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. Author manuscript; available in PMC: 2022 May 23.
Published in final edited form as: J Acoust Soc Am. 2009 Feb;125(2):1197–1201. doi: 10.1121/1.3050310

The dependencies of phase velocity and dispersion on volume fraction in cancellous-bone-mimicking phantoms

Keith A Wear 1
PMCID: PMC9125424  NIHMSID: NIHMS1808082  PMID: 19206892

Abstract

Frequency-dependent phase velocity was measured in 8 cancellous-bone-mimicking phantoms consisting of suspensions of randomly-oriented nylon filaments (simulating trabeculae) in a soft-tissue-mimicking medium (simulating marrow). Trabecular thicknesses ranged from 152 – 356 microns. Volume fractions of nylon filament material ranged from 0 to 10%. Phase velocity varied approximately linearly with frequency over the range from 300 to 700 kHz. The increase in phase velocity (compared with phase velocity in a phantom containing no filaments) at 500 kHz was approximately proportional to volume fraction occupied by nylon filaments. The derivative of phase velocity with respect to frequency was negative and exhibited nonlinear, monotonically-decreasing dependence on volume fraction. The dependencies of phase velocity and its derivative on volume fraction in these phantoms were similar to those reported in previous studies on 1) human cancellous bone and 2) phantoms consisting of parallel nylon wires immersed in water.

Keywords: bone, trabecular, cancellous, velocity, dispersion

PACS code: 4380-Qf

I. INTRODUCTION

Bone sonometry is now an accepted method for diagnosis of osteoporosis (Langton and Njeh, 2004; Laugier, 2008). Speed of sound (SOS) in cancellous bone is highly correlated with bone mineral density (Rossman et al., 1989, Tavakoli and Evans, 1991, Zagzebski et al., 1991, Njeh et al., 1996, Laugier et al., 1997, Nicholson et al., 1998, Hans et al., 1999, Trebacz and Natali, 1999), which is an indicator of systemic osteoporotic fracture risk (Cummings et al., 1993). Calcaneal SOS (in combination with broadband ultrasound attenuation or BUA) has been shown to be predictive of hip fractures in women in prospective studies (Hans et al., 1996, Miller et al., 2002, Hans et al., 2004, Huopio et al., 2004, Schott et al., 2005, Krieg et al., 2006).

Despite the clinical utility of SOS, the mechanisms responsible for variations of SOS in cancellous bone are not well understood yet. Unlike soft tissues, which typically exhibit positive dispersion (phase velocity increasing with ultrasonic frequency) (O’Donnell et al., 1981), cancellous bone exhibits negative dispersion (Nicholson et al., 1996; Strelitzki and Evans, 1996; Droin et al., 1998; Wear, 2000a; Wear, 2000b). This negative dispersion may be explained using a stratified model (Brekhovskikh, 1980, Hughes et al., 1999, Wear, 2001, Qin, 2001), modified Biot-Attenborough theory (Lee et al., 2003), a restricted-bandwidth form of the Kramers-Kronig dispersion relations (Waters and Hoffmeister, 2005) or from the interference of two or more positively-dispersive pulses (Marutyan et al., 2006; Marutyan et al., 2007; Bauer et al., 2008; Anderson et al., 2008).

Measurements on cancellous-bone-mimicking phantoms can provide insight into the determinants of phase velocity and dispersion. The present study complements a previous study, which showed that for phantoms consisting of parallel nylon wires immersed in water, phase velocity and dispersion are primarily determined by volume fraction occupied by nylon wires (Wear, 2005a). In the present study, a new phantom design is utilized. Water is replaced by soft-tissue-mimicking material, which is a more realistic surrogate for marrow than water. Also in the present study, parallel nylon wires are replaced by randomly-oriented nylon filaments. In cancellous bone, the orientation of trabeculae is somewhere in between these two extremes.

II. METHODS

A. Phantoms

Eight phantoms containing nylon filaments (simulating trabeculae) in proprietary soft tissue-mimicking material (simulating marrow) (CIRS Inc., Norfolk, VA) were interrogated. Figure 1 shows a picture of a phantom. A reference phantom containing only soft tissue-mimicking material was also interrogated. Table 1 shows the phantom properties. Three of the phantoms contained nylon filaments with diameter equal to 152 μm, which is reasonably close to the mean trabecular thickness in human calcaneus, 127 μm (Ulrich, 1999). The dimensions for all phantoms were 80 mm X 60 mm X 25 mm. The scanning window dimensions for all phantoms were 60 X 50 mm.

Figure 1.

Figure 1.

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A phantom containing nylon filaments and a reference phantom.

Table 1.

Properties of phantoms.

Filament
Diameter
(μm)		 Filament
Length
(mm)		 Filament Number
Density
(# per cc)		 Volume Fraction (%)
- - 0 0
152 10 100 1.8
203 10 100 3.2
229 10 100 4.1
330 10 100 8.5
356 10 100 9.9
152 12 100 2.2
229 12 100 3.3
152 12 200 4.4
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Nylon is a useful surrogate for cancellous bone material. The longitudinal sound speed in nylon (2600 m/s) is near the low end of the range reported for mineralized bone material (2800 – 4000 m/s, near 500 kHz) (Duck, 1990). The dependencies of phase velocity and dispersion on volume fraction in phantoms consisting of parallel nylon wires in water are similar to those in human cancellous bone (Wear, 2005a). Nylon wires exhibit frequency-dependent scattering similar to that exhibited by cancellous bone (Wear, 1999; Wear, 2004).

A previously reported phantom design, consisting of cubic granules of gelatin immersed in epoxy, has been shown to be useful for the prediction of the dependences of phase velocity, dispersion, and attenuation on porosity of cancellous bone (Clarke et al., 1994; Strelitzki et al., 1997).

B. Ultrasonic Methods

A Panametrics (Waltham, MA) 5800 pulser/receiver was used. Samples were interrogated in through-transmission in a water tank using a pair of coaxially-aligned, Panametrics 500 kHz, broadband, 0.75” (1.9 cm) diameter, 1.5” (3.8 cm) focal length transducers. The propagation path between transducers was twice the focal length. Received radio frequency (RF) signals were digitized (8 bit, 10 MHz) using a LeCroy (Chestnut Ridge, NY) 9310C Dual 400 MHz oscilloscope and stored on computer (via GPIB) for off-line analysis. The transducers were maintained in constant positions. Each phantom was suspended in the water tank by a 2-dimensional stage that enabled movement of the phantom in the two dimensions perpendicular to the beam propagation direction. Attenuation measurements were made at 15 positions throughout each phantom scanning window corresponding to a 3 X 5 grid in which neighboring measurements were separated by 1 cm.

Frequency-dependent phase velocity, cp(f), was computed using

cp(f)=cw1+cwΔϕ(f)2πfd (1)

where f is frequency, Δϕ(f) is the difference in unwrapped phases (see next paragraph) of the received signals with and without the phantom in the water path, d is the phantom thickness (2.5 cm), and cw is the temperature-dependent speed of sound in distilled water given by (Kaye and Laby, 1973)

cw=1402.9+4.835T−0.047016T2+0.00012725T3m∕s (2)

and T is the temperature in degrees Celsius. Temperature, measured with a digital thermometer, was about 20° for these measurements, which meant that cw was about 1482 m/s.

The unwrapped phase difference, Δϕ(f), was computed as follows. Fast Fourier Transforms (FFT’s) of the digitized received signals were taken. The phase of the signal at each frequency was taken to be the inverse tangent of the ratio of the imaginary to real parts of the FFT at that frequency. Since the inverse tangent function yields principal values between -π and π, the phase had to be unwrapped by adding an integer multiple of 2π to all frequencies above each frequency where a discontinuity appeared.

Dispersion was characterized by the slope, dcp/df, of a linear least-squares regression fit of cp(f) vs. f over the range from 300 to 700 kHz, which roughly corresponded to the system −6 dB bandwidth.

III. RESULTS

Figure 2 shows measurements of phase velocity (cp) vs. frequency (f) for one phantom. Phase velocity declined approximately linearly with frequency for all phantoms.

Figure 2.

Figure 2.

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Measurements of phase velocity vs. frequency for one phantom. A linear fit is also shown. Error bars denote standard errors.

Figure 3 shows measurements of cp(500 kHz) vs. volume fraction (VF) on all the phantoms. A linear fit, cp(500 kHz) = 1530 + 3.3 VF m/s (where VF is expressed as a percentage), is in good agreement with the data.

Figure 3.

Figure 3.

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Average phase velocity at 500 kHz vs. volume fraction occupied by nylon filaments. Error bars denote standard deviations.

Figure 4 shows measurements of dcp/df vs. VF for all phantoms. A power law fit, dcp/df = 5.6 – 0.18 VF 2.1 is also shown. Values for dcp/df ranged from +5 to −22 m/sMHz, which is consistent with values reported in human calcaneus in vitro. A greater range for dcp/df has been measured in vivo. See Table 2.

Figure 4.

Figure 4.

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dcp/df vs. volume fraction occupied by nylon filaments. Error bars denote standard deviations.

Table 2.

Estimates of the first derivative of phase velocity with respect to frequency, dcp/df, in human calcaneus from Nicholson et al. (1996, Table 1), Strelitzki and Evans (1996, Table 2), Droin, et al. (1998, Table 1), Wear (2000a, Table 1), and Wear (2007, Table 2). N is the number of calcaneus samples upon which measurements were based.

Author(s) N Frequency
Range
(kHz)		 Age Range
(years)		 dcp/df (mean ± standard
deviation) (m/sMHz)
Nicholson et al. in vitro 70 200 - 800 22 - 76 −40
Strelitzki and Evans in vitro 10 600 - 800 unknown −32 ± 27
Droin et al. in vitro 15 200 - 600 69 - 89 −15 ± 13
Wear (2000) in vitro 24 200 - 600 unknown −18 ± 15
Wear (2007) in vivo 73 300 - 600 21 - 78 −59 ± 52
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IV. DISCUSSION

Phase velocity (cp) in phantoms consisting of randomly-distributed nylon filaments (simulating trabeculae) immersed in soft-tissue-mimicking material (simulating marrow), is an approximately linear, monotonically-increasing function of volume fraction. The derivative of phase velocity with respect to frequency, dcp/df, in these phantoms is a nonlinear, monotonically-decreasing function of volume fraction. Both phase velocity and dcp/df appear to be primarily determined by volume fraction. Since group velocity is determined by cp and dcp/df (Duck, 1990; Wear, 2005a), then group velocity must also be primarily determined by volume fraction.

A previous study showed similar trends for phase velocity and dcp/df in phantoms consisting of parallel nylon wires immersed in water (Wear, 2005a). In Figure 5, the change in phase velocity (compared with the zero volume fraction level) is plotted vs. volume fraction for both the previous and current phantom designs. In Figure 6, the change in dcp/df (compared with the zero volume fraction level) is plotted vs. volume fraction for both phantom designs. Despite significant differences in micro-architecture and fluid media for the two types of phantoms, the effect of volume fraction on phase velocity and dcp/df is remarkably similar. Moreover, it was argued previously that the volume-fraction dependencies of phase velocity and dcp/df in phantoms consisting of parallel nylon wires immersed in water were similar to those reported in cancellous bone in vitro (Wear, 2005a; Wear et al., 2005b). These empirical results, taken collectively, suggest that changes in phase velocity and dcp/df observed in human cancellous bone are primarily determined by volume fraction.

Figure 5.

Figure 5.

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Change in phase velocity (compared with the zero volume fraction level) vs. volume fraction for the present study, which utilized randomly distributed nylon filaments in soft-tissue-mimicking material, and a previous study (Wear, 2005), which utilized phantoms consisting of parallel nylon wires immersed in water. Error bars denote standard deviations.

Figure 6.

Figure 6.

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Change in dcp/df (compared with the zero volume fraction level) vs. volume fraction for the present study, which utilized randomly distributed nylon filaments in soft-tissue-mimicking material, and a previous study (Wear, 2005) which utilized phantoms consisting of parallel nylon wires immersed in water. Error bars denote standard deviations.

Nicholson et al. (2001) measured phase velocity in 69 human calcaneal cancellous bone cubes. They found that, after the data were adjusted for density (which is a measure of bone quantity rather than micro-architecture), there was no significant dependence of phase velocity on micro-architectural parameters. The dominant role of bone quantity (as opposed to micro-architecture) was also seen in a three-dimensional simulation study by Haïat et al. (2007), in which variations in volume fraction of micro CT reconstructions of human cancellous femur explained 94% of variations in SOS.

In measurements on human cancellous lumbar spine, Hans et al. (1999) found SOS to be approximately 2-3% higher in the axial direction compared with the sagittal and coronal directions. Since bone volume fraction is identical in all three orientations, these results suggest that the arrangement of trabeculae, not just the quantity of trabecular material (i.e. volume fraction), does play a role in determining SOS. In these experiments, however, the comparison was between extreme differences in angle between the ultrasound propagation direction and the predominant trabecular direction: approximately parallel (axial) vs. approximately perpendicular (sagittal and coronal). Less dramatic variations in trabecular arrangement, such as those in the phantom experiments and simulation study described above, may not necessarily produce significant variations in SOS or phase velocity.

Anisotropy is more pronounced in bovine cancellous bone. Hosokawa and Otani (1998) found fast wave velocity to be approximately 25% greater in the direction parallel to the predominant trabecular orientation compared with other directions in bovine cancellous tibia. Hughes et al. (1999) found fast wave velocity to be approximately 100% greater in the parallel direction in bovine cancellous tibia and femur. Hoffmeister et al. (2000) found SOS to be approximately 25% higher in the parallel direction in bovine cancellous tibia. Given the dependence of cancellous bone microstructure on loading conditions, and the differences in loading conditions between humans and cows, the enhanced anisotropy in bovine cancellous bone is perhaps not surprising.

V. CONCLUSION

The experiments on phantoms reported here reinforce previous results on phantoms and human cancellous bone in vitro in which volume fraction of trabeculae is the dominant determinant of phase velocity. In bovine cancellous bone, however, both volume fraction and trabecular orientation are significant determinants of phase velocity.

ACKNOWLEDGEMENTS

The author is grateful to Laura Perfetti and 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.

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