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# Constrained Tibial Vibration in Mice: A Method for Studying the Effects of Vibrational Loading of Bone

[+] Author and Article Information
Blaine A. Christiansen1

Department of Orthopaedic Surgery, and Department of Biomedical Engineering,  Washington University in St. Louis, Campus Box 8233, St. Louis, MO 63110bchrist1@bidmc.harvard.edu

Philip V. Bayly

Department of Mechanical Engineering,  Washington University in St. Louis, Campus Box 8233, St. Louis, MO 63110

Matthew J. Silva

Department of Orthopaedic Surgery, and Department of Biomedical Engineering,  Washington University in St. Louis, Campus Box 8233, St. Louis, MO 63110

1

Corresponding author.

J Biomech Eng 130(4), 044502 (May 16, 2008) (6 pages) doi:10.1115/1.2917435 History: Received May 25, 2007; Revised March 10, 2008; Published May 16, 2008

## Abstract

Vibrational loading can stimulate the formation of new trabecular bone or maintain bone mass. Studies investigating vibrational loading have often used whole-body vibration (WBV) as their loading method. However, WBV has limitations in small animal studies because transmissibility of vibration is dependent on posture. In this study, we propose constrained tibial vibration (CTV) as an experimental method for vibrational loading of mice under controlled conditions. In CTV, the lower leg of an anesthetized mouse is subjected to vertical vibrational loading while supporting a mass. The setup approximates a one degree-of-freedom vibrational system. Accelerometers were used to measure transmissibility of vibration through the lower leg in CTV at frequencies from $20Hzto150Hz$. First, the frequency response of transmissibility was quantified in vivo, and dissections were performed to remove one component of the mouse leg (the knee joint, foot, or soft tissue) to investigate the contribution of each component to the frequency response of the intact leg. Next, a finite element (FE) model of a mouse tibia-fibula was used to estimate the deformation of the bone during CTV. Finally, strain gages were used to determine the dependence of bone strain on loading frequency. The in vivo mouse leg in the CTV system had a resonant frequency of $60Hz$ for $±0.5G$ vibration ($1.0G$ peak to peak). Removing the foot caused the natural frequency of the system to shift from $60Hzto70Hz$, removing the soft tissue caused no change in natural frequency, and removing the knee changed the natural frequency from $60Hzto90Hz$. By using the FE model, maximum tensile and compressive strains during CTV were estimated to be on the cranial-medial and caudolateral surfaces of the tibia, respectively, and the peak transmissibility and peak cortical strain occurred at the same frequency. Strain gage data confirmed the relationship between peak transmissibility and peak bone strain indicated by the FE model, and showed that the maximum cyclic tibial strain during CTV of the intact leg was $330±82με$ and occurred at $60–70Hz$. This study presents a comprehensive mechanical analysis of CTV, a loading method for studying vibrational loading under controlled conditions. This model will be used in future in vivo studies and will potentially become an important tool for understanding the response of bone to vibrational loading.

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## Figures

Figure 1

(a) Picture of a mouse in the CTV loading device. The mouse is anesthetized with isoflurane and placed on a stationary platform. The right lower leg of the mouse is constrained by the two posts of the CTV device, with the knee contacting the bottom post (vibration source) and the foot supporting the weight of the top platform (moving mass), which is free to vertically move. (b) MicroCT image of the mouse lower leg in the CTV device. Both the distal femur and the foot are in series with the tibia-fibula and contribute to the vibrational behavior of the mouse leg in the CTV system. (c) Diagram (illustrating the approximate position of the accelerometers used to determine transmissibility) relating the CTV device to a one degree-of-freedom lumped-parameter vibrational system. By using this model, the mass of the system (M) represents the combined mass of the upper platform of the CTV device (the mass of the leg is neglected), while k and c represent the apparent stiffness and damping of the leg, respectively. Transmissibility is calculated as the magnitude of acceleration at the top platform X(t) (output) divided by the magnitude of acceleration at the bottom platform Y(t) (input). Acceleration magnitude was determined by averaging the absolute magnitudes of five maximum acceleration values and five minimum acceleration values (acceleration was approximately centered at zero).

Figure 2

(a) Schematics showing the various dissections used to determine the effect of each component of the mouse leg on the vibrational behavior of the system. (b) Frequency responses of the imaginary component of the transmissibility with standard deviation error bars for a loading magnitude of 0.5G. The “bone only” configuration had very large standard deviations for loading frequencies ≥100Hz; however, all bones tested had peak values for the imaginary component of transmissibility between 110Hzto130Hz. (c)–(e) Graphs showing average transmissibility (c), phase shift (d), and imaginary component of transmissibility (e) of a mouse leg loaded in CTV after the removal of one component of the lower leg (n=3 mice per dissection). In vivo and bone only data are also included for comparison.

Figure 3

FE model of a mouse tibia-fibula in CTV. (a) Assembly of the FE model, showing the end mass, which simulates the top platform of the CTV device, as well as the proximal and distal elastic elements that simulate the elastic contributions of the knee joint and the foot, respectively. (b) Profile of peak longitudinal strain produced during CTV loading. The picture on the left shows the cranial-medial surface of the tibia (the location of peak tensile strain). The picture on the right is a cross section of the tibia showing the relative distribution of peak tensile and compressive strain. (c) Graph showing the frequency dependence of transmissibility (of displacement) and longitudinal strain as estimated by the FE model. Near the natural frequency of the model, both the transmissibility and the longitudinal strain approach infinity since the model does not have a damping component.

Figure 4

Average frequency response of cyclic cortical strain on the tibia of an intact mouse leg subjected to CTV loading, compared to the frequency response of transmissibility of acceleration (0.5G maximum acceleration input magnitude). Strain gages were placed at the location of peak tensile strain (determined by the FE model). The average peak strain value was 330±82με and occurred at 60–70Hz. Error bars represent the standard deviation of dynamic strain magnitude recorded at each loading frequency of interest. Strain and transmissibility values were strongly correlated (R2=0.83). Strain data and transmissibility data were recorded from different animals.

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