Technical note: Generation of a Cerenkov scatter function convolution kernel for a primary proton beam

Abstract Purpose To generate a Cerenkov scatter function (CSF) for a primary proton beam and to study the dependence of the CSF on the irradiated medium. Materials and Methods The MCNP 6.2 code was used to generate the CSF. The CSF was calculated for light‐pigmented, medium‐pigmented, and dark‐pigmented stratified skin, as well as for a homogeneous optical phantom, which mimics the optical properties of human tissue. CSFs were generated by binning all of the Cerenkov photons which escape the back end (end opposite beam incidence) of a 20 × 20 × 20 cm phantom. A 4 × 4 cm, 500 × 500 bin grid was used to create a histogram of the Cerenkov photon flux on the simulated medium’s back surface (surface opposite beam incidence). A triple Gaussian was then used to fit the data. Results From the triple Gaussian fit, the coefficients of the CSF for the four phantom materials was generated. The individual CSF fit coefficient errors, with respect to the Gaussian representation, were found to be between 0.92% and 4.11%. The R2 value for the fit was calculated to be 0.99. The phantom material was found to have a significant effect (63% difference between materials) on the CSF amplitude and full width at half maximum (195% difference between materials). The difference in these parameters for the three skin pigments was found to be small. Conclusions The CSF was obtained for a proton beam using the MCNP 6.2 code for a phantom constructed of light, medium, and dark stratified human skin, as well as for an optical phantom. The CSFs were then fit with a triple‐Gaussian function. The coefficients can be used to generate a radially symmetric CSF, which can then be used to deconvolve a measured Cerenkov image to obtain the dose distribution.

has been demonstrated that there are two secondary mechanisms by which such emissions may occur indirectly: (1) a fast linear component from fast electrons liberated by prompt gamma(99.13%) and neutron (0.87%) emission; and (2) a slow non-linear component arising from the decay of radioactive positron emitters. 4 The present work seeks to develop an in vivo dose measurement method using Cerenkov radiation produced by secondary electrons and positrons within the patient volume. This could be helpful for evaluating the accuracy of the prescribed radiation treatment, monitoring hot spots of dose deposition, and improving the therapeutic outcomes of PT due to the ability to accurately monitor dose in real time. However, there is no method that can perform this type of dosimetry in real time with sufficient accuracy.
Proton therapy beams produce electrons and positrons that are responsible for dose deposition in a medium. Above a certain energy threshold, these charged particles will be superluminal within the medium and emit Cerenkov photons along their path. The Frank-Tamm formula gives the amount and spectrum of Cerenkov photons emitted. 5 The light-to-dose relationship has been approached as a deconvolution problem. 6 Most Cerenkov photons should be emitted in locations proximal to radiation dose deposition, since the proton beam imparts energy due to direct ionization, and therefore deconvolution by a light-transfer function can be used to isolate the corresponding regions of dose.
This approach has been used for photon and electron beams 7,8 , but not for PT beams. In these studies, a Cerenkov scatter function (CSF) was introduced, which represents the collection of all scattered Cerenkov photons that are emitted from the surface of a medium irradiated by a pencil beam of high-energy radiation. The CSF relates Cerenkov photon images taken during treatment to the surface fluence of primary radiation particles through a deconvolution relationship. The beam fluence can then be used to determine the superficial dose deposition using the relationship between beam fluence and the dose scatter function (DSF). 9 To date, no CSF has been produced for a primary proton beam.
In this work, a CSF is generated for a proton therapy beam for four different tissue equivalent phantom material compositions. The MCNP 6.2 code is used to simulate the interactions in the phantom.

2.A | Cerenkov radiation
Cerenkov radiation is electromagnetic radiation which is emitted by charged particles passing through a dielectric material at a speed in excess of the phase velocity of light in that material. When a charged particle moves inside a dielectric material, it excites molecules to higher energy levels and excited states. Upon returning back to their ground state, the molecules re-emit some photons in the form of electromagnetic radiation. According to Huygens principle, the emitted waves move out spherically at the phase velocity of the medium. If the particle motion is slow, the radiated waves bunch up slightly in the direction of motion, but they do not cross. However, if the particle moves faster than the light speed, the emitted waves add up constructively leading to a coherent radiation at an angle with respect to the particle direction, known as Cherenkov radiation.
The result of this effect is a cone of photon emission in the direction of particle motion. Tissue ϕ Camera (x,y,z)

2.B | Cerenkov scatter function
Coordinate system of the Cerenkov radiation creation process and subsequent detection. The symbols represent interaction points.
The CSF can be written as a convolution of the dose scatter function (DSF) kernel and the Cerenkov dose scatter function (CDSF) kernel as follows: The CDSF kernel represents the production of Cerenkov photons by the secondary electrons and positrons from the incident proton beam and the transport of these Cerenkov photons toward the surface of the tissue volume. Equation 2 indicates that the CDSF can be found by deconvolving the CSF with the DSF. 9 The dose deposited in a plane at depth d 0 can be written as: where E is the incident particle energy.

Inserting Equation 2 into Equation 1, and using Equation 3 for
the planar dose distribution D, I can be expressed as a convolution of CDSF and D as follows: The objective now is to determine the 2D dose distribution at depth d 0 in the tissue volume; the problem then becomes one of deconvolution of the Cerenkov photon intensity I, which is captured by the camera, with the CDSF, which can be found by deconvolving with the CSF produced herein and the DSF which will be calculated in a later work. The result is the dose profile at depth d 0 .

2.C | Mathematical assumptions
The convolution statements in Equations 1-4 require a few mathematical assumptions. If it is assumed for modeling simplicity that an ideal imaging system is being considered (100% absorption of escaped Cerenkov photons), the PSF could be represented by a Dirac-delta function, which would allow the PSF term in Equation 1 to be ignored. Also, in order to justify the convolution formulations of Equations 1-4, all of the convolution kernels must be shift invariant. This requirement can be satisfied by requiring the incident proton beam to be normal to the surface of the homogeneous medium.
Therefore, a 0 o beam angle is assumed in the present work.

3.A | MCNP Monte Carlo Simulation
The MCNP 6.2 10 code was used to generate the CSF. The Cerenkov methodology for MCNP is described in detail in Reference 11. The feature is assiduously tested in that work. The CSF was calculated for light-pigmented, medium-pigmented, and dark-pigmented stratified skin, as well as for a homogeneous optical phantom, which mimics the optical properties of human tissue.
The MCNP model consisted of a pencil beam proton radiation source and the phantom being irradiated, with air surrounding. The radiation source was directed at the phantom with a constant beam angle of θ in = 0°, where θ in is the angle of entrance vector defined from the normal surface. As aforementioned, this is to ensure shift invariance. The phantom was modeled as a 20 x 20 x 20 cm 3 box.
The CSF simulations used the spread-out Bragg peak (SOBP) shown in Table 1 below; these data were taken from the University of Florida Proton Therapy Institute treatment planning computer.

3.B | Material properties
Four different phantom materials were used in this study: light-pigmented, medium-pigmented, and darkened-pigmented stratified skin and an optical phantom material. The stratified skin models were

3.C | Generation of Cerenkov scatter function
CSFs were generated by binning all of the Cerenkov photons which escape the back end of the phantom (end opposite beam incidence).
A 4 x 4 cm 2 , 500 x 500 bin grid was used to create a histogram of the Cerenkov photon flux on the simulated medium surface. The histogram was sampled along a line from r = −10 to 10 cm using spoke-sampling which was centered at the origin of the pencil beam on the medium surface. The sampling angle, θ s , was defined on the medium surface as the rotation from the Y-axis about the Z-axis.
Sampling angles of 0-180 degrees in one-degree increments were used. A triple-Gaussian distribution was used to fit the mean of the data: CSF mean ðα, β, γ, δ, ɛ, ζÞ ¼ αe CSF mean is the mean of the CSF cross-sections over the sampling angle θs. The parameter r is the radial distance from the entrance of the proton pencil beam on the optical phantom surface. The parameters α, β, γ, δ, ϵ, and ζ are fit coefficients, generated using the Mathworks MATLAB fitnlm function; they were scaled by the number of primary protons incident on the optical phantom. The fitnlm function also produces the error in the fit coefficients. The triple-Gaussian fit was chose due to it having the best goodness of fit (R 2 ) value of the fits tested, as well as having the lowest error in the fit coefficients.
Single-Gaussian and double-Gaussian fits were also tested but yielded lower R2 values and larger errors in the fit coefficients. The error in CSF mean was found as the standard deviation at each radial point from the CSF cross sections. The large number of histories and the incorporated variance reduction ensured that the error at each point on CSF mean was <1% of the maximum value for each CSF. The Gaussian fit of the CSF data is shown in Figure 2 below. The full width at half maximum (FWHM) and the amplitude was determined for each CSF.

| RESULTS
The coefficients of the CSF for the four phantom materials are shown in Table 2 below. The individual coefficient errors were found to be between 0.92% and 4.11%, indicating that the Gaussian fitting using Equation 6 was appropriate for the data. The R 2 value was calculated to be 0.99.

| DISCUSSION
In the present work, the CSF was calculated for a proton beam incident on phantoms of light, medium, and dark stratified skin, as well as an optical phantom, which mimics the optical properties of human tissue. The SOBP from Table 1 was used for the energy distribution of said beam.
The material was found to have a significant effect on the CSF amplitude and FWHM. The optical phantom CSF amplitude was found to be 63% greater than the light skin phantom; the FWHM for the optical phantom was found to be 195% greater than the light skin phantom material. The skin pigment was found to have a small impact on the CSF amplitude and almost no effect on the FWHM.

| CONCLUSIONS
In the present work, the CSF was obtained for a proton beam using the MCNP 6.2 code for a phantom constructed of light, medium, and dark stratified human skin, as well as for an optical phantom.
The CSFs were then fit with a triple-Gaussian function using the MATLAB fitnlm function. The coefficients for this fit are provided in Table 2 and can be used to generate a radially symmetric CSF, which can then be used to deconvolve a measured Cerenkov image to obtain the dose distribution if the DSF is known beforehand.
The phantom material was found to have a significant effect on the CSF amplitude and FWHM. The difference in these parameters for the three skin pigments was found to be small, meaning that melanin content has a diminutive effect on the CSF. The % errors for the parameters were found to vary from 0.92% to 4.11%, with an R 2 value of 0.99.