An integrated and highly sensitive ultrafast acoustoelectric imaging system for biomedical applications

The acoustoelectric effect, which was first reported by Fox et al (1946), results from the interaction between an acoustic wave propagating through a medium and a coinciding current density. Jossinet et al (1998) have further studied this effect and shown that the acoustic pressure change caused by the wave propagating locally modifies the medium's conductance following the equation below:

$$\begin{align} \frac{\Delta \sigma }{\sigma }=K\Delta P,\nonumber \end{align} \tag{ 1 }$$

with $\sigma$ the conductance, $\Delta P$ the acoustic pressure variation and K an interaction constant (approximately 97 · 10⁻¹¹ Pa⁻¹ in saline (Jossinet et al 1998)).

According to Ohm's law, the voltage change V measured by a remote pair of electrodes can be written as the product of the medium resistivity ρ (inverse of the conductance) with the current density J integrated over the volume measured by the electrodes:

$$\begin{align} V=\iiint{\rho J{\rm d}x{\rm d}y{\rm d}z}\nonumber \end{align} \tag{ 2 }$$

Following equation (1), the resistivity change from its initial value before perturbation by the ultrasound wave is:

$$\begin{align} \rho ={{\rho }_{0}}\left( 1-K\Delta P \right)\nonumber \end{align} \tag{ 3 }$$

Taking into account the contribution of the electrical current density and the electrodes field of view (FOV) into the expression of J, the voltage change V measured by a remote pair of electrodes can therefore be modeled by the following equation:

$$\begin{align} V=\iiint{\left( {{\widetilde{J}}_{{\rm El}}}\cdot {{J}_{{\rm cur}}} \right){{\rho }_{0}}{\rm d}x{\rm d}y{\rm d}z}+\iiint{-\left( {{\widetilde{J}}_{{\rm El}}}\cdot {{J}_{{\rm cur}}} \right)K{{\rho }_{0}}\Delta P{\rm d}x{\rm d}y{\rm d}z},\nonumber \end{align} \tag{ 4 }$$

where ${{\widetilde{J}}_{{\rm El}}}$ is the electrode field density and ${{J}_{{\rm cur}}}$ the current density. The first part of equation (2) corresponds to a low frequency term and can be filtered out in order to retain the high frequency and thus high resolution acoustoelectric signal (Olafsson et al 2008).

UAI is based, like ultrafast ultrasound imaging (UUI), on the use of ultrasound plane waves to form an image. Hence, each emission can be used to map the entire FOV, leading to frame rates that can be equal to the pulse repetition frequencies (PRFs), i.e. up to 10 000 Hz. Plane waves can also be emitted at different angles and resulting echoes are coherently compounded within the beamforming process to produce a high quality image of the object. UUI can produce images with resolution and contrast comparable to focused ultrasound for a limited number of angles, thus maintaining a high frame rate (Montaldo et al 2009).

Similarly, the concept of coherent compounding can be applied to UAI. Ignoring the elevational direction z in equation (2) above, the measured voltage difference due to the acoustic pressure variation can be written as follows:

$$\begin{align} {{V}^{\prime}}=\iint{J\left( x,y \right)P\left( x,y \right){\rm d}x{\rm d}y}\nonumber \end{align} \tag{ 5 }$$

with ${{V}^{\prime}}=\frac{V}{K{{\rho }_{0}}{{P}_{0}}}$, J the current density resulting from the convolution of the electrodes field and the current distribution, P₀ the pressure amplitude and P the pressure field. Considering a plane wave emitted with an angle θ relative to the transducer array and the coordinates q and ct representing the wavefront, as pictured on figure 1(a), equation (3) can be rewritten as:

$$\begin{align} {{V}^{\prime}}=\int{\int_{-\infty }^{+\infty }{J\left( x,y \right)\delta \left( x{\rm sin}\theta +y{\rm cos}\theta -ct \right){\rm d}x{\rm d}y={\rm RT}[J]\left( \theta,ct \right)}},\nonumber \end{align} \tag{ 6 }$$

where RT [J] is the Radon Transform of J, c is the acoustic wave velocity, t is the time, and δ is the Dirac distribution.

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Ultrasound arrays emit waves with a finite bandwidth, which leads to a convolution of the Radon Transform in equation (4) with a term W(ct) representing the emitted waveform:

$$\begin{align} {{V}^{\prime}}=W(ct)*{\rm RT}\left[ J \right]\left( \theta,ct \right)\nonumber \end{align} \tag{ 7 }$$

Therefore, the Radon Transform is limited to regions of the k-space corresponding to the frequency bandwidth, as pictured in figure 1(b). This reduces the axial resolution of the system, which is linked to the ultrasound pulse length, as in focused ultrasound imaging.

In addition, the angular sensitivity of piezoelectric elements and the finite length of the transducer arrays limit the maximum angle at which plane waves can be emitted. This reduces the lateral resolution R_L, which depends on the maximum angle θ_max as follows:

$$\begin{align} {{R}_{{\rm L}}}=\frac{c}{{{f}_{{\rm max}}}}\frac{1}{{\rm 2sin}\left( {{\theta }_{{\rm max}}} \right)}\nonumber \end{align} \tag{ 8 }$$

where c is the speed of sound in the medium and f_max the transducer maximum frequency.

Finally, only a finite number of angles can be emitted in practice, which reduces again the sampling available for the RT as shown on figure 1(b). This affects the system's lateral FOV, defined here as the distance between grating lobes resulting from the pattern of interferences between the combined plane waves. The FOV is directly linked to the step between equally spaced angles $\Delta \theta$ as follow:

$$\begin{align} {\rm FOV}=\frac{c}{{{f}_{{\rm max}}}}\frac{1}{{{}{{{\rm sin}\left( \Delta \theta \right)}}}}\nonumber \end{align} \tag{ 9 }$$

As a consequence, although a larger maximum angle θ_max ensures a better resolution (smaller R_L), for a given PRF (i.e. number of angles), it also results in a smaller FOV.

The integrated UAI scanner is based on a 256-channel Vantage (Verasonics, Kirkland, USA) research platform fitted with a 128-element, 5 MHz linear array US probe (L7-4, ATL Ultrasound Inc., Bothell, USA) on one 128-element connector, and a break-out board allowing for the direct connection of individual channels on the second 128-element connector. The transducer was positioned with its pitch along the Y axis in order to direct the ultrasound pulse between the electrodes aligned along the Z axis, as depicted in figure 2. Channels 1–128 of the system were connected to the transducer and used for the emission of the acoustic wave. The 128 additional channels available via the break-out board were used in receive mode only to sample the UAI signal via measuring copper wire electrodes. The signal from the electrodes was high-pass filtered above 1 MHz and amplified using a high common-mode rejection ratio instrumentation amplifier and fed into the Vantage system via one selected channel of the break-out board with a gain of 1. Within the Vantage system, the signal underwent low-noise pre-amplification of 18 dB, controllable signal reduction at varying depth (set to 0 dB for all depths in the present configuration), and a subsequent amplification of 24 dB.

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The output of the integrated UAI scanner described above was used to form 2D images of the current density using a novel holographic image formation approach. This approach is analogous to optical holography in that it decodes the received signal by comparing it to a set of reference signals of known response by the system. It is based on the explicit construction of the direct problem matrix G, which links the object to image x and the demodulated complex measurement vector y as follows:

$$\begin{align} y=Gx\nonumber \end{align} \tag{ 10 }$$

G is therefore a matrix with a number of columns equal to the number of pixels in the image x and a number of rows equal to the number of samples acquired. Since G approximately amounts to a bandlimited RT, as shown previously, the inverse problem is ill-posed and can be solved in a first approach using the pseudo-inverse formulation:

$$\begin{align} x={{\left( {{G}^{\dagger }}G \right)}^{-1}}{{G}^{\dagger }}y\nonumber \end{align} \tag{ 11 }$$

However, this consists in a deconvolution of the signal which is unstable, and may lead to added noise in the reconstructed image. G^†G is a time-reversal operator and is thus close to identity if no ultrasonic absorption is considered during the building of G matrix (Tanter et al 2000). In more complex media where ultrasonic absorption occurs and results in degraded resolutions, the pseudo-inverse approach corrects for absorption effects through ${{G}^{\dagger }}G$ inversion (Aubry et al 2001, Tanter et al 2001). Hence, the pseudo-inverse is similar to using a back-projection approach which amounts to projecting the time-reversed impulse response function such that

$$\begin{align} x={{G}^{\dagger }}y\nonumber \end{align} \tag{ 12 }$$

This formulation, however, does not take into account the individual pixels' energy. A more accurate approximation of the pseudo-inverse is the following:

$$\begin{align} x={\rm diag}{{\left( {{G}^{\dagger }}G \right)}^{-1}}{{G}^{\dagger }}y\nonumber \end{align} \tag{ 13 }$$

In this case, the back-projection approach is normalized by the individual pixels' energy, without the need for deconvolution.

In the context of this work, the holographic image formation method was used for direct reconstruction of the 2D acoustoelectric image from a single acquisition channel only used in receive mode. Building G for a given emission waveform and imaged medium involves defining a set of basis vectors for the object space and determining the system response for each of these vectors. This was done using the Vantage system tool to simulate the emitted pressure waves and their value for each pixel. The simulation of one pixel allowed for the population of a single column of G. The integrated UAI scanner records the local voltage linked to the direct interaction of the acoustic wave and the current density, so only the transmitted waveform was of interest to construct G. The rows for each column (i.e. pixel) represent the transmission for each of the emitted plane waves along the whole FOV depth. As a consequence, the number of rows in G depended only on the number of plane waves emitted, and the imaging depth. The size of G in this work was therefore sufficiently small to allow for minimal computing time. Similar techniques have been used previously for sonar mapping (Tondin et al 2015), adaptive focusing in therapeutic ultrasound (Seip et al 1994) and image quality enhancement for medical ultrasound (Viola et al 2008), as well as for studies involving compress sensing, but to our knowledge, it is here applied to the reconstruction of UAI images for the first time.

Figure 4(a) shows the resolution measurements of UAI data for 3, 5, 7, 9, 13, 15, 21 and 33 plane waves emitted at different angles, compared to the case of focused acoustoelectric imaging. The maximum steering angle was 16° for 33 plane waves. Examples of reconstructed images are given on figure 4(b) for 9 and 33 plane waves with no averaging. The resolution was measured as the width of the  −10 dB peak, as illustrated for the case of 33 plane waves on figure 4(c). The resolution improved (i.e. the peak width decreased) with the number of plane waves, reaching equivalent values to the focused wave case for 13 plane waves. The resolution improved for higher numbers of angles to reach values of 2.1 mm for 33 plane waves, corresponding to less than a third of the value obtained for focused waves at an F-number of 2.

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Figure 5(a) shows the SNR measured in the UAI images obtained previously for different numbers of plane waves. The SNR increased with the number of plane waves to reach a value 27 dB higher than for focused waves, for an F-number of 2. The SNR for UAI reached a similar value to the focused waves case when 3 plane waves were coherently compounded.

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Both resolution and SNR were measured on 4 different datasets acquired with the same experimental setup and protocol, but with new tinned copper wire electrodes each time. Standard deviations (in percentage of the mean value) between the mean values obtained over 100 frames for each dataset ranged from 7% (mean 19 mm) to 26% (mean 2.4 mm) for resolution and from 3% (0.56) to 13% (24,7) for SNR.