Some representative models of radiation-induced cell death, which is a crucial endpoint in radiobiology, were reviewed. The basic assumptions were identified, their consequences on predicted cell survival were analyzed, and the advantages and drawbacks of each approach were outlined. In addition to “historical” approaches such as the Target Theory, the Linear-Quadratic model, the Theory of Dual Radiation Action and Katz' model, the more recent Local Effect Model was discussed, focusing on its application in Carbon-ion hadrontherapy. Furthermore, a mechanistic model developed at the University of Pavia and based on the relationship between cell inactivation and chromosome aberrations was presented, together with recent results; the good agreement between model predictions and literature experimental data on different radiation types (photons, protons, alpha particles, and Carbon ions) supported the idea that asymmetric chromosome aberrations like dicentrics and rings play a fundamental role for cell death. Basing on these results, a reinterpretation of the TDRA was also proposed, identifying the TDRA “sublesions” and “lesions” as clustered DNA double-strand breaks and (lethal) chromosome aberrations, respectively.

Cell death is a crucial issue in radiation-induced biological damage, not only because it is widely considered as a reference endpoint to characterize the action of ionizing radiation in different biological targets, but also because cell killing is the main aim for any tumour therapy. In particular, the dependence of cell-death RBE (Relative Biological Effectiveness) on the particle energy is a fundamental information for Carbon-ion therapy; with increasing depth in tissue, the primary-beam energy decreases, thus implying an increase in the radiation LET (Linear Energy Transfer) and in its biological effectiveness.

Many experimental data sets on different cell lines cultured

To better understand the mechanisms governing radiation-induced cell death and to allow for predictions where the experimental data are not sufficiently reliable (typically at high doses, where the error bars are generally large due to poor statistics), many theoretical models of radiation-induced cell killing have been proposed [

In this paper, some of these models will be presented and discussed, focusing on the assumptions adopted by the authors and on possible model advantages and drawbacks. Far from being exhaustive, this overview will first deal with three “historical” approaches: Lea’s “Target Theory”, which is one of the earliest interpretive models, the “Linear-Quadratic model”, which overcomes some inconsistencies of the Target Theory with mammalian data and explicitly takes into account DNA damage induction and repair, and the Theory of Dual Radiation Action, which leads as well to a linear-quadratic expression for cell survival but starting from a different background with respect to the LQ model. These three general models will be followed by two approaches (the Katz’ model and the more recent Local Effect Model) that are specific for heavy ions; in particular the LEM approach, which allows calculation of heavy-ion cell survival starting from photon experimental data, is applied at GSI in Germany for the biological optimization of Carbon-ion treatment planning. Finally, an original approach developed at the University of Pavia basing on the link between chromosome aberrations and cell death will be presented, together with recent model predictions on the survival of V79 cells exposed to different radiation types. The peculiarity of this approach consists in being mechanistic, since theoretical survival curves are derived from first principles on the biophysical mechanisms underlying DNA damage induction and repair. Importantly, in contrast with many literature mechanistic models including the Linear-Quadratic approach, only two (semi-)free parameters are adopted. Also, the model presented herein works both for low-, intermediate-, and high-LET radiation.

Lea’s “target theory”, which was first developed in 1946 and published in a refined version in 1955 [

Since the

The target theory makes no assumption about the induction and repair of the initial DNA damage, which is now known to play a fundamental role for radiation-induced clonogenic death. A number of alternative approaches to Lea's theory have been developed to respond to such objection, and several of these approaches are of a linear-quadratic form. In particular Chadwick and Leenhouts in 1981 [

According to Chadwick and Leenhouts, the double helix can undergo a DSB as the result of two different mechanisms: (i) both DNA strands are broken by the same radiation track (or “event"); (ii) each strand is broken independently, but the breaks are close enough in time and space to lead to a DSB. Let

The average number of DSBs per cell is therefore

The Theory of Dual Radiation Action (TDRA) was proposed by Kellerer and Rossi in 1972 [

According to the TDRA, the average number of lesions after a dose

However, an alternative interpretation may be the following: on the basis of the observed relationship between some chromosome aberration types and cell death (see below), sublesions can be thought as DSBs whereas lesions can be thought as lethal chromosome aberrations such as dicentrics and rings; this way a sensitive site of the order of the micrometer becomes consistent with the data, since chromosome aberrations are produced by pairwise interaction of DSBs, and such interaction occurs at the level of interphase chromosome domains, which are known to have linear dimensions of the order of the micrometer.

In the late 1960s [

According to Katz’ approach, the transition from the shouldered curves observed at low LET and the high-LET exponential curves is modelled by attributing the dose to two different inactivation modes, that is the “

A more recent approach, developed and used at GSI for the biological optimization of Carbon-ion treatment planning, is the “Local Effect Model” (LEM) [

The LEM approach has been implemented in the TRiP treatment planning procedure for the Carbon-ion therapy project at GSI. According to LEM, the biological characteristics of the various target tissues are essentially determined by the

At the University of Pavia, a mechanistic model and a Monte Carlo code originally developed for predicting chromosome aberration induction have been recently extended to simulate cell death, starting from the experimentally-observed relationship between some chromosome aberration types (dicentrics, rings, and deletions) and clonogenic inactivation (e.g., [

The model/code for radiation-induced chromosome aberrations, which was initiated more than ten years ago [

The current version of the code can deal either with spherical cell nuclei or with cylindrical nuclei, with dimensions that can be chosen by the user from the input file. The various interphase chromosome territories are modelled as (irregular) intranuclear regions consisting of the union of small adjacent cubic voxels of 0.2

The yield of radiation-induced Cluster Lesions (i.e., average number of CLs per Gy and per cell) is the starting point for the simulation of dose-response curves. In terms of biophysical mechanisms, CLs represent those initial DNA breaks that, being clustered and thus severe, can “evolve” into chromosome aberrations; therefore, the yield of CLs primarily depends on radiation quality (that is radiation type and energy), but it can also be modulated by the repair ability of the specific cell line under consideration. In previous works on chromosome aberration induction in lymphocytes exposed to protons or alpha particles, the CLs yields have been taken from “event-by-event” radiation track-structure simulations in which a CL has been defined as “at least two SSBs on each DNA strand within 30 base pairs” [

For a given irradiated cell, for sparsely-ionizing radiation like X- and gamma-rays an actual number of CLs is extracted from a Poisson distribution, and such lesions are then randomly distributed in the nucleus. For light ions like protons and alpha particles, an actual number of particle tracks traversing the nucleus is extracted from a Poisson distribution with average value

After assigning the spatial positions of each CL in the cell nucleus, the subsequent simulation steps consist of

Up to now, the model has been validated for the induction of the main types of chromosome aberrations in lymphocytes exposed to X- and

The model/code described above has recently been extended to simulate radiation-induced cell death, starting from the experimentally observed one-to-one relationship between the average number of “lethal aberrations” (that is Giemsa-stained dicentrics, rings, and deletions) per cell and

The results described above are reported in Figure

From top to bottom: survival of V79 cells exposed to photons, 0.76 MeV protons, 11.0 MeV/u Carbon ions, and 3.2 MeV alpha particles; the lines are model predictions, the points are experimental data taken from [

Some representative theoretical models of radiation-induced cell inactivation were presented and discussed, outlining the main biophysical assumptions adopted by the various authors and analyzing the consequences of such assumptions in terms of cell survival predictions. More specifically Lea’s target theory, which was developed before the discovery of the DNA double helix for microorganisms exposed to low-LET radiation, in its MTSH version assumes that there exist

Both the Molecular Model and the Theory of Dual Radiation Action lead to a linear-quadratic survival curve, though following different approaches. The Molecular Model considers the DNA double helix as the critical target and takes into account the biophysical mechanisms of DNA damage induction and repair, assuming that the critical damage is a DSB (which can be induced either by a single-track action or by a two-track action) and that the yield of DSBs is proportional to cell inactivation. On the contrary, the TDRA, without specifying the critical target(s) nor the critical damage(s), assumes that radiation induces cellular “sublesions” that interact in pairs to produce “lesions”, which in turn lead to cell death with a certain fixed probability; such interaction can take place only within “sensitive sites” of the order of

After reviewing the literature models mentioned above, an approach developed at the University of Pavia was presented together with recent results. The peculiarity of this approach consists in the fact that on one side it is mechanistic (that is the model predictions are derived from a few basic assumptions on the biophysical mechanisms underlying radiation-induced cell death), but at the same time the number of free parameters is kept at a minimum, since only two semifree parameters are adopted: the yield of DNA “Cluster Lesions” and the value of the threshold distance

The good agreement between model predictions and literature experimental data on low-, intermediate-, and high-LET radiation (photons, protons, alpha particles, and Carbon ions) supported the idea that asymmetric chromosome aberrations, such as dicentrics and rings, play a fundamental role in the mechanisms governing radiation-induced cell death. Furthermore, the model may be considered as a predictive tool for potential applications in radiotherapy including therapy with Carbon ions, which is now adopted in various centres worldwide, including the CNAO centre in Pavia.

This work was partially supported by INFN (project “WIDEST1"). The author is also grateful to Michael Cornforth, Marco Durante, Yoshiya Furusawa, Andrea Ottolenghi, Ben Phoenix, Michael Scholz, Maria Antonella Tabocchini, and Wilma Weyrather for useful discussion and/or data sharing.