Abstract
The atomic force microscopy (AFM) has been widely used to measure the mechanical properties of biological cells through indentations. In most of existing studies, the cell is supposed to be linear elastic within the small strain regime when analyzing the AFM indentation data. However, in experimental situations, the roles of large deformation and surface tension of cells should be taken into consideration. Here, we use the neo-Hookean model to describe the hyperelastic behavior of cells and investigate the influence of surface tension through finite element simulations. At large deformation, a correction factor, depending on the geometric ratio of indenter radius to cell radius, is introduced to modify the force-indent depth relation of classical Hertzian model. Moreover, when the indent depth is comparable with an intrinsic length defined as the ratio of surface tension to elastic modulus, the surface tension evidently affects the indentation response, indicating an overestimation of elastic modulus by the Hertzian model. The dimensionless-analysis-based theoretical predictions, which include both large deformation and surface tension, are in good agreement with our finite element simulation data. This study provides a novel method to more accurately measure the mechanical properties of biological cells and soft materials in AFM indentation experiments.
Highlights
Finite Element SimulationsConsider that we measure the elastic modulus of a cell placing on a substrate by using atomic force microscopy (AFM) indentation methods
Studies of the mechanics of biological cells are crucial for understanding a variety of fundamental cell behaviors, such as motility[1], differentiation[2] and proliferation[3], and have attracted tremendous attention in the fields of tissue engineering, cell biology and cancer treatment[4,5]
When an isolated cell is indented at large deformation, the linear elastic assumption in Hertzian theory is not valid, and the hyperelastic behavior should be taken into account
Summary
Consider that we measure the elastic modulus of a cell placing on a substrate by using AFM indentation methods. The external load P is applied on the cell through the spherical indenter, leading to an indent depth d. The finite element simulations are performed using the commercial finite element methods (FEM) software, ABAQUS. The radius R2 of the spherical indenter is varied from 1 μm to 100 μm[13,14,17], which leads the geometric size ratio β =R2/R1 to be in the range of 0.1–10. The cell is meshed with 4-node bilinear axisymmetric quadrilateral hybrid reduced integration elements and user-defined elements considering surface energy are defined to be attached on the surface. The spherical indenter is treated to be rigid, and the contact between the indenter and the cell is assumed to be frictionless. Convergence tests have been carried out to ensure the accuracy of computational results
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