A Numerical Method to Predict the Effects of Frequency-Dependent Attenuation and Dispersion on Speed of Sound Estimates in Cancellous Bone

Abstract

Many studies have demonstrated that time-domain speed-of-sound (SOS) measurements in calcaneus are predictive of osteoporotic fracture risk. However there is a lack of standardization for this measurement. Consequently, different investigators using different measurement systems and analysis algorithms obtain disparate quantitative values for calcaneal SOS, impairing and often precluding meaningful comparison and/or pooling of measurements. A numerical method has been developed to model the effects of frequency-dependent attenuation and dispersion on transit-time-based SOS estimates. The numerical technique is based on a previously developed linear system analytic model for Gaussian pulses propagating through linearly attenuating, weakly dispersive media. The numerical approach is somewhat more general in that it can be used to predict the effects of arbitrary pulse shapes and dispersion relationships. The numerical technique however utilizes several additional assumptions (compared with the analytic model) which would be required for the practical task of correcting existing clinical databases. These include a single dispersion relationship for all calcaneus samples, a simple linear model relating phase velocity to broadband ultrasonic attenuation, and a constant calcaneal thickness. Measurements on a polycarbonate plate and thirty human calcaneus samples were in good quantitative agreement with numerical predictions. In addition, the numerical approach predicts that in cancellous bone, frequency-dependent attenuation tends to be a greater contributor to variations in transit-time-based SOS estimates than dispersion. This approach may be used to adjust previously acquired individual measurements so that SOS data recorded with different devices using different algorithms may be compared in a meaningful fashion.

Introduction

Speed of sound (SOS) in calcaneus has been demonstrated to be highly useful in the diagnosis of osteoporosis.^1–14 SOS measurements may be performed using two opposing coaxially aligned transducers for transmission and reception, as is done in many commercial bone sonometers. The transit time of ultrasound through a sample (e.g. calcaneus) is compared with the transit time through water. SOS may then be computed using a well-known formula which relates SOS in the sample with SOS in water, the sample thickness, and the difference in ultrasound transit times for the two measurements. In order to measure transit time, a reference point on the pulse waveform such as a zero-crossing is designated. Frequency-dependent attenuation (attenuation varies approximately linearly with frequency in most tissues, including cancellous bone) and dispersion (frequency-dependent phase velocity) however can distort waveforms, shift locations of zero-crossings, and produce artifacts in SOS measurements. This is particularly troublesome for highly attenuating media such as bone.

Ambiguities in zero-crossing-based SOS measurements in cancellous bone were first reported by Laugier et al.¹⁵ and subsequently by Strelitzki and Evans,¹⁶ Nicholson et al.¹⁷ and Wear.¹⁸ There is a considerable variety of choices for designated reference points in the literature including the leading edge (first detectable deviation from zero) of received ultrasonic pulse^6,17,19,20 thresholding at 10% of the maximum value,¹⁷ thresholding at 3 times the noise standard deviation,²¹ first zero crossing,^11,14,17 “specific” zero crossings,¹ first through fourth zero crossing, first and second maxima and minima,¹⁵ and zero crossing of the first negative slope.³ An unfortunate ramification of this lack of standardization is that different investigators using different transducers and analysis algorithms obtain disparate quantitative values for calcaneal SOS, impairing and often precluding meaningful comparison and/or pooling of measurements obtained from different studies.

A mathematical model was previously developed to quantitatively predict these variations for Gaussian pulses propagating through linearly-attenuating, weakly-dispersive media. Good agreement was shown between theoretically predicted variations and averages of experimental measurements on 24 human calcaneus samples in vitro.¹⁸ The aim of the present paper is to investigate the use of a numerical approach in order to model this phenomenon. This numerical technique can be used to compute corrections for previously acquired SOS data so that measurements acquired using different measurement systems and data analysis algorithms may be compared and perhaps pooled meaningfully. The numerical method requires several additional assumptions (see below), which were not required for the analytical model, in order to be practically useful for the task of correcting existing clinical databases. Validation experiments using measurements on a polycarbonate plate and 30 human calcaneus samples in vitro are described. The relative roles of frequency-dependent attenuation and dispersion in producing variations in SOS estimates are investigated.

Theory

One common technique for measuring SOS in a sample (e.g. calcaneus) is as follows. Two opposing coaxially-aligned broadband transducers are used: one transmitter and one receiver. Arrival times of received pulses are measured with and without the sample in the water path. SOS in a sample, c_s, is then computed from

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

where d is the thickness of the sample, Δt is the difference in arrival times, and c_w is the SOS in water.

Figure 1 shows the effects of frequency-dependent attenuation and pulse reference point designation on transit time estimation. A hypothetical calibration (through water) waveform is shown in the top part of the figure. Below is the waveform corresponding to the measurement through a sample with faster SOS than water (hence the earlier arrival time) and linear frequency-dependent attenuation (hence the lower center frequency). The transit time differential, Δt, depends on which reference point pair is used.

Figure 1.

Simulated waveform through water, x(t) (upper), and through bone sample, y(t) (lower). The four zero-crossing pair designations investigated are labeled as A, B, C, and D. The transit-time differential, Δt, (and therefore the SOS estimate) depends on which zero-crossing pair is utilized.

In a previously developed model described in detail elsewhere¹⁸ and summarized here, the effects of attenuation and phase-shifting due to differences in speeds of sound between water and sample are represented as a linear filtering process. The calibration signal through water is denoted by x(t). The impulse response of the linear filtering process is denoted by h(t). The signal recorded with the sample in the water path is y(t).

$$y(t)=h(t)*x(t)$$ (2)

The transfer function, the Fourier transform of h(t), may be modeled as follows.

$$H(f)=e^{-\beta fd}{e}^{-i2\pi f\Delta t(f)}$$ (3)

where f is frequency, β is the slope of the attenuation coefficient of the sample (sometimes referred to as normalized broadband ultrasonic attenuation or nBUA), d is the thickness of the sample, and Δt(f) is the time delay (relative to a water calibration signal) at a given frequency due to a difference in SOS between the sample and calibration given by

$$\Delta t(f)=d\left[\frac{1}{c_s(f)}-\frac{1}{c_w}\right]$$ (4)

where c_w is the SOS in water (assumed to be nondispersive) and c_s(f) is the frequency-dependent SOS in the dispersive medium.

The input or calibration signal, x(t), may often be assumed to be a Gaussian modulated sinusoid. The analytic signal representation is given by

$$x(t)=A{e}^{-t^2/2{\sigma }_t^2}{e}^{i2\pi f_{0}t}{e}^{i{\varphi }_0}$$ (5)

where A is the amplitude, σ_t is a measure of the duration of the pulse, f₀ is the center frequency, and ϕ₀ is the initial phase.

Over a relatively narrow range of frequencies of interrogation, dispersion may be assumed to be approximately linear. This is consistent with published investigations.^16,17,22,23

$$c_s(f)=c_s(f_0)+b_s(f-f_0)$$ (6)

where c_s(f₀) is the phase velocity at the center frequency and b_s is the rate of change of phase velocity with frequency.

Numerical Method

Equations 1 through 6 were incorporated into a program using MATLAB (The Mathworks Inc., Natick, MA, USA). The program computed SOS estimates as functions of attenuation coefficient, β (nBUA), for several different zero-crossing designations on the pulse waveform. The input signal, x(t), was characterized using Equation 5 by f₀ = 500 kHz (typical for clinical bone sonometers), σ_t = 1.1 microsec, and ϕ₀ = 0. (Note that, unlike the analytical model, the numerical method did not require a Gaussian function. Any numerical representation of x(t), including an digitized measurement, could have been used). A Fast Fourier Transform (FFT) was computed to obtain X(f).

In order to compute Y(f) = H(f)X(f), it was necessary to assume values for β, d, and Δt(f). Since the FFT corresponding to any real signal contains negative as well as positive frequency components, the attenuation transfer function had to be modified from e^−βfd to e^−β|f|d. The program was written in the form of a loop with β ranging from 0 to 30 dB/cmMHz (typical range for human calcaneus) in steps of 1 dB/cmMHz. Sample thickness (d) was taken to be 18 mm (the average thickness for 30 calcaneus samples described below). The frequency-dependent time delay Δt(f) is a function of the SOS in water, c_w, and the frequency-dependent phase velocity, c_s(f) = c_s(f₀) + b(|f| − f₀), in the sample. Phase velocity at 500 kHz, c_s(f₀), for a given sample was taken to be a function of attenuation coefficient, β. This relationship was obtained from a linear regression of experimental measurements of c_s(f₀) vs. β. Dispersion was characterized by b = −26.25 m/sMHz (the average of four values found in the literature.^16,17,22,23) The reason for this negative dispersion is currently not well understood.²³ Note that the assumption of weak linear dispersion, which was required in the original analytical model,¹⁸ is not required for the numerical technique. Any functional form or experimental data could have been used in the numerical approach to describe dispersion. The numerical method was repeated for b=0 in order to investigate the importance of dispersion. The digitization rate was 320 MHz.

With the assumptions listed above, a numerical version for the transfer function, H(f), could be computed. The simulated received time domain signal, y(t), was obtained from the inverse Fourier transform of the product of H(f) and X(f). Four SOS computations were performed for each water/sample simulated pulse waveform pair. The four computations were derived from four zero crossing reference point pairs (A, B, C, and D) illustrated in Figure 1.

Experiment

Calcaneus Samples

Thirty excised human calcaneus samples (genders and ages unknown) were defatted using a trichloro-ethylene solution. Since nBUA and SOS of defatted cancellous bone have been found to be only slightly different from their counterparts obtained with marrow left intact^19,21,24 it was assumed that defatting would not substantially affect ultrasonic measurements.

In order to isolate the effects of cancellous bone, the lateral cortical layers were sliced off leaving two parallel surfaces. This produced samples with well-defined uniform thicknesses (d) which facilitated application of the mathematical model. The thicknesses of the samples were measured using calipers and varied from 12 to 21 mm. Prior to ultrasonic interrogation, samples were vacuum degassed underwater in a desiccator. Subsequently, samples were allowed to thermally equilibrate to room temperature. Ultrasonic measurements were performed in distilled water at room temperature. The relative orientation between the ultrasound beam and the calcaneus samples was the same as with in vivo measurements performed with commercial bone sonometers, in which sound propagates in the mediolateral direction.

Ultrasound Measurements

Samples were interrogated in a water tank using a Panametrics (Waltham, MA) 5800 pulser/receiver and Panametrics V301 1” diameter, focussed (focal length = 1.5”), broadband transducers with center frequencies of 500 kHz. A wide-band electrical excitation was applied to the transmitter. Samples were placed in the focal plane. Received 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 through-transmission method described above was used to measure nBUA and SOS. Using two opposing coaxially-aligned transducers (one transmitter and one receiver) separated by twice the focal distance, transmitted signals were recorded both with and without the bone sample (or polycarbonate plate) in the acoustic path. The samples were larger in cross-sectional area than the receiving transducer aperture. Four to eight transmitted signals were recorded for each bone sample. The sample was translated slightly between successive measurements so that a spatial average could be obtained. Attenuation coefficient was then estimated using a log spectral difference technique.²⁵ Attenuation was characterized by the slope of a least-squares linear fit of attenuation coefficient (dB/cm) vs. frequency, resulting in the normalized broadband ultrasonic attnuation²⁴ (nBUA). Since the SOS in calcaneus, approximately 1475–1650 m/s²³ is comparable to that in distilled water at room temperature,²⁶ potential diffraction-related errors^27–29 in this substitution technique may be ignored.²³

To measure SOS, arrival times of received broadband pulses were measured with and without the sample in the water path. Four reference point designations were used. They corresponded to the two zero crossings immediately prior and immediately following the envelope maximum (See Figure 1 – A, B, C, and D). Velocity, c_s, was computed from Equation 1 for each of the four designations. All velocity estimates were expressed as deviations from the group velocity (velocity of the maximum of the pulse envelope). Since the variability among different bone samples tends to be much greater than the variability of repeated measurements on the same sample, only one SOS measurement for each set of four to eight recorded transmitted signals from each sample was generated. For the polycarbonate sample, six SOS measurements (each obtained from eight separate recorded transmitted signals) were performed in order to generate error bars.

Independent measurements of frequency-dependent phase velocity were performed on the calcaneus using a previously reported technique²² in order to obtain an empirical (assumed to be linear) relationship between phase velocity at 500 kHz and nBUA. This relationship was required for the numerical approach. Phase velocity measurements were also performed on the polycarbonate plate in order to obtain estimates of parameters required for the corresponding numerical technique (c_s = 2190 m/s, b_s = 48 m/sMHz). The thickness of the plate was 25.8 mm.

Results

Figure 2 shows numerical predictions and experimental measurements for SOS in the polycarbonate plate for the four zero crossing reference points (A, B, C, and D) utilized. Good agreement between numerical technique and experiment may be seen. Linear fits to numerical predictions and experimental measurements were given by SOS estimate = 2186.8 m/s − 11.1 m/s*cycle (numerical) and SOS estimate = 2190.4 m/s − 10.5 m/s*cycle (experimental) where “cycle” refers to the number of cycles (wavelengths) between the reference point location and the pulse envelope maximum. The average discrepency between these two linear fits over the range of reference point locations considered is 3.6 m/s (0.16%). Diffraction errors may partially account for the disparity between the two.

Figure 2.

Numerical and experimental results for the polycarbonate plate. Speed of sound (SOS) values predicted by numerical analysis are denoted by asterisks. Measurements are denoted by circles. Error bars denote standard deviations of measurements. Measurements and numerical predictions were obtained for zero-crossings denoted by A, B, C, and D. (See Figure 1). The simulated calibration pulse waveform is also shown in order to clarify the relationship between the time axis and zero-crossings of interest.

Figure 3 shows experimental measurements of c_s(f₀) vs. β. The linear regression for this data is given by c_s(f₀) = 1474.9 + 4.46β m/s (where c_s is measured in m/s and β = nBUA is measured in dB/cmMHz). This regression formula was used in the numerical method. Typically, commercial sonometers record broadband ultrasonic attenuation, BUA, which is measured in dB/MHz, rather than the normalized version, nBUA = β = BUA / d. For this data, one could use an average value for d to convert BUA to nBUA.

Figure 3.

Linear regression of experimental measurements of phase velocity at 500 kHz, c_s(f₀), vs. attenuation slope (normalized broadband ultrasonic attenuation or nBUA), β in 30 human calcaneus samples in vitro.

Figure 4 shows simulated and experimental deviations from group velocity for estimates for the four zero crossing designations. There is good agreement between theory and experiment. The average magnitude of the error between simulation and experiment was 0.15 %. As reported previously by others, zero crossing designations on the leading side of the pulse result in estimates which exceed the true group velocity while those on the trailing side result in estimates which are less than the true group velocity.

Figure 4.

Percent deviation from group velocity. Measurements for four reference point designations (A, B, C, and D, see Figure 1). Experimental data points were generated for reference point A (asterisks), B (plus signs), C (circles), and D (squares). Predictions based on the numerical technique were also generated for reference point A (solid line), B (dashed line), C (dotted line), and D (chain dotted line). The independent variable is normalized broadband ultrasonic attenuation (aka attenuation slope).

Figure 5 compares numerical computations which assume no dispersion (b=0, solid lines) with numerical computations which incorporate frequency-dependent attenuation and dispersion (b=−26.25, dashed lines). It can be seen that frequency-dependent attenuation is the factor most responsible for the disparities in SOS estimates.

Figure 5.

Percent deviation from group velocity. Measurements for the four reference point designations. Models excluding (solid lines, b = 0) and including (dashed lines, b=−26.25) dispersion are shown. The independent variable is normalized broadband ultrasonic attenuation (aka attenuation slope).

Discussion

The effects of frequency-dependent attenuation, dispersion, and pulse reference point location on SOS measurements in a polycarbonate plate and in calcaneus have been shown to be accurately modeled using a numerical method based on a linear system theory approach. Quantitative prediction has been provided for the empirical qualitative observation made by others that magnitudes of transit-time-based SOS estimates steadily decrease as the reference point designation is moved from the leading edge to the trailing edge of the pulse waveform. Frequency-dependent attenuation is the primary source of reported variations in SOS estimates based on different reference point locations. Dispersion has a smaller effect and serves to exacerbate variations somewhat for reference points on the leading half of the pulse waveform while mitigating them somewhat for reference points on the trailing half. Although the experimental validation provided here is limited to purely cancellous bone in vitro, it is likely these conclusions are relevant to in vivo applications.

The numerical model is in some respects more flexible than the analytical model previously described.¹⁸ In particular, assumptions of 1) Gaussian pulse shape, 2) linear dispersion, and 3) weak dispersion were required in the analytical model. These approximations were unnecessary for the numerical technique. For example, a digitized calibration pulse waveform could be measured and then used in the numerical method. Likewise, an average (not necessarily linear and not necessarily weak) frequency-dependent phase velocity could be measured and utilized in the numerical approach. In both cases, there is no need to assume a particular functional form. Moreover, in many cases the numerical approach may be far easier to implement than the analytical model which is fairly cumbersome, especially if the effects of dispersion are included.

Variations of measured velocity from one zero-crossing to the next for a calcaneus with a nBUA of 20 dB/cmMHz can be on the order of 2% (30 m/s). (See Figure 4). This is considerable as subtle variations in SOS can convey important diagnostic information. For example, in the EPIDOS prospective study of over five thousand women,⁷ baseline SOS measurements were 1480 ± 24 m/s (mean ± standard deviation) for subjects who experienced a hip fracture over a subsequent two year period compared with 1493 ± 24 m/s for subjects who did not. The difference, 13 m/s, is smaller than the differences that can occur due to different reference point designations. Thus it would be inappropriate to compare or pool this data with data from other studies obtained using different transducers and different data analysis algorithms unless the resulting variations in SOS estimates were properly corrected.

In contrast to the analytical model,¹⁸ the numerical approach implemented here was based on assumptions of 1) a single dispersion relationship (phase velocity vs. frequency) for all calcaneus samples, 2) a simple linear model relating phase velocity at 500 kHz to nBUA, and 3) an average calcaneal thickness for all samples. In reality, dispersion rates vary considerably among different samples;^22,23 there is a lot of scatter about the average trend of the relationship between phase velocity at 500 kHz and nBUA (see Figure 3); and there can be considerable variation in calcaneal thicknesses. While utilization of individual (rather than average) measurements of these parameters in the numerical technique would be expected to improve the level of agreement between numerical method and experiment, this detailed information on individual samples will not generally be available for previously acquired clinical databases. Therefore these simplifying assumptions will generally be required for retroactively correcting existing data. Nevertheless the numerical technique performs rather well.

It has previously been demonstrated that an analytical model performs well in predicting average variations in SOS estimates due to reference point location variations, frequency-dependent attenuation, and dispersion.¹⁸ Figure 4 shows that the numercial approach predicted these variations well for individual bone measurements, even though the numerical approach was based on several additional simplifying assumptions. In addition, the basic linear system approach has been shown in the current paper to be valid in another medium (polycarbonate) with considerably different acoustic properties than cancellous bone (much lower attenuation and positive rather than negative dispersion).

This result indicates promise for using this approach to adjust previously acquired individual measurements so that SOS data recorded with different devices using different algorithms may be compared in a meaningful fashion. In order to implement this technique, investigators would require the following information from manufacturers. 1. Sufficient information to reconstruct x(t) (Equation 5). This could be in the form of a digitized pulse waveform or a set of parameters specifying an analytic approximation (e.g. f₀, σ, ϕ₀ for a Gaussian pulse). 2. The designated reference point location for transit time measurement. In addition, investigators would have to assume a relationship between the midband phase velocity and BUA (see Results section) and a value for the dispersion rate (e.g. b = −26.25 m/sMHz, see Numerical Method section).