A Theoretical Framework for Quantifying Tumour Resistance to Standardized Treatments: A Novel Rudimentary Scalar Mathematical Model with Implications for Breast Cancer Prognosis and Treatment.
Journal:
medRxiv
Published Date:
Jan 15, 2026
Abstract
BackgroundPrecision 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.
MethodsWe 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).
ResultsThe 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 <25%, mathematically defining a requirement for aggressive systemic therapy despite small anatomical size. Conversely, the model identified indolent metastatic phenotypes with stable resistance profiles suitable for de-escalated management.
ConclusionThis scalar model bridges the gap between basic histopathology and precision medicine. By providing a mathematically transparent, calculator-ready method for quantifying tumour resistance, it empowers clinicians in resource-limited settings to make evidence-based decisions on treatment escalation or de-escalation. This framework offers a rigorous, low-cost alternative to genomic profiling and provides a scalable, mathematically transparent scaffold for future AI integration in oncology..
