Modified method to fit cell elastic modulus based on Sneddon model

Patent metadata

Field	Value
Publication number	US-11860187-B2
Application number	US-202017280552-A
Country	US
Kind code	B2
Filing date	Sep 7, 2020
Priority date	Jul 2, 2020
Publication date	Jan 2, 2024
Grant date	Jan 2, 2024

Abstract

This invention belongs to the technical field of cell mechanics and provides a modified method to fit cell elastic modulus based on Sneddon model. The process of the conical atomic force microscope probe compressing into the cell was simulated by ABAQUS. The simulation results are compared with the Sneddon model to get the error caused by Sneddon model. The fitting errors of Sneddon model under different circumstances were obtained by using the method of function fitting, so as to realize the modification of Sneddon model to fit cell elastic modulus. As a modified method to fit cell elastic modulus based on Sneddon model, it can be used to measure the elastic modulus of cells more accurately. The design process is convenient and fast. The design method is easy to master, and the process of use is convenient and simple.

First claim

The invention claimed is:

1. A modified method to fit cell elastic modulus based on a Sneddon model, comprising steps of:
step 1, designing shape parameters of a conical atomic force microscope (AFM) probe,
establishing axisymmetric models of cells and the conical AFM probe; setting the cells as deformable elastomers with an elastic modulus of 5 kPa and a Poisson's ratio of 0.3; setting the conical AFM probe as a rigid body; setting the shape parameters of the conical AFM probe by changing a half angle of a cone a and curvature radius of the cone at tip r, where the half angle of the cone is selected from 20°˜60°, and the curvature radius of the cone at the tip is selected from 20 nm˜60 nm;
step 2, performing finite element simulation analysis of the cells and the models of the conical AFM probe with different shapes designed in step 1;
2.1) performing a simulation of a relationship between a normal force of the conical AFM probe and a compression depth
when the conical AFM probe is under the normal force, a contact area and the compression depth of the conical AFM probe increases with increases of an external force, which is a problem of nonlinear contact and large deformation; using Arbitrary Lagrangian Eulerian (ALE) method to simulate a cell deformation under the external force; setting a contact as a surface to surface contact; a master surface is a side on which the conical AFM probe contacts with a cell, and a slave surface is an upper surface of the cell; analyzing grid convergence, and determining a grid size for calculation; extracting a relationship between the normal force and a displacement of the conical AFM probe;
2.2) performing an error analysis of the simulation results and the Sneddon model,
comparing the normal force of the conical atomic force microscope probe obtained from 2.1) with results of the Sneddon model at the same compression depth, and substituting into the following equation:

δ
=

P
-

P
s

P
s

(
2
)

calculating a relative error in elastic modulus based on the Sneddon model fitting;
where, δ is the relative error in the elastic modulus of the cell based on the Sneddon model fitting, P is the normal force exerted on the conical AFM probe, and P s is the normal force exerted on the conical AFM probe calculated by the Sneddon model;
step 3, function fitting of the relative error calculated in the second step;
fitting the relative error of the conical AFM probe with different shape parameters calculated in step 2 as a function; finding that the relative errors have a linear relationship with a ratio between the curvature radius of the cone at the tip and the compression depth r/δ; further fitting the relative error δ as a polynomial function of r/δ and a, in which the highest exponential of r/δ is 1;
modifying the Sneddon model by the fitted relative error to obtain a modified formula;
step 4, measuring force displacement curves of human osteosarcoma cells (MG63) using the conical AFM probe with two shape parameters, using the Sneddon model and the modified formula to fit the elastic modulus of the human osteosarcoma cells, respectively; increasing the elastic modulus fitted by the Sneddon model with a decrease of the compression depth, wherein the elastic modulus fitted by the modified formula changes marginally with the compression depth; testing force displacement curves of polyvinyl alcohol (PVA) hydrogels with the conical AFM probe with the two shapes parameters, using the Sneddon model and the modified formula to fit the elastic modulus of the PVA hydrogels, respectively, and comparing with a macroscopic elastic modulus of the PVA hydrogels obtained from a macroscopic compression test on a testing machine, wherein the smaller the compression depth is, the greater an error between the elastic modulus fitted by the Sneddon model and the elastic modulus obtained by the macroscopic compression test, and wherein the elastic modulus fitted by the modified formula agrees with the elastic modulus obtained by the macroscopic compression test, and the error is independent of the compression depth.

Assignees

• Univ Dalian Tech

Inventors

• Zhang Wei
• Sun Weihao
• Wu Chengwei
• Ma Jianli

Classifications

• G01Q60/28 Primary

  Adhesion force microscopy · CPC title

• G01N15/10

  Investigating individual particles · CPC title

• G01Q30/04 Primary

  Display or data processing devices · CPC title

• G01Q60/366

  Nanoindenters, i.e. wherein the indenting force is measured · CPC title

• G06F30/23 Primary

  using finite element methods [FEM] or finite difference methods [FDM] · CPC title