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A mathematical model of tumor growth using fractional derivatives

Karen Escutia, Carlos Islas, Pablo Padilla

Abstract

In this work, we investigate a fractional-order tumor growth model aimed at capturing memory effects and nonlocal temporal dynamics inherent to tumor evolution. The model is formulated using Caputo fractional derivatives and incorporates key biological mechanisms related to tumor growth, vascular interaction, and cell clearance. To numerically solve the resulting fractional differential equations, a second-order fractional Runge-Kutta scheme is derived based on a truncated fractional Taylor expansion, providing an accurate and stable computational framework. The proposed model is calibrated using experimental tumor volume data from five patients, and its performance is evaluated through the Root Mean Square Deviation (RMSD) between numerical simulations and experimental observations. The results show that, for all patients considered, the fractional-order model significantly improves the agreement with experimental data compared to the classical integer-order formulation. An optimal fractional order α < 1 is identified in each case, highlighting the relevance of memory effects in tumor growth dynamics and their patient-specific nature. Further insight is obtained through phase-space and projection analyses, which reveal substantial geometric differences between integer-order and fractional-order dynamics. Although the present study is based on a limited number of patient datasets, the results demonstrate the potential of fractional-order modeling as a flexible and powerful framework for describing individualized tumor growth behavior. The proposed approach provides a solid basis for future extensions involving larger datasets, uncertainty quantification, and the incorporation of treatment effects and control strategies.

A mathematical model of tumor growth using fractional derivatives

Abstract

In this work, we investigate a fractional-order tumor growth model aimed at capturing memory effects and nonlocal temporal dynamics inherent to tumor evolution. The model is formulated using Caputo fractional derivatives and incorporates key biological mechanisms related to tumor growth, vascular interaction, and cell clearance. To numerically solve the resulting fractional differential equations, a second-order fractional Runge-Kutta scheme is derived based on a truncated fractional Taylor expansion, providing an accurate and stable computational framework. The proposed model is calibrated using experimental tumor volume data from five patients, and its performance is evaluated through the Root Mean Square Deviation (RMSD) between numerical simulations and experimental observations. The results show that, for all patients considered, the fractional-order model significantly improves the agreement with experimental data compared to the classical integer-order formulation. An optimal fractional order α < 1 is identified in each case, highlighting the relevance of memory effects in tumor growth dynamics and their patient-specific nature. Further insight is obtained through phase-space and projection analyses, which reveal substantial geometric differences between integer-order and fractional-order dynamics. Although the present study is based on a limited number of patient datasets, the results demonstrate the potential of fractional-order modeling as a flexible and powerful framework for describing individualized tumor growth behavior. The proposed approach provides a solid basis for future extensions involving larger datasets, uncertainty quantification, and the incorporation of treatment effects and control strategies.
Paper Structure (6 sections, 3 theorems, 39 equations, 8 figures, 3 tables)

This paper contains 6 sections, 3 theorems, 39 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Formulation of the fractional differential equation model
  4. Second-Order Fractional Runge--Kutta Method
  5. Results and Discussion
  6. Conclusions

Key Result

Theorem 1

Suppose $f(x) \in C[a,b]$ and $^C _aD_t^{\alpha} f(t) \in C(a,b]$ for $0 < \alpha \leq 1$. Then,

Figures (8)

  • Figure 1: Tumor volume evolution for different fractional orders in the case of the: (a) patient one; (b) patient two; (c) patient three; (d) patient four; (e) patient five
  • Figure 2: RMSD evolution for different fractional orders in the case of the: (a) patient one; (b) patient two; (c) patient three; (d) patient four; (e) patient five
  • Figure 3: Phase--space portraits for: (a) patient one; (b) patient two; (c) patient three; (d) patient four; (e) patient five
  • Figure 4: Patient 1
  • Figure 5: Patient 2
  • ...and 3 more figures

Theorems & Definitions (6)

  • definition 1: Riemann--Liouville Fractional Integral Monje2010
  • definition 2: Riemann--Liouville Fractional Derivative Monje2010
  • definition 3: Caputo Fractional Derivative Chakraverty2022
  • Theorem 1: Generalized Mean Value Theorem Ghoreishi2023
  • Theorem 2: Generalized Taylor Formula in the Caputo Sense Ghoreishi2023
  • Theorem 3