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

Adsorption and Permeation Events in Molecular Diffusion

How many times can a diffusing molecule permeate across a membrane or be adsorbed on a substrate? We employ an encounter-based approach to find the statistics of adsorption or permeation events for molecular diffusion in a general confining medium. Various features of these statistics are illustrated for two practically relevant cases: a flat boundary and a spherical confinement. Some applications of these fundamental results are discussed.

Drug polymer conjugates: Average release time from thin films

The reaction-diffusion problem describing the release of drugs conjugated through labile bonds to polymeric thin films has a known analytical solution, when the reaction kinetics is of first order. Using this solution, an exact formula is derived for the average release time of the system. This simple expression provides the characteristic time of release tav as the sum of the corresponding average diffusion time plus the inverse reaction rate constant: tav=(1/12)⋅(L2/D)+(1/k), where L is the slab thickness, D the diffusion coefficient, and k the reaction rate constant. The former term dominates in a diffusion-controlled release, while the latter one in a reaction-controlled delivery. The crossover regime is exactly described by their sum. The obtained result for the average release time is verified by direct numerical integration through the drug release profiles of the analytical solution. The value of fractional drug release at the characteristic average time is between 60-64%. These results can be used for the design of polymer-drug conjugates with a desired delivery time scale, as well as for the experimental determination of the values of microscopic parameters D and k in a conjugated system of interest.

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.

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.

Release of molecules from nanocarriers

Release of drugs or vaccine molecules from macro-, micro-, and nano-sized carriers is usually considered to be limited by diffusion and/or carrier dissolution and/or erosion. The corresponding experimentally observed kinetics are customarily fitted by using the empirical Weibull and Korsemeyer-Peppas expressions. With decreasing size of carriers down to about 100 nm, the timescale of diffusion decreases, and accordingly the release can be kinetically limited, i.e., controlled by jumps of molecules located near the carrier-solution interface. In addition, nanocarriers (e.g., lipid nanoparticles) are often structurally heterogeneous so that the absorption of molecules there can be interpreted in terms of energetic heterogeneity, i.e., distribution of energies corresponding to binding sites and activation barriers for release. Herein, I present a general kinetic model aimed at such situations. For illustration, the deviation of the molecule binding energy from the maximum value was considered to be about 4-8 kcal mol-1. With this physically reasonable (for non-covalent interaction) scale of energetic heterogeneity, the predicted kinetics (i) are linear in the very beginning and then, with increasing time, become logarithmic and (ii) can be nearly perfectly fitted by employing the Weibull or Korsmeyer-Peppas expressions with the exponent in the range from 0.6 to 0.75. Such values of the exponent are often obtained in experiments and customarily associated with non-Fickian diffusion. My analysis shows that the energetic heterogeneity can be operative here as well.

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.

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.

A Bayesian approach to estimate the diffusion coefficient of Rhodamine 6G in breast cancer spheroids

Multicellular spheroids have emerged as a robust platform to model tumor growth and are widely used for studying drug sensitivity. Diffusion is the main mechanism for transporting nutrients and chemotherapeutic drugs into spheroids, since they are typically avascular. In this study, the Bayesian inference was used to solve the inverse problem of determining the light attenuation coefficient and diffusion coefficient of Rhodamine 6G (R6G) in breast cancer spheroids, as a mock drug for the tyrosine kinase inhibitor, Neratinib. Four types of breast cancer spheroids were formed and the diffusion coefficient was estimated assuming a linear relationship between the intensity and concentration. The mathematical model used for prediction is the solution to the diffusion problem in spherical coordinates, accounting for the light attenuation. The Gaussian likelihood was used to account for the error between the measurements and model predictions. The Markov Chain Monte Carlo algorithm (MCMC) was used to sample from the posterior. The posterior predictions for the diffusion and light attenuation coefficients were provided. The results indicate that the diffusion coefficient values do not significantly vary across a HER2+ breast cancer cell line as a function of transglutaminase 2 levels, even in the presence of fibroblast cells. However, we demonstrate that different diffusion coefficient values can be ascertained from tumorigenic compared to nontumorigenic spheroids and from nonmetastatic compared to post-metastatic breast cancer cells using this approach. We also report agreement between spheroid radius, attenuation coefficient, and subsequent diffusion coefficient to give evidence of cell packing in self-assembled spheroids. The methodology presented here will allow researchers to determine diffusion in spheroids to decouple transport and drug penetration changes from biological resistivity.

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.

A Langevin dynamics approach for multi-layer mass transfer problems

We use Langevin dynamics simulations to study the mass diffusion problem across two adjacent porous layers of different transport properties. At the interface between the layers, we impose the Kedem-Katchalsky (KK) interfacial boundary condition that is well suited in a general situation. A detailed algorithm for the implementation of the KK interfacial condition in the Langevin dynamics framework is presented. As a case study, we consider a two-layer diffusion model of a drug-eluting stent. The simulation results are compared with those obtained from the solution of the corresponding continuum diffusion equation, and an excellent agreement is shown.

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.

Intracellular RNA delivery by lipid nanoparticles: Diffusion, degradation, and release

Various RNAs (siRNAs, miRNAs, or mRNAs) can be delivered into cells by lipid nanoparticles (LNPs) of 50-150 nm in diameter. The subsequent RNA release from LNPs may occur via various scenarios. Herein, two related kinetic models are proposed. The first model takes into account that LNPs are often porous so that RNA molecules diffuse in and detach from nanopores. The analysis is focused on RNA diffusion from a pore. The analytical expression obtained for the RNA escape rate constant is used to identify the difference in the release of siRNAs, miRNAs, and mRNAs. The key message here is that the mRNA diffusion from pores appears to be too slow, and accordingly the mRNA release seems to occur primarily via degradation of LNPs. The second coarse-grained model describes the diffusion-mediated release of RNA from a LNP in the situation when this process is accompanied by the LNP degradation at the lipid-solution interface. The corresponding kinetics are shown in detail at different relative rates of the RNA diffusion and LNP degradation. Potentially, this can help to interpret drug plasma levels after various dosing regimens.

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.

Investigation of the Size Distribution for Diffusion-Controlled Drug Release From Drug Delivery Systems of Various Geometries

Various drug delivery systems (DDSs) are often used in modern medicine to achieve controlled and targeted drug release. Diffusional release of drugs from DDSs is often the main mechanism, especially at early times. Generally, average dimensions of DDS are used to model the drug release, but our recent work on drug release from fibers demonstrated that taking into account diameter distribution is essential. This work systematically investigated the effect of size distribution on diffusional drug release from DDSs of various geometric forms such as membranes, fibers, and spherical particles. The investigation clearly demonstrated that the size distribution has the largest effect on the drug release profiles from spherical particles compared to other geometric forms. Published experimental data for drug release from polymer microparticles and nanoparticles were fitted, and the diffusion coefficients were determined assuming reported radius distributions. Assuming the average radius when fitting the data leads to up to 5 times underestimation of the diffusion coefficient of drug in the polymer.