Global linear modeling of AFM indentation curves for soft samples with various indenter geometries

Atomic Force Microscopy (AFM) is a critical technique for probing the mechanical properties of biological and soft materials at the nanoscale, offering high spatial resolution and sensitivity under near-physiological conditions. In biomedical contexts, especially oncology, quantifying cell and tissue stiffness via nanoindentation with Young’s modulus as a key parameter provides essential diagnostic insights. However, the interpretation of force–indentation data remains constrained by the complexity of classical contact mechanics models, particularly for non-ideal indenter geometries such as sphero-conical tips or deep spherical indentations. As a result, oversimplified models like Hertz and Sneddon are often used beyond their ideal regimes. To overcome these limitations, we introduce a unified linearization framework that enables geometry-independent analysis of AFM nanoindentation data. By combining the general nanoindentation differential equation with the weighted mean value theorem for integrals, we derive simplified analytical expressions that allow Young’s modulus to be extracted via straightforward linear fitting. This method is compatible with spherical, parabolic, conical, and sphero-conical indenters, eliminating the need for geometry-specific nonlinear fitting procedures. Unlike conventional methods, the proposed approach allows each indentation curve to be analyzed with the most appropriate contact model, determined by its maximum indentation depth, using linear fitting and a simple correction factor for different indenter geometries, thereby enabling accurate mapping of Young’s modulus. Validation against both simulated and experimental data, from fibroblasts and agarose gels, demonstrates excellent agreement with classical models while significantly streamlining data analysis. This framework paves the way for faster, more accessible, and broadly applicable mechanical characterization of soft materials using AFM, with potential to enhance diagnostic and materials discovery workflows in biomedical and soft-matter research.