Analytical method for reconstructing the stress on a spherical particle from its surface deformation

Abstract

The mechanical forces that cells experience from the tissue surrounding them are crucial for their behavior and development. Experimental studies of such mechanical forces require a method for measuring them. A widely used approach in this context is bead deformation analysis, where spherical particles are embedded into the tissue. The deformation of the particles then allows to reconstruct the mechanical stress acting on them. Existing approaches for this reconstruction are either very time-consuming or not sufficiently general. In this article, we present an analytical approach to this problem based on an expansion in solid spherical harmonics that allows us to find the complete stress tensor describing the stress acting on the tissue. Our approach is based on the linear theory of elasticity and uses an ansatz derived by Love. We clarify the conditions under which this ansatz can be used, making our results useful also for other contexts in which this ansatz is employed. Our method can be applied to arbitrary radial particle deformations and requires a very low computational effort. The usefulness of the method is demonstrated by an application to experimental data. STATEMENT OF SIGNIFICANCE Measurements of mechanical forces acting on cells in a tissue are important for understanding the physical behavior of biological systems, but they are also quite challenging. A common strategy is to place a spherical bead inside the tissue and to then reconstruct the mechanical stress from the bead deformation that this stress causes. Here, we introduce a novel analytical method using which this reconstruction can be achieved. This method is significantly faster than numerical approaches and significantly more general than existing analytical techniques, such that it can be expected to find a broad range of applications in mechanobiology.