Cell culture
Human gingival epithelial and fibroblast cell lines (IBRC D10459C-HUGU) were purchased from the Iranian Biological Resource Center (IBRC). Five Petri dishes with a volume of 35mm were cultured in 1.5mL Dulbecco Modified Eagle Medium (DMEM) containing 20% Fetal Calf serum (FCS) and 7% NMP linked with folic acid manufactured by Nano Zino Co., Iran, at a concentration of 1000ppm over 8 hours in a 37°C incubator with 5% CO2. The cells were subsequently sent to the laboratory, where the creep test was conducted.
Creep Test
A capillary tube was used to calibrate the MTC instrument. The generated force was determined to be 1 ± 299pN using the analogy of an applied magnetic field in this approach. Cells' Petri dishes, utilizing a controlled force technique, were placed inside a magnetic tweezer cytometry and subjected to the creep test (static with initial loading). As a preconditioning test, twenty seconds of loading was applied to the cells before the main experiment. The principal creep test was subsequently conducted by providing a magnetic field to the cells. A magnetic field containing a step wave and a frequency of 0.2Hz was employed for this purpose. Concurrently with the examination, a 5MP microscope-mounted camera with a 1000x magnification acquired photographs of the cells' deformation under the magnetic field. In the subsequent steps, they were employed to examine displacement data. Notably, 15 epithelial cells and 15 fibroblast cells were utilized in this experiment, and the test was performed three times on each cell to confirm the precision of the data. Additionally, the displacement of ten distinct membrane sites per cell was detected.
The experimental video was transmitted to the computer for analysis, and employing the Tracker software, the movements of the cell membrane under the magnetic field were determined using the marker technique in various frames. The output software is the movement data expressed in terms of time, representing the creep response of the cells. For both types of cells, displacement-time data were collected. Each cell's creep response dataset was subsequently averaged to be utilized in viscoelastic modeling computations. The distances were calibrated using a Neubauer chamber.
The creep diagram (displacement-time) was generated for both types of cells in order to select the most suitable viscoelastic model based on the data. It is possible to determine whether the slope of the graph is ultimately fixed or continues ascending based on the rate at which the position of the markers changes over time. Keeping this in mind, searching for the optimal model for the viscoelastic solid or fluid of the target cell was conducted by minimizing the residual error. The fibroblast cell was considered by a three-element solid model (linear viscoelastic model), and the epithelial cell was deemed a four-element fluid model (Berger) based on the type of creep diagram generated from the investigations.
𝜀(𝑡) = \(\frac{\delta ₀}{K₁}\)+ \(\frac{\delta ₀}{\eta ₁}\)t + \(\frac{\delta ₀}{K₂}\left(1-{e}^{-\frac{K₂}{\eta ₂}\mathsf{t}}\right)\) Eq. (1) Berger model’s creep response
In equations (1) and (2), the parameter 𝜀 represents displacement, δ represents constant force (299pN), K1 and K2 denote spring constants, and η1 and η2 represent damper coefficients.
The following describes the creep reaction of the three-element solid model:
𝜀 (𝑡)=δ₀/𝑞₁[1/𝜆(1−𝑒−𝜆𝑡) +𝑝₁𝑒−𝜆], 𝜆=𝑞₀/𝑞₁ Eq. (2)
Where p₁=\(\frac{\eta /K₁}{1+K₂/K₁}\) , q₁=\(\frac{\eta }{1+K₂/K₁}\) and 𝑞₀=\(\frac{K₁}{1+K₂/K₁}\)
Through the curve fit graphic matching procedure on the creep response acquired from each data analysis for both types of fibroblast and epithelial cells with their respective viscoelastic model, the accurate value of the associated coefficients in the viscoelastic model was determined using Mathematica version 11.
Subsequently, the functions of the imaginary portion D'(ω) (loss modulus) and the real portion D'(ω) (storage modulus) were assessed to determine D* (complex creep function) employing the coefficients computed for the viscoelastic model attributable to each kind of fibroblast and epithelial cell.
The following equations are used to calculate the four-element fluid model of Berger.
\({D}^{{\prime }{\prime }}\left(\omega \right)=\frac{{\omega }\left({\text{q}}_{1}+{\text{p}}_{2}{\text{q}}_{1}{{\omega }}^{2}+{\text{p}}_{1}{\text{q}}_{2}{{\omega }}^{2}\right)}{-{{\omega }}^{2}\left({{\text{q}}_{1}}^{2}+{{\text{q}}_{2}}^{2}{{\omega }}^{2}\right)}\) Eq. 3
\({\text{D}}^{{\prime }}\left({\omega }\right)=\frac{(-{\text{p}}_{1}{\text{q}}_{1}+{\text{q}}_{2}-{\text{p}}_{2}{\text{q}}_{2}{{\omega }}^{2})}{-{{\omega }}^{2}({{\text{q}}_{1}}^{2}+{{\text{q}}_{2}}^{2}{{\omega }}^{2})}\) Eq. 4
Where ω indicates the frequency of loading.
The following formulas are used to calculate the Zener solid three-element model:
\({D}^{{\prime }{\prime }}\left(\omega \right)=\frac{\omega \left({q}_{0}{p}_{1}-{q}_{1}\right)}{{{q}_{0}}^{2}+{{q}_{1}}^{2}{\omega }^{2}}\) Eq. 5
\({D}^{{\prime }}\left(\omega \right)=\frac{{q}_{0}+{q}_{1}{p}_{1}{\omega }^{2}}{{{q}_{0}}^{2}+{{q}_{1}}^{2}{\omega }^{2}}\) Eq. 6
Employing Mathematica software, four Berger elements (K1 ƞ1, K2 Ƞ2) for fibroblast cells and three Zener elements (𝐾1, 𝐾2, Ƞ) for epithelial cells were determined, and the resulting values are listed in Table 1. Calculating the coefficients given in equations (3–6) also yielded the values of p and q, and Table 2 demonstrates the estimation of the modules (2). Afterward, the frequency-dependent imaginary and real parameter diagrams were drawn by Mathematica software (diagrams (3) and (4)).
The Anderson-Darling test was utilized to examine the linearity of the data to validate the creep data.
Finite Element Simulation
First, Python 3.7 software's artificial intelligence and 2D microscopic images were used to obtain 3D models of HGE and HGF cells(Waibel et al. 2022) (Blundell et al. 2021)(14) (15). A micrometer-scale two-dimensional image of epithelial and fibroblast cells is shown in Fig. 1.
Three-dimensional computational models of epithelial cells and fibroblasts are depicted as tetrahedral meshed elements in Fig. 2. After verifying the mesh independence study, the fibroblast model group has 629,741 elements, while the epithelial model has 556,302 elements.
Finite element analysis was performed using ANSYS 2021 R1 (ANSYS, Inc., Canonsburg, PA, USA) software. Based on previous studies, a Poisson ratio of 0.5 and a density of \(1040 \frac{Kg}{{m}^{3}}\) were supposed for both epithelial and fibroblast cells(Mokbel et al. 2020)(Kim and Guck 2020). The physical characteristics can be observed based on Table 1 in the numerical simulation for each cell group during investigations. In the shape of an elastic model and according to the cross-sectional region of each cell, the mechanical characteristics of the cells have been computed based on the frequency of loading using experimental data (Table 3).
Ep1 and Ep2 represent the strength of the epithelial cell at 5Hz and 150Hz, respectively. Fi1 and Fi2 also represent the ordering of epithelial cell strength at frequencies of 3 and 150Hz, respectively. The geometrical parameters of epithelial cell length and area are A1 and L1, while fibroblast cell area and length are A2 and L2.
Due to the adhesiveness of these cells, the boundary situations are applied by delivering a uniformly distributed 0.01N load to the upper surface of the cell that is not linked to the EMC. Additionally, it is believed that the attachment between the surface in communication with the cell and the EMC is Rigid.
Table 1
Geometrical and mechanical properties of epithelial and fibroblast cells.
Fibroblast Cell | Epithelial Cell |
Stiffness at gel point \({FI}_{1}\frac{{L}_{2}}{{A}_{2}} \frac{{10}^{-6}}{1.37\times {10}^{-5}}\) Pa | Stiffness at gel point \({EP}_{1}\frac{{L}_{1}}{{A}_{1}} \frac{{10}^{-6}}{7.65\times {10}^{-5}} Pa\) |
Stiffness at 150 Hz frequency \({FI}_{2} \frac{{L}_{2}}{{A}_{2}} \frac{{10}^{-6}}{1.17\times {10}^{-7}} Pa\) | Stiffness at 150 Hz frequency \({EP}_{2}\frac{{L}_{1}}{{A}_{1}} \frac{{10}^{-6}}{2.31\times {10}^{-7}} Pa\) |
Geometers \({A}_{2}=5.2832\times {10}^{-9}{m}^{2}\) \({L}_{2}=3\times {10}^{-5}m\) | Geometers \({A}_{1}=0.0000000032544{m}^{2}\) \({L}_{1}=0.00005m\) |