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Mechanotransduction refers to the mechanisms by which cells sense and respond to local loads and forces. The process of mechanotransduction plays an important role both in maintaining tissue viability and in remodeling to repair damage; moreover, it may be involved in the initiation and progression of diseases such as osteoarthritis and osteoporosis. An understanding of the mechanisms by which cells respond to surrounding tissue matrices or artificial biomaterials is crucial in regenerative medicine and in influencing cellular differentiation. Recent studies have shown that some cells may be most sensitive to low-amplitude, high-frequency (

Mechanotransduction plays an essential role in normal cell function and in a wide range of disease conditions [

To improve our understanding of how cells interact mechanically with surrounding materials and various substrates, experimental studies are typically carried out with cultured cells or tissues, cyclically stimulated in specially designed bioreactors [

In the past several years, advances in finite-element modeling (FEM) capacity have made it possible to model many aspects of cell mechanics [

We report the development of a three-dimensional finite-element model of cyclic strain, simulating adherent osteoblastic cell geometry on an isotropic equibiaxial strain membrane. This model incorporates the viscoelastic properties of the cytoplasm—as well as a basic model of the cytoskeleton—and simulations include sinusoidal strain at frequencies ranging from 1 to 45 Hz, consistent with the range of frequencies used in recent cell and animal experiments. This model is used to investigate the influence of the frequency of mechanical stimulation on peak stress and strain distributions within the cytoplasm and nucleus.

The cell geometry used in this study is representative of an adherent osteoblastic cell as it would be configured ^{3}. The nucleus is modeled as an ellipsoid, with maximum diameter of 5 µm, minimum diameter of 2 µm and volume of 46 µm^{3}. These dimensions are representative of osteoblastic cells observed with confocal microscopy, and are consistent with similar finite-element models reported previously [

The cell model is attached to a thin elastomeric membrane, simulating an experimental configuration that would be used to investigate the effect of cyclic strain. The membrane was specified as a 0.25 mm thick sheet of polydimethylsiloxane (PDMS), with a linear elastic modulus of 1.8 MPa and ν = 0.49, where ν is Poisson’s ratio [

Constitute components considered in this finite-element model were limited to the cytoplasm and nucleus. The material properties of the cell were representative of previous experimental studies [_{0} = 6.5 kPa, E_{∞} = 4.3 kPa, ν = 0.5, τ = 15.4 s), where E_{0} is the instantaneous elastic modulus, E_{∞} is the equilibrium elastic modulus, ν is Poisson’s ratio, and τ is the characteristic relaxation time. Overall, the selected parameters are representative of an osteoblast in a “spread” morphology [

To investigate the importance of viscoelasticity in the model, additional simulations were performed with the time constant τ assigned to the cytoplasm set to 0.1, 1.0, and 10 s (reflecting the wide range of values reported from experiments with atomic force microscopy and magnetic beads). Modeling was also performed in which the cytoplasm was assumed to be a linear elastic material, with elastic modulus of 6.5 kPa and Poisson’s ratio of 0.5. To investigate the importance of the differential elasticity of the cytoplasm and nucleus, an additional numerical simulation was also performed where the nucleus was assigned the same linear elastic modulus and Poisson’s ratio as the cytoplasm (

(

The nucleus in adherent cells is not rigidly embedded within the cytoplasm, but is more likely “tethered” to some extent by a network of actin stress fibers that connect the nucleus to the foot of the cell. To investigate the effect of a simple model of the cytoskeleton, a pre-tensioned cable network of stress fibers was used to model the actin cytoskeleton of the cell, similar in geometric configuration to that in [

Schematic representation of the 3D model geometry used to represent a network of stress fibers, shown in: (

Finite-element modeling was performed using a commercial FE software package (Abaqus CAE v.6.8.3, Simulia, Providence, RI, USA) running on a desktop computer (Intel Core i7 Quad, 4 cores, 2.67 GHz, maximum memory requirements 3 GBytes RAM). The model was discretized using 10-node solid 3D tetrahedral elements for the cell cytoplasm and nucleus, while 3-node planer 3D shell elements were employed for the elastomeric substrate membrane. The total number of elements was 58,656. The model was investigated at strain frequencies of 1, 20, and 45 Hz and computational simulation was carried out at each frequency for 10 complete input strain cycles. Data were recorded at 12 uniformly spaced time intervals per cycle for a total of 120 frames per frequency case. Each finite-element simulation required approximately 2 hours of run time on 2 CPU cores, producing three-dimensional maps of Von Mises stress, strain, and deformation. These quantitative maps were used to determine the maximum values of stress and strain over the strain cycle. The FEM package also produced time-resolved 3D cine loops of local stress and strain over the motion cycle; these movies were used to determine the location of stress concentrations within the cell.

The three-dimensional geometry of the simple cell model is shown in

The stress distribution observed within the viscoelastic cell model is shown in

finite-element modeling (FEM) simulation results, showing peak values of stress in MPa (

Simulations repeated with the time constant τ for the cytoplasm ranging from 0.1 to 10 s showed no impact on the strain values reported within the cell, regardless of the frequency of stimulation. Peak stress values were maintained within 5% at 20 Hz and 45 Hz, regardless of whether τ was set to 0.1 s, 1 s, 10 s or if the cytoplasm was assumed to be linearly elastic. At 1 Hz, peak stress values remained below 2.5% for τ = 10 s and when the cytoplasm was assumed to be linearly elastic. The only significant impact of modeling the cytoplasm as a viscoelastic material was observed with τ = 0.1 s and τ = 1 s for stimulation at 1 Hz; in these cases, the peak stresses reported in the cytoplasm were 28% and 13% lower respectively in a viscoelastic model than in a linear elastic model (

The impact of the assumed viscoelastic relaxation time constant τ at various frequencies. (

The finite element model was repeated with both the nucleus and cytoplasm assumed to be linear elastic materials with identical elastic properties. In this case (

FEM simulation results under the assumption that both cytoplasm and nucleus are linear isoelastic materials, with elastic modulus of 6.5 kPa and Poisson’s ratio 0.5, showing peak values of stress in MPa (

Simulations performed with a model that incorporates a simplified cytoskeleton (

FEM simulation results, showing peak values of stress in MPa (

We have implemented the first finite-element model of a cell undergoing high-frequency equibiaxial strain on an elastomeric substrate. Moreover, we have used this model to determine the influence that the frequency of stimulation may play with respect to local stress and strain within the cell. Our results indicate that high-frequency stimulation (

The inclusion of a simplified cytoskeleton, consisting of 24 stress fibers, produced focal elevations of stress and strain in the peri-nuclear region of the viscoelastic cell model. Focal values observed at the SF attachment points were highly elevated and were not strongly dependent on the frequency of stimulation; however, it is important to note that these values are reported at singularities and may be artifactually amplified. Representative regions of the cytoplasm (distal to the SF attachment points) exhibited frequency-dependent increases in both stress and strain. Our observations support the hypothesis that the cytoskeleton plays an important role in transmitting extracellular mechanical forces to the nucleus, possibly mediating mechanotransduction [

The value of the time constant τ that was assumed in the viscoelastic model did not have a significant impact on the derived stress and strain distributions, with the exception of the model simulating the two lowest time constants (0.1 s and 1 s) and lowest frequency (1 Hz). Under most conditions, at frequencies above 1 Hz, values of τ ranging from 1–15 s will provide acceptable results, as would a purely linear elastic model. Stress concentration was, however, influenced by the differential material properties in the model, such as the assumed mismatch between the properties of the nucleus and cytoplasm. These findings may indicate that in future models, consideration should be given to assigning accurate values to the elastic modulus of the different cellular components, as well as to the determination of viscoelastic parameters.

There are some simplifications in the finite-element model reported here. Adhesion was assumed to be uniform over the entire contact area between cell and substrate; in future studies it will be interesting to model the impact of focal adhesions. The axisymmetric three-dimensional geometry of the cell was not intended to represent

This study indicates the significant influence of the frequency of applied mechanical stimulation, in the case of equibiaxial strain at physiological levels. The elevated levels of stress and strain observed within the cytoplasm and nucleus may indicate a mechanism by which high-frequency mechanical stretch stimulates a response within cells. Future studies will include additional details of internal cytoskeletal structure and investigations of possible resonance conditions within the cell. The model will also be extended to include other mechanisms of cyclic stimulation (such as vibration and shear). Dynamic finite-element models will undoubtedly play an increasingly important role in investigating the way in which cells interact with biomaterial substrates and the local mechanical environment. Such models will also assist in elucidating the mechanisms underlying mechanotransduction.

This work was funded by a grant from the Natural Sciences and Engineering Research Council of Canada (grant number 383395-2010). MWG is supported by a Frederick Banting and Charles Best Canada Graduate Scholarship (CGS) Doctoral Award from the Canadian Institutes of Health Research (CIHR). KLB is supported by a Graduate Student Award from the Canadian Arthritis Network and The Arthritis Society, and by the Joint Motion Program—A CIHR Training Program in Musculoskeletal Health Research and Leadership. DWH is the Sandy Kirkley Chair in Musculoskeletal Research at the Schulich School of Medicine & Dentistry at The University of Western Ontario.