NanOx, a new model to predict cell survival in the context of particle therapy

This section introduces the definitions used in NanOx concerning both local and non-local lethal events. Other more general definitions, such as those concerning radiation impacts and the elements composing the geometry of the cell, are also included here and summarized in table 1 along with the corresponding notation.

Table 1. List of the main notations used in NanOx, their name and definition.

Symbol	Name	Definition
Vs	Sensitive volume(s)	Region(s) of the cell critical for the evaluation of the radiation effects induced by local and/or non-local lethal events
$\Sigma$	Area of influence of the sensitive volume(s)	Area of the volume of influence comprising the radiation impacts that have a non-negligible probability of depositing energy into the sensitive volume(s)
i	Local target	Index of any volume that respects the locality condition placed inside the sensitive volume associated with local lethal events
N	Set of local targets	Number of local targets, for which N  >  1
${{\text{c}}_{i}}$	Configuration of the local target i	Set of parameters that describe the local target i, including for instance its geometry, position and rotation
${{\text{c}}_{N}}$	Configuration of a set of N local targets	Spatial distribution of the N targets, each with an associated configuration ${{\text{c}}_{i}}$
k	Single radiation impact	Index of any set of interactions between the medium and the primary radiation (ion, electron, photon or neutron) and all the secondary particles
K	Set of radiation impacts	Number of radiation impacts, for which K  >  1
${{\text{c}}_{k}}$	Configuration of the radiation impact k	Set of parameters that describe the radiation impact k totally or partially inside the volume of influence, including for instance the spatial distribution of energy-transfer points and the resulting physicochemical events at a given time
${{\text{c}}_{K}}$	Configuration of a set of K radiation impacts	Spatial distribution of the K radiation impacts, each with an associated configuration ${{\text{c}}_{k}}$
x	Local quantity	Physicochemical quantity inside a local target
X	Non-local quantity	Physicochemical quantity inside the cell sensitive volume associated with non-local events

2.1.1. Definitions relative to radiation impacts.

2.1.2. Definitions relative to the cell geometry.

2.1.3. Definitions relative to local lethal events.

2.1.4. Definitions relative to non-local lethal events.

Non-local lethal events.

Non-local lethal events are complementary to local lethal events, i.e. they are harmful to the cell, but are not able to cause its death on their own. As in microdosimetric models, they may correspond to the interaction between two 'distant' sublethal cellular lesions (i.e. non local) (Hawkins 2003), but they may as well represent sublethal damage like DNA single strand breaks (SSBs) and lesions in different cellular structures (e.g. mitochondria, nuclear and cellular membranes). Non-local lethal events may also correspond to a state of oxidative stress, i.e. an imbalance between the amount of antioxidant defenses and chemical reactive species in the cell. The combination of all these events by an effect of accumulation and/or interaction among each other at the (microscopic) cell scale may result in cell death.

Non-local action of radiation.

The non-local action of radiation is represented by X. The choice for a capital letter is in contrast with that set to denote the local action of radiation in order to highlight that X is considered at a much larger scale than x. Although the same letter was selected to denote the action of radiation at both local and non-local scales, it does not imply that the same quantity is used in both cases; it simply represents the fact that it is possible to choose the quantity to input to NanOx.

Global events.

In this first version of NanOx, we chose to represent non-local lethal events by 'global' events, which describe the accumulation of sublethal events in the associated sensitive volume. The production of chemical reactive species seemed to us of particular interest since they are responsible for inducing a significant part of sublethal damages in the DNA (Ravanat et al 2001, von Sonntag 2006) and are directly involved in cellular oxidative stress.

Chemical specific energy and relative chemical effectiveness.

From the production of reactive chemical species, we can conveniently define a chemical specific energy. For that purpose, we established a concept similar to the RBE, the relative chemical effectiveness (RCE):

$$\begin{eqnarray}&&\text{RC}{{\text{E}}_{\text{st}}}={{\left(\frac{{{Z}_{\text{r}}}}{Z}\right)}_{\text{st}}}\end{eqnarray} \tag{ 4 }$$

where RCE_st is the relative chemical effectiveness for a given level st of oxidative stress; Z_r and Z are, respectively, the specific energies of the reference radiation and the radiation under consideration that are required to cause such a level of stress in the sensitive volume associated with non-local lethal events (see figure 2). The chemical specific energy ${{}^{{{\text{c}}_{K}}}}\tilde{Z}$ after a configuration of radiation impacts ${{\text{c}}_{K}}$ is then expressed as:

$$\begin{eqnarray}&&{{}^{{{\text{c}}_{K}}}}\tilde{Z}=\underset{k=1}{\overset{K}{\sum}}\,{{}^{{{\text{c}}_{k}}}}\text{RCE}\centerdot {{}^{{{\text{c}}_{k}}}}Z\end{eqnarray} \tag{ 5 }$$

where ${{}^{{{\text{c}}_{k}}}}Z$ is the specific energy and ${{}^{{{\text{c}}_{k}}}}\text{RCE}$ the relative chemical effectiveness in the sensitive volume associated with non-local lethal events after a single radiation impact with configuration ${{\text{c}}_{k}}$. All these quantities may be estimated by simulations using the LQD (LiQuiD water radiolysis) Monte Carlo code (Gervais et al 2005, 2006). Consequently, the quantity X representing the non-local action of radiation is set to be equal to $\tilde{Z}$ in the present version of NanOx.

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2.1.5. Cell survival versus cell survival to local and to non-local lethal events.

The fundamental premise of NanOx is that the probability of cell survival to an irradiation is characterized by the effect of said irradiation on two sensitive volumes, one associated with local lethal events and another associated with non-local lethal events. Such volumes are a priori different. The probabilities of cell survival to local lethal (${{}^{{{\text{c}}_{K}}}}{{S}_{\text{L}}}$) and non-local lethal (${{}^{{{\text{c}}_{K}}}}{{S}_{\text{NL}}}$) events are assumed to be independent. Thus, the probability of cell survival ${{}^{{{\text{c}}_{K}}}}S$ to a configuration ${{\text{c}}_{K}}$ of impacts within a volume of influence is given by the following expression:

$$\begin{eqnarray}&&{{}^{{{\text{c}}_{K}}}}S={{}^{{{\text{c}}_{K}}}}{{S}_{\text{L}}}\times {{}^{{{\text{c}}_{K}}}}{{S}_{\text{NL}}}.\end{eqnarray} \tag{ 6 }$$

2.1.6. Cell survival to local lethal events.

We postulate that a lethal local event corresponds to inactivating a single local target among N identical local targets distributed uniformly in the sensitive volume associated with local lethal events. Furthermore, the inactivation of a local target is directly and only associated with the quantity $x$ in said local target, through the inactivation probability function $f(x)$, which means that local target responses are independent. Hence, the probability of cell survival to local lethal events for a given configuration of local targets (${{\text{c}}_{N}}$) and radiation impacts (${{\text{c}}_{K}}$) is equal to the probability that no local target is inactivated:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{}^{{{\text{c}}_{N}},{{\text{c}}_{K}}}}{{S}_{\text{L}}} & =\underset{k=1}{\overset{K}{\prod}}\,{{}^{{{\text{c}}_{N}},{{\text{c}}_{k}}}}{{S}_{\text{L}}} \\ {} & =\underset{k=1}{\overset{K}{\prod}}\,\underset{i=1}{\overset{N}{\prod}}\,\left(1-f\left({{}^{{{\text{c}}_{i}},{{\text{c}}_{k}}}}x\right)\right) \end{array}\end{eqnarray} \tag{ 7 }$$

where ${{}^{{{\text{c}}_{i}},{{\text{c}}_{k}}}}x$ is the quantity x in the local target i with configuration ${{\text{c}}_{i}}$ (i.e. position and orientation) after one radiation impact with configuration ${{\text{c}}_{k}}$. Figure 3 shows an example of a radiation impact configuration ${{\text{c}}_{k}}$ for a 12 MeV u⁻¹ carbon ion in water. The track was simulated using LQD.

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The average number of local lethal events, determined over the whole set of possible biological responses in the sequence of a given quantity ${{}^{{{\text{c}}_{i}},{{\text{c}}_{k}}}}x$ in a local target i, corresponds to the probability that said local target is inactivated:

$$\begin{eqnarray}&&{{}^{{{\text{c}}_{i}},{{\text{c}}_{k}}}}n=f\left({{}^{{{\text{c}}_{i}},{{\text{c}}_{k}}}}x\right).\end{eqnarray} \tag{ 8 }$$

We define here a more convenient quantity for the practical implementation of the model, the average effective number of local lethal events (ENLLE) in a local target i:

$$\begin{eqnarray}&&{{}^{{{\text{c}}_{i}},{{\text{c}}_{k}}}}{{n}^{\ast}}=-\ln \left(1-f\left({{}^{{{\text{c}}_{i}},{{\text{c}}_{k}}}}x\right)\right).\end{eqnarray} \tag{ 9 }$$

Given that local lethal events are independent from one another, the average ENLLE in a sensitive volume with N local targets is the sum of the ENLLE in all those local targets:

$$\begin{eqnarray}&&{{}^{{{\text{c}}_{N}},{{\text{c}}_{k}}}}{{n}^{\ast}}=\underset{i=1}{\overset{N}{\sum}}\,{{}^{{{\text{c}}_{i}},{{\text{c}}_{k}}}}{{n}^{\ast}}.\end{eqnarray} \tag{ 10 }$$

Equation (7) can then be written as a function of the ENLLE:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{}^{{{\text{c}}_{N}},{{\text{c}}_{K}}}}{{S}_{\text{L}}} & =\underset{k=1}{\overset{K}{\prod}}\,\exp \left(-{{}^{{{\text{c}}_{N}},{{\text{c}}_{k}}}}{{n}^{\ast}}\right). \\ {} & =\exp \left(-{{}^{{{\text{c}}_{N}},{{\text{c}}_{K}}}}{{n}^{\ast}}\right) \end{array}\end{eqnarray} \tag{ 11 }$$

Assuming that the number of local targets N is large, each particular configuration ${{\text{c}}_{N}}$ can be disregarded and we can thus simplify the estimate of the ENLLE by averaging it over many configurations of local targets:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{}^{{{\text{c}}_{K}}}}{{S}_{\text{L}}}=\underset{k=1}{\overset{K}{\prod}}\,\exp \left(-{{\langle {{}^{{{\text{c}}_{N}},{{\text{c}}_{k}}}}{{n}^{\ast}}\rangle}_{{{\text{c}}_{N}}}}\right). \end{array}\end{eqnarray} \tag{ 12 }$$

Since all local targets are equivalent and homogeneously distributed, we have:

$$\begin{eqnarray}&&{{}^{{{\text{c}}_{K}}}}{{S}_{\text{L}}}=\underset{k=1}{\overset{K}{\prod}}\,\exp \left(-{{\langle F\left({{}^{{{\text{c}}_{i}},{{\text{c}}_{k}}}}x\right)\rangle}_{{{\text{c}}_{i}}}}\right)\end{eqnarray} \tag{ 13 }$$

where the effective local lethal function F is:

$$\begin{eqnarray}&&F(x)=-N\ln \left(1-f(x)\right).\end{eqnarray} \tag{ 14 }$$

The expression chosen for F is given in equation (19). Note that N is contained in F and cannot be explicitly determined.

2.1.7. Cell survival to non-local lethal events.

As discussed in section 2.1.4, non-local lethal events are represented by global events. The probability of cell survival to these events is a function of the chemical specific energy ${{}^{{{\text{c}}_{K}}}}\tilde{Z}$ in the sensitive volume associated with non-local lethal events. As expressed in equation (5), such a chemical specific energy is proportional to the density of chemical reactive species produced in water by a given radiation with respect to a reference radiation. In turn, this density is a function of the time T_RCE that separates the radiation impact and the observation of the production of the reactive chemical species, since they continuously interact and recombine with each other. T_RCE could then be adjusted to obtain a correct description of the dependence of the shoulders in cell survival curves on LET.

The well-known LQ expression was chosen to model the probability of cell survival to global events ${{}^{{{\text{c}}_{K}}}}{{S}_{\text{G}}}$ as a function of the chemical specific energy ${{}^{{{\text{c}}_{K}}}}\tilde{Z}$:

$$\begin{eqnarray}&&{{}^{{{\text{c}}_{K}}}}{{S}_{\text{G}}}\left({{}^{{{\text{c}}_{K}}}}\tilde{Z}\right)={{C}_{\text{norm}}}\centerdot \exp \left(-{{\alpha}_{\text{G}}}\centerdot {{}^{{{\text{c}}_{K}}}}\tilde{Z}-{{\beta}_{\text{G}}}\centerdot {{}^{{{\text{c}}_{K}}}}{{\tilde{Z}}^{2}}\right)\end{eqnarray} \tag{ 15 }$$

where C_norm is a normalization factor ensuring that the average of cell survival over all irradiation configurations leads to the experimental probability of cell survival to an irradiation with a reference radiation characterized by the coefficients ${{\alpha}_{\text{r}}}$ and ${{\beta}_{\text{r}}}$ of the respective LQ fit. C_norm, ${{\alpha}_{\text{G}}}$ and ${{\beta}_{\text{G}}}$ are determined as a function of the ${{\alpha}_{\text{r}}}$ and ${{\beta}_{\text{r}}}$ coefficients through the convolution of the cell survival for a given configuration of radiation impacts ${{\text{c}}_{K}}$ with a Gaussian law knowing that: (1) for an irradiation with low-LET photons, the distribution of energy depositions at a microscopic scale corresponding to that of cell dimensions can be modeled by a Gaussian law (Cunha et al 2016b); and (2) the cell survival to a given configuration ${{\text{c}}_{K}}$ can be determined from the specific energy ${{}^{{{\text{c}}_{K}}}}Z$ in the sensitive volume, i.e. ${{}^{{{\text{c}}_{K}}}}S=S\left({{}^{{{\text{c}}_{K}}}}Z\right)$. It is possible to prove that C_norm can be approximated to 1. In turn, the coefficients ${{\alpha}_{\text{G}}}$ and ${{\beta}_{\text{G}}}$ are to be determined for each cell line from the probability of cell survival to an irradiation with a reference radiation. ${{\alpha}_{\text{G}}}$ is set as 0 in this first version of the model, which allows for a separate adjustment of the local and the global events. Regarding ${{\beta}_{\text{G}}}$, it is possible to prove that:

$$\begin{eqnarray}&&{{\beta}_{\text{G}}}=\frac{{{\beta}_{\text{r}}}}{{{\eta}^{2}}}\end{eqnarray} \tag{ 16 }$$

where ${{\beta}_{\text{r}}}$ corresponds to the β coefficient issued from the LQ fit of cell survival to the reference radiation, and η is defined as:

$$\begin{eqnarray}&&\eta =\frac{\langle Z\rangle}{D}\end{eqnarray} \tag{ 17 }$$

with $\langle Z\rangle$ representing the mean specific energy in the sensitive volume of the cell associated with global events and D the macroscopic irradiation dose (for more details on η see section 2.2).

2.1.8. Average and total cell survival.

In order to compare experimental and theoretical results, the cell survival probability to a macroscopic dose D is determined over all radiation impact configurations that deposit such a dose:

$$\begin{eqnarray}&&S(D)={{\langle {{}^{{{\text{c}}_{K}}}}{{S}_{\text{L}}}\times {{}^{{{\text{c}}_{K}}}}{{S}_{\text{G}}}\rangle}_{{{\text{c}}_{K}}}}\end{eqnarray} \tag{ 18 }$$

The previous sections describing the model have introduced various parameters, however the estimation of the cell survival probability defined in equation (18) does not entail the adjustment and optimization of all of them. For instance, some can be fixed arbitrarily according to considerations made by the authors. These may be either for simplification purposes, it was the case of ${{\alpha}_{\text{G}}}$, or to comply with the definitions of the model, such as that of locality and local targets. The remaining are directly or indirectly set based on available experimental data. In particular, the parameters of the effective lethal function and ${{\beta}_{\text{G}}}$ are optimized through a fit to experimental data specific to a given cell line. On the other hand, the dimensions of the sensitive volume and T_RCE are expected to be less dependent on the cell line, which would allow to set approximated values in the absence of experimental data. Therefore, we can classify the model parameters in function of their dependence on the availability of experimental data specific to the cell line under consideration: not dependent and more or less dependent. Table 2 outlines the parameters of NanOx, the class(es) of events with which they are associated and how they depend on the cell line.

Table 2. Model parameters, the class(es) of events with which they are associated, their description, and their degree of dependence on cell line-specific experimental data (– not dependent;  +  less dependent;  ++ more dependent).

Class	Name	Description	Cell-line dependence
Local	${{V}_{\text{s},\text{L}}}$	Sensitive volume associated with local lethal events	+
	Vt	Local target volume	—
	F(...)	Parameters of the effective local lethal function	++
Global-subclass of non-local lethal events	${{V}_{\text{s},\text{G}}}$	Sensitive volume associated with global lethal events	+
	${{\alpha}_{\text{G}}}$	Coefficients of cell response to global events	—
	${{\beta}_{\text{G}}}$		++
	TRCE	Time after the radiation impact at which the production of reactive chemical species is considered	+

Besides assigning values to each parameter, the implementation of the model also requires making some more general choices that have implications in one or in both classes of events. It is the case of the reference radiation, which we set herein as low-LET photons, more specifically photons from a ⁶⁰Co source. Another example is the type and source of the data input to the model as the choice for x and X. More specifically, it was already defined that $X=\tilde{Z}$ in the current version of NanOx (see section 2.1.4). The chemical specific energy is determined from the specific energy inside the sensitive volume, which we obtained by using LQD. This Monte Carlo tool allows the user to store only the energy transferred to the medium that leads to events relevant for the biological effect of radiation (such as ionizations, atomic excitations and attachment of electrons) and discard the energy that simply causes the heating of the medium. The latter is produced by events such as molecular vibrations, the interaction between electrons and water phonons, the geminate recombination (recombination of the electron with the parent cation) or the attachment of electrons to water molecules (Cunha et al 2016b). The consequence of discarding part of the energy is that the specific energy calculated in the cell sensitive volume does not exactly correspond to the definition of this quantity according to the ICRU (ICRU 2011) since we are not accounting for the totality of the imparted energy. This is the reason for the factor η defined in equation (17), which is not a feature of NanOx but rather connected to the Monte Carlo code used for the calculations of specific energy. In the case of the LQD, η is approximately 80%. To be accurate, we would have to use another term to designate this quantity, for instance, restricted specific energy. However, for the sake of simplicity, we will continue to use just specific energy.

We chose V79 cells (lung fibroblasts from Chinese hamsters) for a first application of the model due to the large amount of data available in the literature. α values extracted from experimentally determined cell survival curves of V79 cells were gathered from the database made available by the PIDE project (Friedrich et al 2012b). Given the high level of variability that characterizes these data, values considered to be representative of the experimental data were chosen for photons and monoenergetic beams of different ions (protons, helium, carbon, neon, and argon ions) and used to optimize the function F. Figure 4 shows both the experimental data pool (open symbols) and the 16 selected data points (full symbols) relative to these radiation qualities.

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In a second stage, experimental data regarding cell survival to photons, protons and carbon ions for a wide range of LET values were collected from the literature and compared with model predictions. Having tested the ability of NanOx to predict the α coefficients in the previous steps, one of the goals of the calculation of cell survival was to evaluate the β coefficient and its dependence on LET. Considering the strong interplay between the α and β coefficients, the representative β coefficients were defined as the ones of the cell survival curves for which the reported α coefficients were close to the representative ones, thereby reducing the impact of the value of α on the evaluation of the performance of NanOx in describing β.

Table 3 lists the values assigned to each of the parameters for an application of NanOx to the V79 cell line.

Table 3. Model parameters, the class of events with which they are associated, the respective value for an application of the model to the V79 cell line, and their degree of dependence on cell line-specific experimental data (–not dependent;  +  less dependent;  ++ more dependent). The cell sensitive volumes associated with local and global lethal events were considered to be the same.

Class	Name	Value	Cell-line dependence
Local and global	Ls	1 μm	+
	ds	9.78 μm	+
Local	Lt	10 nm	—
	dt	20 nm	—
	$F(\ldots )\left\{\begin{array}{c} {{z}_{0}} \\ \sigma \\ h \end{array}\right.$	20 200 Gy	++
		6000 Gy	++
		180 000	++
Global(subclass of non-local lethal events)	${{\alpha}_{\text{G}}}$	0 Gy−1	—
	${{\beta}_{\text{G}}}$	0.0466 Gy−2	++
	TRCE	10−11 s	+

3.2.1. Sensitive volume.

3.2.2. Local targets.

3.2.3. Effective local lethal function.

F was built assuming it increases monotonically with the specific energy in local targets. The outcome of the construction procedure, for which the details are outside the scope of this paper, was close to an error function:

$$\begin{eqnarray}&&F(z)=\frac{h}{2}\left[1+\text{erf}\,\left(\frac{z-{{z}_{0}}}{\sigma}\right)\right].\end{eqnarray} \tag{ 19 }$$

Still another method was implemented, this time for the optimization of F. Again, the details are not described herein, but essentially it consisted in deriving coefficients related to local lethal events from the representative data in order to constrain F and optimize its parameters. A threshold value z₀ of 20 200 Gy, a σ of 6000 Gy and a function maximum h of 180 000 were obtained.

3.2.4. Global events parameters.

In particular, the reproduction of the evolution of α values shows the capability of the model to describe the effectiveness of radiation in function of the LET, including the overkill phenomenon of carbon ions. Regarding protons, one may also wonder about such an overkill effect and about whether NanOx is able to reproduce it. Among the literature that reports experimental α coefficients relative to V79 cells for protons, there is only one data set (Belli et al 1998) indicating a decrease in α values above 30 keV μm⁻¹. This, allied to the fact that the proton range is very short in this LET region, led us to disregard experimental proton data in this LET region when setting the representative data points. It could probably be possible to make the model describe these data, however whether α values actually decrease for protons is debatable. More and reliable data would be necessary to confirm this scenario. Overall, the most remarkable regarding these results is how NanOx can reproduce α over a large range of LET values (from less than 10 keV μm⁻¹ to 1000 keV μm⁻¹) and for ions as different as protons and carbon ions with a function describing local lethal events that relies only on three fitted parameters.

In terms of cell survival curves, the model reproduces the dependence of the shoulder on LET rather well and thus the β coefficient. This was possible due to the introduction of a chemical specific energy based on the cellular level of oxidative stress and represented by the ${{}^{\bullet}}$OH production, which allowed to describe how the role of non-local lethal events changes with LET. As pointed out before, β is less well described in some cases. This is likely due to the value of ${{\beta}_{\text{r}}}$ used as input, which may strongly vary from one set of experimental data to another (Friedrich et al 2012b). Therefore, the ${{\beta}_{\text{r}}}$ coefficient from a given data set may not be the most suitable for all the ions considered. The discrepancies in cell survival data may also be due to differences in the α values. In any case, considering the typical uncertainties on experimental α values and the variability on cell survival data, which may amount up to 50%, we can conclude that the predictions yielded by NanOx are in overall good agreement with the experimental survival curves. Therefore, we can state that, even if it is not possible to establish a straightforward connection between the LQ model coefficients and the parameters characterizing the new model, NanOx is able to predict such coefficients with enough accuracy for clinical purposes thereby demonstrating the potential to be integrated into TPSs in the context of particle therapy. This is supported by the comparison of the calculated α coefficients with NanOx to the experimental data, as shown in figure 5, but also to the predictions by the models that are currently used in clinical practice, MKM and LEM. This is what can be observed in figure 7 for carbon ions. NanOx compares better to MKM, which overall performs better than any of the three versions of LEM at reproducing the experimental data. These are very encouraging results and show the potential of this new framework.

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As already stated, the good results obtained with this first version suggest that the principles upon which NanOx is built are solid and coherent. It is therefore important to continue the development of the model and explore its possibilities. Of particular interest is the analysis of the values assigned to the model parameters and their impact on the outcome. Although such an assessment requires a systematic and thorough study, we can speculate about their sensitivity to cell-line-specific data.

In section 2.2, we have mentioned that some parameters are expected to be more dependent on experimental data relative to the cell line under consideration while others can be arbitrarily fixed. It was the case of ${{\alpha}_{\text{G}}}$ that was set to zero for simplification reasons. In the same way, the dimensions of the local targets were defined to comprise the extension of DSBs and the diffusion of chemical radical species, but as long as the conditions of locality are fulfilled, other dimensions could be used. For instance, the dimensions of larger, more complex structures DNA structures, such as nucleosomes or chromatin fibers would also be suitable.

The remaining parameters were directly or indirectly estimated from experimental data. ${{\beta}_{\text{G}}}$ was obtained from the ${{\beta}_{\text{r}}}$ coefficient of the cell survival to a reference radiation, which in turn was fitted to experimental measurements. As already mentioned, ${{\beta}_{\text{r}}}$ may significantly vary from one data set to another. Indeed, the level of variability in β coefficients is even greater than that in α values (Friedrich et al 2012b). Therefore, the estimation of ${{\beta}_{\text{G}}}$ greatly relies on the availability of experimental data and it is sensitive to the selected data set. The parameters of F, three for V79 cells, were optimized in the present study using 16 representative α values. Also in this case, experimental data are required to perform the fit and the value and even the number of parameters is expected to be susceptible to the specific set of input data used. Therefore, F and its parameters are intrinsically dependent on the experimental α values of the cell line under consideration.

Finally, T_RCE and the sensitive volume dimensions were set based on values reported in the literature. The time for observation of the reactive chemical species production after the radiation impact was set as 10⁻¹¹ s since it is assumed that at this point the primary chemical species are already produced. Then, chemical interaction and diffusion processes follow with the consequent modification of the chemical species distribution (and thus the chemical specific energy) due to their recombination. In a biological complex system, it is not clear when the chemical development of the track stops but dedicated simulation tools consider the range between 10⁻¹² and 10⁻⁶ s (Gervais et al 2005, Kreipl et al 2009). In his work, Gauduel states the importance of the knowledge about the early (up to 10⁻¹⁰ s) short-lived radical processes involved in radiation-induced cell damage (Gauduel 2011). This means that another value corresponding to such events could be set instead. Moreover, the uncertainties in the calculated production of chemical radical species due to the scarcity of experimental data to benchmark simulations should not be disregarded. The dimensions of the sensitive volume were chosen as those of the cell nucleus size reported in Scholz and Kraft (1996), but there is a plethora of values reported in the literature (e.g. Datta et al (1976), Wulf et al (1985) and Maeda et al (2008)) that would as well be valid. We can even argue that, if no value can be found in the literature pertaining to the dimensions of the sensitive volume of a given cell, line it is always possible to employ a value from a different cell line. Indeed, it is our conviction that the dimensions of cellular structures like the nucleus should not vary significantly from one cell line to another. Alternatively, we could consider an 'effective size', in which only the DNA content of the nucleus is taken into account (Scholz and Kraft 1996, Hawkins 2003). As a consequence, both T_RCE and the dimensions of the sensitive volume can be regarded as less dependent on the cell line and less sensitive to the assigned value as well. In any case, as stated before, only through systematic studies we will be able to infer about their impact on the outcome and even to clarify if they can be fixed regardless of the cell line.

Besides the aforementioned assessment of the impact of the values assigned to the model parameters on the outcome, other studies may also be envisaged. Still regarding the parameters, we could for instance introduce two distinct and separated cell sensitive regions to each class of events and make use of a different reactive chemical species in the determination of the chemical specific energy.

More generally, NanOx was initially developed as a flexible framework aimed at testing new ideas. It assumes that the probability of cell survival is the result of cell response to local and non-local lethal events resulting from an irradiation, each described by physical or chemical events at different spatial scales. The model makes no assumptions about the underlying mechanisms, e.g. repair mechanisms or accumulation of reactive chemical species. Introducing assumptions on these mechanisms could be envisaged in future developments, for instance, including the description of dose-rate effects. In addition, NanOx gives the possibility to evaluate other physicochemical quantities to input into the model. In particular, a quantity other than the specific energy could be chosen to represent the local quantity x. Examples are the already mentioned electronic excitations and single or multiple ionizations (section 2.1.3). Experimental data pertaining to cellular DNA damage could be even more interesting since it is closely connected to cell survival at therapeutic doses.

All the studies and possible developments aforementioned will not be limited to V79 cells, but rather to a broad set of cell lines, including human cells. More specifically, the next step will be to apply NanOx to human salivary gland (HSG) tumor cells. The goal is to test how NanOx performs with other cell lines and verify whether the conclusions drawn for V79 cells apply also to other types of cells.