Fractional Modeling of Cancer Cell Signaling Dynamics Under BRAF Inhibition: Numerical Simulations and a Physics-Informed Inverse-Learning Framework

In this paper, a multiscale memory-dependent mathematical model ofcancer cell signaling in the mitogen-activated proteinkinase/extracellular signal-regulated kinase (MAPK/ERK) pathway undertherapy of targeted B-Raf proto-oncogene, serine/threonine kinase (BRAF)is developed. Integer-order models have been found to be effective inmodeling intracellular signaling but lack explicit long-range memorycapabilities. A reaction-diffusion system using the $\Psi$-Hilfer fractionaloperator is employed to incorporate nonlocal memory into the system.Incorporating Hill-type kinetics, stochastic perturbations andpharmacological suppression terms, the model reflects the biologicalprocesses occurring inside the cells. The reduced fractional system isinvestigated numerically using the Adams-Bashforth-Moultonpredictor-corrector (ABM-PECE) method. In addition, a physics-informedneural network (PINN) framework is formulated for inverse parameteridentification from synthetic observations.In the illustrative spatially homogeneous Caputo simulations, thefractional order influenced ERK activation during and after treatment.Lower fractional orders were associated with higher late-time ERKactivation, but did not necessarily produce a greater post-treatmentrebound amplitude. The illustrative simulations demonstrate thatfractional order influences the temporal evolution and persistence ofERK activation. The proposed inverse-learning formulation provides aframework for future parameter estimation from synthetic observations;however, its quantitative reconstruction performance requires furthercomputational verification.