US 11,860,187 B2
Modified method to fit cell elastic modulus based on Sneddon model
Wei Zhang, Liaoning (CN); Weihao Sun, Liaoning (CN); Chengwei Wu, Liaoning (CN); and Jianli Ma, Liaoning (CN)
Assigned to DALIAN UNIVERSITY OF TECHNOLOGY, Liaoning (CN)
Appl. No. 17/280,552
Filed by DALIAN UNIVERSITY OF TECHNOLOGY, Liaoning (CN)
PCT Filed Sep. 7, 2020, PCT No. PCT/CN2020/113672
§ 371(c)(1), (2) Date Mar. 26, 2021,
PCT Pub. No. WO2022/000768, PCT Pub. Date Jan. 6, 2022.
Claims priority of application No. 202010624625.0 (CN), filed on Jul. 2, 2020.
Prior Publication US 2022/0043025 A1, Feb. 10, 2022
Int. Cl. G01Q 60/28 (2010.01); G06F 30/23 (2020.01); G01N 15/10 (2006.01); G01Q 60/24 (2010.01); G01Q 30/04 (2010.01); G01Q 60/36 (2010.01); G06F 119/14 (2020.01)
CPC G01Q 60/28 (2013.01) [G01N 15/10 (2013.01); G01Q 30/04 (2013.01); G01Q 60/366 (2013.01); G06F 30/23 (2020.01); G01Q 60/24 (2013.01); G06F 2119/14 (2020.01)] 1 Claim
OG exemplary drawing
 
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:

OG Complex Work Unit Math
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 Ps 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.