One electron-capture in collisions of fast nuclei with biomolecules of relevance to ion therapy

Abstract

This work is on total cross sections for single electron capture by protons from atomic and molecular targets. Most of the included atomic targets are also the constituents of the examined molecular targets. The latter targets are selected to be biomolecules. Computations are carried out using the continuum distorted wave (CDW) method at intermediate and high impact energies. First computed are atomic cross sections. These are subsequently employed to obtain molecular cross sections in the independent atom model (the Bragg additivity). Atomic targets are the backbone of DNA/RNA, i.e. hydrogen H, carbon C, nitrogen N and oxygen O. Also considered are neon Ne, argon Ar and krypton Kr. The studied molecular targets are H₂, water vapor H₂O, carbon monoxide CO, carbon dioxide CO₂, methane CH₄, ethylene C₂H₄, ethane C₂H₆, butane C₄H₁₀ as well as the DNA/RNA nucleobases, uracil C₄H₄N₂O₂, adenine C₅H₅N₅, guanine C₅H₅N₅O, thymine C₅H₆N₂O₂ and cytosine C₄H₅N₃O. Neon atoms are isoelectronic with water molecules. This permits gaining some useful insights into the role of molecular forces by analyzing the cross sections for neon and water targets. The cross sections for each atomic target shell are summed up and compared with the available experimental data for atomic and molecular targets. The usefulness of the Bragg sum rule is assessed for molecular charge-exchange cross sections. Besides atomic physics, interest in this problem area exists also in heavy ion transport physics of relevance to plasma physics, astrophysics, the fusion program and radiotherapy. For example, in ion therapy, when the primary beam particles reach their deepest penetration distance, they release almost all their energy and this is biologically and clinically of utmost importance. In this targeted region, ions slow down abruptly and move with energies 20 keV/amu ≤ E ≤ 150 keV/amu. Therein, capture and loss of electrons are dominant energy loss mechanisms yielding maximum dose depositions. The present databases for electron capture cross sections can be used in Monte Carlo simulation of ion energy losses in matter, including tissue.

Introduction

The origin of radiotherapy was the measurements by Bragg and Kleeman [1, 2] in 1904/1905. They showed that α-particles exhibit an increased ionization yield near the maximum penetration distance (the so-called range). The measured stopping power, as the distribution of ion energy loss per distance, was almost flat for most of the traversed path and exhibited a sharp maximum at the end of the particles’ track. Such a maximum (later called the Bragg peak) was at variance with the common conception that ion energy loss distribution should decay exponentially with distance.

Regarding electromagnetic interactions of ions with matter, energy losses of ions can occur in various ways. Different channels open up depending of the ion energy. Ionization dominates over e.g. electron capture at high ion energies E. However, when ion energies are considerably reduced, the dominant ion energy loss channels are electron capture and electron loss that occur concomitantly and interchangeably. It is because of this ‘open-channel-switch’ that most of the remaining ion energy is released at the Bragg peak, which corresponds to the lower edge of the intermediate energy region located at 20 keV/amu ≤ E ≤ 150 keV/amu.

In hadron therapy, the entrance energy of the primary beam is selected to reach the target location. The Bragg peak resides at the targeted tumor site. Around the Bragg peak, the radiation dose delivered by ions attains its maximum and this eradicates tumorous cells. Therefore, the immediate surrounding of the Bragg peak is biologically and clinically the most important area for treating the patient's tumor. Moreover, healthy tissue is largely spared in the flat domain prior to the Bragg peak. Such overall favorable dose-depth profile characteristics of ions optimize the net effect of hadron therapy. This is how the physics concept of ion transport in tissue gave impetus to the development of optimized radiotherapy in medicine.

Understandably, due to the unavailability of accelerators, this was not the original intention of Bragg and Kleeman [1, 2]. However, four decades after their experiments, Wilson [3, 4], the former head of the experimental part of the Manhattan Project, was the first to convey to physicians the far-reaching power of the Bragg peak for radiotherapy. To connect to physicians directly, he decided to publish his 1946 article [3] in a prestigious medical journal – Radiology. He explained convincingly that with the advantageous depth-dose profile of ions, physicians would be able to achieve a much better tumor control than with either electrons or photons. He emphasized that because of the optimal conformity of ion dose profiles to the target, tumor would receive maximum radiation dose, with a minimal damage to the surrounding healthy tissue. This Wilson's vision marks the beginning of optimized radiotherapy, the precision ion therapy.

Physicians wasted no time to implement this proposal in collaboration with physicists and already in the 1950s the first patients underwent ion therapy in Berkeley and Uppsala. Nowadays, some seven decades later, ion therapy is the workhorse of radiotherapy with more than 100 accelerators operating worldwide and more than 150000 patients treated by protons (85%) and carbon nuclei (13%) alone [5]. In the accompanying research, much attention has been paid to find the most appropriate ions for tumors at different sites in the human body and the main focus was on comparative analyses of protons and carbon ions [6, 7, 8, 9].

Ions through their direct hits produce about 30% of radiation damage to molecules of the targeted tumor tissue. The remaining damage is made by indirect hits via secondary particles (e.g. δ-electrons), free radicals and reactive oxygen species. In radiobiology, the main radiation targets are considered to be the deoxyribonucleic acid (DNA) molecules. Particularly within the Bragg peak, ions can massively disrupt the internal structure of DNA and inflict lethal, irreparable double strand breaks in these ‘molecules of life’.

Such radiation damage to DNA molecules is caused by energy deposited through electromagnetic and nuclear interactions of ions with tissue. Due to the exceeding complexity of these collisions, deterministic evaluations of energy losses of ions during their passage through tissue are of limited usefulness in practice. For this reasons, comprehensive computations of ion energy losses are usually performed by Monte Carlo (MC) simulations. The main input data to stochastic simulations are deterministically computed electronic and nuclear stopping powers (the former and the latter are concerned with electromagnetic and nuclear interactions, respectively) [9, 10, 11, 12, 13, 14]. On the other hand, the principal ingredient of stopping powers is the appropriate set of atomic, molecular and nuclear cross sections.

There is a number of MC codes in use for hadron therapy, e.g. PTRAN [15, 16, 17, 18], PETRA[19, 20], FLUKA [21], KURBUC [22, 23, 24], SRNA [25, 26, 27, 28, 29, 30], PATRAC [31, 32], SHIELD-HIT [33, 34, 35, 36, 37], SEICS [38], GEANT4-DNA [39], TILDA [40], etc. The first MC code for purposes of radiotherapy was PTRAN [15]. Three MC codes for ion therapy have been developed at the Karolinska Institute in Stockholm: PETRA [19], SHIELD-HIT [33] and KURBUC-carbon [24].

In particular, the MC code SHIELD has originally been designed and implemented by Sobolevsky [41, 42] with no inclination to radiotherapy whatsoever. Later, the first extension of SHIELD to heavy ion therapy (HIT), known as SHIELD-HIT [33], has been developed in a cooperative research project involving the Institute for Nuclear Research of the Russian Academy of Sciences (Moscow), The Nobel Medical Institute – Karolinska Institute, Stockholm (Division of Radiation Physics) and Stockholm University (Department of Medical Physics).

There are two parts that need to be substantially improved in the SHIELD-HIT code. One part is an appropriate and full upgrade of the Bethe-Bloch formula for electronic stopping powers that have, thus far, been empirically extrapolated below the quite high cut-off energy, E_cut-off ∼ 1000 keV/amu. No charge-exchange nor electron loss collisions are included, both of which dominate over ionization and excitation in the clinically most important region around the Bragg peak at 20 keV/amu ≤ E ≤ 150keV/amu. The other part is the essential inclusion of transport of electrons, ejected from the tissue traversed by ions. Especially δ-electrons are crucial as they are by far more effective in imparting radiation damage to the targeted tissue than any ion (primary, secondary or higher-generation ions due to nuclear transmutations of the tissue nuclei and/or the projectile nuclei heavier than protons).

For ion-atom and ion-molecule collisions, the main mechanisms for energy losses of the primary beam particles during their passage through matter are ionization, excitation, electron capture and electron loss. The present focus is on single electron capture by ions from atomic and molecular targets. Both atomic and molecular targets are selected to be of relevance for ion therapy. Thus, the main atomic targets of our interest are hydrogen H, carbon C, nitrogen N and oxygen O. Moreover, these atoms are also the constituents of the presently chosen biomolecules as molecular targets (water, carbon oxides, hydrocarbons and DNA/RNA nucleobases).

The total cross sections for ion-atom/molecule collisions are presently computed using the continuum distorted wave (CDW) method at intermediate and high energies. For charge-exchange and ionization, this most complete distorted wave theory has been introduced by Cheshire [43] and Belkić [44]. For charge-exchange, the CDW method has been shown to be in excellent agreement with measurements above about 50 keV/amu or so on the average [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61].

In the first detailed applications of the CDW method for capture into the ground and excited final states, tedious parametric derivatives of the transition amplitude T_if^(CDW) have been carried out. Therefore, this technique could be done for a handful of transitions. For example, in computations of cross sections Q_if^(CDW), Belkić et al. [49] used partial differentiations of the transition amplitude only for the orbital/quantum numbers 1 ≤ n_i,f ≤ 3 (0 ≤ l_i,f ≤ n_i,f − 1, − l_i,f ≤ m_i,f ≤ +l_i,f) and n_i,fl_i,f = 4s. Later, based on these limited analytical results for T_if^(CDW), a computer program for computations of Q_if^(CDW) has been published [62].

However, the breakthrough in a wider usage of the CDW method came with the availability of the analytical results of the general transition amplitudes T_if^(CDW) in the case of arbitrary i and f. The necessary analytical results for the bound-free matrix elements with any quantum numbers for hydrogenlike and multi-electron atomic systems have been derived in Refs. [63, 64, 65, 66]. These closed expressions for T_if^(CDW) have subsequently been programmed into the robust and automatic computer codes for any Q_if^(CDW) in the case of both hydrogenlike [67] and multi-electron [68] atomic targets. These programs supersede the earlier CDW code [62] based on parametric differentiations.

For multi-electron atomic targets, the available computer program in the CDW method [68] is flexible as it can employ the Roothaan-Hartree-Fock (RHF) wave functions in the analytical forms given by Clementi and Roetti [69] or Bunge et al. [70], both for atomic numbers 2–54, as well as by McLean and McLean [71] for atomic numbers 55–92. These computer codes from Ref. [68] are user-friendly since one can instantly switch from employing the RHF functions from Ref. [69] to Refs. [70, 71] merely by reading the input variational parameters from two different sets of tables. Importantly, the computer codes in the CDW method [67, 68] are remarkably expedient.

Even though the RHF wave functions of Clementi and Roetti [69] are widely used, they are not as accurate as those of Bunge et al. [70] for neutral atoms. The improvements made by Bunge et al. [70] relative to Clementi and Roetti [69] came from using a greater number of the Slater type orbitals (STOs) in the expansion basis set and from partially optimizing the principal quantum numbers n [72, 73].

The CDW computer programs [67, 68] have been built for atomic targets alone. However, they can also be used for molecular targets without any modification. To this end, all that is needed is to resort to the independent atom model (IAM) [74, 75, 76, 77, 78], or equivalently, the Bragg additivity [1, 2]. Atomic cross sections are computed first and then the sought molecular cross sections are deduced by the Bragg sum rule. This rule approximates a molecular cross section by the sum of the corresponding cross sections for each of the constituent atoms multiplied by the number of atoms in the given molecular target.

The underlying assumption of the Bragg additivity is that the molecular degrees of freedom (vibration, rotation, ...) can be ignored. In other words, by neglecting molecular forces, all the atoms in a molecule can be viewed as the independent centers of scattering. This is a reasonable hypothesis at higher impact energies at which the collision time is small which makes the projectiles unable to discern the finer internal structure of a molecular target. The Bragg sum rule has abundantly been used in the literature [79, 80, 81]. It met with success in e.g. charge-exchange involving protons and hydrocarbon molecular targets both at high energies (E ≤ 2500 keV) [82] and at intermediate energies (60–120 keV) [83].

In the case of atomic targets, besides the mentioned hydrogen H, carbon C, nitrogen N and oxygen O, we shall also consider neon Ne, argon Ar and krypton Kr. Molecular targets included in this study are molecular hydrogen H₂, water H₂O, two carbon oxides (carbon monoxide CO, carbon dioxide CO₂), four hydrocarbons (methane CH₄, ethylene C₂H₄, ethane C₂H₆, butane C₄H₁₀) and five DNA/RNA nucleobases (uracil C₄H₄N₂O₂, adenine C₅H₅N₅, guanine C₅H₅N₅O, thymine C₅H₆N₂O₂ and cytosine C₄H₅N₃O). Uracil is from RNA (ribonucleic acid), whereas adenine, guanine, thymine and cytosine are from DNA. Neon atom is isoelectronic with water molecule. Therefore, it may prove instructive to compare the cross sections for these two targets to learn about the relative importance of molecular forces in H₂O as a function of impact energy of projectiles [13, 14, 84, 85].

The atomic and molecular cross section databases to be supplied in this work are deemed to be of versatile use not only in atomic and molecular collision physics, but also in plasma physics and astrophysics [86, 87, 88, 89]. Likewise, they are very important in ion transport physics of relevance to several interdisciplinary areas ranging from the fusion research program for new energy to hadron therapy in medicine [9, 13, 14]. For example, in radiotherapy, dose-planning systems are always in need of improvement, which can come from better estimates of ion energy losses in tissue. For a long time now, much of the cross section input data to MC simulations has been computed using several empirical/phenomenological constructs based on either the asymptotic first Born approximation (as in the Bethe-Bloch formula) or on some mathematical ad hoc expressions (e.g. the Rudd formula) with adjustable parameters fitted to some experimental data from the given measurement. These drawbacks can advantageously become obsolete by employing e.g. the versatile CDW method, which is able to unequivocally provide the needed cross sections from the first principles of physics.

Atomic units will be used throughout unless stated otherwise.