Mathematical modeling of drug release from biodegradable polymeric microneedles
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Abstract
Transdermal drug delivery systems have overcome many limitations of other drug administration routes, such as injection pain and firstpass metabolism following oral route, although transdermal drug delivery systems are limited to drugs with low molecular weight. Hence, new emerging technology allowing high molecular weight drug delivery across the skin—known as ‘microneedles’—has been developed, which creates microchannels that facilitate drug delivery. In this report, drugloaded degradable conic microneedles are modeled to characterize the degradation rate and drug release profile. Since a lot of data are available for polylactic acidcoglycolic acid (PLGA) degradation in the literature, PLGA of various molecular weights—as a biodegradable polymer in the polyester family—is used for modeling and verification of the drug delivery in the microneedles. The main reaction occurring during polyester degradation is hydrolysis of steric bonds, leading to molecular weight reduction. The acid produced in the degradation has a catalytic effect on the reaction. Changes in water, acid and steric bond concentrations over time and for different radii of microneedles are investigated. To solve the partial and ordinary differential equations simultaneously, finite difference and Runge–Kutta methods are employed, respectively, with the aid of MATLAB. Correlation of the polymer degradation rate with its molecular weight and molecular weight changes versus time are illustrated. Also, drug diffusivity is related to matrix molecular weight. The molecular weight reduction and accumulative drug release within the system are predicted. In order to validate and assess the proposed model, data series of the hydrolytic degradation of aspirin (180.16 Da) and albumin (66,000 Da)loaded PLGA (1:1 molar ratio) are used for comparison. The proposed model is in good agreement with experimental data from the literature. Considering diffusion as the main phenomena and autocatalytic effects in the reaction, the drug release profile is predicted. Based on our results for a microneedle containing drug, we are able to estimate drug release rates before fabrication.
Keywords
Mathematical modeling Microneedle Polymer degradation Drug release Poly(lacticcoglycolic acid) Autocatalytic effectIntroduction
Mathematical modeling of drug delivery and predictability of drug release is a subject of steadily increasing academic and industrial importance, with a broad future potential. Because of the significant advances in information technology (IT), the in silico optimization of novel drug delivery systems (DDS) can be assumed to significantly enhance accuracy [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].
Some drugs require parenteral delivery because of instability and enzymatic degradation in the gut in order to avoid firstpass metabolism and gastrointestinal side effects. The difficulty with injections is that they usually have to be administered by professionally trained staff and tend to cause pain [11]. Transdermal drug delivery (TDD) offers an alternative to oral and parenteral routes to avoid gastrointestinal drug degradation, firstpass metabolism and pain, and to prolong drug release and improve patient compliance. The first technology developed and licensed was the transdermal ‘patch’ in 1979. However, these traditional TDD systems are limited to drugs of low molecular weight (M_{W}) (< 500 Da), moderate lipophilicity (log p 1–3), aqueous solubility (> 100 µg/mL) and high potency (daily dose < 10 mg/day) [11]. This is because the stratum corneum (SC), which is the outermost layer of the skin, constitutes the major barrier.
In recent years, microneedle technology, as proposed by Henry et al. (1998), has been developed as an advanced technique for penetration of large M_{W} and hydrophilic compounds into the skin [12]. Microneedles are needlelike structures with diameter in the size order of microns and lengths up to 1 mm. These structures are used to pierce the upper layer of the skin to enable (trans)dermal drug delivery (microneedle technologies). In the research described here, the application of microneedles in drug delivery is focused on both ‘solid’ and ‘hollow’ microneedles as the common types of microneedles.
Solid microneedles are usually manufactured such that they pierce the upper layer of the skin and allow drug passage to the lower layers, where diffusion will be faster. A drug can be coated on the solid microneedle surface such that when they enter the skin, the drug dissolves and the microneedle exits. Also, the drug can be loaded in the microneedle matrix. In nondegradable ones, drugs with low M_{W}s will diffuse to the outer medium, while in degradable matrices drugs with higher M_{W}s can be released as the polymeric matrix degrades. In comparison with hollow microneedles, it is easier to fabricate solid microneedles, which have higher mechanical strength and sharper tips.
A process is defined as a series of operations done on materials. One of the goals of mathematical modeling of a process is to obtain a number of equations that explain the process behavior. Solving these equations illustrates the process response versus different input data. The process type determines whether the equations are algebraic, differential or a combination of both of them [13]. Mathematical modeling of drug delivery and drug release prediction has a profound importance nowadays. In silico optimization of novel drug release systems has a lot of advantages. It can be expected that mathematical modeling can be used for designing new dosage forms. Suitable approximations for geometry, dimensions and polymeric matrix molecular weight, etc., can be employed, so fewer experimental studies are required for developing a product, which leads to saving time and costs. Quantitative analysis of physical, chemical and biological phenomena involved in controlling drug release can explain drug release mechanisms [14].
Experimental procedure
Polymer degradation is a chain scission process in which polymer chains are broken into monomers and oligomers. Degradation of waterinsoluble polymers in aqueous media is a part of their erosion process. Depending on relative water diffusion and polymer chain scission rates in such systems, two types of erosion can be defined: surface erosion and bulk erosion. In the first case, chain scission is much faster than water penetration into the system, so the main degradation process occurs in the outer layers. Polymers with less active groups, such as poly(lacticcoglycolic acid), have more tendency to undergo bulk erosion [14]. Bulk biodegradable polymers are frequently used as drug delivery carriers. For example, one of the applications is microneedle fabrication based on degradable polymers for drug delivery purposes.
In this study, a conical microneedle loaded with drug is considered. Drug release rate and profile from the microneedle is modeled as a function of matrix degradation rate. Simplifying assumptions are required that will be discussed later. Also, model limitations will be described. It is expected that, with knowing a desired drug release rate, we will be able to design a suitable system. At first, matrix degradation and chain shortening is modeled, which is later related to drug diffusion rate changes. The proposed model is a theoretical model based on a mass conservation equation for each component with consideration of physically and chemically involved parameters in polymer degradation and drug release. All these parameters are independently estimated. After deriving the matrix degradation rate, the drug release profile can be predicted. The obtained model led to three simultaneous differential equation systems. The finite difference method for solving partial differential equations and the Runge–Kutta method for ordinary differential equations are employed. Programming code in MATLAB is developed, and experimental data from various articles are utilized for model validation assessment.
The proposed model for polymer degradation—and ultimately, drug release—is based on the involved phenomena and mechanisms. Governing equations for each component’s diffusion and reaction are considered. Since the model is based on governing equations, it is applicable for a wider range of polymer types, provided that assumptions and conditions that are used in deriving the equations are met. In this article, the focus is on aliphatic polyesters, especially poly(lacticcoglycolic acid) (50:50 molar ratio), due to the abundance of experimental data.
Abbreviated signs used in the literature
Abbreviated signs  Definition 

t  Time 
k  Reaction rate constant 
\( _{{M_{n} }} \)  Number average molecular weight 
\( M_{n}^{0} \)  Initial number average molecular weight 
v _{cone}  Cone volume 
v _{ e}  Cylinder volume 
L  Height 
L _{ e}  Equivalent height 
D  Diameter 
D _{ e}  Equivalent diameter 
R  Radius 
\( C_{w} \)  Water concentration 
\( C_{p} \)  Polymer (ester bonds) concentration 
\( C_{p}^{0} \)  Initial polymer (ester bonds) concentration 
\( C_{a} \)  Acid concentration 
\( C_{d} \)  Drug concentration 
\( D_{w} \)  Water diffusion coefficient 
\( D_{a} \)  Acid diffusion coefficient 
\( D_{d} \)  Drug diffusion coefficient 
\( D_{d}^{0} \)  Initial drug diffusion coefficient before degradation 
ρ  Density 
Pseudofirstorder kinetics
When studying the rate of the chain scission reaction in poly(ɛcaprolactone), Pitt et al. [15] suggested that the hydrolysis rate, in the absence of autocatalytic effects, has firstorder kinetics, whereas autocatalytic hydrolysis has secondorder kinetics. In deriving the model, it is assumed that the degradation has pseudofirstorder kinetics. Only the carboxylic acid concentration is considered; the ester concentration is ignored, although the hydrolysis rate depends on both concentrations. The model’s advantage is that only simple calculations are needed to characterize molecular weight reduction [16].
Secondorder kinetics
In the research done by Lyu et al. [17], it was assumed that the degradation rate depends on both water and esteric bond concentrations; hence, secondorder kinetics would apply. However, autocatalytic effects of the degradation products are neglected, and it is assumed that the water concentration remains constant inside the polymer. The rate equation obtained is first order and is a basis for description of polymer bulk degradation, weight loss during surface erosion or degradation with a moving degradation front. The advantage of this model is that the mass loss is predicted for a polymer, which degrades according to a surface process or with a moving erosion front. The disadvantage of the model is that an autocatalytic behavior of degradation in the model derivation is ignored [16].
It can be understood that the assumption of constant water and ester concentrations is only acceptable for the very early periods of degradation initiation. As the polymer chains are broken, it is expected that the water content, ester concentration and system porosity would increase.
Assumptions

Microneedles are considered to be separate, and the overall drug release amount will be achieved by summation of individual microneedle releases.
 Usually, microneedles are conic in order to pierce the skin easily. For simplifying the model and avoiding twodimensional equations, the conic needles are assumed to be cylinders with the same volume (see Fig. 1). Since degradation in this case is a bulk degradation, this assumption can be acceptable; sharp tips of the needle are just designed to facilitate needle entrance into the skin.Thus, the following equations apply:$$ v_{\text{cone}} = \frac{1}{3}\left( {\frac{{\pi D^{2} }}{4}} \right)L $$(3)$$ v_{e} = \left( {\frac{{\pi D_{e}^{2} }}{4}} \right)L_{e} $$(4)$$ v_{\text{cone}} = v_{e} $$(5)$$ {\text{if}} \to L = L_{e} \Rightarrow D_{e} = \frac{\sqrt 3 }{3}D $$(6)

The constitutive monomers of PLGA are lactic acid (LA) and glycolic acid (GA). Thus, there are four different ester bonds: LA–LA, GA–GA, LA–GA and GA–LA. One of the hypotheses is that the ester bonds ratio remains constant during degradation.

The drug is distributed uniformly in the polymeric cylinder matrix and not on its surface.

The main degradation mechanism is hydrolysis. Water enters the bulk of the system in the cylinder and reacts with the ester bonds of the chains, leading to chain scission and conversion of chains into oligomers and monomers.

Edge effects are neglected, and there is no mass transfer from the bottom cross section of the cylinder—so the mass transfer in the cylindrical coordinate will be one dimensional. The results will be more accurate when the \( \frac{L}{D} \) ratio is higher.

Degradation products have good water solubility and can exit the control volume easily.

The reaction rate constant of hydrolysis of the ester bonds during the reaction is assumed to be the same for all of them.
Estimation of model parameters
Required model parameters
Unit  Siegel et al. [18]  Shah et al. [19]  

Cylinder radius  cm  0.6  6.5 
Cylinder height  cm  0.1  0.04 
Polymer molecular weight  g/mol  63,000  18,000 
Polymer density  g/cm^{3}  1  1 
Water diffusivity  cm^{2}/s  3 × 10^{−6}  1.05 × 10^{−5} 
Average acidic monomer diffusivity  cm^{2}/s  1 × 10^{−10}  3.5 × 10^{−10} 
Degradation time  Day  38  35 
Drug  –  Aspirin  Albumin 
Drug diffusivity  cm^{2}/s  1 × 10^{−8}  5.92 × 10^{−10} 
Drug molecular weight  g/mol  180.16  66,000 
Drug weight percent  –  0.2  0.17 
Governing equations
In the first moment, no water has entered the system; its concentration is zero in the whole cylinder. While the time passes, at the center line of the cylinder, there is no concentration change due to symmetry, so it remains zero. At the surface, the concentration will remain constant, the same as solution value.
At first, the acid concentration is a determined value, since the polyester chains have acidic end groups. This concentration can be measured using the end group analysis method or by knowing the number average molecular weight and polymer mass. The mole number of acidic groups can be determined, and by dividing it by the volume, the initial molar concentration of acid can be calculated.
At the cylinder center line, due to symmetry, there are no concentration changes. At the cylinder surface, the acid concentration is equal to zero as the acidic monomers have good water solubility; as soon as exiting the control volume, they dissolve in the medium. This meets the perfect sink condition. Hence, this boundary condition was used.
At first, the drug—with a certain concentration—is distributed uniformly in the polymeric matrix. At the cylinder center line, it is theoretically presumed there are no concentration changes due to symmetry. At the cylinder surface, due to the sink condition, the drug concentration is equal to zero.
Results and discussion
As mentioned previously, data related to the drug and matrix are summarized in Table 2. Two independent series of data for model validation are used. The only way for testing and verifying the model and assumptions accuracy is to compare the results with different and various experimental data. In general, in the figures below, the plotted points represent experimental data and the lines represent the predictions of the model.
In Figs. 9 and 10, model predictions of number average molecular weight versus time, based on the data from the literature, are tested. The data of the systems that are used are as follows:
Aspirin releases from poly(lactic acidcoglycolic acid) with a molar ratio of 1:1, with aspirin having molecular weight of 180.16 grams per mole and the polymer matrix having a molecular weight of 63,000 grams per mole. 100 mg aspirin and 400 mg of PLGA are mixed in solvent casti, ultimately leading to a matrix with 20% by weight Aspirin (data provided in Fig. 9) [16].
As seen, the predicted trend for dimensionless aspirin release (Fig. 11) and the experimental accumulative aspirin release (Fig. 13) is the same, except that the first one is decreasing and the other one is increasing. In other words, as the amount of aspirin in the PLGA is decreased, its total released amount in the medium increases with the same velocity. The same is true about BSA.
Conclusion
Based on several simplifying assumptions, the mass conservation equations for all system components were derived and solved, leading to a predicting model that can predict polymeric matrix degradation and drug release rate of the cylindrical needle. To calculate partial and ordinary differential equations simultaneously, the finite difference and Runge–Kutta methods were employed, respectively. By correlating the polymer degradation rate with the molecular weight, molecular weight changes versus time were illustrated. Also, the drug diffusivity was related to the matrix molecular weight. Molecular weight reduction and accumulative drug release within the system were determined. In order to validate and assess the proposed model, a set of data of hydrolytic degradation of aspirin (180.16 Dalton) and albumin (66,000 Dalton)loaded PLGA (1:1 molar ratio) were used for comparison.
The proposed model is in good agreement with experimental data from the literature. Considering diffusion as the main phenomena and autocatalytic effects in the reaction, the drug release profile is predicted. As a result, for a microneedle containing drug, we will be able to estimate the drug release rate before fabrication. The prediction procedure of the proposed model shows a qualitatively suitable adjustment to experimental data. The main reason for the differences between model and experimental results can be referred to the estimation of some needed parameters of the model, because finding the parameters in exactly the same utilized condition is difficult. Under ideal conditions, it is better to measure the needed parameters for the exact same experiment conditions used in the theory. Also, the simplifying assumptions utilized (e.g., onedimensional diffusion in the cylinder) can add to the error amount.
Notes
Compliance with ethical standards
Conflict of interest
Authors declare no conflict of interest.
Ethical approval
This paper does not contain any studies with human or animal subjects.
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