The dependence of time-domain speed-of-sound measurements on center frequency, bandwidth, and transit-time marker in human calcaneus in vitro.

Abstract

Time-domain speed-of-sound (SOS) measurements in calcaneus are effective predictors of osteoporotic fracture risk. High attenuation and dispersion in bone, however, produce severe distortion of transmitted pulses that leads to ambiguity of time-domain SOS measurements. An equation to predict the effects of system parameters (center frequency and bandwidth), algorithm parameters (pulse arrival-time marker) and bone properties (attenuation coefficient and thickness) on time-domain SOS estimates is derived for media with attenuation that varies linearly with frequency. The equation is validated using data from a bone-mimicking phantom and from 30 human calcaneus samples in vitro. The data suggest that the effects of dispersion are small compared with the effects of frequency-dependent attenuation. The equation can be used to retroactively compensate data. System-related variations in SOS are shown to decrease as the pulse-arrival-time marker is moved toward the pulse center. Therefore, compared with other time-domain measures of SOS, group velocity exhibits the minimum system dependence.

Introduction

Speed of sound (SOS) in calcaneus is very useful for characterization of bone and diagnosis of osteoporosis (Rossman et al., 1989; Tavakoli and Evans, 1991; Zagzebski et al., 1991; Laugier et al., 1993; Njeh et al., 1996; Bouxsien, 1997; Bauer et al., 1997; Laugier et al., 1997; Strelitzki and Evans, 1996; Strelitzki, et al., 1997; Nicholson et al., 1996; Nicholson et al., 1998, Hans et al., 1999, H. Trebacz, and A. Natali, et al., 1999; Hoffmeister et al., 2000; Hoffmeister et al., 2002a, 2002b; Chaffai et al., 2002; Lee et al., 2003a, 2003b; Laugier, 2004; Hakulinen et al., 2005, Yamoto et al., 2006). A landmark prospective clinical study of 5662 women showed that a linear combination of SOS and broadband ultrasound attenuation (BUA) predicts hip fracture risk in elderly women as well as dual energy x-ray absorptiometry (Hans et al., 1996). Other prospective (Huopio et al., 2004; Krieg et al., 2006) and retrospective (Schott et al., 1995; Turner et al., 1995; Mautalen et al., 1995, Thompson et al., 1998; Krieg et al., 2003; Glüer et al., 2004; Maggi et al., 2006) clinical studies have further established the utility of SOS for fracture risk prediction.

In laboratories and in commercial bone sonometers, SOS is usually measured in through-transmission. The ultrasound pulse transit time through the bone is compared with the transit time through a water path. SOS in bone is then computed from SOS in water, bone thickness, and transit-time differential. In order to measure transit time from digitized radio-frequency waveforms, a marker on the waveforms such as a zero-crossing is usually designated. Unfortunately, frequency-dependent attenuation and dispersion can distort waveforms, shift locations of zero-crossings, and produce variations in SOS measurements. See Figure 1. This is particularly troublesome for highly attenuating media such as bone.

Figure 1.

The effect of frequency-dependent attenuation on transit-time differential, Δt. Markers are labeled with an L for leading half or a T for trailing half and are numbered outward from the pulse center. The variation in Δt with marker location is due to the fact that the attenuated pulse is stretched in time as a consequence of the low-pass filtering effect of frequency-dependent attenuation.

Ambiguities in time-domain SOS measurements in cancellous bone were first reported by Laugier et al. (1993) and subsequently by Strelitzki and Evans (1996a), Nicholson et al. (1996) and Wear (2000a). Laugier et al. showed that SOS measurements in cancellous bone are highest for markers near the leading edge of the pulse waveform and lowest for markers near the trailing edge. As previously reported (Wear, 2000a) and also shown in Table I, there is a considerable range of choices for markers. Therefore, different investigators using different measurement systems and different analysis algorithms obtain disparate values for SOS, often precluding meaningful comparison and/or pooling of measurements obtained from different studies. This problem may contribute to the substantial disparity in SOS measurements from different commercial bone sonometers reported by Njeh et al. (2000). See Figure 2. Glüer (2007) recently emphasized that improvement in standardization methods will be required before quantitative ultrasound can play a major role in clinical practice.

Table I.

A non-exhaustive listing of marker designations for SOS measurements in bone.

General Marker Type	Specific location	References
Leading edge (first detectable deviation from zero)		Njeh, et al., 1996, 1997a, 1997b
Thresholding	at 3 times the noise standard deviation	Alves et al., 1996a
	at 20% of amplitude of first half cycle	Hakulinen et al., 2005.
Zero Crossings	first	Nicholson et al., 1998; Trebacz and Natali, 1999; Lee et al., 2003
	“specific”	Rossman et al., 1989.
	of first negative slope	Zagzebski et al., 1991.
	first after threshold of 10%	Haϊat et al., 2006
	first after threshold of 15%	Haϊat et al., 2005
Maximum Absolute Value		Alves et al., 1996b.
Maximum Envelope		Wear, 2000b

Figure 2.

SOS measurements reported by Njeh et al. (2000) in 35 normal post-menopausal women and 35 age-matched women with fractures using six commercial bone sonometers.

Velocity measurement methods that do not rely on designated markers may be more resistant to frequency-dependent-attenuation-induced artifacts. One method estimates relative time delay from the cross correlation of the two waveforms. Unfortunately, this method becomes less robust when signals are severely distorted. In bone, substantial distortion arises from frequency-dependent attenuation, dispersion, multiple scattering, and phase cancellation. Another method utilizes phase velocity, which is measured in the frequency domain, rather than a time-domain SOS measure. Although these alternatives may have some advantages, the ultrasonic bone characterization literature is dominated by the marker-time-shift methodology discussed above (see Table I). Therefore, it is important to explore sources of variability inherent in these methods so that they may be taken into account when interpreting and comparing existing results. Moreover, although companies generally do not publish their SOS-measurement algorithms in the public domain, it is likely that many commercial systems utilize marker-time-shift SOS measurements due to their prevalence in the literature. (Note that a senior author for two of the references in Table I was the founder of Lunar—now GE Lunar—Corporation, a major manufacturer of clinical bone sonometers.) Therefore, these same sources of variability are likely to affect commercial bone sonometer measurements.

A mathematical model was previously developed to predict variations in SOS estimates for Gaussian pulses propagating through media with weak dispersion and attenuation that varies linearly with frequency (Wear, 2000a; Wear, 2001). The model was validated with measurements on 24 human calcaneus samples in vitro. It was shown that in human calcaneus in vitro the effects of attenuation are far greater than the effects of dispersion. Haϊat et al. (2006) used an approach based on this model to investigate SOS measurements in 38 human femur samples in vitro. Rather than using a fixed value for dispersion, however, they compensated SOS measurements based on individual dispersion measurements. They found this approach to be very accurate in numerically predicting disparity between a zero-crossing-based SOS estimate and group velocity. Their analysis was very thorough in delineating the effects of dispersion and attenuation on SOS estimates in femur.

Although the numerical approach accurately predicts system-related and algorithm-related variations in SOS measurements, it requires substantial effort to implement and is unlikely to be widely-adopted by the research community or by industry. In order to make this problem more tractable, a simpler model, which accounts for frequency-dependent attenuation but not dispersion, is derived below. The analysis produces an analytic formula that is easy to use for the adjustment of SOS measurements performed with different measurement systems and different marker locations. The formula provides insight into the effects of system parameters (center frequency and bandwidth), algorithm parameters (marker location) and bone properties (attenuation coefficient and thickness) on time-domain SOS estimates. The formula may be used to retroactively compensate existing data. The model is validated in a bone-mimicking phantom and in 30 human calcaneus samples in vitro.

Theory

SOS is usually measured in through transmission. Two transducers, one transmitter and one receiver, are arranged in a “pitch-catch” orientation. First, a calibration measurement is performed by propagating a pulse from the transmitter to the receiver through a water path. Then a sample (e.g. bone) is placed between the two transducers, and a second measurement is performed. SOS may then be computed from

$$SOS=\frac{c_w}{1+\frac{c_w\Delta t}{d}}$$ (1)

where c_w is the acoustic velocity in water, Δt is the difference in arrival times of the two pulses, and d is the thickness of the sample. Since the pulse has finite duration, a marker (e.g. the maximum or a zero crossing) is designated for the measurements of arrival times. (Of course, for the sake of consistency, the same marker must be chosen for both calibration and sample measurements.) As illustrated in Figure 1, however, frequency-dependent attenuation (which is a low-pass filter) stretches the attenuated signal in time and causes Δt to vary with marker designation.

In this section, a formula for the disparity in SOS measurement due to a disparity in marker designation is derived. Suppose that two markers, n and m, yield Δt values Δt_n and Δt_m, then the resulting disparity in SOS estimates is

$$\Delta SO{S}_{n,m}=\frac{c_w}{1+\frac{c_w\Delta t_n}{d}}-\frac{c_w}{1+\frac{c_w\Delta t_m}{d}}=\frac{c_w^2}{d}\frac{(\Delta t_m-\Delta t_n)}{\left(1+\frac{c_w\Delta t_m}{d}\right)\left(1+\frac{c_w\Delta t_n}{d}\right)}$$ (2)

In order to derive expressions for Δt_n and Δt_m for various marker designations, a model for time-domain signals undergoing frequency-dependent attenuation is taken from a previous paper (Wear, 2000a). Briefly, the water-path calibration signal, x(t), is assumed to be a Gaussian modulated sinusoid, with analytic signal representation given by

$$x(t)=A{e}^{-t^2/2{\sigma }_0^2}{e}^{i2\pi f_{0}t}$$ (3)

where A is the pulse magnitude, σ₀ is a measure of the pulse duration, f₀ is the center frequency, and t is time. (Throughout this section, a subscript of 0 will be used for parameters in the calibration measurement while a subscript of 1 will be used for parameters in the sample—e.g. bone—measurement.) The Fourier transform of x(t) is

$$X(f)=A{\sigma }_0\sqrt{2\pi }{e}^{-{(f-f_0)}^2/2{\sigma }_f^2}$$ (4)

where f is frequency and σ_f = 1/(2πσ₀) is a measure of the bandwidth. A medium with attenuation that varies linearly with frequency can be modeled with the following transfer function,

$$H(f)=e^{-\beta fd}{e}^{-i2\pi fs}$$ (5)

where βf is the attenuation coefficient, d is the thickness of the sample, and s is the time delay (relative to a water reference signal) given by

$$s=d\left[\frac{1}{c_b}-\frac{1}{c_w}\right]$$ (6)

where c_b is the phase velocity in the linearly-attenuating medium. (Note that s < 0 when c_b > c_w.) It can be shown that the attenuated signal, y(t), takes the form (Wear, 2000a)

$$y(t)=B{e}^{-{(t-s)}^2/2{\sigma }_1^2}{e}^{i2\pi f_1(t-s)}$$ (7)

where B is the pulse magnitude, and σ₁ is a measure of the pulse duration. The down-shifted center frequency, f₁, of the attenuated wave is given by (Narayana and Ophir, 1983; Wear, 2000a)

$$f_1=f_0-{\sigma }_f^2\beta d$$ (8)

Locations of zero crossings are determined by the complex exponential factors in Equations 3 and 7. The locations of zero-crossing markers for the symmetric pulses shown in Figure 1 are given in Table II.

Table II.

Locations of zero-crossings for the reference and attenuated pulses shown in Figure 1. The maximum of the reference pulse is halfway between L1 and T1 at t₀ = 0. The period of the reference pulse is 1/f₀. The period of the attenuated pulse is 1/f₁.

Zero Crossing	L3	L2	L1	T1	T2	T3
Reference pulse (t0)	−5/4f0	−3/4f0	−1/4f0	1/4f0	3/4f0	5/4f0
Attenuated pulse (t1)	−5/4f1 + s	−3/4f1 + s	−1/4f1 + s	1/4f1 + s	3/4f1 + s	5/4f1 + s

Suppose that a marker on the calibration signal, x(t), is chosen at t = t_n0. Then, by comparing the complex exponentials in Equations 3 and 7, it can be seen that the corresponding marker on the attenuated waveform, t_n1, will satisfy f₀t_n0 = f₁(t_n1 – s). Therefore, t_n1 = (f₀/f₁)t_n0 + s. The transit time differential is then

$$\Delta t_n=t_{n1}-t_{n0}=\left(\frac{f_0}{f_1}-1\right)t_{n0}+s=a{t}_{n0}+s$$ (9)

where (using Equation 8)

$$a\equiv \frac{f_0}{f_1}-1=\frac{f_0}{f_0-{\sigma }_f^2\beta d}-1=\frac{{\sigma }_f^2\beta d}{f_0-{\sigma }_f^2\beta d}$$ (10)

Equation (2) compares the SOS estimate based on marker n located at t_n0 with the SOS estimate based on marker m located at t_m0. Suppose that the second marker (m) is chosen to be the envelope maximum, so that the second SOS estimate corresponds to group velocity. If the reference pulse is assumed to be centered at the origin, then t_m0 = 0, t_m1 = s, and Δt_m = s. The difference of transit-time differentials for two markers required in the numerator of Equation 2 is then (from Equation 9)

$$\Delta t_m-\Delta t_n=-a{t}_{n0}=-a\frac{\tau }{f_0}$$ (11)

where τ is the interval between calibration marker n and the envelope maximum, measured in units of the calibration waveform period, T₀ = 1/f₀, so that τ ≡ t_n0/T₀ = f₀ t_n0. For example, the central zero crossings on the symmetric calibration waveform in Figure 1 occur at τ = … −5/4, −3/4, −1/4, 1/4, 3/4, 5/4 ….

Equation 11 gives the numerator for the right hand factor in Equation 2. Now the denominator for the right hand factor in Equation 2 must be obtained. Let c_g denote group velocity. Note from Equation (1) that

$$c_g=\frac{c_w}{1+\frac{c_{w}s}{d}}$$ (12)

Now the denominator from Equation 2 may be simplified as follows.

$$\left(1+\frac{c_{w}s}{d}\right)\left[1+\frac{c_w}{d}\left(\frac{a\tau }{f_0}+s\right)\right]=\frac{c_w}{c_g}\left[\frac{c_w}{c_g}+\frac{c_{w}a\tau }{d{f}_0}\right]=\frac{c_w^2}{c_g^2}+\frac{c_w^{2}a\tau }{c_{g}d{f}_0}$$ (13)

Combining Equations 2, 11 and 13 yields

$$SO{S}_n-c_g=-c_g\frac{a\tau }{\frac{f_{0}d}{c_g}+a\tau }$$ (14)

In the ultrasonic characterization of bone, the first term in the denominator of Equation (14) will often dominate the second term. For example, using typical values (f₀ = 500 kHz, σ_f = 100 kHz, normalized broadband ultrasound attenuation or nBUA = 30 dB/cmMHz, β = nBUA / 8.68 = 3.5 1/cmMHz, d = 2 cm, c_g = 1500 m/s → a = 0.16, and τ = −1.25), the two terms become 6.67 and −0.20 respectively. Neglecting the second term in the denominator, Equation (14) simplifies to

$$SO{S}_n-c_g\approx -\frac{c_g^{2}a\tau }{f_{0}d}=-\frac{\tau c_g^2{\sigma }_f^2\beta }{f_0^2}\frac{1}{1-({\sigma }_f^2\beta d/f_0)}$$ (15)

Equation (15) shows that the magnitude of the deviation of the SOS estimate from group velocity has a strong dependence on the fractional bandwidth (σ_f / f₀). The disparity of SOS estimates derived from two arbitrary markers, n and m, separated by Δτ, is given by

$$\Delta SO{S}_{n,m}\approx -\frac{\Delta \tau c_g^2{\sigma }_f^2\beta }{f_0^2}\frac{1}{1-({\sigma }_f^2\beta d/f_0)}$$ (16)

Methods

A. Bone samples and phantom

30 human calcaneus samples (both genders, ages unknown) were defatted using a trichloro-ethylene solution. Defatting was presumed not to significantly affect measurements since SOS of defatted trabecular bone has been measured to be only slightly different from that of bone with marrow left intact (Njeh and Langton, 1998; Alves et al., 1996a). The cortical lateral layers were sliced off leaving two parallel surfaces with direct access to trabecular bone. Since cortical layers are so thin (a few mm), their removal does not alter SOS measurements very much; an experimental investigation of SOS from 20 human calcanea, both before and after cortical plate removal, showed that calcaneal SOS is determined mainly by the cancellous bone component (Njeh and Langton, 1997a). The thicknesses of the samples varied from 12 to 21 mm. In order to remove air bubbles, the samples were vacuum degassed underwater in a desiccator. After vacuum, samples were allowed to thermally equilibrate to room temperature prior to ultrasonic interrogation. Ultrasonic measurements were performed in distilled water at room temperature. The temperature was not actively controlled but was measured for each experiment and ranged between 19°C and 22°C. The relative orientation between the ultrasound beam and the calcanea was the same as with in vivo measurements performed with commercial bone sonometers, in which sound propagates in the mediolateral (or lateromedial) direction.

A quantitative ultrasound phantom (Model 063, CIRS inc., Norfolk, VA) was also interrogated. The phantom was composed of proprietary urethane and had a thickness of 36 mm (http://cirsinc.com/063_ultra.html). The attenuation coefficient slope (nBUA) was measured to be 15.0 dB/cmMHz (at 19.3 degrees Celsius). (The model 063 contains two phantoms labeled “normal” and “osteoporotic.” The “osteoporotic” phantom was measured here.)

B. Ultrasonic measurements

Bone samples and the phantom were interrogated in through-transmission in a water tank using a Panametrics (Waltham, MA) 5800 pulser/receiver with pairs of coaxially-aligned Panametrics transducers. See Table III. Received signals were digitized (8 bits; 10 MHz for bone data, 25 MHz for phantom data) using a LeCroy (Chestnut Ridge, NY) 9310C Dual 400 MHz oscilloscope and stored on computer (via GPIB) for off-line analysis.

Table III.

Properties of ultrasound transducers

Nominal Center Frequency (MHz)	Diameter (mm)	Focal Length (mm)
0.5	25.4	38.1
1.0	25.4	50.8

The temperature-dependent speed of sound in distilled water, c_w, was used as the reference speed and is the given by (Kaye and Laby, 1973)

$$c_w=1402.9+4.835T-0.047016T^2+0.00012725T^3$$ (17)

where T is the temperature in degrees Celsius. Analysis software was written in MATLAB (Natick, MA). Each arrival time was computed as follows. The mean (dc) value of each received signal was subtracted. The signal was filtered with a frequency domain Gaussian filter with center frequency equal to the transducer nominal center frequency and standard deviation equal to 125 kHz (500 kHz data) or 200 kHz (1 MHz data). The envelope was computed using the Hilbert transform. The three zero crossings immediately before (L1, L2, and L3 in Figure 1) and after (T1, T2, and T3) the radio-frequency maximum were measured using linear interpolation between points at which a change in sign occurred.

“Uncompensated” SOS values were computed directly using equation 1. “Compensated” SOS values were computed by subtracting the predicted SOS difference from group velocity—computed using either Equation (14) or (15)—from the uncompensated SOS values. Spectral parameters (f₀ and σ_f), which are required for Equations (14) and (15), were estimated from Gaussian fits to measured water-path spectra. Attenuation coefficients, which are also required in Equations (14) and (15), were computed from linear least squares fits to log spectral differences vs. frequency over the range from 300 to 600 kHz (Narayana and Ophir, 1983; Wear, 2000a).

Generally speaking, this substitution technique can exhibit appreciable error if the speed of sound differs substantially between the sample and the reference (Kaufman et al., 1995). However, one study indicates that this diffraction-related error is negligible in calcaneus (Droin et al., 1998). Apparently, the speed of sound in calcaneus, approximately 1475 – 1650 m/s, (Droin et al., 1998) is sufficiently close to that of distilled water at room temperature, 1487 m/s (from Equation 17) that diffraction-related errors may be ignored.

The effect of system spectral properties (f₀ and σ_f) was investigated by simulating four different systems obtained by applying four different Gaussian filters (instead of the filters described above) to the radio-frequency data acquired after transmission through the CIRS bone-mimicking phantom (see Table IV). SOS was measured using three zero crossings from the leading half of the pulse (L1, L2, and L3). Uncompensated measurements were compared with measurements compensated using Equation 14.

Table IV.

Four systems were simulated by filtering 500 kHz radio-frequency data with Gaussian filters with the characteristics above.

System	Filter f0 (kHz)	Filter σf (kHz)	Signal f0 (kHz)	Signal σf (kHz)	Fractional Bandwidth (σf/f0)
#1	600	100	580	84	15%
#2	600	135	560	102	18%
#3	500	150	500	110	22%
#4	500	200	500	126	25%

Results

A. Simulation

Estimates for SOS in equations (14) and (15) were compared to estimates generated using a previously-published numerical method (Wear, 2000a) in which 1) a simulated Gaussian pulse is generated, 2) an FFT is applied, 3) the dispersive form of the transfer function in Equation 5 is applied (See Wear, 2000a, Equations 3, 4 and 11), 4) an inverse FFT is applied, and 5) zero crossing locations are located on the resulting time-domain waveform. For this simulation, the following parameter values were assumed: f₀ = 500 kHz, σ_f = 100 kHz, d = 2 cm, and c_g = 1550 m/s. Figure 3 shows theoretical (Equation 14) and numerical results for disparity between SOS and group velocity evaluated at leading edge zero crossings L1, L2, and L3 for the non-dispersive case. There is excellent agreement between theoretical and numerical results. Figure 4 shows theoretical results for disparity between SOS and group velocity using Equation (14) and its approximate form, Equation (15). Again, excellent agreement is found.

Figure 3.

Theoretical and numerical results for disparity between SOS and group velocity evaluated at leading edge zero crossings L1, L2, and L3. The range of nBUA considered here is similar to that reported from experimental investigations of human calcaneus in vitro (Langton et al., 1996).

Figure 4.

Theoretical results for disparity between SOS and group velocity using Equation (14) and its approximate form, Equation (15).

Figure 5 shows numerical results for the effect of dispersion on the disparity between SOS and group velocity evaluated at leading edge zero crossing L3. Mean in vitro calcaneal dispersion rates from four published studies (see Table V) range from −15 to −40 m/sMHz, with an average value of −26.25 m/sMHz. According to Figure 5, a typical dispersion rate of −25 m/sMHz has a small effect on the disparity between SOS and group velocity.

Figure 5.

Effect of dispersion on the disparity between SOS and group velocity evaluated at leading edge zero crossing L3.

Table V.

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

Author(s)	N	Frequency Range (kHz)	dcp/df (mean ± standard deviation) (m/sMHz)
Nicholson et al.	70	200 – 800	−40
Strelitzki and Evans	10	600 – 800	−32 ± 27
Droin, Berger, and Laugier	15	200 – 600	−15 ± 13
Wear	24	200 – 600	−18 ± 15

B. Experiments

Figure 6 shows the disparity between SOS and group velocity at 500 kHz for 30 bone samples in vitro evaluated at zero crossings L3, L2, L1, T1, T2, and T3. The steady decline in uncompensated SOS as marker is moved from the leading edge toward the trailing edge, originally reported by Laugier (1993), is evident. The compensation formula (Equation 14) effectively suppresses the dependence of SOS on marker location.

Figure 6.

The disparity between SOS and group velocity at 500 kHz for 30 bone samples in vitro evaluated at zero crossings L3, L2, L1, T1, T2, and T3. The compensated measurements were compensated using Equation 14.

Figure 7 shows the disparity between SOS and group velocity at 1 MHz for 24 bone samples in vitro evaluated at zero crossings L3, L2, L1, T1, T2, and T3. Again, the compensation formula (Equation 14) effectively suppresses the dependence of SOS on marker location.

Figure 7.

The disparity between SOS and group velocity at 1 MHz for 24 bone samples in vitro evaluated at zero crossings L3, L2, L1, T1, T2, and T3. The compensated measurements were compensated using Equation 14.

Figure 8 shows filtered power spectra simulating four different data acquisition systems. Figure 9 shows uncompensated estimates of SOS from the CIRS bone-mimicking phantom evaluated at zero crossings L3, L2, L1 and at the envelope maximum (c_g). Figure 9 shows that system-related variations in SOS decrease as the marker moves toward the pulse center. Relatively subtle changes in system fractional bandwidth can lead to considerable variations in SOS estimate, particularly for markers near the leading edge. Figure 10 shows compensated estimates of SOS from the CIRS bone-mimicking phantom evaluated at zero crossings L3, L2, L1 and at the envelope maximum (c_g). The compensation formula (Equation 14) effectively suppresses the dependence of SOS on marker location and system center frequency and bandwidth.

Figure 8.

Filtered power spectra to simulate four different data acquisition systems.

Figure 9.

Estimates of SOS (uncompensated) from the CIRS phantom evaluated at zero crossings L3, L2, L1 and at the envelope maximum (c_g). The standard deviations for all these values are less than 1.5 m/s.

Figure 10.

Estimates of SOS (compensated using Equation 14) from the CIRS phantom evaluated at zero crossings L3, L2, L1 and at the envelope maximum (c_g). The standard deviations for all these values are less than 1.5 m/s.

Discussion

As Njeh (2000) recently reported, SOS measurements vary widely among commercial systems (see Figure 2). Glüer (2007) recently emphasized that improvement in standardization methods will be required before quantitative ultrasound can play a major role in clinical practice. This paper presents a framework for understanding the effects of system parameters (center frequency and bandwidth), algorithm parameters (marker location) and bone properties (attenuation coefficient and thickness) on time-domain SOS estimates. The framework was validated with measurements in a bone-mimicking phantom and in 30 human calcaneus samples in vitro. The success of this model suggests that accurate predictions of variations in SOS estimates in human calcaneus in vitro may be made even when the effects of dispersion are ignored. Other potential sources of disparity in the SOS measurement—including variations in region of interest location, variations in temperature, assumptions regarding heel thickness—are not considered here.

The variation of SOS with transit-time marker is significant. For example, at 500 kHz, average SOS changes by about 8 m/s when transit-time marker shifts from L3 to L2 (see Figure 6). This is a substantial amount compared with average SOS differences between patients with hip fractures and age-matched controls, which have been reported to be 25 m/s (Schott et al., 1995), 28 m/s (Mautalen et al., 1995), 13.5 m/s (Hans et al., 1996), 20 m/s (Njeh et al., 2000—see Figure 2), and 20 m/s (Krieg et al., 2006). The variation of SOS with transit-time marker is also substantial when compared to many of the inter-system differences reported by Njeh et al. (2000) which are often on the order of 20 – 40 m/s (see Figure 2).

Equation (16) provides a means for comparing SOS estimates, SOS_n and SOS_m, based on different marker times separated by Δτ. In the case of high aτ, it may be necessary to instead use Equation (14) twice (first to compute the difference between SOS_n and c_g, and again to compute the difference between SOS_m and c_g) and then to take the difference between the two differences to yield SOS_n - SOS_m. Therefore, data acquired using different systems and computed using different marker times can be compensated to facilitate comparison or pooling provided that the following variables are known: f₀, σ_f, β, d, and c_g. The first two variables may be estimated by performing spectral analysis on a water-path measurement. Attenuation coefficient slope β may be computed from BUA, which is measured by most calcaneal systems, and d, which may be measured in vitro or even in vivo (Chen et al., 2005; Yi et al., 2007) or if necessary set to an average value (e.g. Nicholson et al., 1997, measured an average value of 25 mm from 28 female calcanea). Finally, although c_g may not be known, a typical value may be assumed. For example, c_g in human calcaneus in vitro has been reported to range from 1475 – 1575 m/s (Wear, 2000b). So a typical value of 1525 m/s may be assumed, producing an error of no more than seven percent in equations (14) – (16).

In order to circumvent problems associated with time-domain SOS measures, some authors have suggested the use of phase velocity, which is measured in the frequency domain, to characterize bone (Alves et al., 1996a; Nicholson et al., 1996; Strelitzki and Evans, 1996b; Droin et al., 1998). The framework presented in this paper provides a method for retroactively estimating phase velocity from time-domain SOS measurements. First, Equation 14 may be used to estimate group velocity, c_g. Then, the following relation may be used to estimate phase velocity, c_pc, at the center frequency, f_c, (Morse and Ingard, 1986, and Duck, 1990)

$$c_g=\frac{c_{pc}}{1-\frac{f_c}{c_{pc}}{\left(\frac{\partial c_p}{\partial f}\right)}_{f=f_c}}$$

The quantity in parentheses is the dispersion rate. If independent dispersion measurements are not available, a typical value, such as those listed in Table V may be assumed.

The framework presented here can provide some insight into why many investigators choose markers near the leading edge of the pulse waveform. Attenuation and sound speed tend to increase with density. From Equation 15, it can be seen that SOS estimates based on markers near the leading edge are boosted by higher attenuation values (recall that τ is negative on the leading half of the pulse). So, for example, in order to increase the sensitivity of the SOS estimate to density, it might be tempting to use a marker near the leading edge. Indeed, Haϊat et al. (2006) found that SOS measurements based on the first zero crossing have a higher correlation with BMD in femur in vitro than group velocity (see Haϊat et al., 2006, Table II). Therefore, a marker near the leading edge might be preferable when a single parameter (SOS) is used to assess the bone, as is the case with current cortical bone systems and may be the case with future femur systems. Most calcaneal systems, however, use a linear combination of BUA and SOS to assess the bone. For these systems, the quantitative diagnostic classifier should be able to properly account for attenuation effects directly by optimally adjusting the weighting coefficient for BUA, without the need to enhance SOS measurements by an amount related to BUA. Indeed, Haϊat et al. (2006) also found that SOS measurements based on the first zero crossing performed comparably to group velocity when either was used in combination with BUA in a multiple regression model (again see Haϊat et al., 2006, Table II).

As elegantly explored by Haϊat et al. (2006), zero-crossing-based measures of SOS are prone to discontinuous jumps as BUA or dispersion varies. These jumps are due to changes in the pulse shape that alter reference points along the pulse in such a way to shift the marker locations by one or more zero-crossing intervals. Since group velocity is derived from the envelope, it is more immune to these effects. Haϊat et al. rightly point out, however, that markers near the leading edge are less sensitive to multi-path interference. In the femur (the bone studied by Haϊat et al.), which has a more complicated shape than the calcaneus, these multi-path interference effects may be sufficiently problematic to justify a preference for a leading-edge zero-crossing marker.

Although calcaneal SOS is an effective predictor of fracture risk, its clinical utility is compromised by the high inter-system variability of the measurement. This paper presents a framework for standardization of SOS measurements. System-related variations in SOS decrease as the pulse-arrival-time marker moves toward the pulse center. Therefore, compared with other time-domain measures of calcaneal SOS, group velocity exhibits the minimum system dependence. When SOS is used in combination with BUA in a linear classifier, the choice of group velocity instead of a zero-crossing-based value should not compromise diagnostic performance.