Probing Long Bones with Ultrasonic Body Waves: Materials and Methods

Supporting Online Material for Probing Long Bones with Ultrasonic Body Waves: Materials and Methods Lawrence H. Le (黎仲勳)1,2,, Yu J. Gu (谷宇)2, Yuping...
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Probing Long Bones with Ultrasonic Body Waves: Materials and Methods Lawrence H. Le (黎仲勳)1,2,, Yu J. Gu (谷宇)2, Yuping Li (李玉平)3 and Chan Zhang (張嬋)1 1

Department of Radiology and Diagnostic Imaging, University of Alberta, Edmonton, AB, T6G2B7 Canada

2

Department of Physics, University of Alberta, Edmonton, AB, T6G2G7 Canada

3

Department of Dentistry, University of Alberta, Edmonton, AB, T6G2N8 Canada

* Corresponding author, email: [email protected]

Sample Preparation and Data Acquisition The bone sample examined in this study is a relatively well-preserved bovine tibia. The overlying tissue has been removed from the bone sample in an effort to maximize the quality of the ultrasound recordings. Despite the initial curvatures and complexities, the sample specimen after end removal and minor polishing can be effectively approximated by a marrow-filled cylindrical tubeS1. We attach two 1 MHz angle beam compressional wave transducers (Panametrics C548, Waltham, MA) to two screw-in wedges (Panametrics ABWM-7T-70 deg, Waltham, MA) to form an incident angle of 51o. The transducer-wedge couples are positioned linearly along the same side of the tibia. One acts as a transmitting source and the other as a receiver. Ultrasound gel is applied on all contacts to ensure proper coupling. The transmitter unit, which is placed at one end of the bone sample, is powered by a Panametrics 5800 P/R with a wide bandwidth (50 KHz-35 MHz). The signal detected by the receiver is digitized by and displayed on a 200 MHz digital storage oscilloscope (LeCroy 422 WaveSurfer, Chestnut Ridge, NY). To increase the signal-to-noise ratio, we store the digitized waveform, which is continuously averaged 128 times, in the internal memory. The signal is further decimated to 4000 samples with a sampling interval of 0.05 s. During the experiment, we position the initial receiver at 24 mm distance from the transmitter, which is the closest distance (or offset) between the source and receiver. After a stable ultrasound signal is recorded, we move the receiver by 1 mm away from the transmitter along the same 1

linear formation and register the second record. This process is repeated 59 more times to obtain a total of 60 ultrasound records; the final receiver is 83 mm from the source location. The recorded signals form a time-distance (x-t) matrix of amplitudes or an (x-t) echogram. Twelve of the sixty records are shown in FIG. S1. The records are not time-gaincompensated (TGC). The amplitude decreases with time and offset (distance) from the source. Two distinct wave groups are visible at small offsets.

FIG. S1. The original (x-t) echogram without TGC and self-normalization.

Reflectivity Method The methodS2,S3 calculates the surface response of stacked layers over a half-space based on a propagator matrix algorithm. The layers and the half-space are homogeneous and isotropic. The physical problem is treated as a boundary value problem in which the wave equation is solved in the Fourier-transformed domain, i.e., the frequency-wavenumber (-k) domain or its variant, the frequency-slowness (-p) domain. The solution corresponding to a monochromatic plane wave of angular frequency  with horizontal wavenumber k (k=p) satisfies all the boundary conditions, which consist of the continuity conditions at the velocity discontinuities of the medium, stress-free condition on the free surface, the source condition at the level of the source and radiation condition in the half-space. By superposition, the total solution can then be expressed as an improper double integral over  and k. The solution gives a complete response including all the possible reflections and transmissions at the

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interfaces within a -k window. The formalism allows a natural implementation of intrinsic attenuation by making the velocities of the media complex. Usually, the evaluation of the integrals can be performed by numerical integration of the k-integral and then a FFT inversion of the -integral.

Separation of Shear and Rayleigh Waves Rayleigh wave propagates along the free surface of the bovine tibia (a free surface is the surface of a body that is subject to neither normal stress nor tangential shear stress). The velocities of Rayleigh wave vR, compressional wave vp, and shear wave vs are related by the following cubic equationS4

  

 3  8 2  8  3 

 vs2 2vs2   16  1  2 v 2p   vp

   0 

(1)

where

v    R  vs 

2

(2)

.

Equation (1) will be solved numerically for vR. As shown in FIG. 3(b) (main text), the arrival times of the direct shear and Rayleigh waves nearly coincide and, as a consequence, the two waves overlap and display as a single event. In order to demonstrate the existence of these two wave types, we reduce the shear wave speed from 1970 m/s to 1000 m/s to increase the speed difference between the shear and Rayleigh waves. The simulated result based on the slower shear wave speed is given in FIG. S2. The difference in arrival times between the two waves is greatly enhanced and the two waves are easily distinguishable in time. This result confirms that the ultrasound moveout identified in FIG. 3(b) are the direct shear and Rayleigh waves. These waves differ from waveguidesS5 (which are trapped waves between bounded solids) since they are mainly associated with a single surface (air-cortical bone interface).

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FIG. S2. The compressional (P), shear (S), and Rayleigh waves. The shear wave speed is reduced to 1000 m/s.

Calculation of Arrival Times

The arrival time of each wave is calculated by the travelled distance divided by the wave speed. The ray path l travelling within a layer of thickness H is given by l

H cos  j

(3)

where j can be the incident (j = 1), reflected (j = 2) or refracted (j = 3) angle and these angles are related by the Snell’s law:

sin 1 sin  2 sin 3   v1 v2 v3

(4)

where v1, v2, and v3 represent the respective speeds (either compressional or shear) of the incident, reflected and refracted waves.

Supplemental References S1

D. Ta, W. Wang, Y. Wang, L. H. Le, Y. Zhou, Ultrasound Med. Biol. 35, 641-652 (2009).

S2

B. L. N. Kennett, Seismic wave propagation in stratified media (Cambridge Univ. Press, 1983). 4

S3

R. Kind, J. Geophys, 42, 191-200 (1976).

S4

L. D. Landau, E. M. Lifshitz, Theory of Elasticity (Pergamon Press, ed. 2, 1984), pp. 109113.

S5

N. R. Joshi, Materials Evaluation, 64, 337-341 (2006).

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