A Theoretical Framework for Quantifying Tumour Resistance to Standardized Treatments: A Novel Rudimentary Scalar Mathematical Model with Implications for Breast Cancer Prognosis and Treatment

Abstract Background Precision oncology relies heavily on genomic profiling and artificial intelligence to predict therapeutic response in breast cancer. However, in low-to-middle-income countries (LMICs), these expensive modalities are inaccessible; forcing clinicians to rely on qualitative TNM staging that often fails to capture individual tumour heterogeneity. There is a critical unmet need fory “frugal innovation”—tools that convert standard histopathological data into quantitative prognostic scores. This study proposes a novel deterministic scalar mathematical model to quantify tumour resistance and predict therapeutic efficacy without high-cost infrastructure. Methods We developed a theoretical framework that transforms qualitative pathological inputs (TNM stage, Grade, and Ki67 status) into quantitative scalar variables. The model is anchored on three derived parameters: (1) Relative Severity of Disease (RSD), a multiplier based on stage-specific survival decay; (2) a graded Tumour Proliferation Index (Ki67); and (3) the Tumour-Dependent Therapeutic Response Coefficient (TmdRxResCoef). Resistance (Res) was modelled as the product of mass (RSD) and velocity (Ki67score), while therapeutic responsiveness was defined as the inverse of resistance (1/Res). Results The model demonstrates a non-linear, inverse relationship between tumour burden and the potential for therapeutic gain. Crucially, the model exposed a “biological equivalence” between Stage I/High-Grade tumours (Resistance Score = 4.0) and Stage IV/Indolent tumours (Resistance Score = 3.8), challenging the dogma that anatomical stage is the sole driver of prognosis. We successfully quantified the “Resistance Gap,” demonstrating that a high-velocity early-stage tumour yields a TmdRxResCoef of