INTRODUCTION

Multi-element ultrasound array transducers are being extensively used for both diagnostic and therapeutic purposes [1, 2]. In contrast with diagnostic ultrasound arrays, therapeutic arrays developed for sonicating abdominal organs typically are strongly focused, have larger overall dimensions and dimensions of the elements, operate at lower frequencies of 1–1.5 MHz and at higher power levels [3, 4]. Such geometry and frequency range are not suitable for good quality imaging compared to the less focused abdominal diagnostic probes of 4–7 MHz, optimized for providing good imaging resolution [2]. While attempts have been made to design therapeutic arrays for dual-mode use [5, 6], most ultrasound-guided therapeutic systems comprise two separate transducer arrays, with a smaller, less focused diagnostic probe placed in the central opening of the high-power large aperture transducer.

Recently, a novel dual-mode diagnostic-type probe comprising a high-frequency imaging array stacked on a low-frequency therapeutic 2D array has been designed for image–guided acoustically-mediated drug delivery to cancerous skin tumors [7]. The proposed treatment is based on the use of contrast agents and thus requires relatively low therapeutic pressures [8, 9]. However, it has been also shown that small-size diagnostic probes are capable of generating focused ultrasound fields with higher pressures sufficient for therapeutic effect, even for generating nonlinear pressure waveforms with shock fronts at the focus [10]. A novel design of such diagnostic-type array has been proposed to combine the capabilities of both inducing de novo cavitation (i.e., cavitation without use of contrast agents) to enhance drug therapy efficacy in abdominal organs and providing ultrasound imaging for treatment visualization [11]. The frequency of 1 MHz was chosen as it has been shown to be effective for enhancing the drug delivery to pancreatic tumors [12, 13]. For such a low frequency, conventional imaging quality would be compromised but for bubble-based sonoporation applications should be sufficient due to the high contrast of bubbles. A computer model of a linear multi-element phased array was developed and numerical simulations of the fields it generates were conducted, confirming the feasibility of the transducer’s operation at the pressure levels and degree of nonlinear waveform distortion, which are sufficient for inducing therapeutic effect of cavitation in tissues [11].

Subsequently, a preclinical prototype of the array, consisting of 64 piezoelectric elements, was manufactured by Sonic Concepts (Bothell, USA) following the specifications of the computer model. A series of experiments have been conducted using this prototype with the goal of characterizing its acoustic field and assessing its suitability for the therapeutic and imaging purposes [14]. The fields produced by the fabricated transducer in various electronic focusing configurations may differ from the results predicted by the ideal computer model. To assess such differences, acoustic holography measurements have been conducted to reconstruct the distributions of the oscillatory velocity on the transducer’s surface for selected focusing configurations of the array [14]. These distributions can serve as boundary conditions for calculating acoustic fields in the space in front of the transducer. However, in order to assess all steering configurations and evaluate the steering range of the array, numerous holograms should be recorded which is time intensive and may not be feasible.

The main goal of this study was to use linear acoustic simulations with experimentally reconstructed boundary conditions to evaluate the performance of the manufactured transducer in comparison to its initial computer design. Acoustic holography data for selected spatial configurations of full array operation and for each individual array element were employed to analyze the variability of the elements in power and spatial field pattern, validate the accuracy of synthetic reconstruction of arbitrary electronic focus steering configurations, and determine the steering range of the array compared to its computer model.

METHODS

For the purposes of computational modeling, the transducer was represented as a rectangular phased array with a total aperture of 14.8 by 51.2 mm and an operating frequency of 1.071 MHz. The array model consisted of 64 elements, each with dimensions of 14.8 by 0.72 mm and a kerf between the elements of 0.08 mm. A uniform distribution of the normal component of the vibrational velocity at the surface of the elements was assumed. During the design of the array, acoustic fields created with two modifications of the array geometry, plane and cylindrically focused (Fig. 1), were evaluated, and the final shape of the array surface was chosen between them. Focusing in the imaging xz plane was achieved electronically in both cases by adjusting the phase at each individual element of the array. This procedure allows for translating the focus and thus increasing the dimensions of the sonicated area without mechanical movement of the array.

Fig. 1.
figure 1

Illustration of (a) planar and (b) cylindrically focused geometries of the array surface.

Full size image

An array transducer was manufactured by Sonic Concepts (Bothell, USA) with nominal parameters corresponding to the computer model (Fig. 2) [11]. Due to complexity of manufacturing cylindrically shaped elements, focusing in the elevational plane was achieved using an acoustic lens. Acoustic holograms were measured in a plane z ≈ 30 mm parallel to the surface of the transducer within the spatial window larger than the diameter of the ultrasonic beam to ensure the complete registration of the generated acoustic field [14].

Fig. 2.
figure 2

Photograph of the manufactured array transducer.

Full size image

All acoustic simulations of the array field were performed using properties of water as the propagation medium. To calculate the phase delays that should be applied to each array element for electronic focusing of the acoustic beam to a specified location, a ray approximation was employed. In order to focus the beam at a point with coordinates (xf, 0, zf) the phase is calculated as:

$${{\varphi }_{i}} = k\left( {\sqrt {x_{f}^{2} + z_{f}^{2}} - \sqrt {\left( {x_{i}^{2} - x_{f}^{2}} \right) + z_{f}^{2}} } \right),$$
(1)

where xi is the coordinate of the center of the i-th array element and k = ω/c is the wave number at angular frequency ω and small-signal sound speed c. In case of a cylindrical modification of the array, additional focusing was achieved in the yz plane due to its curved geometry (Fig. 1b).

Field simulations were performed using the Rayleigh integral when the irradiating surface of the array was cylindrically curved, and the angular spectrum method was used for planar elements [15–17]. For a known distribution of the normal component of the vibrational velocity on the surface of an arbitrary shape, the approximate solution for the pressure field in the form of the Rayleigh integral can be written as:

$$p\left( {x,y,z} \right) = - \frac{{i\omega \rho }}{{2\pi }}\int\limits_\Sigma {{{v}_{n}}({{x}_{1}},{{y}_{1}},{{z}_{1}})\frac{{{{e}^{{ikR}}}}}{R}} d\Sigma .$$
(2)

Here \(p\left( {x,y,z} \right)\) is the complex pressure amplitude at an arbitrary observational point in space with coordinates (\(x,y,z\)), (\({{x}_{1}},{{y}_{1}},{{z}_{1}}\)) are the coordinates of the surface element, \(R = \sqrt {{{{\left( {x - {{x}_{1}}} \right)}}^{2}} + {{{\left( {y - {{y}_{1}}} \right)}}^{2}} + {{{\left( {z - {{z}_{1}}} \right)}}^{2}}} \) is the distance between the surface element and the observation point, ω is the angular frequency, vn is the normal component of the vibrational velocity at the active irradiating surface Σ. In case of an ideal transducer, it is assumed that vn is uniformly distributed along the surface of the array elements: \({{\left. {{{v}_{n}}(x,y,z)} \right|}_{\Sigma }} = {{v}_{n}}_{0}\).

For a known pressure distribution in a plane z = 0, the corresponding angular spectrum is calculated as:

$$\begin{gathered} S({{k}_{x}},{{k}_{y}},z = 0) \\ = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {p\left( {x,y,z = 0} \right){{e}^{{ - i\left( {{{k}_{x}}x + {{k}_{y}}y} \right)}}}} dxdy} , \\ \end{gathered} $$
(3)

and the solution for the spectrum at a distance z can be expressed through the original one using a propagator:

$$\begin{gathered} S({{k}_{x}},{{k}_{y}},z) = S({{k}_{x}},{{k}_{y}},z = 0){{e}^{{i{{k}_{z}}z}}}; \\ {{k}_{z}} \equiv \sqrt {{{k}^{2}} - k_{x}^{2} - k_{y}^{2}} . \\ \end{gathered} $$
(4)

The complex pressure amplitude at an arbitrary point in space located at least several wavelengths away from the source is then given by

$$\begin{gathered} p(x,y,z) = \frac{1}{{4{{\pi }^{2}}}} \\ \times \,\,\iint\limits_{k_{x}^{2} + k_{y}^{2} \leqslant {{k}^{2}}} {S({{k}_{x}},{{k}_{y}},z = 0){{e}^{{i\left( {{{k}_{x}}x + {{k}_{y}}y + {{k}_{z}}z} \right)}}}}d{{k}_{x}}d{{k}_{y}}. \\ \end{gathered} $$
(5)

In processing the experimental holography data acquired in the plane, the Rayleigh integral was used for preliminary numerical processing, which involved rotating and centering the original holograms to eliminate inaccuracies in the relative positioning and alignment of the hydrophone and scanning apparatus and the array [18]. Ideally, the recording plane should be normal to the axis of the array, but experimentally, certain displacements and angular deviations are inevitable. Then, the distribution of the complex pressure amplitude on the array’s surface was reconstructed using the angular spectrum method. To calculate field distributions in the imaging and elevational planes, another form of the Rayleigh integral was used:

$$p\left( {x,y,z} \right) = - \frac{1}{{2\pi }}\int\limits_\Sigma {p\left( {{{x}_{1}},{{y}_{1}},{{z}_{1}}} \right)\frac{\partial }{{\partial {{n}_{1}}}}\left( {\frac{{{{e}^{{ - ikR}}}}}{R}} \right)} d\Sigma ,$$
(6)

where n1 is the normal to the oscillating surface element directed away from the source [17].

Experimental data provided for this numerical study were obtained for different array operating configurations. Three characteristic holograms measured for cases where all array elements were active, as detailed in [14], were used to compare the fields created by the real array and predicted by the initial computer model. In addition, similar to the approach used earlier for another array, 64 holograms were measured for cases where each individual array element was active [19]. These holograms were used here to analyze the variability in power and irradiating pattern of the elements, the mechanical and electrical interactions (cross talk) between the array elements that could alter the field pattern [20], as well as for numerical synthesis of the array’s acoustic field for arbitrary electronic focusing and steering configurations. This allowed for estimating the electronic focus steering range of the array, defined as the region in space where the amplitude of the acoustic pressure at the focus differs by less than 10% from the corresponding value when focused at 50 mm along the acoustic axis of the transducer.

All acoustic characterization simulations were conducted assuming linear wave propagation in water at 20°C with ambient density ρ = 997.6 kg/m3 and small-signal sound speed c = 1482.4 m/s [14]. The spatial steps in Rayleigh integral calculations were 0.65 mm in x and y directions and 0.2 mm for the acoustic axis direction, and acoustic measurement windows were ±58.5 and ±22.75 mm in the x and y directions, respectively, when all transducer elements were active, and ± 55.25 and ±19.5 mm in the x and y directions, respectively, when measuring individual elements. The number of angular spectrum spatial frequency components were 180 by 70 and 160 by 70, correspondingly.

RESULTS

Shown in Fig. 3 are pressure amplitude distributions simulated for the planar (blue) and cylindrically focused (red) array models presented in Fig. 1 along the acoustic axis (a) and transversely in the focal plane (\(z\) = 50 mm) along the x (b) and y (c) axes. The results demonstrate that the cylindrical modification of the transducer geometry yields an enhancement of the focusing gain of the array by 17.6%. In addition, it provides lower pressure levels in the near field of the array for the same focal pressure, and sharper focusing as evidenced by a narrower beam width along the y axis, which could be beneficial for therapeutic sonications. To quantitatively assess the spatial structure of the high-amplitude focal area, an amplitude pressure level of –6 dB from the corresponding pressure amplitude at the focus were chosen as metrics. The cylindrical modification of the array surface led to a reduction in the dimensions of the focal region transversely in the focal plane by 4% along the x axis and by 45% along the y axis. Due to these beneficial features for the therapeutic use, all subsequent simulations were conducted for the case of a cylindrical transducer geometry.

Fig. 3.
figure 3

Pressure amplitude distributions, normalized to the initial pressure amplitude p0, for the planar (1) and cylindrically focused (2) array models along (a) the acoustic axis and transversely in the focal plane along the (b) x axis, and (c) y axis.

Full size image

Figure 4 illustrates characteristic features of the vibrational pattern of the active surface of the array reconstructed based on the holography data. Two-dimensional amplitude and phase distributions of the acoustic pressure on the transducer surface are presented for three different phasing configurations of the array operation: all elements operated in phase (Fig. 4a); electronic focusing of the array on its axis at z = 50 mm (Fig. 4c); the focus was electronically shifted transversely to x = 10 mm at z = 50 mm axially (Fig. 4e). All three holograms were sequentially acquired in the same plane. The original holograms were first rotated and centered, and then backpropagated to the parallel plane of the transducer surface, z = 0, using the angular spectrum method [16, 17]. The rotation angles were –2.00° around the x axis and 0.06° around the y axis showing good positioning of the transducer and holography plane during the measurements. An additional 0.6° rotation around the acoustic axis of the transducer was performed to accurately align the rectangular contours of the transducer with the hologram boundaries [14, 18].

Fig. 4.
figure 4

Boundary conditions for the acoustic pressure amplitude (left column) and phase (right column) on the surface of the array transducer, reconstructed from the measured holograms: (a, b) without electronic focusing (in-phase case); (c, d) focusing at z = 50 mm, x = 0; (e, f) focusing at z = 50 mm, x = 10 mm.

Full size image

In case of in-phase operation (Figs. 4a, 4b), the results demonstrate a good quality of the field pattern along the surface of the array: a visually uniform pressure amplitude distribution, a relatively uniform phase distribution along the x axis, and symmetric phase variation along the y axis corresponding to the cylindrical focusing of the array in the elevational plane. Note that the elements of the array are not visually resolved due to the presence of a lens, potential cross talk between the elements, and relatively large reconstruction grid step of 0.65 mm comparable to the kerf of 0.10 mm. In the case of focusing at 50 mm on the array axis, at the center of the cylindrical curvature of the array (Figs. 4c, 4d), the phase distribution is symmetric in the imaging plane and shows phase variation along the x axis corresponding to additional phase shifts at the elements introduced for electronic focusing. However, some asymmetry in pressure amplitudes of the elements and smaller amplitudes at the peripheral elements are observed presumably caused by longer, less perpendicular, path through dissipative acoustic lens toward the focus compared to the central elements. Similarly, an even stronger variation of the reconstructed source amplitude is observed for the third case, when the focus is additionally steered electronically in the transverse direction to the point z = 50 mm, x = 10 mm (Figs. 4e, 4f): about a twofold smaller amplitude at the elements with larger angles from their axis toward the focus. Note, that reconstruction of the transducer vibration was performed assuming backpropagation in water from the holography plane, therefore the decrease of the amplitude of the peripheral elements for the cases of focus steering is likely an artifact of the lens as it is not observed for the in-phase propagation and correlates with longer path in the lens.

Acoustic pressure distributions in the two axial planes, xz and yz, that correspond to the two focusing array configurations shown in Figs. 4c–4f were calculated using the Rayleigh integral (6) and are presented in Fig. 5. In the imaging plane (Figs. 5a, 5c), the results show translation of the focus to the desired location and tight focusing. Distributions in the elevational plane illustrate the thickness of the high-amplitude focal region of 9 mm at –6 dB level from the maximum where therapeutic effect can be induced by corresponding compensation in the array’s power output. While the field patterns in the yz plane appear visually symmetric, some asymmetry in the amplitude distribution can be noticed in the imaging plane xz in the pre-focal region (Figs. 5a, 5b). In particular, such asymmetry in the field distribution was not expected for the case of focusing on the axis of the array (Fig. 4a), and thus indicates some asymmetry in the array construction or some variances in operation between elements of the array.

Fig. 5.
figure 5

Distributions of the acoustic pressure amplitude in the xz plane (top row) and in the yz plane (bottom row) simulated using boundary conditions reconstructed from the holography data when focusing at (a, b) z = 50 mm, x = 0 and at (c, d) z = 50 mm, x = 10 mm.

Full size image

The results of a more detailed examination of the non-uniformity in the performance of the array elements were obtained by analyzing experimental holography data measured individually for each of them [14]. Similar to the three cases of full array operation, the processed acoustic holograms provided boundary conditions for the pressure amplitude and phase of each element of the transducer. Corresponding differences between the elements were analyzed, quantified, and the results are presented in Fig. 6. Mean values and standard deviations were calculated for the total power emitted by each element of the array (Fig. 6a), as well as the beam widths in xz and yz planes parallel to the corresponding sides of the elements (Figs. 6b and 6c) at a distance of z = 50 mm from their geometric center, which corresponds to the focal length of the acoustic lens. Through the analysis, it was determined that the acoustic power of the array element under the holography measurement conditions is 111 ± 9 mW, the beam width along the x axis is 44.4 ± 2.4 mm, and along the y axis it is 5.23 ± 0.01 mm. Such variations, mostly in the power and directivity in the imaging plane, are potentially responsible for asymmetries observed in the acoustic field when steering the focus electronically. These variations could be partially caused by non-uniformity of the acoustic lens in front of the elements and, for the most peripheral elements, by the casing of the array.

Fig. 6.
figure 6

Characteristics of different array elements computed from their individual holograms: (a) acoustic power; beam width at a distance of z = 50 mm from the transducer’s surface along the (b) x-axis and (c) y-axis. The solid black lines represent the mean values, and the dashed black lines represent the mean ± standard deviations.

Full size image

In Fig. 7, a more detailed examination of the pressure amplitude distribution at the surface of elements with the lowest (element number N = 1 in Fig. 6), highest (N = 5), and closest to the average acoustic power (N = 17) is provided, along with the boundary condition for the uniformly vibrating element of an ideal computer model. A slightly inhomogeneous pressure distribution and small increase in the area of the oscillating surface can be observed in all cases compared to the ideal pattern, which may be attributed to the course spatial sampling used in the measurements, some non-uniformity of lens, and weak cross talk, either mechanical or electrical, with the neighboring elements [20].

Fig. 7.
figure 7

Examples of the pressure amplitude distributions on the surface of the array elements with (a) the lowest, (b) average, and (c) highest power; (d) distribution for the uniformly vibrating element in the ideal transducer model.

Full size image

Directivity properties of the selected elements shown in Fig. 7 are depicted in Fig. 8. Distributions of the acoustic pressure amplitude were calculated in front of the elements in two transverse planes parallel to their sides and passing through their centers. The pressure distributions in the xz plane show that the directivity pattern of each element is noticeably different from the ideal case—for all examined elements, the generated beam width is about twofold smaller at a distance z = 50 mm from the elements. This decrease in directivity of the transducer elements would be expected to affect the steering capabilities of the array, which will be demonstrated below. In the elevational plane, the acoustic field pattern is relatively homogeneous and similar in structure for all transducer elements and an ideal element, as reflected in both the similarity of the directivity patterns and the small dispersion in the beam width along the respective axis.

Fig. 8.
figure 8

Pressure amplitude distributions in the xz plane (top row) and yz plane (bottom row) for the elements with (a), (e) the lowest power; (b), (f) average power; (c), (g) highest power; and (d), (h) for the ideal uniformly vibrating element.

Full size image

In further simulations, the reconstructed holograms of individual elements were used to analyze the electronic focusing range of the array transducer. The synthesis of the boundary conditions using the holograms of individual elements with additional phases corresponding to the case of on-axis focusing demonstrated that the calculated boundary conditions are in good agreement with the same focusing configuration experimentally measured during full-array operation (Figs. 9a, 9b). Consequently, both the mechanical and electrical interaction between elements of the transducer have been completely captured. Comparison of the corresponding acoustic pressure amplitudes calculated in the image plane and the elevational plane at z = 50 mm, also showed excellent agreement (Figs. 9c−9f).

Fig. 9.
figure 9

Pressure amplitude distributions (a), (b) on the surface of the array at z = 0; (c), (d) in the imaging plane; and (e), (f) in the elevational plane. The distributions were reconstructed from holography data when focusing on the axis of the array at z = 50 mm with all operating elements (top row) and as a synthesis from holograms of all individual elements (bottom row).

Full size image

Based on the results that validated the accuracy of the synthetic approach, an analysis of the electronic focusing range of the array transducer was performed using holograms of the individual elements through the synthesis of boundary conditions with the addition of phase delays that correspond to a range of different focusing and steering distances. As previously defined, the acoustic pressure amplitude in the focus, differing by 10% from the corresponding amplitude when focusing at x = 0 and z = 50 mm, was chosen as the boundary of the focusing range. To determine this range, calculations of pressure amplitude maxima were carried out for field configurations focusing over an spatial grid in x, z coordinates with steps between the foci of 0.1 mm.

As depicted in Fig. 10a, the boundaries of the electronic focusing range along the acoustic axis for the transducer array practically coincide with those of an ideal computer model (from 45 to 55 mm). Blue curves in Fig. 10 are obtained by taking the peak pressures from the set of steered beam profiles. When examining the field in the focal plane along the x axis (Fig. 10b), it can be noticed that the electronic focusing range is about twofold smaller, which correlates with the reduced directivity of the array elements. For an ideal computer model, the transverse steering range is 44 (±22 mm), while for the manufactured array transducer, it decreases to 20 (±10 mm).

Fig. 10.
figure 10

Normalized pressure amplitude distributions (a) along the acoustic axis of the array for various focusing distances and (b) along the x axis in the focal plane, z = 50 mm, with different steering angles for the ideal array with uniformly vibrating elements (solid lines) and obtained by synthesizing holograms of the individual elements (crosses). The blue curves represent the envelope of the distributions (peak pressures) as the position of the focus is varies.

Full size image

Finally, two-dimensional distributions showing the pressure amplitude at the peak of the steered focus were also constructed when focusing at each grid point (x, z) in the image plane for the array transducer and the ideal computer model (Fig. 11). The steering range of focusing was determined based on the aforementioned definition of keeping the peak pressure within ±10% of the pressure amplitude at the nominal focus (x = 0, z = 50 mm) and is shown as rectangular contours for both arrays. Even with the reduced range of transverse steering for the array transducer, the total volume that could be irradiated with the array using electronic steering is estimated as 1 cm3, which is clinically relevant for ultrasound-based drug delivery sonications [21].

Fig. 11.
figure 11

Distributions of the normalized peak focal pressure amplitudes in the imaging plane (x, z) when the beam is focused at each point on the computation grid, (a) for the boundary conditions synthesized from individual holograms of the transducer elements and (b) for the ideal array model with uniformly vibrating elements. The green contours represent the boundary of the steering range defined at ±10% from the pressure level when focusing at z = 50 mm.

Full size image

CONCLUSIONS

In this study, the steering performance of a dual-mode 64-element ultrasonic phased array with a center frequency of 1 MHz frequency and manufactured according to the design-stage computational model was assessed for potential use in cavitation-based drug delivery applications in abdominal organs. The benefits of cylindrical focusing of the transducer was demonstrated in design simulations, resulting in a 4% reduction of the focal dimensions along the x axis and a 45% reduction along the y axis, and a 17.6% increase in pressure amplitude at the focus. For the fabricated transducer array, cylindrical focusing was achieved using an acoustic lens.

Acoustic holography measurements were used to obtain boundary conditions, characteristics of the generated acoustic field, and the electronic focusing range of the transducer array. Characteristics of individual array elements were also analyzed based on their individual acoustic holography data. The accuracy of synthesizing various focusing configurations of the array using holograms of its elements and adding corresponding phases to the elements was demonstrated.

The electronic steering range of the array transducer then was assessed using holograms of its individual elements by synthesizing boundary conditions and resulting acoustic fields for a range of different focusing configurations and comparing them with predictions from the design-stage numerical model. It was shown that the overall field structure of the elements and the array transducer generally correspond to those of an ideal array, although with a directivity for each element which is smaller than predicted. As a result, the focusing range in the transverse direction decreased by a factor of about 2 for each element and for the array (from 44 to 20 at 50 mm distance), presumably due to electrical and mechanical cross talk between the elements and absorption in acoustic lens.

The main result of this work is that a low-frequency dual-mode linear ultrasound array with small diagnostic-type elements can be fabricated to follow the properties of the design-stage numerical model and provide a clinically relevant steering range for drug-delivery applications. Steering capabilities of such an array can be examined based on multiple simulations of the fields synthesized from individual holograms of its elements with corresponding phase shifts applied.