A new formalism for modelling parameters α and β of the linear-quadratic model of cell survival for hadron therapy

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.

Accumulation of sublethal radiation damage and its effect on cell survival

Objective.Determine the extent of sublethal radiation damage (SRD) in a cell population that received a given dose of radiation and the impact of this damage on cell survival.Approach.We developed a novel formalism to account for accumulation of SRD with increasing dose. It is based on a very general formula for cell survival that correctly predicts the basic properties of cell survival curves, such as the transition from the linear-quadratic to a linear dependence at high doses. Using this formalism we analyzed extensive experimental data for photons, protons and heavy ions to evaluate model parameters, quantify the extent of SRD and its impact on cell survival.Main results.Significant accumulation of SRD begins at doses below 1 Gy. As dose increases, so does the number of damaged cells and the amount of SRD in individual cells. SRD buildup in a cell increases the likelihood of complex irrepairable damage. For this reason, during a dose fraction delivery, each dose increment makes cells more radiosensitive. This gradual radosensitization is evidenced by the increasing slope of survival curves observed experimentally. It continues until the fraction is delivered, unless radiosensitivity reaches its maximum first. The maximum radiosensitivity is achieved when SRD accumulated in most cells is the maximum damage they can repair. After this maximum is reached, the slope of a survival curve, logarithm of survival versus dose, becomes constant, dose independent. The survival curve becomes a straight line, as experimental data at high doses show. These processes are random. They cause large cell-to-cell variability in the extent of damage and radiosensitivity of individual cells.Significance.SRD is in effect a radiosensitizer and its accumulation is a significant factor affecting cell survival, especially at high doses. We developed a novel formalism to study this phenomena and reported pertinent data for several particle types.

On calculation of the average linear energy transfer for radiobiological modelling

Applying the concept of linear energy transfer (LET) to modeling of biological effects of charged particles usually involves calculation of the average LET. To calculate this, the energy distribution of particles is characterized by either the source spectrum or fluence spectrum. Also, the average can be frequency-or dose-weighted. This makes four methods of calculating the average LET, each producing a different number. The purpose of this note is to describe which of these four methods is best suited for radiobiological modelling. We focused on data for photons (x-rays and gamma radiation) because in this case differences in the four averaging methods are most pronounced. However, our conclusions are equally applicable to photons and hadrons. We based our arguments on recently emerged Monte Carlo data that fully account for transport of electrons down to very low energies comparable to the ionization potential of water. We concluded that the frequency average LET calculated using the fluence spectrum has better predictive power than does that calculated using any of the other three options. This optimal method is not new but is different from those currently dominating research in this area.

A simple model for calculating relative biological effectiveness of X-rays and gamma radiation in cell survival

OBJECTIVES

The relative biological effectiveness (RBE) of X-rays and γ radiation increases substantially with decreasing beam energy. This trend affects the efficacy of medical applications of this type of radiation. This study was designed to develop a model based on a survey of experimental data that can reliably predict this trend.

METHODS

In our model, parameters α and β of a cell survival curve are simple functions of the frequency-average linear energy transfer (L_F) of delta electrons. The choice of these functions was guided by a microdosimetry-based model. We calculated L_F by using an innovative algorithm in which L_F is associated with only those electrons that reach a sensitive-to-radiation volume (SV) within the cell. We determined model parameters by fitting the model to 139 measured (α,β) pairs.

RESULTS

We tested nine versions of the model. The best agreement was achieved with [Formula: see text] and β being linear functions of [Formula: see text] .The estimated SV diameter was 0.1-1 µm. We also found that α, β, and the α/β ratio increased with increasing [Formula: see text] .

CONCLUSIONS

By combining an innovative method for calculating [Formula: see text] with a microdosimetric model, we developed a model that is consistent with extensive experimental data involving photon energies from 0.27 keV to 1.25 MeV.

ADVANCES IN KNOWLEDGE

We have developed a photon RBE model applicable to an energy range from ultra-soft X-rays to megaelectron volt γ radiation, including high-dose levels where the RBE cannot be calculated as the ratio of α values. In this model, the ionization density represented by [Formula: see text] determines the RBE for a given photon spectrum.

Systematic microdosimetric data for protons of therapeutic energies calculated with Geant4-DNA

The purpose of this study was to generate physical data needed for microdosimetry-based models of proton RBE. Our focus was on the frequency and dose average lineal energies, y  F and y  D . We report data for proton energies from 0.1 to 100 MeV, for spherical volumes 2-103 nm in diameter. These data were calculated using Geant4-DNA Monte Carlo software. The physics implemented in Geant4-DNA has been extensively tested for this type of calculations but data on y  F and y  D for protons generated with this code have been very limited. An innovative aspect of our study is that we introduced a straightforward procedure for calculation of y  F and y  D for polyenergetic beams and presented the data in a format that simplifies these calculations. We compared our data with previous studies that used different Monte Carlo codes and with experimental data.

Renormalization of radiobiological response functions by energy loss fluctuations and complexities in chromosome aberration induction: deactivation theory for proton therapy from cells to tumor control

We employ a multi-scale mechanistic approach built upon our recent phenomenological/computational methodologies [R. Abolfath et al., Sci. Rep. 7, 8340 (2017)] to investigate radiation induced cell toxicities and deactivation mechanisms as a function of linear energy transfer in hadron therapy. Our theoretical model consists of a system of Markov chains in microscopic and macroscopic spatio-temporal landscapes, i.e., stochastic birth-death processes of cells in millimeter-scale colonies that incorporates a coarse-grained driving force to account for microscopic radiation induced damage. The coupling, hence the driving force in this process, stems from a nano-meter scale radiation induced DNA damage that incorporates the enzymatic end-joining repair and mis-repair mechanisms. We use this model for global fitting of the high-throughput and high accuracy clonogenic cell-survival data acquired under exposure of the therapeutic scanned proton beams, the experimental design that considers γ-H2AX as the biological endpoint and exhibits maximum observed achievable dose and LET, beyond which the majority of the cells undergo collective biological deactivation processes. An estimate to optimal dose and LET calculated from tumor control probability by extension to ~106 cells per mm-size voxels is presented. We attribute the increase in degree of complexity in chromosome aberration to variabilities in the observed biological responses as the beam linear energy transfer (LET) increases, and verify consistency of the predicted cell death probability with the in vitro cell survival assay of approximately 100 non-small cell lung cancer (NSCLC) cells. The present model provides an interesting interpretation to variabilities in α and β indices via perturbative expansion of the cell survival fraction (SF) in terms of specific and lineal energies, z and y, corresponding to continuous transitions in pair-wise to ternary, quaternary and more complex recombination of broken chromosomes from the entrance to the end of the range of proton beam.

A mechanistic relative biological effectiveness model-based biological dose optimization for charged particle radiobiology studies

In charged particle therapy, the objective is to exploit both the physical and radiobiological advantages of charged particles to improve the therapeutic index. Use of the beam scanning technique provides the flexibility to implement biological dose optimized intensity-modulated ion therapy (IMIT). An easy-to-implement algorithm was developed in the current study to rapidly generate a uniform biological dose distribution, namely the product of physical dose and the relative biological effectiveness (RBE), within the target volume using scanned ion beams for charged particle radiobiological studies. Protons, helium ions and carbon ions were selected to demonstrate the feasibility and flexibility of our method. The general-purpose Monte Carlo simulation toolkit Geant4 was used for particle tracking and generation of physical and radiobiological data needed for later dose optimizations. The dose optimization algorithm was developed using the Python (version 3) programming language. A constant RBE-weighted dose (RWD) spread-out Bragg peak (SOBP) in a water phantom was selected as the desired target dose distribution to demonstrate the applicability of the optimization algorithm. The mechanistic repair-misrepair-fixation (RMF) model was incorporated into the Monte Carlo particle tracking to generate radiobiological parameters and was used to predict the RBE of cell survival in the iterative process of the biological dose optimization for the three selected ions. The post-optimization generated beam delivery strategy can be used in radiation biology experiments to obtain radiobiological data to further validate and improve the accuracy of the RBE model. This biological dose optimization algorithm developed for radiobiology studies could potentially be extended to implement biologically optimized IMIT plans for patients.

RBE Model-Based Biological Dose Optimization for Proton Radiobiology Studies

Purpose

The purpose of the current study was (1) to develop a straightforward and rapid method to incorporate a dose-averaged linear energy transfer (LET _d )-based biological effect model into a dose optimization algorithm for scanned protons; and (2) to apply a novel beam delivery strategy with increased LET _d within the target, thereby enhancing the biological effect predicted using the selected relative biological effectiveness (RBE) model.

Materials and Methods

We first generated pristine dose Bragg curves in water and their corresponding LET _d distributions for 94 groups of proton beams, using experimentally validated Geant4 Monte Carlo simulations. Next, we developed 1-dimensional dose optimization algorithms by using the Python programming language. To calculate the RBE of protons for biological dose optimization, we invoked the McNamara RBE model and applied the radiobiological parameters of the lung cancer H460 cell line with ¹³⁷Cs reference photons.

Results

High-accuracy optimization results were obtained. The relative difference between the delivered dose and the prescribed dose was approximately within ±1.0% in the target. In addition, we obtained the RBE enhancement within the target by applying the LET-painting technique. For example, considering a simple case in which 2 opposed downslope dose fields were superimposed to form a uniform dose in the 5- to 10-cm target region, the center RBE was 1.23 ± 0.01, which was greater than the center RBE of 1.16 ± 0.01 found when using the traditional method of delivering 2 opposed flat dose fields.

Conclusion

We have successfully developed an easy-to-implement method to perform the biological dose optimization procedure by invoking the McNamara RBE model in the iteration process using the Python programming language. According to the RBE model predictions, we conclude that the increased target LET _d enhances the RBE. The accuracy of the RBE model predictions needs to be validated in radiobiological experiments.