Evolutionary dynamics of acid resistance in tumors: A mathematical model

Prithvi Anickode; Fabio Augusto Milner; Prithvi Anickode; Fabio Augusto Milner

doi:10.3934/mbe.2026023

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 3: 594-618. doi: 10.3934/mbe.2026023

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Research article

Evolutionary dynamics of acid resistance in tumors: A mathematical model

Prithvi Anickode ¹,
Fabio Augusto Milner ^{1,2
,
,}

1.
Simon A. Levin Mathematical, Computational and Modeling Sciences Center, Arizona State University, 1031 Palm Walk, Tempe, AZ 85287, USA
2.
School of Mathematical and Statistical Sciences, Arizona State University, 901 Palm Walk Room 216, Tempe, AZ 85281 USA

Received: 21 October 2025 Revised: 03 December 2025 Accepted: 01 January 2026 Published: 27 January 2026

Acidosis in tumors arises from reprogrammed metabolism and compromised vasculature, creating a harsh, acidic microenvironment that drives the evolutionary selection of acid-resistant cell phenotypes. A mathematical model was proposed to integrate phenotypic evolution, microenvironmental acidification, and tumor density dynamics. Three key mechanisms were incorporated in it: frequency-dependent selection favoring acid-resistant cells below a critical pH, stress-induced phenotypic switching, and a positive feedback loop where resistant cells produce excess acid that intensifies selection pressure. The well-posedness of the model was established. Through numerical simulations across biologically relevant parameter regimes, we identified two therapeutically targetable parameters: the baseline acid clearance rate (a proxy for vascular perfusion) and a protection factor (representing acid-resistance "machinery" effectiveness) as critical bifurcation parameters for resistance evolution. The model exhibits qualitatively distinct dynamics depending on phenotypic plasticity levels. In low-plasticity tumors, both parameters exhibit sharp bifurcations with strong parameter interactions: clearance and protection effects are context-dependent, with therapeutic interventions effective only within specific parameter ranges. In high-plasticity tumors, both parameters produce continuous, monotonic responses with independent, additive effects. These regime-dependent dynamics suggest that treatment strategies should adapt to tumor plasticity: in the former, targeting perfusion alone is typically sufficient, though sequential therapy may be required if the perfusion rate approaches or exceeds the bifurcation threshold, whereas in the latter, treatment might benefit from combination therapies addressing both parameters simultaneously. These findings suggest that a low-dimensional model can identify therapeutically targetable parameters governing resistance evolution, suggesting interventions that may prevent or reverse the harmful effect of acid-resistant phenotypes, which are associated with chemotherapy failure, immune evasion, and metastatic progression.
- dynamical system model,
- tumor acidosis,
- parenchyma cells competition
Citation: Prithvi Anickode, Fabio Augusto Milner. Evolutionary dynamics of acid resistance in tumors: A mathematical model[J]. Mathematical Biosciences and Engineering, 2026, 23(3): 594-618. doi: 10.3934/mbe.2026023

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Abstract

Acidosis in tumors arises from reprogrammed metabolism and compromised vasculature, creating a harsh, acidic microenvironment that drives the evolutionary selection of acid-resistant cell phenotypes. A mathematical model was proposed to integrate phenotypic evolution, microenvironmental acidification, and tumor density dynamics. Three key mechanisms were incorporated in it: frequency-dependent selection favoring acid-resistant cells below a critical pH, stress-induced phenotypic switching, and a positive feedback loop where resistant cells produce excess acid that intensifies selection pressure. The well-posedness of the model was established. Through numerical simulations across biologically relevant parameter regimes, we identified two therapeutically targetable parameters: the baseline acid clearance rate (a proxy for vascular perfusion) and a protection factor (representing acid-resistance "machinery" effectiveness) as critical bifurcation parameters for resistance evolution. The model exhibits qualitatively distinct dynamics depending on phenotypic plasticity levels. In low-plasticity tumors, both parameters exhibit sharp bifurcations with strong parameter interactions: clearance and protection effects are context-dependent, with therapeutic interventions effective only within specific parameter ranges. In high-plasticity tumors, both parameters produce continuous, monotonic responses with independent, additive effects. These regime-dependent dynamics suggest that treatment strategies should adapt to tumor plasticity: in the former, targeting perfusion alone is typically sufficient, though sequential therapy may be required if the perfusion rate approaches or exceeds the bifurcation threshold, whereas in the latter, treatment might benefit from combination therapies addressing both parameters simultaneously. These findings suggest that a low-dimensional model can identify therapeutically targetable parameters governing resistance evolution, suggesting interventions that may prevent or reverse the harmful effect of acid-resistant phenotypes, which are associated with chemotherapy failure, immune evasion, and metastatic progression.

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