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Modeling Blood Alcohol Concentration Using Fractional Differential Equations Based on the $ψ$-Caputo Derivative

Om Kalthoum Wanassi, Delfim F. M. Torres

TL;DR

The paper tackles modeling blood alcohol concentration (BAL) by deploying a fractional calculus framework based on the $\psi$-Caputo derivative to capture memory effects in gastric absorption and distribution to the blood. By applying a generalized Laplace transform, it derives explicit analytic solutions for BAL dynamics and demonstrates that the $\psi$-Caputo model encompasses classical and Caputo cases as specializations. Empirical fitting to BAL data shows substantial improvements: the best $\psi$-Caputo setup with an affine kernel $\psi(t)=a_1+a_2 t$ achieves BAL error below $202$ $(\mathrm{mg/L})^2$, a $59\%$ gain over the previous best, with even further gains using non-affine kernels like $\psi(t)=(t+0.5)^{0.97}$. These results underscore the value of kernel choice in fractional models for physiological data and point to broader applications of generalized $\psi$-Caputo operators in biomedical dynamics.

Abstract

We propose a novel dynamical model for blood alcohol concentration that incorporates $ψ$-Caputo fractional derivatives. Using the generalized Laplace transform technique, we successfully derive an analytic solution for both the alcohol concentration in the stomach and the alcohol concentration in the blood of an individual. These analytical formulas provide us a straightforward numerical scheme, which demonstrates the efficacy of the $ψ$-Caputo derivative operator in achieving a better fit to real experimental data on blood alcohol levels available in the literature. In comparison to existing classical and fractional models found in the literature, our model outperforms them significantly. Indeed, by employing a simple yet non-standard kernel function $ψ(t)$, we are able to reduce the error by more than half, resulting in an impressive gain improvement of 59 percent.

Modeling Blood Alcohol Concentration Using Fractional Differential Equations Based on the $ψ$-Caputo Derivative

TL;DR

The paper tackles modeling blood alcohol concentration (BAL) by deploying a fractional calculus framework based on the -Caputo derivative to capture memory effects in gastric absorption and distribution to the blood. By applying a generalized Laplace transform, it derives explicit analytic solutions for BAL dynamics and demonstrates that the -Caputo model encompasses classical and Caputo cases as specializations. Empirical fitting to BAL data shows substantial improvements: the best -Caputo setup with an affine kernel achieves BAL error below , a gain over the previous best, with even further gains using non-affine kernels like . These results underscore the value of kernel choice in fractional models for physiological data and point to broader applications of generalized -Caputo operators in biomedical dynamics.

Abstract

We propose a novel dynamical model for blood alcohol concentration that incorporates -Caputo fractional derivatives. Using the generalized Laplace transform technique, we successfully derive an analytic solution for both the alcohol concentration in the stomach and the alcohol concentration in the blood of an individual. These analytical formulas provide us a straightforward numerical scheme, which demonstrates the efficacy of the -Caputo derivative operator in achieving a better fit to real experimental data on blood alcohol levels available in the literature. In comparison to existing classical and fractional models found in the literature, our model outperforms them significantly. Indeed, by employing a simple yet non-standard kernel function , we are able to reduce the error by more than half, resulting in an impressive gain improvement of 59 percent.
Paper Structure (9 sections, 3 theorems, 32 equations, 1 figure, 7 tables)

This paper contains 9 sections, 3 theorems, 32 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Blood alcohol model and its analytical solution
  5. Application
  6. The classical integer order model
  7. The Caputo fractional order model
  8. The $\psi$-Caputo fractional order model
  9. Conclusion

Key Result

Lemma 7

Let $\alpha >0$, $f \in AC_{\psi}^{n}[a,b]$ for any $b>a$, and $f^{[k]}$, $k=0,1,\ldots n$, be of $\psi(t)$-exponential order. Then,

Figures (1)

  • Figure 1: Blood alcohol level comparison between the real data of Table \ref{['tab1']} and the predictions obtained from the best fractional models \ref{['PL']} with $\psi(t) = t$ (Caputo) and $\psi(t)=0.621767 t$.

Theorems & Definitions (13)

  • Definition 1: The $\psi$-Riemann--Liouville fractional integral MR1347689MR2218073
  • Definition 2: The $\psi$-Riemann--Liouville fractional derivative MR1347689MR2218073
  • Definition 3: The $\psi$-Caputo fractional derivative Almeida1_et_al
  • Definition 4: See MR3244285
  • Definition 5: See MR4577643
  • Definition 6: See MR4577643
  • Lemma 7: See Jarad_et_al
  • Lemma 8: See Jarad_et_al
  • Remark 9
  • Remark 10
  • ...and 3 more