,pore}+Mask\mathrm{*}\left[D_{i,polymer,0}\mathrm{*}{e}^{-{\beta }_{sw}\left(1-\frac{V(t)}{V_0}\right)}+\left(1.3V_{pore}^2-0.3V_{pore}^3\right)\left(D_{i,pore}-D_{i,polymer,0}\mathrm{*}{e}^{-{\beta }_{sw}\left(1-\frac{V(t)}{V_0}\right)}\right)\right]$$ (24c)

The final equations 2a, 16a–b, 18a–b, 22, 23, and 23/24c are solved as a system of PDEs to predict the drug release rate from the release vehicles.

It becomes important to recognize that the diffusivity of the polymer is a function of ${\phi }_i,C_{ol},C_m$, and $C_{drug}$ (acting as a feedback loop) and thereby is a spatially and temporally dependent function. This also means that the PDE governing mass transport is nonlinear and great care must be taken when solving these systems of PDEs to ensure that an accurate and stable solution is reached.

3.4. Verification of Simplified Geometry

In creating the simplified time-dependent geometry of the model it was necessary to verify that each of the following assumptions or simplifications did not introduce significant error: (1) reduction of the dimensionality of the model to one spatial dimension, (2) use of a simplified spatial geometry without the random shell pores, and (3) use of a stochastic model for the internal pores in the 50 μm microsphere.

3.4.1. Reduction of spatial dimensions

COMSOL was used to simulate the diffusion drug out of a solid three dimensional spherical microsphere (Figure 4a), which was directly compared to a Python simulation of the radial drug diffusion out of the microsphere (Figure 4b). The initial spatial gradient in drug concentration was set to high order polynomial depicted by the darkest navy line in Figure 4b. This was necessary in the COMSOL simulation to prevent the software’s necessary treatment of a discontinuous variable. The drug release rates from these two simulations were compared in Figure 4c. The use of a model with a single spatial dimension of radial diffusion introduced a maximum of 0.949% error, and was therefore assumed to be an acceptable simplification to decrease computational cost.

3.4.2. ISFI Geometry Simplification

SEM images of the fully solidified implant were imported into Python where the images were converted into binary, with 1 representing polymer matrix and 0 the pores. The drug released from the simplified geometry (Figure 4d) and the SEM geometry (Figure 4e) where compared to verify that no significant source of error was being introduced (Figure 4f). Relative to the geometry derived from SEM images, utilizing the simplified geometry introduced a maximum error of 0.864% with the inclusion of shell pores, and 8.048% when the shell pores are excluded (Figure 4d–f). Therefore, shell pores were included in the model.

3.4.3. Microsphere Geometry Simplification

Experimental data collected by Pack et al. determined that not only do 50 μm microspheres contain a random distribution of interior pores, but that the loaded drug was heterogeneously distributed in these pores [16]. These previous results were used to build a COMSOL simulation for the diffusion of drug out of 2D slice geometry (Figure 4g). To reduce the complexity of the model the distribution of the pores was analyzed and applied to a Python model for a single spatial dimension, with a random distribution of pores (Figure 4h). The predicted drug release from the 2D COMSOL simulation (Figure 4i, solid grey line) and the stochastic 1D Python simulation (Figure 4i, solid blue line), both in the absence of polymer degradation, are compared to experimental data (Figure 4i, black markers) [16]. These two simulation results are also compared to the drug released from a 1D Python simulation with a homogeneous initial drug distribution (Figure 4i, dashed grey line). The stochastic 1D model with random distribution of pores in the 50 μm microsphere introduced a maximum error of 3.230% relative error to the experimental data, while modeling a homogeneous initial drug distribution produced a maximum error of 20.440% (Figure 4g–i). These results indicate that the initial inhomogeneous distribution of drug recognized by Pack et al. [16] greatly affects the release dynamics and is accounted for in subsequent simulations.

3.5. Model parameterization

In developing this model two sets of unknown parameters values are introduced: (1) the diffusivity of oligomer, monomers, water, solvent, and the loaded drug through the polymer, and (2) the four degradation rates of the polymer.

3.5.1. Model parameterization: diffusivity

Reviewing literature data of drug release profiles from vehicles of varying combinations of drugs and PLGA50:50 Mw allowed us to develop an empirical equation for diffusivity as a function of drug and polymer Mw [21, 29, 57, 76–88] (Figure 5a). The data was found to be best fit by an exponential function (Figure 5b). This function was then used to approximate individual diffusion coefficients for the diffusion of NMP and dichloromethane (DCM); the degradation products oligomers and monomers; and drugs fluorescein and piroxicam through solid polymer, D_i,polymer.

Figure 5. Model Parameterization.

(A) Experimental data collected from a literature review of PLGA 50:50 nanoparticles and small microspheres were used as a training set to derive an empirical relationship describing the polymer diffusivity as a function of the Mw of the polymer and the drug. [21, 29, 57, 76–88]. (B) The equation used in the regression analysis in (A) along with the fit parameters and the R² value. (C-F) The degradation profile of 67, 49, 34, and 21 kDa, PLGA 50:50 polymer 50 μm microspheres, respectively was used to parameterize the degradation rate constants (k_e1, k_e2, k_r1, k_r2) [16].

Fluorescent recovery after photobleaching (FRAP) was performed to measure the diffusivity of fluorescein in 53 kDa dissolved polymer (1.387e-6 m²/day) and water (5.356e-6 m²/day). The arithmetic mean was taken for the diffusion of the loaded drug through polymer pores, $D_{Drug,pore}$ (Table S1). Diffusion weighted imaging (DWI) was used to measure the diffusivity of water in the solidifying polymer. The diffusivity ranged from 3e-10 m²/s (solid polymer matrix) to 2e-9 m²/s (degraded polymer solution). These values were used to parameterize the diffusivity of water, hydroxide, and H+ (which exists as hydronium) through the solidified polymer and soluble polymer pores (${\mathrm{D}}_{\mathrm{i},\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{e}}$ and ${\mathrm{D}}_{\mathrm{i},\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{m}\mathrm{e}\mathrm{r}}$, respectively for each component). The Stokes-Einstein equation was then used to approximate the values of ${\mathrm{D}}_{\mathrm{i},\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{e}}$ for the solvent, monomers, and oligomers.

3.5.2. Model parameterization: degradation rate constants

The degradation rates $\left(k_{e1},k_{e2},k_{r1},k_{r2}\right)$ for the polymer were approximated by fitting the degradation profiles for the 50 μm microspheres with empirical data (Figure 5c–f) [16]. The values for these rates were determined to be: $k_{e1}=4e-3m^3/(mol*day)$, $k_{e2}=3e-6m^6/({mol}^2*day)$, $k_{r1}=6.5e-5m^3/(mol*day)$, and $k_{r2}=7.5e-2m^6/({mol}^2*day)$ (Figure 5c–f).

3.6. Model Verification

With the diffusivities of each component through polymer matrix and pores, and the degradation rates parameterized, the predictive capability of the model was evaluated. Simulations were compared against a host of experimental data including: drug release profiles and encapsulation efficiencies of 10 μm microspheres; drug release and solidification profiles of 5 mm ISFIs; and the pH distributions and encapsulation efficiencies of 50 μm microspheres of varying Mw [16, 21, 27, 89].

3.6.1. Solidification Model Verification

To verify the accuracy and predictive capabilities of the solidification model, simulation results were compared to four sets of empirical data: DWI data for the 5 mm implants, early drug release from the 5 mm implants, and the encapsulation efficiencies of both the 10 and 50 μm microspheres (Figure 6).

Figure 6. Solidification Model Verification.

(A) DWI data for a 15 kDa, 5 mm implant collected over the first 3 days (B) DWI data for a 53 kDa, 5 mm implant collected over the first 3 days. Regions of dark blue indicate the retention of solvent and a slower solidification process due to a lack of water. Regions of dark red indicate areas of high diffusivity of water. (C-E) Ternary phase diagrams overlaid with experimentally derived binodal line (dashed blue) and the predicted volume fractions of 5 mm implants at 0, 1, 6, 24, 48, and 72 hours post-exposure. Gray boxes correspond to 6, 24, 48, and 72 hours post-exposure in increasing lightness. (C)15 kDa, 5mm implant, (D) 29 kDa, 5 mm implant, (E) 53 kDa, 5 mm implant. (F-H) Empirical data for the drug released from a 5mm implants (dots) is compared to model prediction (solid line) over the first 5 days [26]. (F) 15 kDa, 5mm implant, (G) 29 kDa, 5mm implant, (H) 53 kDa, 5mm implant. (I-J) Empirical data for the encapsulation efficiencies for varying polymer Mw (6.5, 21, 34, 49, and 67 kDa) for 10 μm and 50 μm microspheres (grey bars) compared to model prediction (blue bars) [16]. (I) 10 μm microsphere, (J) 50 μm microspheres.

DWI data of 15 and 53 kDa PLGA 5 mm implants was sampled at 0.25, 1, 2, 3, 7, and 10 days post-exposure. Early time points (0.25–3 days) are shown in Figure 6a and 6b. The DWI images show the formation of a solid polymer shell for both Mw polymers within a day, which is maintained over weeks, while the core of the implant solidifies over days but decays within a week. For the 53 kDa polymer, the solvent, which is evident by the dark blue regions of the MR images (Figure 4a), takes much longer to leave the implant thereby delaying the formation of the core over three days and its subsequent degradation. By contrast, the solvent leaves the low Mw polymer implant (15 kDa) within a day, allowing the core of the implant to form within the first day; the core subsequently degrades within the first three days (Figure 4b).

During solidification, the model predicts the time scales of polymer solidification of both the 15 and 53 kDa polymers shown by the rapid transition from liquid to solid polymer in the ternary phase diagram (Figure 6c–e). The grey boxes around the time points in the ternary phase diagrams correspond to the respective DWI images in Figure 6a,b. To verify that the model was able to predict drug release rates, simulations of 15, 29, and 53 kDa 5mm ISFIs were compared to experimental data (Figure 6f–h). The absolute error between experimental measurements and model predictions, averaged over the first 3 days, was 3.342, 1.152, and 3.897%, respectively, giving an average absolute error of 2.797% for the 5 mm implants (Figure 6f–h) [27].

The respective manufacturing processes for producing microspheres was also considered. To produce the microspheres, Pack and colleagues used acoustic excitation to break up a stream of co-dissolved PLGA and drug into uniform spheres, which were collected in a bath for 3 hours to solidify. These microspheres were then lyophilized for an 2 additional days to ensure the removal of solvent and water [16, 17]. This process was simulated by the model to increase the accuracy of the overall simulation and to verify the solidification model by predicting the encapsulation efficiency that was measured experimentally by Pack et al. [16]. The solidification rates used for the ISFIs (Figure 2g) were also used for modeling the microsphere solidification. The model predicted the encapsulation efficiencies of 6.5, 21, 34, 49, and 67 kDa 10 μm microspheres, as well as 21, 34, 49, and 67 kDa 50 μm microspheres (Figure 6i–j), with an average absolute error of 18.594% and 17.740% for the 10 and 50 μm microspheres, respectively.

3.6.2. Degradation Model Verification

To verify the accuracy and predictive capabilities of the polymer degradation model, the simulation results were compared to four sets of empirical data: drug release from 10 μm microspheres, drug release and pH distribution within 50 μm microspheres, and drug release from 5 mm implants.

Predicted drug release rates from 10 μm microspheres of varying MW (6.5, 21, 34, and 67 kDa) were simulated and compared to experimental data [16] (Figure 7a–e). The absolute errors between predicted and measured values were 22.405, 9.313, 8.695, 5.543, and 6.240%, respectively (Figure 7a–e, solid navy lines), giving an average absolute error of 10.439 %. The error in the model prediction of the diffusivity of drug through polymer was then evaluated by allowing D_drug,polymer to vary slightly, without degradation, to match early time points of the experimental data (Figure 7a–e, dashed grey lines). All values for parameters can be found in Table S5. These diffusion coefficients were compared to the predicted values from the equation given in Figure 5a–b with an average relative error of 46.917 %.

Figure 7. Degradation Model Verification.

(A-E) Empirical data for the drug release of 6.5, 21, 34, 49, and 67 kDa, 10 μm microspheres was compared to model predictions using the determined degradation parameters and the diffusivity function (solid navy lines). The diffusivities of drug through polymer were allowed to vary to determine error in the diffusivity function without degradation (dashed grey lines) [16]. (F-I) Empirical data for the drug release of 21, 34, 49, and 67 kDa, 50 μm microspheres is compared to model predictions using the determined degradation parameters and diffusivity function (solid navy lines). Optimizations of pore placement (dashed grey lines), and optimized diffusivity of drug through polymer (dashed blue lines). (J-K) Comparison of model prediction of the spatial average pH in the 34 kDa 50 μm microspheres to data collected by Liu et al. for 7, 14, 21, and 28 days for 40 μm PLGA microspheres [89]. (J) Model prediction from 0 to 60 days compared to data by Liu et al [89]. (K) Same model prediction as in J, but shown from 0 to 30 days compared to data by Liu et al [89].

Predicted drug release rate from 50 μm microspheres of varying MW (21, 34, 49, and 67 kDa) were simulated and compared to experimental data [16] (Figure 7f–i). The absolute errors between the experimental and predicted values were 8.136, 7.384, 2.532, and 2.391%, respectively (Figure 7f–i, solid navy lines) [16]. With an average absolute error of 5.111%. However, with the optimized drug diffusion coefficients (as determined in Figure 7a–e), the average error for the 50 μm microspheres decreased to 3.971% (Figure 7f–i dashed blue lines). To evaluate possible causes for error, the average position of pores for the 21 and 34kDa microspheres were also allowed to vary from 17.99 to 19.50 μm [16] (Figure 7f–g, dashed grey lines). Model parameter values are found in Table S5.

By explicitly modeling the polymer degradation process and acid dissociation, we were able monitor the pH change of the microspheres, which influences the conformation and the potency of the drug [90–92]. Experimental data from Liu et al. was compared to the pH gradient predicted by the model (Figure 7j–k) [89]. Predicted pH values were spatially averaged and compared to measured pH values; resulting in a relative error of 3.688%.

To verify that the model can account for implants of larger size (5 mm ISFIs), model predictions were compared to experimental data of drug release (Figure 8) [27]. These simulations included swelling of the ISFI in the first week post-exposure to water. Phase diagrams (Figure 8a–c) allow us to understand the impact of the solidification process in terms of volume fractions, and also the impact of polymer degradation and swelling on the partial volume fractions. Each individual phenomenon had a different impact on the phase transition which can clearly be seen by the 53 kDa ISFI (Figure 8c, black arrows). Phase inversion caused the volume fraction of water to increase, while the volume fraction of polymer remained constant. Swelling increased this exchange, while also decreasing the polymer volume fraction. Degradation caused the volume fraction of water to increase, while the volume fraction of solvent remained constant.

Figure 8. Degradation Model Verification.

(A-C) Ternary phase diagrams overlaid with experimentally derived binodal line (dashed black) and the predicted volume fractions of a 15, 29 and 53 kDa, 5 mm implant sampled every day for 4 weeks (dots) [29]. Black arrows indicate the impact of solidification, swelling, and degradation on the volume fractions. (D-F) Empirical data for the drug release of 15, 29, 53 kDa, 5 mm implants (dots) compared to model predictions (solid navy lines).

The drug release profiles for the 15, 29, and 53 kDa 5 mm implants were predicted with a time average absolute error of 8.409, 3.647, and 9.680%, respectively (Figure 8d–f, solid navy lines); with an average absolute error of 7.245%.

3.7. Parameter Variation and Model Sensitivity Analysis

To quantify the impact of the variation of each model parameter on the drug release profiles, we performed global parameter sensitivity analysis using Latin Hypercube Sampling to vary model input parameter values and quantified the impact of parameter variation on model output using partial rank correlation coefficient (PRCC) analysis [93, 94]. Two global sensitivity analyses were run for both the microspheres and ISFIs for the entire time course of drug release to evaluate (1) the impact of variation in diffusivity of each mobile molecule in the system and (2) polymer degradation rates (see section 3.5) on drug release, respectively (Figure S3, S4). PRCC analysis results in correlation coefficients that relate the variation in parameter input with the variation in model output (drug release), with 1 being perfect positive correlation, and −1 being perfect negative correlation. The PRCC data showed that for the microspheres (10 μm), the non-catalytic end and auto-catalytic random scission rates (ke1, kr2) had the greatest impact on the drug release profile (Figure S3), while in the ISFIs (5mm), the auto-catalytic degradation rates had a larger impact on the drug release profile (Figure S4). Drug release from the microspheres was profoundly impacted by the diffusion coefficient for the diffusion of drug through the solid polymer, while drug release from the ISFIs was only affected by variations in this parameter for the first day post-exposure. Drug release from the ISFIs were more greatly affected by the diffusion coefficient for oligomers and monomers than the microspheres, Figure S3 and S4.

4. Discussion:

Using the mechanistic model, we were able to investigate the impact of specific aspects of the controlled release vehicle’s geometry, mobile molecule diffusivity, and polymer degradation on drug release profiles. These investigations produced significant insight into the aspects of these polymeric systems that can be manipulated to produce desired drug release dynamics.

4.1. Model Geometry

To accurately predict the drug release from different controlled release vehicles, prior knowledge of the solidified geometry was required. For the 10 μm microspheres, the microspheres were completely solid and had a uniform drug distribution. The 50 μm microspheres were not solid but had randomly positioned and randomly sized pores which followed a skew-normal distribution.

Accounting for the localization of drug in the pores and the heterogeneous distribution of drug through the polymer matrix of the 50 μm microspheres allowed for accurate modeling. Without this non-uniform drug distribution, it would be impossible to unify the 10 and 50 μm microspheres, Figure 4i. Therefore, there is a future need for a mechanistic model to simulate phase inversion and accurately predict the geometry of the solidified release system and the drug distribution throughout the implant or microsphere.

Our DWI data shows that unlike the microspheres, which solidified within 3 hours, the ISFIs require 3 days to solidify (Figure 6a,b), resulting in the formation of a complex structure. Phase inversion is further complicated since the rate of phase inversion is dependent on the Mw of the polymer used to create the ISFI. Upon phase inversion, the most salient features of the 5 mm ISFIs (Figure 2a) were the shell thickness and the radius of the implant. The distribution of interior pores had little impact on the degradation and drug release profiles (Figure 4d–f). However, the presence of shell pores was significant. Simulations of the simplified geometry without shell pores had an average error of 8% relative to simulations using SEM image geometry (Figure 4f). This dependence on shell porosity provides further motivation for future mechanistic modeling of the formation of these pores during the solidification process, as the microstructure of the implant is sensitive to factors such as the polymer Mw and mass of water available to participate in solvent/nonsolvent exchange.

We found that the swelling of these implants, which was quantified by the Exner and colleagues [27], had a significant impact on the diffusivity of the polymer (Eq 24b and Figure 7). The drug release profile cannot be accurately predicted by increasing the degradation rate due to the positive feedback of auto-catalytic degradation; instead, swelling plays a critical role. The swelling slowly increases the diffusivity of the polymer, as compared to the rapid increase in diffusivity due to polymer degradation from acid build up. Swelling also reduces the degradation rate by increasing the diffusion of acidic byproducts out of the polymer matrix. Including swelling in the simulations created a more controlled release profile that was empirically observed by the Exner lab [27]. Here we developed an empirical relationship between the diffusivity of drug and polymer molecular weight. Future mechanistic studies on the relationship between mobile species diffusion, polymer molecular weight, and the impact of swelling on diffusion will provide critical insight into the factors that drive these relationships and will be key in designing implants with specific drug release profiles.

4.2. Model Parameterization Diffusivity/Degradation

Functionalizing the relationship between the PLGA Mw and the polymer diffusivity was key to predicting the drug release from the variety of controlled release systems and highlights the need for future work to approximate diffusion coefficients for a variety of diffusion-controlled release applications.

Previous work by Pack et al. estimated this relationship for the 10 and 50 μm microspheres separately. While the values were close and even identical for the higher Mw polymers, the diffusivity of low Mw polymers was inconsistent between the different microsphere sizes. The goal of our project was to develop a unified model for any size and Mw PLGA based controlled release system. Accounting for the localization of drug concentration in the randomly distributed pores was crucial in unifying the microsphere models (Figure 4g–i).

The function for diffusivity (Figure 5b) was found to predict the diffusivity coefficient of piroxicam in 6.5, 21, 34, 49, and 67 kDa PLGA 50:50 with an average error of 45.950 %. This error, while seemingly large, is reasonably small when considering that the value of diffusivity varies over 11 orders of magnitude from the diffusion of water to the diffusion of a large drug such as clarithromycin. Parameter sensitivity analysis determined that although including the varying polymer diffusivity was critical to the model, the diffusivity of the drug through the polymer was negligible compared to that of the solvent and oligomers. Therefore, there is a need to carefully quantify the transport of polymer byproducts as well as the effect of varying organic solvents on the release of drug from the controlled release systems. For instance, it has been demonstrated that 15 kDa PLGA ISFIs retain the highest mass of organic solvent (NMP), and consequently have elevated levels of residual solvent. This elevated residual solvent would result in pores that have a higher viscosity than the other implant formulations, which would decrease diffusivity through the implant. Interestingly, our model predictions overestimate the rate of release from the 15 kDa implants, indicating a need to account for residual solvent effects (Figure 6f–h). This model also does not account for any potential convection.

Although allowing each of the degradation rates and initial crystallinity to vary based on polymer Mw would have led to a better fit (than in Figure 5c–f), and ultimately a better prediction of drug release in Figs 7–8, it was viewed as unmechanistic and an overparameterization of the model. The relationship between the end and random chain scission rate were consistent with predictions made by Flory [20] who believed that end scission occurred an order of magnitude faster than random scission. The model suggests that between a pH of 5 and 4, end scission occurred 68x to 11x more often than random scission. Interestingly, this mechanistic approach is sufficient to predict release from 50 μm microspheres formulated from PLGA with a Mw of 49 kDa and above. Below this Mw threshold, we observe that our model does not effectively predict the onset of degradation facilitated release. A similar phenomenon is observed for ISFI formulated using PLGA with a Mw greater than 29 kDa. Together these findings indicate a need to introduce a percolation threshold in future studies.

This model neglects convective aspects of transport. Convective forces would primarily occur during phase inversion. During phase inversion the rapid solvent exchange may result in the addition of a transient convective forces due to the rapid loss of solvent to the aqueous bath. These advective forces would result in an elevated burst than predicted by modeling diffusion alone. Interestingly, we observe that for implants formulated with 53 kDa PLGA, our model underestimates the burst release; indicating that convective forces may be a contributing factor for implants formulated using large Mw PLGA polymers. These findings further highlight the need to develop a mechanistic model describing polymer phase inversion dynamics.

4.3. Auto-catalytic degradation

Degradation of PLGA through the cleavage of esters bonds has the important consequence of producing acidic byproducts, which then further enhances chain scission. The impact of auto-catalytic degradation can be seen when comparing the degradation of the polymer shell to that of the core in the 5 mm ISFI (Figure S5), and when observing the polymer degradation in the walls of the interior pores of the implant (Figure S6a). Our experimental DWI and SEM data show extensive degradation and erosion of the interior of implant, while the shell retains structural integrity for the first 10 days (Figure S5). After 10 days, the shell rapidly degrades until its rupture around day 17.

While the core of the implant sustains the most rapid degradation as a whole, the wall between the interior pores have the highest rate of degradation (Figure 14a). Rather than the pores exclusively breaching where it was the thinnest, they broke open where the most bulk polymer was, for example at the intersection of four pores in a hexagonal close-packed structure (Figure S6). This increased degradation is due to the auto-catalytic nature of the PLGA copolymer degradation and the increased concentration of $H^{+}$ ions in the nucleation site (Figure S6a, S6b). The computational model predicts the highest concentration of polymer chain scission to take place in these nucleation sites (Figure 6c) due to the inclusion of the auto-catalytic degradation term in Eq. 1. The importance of the autocatalytic degradation for the large ISFIs is quantitatively reaffirmed by the results of the PRCC analysis where k_e2 had the greatest correlation with drug released from the implant, Figure S4. The microspheres show a more uniform distribution of acid (Figure S3), and therefore are more impacted by the non-catalytic degradation rate k_e1. The PRCC data reaffirmed the need for considering all four unique degradation rates.

Modeling the production and diffusion of H⁺ ions rather than estimating them as a lumped constituent with the oligomer and drug [53, 58–61], allowed us to predict the pH change within the implant (Figure 7j–k) and evaluate the impact of acidification on polymer degradation and drug release. pH change within the implant can be also particularly important parameter to consider when designing patient-specific or treatment-specific controlled release vehicles, as some tissues do not have significant rates of clearance to tolerate highly acidic systems.

Previous literature has modeled polymer degradation with a single first/second order decay term [16, 52, 56, 57, 76]. However, the experimental data and model predictions developed in this work demonstrate that to fully understand, and therefore control the degradation of the implant, it becomes necessary to not only consider end and random scission, but also noncatalytic and autocatalytic degradation [53, 59].

4.4. Modeling error

Numerical solutions of the governing PDEs in Python proved to be a reliable computational model for predicting the drug release rate of the implant for varying PLGA molecular weights, and implant size. Model prediction error, relative to experimentally determined drug release profiles, introduced by reducing the geometry to the simplified model was determined to be an average of 0.864% with shell pores and 8.048% without. A mesh convergence study quantified the numerical error introduced by the algorithm at 0.111%, and 0.787% error was introduced in the stochastic model for the 50 μm microspheres (Figure S7).

The solidification model was shown to be accurate within a 2.480% average error in predicting the drug release from ISFIs in the first 3 days. However, there was 18.594 and 17.740% error in predicting the encapsulation efficiency of the 10 and 50 μm microspheres, respectively. The model’s prediction of higher encapsulation efficiencies of 73.25% for 10 μm and 61.110% for 50 μm microspheres is much closer to previously reported encapsulation efficiencies [76–88]. These differences in model prediction versus experimental data indicate that the solidification model used in this project was much too simple to completely explain this phenomenally complicated process, emphasizing the need for future development of a mechanistic model for polymer solidification.

The degradation model was relatively accurate across the large range in size of the implants, with an average of 7.245 % error in predicting the drug release profile of the 5 mm ISFIs, 10.439% error in predicting the drug release from the 10 μm microspheres, and 5.111% error in predicting the drug release from the 50 μm microspheres. There was, however, notable error in the late-stage release profiles of the 5 mm ISFIs. The divergence from predicted values after 15 days for the 5 mm ISFIs is likely due to the difficulties in collecting experimental data for the later time points. The computational model was developed having a perfect sink 1 mm away from the surface of the implant, however, the empirical data showed that this particular ideal sink condition was not satisfied [27]. This would account for the steepness of the simulated release profile as compared to the experimental data.

4.5. Complexities of in vivo modeling

When expanding the current in vitro model to in vivo predictions we need to consider the impact of injection site on the rate of phase inversion; the final geometry of the implant; the rate of degradation; and the clearance rates of the local tissue. These are outside the scope of the current work but will be addressed in future research. Here we discuss their significance and implications.

Injection site and ISFI geometry:

When ISFIs are formed at the injection site, the impact of mechanical loading is significant. Phase inversion is accelerated, which changes the formation of the shell and pores, while allowing for the rapid escape of the drug [95]. The irregular geometry breaks down symmetry, forcing an increase in model complexity [27]. However, the bounds of the drug release profile can be predicted by modeling the smallest and largest principal axis of the ellipsoid-shaped implant. The local tissue and fluid environment at the injection site would have an increase in density and viscosity, reducing mass transport out of the implant, impacting the phase inversion of ISFIs, the erosion of diffusible polymer, and the release of the drug from the implants.

Tissue clearance rate:

The current model assumes that diffusion of components (cleaved polymer, drug, hydronium ions) is rapid after leaving the implant boundary, causing the respective concentrations to drop to zero at the implant boundary. In physiological conditions where blood and interstitial fluid surround the implant, constituents of these fluids increase the fluid viscosity and decrease the diffusion of drugs, polymer, and ions from the implant. Similarly, the density of the tissue matrix and perfusion of the tissue affect the transport of model components. For example, in highly perfused tissues, the hydrolyzed polymer, drug, and ions are rapidly transported away such that concentrations at the implant surface would be zero, and model results are not affected. However, in highly viscous environments, concentrations at the implant surface would increase over time, and pH and drug concentration would affect polymer degradation and drug release rate.

4.6. Model use and future work

The python code for the model is available at the following DOI on the Purdue University Research Repository (PURR) DOI 10.4231/ASPM-SY16, or directly on the Kinzer-Ursem LabGitHub repository: https://github.itap.purdue.edu/TamaraKinzerursemGroup/DrugReleaseSystemModel.git.

These include a test set of parameter inputs and corresponding model outputs such that the user can validate that the model is working correctly prior to modification for their purposes. Users may be particularly interested in varying polymer properties (Mw, degradation rates) or drug properties (Mw, pH) to predict drug release rates for their system of interest.

In this work we found that for larger implants (2 mm diameter) the shell thickness and shell pores were critical parameters in predicting the release profile; the interior pores had little impact. However in the microparticle systems, the release rate was controlled by the distribution of drug, which is related to the distribution of pores within the particles. Therefore, parameters that effect the distribution of pores would likely significantly impact the release profile in these systems. Interestingly, these parameters are related to the rate of polymer phase inversion. Therefore, factors that affect the solvent/nonsolvent exchange such as the polymer molecular weight, solvent molecular weight, solvent density, solvent miscibility, volume fractions of each constituent, and volume of implant can all be used to alter the release profile from these systems. Future work will be aimed at creating an advanced polymer phase inversion model that can predict the implant geometry from these parameters.

The current model does not account for extra-implant conditions but focuses on the principal mechanisms that dictate the rate of drug release from the implant. Our work focuses on the phenomena of polymer phase inversion, swelling, drug distribution, and polymer degradation of in vitro implants and not their clinical application. However, the model could be expanded to simulate the drug release and biodistribution for in vivo implants. For microspheres and ISFIs, the model boundary conditions would need to be reevaluated. To account for the clearance of the eroded biodegradable polymer and the drug out of the local tissue, a simple physiologically based pharmacokinetic model could be used, which can be implemented as a boundary condition on the implant model [96 – 98]. Future work will consider the effects of tissue-dependent properties on polymer degradation and drug release rate.

5. Conclusion:

Given the high potential for controlled release vehicles to be tailored to individual patient needs, and the ability to increase patient compliance, there is a high potential for their increased and broad use as a replacement for conventional forms of therapeutic administration. As we have shown here, development and validation of mechanistic computational models of these drug delivery systems lends significant insight into the underlying physical and chemical processes that determine drug release dynamics. Furthermore, by incorporating mechanistic detail, these models can be applied to a variety of release vehicle formats, from 10 μm microspheres formed in vitro, to 5mm in situ forming implants with a variety of molecular weights. These results will allow researchers and developers to rapidly screen for conditions (i.e., implant size, polymer molecular weight, polymer ester bond content, drug pH) that produce desired drug release profiles. Thus, we expect this newly developed tool to assist in an acceleration of the iterative design process for producing controlled release vehicles. Immediate future work will be aimed at reducing the model error by collecting experimental data for the diffusion of a variety of drugs in polymers of various molecular weight to further characterize this complex relationship. Another future goal will be to increase the complexity of the solidification model to accurately simulate the phase inversion stage so that prior knowledge of the implant geometry and initial drug distribution are not required by the model.