Reconstruction of thermoacoustic emission sources induced by proton irradiation using numerical time reversal

Abstract

Objective.

Mapping of dose delivery in proton beam therapy can potentially be performed by analyzing thermoacoustic emissions measured by ultrasound arrays. Here, a method is derived and demonstrated for spatial mapping of thermoacoustic sources using numerical time reversal, simulating re-transmission of measured emissions into the medium.

Approach.

Spatial distributions of thermoacoustic emission sources are shown to be approximated by the analytic-signal form of the time-reversed acoustic field, evaluated at the time of the initial proton pulse. Given calibration of the array sensitivity and knowledge of tissue properties, this approach approximately reconstructs the acoustic source amplitude, equal to the product of the time derivative of the radiation dose rate, mass density, and Grüneisen parameter. This approach was implemented using two models for acoustic fields of the array elements, one modeling elements as line sources and the other as rectangular radiators. Thermoacoustic source reconstructions employed previously reported measurements of emissions from proton energy deposition in tissue-mimicking phantoms. For a phantom incorporating a bone layer, reconstructions accounted for the higher sound speed in bone. Dependence of reconstruction quality on array aperture size and signal-to-noise ratio was consistent with previous acoustic simulation studies.

Main Results.

Thermoacoustic source distributions were successfully reconstructed from acoustic emissions measured by a linear ultrasound array. Spatial resolution of reconstructions was significantly improved in the azimuthal (array) direction by incorporation of array element diffraction. Source localization agreed well with Monte Carlo simulations of energy deposition, and was improved by incorporating effects of inhomogeneous sound speed.

Significance.

The presented numerical time reversal approach reconstructs thermoacoustic sources from proton beam radiation, based on straightforward processing of acoustic emissions measured by ultrasound arrays. This approach may be useful for ranging and dosimetry of clinical proton beams, if acoustic emissions of sufficient amplitude and bandwidth can be generated by therapeutic proton sources.

1. Introduction

Proton therapy is a form of radiation therapy in which protons, accelerated upwards of 250 MeV or approximately 60% of the speed of light, are directed at tumors with the purpose of killing malignant cells. Delivery of protons and the resultant ionization in tissue initiates a cascade of biochemical reactions that can result in cell death, with the rate of cell death being higher in rapidly dividing cells such as tumors. For protons and other heavy charged particles, the majority of dose is delivered near the particles’ stopping point, known as the Bragg peak, which is determined by the beam energy and tissue properties. Compared to more common photon radiation therapy, this feature offers substantial advantages in radiation delivery, as a negligible dose is deposited beyond the Bragg peak (ASTRO, 2017). This highly localized energy deposition offers the promise of precise radiation treatments that conform to the shape of the tumor, with minimal dose delivered to the surrounding healthy tissue. This advantage is especially important in cases where the tumor is adjacent to critical organs and tissue structures. For these reasons, proton therapy is the preferred radiation treatment for pediatric cancer (Doyen et al., 2016) and many difficult tumors (Leroy et al., 2016).

Spatial distributions of proton therapy dose are currently planned based on physical modeling incorporating 3D imaging of patient anatomy, primarily mapping of target tissues using magnetic resonance imaging (MR) or positron emission tomography (PET) and general anatomy with x-ray computed tomography (CT) (Pham et al., 2022). CT scans are used as a base model for the detailed 3D anatomy used to estimate proton stopping power (ASTRO, 2017). CT, based on photon interactions rather than proton interactions, retains some uncertainty in estimating stopping power. Because the position (range) of the Bragg peak depends on anatomic details and physical properties of tissues throughout the beam path, estimation of ranges based on anatomic image data is imprecise, and error increases with proton path length through tissues (Pham et al., 2022; Nystrom et al., 2020). In addition, anatomic changes such as patient weight loss or tumor growth can introduce deviations from the planning CT image. Errors are also induced by organ movement on multiple time scales, including cardiac, respiratory, and digestive system motion. For these reasons, proton therapy would benefit from real-time localization of the Bragg peak, as well as mapping of the proton dose in tissue.

One promising approach to real-time ranging for proton therapy employs measurement analysis of thermoacoustic emissions, associated with dissipation of protons’ energy as heat that causes tissue to expand locally. Consequently, for a transient proton beam, time-varying tissue heating within the beam creates a detectable acoustic signature (Askariyan et al., 1979; Sulak et al., 1979; Tada et al., 1991). Localization of proton energy deposition from associated acoustic emissions has been shown feasible (Jones et al., 2014; Assmann et al., 2015; Hickling et al., 2018) and is an active area of current research (van Dongen et al., 2019; Yu et al., 2019; Freijo et al., 2021; Nakamura et al., 2021; Deurvorst et al., 2022). This thermoacoustic approach to mapping of proton dose has potential advantages over other methods for ranging of the Bragg peak, such as positron emission tomography (PET) and prompt gamma imaging (MacKay, 2018; Parodi, 2018), due to the relative simplicity, portability, and inexpensiveness of ultrasound measurement apparatus capable of detecting the emissions.

Several previous studies have reported analysis of ultrasonically measured thermoacoustic emissions from tissue-mimicking phantoms exposed to proton beams, demonstrating that the Bragg peak can be accurately localized by backprojection of thermoacoustic emissions received by an ultrasound linear array (Patch et al., 2016, 2019) or an array of several discrete receiving transducers (Patch et al., 2021). When the same ultrasound array performs B-mode pulse-echo imaging of the same medium, the Bragg peak location can be overlaid on the B-mode image with implicit co-registration (Patch et al., 2016), even in the presence of tissue inhomogeneities (Patch et al., 2019). However, beamforming of emission signals in these studies did not account for sound speed variations in the imaged medium or the finite size and directivity of receiving elements, and concentrated on localizing the position of maximum energy deposition relative to B-mode imaging, rather than spatially mapping sources of thermoacoustic emissions with respect to room coordinates.

Here, proton-induced thermoacoustic emissions from two previously reported experimental studies (Patch et al., 2016, 2019) are beamformed to spatially map thermoacoustic sources, in a manner similar to passive cavitation imaging (Haworth et al., 2012, 2017). These reconstructions incorporate timing information similarly to synchronous passive cavitation imaging methods (Burgess et al., 2018); a similar approach has also been employed to beamform acoustic emissions from proton-induced vaporization of injectable nanodroplets (Heymans et al., 2021). The approach employed here is numerical time reversal, an established method of reconstructing acoustic sources (Bavu and Berry, 2009) which has been employed for photoacoustic imaging (Xu et al., 2004; Treeby et al., 2010; Johnstone et al., 2019), and has the potential advantage of directly reconstructing thermoacoustic sources within inhomogeneous media, without need for numerical optimization or iteration. Numerical time reversal has been studied in simulation for reconstruction of thermoacoustic emissions from inhomogeneous tissue (Yu et al., 2019, 2021; Lascaud et al., 2021; Samant et al., 2022).

In this study, acoustic emission sources induced by proton beams within tissue-mimicking phantoms and measured by linear ultrasound arrays are reconstructed using two different models for the spatiotemporal responses of array elements and accounting for acoustic inhomogeneity of a bone layer. The approach extends previous time-reversal methods for reconstructing thermoacoustic emission sources (Yu et al., 2019, 2021; Lascaud et al., 2021; Samant et al., 2022) by reconstructing sources from limited-aperture measurements, incorporating diffraction effects of standard ultrasound imaging arrays, and relating proton-induced acoustic sources to the amplitude envelope of the numerically time-reversed field. Reconstruction methods are validated by comparison with Monte Carlo simulations of proton dose deposition, including comparison of accuracy for estimating positions of the Bragg peak, and are statistically compared for differences in spatial resolution as assessed from widths of reconstructed Bragg peaks.

2. Theory

In this section, a method is presented for reconstructing thermoacoustic sources from their acoustic emissions as measured by an ultrasound array, using numerical time reversal. This operation is similar to a spatiotemporal matched filter applied to the original acoustic source distribution (Dorme and Fink, 1995), which when implemented numerically can provide high-resolution images of sound source distributions (Bavu and Berry, 2009). Although similar to previous implementations of numerical time reversal (Xu et al., 2004; Treeby et al., 2010; Yu et al., 2019), the approach derived here explicitly accounts for time-dependence of the thermoacoustic source and spatiotemporal response of the array elements. Implementation of source reconstruction, similar to beamforming approaches used in passive cavitation imaging (Haworth et al., 2012, 2017; Burgess et al., 2018), results in a simple representation of the thermoacoustic source from a weighted frequency-domain summation of numerically time-reversed fields from each array element.

The starting point for this reconstruction method is the transient acoustic wave equation with a thermoacoustic source term, which can be written for an acoustically homogeneous, unbounded medium as (Wang and Wu, 2007)

$$\begin{array}{c}\frac{{\partial }^{2}p(\mathbf{r},t)}{\partial t^2}-v_s^2{\nabla }^{2}p(\mathbf{r},t)=\frac{v_s^2\beta }{C_p}\frac{\partial H(\mathbf{r},t)}{\partial t},\\ =\mathrm{\Gamma }(\mathbf{r})\frac{\partial H(\mathbf{r},t)}{\partial t},\end{array}$$ (1)

where p is the acoustic pressure (Pa), v_s is the speed of sound (m/s), β is the thermal expansion coefficient (K⁻¹), C_P is the specific heat capacity (J/kg/K), Γ is the Grüneisen parameter (dimensionless), and H is the thermal power deposited per unit volume (W/m³), equal to the radiation dose rate (Gy/s) multiplied by the mass density (kg/m³). The form of the source term in Eq. (1) assumes thermal confinement, such that duration of the proton pulse is much smaller than the thermal relaxation time of the medium (Wang and Wu, 2007).

The source of thermoacoustic emissions, given by the right-hand side of Eq. (1), is assumed to take the form

$$\begin{gathered} \frac{\beta }{C_p}\frac{\partial H(\mathbf{r},t)}{\partial t}=q(\mathbf{r})\frac{\partial w(t)}{\partial t} \\ =q(\mathbf{r})a(t) \end{gathered}$$ (2)

This expression includes a position-dependent source amplitude q(r) (units Pa·s/m²) that is assumed to be positive everywhere (q(r) > 0), as well as the temporal derivative a(t) (s⁻¹) of the proton radiation’s time-dependent pulse shape w(t) (dimensionless), assumed to be independent of the spatial position r. The pulse shape w(t) may be assumed proportional to the time-dependent proton beam current (Patch et al., 2016). The theory outlined below shows how the source amplitude q(r) can be approximately mapped by numerical time reversal of received emissions.

Measured emissions, from which the thermoacoustic source distribution is to be reconstructed, are received by an N-element ultrasound array with element-wise impulse responses h_n(r, t), i.e., measured signals resulting from a spatiotemporally impulsive source strength δ(r) δ(t), where δ represents the Dirac delta function. Due to causality, the elements’ impulse responses h_n(r, t) are identically zero for t < 0. In general, these impulse responses may incorporate effects of an inhomogeneous medium with acoustic properties dependent on the spatial position r, including multiple scattering, refraction, and other inhomogeneous-medium effects.

The signal received by each array element is taken to have units of pressure (Pa), assuming the elements have been calibrated to measure the time-dependent acoustic pressure, averaged over the element surface. The resulting calibrated pressure signal p_n(t) received by each element is then given by a temporal convolution of the source strength and the element’s impulse response, integrated over a volumetric region V₀ containing all sources,

$$p_n(t)={\int }_{V_0}\left[{\int }_{-\infty }^{\infty }q\left({\mathbf{r}}_0\right)a(\tau )h_n\left({\mathbf{r}}_0,t-\tau \right)d\tau \right]d{V}_0.$$ (3)

The received pressure given by Eq. (3) is an adaptation of the Green function solution for the radiated pressure (Wang and Wu, 2007), with the Green function (i.e., impulse response of an ideal point pressure receiver) replaced by the impulse response for each element (m⁻¹·s⁻¹). For an ideal point receiver, this impulse response is equivalent to the Green function for the 3D wave equation,

$$h_{\delta ,n}(\mathbf{r},t)=\frac{\delta \left(t-\left|\mathbf{r}-{\mathbf{r}}_n\right|/{\upsilon }_s\right)}{4\pi \left|\mathbf{r}-{\mathbf{r}}_n\right|},$$ (4)

where δ is the Dirac delta function (s⁻¹) and r_n is the position of the nth point receiver.

In the derivation below, a numerically time-reversed field is defined and related directly to the acoustic emission source amplitude q(r). Notably, this derivation applies for general inhomogeneous media, as long as the emission signals recorded by array elements follow Eq. (3), with known forms for the impulse response of each element within the medium of interest. The array element impulse responses are assumed identical for operation as a receiver or a transmitter, due to acoustic reciprocity.

A numerically time-reversed field is defined as a convolution of the time-reversed received signals with the corresponding element impulse responses, summed over all array elements:

$$u_{\text{TR}}(\mathbf{r},t)=\sum _1^N{\int }_{-\infty }^{\infty }{p}_n\left(-t_0\right)h_n\left(\mathbf{r},t-t_0\right)d{t}_0.$$ (5)

This field can be written in terms of the source parameters and array element responses by inserting Eq. (3) into Eq. (5), yielding

$$u_{\text{TR}}(\mathbf{r},t)=\sum _1^N{\int }_{-\infty }^{\infty }{\int }_{V_0}\left[{\int }_{-\infty }^{\infty }q\left({\mathbf{r}}_0\right)a(\tau )h_n\left({\mathbf{r}}_0,-t_0-\tau \right)d\tau \right]d{V}_0{h}_n\left(\mathbf{r},t-t_0\right)d{t}_0.$$ (6)

This time-reversed field can be related more compactly with the source distribution q(r) by re-ordering operations and making the changes of variables τ → −τ, t₀ → −t₀ + τ, resulting in the expression

$$\begin{gathered} u_{\text{TR}}(\mathbf{r},t)={\int }_{V_0}q\left({\mathbf{r}}_0\right){\int }_{-\infty }^{\infty }a(-\tau )\left[\sum _1^N{\int }_{-\infty }^{\infty }{h}_n\left({\mathbf{r}}_0,t_0\right)h_n\left(\mathbf{r},t-\tau +t_0\right)d{t}_0\right]d\tau d{V}_0 \\ ={\int }_{V_0}q\left({\mathbf{r}}_0\right){\int }_{-\infty }^{\infty }a(-\tau )R\left({\mathbf{r}}_0,\mathbf{r},t-\tau \right)d\tau d{V}_0, \end{gathered}$$ (7)

where

$$R\left({\mathbf{r}}_0,\mathbf{r},t\right)=\sum _1^N{\int }_{-\infty }^{\infty }{h}_n\left({\mathbf{r}}_0,t_0\right)h_n\left(\mathbf{r},t+t_0\right)d{t}_0$$ (8)

can be regarded as a spatiotemporal cross-correlation (s⁻¹ · m⁻²) of the array elements’ impulse responses in the medium of interest. This cross-correlation comprises a matched filter for the element impulse response functions, incorporating diffraction, any modeled acoustic inhomogeneities, and the elements’ electroacoustic responses (Dorme and Fink, 1995).

For a receiving array with a sufficiently large aperture and a sufficiently broadband frequency response, the spatiotemporal autocorrelation function (8) approaches zero except in the vicinity r = r₀, t = 0. Thus, the numerically time-reversed field can be approximated as

$$u_{\text{TR}}(\mathbf{r},t)\approx \left[{\int }_{-\infty }^{\infty }{\int }_{V_0}R\left({\mathbf{r}}_0,\mathbf{r},\tau \right)d{V}_{0}d\tau \right]q(\mathbf{r})a(-t).$$ (9)

Under the assumption that the bracketed integral is weakly dependent on the field position r (e.g., that the elements’ impulse responses are approximately position-independent over the source volume V₀, which for proton radiation is a confined region near the Bragg peak), Eq. (9) comprises a scaled, time-reversed representation of the original thermoacoustic source q(r)a(t).

To approximate the spatial distribution of the thermoacoustic source amplitude q(r) from the numerically time-reversed field u_TR(r, t) without influence from oscillations in the emission waveform a(t) and the element impulse responses h_n(t), it is useful to consider the amplitude envelope of the time-reversed field, which can be determined from the complex analytic-signal form of time-dependent signals,

$$\widehat{x}(t)=x(t)+j\mathscr{H}[x(t)],$$ (10)

where $\mathscr{H}$ denotes the Hilbert transform. In this form, Eq. (9) can be written

$${\widehat{u}}_{\text{TR}}(\mathbf{r},t)\approx B(\mathbf{r})\widehat{a}(-t)q(\mathbf{r}),$$ (11)

where B (m) is the bracketed term in Eq. (9), assumed to depend weakly on the position r. Since the source amplitude q(r) is constrained to be positive, it can then be approximately reconstructed as

$$q(\mathbf{r})\approx \frac{\left|{\widehat{u}}_{\text{TR}}\left(\mathbf{r},t_1\right)\right|}{|B(\mathbf{r})|\left|\widehat{a}\left(-t_1\right)\right|},$$ (12)

where t₁ is a specified time at which the source waveform a(−t₁) has substantial amplitude (e.g., near the time of the initial thermoacoustic emission). Thus, within multiplicative constants, the source amplitude q(r) is approximated by the magnitude of the time-reversed field $\left|{\widehat{u}}_{\text{TR}}\right|$, evaluated at a time t₁ corresponding to the acoustic emission. Given calibration of the array element impulse responses h_n(r, t) in units of pressure and numerical evaluation of the integral term B(r), Eq. (12) yields a quantitative estimate of the position-dependent thermoacoustic source amplitude q(r) (Pa · s/m²).

This reconstruction of the thermoacoustic source amplitude can be conveniently computed numerically in the frequency domain. Applying the convolution theorem to Eq. (5), the Fourier transform of the time-reversed field u_TR(r, t) is written as

$$\begin{gathered} U_{\text{TR}}(\mathbf{r},f)={\int }_{-\infty }^{\infty }{u}_{\text{TR}}(\mathbf{r},t)e^{-j2\pi ft}dt \\ =\sum _1^N{P}_n{(f)}^{*}{H}_n(\mathbf{r},f), \end{gathered}$$ (13)

where capital letters denote Fourier transformed quantities and the star denotes phase conjugation (i.e., time reversal of each frequency component). This frequency-domain expression can then be inverted at the specified time t₁ as

$${\widehat{u}}_{\text{TR}}\left(\mathbf{r},t_1\right)=2\sum _1^N{\int }_0^{\infty }{P}_n{(f)}^{*}{H}_n(\mathbf{r},f)e^{j2\pi f{t}_1}df,$$ (14)

where restriction of integration to positive frequencies (f ≥ 0) and multiplication by 2 results in the analytic-signal form of the time-reversed field. From Eq. (12), the thermoacoustic source amplitude q(r) is then approximated, within a multiplicative constant, by this field’s magnitude $\left|{\widehat{u}}_{\text{TR}}\left(\mathbf{r},t_1\right)\right|$.

This approach to numerical time reversal was implemented here in two versions, incorporating two different models for the array element impulse response h_n(r, t). The first approach models a linear array as ideal line sources and receivers at positions r_n = (x_n, y_n, z_n), where x is the elevation direction, y is the azimuth (array) direction and z is the depth (axial) direction (see Figure 1(a) for a sketch of the beamforming geometry). For a homogeneous medium, the element impulse response is then written in the frequency domain, based on the two-dimensional Green function for the wave equation, as

$$H_{\text{line},n}(\mathbf{r},f)=-\frac{i}{4L}{H}_0^{(2)}\left(k\left|\mathbf{r}-{\mathbf{r}}_n\right|\right)$$ (15)

$$\approx \frac{e^{-i\pi /4}}{\sqrt{8\pi k}L}\frac{e^{-ik\left|\mathbf{r}-{\mathbf{r}}_n\right|}}{\sqrt{\left|\mathbf{r}-{\mathbf{r}}_n\right|}},$$ (16)

where $H_0^{(2)}$ is the Hankel function of the second kind, corresponding to outgoing waves from an ideal, infinite-length line source (Morse and Ingard, 1968), $\left|\mathbf{r}-{\mathbf{r}}_n\right|=\sqrt{{\left(y-y_n\right)}^2+{\left(z-z_n\right)}^2}$ is the distance from an array element to a field point within the image plane (x = 0), k is the wavenumber 2πf/v_s, and L (m) is a constant associated with the finite length of physical line elements. The approximation of Eq. (16) is valid for distances large compared to the acoustic wavelength, and its denominator corresponds to cylindrical spreading of wavefronts from each element. The resulting numerical time reversal operation, i.e., Eq. (14) incorporating the element impulse response from Eq. (16), is similar to delay-and-sum beamforming or backprojection of the received emissions (Patch et al., 2016, 2019).

Figure 1:

49 MeV protons stopping in an oil cavity (depth approximately 47–82 mm) within a tissue-mimicking gelatin phantom. (a) Acoustic source reconstruction by simulated time reversal employing ideal line sources, with inset showing definition of coordinates used in source reconstructions. In this and later figures, the relative reconstructed source amplitude is overlaid using a linear scale and the Matlab “hot” colormap on the underlying B-mode image from the same linear array, plotted using a (b) Acoustic source reconstruction by simulated time reversal employing rectangular sources. (c) Proton dose calculated by TRIM Monte Carlo simulation. (d) Maximum-amplitude projections of the Bragg peak along the azimuth (top) and depth (bottom) directions for reconstructions using rectangular sources (RS) and line sources (LS). The projection plotted vs. depth also shows positions of the proton beam entry point (EP) and Bragg peak (BP) estimated from the Monte Carlo simulation.

The second numerical time-reversal method incorporates the modeled time-dependent, three-dimensional acoustic field generated by each array element, excited by a time-reversed copy of its received acoustic emission signal. For rectangular array elements, the impulse response can be conveniently computed in the frequency domain using a Fresnel approximation for rectangularly symmetric sources (Mast, 2007).

For either of these approaches, numerical time reversal can implemented in the frequency domain from received signals p_n(t_m) sampled at a rate f_s, computing their discrete Fourier transforms P_n(f_m) and implementing a discrete version of Eq. (14) as

$$\left|{\widehat{u}}_{\text{TR}}\left(\mathbf{r},t_1\right)\right|=\left|\frac{2}{T}\sum _{f_m=0}^{f_{\max }}\sum _{n=1}^N{P}_n{\left(f_m\right)}^{*}{H}_n\left(\mathbf{r},f_m\right)e^{j2\pi f_m{t}_1}\right|,$$ (17)

where T is the duration of the sampled signals p_n(t_m) and the summation is carried out over non-negative frequencies up to f_max, a cutoff not greater than the Nyquist frequency f_s/2. Under the assumptions described above, the reconstructed field amplitude $\left|{\widehat{u}}_{\text{TR}}\left(\mathbf{r},t_1\right)\right|$ can then be regarded as a relative map of the thermoacoustic source amplitude q(r).

3. Methods

Measured data

Beamformed acoustic emissions analyzed here were measured data, recorded in two previously reported studies. The first study (Patch et al., 2016) employed an 88” cyclotron to form a proton beam with energy 49±0.15 MeV, focused to 4 mm diameter and modulated by an electrostatic chopper to generate pulses of 1.76 μs length (full temporal width at half maximum). The proton beam was directed into a tissue-mimicking, gelatin-based ultrasound phantom with a cavity mimicking the intestine that could be filled with air or fluid.

The second study (Patch et al., 2019) employed 12 MHz superconducting resonators to accelerate protons into a beam with energy 15.99 ± 0.15 MeV and temporal pulse length 0.25 μs. In this case, the proton beam was directed into a water target within an acrylic tank, in some cases also including a tissue-mimicking gelatin phantom and a 5 mm layer of cortical bone.

In both studies, acoustic emissions were received by a 96-channel linear ultrasound array with nominal bandwidth 1–4 MHz (P4-1, ATL), centered approximately on the proton beam axis. Emissions received by each array element were recorded by a programmable ultrasound platform (Verasonics V1), with sampling rates of 30 MHz in the first study (49 MeV protons) and 10 MHz in the second study (16 MeV protons). Further details of the measurements are provided in previous publications (Patch et al., 2016, 2019).

Center frequencies of measured acoustic emissions were estimated from the power spectra of filtered, averaged emissions. Power spectra were measured as the mean squared magnitude of the discrete Fourier transform of emission signals, averaged over all array elements and all repeated measurements performed for each case. The center frequency of measured emissions for each case was then estimated as the first spectral moment of the measured power spectrum.

Signal-to-noise ratios (SNR) of measured acoustic emissions were estimated as the ratio of maximum envelope magnitude of filtered, averaged emission signals to the root-mean-square value of filtered, averaged emissions within a time span not observed to include any substantial proton-induced emissions. For 49 MeV protons, this analysis was performed only for synchronous averaging of 1024 emission measurements, because only signals averaged in this manner had been recorded. For 16 MeV protons, SNR was estimated for different numbers of synchronously averaged emission signals (128, 64, 32, 16, 8, 4, 2, and 1).

Acoustic source reconstructions

Before numerical reconstructions, received emissions were averaged synchronously, with 1024 averages for emissions from 49 MeV protons and 128 averages for emissions from 16 MeV protons, as employed in previously published analyses (Patch et al., 2016, 2019). After subtraction of the average signal level to remove any zero-frequency (DC) components, measured emissions were low-pass filtered by frequency-domain multiplication with the function

$$H_{\text{LP}}(f)=\frac{1-\mathrm{erf}\left(\left(f-f_c\right)/(\sqrt{2}\sigma )\right)}{2},f>0,$$ (18)

where the cutoff frequency was f_c = 1 MHz for 49 MeV protons and f_c = 3.5 MHz for 16 MeV protons and the width parameter σ was 0.1 MHz. These cutoff values were empirically found to reduce noise while not substantially impacting spatial resolution.

Reconstructions were performed using the numerical time-reversal method of Eq. (17), implemented in Matlab (The MathWorks). Acoustic fields of array elements were computed both using the idealized line source model of Eq. (16) and a modified Fresnel approximation that represents the fields of rectangular apertures as diffracted spherical waves (Mast, 2007). For the rectangular-source model, elements of the P4-1 array were assumed to be flat rectangles with width 0.245 mm in the azimuthal direction, height 14 mm in the elevation direction, and pitch 0.295 mm, consistent with manufacturer-provided dimensions and previously published characterization of this array type (Vogt et al., 2017).

Cases compared for 49 MeV protons (Patch et al., 2016) included protons stopping in an oil-filled cavity within a tissue-mimicking gelatin phantom for a sagittal view (Figure 6(a) in Patch 2016; N = 27 repeated measurements) and protons stopping in a tissue-mimicking gelatin phantom distal to an air-filled cavity for a coronal view (Figure 6(c) in Patch 2016; N = 25). Cases compared for 16 MeV protons (Patch et al., 2019) included protons stopping in water (Figure 3(a)–(b) in Patch 2019; N = 14), and stopping in a 5 mm bone layer (Figure 5(a)–(b) in Patch 2019; N = 6). For consistency with previous published analysis, the speed of sound v_s was assumed to be a constant 1.48 mm/μs for the cases with 49 MeV protons (Patch et al., 2016), and a constant 1.54 mm/μs for water and gelatin in the cases with 16 MeV protons (Patch et al., 2019).

For the case with protons stopping in the 5 mm bone layer (Patch et al., 2019), aberration-corrected thermoacoustic images were also obtained by applying the phase shift

$$\varphi \left(\mathbf{r},{\mathbf{r}}_n\right)=2\pi f_m{l}_{\text{bone}}\left(\mathbf{r},{\mathbf{r}}_n\right)\left(\frac{1}{{\upsilon }_s}-\frac{1}{{\upsilon }_{\text{bone}}}\right)$$ (19)

to each signal before summation, where l_bone(r, r_n) is the line-of-sight path length through a bone specimen, assumed to comprise a parallel layer with sound speed v_bone = 3.17 mm/μs (Patch et al., 2019) spanning the depths 60.2 ≤ z ≤ 65.2 mm. Changes in travel time due to refraction at the water-bone interface were found not to substantially affect reconstructions, and were thus neglected for simplicity.

For display, reconstructed acoustic source maps are overlaid using a linear scale on grayscale B-mode images of each phantom configuration, obtained using the same array. Because the same array elements, array and phantom positions, and assumed sound speeds were used for both B-mode imaging and time-reversal reconstructions, these two images are intrinsically co-registered for each case (Patch et al., 2016). Thus, acoustic emission sources localized by numerical time reversal appear at their original positions, relative to structures depicted in the B-mode image.

In order to test differences between the two considered versions of numerical time reversal, reconstructions were repeated for multiple realizations of each measured configuration, each comprising an independent measurement of all averaged emission signals. Reconstructed acoustic sources were spatially compared based on their azimuthal and axial widths, corresponding respectively to transverse and axial dimensions of the proton beam. Maximum-amplitude projections of acoustic source reconstructions were computed in the azimuthal and axial directions. Corresponding beamwidths, defined as the distance between −3 dB amplitudes in each direction at the detected Bragg peak, were found using linear interpolation of the reconstructed sources and statistically compared using the paired Student t test between the two numerical time-reversal methods for each configuration.

To assess the effects of noise level and aperture size on numerical time-reversal reconstructions, analysis of emissions from 16 MeV protons stopping in water was repeated for multiple averaging conditions and receiving aperture sizes. Reconstructions modeling array elements as either line sources or rectangular sources were completed and analyzed using different numbers of synchronously averaged signals (128, 64, 32, 16, 8, 4, 2, and 1) and different array apertures (96, 64, 48, 32, 16, and 8 elements), each centered at the array’s azimuthal midpoint. Estimated Bragg peak depth and −6 dB width of the corresponding reconstructed acoustic source distribution were determined for a total of N = 14 iterations for each combination.

Monte Carlo simulations

For verification of accuracy and spatial resolution, acoustic source reconstructions were compared with Monte Carlo simulations using the TRIM (Transport of Ions in Matter) algorithm as implemented in the software SRIM (Stopping and Range of Ions in Matter; version SRIM-2013, available from SRIM.org). The TRIM algorithm computes three-dimensional paths of ions in layered amorphous materials using a binary collision approximation (Ziegler et al., 2007).

To accurately represent proton beam profiles within the TRIM algorithm, initial proton energies, starting positions, and trajectories were specified by a Matlab script. Beams were simulated using a total of 5 × 10⁵ ions. Each ion’s initial trajectory and position were randomly sampled from weighted Gaussian distributions that mimicked measured beam fluences and profiles, as described previously (Patch et al., 2016, 2019). Parallel material layers following the beam exit window were modeled within TRIM to mimic the geometry of the measurements analyzed (Patch et al., 2016, 2019), following descriptions of material properties and thicknesses for each layer.

Output data from TRIM simulations specified each ion’s energy, position, and trajectory at every space-time point corresponding to a differential energy loss of 100keV. A Matlab script computed the total position-dependent dose by calculating the change in ion energy vs. depth (dE/dz) for each simulated history. The final dose deposition map was produced by summing the position-dependent deposited energy E across all particle histories. A cross-section of this simulated dose map was recorded in the ultrasound image plane with a step size of 0.1 mm in the azimuth direction and 0.01 mm in the depth direction, then smoothed using the Matlab function imgaussfilt3 with a width parameter of 1 pixel in each direction. The resulting simulated dose map was overlaid on corresponding B-mode ultrasound images for comparison with thermoacoustic source reconstructions. To register TRIM simulation output with B-mode ultrasound images, depths of specific interfaces between layers were aligned with their apparent depths on the B-mode images.

In order to compare Bragg peak positions simulated and estimated from time-reversal reconstructions, Bragg peak depths relative to the ultrasound array position (z = 0) was estimated as the z coordinate of the grid point with maximum simulated proton dose. These depths were compared with those estimated from the acoustic source reconstruction, defined as the z coordinate at which the maximum-amplitude projection of the mapped source was largest. Similarly, the primary entry point of the proton beam into its stopping layer was estimated as the maximum amplitude of the first derivative of the simulated proton deposition vs. depth, computed by centered differences. Widths of simulated Bragg peaks in the azimuthal and depth directions, relative to the ultrasound array, were quantified as the $-3\text{dB}(1/\sqrt{2}\text{amplitude}\text{ratio})$ amplitude ratio) widths of the simulated proton dose distribution in the respective directions.

4. Results

Acoustic emissions measured for 49 MeV protons were analyzed for two cases, the first with protons stopping in an oil-filled cavity with a tissue-mimicking ultrasound phantom, and the second with protons stopping in gelatin distal to an air-filled cavity. Measured signal-to-noise ratios for emissions measured with 1024 synchronous averages were 11.9 ± 1.3 dB (mean ± standard deviation for protons stopping in oil and 7.0±1.2 dB for protons stopping in gelatin. Measured center frequencies of recorded emissions were 333 kHz for protons stopping in oil and 386 kHZ for protons stopping in gelatin.

Representative thermoacoustic source reconstructions for 49 MeV protons stopping in the oil-filled cavity (Patch et al., 2016) are shown in Figure 1. As observed previously from backprojection of the same measured data (Patch et al., 2016), substantial source amplitude is seen both at the Bragg peak (depth 65 mm) and at the point of the proton beam’s entry into the oil-filled cavity (depth 83 mm). For the reconstructions employing rectangular sources, appreciable improvements are seen relative to reconstructions employing ideal line sources, observed as narrowing of the reconstructed Bragg peak in both the lateral (azimuth) direction (mean ± standard deviation of −3 dB widths 10.5±0.6 mm for rectangular sources, 12.1±0.5 mm for line sources) and the axial (depth) direction (3.6±0.3 mm for rectangular sources, 3.8 ± 0.2 mm for line sources). For comparison, Figure 1 also shows the map of proton energy deposition simulated by the Monte Carlo method. The Monte Carlo-simulated Bragg peak had a −3 dB width of 3.2 mm along the azimuth (array) direction and 0.98 mm along the depth (axial) direction. Simulated and reconstructed positions of the Bragg peak agreed well, although the reconstructed acoustic source width was wider than the Monte Carlo-simulated Bragg peak in both directions.

Figure 2 shows an acoustic source reconstruction for 49 MeV protons stopping in gelatin distal to the air-filled cavity in the same tissue-mimicking ultrasound phantom (Patch et al., 2016). As in Figure 1, substantial reconstructed source amplitude is seen both at the Bragg peak and the point of entry into the gelatin. Similar improvements in resolution are evident for the reconstruction using rectangular sources, compared to the reconstruction using line sources, in both the azimuth direction (peak widths 7.8 ± 1.8 mm for rectangular sources, 9.3±0.7 mm for line sources) and the depth direction (4.0±1.6 mm for rectangular sources, 5.5 ± 0.6 mm for line sources). In comparison, the Monte Carlo-simulated Bragg peak had a −3 dB width of 3.3 mm along the azimuth (array) direction and 0.89 mm along the depth (axial) direction. Both reconstructions show greater spatial variability compared to Figure 1, likely caused by the different distances from the Bragg peak to the ultrasound array, about 65 mm for protons stopping in the oil-filled cavity (Figure 1) and 30 mm for protons stopping in gelatin (Figure 2). Source reconstruction by numerical time reversal at smaller depths is likely more sensitive to diffraction effects in the near field of the ultrasound array aperture. In addition, the cavity’s cylindrical shape, not accounted for in the Monte Carlo simulation, may have increased proton range straggle, both upon entry into the cavity (Figure 1) and exit into the gelatin (Figure 2).

Figure 2:

49 MeV protons stopping in gelatin for the phantom from Figure 1 with an air-filled cavity (depth approximately 45–80 mm), in the same format as Figure 1. For visualization of the entire image plane, reconstructed acoustic source distributions are overlaid on co-registered B-mode images of the same phantom with an oil-filled cavity. (a) Reconstruction using line sources. (b) Reconstruction using rectangular sources. (c) Proton dose from Monte Carlo simulation. (d) Maximum-amplitude projections with estimated positions of proton beam entry point and Bragg peak.

Acoustic emissions measured for 16 MeV protons were analyzed for two cases, the first with protons stopping in water, and the second with protons stopping in a 5 mm bone layer. Measured signal-to-noise ratios for emissions measured with 128 synchronous averages were 14.0 ± 0.7 dB for protons stopping in water and 14.6 ± 0.6 dB for protons stopping in bone. SNR was found to decrease by about 3 dB for each twofold decrease in the number of synchronously averaged measurements (e.g., 11.0 ± 0.6 dB for 64 averages), consistent with the expectation that variance of measured noise spectra is inversely proportional to the number of averages. Measured center frequencies of recorded emissions were 1.76 MHz for protons stopping in water and 1.74 MHz for protons stopping in bone.

A reconstruction for 16 MeV protons stopping in water (Patch et al., 2019) is shown in Figure 3. In addition to reconstructed thermoacoustic emission sources at the Bragg peak and the entry point from the bone layer into water, an additional image source was reconstructed, as seen previously using backprojection (Patch et al., 2019). This image source was caused by strong reflection of the Bragg peak thermoacoustic emission from the interface between bone and water layers. Reconstructed acoustic sources associated with the Bragg peak were similar in width between the two reconstruction methods. The reconstruction employing rectangular sources shows a slightly narrower reconstructed peak in the azimuthal direction (5.1±0.5 mm for rectangular sources, 5.3±0.5 mm for line sources), while the reconstruction employing line sources shows a slightly narrower reconstructed peak in the depth direction (0.70 ± 0.04 mm for rectangular sources, 0.59 ± 0.06 mm for line sources). In comparison, the Monte Carlo-simulated Bragg peak had a −3 dB width of 2.5 mm along the azimuth (array) direction and 0.32 mm along the depth (axial) direction.

Figure 3:

Reconstructions for 16 MeV protons stopping in water. An air-water interface, bounded by a 60 μm membrane, is seen at approximately 67 mm depth. (a) Line sources. (b) Rectangular sources. (c) Proton dose from Monte Carlo simulation. (d) Maximum-amplitude projections of acoustic reconstructions.

A reconstruction for 16 MeV protons stopping in the 5 mm bone layer (Patch et al., 2019) is shown in Figure 4 (a)–(b). Similar to the reconstruction for 16 MeV protons stopping in water (Figure 3), widths of the reconstructed Bragg peak acoustic source were slightly narrower in azimuth for the reconstruction employing rectangular sources (4.6 ± 0.5 mm for rectangular sources, 4.7 ± 0.5 mm for line sources) and slightly narrower in depth for line sources (0.88 ± 0.08 mm for rectangular sources, 0.80 ± 0.10 mm for line sources). The Monte Carlo-simulated Bragg peak had a −3 dB width of 3.2 mm along the azimuth (array) direction and 0.20 mm along the depth (axial) direction. For this reconstruction, assumption of a homogeneous sound speed resulted in an inaccurate estimate of the Bragg peak’s position in the depth direction, relative to room coordinates.

Figure 4:

Reconstructions for 16 MeV protons stopping in a 5 mm bone layer. Bone layer boundaries at about 60 and 65 mm depths are indicated by red dashed lines in each image panel. At 52–53 mm depth, a reflection is seen from the oblique lower boundary of a tissue-mimicking gelatin phantom that extended over depths of approximately 15–55 mm. (a) Line sources, numerical time reversal into water. (b) Rectangular sources, numerical time reversal into water. (c) Line sources, corrected for travel time through bone. (d) Rectangular sources, corrected for travel time through bone. (e) TRIM simulation of proton dose. (f) Maximum-amplitude projections, corrected for travel time through bone.

An aberration-corrected reconstruction for 16 MeV protons stopping in bone, accounting for the higher speed of sound within bone, is shown in Figure 4 (a)–(d). With correction, the depth position of the Bragg peak shifted to that expected from the Monte Carlo simulation. This correction caused no substantial change in azimuthal resolution (4.6 ± 0.5 mm for rectangular sources, 4.8 ± 0.5 mm for line sources), but apparent depth resolution worsened (1.8±0.2 mm for rectangular sources, 1.6±0.2 mm for line sources) because the reconstructed Bragg peak widened along the depth direction, due to the longer acoustic wavelength in the higher-speed bone.

Depth positions of the Bragg peak estimated from acoustic source reconstructions, defined relative to the array surface, are summarized as means and standard deviations in Table 1, with a comparison to Bragg peak depths from TRIM simulations for the same cases (red dashed lines in panels (d) of Figures 1–4). These results are consistent with those reported previously using backprojection of measured acoustic emissions (Patch et al., 2016, 2019), agreeing with the Bragg peak depth within 0.5 mm in all cases. For 49 MeV protons stopping in a tissue-mimicking phanton, the maximum difference between mean estimated Bragg peak depths and Monte Carlo simulated depths was 1.86 mm for numerical time reversal employing idealized line sources and 0.47 mm for numerical time reversal employing rectangular sources. For 16 MeV protons stopping in water, Bragg peak depth error was < 0.5 mm for both numerical time reversal methods. For 16 MeV protons stopping in bone, Bragg peak depth error was > 2 mm for both uncorrected numerical time reversal methods, due to the higher speed of sound in the bone layer. Correction for this sound speed difference resulted in depth errors < 0.5 mm for both methods. In all cases, these errors were smaller than the −3 dB width of the Bragg peak reconstruction in either the azimuth or depth directions, as plotted in Figure 5. The differences are also comparable to the observed depth resolution of the B-mode images, seen in Figures 1–4 to be about 1 mm, and thus are comparable to the achievable accuracy of registration between TRIM simulations and acoustic source reconstructions.

Table 1:

Bragg peak positions (depths of the maximum reconstructed thermoacoustic source amplitude, relative to the array surface) estimated from acoustic source reconstructions by numerical time reversal using rectangular sources and idealized line sources, shown as mean ± standard deviation in mm. Also shown for comparison are corresponding Bragg peak depths computed by TRIM simulations.

Configuration	Line sources	Rectangular sources	TRIM simulation
49 MeV protons stopping in oil, N = 27 realizations	65.02 ± 0.20	64.68 ± 0.45	64.51
49 MeV protons stopping in gelatin, N = 25	30.17 ± 1.08	28.78 ± 1.44	28.31
16 MeV protons stopping in water, N = 14	63.99 ± 0.03	63.95 ± 0.04	64.36
16 MeV protons stopping in bone (uncorrected), N = 6	62.12 ± 0.04	62.03 ± 0.03	64.37
16 MeV protons stopping in bone (corrected), N = 6	64.13 ± 0.06	63.96 ± 0.06	64.37

Figure 5:

Means and standard deviations for −3 dB azimuthal (array direction) and depth (axial direction) widths of reconstructed acoustic emission sources in the vicinity of the Bragg peak. Asterisks denote significant differences between time-reversal reconstructions employing line sources (dark bars) and rectangular sources (light bars). Asterisks denote statistical significance of two-tailed, paired t tests for p < 0.05 (*), p < 10⁻³ (**), or p < 10⁻⁶ (***). (a) 49 MeV protons stopping in oil (N = 27), corresponding to Figure 1. (b) 49 MeV protons stopping in gelatin (N = 25), corresponding to Figure 2. (c) 16 MeV protons stopping in water (N = 14), corresponding to Figure 3. (d) 16 MeV protons stopping in bone with corrected travel time (N = 6), corresponding to Figure 4(c–d).

Results of Student t tests comparing −3 dB beamwidths for the two numerical time-reversal methods are shown in Figure 5. For the cases with 49 MeV protons stopping in oil and gelatin, substantially smaller azimuthal peak widths, indicating higher azimuthal image resolution (p < 10⁻⁴), are seen for the reconstructions employing rectangular sources. Significantly smaller peak widths along the depth direction are also apparent (p < 10⁻⁴). For the cases with 16 MeV protons stopping in water and bone, differences between azimuthal and depth resolution were small but statistically significant between the two numerical time-reversal methods, with slightly better azimuthal resolution for reconstructions with rectangular sources and slightly better depth resolution for reconstructions with line sources (p < 0.05). Similar Student t tests comparing corrected vs. uncorrected reconstructions showed no significant changes in azimuthal resolution (p > 0.3) but a significant worsening of depth resolution (p < 10⁻⁴).

Notably, apparent spatial resolution of acoustic emission maps was substantially better for the measurements with 16 MeV protons shown in Figures 3–4, primarily due to the shorter temporal pulse length of the recorded acoustic emissions in these experiments (Patch et al., 2019), compared to the measurements with 49 MeV protons shown in Figures 1–2 (Patch et al., 2016). This shorter pulse length, associated with higher center frequency and greater bandwidth, resulted in source reconstructions with widths comparable to simulated peaks of proton dose deposition along both the azimuth (array) and depth (axial) directions. In comparison, the lower center frequency and bandwidth for the reconstructions depicted in Figures 1–2 resulted in reconstructed source maps considerably wider than simulated proton dose distributions in both the azimuth and depth directions.

Results for thermoacoustic reconstruction as a function of emission signal averaging and array aperture size are shown in Figures 6–8 for the case with 16 MeV protons stopping in water. Figure 6 shows representative reconstructions employing 128, 32, and 8 synchronous averages with aperture sizes of 96, 32, and 16 array elements, each shown a format analogous to Figure 3. For numerical time-reversal reconstructions modeling array elements either as ideal line sources or diffracting rectangular sources, trends are similar, including a decrease of azimuthal resolution with decreased aperture size and increased artifacts (spuriously reconstructed acoustic sources) with fewer averages.

Figure 6:

Numerical time-reversal reconstructions for 16 MeV protons stopping in water. Representative reconstructions are shown for 8–128 coherent averages of acoustics emissions received by apertures with 16–96 elements, modeled as (a) ideal line sources and (b) rectangular sources.

Figure 8:

Effects of receiving array aperture (8–96 elements) on estimation of Bragg peak (BP) position for 16 MeV protons stopping in water with measurements employing 128 coherent averages. Mean ± standard deviation of estimated Bragg peak depth position (top) and −6 dB width of reconstructed acoustic source distribution (bottom) are shown for time-reversal reconstructions employing (a) ideal line elements and (b) rectangular array elements.

Corresponding effects of signal averaging on estimation of the Bragg peak position are quantified in Figure 7, showing the estimated peak position in the depth direction and the estimated −3 dB peak width vs. the number of synchronous averages for reconstructions with a full (96-element) array aperture. Substantial degradation in ranging accuracy is seen for estimates employing less than 16 averages, corresponding to SNR less than 5 dB, while width of the reconstructed source distribution varies only slightly vs. the number of averages. Figure 8 shows corresponding statistics for dependence on the array aperture size for reconstructions with 128 synchronous averages. Here, ranging accuracy is seen to depend only weakly on aperture size. In contrast, width of the reconstructed acoustic source distribution increased as the number of array elements decreased, consistent with the expected inverse proportionality between acoustic beam width and aperture size.

Figure 7:

Effects of coherent acoustic emission averaging (1–128 averages) on estimation of Bragg peak (BP) position for 16 MeV protons stopping in water with measurements by a 96-element array aperture. Mean ± standard deviation of estimated Bragg peak depth position (top) and −6 dB width of reconstructed acoustic source distribution (bottom) are shown for time-reversal reconstructions employing (a) ideal line elements and (b) rectangular array elements.

5. Discussion

The results reported here suggest that reconstruction of thermoacoustic sources based on measurements by standard clinical ultrasound arrays can be useful for mapping of dose distribution delivered by proton beams. This approach provides a map of the thermoacoustic source amplitude, which is proportional to the time derivative of the delivered radiation dose rate. Results suggest that improved reconstruction of thermoacoustic sources (e.g., improved lateral resolution) is attained by numerical time reversal that models array elements as diffracting rectangular sources, relative to models employing ideal line sources. Numerical time reversal modeling array elements as ideal point sources, as performed in previous studies (Yu et al., 2019; Lascaud et al., 2021), results in spatial resolution similar to that obtained here using ideal line sources, since the modeled acoustic field of each element (Equation 16) differs only by multiplicative constants and the slowly varying factor $1/\sqrt{\left|\mathbf{r}-{\mathbf{r}}_n\right|}$.

In principle, Eq. (12) can be employed to estimate the thermoacoustic source term from Eq. (1), which relates directly to the delivered dose. However, accurate estimation the delivered proton dose by this method requires several additional steps not undertaken here, including calibration of array elements to measure spatially averaged acoustic pressure and accurate accounting of medium acoustic and thermal properties. Dosimetry based on reconstruction of thermoacoustic sources has been shown feasible in simulations (Alasnea et al., 2015; van Dongen et al., 2019; Yu et al., 2019, 2021; Samant et al., 2022), but is beyond the scope of the work reported here. Alternatively, the reconstructed thermoacoustic source amplitude could be calibrated to proton dose measured in phantoms, similar to a technique implemented to estimate ionizing radiation dose from acoustic emission amplitude in FLASH radiotherapy (Oraiqat et al., 2020).

Use of numerical time reversal for thermoacoustic source reconstruction allows straightforward accounting for acoustic inhomogeneities in the measured medium (Yu et al., 2019; Xu et al., 2004; Treeby et al., 2010). Although the only acoustic inhomogeneity considered here was the higher speed of sound within a layer of bone, the approaches described here can be applied to general inhomogeneous media. This requires only application of Eq. (17), with the acoustic field of each element H_n(r, f_m) computed within the inhomogeneous medium. One convenient means appropriate for numerically simulating ultrasound propagation within soft tissues is the k-space method (Tabei et al., 2002; Treeby and Cox, 2010). To maintain co-registration of thermoacoustic source reconstructions, the same aberration correction would need to be applied to the anatomic ultrasound image, which can potentially be performed using available methods for pulse-echo B-mode imaging (Fontanarosa et al., 2011; Jaeger et al., 2015) and ultrasound tomography (Mast, 2002; Malik et al., 2018; Javaherian and Cox, 2021).

For effective mapping of proton doses from their thermoacoustic emissions, sufficient signal-to-noise ratio (SNR) must be attained from an appropriate combination of measurement hardware, emission amplitude and bandwdith, and averaging of measurements. The reconstructions shown in Figures 1–4 were attained for relatively low SNR within the interval 7–15 dB, while the minimum SNR required for accurate Bragg peak position estimation was found to be approximately 5 dB, as illustrated in Figures 6 and 7. This result is consistent with trends for the dependence of ranging accuracy on SNR reported in a previous simulation study (Lascaud et al., 2021), in that accuracy is markedly decreased below a certain SNR threshold; however, the specific threshold differed, likely because of differences in the studies’ respective definitions of SNR.

Altering the numbers of synchronously averaged emission signals (1024 for 49 MeV protons and 128 for 16 MeV protons) will correspondingly alter the SNR of reconstructed acoustic source maps, in approximate proportion to the square root of the number of averages (e.g., a 3 dB gain in SNR for each doubling in number of averaged signals), consistent with trends seen in Figures 6 and 7 for 16 MeV protons stopping in water. SNR would also be increased by use of receiving arrays that are highly sensitive within the frequency range of the measured emissions. For both sets of measured data employed here, emissions had substantial energy in frequency components below the array’s nominal frequency band of 1–4 MHz, which tended to decrease SNR, thus requiring greater signal averaging. More precise matching between the frequency ranges of proton-generated acoustic emissions and ultrasound array sensitivity would further decrease the amount of averaging required.

In addition, spatial resolution of reconstructed sources is limited in the azimuthal direction (relative to the ultrasound array) by the finite size of the array aperture, with higher azimuthal resolution attained by larger apertures. This effect is illustrated in Figure 6, which shows azimuthal broadening of reconstructed source distributions for decreased aperture sizes, and Figure 8, which demonstrates inverse proportionality between aperture size and azimuthal width of the reconstructed source distribution. Azimuthal resolution also increases in general with the center frequency of measured emissions, consistent with the narrower peak widths seen in Figures 3 and 4 for 16 MeV protons (center frequency > 1.7 MHz), compared to 49 MeV protons (center frequency < 0.4 MHz, Figures 1 and 2).

In the depth direction of an ultrasound array, spatial resolution of acoustic source reconstructions is limited primarily by the bandwidth of measured thermoacoustic emissions, determined by factors including the beam’s temporal pulse length or modulation, as well as properties of the receiving ultrasound array. In addition, emission bandwidth is limited by spatial spreading of the thermoacoustic source (Anastasio et al., 2007), which can increase with proton beam energy due to greater range straggle. For these reasons, accuracy of Bragg peak ranging along the depth direction of the ultrasound array is influenced only slightly by the array aperture size, as seen in Figure 8; similar results were obtained in a previous simulation study (Lascaud et al., 2021). In general, reconstructions of acoustic sources will be improved by increased intrinsic bandwidth of the thermoacoustic source and receiving transducers, consistent with observations from simulation studies of proton range verification (Jones et al., 2016).

For emissions measured by linear arrays, increases in the bandwidth of measured signals will particularly result in increased spatial resolution along the transducer’s depth (axial) direction. The present results suggest that for range verification accuracy on the order of 1 mm, bandwidth of approximately 1 MHz, corresponding to pulse lengths on the order of 2 μs, may be sufficient. More modest accuracy could be attained for emissions with lower bandwidths, with range uncertainty approximately inversely proportional to the bandwidth, although increased accuracy can potentially be obtained by incorporation of appropriate a priori information (Patch et al., 2021). Modulation of proton beams to decrease temporal pulse lengths and increase the center frequency of thermoacoustic emissions would potentially increase spatial resolution of source reconstructions, while decreasing the corresponding delivered proton dose. Future advancements of FLASH proton therapy (Wu et al., 2021), if engineered to increase the center frequency and bandwidth of associated acoustic emissions, may enable clinically practical acoustic mapping of proton dose by numerical time reversal-based reconstruction of thermoacoustic sources.

6. Conclusion

An approach for numerical time reversal of measured thermoacoustic emissions induced by proton beams has been introduced and applied to reconstruction of in tissue-mimicking phantoms. Two implementations of numerical time reversal, modeling array elements as either ideal line sources or diffracting rectangular sources, both successfully reconstructed acoustic source amplitude maps consistently with proton dose deposition predicted by Monte Carlo simulations. Resolution of acoustic source maps in the azimuth (array) direction was significantly improved by incorporation of array element diffraction in the numerical time-reversal algorithms. For a case with protons stopping within a bone layer, spatial accuracy of the reconstructed Bragg peak position was improved by aberration correction accounting for the layer’s higher speed of sound. Both implementations were limited in spatial resolution by the available array aperture dimensions and bandwidth of recorded acoustic emission signals. Numerical time reversal incorporating accurate transducer models and aberration correction may be useful for range verification and dosimetry of clinical proton beams, if acoustic emissions of sufficient amplitude and bandwidth can be generated by therapeutic proton sources.