^{1}, Amber M. Nelson

^{1}, Mark R. Holland

^{1}and James G. Miller

^{1,a)}

### Abstract

A Bayesianprobability theory approach for separating overlapping ultrasonic fast and slow waves in cancellous bone has been previously introduced. The goals of this study were to investigate whether the fast and slow waves obtained from Bayesian separation of an apparently single mode signal individually correlate with porosity and to isolate the fast and slow waves from medial-lateral insonification of the calcaneus. The Bayesian technique was applied to trabecular bone data from eight human calcanei insonified in the medial-lateral direction. The phase velocity, slope of attenuation (nBUA), and amplitude were determined for both the fast and slow waves. The porosity was assessed by micro-computed tomography (microCT) and ranged from 78.7% to 94.1%. The method successfully separated the fast and slow waves from medial-lateral insonification of the calcaneus. The phase velocity for both the fast and slow wave modes showed an inverse correlation with porosity ( = 0.73 and = 0.86, respectively). The slope of attenuation for both wave modes also had a negative correlation with porosity (fast wave: = 0.73, slow wave: = 0.53). The fast wave amplitude decreased with increasing porosity ( = 0.66). Conversely, the slow wave amplitude modestly increased with increasing porosity ( = 0.39).

The authors wish to thank Christian C. Anderson and G. Larry Bretthorst for their invaluable assistance with Bayesian analysis, Tarpit Patel for his help making measurements, and Dan Loesche for his help acquiring tissue samples. This work was supported, in part, by NIH/NIAMS Grant Nos. R01AR057433 and P30AR057235.

I. INTRODUCTION

II. METHODS

A. Bone samples

B. X-ray microCT measurements

C. Ultrasonic data acquisition

D. Bayesiananalysis

1. The model

2. Parameter estimation

3. Markov chain Monte Carlo

III. RESULTS

A. Phase velocity

B. Slope of attenuation

C. Fast and slow wave amplitude

IV. DISCUSSION

A. Fast and slow waves identified by the Bayesian method

B. Interpretation of fast and slow waves

C. Fast and slow wave amplitudes

D. Comparison with Biot theory

E. Limitations

V. CONCLUSION

### Key Topics

- Probability theory
- 22.0
- Wave attenuation
- 17.0
- Acoustic waves
- 16.0
- Ultrasonics
- 9.0
- Transducers
- 6.0

##### A61B6/03

## Figures

Ultrasonic rf signals having propagated through a water path (gray circles) and the same path with a cancellous bone sample inserted (black circles).

Ultrasonic rf signals having propagated through a water path (gray circles) and the same path with a cancellous bone sample inserted (black circles).

Experimental data from human calcaneus, Bayesian model, and residual. The top panel shows the experimental data that are input into the algorithm. The Bayesian model is generated as described in Sec. II D. The bottom plot shows the residual, defined as the difference between the experimental data and the Bayesian model, on a substantially expanded vertical scale.

Experimental data from human calcaneus, Bayesian model, and residual. The top panel shows the experimental data that are input into the algorithm. The Bayesian model is generated as described in Sec. II D. The bottom plot shows the residual, defined as the difference between the experimental data and the Bayesian model, on a substantially expanded vertical scale.

A representative model waveform (same waveform as in Fig. 2) and the fast and slow waves that comprise it.

A representative model waveform (same waveform as in Fig. 2) and the fast and slow waves that comprise it.

Fast wave speed (left panel) and slow wave speed (right panel) plotted against sample porosity. Porosity was determined by . Each value shown is the mean of six runs. The vertical bar on each point represents the standard deviation of these six runs.

Fast wave speed (left panel) and slow wave speed (right panel) plotted against sample porosity. Porosity was determined by . Each value shown is the mean of six runs. The vertical bar on each point represents the standard deviation of these six runs.

Slope of attenuation for the fast wave (left panel) and slow wave (right panel) plotted against sample porosity. Porosity was determined by . Each value shown is the mean of six runs. The vertical bar on each point represents the standard deviation of these six runs. The solid lines are linear best fit lines. The *R*-squared value for the fit is shown in each panel.

Slope of attenuation for the fast wave (left panel) and slow wave (right panel) plotted against sample porosity. Porosity was determined by . Each value shown is the mean of six runs. The vertical bar on each point represents the standard deviation of these six runs. The solid lines are linear best fit lines. The *R*-squared value for the fit is shown in each panel.

Fast wave amplitude (left panel) and slow wave amplitude (right panel) plotted against sample porosity. Porosity was determined by . Each value shown is the mean of six runs. The vertical bar on each point represents the standard deviation of these six runs. The solid lines are linear best fit lines. The *R*-squared value for the fit is shown in each panel.

Fast wave amplitude (left panel) and slow wave amplitude (right panel) plotted against sample porosity. Porosity was determined by . Each value shown is the mean of six runs. The vertical bar on each point represents the standard deviation of these six runs. The solid lines are linear best fit lines. The *R*-squared value for the fit is shown in each panel.

The ratio of the fast wave amplitude to the slow wave amplitude plotted against porosity. Porosity was determined by . Each value shown is the mean of six runs. The vertical bar on each point represents the standard deviation of these six runs. The solid line is a linear best fit line. The *R*-squared value for the fit is shown.

The ratio of the fast wave amplitude to the slow wave amplitude plotted against porosity. Porosity was determined by . Each value shown is the mean of six runs. The vertical bar on each point represents the standard deviation of these six runs. The solid line is a linear best fit line. The *R*-squared value for the fit is shown.

## Tables

Physical characteristics of defatted calcaneus samples. Direct thickness measurements are reported as mean plus or minus the standard deviation of five independent manual measurements.

Physical characteristics of defatted calcaneus samples. Direct thickness measurements are reported as mean plus or minus the standard deviation of five independent manual measurements.

Derived parameters of defatted calcaneus bone samples. Apparent density was determined by dividing the mass of the dehydrated samples (mass column of Table I) by the product of the mean linear dimensions (direct thickness measurement columns in Table I). The micro-computed tomography system provides a bone volume per total volume measurement. Porosity is .

Derived parameters of defatted calcaneus bone samples. Apparent density was determined by dividing the mass of the dehydrated samples (mass column of Table I) by the product of the mean linear dimensions (direct thickness measurement columns in Table I). The micro-computed tomography system provides a bone volume per total volume measurement. Porosity is .

Prior probability distributions used for Bayesian calculations. Each prior distribution is a bounded Gaussian described by the four parameters in this table. For each parameter, the upper and lower bounds were chosen to be wide enough so as to affect the final result only minimally. The mean was chosen to be the midpoint between the upper and lower bounds, and the standard deviation was chosen to be one half of the difference between the upper and lower bound.

Prior probability distributions used for Bayesian calculations. Each prior distribution is a bounded Gaussian described by the four parameters in this table. For each parameter, the upper and lower bounds were chosen to be wide enough so as to affect the final result only minimally. The mean was chosen to be the midpoint between the upper and lower bounds, and the standard deviation was chosen to be one half of the difference between the upper and lower bound.

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