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Numerical simulation of hybrid deterministic-stochastic PDE models for cancer tumor growth
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DOI：10.1080/27684830.2025.2567717.png)
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Abstract
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Mathematical modeling plays a vital role in advancing our understanding of cancer dynamics and guiding therapeutic strategies (Barbero et al., 2024; Lokenath, 2003). Classical integer-order models, while insightful, often fail to capture memory effects, anomalous diffusion, and stochastic variability observed in tumor-immune interactions. Fractional calculus, with its nonlocal and memory-preserving properties, has emerged as a powerful alternative in biomedical modeling (Toufik & Atangana, 2017; Kilbas et al., 2006; Luchko, 2009). This study proposes a hybrid crossover model for simulating cancer tumor dynamics by integrating integer-order, fractal-fractional, and stochastic differential operators across distinct subsystems. The model captures the complex interactions among tumor cell density, immune response, tissue temperature, and nutrient concentration, each governed by differential laws best suited to its physiological characteristics. Integer-order partial differential equations describe classical diffusion and growth behaviors; fractal-fractional differential equations capture memory-dependent and nonlocal biological effects; and stochastic partial differential equations account for random environmental fluctuations through multiplicative space-time white noise. For the numerical treatment of this coupled system, the Atangana-Baleanu fractal-fractional Lagrange two-step polynomial method is employed for the fractal-fractional subsystem, while the Milstein method is used for the stochastic components, and a stable finite difference scheme is applied to the deterministic parts. The model ensures the positivity and boundedness of all biological variables, preserving physical realism. The model ensures the positivity and boundedness of all biological variables, preserving physical realism. Numerical simulations reproduce qualitatively consistent tumor regression patterns observed in clinical oncology, such as partial shrinkage, slowed response under memory effects, and heterogeneous suppression due to fractal tissue structure.
Keywords:
Fractal fractional calculus
cancer tumor modeling
Atangana-Baleanu derivative
numerical simulation
stochastic differential equation
Milstein method
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1.1
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72
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