# A Computational Model for Drug Release from PLGA Implant

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## Abstract

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## 1. Introduction

_{1}(65:35) and PLGA

_{2}(50:50) nanofiber mats are investigated. Next, we present fundamental equations of the radial 1D element and the equation of degradation implemented into our model, including the hydrophobic effects. This is followed by the formulation of the smeared model for the fiber network. Finally, we demonstrate the applicability and accuracy of the computational models by presenting both numerical and experimental results.

## 2. Materials and Methods

#### 2.1. Materials

_{1}(average molecular weight MW 40,000–75,000 g/mol) with the mass ratio of lactide:glycolide units being (65:35), poly(d,l-lactide-co-glycolide) PLGA

_{2}(average molecular weight MW 30,000–60,000 g/mol), with the mass ratio of lactide:glycolide units being (50:50), Rhodamine B (RhB), span-80, N,N-dimethylformamide (DMF) and chloroform (CHCl

_{3}), were purchased from Sigma-Aldrich Co. (Milwaukee, WI, USA). The chemicals were used without further purification. Phosphate buffered saline solution (PBS) was made by dissolving one tablet of PBS, supplied by Fisher Scientific (Hampton, NH, USA) in 200 mL of distilled water.

#### 2.1.1. Preparation of PLGA Nanofibers Produced via Emulsion Electrospinning

_{1}and PLGA

_{2}(3.0 g) were initially dissolved in the mixture of solvents chloroform/DMF (8.25/2.75 g) and magnetically stirred at 200 rpm at room temperature for 24 h. After that, span-80 (50.0 mg) was added to this polymer solutions, followed by the addition of 5 wt.% of RhB aqueous solution (60 µL). The mixture was additionally stirred for 2 h.

_{2}nanofiber scaffold is highly porous and has a dense mesh structure with bead-free and randomly arranged nanofibers. For the determination of the implant porosity, the nanofiber sheet’s apparent density was firstly estimated by the measurement of volume and mass of samples [26]. The results showed that for PLGA

_{1}and PLGA

_{2}, the porosities were ~59% and ~78%, respectively. A numerical investigation regarding the influence of porosity was carried out in our previous paper. There, a radial 1D finite element for drug release from drug loaded nanofibers was introduced, where we confirmed that our model can predict the drug release for different porosities of the model [22].

#### 2.1.2. Drug Loading Efficiency

_{1}and PLGA

_{2}were 99% and 0.09%, respectively.

#### 2.1.3. In Vitro Drug Release Studies

_{1}and PLGA

_{2}nanofiber mats were cut into small pieces, and approximately 40 mg (2.5 cm × 2.5 cm) of each sample was immersed in 20 mL phosphate buffer solution (PBS, pH = 7.4) at 37 °C. At certain time intervals, 1 mL of sample solution taken for the analysis was replaced with 1 mL of fresh PBS solution. This frequent process lasted for 2 months. The amount of RhB released in PBS at each time point was monitored by measuring the UV absorbance of the maximum peak for RhB (at an optical wavelength of 554 nm). The accumulated release of RhB was calculated based on a standard RhB absorbance-concentration calibration curve.

_{1}and PLGA

_{2}nanofiber mats was investigated and is presented in Figure 2. Within the first 24 h, 0.3% was released from PLGA

_{2}, while release from PLGA

_{1}was not observed. After two weeks, the total of ~34% of RhB was released from PLGA

_{2}and ~14% from PLGA

_{1}. After 30 days, nearly 47% and 18% of the RhB was released from PLGA

_{2}and PLGA

_{1}, respectively. At the end of the observed profile release period, PLGA

_{2}nanofiber mat released ~60% and PLGA

_{1}nanofiber mat released ~30% RhB. Based on the obtained release profiles, it can be concluded that PLGA

_{2}has a faster release profile when compared to PLGA

_{1}. The release kinetics of incorporated RhB from PLGA electrospun nanofibers and the time required for hydrolytic degradation of PLGA depend on a molecular weight and chemical composition of polymers, porosity, crystallinity, hydrophobic/hydrophilic nature, as well as on a lactide/glycolide ratio. This is because low molecular weight PLGA generally leads to faster polymer degradation and a more rapid drug release. Additionally, as lactide is more hydrophobic than glycolide, an increase in lactide content in PLGA copolymers decreases the polymer degradation rate, followed by a slower drug release. The PLGA with 50:50 lactide to glycolide ratio had the fastest degradation rate (1–2 months), while the PLGA with 65:35 lactide to glycolide ratio degraded after 3–4 months.

## 3. Computational Models

#### 3.1. Fundamental Equations

_{w}, φ), where M

_{w}is PLGA average molecular weight (MW) and ϕ is porosity. The function D = D(M

_{w}, φ) can be expressed as:

_{s}

_{0}is diffusivity for the initial molecular weight M

_{w,0}and α = 1.714 the experimentally determined coefficient. The molecular weight M

_{w}and porosity ϕ are functions of time t, described as:

_{w}and k are degradation rate constants, taken as 2.5 × 10

^{−7}s

^{−1}, and φ

_{0}(= 0) is the initial porosity.

#### 3.2. Diffusion within Fibers

#### 3.2.1. Axial Diffusion

_{I}, N

_{J}are the interpolation functions, A is fiber cross-sectional area, and L is element length;

**C**and

**C**

^{t}are nodal concentrations at the end and start of time step, respectively. Note that the balance equation, Equation (7), can be written for the continuum, using Equation (1), with matrices:

#### 3.2.2. Radial Diffusion

_{1}+ L

_{2})/2 in Figure 3a).

_{i}is the radius of node 1 of the current subelement (closer to the axis of symmetry) and L is the element length. The value of D

_{fiber}is the mean diffusivity for the element, calculated according to Equation (4) in cases when degradation and erosion are present. An expression for the initial concentration distribution in radial subelements is given in the Appendix A.

#### 3.3. Fundamental Equations for CSFE

## 4. Numerical and Experimental Results of Drug Release

_{1}and PLGA

_{2}are taken from References [12,13,14,15,16,17,18]. We adopted the following conditions in our FE models:

- Hydrophobicity (partitioning) of drug transport within PLGA
_{1}is lower than for PLGA_{2}; - Degradation of PLGA
_{1}is much slower than degradation of PLGA_{2}.

#### 4.1. Preparation and Numerical Simulation of Detailed FE Models

_{fiber}= 4 × 10

^{−10}cm

^{2}/s, which is taken from Reference [31], approximately 104 times lower than in water. The time period of simulation was 75 days (15 time steps with 5 days each). The boundary conditions of the model are: Prescribed concentration C

_{0}in fibers and C = 0 at outer boundary of implant (boundary where mass release is measured).

- Dimensions: 80 µm × 90 µm × 90 µm;
- FE mesh: 40 × 48 × 48 divisions;
- Diffusion coefficient within fibers: D
_{fiber}= 0.04 µm^{2}/s; - Diffusion coefficient in between fibers: D
_{liquid}= 0.04 µm^{2}/s; - Mean diameter: D = 2.5 µm.

#### 4.2. Application of Smeared Modeling for Drug Transport in PLGA Implant

- Fiber domain—equivalent domain of fibers;
- Surrounding domain—equivalent “pore” space surrounding fibers.

- Volume fraction of fibers in PLGA layers;
- Diffusion coefficient within PLGA fiber, for either 24 wt.% 50:50 and 65:35 emulsion;
- Diffusion coefficient of drug within the surrounding domain. Coefficient of hydrophobicity (partitioning);
- Mean diameter of PLGA fibers.

_{liquid}= 0.004 µm

^{2}/s. The parameters used in our composite smeared finite element (CSFE) model are:

- Volume fraction of fibers: ${r}_{V}$= 0.4223;
- Mean diameter of fibers: D = 2.5 µm;
- Diffusion coefficient within fibers: D
_{wall}= 0.04 µm^{2}/s; - Equivalent diffusion coefficient in surrounding domain: D
_{liquid}= 0.004 µm^{2}/s; - Partitioning: P = 1.

#### 4.3. Comparation of Numerical and Experimental Results

_{1}are shown in Figure 9 and Figure 10, for domains with fibers and the surrounding domain, in a period of 75 days. It can be seen that there are small differences between the two models; hence, the smeared modeling concept can be used for the prediction of drug transport from drug impregnated nanofibers.

_{1}and PLGA

_{2}(both detailed and smeared model) are given in Figure 11 and Figure 12. In a numerical simulation, for PLGA

_{1}, we used partitioning coefficient P = 2 × 10

^{5}and degradation coefficient ${\kappa}_{w}$ = 2.5 × 10

^{−7}s

^{−1}, while for PLGA

_{2}, the partitioning coefficient was P = 5 × 10

^{5}and degradation coefficient ${\kappa}_{w}$ = 2.0 × 10

^{−7}s

^{−1}. Coefficients used in simulations are in accordance with experimental studies, where it is stated that PLGA 50:50 (PLA/PGA) exhibits a faster degradation than PLGA 65:35, and higher hydrophilicity (which means a smaller partitioning coefficient).

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Initial Concentrations for Radial Subelements

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**Figure 1.**SEM images of drug loaded 24 wt.% 50:50 PLGA nanofibers (PLGA

_{2}).Images of same PLGA mat with different scale bars 100 µm (

**a**), 50 µm (

**b**), 10 µm (

**c**) and 10 µm (

**d**).

**Figure 2.**The experimental release of Rhodamine B (RhB) from poly(d,l-lactide-co-glycolide) PLGA

_{1}(

**a**) and PLGA

_{2}(

**b**) nanofiber mats.

**Figure 3.**Definition of 1D radial finite element (

**a**) and (

**b**), and 1D radial element with subelements (

**c**), according to Reference [22]. Copyright 2017. Reproduced with permission from the Journal of the Serbian Society for Computational Mechanics.

**Figure 4.**Network of fibers and a domain of finite element with indicated fiber directions (domain is 90 μm × 90 μm).

**Figure 5.**(

**a**) Diffusion from fiber surface dA

_{fib}

_{,}which corresponds to the fiber volume dV

_{fib}and total volume dV; dV

_{sur}is the volume occupied by the surrounding medium; (

**b**) Corresponding smeared model.

**Figure 6.**Composite smeared finite element (CSFE) with fiber and surrounding domain and connectivity elements at each node; a list of nodal parameters is given in the figure.

**Figure 7.**Configuration of the finite element (FE) model of PLGA implant, (

**a**) SEM imaging of one layer of PLFA fibers, (

**b**) reconstructed 1D mesh of fibers (scan bar 20 μm), (

**c**) reconstructed FE model, (

**d**) configuration and geometry of FE model [22]. Copyright 2017. Reproduced with permission from the Journal of the Serbian Society for Computational Mechanics.

**Figure 8.**PLGA domain modeled using a smeared composite finite element or detailed model with the mesh of fibers.

**Figure 9.**PLGA implant—concentration field for the detailed and smeared model, for the diffusion of Span-80/RhB complex within the PLGA implant.

**Figure 10.**PLGA implant—concentration filed in the surrounding domain, for the detailed and smeared model, for the diffusion of Span-80/RhB complex within the PLGA implant.

**Figure 11.**Cumulative Release vs. Time for Span-80/RhB complex impregnated and for 24 wt.% 65:35 PLGA. Experimental curve and computational results obtained using the true (detailed) and smeared model of PLGA nanofibers.

**Figure 12.**Diagram Cumulative Release vs. Time. Comparison of results for the true (detailed) and smeared model of PLGA nanofibers, with Span-80/RhB complex impregnated and for 24 wt.% 50:50 PLGA. Experimental curve and computational results obtained using true (detailed) and smeared model of PLGA nanofibers.

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**MDPI and ACS Style**

Milosevic, M.; Stojanovic, D.; Simic, V.; Milicevic, B.; Radisavljevic, A.; Uskokovic, P.; Kojic, M.
A Computational Model for Drug Release from PLGA Implant. *Materials* **2018**, *11*, 2416.
https://doi.org/10.3390/ma11122416

**AMA Style**

Milosevic M, Stojanovic D, Simic V, Milicevic B, Radisavljevic A, Uskokovic P, Kojic M.
A Computational Model for Drug Release from PLGA Implant. *Materials*. 2018; 11(12):2416.
https://doi.org/10.3390/ma11122416

**Chicago/Turabian Style**

Milosevic, Miljan, Dusica Stojanovic, Vladimir Simic, Bogdan Milicevic, Andjela Radisavljevic, Petar Uskokovic, and Milos Kojic.
2018. "A Computational Model for Drug Release from PLGA Implant" *Materials* 11, no. 12: 2416.
https://doi.org/10.3390/ma11122416