Machine learning (ML) -based algorithms are playing an important role in cancer diagnosis and are increasingly being used to aid clinical decision-making . However, these commonly operate as ‘ black boxes ’ and it is unclear how decisions are derived . Recently, techniques have been applied to help us understand how specific ML models work and explain the rational for outputs . This study aims to determine why a given type of cancer has a certain phenotypic characteristic . Cancer results in cellular dysregulation and a thorough consideration of cancer regulators is required . This would increase our understanding of the nature of the disease and help discover more effective diagnostic, prognostic, and treatment methods for a variety of cancer types and stages . Our study proposes a novel explainable analysis of potential biomarkers denoting tumorigenesis in non-small cell lung cancer . A number of these biomarkers are known to appear following various treatment pathways . An enhanced analysis is enabled through a novel mathematical formulation for the regulators of mRNA, the regulators of ncRNA, and the coupled mRNA–ncRNA regulators . Temporal gene expression profiles are approximated in a two-dimensional spatial domain for the transition states before converging to the stationary state, using a system comprised of coupled-reaction partial differential equations . Simulation experiments demonstrate that the proposed mathematical gene-expression profile represents a best fit for the population abundance of these oncogenes . In future, our proposed solution can lead to the development of alternative interpretable approaches, through the application of ML models to discover unknown dynamics in gene regulatory systems.