Enhancement of accuracy and precision of cell Young’s modulus measurement: FEM optimization and algebraic compensation

Abstract

This study addresses the inaccuracy of the Hertz model in measuring single-cell Young’s modulus via probe-based indentation. We propose two complementary correction strategies: finite element model optimization and an algebraic compensation algorithm. Both methods significantly enhance measurement accuracy and consistency within the biologically relevant 400–1200 Pa range (typical for adherent cancer cells). The compensation algorithm achieves a relative error below 5.2%. This work establishes a robust framework for cellular mechanical characterization, paving the way for incorporating greater biological complexity in future models.

1. Introduction

In the past decades, the quantitative measurement technologies of cell’s mechanical characteristics have formed a complementary multi-method system. As one of classic measurement techniques, the atomic force microscopy (AFM) plays an important role in nanoscale mechanical imaging [1–3]. In AFM, the single-point force measurement and mechanical imaging with nanoscale spatial resolution are achieved by bringing a microcantilever probe into contact with the cell surface and using the probe's bending deformation to feedback mechanical responses. Since Ashkin first demonstrated the stably trapping of microparticles [4], the optical tweezers (OTs) technique has provided a new prospect for cell mechanics research, as one of non- or flexible contact measurement techniques. The OTs technology relying on the combined effect of optical field gradient force and scattering force could enable the micron-sized microspheres as “mechanical probes”. Both AFM and OTs could be used to indent the cell by using the probe while precisely measuring the mechanical responses of the cell deformation.

The probe-based local indentation technique, capable of applying of pico- to nano-newton forces while detecting nanometer-scale deformations, has become an essential tool for assessing these cellular mechanical properties [5–7]. However, the elastic modulus of cells cannot be measured directly. Instead, it is typically derived from indentation data using the Hertz contact model [8,9]. Notably, reported mechanical properties of cells often vary considerably due to factors such as experimental methods [10], cellular morphology [11] and internal structure [12], contact force applied [13], and even the probe geometry [10,14]. This significant variability complicates the quantification of cellular mechanical characteristics, as both the precision and accuracy of the measurement are critical. In practice, precision refers to the consistency of repeated measurements under identical conditions. For example, the elastic modulus of MCF-7 breast cancer cells measured by atomic force microscopy (AFM) using a spherical probe (radius: 2.5 µm) was approximately 0.53 ± 0.52 kPa at a loading rate of 10 µm/s and 37 °C, but about 1.31 ± 0.54 kPa at 6 µm/s and 25 °C [10]. The smaller standard deviation in the latter case indicates higher measurement precision. Assessing accuracy, the closeness of a measurement to the true value, presents a greater challenge in biological experiments, because the true elastic modulus is often unknown. For example, measurements of MCF-7 cells’ elastic modulus using an AFM with the conical probe yield values nearly three orders of magnitude larger than those obtained by particle-tracking microrheology. Such discrepancies highlight that the apparent mechanical properties of a cell depend strongly on the measurement technique, obscuring the true underlying value. To address this issue, the finite element method (FEM)-based simulation work offers a promising approach [15–17]. By assigning a known elastic modulus to a computational cell model and simulating its mechanical response to an applied load, this method provides a controlled framework for interpreting experimental data and evaluating measurement validity [18].

In this paper, we firstly constructed a computational mechanical model in FEM, in which the elasticity of a cell was assigned a preset Young's modulus. This model was used to simulate the deformation of the cell under indentation by a spherical probe, generating a simulated force-indentation depth relationship analogous to that obtained in physical experiments. Secondly, the local refinement mesh and adjustment of the penalty factor multiplier were implemented in the optimized simulation. Thirdly, an algebraic compensation algorithm based on the curve fitting method and formula derivation was conducted. Last but not least, the results of original model, the optimization simulation and the compensation algorithm were validated and compared quantitatively, which demonstrates solutions for accurate/precise measurements of the Young's modulus of living cells manipulated by the probe with OTs or AFM.

2. Methods

2.1. Cell model

To improve the measurement accuracy and consistency of measuring the elastic modulus of cells by manipulating the probe, here we chose to build up the mechanical model and to simulate the cell deformation with the ideal conditions to investigate the dominate factors inducing discrepancy of the measured elastic modulus. Figure 1 shows the schematic diagram of the model that the cell surface could be indented by the spherical probe with the downward force F. The cell geometry is a spherical crown with the height of H_C and the sphere radius of R_C, while the probe has the radius of R_P. In this model the cell lies down on the substrate, and the stress is loaded to deform the cell surface through the rigid probe while the cell bottom is constrained to be fixed. The contact pair between the cell surface and the probe was employed to solve the contact mechanics problem approximately through the penalty method. The whole model was meshed with totally 52001 elements. We chose the linear elastic material for the cell with the Young’s modulus E_C and Poisson’s ratio v_C in the condition of slight deformation of which the strain fully complies with the magnitude less than 10% of the sample thickness threshold [19,20]. The final deformation in the state of equilibrium was calculated through the stationary solver. Because of the significant discrepancy of elastic modulus between the probe and the cell, the deformation of the probe is ignored.

Fig. 1.

Schematic diagram of contact model between cell and probe.

2.2. Prediction of Young’s modulus

A series of the cell deformations were simulated by sweeping the loading force F from 0 to 300 pN with 15 steps. With mechanical properties given by preset parameters for the cell and the probe in Table 1, the deformation of adherent cancer cells under a series of pressing forces from the spherical probe were simulated in COMSOL software. The strain distributions of the internal part of the deformed cell are presented in Fig. 2(a-c) when the loading force is 20, 160, and 300 pN, respectively. The probe contours represent the initial position, and the position of the probe gradually descends down. The cell strain magnitude and the deformation volume augment with the rising loading force. The vertical displacement i of the cell at the contact point (the cell apex) as a function of the loading force F is shown in blue squares of Fig. 2(d). It can be clearly found that the indentation depth of the cell surface gradually raises with the increase of the loading force, and the indentation depth of the model reaches 0.4 µm corresponding to the loading force of 300 pN. In practice, the Young’s modulus for the cell are often obtained based on the Hertz model, as well as the parameters (R_C and R_P) and the measured variables (F and i). The Hertz model was still chosen to raise the error problem, along with the optimization method and the algebraic compensation algorithm, although the maximum strain is beyond the Hertz condition (strain magnitude less than 10% of the cell height). Instead of the experimentally measured F and i, the simulated results of F and i were used to calculate the estimated Young’s modulus E_cal for the cell in the Hertz contact model as [23]

$$E_{\text{cal}}=\frac{3}{4}(1-{\nu }_{\mathrm{C}}^2)F\sqrt{\frac{R_{\mathrm{C}}+R_{\mathrm{P}}}{R_{\mathrm{C}}{R}_{\mathrm{P}}}}{i}^{{-}^{3/2}}$$ (1)

Table 1. Parameters in model.

RC(µm)	HC(µm)	ρC(g/cm3)	EC(Pa)	vC	RP(µm)	EP(GPa)	vP
62	6	1.1	600 [21]	0.48	2.5	70	0.24 [22]

Fig. 2.

Strain distribution and mechanical response of cells under different loading forces. Three-quarter section of strain distribution of deformed cell under loading force of 20 pN (a), 160 pN (b), and 300 pN (c), respectively; simulated results of apex displacement of cell versus loading force, and the estimated Young's modulus calculated by Hertz contact model (d); Apex displacement of cell with a series of preset Young's moduli versus loading forces (e); Estimated Young's modulus versus loading force calculated by Eq. (1) with preset Young's moduli (f).

Figure 2(d) also shows the variation of the estimated Young’s modulus E_cal, which exhibits a negative exponential decay trend, with respect to the loading force F. It is obvious to see that the estimated Young’s modulus E_cal changes from 1554 to 419 Pa as the loading force F from 20 pN to 300 pN, whereas there is only one Young’s modulus E_set of 600 Pa defined for the cell in the simulation model. For the universality of the solution, we calculated a series of results under a series of preset Young's moduli E_set from 400 to 1200 Pa, and the displacement of its apex under different forces for each E_set is shown in Fig. 2(e). The displacement increases with the increasing loading force, but the average displacement decreases with the increase of the preset Young's modulus. We substitute the force and displacement simulation data into Eq. (1) to obtain a series of estimated Young's moduli, as shown in Fig. 2(f). It can be seen that the estimated Young's modulus decreases with the increase of the loading force for each preset Young's modulus. This indicates the contact force will have an inverse impact on the measurement results of Young's modulus. The estimated Young's modulus has large errors in precision and accuracy, which is far from the true value of Young's modulus. Therefore, it is necessary to optimize the model or improve the algorithm to reduce the error.

2.3. Optimization simulation

Due to the use of finite element analysis to simulate the contact model, the refinement level of the mesh has a significant impact on the simulation results [24]. As the refinement level increases, the number of grids in the simulation model greatly increases, resulting in a sudden increase in computational complexity. In order to significantly reduce the computational load without significantly affecting the calculation results, we locally refined the region close to the contact apex for the cell. We propose a grid refinement formula as

$$L=q\sqrt{x^2+y^2+{(z-z_0)}^2}$$ (2)

where x, y and z (unit: m) represent the spatial positions of the grid. This formula is derived for the mesh optimization needs of the cell-probe contact area, with the contact apex as the refinement center. The mesh size increases radially outward, which not only ensures the calculation accuracy of the contact area but also avoids computational redundancy caused by excessively dense meshes in non-contact areas. Converting it into COMSOL code and inputting it into the mesh size expression input box can complete the local refinement of the model mesh. This formula starts from the contact apex between the cell and the probe, and the grid size L increases radially outward. To prevent excessive coarsening in the place away from the contact apex, a maximum limit on grid size is defined, and the maximum grid size is set to be 1.86 µm. The constant z₀ is defined to be the height of the cell apex Hc (6 µm) for coordinate transformation in order to set the contact point as the refined center point. And q (unit: 1) is a coefficient for adjusting the refinement degree. The top view, three-quarter view and zoomed picture for local refinement mesh are shown in Fig. 3(a-c). We conducted simulation tests on the relationship between the estimated Young’s modulus and the loading force, aiming to explore the suitable refinement coefficient that both minimizes the consumption of computing resources and meets the accuracy requirements.

Fig. 3.

Effects of mesh refinement and penalty factor optimization. Grid diagram of the cell model with refinement parameter q = 0.15 in top view (a), three-quarter view (b) and zoomed view (c), estimated Young's modulus versus loading force as q is 0.13, 0.15, 0.17, 0.2, 0.3, and 0.4 respectively (d), estimated Young's modulus versus loading force with preset penalty factor multipliers of 1, 5, 10, 20, 40, 60, and 80 (e), and error analysis of averaged Young's modulus before mesh refinement, after mesh refinement, and after penalty factor optimization with black bar denoting error range of E_cal (f).

When the value of refinement coefficient q is bigger, 0.4 or 0.3, the simulation model could not converge well and the estimated Young’s modulus has significant fluctuations as shown in Fig. 3(d). As the refinement coefficient decreases, the model gradually converges well. When q drops to 0.15 or lower, both sets of estimated Young’s moduli almost coincide. Considering that further refinement does not significantly improve the precision and accuracy and requires more computing resources, the refinement coefficient is set to be 0.15.

Other than the mesh quality, we found that the accuracy of the estimated Young’s modulus could be seriously influenced by the bigger initial step size of the loading force. This is because the initial contact between the probe and the cell is a point contact. If the initial step size of the loading force is too big, the following deformation process based on the result of the initial value might be seriously inconsistent with the actual contact process. To solve this problem, we firstly simulated a series of small values of the loading force, namely 0.1, 0.5, 1.2, 3, 5, 10, and 16 pN, before solving the other loading forces from 20 pN to 300 pN. After refining the initial step sizes, the nonconvergent problem in simulation was successfully solved.

Aiming to optimize the solution process, we additionally employed the penalty method [25] to confine the solution space and enhance accuracy. In optimization algorithms, the penalty method converts constrained problems into unconstrained ones by assigning a large objective function value to any iteration point that violates the constraints. This drives the minimum of the unconstrained problem to approach or lie within the feasible region. The function of penalty method is to ensure that the mechanical behavior between contacting objects conforms to the actual physical laws and avoid unreasonable penetration. The penalty method is to punish the violation of the nonpenetration constraint (i.e. the occurrence of penetration), and adjust the state of the object by applying stress, so that the penetration gradually decreases, but it does not strictly guarantee the complete non penetration, but only controls the penetration within a very small range. In the penalty method, the contact pressure as a function of the physical gap is expressed as [26]

$$T_n=\{\begin{array}{cc}{f}_p{p}_n{g}_n+p_0& g_n<p_0/p_n\\ 0& \text{else}\end{array}$$ (3)

where T_n is the contact pressure assumed positive in compression, g_n is the physical gap distance between the probe and the cell, P₀ is the pressure at zero gap, P_n is the penalty factor, and f_p is penalty factor multiplier tuning the value of penalty factor. The default value of P_n is defined by [26]

$$p_n=\frac{E_{\text{set}}}{h_{min}}$$ (4)

where h_min is the minimum element size on the cell. The penalty factor is usually large enough for the formulation to be coercive. The higher the value, the better the boundary condition is fulfilled, but a too high value results in an ill-conditioned equation system. We tuned the value of the penalty factor by varying the magnitude of the penalty factor multiplier. A series of penalty factor multipliers were tested as shown in Fig. 3(e). It can be observed that as the penalty factor multiplier increases, the estimated Young's modulus gradually converges toward a stable value. When f_p reaches 60, further increases have negligible impact on the simulation results. Therefore, the penalty factor multiplier was ultimately set to 60. Compared with E_set in yellow squares when f_p = 1, the E_set in blue triangles when f_p = 60 seems more accurate and reliable.

To verify the effect of these methods, the comparison before and after mesh refinement and optimization is shown in Fig. 3(f). Prior to model optimization, the error in the estimated Young's modulus was directly proportional to the preset value, reaching a maximum of ±280 Pa at E_set = 1200 Pa. Following mesh refinement, the maximum error was significantly reduced to ±47 Pa. Subsequent optimization of the penalty factor further enhanced accuracy and precision, lowering the maximum error to ±15 Pa. It is obvious to see that the simulation errors might result from the multiple factors, such as contact geometry, boundary conditions and numerical noise, etc.

2.4. Compensation algorithm

Experimental measurements of cellular elasticity typically rely on the Hertz contact model, which is difficult to accurately replicate in FEM simulations. To address this discrepancy, we developed a compensation algorithm. The derivation of this algorithm first establishes the relationship between the estimated Young's modulus and the contact force, yielding a curve-fitting equation. As shown in Fig. 4(a), the contact force F exhibits a phenomenologically exponential relationship with the fitted Young's modulus as

$$E=E_0+A{\mathrm{e}}^{B\cdot F}$$ (5)

where E is the dependent variable of the above equation, representing the fitted Young's modulus, F is the independent variable representing the indentation force, E₀, A and B are exponential fitting parameters. Following the exponential fitting of five datasets (corresponding to preset Young’s moduli of 400, 600, 800, 1000, and 1200 Pa) relating the estimated Young’s modulus to the loading force, five sets of fitting parameters (E₀, A, and B) were obtained in Eq. (5). Notably, parameter B was found to remain approximately constant across all fits. To simplify subsequent calculations, B was therefore assigned a fixed value of -0.02 pN⁻¹.

Fig. 4.

Fitting and correction effects of the compensation algorithm. Five fitting curves for estimated Young's modulus versus loading force in conditions of preset Young’s moduli of 400, 600, 800, 1000, and 1200 Pa respectively (a), five fitting lines for scatter points of parameters E₀ & A versus E_set (b), and corrected Young's modulus versus loading force in conditions of five preset Young’s moduli (c).

The discrete values of E₀ and A exhibit the linear relationship with E_set, respectively. Thus, we performed the linear fitting for E₀ and A versus E_set, which is expressed as

$${E_0}_i=\mathrm{a}+\mathrm{b}{E}_{\text{set}i}$$ (6)

$$A_i=\mathrm{c}+\mathrm{d}{E}_{\text{set}i}$$ (7)

with fitting parameters a, b, c and d. The subscript i means group number of the preset Young’s modulus. The fitted values of parameters a, b, c and d are -244.76 Pa, 1.22, -1268.36 Pa, and 4.99, respectively. Substitution of these values into Eqs. (6) and (7) yields the fitting curve presented in Fig. 4(b). By substituting Eqs. (1), (6), and (7) into (5), we obtained the corrected Young's modulus written by

$$E_x=\frac{\frac{3}{4}(1-{\nu }_{\mathrm{C}}^2)F\cdot i^{{-}^{3/2}}\sqrt{\frac{R_{\mathrm{C}}+R_{\mathrm{P}}}{R_{\mathrm{C}}{R}_{\mathrm{P}}}}-\mathrm{a}-\mathrm{c}{\mathrm{e}}^{B\cdot F}}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F}}$$ (8)

with x = 1, 2, …, m, and m denoting the steps number of loading forces.

To derive the matrix form of the Young’s modulus correction formula, we expressed the variables in Eq. (8) in the one dimensional array form:

$${\mathbf{F}}_1=\left[\begin{array}{cccc}\frac{F_1}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F_1}}& \frac{F_2}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F_2}}& \cdots & \frac{F_m}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F_m}}\end{array}\right]$$ (9)

$${\mathbf{F}}_2=\left[\begin{array}{cccc}\frac{1}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F_1}}& \frac{1}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F_2}}& \cdots & \frac{1}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F_m}}\end{array}\right]$$ (10)

$${\mathbf{F}}_3=\left[\begin{array}{cccc}\frac{{\mathrm{e}}^{B\cdot F_1}}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F_1}}& \frac{{\mathrm{e}}^{B\cdot F_2}}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F_2}}& \cdots & \frac{{\mathrm{e}}^{B\cdot F_m}}{\mathrm{b}+\mathrm{d}{\mathrm{e}}^{B\cdot F_m}}\end{array}\right]$$ (11)

$$\mathbf{D}=\left[\begin{array}{lll}{i}_1^{{-}^{3/2}}& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & i_m^{{-}^{3/2}}\end{array}\right]$$ (12)

$${\mathbf{E}}_{\text{com}}=\left[\begin{array}{cccc}{E}_1& E_2& \cdots & E_m\end{array}\right]$$ (13)

where F₁, F₂ and F₃ are one dimensional arrays with respect to loading force F₁-F_m, D is the m × m matrix containing displacement i₁-i_m, and E_com is the matrix form of the compensation values of Young's modulus obtained by a cell under the application of a series of forces. The matrix expression of the corrected Young's modulus is deduced as

$${\mathbf{E}}_{\text{com}}=\frac{3}{4}(1-{\nu }_{\mathrm{C}}^2){\mathbf{F}}_1\mathbf{D}\sqrt{\frac{R_{\mathrm{C}}+R_{\mathrm{P}}}{R_{\mathrm{C}}{R}_{\mathrm{P}}}}-\mathrm{a}{\mathbf{F}}_2-\mathrm{c}{\mathbf{F}}_3$$ (14)

It is obvious to see that not only a series of force F and corresponding indentation i values, but also the parameters of a, b, c and d are supposed to be obtained before solving Eq. (14). It means the compensation algorithm still needs the preset/empirical values of the cell Young’s modulus. Discrete Young's modulus of compensation versus the loading force in conditions of five preset Young’s moduli is shown in Fig. 4(c). It can be seen from the figure that the corrected Young's modulus is further close to the preset one after compensation, indicating that the method of formula compensation is effective.

3. Discussion

It is clear to see that simulation results exhibited limitations in both precision and accuracy as shown in Fig. 5(a), although the model was simulated in the FEM and the elasticity was deduced by using the Hertz contact model. Given that a range of Young's modulus values were assigned to the model to calculate the actual modulus from cellular deformation under varying loads, it is necessary to validate the simulation results obtained under these different modulus settings.

Fig. 5.

Data distribution and error analysis. Distribution characteristics, error range, and significance analysis results of raw data of E_cal from original model with black dots representing a series of raw data, the color filled area indicating the distribution range and density of the data, and p value indicating the significance between two adjacent groups of data (a), comparison of E_cal of original model, optimization simulation, and compensation algorithm with small dots of three different colors representing the data distribution and asterisk indicates the significance of the data between two groups (b), data distribution and error analysis of optimization simulation and compensation algorithm with small dots of two colors represent the data distribution obtained by the two methods(c). The boxplot shows the quartile range, median and error range of the data.

The compensation algorithm is proposed to rapidly obtain high-precision results via low-cost rough simulation combined with formula-based compensation. Due to the simplified parameter settings, rough simulation has obvious numerical noise, so it is necessary to verify through statistical tests (Mann-Whitney test, Levene test): (1) Whether the simulation results under different preset moduli have significant differences to ensure the discriminability of the model; (2) Whether the data dispersion after optimization and compensation is significantly reduced to verify the algorithm's noise suppression effect. The results of the statistical tests quantify the data discrepancies before and after optimization, providing statistical support for the effectiveness of the two methods of model optimization and algorithm compensation.

First, a normality test was conducted on the simulated Young’s modulus data from the initial model, which indicated that the data were not normally distributed. Therefore, the non-parametric Mann–Whitney test was employed. As shown in Fig. 5(a), the test revealed statistically significant differences between the simulation results obtained under different Young’s modulus values, thereby confirming the validity of the initial modeling approach. The wider the colorful area, the denser the data distribution. Then, in order to prove whether the two methods of model optimization and formula compensation can effectively improve the accuracy of the model, we conduct Levene test on the initial modeling data, the data after model optimization and the data after formula compensation, and the inspection results are shown in Fig. 5(b). The test results show that both methods significantly reduce the degree of data discretization and greatly improve the accuracy of simulation results. Finally, in order to compare the optimization effects of two methods on the simulation model, we calculated their median, interquartile distance and standard deviation respectively, and the comparison results are shown in Fig. 5(c). The comparison reveals that both methods significantly reduce the data dispersion, while demonstrating a high accuracy and precision.

To quantitatively compare the effects of model optimization and formula compensation, we use the relative error and relative uncertainty for the error analysis. The relative error represents the degree of deviation from the true value of the calculation result, and the relative uncertainty characterizes the degree of dispersion of the calculated results. The calculation formulas for relative error (δ), uncertainty (u) and relative uncertainty (γ) are expressed by [27]

$$\delta =\frac{|E_{\text{cal}}-E_{\text{set}}|}{E_{\text{set}}}\times 100\%$$ (15)

$$u=\frac{\sqrt{\left[{\sum }_{i=1}^n{(E_{\text{cal}}-E_{\text{set}})}^2\right]/n}}{\sqrt{n}}$$ (16)

$$\gamma =\frac{u}{E_{\text{set}}}$$ (17)

It can be seen from Table 2 that both the relative error and relative uncertainty of the calculated Young's modulus E_cal of the raw data increase as the cell stiffness increases. The relative errors greater than 30% indicates that E_cal of the original model is obviously deviated from the true value. The relative uncertainties greater than 10% in all cases of the original model indicates that the calculated Young's modulus values manifest a large degree of dispersion. With the optimization and compensation, the relative error and relative uncertainty are significantly reduced, indicating that the precision and accuracy of the calculated value of Young's modulus are significantly improved. The results obtained through formula compensation are closer to the true value and have a smaller degree of dispersion by contrast with the optimization.

Table 2. Comparison of relative error and relative uncertainty for original model, model optimization and formula compensation.

Truth value Eset (Pa)	Original model		Optimization simulation		Compensation algorithm
	Relative error δori (%)	Relative uncertainty γori (%)	Relative error δopt (%)	Relative uncertainty γopt (%)	Relative error δcom (%)	Relative uncertainty γcom (%)
400	31	10	9.0	2.5	4.3	1.3
600	30	13	6.9	1.8	4.9	1.6
800	35	16	5.7	1.6	5.2	1.6
1000	43	20	5.0	1.4	3.8	1.2
1200	53	23	4.5	1.2	2.0	0.67

Although the Hertz model is reasonable under the small strain conditions of this study, it is necessary to pay attention to the difference between its core assumption of “semi-infinite elastic body” and the actual cell geometry. In this study, the cell height is only 6 µm, and the maximum indentation depth is 0.4 µm. The ratio of indentation depth to cell height is approximately 6.7%, which exceeds the applicable threshold of the “semi-infinite” assumption (usually ≤5%). Referring to the study by Guz et al. [7] in 2014, when the ratio of indentation depth to sample thickness exceeds 5%, the finite thickness will lead to an overestimation of the Young’s modulus. The cell bottom is adherently fixed, and the substrate effect is negligible. The adhesion force between the probe and the cell is extremely small, so adhesion correction is unnecessary. The strain at the contact point is less than 6.7%, and thus the large deformation correction is not required. Therefore, the main error source is the finite thickness effect. In this regard, referring to the correction strategies proposed by Dimitriadis [28], Sun [29] et al., we specifically corrected this error through the formula compensation framework derived in this study (i.e. Eqs. (8)–(14)). Since numerical errors are corrected through model optimization, Young’s modulus varies linearly with applied force. Thus, we replaced the original exponential fitting with linear fitting and adjusted the parameters of the compensation formula accordingly. The corrected results are shown in Table 3, which effectively reduces the error caused by finite thickness. This study verifies the effectiveness of the Hertz model under simplified conditions, laying a foundation for subsequent extended research combining the mechanical characteristics of real cells.

Table 3. Comparison of finite thickness error correction results.

Truth value Eset (Pa)	Before correction		After correction
	Relative error δopt (%)	Relative uncertainty γopt (%)	Relative error δcom (%)	Relative uncertainty γcom (%)
400	9.0	2.5	6.5	1.9
600	6.9	1.8	4.8	1.5
800	5.7	1.6	3.9	1.3
1000	5.0	1.4	3.3	1.1
1200	4.5	1.2	2.9	1.0

The above error analysis proves the validity of modeling and accuracy of simulation with the spherical probe in the cell contact model. Because this is the process of measuring the Young's modulus of cells simulated by simulation software, the main sources of error mainly come from the Hertz model and simulation calculations. Since the first method only involves optimizing the model, the error caused by the Hertz contact model has not been optimized. The second method of compensation formula optimizes the overall error, including the error caused by Hertz model. However, the discrepancy between the optimization results of the two methods is negligible, which suggests that the approximation of the Hertz model induces only a minor error within the small deformation range. The feasibility of using the Hertz model for measuring the Young's modulus of cells with spherical probes was verified through simulation. It was also shown that the errors in measuring the Young's modulus of cells are mainly due to the heterogeneity of cell structure and the complexity of cell mechanical properties, such as the distribution of the cytoskeleton [30,31], and the cellular mechanoresponse [32] of living cells under external forces.

4. Conclusion

This study aims to address a key challenge: how to achieve accurate measurement of the Young's modulus of living cells using two major indentation technologies with probes by OTs and AFM. By constructing an FEM mechanical contact model to simulate the mechanical behavior of the cell, we found that there are significant errors in the calculation results of Young's modulus when using the traditional Hertz contact model.

To solve this problem, we applied two complementary strategies: first, a finite element model optimization strategy that integrates local mesh refinement of the contact area and penalty factor adjustment, forming a standardized optimization process; second, an algebraic compensation algorithm based on the least squares principle and curve fitting, which synergistically corrects system errors from both the Hertz model and FEM simulation. These two complementary strategies form independent frameworks for error correction, which significantly improve the accuracy and consistency of cellular Young's modulus measurements. Within the range of Young's modulus relevant to biological research (400–1200 Pa), both strategies significantly improved measurement performance: The relative error of the optimized finite element model was reduced to the value below 6.5% and the relative uncertainty was reduced to the value below 1.9%. The compensation algorithm performed even better, with a relative error below 5.2% and relative uncertainty below 1.6%. The compensation method enabled consistent calculation of Young's modulus for cells with equivalent stiffness under different loading forces, resulting in estimates that closely matched the preset values. Yet the limitation of the requirement for a few preset values still needs to be overcome. The method introduced in this paper establishes a robust framework for the cellular mechanical characterization. Future work will focus on expanding this framework to incorporate critical factors such as structural inhomogeneity (e.g., cytoskeletal distribution) and time-dependent viscoelastic responses. This evolution will enhance the model's biological fidelity and broaden its applicability in probing sophisticated mechanobiological phenomena.

Funding

National Natural Science Foundation of China https://ror.org/01h0zpd94 ( 12564044); Guangxi Natural Science Foundation ( 2023GXN SFBA026071); Guangxi Science and Technology Program Project ( AD21220086, AD25069073); Innovation Project of GUET Graduate Education ( 2024YCXS235).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are available from the corresponding author upon reasonable request.