A study of a fractional-order cancer biomarker model theoretical analysis and numerical simulation

© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )

Download PDF

XML

Cite

Abstract

This paper presents a fractional-order mathematical model for describing the dynamics of cancer biomarkers based on the Caputo fractional derivative. The proposed model incorporates memory effects inherent in biological systems, which provides a more realistic framework than the corresponding classical integer-order model. The existence and uniqueness of solutions are established using fixed-point theory, while the local stability of the equilibrium point is investigated through suitable stability criteria. Numerical solutions are obtained using the Generalized Fractional Taylor Expansion (GFTE), and the effectiveness and accuracy of the proposed numerical scheme are demonstrated. Furthermore, Particle Swarm Optimization (PSO) is employed for determining the optimal model parameters and nanoparticle dosage to improve tumor size estimation. The numerical results reveal the influence of the fractional-order parameter on biomarker dynamics and show that the proposed fractional-order model offers a flexible and robust framework for cancer biomarker analysis, early tumor detection, and optimization-based parameter estimation.

Keywords

Fractional-order cancer model

Cancer biomarkers

Existence and uniqueness

Local stability

Generalized fractional taylor expansion

Particle swarm optimization

Funding

None.

Conflict of interest

The authors declare that they have no conflict of interest.

References

Hamadneh T, Abu Falahah I, Batiha B. On the computation of the general simplicial Bernstein inclusion-isotone property for stability analysis of least-squares polynomials. Mathematics. 2026;14(3):517. https://doi.org/10.3390/math14030517
Shukla VK, Joshi MC, Mishra PK, Xu C. Generalized synchronization between coupled integer, fractional and delay chaotic systems. Evolving Systems. 2025;16(4):132. https://doi.org/10.1007/s12530-025-09758-x
Xua C, Liao M, Farman M, Shehzad A. Hydrogenolysis of glycerol by heterogeneous catalysis: A fractional order kinetic model with analysis. MATCH Commun Math Comput Chem. 2024;91(3):635-664. https://doi.org/10.46793/match.91-3.635X
Baber MZ, Yasin MW, Xu C, Ahmed N, Iqbal MS. Numerical and analytical study for the stochastic spatial dependent prey-predator dynamical system. J Comput Nonlinear Dyn. 2024;19(10):101003. https://doi.org/10.1115/1.4066038
Ali G, Marwan M, Rahman U, Hleili M. Investigation of fractional-ordered tumor-immune interaction model via fractional-order derivative. Fractals. 2024;32(6):2450119. https://doi.org/10.1142/S0218348X24501196
Lin J, Xu C, Zhao Y, Deng Q. Hopf bifurcation and controller design for a predator-prey model with double delays. AIP Advances. 2025;15(7):075329. https://doi.org/10.1063/5.0283543
Deng Q, Xu C, Lin J, Zhao Y. Bifurcation mechanism, speed feedback controller, and hybrid controller design in a delayed tumor-immune competitive model. AIP Advances. 2025;15(9):095009. https://doi.org/10.1063/5.0292455
Xu C, Balci E. Hunting cooperation and gestation delay in a prey-predator model with fractional derivative. J Appl Anal Comput. 2026;16(2):1035-1053. https://doi.org/10.11948/20250147
Lin J, Xu C, Zhao Y, Pang Y, Liu Z, Shen J. Bifurcation and control of a predator-prey system with two time delays. J Appl Anal Comput. 2026;16(2):807-859. https://doi.org/10.11948/20250164
Kwong GA, Dudani JS, Carrodeguas E, Mazumdar EV, Zekavat SM, Bhatia SN. Mathematical framework for activity-based cancer biomarkers. Proc Natl Acad Sci USA. 2015;112(41):12627-12632. https://doi.org/10.1073/pnas.1506925112
Kwon EJ, Dudani JS, Bhatia SN. Ultrasensitive tumour-penetrating nanosensors of protease activity. Nat Biomed Eng. 2017;1(4):0054. https://doi.org/10.1038/s41551-017-0054
Bergstrom E, Gutierrez N, Masson H. Modifying a "Mathematical Framework for Activity-Based Cancer Biomarkers": Correlating Tumor Size and Reporter Concentration. BENG 221 Course Project Report. La Jolla, CA: University of California San Diego; 2017. Accessed July 30, 2026. https://isn.ucsd.edu/courses/beng221/problems/2017/Modifying%20a%20Mathematical%20Framework%20for%20Activity-Based%20Cancer%20Biomarkers.pdf
Muhammad G, Akram M. Fuzzy fractional generalized Bagley-Torvik equation with fuzzy Caputo gH-differentiability. Eng Appl Artif Intell. 2024;133:108265. https://doi.org/10.1016/j.engappai.2024.108265
Muhammad G, Akram M, Alsulami H, Hussain N. Granular fuzzy fractional continuous-time linear systems: Roesser and Fornasini-Marchesini models. Fractal Fract. 2025;9(7):398. https://doi.org/10.3390/fractalfract9070398
Muhammad G, Akram M. Fuzzy fractional epidemiological model for Middle East respiratory syndrome coronavirus on complex heterogeneous network using Caputo derivative. Inf Sci. 2024;659:120046. https://doi.org/10.1016/j.ins.2023.120046
Rudin W. Principles of Mathematical Analysis. 3rd ed. New York, NY: McGraw-Hill; 1976. Accessed July 30, 2026. https://david92jackson.neocities.org/images/Principles_of_Mathematical_Analysis-Rudin.pdf
Kreyszig E. Introductory Functional Analysis with Applications. 1st Wiley ed. New York, NY: John Wiley & Sons; 1991. Accessed July 30, 2026. https://physics.bme.hu/sites/physics.bme.hu/files/users/BMETE15AF53_kov/Kreyszig%20-%20Introductory%20Functional%20Analysis%20with%20Applications%20(1).pdf
Coddington EA. An Introduction to Ordinary Differential Equations. New York, NY: Courier Corporation; 2012. Accessed July 30, 2026. https://personal.math.ubc.ca/~cass/research/pdf/DE.pdf
Podlubny I. Fractional Differential Equations. San Diego, CA: Academic Press; 1999. Accessed July 30, 2026. https://shop.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9
Perko L. Differential Equations and Dynamical Systems. 3rd ed. New York, NY: Springer; 2001. https://doi.org/10.1007/978-1-4613-0003-8
Hirsch MW, Smale S, Devaney RL. Differential Equations, Dynamical Systems, and an Introduction to Chaos. 3rd ed. Oxford, UK: Academic Press; 2013. Accessed July 30, 2026. https://eclass.uoa.gr/modules/document/file.php/MATH626/Morris%20W.%20Hirsch%2C%20Stephen%20Smale%20and%20Robert%20L.%20Devaney%20(Auth.)%20-%20Differential%20Equations%2C%20Dynamical%20Systems%2C%20and%20an%20Introduction%20to%20Chaos-Academic%20Press%20(2012).pdf
Odibat ZM, Momani S. An algorithm for the numerical solution of differential equations of fractional order. J Appl Math Inform. 2008;26:15-27. Accessed July 30, 2026. https://www.researchgate.net/profile/Shaher-Momani/publication/264059551_An_Algorithm_for_the_numerical_solution_of_differential_equations_of_fractional_order/links/548f3e100cf2d1800d8623f0/An-Algorithm-for-the-numerical-solution-of-differential-equations-of-fractional-order.pdf
Podlubny I, Dorcak L, Kostial I. On fractional derivatives, fractional-order dynamic systems and PI^λD^μ-controllers. In: Proceedings of the 36th IEEE Conference on Decision and Control. IEEE; 1997;5:4985-4990. https://doi.org/10.1109/CDC.1997.649841
Caputo M. Linear models of dissipation whose Q is almost frequency independent -- II. Geophys J Int. 1967;13(5):529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
Horn RA, Johnson CR. Matrix Analysis. 2nd ed. Cambridge, UK: Cambridge University Press; 2013. https://doi.org/10.1017/CBO9780511810817

Previous article in this issue

Next article in this issue

An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing