Theoretically proposed optimal frequency for ultrasound induced cartilage restoration

Abstract

Background: Matching the frequency of the driving force to that of the system's natural frequency of vibration results in greater amplitude response. Thus we hypothesize that applying ultrasound at the chondrocyte's resonant frequency will result in greater deformation than applying similar ultrasound power at a frequency outside of the resonant bandwidth. Based on this resonant hypothesis, our group previously confirmed theoretically and experimentally that ultrasound stimulation of suspended chondrocytes at resonance (5 MHz) maximized gene expression of load inducible genes. However, this study was based on suspended chondrocytes. The resonant frequency of a chondrocyte does not only depend on the cell mass and intracellular stiffness, but also on the mechanical properties of the surrounding medium. An in vivo chondrocyte's environment differs whether it be a blood clot (following microfracture), a hydrogel or the pericellular and extracellular matrices of the natural cartilage. All have distinct structures and compositions leading to different resonant frequencies. In this study, we present two theoretical models, the first model to understand the effects of the resonant frequency on the cellular deformation and the second to identify the optimal frequency range for clinical applications of ultrasound to enhance cartilage restoration.

Results: We showed that applying low-intensity ultrasound at the resonant frequency induced deformation equivalent to that experimentally calculated in previous studies at higher intensities and a 1 MHz frequency. Additionally, the resonant frequency of an in vivo chondrocyte in healthy conditions, osteoarthritic conditions, embedded in a blood clot and embedded in fibrin ranges from 3.5 - 4.8 MHz.

Conclusion: The main finding of this study is the theoretically proposed optimal frequency for clinical applications of therapeutic ultrasound induced cartilage restoration is 3.5 - 4.8 MHz (the resonant frequencies of in vivo chondrocytes). Application of ultrasound in this frequency range will maximize desired bioeffects.

Keywords: Cellular deformation; Mechanical energy density; Resonant frequency.

Fig. 1

Model Geometry: a) A suspended cell (indicated by the small sphere) immersed in growth media. The ultrasound source (14 kPa) is positioned at the bottom and is indicated by the blue dotted line. The cell position is frequency dependent to ensure the position remains at an antinode. b A chondron (indicated by the sphere) embedded in the extracellular matrix (indicated by the cylinder) and immersed in growth media. The ultrasound source is positioned at the bottom (blue dotted line). The cell position is frequency dependent

Fig. 2

Ultrasound induced cellular deformation, the color represents displacement in nanometers. The frequency and pressure amplitude was varied a) 1 MHz; 14 kPa, b) 5 MHz; 14 kPa, c) 6.5 MHz; 14 kPa, d) 1 MHz; 170 kPa, e) 5 MHz; 170 kPa, f) 6.5 MHz; 170 kPa. (The knobby appearances in A and D are exaggerated to visually see the displacement. The displacement magnitude is depicted by color and is not depicted to scale in the figures)

Fig. 3

Resonant frequency of a chondrocytes in a blood clot and fibrin. a A suspended cell, a chondrocyte surrounded by a PCM with a thickness of 2.5 μm and a Young’s modulus of 1 kPa and a chondrocyte surrounded by a PCM with a thickness of 2.5 μm and a Young’s modulus of 500 kPa embedded in a blood clot. b A suspended cell, a chondrocyte surrounded by a PCM with a thickness of 2.5 μm and a Young’s modulus of 1 kPa and a chondrocyte surrounded by a PCM with a thickness of 2.5 μm and a Young’s modulus of 500 kPa embedded in fibrin

Fig. 4

Resonant frequency of chondrons with varying parameters. a PCM thickness of 2.5 μm with a Young’s modulus of 40 kPa, 300 kPa and osteoarthritic conditions. b PCM thickness of 6 μm with a Young’s modulus of 40 kPa, 300 kPa and osteoarthritic conditions

Fig. 5

Resonant frequency of chondrons embedded in an ECM with Young’s modulus of 500 kPa with PCM varying parameters. a PCM thickness of 2.5 μm with a Young’s modulus of 40 kPa, 300 kPa and osteoarthritic conditions. b PCM thickness of 6 μm with a Young’s modulus of 40 kPa, 300 kPa and osteoarthritic conditions

Fig. 6

Resonant frequency of chondrons embedded in an ECM with Young’s modulus of 2 MPa with PCM varying parameters. a PCM thickness of 2.5 μm with a Young’s modulus of 40 kPa, 300 kPa and osteoarthritic conditions. b PCM thickness of 6 μm with a Young’s modulus of 40 kPa, 300 kPa and osteoarthritic conditions

Fig. 7

Parameter Analysis. a Varying the porosity in an osteoarthritic environment with a PCM thickness of 3.75 μm and a PCM Young’s modulus of 25 kPa and ECM Young’s modulus of 300 kPa. b Varying the radius in an osteoarthritic environment with a PCM thickness of 2.5 μm and a PCM Young’s modulus of 40 kPa and ECM Young’s modulus of 500 kPa