1 Introduction

Medical Ultrasound Imaging is the most widespread real-time non-invasive imaging system and it exploits the ability of human tissue to reflect ultrasound signals. Various ultrasound modalities process the reflected echoes differently, generating either morphological images or measures of tissue properties. While the large majority of ultrasound modalities deals with wideband beam patterns, the quantitative application Acoustic Radiation Force Impulse (ARFI) elastography employs high frequency pulses consisting of hundreds of cycles with an extremely small fractional band, a scenario that can be referred to as narrowband [31]. To be more precise, in ARFI the probe emits a narrowband pulse to generate a shear wave in the tissue; several tracking pulses are then emitted to sample the propagation of the shear wave, and to estimate the local shear wave velocity; such an estimation allows one to compute the tissue elasticity map, displayed as a color-coded image [7, 9, 16, 22, 32].

The performance of each modality significantly depends on how the transmission and reception steps are tuned, reflecting both on the image quality and the accuracy of measurements. The two steps are characterized by different parameters such as pulse shape, central frequency, transmission focus, transmit and receive apodization. As for the transmission step, the effect of the parameters is summarized by the transmit beam pattern (TBP), which encodes the information about energy distribution across the image field. Generally, the ideal transmit beam pattern, both narrowband or wideband, should have a main lobe width as uniform as possible along the depth of interest and a side lobe level as low as possible (Fig. 1). Moreover, the main lobe intensity along depth should be as uniform as possible, since its maximum peak is bounded by both mechanical and thermal limits assuring the safety of the insonification. These requirements are in contrast with the typical hourglass-shaped beam obtained with a fixed focalization depth [11, 18], in which the main lobe width has its minimum close to the focal depth and it gets wider moving away from it. Furthermore, the near field region proximal to the probe is often characterized by side lobes whose intensity is even higher than the value along the central line [4, 33]. Finally, the combined effect of geometric and thermoviscous attenuation causes a quick energy drop-out in the adjacent post focal region. These drawbacks have a negative impact on both image quality and measurement accuracy. For example, in ARFI elastography a non uniform main lobe width generates shear waves of non uniform temporal length along depth, causing bias in the estimation of shear wave times of arrival [7, 8]. The presence of side lobes with high intensity generates instead spurious shear waves that hinder the effect of the desired one, while the quick energy drop-out in the post-focal region causes a rapid decrease of shear wave energy along depth, thus dramatically decreasing the SNR and the accuracy of the estimated shear wave time of arrival [35]. Tuning the high level parameters of a standard beam pattern, for instance focal depth, aperture, and transmit frequency, may improve some beam pattern features but often at the expense of others. For example, increasing the aperture and deepening the focal depth reduces the energy drop-out but decreases the main lobe width uniformity.

Fig. 1
figure 1

Picturing of ideal and typical transmit beam patterns

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One approach addressed in the literature [3, 5, 6, 12, 19, 24, 34] consists in optimizing a set of low level parameters, such as the transmit intensity at each channel, i.e., the apodization window, by minimizing a suitable cost function in the wideband case. In this framework the solution of this problem can have a closed form or may be easily computed with optimization algorithms, being the cost function convex. A few other methods optimize the delay patterns solely regarding the intensity at focus [15, 25, 26], while other strategies aim at calculating the weight vector of a transducer array to obtain a low-sidelobe transmitting beam pattern [13, 14]. Moreover, diffraction-free beams or limited diffraction beams (e.g., the Bessel beam and the Axicon beam) should achieve the desired pencil-like shape (Fig. 1, left panel). However, neither of the techniques meets our requirements since generating Bessel beams [20] requires apodization, which is not considered in this work, while using Axicon beams [2] typically results in significant side lobes, making them unsuitable for our purposes. To the best of our knowledge the optimization of transmit delays has been relatively underexplored, with a greater emphasis placed on advancing image reconstruction techniques. Meanwhile, the challenges associated with parameter tuning are presently addressed through manual tuning methods.

In this work we propose to optimize the narrowband beam pattern, suited for ARFI elastography, by considering each single transmit delay as an independent variable and keeping the transmit intensity fixed. The benefit of this method would be to streamline the calibration process of imaging equipment by providing a set of parameters to serve as a starting point for manual calibration, a step that occurs systematically for each signal acquisition mode. The choice of optimizing apodization-free beam patterns, in which all the active elements transmit at the same voltage, widens the scope of application of this method also to low-price commercial devices, equipped with simple two-level transmit pulsers instead of linear amplifiers. The much higher number of degrees of freedom, in comparison to the classical single-focus delay parametrization, allows the energy to be distributed over a greater depth range, thus improving the beam uniformity along depth. To quantitatively compare the optimized beam patterns with a large set of standard ones, we define three novel ad-hoc quality metrics, specifically designed to measure how much a beam pattern approaches the ideal behaviour required by the applications described above. Existing beam pattern metrics, like peak Side Lobe Level, integrated Side Lobe Level, or Full Width at Half Maximum of main lobe are quite effective for evaluating far field beam patterns or near field beam pattern at a single depth, but they fail to capture the global beam pattern quality along a depth range. Our numerical experiments demonstrate that the optimized Beam Patterns (BPs) are indeed thinner and more uniform along depth, with low side lobes and a uniform main lobe intensity. The application of the method to ARFI elastography results in higher intensity and uniformity for shear waves, which may ultimately lead to more precise reconstruction of elasticity maps. It should be emphasized, however, that we hereby only consider the optimization of transmission step related to the shock pulse, leaving the tracking transmission and reception for future studies.

The paper is organized as follows. Section 2 briefly describes the mathematical formulation of the problem including the derivation of the transmit BP equation, the definition of the optimization problem, the metrics for evaluating the goodness of the results, and the simulation setting. Section 3 presents both the qualitative and the quantitative results. Finally, in Sect. 4 our conclusions are drawn.

2 Materials and methods

2.1 Biomedical ultrasound model formulation

We proceed with describing the simulation of TBP focusing more specifically on the narrowband case, which allows to approximate the waveform as an infinite pulse. We refer to [31] for a full explanation of the rationale behind this reasoning, and we will give here only a brief description. We consider a linear probe composed of 2N piezoelectric elements of width \(\Delta x\) that convert the electric pulse into a pressure wave and vice-versa. For convex probes the treatment is analogous. We consider a Cartesian coordinate system whose origin corresponds to the geometrical center of the probe (see Fig. 2), so that the center of each probe element is at the position \(c_i = (i + 1/2) \Delta x\) with \(i=-N,\ldots ,N-1\). Each point \(\vec {x} = (x,z)\) of the image plane is identified by a positive value of the z coordinate.

2.1.1 The BP equation

The TBP is the pattern of radiation generated from a beam and it can be described as the energy or the power, for a periodic signal, of the propagating wavefront, evaluated at every spatial point of the field. The propagating wavefront can be computed from the signal emitted by the probe and the spatial-temporal impulse response characterising the probe itself. In particular, the impulse response function \(h_i(\vec {x},t)\) of each i-th element is the integral over the element surface \(A_i\) of a spherical wave starting from each point of the element surface [11]:

$$\begin{aligned} h_i(\vec {x},t) = \int _{A_i} \frac{\delta \left( t - \frac{\vert \vert \vec {x}-\vec {y}\vert \vert }{c}\right) }{2\pi \vert \vert \vec {x}-\vec {y}\vert \vert } d\vec {y}, \end{aligned}$$
(1)

where c is the ultrasound speed propagation. We assume that every active element emits the same temporal transmit waveform I(t) [17]. Thus, the resulting signal is obtained at every point \(\vec {x}\) by convolving I(t) with the sum of all the impulse responses of the active elements and with a term accounting for the tissue thermoviscous attenuation \(a(\vec {x},t)\). The waveform I(t) can be trasmitted with a different delay for each active element. The BP shape depends on the set of transmission times because of the various destructive and constructive interferences which are generated during the propagation. We introduce the transmission delay \(D_i\), associated with the i-th element, and we describe the delay in the signal emission as a translation in time \(\delta (t - D_i)\). By defining

$$\begin{aligned} H_i(\vec {x},\cdot ): = a(\vec {x},\cdot ) *h_i(\vec {x},\cdot ), \end{aligned}$$
(2)

we can describe the propagating wavefront in time as:

$$\begin{aligned} S(\vec {x},\cdot ) = I *\sum _{i=-N}^{N-1} H_{i}(\vec {x},\cdot ) *\delta (\cdot - D_i). \end{aligned}$$
(3)

In the narrowband case one can assume I(t) to be an infinite pulse with given frequency \(f_0\), i.e., \(I(t) = e^{-2 \pi j f_0 t}\). Thus, the power of the BP is

$$\begin{aligned} P(\vec {x}): = \frac{1}{T}\int _{[0,T]} \left| S(\vec {x},t) \right| ^2\, d t , \end{aligned}$$
(4)

where \(T = \frac{1}{f_0}\). Moving to the time frequency domain and relying on Plancherel’s theorem, stating the equivalence of signal power in time and frequency domain, the BP power can be written as the sum of the squared absolute values of the Fourier series coefficients:

$$\begin{aligned} P(\vec {x}) = \sum _{s = -\infty }^{+\infty } \Big \vert \sum _{i=-N}^{N-1} {\hat{H}}_{i}(\vec {x}, s) e^{-2\pi j D_i f_0 s} \Big \vert ^2, \end{aligned}$$
(5)

where \({\hat{H}}_{i}(\vec {x}, s)\) is the Fourier Series of \(H_{i}(\vec {x}, t)\). Since we have assumed I(t) to be a pure tone, only one coefficient of the series is different from zero. Therefore, Eq. 5 reduces to:

$$\begin{aligned} P(\vec {x}) = \Big \vert \sum _{i=-N}^{N-1} {\hat{H}}_{i}^{f_0}(\vec {x}) e^{-2\pi j D_i f_0} \Big \vert ^2, \end{aligned}$$
(6)

where we denote \({\hat{H}}_{i}^{f_0}(\vec {x}): = {\hat{H}}_{i}(\vec {x}, 1)\). In particular, we can exploit the modulus properties obtaining:

$$\begin{aligned} P(\vec {x}) = \left( \sum _{m=-N}^{N-1} {\hat{H}}_{m}^{f_0}(\vec {x}) e^{-2\pi j D_m f_0}\right) \overline{\left( \sum _{n=-N}^{N-1} {\hat{H}}_{n}^{f_0}(\vec {x}) e^{-2\pi j D_n f_0} \right) }. \end{aligned}$$
(7)

Finally, making use of the exponential trigonometric form and using the cosine addition formula, the BP takes the form

$$\begin{aligned} P(\vec {x}) = \sum _{m,n=-N}^{N-1} {{\hat{H}}}_{m,n}^{f_0}(\vec {x}) \cos (2\pi f_0 (D_{m} - D_{n})), \end{aligned}$$
(8)

where

$$\begin{aligned} {\hat{H}}_{m,n}^{f_0}(\vec {x}): = {\hat{H}}_m^{f_0}(\vec {x}) \cdot \overline{{\hat{H}}_n^{f_0}(\vec {x})}, \end{aligned}$$

is obtained as the product between the Fourier Series of \(H_m(\vec {x}, t)\) and the conjugate of the Fourier transform of \(H_n(\vec {x},t)\) at frequency \(f_0\).

2.2 Optimization of beam pattern

This subsection introduces the optimization problem and three novel metrics that are based on the desired outcome for the application of the method to ARFI elastography.

2.2.1 Delay optimization

The aim of the optimization problem is to make \(P(\vec {x})\) as similar as possible to a prescribed BP shape by varying the frequency \(f_0\), the number of active elements N, and their corresponding delays \(D_i\), with \(i=0,\dots , N-1\). Since we are interested in BP shapes which are symmetric with respect to the probe central transmit line, we impose \(D_{-i}=D_{i-1}\). We consider the delays as free variables so that the BP changes according to \(N + 2\) parameters:

$$\begin{aligned} p:= (N, f_0, D_0, \dots , D_{N-1}). \end{aligned}$$

In order to compute the BP in Eq. (8), we sample \(\vec {x}\) by taking a grid of points \((x_u,z_v)\) with \(x_u = u \Delta x/4\) where \(u=-4N,\ldots ,4N\) and \(v=0,\ldots ,M\) (see Fig. 2).

Fig. 2
figure 2

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

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From a computational point of view, we implement \(h_i(x_u,z_v)\), the integral over a single element surface, according to the model in [31]. The idea is to consider each point on the probe surface as a source of a spherical wave and compute the impulse response with respect to a certain field point as the sum of the spherical waves, each one weighted by the distance between the probe surface point and the field point. Moreover, by fixing the central frequency \(f_0\), we pre-compute the attenuation coefficients and the impulse response function for each element. An example of impulse response function for a linear probe with pulse central frequency \(f_0=4.5\) MHz is given on the left panel of Fig. 2. Then we compute \({{\hat{H}}}_{i, j}^{f_0}(x_u,z_v)\) by using the discrete Fourier transform over time at frequency \(f_0\). The sampled BP is a vector X(p) depending non-linearly on p, whose components are given by \(X_k:= P(\vec {x}_k)\) where k is an unrolled index, namely \(k=u + (8N+1) v\). Therefore, we can formulate a Least Squares (LS) problem as follows

$$\begin{aligned} p^* = \arg \min _{p} \Vert X(p) - G \Vert ^2_2, \end{aligned}$$
(9)

where G is the discretization of the prescribed BP rectangular shape we aim to approximate. The width of G has been chosen (experimentally) to be 4 pixels (\(=\Delta x\)), i.e., the minimum width possible so that the optimization algorithm would give non trivial results (limit case of no transmission). More specifically, our aim is to establish the narrowest width achievable for the rectangle, for two reasons mainly: first, to ensure the main lobe is as tight as possible, and second, based on our experiments, increasing the rectangle’s size generally results in elevated side lobes near the probe. Since the vector X(p) is separately periodic with respect to the delays \(D_0, \ldots , D_{N-1}\), the optimization problem is non linear and non convex as the parameter domain contains an N-dimensional torus (cf. Appendix in [31]), therefore not suitable for standard optimization techniques. Among the many methods developed to deal with non convex optimization, e.g., genetic algorithms [10], simulated annealing [23], manifold optimization [1] to mention a few, we propose to use the Particle Swarm Optimization (PSO) [21]. PSO is a Swarm Intelligence technique inspired by self-organizing interactions happening during bird flocking. The swarm is a collection of possible solutions (particles) with which the algorithm is initialized. At each step the positions of the particles are updated according to their velocity and their descent direction, and each particle solution (personal best) is adjusted taking into account the swarm best position (global best). One advantage of PSO is its dependence on a small number of parameters to tune, but in high dimensional spaces it tends to converge slowly to a global optimum. This method has been proven to be particularly efficient in solving this problem, after a number of attempts with different strategies.

Remark 1

We acknowledge that a hybrid method is frequently utilized for fine-tuning. We experimented with diverse approaches and penalty functions to optimize delays, and found that this particular solution is currently the most effective. Given the complexity of the domain (a manifold as described in [30]), we deemed an entirely gradient-based approach inadequate. However, we intend to explore a hybrid approach to see if it yields improved results.

Remark 2

Observe that in this work we do not consider optimizing the apodization weights [3, 5, 6, 12, 19, 24, 34]. On the one hand, this widens the scope of application of this method also to mid-low level commercial devices where the transmit signals are often generated by three-level pulsers that prevent transmit apodization from being implemented. Moreover, optimizing the weights as free variables would introduce N additional parameters to the algorithm, hence leading to a significant slowdown of the computational time, already non negligible. A possible workaround could be to adopt a two-step optimization, by first optimizing the delays and then, fixing the delay curve, optimizing the apodization weights.

2.2.2 Evaluation metrics

In order to allow a quantitative evaluation of optimized BPs with respect to standard ones, we devised a set of metrics aimed at providing a synthetic evaluation of the overall BP quality. The first metric aims at measuring the main lobe width average value and uniformity along depth. To do so, we make the assumption that the Main Lobe (ML) at each depth consists of the BP portion centered at the transmission line that differs from the central value less than 6 dB (Fig. 3). The choice of this threshold is coherent with the usual definition of Full Width at Half Maximum. Moreover, this assumption penalizes those patterns whose maximum value is not on the central line.

Fig. 3
figure 3

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

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Therefore, we define the Main Lobe Width (MLW) at a certain fixed depth z as:

$$\begin{aligned} MLW(z)= & {} \max _{x,y \in \Psi _z}\vert x-y\vert \\{} & {} \quad \text { where } \quad \Psi _z:= \left\{ x \quad \text {s.t.} \quad P(x, z) \ge \frac{P({\bar{x}}, z)}{\epsilon } \right\} , \end{aligned}$$

where \({\bar{x}}\) is the azimuth coordinate of the beam central line and \(\epsilon = 10^{\frac{3}{5}} \approx 4\).

The second metric evaluates the overall side lobe level. Let us define the average Side Lobe Level (aSLL) at a fixed depth as the average over x of all the BP values outside the region of the main lobe (as defined above):

$$\begin{aligned} aSLL(z) = \frac{1}{P({\bar{x}},t)} \frac{1}{|\Psi _z^C|} \int _{\Psi _z^C} P(x,z) \, dx, \end{aligned}$$

where \(|\Psi _z^C|\) denotes the measure of the complementary set \(\Psi _z^C\) of \(\Psi _z\). Note that the aSLL is normalized with respect to the central line value at the each depth z, in order to cope with the overall decrease trend of the BP along depth. By computing mean and standard deviation of MLW(z) and aSLL(z) over a set of depths of interest, we can evaluate the main lobe width uniformity and narrowness, and the side lobes uniformity and average level.

Finally, to evaluate the drop-out of main lobe power along depth we compute the mean integral of the central line values, namely the Central Line Power (CLP):

$$\begin{aligned} CLP= & {} \frac{1}{|\zeta |}\int _{\zeta } \frac{P({\bar{x}}, z)}{\max _{{\hat{x}}, {\hat{z}}}P({\hat{x}}, {\hat{z}})} \, dz\\{} & {} \quad \text { where } \quad \zeta = \left\{ z \quad \text {s.t.} \quad z \in [z_{\min }, z_{\max }] \right\} , \end{aligned}$$

where \([z_{min}, z_{max}]\) denotes a range of depths of interest. Contrary to the typical assessment of the depth of field (DOF) measured as the \(-6\) dB length of the beam, this metric provides also useful information about the power uniformity along depth.

Remark 3

The metrics are defined with the aim of providing a tool to facilitate the comparison of BPs. Particularly in our case study, it holds that:

  • the lower the MLW value the better, as it means having on average a narrower main lobe,

  • the lower the aSLL value the better, as it signifies having on average less intense side lobes,

  • the higher the CLP value the better, this means that the power drop occurs deeper and that, on average, the maximum power is maintained for a longer period along the central transmission line.

Remark 4

We also assessed the canonical SLL. However, we would like to point out that SLL is the value of side lobe level at a single point (corresponding to its maximum value), whereas our metric aSLL provides a more comprehensive evaluation of the entire side lobe area. Additionally, conventional metrics such as integrated Side Lobe Level or Full Width at Half Maximum of the main lobe are quite effective in assessing far field beam patterns or near field beam patterns at a specific depth. However, they fall short in comprehensively capturing the overall quality of the beam pattern along a depth range.

2.3 Simulation setting and coding strategy

We briefly explain how realistic measurement data are simulated and review the coding strategy that is based on a greedy approach and Particle Swarm Optimization.

Let us consider two commercial linear probes, L 4–15 and LX 3–15 from Esaote S.p.A,Footnote 1 hereafter denoted by Probe 1 and Probe 2, with the same depth range of 4.2 cm and different pitch \(\Delta x\) (smaller for Probe 2). Our purpose is to obtain a beam shape as uniform as possible along depth, with the lowest possible level of side lobes. The goal is to increase the DOF of the transmission beam with respect to a standard hourglass-shaped beam pattern related to a fixed focal depth. To do this, we let the delays free, breaking their dependence on the standard focalization law related to a single focal depth. We define the desired BP as a rectangular shape centered at depth 1.75 cm with length of 2.5 cm, with value 1 inside and 0 outside the rectangle (right panel of Fig. 2). The choice of this particular rectangular shape is reasonable under different aspects. First, it forces the BP uniformity along depth, while the zero values at its sides enforce a low level of side lobes. Also, it should be noted that the rectangle does not fully cover the whole depth range, so as to disregard the BP behavior before the first depth of interest, where energy levels are consistently high, as well as beyond a specific depth. This is due to the attenuation factor (cf. Eq. (2)) hindering signal transmission with high frequencies, which makes optimizing the final field portion challenging for linear probes. Finally, it is assumed that the average propagation velocity is homogeneous and equal to \(1540 \,\ m/s\). The study of any BP degradation effects due to changes in propagation velocity is beyond the scope of this paper.

As for the optimization algorithms, we first employ a greedy approach to optimize both the frequency of the narrowband pulse and the aperture. Subsequently, we employ the Particle Swarm Optimization (PSO) Algorithm [21] to optimize the vector of delays, driven by the following rationale. By choosing a finite set of frequencies suitable for the probe, we can generate the corresponding impulse response map set and rapidly simulate a beam. The aperture is an integer and it is reasonable to assume the active surface length is greater than the smallest dimension of the rectangle. The choice of PSO algorithm is justified by the nature of the problem: the free parameters’ domain contains a N-dimensional torus (cf. [31] for the reasoning behind this assertion), therefore every gradient-based algorithm would work in a fixed local chart, thus being very sensitive to the initialization. Conversely, PSO adopts multiple random initialization, making it less affected by the manifold domain structure.

To generate the beam images we use our simulator parUST [31], developed in Python™and freely available on GitHub. The simulator implementation is based on a two-step technique that allows one to pre-compute the system frequency response for a given probe model and region of interest: this methodology is particularly suited when the simulator is used as a kernel in gradient-based or non gradient-based optimization problems to the effect that many different sets of transmit parameters on the beam pattern shape have to be tested. Finally, we usePySwarmsFootnote 2 free software research toolkit to apply PSO algorithm in Python™. At each iteration the algorithm generates a BP for each particle corresponding to a vector of transmit delays, and it measures the adherence between the actual and the prescribed beam. Afterwards, it updates each particle by taking into account its adherence value and the overall direction of the swarm. Regarding the algorithm parameters, we set the cognitive parameter to 0.5, the social parameter to 0.7, and the inertia parameter to 0.9, thus giving more importance to the group behaviour rather than to the single particle movement. It should be pointed out that the overall optimization computational cost can be high (around 8 hours for one experiment), but, on the other hand, the idea is to provide an initial suboptimal solution for conducting practical tests with the machines and then build upon them using a solid set of parameters. In particular, the method proposed and applied in a specific set could be extended to other case studies. The benefit of this would be to streamline the calibration process by providing a set of parameters to serve as a starting point for manual calibration, a step that occurs systematically for each signal acquisition mode.

3 Numerical results and discussion

In this section we show the BPs optimized by means of the proposed technique and we compare them to standard ones for the two linear probes, along with some qualitative and quantitative considerations. Our simulation studies focus on pure tone narrowband signals with the aim of optimizing ARFI shock pulse. The optimization has been carried out by exploiting the in-house developed software parUST [31]. This software allows to use precomputed impulse responses for each element with a sampling over the x-axis corresponding to the finest resolution available on commercial devices. However, using precomputed impulse responses forces the optimization of the frequency of the pure tone and the number of active elements to be performed with a grid search approach. The main disadvantage is that we need to repeat the optimization process for each pairs of frequency and number of active elements. On the other hand, each problem can be solved by using a global optimization approach such as the PSO. In this setting, grid search works as a metaheuristic strategy to identify the optimal regions (frequencies and active elements) while the PSO algorithm plays the role of the fine-tuning optimizer for the delays. PSO solutions are accurate enough to make local optimization methods, e.g., gradient based algorithms, not useful to significantly improve their quality.

For visualization and evaluation of the results, each BP was normalized with respect to its maximum intensity, followed by conversion to dB and truncation of the range of values into \([0, -60]\). This choice was made to ensure that the spatial peak of energy in the field of view is the same for all the solutions. This allows a fair comparison taking into account safety regulations to facilitate future testing for commercial use of the results.

3.1 Qualitative results

First, we qualitatively analyze a set of narrowband BPs obtained with a standard focalization law as a function of focal depth, aperture, and frequency, and compare their shapes with the optimized beam patterns for the two different probes described in Sect. 2.3 (Figs. 4 and 5).

Fig. 4
figure 4

Standard and optimized beam patterns for Probe 1 for two different frequency values and with three different numbers of active elements. For the standard beam patterns we display two different depths of focus

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Fig. 5
figure 5

Standard and optimized beam patterns for Probe 2 for two different frequency values and with three different numbers of active elements. For the standard beam patterns we display two different depths of focus

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One can observe the well-known drawbacks of standard delay profiles. For small apertures, the main lobe is considerably large and its intensity profile rapidly decreases over depth. By increasing the aperture, the main lobe width shrinks around the focal depth but it is enlarged considerably in the near field region, where the intensity peak is not even aligned with the main lobe axis (however, this is also probably due to the lack of apodization). These drawbacks can severely affect the accuracy of shear wave propagation estimation in ARFI elastography [35]. The only difference for Probe 2 is the reduced probability of producing grating lobes, primarily because of the smaller pitch. Figures 4 and 5 display also some examples of the BP obtained by optimizing aperture, frequency, and delay profile. Focusing the analysis on the depth range of interest given by the rectangle of the prescribed beam pattern (right panel in Fig. 2), it can be observed that the intensity peak is located on the main lobe axis at each depth. Moreover, the main lobe width is considerably uniform over the whole depth range especially in the near field in comparison with the BP obtained with the same aperture (28 elements) and a standard delay profile.

Fig. 6
figure 6

Axial profiles of the standard and optimized BPs central line for the two probes. In each plot the dotted lines represent the profiles of standard BPs focalized at two different depths, whereas the continuous black line refers to the corresponding optimized BP

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We can appreciate the optimized BPs having a more uniform main lobe in Fig. 6, displaying the axial profiles for different apertures with fixed frequency \(4 \, MHz\). Standard BPs are represented by the dotted lines: if the transmission focus is closer to the probe, the power drop occurs soon and fast, whereas if the transmission focus is deeper, the maximum power is achieved deeper in depth but it drops soon. The continuous black line represents instead the corresponding optimized profile. We can notice that the power drop is slower and it occurs more deeply. Overall, the profile corresponding to a deeper focus seems closer to the optimized solution. This could be attributed to the fact that our desired BP is precisely aimed at preferring shapes that achieve and maintain a maximum energy as deep as possible. However, on the depths before the peak of the focused profile at \(3 \, cm\), the optimized profile more closely resembles the focused profile at \(1.5 \, cm\) i.e., it reaches maximum intensity earlier.

The optimized delay profile are displayed in Fig. 7 together with the standard profiles for three different focal depths. Some observations are in order. First, in our approach the delay variables are completely independent, meaning the delay profiles are not classical downward-oriented parabolas, but they may exhibit sharp inter-element transitions. The rationale behind it is that each couple of elements could focalize at a different depth, without any constraints, thus exploiting the constructive and destructive interference in a different way with respect to classical methods. Consequently, the delay values are not always decreasing from the central element to the most lateral ones. The relationship of the optimized profile to the family of standard profiles confirms the effectiveness of the proposed approach in optimizing the narrowband TBP. Considering each delay as a free variable has increased significantly the number of delays combinations, therefore augmenting the possible BP shapes. In particular, the optimized profile can be seen as a combination of standard delay profiles at different focal depths.

Fig. 7
figure 7

Comparison between standard and optimized delay profiles for both Probe 1 and 2. Standard profiles are depicted with the dashed grey curves and they get higher with increasing focal depth (from 0.5 to 3.5 cm). Optimized curves are depicted in red, blue, and green, corresponding to minimum MLW value, minimum aSLL value, and maximum CLP value, respectively (see Sect. 2.2.2)

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3.2 Quantitative results

In this paragraph we use the above-introduced metrics to quantitatively evaluate the obtained results in the case of BP simulations and the impact on ARFI elastography.

3.2.1 Evaluation metrics outcomes

Figures 8, 9 and 10 provide a graphical depiction of values for MLW, aSLL, and CLP, both in the standard case and for the optimized beam patterns, computed in the setting of the above introduced probe types. In both cases, metrics values refer to numerical experiments with 5 different frequency values and 5 apertures, while for standard BPs also 3 focus values are taken into account. On each panel’s x axis the different experiments are listed, divided by grey and white bands according to the chosen frequency. As an example, on the leftmost panel of Fig. 8, the first grey band gathers together all the experiments performed at 4 MHz (15 experiments with constant frequency 4 MHz), with 5 different values for the aperture and 3 focus values, while the first white band groups together all the experiments at 4.5 MHz, and so on. For the optimized case (second and fourth panels from the left), each band corresponds to 5 experiments with different aperture values. The black lines in the first two figures represent the mean values of the metrics along depth, while the colored shadows display the corresponding standard deviations. Metrics values are computed over the depth interval defined from \(z_{min} = 0.5\) cm until the end of the field of interest \(z_{max} = 4.2\) cm (cf. Fig. 2), hence including also the area from the end of the chosen rectangle to the end of the field of interest. This indicates that the chosen rectangle length does not severely affect the outcome of the optimization algorithm, which remains better than the standard cases even outside the rectangle limits. In fact, performing the optimization with a longer rectangle does not improve the results as no parameter set is able to reduce the error far beyond the geometrical focus of the probe. From a physical point of view this is reasonably due to attenuation effects, and it has been confirmed by additional tests we do not show here for the sake of brevity. It can be observed that MLW values for the optimized BPs are always lower than the standard case, and with a significantly smaller standard deviation, in line with what one should expect from the optimization algorithm. This indicates that all the 25 optimized BPs have a thinner and more uniform MLW with respect to all the \(5\cdot 5\cdot 3 = 75\) evaluated standard BPs (cf. Fig. 8).

As for aSLL values, they are on average slightly higher than the standard ones, but considerably more stable throughout all the experiments (cf. Fig. 9). Since in ARFI the shear wave velocity is usually estimated detecting the peak of the shear wave in time (cf. Sect. 3.2.2), the presence of a more uniform side lobe pattern should cause less spurious peaks than a more variable pattern. Indeed, one can observe that aSLL for the optimized BPs achieve values in between the best and the worst values of standard BP, both for mean and standard deviation. However, the lowest average aSLL for standard BP is just 1.5 dB lower than the lowest average aSLL for optimized BP and is achieved at the price of a much higher MLW.

Finally, our investigation confirms that CLP values are higher in the optimized case for both probes, once again in line with the desired criteria for optimization (cf. Fig. 10). More significantly, the best values of CLP for standard BP are in general achieved with large apertures, but this condition is associated with worse values of MLW.

Fig. 8
figure 8

Mean values along depth and standard deviations of MLW (mm) over 2 probes (red and blue), 5 frequencies, 5 apertures. For standard BPs, 3 focus values are considered (1.5, 2, 3 cm), hence 3 points in the standard BP plot corresponds to 1 point in the optimized version

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Fig. 9
figure 9

Mean values along depth and standard deviations for aSLL (dB) over 2 probes, 5 frequencies, 5 apertures. For standard BPs, 3 focus values are considered (1.5, 2, 3 cm), hence 3 points in the standard BP plot corresponds to 1 point in the optimized version

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Remark 5

We attempted to use total variation regularization along depth to stretch the main lobe shape, but this approach did not yield any improvements in the results, likely due to attenuation. We refrained from imposing constraints on the frequencies since we relied on the narrowband approximation outlined in [30], but we have exploited the periodicity of the delays in \([0, 1/f_0]\) for better visualization.

Fig. 10
figure 10

Values of CLP (dB) over 2 probes, 5 frequencies, 5 apertures. For standard BPs, 3 focus values are considered (1.5, 2, 3 cm), hence 3 points in the standard BP plot corresponds to 1 point in the optimized version

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To quantitatively evaluate the enhancement of the main lobe width in terms of intensity and uniformity over the whole depth range, with respect to classical focalized Beam Patterns, we compute the average percentage gain over all the conducted experiments. The main lobe width turns out to be considerably more intense (\(+31.5\%\) on average) and uniform (\(+28.7\%\) on average) than the one obtained by using the standard focalized Beam Patterns.

3.2.2 ARFI evaluation

In the context of ARFI elastography, a single ultrasound transducer serves the dual purpose of both inducing and monitoring a deformation response [28]. The distribution of the acoustic radiation force field is spatially governed by the active transducer aperture, the material properties, and characteristics of the transmitted beam, while its magnitude is influenced by factors such as attenuation and intensity. Since attenuation varies with frequency and depth, selecting the optimal frequency to generate an acoustic radiation force becomes an application-specific task, and entails a trade-off between minimizing attenuation losses in the near field and maximizing focal point gain [29]. The configuration of the transmitted excitation beam parameters can be customized to account for these effects.

In order to assess that the optimization of the transmitted beam profile directly affects the desired characteristics of the generated shear waves, the following procedure has been implemented. First, the intensity of the transmitted field was simulated in Matlab with a code based on Field II model both with standard parameters and with optimized parameters. Then, the intensity distribution of the transmitted fields was used as a shock source for shear wave generation with Green’s functions of the Voigt-model-based Navier’s equation, where the Green function is estimated in the three dimensional spatial domain and in the temporal frequency domain [27]. For each shock, we consider the same set of tracking coordinates where to estimate the shear wave. The consistency between our simulator and FIELD II has been assessed while presenting the software in [30].

We set as tracking coordinates an azimuth distance of 0.7, 1, 1.3, 1.5 cm from the shock pulse central line, and depth from the probe surface 0.5, 1, 1.5, 2, 2.5, 3, 3.5 cm. We evaluate the precision of ARFI estimation using three distinct measures. For every combination of depth and azimuth distance, we calculate 1) maximum peak intensity (Fig. 11), 2) the shear wave time of flight (Fig. 12), and for each depth we estimate 3) the shear velocity along azimuth performing a linear regression of times of flight (Fig. 13).

Fig. 11
figure 11

Range of shear wave peak intensities, the lower the range the better

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Fig. 12
figure 12

For each considered tracking distance from the center, we compute the standard deviation of times of flights along depth. We display the mean of these standard deviations. The lower the mean, the more uniform the time of flights

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Fig. 13
figure 13

For each considered depth, we estimate the shear wave velocity along the tracking distance. In the graphs we report the absolute difference between the mean of the seven estimated velocities and the “theorical” one. The lower this difference the better

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As for 1), a high peak intensity produces a higher Signal to Noise Ratio (SNR), which states a lower bound for the shear wave detection. Because of this, the best condition is achieved when the intensity of the peaks is uniform over all the tracking points. For this reason we use as a quality metric the ratio between maximum and minimum values of the peaks. The obtained results, shown in Fig. 11, demonstrate that optimized and standard BPs are comparable under this criterion. Once again, grey and white bands group together all the experiments at a constant frequency value and with varying aperture and focus values. On the other hand, the uniformity of times of flight across depth fits the propagation model of an ideal cylindrical wave, which is at the base of shear velocity estimation methods. In Fig. 12 we show the mean standard deviation of times of flight in the optimized and the standard case, demonstrating the improvements achieved with the optimized BPs. Lastly, for every depth we assess the precision of shear velocity estimation compared to the theoretical value imposed for simulation (we suppose the shear wave to have a propagation velocity of approximately 5.77 m/s and we neglect the viscosity), and we display in Fig. 13 the mean value along depth. Once again, the acquired results confirm that the optimized cases are at least comparable with the standard ones and that, by using the optimized beam patterns, the range between minimum and maximum error is significantly lower with respect to the standard case.

4 Conclusions

Our work provides an effective approach for optimizing the narrowband transmit beam pattern, that aims to ultimately improve accuracy and reduce artifacts in important medical ultrasound applications with a focus on ARFI elastography imaging. It should be noted that we do not hereby consider the imaging improvements, but we provide the necessary tools to quantitatively assess the strengths of our optimization method in the case of ARFI elastography. Indeed, the method provides sets of BPs that are extremely uniform along depth in terms of both BP width and intensity, with an improvement of \(+31.5\%\) on average of the intensity and of \(+28.7\%\) on average in uniformity, and with low side lobe levels, as required by usual biomedical ultrasound applications. Moreover, the direct application to ARFI elastography provides comparable results in terms of shear wave peak intensity and even better results in terms of standard deviation of time of flights and velocity estimation. The adopted strategy makes it easy to extend the calculations to convex probes, by a simple change from Cartesian to polar coordinates, as well as to steered BP.

Our work has some limitations nonetheless. The advantages of this method are not demonstrated with optimized BPs with a different number of active elements, nor with a different BP shape. In this regard, a potential expansion of the approach could involve experimenting Neural Networks with various ground truth shapes as inputs and the optimized delays as outputs. Future developments will also include apodization weights in the set of to-be-optimized variables, thus compensating the slight deterioration of side lobe level observed with the present method with respect to standard delay laws [33]. Finally, the extension to wideband beam patterns will be investigated, as well as the extension of the method to imaging experiments.