Modelling Cellular Response to Ionizing Radiation: Mechanistic, Semi-Mechanistic, and Phenomenological Approaches - A Historical Perspective.

Radiation research is a multidisciplinary field, and among its many branches, mathematical and computational modelers have played a significant role in advancing boundaries of knowledge. A fundamental contribution is modelling cellular response to ionizing radiation as that is the key to not only understanding how radiation can kill cancer cells, but also cause cancer and other health issues. The invention of microdosimetry in the 1950s by Harold Rossi paved the way for brilliant scientists to study the mechanism of radiation at cellular and sub-cellular scales. This paper reviews some snippets of ingenious mathematical and computational models published in microdosimetry symposium proceedings and publications of the radiation research community. Among these are simulations of radiation tracks at atomic and molecular levels using Monte Carlo methods, models of cell survival, quantification of the amount of energy required to create a single strand break, and models of DNA-damage-repair. These models can broadly be categorized into mechanistic, semi-mechanistic, and phenomenological approaches, and this review seeks to provide historical context of their development. We salute pioneers of the field and great teachers who supported and educated the younger members of the community and showed them how to build upon their work.

Modelling DNA damage-repair and beyond.

The paper presents a review of mechanistic modelling studies of DNA damage and DNA repair, and consequences to follow in mammalian cell nucleus. We hypothesize DNA deletions are consequences of repair of double strand breaks leading to the modifications of genome that play crucial role in long term development of genetic inheritance and diseases. The aim of the paper is to review formation mechanisms underlying naturally occurring DNA deletions in the human genome and their potential relevance for bridging the gap between induced DNA double strand breaks and deletions in damaged human genome from endogenous and exogenous events. The model of the cell nucleus presented enables simulation of DNA damage at molecular level identifying the spectrum of damage induced in all chromosomal territories and loops. Our mechanistic modelling of DNA repair for double stand breaks (DSB), single strand breaks (SSB) and base damage (BD), shows the complexity of DNA damage is responsible for the longer repair times and the reason for the biphasic feature of mammalian cells repair curves. In the absence of experimentally determined data, the mechanistic model of repair predicts the in vivo rate constants for the proteins involved in the repair of DSB, SSB, and of BD.

Verification of KURBUC-based ion track structure mode for proton and carbon ions in the PHITS code.

The particle and heavy ion transport code system (PHITS) is a general-purpose Monte Carlo radiation transport simulation code. It has the ability to handle diverse particle types over a wide range of energy. The latest PHITS development enables the generation of track structure for proton and carbon ions (1H+, 12C6+) based on the algorithms in the KURBUC code, which is considered as one of the most verified track-structure codes worldwide. This ion track-structure mode is referred to as the PHITS-KURBUC mode. In this study, the range, radial dose distributions, and microdosimetric distributions were calculated using the PHITS-KURBUC mode. Subsequently, they were compared with the corresponding data obtained from the original KURBUC and from other studies. These comparative studies confirm the successful inclusion of the KURBUC code in the PHITS code. As results of the synergistic effect between the macroscopic and microscopic radiation transport codes, this implementation enabled the detailed calculation of the microdosimetric and nanodosimetric quantities under complex radiation fields, such as proton beam therapy with the spread-out Bragg peak.

Monte Carlo Electron Track Structure Calculations in Liquid Water Using a New Model Dielectric Response Function.

Monte Carlo track structure codes provide valuable information for understanding radiation effects down to the DNA level, where experimental measurements are most difficult or unavailable. It is well recognized that the performance of such codes, especially at low energies and/or subcellular level, critically depends on the reliability of the interaction cross sections that are used as input in the simulation. For biological media such as liquid water, one of the most challenging issues is the role of condensed-phase effects. For inelastic scattering, such effects can be conveniently accounted for through the complex dielectric response function of the media. However, for this function to be useful it must fulfill some important sum rules and have a simple analytic form for arbitrary energy- and momentum-transfer. The Emfietzoglou-Cucinotta-Nikjoo (ECN) model offers a practical, self-consistent and fully analytic parameterization of the dielectric function of liquid water based on the best available experimental data. An important feature of the ECN model is that it includes, in a phenomenological manner, exchange and correlation effects among the screening electrons, thus, going beyond the random-phase approximation implicit in earlier models. In this work, inelastic cross sections beyond the plane wave Born approximation are calculated for low-energy electrons (10 eV-10 keV) based on the ECN model, and used for Monte Carlo track structure simulations of physical quantities relevant to the microdosimetry of low-energy electrons in liquid water. Important new developments in the physics of inelastic scattering are discussed and their effect on electron track structure is investigated by a comparison against simulations (under otherwise identical conditions) using the Born approximation and a simpler form of the dielectric function based on the Oak Ridge National Laboratory model. The results reveal that both the dielectric function and the corrections to the Born approximation may have a sizeable effect on track structure calculations at the nanometer scale (DNA level), where the details of inelastic scattering and the role of low-energy electrons are most critical.

A stochastic cascade model for Auger-electron emitting radionuclides.

To benchmark a Monte Carlo model of the Auger cascade that has been developed at the Australian National University (ANU) against the literature data. The model is applicable to any Auger-electron emitting radionuclide with nuclear structure data in the format of the Evaluated Nuclear Structure Data File (ENSDF). Schönfeld's algorithms and the BrIcc code were incorporated to obtain initial vacancy distributions due to electron capture (EC) and internal conversion (IC), respectively. Atomic transition probabilities were adopted from the Evaluated Atomic Data Library (EADL) for elements with atomic number, Z = 1-100. Atomic transition energies were evaluated using a relativistic Dirac-Fock method. An energy-restriction protocol was implemented to eliminate energetically forbidden transitions from the simulations. Calculated initial vacancy distributions and average energy spectra of 123I, 124I, and 125I were compared with the literature data. In addition, simulated kinetic energy spectra and frequency distributions of the number of emitted electrons and photons of the three iodine radionuclides are presented. Some examples of radiation spectra of individual decays are also given. Good agreement with the published data was achieved except for the outer-shell Auger and Coster-Kronig transitions. Nevertheless, the model needs to be compared with experimental data in a future study.

Nanodosimetry and RBE values in radiotherapy.

In a recent paper, the authors reported that the dose mean lineal energy, [Formula: see text] in a volume of about 10-15 nm is approximately proportional to the α-parameter in the linear-quadratic relation used in fractionated radiotherapy in both low- and high-LET beams. This was concluded after analyses of reported radiation weighting factors, WisoE (clinical RBE values), and [Formula: see text] values in a large range of volumes. Usually, microdosimetry measurements in the nanometer range are difficult; therefore, model calculations become necessary. In this paper, the authors discuss the calculation method. A combination of condensed history Monte Carlo and track structure techniques for calculation of mean lineal energy values turned out to be quite useful. Briefly, the method consists in weighting the relative dose fractions of the primary and secondary charged particles with their respective energy-dependent dose mean lineal energies. The latter were obtained using a large database of Monte Carlo track structure calculations.

Spectrum of Radiation-Induced Clustered Non-DSB Damage - A Monte Carlo Track Structure Modeling and Calculations.

The aim of this report is to present the spectrum of initial radiation-induced cellular DNA damage [with particular focus on non-double-strand break (DSB) damage] generated by computer simulations. The radiation types modeled in this study were monoenergetic electrons (100 eV-1.5 keV), ultrasoft X-ray photons Ck, AlK and TiK, as well as some selected ions including 3.2 MeV/u proton; 0.74 and 2.4 MeV/u helium ions; 29 MeV/u nitrogen ions and 950 MeV/u iron ions. Monte Carlo track structure methods were used to simulate damage induction by these radiation types in a cell-mimetic condition from a single-track action. The simulations took into account the action of direct energy deposition events and the reaction of hydroxyl radicals on atomistic linear B-DNA segments of a few helical turns including the water of hydration. Our results permitted the following conclusions: a. The absolute levels of different types of damage [base damage, simple and complex single-strand breaks (SSBs) and DSBs] vary depending on the radiation type; b. Within each damage class, the relative proportions of simple and complex damage vary with radiation type, the latter being higher with high-LET radiations; c. Overall, for both low- and high-LET radiations, the ratios of the yields of base damage to SSBs are similar, being about 3.0 ± 0.2; d. Base damage contributes more to the complexity of both SSBs and DSBs, than additional SSB damage and this is true for both low- and high-LET radiations; and e. The average SSB/DSB ratio for low-LET radiations is about 18, which is about 5 times higher than that for high-LET radiations. The hypothesis that clustered DNA damage is more difficult for cells to repair has gained currency among radiobiologists. However, as yet, there is no direct in vivo experimental method to validate the dependence of kinetics of DNA repair on DNA damage complexity (both DSB and non-DSB types). The data on the detailed spectrum of DNA damage presented here, in particular the non-DSB type, provide a good basis for testing mechanistic models of DNA repair kinetics such as base excision repair.

DSB repair model for mammalian cells in early S and G1 phases of the cell cycle: application to damage induced by ionizing radiation of different quality.

The purpose of this work is to test the hypothesis that kinetics of double strand breaks (DSB) repair is governed by complexity of DSB. To test the hypothesis we used our recent published mechanistic mathematical model of DSB repair for DSB induced by selected protons, deuterons, and helium ions of different energies representing radiations of different qualities. In light of recent advances in experimental and computational techniques, the most appropriate method to study cellular responses in radiation therapy, and exposures to low doses of ionizing radiations is using mechanistic approaches. To this end, we proposed a 'bottom-up' approach to study cellular response that starts with the DNA damage. Monte Carlo track structure method was employed to simulate initial damage induced in the genomic DNA by direct and indirect effects. Among the different types of DNA damage, DSB are known to be induced in simple and complex forms. The DSB repair model in G1 and early S phases of the cell cycle was employed to calculate the repair kinetics. The model considers the repair of simple and complex DSB, and the DSB produced in the heterochromatin. The inverse sampling method was used to calculate the repair kinetics for each individual DSB. The overall repair kinetics for 500 DSB induced by single tracks of the radiation under test were compared with experimental results. The results show that the model is capable of predicting the repair kinetics for the DSB induced by radiations of different qualities within an accepted range of uncertainty.

Radiation induced base excision repair (BER): a mechanistic mathematical approach.

This paper presents a mechanistic model of base excision repair (BER) pathway for the repair of single-stand breaks (SSBs) and oxidized base lesions produced by ionizing radiation (IR). The model is based on law of mass action kinetics to translate the biochemical processes involved, step-by-step, in the BER pathway to translate into mathematical equations. The BER is divided into two subpathways, short-patch repair (SPR) and long-patch repair (LPR). SPR involves in replacement of single nucleotide via Pol ß and ligation of the ends via XRCC1 and Ligase III, while LPR involves in replacement of multiple nucleotides via PCNA, Pol δ/ɛ and FEN 1, and ligation via Ligase I. A hallmark of IR is the production of closely spaced lesions within a turn of DNA helix (named complex lesions), which have been attributed to a slower repair process. The model presented considers fast and slow component of BER kinetics by assigning SPR for simple lesions and LPR for complex lesions. In the absence of in vivo reaction rate constants for the BER proteins, we have deduced a set of rate constants based on different published experimental measurements including accumulation kinetics obtained from UVA irradiation, overall SSB repair kinetic experiments, and overall BER kinetics from live-cell imaging experiments. The model was further used to calculate the repair kinetics of complex base lesions via the LPR subpathway and compared to foci kinetic experiments for cells irradiated with γ rays, Si, and Fe ions. The model calculation show good agreement with experimental measurements for both overall repair and repair of complex lesions. Furthermore, using the model we explored different mechanisms responsible for inhibition of repair when higher LET and HZE particles are used and concluded that increasing the damage complexity can inhibit initiation of LPR after the AP site removal step in BER.

Microdosimetry of proton and carbon ions.

PURPOSE: To investigate microdosimetry properties of 160 MeV/u protons and 290 MeV/u(12)C ion beams in small volumes of diameters 10-100 nm. METHODS: Energy distributions of primary particles and nuclear fragments in the beams were calculated from simulations with the general purpose code SHIELD-HIT, while energy depositions by monoenergetic ions in nanometer volumes were obtained from the event-by-event Monte Carlo track structure ion code PITS99 coupled with the electron track structure code KURBUC. RESULTS: The results are presented for frequencies of energy depositions in cylindrical targets of diameters 10-100 nm, dose distributions yd(y) in lineal energy y, and dose-mean lineal energies yD. For monoenergetic ions, the yD was found to increase with an increasing target size for high-linear energy transfer (LET) ions, but decrease with an increasing target size for low-LET ions. Compared to the depth dose profile of the ion beams, the maximum of the yD depth profile for the 160 MeV proton beam was located at ∼ 0.5 cm behind the Bragg peak maximum, while the yD peak of the 290 MeV/u (12)C beam coincided well with the peak of the absorbed dose profile. Differences between the yD and dose-averaged linear energy transfer (LETD) were large in the proton beam for both target volumes studied, and in the (12)C beam for the 10 nm diameter cylindrical volumes. The yD determined for 100 nm diameter cylindrical volumes in the (12)C beam was approximately equal to the LETD. The contributions from secondary particles to the yD of the beams are presented, including the contributions from secondary protons in the proton beam and from fragments with atomic number Z = 1-6 in the (12)C beam. CONCLUSIONS: The present investigation provides an insight into differences in energy depositions in subcellular-size volumes when irradiated by proton and carbon ion beams. The results are useful for characterizing ion beams of practical importance for biophysical modeling of radiation-induced DNA damage response and repair in the depth profiles of protons and carbon ions used in radiotherapy.

Inelastic cross sections for low-energy electrons in liquid water: exchange and correlation effects.

Low-energy electrons play a prominent role in radiation therapy and biology as they are the largest contributor to the absorbed dose. However, no tractable theory exists to describe the interaction of low-energy electrons with condensed media. This article presents a new approach to include exchange and correlation (XC) effects in inelastic electron scattering at low energies (below ∼10 keV) in the context of the dielectric theory. Specifically, an optical-data model of the dielectric response function of liquid water is developed that goes beyond the random phase approximation (RPA) by accounting for XC effects using the concept of the many-body local-field correction (LFC). It is shown that the experimental energy-loss-function of liquid water can be reproduced by including into the RPA dispersion relations XC effects (up to second order) calculated in the time-dependent local-density approximation with the addition of phonon-induced broadening in N. D. Mermin's relaxation-time approximation. Additional XC effects related to the incident and/or struck electrons are included by means of the vertex correction calculated by a modified Hubbard formula for the exchange-only LFC. Within the first Born approximation, the present XC corrections cause a significantly larger reduction (∼10-50%) to the inelastic cross section compared to the commonly used Mott and Ochkur approximations, while also yielding much better agreement with the recent experimental data for amorphous ice. The current work offers a manageable, yet rigorous, approach for including non-Born effects in the calculation of inelastic cross sections for low-energy electrons in liquid water, which due to its generality, can be easily extended to other condensed media.

Ionizing radiation and genetic risks. XVII. Formation mechanisms underlying naturally occurring DNA deletions in the human genome and their potential relevance for bridging the gap between induced DNA double-strand breaks and deletions in irradiated germ cells.

While much is known about radiation-induced DNA double-strand breaks (DSBs) and their repair, the question of how deletions of different sizes arise as a result of the processing of DSBs by the cell's repair systems has not been fully answered. In order to bridge this gap between DSBs and deletions, we critically reviewed published data on mechanisms pertaining to: (a) repair of DNA DSBs (from basic studies in this area); (b) formation of naturally occurring structural variation (SV) - especially of deletions - in the human genome (from genomic studies) and (c) radiation-induced mutations and structural chromosomal aberrations in mammalian somatic cells (from radiation mutagenesis and radiation cytogenetic studies). The specific aim was to assess the relative importance of the postulated mechanisms in generating deletions in the human genome and examine whether empirical data on radiation-induced deletions in mouse germ cells are consistent with predictions of these mechanisms. The mechanisms include (a) NHEJ, a DSB repair process that does not require any homology and which functions in all stages of the cell cycle (and is of particular relevance in G0/G1); (b) MMEJ, also a DSB repair process but which requires microhomology and which presumably functions in all cell cycle stages; (c) NAHR, a recombination-based DSB repair mechanism which operates in prophase I of meiosis in germ cells; (d) MMBIR, a microhomology-mediated, replication-based mechanism which operates in the S phase of the cell cycle, and (e) strand slippage during replication (involved in the origin of small insertions and deletions (INDELs). Our analysis permits the inference that, between them, these five mechanisms can explain nearly all naturally occurring deletions of different sizes identified in the human genome, NAHR and MMBIR being potentially more versatile in this regard. With respect to radiation-induced deletions, the basic studies suggest that those arising as a result of the operation of NHEJ/MMEJ processes, as currently formulated, are expected to be relatively small. However, data on induced mutations in mouse spermatogonial stem cells (irradiation in G0/G1 phase of the cell cycle and DSB repair presumed to be via NHEJ predominantly) show that most are associated with deletions of different sizes, some in the megabase range. There is thus a 'discrepancy' between what the basic studies suggest and the empirical observations in mutagenesis studies. This discrepancy, however, is only an apparent but not a real one. It can be resolved by considering the issue of deletions in the broader context of and in conjunction with the organization of chromatin in chromosomes and nuclear architecture, the conceptual framework for which already exists in studies carried out during the past fifteen years or so. In this paper, we specifically hypothesize that repair of DSBs induced in chromatin loops may offer a basis to explain the induction of deletions of different sizes and suggest an approach to test the hypothesis. We emphasize that the bridging of the gap between induced DSB and resulting deletions of different sizes is critical for current efforts in computational modeling of genetic risks.

Biochemical DSB-repair model for mammalian cells in G1 and early S phases of the cell cycle.

The paper presents a model of double strand breaks (DSB) repair in G1 and early S phases of the cell cycle. The model is based on a plethora of published information on biochemical modification of DSB induced by ionizing radiation. So far, three main DSB repair pathways have been identified, including nonhomologous end-joining (NHEJ), homologous recombination (HR), and microhomology-mediated end-joining (MMEJ). During G1 and early S phases of the cell cycle, NHEJ and MMEJ repair pathways are activated dependent on the type of double strand breaks. Simple DSB are a substrate for NHEJ, while complex DSB and DSB in heterochromatin require further end processing. Repair of all DSB start with NHEJ presynaptic processes, and depending on the type of DSB pursue simple ligation, further end processing prior to ligation, or resection. Using law of mass action the model is translated into a mathematical formalism. The solution of the formalism provides the step by step and overall repair kinetics. The overall repair kinetics are compared with the published experimental measurements. Our calculations are in agreement with the experimental results and show that the complex types of DSBs are repaired with slow repair kinetics. The G1 and early S phase model could be employed to predict the kinetics of DSB repair for damage induced by high LET radiation.

The non-homologous end-joining (NHEJ) mathematical model for the repair of double-strand breaks: II. Application to damage induced by ultrasoft X rays and low-energy electrons.

We investigated the kinetics of simple and complex types of double-strand breaks (DSB) using our newly proposed mechanistic mathematical model for NHEJ DSB repair. For this purpose the simulated initial spectrum of DNA DSB, induced in an atomistic canonical model of B-DNA by low-energy single electron tracks, 100 eV to 4.55 keV, and the electrons generated by ultrasoft X rays (CK, AlK and TiK), were subjected to NHEJ repair processes. The activity elapsed time of sequentially independent steps of repair performed by proteins involved in NHEJ repair process were calculated for separate DSB. The repair kinetics of DSBs were computed and compared with published data on repair kinetics obtained by pulsed-field gel electrophoresis method. The comparison shows good agreement for V79-4 cells irradiated with ultrasoft X rays. The average times for the repair of simple and complex DSB confirm that double-strand break complexity is a potential explanation for the slow component of DSB repair observed in V79-4 cells irradiated by ultrasoft X rays.

The non-homologous end-joining (NHEJ) pathway for the repair of DNA double-strand breaks: I. A mathematical model.

This article presents a biochemical kinetic model for the non-homologous end joining (NHEJ) of DNA double-strand break (DSB) repair pathway. The model is part of a theoretical framework to encompass all cellular DSB repair pathways. The NHEJ model was developed by taking into consideration the biological characteristics of the repair processes in the absence of homologous recombination (HR), the major alternative pathway for DSB repair. The model considers fast and slow components of the repair kinetics resulting in a set of differential equations that were solved numerically. In the absence of available published data for reaction rate constants for the repair proteins involved in NHEJ, we propose reaction rate constants for the solution of the equations. We assume as a first approximation that the reaction rate constants are applicable to mammalian cells under same conditions. The model was tested by comparing measured and simulated DSB repair kinetics obtained with HR-deficient cell lines irradiated by X rays in the dose range of 20-80 Gy. Measured data for initial protein recruitment to a DSB were used to independently estimate rate constants for Ku70/Ku80 and DNA-dependent protein kinase catalytic subunit (DNA-PKcs). We show here based on the model of DSB repair described in this article, application of the model in the accompanying article (Taleei et al., Radiat. Res. 179, 540-548, 2013) and by simulation of repair times for each individual DSB produced by individual tracks of electrons, that the complexity of damage may explain the slow kinetics of DNA DSB repair.

Cross sections for bare and dressed carbon ions in water and neon.

The paper presents calculated cross sections for bare and dressed carbon projectiles of charge states q (0 to 6) with energies 1-10(4) keV u(-1) impacting on molecular water and atomic neon targets. The cross sections of water are of interest for radiobiological studies, but there are very few experimental data for water in any phase, while those for liquid water are non-existent. The more extensive experimental database for the neon target made it possible to test the reliability of the model calculations for the many-electron collision system. The current calculations cover major single and double electronic interactions of low and intermediate energy carbon projectiles. The three-body classical trajectory Monte Carlo (CTMC) method was used for the calculation of one-electron transition probabilities for target ionization, electron capture and projectile electron loss. The many-electron problem was taken into account using statistical methods: a modified independent event model was used for pure (direct) and simultaneous target and projectile ionizations, and the independent particle model for pure electron capture and electron capture accompanied by target ionization. Results are presented for double differential cross sections (DDCS) for total electron emission by carbon projectile impact on neon. For the water target, we present the following: single differential cross sections (SDCS) and DDCS for single target ionization; total cross sections (TCS) for electron emission; TCS for the pure single electronic interactions; equilibrium charge state fractions; and stopping cross sections. The results were found to be in satisfactory agreement with the experimental data in many cases, including DDCS and SDCS for the single target ionization, TCS for the total electron emission and TCS for the pure single electron capture. The stopping cross sections of this work are consistent with the other model calculations for projectile energies ≥800 keV u(-1), but smaller than the other calculations at lower energies. The discrepancy arises from the inclusion of all carbon charge states and coupling between electron capture and target ionization channels, while other models use an average projectile charge. The CTMC model presented here provides a tool for cross section calculations for low and intermediate energy carbon projectiles. The calculated cross sections are required for Monte Carlo track structure simulations of full-slowing-down tracks of carbon ions. The work paves the way for biophysical studies and dosimetry at the cellular and subcellular levels in the Bragg peak area of a therapeutic carbon ion beam.