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# Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

Muath Awadalla1,*, Kinda Abuasbeh1, Yves Yannick Yameni Noupoue2, Mohammed S. Abdo3
1 Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf, Al Ahsa, 31982, Saudi Arabia
2 Institute of Statistics, Biostatistics and Actuarial Science, Faculty of Science, Universite Catholique de Louvain, Louvain-La-Neuve, 1348, Belgium
3 Department of Mathematics, Hodeidah University, Al-Hudaydah, 3114, Yemen
* Corresponding Author: Muath Awadalla. Email:
(This article belongs to this Special Issue: Applications of Fractional Operators in Modeling Real-world Problems: Theory, Computation, and Applications)

Computer Modeling in Engineering & Sciences 2023, 135(3), 2767-2785. https://doi.org/10.32604/cmes.2023.024036

Received 22 May 2022; Accepted 12 August 2022; Issue published 23 November 2022

### Abstract

This study focuses on the dynamics of drug concentration in the blood. In general, the concentration level of a drug in the blood is evaluated by the mean of an ordinary and first-order differential equation. More precisely, it is solved through an initial value problem. We proposed a new modeling technique for studying drug concentration in blood dynamics. This technique is based on two fractional derivatives, namely, Caputo and Caputo-Fabrizio derivatives. We first provided comprehensive and detailed proof of the existence of at least one solution to the problem; we later proved the uniqueness of the existing solution. The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques. Stability via the Ulam-Hyers (UH) technique was also investigated. The application of the proposed model on two real data sets revealed that the Caputo derivative was more suitable in this study. Indeed, for the first data set, the model based on the Caputo derivative yielded a Mean Squared Error (MSE) of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360. Caputo-Fabrizio-basedderivative appeared to be the second-best method for the problem, with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532. For the second experiment, Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066, whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299. Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach.

### Keywords

Pharmacokinetics; caputo fractional derivative; stability study; Caputo-Fabrizio fractional derivative

Awadalla, M., Abuasbeh, K., Yannick, Y., Abdo, M. S. (2023). Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives. CMES-Computer Modeling in Engineering & Sciences, 135(3), 2767–2785.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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