Mathematical modeling of drug delivery

Due to the significant advances in information technology mathematical modeling of drug delivery is a field of steadily increasing academic and industrial importance with an enormous future potential. The in silico optimization of novel drug delivery systems can be expected to significantly increase in accuracy and easiness of application. Analogous to other scientific disciplines, computer simulations are likely to become an integral part of future research and development in pharmaceutical technology. Mathematical programs can be expected to be routinely used to help optimizing the design of novel dosage forms. Good estimates for the required composition, geometry, dimensions and preparation procedure of various types of delivery systems will be available, taking into account the desired administration route, drug dose and release profile. Thus, the number of required experimental studies during product development can be significantly reduced, saving time and reducing costs. In addition, the quantitative analysis of the physical, chemical and potentially biological phenomena, which are involved in the control of drug release, offers another fundamental advantage: The underlying drug release mechanisms can be elucidated, which is not only of academic interest, but a pre-requisite for an efficient improvement of the safety of the pharmaco-treatments and for effective trouble-shooting during production. This article gives an overview on the current state of the art of mathematical modeling of drug delivery, including empirical/semi-empirical and mechanistic realistic models. Analytical as well as numerical solutions are described and various practical examples are given. One of the major challenges to be addressed in the future is the combination of mechanistic theories describing drug release out of the delivery systems with mathematical models quantifying the subsequent drug transport within the human body in a realistic way. Ideally, the effects of the design parameters of the dosage form on the resulting drug concentration time profiles at the site of action and the pharmacodynamic effects will become predictable.     

Biodegradable polymeric drug delivery systems

The use of biodegradable polymeric materials as drug carriers is a relatively new dimension in polymeric drug delivery systems. A number of biodegradable or bioerodible polymers, such as poly (lactic/glycolic acid) copolymer, poly(α-amino acid), polyanhydride, and poly (ortho ester) are currently being investigated for this purpose. These polymers are useful for matrix and reservoir-type delivery devices. In addition, when chemical functional groups are introduced to the biodegradable polymer backbone, such as poly (N-(2-hydroxypropy) methacrylamide), the therapeutic agent can be covalently bound directly orvia spacer to the backbone polymer. These polymer/drug conjugates represent another new dimension in biodegradable polymeric drug delivery systems. In this paper, major emphasis is placed on clinical applications of biodegradable polymeric delivery systems. In addition, examples of biodegradable polymeric durg delivery systems currently being investigated will be discussed for the purpose of demonstrating the potential importance of this new field.     

Implantable,polymeric systems for modulated drug delivery

The ability to deliver therapeutic agents to a patient in a pulsatile or staggered release profile has been a major goal in drug delivery research over the last two decades. This review will cover methods that have been developed to control drug delivery profiles with implantable polymeric systems. Externally and internally controlled systems will be discussed, spanning a range of technologies that include pre-programmed systems, as well as systems that are sensitive to modulated enzymatic or hydrolytic degradation, pH, magnetic fields, ultrasound, electric fields, temperature, light and mechanical stimulation. Implantable systems have the potential to improve the quality of life for patients undergoing therapy with a variable dosing regime by eliminating the need for multiple intravenous injections. Ideally, these systems would also result in increased patient compliance with a given therapy due to the relative ease of self-dosing.     

Mathematical modeling of drug release from bioerodible microparticles: effect of gamma-irradiation.

Bioerodible polymers used in controlled drug delivery systems, such as poly(lactic-co-glycolic acid) (PLGA) undergo radiolytic degradation during gamma-irradiation. In spite of the considerable practical importance, yet only little knowledge is available on the consequences of this sterilization method on the resulting drug release patterns in a quantitative way. The major objectives of the present study were: (i) to monitor the effects of different gamma-irradiation doses on the physicochemical properties of drug-free and drug-loaded, PLGA-based microparticles; (ii) to analyze the obtained experimental results using adequate mathematical models; (iii) to get further insight into the occurring physical and chemical phenomena; and (iv) to relate the applied gamma-irradiation dose in a quantitative way to the resulting drug release rate. 5-Fluorouracil-loaded, PLGA-based microparticles were prepared with an oil-in-water solvent extraction method and exposed to gamma-irradiation doses ranging from 0 to 33 kGy. Size exclusion chromatography, differential scanning calorimetry, scanning electron microscopy, particle size analysis, determination of the actual drug loading and in vitro drug release kinetics were used to study the effects of the gamma-irradiation dose on the physicochemical properties of the microparticles. Two mathematical models-a simplified and a more comprehensive one-were used to analyze the experimental results. The simplified model considers drug diffusion based on Fick's second law for spherical geometry and a Higuchi-like pseudo-steady-state approach. The complex model combines Monte Carlo simulations (describing polymer erosion) with partial differential equations quantifying drug diffusion with time-, position- and direction-dependent diffusivities. Interestingly, exponential relationships between the gamma-irradiation dose and the initial drug diffusivity within the microparticles could be established. Based on this knowledge both models were used to predict the resulting drug release kinetics as a function of the gamma-irradiation dose. Importantly, the theoretical predictions were confirmed by experimental results.     

Mathematical modeling and simulation of drug release from microspheres: Implications to drug delivery systems

This article aims to provide a comprehensive review of existing mathematical models and simulations of drug release from polymeric microspheres and of drug transport in adjacent tissues. In drug delivery systems, mathematical modeling plays an important role in elucidating the important drug release mechanisms, thus facilitating the development of new pharmaceutical products by a systematic, rather than trial-and-error, approach. The mathematical models correspond to the known release mechanisms, which are classified as diffusion-, swelling-, and erosion-controlled systems. Various practical applications of these models which explain experimental data are illustrated. The effect of gamma-irradiation sterilization on drug release mechanism from erosion-controlled systems will be discussed. The application of existing models to nanoscale drug delivery systems specifically for hydrophobic and hydrophilic molecules is evaluated. The current development of drug transport modeling in tissues utilizing computational fluid dynamics (CFD) will also be described.     

生物可降解纳米药物转运系统研究进展

纳米药物载体有延长药物作用时间、增加疗效、降低毒副作用、缓控释给药等优点。而生物可降解高分子材料因其良好的生物利用度、载药能力和控释能力以及较低的毒性而被广泛用于纳米药物。本文综述了聚乳酸-羟基乙酸共聚物(PLGA)、聚乳酸(PLA)、聚己内酯(PCL)高分子化合物制备纳米粒的合成和载药方法及应用。     

Intelligent drug delivery systems: polymeric micelles and hydrogels

Advanced drug delivery systems try to adjust the site and/or the rate of the release to the physiological conditions of the patient, to the progression of the illness, or to the circadian rhythms. Being different from classical pre-programmed controlled release dosage forms, the new devices aim to provide the drug release profile best for the needs of each patient. Intelligent drug delivery systems are mostly based on stimuli-responsive polymers which sense a change in a specific variable and activate the delivery; this phenomenon being reversible. This review reports on recent advances in the development of open-loop and closed-loop control systems based on stimuli-responsive polymers and their application in the drug delivery field as pulsatile and self-regulated devices. The aim of this review is to describe the most recent advances in the development of intelligent micelles and hydrogels which are sensitive to pH, specific molecules (with a mention to the molecular imprinting), temperature, irradiation or electric field, and the applications of which these mechanisms are intended.     

Mathematical modeling of polymer erosion: consequences for drug delivery

Bioerodible polymers have been extensively used as carriers for drug delivery and as scaffolds for tissue engineering. The ability to model and predict erosion behavior can enable the rational design and optimization of biomaterials for various biomedical applications in vivo. This review examines critically the current approaches in mathematical modeling of the erosion of synthetic polymers. The models are classified broadly based on whether they use phenomenological, probabilistic, or empirical approaches. An analysis of the various physical, chemical, and biological factors affecting polymer erosion and the classes of bioerodible polymers to which these analyses have been applied are discussed. The key features and assumptions associated with each of the models are described, and information is provided on the limitations of the models and the various approaches. The review concludes with several directions for future models of polymer erosion.     

Applications of polymeric nanocapsules in field of drug delivery systems

Drug-loaded polymeric nanocapsules have exhibited potential applications in the field of drug delivery systems in recent years. This article entails the biodegradable polymers generally used for preparing nanocapsules, which include both natural polymers and synthetic polymers. Furthermore, the article presents a general review of the different preparation methods: nanoprecipitation method, emulsion-diffusion method, double emulsification method, emulsion-coacervation method, layer-by-layer assembly method. In addition, the analysis methods of nanocapsule characteristics, such as mean size, morphology, surface characteristics, shell thickness, encapsulation efficiency, active substance release, dispersion stability, are mentioned. Also, the applications of nanocapsules as carriers for use in drug delivery systems are reviewed, which primarily involve targeting drug delivery, controlled/sustained release drug delivery systems, transdermal drug delivery systems and improving stability and bioavailability of drugs. Nanocapsules, prepared with different biodegradable polymers, have received more and more attention and have been regarded as one of the most promising drug delivery systems.     

Mechanisms by which cyclodextrins modify drug release from polymeric drug delivery systems

The objective of this research was to examine the human sub-bronchial gland cell line, Calu-3, and assess its potential as a metabolic and transport model to study drug delivery to the respiratory epithelium. The present studies were conducted using Calu-3 cells grown in Transwells(R) or in multiwell cluster plates. TEER values for Calu-3 monolayers were determined using the World Precision Instrument Voltohmmeter and STX-2 electrode. The results confirmed that Calu-3 cells form tight monolayers and give appreciable TEER values in culture when grown under air-interface conditions. Permeability data for small lipophilic molecules across Calu-3 monolayers suggested that the cell line is a suitable model to examine the transport of low molecular weight substances and xenobiotics. Calu-3 cells were also found to efflux FITC-transferrin (MW 80000) in a polarized manner. The metabolic capacity of Calu-3 cells was also examined. The P4501A1 and P4502B isozymes were determined to be functional, but not inducible, with fluorescent resorufin assays. The data indicated that the Calu-3 cell line may be useful for studying the contributions of bronchial epithelial cells to mechanisms of drug delivery at the respiratory epithelium.     

Responsive polymeric delivery systems

This paper discusses the state of the art in a relatively new approach in the field of controlled drug delivery-responsive polymeric drug delivery systems. Such systems are capable of adjusting drug release rates in response to a physiological need. The fundamental principles of externally and self-regulated delivery systems are examined. Special attention is paid to specific clinical settings such as diabetes, presenting the advantages and disadvantages of different approaches.     

Responsive polymeric delivery systems

This paper discusses the state of the art in a relatively new approach in the field of controlled drug delivery–responsive polymeric drug delivery systems. Such systems are capable of adjusting drug release rates in response to a physiological need. The fundamental principles of externally and self-regulated delivery systems are examined. Special attention is paid to specific clinical settings such as diabetes, presenting the advantages and disadvantages of different approaches.     

Mathematical modeling of surface-active and non-surface-active drug transport in emulsion systems

Mathematical models were developed for the prediction of surface-active and non- surface-active drug transport in triphasic (oil, water, and micellar) emulsion systems as a function of micellar concentration. These models were evaluated by comparing experimental and simulated data. Fick's first law of diffusion with association of the surface-active or complexation nature of the drug with the surfactant was used to derive a transport model for surface-active drugs. This transport model assumes that the oil/water (O/W) partitioning process was fast compared with membrane transport and therefore drug transport was limited by the membrane. Consecutive rate equations were used to model transport of non- surface-active drugs in emulsion systems assuming that the O/W interface acts as a barrier to drug transport. Phenobarbital (PB) and barbital (B) were selected as surface-active model drugs. Phenylazoaniline (PAA) and enzocaine (BZ) were selected as non- surface-active model drugs. Transport studies at pH 7.0 were conducted using side-by-side diffusion cells and bulk equilibrium reverse dialysis bag techniques. According to the surface-active drug model, an increase in micellar concentration is expected to decrease drug-transport rates. Using the Microft EXCEL program, the non- surface-active drug model was fitted to the experimental data for the cumulative amount of the model drug that disappeared from the donor chamber. The oil/continuous phase partitioning rates (k₁) and the membrane transport rates (k₂) were estimated. The predicted data were consistent with the experimental data for both the surface-active and non- surface-active models.     

Mathematical modeling of drug dissolution

The dissolution of a drug administered in the solid state is a pre-requisite for efficient subsequent transport within the human body. This is because only dissolved drug molecules/ions/atoms are able to diffuse, e.g. through living tissue. Thus, generally major barriers, including the mucosa of the gastro intestinal tract, can only be crossed after dissolution. Consequently, the process of dissolution is of fundamental importance for the bioavailability and, hence, therapeutic efficacy of various pharmaco-treatments. Poor aqueous solubility and/or very low dissolution rates potentially lead to insufficient availability at the site of action and, hence, failure of the treatment in vivo, despite a potentially ideal chemical structure of the drug to interact with its target site. Different physical phenomena are involved in the process of drug dissolution in an aqueous body fluid, namely the wetting of the particle's surface, breakdown of solid state bonds, solvation, diffusion through the liquid unstirred boundary layer surrounding the particle as well as convection in the surrounding bulk fluid. Appropriate mathematical equations can be used to quantify these mass transport steps, and more or less complex theories can be developed to describe the resulting drug dissolution kinetics. This article gives an overview on the current state of the art of modeling drug dissolution and points out the assumptions the different theories are based on. Various practical examples are given in order to illustrate the benefits of such models. This review is not restricted to mathematical theories considering drugs exhibiting poor aqueous solubility and/or low dissolution rates, but also addresses models quantifying drug release from controlled release dosage forms, in which the process of drug dissolution plays a major role.     

Microneedle-based drug delivery systems: Microfabrication,drug delivery,and safety

Many promising therapeutic agents are limited by their inability to reach the systemic circulation, due to the excellent barrier properties of biological membranes, such as the stratum corneum (SC) of the skin or the sclera/cornea of the eye and others. The outermost layer of the skin, the SC, is the principal barrier to topically-applied medications. The intact SC thus provides the main barrier to exogenous substances, including drugs. Only drugs with very specific physicochemical properties (molecular weight < 500?Da, adequate lipophilicity, and low melting point) can be successfully administered transdermally. Transdermal delivery of hydrophilic drugs and macromolecular agents of interest, including peptides, DNA, and small interfering RNA is problematic. Therefore, facilitation of drug penetration through the SC may involve by-pass or reversible disruption of SC molecular architecture. Microneedles (MNs), when used to puncture skin, will by-pass the SC and create transient aqueous transport pathways of micron dimensions and enhance the transdermal permeability. These micropores are orders of magnitude larger than molecular dimensions, and, therefore, should readily permit the transport of hydrophilic macromolecules. Various strategies have been employed by many research groups and pharmaceutical companies worldwide, for the fabrication of MNs. This review details various types of MNs, fabrication methods and, importantly, investigations of clinical safety of MN.     

Mathematical modeling of intraperitoneal drug delivery: simulation of drug distribution in a single tumor nodule

The intraperitoneal (IP) administration of chemotherapy is an alternative treatment for peritoneal carcinomatosis, allowing for higher intratumor concentrations of the cytotoxic agent compared to intravenous administration. Nevertheless, drug penetration depths are still limited to a few millimeters. It is thus necessary to better understand the limiting factors behind this poor penetration in order to improve IP chemotherapy delivery. By developing a three-dimensional computational fluid dynamics (CFD) model for drug penetration in a tumor nodule, we investigated the impact of a number of key parameters on the drug transport and penetration depth during IP chemotherapy. Overall, smaller tumors showed better penetration than larger ones, which could be attributed to the lower IFP in smaller tumors. Furthermore, the model demonstrated large improvements in penetration depth by subjecting the tumor nodules to vascular normalization therapy, and illustrated the importance of the drug that is used for therapy. Explicitly modeling the necrotic core had a limited effect on the simulated penetration. Similarly, the penetration depth remained virtually constant when the Darcy permeability of the tissue changed. Our findings illustrate that the developed parametrical CFD model is a powerful tool providing more insight in the drug transport and penetration during IP chemotherapy.