Mathematical modeling of tumor nanomechanical fingerprints: A weighted skew-normal distribution approach for cancer diagnosis and treatment monitoring

Atomic Force Microscopy (AFM) is a key method for nanomechanical characterization of cells and tissues, with AFM-derived fingerprints proposed as biomarkers for cancer diagnosis and treatment monitoring. These signatures typically include a higher elasticity peak (HEP), reflecting extracellular matrix stiffening due to collagen overproduction, and a lower elasticity peak (LEP), indicative of cancer cell softening. Despite their potential, AFM elasticity spectra are often assessed qualitatively, and a standardized mathematical framework for quantitative analysis is lacking. Here, we provide a rigorous mathematical characterization of Young’s modulus distribution in normal and cancerous tissues, aiming to improve cancer diagnosis and treatment monitoring. Previously published AFM elasticity spectra from murine tumors were employed and analyzed to evaluate tumor nanomechanical changes at different time points, 14, 21, and 28 days after cell implantation, including both untreated controls and a tranilast-treated group, with tranilast being a drug known to reduce collagen levels. The weighted skew-normal distribution was employed to model AFM data due to its ability to capture the two-peak structure of cancerous tissue, reflecting a mixture of soft cancer cells and stiffer components. We hypothesized that tranilast treatment would progressively shift the HEP to lower values. Model accuracy was confirmed by high R² values and low Cramér–von Mises (CvM) criteria. Results revealed a transition from a two-peak distribution in controls (HEP and LEP) to peak convergence in tranilast-treated tissue at 28 days. We conclude that the weighted skew-normal distribution offers a robust method for quantifying tumor nanomechanics, which is related to therapeutic outcomes.