Monte Carlo calculations of target fragments from helium and carbon ion interactions with water

Abstract

When high energetic heavy ions interact with any target, short range, high linear energy transfer (LET) target fragments are produced. These target fragments (TFs) can give a significant dose to the healthy tissue during heavy ion cancer therapy, and when cosmic radiation interacts with astronauts. This paper presents Monte Carlo simulations, using the Particle and Heavy Ion Transport code System (PHITS), to characterize target fragments from reactions of helium and carbon ions with water. The calculated ranges, LET, doses, and production cross sections are presented. It is shown that protons, deuterons, tritons, alpha particles, ³He, ⁶He, nitrogen, oxygen, and fluorine ions are the most probable target fragments when carbon and helium ions collide with water. Among the produced target fragments, alpha particles and nitrogen ions give the highest dose to the targets, since the combination of fluence and LETs of these TFs are highest among the produced fragments. The production cross sections of proton and oxygen are the highest among the target fragments cross sections when helium and carbon ions imping on water, because these TFs can be produced through more reaction channels compared to other fragments. These findings are helpful for accurate dose measurement during heavy ion cancer therapy and for shielding of space radiation.

Introduction

Charged particle therapy (CPT) is a growing and favorable cancer treatment modality, especially for deep seated radioresistant tumors due to its higher therapeutic ratio for various cancer types and greater efficiency when compared to conventional photon therapy [1]. Proton and carbon ion therapies have been employed for cancer treatment since 50s and 70s, respectively [2]. Protons and heavier ions have less lateral and longitudinal scattering [3], and a better dose deposition characteristics [4] compared to photons. Currently there are over 108 proton therapy and 16 carbon ion therapy facilities in operation worldwide [5]. Since the energy lost by charged particles is inversely proportional to the square of their velocities, most energy is transferred to the target just before the particle comes to a complete stop. This characteristic depth dose distribution is known as the Bragg curve [6], and the peak is called the Bragg peak. During the deceleration of the projectile inside the target, some of the projectiles break up into projectile fragments (PFs) and some of the target nuclei into target fragments (TFs). During heavy ion interactions, nuclear reactions can occur when the nuclear densities (average kinetic energy per nucleon) of the target and projectile begin to overlap. At minimal overlaps, elastic or inelastic scattering and few-nucleon transfer through direct reactions may happen. At small impact parameters, new and unusual features emerge. If the impact parameter is small enough for complete overlap of the interacting nuclei, a compound nucleus—representing the complete fusion of the two nuclei—can be formed as an intermediate state. However, to overcome the repulsive Coulomb barrier, the incident ion must be highly energetic, resulting in the formation of a compound nucleus with considerable excitation energy. Due to the high incident energy, the compound nucleus can achieve a high energy density. Analyzing these compound states and their decay modes poses a significant challenge for nuclear theory. Once the excited compound state is formed, there are numerous of decay channels available. It can split roughly in half, either through the original entrance channel or a closely related one. A more probable decay mode for the compound nucleus is particle emission. Emission of charged particles is inhibited by the Coulomb barrier. In reactions involving heavy nuclei, the compound nucleus is extremely proton-rich, but neutron emission remains the preferred decay mode, even for heavy nuclei with a proton excess of 10–20 or more. Thus, it is possible to study nuclei far from stability on the proton-rich side through (HI, xn) reactions, where HI represents any heavy ion and x may range from 5 to 10 [7].

The relative biological effectiveness (RBE), which is defined as the ratio of a dose of photons to a dose of any other particle to produce the same biological effect, for the same endpoint [8], is an important property estimating the biological damage resulting from the interaction of high energetic ions with DNA or any other cellular components . The RBE depends on various physical parameters including linear energy transfer (LET), dose, particle characteristics, fractionation, and biological parameters such as tissue type, cell cycle phase, oxygenation level [9]. The RBE increases with LET until around 100–200 keV/µm, depending on cell line, end point, etc., after which it decrease again due to so called “over kill” [10]. PFs are lighter (lower z) than the projectile with lower LETs, RBEs, and might have long ranges [11] whereas some of the TFs are heavy and have higher LETs, RBEs, and shorter ranges [10], [11] in the order of 10–100 µm [12], [13], [14], [15], [16], [17], and their dose distribution aligns with the Bragg curve and can cause an unwanted dose enhancing to the healthy tissues. During the assessment of the dose received by the healthy tissues and for improving treatment planning systems, the dependence of RBE with the produced target fragments much be taken into consideration. This depends on the production cross sections, LETs, TFs ranges, and dose fractionation [18].

Since the PFs are lighter than the projectile and can have longer ranges than the projectiles, they make a tail beyond the Bragg peak. The heavier the projectile is, the more PFs will be produced [11], [17], [19]. To address this issue, researchers are considering helium ion as an alternative to carbon ion due to their smaller fragmentation tail [20] compared to carbon ions, but still higher RBE and less scattering than protons. At the Lawrence Berkeley National Laboratory (LBNL) 2054 patients were treated with helium ions during 1957–92 [21]. Recently Heidelberger Ionenstrahl-Therapiezentrum (HIT) [22] has also started to treat patients with helium beams [23].

There is very limited literature on TFs from the interactions of ion beams used in heavy ion cancer therapy. However, accurate dose measurement, proper handling of ion beams during heavy ion therapy, and effective shielding against space radiation, require a thorough understanding of the characteristics of these TFs. This paper presents LETs, production cross sections, and dose depositions of TFs resulting from the interactions of helium and carbon ions with water calculated by the Particle and Heavy Ion Transport code Systems (PHITS) [24]. The result of this study is important for heavy ion cancer therapy and radiation dosimetry and radiation protection in space.

Materials and methods

Models used in PHITS

For the simulations, PHITS (version 3.22) [25] was used. For the interactions of the helium ion beams, we used the Liege intranuclear cascade model (INCL4.6) [26] followed by the statistical decay model (SDM), and for the interactions of the carbon ion beams, we used the relativistic JAERI Quantum Molecular Dynamics (JQMD) [26], [27], [28], followed by SDM. The core components of the QMD model include the self-consistent nuclear mean field, calculated from effective nucleon-nucleon (N-N) interactions, and a stochastic N-N collision term that accounts for Pauli blocking. Due to these mechanisms, QMD can provide a unified description of the various reaction mechanisms in heavy ion (HI)-HI reactions over projectile energies ranging from several MeV to GeV/nucleon. In this energy regime, the entire process is divided into two parts: the dynamical process, where direct reactions, non-equilibrium reactions, and the dynamical formation of highly excited fragments occur; and the statistical process, where evaporation and fission decay occur over longer time scales than the dynamical process. Both processes are combined in the JQMD model. In the QMD part, each nucleon state is represented by a Gaussian wave function of width L, as follows:

$${\varphi }_i\left(r\right)=\frac{1}{{\left(2\pi L\right)}^{3/4}}{\mathit{exp}}^{\left[-\frac{{\left(\mathit{r}-{\mathit{R}}_{\mathit{i}}\right)}^2}{4L}+\frac{i}{\hslash }\mathit{r}·{\mathit{P}}_{\mathit{i}}\right]}$$ (1)

here ${\mathit{R}}_{\mathit{i}}$ and ${\mathit{P}}_{\mathit{i}}$ represent the center of position and momentum for the ith nucleon, respectively. The evolution of ${\mathit{R}}_{\mathit{i}}$ and ${\mathit{P}}_{\mathit{i}}$ over time is governed by Newtonian equations and a stochastic two-body collision term, based on the time-dependent variational principle, as follows:

$$\dot{{\mathit{R}}_{\mathit{i}}}=\frac{\partial H}{\partial {\mathit{P}}_{\mathit{i}}},\dot{{\mathit{P}}_{\mathit{i}}}=\frac{-\partial H}{\partial {\mathit{R}}_{\mathit{i}}}$$ (2)

In this model, nuclear cascade calculations consider nucleon-nucleon collisions when the impact parameter between two nucleons is smaller than $\sqrt{\frac{\sigma }{\pi }}$, where σ represents the energy-dependent cross section for both elastic and inelastic collisions. The cross sections are parameterized according to the initial and final total isospin states $i$ and $f$ of the two-nucleon system, following the parametrization by VerWest and Arndt, as:

$${\sigma }_{\mathit{if}}\left(s\right)=\frac{\pi {\left(\hslash c\right)}^2}{2p^2}\alpha {\left(\frac{p_r}{p_0}\right)}^{\beta }\frac{{m_0}^2{\mathrm{\Gamma }}^2{\left(\frac{q}{q_0}\right)}^3}{{\left(s^{\ast }-m_0^2\right)}^2+m_0^2{\mathrm{\Gamma }}^2}$$ (3)

where, ${p_0}^2=\frac{s_0}{4}-{m_N}^2,s_0={\left(m_N+m_0\right)}^2$, ${p_r}^2\left(s\right)=\frac{\left[s-{\left(m_N-\langle M\rangle \right)}^2\right]\left[s-{\left(m_N+\langle M\rangle \right)}^2\right]}{4s},$

$$\begin{array}{c}{q_r}^2\left(s^{\ast }\right)=\frac{\left[s^{\ast }-{\left(m_N-m_{\pi }\right)}^2\right]\left[s^{\ast }-{\left(m_N+m_{\pi }\right)}^2\right]}{4s^{\ast }},\\ s^{\ast }={\langle M\rangle }^2,q_0=q(m_0^2)\end{array}$$

$\langle M\rangle$ represents the mean mass of the resonance, and $q$ denotes the center-of-mass momentum in the $\pi N$ channel. Fragments and their corresponding excitation energies were identified using the Statistical Decay Model (SDM) at the end of the dynamic stage. Each fragment was identified using a minimum distance chain procedure, where two nucleons are considered part of the same fragment if the distance between their centroids is less than 4 fm. The total energy of the fragment is then calculated in its rest frame, and the excitation energy is estimated by subtracting the ground state energy. The emission probability $P_x$ of particles such as p, n, d, t, ³He and alpha particles via evaporation, is determined according to the Fermi gas model as follows:

$$P_x=\left(2J_x+1\right)m_x∊{\sigma }_x\left(∊\right)\rho \left(E\right)d∊$$ (4)

where $J_x,m_x,\mathrm{and}∊$ represent the spin, mass, and kinetic energy of particle $x$, respectively. Mean while ${\sigma }_x\left(∊\right)$ denotes the inverse cross section for the absorption of the particle with energy $∊,$ and $\rho \left(E\right)$ signifies the level density of the residual nucleus with excitation energy $E$. The level density is calculated as follows:

$$\rho \left(E\right)=w_0\exp \left(2\sqrt{\mathit{aE}}\right)$$ (5)

where $a=\frac{A}{8}{\mathit{MeV}}^{-1}$ and $w_0$ is constant. The inverse cross section is assumed to take the form

$${\sigma }_x\left(∊\right)=\left\{\begin{array}{c}\begin{array}{c}\left(1-\frac{U_x}{∊}\right)\end{array}\pi R^2,{∊>U}_x\\ 0{∊\le U}_x\end{array}\right)$$ (6)

here $R$ represents the absorption radius, and $U_x$ is the Coulomb barrier for particle $x$. The excitation energy $E$ is given by

$$E=E_0-∊-Q$$ (7)

where $E_0$ represents the excitation energy of the parent nucleus, and $Q$ is the reaction $Q$ value calculated using the semi-empirical mass formula. The total emission probability $R_x$ for particle $x$ is obtained by integrating over the available energy as follows:

$$R_x=\left(2J_x+1\right)m_x{\int }_{U_x}^{E_0-Q}∊{\sigma }_x\left(∊\right)\rho \left(E_0-Q_x-∊\right)d∊$$ (8)

Simulation set up

Circular helium and carbon ion beams with a diameter of 5 cm, at various energies ranging from 30 to 500 MeV/n, were transported through a water target with a thickness 0.05 cm. Figure 1(a) illustrates the simulation setup, with a 5 cm × 5 cm × 0.05 cm water target, sandwiched between two 5 cm × 5 cm × 5 cm void regions. Figure 1(b) shows the simulated 200 MeV/n circular carbon ion beam with a diameter of 5 cm passing through the water target. The beam originates from the beam port at 0 cm on the z-axis. Helium and carbon ion interactions with water were observed for 10 million histories at each aforementioned energy point.

Figure 1.

(a) Simulation set up for carbon interactions with water. The target thickness is 0.05 cm. The figure is not to scale for better visualization purpose. There are two 5 cm thick vacuum regions before and after the target. (b) Illustrations of 200 MeV/n carbon ion beam passing through the target.

After the ion beams passed through the water, the energies and production rates of TFs were calculated using the [T-Product] tally in PHITS. Dose and production cross sections of the TFs were calculated using the [T-Deposit] and [T-Yield] tallies.

Results and discussion

Energy and production rate of target fragments (TFs)

We simulated the TFs produced during the interactions of helium and carbon ions with water using PHITS. It is well established that PFs are mostly travelling in the forward direction with nearly the same energy as the projectile [29], and that the TFs are getting a fraction of the projectile energy. PHITS can calculate the kinetic energy and mass distributions of all fragments with the JQMD model, followed by SDM. PFs have larger kinetic energies than the TFs, so based on kinetic energies of the fragments, the TFs can be separated from the PFs as can be seen in Figure 2. We divided the TFs into two categories, lighter and heavier ones. TFs, having Z < 3 was defined as lighter TFs and TFs with Z ≥ 3 was heavier TFs. In the Figure 2, the production rate and energy spectra of different lighter TFs from the interaction of carbon beam with water at the energies 30 (a), 200 (b), and 400 (c) MeV/n, respectively are presented. Figures 2(d), (e), and (f) illustrate the same for the interaction of helium ion beam with water at the mentioned energies, respectively. Figures 2(g), (h) and 2(i) show the energy distribution of helium and its isotopes (²He, ³He, and ⁶He) for carbon interactions with water. Figures 2(j), (k), and (l) show the same for helium interactions with water. The energy distributions of fragments have two peaks. The lower energy peak represents the TFs since they have lower kinetic energies than the PFs, while the higher energy peak, closer to the kinetic energy of the projectile, corresponds to the PFs [30]. From the figures 2(a)–(f), it can be seen that the energy of proton and deuteron is not sharp as other fragments. The energies of these two types of fragments are distributed over a larger energy region; from lower to higher energies since these fragments are produced from both the projectiles and the targets. When analyzing the energy distribution of TFs in Figure 2, it is evident that protons, deuterons, tritons, and alpha particles are the dominant ones among lighter TFs, with protons having the highest production rate.

Figure 2.

The production rate of lighter fragments at 30, 200, and 400 MeV/n are shown in figures (a), (b), and (c), respectively, for the interactions of carbon with water. Figures (d), (e), and (f) show the production rates of the fragments from the interactions of helium with water at the same energies. From these figures, it is evident that most of the lighter TFs are protons and tritons in both of the interactions. At higher projectile energies the distributions of TFs and PFs can easily be separated as shown in Figures 2(b), (c), (e) and (f). Figures 2(g), (h) and (i) show the energy distribution of alpha and its isotopes (²He, ³He and ⁶He) for the carbon interaction with water. Figures 2(j), (k) and (l) show the same for the helium interaction with water.

Figure 3 shows the spectra of heavier TFs, which include nitrogen, oxygen fluorine, neon, sodium, and magnesium for the energies 30, 200, and 400 MeV/n in Figures 3(a), (b), and (c), respectively, for the interactions of carbon with water. Figures 3(d), (e) and (f) show the TFs for helium interactions with water. Similar to the previous case, we do not observe any peaks for PFs since these fragments (nitrogen, oxygen, fluorine, neon, sodium and magnesium) cannot be produced from the carbon and helium projectiles. These heavier TFs originate from the oxygen atoms of the water target. Among the heavier TFs, nitrogen, oxygen, fluorine, and neon are produced with a significantly higher production rate at the lower energies for both interactions, as shown in Figures 3(a) and (d). The reason for the production of TFs heavier than the oxygen is that there are numerous decay modes of the compound nucleus formed by fusion of carbon and oxygen. This compound nucleus can emit particles within the limit of total charge (Z number) within the combined limit of carbon (Z = 6) and oxygen (Z = 8) [7], as long as it has excitation energy enough to do so. As shown in Figure 3, nitrogen has a highest production rate compared to the other produced TFs.

Figure 3.

The production rates of heavier TFs from oxygen atoms of water at 30, 200, and 400 MeV/n are shown in the figures (a), (b), and (c), respectively, for the interaction of carbon with water. The figures (d), (e), and (f) show the same for the interaction of helium with water at the same energies.

Ranges and LETs of TFs

TFs can scatter during their travel inside the target material [31]. Fragments with higher masses experience less scattering. From the energy distribution of the TFs, we can observe that the TFs have energies in the range of several tens MeV. In Figure 4(a), the ranges in water are shown for significant TFs, which include protons, alpha particles, nitrogen, oxygen, and fluorine fragments. As can be seen from Figure 4, at several tens MeV the TFs can travel several tens of μm inside water and healthy tissues. In Figure 4(b), the LETs (keV/μm) are shown for the same fragments as shown in Figure 4(a). The ranges and the LETs were calculated using SRIM [32]. These ranges and LETs help us assessing the area around which the healthy tissue may receive unwanted dose from TFs. Among these TFs, proton, deuteron, and tritons have ranges thousand times longer than the other TFs at the total energy of 100 MeV. From the range calculations, we can see that a few healthy cells beyond the Bragg peak may receive unwanted dose from these TFs. At the same time, from the graph of LET, we can see the LETs of these TFs. Among these TFs, fluorine has the highest LET (for example, 1470 keV/μm at 8 MeV), nitrogen has a lower LET (for example, 1066 at 6.5 MeV) than fluorine and oxygen, whereas oxygen has the LET in between them (for example, 1261 at 7 MeV) and proton has the lowest LET.

Figure 4.

(a) Ranges of protons, alphas, nitrogen, oxygen and fluorine in water at energies below 100 MeV, calculated using SRIM. (b) The LETs of these TFs show how high the LETs of the heavier TFs are, which is significant for dose calculation beyond the Bragg peak.

The dose will be highest for those TFs that have the highest production rate and relatively high LET. Figures 5(a), (b), and (c) show the dose of heavier TFs from oxygen atoms for the carbon interaction with water. Figures 5(d), (e), and (f) show the same for helium interaction with water. In these figures, the calculated doses specifically for the target region (water) are shown.

Figure 5.

Figures 5(a), (b) and (c) show the dose of heavier TFs originating from oxygen atoms from the interactions of carbon with water. Figures 5(d), (e), and (f) show the same for the interactions of helium with water. As can be seen in these figures, nitrogen and oxygen contribute to the highest dose to the target.

From Figure 5, we can see that nitrogen and oxygen fragments give the maximum dose to the target for both carbon and helium interactions with water. Figure 6 illustrates the dose for lighter TFs for both interactions. Figures 6(a), (b) and (c) show the dose of lighter TFs at 30, 200 and 400 MeV/n energies, respectively for the C interactions with water. Figures 6(d), (e), and (f) show the same for helium interactions with water at the mentioned energies, respectively. It is observed that proton, deuteron, and alpha provide the highest dose to the target as depicted in Figure 6.

Figure 6.

The figure shows the dose distributions of lighter TFs at 30 (a), 200 (b), and 400 (c) MeV/n for the interactions of carbon with water. Figures 6(d), (e), and (f) show the same for helium interactions with water at the mentioned energies, respectively. From this figure, it can be seen that the proton, deuteron, and alpha have higher doses among others.

The normalized doses from significant TFs, such as alpha, proton, deuteron, and nitrogen are presented in Figure 7. This figure illustrates that nitrogen and alpha particles provide the highest doses to the targets among the other TFs. In the lower energy range, alpha, nitrogen, and oxygen have nearly identical doses, but at higher energies, nitrogen and alpha are the main dose giving TFs.

Figure 7.

This figure shows the normalized dose of significant TFs from water, which are alpha, proton, deuteron, nitrogen and oxygen. Nitrogen gives the highest dose among the other heavier TFs and alpha possess the highest dose among lighter TFs.

Production cross sections of TFs

The production cross sections of TFs resulting from the inelastic interactions caused by the carbon and helium ions interacting with water was also analyzed. It was found that among all TFs, alpha particles have the highest production cross-section for the carbon interactions with water. In the case of He beams interacting with water, proton and alpha particles have the largest cross sections, which are nearly identical, as shown in Figure 8(a) and Figure 8(b) for C+water and He+water interactions, respectively. Nitrogen and oxygen, as shown in Figure 8, have production cross sections that are about ten times lower than those of proton and alpha. The production cross sections for protons and alpha particles are high since both carbon and helium projectiles can break up into alpha particles, protons and neutrons. Oxygen can be produced from the pickup reactions by carbon ions. No other fragments have these multiple channels to be produced, which made the highest cross section for proton and oxygen in the reaction of He+water and C+Water respectively.

Figure 8.

This figure shows the total production cross sections of proton, alpha, nitrogen and oxygen fragments for the reactions of C+water (a) and He+water (b). It can be seen that alpha particles and protons have higher production cross sections than nitrogen and oxygen.

Conclusion

Understanding the characteristics of TFs is of great importance for being able to accurately calculating the dose to patients during cancer therapy, for estimating the dose to astronauts exposed to the space radiation, for designing shielding against heavy ion exposure, and for comprehending the interaction mechanism of heavy ions with matter. There is a limited existing literature data on TFs characteristics. In this study, we characterized the TFs for carbon and helium ion interactions with water based on Monte Carlo simulations using PHITS [23]. The dose distributions, along with production rate, ranges, and production cross sections have been analyzed for the most probable TFs. The TFs from water targets have been classified into two categories, lighter (Z<3) and heavier (Z≥3) TFs. The observed lighter TFs include proton, deuteron, triton, alpha, ³He, and ⁶He while the heavier TFs are nitrogen, oxygen, neon, sodium, and magnesium fragments. The lighter TFs are produced alongside some PFs of the same particle types (proton, deuteron, triton, alpha). However, the heavier TFs originate exclusively from the oxygen atoms of water. Our calculations reveal that the production rates are significant for protons, deuterons, tritons, alpha, and its isotopes, nitrogen, and oxygen. The ranges of these TFs are several μm in water, except for protons and alphas, which have much longer ranges. Protons, deuteron, triton and alpha along with its isotopes, have ranges that are around 1000 times higher than that of nitrogen and oxygen. Among all the TFs, nitrogen and oxygen deliver higher doses than the other fragments. On the other hand, helium (together with all its isotopes) and protons, deuterons, tritons have larger production cross sections than nitrogen and oxygen. This study confirmed that heavy ion interactions with composite targets, containing oxygen atoms, will produce significant amounts of nitrogen and oxygen TFs. Therefore, for accurate dose calculations for heavy ion therapy and space radiation protection, it is essential to consider the dose from the protons, alpha, nitrogen, and oxygen TFs. This study also highlights the importance of measuring the characteristics of TFs for various types of ion beam interactions, including those involving in heavy ion therapy.

CRediT authorship contribution statement

Quazi Muhammad Rashed Nizam: Writing – review & editing, Writing – original draft, Supervision, Software, Methodology, Investigation, Conceptualization. Asif Ahmed: Writing – review & editing, Visualization, Validation, Software, Formal analysis, Data curation. Iftekhar Ahmed: Writing – review & editing, Visualization, Validation, Software, Formal analysis, Data curation. Lembit Sihver: Writing – review & editing, Validation, Supervision, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.