WO2022000768A1 - 一种Sneddon模型拟合细胞弹性模量的修正方法 - Google Patents

一种Sneddon模型拟合细胞弹性模量的修正方法 Download PDF

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WO2022000768A1
WO2022000768A1 PCT/CN2020/113672 CN2020113672W WO2022000768A1 WO 2022000768 A1 WO2022000768 A1 WO 2022000768A1 CN 2020113672 W CN2020113672 W CN 2020113672W WO 2022000768 A1 WO2022000768 A1 WO 2022000768A1
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conical
cell
elastic modulus
model
sneddon
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张伟
孙伟皓
吴承伟
马建立
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大连理工大学
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N15/00—Investigating characteristics of particles; Investigating permeability, pore-volume or surface-area of porous materials
    • G01N15/10—Investigating individual particles
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01Q—SCANNING-PROBE TECHNIQUES OR APPARATUS; APPLICATIONS OF SCANNING-PROBE TECHNIQUES, e.g. SCANNING PROBE MICROSCOPY [SPM]
    • G01Q30/00—Auxiliary means serving to assist or improve the scanning probe techniques or apparatus, e.g. display or data processing devices
    • G01Q30/04—Display or data processing devices
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01Q—SCANNING-PROBE TECHNIQUES OR APPARATUS; APPLICATIONS OF SCANNING-PROBE TECHNIQUES, e.g. SCANNING PROBE MICROSCOPY [SPM]
    • G01Q60/00—Particular types of SPM [Scanning Probe Microscopy] or microscopes; Essential components thereof
    • G01Q60/24—AFM [Atomic Force Microscopy] or apparatus therefor, e.g. AFM probes
    • G01Q60/36—DC mode
    • G01Q60/366—Nanoindenters, i.e. wherein the indenting force is measured
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01Q—SCANNING-PROBE TECHNIQUES OR APPARATUS; APPLICATIONS OF SCANNING-PROBE TECHNIQUES, e.g. SCANNING PROBE MICROSCOPY [SPM]
    • G01Q60/00—Particular types of SPM [Scanning Probe Microscopy] or microscopes; Essential components thereof
    • G01Q60/24—AFM [Atomic Force Microscopy] or apparatus therefor, e.g. AFM probes
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01Q—SCANNING-PROBE TECHNIQUES OR APPARATUS; APPLICATIONS OF SCANNING-PROBE TECHNIQUES, e.g. SCANNING PROBE MICROSCOPY [SPM]
    • G01Q60/00—Particular types of SPM [Scanning Probe Microscopy] or microscopes; Essential components thereof
    • G01Q60/24—AFM [Atomic Force Microscopy] or apparatus therefor, e.g. AFM probes
    • G01Q60/28—Adhesion force microscopy
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

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  • the invention belongs to the technical field of cell mechanics, and particularly relates to a correction method for fitting cell elastic modulus using Sneddon model when using atomic force microscope to measure cell elastic modulus, which can more accurately test cell elastic modulus.
  • the elastic modulus of cells changes significantly during cell growth, maturation, proliferation, senescence, death, and pathology.
  • the elastic modulus of healthy red blood cells is much larger than the elastic modulus and viscoelasticity of red blood cells in patients with sickle cell anemia
  • the elastic modulus of healthy breast cells is more than twice that of breast cancer cells
  • the elastic modulus of benign breast tumor cells The elastic modulus is 1.4-1.8 times higher than that of malignant breast cancer cells.
  • the Sneddon model is commonly used for fitting when testing cellular elastic moduli using conical AFM probes.
  • the formula for the Sneddon model is:
  • P is the pressure applied in the normal direction
  • ⁇ is the half-opening angle of the rigid cone tip
  • d is the depth of the rigid cone tip pressed into the elastic half-space
  • E is the elastic modulus of the elastic half-space
  • ⁇ is the elastic half-space Poisson's ratio.
  • the contact deformation is a small deformation.
  • the radius of curvature of the rigid tip is 0.
  • the depth of the AFM probe pressed into the cell is tens to hundreds of nanometers, which is not a small deformation category relative to the thickness of several micrometers in the adherent cells.
  • the radius of curvature of the tip of the AFM probe cannot be 0, generally 20-60 nm.
  • the conical AFM probe Under the action of normal pressure, the conical AFM probe (hereinafter referred to as the conical AFM probe) increases with the increase of the external pressure due to the contact area and indentation depth between the cell and the conical AFM probe. , which is a nonlinear contact and large deformation problem.
  • ABAQUS has strong mechanical simulation capabilities and can solve complex nonlinear problems. Therefore, ABAQUS is selected to simulate the process of the conical atomic force microscope probe being pressed into the cell.
  • the present invention uses ABAQUS to simulate the process of pressing the conical atomic force microscope probe into the cell, and considers the influence of the radius of curvature of the conical probe of the atomic force microscope, the half angle of the cone and the probe pressing depth on the elastic modulus of the test cell, and simulates the pressure P and Press the relationship of the depth d to get the error that will be generated by fitting the Sneddon model. Finally, by numerically fitting the error and the function of the geometric parameters of the conical AFM probe and the indentation depth, a modification method of the Sneddon model is proposed, which can more accurately characterize the elastic modulus of cells.
  • the invention provides a correction method for fitting cell elastic modulus with Sneddon model.
  • the process of pressing the conical atomic force microscope probe into the cell is simulated by ABAQUS, and the error generated by using the Sneddon model is obtained by comparing with the Sneddon model, and the error of using the Sneddon model in different situations is obtained by the method of function fitting, so as to realize Correction for fitting cell elastic moduli using the Sneddon model.
  • a correction method of Sneddon model fitting cell elastic modulus the steps are as follows:
  • the first step is to design the shape parameters of the conical AFM probe
  • the axisymmetric model of the cell and the conical AFM probe was established in ABAQUS, in which the cell was set as a deformable elastic body with an elastic modulus of 5 kPa and a Poisson's ratio of 0.3; the conical AFM probe was set as a rigid body; the conical AFM probe was set as a rigid body;
  • the shape parameters of the atomic force microscope probe are realized by changing the cone half angle ⁇ and the curvature radius r of the cone head, where the cone half angle is selected from 20° to 60°, and the head curvature radius is selected from 20nm to 60nm;
  • the second step is to perform finite element simulation analysis on the cells and the models of the conical AFM probes designed in the first step.
  • the contact setting is surface-to-surface contact, the main surface is selected as the side where the conical atomic force microscope probe contacts the cell, and the secondary surface is selected as the upper surface of the cell.
  • conduct mesh convergence analysis and then select the mesh size with higher calculation accuracy and efficiency for calculation. The relationship between the normal pressure and displacement of the conical AFM probe in the extraction results.
  • the relative error can effectively explain the degree of deviation between the Sneddon model fitting results and the ABAQUS simulation results.
  • the normal pressure of the conical AFM probe calculated in 2.1) at the same indentation depth is compared with the results of the Sneddon model and brought into the following formula
  • the relative error of fitting elastic modulus of Sneddon model is calculated.
  • ⁇ is the relative error of fitting the cell elastic modulus with the Sneddon model
  • P is the normal pressure of the conical AFM probe simulated by ABAQUS
  • P s is the normal pressure of the conical AFM probe calculated using the Sneddon model , because there is a primary relationship between the normal pressure of the conical AFM probe and the elastic modulus of the cell in the formula of the Sneddon model, the relative error of the normal pressure is equal to the relative error of the elastic modulus.
  • the third step is to perform function fitting on the relative error results calculated in the second step
  • the relative error of the conical AFM probe with enough shape parameters calculated in the second step is fitted by a function. Based on the fact that the relative error is a dimensionless quantity and the law found during data sorting, that is: the relative error has a linear relationship with the ratio r/d of the radius of curvature of the cone head and the indentation depth. Considering the simplicity of the fitting formula results, the relative The error ⁇ is fitted as a polynomial function of r/d and ⁇ rather than a Fourier series. The highest exponent of r/d in the polynomial is 1.
  • the present invention provides a correction method for fitting cell elastic modulus with Sneddon model.
  • ABAQUS which can solve complex nonlinear problems
  • the process of the conical atomic force microscope probe being pressed into the cell is simulated by the finite element method, and the mesh convergence analysis is carried out.
  • shape parameters curvature radius of cone head and cone half angle
  • indentation depth of cone-shaped atomic force microscope probes By changing the shape parameters (curvature radius of cone head and cone half angle) and indentation depth of cone-shaped atomic force microscope probes, the relative error of fitting with Sneddon model was calculated when cones of different shapes were indented in different depths. And the error generated when using Sneddon model to fit cell elastic modulus was fitted by function fitting.
  • As a correction method for using Sneddon model to fit cell elastic modulus it can be used to measure cell elastic modulus more accurately.
  • the design process is convenient and fast, the design method is easy to master, and the use process is convenient and simple.
  • Fig. 1 is the design flow chart of the modification of the Sneddon model fitting method of cell elastic modulus in the technical realization scheme
  • Figure 2 is a diagram of the numerical simulation process
  • (a) is the deformation solution model established in ABAQUS
  • (b) is the deformation cloud map of the cell
  • (c) is the change of the normal pressure of the conical atomic force microscope probe with the indentation depth Figure
  • (d) is the comparison between the simulation results and the Sneddon model calculation results.
  • Figure 3 shows the results of the relative error of the cell elastic moduli fitted by the Sneddon model when changing the radius of curvature of the head of the conical AFM probe and the indentation depth for a fixed half-angle conical AFM probe.
  • Figure 4 shows that when r/d is the abscissa, r/d and the relative error ⁇ are approximately linearly related.
  • Figure 5 shows the error generated when using the Sneddon model to calculate the elastic modulus of cells when using a polynomial function to fit conical AFM probes with different shape parameters into different depths.
  • Figure 6 shows the verification of the modified model used in the elastic modulus test of human osteosarcoma cells, and the results of using the Sneddon model and the modified model to fit the force-displacement curve obtained by testing human osteosarcoma cells with an atomic force microscope probe are compared.
  • Figure 7 shows the verification of the modified model used in the elastic modulus test of PVA hydrogel.
  • the Sneddon model and the modified model are used to fit the force-displacement curve obtained by testing the PVA hydrogel with an atomic force microscope probe and compare it with the macroscopic compression test obtained. Comparative Results.
  • Figure 2 shows the process of numerical simulation, in which a is the deformation solution model established in ABAQUS, b is the deformation cloud map of the cell, c is the change of the normal pressure of the conical atomic force microscope probe with the indentation depth, and d is the simulation The results are compared with those calculated by the Sneddon model.
  • Figure 3 is the result of the relative error of the cell elastic modulus fitted by the Sneddon model when changing the radius of curvature of the head of the conical AFM probe and the indentation depth for a conical AFM probe with a fixed half angle.
  • Figure 4 shows a linear relationship between r/d and the relative error ⁇ when r/d is the abscissa.
  • Figure 5 is a graph of the error generated when the Sneddon model is used to calculate the cell elastic modulus when conical AFM probes with different shape parameters are fitted with polynomial functions, where P is the normal direction of the conical AFM probe pressure, ⁇ is the half angle of the conical AFM probe, r is the radius of curvature of the conical AFM probe head, d is the indentation depth of the conical AFM probe, and ⁇ is the elastic modulus of the cell fitted with the Sneddon model Relative error from simulation results.
  • P is the normal direction of the conical AFM probe pressure
  • ⁇ is the half angle of the conical AFM probe
  • r is the radius of curvature of the conical AFM probe head
  • d is the indentation depth of the conical AFM probe
  • ⁇ is the elastic modulus of the cell fitted with the Sneddon model Relative error from simulation results.
  • FIG. 6 is the verification of the modified model used in the elastic modulus test of human osteosarcoma cells, and the results of using the Sneddon model and the modified model to fit the force-displacement curve obtained by testing human osteosarcoma cells with an atomic force microscope probe are compared.
  • Figure 7 is the verification of the modified model used in the elastic modulus test of PVA hydrogel. The Sneddon model and the modified model are used to fit the force-displacement curve obtained by testing PVA hydrogel with an atomic force microscope probe, and the macroscopic compression test obtained. Comparative Results.
  • a is the comparison of the absolute value of the elastic modulus of the hydrogel, the shaded part is the elastic modulus obtained by the macroscopic test
  • b is the comparison of the relative value of the elastic modulus of the hydrogel
  • the relative value is the elastic modulus measured by the atomic force microscope divided by the elastic modulus Modulus of elasticity measured macroscopically.

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Abstract

一种Sneddon模型拟合细胞弹性模量的修正方法。通过ABAQUS模拟圆锥形原子力显微镜探针压入细胞的过程，与Sneddon模型对比得出使用Sneddon模型产生的误差，并通过函数拟合的方法得出不同情况下使用Sneddon模型拟合的误差，从而实现对Sneddon模型拟合细胞弹性模量的修正。该Sneddon模型拟合细胞弹性模量的修正方法可以用于更准确地测量细胞弹性模量，设计过程方便快捷，设计方法易于掌握，使用过程方便简单。

Description

一种Sneddon模型拟合细胞弹性模量的修正方法 技术领域
本发明属于细胞力学技术领域，特别涉及使用原子力显微镜测量细胞弹性模量时，对使用Sneddon模型拟合细胞弹性模量的修正方法，可更准确地测试出细胞的弹性模量。
背景技术
细胞的弹性模量在细胞生长发育、成熟、增殖、衰老、死亡和病变过程中会发生显著改变。例如健康红细胞的弹性模量要远远大于镰状细胞贫血症患者红细胞的弹性模量和粘弹性，健康乳腺细胞的弹性模量是乳腺癌细胞弹性模量的2倍以上，良性乳腺肿瘤细胞的弹性模量是恶性乳腺癌细胞弹性模量的1.4‐1.8倍。精准测量细胞的弹性模量对于细胞力学的发展以及研发基于力学原理的疾病诊疗新方法具有重要指导意义。
Sneddon模型被普遍用于使用锥形原子力显微镜探针测试细胞弹性模量时的拟合。Sneddon模型的公式为：
Figure PCTCN2020113672-appb-000001
式中，P为法向施加的压力，α为刚性锥尖的半开角，d为刚性锥形针尖压入弹性半空间的深度，E为弹性半空间的弹性模量，υ为弹性半空间的泊松比。使用原子力显微镜测量细胞弹性模量时通过获得法向压力P和压入深度d的关系曲线，并通过标准最小二乘法将该曲线与Sneddon公式拟合得到细胞的弹性模量。
但是，Sneddon模型中存在两个重要的假设：(1)接触变形为小变形。(2)刚性针尖的曲率半径为0。在实际实验中，原子力显微镜探针压入细胞的深度为几十到几百纳米，相对于贴壁细胞的几微米厚度不属于小变形范畴。此外，由于加工精度所限，原子力显微镜探针针尖的曲率半径不可能为0，一般为20‐60 nm。这种实验与Sneddon模型假设的不一致性决定了使用该模型拟合实验数据时会带来明显误差，使获得的弹性模量偏离本征值，这给准确测量细胞的弹性模量提出了巨大的挑战。
圆锥形原子力显微镜探针(以下简称圆锥形原子力显微镜探针)在法向压力的作用下，由于细胞与圆锥形原子力显微镜探针的接触面积和压入深度随着外压的增大而增大，这是非线性接触和大变形问题，ABAQUS有强大的力学仿真能力，可解决复杂非线性问题，因此选用ABAQUS来模拟圆锥形原子力显微镜探针压入细胞的过程。
本发明使用ABAQUS模拟圆锥形原子力显微镜探针压入细胞的过程，考虑了原子力显微镜锥形探针曲率半径、圆锥半角和探针压入深度对测试细胞弹性模量的影响，模拟出压力P和压入深度d的关系，得出使用Sneddon模型拟合会产生的误差。最后，通过数值拟合出误差和圆锥形原子力显微镜探针几何形状参数及压入深度的函数，提出了一种Sneddon模型的修正方法，可更准确的表征细胞的弹性模量。
发明内容
本发明提供了一种用Sneddon模型拟合细胞弹性模量的修正方法。通过ABAQUS模拟圆锥形原子力显微镜探针压入细胞的过程，与Sneddon模型对比得出使用Sneddon模型产生的误差，并通过函数拟合的方法得出不同情况下使用Sneddon模型拟合的误差，从而实现对使用Sneddon模型拟合细胞弹性模量的修正。
本发明的技术方案：
一种Sneddon模型拟合细胞弹性模量的修正方法，步骤如下：
第一步，设计圆锥形原子力显微镜探针的形状参数
在ABAQUS中建立细胞和圆锥形原子力显微镜探针的轴对称模型，其中细胞设置为可变形的弹性体，弹性模量5kPa，泊松比0.3；圆锥形原子力显微镜探针设置为刚体；设置圆锥形原子力显微镜探针的形状参数，通过改变圆锥半角α和圆锥头部曲率半径r来实现，其中圆锥半角选取20°～60°，头部曲率半径选取20nm～60nm；
第二步，对第一步设计的细胞和不同形状的圆锥形原子力显微镜探针的模型进行有限元模拟分析
2.1)圆锥形原子力显微镜探针法向压力和压入深度的关系模拟
圆锥形原子力显微镜探针在法向压力的作用下，由于细胞与圆锥形原子力显微镜探针的接触面积和压入深度随着外压的增大而增大，这是非线性接触和大变形问题，选用ABAQUS中的ALE(Arbitrary Lagrangian Eulerian)方法进行外压作用下的细胞的变形模拟。接触设置为面面接触，主面选取为圆锥形原子力显微镜探针与细胞接触的那一侧，从面选取为细胞的上表面。并进行网格收敛性分析，然后选择计算精度和效率较高的网格尺寸进行计算。提取结果中圆锥形原子力显微镜探针法向压力和位移的关系。
2.2)对模拟结果和Sneddon模型进行误差分析
相对误差能够有效说明Sneddon模型拟合结果与ABAQUS模拟结果的偏离程度。将相同压入深度下2.1)中算得的圆锥形原子力显微镜探针法向压力与Sneddon模型结果对比并带入下式
Figure PCTCN2020113672-appb-000002
计算得到Sneddon模型拟合弹性模量的相对误差。其中δ是用Sneddon模型拟合细胞弹性模量的相对误差，P是用ABAQUS模拟得到的圆锥形原子力显微镜探针法向压力，P s是用Sneddon模型计算的圆锥形原子力显微镜探针法向压力， 因为Sneddon模型的公式中圆锥形原子力显微镜探针法向压力和细胞弹性模量之间存在一次关系，所以法向压力的相对误差就等于弹性模量的相对误差。
第三步，对第二步计算出的相对误差结果进行函数拟合
将第二步计算得到的足够多的形状参数的圆锥形原子力显微镜探针的相对误差进行函数拟合。基于相对误差是无量纲量以及在数据整理时发现的规律，即：相对误差和圆锥头部曲率半径和压入深度的比值r/d呈线性关系，考虑到拟合公式结果的简洁，将相对误差δ拟合为r/d和α的多项式函数的形式而不是傅里叶级数的形式。多项式中r/d的最高次指数为1。
对上述得出的拟合函数进行实验验证：
首先，用第三步得出的相对误差的拟合函数对Sneddon模型进行修正得到修正公式。其次，使用两种形状参数的原子力显微镜探针测试人骨肉瘤细胞(MG63)的力位移曲线，分别用Sneddon模型和修正公式拟合细胞的弹性模量。比较发现用Sneddon模型拟合得到的弹性模量随着压入深度减小大幅度增加，用修正公式拟合得到的弹性模量随着压入深度的变化几乎没有改变。然后，使用两种形状的原子力显微镜探针测试PVA水凝胶的力位移曲线，分别用Sneddon模型和修正公式拟合水凝胶的弹性模量，再与用万能试验机进行宏观压缩测试得到的水凝胶的宏观弹性模量对比。比较发现用Sneddon模型拟合得到的弹性模量与宏观测试得到的弹性模量存在误差，且压入深度越小误差越大，用修正公式拟合得到的弹性模量与宏观测试得到的弹性模量基本吻合，且误差随压入深度没有明显的变化。以上对于PVA水凝胶和人骨肉瘤细胞实验验证了第三步得到的拟合函数的准确性。
本发明的有益效果：本发明提供一种用Sneddon模型拟合细胞弹性模量的修正方法。使用可解决复杂非线性问题的ABAQUS，用有限单元法模拟了圆锥形 原子力显微镜探针压入细胞的过程，并进行网格收敛性分析。通过改变圆锥形原子力显微镜探针的形状参数(圆锥头部曲率半径和圆锥半角)和压入深度，计算不同形状的圆锥压入不同深度时使用Sneddon模型拟合会产生的相对误差。并用函数拟合的方式拟合出使用Sneddon模型拟合细胞弹性模量时产生的误差。作为使用Sneddon模型拟合细胞弹性模量的修正方法，可以用于更准确地测量细胞弹性模量，设计过程方便快捷，设计方法易于掌握，使用过程方便简单。
附图说明
图1为技术实现方案中细胞弹性模量Sneddon模型拟合方法修正的设计流程图；
图2为数值模拟过程图，(a)是在ABAQUS中所建立的变形求解模型，(b)是细胞的变形云图，(c)是圆锥形原子力显微镜探针法向压力随压入深度的变化图，(d)是模拟结果与Sneddon模型计算结果对比图。
图3为对于固定半角圆锥形原子力显微镜探针，改变圆锥形原子力显微镜探针的头部曲率半径和压入深度时用Sneddon模型拟合的细胞弹性模量的相对误差的结果。
图4为以r/d为横坐标时，r/d与相对误差δ近似呈线性关系。
图5为使用多项式函数拟合不同形状参数的圆锥形原子力显微镜探针压入不同深度时使用Sneddon模型计算细胞弹性模量时会产生的误差。
图6为修正模型用在人骨肉瘤细胞弹性模量测试的验证，分别使用Sneddon模型和修正模型拟合用原子力显微镜探针测试人骨肉瘤细胞得到的力位移曲线的结果对比。
图7为修正模型用在PVA水凝胶弹性模量测试的验证，分别使用Sneddon模型和修正模型拟合用原子力显微镜探针测试PVA水凝胶得到的力位移曲线并与宏观压缩测试得到的宏观结果对比。(a)水凝胶弹性模量绝对值的对比，其中阴 影部分为宏观测试得到的弹性模量。(b)水凝胶弹性模量相对值的对比，相对值为原子力显微镜测出的弹性模量除以宏观测试的弹性模量。
具体实施方式
下面结合附图和技术方案，进一步说明本发明的具体实施方式。
图2是数值模拟的过程，其中a是在ABAQUS中所建立的变形求解模型，b是细胞的变形云图，c是圆锥形原子力显微镜探针法向压力随压入深度的变化图，d是模拟结果与Sneddon模型计算结果对比图。图3是对于固定半角的圆锥形原子力显微镜探针，改变圆锥形原子力显微镜探针的头部曲率半径和压入深度时用Sneddon模型拟合的细胞弹性模量的相对误差的结果。图4是以r/d为横坐标时，r/d与相对误差δ呈线性关系。图5是使用多项式函数拟合不同形状参数的圆锥形原子力显微镜探针压入不同深度时使用Sneddon模型计算细胞弹性模量时会产生的误差图，其中P是圆锥形原子力显微镜探针的法向压力，α是圆锥形原子力显微镜探针的半角，r是圆锥形原子力显微镜探针的头部曲率半径，d是圆锥形原子力显微镜探针压入深度，δ是用Sneddon模型拟合细胞弹性模量与模拟结果的相对误差。图6是修正模型用在人骨肉瘤细胞弹性模量测试的验证，分别使用Sneddon模型和修正模型拟合用原子力显微镜探针测试人骨肉瘤细胞得到的力位移曲线的结果对比。图7是修正模型用在PVA水凝胶弹性模量测试的验证，分别使用Sneddon模型和修正模型拟合用原子力显微镜探针测试PVA水凝胶得到的力位移曲线并与宏观压缩测试得到的宏观结果对比。其中a是水凝胶弹性模量绝对值的对比，阴影部分为宏观测试得到的弹性模量，b是水凝胶弹性模量相对值的对比，相对值为原子力显微镜测出的弹性模量除以宏观测试的弹性模量。
实施例1
(1)首先在ABAQUS中建立圆锥形原子力显微镜探针和细胞的模型，如图2(a)，设置圆锥形原子力显微镜探针的形状参数，形状参数设计通过改变圆锥半角α，圆锥头部曲率半径r来实现。使压入深度d为600nm，圆锥半角α为60°，改变圆锥头部曲率半径r为20nm，30nm，40nm，50nm，60nm。
(2)细胞在圆锥形原子力显微镜探针外压的作用下，细胞沿着圆锥形原子力显微镜探针的形状而变形，细胞与圆锥形原子力显微镜探针的接触面积逐渐增大，这是非线性接触和大变形问题，ABAQUS可解决复杂的非线性问题，故选用ABAQUS来进行外压作用下的电容传感器的变形模拟。在ABAQUS中进行外压作用下的变形响应，如图2(b)，提取圆锥形原子力显微镜探针法向压力和压入深度的关系，如图2(c)。进行网格收敛性分析验证算法有效性。模拟结果如图2(d)所示
(3)将模拟的圆锥形原子力显微镜探针法向压力和压入深度关系和Sneddon模型对比计算出Sneddon模型拟合细胞弹性模量的相对误差。如图3所示。
(4)基于数据整理时发现的r/d与相对误差δ之间的线性关系，如图4所示。使用多项式函数拟合不同形状参数的圆锥形原子力显微镜探针压入不同深度时使用Sneddon模型计算细胞弹性模量时会产生的误差。如图5所示。
(5)首先，用相对误差的拟合函数对Sneddon模型进行修正得到修正公式。其次，使用两种形状参数的原子力显微镜探针测试人骨肉瘤细胞(MG63)的力位移曲线，分别用Sneddon模型和修正公式拟合细胞的弹性模量。比较发现用Sneddon模型拟合得到的弹性模量随着压入深度减小大幅度增加，用修正公式拟合得到的弹性模量随着压入深度的变化几乎没有改变，如图6所示。然后，使用两种形状的原子力显微镜探针测试PVA水凝胶的力位移曲线，分别用Sneddon 模型和修正公式拟合水凝胶的弹性模量，再与用万能试验机进行宏观压缩测试得到的水凝胶的宏观弹性模量对比。比较发现用Sneddon模型拟合得到的弹性模量与宏观测试得到的弹性模量存在误差，且压入深度越小误差越大，用修正公式拟合得到的弹性模量与宏观测试得到的弹性模量基本吻合，且误差随压入深度没有明显的变化，如图7所示。以上对于PVA水凝胶和人骨肉瘤细胞实验验证了第三步得到的拟合函数的准确性。
以上所述实施例仅表达了本发明的实施方式，但并不能因此而理解为对本发明专利的范围的限制，应当指出，对于本领域的技术人员来说，在不脱离本发明构思的前提下，还可以做出若干变形和改进，这些均属于本发明的保护范围。

Claims (1)

  1. 一种Sneddon模型拟合细胞弹性模量的修正方法，其特征在于，步骤如下：
    第一步，设计圆锥形原子力显微镜探针的形状参数
    在ABAQUS中建立细胞和圆锥形原子力显微镜探针的轴对称模型，细胞设置为可变形的弹性体；圆锥形原子力显微镜探针设置为刚体；设置圆锥形原子力显微镜探针的形状参数，通过改变圆锥半角α和圆锥头部曲率半径r来实现，其中圆锥半角选取20°～60°，头部曲率半径选取20nm～60nm；
    第二步，对第一步设计的细胞和不同形状的圆锥形原子力显微镜探针的模型进行有限元模拟分析
    2.1)圆锥形原子力显微镜探针法向压力和压入深度的关系模拟
    圆锥形原子力显微镜探针在法向压力的作用下，由于细胞与圆锥形原子力显微镜探针的接触面积和压入深度随着外压的增大而增大，这是非线性接触和大变形问题，选用ABAQUS中的ALE方法进行外压作用下的细胞变形模拟；接触设置为面面接触，主面选取为圆锥形原子力显微镜探针与细胞接触的那一侧，从面选取为细胞的上表面；并进行网格收敛性分析，然后确定网格尺寸进行计算；提取结果中圆锥形原子力显微镜探针法向压力和位移的关系；
    2.2)对模拟结果和Sneddon模型进行误差分析
    将相同压入深度下2.1)中得到的圆锥形原子力显微镜探针法向压力与Sneddon模型结果对比并带入下式：
    Figure PCTCN2020113672-appb-100001
    计算得到Sneddon模型拟合弹性模量的相对误差；
    其中，δ是用Sneddon模型拟合细胞弹性模量的相对误差，P是用ABAQUS模拟得到的圆锥形原子力显微镜探针法向压力，P s是用Sneddon模型计算的圆锥形原子力显微镜探针法向压力；
    第三步，对第二步计算出的相对误差结果进行函数拟合
    将第二步计算得到的形状参数的圆锥形原子力显微镜探针的相对误差进行函数拟合，得出相对误差和圆锥头部曲率半径和压入深度的比值r/d呈线性关系，进一步将相对误差δ拟合为r/d和α的多项式函数的形式，多项式中r/d的最高次指数为1。
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