NORMAS GENERALES

varying attenuation coefficient as input to Eq. 4.3. The attenuation of the phantoms used in this study was modelled assuming a linear dependence on frequency (n = 1 in Eq. 3.25). Because the attenuation coefficient of soft tissue as well as of the currently studied tissue-mimicking phantoms is not expected to exceed 2 dB/cm/MHz [71], the input attenuation coefficient in the simulation was discretely changed in the interval between 0 and 2 dB/cm/MHz with a step of 0.05 dB/cm/MHz. The resulting estimated spectra were compared to the simulated or real experimental spectra obtained at the corresponding depth using the above outlined sliding window approach after a calibration in the first window. An example of the spectral comparison for both simulated and experimental data can be seen in Fig. 4.4. A -20 dB frequency range of all spectra was selected for the comparison in order to operate above the noise level. The distance between the spectra of the windowed signals at every depth and for every attenuation input was calculated using Eq. 4.8, and, finally, the global attenuation coefficient of the phantom was determined using Eq. 4.9.

In this way, the attenuation coefficient of each phantom was estimated using data sets recorded at different “transducer-phantom” distances, namely in the near-field, focal zone and far-field for the simulation study and in the focal zone and far-field for the experimental study. As mentioned before, the re- construction was performed considering two conditions. First, the attenuation estimates were obtained under the assumption of the joint action of diffraction and attenuation, and afterwards under the assumption of plane-wave propaga- tion neglecting the diffraction effects.

Some additional analysis were performed to verify the accuracy and the vari- ance of the local attenuation estimates and to check the feasibility of increasing the spatial resolution of the proposed method by decreasing the number of win- dows used during the spectral comparison. For this purpose, local attenuation estimates were determined in every window as:

α(zi) = arg min

α D(zi, α). (4.11)

4.4

Results

The total attenuation estimates obtained in the simulation study are presented in Table 4.2 and visualized in Fig. 4.5a. When the effects of diffraction were in- corporated in the model, the method allowed to retrieve the exact attenuation coefficient of the generated signals regardless of the position of the phantom with respect to the transducer focus. As expected, when the effects of diffrac- tion are not accounted for, the attenuation coefficient is underestimated in

4. Diffraction - independent attenuation estimation

Figure 4.4: Example of the spectral fit between the measured and simulated Fourier spectra in the simulation study (on the left) and in the experimental phantom study (on the right).

the near-field of the transducer, where the amplitude of the ultrasound beam experiences increase due to the focusing of the beam, and is overestimated in the far-field region, where its amplitude decreases due to diffraction. Fig. 4.5b presents the average of local estimates obtained for every data set. It can be seen that the average local estimates do not significantly deviate from the total attenuation estimates in Fig. 4.5a. When the diffraction was taken into account, the standard deviation of the local estimates did not exceed 0.05 dB/cm/MHz. In Fig. 4.6a, the local attenuation estimates are shown as a function of depth of the estimation window. It can be seen that the estimates start converging to a certain value after the first 15- 20 mm inside of the ROI. Finally, Fig. 4.6b presents the distance measure of Eq. 4.8 computed in the focal zone of the transducer, which appears to have a clear minimum for all three data sets.

Estimated attenuation coefficient, dB/cm/MHz (relative error, %) With diffraction correction Without diffraction correction “Ground

truth” Near-field Focal zone Far-field Near-field Focal zone Far-field 0.3 0.30 (0) 0.30 (0) 0.30 (0) 0.20 (33) 0.40 (33) 0.45 (50) 0.5 0.50 (0) 0.50 (0) 0.50 (0) 0.35 (30) 0.60 (20) 0.65 (30) 0.7 0.70 (0) 0.70 (0) 0.70 (0) 0.60 (14) 0.85 (21) 0.85 (21)

Table 4.2: Attenuation estimates obtained in the simulation study, with and without diffraction correction at different positions along the transducer beam.

4.4. Results

(a)

(b)

Figure 4.5: Total (a) and averaged local (a) attenuation estimation results in the simulation study based on three data sets with attenuation coefficients equal to 0.3, 0.5 and 0.7 dB/cm/MHz. “ND” denotes the estimates obtained without diffraction correction (plane-wave propagation model).

4. Diffraction - independent attenuation estimation

(a) (b)

Figure 4.6: (a) Local attenuation estimates obtained at different depths for three data sets with attenuation coefficients equal to 0.3, 0.5 and 0.7 dB/cm/MHz generated in the near-field, focal zone and far-field of the transducer. (b) Dis- tance measure (Eq. 4.8) versus the input attenuation coefficient values of the exhaustive search for the three data sets computed at the focal zone of the trans- ducer.

The results of the experimental phantom study are presented in Table 4.3 and visualized in Fig. 4.7a. The proposed method provided close to the “ground- truth” attenuation estimates both in the focal zone of the transducer as well as in the far-field with average relative errors of 9.2% and 10.0%, respectively. As expected, upon neglecting diffraction effects, the attenuation estimate in the focal zone were acceptable (average relative error of 7.2%). However, the attenuation estimates in the far-field were considerably overestimated (aver- age relative error of 41.3%). Although the average error of the attenuation estimates obtained in the focal zone with the diffraction correction is slightly higher than the error obtained when the diffraction effects were not accounted for, these results are not statistically different. On the contrary, the average estimates obtained in the far-field of the transducer under the plane wave ap- proximation are statistically different from those obtained with the diffraction correction (p-value < 0.001). Fig. 4.7b presents the averaged local attenuation estimates for the same data sets. Again, it can be seen that these values do not significantly differ from the total attenuation estimates. However, for some data sets high values of the standard deviation can be noticed. In Fig. 4.8, the local attenuation estimates are shown with depth inside the ROI obtained in the focal zone (a) and in the far-field (b) of the transducer. Again, the estimates seem to converge to a particular value at a depth of 15 - 20 mm. However, it can be noticed that the far-field estimates possess a higher vari-