Phase aberration in transcranial ultrasound imaging (TUI) caused by the human skull leads to an inaccurate image reconstruction. In this article, we present a novel method for estimating the speed of sound and an adaptive beamforming technique for phase aberration correction in a flat polyvinylchloride (PVC) slab as a model for the human skull. First, the speed of sound of the PVC slab is found by extracting the overlapping quasi-longitudinal wave velocities of symmetrical Lamb waves in the frequency-wavenumber domain. Then, the thickness of the plate is determined by the echoes from its front and back side. Next, an adaptive beamforming method is developed, utilizing the measured sound speed map of the imaging medium. Finally, to minimize reverberation artifacts caused by strong scatterers (i.e., needles), a dual probe setup is proposed. In this setup, we image the medium from two opposite directions, and the final image can be the minimum intensity projection of the inherently co-registered images of the opposed probes. Our results confirm that the Lamb wave method estimates the longitudinal speed of the slab with an error of 3.5% and is independent of its shear wave speed. Benefiting from the acquired sound speed map, our adaptive beamformer reduces (in real time) a mislocation error of 3.1, caused by an 8 mm slab, to 0.1 mm. Finally, the dual probe configuration shows 7 dB improvement in removing reverberation artifacts of the needle, at the cost of only 2.4-dB contrast loss. The proposed image formation method can be used, e.g., to monitor deep brain stimulation procedures and localization of the electrode(s) deep inside the brain from two temporal bones on the sides of the human skull.

In the above article [1], we mentioned that the superposition of the different symmetric (S) modes in the frequencywavenumber (f-k) domain results in a high-intensity region where its slope corresponds to the longitudinal wave speed in the slab. However, we have recently understood that this highintensity region belongs to the propagation of a wave called lateral wave or head wave [2]-[5]. It is generated if the longitudinal sound speed of the aberrator (i.e., the PVC slab) is larger than that of water and if the incident wavefront is curved. When the incidence angle at the interface between water and PVC is near the critical angle, the refracted wave in PVC reradiates a small part of its energy into the fluid (i.e., the head wave). As discussed in [4], if the thickness of the waveguide is larger than the wavelength, the first arriving signal is the head wave. This is also the case in our study [1] where the ultrasound wavelength of a compressional wave in PVC was close to 1 mm, and a PVC slab with a thickness of 8 mm was used. In this Erratum, numerical simulations (with SimSonic solver [5]) and experimental measurements (with the same PVC slab used in [1]) are conducted to investigate the propagation of the Lamb waves and head wave in detail, for the specific configuration studied in [1]. The pitch and element width of the P4-1 probe were used to assemble the numerical signals [see Fig. 1(a)]. If all the data simulated for the P4-1 probe is used, there indeed is a region with a slope [see Fig. 1(b)], but this has a low intensity, meaning that the head wave has a relatively low amplitude compared to the specular reflections. Once the head wave is isolated, the sound speed can be estimated with a 0.3% error from the f-k domain plot [see Fig. 1(c)]. No significant difference is observed between Fig. 1(b) and (d), in which the head wave is muted. Our experimental results show that if only the head wave (the first arriving signal) is used [see Fig. 2(b)], the slope of the linear fitting in the f-k domain also yields the longitudinal sound speed of the PVC with a 0.3% error. Of note, the signal processing (i.e., linear fitting in the f-k domain) used in our study [1] still works for the head wave and is correct provided that the aberrator is parallel to the probe [6]. Also, in [1, p. 6], it is mentioned that “the curved structure of the skull might lead to other types of modes, such as the torsional modes.” Here, we acknowledge that this sentence is not correct, as torsional modes only exist in cylindrical waveguides or rectangular bars. We would like to mention that Guillaume Renaud is added as a coauthor to acknowledge his contribution to the findings reported in this Erratum.

Deep brain stimulation (DBS) is used to modify the brain function. Localization of the electrode(s) used in DBS is an important mater since the success of the treatment highly depends on it. Transcranial ultrasound imaging (TUI) can be a proper candidate to monitor the electrode(s), but it is affected by the phase aberration caused by the skull bone. To address this issue, in this paper, we propose a novel beamforming method based on the sound velocity map of the imaging medium. For each combination of the imaging grid and element of the array, an averaged sound speed (ASD) is calculated. Then, the ASD is used inside a delay-and-sum beamforming method. The numerical results show that the proposed method compensates the phase aberration caused by the skull having a thickness of 5 mm with a sound speed twice of the imaging medium (i.e., water). The proposed method can be implemented in a real-time manner, which makes it a great candidate to be used in operation rooms for surgeries.

Phase aberration of focused ultrasound by tissue structure causes focus degradation and reduces the quality of B-mode images. Refraction at the boundary between subcutaneous fat and muscle is one of the dominant factors behind such degradation. To correct this, we propose a refraction compensation method in which ultrasound is transmitted and received twice. The boundary shape between different tissues is detected by the first ultrasound transmission. Next, ultrasound rays from probe elements to the target are calculated taking refraction into account. Corrected delay times are calculated from the length of the rays and the sound velocity of the medium. Finally, ultrasound is transmitted a second time using the corrected delay time and a B-mode image is created. We evaluate the correction effect of the proposed method by numerical simulation and experiments with non-compensated and refraction-compensated cases of intensity distribution of the focused ultrasound. Results show that focus degradation is effectively corrected by the proposed method.