Representation Lamb Waves In The Time Frequency

Published Date: 02 Nov 2017

In this section numerical simulations of wave propagation in healing long bones are presented. First, we developed 2D healing bone models with the original material properties of callus segments derived using the SAM images of Figs. 3b, 3c, and then the equivalent homogeneous and isotropic numerical models were constructed with the effective material properties derived from IEMA (Figs. 6a-e). The effectiveness of IEMA as a homogenization method is investigated numerically by conducting FAS velocity measurements and guided wave analysis. Secondly, the effective material properties and the attenuation coefficient derived from IEMA are incorporated in BEM computational models of healing long bones corresponding to the third, sixth and ninth week of consolidation. The dispersion of guided waves is investigated in the time-frequency domain.

Ultrasound excitation

The numerical solution of the 2D wave propagation problem was performed by using the boundary element method based on ISoBEM (BEM S&S). A Hanning sine pulse was used as the excitation signal expressed via the equation:

(13)

where A=1 MPa is the amplitude of the excitation, f0 is the central angular frequency and n0 is the number of the sinusoidal cycles in the pulse. The examined central angular frequencies were 300 kHz, 500 kHz and 1 MHz, while the excitation signal included four sinusoidal cycles in the pulse. The duration of the simulated signals was 40 μs. The accuracy of the solution depends on the relation between the element size and the wavelength. We used at least four three-noded quadratic line elements per smallest wavelength.

Representation of Lamb Waves in the Time-Frequency Domain

Time-frequency (t, f) analysis was conducted in order to represent the propagating wave modes. The reassigned smoothed-pseudo Wigner-Ville (RSPWV) distribution function was used as it was found to be effective in representing and isolating guided modes [6]. In the case of a homogeneous isotropic elastic plate with traction-free upper and lower surfaces, the guided waves are plane strain waves called plate waves or Lamb waves. Frequency-group velocity (f, cg) dispersion curves were computed for the plate model based on the Lamb wave theory. Plate thickness was set to 4 mm, which is an average value of multiple, random measurements on the upper and lower cortical bone segments of the SAM images. The (f, cg) dispersion curves were super-

imposed to the (t, f) representations in order to investigate guided wave propagation. Compared to other techniques [23-26] the main advantage of this one is that a single broadband excitation is needed to represent the dispersion of multiple wave modes. In Fig. 5, analytically derived Lamb dispersion curves of symmetric and asymmetric modes are illustrated.

5.3 Measurements of the velocity of the first arriving signal

Axial-transmission measurements were performed by keeping constant the center-to-center distance between the transducers to 25 mm. The transducers size was set to 3mm and they were placed equidistant from the osteotomy gap directly onto the cortical bone surface. The ultrasound propagation velocity was determined by dividing the transducers’ in-between distance to the transition time of the FAS. In order to detect the FAS a threshold was used, corresponding to 10% of the amplitude of the first signal extremum.

5.4 Numerical simulations in cortical bone and callus with effective properties

In this section we performed two sets of simulations of wave propagation in healing bone models to investigate whether IEMA could be used as a numerical tool of modeling media with a complicated porous nature in a simple manner.

The callus segments of Figs. 6a, 6d (length = 3 mm and width = 4 mm) were incorporated in two simple computational cortical bone geometries. In the first case (Figs. 6b, 6e), the original material properties and the nonhomogeneous geometries of the callus were used derived from the SAM images. In the second case (Figs. 6c, 6f), the complex callus regions were replaced by a homogeneous and isotropic medium having the effective material properties calculated using IEMA. Osseous tissue was considered as the matrix of the composite and blood as the material of the spherical inclusions. Blood volume concentrations were measured 25.78% and 28.68% in the

callus segments of Figs. 6a and 6d while the radius of the scatterer was set to 200 μm in both cases. The material properties of cortical bone were derived from Table II and correspond to the ninth postoperative week. Tables V and VI present the average material properties of the callus segments in Figs. 6a and 6d calculated through the Eqs. (10), (11), (12), as well as the effective material properties derived using IEMA for the excitation frequency of 300kHz. First, axial transmission measurements of the FAS velocity were conducted for the excitation frequency of 300 kHz. Then, guided wave analysis was performed.

Concerning the computational models of Figs. 6b, 6c the FAS velocity was calculated as 3928 m/s, while for the computational model of Figs. 6e, 6f the FAS velocity was 3985 m/s in both the nonhomogeneous and homogeneous geometries showing the significant accuracy of the methodology. Figure 7 shows the signal waveforms obtained from the computational models of Figs. 6e, 6f.

Figure 8 shows the (t, f) representations corresponding to the Figs. 6b, c and 6e, f. The A1 mode was identified in all the examined cases. The t-f representations are significantly attenuated (Figs. 8b, 8d) when IEMA is used in comparison to the original signal (Figs. 8a, 8c). Moreover, the time frequency diagrams for the two homogenized models of week 6 and week 9 are almost similar (Figs. 8b, 8d).

5.5. Simulations in different healing stages

Two-dimensional computational models of healing long bones were developed corresponding to the third, sixth and ninth postoperative week (Fig. 9). The complex callus region was replaced by a homogeneous and isotropic medium having the effective material properties derived from IEMA (Tables IV, VII, VIII). An intact bone model was also developed having the material properties of the cortical bone at week 9 (Table II). The guided mode features derived from the osteotomy numerical models were compared to the intact bone estimations.

The (t, f) representations of the signals obtained from the intact bone model and the osteotomy bone models after three, six and nine weeks of consolidation are illustrated in Figs. 10-12. The (t, f) representations are shown in the form of pseudo-color two-dimensional images, where the color of a point corresponds to the amplitude (in dB) of the energy distribution. At each healing stage the (f, cg) dispersion curves presented in Fig. 5 were superimposed to the (t, f) representations to investigate the propagation of guided waves during healing. Figures 10a, 11a, 12a correspond to the intact bone model, Figs. 10b, 11b, 12b represent week 3, Figs. 10c, 11c, 12c represent week 6, and Figs. 10d, 11d, 12d represent week 9.

For the central frequency of 300 kHz (Figs. 10a-d) the S0 and A0 modes were found to be the dominant modes, while the A1 mode was also identified. In particular, for the intact bone model the S0 and A0 modes were detected in the frequency ranges 0.28 – 0.42 MHz and 0.19 – 0.38 MHz. In week 3 the S0 mode was detected from 0.20 – 0.42 MHz. In weeks 6 and 9 the S0 and A0 were the dominant modes in the frequency ranges 0.28– 0.38 MHz, 0.21– 0.38 MHz and 0.31– 0.38 MHz, 0.21– 0.38 MHz, respectively. In week 9 the A1 was also identified from 0.30 – 0.38 MHz. The significant restoration of the dominant modes during the final healing stages is shown in Figs. 10 c, d approaching the intact bone observations (Fig. 10a).

For the central frequency of 500 kHz (Figs.11 a-d), the dominant modes were S0 and S1 during the whole healing process, while the mode A1 was also detected in weeks 6 and 9. Specifically, for the intact bone model the S0 and S1 modes were detected for frequencies from 0.43 – 0.66 MHz and 0.52 – 0.56 MHz, respectively. In week 3 the S0 and S1 modes were identified in the frequency ranges 0.38 – 0.55 MHz and 0.52 – 0.59 MHz, respectively. In week 6 the S0 modes is identified for frequencies from 0.34 – 0.51 MHz, while in week 9 the modes S0 and S1 were mainly detected in the frequency ranges 0.42 – 0.55 MHz and 0.52 – 0.55 MHz, respectively, showing a significant restoration in comparison to the intact bone model.

Finally, some high-frequency modes were also supported when the central frequency of 1000 kHz was applied (Figs.12 a-d). The dominant modes were S1 and