Mechanistic modelling of drug release from multi-layer capsules

Mathematical Modeling of Drug Delivery from Bi-Layered Core-Shell Polymeric Microspheres

Chronic diseases usually require repetitive dosing. Depending on factors such as dosing frequency, mode of administration, and associated costs this can result in poor patient compliance. A better alternative involves using drug delivery systems to reduce the frequency of dosing and extend drug release. However, reaching the market stage is a time-consuming process. In this study, we used two numerical approaches for estimating the values of the critical parameters that govern the diffusion-controlled drug release within bilayered core-shell microspheres. Specifically, the estimated parameters include burst release, drug diffusion coefficient in two polymers, and the drug partition coefficient. Estimating these parameters provides insight for optimizing device design, guiding experimental efforts, and improving the device's effectiveness. We obtained good agreement between the models and the experimental data. The methods explored in this work apply not only to bi-layered spherical systems but can also be extended to multi-layered spherical systems.

On the equivocal nature of the mass absorption curves

The idea behind the research presented is based upon apparently contradictory experimental results obtained here by means of photoacoustics modalities for the same drug donor/acceptor membrane system, serving as a surrogate for a transdermal delivery system. The first modality allowed for the monitoring of the total amount of mass uptake (m(t)-type data), while the second technique allowed for the quantification of time-dependent concentration distribution within the acceptor membrane (c(x,t)-type data). Despite of a very good agreement between the mt data and the 1st-order uptake fitting model (standard Fickian diffusion with constant source boundary condition), the standard approach failed during the c(x,t) data analysis. The results led to the analysis of the interfacial transfer contribution to the overall mass transfer efficiency, which eventually allowed to question reliability of the mt data analysis for the determination and quantification of the mass transport parameters. A more detailed analysis of the c(x,t) by means of the newly introduced transport rate number parameter revealed, that the mass uptake by the acceptor is almost equally influenced by interfacial and bulk transport processes. The analyses performed were translated into a model-free characteristic times, i.e. parameters independent of the model scheme used.

Experimental and mathematical approaches for drug delivery for the treatment of wet age-related macular degeneration

Several chronic eye diseases affect the posterior segment of the eye. Among them age-related macular degeneration can cause vision loss if left untreated and is one of the leading causes of visual impairment in the world. Most treatments are based on intravitreally injected therapeutics that inhibit the action of vascular endothelial growth factor. However, due to the need for monthly injections, this method is associated with poor patient compliance. To address this problem, numerous drug delivery systems (DDSs) have been developed. This review covers a selection of particulate systems, non-stimuli responsive hydrogels, implants, and composite systems that have been developed in the last few decades. Depending on the type of DDS, polymer material, and preparation method, different mechanical properties and drug release profiles can be achieved. Furthermore, DDS development can be optimized by implementing mathematical modeling of both drug release and pharmacokinetic aspects. Several existing mathematical models for diffusion-controlled, swelling-controlled, and erosion-controlled drug delivery from polymeric systems are summarized. Compartmental and physiologically based models for ocular drug transport and pharmacokinetics that have studied drug concentration profiles after intravitreal delivery or release from a DDS are also reviewed. The coupling of drug release models with ocular pharmacokinetic models can lead to obtaining much more efficient DDSs for the treatment of age-related macular degeneration and other diseases of the posterior segment of the eye.

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.

Fractional modeling of drug diffusion from cylindrical tablets based on Fickian and relaxed approaches with in vivo validation

Mathematical simulation of drug diffusion is a significant tool for predicting the bio-transport process. Moreover, the reported models in the literature are based on Fick's approach, which leads to an infinite propagation speed. Consequently, it is essential to construct a mathematical model to represent the diffusion processes for estimating drug concentrations at different sites and throughout the circulation. Thus, in this article, the diffusion process is employed to propose three models for estimating the drug release from multi-layer cylindrical tablets. A fractional model is presented based on Fick's approach, while classical and fractional Cattaneo models are presented using the relaxed principle. Various numerical methods are used to solve the specified problem. The numerical scheme's stability and convergence are demonstrated. Drug concentration and mass profiles are presented for the tablet and the external medium and compared with the in vivo plasma profiles. The results show the efficiency and precision of the proposed fractional models based on the fourth-order weighted-shifted Grünwald-Letnikov difference operator approximation. These models are compatible with the in vivo data compared with the classical Fick's one.

A decade of developing applications exploiting the properties of polyelectrolyte multilayer capsules

Transferring the layer-by-layer (LbL) coating approach from planar surfaces to spherical templates and subsequently dissolving these templates leads to the fabrication of polyelectrolyte multilayer capsules. The versatility of the coatings of capsules and their flexibility upon bringing in virtually any material into the coatings has quickly drawn substantial attention. Here, we provide an overview of the main developments in this field, highlighting the trends in the last decade. In the beginning, various methods of encapsulation and release are discussed followed by a broad range of applications, which were developed and explored. We also outline the current trends, where the range of applications is continuing to grow, including addition of whole new and different application areas.

Characterization of the Shells in Layer-By-Layer Nanofunctionalized Particles: A Computational Study

Drug delivery carriers are considered an encouraging approach for the localized treatment of disease with minimum effect on the surrounding tissue. Particularly, layer-by-layer releasing particles have gained increasing interest for their ability to develop multifunctional systems able to control the release of one or more therapeutical drugs and biomolecules. Although experimental methods can offer the opportunity to establish cause and effect relationships, the data collection can be excessively expensive or/and time-consuming. For a better understanding of the impact of different design conditions on the drug-kinetics and release profile, properly designed mathematical models can be greatly beneficial. In this work, we develop a continuum-scale mathematical model to evaluate the transport and release of a drug from a microparticle based on an inner core covered by a polymeric shell. The present mathematical model includes the dissolution and diffusion of the drug and accounts for a mechanism that takes into consideration the drug biomolecules entrapped into the polymeric shell. We test a sensitivity analysis to evaluate the influence of changing the model conditions on the total system behavior. To prove the effectiveness of this proposed model, we consider the specific application of antibacterial treatment and calibrate the model against the data of the release profile for an antibiotic drug, metronidazole. The results of the numerical simulation show that ∼85% of the drug is released in 230 h, and its release is characterized by two regimes where the drug dissolves, diffuses, and travels the external shell layer at a shorter time, while the drug is released from the shell to the surrounding medium at a longer time. Within the sensitivity analysis, the outer layer diffusivity is more significant than the value of diffusivity in the core, and the increase of the dissolution parameters causes an initial burst release of the drug. Finally, changing the shape of the particle to an ellipse produces an increased percentage of drugs released with an unchanged release time.

Mathematical Modelling Highlights the Potential for Genetic Manipulation as an Adjuvant to Counter Efflux-Mediated MDR in Salmonella

Bacteria have developed resistance to antibiotics by various mechanisms, notable amongst these is the use of permeation barriers and the expulsion of antibiotics via efflux pumps. The resistance-nodulation-division (RND) family of efflux pumps is found in Gram-negative bacteria and a major contributor to multidrug resistance (MDR). In particular, Salmonella encodes five RND efflux pump systems: AcrAB, AcrAD, AcrEF, MdsAB and MdtAB which have different substrate ranges including many antibiotics. We produce a spatial partial differential equation (PDE) model governing the diffusion and efflux of antibiotic in Salmonella, via these RND efflux pumps. Using parameter fitting techniques on experimental data, we are able to establish the behaviour of multiple wild-type and efflux mutant Salmonella strains, which enables us to produce efflux profiles for each individual efflux pump system. By combining the model with a gene regulatory network (GRN) model of efflux regulation, we simulate how the bacteria respond to their environment. Finally, performing a parameter sensitivity analysis, we look into various different targets to inhibit the efflux pumps. The model provides an in silico framework with which to test these potential adjuvants to counter MDR.

Numerical method to characterise capsule membrane permeability for controlled drug delivery

Design and characterisation of capsules is not an easy task owing to the multiple involved preparation factors and parameters. Here, a novel method to characterise capsule membrane permeability to solute molecules by an inverse approach is proposed. Transport of chemical species between the capsule core and the surrounding medium through the membrane is described by the Fick's second law with a position-dependent diffusion coefficient. Solutions are computed in spherical coordinates using a finite difference scheme developed for diffusion in multilayer configuration. They are validated using semi-analytical solutions and fully three-dimensional lattice Boltzmann simulations. As a proof of concept, the method is applied to experimental data available in the literature on the kinetics of glucose release and absorption to determine the membrane permeability of capsules. The proposed method is easy to use and determines correctly the permeability of capsule membranes for controlled drug release and absorption applications.

Mathematical modelling of drug-diffusion from multi-layered capsules/tablets and other drug delivery devices

In this paper, two mathematical models have been formulated by extending the basic reaction-diffusion model, along with suitable initial and boundary conditions to study the drug delivery and its diffusion in biological tissues from multi-layered capsules/tablets and other drug delivery devices (DDDs), respectively. These devices are either taken orally or through other drug-administration routes. The formulated models are solved using the variational finite element method followed by the fundamental matrix method, to study the drug delivery and its diffusion more efficiently. The main aim of this work is to provide an effective model, using optimal mathematical techniques to help researchers and biologists in medicine in decreasing the endeavours and expenses in designing DDDs. The outcomes obtained are compared with the experimental data to demonstrate the validity and the feasibility of the proposed work.

Polyelectrolyte Multilayer Capsule (PEMC)-Based Scaffolds for Tissue Engineering

Tissue engineering (TE) is a highly multidisciplinary field that focuses on novel regenerative treatments and seeks to tackle problems relating to tissue growth both in vitro and in vivo. These issues currently involve the replacement and regeneration of defective tissues, as well as drug testing and other related bioapplications. The key approach in TE is to employ artificial structures (scaffolds) to support tissue development; these constructs should be capable of hosting, protecting and releasing bioactives that guide cellular behaviour. A straightforward approach to integrating bioactives into the scaffolds is discussed utilising polyelectrolyte multilayer capsules (PEMCs). Herein, this review illustrates the recent progress in the use of CaCO3 vaterite-templated PEMCs for the fabrication of functional scaffolds for TE applications, including bone TE as one of the main targets of PEMCs. Approaches for PEMC integration into scaffolds is addressed, taking into account the formulation, advantages, and disadvantages of such PEMCs, together with future perspectives of such architectures.

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.

Spatiotemporal antiresonance in coupled reaction-diffusion systems

We present a theoretical study of the spatiotemporal antiresonance in a system of two diffusively coupled chemical reactions, one of which is driven by an external periodic forcing. Although antiresonance is well known in various physical systems, the phenomenon in coupled chemical reactions has largely been overlooked. Based on the linearized dynamics around the steady state of the two-component coupled reaction-diffusion systems we have derived the general analytical expressions for the amplitude-frequency response functions of the driven and undriven components of the system. Our theoretical analysis is well corroborated by detailed numerical simulations on coupled Gray-Scott reaction-diffusion systems exhibiting antiresonance dip in the amplitude-frequency response curve as a result of destructive interference between the coupling and the periodic external forcing imparting differential stability of the two subsystems. This leads to the emergence of spatiotemporal patterns in an undriven subsystem, while the driven one settles down to a homogeneously stable steady state.

Nanohybrid carriers: the yin–yang equilibrium between natural and synthetic in biomedicine

Nanocarriers are key players in biomedicine applications. The development of hybrid nanoparticles stems from the need to enhance their quality by lowering disadvantages and fusing the positive qualities of both natural and synthetic materials.

Drug delivery from microcapsules: How can we estimate the release time?

Predicting the release performance of a drug delivery device is an important challenge in pharmaceutics and biomedical science. In this paper, we consider a multi-layer diffusion model of drug release from a composite spherical microcapsule into an external surrounding medium. Based on this model, we present two approaches for estimating the release time, i.e. the time required for the drug-filled capsule to be depleted. Both approaches make use of temporal moments of the drug concentration at the centre of the capsule, which provide useful insight into the timescale of the process and can be computed exactly without explicit calculation of the full transient solution of the multi-layer diffusion model. The first approach, which uses the zeroth and first temporal moments only, provides a crude approximation of the release time taking the form of a simple algebraic expression involving the various parameters in the model (e.g. layer diffusivities, mass transfer coefficients, partition coefficients) while the second approach yields an asymptotic estimate of the release time that depends on consecutive higher moments. Through several test cases, we show that both approaches provide a computationally-cheap and useful tool to quantify the release time of composite microcapsule configurations.

Monte Carlo simulations in drug release

We present methods based on simple sampling Monte Carlo simulations that are used in the study of controlled drug release from devices of various shapes and characteristics. The manuscript is part of a special tribute issue for Prof. Panos Macheras and we have chosen applications of the Monte Carlo method in the field of drug release that were pioneered by him and his research group. Thus, we focus on the investigation of diffusion based release and we present methods that go beyond the application of the classical fickian diffusion equation. We describe methods that have proven to be effective in illuminating the profound effects of the substrate heterogeneity on the drug release profiles and demonstrate some of the most powerful applications of agent based simulations and numerical methods in the field of pharmacokinetics.

Polymer Capsules for Enzymatic Catalysis in Confined Environments

Catalysis is at the base of a series of biological and technological application processes. In recent years, the tendency has developed to carry out catalyzed reactions within confined structures, thus forming systems called micro or nanoreactors. Compartmentalized structures are cavities delimited by a wall where specific functions are introduced with a defined concentration and in the desired sites. These containers are generally referred to as nano or microcapsules, assuming the function of reactors in the presence of chemical reactions. Among the various types of existing structures, one of the most interesting is represented by systems made with polymers. This review aims to highlight some of the current advances in the use of functionalized structures that are useful for catalysis reactions, paying particular attention to polymer capsules and enzymes. The built-up methods used for the production of polymer capsules, as well as the aspects that influence membrane permeability and reactivity to environmental conditions, are discussed. Recent advances on biocatalysis confined in polymeric capsules are illustrated, and the strengths and weaknesses of the principal nanoreactors are considered.

ATP Release by Red Blood Cells under Flow: Model and Simulations

ATP is a major player as a signaling molecule in blood microcirculation. It is released by red blood cells (RBCs) when they are subjected to shear stresses large enough to induce a sufficient shape deformation. This prominent feature of chemical response to shear stress and RBC deformation constitutes an important link between vessel geometry, flow conditions, and the mechanical properties of RBCs, which are all contributing factors affecting the chemical signals in the process of vasomotor modulation of the precapillary vessel networks. Several in vitro experiments have reported on ATP release by RBCs due to mechanical stress. These studies have considered both intact RBCs as well as cells within which suspected pathways of ATP release have been inhibited. This has provided profound insights to help elucidate the basic governing key elements, yet how the ATP release process takes place in the (intermediate) microcirculation zone is not well understood. We propose here an analytical model of ATP release. The ATP concentration is coupled in a consistent way to RBC dynamics. The release of ATP, or the lack thereof, is assumed to depend on both the local shear stress and the shape change of the membrane. The full chemo-mechanical coupling problem is written in a lattice-Boltzmann formulation and solved numerically in different geometries (straight channels and bifurcations mimicking vessel networks) and under two kinds of imposed flows (shear and Poiseuille flows). Our model remarkably reproduces existing experimental results. It also pinpoints the major contribution of ATP release when cells traverse network bifurcations. This study may aid in further identifying the interplay between mechanical properties and chemical signaling processes involved in blood microcirculation.

Characteristic time scales for diffusion processes through layers and across interfaces

This paper presents a simple tool for characterizing the time scale for continuum diffusion processes through layered heterogeneous media. This mathematical problem is motivated by several practical applications such as heat transport in composite materials, flow in layered aquifers, and drug diffusion through the layers of the skin. In such processes, the physical properties of the medium vary across layers and internal boundary conditions apply at the interfaces between adjacent layers. To characterize the time scale, we use the concept of mean action time, which provides the mean time scale at each position in the medium by utilizing the fact that the transition of the transient solution of the underlying partial differential equation model, from initial state to steady state, can be represented as a cumulative distribution function of time. Using this concept, we define the characteristic time scale for a multilayer diffusion process as the maximum value of the mean action time across the layered medium. For given initial conditions and internal and external boundary conditions, this approach leads to simple algebraic expressions for characterizing the time scale that depend on the physical and geometrical properties of the medium, such as the diffusivities and lengths of the layers. Numerical examples demonstrate that these expressions provide useful insight into explaining how the parameters in the model affect the time it takes for a multilayer diffusion process to reach steady state.

Modelling mass diffusion for a multi-layer sphere immersed in a semi-infinite medium: application to drug delivery

We present a general mechanistic model of mass diffusion for a composite sphere placed in a large ambient medium. The multi-layer problem is described by a system of diffusion equations coupled via interlayer boundary conditions such as those imposing a finite mass resistance at the external surface of the sphere. While the work is applicable to the generic problem of heat or mass transfer in a multi-layer sphere, the analysis and results are presented in the context of drug kinetics for desorbing and absorbing spherical microcapsules. We derive an analytical solution for the concentration in the sphere and in the surrounding medium that avoids any artificial truncation at a finite distance. The closed-form solution in each concentric layer is expressed in terms of a suitably-defined inverse Laplace transform that can be evaluated numerically. Concentration profiles and drug mass curves in the spherical layers and in the external environment are presented and the dependency of the solution on the mass transfer coefficient at the surface of the sphere analyzed.