Simple non-invasive in vivo measurement of stress difference in the skin

Accurate, non-invasive quantification of in vivo skin stress would improve surgical planning and objective assessment of skin mechanics. We present a simple acousto-elastic approach that estimates the in vivo in-plane stress difference from surface wave speeds measured along two orthogonal directions on the skin. For a large class of incompressible hyperelastic anisotropic models with one family of parallel fibres aligned with the skin tension lines, we show that the principal Cauchy stress difference satisfies: ${\sigma }_1-{\sigma }_2\simeq \rho (v_{\text{max}}^2-v_{\text{min}}^2)$ , where $\rho$ is the tissue mass density, and $v_{\text{max}}$ , $v_{\text{min}}$ are the Rayleigh wave speeds along, and orthogonal to, the direction of greatest tension, respectively, with a relative error below 9 percent. We validate the formula with finite element simulations in plane strain using the neo-Hookean and Holzapfel-Gasser-Ogden materials. In the neo-Hookean case, stress differences calculated from simulated wave speeds agree with the ground truth within 2.3 to 7.9 percent across pre-stretches ${\lambda }_1=1.05$ to 1.20. In the anisotropic case with fibres parallel to the principal pre-stress, the error is 0.25 to 2.81 percent over the same range. These results provide a proof of concept for estimating the in-plane stress difference in skin from orthogonal Rayleigh-wave speeds, within the assumptions of a homogeneous fibre-reinforced hyperelastic model.