Dissolution research started to develop about 100 years ago as a field of physical chemistry and since then important progress has been made. However, explicit interest in drug related dissolution has grown only since the realisation that dissolution is an important factor of drug bioavailability in the 1950s. This review attempts to account the most important developments in the field, from a historical point of view. It is structured in a chronological order, from the theoretical foundations of dissolution, developed in the first half of the 20th century, and the development of a relationship between dissolution and bioavailability in the 1950s, going to the more recent developments in the framework of the Biopharmaceutics Classification System (BCS). Research on relevant fields of pharmaceutical technology, like sustained release formulations, where drug dissolution plays an important role, is reviewed. The review concludes with the modern trends on drug dissolution research and their regulatory implications.
We explore the use of fractional order differential equations for the analysis of datasets of various drug processes that present anomalous kinetics, i.e. kinetics that are non-exponential and are typically described by power-laws. A fractional differential equation corresponds to a differential equation with a derivative of fractional order. The fractional equivalents of the "zero-" and "first-order" processes are derived. The fractional zero-order process is a power-law while the fractional first-order process is a Mittag-Leffler function. The latter behaves as a stretched exponential for early times and as a power-law for later times. Applications of these two basic results for drug dissolution/release and drug disposition are presented. The fractional model of dissolution is fitted successfully to datasets taken from literature of in vivo dissolution curves. Also, the proposed pharmacokinetic model is fitted to a dataset which exhibits power-law terminal phase. The Mittag-Leffler function describes well the data for small and large time scales and presents an advantage over empirical power-laws which go to infinity as time approaches zero. The proposed approach is compared conceptually with fractal kinetics, an alternative approach to describe datasets with non exponential kinetics. Fractional kinetics offers an elegant description of anomalous kinetics, with a valid scientific basis, since it has already been applied in problems of diffusion in other fields, and describes well the data.
Fractional calculus, the branch of calculus dealing with derivatives of non-integer order (e.g., the half-derivative) allows the formulation of fractional differential equations (FDEs), which have recently been applied to pharmacokinetics (PK) for one-compartment models. In this work we extend that theory to multi-compartmental models. Unlike systems defined by a single ordinary differential equation (ODE), considering fractional multi-compartmental models is not as simple as changing the order of the ordinary derivatives of the left-hand side of the ODEs to fractional orders. The latter may produce inconsistent systems which violate mass balance. We present a rationale for fractionalization of ODEs, which produces consistent systems and allows processes of different fractional orders in the same system. We also apply a method of solving such systems based on a numerical inverse Laplace transform algorithm, which we demonstrate that is consistent with analytical solutions when these are available. As examples of our approach, we consider two cases of a basic two-compartment PK model with a single IV dose and multiple oral dosing, where the transfer from the peripheral to the central compartment is of fractional order α < 1, accounting for anomalous kinetics and deep tissue trapping, while all other processes are of the usual order 1. Simulations with the studied systems are performed using the numerical inverse Laplace transform method. It is shown that the presence of a transfer rate of fractional order produces a non-exponential terminal phase, while multiple dose and constant infusion systems never reach steady state and drug accumulation carries on indefinitely. The IV fractional system is also fitted to PK data and parameter values are estimated. In conclusion, our approach allows the formulation of systems of FDEs, mixing different fractional orders, in a consistent manner and also provides a method for the numerical solution of these systems.
An algorithm for automatic order reduction of models defined by large systems of differential equations is presented. The algorithm was developed with systems biology models in mind and the motivation behind it is to develop mechanistic pharmacokinetic/pharmacodynamic models from already available systems biology models. The approach used for model reduction is proper lumping of the system's states and is based on a search through the possible combinations of lumps. To avoid combinatorial explosion, a heuristic, greedy search strategy is employed and comparison with the full exhaustive search provides evidence that it performs well. The method takes advantage of an apparent property of this kind of systems that lumps remain consistent over different levels of order reduction. Advantages of the method presented include: the variables and parameters of the reduced model retain a specific physiological meaning; the algorithm is automatic and easy to use; it can be used for nonlinear models and can handle parameter uncertainty and constraints. The algorithm was applied to a model of NF-B signalling pathways in order to demonstrate its use and performance. Significant reduction was achieved for the example model, while agreement with the original model was proportional to the size of the reduced model, as expected. The results of the model reduction were compared with a published, intuitively reduced model of NF-B signalling pathways and were found to be in agreement, in terms of the identified key species for the system's kinetic behaviour. The method may provide useful insights which are complementary to the intuitive reduction approach, especially in large systems.
We are witnessing the birth of a new variety of pharmacokinetics where non-integer-order differential equations are employed to study the time course of drugs in the body: this is dubbed "fractional pharmacokinetics". The presence of fractional kinetics has important clinical implications such as the lack of a half-life, observed, for example with the drug amiodarone and the associated irregular accumulation patterns following constant and multiple-dose administration. Building models that accurately reflect this behaviour is essential for the design of less toxic and more effective drug administration protocols and devices. This article introduces the readers to the theory of fractional pharmacokinetics and the research challenges that arise. After a short introduction to the concepts of fractional calculus, and the main applications that have appeared in literature up to date, we address two important aspects. First, numerical methods that allow us to simulate fractional order systems accurately and second, optimal control methodologies that can be used to design dosing regimens to individuals and populations.
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