A stochastic approach to delays optimization for narrowband transmit beam pattern in medical ultrasound

TL;DR

The paper tackles the problem of non-uniform and energy-dropping transmit beam patterns in narrowband ultrasound, particularly for ARFI elastography, by optimizing per-element delays as free variables within a non-linear least-squares framework to match a 2D rectangular target beam described by $P(\vec{x})$. The authors formulate the objective with a depth-spanning rectangular prescription $G$ and solve the resulting non-convex problem using Particle Swarm Optimization, supported by three novel metrics: $MLW(z)$, $SLL(z)$, and $CLP$. Results from synthetic simulations show thinner, more uniform main lobes and better depth concentration, with ARFI-specific benefits including improved uniformity of shear-wave timing and velocity estimates. The approach offers practical improvements for ARFI elastography and can be extended to convex or steered beams, with future work proposed on apodization, wideband patterns, and data-driven shape learning.

Abstract

Ultrasound imaging is extensively employed in clinical settings due to its non-ionizing nature and real-time capabilities. The beamformer represents a crucial component of an ultrasound machine, playing a significant role in shaping the ultimate quality of the reconstructed image. Therefore, Transmit Beam Pattern (TBP) optimization is an important task in medical ultrasound, but state-of-the-art TBP optimization has well-known drawbacks like non-uniform beam width over depth, presence of significant side lobes, and quick energy drop out after the focal depth. To overcome these limitations, we developed a novel optimization approach for TBP by focusing the analysis on its narrowband approximation, particularly suited for Acoustic Radiation Force Impulse (ARFI) elastography, and considering transmit delays as free variables instead of linked to a specific focal depth. We formulate the problem as a non linear Least Squares problem to minimize the difference between the TBP corresponding to a set of delays and the desired one, modeled as a 2D rectangular shape elongated in the direction of the beam axis. In order to quantitatively evaluate the results, we define three quality metrics based on main lobe width, side lobe level, and central line power. Results obtained by our synthetic software simulation show that the main lobe width is considerably more intense and uniform over the whole depth range with respect to classical focalized Beam Patterns, and our optimized delay profile results in a combination of standard delay profiles at different focal depths. The application of the proposed method to ARFI elastography shows improvements in the concentration of the ultrasound energy along a desired axis.

Figures (13)

• Figure 1: Left: Scheme of the adopted geometry, in the background the simulated energy pattern for elements $c_0$ and $c_{-1}$. Right: Prescribed rectangular BP to approximate.
• Figure 2: Two examples of BP profile to clarify the rationale for the normalization with respect to the central line in the metrics definition. The red dot identifies the BP central line, the green dashed line highlights the -6 dB level under the central value, the grey area is the Side Lobe area and the central white area is the main lobe area. The main lobe width increases when the maxima are not on the central line, thus penalizing BPs with this undesired behaviour.
• Figure 3: Narrowband beam shapes obtained with standard focal law (single depth focus) at frequency $4.5$ MHz with Probe 1. The focal depth is constant along the rows, while the aperture is constant along the columns.
• Figure 4: Narrowband beam shapes obtained with standard focal law (single depth focus) at frequency $4.5$ MHz with Probe 2. The focal depth is constant along the rows, while the aperture is constant along the columns.
• Figure 5: Optimized beam patterns for Probe 1 for three different frequencies and with five different numbers of active elements.