Monte Carlo simulation methods-based models for analyzing the kinetics of drug delivery from controlled release systems

Abstract

Pharmaceutical controlled-release formulations are systems developed by a set of unit operations to achieve a satisfactory combination between a drug and excipients to allow its gradual release. These devices must simultaneously meet criteria for stability, biocompatibility, safety, efficacy, scalability at industrial volumes, and technological efficiency for drug release. Controlled-release systems (CRSs) must release drugs in a way that maintains an adequate concentration in the organism, a requirement that is challenging to meet in practice. Even though novel CRSs may be designed with new materials as excipients, new drugs, or emerging manufacturing technologies, the mechanisms for drug release continue to be governed by a set of similar physicochemical phenomena such as diffusion, swelling, or erosion. These phenomena are too complex to be analyzed by numerical methods; however, they are relatively accessible by probabilistic models especially the Monte Carlo simulation. In this review, we discuss key findings related to the use of this probabilistic method for analyzing the drug-controlled release process in different pharmaceutical devices. Based on this evidence, we propose their potential application in the characterization of new drug-controlled release systems, synergy with other computational methods, and their capability to be adapted for in vivo or in vitro kinetic analysis.

Keywords:
Monte Carlo simulation; Diffusion; Drug-controlled release; Percolation threshold; Nanotechnology; Simulation; Swelling

INTRODUCTION

Pharmaceutical dosage forms are the result of a combination of excipients and drug(s) that are technologically efficient in facilitating administration in humans and animals. In the development stage, an optimal formulation is designed using trial-and-error methods; its final composition is limited by the number of tests or the expertise of the formulation specialist. These empirical formulations can be time-consuming and expensive and may have an impact on Critical Quality Attributes (CQAs) such as potency, release kinetics, process-related impurities, etc. To minimize this impact, before scaling up pharmaceutical manufacturing for a selected formulation, several assessment strategies can be used to estimate and reduce the risk for a specific formulation. For example, a Risk Assessment Analysis based on the ICH scientific guideline Q9 on Quality Risk Management could be employed to determine the failure risk concerning the critical attributes of quality in a marketed product (Coleman, 2023). Another useful approach is development based on Quality by Design (QbD), where development is initiated with predefined objectives for the relationship of the product quality to the desired clinical performance. Essentially, it emphasizes understanding of the product and process based on scientific principles, and quality risk management to decrease the failure risk during the design of the product (Davanço, Campos, de Oliveira Carvalho, 2020; Pramod et al., 2016) (Figure 1).

FIGURE 1
Overview of the QbDs development cycle. Quality Target Product Profile (QTPP), Critical Quality Attributes (CQAs), Critical Process Parameters (CPPs), Critical Material Attributes (CMAs).

To implement this strategy, the first step is to establish a Quality Target Product Profile (QTPP) which shows the requirements of the patients and to translate it into design features in the pharmaceutical formulation. For example, a formulation may require a specific drug release profile, long-term stability, or a particular pharmaceutical dosage form, etc. In this step, between the theoretical support and pre-formulation stages, in-silico methods are useful to propose pre-formulation conditions and to simultaneously reduce the design space and the number of tests required achieve an optimal formulation (Yu et al., 2014). Other tools such as Design of Experiments (Politis et al., 2017), in vitro-in vivo correlation (Huang, Goolcharran, Ghosh, 2011; Kaur et al., 2015; Peraman, Bhadraya, Padmanabha Reddy, 2015; Yekpe et al., 2018), 3D-printing technology (Tracy et al., 2023), photolithography and computer-aided design (Andreadis et al., 2022; Geraili, Xing, Mequanint, 2021), the Biopharmaceutical Classification System (Davit et al., 2016) and modeling by simulation constitute novel strategies for generating an optimal CRS. Moreover, these trends have been integrated as part of regulatory affairs dossiers used to support the reproducibility of therapeutic benefits. In this context, a new line of research is emerging that utilizes computational simulation to anticipate the design of new formulations, with the aim of maximizing compliance with specifications and avoiding negative impacts on CQAs. For example, simulation models have opened possibilities to explore new relationships in silico-in vivo, which is highly valuable for the prediction of an in-vivo drug delivery profile; to establish dissolution specifications for finished products; or to support long-term stability studies (Al-Tabakha, Alomar, 2020; Casalini, 2021; Kambayashi, 2023). To achieve all of these objectives, new challenges in computational simulation have emerged, especially in the interpretation and transference of physicochemical properties, biological properties, or descriptions of hydrodynamic phenomena to computational algorithms (Adepu, Ramakrishna, 2021). In this review, we identify some key findings generated by probabilistic simulation models used to analyze drug controlled release from different systems. Based on these reports, we propose their potential application in the characterization of new drug controlled release systems, their synergy with other computational methods, and their capability to be adapted for in vivo or in vitro kinetic analysis.

Controlled Release Systems

Controlled release systems (CRSs) are formulations designed to release active pharmaceutical ingredients of different molecular sizes, from small molecules to peptides or nucleic acids, at a constant rate over long periods of time. CRSs also allow dose frequency to be reduced, increase adherence to treatments and reduce adverse effects (Park, 2014). These formulations continue to be valid and are highly desirable for new developments; in fact, a large number of formulation proposals, including diffusion-controlled CRSs, are in advanced clinical phases. Several CRSs have been developed for specific treatments, including asthma, arthritis, Alzheimer’s disease, duodenal ulcer, cancer, cardiovascular diseases, diabetes, systemic arterial hypertension, and hypercholesterolemia, among others (Gao et al., 2023). Some of these CRSs show drug release patterns described by zero-or first-order kinetics, or a combination of the two, such as burst or sustained release (Adepu, Ramakrishna, 2021).

Diffusion as key mechanism involved in the drug-controlled release

Diffusion is the main mass transfer phenomenon associated with diffusion-controlled CRSs but its description in physical terms and mathematical expressions is complex. Diffusion phenomena can be analyzed by Fick’s law and expressed by the equation (1):

$J=-D\frac{\partial c}{\partial x}$ (1)

Where J is the rate of transfer per sectional unit area (cm ² / s); ∂c is the change in concentration of the diffusing species (g/cm ³ ); D denotes the diffusion coefficient (also called diffusivity, (cm ² /s) and ∂c is the change in distance (cm). Fick’s second law of diffusion can be derived from Fick’s first law considering mass balance. For example, for three dimensions and different diffusion coefficients that vary with position, time, and/or solute concentration, the relationship generalizes to the following equation (2):

$\frac{\partial c}{\partial t}=\frac{\partial }{\partial x}\left(D\frac{\partial c}{\partial x}\right)+\frac{\partial }{\partial y}\left(D\frac{\partial c}{\partial y}\right)+\frac{\partial }{\partial z}\left(D\frac{\partial c}{\partial z}\right)$ (2)

Where c is the concentration of the diffusing species (g/ cm ³ ), D is the diffusion coefficient (cm ² s ^-1 ) and x,y, z are the three spatial coordinates. The resolution of Fick’s equations should be adapted for each case, considering the geometry of the device, the size of the system, dimensionality, etc., and implying the establishment of specific conditions denoted as “initial and boundary conditions”. For example, for a cylinder, which could be described as the classical geometry for a pharmaceutical device, Fick’s second law of diffusion can be solved considering these initial and boundary conditions, leading to the following expression for the drug released at time t,

$\frac{M_t}{M_{\infty }}=1-\frac{32}{{\pi }^2}\sum _{n=1}^{\infty }\frac{1}{q_n^2}\exp \left(-\frac{q_n^2}{R^2}Dt\right)·\sum _{p=0}^{\infty }\frac{1}{{\left(2p+1\right)}^2}·\exp \left(-\frac{{\left(2p+1\right)}^2{\pi }^2}{H^2}Dt\right)$ (3)

where R and H denote the radius and height of the cylinder, n and p are summation indices, and q _n is the n-th root of the zero-order Bessel function of the first kind [J ₀ (q _n )=0]. On the one hand, the “initial conditions” concern the initial movement of particles inside a system with a particular solution when c(x,t)=0. In fact, equation (1) makes sense under initial and boundary conditions, limiting their solving to analytical solutions whose interpretation is only limited to the initial and boundary conditions. On the other hand, “boundary conditions” in non-homogeneous systems refer to constraints imposed by the system, such as critical changes of the physicochemical properties in the excipients or drugs as a transition phase or relaxation of polymers whose mathematical interpretation can be described by discontinuities that are difficult to analyze (Siepmann et al., 2012). One of the restrictions of these determinist models is that they describe the movement of drug particles along stationary boundaries, and thus do not explain completely the kinetic behavior in dynamic conditions such as swelling and erosion, among others.

One of the alternatives that make statistical physics and probabilistic theory available for the comprehensive study of the movement of drug particles during dynamic and simultaneous events is modeling using stochastic methods such as the Monte Carlo simulation and cellular automata. We conclude this section by showing that a deterministic solution for the diffusion phenomenon is not practical for analysis of the kinetics of drug release from CRSs, and therefore that alternative approaches could facilitate or aid their analysis.

What is the Monte Carlo simulation?

Monte Carlo simulation comprises a broad class of computational procedures that use randomly generated numbers to describe phenomena of interest and provide a statistical approximation when the exact values are unavailable or the solution of the problem is complex (Figure 2).

FIGURE 2
(Left side) Scheme of diffusion-controlled drug delivery systems with the drug homogeneously dispersed in a polymeric, swellable, and hydrophilic excipient. As the swelling process progresses, the gel layer gradually becomes thicker, resulting in progressively slower drug release rates. However, due to continued hydration, the polymer disentangles from the matrix surface, resulting in a gradual dissolution decreasing the thickness of the gel layer and increased dissolution rate (Right side). The figure shows an example of a random walk in a 2D network passing through a boundary. Once it crosses the boundary, the particle is counted as a released particle. The sites visited by the walker are marked in red. The walker can be water particles (blue) or hydrated drug particles (red).

The Monte Carlo method is founded on the principles of statistical physics, a relatively unexplored field among formulation scientists; consequently, the applications of these concepts are sometimes limited. In statistical mechanics, through the partition function Z, we can determinate all the thermodynamic properties of a system with many degrees of freedom (Landau, Binder, 2021; Reichl, 2016). In particular, one application of the Monte Carlo simulation has allowed simulation of the motion of a set of particles using the random walker concept (Du et al., 2022; Weiss, 1983; Wu et al., 2022). The classical problem of the random walker, which is one of the most studied problems in the statistical mechanics of nonequilibrium systems, postulates that the walker will take an average of a certain number of steps in one direction or another, with each step being statistically independent of the others (Figure 3).

FIGURE 3
A. A Monte Carlo step: the figure show that each attempt of displacing, whether accepted or not, the time increases by a value equal to 1/Nt, where Nt is the number of drug particles remaining inside the matrix at time t and when Nt particles have been chosen, it is considered an arbitrary time unit called a Monte Carlo Step. Occasionally, when the random walker encounters barriers or obstacles on the path, new rules or conditions are implemented, for example, when the random walker is close to an obstacle and can only take one step to the right, one step to the left, or one step backwards but not to the forward. Physical barriers can be simulated considering the properties of the materials, such as the geometry of the system, the presence/absence of coatings, porosity, or inclusive environmental conditions. B. Different pharmaceutical systems can be simulated using the random walker model.

The random walker randomly chooses one of the different options to carry out a movement to a neighboring site; if the requirements are satisfied, the walker migrates to this new position, but if the rules for movement are not satisfied, the walker remains static in the same place. With each attempted displacement, whether accepted or not, the time increases by a value equal to 1/Nt, where Nt is the number of drug particles remaining inside the matrix. The time t when Nt particles have been assessed is considered an arbitrary time unit called a Monte Carlo Step (MCS) (Figure 3) (Gomes Filho, Oliveira, Barbosa, 2016).

$X_{n+1}=X_n+{\in }_n$ (4)

Depending on the trajectory generated by the random walker, it is possible to determine the complete route and the final location of each particle, for instance, when a particle is either inside or outside of a pharmaceutical device (Feller, 1991; Hughes, 1995; Weiss, 1983 ). Formally, a random walker is a Markov chain with independent additive increments (Fishman, 2013). In a series of random variables X _n , a random walk satisfies: where ϵ ₁ , ϵ ₂ ,.. are a sequence of identically distributed random variables and are generated independently of X _n (Kroese, Taimre, Botev, 2013; Robert, Casella, Casella 1999). However, while the random walker model maintains a microscopic approach, the results obtained can also be seen and are measurable at the macroscopic level. For example, the movement of a molecule that diffuses in a porous solid can be explained at the macroscopic level as the amount of drug released. The particles can be used to refer to molecules, nanoparticles, granules, or aerosols, etc., and are unitary entities moving freely in a specific direction for a distance roughly equal to their mean free path. To ensure the success of the implementation of a Monte Carlo simulation algorithm, several factors must be considered, especially the use of an appropriate sampling technique, the incorporation of a pseudo-random number generator, and the generation of an ensemble of configurations with statistical value, all of which must be able to describe the phenomenon of interest (Sawilowsky, 2003). Monte Carlo simulations calculate the number and position of particles as a function of time. If the position of the particle is outside of the device and is labeled as “released” it is added to an accumulative counter. Analogously, another types of particle that moves also can be tracked during all simulations, for example, excipients. Finally, these simulation data can be correlate with the drug amount, volume, erosion percentage, gel particles generated during a phase transition, number of internal particles, hydrated particles, water amount, etc. Afterwards, it is possible to build different correlations between each simulation data and time to generate empirical or semi-empirical models that describe the kinetic patterns (Bruschi, 2015). For example, using Monte Carlo Simulation, it is possible to compare hypothetical conditions and predict their impact on critical quality attributes, such as the drug/excipient ratio needed for appropriate gel formation, which is critical for a drug-controlled release; or the calculation of the percolation threshold required to allow the total release of the drug load from a pharmaceutical device. In conclusion, to develop a computational model, it is necessary to know the drug-release mechanisms of the pharmaceutical dosage form to translate them adequately to a probabilistic simulation algorithm.

Requirements to develop a simulation model using the Monte Carlo method

To simulate a drug delivery process, it is necessary to establish the hydrodynamics implicit in the release of drug particles into the release medium or biological environment. These models should be in agreement with their classification, especially with the limiting steps that condition the drug dissolution process. (Fu, Kao, 2010; Langer, 1990).

According to the classical works by Peppas (Peppas, Franson, 1983) and Robert Langer (1990), different pharmaceutical devices can be divided into classes depending on the set of mechanisms for drug release, including diffusion-controlled, chemically controlled, solvent-activated and magnetically controlled delivery systems. Particularly, the Monte Carlo simulation method can be easily adapted to diffusion-controlled systems including novel pharmaceutical devices of nanometric size (Figure 4).

FIGURE 4
Diffusion-based controlled release systems. Some representative nanosystems available to be modeled include liposomes, dendrimers, hydrogels, nanogels, carbon nanotubes, and gold nanoparticles.

However, while this classification continues to be valid, with the emergence of new materials, new chemical classes of drugs, and new manufacturing processes, it is necessary to include additional criteria for a robust classification (Han et al., 2010; Langer, 1990; Mandal et al., 2010; Peppas, Franson, 1983). For example, in innovative pharmaceutical formulations whose release is based on erosion, pulsative systems, rupturable membranes, or hydrolysis-mediated systems, the classical diffusioncontrolled mechanism does not completely explain the kinetics of drug release (Beugeling et al., 2018; Mandal et al., 2010). In this frame, the first challenge for drug release modeling from CRSs should be to understand all, or at least the main mechanisms for drug release implicit in each class of CRS that are not yet fully characterized (Mandal et al., 2010). For example, experiments carried out in polymer-based nanosystems have shown changes in the diffusion coefficients due to modifications in the polymer biodegradation process that occurs in parallel to the drug release indicating as additional requirement the chemical reactivity or stability of the excipients (Macha et al., 2019). Interestingly, other mechanisms, such as the staggered profile, which is regulated by the biochronology of each patient, to allow regulating the discharge of drugs in a pulsatile profile could be analyzed by an in silico approach to explain the effect of the pulsatile discharge of drugs under different biological conditions (Mandal et al., 2010).

Monte Carlo simulations in fractal and Euclidean systems

In physical formalism, the drug-controlled release process could be described as the escape of particles from a system of fractal geometry, i.e. a system characterized by irregular spaces. In contrast to a Euclidean space in which all the systems are homogeneous and conserve a single value for the diffusivity of their particles, a fractal system is a disordered medium in which the diffusivity is dynamic. This problem was initially raised by Bunde et al., in 1985, who concluded that there is an empirical relationship between release rate and time. This relationship is known as the Power law or KorsmeyerPeppas Equation:

$\frac{M_T}{M_{\infty }}=k{t}^n$ (5)

where, M_t and M_∞ are the amounts of drug released at times t and infinite, respectively. k is a parameter determined under in vitro or in vivo conditions (O’Farrell et al., 2022) and n is an exponent that depends on the geometry of the boundary of the system, which can have an interpretative sense for the drug release mechanisms (Kosmidis, Argyrakis, Macheras, 2003a). Following Bunde’s work, several reports have developed this theory and improved the study of Fickian diffusion in fractal and Euclidian spaces (Kosmidis, Argyrakis, Macheras, 2003a, b ). In both cases, it was also found that the Weibull function appropriately describes the complete drug release curve when the drug release mechanism is Fickian diffusion. The Weibull model is represented by Equation 6:

$\frac{M_T}{M_{\infty }}=1-e^{-\alpha t^{\beta }}$ (6)

where is a scale parameter, and β is the curve shape factor. In the case of release from Euclidian matrices (Kosmidis, Argyrakis, Macheras, 2003a) the value of the exponent b was found to be in the range 0.69-0.75. Experimental approaches have used this mathematical relationship to understand when a system obeys Fick’s first law (Corsaro et al., 2021; Kobryń et al., 2017; Martín-Camacho et al., 2023; Ranjan, Jha, 2021; Yu et al., 2014). Kosmidis and colleagues (2003b) were the pioneers of the introduction of Monte Carlo kinetic simulations using the concept of a random walker to calculate the diffusion coefficient in inert tortuous systems (Kosmidis, Dassios, 2019; Kosmidis, Macheras, 2007; Papadopoulou et al., 2006). In this first publication, two-dimensional networks (d = 2) were analyzed to describe the release behavior from a Euclidean and a fractal matrix and the percolating aggregate at the percolation threshold in square lattices (Kosmidis, Argyrakis, Macheras, 2003b). In the case of release from the two-dimensional percolation fractal with fractal dimension d _f = 91/48 the values of b ranged from 0.35 to 0.39 (Kosmidis, Argyrakis, Macheras, 2003b). In the same way, when they studied the fractal matrix, they found that the release profile of the drug is adjusted to the Weibull equation.

On the other hand, in Euclidean matrices, the validity of the Higuchi equation for a one-dimensional matrix system has been demonstrated by Monte Carlo simulation. Moreover, Monte Carlo simulations have been applied to analyze the effect on a controlled release profile when a drug is arranged in random mixtures with areas of high and low diffusivity. This evokes matrices with a coating-like structure that are identified as following a Weibull distribution, suggesting that thin-layer enteric-coated systems could modify the release profile resulting in diffusion coefficients three orders of magnitude lower than the drug release at a constant rate and without coating-like properties (Kosmidis, Macheras, 2007). These Monte Carlo simulation results apparently point to a generalization between the Weibull model and the system geometry (Kosmidis, Argyrakis, Macheras, 2003a). Villalobos and colleagues have studied inert spherical matrices using the Monte Carlo methods and the implication of the percolation phenomenon exerted by the basic components of this type of matrix (drug and excipient) on drug release and showing that the fractal dimension value is useful to estimate the complexity of porous microstructures and to define when a specific medium is homogenous or heterogeneous (Villalobos et al., 2017). . Finally, these contributions conclude that fractal and Euclidean matrices can be used as models for drug controlled released and they can serve as a reference point for the creation of more complex models.

Monte Carlo simulations in new drugs, excipients, and dissolution mediums

The mechanisms of drug-controlled release are intimately associated with the properties of the drugs used, such as their hydrophobicity, electric charge, molecular size, isoelectric point, etc. Particularly, in biotechnological products such as proteins, DNA, mRNA, and siRNA these properties demand consideration (Jiang, Abedi, Shi, 2021; Park, Otte, Park, 2022; Poornima et al., 2022) including their retention properties on substrates or polymeric materials (De Izarra et al., 2021; Senapati et al., 2021). Eventually, a diffusion/erosion mechanism for the drug or polymers is required to release these particles at a constant rate. For example, in substrate-mediated delivery the DNA is immobilized to a material that functions to support cell adhesion and migration and places DNA directly in the cellular microenvironment. In this type of system, molecular interactions play a key role because charge could bind other components to the delivery vehicle. Furthermore, delivery from most polymeric devices likely occurs through a combination between binding and release mechanisms, and both the vector and the polymer can be adapted to regulate these interactions (Pannier, Shea, 2004). From this perspective, a significant number of properties of macromolecules, polymeric materials and their interactions with their biological targets can be modeled. Additionally, the release profile of a drug may be affected by various proteins, cell types, and in vivo enzymes; thereby, they also become an attractive system for probabilistic modeling (Seo, Mittal, 2021).

In vitro experiments use biorelevant dissolution media to replicate the impact of biological fluids on drug-controlled release in various pharmaceutical devices. Some of them have been characterized according to their physicochemical properties, representing a potential niche for developing simulation models and assessing the effect of the dissolution medium on the drug-controlled release kinetics. Some examples of biorelevant dissolution media are FaSSIF (fasted-state simulated intestinal fluid), FeSSIF (fed-state simulated intestinal fluid), FaSSGF (fasted-state simulated gastric fluid) and FeSSGF (fed-state simulated gastric fluid), etc.(Lemos, Prado, Rocha, 2022). It is likely that phenomena such as hydrolytic activity, pH, temperature, viscosity, and enzymatic activity are excellent physicochemical parameters accessible for an assessment by simulation approaches (Fernandez-Lopez et al., 2023; Gao et al., 2019; Liang et al., 2022; Martín-Camacho et al., 2023; Russo et al., 2022; Vernon-Carter et al., 2022). Furthermore, another line of applications for Monte Carlo simulations could be related to create in vitro conditions to establish the experimental conditions for a dissolution test. For example, simulations based on the Finite Element Method through the software COMSOL Multiphysics® have shown the impact of the perfect sink assumption and assess the effect depending on stirring speed, drug loading, and erosion and swelling properties (Ranjan, Jha, 2021).

With this approximation, it is possible to identify the initial parameters for the development of dissolution analytical methods and predict dissolution profiles and to study how these are affected by the stirring speed and volume of dissolution. However, this deterministic simulation method is distant from stochastic procedures, it could be used as a reference to validate theoretical data generated by Monte Carlo simulation.

We conclude this section by demonstrating the potential of probabilistic simulation methods to analyze the effect of biorelevant dissolution media on the drug-controlled release and in the construction of dissolution analytical methods and to evaluate new biological drugs or excipients using specific physicochemical parameters which directly affects drug-controlled release kinetics.

Monte Carlo simulations in polymeric and inert systems

CRSs designed with polymeric matrices have gained popularity for establishing simulation models because they allow a study systematic for drug release (Colombo et al., 1995). The physicochemical events that take place during the drug release initiate when a swellable polymeric CRS contacts water, forming a hydration front. As the hydration front advances, the particles begin to move (Vaitukaitis et al., 2020). Simultaneously, water interacts with the polymer promoting swelling. However, this class of polymers shows different swelling patterns; most of them are similar in layer formation, but their thickness or formation dynamic is variable. When a tablet undergoes swelling and water permeates into its core, various fronts are created. Several reports have identified three distinct zones: the diffusion front, the erosion front and the swelling front which have been characterized by highly sensitive equipment such as IR, NIR, MNR, etc. (Avalle et al., 2013; Yassin et al., 2015). The increased resolution of such analytical equipment presents new benefits in the field of pharmaceutical technology since systems with heightened sensitivity improve the quality of process description, facilitating a more precise comparison among various pharmaceutical formulations that exhibit subtle variations. Simultaneously, these systems enhance the potential for generating data that can be effectively utilized to develop computational models (Manaia et al., 2017). A recent study demonstrated that a gel formulation can produce a novel irregular region characterized by micropores and microbubbles. This region, with a thickness of 300 µm, is likely formed by internal structural rearrangements within the gel, which are induced by the dissolution of acid compounds and modifications in the pH microenvironment. This is an anomalous region which is very attractive for simulation by probabilistic methods (Juszczyk et al., 2021). In the same report, the presence of intricate spatiotemporal hydration patterns generated by swellable polymer-based matrices such as sodium alginate is highlighted. In this experiment, both the composition and the molecular properties of the constituents play a significant role in the observed behavior (Juszczyk et al., 2021). However, the diffusion mechanism is seen as the main phenomenon that orchestrates the movement of dissolved drug particles through the polymer network. There is also evidence that other zones could be formed during the swelling of tablets, suggesting new control mechanisms for drug release. This example illustrates a distinctive hydrodynamic phenomenon that emerges as a result of the implementation of new polymeric materials. To date, we know that the mechanisms that affect the release of a drug from a polymeric matrix involve diffusion, water penetration, hydration, solvation, gel formation, erosion, swelling front formation, and the formation of new zones. Furthermore, some studies have shown that also the design properties of CRSs, such as their shape and size, have an evident impact on the drug release kinetics (Kulinowski et al., 2014, 2016). Sanjeev Garg and colleagues employed predictive quantitative structure-property relationship (QSPR) models to identify an optimal diffusion-controlled drug delivery system using Paclitaxel as a model drug. Garg’s team also concluded that the type of polymer, the diffusivities, the initial loading, and the number of nanoparticles, as well as the shape and size of the system play a crucial role in regulating the effectiveness of the drug for different dosage regimens (Pramanik, Garg, 2019).

Finally, the calculation of the percolation threshold value in inert systems is another intriguing use of the Monte Carlo simulation in pharmaceutical development. This property indicates the concentration at which the polymer or drug can form an infinite cluster and percolative. Above the excipient percolation threshold value, a continuous cluster of this component is produced and is capable of regulating the hydration and release rate (Caraballo, 2009). Below this threshold value, the excipient does not regulate the release of the drug by directly affecting its release. This critical value can be predictable and analyzed by Monte Carlo Simulation (Adamczyk, Polanowski, Sikorski, 2009; Pawłowska, Sikorski, 2013; Polanowski, Sikorski, 2017) and is an indicator of the dose amount trapped at infinite time considering the initial composition of the matrix (Martínez et al., 2009; Villalobos et al., 2017). Examples concerning percolation threshold values are shown in Table I. Nonetheless, in polymeric drug controlled release devices this concentration should be assessed with caution. For example, in swellable CRSs, the drug-release is regulated by a polymer network that forms a gel layer, which alters the release kinetics. Consequently, the actual percolation threshold values can differ from the values observed in an inert and porous system. Recent experiments using fluorescent microscopy have demonstrated the feasibility of observing the gel layer formation during the initial stage of polymer hydration and revealing a potential connection between inert and swellable systems through of the concept of percolation threshold (Mason et al., 2015).

TABLE I
Some Percolation Threshold values determined for some excipients and drugs in different pharmaceuticals devices.

Number	Objective	Main findings	Reference
1	Study the role played for mannitol (excipient) in the dissolution of drug tablets and their impact in the percolation threshold value.	The particle size is the fundamental determinant of the percolation threshold. Specific surface area is a critical property of mannitol influencing the dissolution behavior of binary mixture tablets containing poorly water-soluble drugs.	Zhang et al., 2024
2	Generation to evidence to support the use of PBS-CL as pharmaceutical matrix using hot processing techniques.	An excipient percolation threshold value around 23% v/v in agreement with a continuum percolation model.	Galdón et al.,2021
3	Description of the design of a matrix tablet composed of polyethylene oxides (PEOs) with relative molecular masses.	Percolation thresholds values were determined for all the selected PEO formulations (18, 16, and 12 %, w/w)	Draksler et al., 2021
4	Effect of the drug solubility on the critical concentration threshold.	Percolation threshold values for matrices of okra gum (Hibiscus esculentus) biopolymer containing flurbiprofen (poorly soluble drug), theophylline (slightly soluble) and metformin (soluble) were 20.27-20.84% (v/v), 23.19-23.65% (v/v) and 25.24-26.14% (v/v), respectively. Percolation threshold of the biopolymer increase with the solubility of the drug.	Khizer et al., 2020
5	Determination whether a percolation threshold value, previously determined for ibuprofen/ microcrystalline cellulose (MCC) blends using percolation theory and compression data	The application of percolation threshold theory to predict compaction behavior of pharmaceutical powder blends.	Queiroz et al., 2019
6	Identification of the percolation threshold of a poorly water-soluble drug.	A percolation threshold value for mefenamic acid is approximately 20% v/v	Wenzel et al., 2017
7	Development and characterization of a highly porous floating tablet containing cilostazol	Estimation of percolation threshold value or a volume concentration of HPMC at which it can form a continuous cluster exists around 16.33% v/v HPMC	Hwang et al., 2017
8	Identification of a new way for monitoring dissolution data using confocal laser scanning fluorescence microscopy (CLSM).	The HPMC percolation threshold value estimated this way was found to be 12.8% v/v, which was equivalent to a matrix polymer content of 11.5% w/w.	Mason et al., 2015
9	Characterization of a new linear functional biodegradable polythiourethane-d,l-1,4- dithiothreitol-hexamethylene diisocyanate [PTU(DTT-HMDI)] using theophylline anhydrous as model drug.	The excipient percolation threshold values have been estimated between 20% and 30% w/w of polymer.	Campiñez et al., 2015
10	Identification and assessment of the polymer properties that affect the direct compression and the release of drug since water-insoluble matrices.	A similar site percolation threshold value of 65% v/v was found for Kollidon® SR, Eudragit® RS and ethyl cellulose in dry state and particularly a critical concentration less than 55% v/v polymer was reported for Kollidon® SR.	Grund et al., 2014
11	Development of a potential formulation based on thermoplastic polymer to reduce the number administrations per day.	Carbopol 974P and 971P have been selected as formulation polymer. The polymer percolation threshold value has been exceeded with 15% w/w of polymer. Therefore, sustained release tablets have been developed with only 15% of excipient. This implies that matrix tablets containing 750 mg of drug, intended for two administrations a day, can be obtained with a similar weight to those existing in the market containing 500 mg of drug for three administrations a day.	Aguilar-De-Leyva et al., 2014
12	Generation a new in silico simulation program that describes the mechanical structure of binary compacts formed from an excipient with excellent compatibility and a drug with null compatibility.	The excipient percolation threshold for the simulated system was 0.3395, indicating that over this excipient fraction, a compact with defined mechanical properties will be formed.	Martínez Lizbeth et al., 2013
13	Matrix tablets of a model drug acetaminophen (APA) using a highly compressible low glass transition temperature polymer silicone pressure sensitive adhesive (PSA) at various binary mixtures of silicone PSA/APAP ratios	The critical points attributed to both silicone PSA and EC tablet percolation thresholds values were found to be between 2.5% and 5% w/w.	Toila and Kevin Li 2012
14	Estimation of percolation threshold in a multi-component system constituted by HPMC K4M/lactose as hydrophilic matrix, phenobarbital as drug and magnesium stearate as lubricant.	HPMC/lactose and HPMC/EC/lactose matrices, from the point of view of the percolation theory, the optimum concentration for HPMC, to obtain a hydrophilic matrix system for the controlled release of phenobarbital is higher than 18.1% (v/v) HPMC	Maghsoodi and Barghi 2011
15	Characterization of pharmaceutical systems constituted by HPMC containing carbamazepine and verapamil HCl.	The critical points for both formulations are between 10% and 20% w/w of HPMC K100M CR (10.0% and 19.5% v/v HPMC K100M CR for carbamazepine tablets and 9.9% and 19.7% v/v for verapamil HCl tablets).	Goncalves-Araujo et al., 2010
16	Identification of critical concentration of a ternary system containing starch-based disintegrant using the principles of the percolation theory and using caffeine and mefenamic acid as model drugs.	The percolation threshold value can be described by the volumetric ratio of the disintegrant to the drug substance being equal to 0.2 (v/v) in in which both components have similar average particle sizes	Kimura et al., 2007
17	Estimation the percolation threshold of HPMC K4M in matrices of acyclovir and to apply the obtained result to the design of hydrophilic matrices for the controlled delivery of this drug.	Acyclovir as drug and HPMC K4M as matrix. This percolation threshold value was situated between 20.76% and 26.41% v/v of excipient plus initial porosity.	Fuertes et al., 2006
18	Estimation the excipient percolation threshold in HPMC K4M hydrophilic matrices containing acetaminophen, theophylline and ranitidine HCl.	Excipient percolation thresholds value were situated between 24.8-25.8, 14.7-18.4 and around 31.2% (v/v) HPMC in theophylline, ranitidine and acetaminophen matrices, respectively.	Fuertes et al., 2010
19	Estimation of the percolation thresholds in a system of Eudragit ® RS-PM and naltrexone hydrochloride.	The method of Leuenberger and Bonny gives 31.11+/-7.95% v/v as the critical porosity, which corresponds to a percolation range from 12 to 20% (w/w) of drug content. The excipient percolation threshold value is estimated between 25.4 and 31.1% (v/v) based on the release profiles and the analysis of the release kinetics.	Caraballo et al., 1999

Monte Carlo simulations in hydrogels and nanogels

Hydrogels are novel pharmaceutical devices that have exceptional swellable and water-uptake properties because they are composed of a polymeric network with excellent mechanical strength (Hu et al., 2022; Romischke et al., 2022). Other properties, such as biodegradability, sensitivity to stimulus, and the ability to absorb fluids, have generated a new field for the study of these new formulations. Some properties involved in the release of drugs, such as their permeability, diffusion, and erosion characteristics make hydrogels excellent systems for controlled drug delivery and for their modeling by probabilistic simulation (Liao, Hou, Huang, 2022). For example, the diffusion of drugs in hydrogels is generally modeled by one of the following three theoretical approaches: (a) hydrodynamic theory, which considers interactions between the drug and the surrounding hydrogel matrix; (b) free volume theory, which assumes that the solute is transported via dynamic empty spaces between molecules without effects caused by interaction; and (c) obstruction theory, which models the polymer net as a barrier for the diffusion of the drug within the liquid (Turnbull, Cohen 1970; Cukier, 1984; Mackie, Meares, 1955). Recently, an analysis of diffusion in hydrogels by positron annihilation lifetime spectroscopy (PALS), a technique capable of measuring the molecular pores in biomaterials under wet conditions, showed the relevance of interactions between the hydrogel and the drug (Axpe et al., 2019). In addition to the study of diffusive phenomena, hydrogels have also been used as excellent models to analyze the hydrodynamics of gel expansion or water-uptake properties (Benkő et al., 2022; Jagusiak et al., 2020; Kashkool, Soltani, Souri, 2020; Kocaaga, Guner, Kurkcuoglu, 2022; Pérez-Mas et al., 2018). For example, Perez-Mas and colleagues (2018) analyzed molecular properties in neutral hydrogels by means of coarse-grained grand canonical Monte Carlo Simulations. In this work, it was possible to establish theoretical conditions in which the materials could maximize the adsorption of solute. Recent studies showed that the size of cosolutes, together with the magnitude of hydrophobic attractions and of steric repulsion play a key role in improving the absorption of cosolutes in neutral hydrogels (Pérez-Mas et al., 2018). However, hydrogels continue to be studied, and several theoretical models can be used to understand their mechanical movements at the nanometric level (Ahualli et al., 2017; Quesada-Pérez, Martín-Molina, 2013). Recently, the same research group reported the effect of different molecular parameters of hydrogels on solute diffusion using coarse-grained simulations, concluding that chain flexibility inside the polymeric network is relevant for the movement of particles (Quesada-Pérez et al., 2022). Additionally, Quesada-Perez and Martin-Molina published a historical overview that showed the progress made in coarse-grain simulations of solute diffusion in hydrogels and nanogels. These results have been crucial for understanding how drug-controlled release is carried out in this type of pharmaceutical device (Quesada-Pérez, Martín-Molina, 2021). Finally, Monte Carlo simulations are establishing a synergy with other tools such as network analysis and artificial intelligence with the aim of enhancing the predictability of outcomes generated by pharmaceutical developments. Unsupervised methodologies possess the ability to discern patterns; nevertheless, their limitation is in the necessity of having enough data to correctly describe each event. It is feasible to employ this probabilistic method to generate data that can be exploited by these approaches (Bannigan et al., 2023; Han et al., 2019; Manolis, Lagaros, 2002; Santana et al., 2020; Sarrut, Krah, 2021).

Another interesting application for Monte Carlo simulations has been the analysis of ideal conditions for transdermal delivery in epidermal membranes of hair follicles, suggesting the successful application of these models in more complex systems (Barry, 2002). For example, currently, DNA immunization can be implemented through topical administration, suggesting follicular transport, so that the hair follicle promises to be a target for gene therapy. This is interesting because several formulations for controlled release could be ideal for alopecia therapy; however, this type of model has not been completely explored by simulation (Fan et al., 1999; Hoffman, 2000; Ryu et al., 2020). Another interesting model that explores the interactions between drugs and human tissue by stochastic procedures is reported by Islam and colleagues. Their report conducted an analysis using a time-adaptive Brownian Dynamics algorithm to investigate the impact of particle size on the dispersion and penetration of drugs within intracellular tissues (Islam, Barua, Barua, 2017).

Monte Carlo simulations in liposomes

Liposomes are spherical vesicles constituted by phospholipids and steroids, usually in the size range that can be used as a model for are analogous to cell plasma membranes (Bozzuto, Molinari, 2015; Sercombe et al., 2015). Here, compatibility with drugs and physical barriers generated by the lipid bilayer are key features when considering liposomes as drug delivery systems. For example, liposomes are effective systems in containing and protecting labile molecules such as RNAs or proteins, but there are still several challenges to improve their stability, loading mechanism, and reproducibility, which limit their use to specific liposome-drug pairs. Moreover, these nanosystems employ a set of physical mechanisms for drug-controlled release, which include Fickian diffusion and erosion of the bilayer (Figure 5).

FIGURE 5
A. Schematic overview of extracellular matrix and its major components. Although the extracellular matrix varies depending on the tissue, the matrix is mainly composed of a variety of proteins such as collagen, elastin, fibronectin, among others and polysaccharides assembled into an organized meshwork. B. Schematization of the extracellular matrix can be considered as a porous system through which particles with different diffusive properties can transit.

Therefore, these pharmaceutical devices present potential subjects for computational modeling, yet remain unexplored to date (Liu, Bravo, Liu, 2021; Wu et al., 2019). Analyzing a liposome as a model system, we observe that drugs always maintain a probability to diffuse by active transport or diffusion through pores.

If the encapsulated drug or the bilayer is altered in any way, for instance by biodegradation, the load of the liposomes can be accelerated, leading to changes in the drug-controlled release kinetics. Due to their analogy with the cell membrane, liposomes facilitate the incorporation of some drugs, enhancing their bioavailability (Wahab, Mögel, Schiller, 2011). Design of drug-controlled release by liposomes involves new materials, including PEGylated materials based on polyethylene glycol (PEG), surface-anchored ligands like antibodies, carbohydrates, peptides and systems that amalgamate with nanomaterials. A recent report predicted the release of drugs from liposomes into the bloodstream, modeling the drug release and the collective motion of the liposome and its surrounding blood cells with elegance (Kaoui, 2018). In the context of a single liposome and red blood cells within a limited area of the blood vessel, the targeted site can be defined as a specific region within which the flow closely approximates a simple shear flow. This results in the vessel walls becoming rigid and therefore resistant to deformation under hydrodynamic pressures (Kaoui, 2018).

Some mechanisms manipulate electrical or ultrasound field-induced membrane permeation to lead to drug release. Using this mechanism, Nily Dan (2015) analyzed a model for liposomal release induced by low-frequency ultrasound by probabilistic simulation. With this assumption, the release is largely dominated by diffusion through the membrane. Using Monte Carlo simulations, Dan concluded that pore properties in the liposomes, such as distribution, pore size, and relative pore surface, are key to an adequate drug-controlled release profile. Here, a complex combination of different diffusive processes of the drug can be represented using liposomes as model system, but these have not yet been explored by simulation.

Monte Carlo simulations in exosomes

Exosomes are extracellular vesicles that are promising pharmaceutical dosage systems for drug-controlled release. These systems, which are conserved across all kingdoms of life and feature in biological processes such as cell-to-cell communication, maintain biophysical properties that can be modeled and considered as novel biopharmaceutical vehicles (Jimenez-Jimenez et al., 2019a). Moreover, exosomes can be used for different biological targets, including different tissues or species such as parasites, bacteria, and plants, so that, in each new environment, different simulation parameters are needed (Koomullil et al., 2021). Recently, Lenzini and colleagues (2020) showed that at the cellular level, exosomes must traverse the extracellular matrix which is composed of nanopores that are highly selective to particle size. A formulation designed by pharmaceutical criteria remains challenging based on several characteristics, such as efficiency of loading methods, reproducibility, pharmacological targets, and long-term stability (Herrmann, Wood, Fuhrmann, 2021). A strategy based on hydrogels can regulate the transport of EVs through the extracellular matrix. This combined hydrogel - exosome system facilitates motion leading to free diffusion and enhancing water permeation (Lenzini et al., 2020). Interestingly, this type of phenomenon arguably has potential for computational modeling by Monte Carlo simulation.

Simulations by cellular automata

Cellular automata (CA) models are discrete computational models for analyzing dynamic systems in which the elements are in close interaction. Simulation using cellular automata has been used in different fields to resolve issues related to diffusive phenomena (Cao et al., 2023; Fukś, Mudiyanselage, 2022; Kulagin, Shapovalov, 2023; Liang et al., 2023; Ma, Lin, 2022; Medina et al., 2022). This model uses identically programmed units denominated “cells” which are in direct contact with their neighbors and subject to a finite set of prescribed rules for local transitions. One of the main similarities between MCS and CA is their integration within regular spatial lattices. The movement of each cell follows a rule and is assessed in each step. When the system is updated synchronously, only if the state of the cell at the time is satisfied is the transition carried out (El Yacoubi, El Jai, 2002) (Figure 6).

FIGURE 6
Cell states and transitions in a cellular automaton model is composed as a set of cells forming 3D rectangular lattice. Molecular motion is modeled for each block independently.

Several controlled delivery systems have been modeled by cellular automata and we will describe some of them to exemplify the potential of cellular automata in the field. Another similarity between Monte Carlo simulation and cellular automata is that the diffusion mechanism based on random walking avoids the need to solve Fick’s law for these systems. For instance, Laaksonen and colleagues presented one of the first models for swelling drug-controlled release using a cellular automata approach. In this model, for different defined cell types with water, drug, erodible polymer and wet polymer in each, established transition rules operate when the tablet interacts physically with water molecules. Using the solid drug/saturated drug ratio as a parameter in the model, it is possible to simulate two compounds with different solubilities (Laaksonen,

Hirvonen, Laaksonen, 2009). Recently, García-Fandiño and colleagues showed the impact of a synergistic in silico-in vitro approach, demonstrating the predictive power of computational simulations for vitamin E−sphingomyelin nanosystems (Bouzo et al., 2020). Fathi and colleagues developed a stochastic model using cellular automata to simulate encapsulant release from lipid nanocarriers containing hespertin, which considered diffusion as the predominant mechanism, using MATLAB®. In this simulation, the impact of the load of drug, the type of neighborhood defined, and the solubility, encapsulant distribution, and type of release medium were assessed in two and three dimensions. Here, it was possible to identify concordance between real experimental data for hespertin release and the simulation results (Fathi et al., 2013). Finally, some material biosorbents have been used for the pharmaceutical design of microspheres, and they have been characterized by modeling with a three-dimensional cellular automaton showing a good correlation with experimental data (Bertrand, Leclair, Hildgen, 2007).

CONCLUSION

We have conducted a brief review and discussion on the use of probabilistic methods, such as Monte Carlo and Cellular Automata simulations to describe their impact in the analysis of the kinetic patterns generated during drug-controlled release. We identified some key points:

• -Systems that can be simulated must maintain a basic set of physicochemical phenomena that orchestrate the drug-controlled release, especially the diffusion of the drug and dissolution medium.

• - Simulation of drugs mobilized in biological systems such as the extracellular matrix, skin tissue, blood tissue, synovial fluid, etc., can be studied as fractal systems.

• - Drug properties such as their solubility, molecular weight, coefficients of partition, topology, or molecular interactions can be useful parameters for design of a Monte Carlo simulation algorithm with greater capability to predict the kinetic phenomena associated with the drug-controlled release. Moreover, enzymatic activity and heterogeneous environments can also be simulated.

• - Nanopharmaceutical devices can be seen as new models for probabilistic simulations and analysis of drug release kinetics under different design conditions.

• - Dynamic events that occur during drug-controlled release in different pharmaceutical devices, such as swelling, erosion, water diffusion, changes of viscosity, enzymatic action, and the effect of new components in the formulation, may also be modeled or coupled in Monte Carlo simulations. To improve the efficiency and prediction of Monte Carlo simulations, it is necessary to integrate new in silico approaches such as machine learning, artificial intelligence, deterministic methods, or novel computational strategies.

• - Simulation findings exhibit significant agreement with experimental data, indicating that this approach has he capability to either predict or validate kinetic data for various pharmaceutical devices.

Finally, despite the limited number of reports on these models, we are optimistic about their potential in the early stages of formulation of new pharmaceutical dosages.

ACKNOWLEDGMENTS

We thank to the PhD Homero Gomez Velasco for his generous support and helpful discussions. We also thank to Silvia Lizbeth Reyes Malagón for her valuable support in the design of some figures for this report. This paper presents part of the PhD thesis of Saúl Jiménez-Jiménez, who is a doctoral student from Programa de Doctorado en Ciencias Químicas, Universidad Nacional Autónoma de México (UNAM). Saul Jiménez-Jiménez thanks for the fellowship 314168 from CONAHCYT previously received. José-Gerardo Mejía-Hernández thanks for the fellowship 1313023 received from CONAHCYT.

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• Funding
  Rafael Villalobos-García would like to thank the Programa de Cátedras FESC under Grant Number 2.13.18.23.

Edited by

• Associated Editor:
  Gabriel Lima Barros de Araújo

Publication Dates

• Publication in this collection
  20 Jan 2025
• Date of issue
  2025

History

• Received
  08 May 2024
• Accepted
  24 Sept 2024