Creating uniform cluster dose spread-out Bragg peaks for proton and carbon beams

Abstract

Background:

Preliminary data have shown a close association of the generalized ionization cluster size dose (in short, cluster dose) with cell survival, independent of particle type and energy, when cluster dose is derived from an ionization detail parameter preferred for its association with cell survival. Such results suggest cluster dose has the potential to replace RBE-weighted dose in proton and ion beam radiotherapy treatment plan optimization, should a uniform cluster dose lead to comparable biological effects. However, further preclinical investigations are warranted to confirm this premise.

Purpose:

To present an analytical approach to create uniform cluster dose spread-out Bragg peaks (SOBP) for evaluation of the potential of cluster dose to result in uniform biological effect.

Methods:

We modified the coefficients of the Bortfeld and Schlegel weight formula, an analytical method typically used for the creation of radiation dose SOBP in particle therapy, to produce uniform cluster dose SOBP of different widths (1 to 5 cm) at relevant clinical proton and carbon beam energies. Optimum parameters were found by minimization of the ratio between the maximum and minimum cluster dose in the SOBP region using the Nelder-Mead method.

Results:

The coefficients of the Bortfeld and Schlegel weight formula leading to uniform cluster dose SOBPs were determined for each combination of beam energy and SOBP width studied. The uniformity of the resulting cluster dose SOBPs, calculated as the relative difference between the maximum and minimum cluster dose within the SOBP, was within 0.3% to 3.5% for the evaluated proton beams and 1.3% to 3.4% for the evaluated carbon beams.

Conclusions:

The modifications to the analytical approach to create radiation dose SOBPs resulted in uniform cluster dose proton and carbon SOBPs over a wide range of beam energies and SOBP widths. Such SOBPs should prove valuable in preclinical investigations for the selection of nanodosimetric quantities to be used in proton and ion therapy treatment planning.

1. Introduction

Nanodosimetry is the study of the spatial distribution of individual energy deposition events (mainly ionizations) in biologically relevant volumes¹. This typically uses probability or frequency distributions of ionization cluster sizes and derived quantities such as statistical moments of these distributions^2–6. Contrary to microdosimetric quantities such as LET, which focus on variations of energy deposits at micrometer scale⁷, nanodosimetry accounts for nanoscopic ionization patterns. A strong association has been found in previous works^8–11 between these quantities and different biological effects of relevance to cancer therapy. This suggests a high potential for using nanodosimetric quantities to improve biologically optimized charge particle treatment planning.

Several attempts have been made for incorporating nanodosimetric considerations in treatment planning strategies, for instance including nanodosimetric quantities in RBE models¹² or simultaneously optimizing radiation dose and nanodosimetric quantities^13,14. In a recent work, Faddegon et al.¹¹ presented an ionization detail (ID)-based mathematical formalism for the definition and selection of ionization detail parameters ($I_p$) and their use in charge particle radiotherapy treatment planning. This formalism is based on (i) a generalized definition of ionization cluster size, $v$, ionization cluster size frequency distribution, $f(v)$ and $I_p$, (ii) the transition from nanodosimetric quantities calculated at nanoscale to macroscale by defining a voxel averaged $I_p$, and (iii) the concept of generalized ionization cluster size dose (or cluster dose). In short, a $I_p$, is a collapsed representation of ionization cluster size distributions at nanoscale for a given particle type and energy¹². An example of $I_p$ is $F_k$, defined as the number of clusters of $k$ or more ionizations per unit track length. Contrary to other approaches that compute probability distributions, this formalism considers the absolute frequency of clusters per particle and average track length. In a macroscopic voxel, the voxel-averaged $I_p$, $I_p^{\phi }$, is defined as the track weighted average of the nanoscopic $I_p$ of the set particles, $\phi$, of different type and energy interacting within that volume. The cluster dose is defined as the fluence-weighted sum of the $I_p^{\phi }$ of the charged particles that result in ionization in a macroscopic volume (i.e. proportional to the product of fluence and $I_p^{\phi }$), analogous to physical dose (i.e., the product of the charged particle fluence and the mean mass stopping power). See a detailed description of these quantities in Section 2.1. Contrary to $I_p$, cluster dose is dependent on the local fluence, allowing the optimization of the source fluence of individual pencil beams to obtain a prescribed cluster dose in treatment planning, as conventionally done for optimizing plans based on RBE-weighted dose. Preliminary data¹¹ has shown that the cluster dose derived from $I_p$ quantities ${\text{F}}_5$ and ${\text{F}}_7$ strongly associates with cell survival in aerobic and hypoxic conditions, respectively, even more strongly than the linear energy transfer (LET). In other words, the same cluster dose is expected to lead to comparable biological effects independent of particle type and energy. The requirement of larger cluster sizes in hypoxic cells can be explained by the lower probability of an ionization to be converted to a DNA strand break, compared to aerobic cells. Such results provide evidence for using cluster dose in proton and ion beam radiotherapy treatment plan optimization. Forthcoming preclinical investigations are warranted to evaluate more fully these and other preferred $I_p$. For this purpose, methods to create uniform cluster dose in the spread-out Bragg peak (SOBP) will provide the means to experimentally validate $I_p$ and the derived cluster dose. A uniform radiation dose SOBP does not result in a uniform cluster dose. As shown with the uniform radiation dose SOBPs plotted in Figure 1, the cluster dose increases with depth due to the rise in $I_p$ (${\text{F}}_5$ in this case), i.e., due to the increased number of ionization clusters per unit track length as the charged particle slows down.

Figure 1.

F₅ (right-side scale), radiation dose and cluster dose (left-side scale) as a function of depth for 3 cm width proton (100 MeV) and carbon (188 MeV/u) SOBPs.

In this work, we propose a modification to the coefficients of the Bortfeld and Schlegel¹⁵ weight formula, an analytical method for the creation of radiation dose SOBP in particle therapy, to create proton and carbon uniform cluster dose SOBP.

2. Methods and material

2.1. Ionization detail parameter and cluster dose

In this work we consider nanodosimetric quantities defined in our previous work¹¹, i.e., ionization detail (ID) parameters and cluster dose.

An ID parameter ($I_p$) is a collapsed representation of the frequency distribution, $f(v)$, of a generalized ionization cluster size $v$ at nanoscale for a given particle type and energy. The $I_p$ considered in this study, from our previous work, is $F_k$:

$$F_k={\sum }_{v=k}f(v),$$ (1)

the number of clusters of $k$ or more ionizations per unit track length, where $v$ is the number of ionizations in randomly distributed nanometer-sized volumes of a specific size, shape and orientation. In a macroscopic (millimeter size) volume, where the radiation field may be composed of a set of different particle types and energies, the voxel-averaged $I_p$, $I_p^{\phi }$, is defined as the track weighted average of the nanoscopic $I_p$ of the set of particles, $\phi$, that produce ionization clusters in that volume. In the case of $F_k$:

$$F_k^{{\phi }_j}={\sum }_{v=k}\frac{{\sum }_{c\in {\phi }_j}{t}_j^c{f}^c(v)}{{\sum }_{c\in {\phi }_j}{t}_j^c},$$ (2)

where $j$ is the voxel index and $c$ the particle class (i.e., particle type and energy). The density of ionization scales with the density of the medium. To account for this, the track length is scaled with the density of the material in voxel $j$ as compared to the density of the medium in the calculation of $f(v)$ for each particle class. The cluster dose in a macroscopic volume is then defined as the density of ionization clusters in that volume, equal to the product of the fluence, $\varphi$, and a selected $I_p^{\phi }$ divided by the density, ${\rho }_0$, of the medium used to calculate the nanoscopic $I_p$:

$$g_j^{\left(I_p\right)}={\varphi }_j{I}_p^{{\phi }_j}\text{/}{\rho }_0,$$ (3)

The $I_p$ considered for the cluster dose calculation in this work was $F_5$, i.e., the number of clusters of five or more ionizations in biologically relevant nanoscopic volumes. The rationale behind the selection of this $I_p$ is that the bulk of our upcoming experiments are to be performed in aerobic conditions and, in our preliminary assessment¹¹, $F_5$ is the parameter that better associates with cell survival at this level of oxygenation.

2.2. The Bortfeld and Schlegel weight formula

The modified version of the weight formula of Bortfeld and Schlegel¹⁵ proposed by Jette and Chen¹⁶ was adopted for this study. This analytical method is used in particle therapy to compute the weights of a set of monoenergetic beams for creating uniform radiation dose SOBP. In this approach, the SOBP depth curve is described by the range, ${\text{R}}_0$, of the monoenergetic beam of maximum energy, ${\text{E}}_0$, and the SOBP width, $\text{χ}$, defined as a fraction of ${\text{R}}_0$. The SOBP is created by $n$ monoenergetic beams of range $r_k$ and energy $e_{\text{k}}$,

$$r_k=\left[1-\left(1-\frac{k}{n}\right)\chi \right]R_0,$$ (4)

$$e_k={\left(\frac{r_k}{ɑ}\right)}^{1/p_0},$$ (5)

where $\text{α}$ is the proportionality factor and $p_0$ is the exponent of the range-energy relation for each charged particle. The $\text{α}$ and $p_0$ values considered in this study are 2.2×10⁻³ cm/MeV and 1.77 for proton beams and 1.14×10⁻⁵ cm/MeV and 1.738 for carbon beams. The weight of each monoenergetic beam $k$ is given by,

$$w_k=1-{\left(1-\frac{1}{2n}\right)}^{1-\frac{1}{p}}\hspace{0.33em}\text{for}\hspace{0.33em}\text{the}\hspace{0.33em}\text{minimum}\hspace{0.33em}\text{energy}\hspace{0.33em}(k=0),$$

$$w_k={\left[1-\frac{1}{n}\left(k-\frac{1}{2}\right)\right]}^{1-\frac{1}{p}}-{\left[1-\frac{1}{n}\left(k+\frac{1}{2}\right)\right]}^{1-\frac{1}{p}}\hspace{0.33em}\text{for}\hspace{0.33em}\text{the}\hspace{0.33em}\text{maximum}\hspace{0.33em}\text{energy}\hspace{0.33em}(k=n),$$ (6)

$$w_k={\left(\frac{1}{2n}\right)}^{1-\frac{1}{p}}\hspace{0.33em}\text{for}\hspace{0.33em}\text{the}\hspace{0.33em}\text{other}\hspace{0.33em}\text{energies}\hspace{0.33em}(k=1,\dots ,n-1),$$

where the range-energy parameter $p$ that leads to a uniform SOBP is dependent on the SOBP width and beam energy.

In this work, the value of $p$ that leads to uniform cluster dose in the SOBP was determined using the Nelder-Mead optimization algorithm¹⁷. The optimum $p$-value was found by minimization of the ratio between the maximum and minimum cluster dose in the SOBP region.

2.3. Monte Carlo simulations

Radiation dose and cluster dose distributions were calculated using the Monte Carlo TOPAS toolkit version 3.9 (based on Geant4.10.7)^18,19. For the optimization of p-values, the target volume used for scoring was a 20-mm radius water cylinder divided into 1-mm bins in depth. It was irradiated along its z-axis by a uniform, normally incident, circular beam of 10 mm radius. The cluster dose was calculated in those macroscopic (mm-sized) volumes using eq. 3.

In addition, a constant depth cluster dose distribution was computed in a 3D mouse head model using the mouse phantom provided as a TOPAS example. Both radiation dose and cluster dose were computed. The resolution of the phantom was 0.5×0.5×0.5 mm³. Three different materials were used: “G4_BRAIN_ICRP” for the brain tissue, “G4_BONE_COMPACT_ICRU” for the skull, and “G4_TISSUE_SOFT_ICRU” for the rest of the head. These three materials are included in the Geant4 material database of NIST compounds²⁰. A 36.3 mm water block was used as a range shifter to place the center of the SOBP at the center of the mouse head. The scoring volume was a cylinder of 2.5 mm radius centered in the mouse brain and divided in 1 mm voxels in the beam direction. The beam source was circular carbon beam of 10 mm radius.

2.4. Beam configuration

The SOBPs studied in this work were defined by a beam energy (i.e., particle range) and SOBP widths, denoted as the percentage of the maximum range. Relevant clinical beam energies were used, i.e., 70, 100, 150 and 200 MeV for protons and 150, 200, 250 and 300 MeV/u for carbon. Published energy spread values for clinical proton beams reported were used²¹. Carbon beams were nearly monoenergetic with 0.1% energy spread. SOBP widths from 1 to 5 cm in steps of 5% of the maximum range were evaluated. The number of monoenergetic beams to create the SOBP was chosen to be equally spaced by 1 mm in range.

For the mouse head irradiation, a carbon SOBP characterized by a maximum energy of 150 MeV/u and a width of 30% of its maximum range (i.e., 1.2 cm wide SOBP) was considered. The size of the SOBP was chosen to conform to the width of the mouse brain in the beam direction. The same spot configuration that resulted from the uniform cluster dose SOBP optimization in water was used.

3. Results

The $p$-values leading to uniform cluster dose SOBPs for different proton and carbon beam energies and SOBP widths are presented in Table 1 and Table 2.

Table 1.

The $p$-values leading to uniform cluster dose SOBPs for a range of proton beam energies and SOBP widths.

Max. energy (MeV)	SOBP width (% of range)
	10	15	20	25	30	35	40	45	50
70	-	-	-	1.83	1.83	1.81	1.79	1.79	1.78
100	-	1.84	1.80	1.78	1.76	1.74	1.73	1.72	1.70
150	1.76	1.71	1.68	1.65	1.64	-	-	-	-
200	1.69	1.63	1.61	-	-	-	-	-	-

Table 2.

The $p$-values leading to uniform cluster dose SOBPs for a range of carbon beam energies and SOBP widths.

Max. energy (MeV/u)	SOBP width (% of range)
	10	15	20	25	30	35	40	45	50
150	-	-	-	2.50	2.49	2.47	2.47	2.46	2.46
200	-	2.46	2.46	2.44	2.41	2.40	2.39	2.37	2.33
250	-	2.41	2.37	2.31	2.30	2.29	-	-	-
300	2.37	2.33	2.29	2.25	-	-	-	-	-

The cluster dose uniformity within the SOBP, calculated as the relative difference between the maximum and minimum cluster dose within this region, ranges from 0.3% to 3.5% and 1.3% to 3.4% for the evaluated proton and carbon beams, respectively. Figure 2 and Figure 3 show the respective depth cluster dose curves.

Figure 2.

Radiation dose and cluster dose proton depth curves for the beam energies and SOBP widths listed in Table 1.

Figure 3.

Radiation dose and cluster dose carbon depth curves for the beam energies and SOBP widths listed in Table 2.

The linear interpolation of the MAT $p$-value for intermediate energies and SOBP widths leads to uniform cluster dose SOBP. For instance, for a proton beam SOBP characterized by ${\text{E}}_0=125$ MeV and $\text{χ}=23\%$, a $p$-value of 1.725 resulted in a uniformity within 1.5%. Similarly, for a carbon beam SOBP characterized by ${\text{E}}_0=175$ MeV/u and $\text{χ}=42\%$, a $p$-value of 2.42 resulted in a uniformity within 1.4%.

To show the impact of applying the optimized beam configuration in homogeneous media to an example of a preclinical setting involving heterogeneities, the depth cluster dose distribution in a mouse head irradiation was evaluated. As described in Section 2, the SOBP configuration obtained from the optimization of the $\text{p}$-value in water was used. This configuration was a carbon SOBP described by ${\text{E}}_0=150$ MeV/u, $\text{χ}=30\%$, and $p$-value of 2.49. This setting was chosen to deliver a uniform cluster dose SOBP of the size of the mouse brain. The SOBP uniformity in depth within the mouse brain was within 1.7 %. Figure 4 shows the computed radiation dose and cluster dose profile in depth.

Figure 4.

(a) Schematic representation of the mouse head irradiation in TOPAS, showing the beam (yellow arrow) normally incident on a water block (blue) placed upstream of the mouse (brain in grey, skull bone in white, remainder of the head in orange). (b) Radiation dose and cluster dose as a function of depth along the dotted red line beam axis in panel (a). The radiation dose was normalized to 1 Gy at the center of the brain. The cluster dose was normalized by the fluence required to deliver 1 Gy at the center of the brain.

4. Discussion

The approach presented in this work provides the means to create uniform depth cluster dose distributions over the SOBP in water. This methodology is expected to be used in forthcoming in vitro preclinical investigations in the context of further investigating the association of cluster dose with relevant biological endpoints (e.g., cell survival). In those studies, homogeneous water-equivalent media are used. In this work, we also evaluated the impact of applying the results of the optimization obtained in water in a mouse head irradiation, which involves medium heterogeneities. A carbon irradiation was chosen for this evaluation since dose distributions of carbon ion beams are more sensitive to heterogeneities than proton beams. The resulting cluster dose SOBP was uniform within 1.7 % across the mouse brain in depth.

Cluster dose as defined is proportional to the local fluence, as is radiation dose. Therefore, for a given beam quality (type and energy), cluster dose can be modified by modifying the incident fluence. In this work, we have shown that the optimization of the beamlet weight (i.e., local fluence along the beam direction) leads to uniform cluster dose distributions in homogeneous media. Therefore, cluster dose treatment plan optimization may be performed similarly to current dose (or RBE-weighted dose) optimization strategies. In addition, simultaneous optimization of these quantities along with voxel-averaged ID parameters may be considered, analogous to the optimization approaches including dose and microdosimetric quantities such as LET^22–24. The application of cluster dose to clinical treatment planning is discussed in more detail in our previous work¹¹.

In this study, the cluster dose derived from the $I_p$ parameter $F_5$ was considered. The optimization considering $F_5$ also leads to uniform (within 3.5 %) cluster dose SOBP for $F_7$ for proton beams (see Figure 5a). However, the uniformity of depth cluster dose distributions is highly dependent on the $I_p$ for carbon beams (see Figure 5b). For experiments with carbon beams performed in other conditions, e.g., hypoxic media where $F_7$ better associates with cell survival, the $\text{p}$-values leading to uniform SOBPs may be determined following the approach presented in this paper.

Figure 5.

Cluster dose depth curves for (a) proton and (b) carbon beams considering different ionization detail parameters (F₅ and F₇) for the computation of the cluster dose.

5. Conclusions

This work provides an analytical approach to calculate the set of monoenergetic beam weights required to deliver constant cluster dose to biological samples in preclinical studies. The approach serves as a surrogate for treatment planning system modules including the cluster dose concept for optimization objectives.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.