A Human-Compatible Gradient Coil for Visualizing Ultrasound Propagation

Abstract

In pursuit of a magnetic resonance imaging (MRI) based means to tailor transcranial focused ultrasound neuromodulation to a patient’s unique skull morphology, this study presents a specialized gradient coil that sensitizes MRI images to ultrasonic vibrations at depths equivalent to the human cortex in a technique called the magnetic resonance hydrophone. The coil comprises a 60 mm diameter, pancake-style design that encodes acoustic displacements into MR images at the cost of an inhomogeneous encoding field. The coil was coupled with a 500 kHz, custom built, ultrasonic transducer. Both the magnetic field gradient of the coil and the acoustic field of the transducer were characterized in benchtop experiments. Acoustic standing waves were estimated in silico. Resulting MR images displayed a sinusoidal phase pattern modulated by both the transducer’s acoustic field and the coil’s magnetic field gradient. Acoustic pressures were estimated from the resulting images and compared to hydrophone measurements. The pancake-style coil produced a pressure measurement uncertainty pattern due to electronic noise that increased exponentially with depth. Uncertainty at locations between 0 and 30 mm of depth within a region approximately 10 mm wide scaled between approximately 20 kPa and 100 kPa. On average, the MRH underestimated the hydrophone by 12 kPa with the difference between the two following a standard deviation of 21 kPa.

1. Introduction

Transcranial ultrasound neuromodulation (TUSN) promises to treat mental health disorders by transmitting low intensity ultrasound waves through the intact scalp and skull into the brain. These waves carry thermal and mechanical energy that then modulate neural activity[1]. Unlike their electromagnetic counterparts, acoustic fields can be shaped to produce foci at nearly any depth within the patient, resulting in flexible and focal treatment throughout the human brain[2,3]. TUSN promises to combine non-invasiveness, depth penetration, focal effects, and flexible steering to modulate both individual nuclei and distributed neural networks.

Despite positive anecdotal treatment outcomes [4–8], TUSN, as is currently practiced, may fail to treat a large, diverse patient population. Human skulls possess incredible diversity in acoustic properties[9–11]. This diversity conspires to produce, between patients, up to four-fold deviations in expected acoustic transmission [12]. Without some mechanism to detect and compensate for these alterations, large deviations limit treatment benefits to patients with transmissive skulls.

Magnetic resonance imaging (MRI) has historically played an important role helping clinicians deal with skull-induced acoustic aberration during transcranial focused ultrasound surgeries (TUSS), where the intensity of the acoustic field is sufficiently strong to induce permanent thermal necrosis or mechanical fractionation. For example, ultrasound-induced temperature increases can be mapped via the proton-resonance-frequency shift effect[13,14] and thermal necrosis can be observed via T2-weighted[15–18] and diffusion-weighted[19,20] imaging. Mechanical fractionation can be observed via diffusion-weighted imaging[21,22] and acoustic cavitation can be observed using intravoxel incoherent motion imaging[23–25]. Clinicians can, in a patient-specific manner, use these metrics to localize the transducer focus, gauge whether acoustic output is sufficient to achieve desired effects, and estimate the tissue’s likely response to treatment.

In comparison to TUSS, TUSN utilizes very small acoustic intensities to reversibly perturb neurofunction. These intensities are too small to induce a substantial local temperature rise, produce an acoustic radiation force displacement, or produce incoherent water motion via cavitation. For these reasons, existing MRI guidance techniques cannot provide quantitative measures of skull-induced attenuation during TUSN. For example, magnetic resonance acoustic radiation force imaging can identify the location of the acoustic focus within the patient [26,27]. However, the technique is non-quantitative and deposits energy at a rate roughly equal to the actual TUSN treatment. Under a TUSN regime, existing MRI guidance techniques remain too insensitive to provide patient-specific estimates of trans-skull acoustic behavior.

Our scientific premise holds that a magnetic resonance hydrophone (MRH) will provide sensitive measurements of very low intensity acoustic beams in the brain cortex during insonation. These measurements will inherently include the treatment-specific effects of the patient’s skull at smaller intensities than currently used during TUSN.

In pursuit of this premise, this study develops a human-compatible MRH prototype [28] that can visualize acoustic pressures at depths equivalent to the human cortex at sub-therapeutic TUSN acoustic amplitudes. We describe the design and fabrication of the prototype and present measurements of the accuracy and sensitivity of our design when measuring acoustic pressures in tissue mimicking phantoms.

1.2. The Magnetic Resonance Hydrophone

The MRH was first introduced by Walker and Plewes[29,30], who applied low-frequency, shear-wave encoding concepts introduced by Muthupillai et al.[31] to 500 kHz, longitudinal waves. The device works by placing an oscillating gradient coil near an acoustic field. When operated synchronously in an (MRI) scanner, the MRH’s magnetic field gradient encodes microscopic acoustic displacements into MR image voxel phase. Pressure, p, can then be measured from accrued voxel phase through the equation

$$p=\frac{2\rho c\omega \varphi (\overrightarrow{r})}{\gamma G(\overrightarrow{r})T\text{sin}\left(\overrightarrow{k}\cdot \overrightarrow{r}+{\theta }_o\right)}.$$ (1)

Where $\varphi (\overrightarrow{r})$ is the voxel phase at voxel location $\overrightarrow{r}$, ρ is the density of the voxel, c is the sound speed of the voxel, ω is the angular frequency of both the ultrasound beam and the encoding magnetic field, $\overrightarrow{k}$ represents the wavenumber of the ultrasound beam, θ_o is the difference in initial phase between the gradient and ultrasound waveforms, γ is the gyromagnetic ratio of hydrogen, T is the time during which the coil and transducer are activated, and $G(\overrightarrow{r})$ is the magnetic field gradient produced by the coil. In this manner, MR images may be used to non-invasively map acoustic pressures in regions where simultaneous acoustic propagation and the oscillating magnetic field gradient are co-linear.

In their seminal work, Plewes and Walker constructed a gradient coil insert with an independent driving and amplification system that resonated at ultrasonic frequencies. To maximize gradient homogeneity throughout the acoustic field, they chose to implement a Maxwell pair coil design. The pair were 42 mm in diameter, separated by 60 mm and were able to achieve gradient values of 0.4 T m⁻¹ within an 83 cm³ volume[30]. While the accuracy and precision of this MRH design were never published in a peer reviewed venue, a later Master’s thesis by Evans reported that the MRH produced nearly zero mean error and an uncertainty in measurement of 12% for a pressure range of 200 to 1200 kPa when compared to a calibrated hydrophone[32].

While the Maxwell pair concept was effective at detecting acoustic waves, its geometry rendered the resulting images sensitive to ultrasound only in the volume between the two small coils comprising the Maxwell pair. This geometry precludes use in a human head where the encoding magnetic field gradient must extend outside the volume of the coil windings and into the patient. In this paper, we introduce a pancake-style MRH coil design whose field gradient can extend into human neuroanatomy. The design sacrifices gradient uniformity for improved penetration into the human brain. The following work describes the construction and characterization of this MRH form factor.

2. Methods

2.1. Coil Design and Construction

The pancake-style MRH coil form factor design followed three requirements: 1) compatibility with a single-element focused ultrasound neuromodulation device similar to that discussed in Legon et al.[33]; 2) instantaneous power consumption less than 500 W (a limitation set by available amplifiers); 3) equivalent sensitivity to the Maxwell pair design by Plewes and Walker at a 20 mm depth from the coil surface, which is a depth we estimate to be within the cortex of the brain for most patients. All other design choices were made in subservience to these three constraints.

MRH sensitivity to ultrasound maximizes when acoustic waves propagate along the direction that also maximizes G in Equation (1). Therefore, the acoustic propagation axis was chosen to be colinear to both the MRH winding axis and the MRI scanner’s bore axis, hereafter labeled as the z-axis. Meanwhile, the inner radius of the coil was set to 15 mm to make space for an ultrasound transducer. This geometry would allow acoustic propagation through the region of maximum magnetic field gradient. The frequency of both the ultrasound transducer and the coil was chosen to be 500 kHz to approximate the frequencies used in TUSN studies[33–35]. At this frequency range, resistive skin and proximity effects significantly increased resistive losses in the coil. To counteract these effects, we chose to wind the coil with threads of 1050 strands of woven 44 AWG Litz wire.

Before construction, coil designs of varying outer radii and thicknesses were simulated using a finite element software (COMSOL, Burlington, MA), assuming a peak current of 20 A, until the simulated gradient at 20 mm distance from the coil reached the 0.4 T m⁻¹ threshold reported by Plewes and Walker[30]. This threshold guaranteed that the pancake-style coil will perform similarly to the MRH produced by Plewes and Walker. The MRH coil was then wound to match the dimensions of the first simulated design to meet this requirement, resulting in 6 planes of 7 turns of Litz wire threads. The final build was housed in Teflon and perfused with epoxy (Loctite EA M-31CL, Henkel North America, Rocky Hill, CT) and is shown in Figure 1A. The coil produced an inductance of 53.4 μH and a series resistance at 500 kHz of 0.53 Ω.

Figure 1:

(A) Photograph of the gradient coil prototype. (B) Gel form, MRH gradient coil, and transducer (PZT) along with accompanying housing used during experimentation.

2.2. Coil Tuning and Matching

To minimize coil impedance at 500 kHz and match the circuit to the 50 Ω output impedance of the amplifier system, the coil was connected in series to a network of high voltage, ceramic capacitors with an equivalent capacitance of 1800 pF. This network was connected to a custom transformer wound with Litz wire about a ferrite core (3F36, Ferroxcube, Taipei, Taiwan) to match the coil impedance to the amplifier. The final circuit build, which combined the transformer, cables, capacitors and coil, had a bandwidth of 26 Hz, or a Q-factor of over 19,000.

2.3. Coil Characterization

The field gradient, G, of the coil was characterized in free field using a multi-axis Teslameter (F71, Lake Shore Cryotronics, Westerville, USA) placed on an automated 3 axis stage. The sensitive element of the Teslameter’s probe was cantilevered such that it was suspended 150 mm from the stage’s movement arms in order to minimize field perturbations from ferrous objects in the stage. Since the Teslameter was band limited to 20 kHz or less, the coil was activated with a direct current of 10 A that bypassed the transformer and capacitor network. The peak intensity and spatial distribution of the electromagnetic field produced by the coil was assumed to behave similarly under DC and 500 kHz conditions. The Teslameter recorded the z-component of the coil’s field by rastering through a 140 × 140 mm grid whose resolution and orientation matched the MR imaging plane described below (0.55 mm isotropic resolution, 0.2 s dwell time per measurement). A secondary, control acquisition was acquired while the coil was deactivated and then was subtracted, elementwise, from the primary acquisition. The measurements were then normalized to 1 A of current through the coil. The gradient field with respect to the anticipated direction of acoustic propagation was computed using a fourth-order central difference scheme with a second-order difference scheme on the boundaries of the acquired field.

2.4. Transducer Design, Construction, and Characterization

An ultrasound transducer was constructed by mounting a 10 mm diameter, 500 kHz resonant piezoelectric ceramic disc (SMD10T4R111, Steiner & Martins Inc, Davenport, FL, USA) inside of a stereolithography printed resin (SLA) enclosure. On one end of the enclosure protruded the coaxial attachment, with air-backing for the disc, and, on the other, a 25 mm extruded waveguide terminated with a lens with a 20 mm radius of curvature. The waveguide distanced the piezoceramic disc from the oscillating magnetic field and, thereby, limited inductive heating in the piezoceramic. The design was sealed with epoxy. The resulting transducer was electrically matched to 50 Ω using a variable capacitor and inductor network.

The transducer output, with the accompanying waveguide, was characterized in a large tank of degassed water using a calibrated hydrophone (HNR-1000, Onda, Sunnyvale, CA, effective diameter: 2.8 mm compared to the ~ 3mm expected acoustic wavelength, which may allow some blur of the acoustic field). The hydrophone was mounted on a same tri-axial stage that mounted the Teslameter and rastered, underwater, through the same grid as that which characterized the coil in section 2.3 (0.55 mm resolution, 0.2 s dwell time). The hydrophone was mounted such that its sensitive surface area was orthogonal to the propagation direction. For each location, the transducer was excited by a function generator (SDG 2042X, Siglent Technologies, Solon, Ohio) and amplifier (240L, E&I, Rochester, New York) paired with the matching network at 500 kHz using 110 sinusoidal cycles. Peak-to-peak driving voltages were 40 and 80 Volts.

Hydrophone voltages were recorded using an oscilloscope (2104X, Siglent Technologies, Solon, Ohio). To compensate for the wideband response of the hydrophone compared to the narrowband response of the MRH coil, the received waveform was convolved with a 35 kHz bandpass Hamming filter. To avoid the errors from transducer ring-up, pressure was estimated from the last 70 cycles of the sampled waveform. Attenuation in the waveguide limited the transducer’s peak focal pressure to less than 200 kPa. Acoustic pressures associated with peak-to-peak voltages between 40 and 80 Volts were inferred using linear interpolation.

2.5. Imaging Gel Construction.

A 50 mm diameter and 75 mm length cylindrical tube with mylar capped ends (0.1 mm thickness, <1 dB attenuation) was filled with a tissue-mimicking, 2% agar gel mixture. The agar mixture cooled under vacuum to remove bubbles. The density of the gel was estimated by measuring the mass of the gel and dividing by volume and found to be 1028 kg m⁻³. The sound speed of the gel was estimated using the time-of-flight method and found to be 1577 m s⁻¹. Briefly, a 500 kHz transducer of identical construction as described above but lacking the 25 mm waveguide was placed under degassed water such that a 20-cycle acoustic pulse impinged orthogonally on the surface of the gel cylinder. The center of the cylinder was placed equidistant from the transducer surface and a hydrophone (HNR-1000, Onda, Sunnyvale, CA). The transducer and hydrophone were placed 15 mm apart. Time-of-flight was determined by the duration, as measured on an oscilloscope, between the leading edge of the driving voltage and the received leading edge on the hydrophone. As a control, measurement was repeated with the cylinder filled with degassed water instead of gel. We assumed an attenuation, which was too small to measure with precision using our instruments, of 0.25 dB cm⁻¹[36].

2.6. System Assembly.

The overall assembly consisting of the gel medium, the ultrasound transducer and the coil were mounted together inside of an acrylic hollow cylinder. The agar gel had a 2 mm layer of insulating foam placed between it and the coil to help prevent the coil from melting the gel. The ultrasound transducer was acoustically coupled to the agar by removing a mylar cap and applying coupling gel (#4963, McKesson, Irving, Texas). All of this was then wrapped with a Body 18 MRI receiver coil (Model: 10496515, Siemens Healthcare, Erlangen, Germany). Finally, this bundle was centered inside a 3T MRI scanner (Vida, Siemens Healthcare, Erlangen, Germany). A photograph of the setup, without the MRI coils, is shown in Figure 1B.

The coil was driven by a class AB amplifier (TPO102_400, Sonic Concepts Inc, Bothell, Washington) which acted as both the frequency source and power amplifier. This was transferred through a pass-through plate in the MRI room’s Faraday cage to the matching network inside the scanner room. The ultrasound driving system was also modified from what is described in section 2.4 by inserting a custom logic device between the signal generator and amplifier that would alternately select between the driving waveform or a waveform with identical amplitude and frequency but shifted in phase by 180°. This alternating shift in phase produces a net phase under spin echo MR imaging[29,31]. The amplified waveform was then passed to the transducer through the pass-through plate.

During experimentation, the driving frequency produced by the function generators and then sent to the coil and the transducer was shifted to match the coil network to as near to 50 Ω as possible. The implemented frequency shift, depending on the placement of the transformer in the MRI room, varied between 492 and 501 kHz. The driving system produced a peak-to-peak current in the coil between 47 and 50 A and consumed approximately 312 W RMS power. The transducer consumed approximately 16 W RMS power at maximum driving voltage. During experimentation, the 4% duty cycle imposed by the repetition time (TR) of the MRI pulse sequence discussed in the next section reduced these values to 12 W and 0.6 W, respectively.

2.7. Acoustic Simulation

To assess the possibility of standing waves and interference patterns from reflections off of the air-boundary interfaces in the gel setup, a 2D, k-space, pseudo-spectral numerical solver (K-Wave, version 1.4, Downloaded July 2024, k-wave.org) was used to estimate acoustic propagation in the free field and within the gel container. Figure 2 displays a simulation mesh of size 90 × 130 mm and resolution 0.1 mm with features corresponding to the geometry of the gel system in the same plane discussed in sections 2.4 and 2.5. The mesh includes air surrounding the gel holder and a mylar cap. Features were assigned the following acoustic sound speed and density: SLA: 2550 m/s, 1100 kg/m^3; Acrylic: 2610 m/s, 1180 kg/m^3; gel: 1500 m/s, 998 kg/m^; air: 343 m/s, 1.2 kg/m^3; mylar: 2540 m/s, 1180 kg/m^3. The properties of SLA were measured on site while other properties were taken from the literature[37]. The entire mesh was assigned attenuation of 0.25 dB cm⁻¹ as found in the literature[36]. Attenuation in the plastic and air structures were not deemed critical due to the large impedance mismatch between the gel and the surrounding plastic barriers which reflects more than 90% of the acoustic energy.

Figure 2:

Simulation geometry describing the piezoelectric source, waveguide material, lens, gel target, and boundary materials. K-space pseudospectral simulation propagated acoustic waves from the source into the target using a 2 ms pulse at 500 kHz.

A cylindrical piston pressure source was placed at the end of the waveguide and initiated a 2 ms duration, tone burst with a frequency of 500 kHz, which was then allowed to propagate through the simulation mesh. A 2 ms tone burst allows the acoustic waves to traverse approximately 3 m of gel material, producing approximately −70 dB attenuation. This attenuation level which was deemed sufficient to approximate a steady-state solution. The simulation used a time step of 2 ns for the full duration of the tone burst. Peak pressures, achieved in the steady state, was collected at each mesh location. To compare the simulation to hydrophone measures, the simulation was repeated while assigning the acoustic properties of water to all materials within and surrounding the gel holder.

2.8. MRI Pulse Sequence.

To acquire pressure-sensitized MR images, the MRI scanner implemented a spin-echo pulse sequence similar to that used by Plewes and Walker[29]. In addition, the pulse sequence emitted electrical trigger pulses immediately after the sequence’s excitation and refocusing radio frequency pulses. Each trigger caused the function generators to produce a 500 kHz tone burst of duration 20.8 ms in both the transducer and gradient coil. As described above, and necessary to prevent phase accrued after the MRI refocusing pulse from rewinding phase accrued before the pulse, every even trigger (which occurred at the refocusing pulse in the sequence) inverted the tone burst produced by the transducer and reversed the direction of ultrasound-specific voxel phase accrual[31]. Pulse sequence timing, including the inversion of the tone burst, are shown in Figure 3. The sequence scanned a coronal plane that transected the central diameter of the transducer and gel where we anticipated the acoustic field and magnetic field gradient to both reach their maximum. This plane matched the same plane used to acquire magnetic and acoustic fields in sections 2.3 and 2.4, above. After acquiring 3 images, a second set of 3 images were acquired with the phase of the tone burst driving the coil shifted by 180 degrees. Additional pulse sequence parameters for all acquisitions were TR/TE: 1000/60 ms; field of view: 140 × 140 mm × 5 mm; resolution 0.55 × 0.55 × 5 mm; matrix size: 256 × 256 × 1 pixels; echo train length: 1; flip angles: 90°/180°; receiver bandwidth: 20.3 kHz; time per single image: 256 seconds; total time per pressure map: 24 minutes. A total of five pressure maps were acquired while the peak-to-peak driving voltage across the ultrasound transducer was increased from 40 V to 80 V.

Figure 3:

Pulse sequence diagram for the spin-echo sequence used in the study. A trigger signal (Trig) activates both the encoding gradient coil $\left(G_{M_E}\right)$ and the tone burst of ultrasound (US) during timing gaps between the 90° and 180° RF pulses. The second tone burst of ultrasound has inverted phase relative to the first tone burst.

2.8. Pressure Calculation.

Acoustic pressure was calculated from the resulting MR images by first averaging complex pixel values across the three images acquired under identical coil driving phase and then subtracting the resulting voxel phases from data sets acquired with opposite coil driving phase and dividing the result by 2. This subtraction removes background sources of voxel phase not associated with ultrasound. Then, voxel phases within a 4×25 mm region of interest located in the center of the image at 3.3 mm from the focusing lens were retained. This region comprised the majority of ultrasound-related image contrast in the imaging plane while also avoiding complex acoustic and magnetic structures at the transducer/coil/agar interface. A custom script (MATLAB, MathWorks, Natick, MA) detected the peaks and troughs in the now averaged and subtracted phase image, took the absolute values of voxel phase at these locations, and then calculated pressure using Equation (1) while assuming that the absolute value of the term $\text{sin}\left(\overrightarrow{k}\cdot \overrightarrow{r}+{\theta }_o\right)$ at these locations was unity. The resulting pressure values were then compared to those obtained using the hydrophone.

2.9. MRH Accuracy and Uncertainty.

The accuracy of the MRH was assessed by repeating MRH acquisitions while increasing the driving voltage across the transducer from 40 to 80 peak-to-peak V. Differences in pressure within the region of interest between the MRH and hydrophone measurement were then calculated. Uncertainty in MRH measurement due to electrical noise was assessed by acquiring a total of 5 MRH images with the ultrasound transducer deactivated. Because these images lacked a peak-trough structure, acoustic pressure was then calculated for all voxels within the region of interest. Uncertainty was estimated by taking the per-pixel standard deviation taken across the 5 MRH images.

3. Results

3.1. Magnetic Field Gradient.

The measured magnetic field gradient produced by the coil is shown in Figure 4A. As discussed in section 2.1, the axis of this plane colinear with the MRI bore axis was labeled the z-axis. The other axis of the plane was labeled the y-axis. The field is reported in units of milli-Tesla per meter per Ampere of current driven through the coil. Figure 4B displays a cut line through the center of both the gradient field and the originating magnetic field, each reported in units of milli-Tesla per unit Ampere of current driven through the coil. All plots are normalized to 1 A of current. Both the magnetic field and its gradient decay swiftly with distance from the coil surface. Very strong magnetic field gradients are found close to the inner radius of the coil. The gradient field satisfies our requirement that, when scaled by 28 A during operation, it will produce a slope greater than 0.4 T/m at a distance of 20 mm from the surface of the coil. Finally, the figure indicates a central region of width ~10 mm and depth of 20 mm where the gradient decays smoothly with distance (z) and has little horizontal (x) variation.

Figure 4:

(A) Spatial map of the magnetic field gradient produced by the gradient coil. (B). A cutline through the center of (A) (as indicated by the red dotted line in A) showing both the magnetic field and its gradient. All values are normalized to 1 A through the coil.

3.2. Acoustic Fields

The acoustic free-field produced by the filtered hydrophone measurement is shown in Figure 5A for the case when driving the transducer with a peak-to-peak voltage of 80 V. The field displays a focus at 11 mm distance from the surface of the lens with peak pressures near 100 kPa. The deviation in focal location from lens curvature likely occurred due to multiple propagation modes within the waveguide. The −3 dB widths of the focus were 5 mm (lateral) and 30 mm (depth). Side lobes can also be seen in the plot. We observed the waveguide to produce significant attenuation, limiting peak acoustic output to 120 kPa or less. We verified that the filtered focal pressure behaved linearly with driving voltage for this study by multiplying the 40 V acoustic field by two and measuring the average pressure difference with the 80 V measurement in voxels in the central focus, resulting in an average difference of 1.7 ± 4.7 kPa. We considered these differences to be below the precision of our measurement instrumentation.

Figure 5:

(A) A map of the maximum acoustic free-field produced by the transducer when driven with 80 V peak-to-peak and acquired by the hydrophone. The transducer produces a 5 mm × 30 mm focus with peak pressures near 100 kPa. Side lobes are also visible. (B) Simulated maximum acoustic pressure (normalized) in free-field water. The simulation exaggerates near field structures, but the main focus maintains similar geometry to the measured acoustic free field. (C) Simulated maximum acoustic pressure (normalized) in the gel with reflections off of the acrylic and mylar surfaces. A standing wave pattern introduces nodes and anti-nodes but the focus geometry remains consistent with the free field.

The simulated maximum acoustic free-field in water is displayed in Figure 5B. The simulated pressure is normalized to 1 kPa because the simulated surface pressure of the transducer was arbitrarily chosen. The simulation amplified near field structures relative to the hydrophone measure. The focus of the simulated field was found to be at 12 mm. The −3 dB beam widths of the simulated field were 6.1 mm (lateral, x) and 42 mm (depth, z). The simulated maximum acoustic field in the gel is shown in Figure 5C and is also normalized to 1 kPa. Reflections within the gel structure from the impedance mismatches between the gel and the plastic housing and air boundaries produce a standing wave effect with nodes and anti-nodes. The reflections lengthen out the focus in the depth direction and shorten the beamwidth in the lateral direction with −3 dB beam widths of 5.2 mm (lateral, x) and 49 mm (depth, z). Additionally, the locus of peak pressure shifted from 12 mm to 14 mm.

3.3. Magnetic Resonance Images

Example magnitude and phase magnetic resonance images are shown in Figure 6. The voxel magnitude image shows a thin layer of coupling gel followed by a uniform agar gel modulated smoothly by coupling between the phantom and the MR excitation field[38]. However, voxel phases near the transducer and coil at the bottom of the image show a repeating wave pattern that corresponds to the sine term in Equation (1) and have been reported elsewhere[28–30]. The sinusoidal pattern of phase decays rapidly with distance from the transducer and surface coil. The decay in amplitude of the repeating phase pattern can be explained by noting that the sensitivity of voxel phase to acoustic motion scales directly with gradient strength. Because the magnetic field gradient shown in Figure 4A decays rapidly with distance from the coil, the magnitude of the phase also decays rapidly.

Figure 6:

Example magnitude (A) and phase (B) images acquired during experimentation. Phase images show a sinusoidal pattern propagating along the axis of the gel (image bottom) that decays with distance.

This sinusoidal pattern does not represent the peak and negative pressure phases of the acoustic beam and, instead, as formulated in, [28–30], represents the magnitude and sign of the correlation between the acoustic displacement pattern and the magnetic field gradient. For monochromatic plane waves, this correlation modulates sinusoidally with identical wavelength and direction of propagation to the acoustic field according to the initial phase pattern of the acoustic wave, as described in Equation (1). Therefore, according to Equation (1), the magnitude of the peaks and troughs of this pattern scale with both the acoustic and magnetic gradient fields and decays quickly with distance from the coil and transducer surfaces.

3.4. Pressure Calculations

Figure 7A displays an example phase image that has been converted to pressure using Equation (1) and assuming $\text{sin}\left(\overrightarrow{k}\cdot \overrightarrow{r}+{\theta }_o\right)$ equals unity for all voxels. Due to rapid decay of the magnetic field gradient, the calculation produces obvious noise at depths greater than 30 mm. Within the figure, the $\text{sin}\left(\overrightarrow{k}\cdot \overrightarrow{r}+{\theta }_o\right)$ term continues to modulate the estimated pressure field. The red box in the figure indicates the region of interest where the MRH measurement was assumed to be reliable and from which peaks and troughs within the phase pattern were selected for comparison with the hydrophone. Intermediate pixels where the $\text{sin}\left(\overrightarrow{k}\cdot \overrightarrow{r}+{\theta }_o\right)$ term dominates the reported pressure values were ignored.

Figure 7:

(A) Example pressure map calculated using Equation 1 in the text. Noise increases quickly with distance from the coil (at image bottom). The red box indicates the region of interest used for further analysis. (B) The region within the red box with pixels labeled as a peak or trough marked with a red dot. Noise far from the coil surface adds some jitter to the pixel labeling. (C) cutlines through pressure maps obtained when driving the transducer by 40 and 60 V peak-to-peak. The free field acoustic focus is marked in red. Accounting for the sinusoidal pattern, pressure and noise both increase with distance from the focus.

Figure 7B displays pixels within the region of interest that were labeled by the Matlab script as a peaks and troughs and included for analysis. Most of the labeled pixels reside on colinear rows in the region of interest. However, noise increasingly influences where peaks and troughs were detected as distance from the coil increases. This jitter introduces some additional noise into the resulting pressure estimates. However, the jitter exceeds no more than a single pixel-width and, assuming a smoothly varying acoustic field, we therefore expect the resulting noise in the pressure estimates to be small.

For easier comparison, Figure 7C, compresses pixels within this region into scatter plots for images acquired with 40 and 60 V peak-to-peak across the transducer. Similar to both the free field and simulation, acoustic pressure increases as distance from the surface of the transducer approaches the focus, marked by a red dashed line in the plots. However, estimated acoustic pressure does not quickly decrease with increased depth and even appears to increase in the 60 V and 80 V peak-to-peak cases (not shown in figure).

3.5. MRH Accuracy

Figure 8A displays a Bland-Altman[39] plot comparing measured pressures by both the hydrophone and the MRH across all transducer driving voltages and all phase peaks and troughs within the region of interest. The plot indicates that the MRH, on average, underestimates the free-field hydrophone pressure by 12 kPa with an average standard deviation of 21 kPa, resulting in a 95% confidence interval spanning 82 kPa. Differences between the MRH and hydrophone appear uniformly distributed across the −12 kPa line across almost the entire range of pressures tested in this study. However, points at higher pressures tend to show greater underestimation. The variance of the differences appears to scale with pressure.

Figure 8:

(A) Bland-Altman plot comparing the MRH and hydrophone pressures within the region of interest. The MRH underestimates the hydrophone by 12 kPa on average with a standard deviation of 21 kPa. (B). Standard deviation of the MRH measurement with no acoustic signal present as a function of depth from the MRH. Note that the y-axis follows a log scale. Uncertainty increases exponentially with depth. Uncertainty remains bounded within 100 kPa at 30 mm from the MRH surface.

3.6. MRH Electronic Noise Uncertainty

Figure 8B displays on a logarithmic scale an image of the pixel-wise standard deviation of MRH pressure measurements taken across all 5 acquisitions with no ultrasound present. While MRH pressure measurements should ideally return 0 kPa in this case, electronic and sampling noise confounds the measurement. The figure shows that uncertainty due to noise scales exponentially with depth. For locations between 0 and 30 mm of depth within a region approximately 10 mm wide, uncertainty scales between approximately 20 kPa and 100 kPa. These results can be explained by noting that a pixel-wise calculation of pressure in Equation (1) requires dividing the pixel phase by the gradient at that pixel. As the gradient decreases in magnitude, the scaling factor amplifies both the pixel phase and any noise associated with the measurement. Therefore, pressure values located at distance from the coil experience noise amplification.

4. Discussion

This study presents a human-compatible magnetic resonance hydrophone design. The presented coil form factor supports a magnetic field gradient that reaches a magnitude of 0.4 T m⁻¹ at a depth of 20 mm. This depth would reach the cortex of the brain for many adults. The study also presents a technique for extracting pressure values from the resulting MR images. The resulting MR images, as shown in Figure 6, display phase contrast proportional to acoustic vibrations even though the region of measurement extends beyond the windings of the sensitizing coil. The amplitude of the phase pattern decays with distance from the coil as the magnitude of the encoding gradient also decays. Computed pressure values, as shown in Figure 7, between the coil surface and the transducer focus follow the predicted behavior of swiftly increasing in pressure amplitude.

This study also shows that the measurement uncertainty of this MRH form factor increases exponentially with distance from the coil surface, as shown in Figure 8. Nonetheless, uncertainty remains less than 100 kPa at distances as large as 30 mm. We assume the circular symmetry of the coil permits circular symmetry in this sensitivity pattern. We find these results highly encouraging given that many neuromodulation studies use pressures of 500 kPa or higher[34,40–42]. In the current state of the art, skull transmission can vary between 3% and 80% of the incident beam, depending on the subject[43]. Should the 12 kPa bias remain consistent at these higher pressures, then this bias will be considered negligible.

Similar to the Maxwell pair format reported by Plewes and Walker[29], the presented pancake-style coil form factor produces phase images sensitized to acoustic displacement that are modulated by a sinusoidal pattern with an identical wavenumber to the underlying acoustic wave. The presented coil consumed less power (13 W) than the reported power consumption of the Maxwell pair (125 W)[30]. However, unlike the Maxwell pair, sensitivity degraded with distance from the coil. At 20 mm of depth from the surface, the presented coil demonstrated the same order of pressure uncertainty (~50 kPa) as that reported by Evans (20 kPa)[32]. Further direct comparison between the two coil designs is limited by the difference in pressure magnitudes examined (20–100 kPa, here, vs 200–1200 kPa in Evans [32]). Finally, Baril et al. produced an MRH design that leverage an oscillating gradient in the MRI scanner’s excitation field [44,45]. However, the method has only been tested for gradient oscillations in the audio range and cannot be compared here.

When compared to a calibrated hydrophone, the MRH, on average, underestimated the hydrophone by 12 kPa with a standard deviation of 21 kPa. However, at higher pressures, the MRH begins to overestimate the hydrophone. This behavior can be explained by observing that the pressure pattern estimated by the MRH beyond the free-field focus in Figure 7C does not closely match that measured in free field, with pressure increasing at depths of 20 mm in the gel as opposed to decreasing pressure in the free field. This phenomenon contributes to most of the MRH overestimation in Figure 8A.

Standing waves might explain the increase in observed post-focal pressure in the MRH because reflections could constructively interfere within the gel. It is difficult to measure standing waves within the gel geometry due to the stiffness of the gel and the directivity of a hydrophone. However, Figure 5 demonstrates that acoustic simulations mimicking the free field transducer geometry can nearly replicate the geometry of the observed free-field measurements. Application of the same simulation techniques to the gel geometry, as shown in Figure 5C, suggest that interference patterns do shift and lengthen the focus width in the z direction, though the effect is small. They also superimpose the familiar pattern of nodes and antinodes.

Additionally, the derivation of Equation Equation (1) in Plewes and Walker [29]assumes a monochromatic plane wave. The addition of a backwards propagating wave into the derivation will modify the sine term in Equation (1) to become sin(kx) sin (θ_o). Inadvertent attenuation of the derived pressure can occur under standing wave conditions when the true phase shift, θ_o, between the standing wave and the gradient field is not equal to an odd multiple of π/2. The term sin (θ_o) would then have magnitude less than 1. By assuming that all sine terms in Equation (1) have magnitude 1, the pressure calculation may divide the voxel phase by too large of a number and effectively attenuate the pressure estimated by the MRH. While our team made careful efforts to choose θ_o that maximizes the observed voxel phase, small deviations in θ_o from odd multiples of π/2 may explain the MRH’s underestimation of pressure.

The amplification of MRH measures of post-focal pressure could also be explained by an underestimation of the non-uniform field gradient of the encoding coil. In this work, the field gradient was measured with a Teslameter on a test bench. A mis-registration in the test geometry may artificially attenuate the measured gradient and, thereby, amplify the post-focal pressure. However, considering the steep slope of the prefocal gradient in Figure 4 compared to the post-focal gradient, a mis-registration should amplify more strongly the pre-focal pressures compared to the post-focal pressures. Additionally, the field measured on the test bench may not match the field used during experimentation because the field produced during experimentation will be altered by the magnetic susceptibility of the gel phantom. However, we expect this effect to be small because water’s volume susceptibility is on the order of 9 ppm.

Additional sources measurement differences may include that either the acoustic pressure or the sinusoidal phase pattern in Equation (1) may not hold constant over the MR voxel volume. The MRI voxel sensitivity function for Cartesian scanning approximately follows a sinc function [46]. As this function convolves across the image, it may average over both the sinusoidal phase pattern observed in Figure 6 as well as the underlying pressure field. Averaging over both phenomena will result in underestimation.

Additional sources of potential error include that the acoustic wave may not be entirely co-linear with the coil’s magnetic field gradient, the hydrophone’s large effective diameter may blur the measured acoustic free field, and possible error within the hydrophone’s calibration. While the pancake-style MRH coil was able to report acoustic pressures at locations within the beam path, it was not able to resolve important features such as the 3dB widths of the beam. Sidelobe were not immediately distinguishable. This lack of resolution might be driven by the limited spatial extent of the MRH’s sensitizing magnetic field rather than the actual acoustic field.

While the presented results are promising, this study possesses several important limitations. As described above, this study does not measure the standing wave field inside the gel due to difficulty in mounting and steering a hydrophone in an enclosed environment with complex reflection geometry. The study also does not examine the sensitivity of the pancake-style form factor as a function of angle relative to the MRI bore-axis. Indeed, no preexisting literature has studied MRH behavior at oblique angles. However, at 20 mm depths and along the central axis, the majority of the coil’s magnetic field orients parallel to the winding axis. In clinical practice, skull curvature may force the acoustic field to propagate at an oblique angle. We would expect the MRH’s sensitivity to oblique fields to decrease with the cosine of the angle between the acoustic field and gradient field. Future studies might confirm this effect.

The study also uses a relatively large diameter hydrophone which likely blurs features in the focal field. At 500 kHz, the effective diameter of the hydrophone is 2.8 mm, which is large compared to the focal width of 5 mm. Meanwhile, the voxel sensitivity function of the MR images acquired here follows a 3D sinc-like pattern[46] whose nominal widths along the plane parallel to the plane of the hydrophone’s surface are 0.5 mm and 5 mm. This sensitivity pattern, which is large compared to the beam width in one dimension, may contribute to the concordance found between the MRH and hydrophone. Smaller diameter hydrophones may produce larger deviations from the MRH. Additionally, we would expect the MRH to less reliably measure pressures of oblique acoustic fields or fields with complex structures in the through-plane direction.

This study, due to limitations in available hardware, compares the performance of the MRH to a single hydrophone. However, hydrophones possess error ranging as large as 20% [47]which encompasses the 12 kPa average error between both measurement devices observed in this study. A larger cohort of hydrophones, with differing manufacture, may better triangulate the absolute error of the MRH.

Finally, due to large losses in the waveguide used in this study, measures of pressure after insonation through a cadaver skull, which attenuates near 80% at 500 kHz, would produce results below 20 kPa. The resulting signal would very close to the noise floor observed in this work. The waveguide was necessary to limit inductive heating in the PZT ceramic of the transducer. In clinical use, the transducer, to avoid heating, would also need to use a waveguide or some other method to limit inductive heating.

While the results of this study are promising, several challenges remain to use the MRH to correct attenuation produced by the skull. The presented MRH form factor visualizes acoustic waves at depths up to 30 mm, limiting its usefulness to only the cortical surfaces of the brain. New techniques will be required to guide TUSN at deeper targets. Transducer designs must be able to avoid inductive heating while transmitting sufficient power through the skull to exert a potential bioeffect. Meanwhile, the MRH is currently designed to measure pressures produced by nearly continuous waves under low amplitude, linear propagation. Under low power conditions, where the waveform is expected to be linear, the MRH’s pressure measures should predict both peak positive and peak negative pressure. Under high power conditions, the acoustic waveform may develop shocks where the peak and negative pressures are different. The narrow bandwidth of the MRH limits its ability to capture these high bandwidth waveforms. Finally, the presented design intends to measure propagation along the cranial-caudal direction, which limits acoustic access to the superior surface of the skull while many targets are more easily accessed more obliquely through the temporal or parietal bones. The MRH coil presented here will have limited utility under these cases.

Future studies may also examine extending the MRH’s region of less than 100 kPa uncertainty beyond 30 mm as well as estimating pressure in more voxels than just the peaks and troughs of the sinusoidal phase pattern.

5. Conclusion

This study presents a human-compatible gradient coil for visualizing acoustic fields. The coil produces a sensitizing field gradient that can extend into depths equivalent to the cortex of the human brain and produce images sensitive to acoustic propagation. The device, when compared to a calibrated hydrophone, underestimates the hydrophone by 12 kPa with a standard deviation of 21 kPa when insonating agar gel, though standing waves may have introduced error in this measurement.