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. 2025 Apr 12;15(1):12580.
doi: 10.1038/s41598-025-96737-6.

Fractional order modeling of prostate cancer with pulsed treatment and the impact of effector cell killing and cell competition

Hicham Saber  1 , Hussien Albala  2 , Tariq Aljaaidi  3 , Yousef Jawarneh  1 , Abdelkader Moumen  1 , Khaled Aldwoah  4
Affiliations

Fractional order modeling of prostate cancer with pulsed treatment and the impact of effector cell killing and cell competition

Hicham Saber et al. Sci Rep. .

Abstract

This manuscript illustrates a fractional-order mathematical model for prostate cancer (PC) growth under pulsed treatment, incorporating the effects of effector cell killing and competition between androgen-dependent (AD) and androgen-independent (AI) PC cells. We establish the existence and uniqueness of solutions using fixed-point theorems (Leray-Schauder and Banach) and investigate Ulam-Hyers stability to assess the model's solution stability under the fractional order. Numerical results are obtained via the fractional Euler method for simulation of the model at various fractional orders. Graphical results illustrate the model's dynamics, and the impact of key parameters, including the effector cell killing rate and inter-cell competition, is investigated. These findings provide insights into the complex interplay between treatment, immune response, and cancer cell dynamics, potentially informing therapeutic strategies for PC.

Keywords: Caputo operator; Euler method; Prostate cancer; Stability.

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Conflict of interest statement

Declarations. Competing interests: The authors declare no competing interests. Declaration of generative AI: The researchers did not use artificial intelligence in preparing this paper.

Figures

Figure 1
Figure 1
The demonstration of the state variable formula image with different values of formula image of model (2).
Figure 2
Figure 2
The demonstration of the state variable formula image with different values of formula image of model (2).
Figure 3
Figure 3
The demonstration of the state variable formula image with different values of formula image of model (2).
Figure 4
Figure 4
The demonstration of formula image on the state variables formula image and formula image with formula image of the model (2).
Figure 5
Figure 5
The demonstration of formula image on the state variables formula image and formula image with formula image of the model (2).
Figure 6
Figure 6
The demonstration of formula image on the state variables formula image and formula image with formula image of the model (2).
See this image and copyright information in PMC

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