Modeling Drug Concentration Level in Blood Using Fractional Differential Equation Based on Psi‐Caputo Derivative

Abstract

This article studies a pharmacokinetics problem, which is the mathematical modeling of a drug concentration variation in human blood, starting from the injection time. Theories and applications of fractional calculus are the main tools through which we establish main results. The psi-Caputo fractional derivative plays a substantial role in the study. We prove the existence and uniqueness of the solution to the problem using the psi-Caputo fractional derivative. The application of the theoretical results on two data sets shows the following results. For the first data set, a psi-Caputo with the kernel ψ = x + 1 is the best approach as it yields a mean square error (MSE) of 0.04065 . The second best is the simple fractional method whose MSE is 0.05814 ; finally, the classical approach is in the third position with an MSE of 0.07299 . For the second data set, a psi-Caputo with the kernel ψ = x + 1 is the best approach as it yields an MSE of 0.03482 . The second best is the simple fractional method whose MSE is 0.04116 and, finally, the classical approach with an MSE of 0.048640 .

Modeling blood alcohol concentration using fractional differential equations based on the ψ‐Caputo derivative

“…Such result can be improved using our fractional model (9). Indeed, by other choices of function 𝜓 and 𝛼 and 𝛽 in ( 9), the solution of our fractional model ( 9) can be closer to the real data of Table 1 than the solution obtained by the classical model (14). To measure that, we follow Rosales et al [30] and define the gain  of the efficiency of our model, comparing the error (19) of the classical model, E classical , with the error (13) associated to a particular fractional instance of our model ( 9):…”

“…The error (13) between the real data of Table 1 and the values of Table 2 obtained by the classical model ( 14) is 775 (mg∕L) 2 . However, as shown in Qureshi et al [12], these results can be improved by choosing A 0 = 261.721, k 1 = 0.111946, k 2 = 0.0186294, (18) for which model (14) gives the values of Table 3, decreasing the error from 775 (mg/L) 2 to…”

“…Using the real experimental data of blood alcohol level of Table 1, Almeida et al [11] used a numerical optimization approach based on the least squares approximation to determine the orders 𝛼 and 𝛽 of the fractional Caputo operator that better describes the real data. They proved that the Caputo fractional model ( 21) fits better the available data when compared with the classical one given by (14). Moreover, in 2019, Qureshi et al [12] considered not only the Caputo fractional operator, which has a singular kernel, but also nonsingular kernels: They investigated the use of the ABC and the CF kernels to fractionalize the classical model.…”

“…Ludwin [28] tried to fit the experimental data given in Table 1 using the classical model (14). Using an Excel solver, Ludwin found the values…”

On Coupled System of Langevin Fractional Problems with Different Orders of μ-Caputo Fractional Derivatives

“…As indicated in the above system, we present this study with a µ-Caputo fractional derivative operator (FDO), which is a generalization of the Riemann-Liouville FDO. Below, we highlight some of its advantages, which have been discussed in various research papers and articles in the field of fractional calculus and its applications (see e.g., [1,2,[35][36][37]):…”