Toroidal indentation for measuring cell and tissue mechanical anisotropy

Indentation-based mechanical tests are advantageous for measuring tissue and cell stiffness due to their simplicity and ability to probe samples non-destructively. Most commonly, spherical or pyramidal probes are used, and Hertzian analysis is applied to calculate the modulus. This technique assumes material isotropy, which ignores the direction-dependent properties of fibrous tissues and polarized cells. In this study, we aimed to develop a generalized indentation method to estimate the anisotropic elastic moduli of biomaterials across scales from macroscopic tissues to single cells. Torus-shaped indenter probes were created with radii ranging from millimeters to microns and aspect ratios up to 10:1. Anisotropic muscle tissue, cell monolayers, and single cells were indented in directions perpendicular and parallel to their preferred fiber orientations. To determine intrinsic anisotropic moduli (E₁ and E₂), a linear incompressible transversely isotropic material model was developed, and finite element modeling was used to simulate normalized loading curve pairs for a wide range of E₁:E₂ parameter sets. A deep learning model was then trained with the simulated data and used to calculate the moduli. We found that the degree of anisotropy (E₁:E₂) was comparable to published results for muscle (∼1:3.7), aligned cell monolayers (∼1:3.6), and polarized single cells (∼1:1.7). This method generalizes the isotropic Hertzian indentation approach for anisotropic biomaterials and cells across length scales. The toroidal probes can be fabricated for less than $30 and are compatible with commercial indenters, making the method widely accessible to researchers.

Statement of significance

The toroidal indentation method and the finite element based deep learning model developed in this study provides a readily available, accessible, and low-cost method to measure anisotropic stiffness of broad biological materials from macroscopic tissues to microscopic cells using conventional laboratory devices, which was, to the best of our knowledge, impossible without using imaging and inverse finite element model or highly specialized customized equipment. By providing a tool, this study has the potential to facilitate the understanding of mechanical properties of anisotropic biological systems such as polarized cells, multicellular aggregates, and micro tissues, hence expedite the studies in tissue engineering, biomechanics, and mechanobiology.