Finite element analysis of the release of slowly dissolving drugs from cylindrical matrix systems

Modelling the controlled drug release of push-pull osmotic pump tablets using DEM

The push-pull osmotic pump tablet is a promising drug delivery approach, offering advantages over traditional dosage forms in achieving consistent and predictable drug release rates. In the current study, the drug release process of push-pull osmotic pump tablets is modelled for the first time using the discrete element method (DEM) incorporated with a microscopic diffusion-induced swelling model. The effects of dosage and formulation design, such as delivery orifice size, drug-to-polymer ratio, tablet surface curvature, friction between particles and cohesion of polymer particles, on the drug release performance are systematically analysed. Numerical results reveal that an enlarged delivery orifice significantly increases both the total drug release and the drug release rate. Moreover, the larger the swellable particle component in the tablet, the higher the drug release rate. Furthermore, the tablet surface curvature is found to affect the drug release profile, i.e. the final drug release percentage increases with the increasing tablet surface curvature. It is also found that the drug release rate could be controlled by adjusting the inter-particle friction and the cohesion of polymer particles in the formulation. This DEM study offers valuable insights into the mechanisms governing drug release in push-pull osmotic pump tablets.

The Effect of Polymer-Solvent Interaction on the Swelling of Polymer Matrix Tablets: A Magnetic Resonance Microscopy Study Complemented by Bond Fluctuation Model Simulations

Polymer matrix tablets are an important drug-delivery system widely used for oral drug administration. Understanding the tablet hydration process, both experimentally and theoretically, is, thus, very important for the development of drug delivery systems that exhibit high drug loading capacity and controlled release potential. In this study, we used magnetic resonance microscopy (MRM) to nondestructively and dynamically analyze the water hydration process of xanthan-based tablets. The swelling process was characterized by well-resolved fronts of erosion, swelling, and penetration. The experimental results were complemented by numerical simulations of the polymer matrix hydration process. In the simulations, the polymer tablet matrix was modeled as an assembly of interacting chains with embedded drug particles, while its hydration process was mediated by interaction with solvent particles. The swelling dynamics were modeled within a Monte Carlo-based bond fluctuation model (BFM) that elegantly accounted for steric and nearest-neighbor interactions. This study provides an efficient experimental-theoretical approach for the study of polymer matrix swelling processes.

A new modelling approach for dissolution of polydisperse powders

A new modelling approach for dissolution of polydisperse powders is developed within the framework of the classical Noyes-Whitney/Nernst-Brunner analysis. Its distinguishing feature is that the underlying continuous particle-size distribution is retained. Two different but related dependencies of the diffusion-layer thickness on particle size are considered. First, a power-law dependence that interpolates between a thickness that is proportional to (or equals) the particle radius (obtained when the exponent equals 1) and a constant thickness (obtained when the exponent is 0). Second, a piecewise linear function such that the thickness equals the particle radius for sufficiently small particles and is constant for larger ones. The modelling approach is exemplified by consideration of a lognormal particle-size distribution. Highly accurate closed-form expressions for the fraction of dissolved drug are obtained for dissolution under sink conditions (which are exact if the diffusion-layer thickness is radius-independent). Moreover, it is demonstrated that any result derived under sink conditions can be reused to determine the fraction of dissolved/absorbed drug under non-sink conditions, using the concept of a retarded time. Comparison with literature data and experiments are used to validate the modelling approach and to demonstrate its usefulness in a practical context.

Triggered and controlled release of bioactives in food applications

Bioactive compounds (e.g., nutraceuticals, micronutrients, antimicrobial, antioxidant) are added to food products and formulations to enhance sensorial/nutritional attributes and/or shelf-life. Many of these bioactives are susceptible to degradation when exposed to environmental and processing factors. Others involve in undesirable interactions with food constituents. Encapsulation is a useful tool for addressing these issues through various stabilization mechanisms. Besides protection, another important requirement of encapsulation is to design a carrier that predictably releases the encapsulated bioactive at the target site to elicit its intended functionality. To this end, controlled release carrier systems derived from interactive materials have been developed and commercially exploited to meet the requirements of various applications. This chapter provides an overview on basic controlled and triggered release concepts relevant to food and active packaging applications. Different approaches to encapsulate bioactive compounds and their mode of release are presented, from simple blending with a compatible matrix to complex multiphase carrier systems. To further elucidate the mass transport processes, selected diffusion and empirical release kinetic models are presented, along with their brief historical significance. Finally, interactive carriers that are responsive to moisture, pH, thermal and chemical stimuli are presented to illustrate how these triggered release mechanisms can be useful for food applications.

Model for the Analysis of Membrane-Type Dissolution Tests for Inhaled Drugs

Impactor-type dose deposition is a common prerequisite for dissolution testing of inhaled medicines, and drug release typically takes place through a membrane. The purpose of this work is to develop a mechanistic model for such combined dissolution and release processes, focusing on a drug that initially is present in solid form. Our starting points are the Noyes–Whitney (or Nernst–Brunner) equation and Fick’s law. A detailed mechanistic analysis of the drug release process is provided, and approximate closed-form expressions for the amount of the drug that remains in solid form and the amount of the drug that has been released are derived. Comparisons with numerical data demonstrated the accuracy of the approximate expressions. Comparisons with experimental release data from literature demonstrated that the model can be used to establish rate-controlling release mechanisms. In conclusion, the model constitutes a valuable tool for the analysis of in vitro dissolution data for inhaled drugs.

Drug Release from a Spherical Matrix: Theoretical Analysis for a Finite Dissolution Rate Affected by Geometric Shape of Dispersed Drugs

Amending the neglect of finite dissolution in traditional release models, this study proposed a more generalized drug release model considering the simultaneous dissolution and diffusion procedure from a drug-loaded spherical matrix. How the shape factor (n = 0, 1/2, and 2/3 for the planar, cylindrical, and spherical geometry, respectively) of dispersed drug particles affected the release from the matrix was examined for the first time. Numerical solutions of this generalized model were validated by consensus with a short-time analytical solution for planar drugs and by the approach of the diffusion-controlled limits with Higuchi’s model. The drug release rate increases with the ratio of dissolution/diffusion rate (G) and the ratio of solubility/drug loading (K) but decreases with the shape factor of drug particles. A zero-order release profile is identified for planar drugs before starting the surface depletion layer, and also found for cylindrical and spherical dispersed drugs when K and G are small, i.e. the loaded drug is mainly un-dissolved and the drug release rate is dissolution-controlled. It is also shown that for the case of a small G value, the variation of drug release profile, due to the drug particle geometry, becomes prominent. Detailed comparison with the results of the traditional Higuchi’s model indicates that Higuchi’s model can be applied only when G is large because of the assumption of an instantaneous dissolution. For K = 1/101–1/2, the present analysis suggests an error of 33–85% for drug release predicted by Higuchi’s model for G = 10⁰, 14–44% error for G = 10¹, while a less than 5% error for G ≧ 10³.

Continuum Modeling and Simulation in Bone Tissue Engineering

Bone tissue engineering is currently a mature methodology from a research perspective. Moreover, modeling and simulation of involved processes and phenomena in BTE have been proved in a number of papers to be an excellent assessment tool in the stages of design and proof of concept through in-vivo or in-vitro experimentation. In this paper, a review of the most relevant contributions in modeling and simulation, in silico, in BTE applications is conducted. The most popular in silico simulations in BTE are classified into: (i) Mechanics modeling and scaffold design, (ii) transport and flow modeling, and (iii) modeling of physical phenomena. The paper is restricted to the review of the numerical implementation and simulation of continuum theories applied to different processes in BTE, such that molecular dynamics or discrete approaches are out of the scope of the paper. Two main conclusions are drawn at the end of the paper: First, the great potential and advantages that in silico simulation offers in BTE, and second, the need for interdisciplinary collaboration to further validate numerical models developed in BTE.

An overview on the mathematical modeling of hydrogels' behavior for drug delivery systems

Hydrogels-based systems (HBSs) for drug delivery are nowadays extensively used and the interest in modeling their behavior is dramatically increasing. In this review a critical overview on the modeling approaches is given, quantitatively and qualitatively analyzing the publications on the subject, the trend of the publications per year and the type of modeling approaches. It was found that, despite the drug release fitting models (i.e. Higuchi's equation) are the most abundant, their use for HBSs is decreasing in the last years and luckily, considering the limiting assumption on which they were built, they will be confined to simple mathematical fitting equations. Within the mechanistic models the "multi-component" with the swelling approximation (mass transport only) and with the mechanics (fully coupled) are experiencing the highest growth rate, with much more interest toward the last one that, in the next years could be able to provide a first principles model. Statistical models, especially based on the response surface methodology, are rapidly spreading in the scientific community mainly thanks to their ability to be predictive, regardless of the phenomenology, in the analyzed design space with very low efforts. Neural Networks models for HBSs, in countertrend with their use in the pharmaceutical industry, have never take off preferring less data demanding statistical models.

Model of dissolution in the framework of tissue engineering and drug delivery

Dissolution phenomena are ubiquitously present in biomaterials in many different fields. Despite the advantages of simulation-based design of biomaterials in medical applications, additional efforts are needed to derive reliable models which describe the process of dissolution. A phenomenologically based model, available for simulation of dissolution in biomaterials, is introduced in this paper. The model turns into a set of reaction-diffusion equations implemented in a finite element numerical framework. First, a parametric analysis is conducted in order to explore the role of model parameters on the overall dissolution process. Then, the model is calibrated and validated versus a straightforward but rigorous experimental setup. Results show that the mathematical model macroscopically reproduces the main physicochemical phenomena that take place in the tests, corroborating its usefulness for design of biomaterials in the tissue engineering and drug delivery research areas.

Advances in mechanistic understanding of release rate control mechanisms of extended-release hydrophilic matrix tablets

Approaches to characterizing and developing understanding around the mechanisms that control the release of drugs from hydrophilic matrix tablets are reviewed. While historical context is provided and direct physical characterization methods are described, recent advances including the role of percolation thresholds, the application on magnetic resonance and other spectroscopic imaging techniques are considered. The influence of polymer and dosage form characteristics are reviewed. The utility of mathematical modeling is described. Finally, how all the information derived from applying the developed mechanistic understanding from all of these tools can be brought together to develop a robust and reliable hydrophilic matrix extended-release tablet formulation is proposed.

Fickian-Based Empirical Approach for Diffusivity Determination in Hollow Alginate-Based Microfibers Using 2D Fluorescence Microscopy and Comparison with Theoretical Predictions

Hollow alginate microfibers (od = 1.3 mm, id = 0.9 mm, th = 400 µm, L = 3.5 cm) comprised of 2% (w/v) medium molecular weight alginate cross-linked with 0.9 M CaCl₂ were fabricated to model outward diffusion capture by 2D fluorescent microscopy. A two-fold comparison of diffusivity determination based on real-time diffusion of Fluorescein isothiocyanate molecular weight (FITC MW) markers was conducted using a proposed Fickian-based approach in conjunction with a previously established numerical model developed based on spectrophotometric data. Computed empirical/numerical (D_empiricial/D_numerical) diffusivities characterized by small standard deviations for the 4-, 70- and 500-kDa markers expressed in m²/s are (1.06 × 10⁻⁹ ± 1.96 × 10⁻¹⁰)/(2.03 × 10⁻¹¹), (5.89 × 10⁻¹¹ ± 2.83 × 10⁻¹²)/(4.6 × 10⁻¹²) and (4.89 × 10⁻¹² ± 3.94 × 10⁻¹³)/(1.27 × 10⁻¹²), respectively, with the discrimination between the computation techniques narrowing down as a function of MW. The use of the numerical approach is recommended for fluorescence-based measurements as the standard computational method for effective diffusivity determination until capture rates (minimum 12 fps for the 4-kDa marker) and the use of linear instead of polynomial interpolating functions to model temporal intensity gradients have been proven to minimize the extent of systematic errors associated with the proposed empirical method.

Numerical modelling of transdermal delivery from matrix systems: parametric study and experimental validation with silicone matrices

A model is presented for transdermal drug delivery from single-layered silicone matrix systems. The work is based on our previous results that, in particular, extend the well-known Higuchi model. Recently, we have introduced a numerical transient model describing matrix systems where the drug dissolution can be non-instantaneous. Furthermore, our model can describe complex interactions within a multi-layered matrix and the matrix to skin boundary. The power of the modelling approach presented here is further illustrated by allowing the possibility of a donor solution. The model is validated by a comparison with experimental data, as well as validating the parameter values against each other, using various configurations with donor solution, silicone matrix and skin. Our results show that the model is a good approximation to real multi-layered delivery systems. The model offers the ability of comparing drug release for ibuprofen and diclofenac, which cannot be analysed by the Higuchi model because the dissolution in the latter case turns out to be limited. The experiments and numerical model outlined in this study could also be adjusted to more general formulations, which enhances the utility of the numerical model as a design tool for the development of drug-loaded matrices for trans-membrane and transdermal delivery.

Combinatorial approach in the design of multifunctional polymeric nano-delivery systems for cancer therapy

This update summarizes the recent advances in combinatorial design of polymeric material for developing multifunctional nanovectors to deliver nucleic acids and chemodrugs for cancer therapy.

Numerical modelling and experimental investigation of drug release from layered silicone matrix systems

Medical devices and polymeric matrix systems that release drugs or other bioactive compounds are of interest for a variety of applications. The release of the drug can be dependent on a number of factors such as the solubility, diffusivity, dissolution rate and distribution of the solid drug in the matrix. Achieving the goal of an optimal release profile can be challenging when relying solely on traditional experimental work. Accurate modelling complementing experimentation is therefore desirable. Numerical modelling is increasingly becoming an integral part of research and development due to the significant advances in computer simulation technology. This work focuses on numerical modelling and investigation of multi-layered silicone matrix systems. A numerical model that can be used to model multi-layered systems was constructed and validated by comparison with experimental data. The model could account for the limited dissolution rate and effect of the drug distribution on the release profiles. Parametric study showed how different factors affect the characteristics of drug release. Multi-layered medical silicone matrices were prepared in special moulds, where the quantity of drug in each layer could be varied, and release was investigated with Franz-diffusion cell setup. Data for long-term release was fitted to the model and the full depletion of the system predicted. The numerical model constructed for this study, whose input parameters are the diffusion, effective dissolution rate and dimensional solubility coefficients, does not require any type of steady-state approximation. These results indicate that numerical model can be used as a design tool for development of controlled release systems such as drug-loaded medical devices.

Nanocarrier Hydrodynamics and Binding in Targeted Drug Delivery: Challenges in Numerical Modeling and Experimental Validation

This review discusses current progress and future challenges in the numerical modeling of targeted drug delivery using functionalized nanocarriers (NC). Antibody coated nanocarriers of various size and shapes, also called functionalized nanocarriers, are designed to be injected in the vasculature, whereby they undergo translational and rotational motion governed by hydrodynamic interaction with blood particulates as well as adhesive interactions mediated by the surface antibody binding to target antigens/receptors on cell surfaces. We review current multiscale modeling approaches rooted in computational fluid dynamics and nonequilibrium statistical mechanics to accurately resolve fluid, thermal, as well as adhesive interactions governing nanocarrier motion and their binding to endothelial cells lining the vasculature. We also outline current challenges and unresolved issues surrounding the modeling methods. Experimental approaches in pharmacology and bioengineering are discussed briefly from the perspective of model validation.

A retrospective mathematical analysis of controlled release design and experimentation

The development and performance evaluation of new biodegradable polymer controlled release formulations relies on successful interpretation and evaluation of in vitro release data. However, depending upon the extent of empirical characterization, release data may be open to more than one qualitative interpretation. In this work, a predictive model for release from degradable polymer matrices was applied to a number of published release data in order to extend the characterization of release behavior. Where possible, the model was also used to interpolate and extrapolate upon collected released data to clarify the overall duration of release and also kinetics of release between widely spaced data points. In each case examined, mathematical predictions of release coincide well with experimental results, offering a more definitive description of each formulation's performance than was previously available. This information may prove particularly helpful in the design of future studies, such as when calculating proper dosing levels or determining experimental end points in order to more comprehensively evaluate a controlled release system's performance.

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.

An approach to give prospective life-span of the copper/low-density-polyethylene nanocomposite intrauterine device

As a novel copper-containing intrauterine device (IUD), the prospective life-span of the copper/low-density-polyethylene (Cu/LDPE) nanocomposite IUD is very important for the future clinical use and should be given in advance. Here a novel approach, cupric ions accelerated release in diluted nitric acid solution and cupric ions concentration release in various volume of simulated uterine solution (SUS), is reported to verify the type of cupric ions release model of the cylindrical matrix-type nanocomposite IUD, and to obtain the minimal cupric ions release rate that need to ensure contraceptive efficacy and the thickness of copper particles exhausted layer of the cylindrical matrix-type nanocomposite IUD within two difficult immersion durations in experimental volume of SUS, respectively. Using these results, the prospective life-span of the cylindrical matrix-type nanocomposite IUD can be obtained. For instance, the prospective life-span of the novel γ-shape nanocomposite IUD with 25 wt% of copper nanoparticles and 2 mm of diameter and a total weight of 285 mg can be given in advance and it is about 5 years in the future clinical use.

Modelling of diclofenac sodium diffusion from swellable and water-soluble polyethylene oxide matrices

Objectives

The main objective of this study was to develop a mathematical model for the characterization of diclofenac sodium diffusion from polyethylene oxide (PEO) matrices. A model was developed on the basis of the diffusion theory accounting for the characteristics of the polymer: swelling with subsequent dissolution in water. The concentration-dependent diffusion of drug and water was taken into account. Experimental data were analysed using a computer software program as an aid for solving partial differential equations.

Methods

Six formulations of matrix tablets with different drug-excipient ratios were prepared using low-molecular-weight PEO as a matrix-forming material. For obtaining drug release data, dissolution studies were performed and water uptake by pure PEO matrices was studied as well.

Key findings

A good agreement of the developed model with experimental results was demonstrated. Some anomalies in drug diffusion were observed and their origin was questioned. Changes in the parameters characterizing the process of diffusion are attributed to glassy-rubbery polymer transitions. Additional interpretation of this phenomenon on the basis of percolation theory is also provided.

Conclusions

The obtained model has the ability to predict the required characteristics of matrices for desired drug release. The composition of batches with undesirable release properties can be predetermined and avoided in manufacturing.

A unified mathematical model for the prediction of controlled release from surface and bulk eroding polymer matrices

A unified model has been developed to predict release not only from bulk eroding and surface eroding systems but also from matrices that transition from surface eroding to bulk eroding behavior during the course of degradation. This broad applicability is afforded by fundamental diffusion/reaction equations that can describe a wide variety of scenarios including hydration of and mass loss from a hydrolysable polymer matrix. Together, these equations naturally account for spatial distributions of polymer degradation rate. In this model paradigm, the theoretical minimal size required for a matrix to exhibit degradation under surface eroding conditions was calculated for various polymer types and then verified by empirical data from the literature. An additional set of equations accounts for dissolution- and/or degradation-based release, which are dependent upon hydration of the matrix and erosion of the polymer. To test the model's accuracy, predictions for agent egress were compared to experimental data from polyanhydride and polyorthoester implants that were postulated to undergo either dissolution-limited or degradation-controlled release. Because these predictions are calculated solely from readily attainable design parameters, it seems likely that this model could be used to guide the design controlled release formulations that produce a broad array of custom release profiles.

Percolative drug diffusion from cylindrical matrix systems with unsealed boundaries

Release of NaCl in both the axial and radial directions from cylindrical ethyl cellulose tablets were investigated by the alternating ionic current method. The pore structure of the investigated binary mixtures was examined by mercury porosimetry and scanning electron microscopy, and the nm range fractal surface dimension of tablet pore walls was extracted from krypton gas adsorption isotherms. The drug release was shown to consist of two overlapping processes of which the first was ascribed to dissolution of NaCl close to the tablet boundary followed by subsequent diffusion through a thin ethyl cellulose layer and a second from which a porosity percolation threshold of 0.22 could be extracted. As well, a cross-over to effective-medium behaviour at a porosity of approximately 0.44 was observed. The presented findings showed that drug release from matrix tablets with unsealed tablet walls substantially differs from earlier investigated release processes for which the drug has only been allowed to escape through one of the flat tablet surfaces. Thus, the present study brings forward knowledge important for the tailoring of controlled drug delivery vehicles with optimum release patterns.

Modeling doxorubicin transport to improve intratumoral drug delivery to RF ablated tumors

A mathematical model of drug transport provides an ideal strategy to optimize intratumoral drug delivery implants to supplement radiofrequency (RF) ablation for tumor treatment. To simulate doxorubicin transport in non-ablated and ablated liver tumors, a one-dimensional, cylindrically symmetric transport model was generated using a finite element method (FEM). Parameters of this model, the diffusion (D) and elimination (gamma) coefficients for doxorubicin, were estimated using drug distributions measured 4 and 8 days after placing biodegradable implants in non-ablated and ablated rabbit VX2 liver carcinomas. In non-ablated tumor, values of diffusion and elimination parameters were 25% and 94% lower than normal liver tissue, respectively. In ablated tumor, diffusion near the ablation center was 75% higher than non-ablated tumor but decreased to the non-ablated tumor value at the ablation periphery. Drug elimination in ablated tumor was zero for the first four days, but by day 8 returned to 98% of the value for non-ablated tumor. Three-dimensional (3-D) simulations of drug delivery from implants with and without RF thermal ablation underscore the benefit of using RF ablation to facilitate local drug distribution. This study demonstrates the use of computational modeling and optimal parameter estimation to predict local drug pharmacokinetics from intratumoral implants after ablation.