Fractional kinetics in drug absorption and disposition processes

Abstract

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-Order Closed-Loop Model Reference Adaptive Control for Anesthesia

“…We can observe that all schemes met the control objective-taking all of the patients to BIS = 50. It can be seen that the controllers based on the second-order (14) and third-order models (15) had better performances than the controllers based on the first-order structure (13). Figure 10 shows the control input of the four implemented controllers and it a similar behavior can be seen between the controllers.…”

“…Therefore, with the identification scheme and the simulation presented, it is shown that the proposed models were able to capture the input-output behavior of the presented example. Moreover, it was shown that the simple fractional-order models (13)(14)(15) are able to capture the responses of the patients, as illustrated in Figure 4.…”

“…In reference [17], the authors present the relationship between the diffusion process and fractional-order models. Also, the introduction of fractional-order pharmacokinetic models are presented in references [15,16,18,19] which represent the experimental data in a more precise way, thanks to the t −α decay of the fractional operators, and with this, a new line of investigation was opened ion the area of drug delivery systems.…”

“…In [15,70,71], it was suggested that biological systems (like in pharmacology and bioengineering) could be represented with a fractional-order model with a more simple structure compared with the integer-order counterpart, simplifying the control design by using a less complex model.…”

“…In the case of the FOCMRAC scheme, three controllers were implemented, each one based on the models (13)(14)(15) respectively, and applied to the 30 virtual patients. Figure 9 shows the response of the 30 virtual patients with the four control schemes.…”