Mathematical modeling of simultaneous drug release and in vivo absorption

A Comparative Mathematical Analysis of Drug Release from Lipid-Based Nanoparticles

Mathematical modeling of drug release from drug delivery systems is crucial for understanding and optimizing formulations. This research provides a comparative mathematical analysis of drug release from lipid-based nanoparticles. Drug release profiles from various types of lipid nanoparticles, including liposomes, nanostructured lipid carriers (NLCs), solid lipid nanoparticles (SLNs), and nano/micro-emulsions (NEMs/MEMs), were extracted from the literature and used to assess the suitability of eight conventional mathematical release models. For each dataset, several metrics were calculated, including the coefficient of determination (R2), adjusted R2, the number of errors below certain thresholds (5%, 10%, 12%, and 20%), Akaike information criterion (AIC), regression sum square (RSS), regression mean square (RMS), residual sum of square (rSS), and residual mean square (rMS). The Korsmeyer-Peppas model ranked highest among the evaluated models, with the highest adjusted R2 values of 0.95 for NLCs and 0.93 for other liposomal drug delivery systems. The Weibull model ranked second, with adjusted R2 values of 0.92 for liposomal systems, 0.94 for SLNs, and 0.82 for NEMs/MEMs. Thus, these two models appear to be more effective in forecasting and characterizing the release of lipid nanoparticle drugs, potentially making them more suitable for upcoming research endeavors.

Modelling smart drug release with functionally graded materials

Functionally graded materials (FGMs), possessing properties that vary smoothly from one region to another, have been receiving increasing attention in recent years, particularly in the aerospace, automotive and biomedical sectors. However, they have yet to reach their full potential. In this paper, we explore the potential of FGMs in the context of drug delivery, where the unique material characteristics offer the potential of fine-tuning drug-release for the desired application. Specifically, we develop a mathematical model of drug release from a thin film FGM, based upon a spatially-varying drug diffusivity. We demonstrate that, depending on the functional form of the diffusivity (related to the material properties) a wide range of drug release profiles may be obtained. Interestingly, the shape of these release profiles are not, in general, achievable from a homogeneous medium with a constant diffusivity.

Stability and release mechanisms of double emulsions loaded with bioactive compounds; a critical review

Double emulsions (DEs), known as emulsions of emulsions, are dispersion systems in which the droplets of one dispersed liquid are further dispersed in another liquid, producing double-layered liquid droplets. These systems are widely used in the food and pharmaceutical industries due to their ability to co-encapsulate both hydrophilic and hydrophobic bioactive compounds. However, they are sensitive and unstable and their controlled release is challenging. In this study, first, the stability of DEs and their release mechanisms are reviewed. Then, the factors affecting their stability, and the release of bioactive compounds are studied. Finally, modeling of the release in DEs is discussed. This information can be useful to optimize the formulation of DEs in order to utilize them in different industries.

Mathematical modelling of drug delivery from pH-responsive nanocontainers

Targeted drug delivery systems represent a promising strategy to treat localised disease with minimum impact on the surrounding tissue. In particular, polymeric nanocontainers have attracted major interest because of their structural and morphological advantages and the variety of polymers that can be used, allowing the synthesis of materials capable of responding to the biochemical alterations of the environment. While experimental methodologies can provide much insight, the generation of experimental data across a wide parameter space is usually prohibitively time consuming and/or expensive. To better understand the influence of varying design parameters on the release profile and drug kinetics involved, appropriately-designed mathematical models are of great benefit. Here, we developed a continuum-scale mathematical model to describe drug transport within, and release from, a hollow nanocontainer consisting of a core and a pH-responsive polymeric shell. Our two-layer mathematical model accounts for drug dissolution and diffusion and includes a mechanism to account for trapping of drug molecules within the shell. We conduct a sensitivity analysis to assess the effect of varying the model parameters on the overall behaviour of the system. To demonstrate the usefulness of our model, we focus on the particular case of cancer treatment and calibrate the model against release profile data for two anti-cancer therapeutical agents. We show that the model is capable of capturing the experimentally observed pH-dependent release.

Understanding the burst release phenomenon: toward designing effective nanoparticulate drug-delivery systems

Burst release of encapsulated drug with release of a significant fraction of payload into release medium within a short period, both in vitro and in vivo, remains a challenge for translation. Such unpredictable and uncontrolled release is often undesirable, especially from the perspective of developing sustained-release formulations. Moreover, a brisk release of the payload upsets optimal release kinetics. This account strives toward understanding burst release noticed in nanocarriers and investigates its causes. Various mathematical models to explain such untimely release were also examined, including their strengths and weaknesses. Finally, the account revisits current techniques of limiting burst release from nanocarriers and prioritizes future directions that harbor potential of fruitful translation by reducing such occurrences.

Predicting Transdermal Fentanyl Delivery Using Mechanistic Simulations for Tailored Therapy

Transdermal drug delivery is a key technology for administering drugs. However, most devices are “one-size-fits-all”, even though drug diffusion through the skin varies significantly from person-to-person. For next-generation devices, personalization for optimal drug release would benefit from an augmented insight into the drug release and percutaneous uptake kinetics. Our objective was to quantify the changes in transdermal fentanyl uptake with regards to the patient’s age and the anatomical location where the patch was placed. We also explored to which extent the drug flux from the patch could be altered by miniaturizing the contact surface area of the patch reservoir with the skin. To this end, we used validated mechanistic modeling of fentanyl diffusion, storage, and partitioning in the epidermis to quantify drug release from the patch and the uptake within the skin. A superior spatiotemporal resolution compared to experimental methods enabled in-silico identification of peak concentrations and fluxes, and the amount of stored drug and bioavailability. The patients’ drug uptake showed a 36% difference between different anatomical locations after 72 h, but there was a strong interpatient variability. With aging, the drug uptake from the transdermal patch became slower and less potent. A 70-year-old patient received 26% less drug over the 72-h application period, compared to an 18-year-old patient. Additionally, a novel concept of using micron-sized drug reservoirs was explored in silico. These reservoirs induced a much higher local flux (µg cm^-2 h^-1) than conventional patches. Up to a 200-fold increase in the drug flux was obtained from these small reservoirs. This effect was mainly caused by transverse diffusion in the stratum corneum, which is not relevant for much larger conventional patches. These micron-sized drug reservoirs open new ways to individualize reservoir design and thus transdermal therapy. Such computer-aided engineering tools also have great potential for in-silico design and precise control of drug delivery systems. Here, the validated mechanistic models can serve as a key building block for developing digital twins for transdermal drug delivery systems.

Controlled anti-cancer drug release through advanced nano-drug delivery systems: Static and dynamic targeting strategies

Advances in nanomedicine, including early cancer detection, targeted drug delivery, and personalized approaches to cancer treatment are on the rise. For example, targeted drug delivery systems can improve intracellular delivery because of their multifunctionality. Novel endogenous-based and exogenous-based stimulus-responsive drug delivery systems have been proposed to prevent the cancer progression with proper drug delivery. To control effective dose loading and sustained release, targeted permeability and individual variability can now be described in more-complex ways, such as by combining internal and external stimuli. Despite these advances in release control, certain challenges remain and are identified in this research, which emphasizes the control of drug release and applications of nanoparticle-based drug delivery systems. Using a multiscale and multidisciplinary approach, this study investigates and analyzes drug delivery and release strategies in the nanoparticle-based treatment of cancer, both mathematically and clinically.

Modeling drug delivery from multiple emulsions

We present a mechanistic model of drug release from a multiple emulsion into an external surrounding fluid. We consider a single multilayer droplet where the drug kinetics are described by a pure diffusive process through different liquid shells. The multilayer problem is described by a system of diffusion equations coupled via interlayer conditions imposing continuity of drug concentration and flux. Mass resistance is imposed at the outer boundary through the application of a surfactant at the external surface of the droplet. The two-dimensional problem is solved numerically by finite volume discretization. Concentration profiles and drug release curves are presented for three typical round-shaped (circle, ellipse, and bullet) droplets and the dependency of the solution on the mass transfer coefficient at the surface analyzed. The main result shows a reduced release time for an increased elongation of the droplets.

Preparation and Characterization of PVA-Co-PE Drug-Loaded Nanofiber Membrane by Electrospinning Technology

A new transdermal drug delivery system of nanofiber membrane with good biocompatibility and high drug loading was developed by electrospinning technology in this study. Using vinyl alcohol-co-ethylene (PVA-co-PE) polymer as a spinning matrix and non-steroidal anti-inflammatory drug (NSAID) sulindac (SUL) as a model drug, the SUL@PVA-co-PE nanofiber membrane was prepared and characterized systematically. The morphology, molecular vibrational transitions, thermogravimetric attributes, and in vitro drug release and transdermal characteristics of drug-loaded nanofiber membranes were analyzed. The results indicated that the surface of PVA-co-PE nanofibers was uniform and smooth with the diameter ranged from 461 to 696 nm. Notably in vitro simulation experiments demonstrated that SUL@PVA-co-PE nanofiber membrane could provide a continuous drug release to reach the effective concentration of the drug, and exhibited significantly higher cumulative drug permeability compared to commercially available patches, Taken together, PVA-co-PE nanofiber membranes exhibited the characteristics of high drug loading and stability, and represented the potential to be utilized as a new transdermal drug delivery carrier with pronounced development prospect.

Percutaneous delivery of levetiracetam as an alternative to topical nonsteroidal anti-inflammatory drugs: formulation development, in vitro and in vivo characterization

The study focused on formulation of carmellose sodium hydrogels and nonionic microemulsions with 5% and 10% of levetiracetam and investigation of drug concentration influence on their physicochemical characteristics and in-use stability as well as influence of drug concentration and carrier type on in vitro drug release and in vivo antihyperalgesic/antiedematous activity in a rat model of localized (intraplantar) carrageenan-induced inflammation. Hydrogels were pseudoplastic semisolids with thixotropy and pH 7.37-7.58. Microemulsions were low viscous Newtonian nanodispersions of oil droplets (13.11-15.11 nm) in water, with pH 4.01-4.64. Physical stability of the investigated systems was preserved over the 3-month storage under ambient conditions. Levetiracetam release followed zero order and Korsmeyer-Peppas models (R² ≥ 0.99) reflecting the combined effects of drug concentration and carrier viscosity. All levetiracetam-loaded formulations produced significant reduction of hyperalgesia and paw swelling induced by carrageenan (p < 0.001). Their efficacy in exerting antihyperalgesic activity was significantly higher than that observed with the reference 5% ibuprofen hydrogel preparation (up to 6 h) (p < 0.001), while antiedematous activity was comparable with the reference product. No erythema and visible blood vessels were observed in a rat ear test. The study demonstrated percutaneous delivery of levetiracetam as useful and safe therapeutic option for localized inflammatory pain with potential to overcome the insufficient efficacy of topically applied nonsteroidal anti-inflammatory drugs in the form of a hydrogel. Graphical abstract.

Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems

Embedding of active substances in supramolecular systems has as the main goal to ensure the controlled release of the active ingredients. Whatever the final architecture or entrapment mechanism, modeling of release is challenging due to the moving boundary conditions and complex initial conditions. Despite huge diversity of formulations, diffusion phenomena are involved in practically all release processes. The approach in this paper starts, therefore, from mathematical methods for solving the diffusion equation in initial and boundary conditions, which are further connected with phenomenological conditions, simplified and idealized in order to lead to problems which can be analytically solved. Consequently, the release models are classified starting from the geometry of diffusion domain, initial conditions, and conditions on frontiers. Taking into account that practically all solutions of the models use the separation of variables method and integral transformation method, two specific applications of these methods are included. This paper suggests that "good modeling practice" of release kinetics consists essentially of identifying the most appropriate mathematical conditions corresponding to implied physicochemical phenomena. However, in most of the cases, models can be written but analytical solutions for these models cannot be obtained. Consequently, empiric models remain the first choice, and they receive an important place in the review.

Mechanistic modelling of drug release from multi-layer capsules

We propose a novel in silico model for computing drug release from multi-layer capsules. The diffusion problem in such heterogeneous layer-by-layer composite medium is described by a system of coupled partial differential equations, which we solve analytically using separation of variables. In addition to the conventional partitioning and mass transfer interlayer conditions, we consider a surface finite mass transfer resistance, which corresponds to the case of a coated capsule. The drug concentration in the core and through all the layers, as well as in the external release medium, is given in terms of a Fourier series that we compute numerically to describe and characterize the drug release mechanism.

A physiologically-based model to predict individual pharmacokinetics and pharmacodynamics of remifentanil

Remifentanil based anesthesia is nowadays spread worldwide. This drug is characterized by a rapid onset of the analgesic effects, but also by a rapid onset of the side effects. For this reason, the knowledge of the remifentanil concentration in the human body is a key topic in anesthesiology. The aims of this work are to propose and validate a physiologically based pharmacokinetic model capable to predict both the pharmacokinetics and pharmacodynamics of remifentanil, and to take into account the inter-individual differences among the patients (such as height and body mass). The blood concentration of remifentanil has been successfully simulated and compared with experimental literature data. The pharmacodynamics, in terms of effect of remifentanil on minute ventilation and electroencephalogram, has been implemented in this model. Moreover, the remifentanil concentration in various organs and tissues is predicted, which is a significant improvement with respect to the traditional compartmental models. The availability of the model makes possible the prediction of the effects of remifentanil administration, also accounting for individual parameters.

Modeling of the Behavior of Natural Polysaccharides Hydrogels for Bio-pharma Applications

Hydrogels, even if not exclusively obtained from natural sources, are widely used for pharmaceuticals and for biomedical applications. The reasons for their uses are their biocompatibility and the possibility to obtain systems and devices with different properties, due to variable characteristics of the materials. In order to effectively design and produce these systems and devices, two main ways are available: i) trial-and-error process, at least guided by experience, during which the composition of the system and the production steps are changed in order to get the desired behavior; ii) production process guided by the a-priori simulation of the systems’ behavior, thanks to proper tuned mathematical models of the reality. Of course the second approach, when applicable, allows tremendous savings in term of human and instrumental resources.

In this mini-review, several modeling approaches useful to describe the behavior of natural polysaccharide-based hydrogels in bio-pharma applications are reported. In particular, reported case histories are: i) the size calculation of micro-particles obtained by ultrasound assisted atomization; ii) the release kinetics from core-shell micro-particles, iii) the solidification behavior of blends of synthetic and natural polymers for gel paving of blood vessels, iv) the drug release from hydrogel-based tablets. This material can be seen as a guide toward the use of mathematical modeling in bio-pharma applications.

In silico-in vitro-in vivo studies of experimentally designed carvedilol loaded silk fibroin-casein nanoparticles using physiological based pharmacokinetic model

The study aimed to design and develop carvedilol loaded silk fibroin-casein nanoparticles using 3² full factorial design. Silk fibroin and casein concentration were selected as the independent variables and their effect were observed on dependent variables: particle size, polydispersity index, encapsulation efficiency, drug release, and dissolution efficiency. The developed optimized formulation was characterized using fourier transform infrared spectroscopy, differential scanning calorimetry, and Powder X-ray diffraction. Surface morphology of optimized formulation using scanning electron microscopy, transmission electron microscopy, and atomic force microscopy revealed spherical nature of particles without any evidence of aggregation. The optimized formulation showed a 2.04-fold increase in C_max, and 6.87-fold increase in bioavailability as compared to aqueous suspension. The formulation showed sustained release as confirmed by increases in mean residence time. The in vivo in silico simulation using physiologically based pharmacokinetics (PBPK) model and population simulation (100 subjects) revealed a reasonable degree of superimposition of simulated and observed pharmacokinetic parameters based on overall fold error (≤2.0). The enhanced bioavailability with sustained effect demonstrates potential of silk fibroin as an alternative carrier for drug delivery and presents Gastoplus™ as efficient tool for in vivo in silico simulations.

Polymer-Free Drug-Eluting Stents: An Overview of Coating Strategies and Comparison with Polymer-Coated Drug-Eluting Stents

Clinical evaluations have proven the efficacy of drug-elution stents (DES) in reduction of in-stent restenosis rates as compared to drug-free bare metal stents (BMS). Typically, DES are metal stents that are covered with a polymer film loaded with anti-inflammatory or antiproliferative drugs that are released in a sustained manner. However, although favorable effects of the released drugs have been observed, the polymer coating as such has been associated with several adverse clinical effects, such as late stent thrombosis. Elimination of the polymeric carrier of DES may therefore potentially lead to safer DES. Several technologies have been developed to design polymer-free DES, such as the use of microporous stents and inorganic coatings that can be drug loaded. Several drugs, including sirolimus, tacrolimus, paclitaxel, and probucol have been used in the design of carrier-free stents. Due to the function of the polymeric coating to control the release kinetics of a drug, polymer-free stents are expected to have a faster drug elution rate, which may affect the therapeutic efficacy. However, several polymer-free stents have shown similar efficacy and safety as the first-generation DES, although the superiority of polymer-free DES has not been established in clinical trials.

Topological characterization of a bacterial cellulose-acrylic acid polymeric matrix

This paper focuses on the micro- and nano-topological organization of a hydrogel, constituted by a mixture of bacterial cellulose and acrylic acid, and intended for biomedical applications. The presence of acrylic acid promotes the formation of two interpenetrated continuous phases: the primary "pores phase" (PP) containing only water and the secondary "polymeric network phase" (PNP) constituted by the polymeric network swollen by the water. Low field Nuclear Magnetic Resonance (LF NMR), rheology, Scanning Electron Microscopy (SEM) and release tests were used to determine the characteristics of the two phases. In particular, we found that this system is a strong hydrogel constituted by 81% (v/v) of PP phase the remaining part being occupied by the PNP phase. Pores diameters span in the range 10-100 μm, the majority of them (85%) falling in the range 30-90 μm. The high PP phase tortuosity indicates that big pores are not directly connected to each other, but their connection is realized by a series of interconnected small pores that rend the drug path tortuous. The PNP is characterized by a polymer volume fraction around 0.73 while mesh size is around 3 nm. The theoretical interpretation of the experimental data coming from the techniques panel adopted, yielded to the micro- and nano-organization of our hydrogel.

Controlled release for local delivery of drugs: barriers and models

Controlled release systems are an effective means for local drug delivery. In local drug delivery, the major goal is to supply therapeutic levels of a drug agent at a physical site in the body for a prolonged period. A second goal is to reduce systemic toxicities, by avoiding the delivery of agents to non-target tissues remote from the site. Understanding the dynamics of drug transport in the vicinity of a local drug delivery device is helpful in achieving both of these goals. Here, we provide an overview of controlled release systems for local delivery and we review mathematical models of drug transport in tissue, which describe the local penetration of drugs into tissue and illustrate the factors - such as diffusion, convection, and elimination - that control drug dispersion and its ultimate fate. This review highlights the important role of controlled release science in development of reliable methods for local delivery, as well as the barriers to accomplishing effective delivery in the brain, blood vessels, mucosal epithelia, and the skin.

Application of mathematical modeling in sustained release delivery systems

INTRODUCTION

This review, presenting as starting point the concept of the mathematical modeling, is aimed at the physical and mathematical description of the most important mechanisms regulating drug delivery from matrix systems. The precise knowledge of the delivery mechanisms allows us to set up powerful mathematical models which, in turn, are essential for the design and optimization of appropriate drug delivery systems.

AREAS COVERED

The fundamental mechanisms for drug delivery from matrices are represented by drug diffusion, matrix swelling, matrix erosion, drug dissolution with possible recrystallization (e.g., as in the case of amorphous and nanocrystalline drugs), initial drug distribution inside the matrix, matrix geometry, matrix size distribution (in the case of spherical matrices of different diameter) and osmotic pressure. Depending on matrix characteristics, the above-reported variables may play a different role in drug delivery; thus the mathematical model needs to be built solely on the most relevant mechanisms of the particular matrix considered.

EXPERT OPINION

Despite the somewhat diffident behavior of the industrial world, in the light of the most recent findings, we believe that mathematical modeling may have a tremendous potential impact in the pharmaceutical field. We do believe that mathematical modeling will be more and more important in the future especially in the light of the rapid advent of personalized medicine, a novel therapeutic approach intended to treat each single patient instead of the 'average' patient.

Modeling of the reticulation kinetics of alginate/pluronic blends for biomedical applications

In this work, blends of alginate/pluronic (F127) for biomedical applications were investigated. In particular, the kinetics of alginate chain reticulation by bivalent cations was studied by experimental and modeling approaches. Two kinds of sodium alginate were tested to obtain hard gel films. The thicknesses of the reticulated alginate films were measured as function of the exposure time and of the reticulating copper (Cu(2+)) solution concentration. The kinetics was described by a proper model able to reproduce the experimental data. The model parameters, evaluated based on the measurements of thicknesses as function of Cu(2+) concentration and exposure time, were further validated by comparing the prediction of the model with another set of independent measurement; here, the depletion of Cu(2+) ions in the conditioning solution above the reacting gel is measured as function of time. The tuned model could be used in the description of the future applications of the blends.

Mathematical modeling of drug delivery from autocatalytically degradable PLGA microspheres--a review

PLGA microspheres are widely studied for controlled release drug delivery applications, and many models have been proposed to describe PLGA degradation and erosion and drug release from the bulk polymer. Autocatalysis is known to have a complex role in the dynamics of PLGA erosion and drug transport and can lead to size-dependent heterogeneities in otherwise uniformly bulk-eroding polymer microspheres. The aim of this review is to highlight mechanistic, mathematical models for drug release from PLGA microspheres that specifically address interactions between phenomena generally attributed to autocatalytic hydrolysis and mass transfer limitation effects. Predictions of drug release profiles by mechanistic models are useful for understanding mechanisms and designing drug release particles.

Predictability of drug release from cochlear implants

A simplified mathematical theory is presented allowing for in silico simulation of the effects of key parameters of miniaturized implants (size and composition) on the resulting drug release kinetics. Such devices offer a great potential, especially for local drug treatments, e.g. of the inner ear. However, the preparation and characterization of these systems is highly challenging, due to the small system dimensions. The presented mathematical theory is based on Fick's second law of diffusion. Importantly, theoretical predictions do not require the knowledge of many system-specific parameters: Only the "apparent" diffusion coefficient of the drug within the implant matrix is needed. This parameter can be easily determined via drug release measurements from thin, macroscopic films. The validity of the theoretical model predictions was evaluated by comparison with experimental results obtained with a cochlear implant. The latter consisted of miniaturized electrodes, which were embedded in a silicone matrix loaded with various amounts of dexamethasone. Importantly, independent experimental results confirmed the theoretical predictions. Thus, the presented simplified theory can help to significantly speed up the optimization of this type of controlled drug delivery systems, especially if long release periods are targeted (e.g., several months or years). Straightforward experiments with thin, macroscopic films and computer simulations can allow for rapid identification of optimal system design.

A computational procedure for assessing the dynamic performance of diffusion-controlled transdermal delivery devices

The dynamic performances of two different controlled-release systems were analyzed. In a reservoir-type drug-delivery patch, the transdermal flux is influenced by the properties of the membrane. A constant thermodynamic drug activity is preserved in the donor compartment. Monolithic matrices are among the most inexpensive systems used to direct drug delivery. In these structures, the active pharmaceutical ingredients are encapsulated within a polymeric material. Despite the popularity of these two devices, to tailor the properties of the polymer and additives to specific transient behaviors can be challenging and time-consuming. The heuristic approaches often considered to select the vehicle formulation provide limited insight into key permeation mechanisms making it difficult to predict the device performance. In this contribution, a method to calculate the flux response time in a system consisting of a reservoir and a polymeric membrane was proposed and confirmed. Nearly 8.60 h passed before the metoprolol delivery rate reached ninety-eight percent of its final value. An expression was derived for the time it took to transport the active pharmaceutical ingredient out of the polymer. Ninety-eight percent of alpha-tocopherol acetate was released in 461.4 h following application to the skin. The effective time constant can be computed to help develop optimum design strategies.