^{1,a)}

### Abstract

Frequency-dependent phase velocity was measured in trabecular-bone-mimicking phantoms consisting of two-dimensional arrays of parallel nylon wires (simulating trabeculae) with thicknesses ranging from 152 to and spacings ranging from 700 to . Phase velocity varied approximately linearly with frequency over the range from 400 to 750 kHz. Dispersion was characterized by the slope of a linear least-squares regression fit to phase velocity versus frequency data. The increase in phase velocity (compared with that in water) at 500 kHz was approximately proportional to the (1) square of trabecular thickness, (2) inverse square of trabecular spacing, and (3) volume fraction occupied by nylon wires. The first derivative of phase velocity with respect to frequency was negative and exhibited nonlinear, monotonically decreasing dependencies on trabecular thickness and volume fraction. The dependencies of phase velocity and its first derivative on volume fraction in the phantoms were consistent with those reported in trabecular bone.

The author is grateful to Heather Miller, C.I.R.S., Norfolk, VA, for assistance in phantom design and construction. The mention of commercial products, their sources, or their use in connection with material reported herein is not to be construed as either an actual or implied endorsement of such products by the Food and Drug Administration.

I. INTRODUCTION

II. METHODS

A. Phantoms

B. Ultrasonic methods

III. RESULTS

IV. DISCUSSION

### Key Topics

- Speed of sound
- 18.0
- Velocity measurement
- 11.0
- Tissues
- 4.0
- Ultrasonic velocity
- 3.0
- Ultrasonics
- 3.0

## Figures

(a) Side view and (b) top view of experimental setup.

(a) Side view and (b) top view of experimental setup.

Measurements (*) of phase velocity vs frequency for phantom with and . A linear least-squares regression fit to the data is also shown (solid line).

Measurements (*) of phase velocity vs frequency for phantom with and . A linear least-squares regression fit to the data is also shown (solid line).

Phase velocity at 500 kHz vs Tb.Th for four phantoms with . Error bars denote standard deviations. A quadratic fit, , is also shown.

Phase velocity at 500 kHz vs Tb.Th for four phantoms with . Error bars denote standard deviations. A quadratic fit, , is also shown.

Phase velocity at 500 kHz vs for four phantoms with . Error bars denote standard deviations. A curve fit, , is also shown.

Phase velocity at 500 kHz vs for four phantoms with . Error bars denote standard deviations. A curve fit, , is also shown.

Phase velocity at 500 kHz vs volume fraction for all seven phantoms. A linear fit, , is also shown.

Phase velocity at 500 kHz vs volume fraction for all seven phantoms. A linear fit, , is also shown.

The first derivative of phase velocity with respect to frequency, , vs Tb.Th for four phantoms with . Error bars denote standard deviations. A power law fit, is also shown.

The first derivative of phase velocity with respect to frequency, , vs Tb.Th for four phantoms with . Error bars denote standard deviations. A power law fit, is also shown.

The first derivative of phase velocity with respect to frequency, , vs for four phantoms with . Error bars denote standard deviations.

The first derivative of phase velocity with respect to frequency, , vs for four phantoms with . Error bars denote standard deviations.

The first derivative of phase velocity with respect to frequency, , vs volume fraction for all seven phantoms. A power law fit, is also shown.

The first derivative of phase velocity with respect to frequency, , vs volume fraction for all seven phantoms. A power law fit, is also shown.

The first derivative of phase velocity with respect to frequency, , vs volume fraction in 30 human calcaneus samples.

The first derivative of phase velocity with respect to frequency, , vs volume fraction in 30 human calcaneus samples.

## Tables

Phantom properties. Tb.Th is trabecular (nylon wire) thickness. The variable is the interwire spacing, which is equal to the sum of Tb.Th and Tb.Sp (trabecular separation). The volume fraction (VF) is the fraction of volume occupied by nylon wire. .

Phantom properties. Tb.Th is trabecular (nylon wire) thickness. The variable is the interwire spacing, which is equal to the sum of Tb.Th and Tb.Sp (trabecular separation). The volume fraction (VF) is the fraction of volume occupied by nylon wire. .

Estimates of the first derivative of phase velocity with respect to frequency, , in human calcaneus from Nicholson *et al.* (1996, Table I), Strelitzki and Evans (1996, Table II), Droin *et al.* (1998, Table I), and Wear (2000a, Table I). is the number of calcaneus samples upon which measurements were based.

Estimates of the first derivative of phase velocity with respect to frequency, , in human calcaneus from Nicholson *et al.* (1996, Table I), Strelitzki and Evans (1996, Table II), Droin *et al.* (1998, Table I), and Wear (2000a, Table I). is the number of calcaneus samples upon which measurements were based.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content