Mathematical models for estimating effective diffusion parameters of spherical drug delivery devices

Mathematical modeling of drug delivery is of increasing academic and industrial importance in many aspects. In this paper, we propose an optimization approach for the estimation of the parameters characterizing the diffusion process of a drug from a spherical porous polymer device to an external finite volume. The approach is based on a nonlinear least-squares method and a novel mathematical model which takes into consideration both boundary layer effect and initial burst phenomenon. An analytical solution to the model is derived and a formula for the ratio of the mass released in a given time interval and the total mass released in infinite time is also obtained. The approach has been tested using experimental data of the diffusion of prednisolone 21-hemisuccinate sodium salt from spherical devices made of porous poly(2-hydroxyethyl methacrylate) hydrogels. The effectiveness and accuracy of the method are well demonstrated by the numerical results. The model was used to determine the diffusion parameters including the effective diffusion coefficient of the drug from a series of devices that vary in both the porous structure and the drug loading levels. The computed diffusion parameters are discussed in relation to the physical properties of the devices.     

A simple spectroscopic method for the determination of the release kinetics of drugs from PHB

A simple method is proposed for the determination of the release kinetics of small molecular weight drugs from amorphous PHB. The method uses the hipsochromic shift of the absorbance of active molecules caused by changes in the UV–Vis spectra as an effect of changing environment. Fuchsine with a strong hypsochromic shift was used as model drug in the experiments. A simple experimental setup was created which consists of the positioning of a thin PHB film into the center of the cell of a spectrometer. The light goes through the film and the surrounding solution and records their spectra simultaneously. The arrangement makes possible the quantitative determination of the dissolution of the drug without any further interference. The solution of Fick's second law under the initial and boundary conditions of the experimental setup and the numerical solution of the equation allow the quantitative analysis of the experimental results and the prediction of release kinetics. Excellent agreement was found between prediction and the experimental results. The approach made possible also the determination of the diffusion coefficient of the model drug in amorphous PHB. The developed method can be used for all polymers and with all drugs, which show sufficiently strong hypsochromic shift during their transfer from the polymer to the solution phase.     

Abrasively modified electrodes: mathematical modelling and numerical simulation of electrochemical dissolution/growth processes under cyclic voltammetric conditions

An approximate mathematical model for electrochemical dissolution/growth processes of diffusionally independent and well-separated particles randomly dispersed on an inert conducting electrode surface is presented and solved using numerical simulation. The model, mimicking abrasively modified electrodes where particles of electroactive voltammograms solid are immobilised on an electrode surface, provides clear insights into the effects of different parameters on the voltammetric response of such systems and permits the exploration of the competition taking place between mass transport and surface processes. The mathematical model is then compared with experimental data obtained with basal plane pyrolytic graphite electrode abrasively modified with solid particles of perinaphthenone and studied in aqueous solution.Dedicated to Professor Dr. Alan M. Bond on the occasion of his 60th birthday.     

Mass dissolution of polydisperse set of particles in a flow system

A mathematical model of continuous dissolution of salts based on a dynamic equation for crystal size distribution function was considered. A stationary solution was obtained by the moments method. For a nonstationary state, an evolution equation for the undersaturation of solution was derived. An approximate analytical method of solving this equation, which is in good agreement with the numerical solution, was proposed. The notion of the coefficient of continuous dissolution efficiency was introduced and analytical expressions for this value were obtained. The stability of the stationary solution was analyzed.Translated from Kolloidnyi Zhurnal, Vol. 66, No. 6, 2004, pp. 793–801.Original Russian Text Copyright © 2004 by Moshinskii.     

Novel Supports Based on Polysaccharides for Sustained‐Release of Isosorbide Dinitrate

Novel supports based on carboxymethylcellulose (CMC), crosslinked with epichlorohydrin (EPC), and microparticles based on acetylphthalylcellulose (APC), for sustained‐release of isosorbide dinitrate (Isoket, Ik), were obtained. The drug has been included into CMC hydrogels through diffusion from ethanol‐water solution. Studies about the ethanol–water ratio influence on including the drug have shown an increased amount of included drug at higher content of water in the alcohol‐water mixture. Isoket–ACP microparticles have been obtained by drug and polymer co‐precipitation from emulsified aqueous solution.

The kinetics for “in vitro” release of Ik from polymeric materials, in simulated conditions for intestinal tract medium, where the drug is preferentially absorbed, has been analyzed. The experimental data have shown a “zero” order kinetic for drug release, which is characteristic for systems controlled by diffusion.     

Enhanced dissolution of oxcarbazepine microcrystals using a static mixer process

The purpose of this study was to form micronized powders of Oxcarbazepine (OXC), a poorly water-soluble drug, using a static mixer technique to enhance the dissolution rate. Controlled precipitation was achieved injecting the organic OXC solution rapidly into an aqueous methylcellulose (MC) protective solution by means of a static mixer thus providing turbulent and homogeneous mixing. Furthermore, a factorial design was implemented for data analysis. The physicochemical properties of the freeze-dried dispersions were evaluated by differential scanning calorimetry (DSC), infrared spectroscopy (FTIR) and X-ray diffraction (XRD). Drug microcrystals showed a narrow size distribution with approximately 2 μm mean particle size and high drug loading. DSC and FTIR studies revealed that the drug remained in crystalline state and no drug–polymer interaction occurred. The dissolution studies showed enhanced dissolution of OXC microcrystals compared to the pure drug. The static mixer technique was proved capable for micro-sized polymeric particles. This is an inexpensive, less time consuming and fully scalable process for development of poorly soluble drugs.     

Preparation of Losartan Potassium Controlled Release Matrices and In-Vitro Investigation Using Rate Controlling Agents

Controlled release matrices have predictable drug release kinetics, provide drugs for an extended period of time, and reduce dosing frequency with improved patient compliance as compared with conventional tablet dosage forms. In the current research work, losartan potassium controlled release matrix tablets were fabricated and prepared with rate altering agents; that is, Ethocel grade 100 combined with Carbopol 934PNF. Various drug to polymer ratios were used. HPMC, CMC, and starch were incorporated in some of the matrices by replacing some amount of filler (5%). The direct compression method was adopted for the preparation of matrices. In phosphate buffer (pH 6.8), the dissolution study was conducted by adopting the USP method-I as the specified method. Drug release kinetics was determined and dissolution profiles were also compared with the reference standard. Prolonged release was observed for all matrices, but those with Ethocel 100FP Premium showed more extended release. The co-excipient (HPMC, CMC, and starch) exhibited enhancement in the drug release rates, while all controlled release matrices released the drug by anamolous non-Fickian diffusion mechanism. This combination of polymers (Ethocel grade 100 with Carbopol 934PNF) efficiently extended the drug release rates up to 24 h. It is suggested that these matrix tablets can be given in once a day dosage, which might improve patient compliance, and the polymeric blend of Ethocel grade 100 with Carbopol 934PNF might be used in the development of prolonged release matrices of other water-soluble drugs.     

Application of Surface Diffusion Model to the Adsorption of Dyes on Bagasse Pith

A homogeneous solid phase diffusion model (HSDM) has been developed using a computer to predict the performance of a batch adsorber. The computer program utilises a semi-analytical solution for a two resistance model based on external mass transfer and homogeneous solid phase diffusion. The model has been successfully applied to four adsorption systems, namely, the adsorption of AB25, AR114, BB69 and BR22 onto pith. The method produces excellent correlations between experimental and theoretical concentration decay curves in batch adsorbers. The model developed presents a solution using a single solid diffusion coefficient and a single external mass transfer coefficient which are sufficient to characterise the system within a range of initial dye concentration, 25–300 mg · dm3 and solid/liquid ratios (w/v) 0.25–2.     

Enzymatic erosion of bioartificial membranes to control drug delivery

The preparation of an enzymatic controlled drug release system from blends of PVA/starch/alphaA, in the form of films, is described. It was shown that alphaA hydrolyses the starch within these films, resulting in a time-dependent change of the porosity in the matrix. Films were characterized by calorimetric analysis to study the interactions between the enzyme and the polymeric constituents at the molecular level. The presence of alphaA, in fact, influenced the PVA crystallization in the blends. Release tests and permeability experiments were carried out to evaluate the transport properties of the films. An increase in porosity and permeability was observed by increasing alphaA content (16-28 wt.-%). Films loaded with theophylline and caffeine were also prepared to analyze drug release properties of the matrix. Drug release kinetics were coherent with the measured changes in porosity: at higher alphaA concentrations the amount of released drug increased under the influence of diffusion and erosion processes. The results obtained are promising for the realization of drug delivery devices for a rapid release or for the release of poorly soluble drugs which usually remain entrapped in the matrix.SEM images of a PVA/starch/alphaA film before (A) and after (B) the erosion.     

高分子药物－可乐定阴离子丙烯酸树脂盐的研制及释药性能探索

研制了高分子药物－可乐定阴离子丙烯酸树脂盐并在模拟胃和小肠ｐＨ的缓冲液中进行了高分子药物的溶解释放试验。结果表明其中某些试样能够在微碱性的缓冲液中缓慢溶解释放，可望达到减少用药量。降低副作用和延长药效的目的。     

Immobilization of camptothecin with surfactant into hydrogel for controlled drug release

A new strategy has been developed for the controlled release of a hydrophobic anticancer drug, camptothecin (CPT), which suffers a limited therapeutical utility because of its poor water solubility. CPT was first solubilized in the solution of a cationic surfactant, dodecyltrimethylammonium bromide (DTAB). It has been demonstrated that the presence of DTAB has increased the solubility of CPT significantly. In a 50 mM DTAB solution, the drug’s solubility was enhanced to 85 μM, 22 times of its solubility in pure water. The micellar drug solution of CPT-DTAB was subsequently used to prepare agarose hydrogels, which act as the drug carriers in the release studies. To fully take advantage of the cationic property of DTAB, negatively charged κ-carrageenan was added as a guest polymer in some hydrogel samples. The release of CPT from these hydrogel-surfactant systems was performed at 37 °C and the effects of DTAB and κ-carrageenan on the release of CPT were studied respectively. By fitting to the well-known Fickian diffusion model, the diffusion coefficients of CPT were obtained.     

In vitro release of metoclopramide from hydrophobic matrix tablets. influence of hydrodynamic conditions on kinetic release parameters

There has been growing interest in the subject of drug delivery and the design and evaluation of controlled-release systems. The simplest way to control the release of an active agent is to disperse it in an inert polymeric matrix. Controlled-release systems are of interest because they are technologically simple, relatively cheap, and practically unaffected by physiological changes. In this study, a new matrix system was formed by an active principle, metoclopramide hydrochloride, scattered into a biocompatible hydrophobic polymerical mesh, polyamide 12, to achieve sustained and controlled delivery of metoclopramide hydrochloride. This research was conducted to investigate the in vitro drug release behavior from these new inert polymeric matrix tablets. The drug release process was investigated both experimentally and by means of mathematical models. Different models were applied for the evaluation of drug release data. On the basis of our results, a biexponential equation was proposed, Q=Qfast(1)(1 - e(-Kfast t)) + Qslow(2)(1 - e(-Kslow t)), in an attempt to explain the mechanism responsible for the release process. Additionally, the influence of the experimental conditions of the dissolution devices, such as rate of flow and pH of dissolution medium, on the parameters that characterize the release mechanism was studied, and it was found that the main factor was the hydrodynamic condition of rate of flow.     

Diffusion relaxation times of nonequilibrium isolated small bodies and their solid phase ensembles to equilibrium states

The possibility of obtaining analytical estimates in a diffusion approximation of the times needed by nonequilibrium small bodies to relax to their equilibrium states based on knowledge of the mass transfer coefficient is considered. This coefficient is expressed as the product of the self-diffusion coefficient and the thermodynamic factor. A set of equations for the diffusion transport of mixture components is formulated, characteristic scales of the size of microheterogeneous phases are identified, and effective mass transfer coefficients are constructed for them. Allowing for the developed interface of coexisting and immiscible phases along with the porosity of solid phases is discussed. This approach can be applied to the diffusion equalization of concentrations of solid mixture components in many physicochemical systems: the mutual diffusion of components in multicomponent systems (alloys, semiconductors, solid mixtures of inert gases) and the mass transfer of an absorbed mobile component in the voids of a matrix consisting of slow components or a mixed composition of mobile and slow components (e.g., hydrogen in metals, oxygen in oxides, and the transfer of molecules through membranes of different natures, including polymeric).     

Nanosizing drug particles in supercritical fluid processing

The supercritical fluid-processing technique, rapid expansion of a supercritical solution into a liquid solvent (RESOLV), was applied to the nanosizing of water-insoluble drug particles. Selected for demonstration were antiinflammatory drugs Ibuprofen and Naproxen, for which CO2 and CO2-cosolvent systems were used. The RESOLV process produces exclusively nanoscale (less than 100 nm) Ibuprofen and Naproxen particles suspended in aqueous solutions, and the aqueous suspensions of the drug nanoparticles are protected from particle agglomeration and precipitation by using common polymeric and oligomeric stabilization agents.     

Crystalline inclusion complexes formed between the drug diflunisal and block copolymers

The solid form of drugs plays a central role in optimizing the physicochemical properties of drugs, and new solid forms will provide more options to achieve the desirable pharmaceutical profiles of drugs. Recently, certain drugs have been found to form crystalline inclusion complexes (ICs) with multiple types of linear polymers, representing a new subcategory of pharmaceutical solids. In this study, we used diflunisal (DIF) as the model drug host and extended the guest of drug/polymer ICs from homopolymers to block copolymers of poly(ethylene glycol) (PEG) and poly(ε-caprolactone) (PCL). The block length in the guest copolymers showed a significant influence on the formation, thermal stability and dissolution behavior of the DIF ICs. Though the PEG block could hardly be included alone, it could indeed be included in the DIF ICs when the PCL block was long enough. The increase of the PCL block length produced IC crystals with improved thermal stability. The dissolution profiles of DIF/block copolymer ICs exhibited gradually decreased aqueous solubility and dissolution rate with the increasing PCL block length. These results demonstrate the possibility of using drug/polymer ICs to modulate the desired pharmaceutical profiles of drugs in a predictable and controllable manner.     

Anticancer drug release from poly(N-isopropylacrylamide/itaconic acid) copolymeric hydrogels

The drug uptake and release of anticancer drug from N-isopropylacrylamide/itaconic acid copolymeric hydrogels containing 0–3 mol% of itaconic acid irradiated at 48 kGy have been investigated. 5-Fluorouracil (5-FU) is used as a model anticancer drug. The effect of 5-FU solution on swelling characteristics of PNIPAAm and P(NIPAAm/IA) copolymeric hydrogels have also been studied. The percent swelling, equilibrium swelling, equilibrium water/5-FU content and diffusion constant values are evaluated for poly(N-isopropylacrylamide) (PNIPAAm) and poly(N-isopropylacrylamide/itaconic) (P(NIPAAm/IA)) hydrogels at 130 ppm of 5-FU solution at room temperature. Diffusion of 5-FU solution into the hydrogels has been found to be the non-Fickian type. Finally, the kinetics of drug release from the hydrogels are examined.     

Preparation of thermosensitive polymeric organogels and their drug release behaviors

Stimuli-sensitive drug delivery systems—in particular, stimuli-sensitive polymeric hydrogels swollen with water—have attracted considerable attention in medical and pharmaceutical fields. This study concerns with the synthesis of thermosensitive polymeric organogels for controlled drug release; a copolymerization of stearyl acrylate (SA) with a cross-linker and the loading of indomethacin as a model lipophilic drug were accomplished in oleyl alcohol. The pulsatile (on-off) drug release was successfully conducted: release was halted at 36 °C and release occurred at over ca. 40 °C. This drug release pattern is suitable for thermochemotherapy combined with hyperthermia. The differential scanning calorimetric measurement suggests the following mechanism: the ordered crystalline structure, i.e., the alignment of hydrophobic alkyl side chains, works to prevent indomethacin diffusion from the organogel below the crystallization temperature, while the disordered amorphous structure above the melting temperature allows indomethacin to diffuse.     

Mass transfer coupled with the relaxation of a polymeric matrix: A controlled release process

A solution to Fick's equation is presented which accurately predicts the transfer of mass out of a polymeric rod or sheet undergoing relaxation by a solvent permeating it by Case II transport. There is a critical length. Before the solvent permeates to this length the diffusible material can diffuse away from the moving boundary faster than it is becoming available at the boundary. Afterward the reverse is true. Five sets of experimental data from three different sources have been used to test the model. The agreement is excellent.     

Improved dissolution of an insoluble drug using a 4-fluid nozzle spray-drying technique

A solid dispersion of the drug can be made using a polymer carrier to improve solubility. Generally, drugs become amorphized when solid dispersion is formed using a polymer carrier. In such high energy conditions, the solubility of the drug molecule is increased. We previously prepared solid dispersion using a spray-drying technique and reported its solubility and crystallinity. In this study, hydroxypropylmethylcellulose (HPMC) was used as the carrier, and tolubutamide was the model drug, which is water-insoluble. Solubility was evaluated by preparing a solid dispersion using a newly developed 4-fluid nozzle spray dryer. Observation of particle morphology by scanning electron microscopy (SEM) revealed that the particles from the spray drying were atomized to several microns, and they had also become spherical. Assessment of the crystallinity of the spray-dried particles by powder X-ray diffraction and differential scanning calorimetry demonstrated that the tolbutamide had been amorphized, forming a solid dispersion. The apparent release rate constant K of the drug from the spray-dried particles was 4 to 6 times faster than the original drug in pH 1.2, and it was also 1.5 to 1.9 times faster than the original drug in pH 6.8. The 70% release time (T(70)) of the drug from the spray-dried particles was 20 to 30 times faster than the original drug in pH 1.2 solution as well as 2 to 3 times faster than the original drug in pH 6.8 solution. Pharmaceutical preparations prepared in this way using the 4-fluid nozzle system spray dryer formed composite particles, resulting in a remarkably improved dissolution rates of the drug.     

Diffusivity of amorphous drug in solid dispersion

Filmy solid dispersion of terfenadine (TFD), fenofibrate (FFB), and carbamazepine (CBZ) and methacrylic acid methyl methacrylate copolymer (Eudragit^®) was prepared by evaporating their solution. Raman and IR measurements for the filmy samples were performed. Concentration profile of TFD, FFB, and CBZ in solid dispersions was evaluated by their characteristic peaks, and then their diffusion rate constants were calculated. The start point of the crystallization peak under isothermal condition was determined by XRD–DSC. Viscoelastic character of Eudragit^® was evaluated by dynamic mechanical analysis (DMA). The distribution map of drugs in their solid dispersions showed the diffusion state of drugs during storage. The concentration profile of TFD, FFB, and CBZ in the solid dispersion was calculated from obtained mapping data. The diffusion rate constant of both drug in Eudragit^® EPO was higher than that in Eudragit^® RLPO. The induction period of crystallization from amorphous CBZ was gradually delayed with increasing amounts of Eudragit^®. The IR peak due to C=O was shifted to higher wave number; it suggested that there were some molecular interactions between CBZ and Eudragit^®. From the results of the change in the interaction of drug-Eudragit^®, it may be concluded that the diffusivity of drug molecule in polymer closely related to the delay of the induction period of crystallization of amorphous. DMA measurement clarified the difference in the viscosity of Eudragit^® having different functional groups and molecular mass. These results suggested that the retardation of crystallization by Eudragit^® could be related to the sample viscosity.