Aristides Dokoumetzidis

Analysis of Dissolution Data Using Modified Versions of Noyes–Whitney Equation and the Weibull Function

PurposeThe aim of the study is to develop modified, branched versions of the Noyes–Whitney and the Weibull equations, including explicitly the solubility/dose parameter, for the analysis of dissolution data, which reach the plateau either at infinite or finite time.MethodsThe modified Weibull function is applied to the analysis of experimental and literature dissolution data. To demonstrate the usefulness of the mathematical models, two model drugs are used: one highly soluble, metoprolol, and one relatively insoluble, ibuprofen.ResultsThe models were fitted successfully to the data performing better compared with their classic versions. The advantages of the use of the models presented are several. They fit better to a large range of datasets, especially for fast dissolution curves that reach complete dissolution at a finite time. Also, the modified Weibull presented can be derived from differential equations, and it has a physical meaning as opposed to the purely empirical character of the original Weibull equation. The exponent of the Weibull equation can be attributed to the heterogeneity of the process and can be explained by fractal kinetics concepts. Also, the solubility/dose ratio is present explicitly as a parameter and allows to obtain estimates of the solubility even when the dissolution data do not reach the solubility level.ConclusionThe use of the developed branched equations gives better fittings and specific physical meaning to the dissolution parameters. Also, the findings underline the fact that even in the simplest, first-order case, the speed of the dissolution process depends on the dose, a fact of great importance in biopharmaceutic classification for regulatory purposes.

Identification of biowaivers among Class II drugs: Theoretical justification and practical examples

AbstractPurpose. To set up a theoretical basis for identifying biowaivers among Class II drugs and apply the methodology developed to nonsteroidal anti-inflammatory drugs (NSAIDs). Methods. The dynamics of the two consecutive drug processes dissolution and wall permeation are considered in the time domain of the physiologic transit time using a tube model of the intestinal lumen. The model considers constant permeability along the intestines, a plug flow fluid with the suspended particles moving with the fluid, and dissolution in the small particle limit. The fundamental differential equation of drug dissolution-uptake in the intestines is expressed in terms of the fraction of dose dissolved. Results. The fundamental parameters, which define oral drug absorption in humans resulting from this analysis, are i) the formulation-related factors, dose, particle radius size, and ii) the drug-related properties, dimensionless solubility/dose ratio (1/q), and effective permeability. Plots of dose as a function of (1/q) for various particle sizes unveil the specific values of these meaningful parameters, which ensure complete absorption for Class II drugs [(1/q) < 1]. A set of NSAIDs were used to illustrate the application of the approach in identifying biowaivers among the NSAIDs. Conclusions. The underlying reason for a region of fully absorbed drugs in Class II originates from the dynamic character of the dissolution-uptake processes. The dynamic character of the approach developed allows identification of biowaivers among Class II drugs. Several biowaivers among the NSAIDs were identified using solubility data at pH 5.0 and in fed-state-simulated intestinal fluid at pH 5.0. The relationships of formulation parameters, dose, particle radius, and the drug properties, dimensionless solubility/dose ratio (1/q), and permeability with the fraction of dose absorbed for drugs with low 1/q values [(1/q) < 1] can be used as guidance for the formulation scientist in the development phase.

The mean dissolution time depends on the dose/solubility ratio

AbstractPurpose. To investigate the relationship between mean dissolution time (MDT) and dose/solubility ratio (q) using the diffusion layer model. Methods. Using the classic Noyes-Whitney equation and considering a finite dose, we derived an expression for MDT as a function of q under various conditions. q was expressed as a dimensionless quantity by taking into account the volume of the dissolution medium. Our results were applied to in vitro and in vivo data taken from literature. Results. We found that MDT depends on q when q > 1 and is infinite when q > 1 and that the classic expression of MDT = 1/k, where k is the dissolution rate constant, holds only in the special case of q = 1. For the case of perfect sink conditions, MDT was found to be proportional to dose. Using dissolution data from literature with q < 1, we found better estimates of MDT when dependency on dose/solubility ratio was considered than with the classic approach. Prediction of dissolution limited absorption was achieved for some of the in vivo drug examples examined. Conclusion. The mean dissolution time of a drug depends on dose/solubility ratio, even when the model considered is the simplest possible. This fact plays an important role in drug absorption when absorption is dissolution limited.

Nonlinear dynamics and chaos theory: concepts and applications relevant to pharmacodynamics.

The theory of nonlinear dynamical systems (chaos theory), which deals with deterministic systems that exhibit a complicated, apparently random-looking behavior, has formed an interdisciplinary area of research and has affected almost every field of science in the last 20 years. Life sciences are one of the most applicable areas for the ideas of chaos because of the complexity of biological systems. It is widely appreciated that chaotic behavior dominates physiological systems. This is suggested by experimental studies and has also been encouraged by very successful modeling. Pharmacodynamics are very tightly associated with complex physiological processes, and the implications of this relation demand that the new approach of nonlinear dynamics should be adopted in greater extent in pharmacodynamic studies. This is necessary not only for the sake of more detailed study, but mainly because nonlinear dynamics suggest a whole new rationale, fundamentally different from the classic approach. In this work the basic principles of dynamical systems are presented and applications of nonlinear dynamics in topics relevant to drug research and especially to pharmacodynamics are reviewed. Special attention is focused on three major fields of physiological systems with great importance in pharmacotherapy, namely cardiovascular, central nervous, and endocrine systems, where tools and concepts from nonlinear dynamics have been applied.

A commentary on fractionalization of multi-compartmental models

Fractional calculus, the branch of calculus that deals with derivatives of non-integer order, e.g., a half derivative, allows the formulation of fractional differential equations (FDEs), capable of describing a range of phenomena, most of them related in one way or another to anomalous diffusion processes [1, 2]. FDEs have recently found application in the field of pharmacokinetics (PK), since the presence of non-classical, anomalous kinetics has been established years ago and many articles have appeared in the literature trying to quantify these processes by the use of either empirical power-laws or fractal kinetics [3, 4]. Fractional pharmacokinetics (fPK) was first described by Dokoumetzidis and Macheras in [5] where the concept was introduced for a simple ‘‘one-compartment’’ model that gave rise to a Mittag-Leffler function (MLF). The MLF has very nice properties since it behaves as a power law for large time scales but as an exponential for small times, hence the MLF can describe kinetic data that follows power law terminal kinetics without presenting problems for t = 0. However, if we want to write models in more physiological terms then eventually we will need a formulation with more than one compartments. The first attempt to write multicompartmental fPK models was done by Popovic et al. [6], who basically fractionalized the classic compartmental pharmacokinetics in a straightforward manner, i.e., they generalized the classic first order derivatives found on the left hand side of ordinary differential equations (ODEs) by replacing them with fractional derivatives. But is this change in the order of the derivatives all that is needed to establish correct and consistent fPK models? In this note we will demonstrate that it is not.

Development of a reaction-limited model of dissolution: application to official dissolution tests experiments.

A reaction-limited model for drug dissolution is developed assuming that the reaction at the solid-liquid interface is controlling the rate of dissolution. The dissolution process is considered as a bidirectional chemical reaction of the undissolved drug species with the free solvent molecules, yielding the dissolved species of drug complex with solvent. This reaction was considered in either sink conditions, where it corresponds to the unidirectional case and the entire amount of the drug is dissolved, or reaching chemical equilibrium, which corresponds to saturation of the solution. The model equation was fitted successfully to dissolution data sets of naproxen and nitrofurantoin formulations measured in the paddle and basket apparatuses, respectively, under various experimental conditions. For comparative purposes these data were also analyzed using three functions based on the diffusion layer model. All functions failed to reveal the governing role of saturation solubility in the dissolution process associated with the diffusion layer model when the conditions for the valid estimation of saturation solubility, established theoretically in this study, were met by the experimental set up employed. Overall, the model developed provides an interesting alternative to the classic approaches of drug dissolution modeling, quantifying the case of reaction-limited dissolution of drugs.

Predictive models for oral drug absorption: from in silico methods to integrated dynamical models.

Poor oral absorption is one of the most common reasons for a drug to be terminated during development. Oral drug absorption is a complex process affected by many competing factors related to the compound, the formulation and the gastrointestinal physiology. Throughout drug development, in silico, computational and mathematical models play important roles in the support of drug development and decision making in absorption-related issues. These models range from simple empirical rule of thumb tools to sophisticated dynamic systems. This article reviews the different computational methods for oral drug absorption for the various processes, with emphasis on solubility, permeability, dissolution and release rates, and gastrointestinal transit, but also on the modern integrated absorption prediction systems and computer software.

A Model for Transport and Dispersion in the Circulatory System Based on the Vascular Fractal Tree

AbstractMaterials are distributed throughout the body of mammals by fractal networks of branching tubes. Based on the scaling laws of the fractal structure, the vascular tree is reduced to an equivalent one-dimensional, tube model. A dispersion–convection partial differential equation with constant coefficients describes the heterogeneous concentration profile of an intravascular tracer in the vascular tree. A simple model for the mammalian circulatory system is built in entirely physiological terms consisting of a ring shaped, one-dimensional tube which corresponds to the arterial, venular, and pulmonary trees, successively. The model incorporates the blood flow heterogeneity of the mammalian circulatory system. Model predictions are fitted to published concentration-time data of indocyanine green injected in humans and dogs. Close agreement was found with parameter values within the expected physiological range.

On the ubiquitous presence of fractals and fractal concepts in pharmaceutical sciences: a review.

Fractals have been very successful in quantifying natures geometrical complexity, and have captured the imagination of scientific community. The development of fractal dimension and its applications have produced significant results across a wide variety of biomedical applications. This review deals with the application of fractals in pharmaceutical sciences and attempts to account the most important developments in the fields of pharmaceutical technology, especially of advanced Drug Delivery nano Systems and of biopharmaceutics and pharmacokinetics. Additionally, fractal kinetics, which has been applied to enzyme kinetics, drug metabolism and absorption, pharmacokinetics and pharmacodynamics are presented. This review also considers the potential benefits of using fractal analysis along with considerations of nonlinearity, scaling, and chaos as calibration tools to obtain information and more realistic description on different parts of pharmaceutical sciences. As a conclusion, the purpose of the present work is to highlight the presence of fractal geometry in almost all fields of pharmaceutical research.

The Changing Face of the Rate Concept in Biopharmaceutical Sciences: From Classical to Fractal and Finally to Fractional

The time course of drug in the body is dynamic. A large number of processes, based on fundamental physicochemical principles, are involved from the initial disintegration of the tablet and the dissolution of the active ingredient, to the pharmacological effect of drug. However, irrespective of the detailed characteristics, the common and most principal component of the underlying mechanism of numerous drug processes is diffusion. The diffusion of molecules at the microscopic level results in the observed “flux” at the macroscopic level and further to the “ rate” of the process, which is the crux of the matter for the present commentary.