Method for calculating viscoelastic mechanical parameters of cells based on indentation experiment
Through the modified model based on finite element data, the failure problems of the Hertz model in the calculation of cell viscoelastic mechanical parameters is solved, and more accurate extraction of cellular mechanical parameters is achieved, supporting more accurate biological behavior analysis.
Patent Information
- Application Number
- CN202411885594.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-20
- Publication Date
- 2025-04-15
AI Technical Summary
In the prior art, when calculating cell viscoelastic mechanical parameters using Hertz model, it is difficult to satisfy the small deformation hypothesis and the semi-infinite space hypothesis, resulting in errors in the calculation results.
Using a correction model based on finite element data, the deformation process of cells in AFM indentation experiments was carried out through ABAQUS software, and combined with the Hertz model and viscoelastic theory, an explicit correction model was established to eliminate errors.
This method can accurately extract the elastic modulus and apparent viscosity of cells, eliminate errors caused by failure of small deformation assumptions and semi-infinite space assumptions, provide more accurate cellular mechanics parameters and support better understanding of biological behavior.
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Abstract
Description
Technical Field
[0001] The invention belongs to the field of cell mechanics, and relates to a calculation method for fitting cell viscoelastic mechanical parameters, and in particular to a method for calculating cell viscoelastic mechanical parameters based on an indentation experiment. Background Art
[0002] Cells are typical viscoelastic materials. Viscosity and elasticity parameters can not only reflect the changes in cell spreading, adhesion, differentiation and migration, but also serve as indicator parameters for evaluating disease progression and treatment effects. For example, human umbilical vein endothelial cells with high elastic modulus show stronger diffusion ability and adhesion. During the differentiation of myoblasts into muscle cells, their elastic modulus increases from 11.5 kPa to 45.3 kPa. The elastic modulus of normal breast cells is 33% higher than that of breast cancer cells, and the viscosity is 66% higher. This difference enhances the migration and invasion ability of cancer cells. Resveratrol treatment can significantly reduce the elasticity and viscosity of breast cancer cells, thereby reducing their flexibility and deformation ability, and then inhibiting their invasive behavior. Compared with MCF-7 breast cancer cells, MDA-MB-231 breast cancer cells have lower viscosity, but stronger migration and invasion ability. Although there is still controversy about the influence of cell elasticity and viscosity on migration and invasion ability, the general consensus is that the viscoelastic parameters of cells can be used as quantitative indicators to characterize their biological behavior.
[0003] Atomic force microscopy (AFM) has become the main tool for measuring the viscoelastic properties of cells due to its precise control ability during loading, wide force measurement range, and real-time feedback of force and deformation. The Hertz model and viscoelastic model (as shown in Formula 1) are used to fit the force-displacement curve after the AFM probe is pressed into the cell, and the corresponding viscoelastic parameters can be obtained.
[0004]
[0005] In the formula, P H represents the indentation force, E is the elastic modulus, μ is the Poisson's ratio, r represents the probe radius, and d is the indentation depth. When the indentation process involves a delay time, the elastic modulus E(t) will change with time and become a time-dependent function. The Maxwell model and power law model (PRL) are often used to describe the attenuation behavior of E(t).
[0006] In cell indentation experiments, the basic assumptions of the Hertz model are often difficult to meet, which may lead to errors in the calculated E(t). (1) Violation of the small deformation assumption: In AFM cell indentation experiments, the indentation depth is usually close to the probe radius. In order to avoid excessive influence of the contact point on the calculation, the ratio of the compression depth to the cell thickness is required to reach a certain threshold, which violates the assumption of small deformation. (2) Deviation from the semi-infinite space assumption: The cell radius is close to the probe size, resulting in the radial deformation of the cell affecting the calculation of its mechanical properties during the indentation process, making the semi-infinite space assumption difficult to hold.
[0007] In order to reduce the error caused by the Hertz model and reveal the mapping relationship between cell structure, viscoelastic mechanical properties and their biological behavior, researchers have conducted a lot of exploration. For example, Darling introduced the Dimitriadis model in the paper "Athin-layer model for viscoelastic, stress-relaxation testing of cells using atomic force microscopy: do cell properties reflect metastatic potential? [J]. Biophysical Journal, 2007.", and obtained elastic modulus and apparent viscosity that are independent of indentation depth. Garcia constructed the constitutive equation through the Kelvin-Voigt viscoelastic theory in the paper "Determination of the viscoelastic properties of a single cell cultured on a rigid support by force microscopy [J]. Nanoscale, 2018.", which can extract viscoelastic parameters from the force-indentation curve and eliminate the influence of indentation depth. However, current research mainly focuses on correcting the small deformation assumption, ignoring the impact of the semi-infinite space assumption in the Hertz model. In AFM indentation experiments, the failure of the semi-infinite space assumption is mainly attributed to the limited area of the cell, especially the area where the probe indentation force acts cannot be significantly smaller than the overall area of the cell. The spreading area of the cell is a key indicator in biology, which often reflects the ability of the cell to adapt to the environment and is related to a variety of biological behaviors. For example, in a high viscosity environment, the spreading area of breast cancer cells increases, and the migration rate increases accordingly; after the addition of the F-actin polymerization inhibitor LAT-A, the area change is limited and the cell migration is effectively inhibited. Similarly, bone marrow mesenchymal stem cells (MSCs) have an increased spreading area in a high viscosity environment and are more likely to differentiate into osteoblasts. In addition, matrix hardening increases the spreading area of breast cancer cells, thereby promoting their autophagic behavior. Compared with semi-infinite materials, finite area materials will undergo significant radial deformation when compressed by the probe, which will lead to errors in the extraction of cell mechanical parameters. Therefore, in AFM indentation experiments, the effect of cell area must be considered. Summary of the invention
[0008] In view of the problem that existing models are difficult to accurately extract cell viscoelastic parameters, this patent develops a revised model based on finite element data. The model adopts an explicit form, is simple and easy to use, and is suitable for wide promotion. The model was verified by finite element simulation, hydrogel and cell indentation experiments. The revised model can simultaneously solve the error problems caused by the small deformation assumption and the semi-infinite space assumption, and can accurately extract viscoelastic parameters such as cell elastic modulus and apparent viscosity, providing strong support for understanding the mechanical biological behavior of cells.
[0009] In order to achieve the above object, the technical solution adopted by the present invention is:
[0010] A method for calculating the viscoelastic mechanical parameters of cells based on indentation experiments, wherein the method first establishes a finite element analysis (FE) model of the cell, and performs finite element simulation on the deformation process of cells of different sizes in the AFM indentation experiment through ABAQUS to obtain FE calculation data. Secondly, the FE calculation data is compared with the Hertz model (Formula 1) calculation data, and the relative error of the two sets of data is calculated when the indentation conditions are the same (the same cell thickness t, cell radius R, probe radius r, indentation depth d and cell elastic modulus E). Thirdly, by reasonably setting the equation form, the relative error data is effectively fitted, and the Hertz model is corrected in the form of an explicit model to establish a corrected model. Finally, the corrected model is combined with a mechanical model used to describe the viscoelasticity of the material, which can correct the errors caused by the failure of the small deformation assumption and the semi-infinite space assumption, thereby accurately extracting the viscoelastic mechanical parameters of the cell. The following steps are included:
[0011] The first step is to build a FE model of the cell and collect data, specifically:
[0012] The FE model of the cell was constructed, and the deformation process of cells of different sizes in the AFM indentation experiment was numerically simulated using the finite element software ABAQUS to obtain the FE calculation data.
[0013] The schematic diagram of FE modeling is as follows Figure 2 As shown. Specifically, the cell FE model is a homogeneous cylinder, the height of the cylinder is the cell thickness t, and the bottom radius of the cylinder is the cell radius R. A sphere with a radius r is established at the center of the cylinder to simulate the AFM probe, and the given spherical displacement is the probe displacement d. The purpose of setting the above parameters is to obtain FE indentation data that does not conform to the small deformation assumption and the semi-infinite space assumption, so as to correct the Hertz model. The elastic modulus of the cell is set to E. Since the ball of the probe deforms slightly during the indentation process, the probe is set to a rigid body (no deformation occurs relative to the cell).
[0014] The second step is to calculate the relative error between the FE calculation data and the Hertz model calculation data and quantify the relative error, specifically:
[0015] When the parameters are the same (same t, R, r, d), there is a significant error between the indentation force P calculated by FE and the Hertz model. Therefore, in order to quantify the gap between the Hertz model and FE data, a relative error is defined, as shown in Formula 2.
[0016]
[0017] Among them, P H represents the indentation force of the Hertz model, P FE represents the indentation force of FE simulation.
[0018] The third step is to establish a correction model.
[0019] By setting the equation form reasonably, the relative error data is effectively fitted, the Hertz model is corrected in the form of an explicit model, and a correction model is established to correct the errors from the small deformation assumption and semi-infinite space. Specifically:
[0020] In order to eliminate the error caused by the failure of the small deformation assumption and the semi-infinite space assumption, the relative error δ shown in Formula 2 needs to be fitted as a function with t, R, r, and d as independent variables. Therefore, the relative error shown in Formula 3 is fitted by the product of two correction factors λ1 and λ2, while correcting the error caused by the failure of the deformation assumption and the semi-infinite space assumption. λ1 is defined as The polynomial function of is shown in Formula 4. This form is derived from the Dimitriadis equation ("Determination of Elastic Moduli of Thin Layers of Soft Material Using the Atomic Force Microscope [J]. Biophysical Journal, 2002."), which has been widely verified to correct the error of the small deformation assumption. λ2 is set as a power function containing dimensionless parameters R and r to correct the error of the semi-infinite space assumption, as shown in Formula 5. These parameters are fitted by the differential evolution algorithm of the Scipy package in Python.
[0021] δ=λ1λ2#(3)
[0022]
[0023] Among them, k i (i=0,1,2,...,12) represents the fitting parameters of the correction model.
[0024] The fourth step is to quantify the fitting accuracy of the modified model, specifically:
[0025] The relative error data obtained by formula 2 is compared with the relative error calculated by the modified model shown in formula 3 to calculate the degree of fit. It is generally believed that a degree of fit exceeding 95% indicates that the modified model has a good fit effect on the data.
[0026] The fifth step is the finite element verification of the modified model, specifically:
[0027] The correction model is combined with stress relaxation models, such as the indentation force calculated by Maxwell and power law models, and compared with the calculation results of FE to analyze the prediction ability of the correction model (it is generally believed that an error rate less than 5% indicates that the prediction effect of the correction model is good).
[0028] The present invention has the following beneficial effects:
[0029] (1) The present invention uses FE to calculate indentation data that does not conform to the small deformation assumption and the semi-infinite space assumption, and establishes a correction model by fitting the data, so as to accurately extract the elastic parameters (elastic modulus) and the viscosity parameters (such as apparent viscosity, power law exponent, etc.) of the cells, and eliminate the influence of the small deformation assumption and the semi-infinite space assumption on the calculation results in the process of viscoelastic mechanical parameter extraction.
[0030] (2) The process of establishing the correction model by the method of the present invention is simple, avoiding the complex equation solving. Moreover, when establishing the correction model, the influence of t, R, r, and d on the relative error can be fitted to simultaneously consider the influence of the failure of the small deformation assumption and the semi-infinite space assumption.
[0031] (3) The modified model established in the present invention is an explicit model, which is convenient for fitting viscoelastic parameters.
[0032] (4) The correction model established by the present invention is universal and can be used in conjunction with viscoelastic mechanical models such as the Maxwell model, power law model, and Kelvin-Voigt model to effectively calculate the viscoelastic parameters in the model.
[0033] In summary, the present invention can eliminate the influence of the small deformation assumption and semi-infinite space assumption of the traditional Hertz contact mechanics model in the process of pressing the cell with the atomic force microscope probe; the modeling process is simple and avoids solving complex equations. At the same time, the modified model is convenient and simple to apply as an explicit model. BRIEF DESCRIPTION OF THE DRAWINGS
[0034] Figure 1 A flow chart for modeling the modified model in the present invention;
[0035] Figure 2 Schematic diagram of the finite element calculation and all considered variables;
[0036] Figure 3Comparison between the Hertz model and FE calculation results. Figure 3 Figure A shows the variation of indentation force with indentation depth, where the r / t ratio is 1; Figure 3 Figure B shows the variation of indentation force with indentation depth, where the r / t ratio is 3; Figure 3 Figure C shows the variation of indentation force with cell radius; Figure 3 Figure D shows the variation of indentation force with probe radius.
[0037] Figure 4 This is a quantitative analysis of the relative error. Figure 4 Figure A shows the variation of relative error with indentation depth and probe radius; Figure 4 Figure B shows the variation of relative error with indentation depth and cell radius; Figure 4 Figure C shows the change of relative error with probe radius and cell radius.
[0038] Figure 5 To analyze the fitting effect of the modified model. Figure 5 Figure A is the fitting result of relative error when d / t=0.1. Figure 5 Figure B shows the fitting results of relative error when d / t=0.2. Figure 5 Figure C is the fitting result of relative error when d / t=0.3. Figure 5 Figure D in the figure is the comparison between the relative error and the relative error calculated by the modified model;
[0039] Figure 6 To analyze the prediction effect of the revised model. Figure 6 Figures A and B are comparisons of the indentation forces calculated by the FE, Hertz model, and the modified model when d / t=0.1, r / t=0.5, and R / t=1, where Figure B is a partial enlarged view of Figure A from 0 to 2s. Figure 6 Figures C and D are comparisons of the indentation forces calculated by the FE, Hertz model, and the modified model when d / t=0.2, r / t=0.5, and R / t=1, where Figure D is a partial enlarged view of Figure C at 0-2s. DETAILED DESCRIPTION
[0040] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions. Figure 2 It is a schematic diagram of the FE model. Figure 3 Comparison of Hertz model and FE calculation results. Figure 4 Quantitative analysis of relative errors. Figure 5 It is the analysis of the fitting effect of the modified model. Figure 6 It is the analysis of model prediction effect.
[0041] Example 1
[0042] The first step is to build a FE model of the cell and collect data, specifically:
[0043] The FE model of the cell was constructed, and the deformation process of cells of different sizes in the AFM indentation experiment was numerically simulated using the finite element software ABAQUS to obtain the FE calculation data.
[0044] The schematic diagram of FE modeling is as follows Figure 2 As shown. Specifically, the cell FE model is a homogeneous cylinder, the thickness of the cylinder is set to 8μm, and the radius R of the cylinder is set to 8μm~80μm. A spherical simulation AFM probe is established at the center of the cylinder, and the radius r is set to 4μm~24μm. The given spherical displacement is the probe displacement d, which is set to 0.08μm~2.4μm. The purpose of setting the above parameters is to obtain FE indentation data that does not conform to the small deformation assumption and the semi-infinite space assumption, so as to correct the Hertz model. The cell elastic modulus E is set to 2kPa, and the Poisson's ratio is set to 0.49. Since the ball of the probe deforms slightly during the indentation process, the probe is set to a rigid body (no deformation occurs relative to the cell).
[0045] The second step is to calculate the relative error between the FE calculation data and the Hertz model calculation data and quantify the relative error, specifically:
[0046] When the parameters are the same (same t, R, r, d), there is a significant error between the indentation force P calculated by FE and Hertz model, such as Figure 3 Therefore, in order to quantify the gap between the Hertz model and the FE data, the relative error shown in Formula 2 is defined. The result of the relative error is as follows: Figure 4 shown.
[0047] The third step is to establish a correction model.
[0048] By setting the equation form reasonably, the relative error data is effectively fitted, the Hertz model is corrected in the form of an explicit model, and a correction model is established to correct the errors from the small deformation assumption and semi-infinite space. Specifically:
[0049] In order to eliminate the error caused by the failure of the small deformation assumption and the semi-infinite space assumption, the relative error δ shown in Formula 2 needs to be fitted as a function with t, R, r, and d as independent variables. Therefore, the relative error shown in Formula 3 is fitted by the product of two correction factors λ1 and λ2, while correcting the error caused by the failure of the deformation assumption and the semi-infinite space assumption. λ1 is defined as The polynomial function of is shown in Formula 4. This form is derived from the Dimitriadis equation ("Determination of Elastic Moduli of Thin Layers of Soft Material Using the Atomic Force Microscope [J]. Biophysical Journal, 2002."), which has been widely verified to correct the error of the small deformation assumption. λ2 is set as a power function containing dimensionless parameters R and r to correct the error of the semi-infinite space assumption, as shown in Formula 5. These parameters are fitted by the differential evolution algorithm of the Scipy package in Python. The fitting parameters are k0 = -0.01656146, k1 = -0.46342625, k2 = -1.6636559, k3 = -0.39308717, k4 = -1.8701126, k5 = -0.51146411, k6 = 1.53442851, k7 = 0.16639311, k8 = 0.07352289, k9 = 0.04819098, k 10 =-0.63871273, k 11 =0.20051321, k 12 =-1.13294752.
[0050] The fourth step is to quantify the fitting accuracy of the modified model, specifically:
[0051] Comparing the relative error data obtained by formula 2 with the relative error calculated by the modified model shown in formula 3, it is found that the two sets of data are basically overlapped, and the calculated fit is greater than 99%, as shown in Figure 5 shown.
[0052] The fifth step is the finite element verification of the modified model, specifically:
[0053] The indentation force calculated by the modified model combined with the stress Maxwell model was compared with the calculation results of FE to analyze the prediction ability of the modified model. It can be seen that the indentation force calculated by the Hertz model is obviously different from the result calculated by FE, but the modified model can effectively describe the magnitude of the relaxation force under three different sets of parameters, and the prediction error rate is less than 5%. Figure 6 shown.
[0054] The above-mentioned embodiments only express the implementation methods of the present invention, but they cannot be understood as limiting the scope of the patent of the present invention. It should be pointed out that for those skilled in the art, several modifications and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.
Claims
1. A method for calculating cell viscoelastic parameters based on indentation experiment, characterized in that: The method described herein firstly establishes a finite element analysis model of the cell, which is defined as a FE model, and performs finite element simulation to obtain FE calculation data; secondly, the FE calculation data is compared with the Hertz model calculation data, and the relative error of the two sets of data when the indentation conditions are the same is calculated; thirdly, by reasonably setting the equation form, the relative error data is effectively fitted, and the Hertz model is corrected in the form of an explicit model to establish a corrected model; finally, the corrected model is combined with a mechanical model used to describe the viscoelasticity of the material to correct the errors caused by the failure of the small deformation assumption and the semi-infinite space assumption, thereby accurately extracting the viscoelastic mechanical parameters of the cell.
2. The method for calculating cell viscoelastic parameters based on indentation experiment according to claim 1, characterized in that: The following steps are involved: The first step is to build a FE model of the cell and collect data, specifically: The FE model of the cell was constructed, and the deformation process of cells of different sizes in the AFM indentation experiment was numerically simulated using the finite element software ABAQUS to obtain the data for FE calculation; The second step is to calculate the relative error between the FE calculation data and the Hertz model calculation data and quantify the relative error, specifically: When the parameters of the FE model and the Hertz model are the same, in order to quantify the gap between the Hertz model and the FE data, the relative error is defined as shown in Formula 2; Among them, P H represents the indentation force of the Hertz model, P FE represents the indentation force simulated by FE; The third step is to establish a correction model; In order to eliminate the error caused by the failure of the small deformation assumption and the semi-infinite space assumption, the relative error δ shown in Formula 2 is fitted as a function with t, R, r, and d as independent variables; the relative error shown in Formula 3 is fitted by the product of two correction factors λ1 and λ2, and the error caused by the failure of the deformation assumption and the semi-infinite space assumption is corrected at the same time; λ1 is defined as The polynomial function of λ2 is set as a power function containing dimensionless parameters R and r to correct the error of the semi-infinite space assumption. The fitting is performed using the differential evolution algorithm of the Scipy package in Python. δ=λ1λ2#(3) The fourth step is to quantify the fitting accuracy of the modified model, specifically: The relative error data obtained by formula 2 is compared with the relative error calculated by the modified model shown in formula 3 to calculate the fit; The fifth step is the finite element verification of the modified model, specifically: The corrected model was combined with the stress relaxation model, and the calculated indentation force was compared with the FE calculation results to analyze the predictive ability of the corrected model.
3. The method for calculating cell viscoelastic parameters based on indentation experiment according to claim 2, characterized in that: In the first step, the cell FE model is a homogeneous cylinder, the height of the cylinder is the cell thickness t, and the bottom radius of the cylinder is the cell radius R; a sphere with a radius r is established at the center of the cylinder to simulate the AFM probe, and the given sphere displacement is the probe displacement d; the purpose of setting the above parameters is to obtain FE indentation data that does not conform to the small deformation assumption and the semi-infinite space assumption, and to correct the Hertz model; the elastic modulus of the cell is set to E; and the probe is set to a rigid body.
4. The method for calculating cell viscoelastic parameters based on indentation experiment according to claim 2, characterized in that: In the second step, the same parameters means that the FE model and the Hertz model have the same cell thickness t, cell radius R, probe radius r, indentation depth d and cell elastic modulus E.
5. The method for calculating cell viscoelastic parameters based on indentation experiment according to claim 2, characterized in that: In the third step, the correction factor λ1 is as shown in Formula 4, and the correction factor λ2 is as shown in Formula 5: Among them, k i (i=0,1,2,...,12) represents the fitting parameters of the correction model.
6. The method for calculating cell viscoelastic parameters based on indentation experiment according to claim 2, characterized in that: In the fourth step, when the degree of fit exceeds 95%, it means that the modified model has a good fit effect on the data.
7. The method for calculating cell viscoelastic parameters based on indentation experiment according to claim 2, characterized in that: In the fifth step, when the error rate is less than 5%, it indicates that the prediction effect of the modified model is good.
8. The method for calculating cell viscoelastic parameters based on indentation experiment according to claim 2, characterized in that: In the fifth step, the stress relaxation model includes a Maxwell model and a power-rate model.