Evolutionary Dynamics of Acid Resistance in Tumors: A Mathematical Model

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 is proposed to integrate phenotypic evolution, microenvironmental acidification, and tumor density dynamics. Three key mechanisms are 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. Well-posedness is established. Numerical simulations across biologically relevant parameter regimes lead to identifying two therapeutically targetable parameters as critical bifurcation parameters for resistance evolution: baseline acid clearance rate and a protection factor representing acid-resistance machinery effectiveness. In low-plasticity tumors, both parameters lead to 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 to prevent or reverse the harmful effect of acid-resistant phenotypes.