*3.1. Bragg Peak*

The main physical advantage of heavy particles as compared to photons is their characteristic depth-dose profile, the known Bragg curve in honor of Sir William Henry Bragg who investigated the energy deposition of alpha particles, which form a Radium source in the air at the beginning of the last century [12]. While the photon dose decreases exponentially with penetration depth according to the absorption law for electromagnetic radiation, the depth-dose of heavy charged particles exhibits a flat plateau region with a low range of the particles [13]. This paper uses an analytical approach presented by T. Bortfiel in 1996 [4], and the numerical representation valid for protons with energies between 10 MeV and 200 MeV. Thus, the energy of single protons along a Z-axis in a homogeneous medium (water) is considered. The total energy released in the medium per unit mass in the Z-axis is shown below.

$$T(z) = -\frac{1}{\rho} \left( \phi(z) \frac{dE(z)}{dz} + \frac{d\phi(z)}{dz} E(z) \right) \tag{10}$$

where φ(*z*) is the proton flow, that is, the number of protons per cm2, *E*(*z*) is the energy deposited on the Z-axis, and ρ represents the mass density of the medium. The method makes use of a midpoint where a certain fraction γ of the energy released in nuclear interactions is absorbed locally while the rest is ignored. Then, the total absorbed energy *D*ˆ (*z*), will be given by:

$$\mathcal{D}(z) = -\frac{1}{\rho} \Big( \phi(z) \frac{dE(z)}{dz} + \gamma \frac{d\phi(z)}{dz} E(z) \Big) \tag{11}$$

A relationship between the initial energy *E*(*z* = 0) = *E*<sup>0</sup> and the range *z* = *R*<sup>0</sup> in the medium can be approximated as *<sup>R</sup>*<sup>0</sup> = <sup>α</sup>*Ep* <sup>0</sup> for *p* = 1.5. This relation is valid for protons with energy close to 250 MeV. The factor α is proportional to the square root of the effective atomic mass of the medium [4]. Using the inverse for *Ro* ≤ 0.5 cm and assuming *Eo* to be given in units of MeV, the best fit parameters for *Eo*(*Ro*) are *<sup>p</sup>* = 1.77, <sup>α</sup> = 2.2 <sup>×</sup> 10−<sup>3</sup> for the proton in water [4]. The remaining energy *<sup>E</sup>*(*z*) at an arbitrary depth *z* ≤ *Ro* fails to travel the distance *Ro* − *z* according to the range-energy relationship shown below.

$$E(z) = \frac{1}{\alpha^{1/p}} (R\_0 - z)^{1/p} \tag{12}$$

For energies above 20 MeV, there is non-negligible probability that protons may be lost from the beam due to nuclear interactions. This non-elasticity was studied and tabulated by Janni [14] as a function of the residual range (*R*<sup>0</sup> − *z*). The proton flow φ(*z*) can be written by using the equation below.

$$\phi(z) = \phi\_0 \frac{\beta}{1 + \beta R\_0} \tag{13}$$

where φ<sup>0</sup> is the primary fluence and the slope parameter β was determined to be β = 0.012 cm−<sup>1</sup> [4]. Thus, the distribution of the deposition of energy along the depth range can be expressed as the equation below.

$$D(z) = \Phi\_0 \frac{e^{\zeta^2/4} \sigma^{\frac{1}{\overline{\rho}}} \Gamma\left(\frac{1}{p}\right)}{\sqrt{2\pi} \rho p \alpha^{\frac{1}{\overline{\rho}}} (1 + \beta R\_0)} \times \left[\frac{1}{\sigma} L\_{-1/p}(-\zeta) + \left(\frac{\beta}{p} + \gamma \beta + \frac{\varepsilon}{R\_0}\right) L\_{-1/p - 1}(-\zeta)\right] \tag{14}$$

where <sup>Γ</sup> represents the gamma function, <sup>ζ</sup> = (*R*<sup>0</sup> <sup>−</sup> *<sup>z</sup>*)/σ, with a <sup>σ</sup> value of 0.012*R*0.935 <sup>0</sup> , and ε represents a relatively small fraction of the fluence Φ<sup>0</sup> in the peak. Figure 3 shows the distribution of the dose as a function of the range for a different proton energy.

**Figure 3.** Bragg peak for different energies. (**a**) The deposition of the dose varies according to the energy of the proton. The maximum of the Bragg peak varies according to the energy; (**b**) The relationship Range–Energy for protons in water is shown.

### *3.2. Thermoacoustic Model*

In the thermoacoustic case, an excited point source emits a pressure wave proportional to the first time derivate of the excitation pulse [15]. Hence, a Gaussian excitation pulse leads to a bipolar acoustic emission consisting of a positive compression, which results in an increase in pressure. This is followed by a negative rarefaction, which is a decrease of pressure. The positive and negative pressure peaks are not only due to the heating and cooling of the medium, but the variation of the heating rate also plays a role. The medium expands or contracts according to its coefficient of thermal volumetric expansion α'. As a result, a pressure wave is observed. The pressure wave from an energy deposition in a region can be understood as the sum of the individual responses that would be observed from decomposing the spatial deposition into point sources. The resulting pressure signal depends on the time derivative of the excitation pulse. The amplitude of the wave depends on the energy deposited, the number of protons per pulse of the beam, and the temporal shape of the excitation pulse. A dose of 1 Gy generates a ∼ 240 μK temperature increase in water [15]. Ignoring heat diffusion and cinematic viscosity, the wave equation that describes the pressure *<sup>p</sup>* at a time *<sup>t</sup>* and position *r*, is shown below [16–19].

$$\overrightarrow{\nabla}^{2}p(\overrightarrow{r},t) - \frac{1}{c\_{\rm s}^{2}} \cdot \frac{\partial^{2}p(\overrightarrow{r},t)}{\partial t^{2}} = -\frac{\alpha'}{\mathbb{C}\_{p}} \cdot \frac{\partial^{2}\varepsilon(\overrightarrow{r},t)}{\partial t^{2}}\tag{15}$$

where *cs* (ms<sup>−</sup>1) represents the speed of sound in the middle, *Cp* (J kg<sup>−</sup>1K−1) is the specific heat capacity, and → *r*, *t* (J s<sup>−</sup>1m−3) is the energy density deposited in the medium. Equation (15) can be solved using the Kirchhoff integral as shown below.

$$p\left(\overrightarrow{r},t\right) = \frac{1}{4\pi} \frac{\alpha'}{\mathbb{C}\_p} \int\_V \frac{dV'}{\left|\overrightarrow{r} - \overrightarrow{r}'\right|} \cdot \frac{\partial^2}{\partial t^2} \varepsilon \left(\overrightarrow{r}', t - \frac{\left|\overrightarrow{r} - \overrightarrow{r}'\right|}{c\_s}\right) \tag{16}$$

where *p* → *r*, *t* denotes the hydrodynamic pressure at a given place and time. The values for the thermoacoustic model were an energy of 100 MeV, a temporal profile of 1 <sup>μ</sup>s, 3.4 <sup>×</sup> 106 protons per pulse, the beam with a size of 1 mm, and a sensor located 40 mm from the Bragg peak. The characteristics of the simulation are given by simulation results from different studies, as well as their application in clinical cases [7,15,20–25]. The values for this case are shown in Figure 4. As a result, the pressure obtained at the reception point will be the signal that will be emitted by the piezoelectric [26] transducer to simulate a bipolar source that will be located by the sensor array.

**Figure 4.** (**a**) Bragg curves with an initial energy of 100 MeV protons in water. The line represents the dose contribution from the fraction of protons that have nuclear interactions; (**b**) Pressure for a sensor located 4 cm from the Bragg peak on the axis of symmetry of the emission.

### **4. Experimental Setup**

The experimental data measurements were made in the laboratories of the physics department at the Universitat Politècnica de València (Spain). There is a water tank with a volume of 0.64 m<sup>3</sup> with a programmable 3D axis system MOCO PI MICOS arm that was programmed to move the Reson TC4014 receiver hydrophone in the tank. The hydrophone has a receiving sensitivity of −186 ± 3 dB @1 V/μPa and a frequency response from 15 kHz to 480 kHz. The emitter hydrophone is a Reson TC4038 with a transmitting response of 110 dB @1 μPa/V @ 1m and a frequency response from 50 kHz to 800 kHz. Figure 5 shows the experimental setup with the transmitter and receiver inside the tank. A National Instruments data acquisition system was used with PXI type cards to generate the signal used as input of the linear E&I A150 amplifier that feeds the transmitter. Both the receiving and the feeding signal were captured. The latter was captured with an ×100 probe to avoid overloads in the system. All signals were stored at 10 Ms/s with a duration of 500 μs.

**Figure 5.** (**a**) The emitter and receiver are located as close as possible to each other to calibrate the motors; (**b**) The first position for measurements of sound speed and location; (**c**) System of generation and capture of signals.

Two experiments have been performed. First, a calibration is done to reduce the uncertainty due to the time of arrival from which the speed of the sound has been measured. For this, 12 different reception positions were assessed in a straight line on the emitter axis. By having the 12 measurements along the line, a time-distance linear adjustment is made whose slope value corresponds to the speed of sound. In addition, the calibration allows a time correction that results in a decrease of the arrival time of the signal, according to the linear adjustment obtained in the fit.

In the second part of the experiment, 12 reception points have been set that will correspond to the 12 sensor positions of the array. The mechanical axis moves the hydrophone to each of the points and then the signal is emitted and recorded 10 times. Figure 5b shows one of the measuring points where the Reson TC4014 sensor is fixed to the mechanical arm.
