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Bone cells are deformed according to mechanical stimulation they receive and their mechanical characteristics. However, how osteoblasts are affected by mechanical vibration frequency and acceleration amplitude remains unclear. By developing 3D osteoblast finite element (FE) models, this study investigated the effect of cell shapes on vibration characteristics and effect of acceleration (vibration intensity) on vibrational responses of cultured osteoblasts. Firstly, the developed FE models predicted natural frequencies of osteoblasts within 6.85–48.69 Hz. Then, three different levels of acceleration of base excitation were selected (0.5, 1, and 2 g) to simulate vibrational responses, and acceleration of base excitation was found to have no influence on natural frequencies of osteoblasts. However, vibration response values of displacement, stress, and strain increased with the increase of acceleration. Finally, stress and stress distributions of osteoblast models under 0.5 g acceleration in

It is widely accepted that bone is a dynamic tissue, because bone remodelling cells (including bone formation cells (osteoblasts) and bone degrading cells (osteoclasts)) can be activated under mechanical stimuli [

Human skeleton and bone cells are frequently subjected to vibration force experienced through activities or exercise, and the vibration is described often by frequency and by acceleration (acceleration < 1 g as low intensity and acceleration ≥ 1 g as high intensity) [

In the present work, we aimed to investigate the biomechanical responses (displacement, von Mises stress, and strain) of osteoblasts of various shapes to mechanical vibration of different levels of acceleration. The main objectives of this study were fivefold: (1) to develop the idealized continuum FE models of osteoblasts of six different shapes; (2) to obtain the natural frequencies and mode shapes of all osteoblast FE models; (3) to determine the harmonic responses (like displacement and von Mises stress of nucleus centre) to base excitation vibration of osteoblast FE models; (4) to investigate the effect on osteoblast responses of base excitation under three levels of acceleration, that is, 0.5 g, 1 g, and 2 g, respectively; and (5) to investigate the effect factor on resonance frequency.

The shapes of osteoblasts used for the FE modelling in the current study were from the results of experimental investigation [

Geometry property and element data for osteoblast finite element models.

Models | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Cell height ( | 15.45 | 12.75 | 10.96 | 9.41 | 8.03 | 6.82 |

Surface (^{2}) | 898.98 | 835.24 | 799.52 | 825.35 | 891.97 | 985.89 |

Bottom area (^{2}) | 125.09 | 243.84 | 421.28 | 549.88 | 696.53 | 842.90 |

Volume (^{3}) | 2988.40 | 2998.36 | 3000.60 | 3000.49 | 3010.90 | 3019.02 |

Nucleus volume (^{3}) | 104.72 | 104.72 | 104.72 | 104.72 | 104.72 | 104.72 |

Number of elements | ||||||

Nucleus | 20472 | 20092 | 20668 | 25248 | 23424 | 17388 |

Cytoplasm | 148020 | 127884 | 115060 | 130880 | 111808 | 88592 |

Membrane | 7480 | 7288 | 8056 | 9856 | 10048 | 8932 |

Geometry of idealized finite element models of osteoblasts of different shapes.

3D osteoblast finite element modelling.

The geometry of the osteoblasts is selected as part of a sphere. The nucleus is modelled as ellipsoid because the nucleus is normally modelled as sphere [^{3} and 104.5 ^{3}, respectively, which are based on the study of McGarry et al. [

Here, the 3D osteoblast FE models were developed based on the corresponding geometry data. The FE models of different shapes are shown in Figure

In this study, the materials were assumed isotropic/linear/elastic for the osteoblast FE models; and their properties and density for the osteoblast are given in Table ^{3} : 1500 kg/m^{3} : 1800 kg/m^{3}) of membrane, cytoplasm, and nucleus was assumed [

Material properties and density for the osteoblast finite element models.

Components | Young’s modulus | Poisson’s ratio | Density |
---|---|---|---|

Membrane | 1 kPa | 0.3 | 600 kg/m^{3} |

Cytoplasm | 1.5 kPa | 0.37 | 1500 kg/m^{3} |

Nucleus | 6 kPa | 0.37 | 1800 kg/m^{3} |

For endothelial cells, Young’s modulus of membrane, cytoplasm, and nucleus was set at 775 Pa, 775 Pa, and 5.1 kPa, respectively. Poisson’s ratio of membrane, cytoplasm, and nucleus was set at 0.33 [

Density is an important parameter in vibration simulation. The initial cellular density was assumed as 1000 kg/m^{3} [^{3} was used as the density of cytoplasm, nucleus, and membrane in the endothelial cell [^{3} [^{3}, 1800 kg/m^{3}, and 600 kg/m^{3} were set as the densities of cytoplasm, nucleus, and membrane, respectively [

Natural frequency extraction is an eigenvalue analysis procedure, which determines the natural frequencies and shapes of mode for a structure. In this study, software ABAQUS was used to conduct the natural frequency extraction. The governing dynamic equation of the response in ABAQUS can be expressed as follows [

The free vibration structure without damping may be represented as

The solution of

Then, (

Here, the natural frequencies of the six osteoblast models (Model I, Model II, Model III, Model IV, Model V, and Model VI) were obtained through the natural frequency extraction, and the corresponding mode shapes of the six osteoblast models were presented by the FE analysis.

In this study, the free vibration of the system was considered with only one degree of freedom. In the simulation, the different levels of acceleration of base excitation were applied to analyse the effect of acceleration on the cell, that is, 0.5 g, 1 g, and 2 g (g = 9.8 m/s^{2}), respectively. It is well known that acceleration (g forces, g = 9.8 m/s^{2}) is the best term to describe vibration intensity [

A previous human vibration test was conducted under the acceleration of 0.04–19.3 g, and it was found that the resonant frequency values of ankle, knee, hip, and spine are 10–40 Hz, 10–25 Hz, 10–20 Hz, and 10 Hz, respectively [

The natural frequencies and vibration mode shapes of the six osteoblast models were obtained after completing the FE modal analyses. Figure

Resonance frequencies of the first ten modes for the 6 finite element models of osteoblasts.

Mode shapes of the first ten vibration modes for Model VI (from top view). (a) 1st mode at 28.57 Hz; (b) 2nd mode at 28.61 Hz; (c) 3rd mode at 33.79 Hz; (d) 4th mode at 41.13 Hz; (e) 5th mode at 41.22 Hz; (f) 6th mode at 42.60 Hz; (g) 7th mode at 42.71 Hz; (h) 8th mode at 43.53 Hz; (i) 9th mode at 48.68 Hz; and (j) 10th mode at 48.69 Hz.

Based on the natural frequencies and the assumed uniform acceleration of base excitation within the frequency range between 1 and 50 Hz, the harmonic responses of the six FE models were computed. To investigate the responses of the different directions (

The displacement of the centre of the nucleus versus vibration frequency of osteoblasts. (a) 0.5 g acceleration in

The mode shapes of the FE models at the peak frequency under 0.5 g acceleration in

Mode shapes at the peak frequency under 0.5 g acceleration (from top view). (a) In

Displacement distribution at the peak frequency under different levels of base excitation acceleration. (a) In

In addition, the von Mises stress values of the centre of the nucleus under the different levels of base excitation in

The von Mises stress value of the centre of nucleus versus frequency under different levels of acceleration. (a) 0.5 g acceleration in

Moreover, the von Mises stress contours of the FE models under 0.5 g base excitation in

von Mises stress contours of the different cell models in

Similarly, the strain values of the centre of the nucleus under the different base excitation in

The strain value of the centre of nucleus versus frequency of the six different cell models at different levels of acceleration. (a) 0.5 g acceleration in

Furthermore, the strain contours of the 6 FE models under 0.5 g base excitation in

Strain contours of the cell in

Resonance frequency responses to the osteoblast models in different directions. (a) Resonance frequency versus cell height; (b) resonance frequency versus bottom area of the cell; and (c) resonance frequency versus different models.

The relationships of resonance frequency with cell height and with bottom area were analysed when the volume and density of the cell were assumed as constants. The results are shown in Figures

Values of

Independent variable | Coefficient of | Direction | ||
---|---|---|---|---|

| | | ||

| | 852.63 | 856.51 | 607.60 |

| −874.73 | −878.29 | −628.97 | |

| 275.97 | 277.04 | 190.94 | |

| −234.36 | −235.70 | −136.72 |

Values of

Independent variable | Coefficient of | Direction | ||
---|---|---|---|---|

| | | ||

^{2}) | | −50.63 | −55.41 | 61.23 |

| 10.61 | 12.57 | −34.011 | |

| 1.23 | 0.96 | 7.36 | |

| 55.01 | 58.90 | −29.06 |

There have been many previous studies using various models that were developed to reveal bone cell mechanical behaviours. Some numerical computational models of cultured cells have been developed to examine the responses of cultured cells to various mechanical stimulations. For example, to study the universal dynamic behaviours of osteoblasts, a three-dimensional (3D) soft matter cell model was developed using the multiscale moving contact line theory [

In this present study, to investigate the vibrational responses of the different shapes of an osteoblast subjected to vibration of base excitation, six idealized FE models of an osteoblast were created. Firstly, cell geometry was created and FE models were developed accordingly. For these models, the initial volumes of the cell were basically the same and the densities were constant in the simulation. Secondly, natural frequency (resonance frequency) of the different models was extracted by using FE analysis. Then, harmonic vibration of the osteoblast models was analysed with three different base excitation acceleration values, namely, 0.5 g, 1 g, and 2 g. The response results were obtained for the harmonic vibration including displacement, von Mises stress, and strain of the centre of the nucleus under different acceleration values. Finally, the effects of cell height and bottom area on resonance frequency were analysed, and the fitted curves of resonance frequency versus cell height and resonance frequency versus bottom area were obtained.

Based on the previous studies, the vibration frequency is crucial for bone cells to complete the bone resorption and bone formation [

In this study, the osteoblast models were assumed as one-degree-of-freedom vibrational system. The resonance phenomenon can occur at some natural frequency and can be observed from vibrational responses like displacement, von Mises stress, and strain. The cell biomechanical responses (e.g., displacement, von Mises stress, and strain of the centre of the nucleus models of osteoblasts) were obtained when the models were subjected to three different acceleration values of base excitation vibration (0.5 g, 1 g, and 2 g). The current study observed that the resonance frequency does not change with the acceleration, which suggests that the natural frequency of the bone cell is determined by the intrinsic factors and is not affected by the external factors. The current study has also examined the vibration responses of different FE models of the bone cell under 0.5 g acceleration in

The current study has investigated the displacement, von Mises stress, and strain responses of bone cell models under different acceleration values and in different directions. The results showed that the values of displacement, von Mises stress, and strain increased with the acceleration. The values in

Our FE models also indicate that von Mises stress is concentrated in the nucleus and strain is basically concentrated around the nucleus for the harmonic response under 0.5 g acceleration in

Based on the simulation results analysis, there is a relationship between resonance frequency and cell height or between resonance frequency and bottom area. The resonance frequency can be expressed as the function of cell height (

It must be noted that, in this study, the geometry of the bone cell was assumed to be comprised of three components, that is, membrane, cytoplasm, and nucleus. In the previous studies, the tensegrity structure was used to simulate the fibres of cells like microtubules and microfilaments [

In the current study, different osteoblast FE models were developed and were used to extract the natural frequencies and to analyse the harmonic responses under different acceleration values (0.5 g, 1 g, and 2 g). It was found that the natural frequencies do not change with the variation of acceleration of base excitation. The response values of displacement, von Mises stress, and strain increase with the increase of acceleration, and the response values in

The authors declare that they have no competing interests.

Liping Wang and Cory J. Xian were responsible for study design. Cory J. Xian conducted the study. Data collection was the responsibility of Liping Wang. Data analysis was carried out by Liping Wang and Hung-Yao Hsu. Data interpretation was done by Liping Wang and Hung-Yao Hsu. Liping Wang and Cory J. Xian drafted the manuscript. And Liping Wang, Xu Li, and Cory J. Xian revised the manuscript content. All authors have read and approved the final submitted manuscript.

Liping Wang is supported by the Australian National Health and Medical Research Council (NHMRC) Postgraduate Research Scholarship grant, and Cory J. Xian is supported by the NHMRC Senior Research Fellowship. This work was supported by the National Natural Science Foundation of China (Project Grant no. 81671928).