Estimation of fast and slow wave properties in cancellous bone using Prony’s method and curve fitting

Abstract

The presence of two longitudinal waves in poroelastic media is predicted by Biot’s theory and has been confirmed experimentally in cancellous bone. The Modified Least Squares Prony’s (MLSP) method in conjunction with curve-fitting (MLSP+CF) is tested using simulations based on published values for wave parameters in cancellous bone from several studies in bovine femur, human femur, and human calcaneus. The search algorithm is accelerated by exploiting correlations among search parameters. The performance of the algorithm is evaluated as a function of signal-to-noise ratio (SNR). For a typical experimental SNR (40 dB), the root-mean-square errors (RMSEs) for one example (human femur) with fast and slow waves separated by approximately half of a pulse duration were 1 m/s (slow wave velocity), 4 m/s (fast wave velocity), 0.4 dB/cmMHz (slow wave attenuation slope), and 1.7 dB/cmMHz (fast wave attenuation slope). The MLSP+CF method is fast (requiring less than 2 seconds at SNR = 40 dB on a consumer-grade notebook computer) and is flexible with respect to the functional form of the parametric model for the transmission coefficient. The MLSP+CF method provides sufficient accuracy and precision for many applications such that experimental error is a greater limiting factor than estimation error.

I. INTRODUCTION

Quantitative ultrasound is emerging as an important modality for characterization of bone (Langton et al., 1984; Laugier, 2008; Laugier, 2011; Barkmann and Glüer, 2011). In 2007, the International Society for Clinical Densitometry officially endorsed quantitative ultrasound in the management of osteoporosis (Krieg et al., 2008).

Cancellous bone consists of a porous, mineralized trabecular network filled with fluid (marrow in vivo or water in vitro). The propagation of two longitudinal waves in poro-elastic media is predicted by Biot theory (Biot 1956a, 1956b, 1956c, 1962, 1963), which has been reviewed by Haire and Langton (1999). After early applications of Biot theory to cancellous bone (McKelvie and Palmer 1991; Williams 1992), the presence of two waves was confirmed experimentally by Hosokawa and Otani (1997, 1998), Kaczmarek et al. (2002), and Cardoso et al. (2003). Theoretical investigations of Biot theory in the context of cancellous bone have been reported by Cowin (1999), Pakula and Kubik, (2002), Lee et al. (2003, 2006), Hughes et al. (2003, 2007), Fellah et al. (2004, 2008), Sebaa et al., (2006, 2008), Aygun et al. (2009), Cardoso and Cowin (2011), Cowin and Cardoso (2011), and Buchanan et al. (2012). Additional experimental confirmations of Biot theory and two-wave phenomena in cancellous bone have been reported by Mohamed et al. (2003), Wear et al. (2005), Pakula et al. (2008), Nagatani et al. (2008, 2009), Cardoso et al. (2008), Mizuno et al., (2009, 2010); and Yamamoto et al. (2009). Numerical applications in cancellous bone have been reported by Hosokawa (2005, 2008) and Nagatani et al. (2008, 2009). In Biot theory, two longitudinal waves correspond to the fluid and solid moving in-phase (fast wave) and out-of-phase (slow wave) with each other. Although many experiments are conducted in purely cancellous bone, two longitudinal waves may be detected even when cortical plates surround the cancellous bone (Mizuno et al., 2011). Additional information may be found by investigating refraction at oblique orientation to trabecular network (Hosokawa, 2011) and anisotropy of fast and slow waves (Yamashita et al., 2012). Fast and slow waves may also be observed in stereo-lithographical bone replicas (Aygun et al., 2010) and water-saturated aluminum foams (Zhang et al., 2011). One clinical device evaluates separately propagation speed and amplitude of fast and slow waves at the distal radius (Breban et al., 2010).

Through-transmission ultrasound measurements on cancellous bone in vitro suggest that the fast and slow waves have different frequency-dependent attenuation coefficients (Hosokawa and Otani 1997; Kaczmarek et al. 2002; Cardoso et al. 2003). Fast wave velocity in bovine cancellous bone depends on structural anisotropy and is maximum when propagation is parallel to the predominant trabecular orientation (Mizuno et al., 2010). Theoretical analysis suggests that the relative amplitudes of fast and slow waves are sensitive to bone volume fraction, sample thickness, tortuosity, and viscous characteristic length (Fellah et al. 2004). Three-dimensional finite-difference time-domain simulations suggest that fast wave velocity is much more sensitive to bone volume fraction than slow wave velocity (Haïat et al. 2008; Hosokawa 2008) and that fast and slow wave attenuation coefficients exhibit markedly different dependencies on bone volume fraction (Hosokawa, 2008).

Separation of fast and slow waves can be challenging because the two waves often overlap due to similar propagation delays through bone samples (arising either from similar velocities or insufficient sample thickness or both). The fast/slow wave velocity difference is maximum when the ultrasound propagates parallel to the predominant trabecular orientation (Hosokawa and Otani 1997, 1998). The degree of structural anisotropy can influence separability (Haïat et al. 2008). A low correlation coefficient between transmission signal loss and frequency may indicate the simultaneous presence of multiple waves (Haïat et al. 2008).

A Bayesian method has been shown to be effective for separation of overlapping time-domain pulses for simulated signals mimicking measurements from cancellous bone (Marutyan et al. 2007). This method involves calculations using Markov chain Monte Carlo with simulated annealing. The Bayesian method has been shown to produce good agreement between simulations and data acquired from a human femur condyle specimen (Anderson et al., 2010). The Bayesian method has been applied to through-transmission measurements on bovine cancellous femur at 1 MHz reported by Nagatani et al. (2008), and it produced consistent estimates of attenuation coefficients and phase velocities for overlapping fast and slow waves for measures obtained for bone thicknesses ranging from 6 to 15 mm (Nelson et al., 2011). The Bayesian method has also been shown to produce estimates of attenuation coefficients and phase velocities for fast and slow waves that correlate with porosity in human calcaneus (Hoffman et al, 2012).

The space alternating generalized expectation maximization algorithm (SAGE) has been applied to simulated data based on measurements from proximal femur and has been shown to provide rapid and accurate estimates of attenuation coefficients and phase velocities for overlapping fast and slow waves (Dencks et al., 2009). Coded excitation has been shown to enhance fast and slow wave identification in human calcaneus (Lashkari et al., 2012).

The Modified Least-Squares Prony’s (MLSP) method has been shown to be effective for estimation of attenuation coeficients and phase velocities of fast and slow waves, but can overestimate phase velocities by up to about 4% (fast wave) and 1% (slow wave) in data from phantoms and simulations (Wear, 2010a). Filtering of data with a chirp filter prior to application of the MLSP method, however, can greatly improve accuracy of phase velocity estimates (Wear, 2011b). Like the Bayesian approach, the MLSP method models the ratio of through-transmission spectra with and without cancellous bone in a water path (i.e., the transmission coefficient) as a sum of exponentially-damped sinusoids and recovers amplitude, attenuation coefficient, and phase velocity for each component.

The present investigation extends the MLSP approach by taking parameter estimates generated from the MLSP method and using them as initial guesses in a curve-fitting (CF) algorithm. The algorithm searches parameter space for the parameter combination that minimizes the root-mean-square (RMS) error between the data (i.e., measured or simulated transmission coefficient) and a parametric model function. The MLSP+CF method is tested using simulations based on several sets of parameter values for cancellous bone reported in the literature. The simulation is conducted over a range of signal-to-noise ratios (SNRs).

II. METHODS

A. Two-wave model

The model used here is adapted from a previous model (Marutyan et al., 2006) for composite media such as cancellous bone that exhibit two waves propagating simultaneously through a linear-with-frequency attenuating medium. The model assumes two co-axial transducers in a through-transmission or “pitch/catch” orientation. If X(ω) is the spectrum of the signal passing through a water-only path and Y(ω) is the spectrum of the signal passing through a water-sample-water path, then,

$$Y(\omega )=X(\omega )\left[H_{\mathit{\text{fast}}}(\omega )+H_{\mathit{\text{slow}}}(\omega )\right]$$ (1)

where

$$H_j(\omega )=A_j\text{exp}[-{\alpha }_j(\omega )d]\text{exp}\left[i\omega \frac{d}{v_j(\omega )}\right]\text{exp}\left[-i\omega \frac{d}{c}\right],$$ (2)

ω = 2πf	and f is the ultrasound frequency,
Aj	includes the effects of transmission through boundaries,
αj(ω) =	attenuation coefficient = βj ω / 2π,
βj =	attenuation coefficient slope
vj(ω) =	phase velocity,
d =	sample thickness,
c =	speed of sound in water,

and j stands for either fast or slow. The factor in brackets in Equation (1) is sometimes referred to as the transmission coefficient.

This expression includes an exponential factor, exp[-i ωd/c], to explicitly account for the fact that, in a substitution experiment, the attenuating sample replaces an equivalent length of water in the acoustic beam path. Note that Marutyan et al. (2006) used H_j(ω) in a slightly different sense—to describe the effect of sample itself (not necessarily the spectral ratio from a substitution experiment) and therefore omitted this exponential factor. Equation 2 neglects attenuation due to water and diffraction effects, which can be significant when the thickness and sound speed contrast between the reference and experimental media are sufficiently great (Kaufman et al., 1995). The two-wave model can predict negative dispersion (Anderson et al., 2008), which has been reported in human cancellous bone (Nicholson et al., 1996; Stelitzki et al., 1996; Droin et al., 1998; Wear, 2000a). In the event that the fast wave attenuation coefficient is a function of propagation distance, as has been suggested by a numerical and experimental study of wave attenuation in cancellous bone (Nagatani, et al., 2008), the fast wave attenuation coefficient in Equations 1 and 2, α_fast(ω), could represent a spatially-averaged effective fast wave attenuation coefficient.

The nearly local form of the Kramers-Kronig relations for media with linear-with-frequency attenuation predicts the following dispersion relation (O’Donnell et al., 1981, Waters et al., 2000, Waters et al., 2005; Anderson et al., 2008)

$$v_j(\omega )=v_j({\omega }_0)+v_j{({\omega }_0)}^2\frac{{\beta }_j}{{\pi }^2}\text{ln}\left(\frac{\omega }{{\omega }_0}\right)$$ (3)

This expression is valid for small dispersion. Models based on fast and slow waves obeying this dispersion relation have been shown to be consistent with measurements of frequency-dependent phase velocity in cancellous bone (Marutyan, et al., 2006; Anderson et al., 2008).

B. Prony’s Method and Variants

Prony’s method and its variants model a digitized complex signal, x[1], x[2], x[3], …, x[N], as the sum of p exponentially-damped sinusoids as follows (Marple, 1987):

$$x[n]=\sum _{j=1}^p{A}_j\text{ exp}[(s_j+i2\pi q_j)(n-1)\Delta \omega +i{\theta }_j]$$ (4)

where A_j is an amplitude, s_j is a damping rate, q_j is an oscillation rate, and θ_j is an initial phase of the j^th complex exponential. Δω is the sample interval. In Equation 3, Marple’s notation has been modified in order to reduce confusion that may arise from the fact that while the most common application of Prony’s method and its variants is modeling a time domain signal, the present application is modeling a frequency domain signal.

By comparing Equation 4 with Equations 1 and 2, the following correspondences can be made:

• p = 2,

• ω = (n-1) Δω,

• s_j = − β_j d / 2π,

• $q_j=\frac{d}{2\pi }\left[\frac{1}{v_j({\omega }_0)}-\frac{1}{c}\right],$ and

• θ_j = 0 (with no loss in generality if A_j are allowed to be complex).

In Equation 4, the frequency-dependence of phase velocities, v_j(ω), has been ignored so that v_j(ω) ≈ v_j(ω₀) where ω₀ = 2πf₀ and f₀ is a reference frequency such as the transducer center frequency. However, as mentioned in the previous section, wave velocities will be expected to exhibit dispersion (O’Donnell et al., 1981; Anderson et al., 2008). Prony’s method may be extended to dispersive signals through the application of chirp filters (Wear, 2010b) or through the use of curve fitting as will be explained in the remainder of this paper.

Prony’s method and its variants have been described elsewhere (Marple, 1987). Prony’s original method was designed for cases when the number of data points equals the number of parameters to be estimated. In this case, an exact fit of the model to the data may be performed. When the number of data points exceeds the number of parameters to be estimated, the MLSP method may be used to generate an approximate fit of the model to the data. Prony’s method and its variants consist of three steps. First, the data are fit to a linear prediction model. Then, estimates for damping rates and oscillation rates are obtained from the roots of a polynomial formed from the linear prediction coefficients obtained in step 1. Finally, estimates for amplitudes and initial phases are obtained from the solution of a set of linear equations. For more detail, see (Marple 1987).

C. Simulated Data

A simulation was performed to investigate the dependencies of the MLSP and MLSP+CF estimates of six parameter values—A_fast, A_slow, β_fast, β_slow, v_fast(ω₀), and v_slow(ω₀)—on signal-to-noise ratio (SNR) using parameters reported for cancellous bone shown in Table 1. Noise-free transmission coefficients—H_total(ω)=H_fast(ω)+H_slow(ω)—were generated using Equations 1 and 2. X(ω) was a Gaussian function, exp[-(f-f₀)² / 2σ_f²] with center frequency f₀ shown in Table I and σ_f = f₀ / 4. Noise-free spectra, Y(ω), were obtained using Equation 1. Time-domain radio-frequency (RF) waveforms were obtained by taking inverse Fast Fourier Transforms (FFTs) of the noise-free spectra. Gaussian white noise was added to time-domain RF waveforms to generate signals with SNR’s ranging from 30 to 50 dB. SNR was defined as the ratio of the maximum time-domain pulse amplitude to the root-mean-square (RMS) time-domain amplitude of the Gaussian white noise. After adding noise, time-domain signals were transformed back to frequency domain by applying FFT in order to obtain noisy transmission coefficients. The MLSP and MLSP+CF methods were applied to noisy, frequency-domain transmission coefficients. 100 trials with different random noise realizations were generated for each value of SNR so that means and standard deviations of all six parameter estimates could be obtained.

Table I.

Parameters used for fast and slow wave amplitudes, attenuation coefficients, and velocities. Parameter values reported by Nelson et al. (2011) correspond to data from Nagatani et al. (2008). Parameter values for human calcaneus (right column) correspond to mean values from Hoffman et al. (2012), in which numerical values for mean v_fast and v_slow were reported numerically and estimates for mean values for A_fast, A_slow, β_fast, and β_slow may be estimated from Figures 6 and 5 (Hoffman et al., 2012).

Reference	Nelson et al., 2011, Nagatani et al., 2008	Anderson et al., 2010	Hoffman et al., 2012
Source	Bovine femur	Human femur	Human calcaneus
Center frequency f0 (MHz)	1.0	0.5	0.5
Thickness (mm)	6,7,8,…,15	16.8	11.9
Afast	1.00	0.82	0.3
Aslow	0.13	0.23	0.6
βfast (dB/cmMHz)	49.2	42.8	4
βslow (dB/cmMHz)	7.1	5.2	4
vfast (ω0) (m/s)	1933	2036	1675
vslow (ω0) (m/s)	1475	1511	1527
Separation (theory)	0.38 – 0.95	0.56	0.13
Separation (numerical)	0.45 – 0.89	0.65	0.12
Frequency Analysis Range (MHz) (flow − fhigh)	0.25 – 1.6	0.2 – 0.8	0.2 – 0.8

The difficulty of measurement of fast and slow waves is influenced by 1) the degree of separation between the fast and slow waves, and by 2) noise. The degree of separation, s, may be characterized by the dimensionless ratio of the time lag between the fast and slow wave envelope maxima, d(1/v_slow – 1/v_fast), and an index of the fast and slow pulse durations. An approximate formula for the pulse durations may be obtained under the assumption of no dispersion. In this case, the time-domain envelope is Gaussian and is proportional to exp[-(t-t₀)² / 2σ_t²] where σ_t = 1 / (2πσ_f) (Wear, 2000b). An approximate measure of the pulse duration is 4σ_t, as 95% of the area under a Gaussian function with mean μ and standard deviation σ is contained within μ ± 2 σ (Walpole and Myers, 1978). Therefore, an approximate separation index may be given by

$$s=\frac{\pi {\sigma }_{f}d}{2}\left(\frac{1}{v_{\mathit{\text{slow}}}}-\frac{1}{v_{\mathit{\text{fast}}}}\right)$$ (5)

The separation was evaluated two ways for each simulation: 1) calculation using Equation (5), which ignores dispersion, and 2) computation from Gaussian fits to envelopes of simulated dispersive fast and slow waves and using the average of fast and slow pulse durations. Since the fast wave is typically smaller than the slow wave for bone applications, the task of measuring the fast wave is typically more difficult than the task of measuring the slow wave. Therefore, the fast wave SNR, SNR_fast, was often a more meaningful indicator of task difficulty than SNR. SNR_fast was defined as the ratio of the maximum time-domain fast wave amplitude to the root-mean-square (RMS) time-domain amplitude of the Gaussian white noise.

D. Data Analysis

The data analysis consisted of three stages. In the first stage of analysis, the Modified Least Squares Prony (MLSP) method was applied to noisy transmission coefficients to derive quick initial guesses for parameters to be used in subsequent curve-fitting. The routine mprony.m from StatBox 4.2 (StatSci.org, 2010) was used to perform MLSP method computations in Matlab (Mathworks, Natick, MA). (Note: The adjective “modified” has been used in conjunction with Prony’s method in multiple ways. Here the term is used in the sense of Osborne and Smyth (1995).) Model orders for values of p = 2, 3, 4, 5, and 6 were applied to each waveform. Previous investigators have shown that in the presence of noise, biases in parameter estimates may be reduced by selecting model orders higher than the number of exponentials actually in the signal (Van Blaricum and Mittra, 1975; Kumaresan and Tufts, 1982; Marple, 1987). This is reasonable because the model should be able to accommodate not only the energy in the signal but also the energy in the noise as well. For p > 2, the two waves with maximum total energy were designated as the two signal waves. The wave with the faster velocity was designated as the fast wave, H_fast(ω), and the wave with the slower velocity was designated as the slow wave, H_slow(ω). The model estimate was the sum of the two highest energy waves, H_fast(ω) + H_slow(ω). The order (p) that produced the minimum final prediction error (FPE) between the model and the simulated H(ω) was regarded as the optimum order, where

$$FPE={\rho }_p\left(\frac{N+(p+1)}{N-(p+1)}\right),$$ (6)

ρ_p is the prediction error power, and N is the number of samples in the estimate for H(ω) (Marple, 1987).

Initial estimates of the six wave parameters—A_fast, A_slow, β_fast, β_slow, v_fast(ω₀), and v_slow(ω₀)—were obtained by applying the MLSP method directly to simulated noisy transmission coefficients. In order to reduce variance of MLSP-based estimates, the MLSP method was performed 9 times with 9 slightly different frequency ranges, and the medians of the 9 parameter estimates were taken. The mean lower and upper limits of the frequency ranges (f_low and f_high) depended on center frequency (f₀) and are shown in Table I. Three lower limits were used: f_low – δf, f_low, and f_low + δf, where δf = 10 kHz. Three upper limits were used: f_high – δf, f_high, and f_low + δf. This led to 3 × 3 = 9 possible combinations.

In the second stage of analysis, six-parameter space was searched over a region surrounding the initial MLSP estimates for the combination that minimized the frequency integral (over the range from f_low to f_high) of the root-mean-square (RMS) difference between the model function computed from the six parameters—H_total(ω)=H_fast(ω)+H_slow(ω)—and the simulated noisy frequency-dependent transmission coefficient.

Searching six-parameter space could require a lot of computation time. However, if two parameters are highly interdependent, then once one parameter estimate is chosen, the search range for plausible values for the other may be narrowed and therefore the search may be accelerated. In order to investigate the possibility of accelerating the six-parameter search, two postulates were made regarding the application of MLSP algorithm to the simulated data generated as described above. First, it was postulated that high correlations might occur between MLSP-based estimates of A_fast and β_fast and between MLSP-based estimates of A_slow and β_slow. This is because (from Equation 2) overestimates of A_j would be expected to be accompanied by overestimates of β_j while underestimates of A_j would be expected to be accompanied by underestimates of β_j. Second, it was postulated that the MLSP algorithm would produce pairs of estimates—${\widehat{A}}_j$, ${\widehat{\beta }}_j$—that would be consistent in the sense that their combination would produce a magnitude of the estimate of H_j(2πf_cj) that would be roughly comparable with the true value of the magnitude of H_j(2πf_cj) at the pulse center frequency, f = f_cj. Thus,

$${\widehat{A}}_j\text{ exp}(-{\widehat{\beta }}_j{f}_{cj}d)\approx A_j\text{ exp}(-{\beta }_j{f}_{cj}d)$$ (7)

where ${\widehat{A}}_j$ is the MLSP estimate of A_j, and ${\widehat{\beta }}_j$ is the MLSP estimate of β_j. An approximate formula for f_cj may be obtained under the assumption of no dispersion. For a nondispersive Gaussian-shaped pulse—with spectrum proportional to $\text{exp}[-{(f-f_{cj})}^2/2{\sigma }_f^2]$—that propagates through a medium with linear-frequency-dependent attenuation, the center frequency is equal to the difference between the center frequency of the input pulse, f₀, and the downshift due to linear-frequency-dependent attenuation (Narayana and Ophir, 1983)

$$f_{cj}=f_0-{\sigma }_f^2{\beta }_{j}d$$ (8)

Note that the bandwidth of a nondispersive Gaussian pulse (which is proportional to σ_f) remains unchanged as it propagates through a linearly-attenuating medium (Narayana and Ophir, 1983). The Gaussian shape is an excellent approximation for broadband pulses used in many ultrasound applications (see, for example, Wear, 2000b, Figure 3). The rate of change of ${\widehat{\beta }}_j$ with respect to ${\widehat{A}}_j$ may be obtained by taking the logarithm of Equation 7 and then differentiating with respect to ${\widehat{A}}_j$ (assuming that the right hand side is a constant). The result is

$$\frac{\partial {\widehat{\beta }}_j}{\partial {\widehat{A}}_j}\approx \frac{1}{{\widehat{A}}_j{f}_{cj}d}.$$ (9)

Figure 3.

(color online) Gaussian fits to distributions of parameter estimates for all three stages of the algorithm—Stage 1 (MLSP), Stage 2 (MLSP + 4D search), and Stage 3 (MLSP + 4D search + sequential high resolution 1D searches)—on simulated data for parameter values from Nelson et al. (2011) with bone sample thickness = 6 mm and SNR = 40 dB. Bias and variance of the estimates are reduced as the algorithm progresses from Stage 1 to Stages 2 and 3. As has been observed previously (Wear, 2010), the MLSP method (Stage 1) tends to overestimate the fast velocity (see lower left panel of Figure 5). However, this initial bias is reduced dramatically in Stages 2 and 3.

Under these assumptions, the search through six-parameter space—A_fast, A_slow, β_fast, β_slow, v_fast(ω₀), and v_slow(ω₀)—could be reduced to a four-dimensional search, resulting in an improvement in speed. The two velocities were free parameters in the search. A_j, β_j spaces for j=1, 2 were searched along contours given by Equation 7 in which the left side was computed using the estimates from the MLSP method. Therefore, the second stage of analysis was effectively a four-dimensional search.

Finally, after the four-dimensional search (Stage 2), parameter estimates could be refined by varying one parameter at a time (while holding the other five parameters fixed) and searching at high resolution to find the value that minimized the frequency integral (over the range from f_low to f_high) of the root-mean-square (RMS) difference between the model function computed from the six parameters—H_total(ω)=H_fast(ω)+H_slow(ω)—and the simulated noisy frequency-dependent transmission coefficient (Stage 3). For each single-parameter search, the new estimate was determined from a parabolic fit in the region of minimum RMS difference (Bevington and Robinson, 1992). Since the estimate of each parameter depended on the estimates of the remaining five parameters, the sequence of six single-parameter searches was repeated until both velocities stabilized to within 0.05 m/s, both attenuations stabilized to within 0.005 dB/cmMHz, and both magnitudes stabilized to within 0.0005.

Because the variances of MLSP estimates increased as SNR decreased, the search ranges for the 4D search (Stage 2) were expanded as SNR decreased. Table II shows 4D search ranges and 4D search step sizes for Stage 2 analysis of data generated using parameters from studies in which trabecular alignment was approximately parallel or mixed relative to the ultrasound propagation direction (Nelson et al., 2011; Anderson et al., 2010). Table III shows 4D search ranges and 4D search step sizes for Stage 2 analysis of data generated using parameters from studies in which trabecular alignment was approximately perpendicular to the ultrasound propagation direction (Hoffman et al., 2012). The difference between fast and slow velocity was relatively small for perpendicular alignment. Therefore, pulse separation was also relatively small, 0.12 (see Table I). This led to increased variances in MLSP estimates and therefore required larger search ranges for the 4D search.

Table II.

4D search ranges, 4D search step sizes for Stage 2 analysis of data generated using parameters from studies in which trabecular alignment was approximately parallel or mixed relative to the ultrasound propagation direction (Nelson et al., 2011; Anderson et al., 2010).

SNR (dB)	Parameter	4D Search Range	4D Search Step
30	Afast		0.10
	Aslow		0.02
	βfast (dB/cmMHz)	βMLSP,fast – 30 → βMLSP,fast + 45	2.0
	βslow (dB/cmMHz)	βMLSP,slow – 50 → βMLSP,slow + 10	2.0
	vfast(f0) (m/s)	vMLSP, fast – 500 → vMLSP, fast + 150	25
	vslow(f0) (m/s)	vMLSP, slow – 140 → vMLSP, slow + 80	25
35	Afast		0.10
	Aslow		0.02
	βfast (dB/cmMHz)	βMLSP,fast – 30 → βMLSP,fast + 45	2.0
	βslow (dB/cmMHz)	βMLSP,slow – 15 → βMLSP,slow + 8	2.0
	vfast(f0) (m/s)	vMLSP, fast – 500 → vMLSP, fast + 150	25
	vslow(f0) (m/s)	vMLSP, slow – 30 → vMLSP, slow + 80	25
40	Afast		0.10
	Aslow		0.02
	βfast (dB/cmMHz)	βMLSP,fast – 25 → βMLSP,fast + 30	2.0
	βslow (dB/cmMHz)	βMLSP,slow – 12 → βMLSP,slow + 8	2.0
	vfast(f0) (m/s)	vMLSP, fast – 200 → vMLSP, fast + 100	25
	vslow(f0) (m/s)	vMLSP, slow – 30 → vMLSP, slow + 40	25
45	Afast		0.10
	Aslow		0.02
	βfast (dB/cmMHz)	βMLSP,fast – 20 → βMLSP,fast + 10	2.0
	βslow (dB/cmMHz)	βMLSP,slow – 12 → βMLSP,slow + 7	2.0
	vfast(f0) (m/s)	vMLSP, fast – 180 → vMLSP, fast + 70	25
	vslow(f0) (m/s)	vMLSP, slow – 25 → vMLSP, slow + 30	25
50	Afast		0.10
	Aslow		0.02
	βfast (dB/cmMHz)	βMLSP,fast – 15 → βMLSP,fast + 10	2.0
	βslow (dB/cmMHz)	βMLSP,slow – 12 → βMLSP,slow + 4	2.0
	vfast(f0) (m/s)	vMLSP, fast – 150 → vMLSP, fast + 30	25
	vslow(f0) (m/s)	vMLSP, slow – 25 → vMLSP, slow + 30	25

Table III.

4D search ranges, 4D search step sizes for Stage 2 analysis of data generated using parameters from studies in which trabecular alignment was approximately perpendicular to the ultrasound propagation direction (Hoffman et al., 2011).

SNR (dB)	Parameter	4D Search Range	4D Search Step
40	Afast		0.10
	Aslow		0.02
	βfast (dB/cmMHz)	βMLSP,fast – 40 → βMLSP,fast + 10	2.0
	βslow (dB/cmMHz)	βMLSP,slow – 20 → βMLSP,slow + 10	2.0
	vfast(f0) (m/s)	vMLSP, fast – 500 → vMLSP, fast + 200	25
	vslow(f0) (m/s)	vMLSP, slow – 100 → vMLSP, slow + 200	25
45	Afast		0.10
	Aslow		0.02
	βfast (dB/cmMHz)	βMLSP,fast – 35 → βMLSP,fast + 10	2.0
	βslow (dB/cmMHz)	βMLSP,slow – 20 → βMLSP,slow + 10	2.0
	vfast(f0) (m/s)	vMLSP, fast – 400 → vMLSP, fast + 175	25
	vslow(f0) (m/s)	vMLSP, slow – 75 → vMLSP, slow + 175	25
50	Afast		0.10
	Aslow		0.02
	βfast (dB/cmMHz)	βMLSP,fast – 30 → βMLSP,fast + 10	2.0
	βslow (dB/cmMHz)	βMLSP,slow – 20 → βMLSP,slow + 10	2.0
	vfast(f0) (m/s)	vMLSP, fast – 350 → vMLSP, fast + 150	25
	vslow(f0) (m/s)	vMLSP, slow – 50 → vMLSP, slow + 150	25

The step sizes for the sequential high-resolution one-dimensional searches (Stage 3) were equal to the 4D search step sizes divided by 5 (parallel/mixed) or 10 (perpendicular). The search ranges for the sequential high-resolution one-dimensional searches extended ± 10 steps about the estimates produced by the 4D search. The final estimates of the sequential high-resolution one-dimensional searches were obtained by parabolic fits to the 5 points with minimum RMS difference between the parametric model—H_total(ω)=H_fast(ω)+H_slow(ω)—and the simulated noisy frequency-dependent transmission coefficient data.

Search spaces were also limited by knowledge of possible parameter values from the literature. These limits were 0 – 75 dB/cmMHz (fast and slow wave attenuation slopes), 1400 – 3000 m/s (fast wave velocity at the center frequency) and 1400 – 1600 m/s (slow wave velocity at the center frequency). For the study in which trabecular alignment was perpendicular to the ultrasound propagation direction (Hoffman et al., 2012), the upper limit of the fast wave velocity search space was further restricted to v_slow + 500 m/s. This is reasonable because the difference between fast and slow wave velocities is known to decrease as the trabecular alignment approaches perpendicular orientation with respect to the ultrasound propagation direction. Restricting searches to these ranges accelerated the computation without sacrificing accuracy. Similar restriction of parameter values has been employed by other investigators (Hoffman et al., 2012).

Data analysis was performed using a Dell notebook computer with Intel® Core^™ i5–2520M CPU @ 2.5 GHz with 4 GB RAM.

III. RESULTS

Table 1 shows theoretical (Equation 5) and numerical estimates for the separation parameter for the fast and slow wave parameters considered. Equation (5), which neglects dispersion, was accurate to within 16% for the problems considered. The range of separations considered, 0.12 – 0.89, shows substantial overlap between the two waves, which indicates the difficulty of the parameter estimation problem.

Figure 1 shows a scatter plot of ${\widehat{A}}_{\mathit{\text{fast}}}$, ${\widehat{A}}_{\mathit{\text{slow}}}$, ${\widehat{\beta }}_{\mathit{\text{fast}}}$ and ${\widehat{\beta }}_{\mathit{\text{slow}}}$ generated by the MLSP algorithm on simulated data for parameter values from Nelson et al. (2011) and Nagatani et al. (2008) with bone sample thickness = 6 mm and SNR = 40 dB. For reference, a measurement of mean SNR from a previous investigation involving through-transmission measurements on 43 human calcaneus samples in vitro (Wear and Laib, 2003) is 40.03 dB. Linear fits to the fast and slow wave data are also shown. As postulated, ${\widehat{A}}_{\mathit{\text{fast}}}$ and ${\widehat{\beta }}_{\mathit{\text{fast}}}$ are highly correlated, and ${\widehat{A}}_{\mathit{\text{slow}}}$ and ${\widehat{\beta }}_{\mathit{\text{slow}}}$ are highly correlated. Furthermore, the linear fits pass through the true values for the parameters (indicated by the large + signs). Therefore, although there is considerable scatter in the MLSP-based estimates, parameter pairs appear to be roughly consistent with Equation 7. The slopes of the linear fits in Figure 1, 25.7 and 103.3 dB/cmMHz per unit amplitude for the fast and slow waves respectively, are comparable to theoretical predictions from Equation 9, 18.4 and 114.8 dB/cmMHz per unit amplitude.

Figure 1.

(color online) Scatter plot of initial guesses for A_fast, A_slow, β_fast and β_slow generated by the MLSP algorithm for parameter values from Nelson et al. (2011) using a bone sample thickness = 6 mm for 100 trials and SNR = 40 dB. True parameter values for the fast and slow waves are shown by the large + signs. Linear fits to the fast and slow wave data are also shown.

Figure 2 illustrates one plane from the four-dimensional search on simulated data for parameter values from Nelson et al. (2011) with bone sample thickness = 6 mm and SNR = 40 dB. The x indicates the true values for A₂ (0.13) and β₂, (7.1 dB/cmMHz) where 2 corresponds to the slow wave. The circle (○) shows the initial estimate provided by the MLSP algorithm. The curve shows the search contour for which A₂exp(−β₂ f_c2d) = constant. The box (□) shows the point in along the search contour with the minimum RMS difference between the noisy simulated frequency-dependent transmission coefficient data and the dispersive parametric model fit. This point (□) is closer to the true values (x) than the initial MLSP estimate (○).

Figure 2.

Search in the β₂, A₂ plane for the values yielding the minimum mean square error between the simulated data and the dispersive parametric model fit for parameter values from Nelson et al. (2011) with bone sample thickness = 6 mm and SNR = 40 dB. The x indicates the true values for the slow wave magnitude and attenuation slope. The circle (○) shows the initial estimate provided by the MLSP algorithm. The curve shows the search contour for which A₂exp(−β₂ f_c2d) = constant. The box (□) shows the point in along the search contour with the minimum RMS difference between the noisy simulated frequency-dependent transmission coefficient data and the dispersive parametric model fit. This point (□) is closer to the true values (x) than the initial MLSP estimate (○).

Figure 3 shows Gaussian fits to distributions of parameter estimates (obtained from 100 trials with 100 different Gaussian noise realizations) for all three stages of the algorithm—Stage 1 (MLSP), Stage 2 (MLSP + 4D search), and Stage 3 (MLSP + 4D search + sequential high resolution 1D searches)—on simulated noisy frequency-dependent transmission coefficient data for parameter values from Nelson et al. (2011) with bone sample thickness = 6 mm and SNR = 40 dB. The biases and variances of estimates are reduced as the algorithm progresses from Stage 1 to Stage 2 to Stage 3. As has been observed previously (Wear, 2010a; Wear, 2010b), the MLSP method (Stage 1) tends to overestimate the fast velocity (see lower left panel of Figure 3). This bias may be due to the fact that the MLSP method ignores dispersion of the fast wave (Wear, 2010a; Wear, 2010b). However, the initial bias is reduced dramatically in Stages 2 and 3.

Figure 4 shows performance of the MLSP+CF algorithm for estimating fast wave (left column) and slow wave (right column) parameters on simulated noisy frequency-dependent transmission coefficient data for parameter values from Nagatani et al. (2008) and Nelson et al. (2011) for bone thicknesses of 6 mm and 15 mm. The horizontal dotted lines show the true values for the parameters. For SNR greater than or equal to 40 dB, mean values of estimated parameters were close to true values. Recall that a measurement of mean SNR from a previous investigation involving through-transmission measurements on 43 human calcaneus samples in vitro (Wear and Laib, 2003) is 40.03 dB. Table IV gives means, standard deviations, and root-mean-square errors (RMSEs) for SNR = 40 dB. SNR_fast in the simulation was computed to be SNR – 4.9 dB (thickness = 6 mm) and SNR – 27.3 dB (thickness = 15 mm). The fact that SNR_fast at 15 mm was smaller by about 22.4 dB than SNR_fast at 6 mm helps to explain the reduction in performance of the MLSP+CF method for thickness = 15 mm for SNR < 40 dB in Figure 4.

Figure 4.

Performance of the complete algorithm for estimating fast wave (left) and slow wave (right) parameters on simulated data for parameter values from Nelson et al. (2011) for bone thicknesses of 6 mm and 15 mm. The horizontal dotted lines show the true values.

Table IV.

Means, standard deviations, and root-mean-square errors (RMSE) for parameter estimates from simulations with SNR = 40 dB based on parameter values reported by Nelson et al. (2011), which correspond to data from Nagatani et al. from bovine femur (2008).

		Mean ± Std. Deviation (RMSE)	Mean ± Std. Deviation (RMSE)
Frequency (MHz)	1.0
Frequency Analysis Range (MHz) (flow − fhigh)	0.25 – 1.6
Separation (theory)		0.38	0.95
Separation (numerical)		0.45	0.89
Thickness (mm)		6	15
Afast	1.00	1.00 ± 0.07 (0.07)	1.01 ± 0.19 (0.19)
Aslow	0.13	0.13 ± 0.01 (0.01)	0.13 ± 0.01 (0.01)
βfast (dB/cmMHz)	49.2	49.0 ± 1.6 (1.6)	48.7 ± 3.0 (3.0)
βslow (dB/cmMHz)	7.1	7.3 ± 0.6 (0.6)	7.2 ± 0.3 (0.4)
vfast (ω0) (m/s)	1933	1933 ± 5 (5)	1930 ± 16 (16)
vslow (ω0) (m/s)	1475	1475 ± 1 (1)	1475 ± 1 (1)

Figure 5 shows performance of the MLSP+CF algorithm for estimating fast wave (left column) and slow wave (right column) parameters on simulated data for parameter values from Anderson et al. (2010). The horizontal dotted lines show the true values. The mean values of estimated parameters were close to the true values. Table V gives mean, standard deviations, and root-mean-square errors (RMSE) for SNR = 40 dB. SNR_fast was computed to be SNR – 17.7 dB.

Figure 5.

Performance of the complete algorithm for estimating fast wave (left) and slow wave (right) parameters on simulated data for parameter values from Anderson et al. (2010). The horizontal dotted lines show the true values.

Table V.

Means, standard deviations, and root-mean-square errors (RMSE) for parameter estimates from simulations with SNR = 40 dB based on parameter values reported by Anderson et al. (2010) for data on human femur.

		Mean ± Std. Deviation (RMSE)
Frequency (MHz)	0.5
Frequency Analysis Range (MHz) (flow − fhigh)	0.2 – 0.8
Separation (theory)	0.56
Separation (numerical)	0.65
Thickness (mm)	16.8
Afast	0.82	0.85 ± 0.09 (0.09)
Aslow	0.23	0.24 ± 0.01 (0.01)
βfast (dB/cmMHz)	42.8	43.2 ± 1.6 (1.7)
βslow (dB/cmMHz)	5.2	5.5 ± 0.3 (0.4)
vfast (ω0) (m/s)	2036	2037 ± 4 (4)
vslow (ω0) (m/s)	1511	1511 ± 1 (1)

Figure 6 shows performance of the MLSP+CF algorithm for estimating fast wave (left column) and slow wave (right column) parameters on simulated data for parameter values from Hoffman et al. (2012). The horizontal dotted lines show the true values. The data generated using these parameters were particularly challenging because the separation between the fast and slow waves was only 0.12, as opposed to 0.45 – 0.89 for the data from Nagatani et al. (2008) and Nelson et al. (2011) or 0.65 for the data from Anderson et al. (2010) (See Table I). Consequently, the MLSP+CF did not perform as well at low values of SNR. The mean values of v_fast, v_slow and β_slow were close to the true values. However, β_fast was overestimated for SNR ranging from 40 to 50 dB. Variance for β_slow was high for SNR ≤ 42 dB but much improved for SNR ≥ 45 dB. SNR_fast was computed to be SNR – 2.4 dB.

Figure 6.

Performance of the complete algorithm for estimating fast wave (left) and slow wave (right) parameters on simulated data for parameter values from Hoffman et al. (2012). The horizontal dotted lines show the true values.

Figure 7 shows mean computation times for simulations based on parameters from Nagatani et al. (2008) and Nelson et al. (2011) (thicknesses of 6 and 15 mm) and Anderson et al. (2010) (thickness of 16.8 mm), which ranged from less than 4 sec at SNR = 30 dB to less than 2 sec at SNR ≥ 40 dB. For the simulations based on parameters from Hoffman et al. (2012), mean computation time ranged from 2.7 to 2.1 sec as SNR varied from 40 to 50 dB (not shown).

Figure 7.

Mean computation times for simulations based on parameters from Nagatani et al. (2008) and Nelson et al. (2011) (thicknesses of 6 and 15 mm) and Anderson et al. (2010) (thickness of 16.8 mm).

IV. DISCUSSION

Pulses transmitted through cancellous bone samples contain important information regarding material and structural properties. This information can be enhanced when pulses are decomposed into fast and slow wave components. This decomposition is often difficult, however, due to temporal overlap between the two components.

The MLSP method is an easy-to-implement, fast method for producing good initial guesses for curve-fitting algorithms and models for detecting fast and slow waves in experiments relevant to cancellous bone. The present paper has shown that augmenting the MLSP method with curve fitting (MLSP+CF) can significantly enhance accuracy and precision of estimates of fast and slow wave parameters, while still remaining fast (< 2 sec for SNR = 40 dB on a consumer-grade notebook computer). Performance has been evaluated over a broad range of SNRs using realistic fast and slow wave attenuation coefficients and phase velocities taken from literature on experimental measurements in cancellous bone samples from bovine femur, human femur, and human calcaneus. These problems are challenging, with fast and slow waves separated by only 0.12 – 0.89 pulse lengths. The MLSP+CF method was not directly applied to measurements reported by Lashkari et al. because these authors did not report values for β_fast and β_slow. However, values for fast and slow wave velocities for 15.7-mm-thick human calcaneus reported by Lashkari et al. (2058 m/s and 1477 m/s respectively) are similar to those reported by Anderson et al. (2010) (see Table I) and would seem therefore to be also within the capabilities of the MLSP+CF method.

The MLSP+CF method is flexible with respect to the functional form of the parametric model of the transmission coefficient and can accommodate dispersion of fast and slow waves. Although the initial MLSP stage (Stage 1) requires that a frequency-domain transmission coefficient model consisting of a sum of damped sinusoids is approximately correct, the second two stages (Stages 2 and 3) can be programmed using any user-specified parametric model that may deviate from the sum-of-damped-sinusoids model.

The MLSP+CF method provides sufficient accuracy and precision for most applications such that experimental error is likely to be a greater limiting factor than estimation error. Experimental errors in measurements on cancellous bone arise from many factors, including phase cancellation (Wear, 2007; Wear, 2008; Cheng et al., 2011), refraction artifacts, multipath interference, and errors in diffraction correction in substitution experiments due to mismatch in velocity between sample and water. This last effect, which has been studied extensively (Xu and Kaufman, 1993; Kaufman et al., 1995; Droin et al., 1998) is especially problematic for the fast wave, for which the mismatch can be approximately 500 m/s. For example, Droin et al. (1998) showed that between 200 and 800 kHz, a velocity mismatch of approximately 720 m/s can result in errors on the order of −0.4 dB/cm for attenuation slope and −20 m/s for phase velocity.

The uncertainties of attenuation slopes of the MLSP+CF method may be viewed in the context of the wide range of attenuation coefficient slopes reported for fast waves (bovine: 49 – 102 dB/cmMHz; human: 0 – 140 dB/cmMHz) and slow waves (bovine: 7 – 26 dB/cmMHz; human: 0 – 40 dB/cmMHz) in cancellous bone (Cardoso, et al., 2003; Nelson et al., 2011; Anderson et al., 2010; Hoffman et al., 2012). The accuracies of velocities may be viewed in the context of the wide range of phase velocities reported for fast waves (bovine: 1710 – 3500 m/s; human: 1480 – 2900 m/s) and slow waves (bovine: 1175 – 1480 m/s; human: 1210 m/s – 1485 m/s) in cancellous bone (Hosokawa and Otani, 1997; Hughes et al., 1999; Cardoso, et al., 2003; Mizuno et al., 2009; Nelson et al., 2011; Anderson et al., 2010; Hoffman et al., 2012). The uncertainties in Tables IV and V and Figures 4–6 are small compared with these wide ranges.

The MLSP+CF method produces excellent initial estimates that could be used as inputs to other algorithms, such as the Bayesian method (Marutyan et al., 2007) or the SAGE algorithm (Dencks and Schmitz, 2009), in order to accelerate their convergence to final estimates for fast and slow wave parameters.

For problems typical for cancellous bone, the MLSP+CF method tends to produce better estimates for the slow wave parameters than for the fast wave parameters. This is likely due to the fact that the slow wave typically is bigger than the fast wave. Since the MLSP and MLSP+CF methods perform functional fits in frequency domain, their performances tend to improve with bandwidth. The most time-consuming step in the MLSP+CF method is the 4D search. Greater accuracy and precision may be possible by making finer steps in the 4D search, but this would come at a cost of greater computation time.

V. CONCLUSION

The MLSP+CF method is a straightforward, fast method for estimating fast and slow wave parameters from ultrasound experiments on cancellous bone. The MLSP+CF method has been validated in simulations based on parameters using a wide range of values for wave parameters drawn from literature on ultrasound measurements on cancellous bone samples from bovine femur, human femur, and human calcaneus. The MLSP+CF method has been shown to be effective even when the fast and slow waves are separated by less than two tenths of one pulse duration.