Propagation of time-reversed Lamb waves in bovine cortical bone in vitro

In recent years, a number of studies have used the axial transmission technique to characterize long cortical bones.^1–3 This method was initially developed in the 1950s to determine fracture healing by measuring the sound velocity across the fracture site in long bones.⁴ The axial transmission technique typically measures the velocity of the first arriving signal (FAS) along the axial direction of long bones (typically the tibia), using a transmitter and a receiver placed on the same side of the bone axis. When the cortical thickness is comparable with, or greater than, the wavelength, the FAS propagates at the velocity of the lateral longitudinal wave (or P-head wave) along the cortical surface, whereas when the thickness is less than the wavelength, the FAS velocity tends to decrease with decreasing thickness toward the velocity of the lowest order symmetrical (S0) Lamb wave guided by the cortical thickness.^5,6 It has been reported that the FAS velocity can characterize both material properties and thickness of cortical bones.⁷ It has also been found that the slow guided wave (SGW) arriving after the FAS can provide enhanced characterization of both material properties and thickness of cortical bones.⁸ In contrast to the FAS, the SGW has shown to essentially behave as the lowest order antisymmetrical (A0) Lamb wave independently of the cortical thickness-to-wavelength ratio with the velocity increasing with increasing thickness.⁹ 

Lamb waves are two-dimensional elastic waves that propagate in a free solid plate with finite thickness in a vacuum.¹⁰ They result from multiple reflections and mode conversions of the longitudinal and the shear waves from the upper and lower surfaces of the plate. Related technologies based on guided Lamb waves are widely used in the field of nondestructive testing for the assessment of plates, pipes, and more complex structures.¹¹ The use of Lamb waves in long cortical bones is also attractive because the entire thickness of cortical bones can be interrogated.¹² However, the dispersive nature of Lamb waves makes signal interpretation difficult and leads to signal-to-noise problems because the peak amplitude in the signal envelope rapidly decreases with the distance if the dispersion is strong. It has been demonstrated that a time reversal process of Lamb waves can automatically compensate for the dispersive nature of Lamb waves, leading to time recompression and self-focusing of Lamb waves.¹³ Recent studies have shown that the time-reversed Lamb waves generated by the time reversal process can be usefully applied to health monitoring of plate-like structures.^14,15 The present study aims to investigate the propagation of time-reversed Lamb waves in bovine cortical bone in vitro. The group velocities of the time-reversed Lamb waves in 18 bovine tibiae were measured using the axial transmission technique. Their correlation with the cortical thickness was also examined.

Eighteen fresh bovine tibiae were purchased from a local slaughter house. Their proximal ends were removed using a rotary electric saw to make hollow tube-shaped tibiae without any soft tissue and marrow. All the tibiae were defatted with water jetting and kept frozen at −16 °C before ultrasound measurements. The mean cortical thickness of each tibia was measured using calipers on ten different locations at the tibial mid-shaft with a least irregularly shaped area where the cortical thickness was relatively well defined. The mean cortical thickness of the 18 tibiae ranged from 1.54 to 4.26 mm and their standard deviations were within 0.5 mm.

Figure 1 shows a schematic diagram of the experimental setup for ultrasound measurements using the axial transmission technique with a time reversal acoustics (TRA) electronic system. Ultrasound measurements were performed at the tibial mid-shaft where the cortical thickness was measured. A pair of custom-made cylindrical transducers with a diameter of 12.7 mm and a center frequency of 200 kHz (one acting as a transmitter and the other as a receiver) was positioned on the same side of the bone axis perpendicularly to the surface. The transducer aperture was small enough to contact with the curved surface of the tibia. A standard ultrasound gel was applied for acoustic coupling between the transducer and the tibia. The transmitter was excited by a one-cycle tone burst with a center frequency of 200 kHz with a TRA electronic system developed at Artann laboratories.¹⁶ The signal processing and the time reversal were also performed using the TRA electronic system.¹⁷ Received radio-frequency (RF) signals were digitized using a digital oscilloscope (WS44Xs, LeCroy, Chestnut Ridge, NY) and stored on a personal computer for off-line analysis. As shown in Fig. 1, the initial separation of the transducers was 30 mm, which was measured from the center of both transducers. The receiver was manually moved away from the fixed transmitter along the long axis of the tibia. The transmitter-receiver distance was increased from 30 to 60 mm in 2-mm steps such that the responses were recorded at 16 discrete distances during the scan. Received RF signals at each distance were averaged over 100 waveforms.

The time reversal process of Lamb waves used here is based on the reciprocity principle and on the invariance of the wave equation with the sign of the time variable.¹³ The procedure is as follows. First, a one-cycle tone burst with a center frequency of 200 kHz is radiated by a transmitter and recorded by a receiver. Figure 2(a) shows the initial Lamb wave recorded by the receiver in a bovine tibia with a thickness of 2.74 mm, which seems to elongate due to the dispersive characteristics of Lamb waves. Second, the initial Lamb wave is reversed in time and applied to the transmitter again. The time-reversed waveform of the initial Lamb wave is presented in Fig. 2(b). Third, this time-reversed signal is radiated by the transmitter and focused at the receiver.

Figure 2(a) shows the response signal recorded at the transmitter-receiver distance of 30 mm in a bovine tibia with a thickness of 2.74 mm. The shape of the response signal indicates the presence of two distinct waves, called the FAS and the energetic SGW, with different velocities, which were consistently observed from the response signals measured in the 18 bovine tibiae used here.¹⁸ In clinical examination of long cortical bones such as the radius and the tibia using the axial transmission technique, the parameter currently used as an indicator of fracture risk is the FAS velocity.¹⁹ This is attributable to the fact that the FAS is not strongly dispersive, and its velocity can be consistently determined from the known transmitter-receiver distance and the time of flight. However, the FAS tends to be guided by the wall thickness only at wavelengths greater than the bone wall thickness and its velocity is strongly influenced by the cortical thickness-to-wavelength ratio.²⁰ It is notable that the SGW seems more sensitive (over the FAS) to any change of material properties inside a cortical bone plate due to cortical thickness change with aging and osteoporosis.⁸ The problem arises because the SGW arrives after the FAS and typically interferes with other contributions, particularly at short transmitter-receiver distances, so that special signal processing techniques must be implemented for consistency of mode identification and velocity estimation.

The present study applies the time-reversed Lamb waves generated by a time reversal process of Lamb waves to overcome these difficulties. Figure 3 shows the time-domain signal and the frequency spectrum of the time-reversed Lamb wave recorded at the transmitter-receiver distance of 30 mm in a bovine tibia with a thickness of 2.74 mm. In contrast to the initial Lamb wave in Fig. 2(a), the time-reversed Lamb wave clearly shows the temporal focusing, which was consistently observed throughout the time-reversed Lamb waves recorded at the 16 discrete distances during the scan. It should also be noted that the frequency spectrum shows a single peak frequency at approximately 200 kHz, thus confirming that a pure Lamb wave can be successfully launched through the time reversal process. This is consistent with the fact that the time reversal process of Lamb waves can automatically compensate for the dispersive nature of Lamb waves.¹³ 

Figure 4 shows the experimental group velocities of the time-reversed Lamb waves as a function of the cortical thickness for the 18 bovine tibiae. The gray solid line represents the linear fit to the measurements. The experimental group velocities of the time-reversed Lamb waves were determined from the slope of the graph of the transducer separation vs the arrival time of envelope maximum. The envelope of the time-reversed Lamb waves was calculated using the Hilbert transform.²¹ The theoretical group velocity of the 200-kHz A0 Lamb wave as a function of the cortical thickness for a cortical bone plate model was also plotted in Fig. 4. The theoretical group velocity was predicted using the Lamb wave theory for elastic wave propagation in a free solid plate with finite thickness in a vacuum¹⁰ based on the assumption that the axial Lamb waves in a tube-shaped tibia are similar in behavior to those in a plate provided that the cortical thickness of the tibia is small compared to its diameter.¹² The longitudinal and the shear wave velocities assumed for cortical bone in calculating the group velocity were 4000 and 1800 m/s, respectively.²² As found in Fig. 4, the experimental group velocities of the time-reversed Lamb waves showed a significant correlation with the cortical thickness (Pearson's correlation coefficient r = 0.77) and tended to follow the theoretical group velocity of the A0 Lamb wave fairly well, consistent with the behavior of the SGW in long cortical bones.

In summary, the present study has investigated the propagation of time-reversed Lamb waves in bovine cortical bone in vitro. The time-reversed Lamb waves were successfully launched at 200 kHz in 18 bovine tibiae through a time reversal process of Lamb waves, and their group velocities were measured using the axial transmission technique. For the clinical application of the time-reversed Lamb waves, one important factor is the layer of soft tissue on top of cortical bone because Lamb waves are generally strongly sensitive to conditions at waveguide boundaries. Therefore this source of artifact warrants further attention.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (Grant No. 2014R1A1A1A05002187).