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Estimation of fast and slow wave properties in cancellous bone using Prony's method and curve fitting
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10.1121/1.4792935
/content/asa/journal/jasa/133/4/10.1121/1.4792935
http://aip.metastore.ingenta.com/content/asa/journal/jasa/133/4/10.1121/1.4792935

Figures

Image of FIG. 1.
FIG. 1.

(Color online) Scatter plot of initial guesses for Af ast, 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.

Image of FIG. 2.
FIG. 2.

Search in the β2 , A 2 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 × 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 A 2 exp(−β2 fc2d) = const. 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 (×) than the initial MLSP estimate (○).

Image of FIG. 3.
FIG. 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). However, this initial bias is reduced dramatically in Stages 2 and 3.

Image of FIG. 4.
FIG. 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 and 15 mm. The horizontal dotted lines show the true values.

Image of FIG. 5.
FIG. 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.

Image of FIG. 6.
FIG. 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.

Image of FIG. 7.
FIG. 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).

Tables

Generic image for table
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 vf ast 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 Figs. 5 and 6 ( Hoffman et al., 2012 ).

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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 ).

Generic image for table
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., 2012 ).

Generic image for table
TABLE IV.

Means, standard deviations, and 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)

Generic image for table
TABLE V.

Means, standard deviations, and 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.

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/content/asa/journal/jasa/133/4/10.1121/1.4792935
2013-04-03
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Estimation of fast and slow wave properties in cancellous bone using Prony's method and curve fitting
http://aip.metastore.ingenta.com/content/asa/journal/jasa/133/4/10.1121/1.4792935
10.1121/1.4792935
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