Electromagnetic Signal of a Proton Beam in Biological Tissues for a Potential Range-Verification Approach in Proton Therapy

In the present work we investigate the fundamental properties of a proton beam’s electromagnetic signal as a candidate for range verification in proton therapy. We compute the electric and magnetic fields from a pulsed proton beam in biological tissues in a cylindrical geometry via a dedicated analytical solution of the Maxwell equations. The tissues are specified through their dielectric response and stopping power, where we consider a straight trajectory with a range that corresponds to protons with an initial energy of 150 MeV. We analyze the characteristics of the pulse and the impact of the pulse shape. The effect of the tissue’s conductivity on the deposited charges is specifically taken into account. In addition, we calculate and analyze the radiative pulse through the Liénard-Wiechert approach. We find that the conductivity charge relaxation has a big influence on the electric field strength, even in the presence of boundaries. It strongly depends on its environment, i.e., the permittivity, conductivity, and boundaries of the surrounding tissues. The electric field that originates from the primary protons vanishes in a time scale of ns. Conversely, the magnetic field is independent of its environment and constant in strength during a rectangular pulse ($10\mu \mathrm{s}$). For $\mu \mathrm{A}$ peak beam currents it lies in the pT range, 10 cm away from the beam axis, and is therefore detectable through, e.g., optical magnetometry. Its spatial profile, however, does not exhibit a distinct feature with respect to the range, but decreases rather smoothly and characteristically along the beam axis. The typical velocity dependence of the magnetic field strength is suppressed, due to the increasing charge density downstream. Finally, the primary protons give rise to an electromagnetic pulse in the radio spectrum (it peaks at approximately 9 GHz). It originates spherically around the range and consists partially of bremsstrahlung and Cherenkov radiation. Yet, a broader comparison to other radiation sources, e.g., secondary electron bremsstrahlung, is required. Based on our findings, we conclude that the magnetic field signal has favorable properties that motivate its detection experimentally for range-verification purposes.

Figure 1

Point-particle proton dynamics within water for an initial kinetic energy of $Q_{\mathrm{in}}=150$ MeV. The dashed lines are the extrapolated sections. Note that the close-up plot of the deceleration required a different scale.

Figure 2

Cylindrical geometrical setup, with radius $R$, and coordinate system. $E_{\rho }$ and $E_z$ are the electric field components along the $\rho$ and $z$ axis. $B_{\phi }$ is the azimuthal component of the magnetic field. The gray shaded volume has the tissue- and frequency-dependent permittivity ${\epsilon }_{\mathrm{tissue}}(\omega )$, surrounded by air (${\epsilon }_{\mathrm{air}}\approx {\epsilon }_0$).

Figure 3

Shape evolution of an initially $\beta$-distributed pulse, which is given by $B(z,\alpha ,\beta )=z^{\alpha -1}(1-z{)}^{\beta -1}/[\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )/\mathrm{\Gamma }(\alpha +\beta )]$, where $\alpha =\beta =6$ has a strong resemblance to a normal distribution, and simultaneously a favorably finite support. The initial pulse has been scaled to $\mathrm{\Delta }\zeta =10\mathrm{mm}$. For the initial pulse, it is scaled to $\mathrm{\Delta }\zeta =10\mathrm{mm}$. One can observe how $\zeta (z;t)$ traverses across space and time and eventually culminates in a $\delta$ pulse at the range of 150-MeV protons (approximately 15.8 cm). Scattering and energy straggling are neglected.

Figure 4

The two surfaces, cylinder (black) and hemisphere (red), through which we calculate the energy flow.

Figure 5

Magnetic field of a rectangular beam ($T=10$ ns) with a peak beam current of $1.6\mu \mathrm{A}$ along the cylinder axis at $\rho =10$ cm in various tissues. The field extends smoothly beyond the range at $z_s=15.84$ cm.

Figure 6

Vertical profiles of Fig. 5 across time at $z_s=15.84$ cm. The profile for water goes beyond the scale and is therefore plotted separately in the insert.

Figure 7

Logarithmic spatial profile of the magnetic field for a constant beam with a current of $2\mu \mathrm{A}$ (tissue independent).

Figure 8

Logarithmic temporal charge relaxation in various tissues. [Eq. (63) with $\gamma (t)=\delta (t)$]. $g(t)$ can be interpreted as the percentage of charge in the center relative to its initial value.

Figure 9

Semilogarithmic plot of the charge distribution $f(z,t)$ on the cylinder surface with a radius of $R=10$ cm in various tissues. The color scale for the $\delta$ pulses (left) differs from the color scale of the rectangular pulses (right).

Figure 10

Semilogarithmic plot of the $\rho$ component of the electric field outside of the cylinder ($R=10$ cm). The color scale for the $\delta$ pulses (left) differs from the color scale of the rectangular pulses (right).

Figure 11

Vertical profiles of Fig. 10 at $z_s=0$ across time of both the $\delta$ and rectangular pulse. The insert has on the vertical axis the same units but a different scale. The different colors correspond to the different tissues (just as in Fig. 8), while the different line styles are different electric field contributions. The dotted lines correspond to the electric field that originates from the center ($◯\mathrm{a}$), while the dashed lines are the surface contributions ($◯\mathrm{b}$). The solid lines are the total electric field on the outside [Eq. (74)].

Figure 12

$z$ (left) and $\rho$ components (right) of the electric field at $\rho =10$ cm along $z$ and $t$ without boundary conditions for a rectangular pulse with a duration of $T=10$ ns.

Figure 13

Vertical profile of $E_{\rho }(\rho =10\mathrm{cm},z_s,t)$ (right side of Fig. 12) across time at $z_s=15.84$ cm. We added the two additional tissues from Fig. 8. Gray matter has been scaled down by a factor of 10.

Figure 14

$E_{\rho }$ and $E_z$ in water (without conductivity and boundaries) at $t=2.5$ ns. The white dot on the central axis ($\rho =0$) belongs to the protons’ final position, i.e., range, at $z=15.84$ cm. The dashed line outlines the radial pulse of bremsstrahlung, the origin of which is the range. The solid line follows the Cherenkov radiation wave front. The fluctuations along the path of the proton (close to $\rho =0$ and for $z\approx [0.1,0.15]$) are numerical noise.

Figure 15

Temporal profile of $E_{\rho }$ and $E_z$ in water (without conductivity and boundaries) at ${\mathbf{r}}_0=(\rho ,z)=(5\mathrm{cm},z_s)$ with a comparison to the inclusion of the path for $t<0$ and the appearance radiation ${\tilde{\mathbf{E}}}_s$ (see Appendix 10). The dashed lines of the same colors are the respective asymptotes. The superscripts $◯1$ and ± are defined in Sec. 2 4ca.

Figure 16

Frequency spectrum of Fig. 15 at ${\mathbf{r}}_0=(\rho ,z)=(5\mathrm{cm},z_s)$. The gray lines in the background are frequency spectra of Cherenkov radiation at different velocities $\beta =v/c_0$. See text for details.

Figure 17

Radiation energy distribution $p_{\mathbf{s}}(\mathbf{r})$ along $z$, split into cylinder (left) and hemisphere (right) at $R_1=10$ cm. It is defined in Sec. 3 4c.

Figure 18

Connection between the pulse distribution $\gamma (t;T)$ and the launch time ${\tau }_{\gamma }(n/N)$ in the case of an exemplary raised cosine distribution: $\gamma (t;T)=2/T{\sin }^2(\pi t/T)$, for $t\in [0,T]$.