In response to the comments to the paper ‘Modelling of cell killing due to sparsely ionizing radiation in normoxic and hypoxic conditions and an extension to high LET radiation’ by T. Friedrich et al.

Sir: We appreciate the interest in our paper ‘Modelling of cell killing due to sparsely ionizing radiation in normoxic and hypoxic conditions and an extension to high LET radiation’. We were surprised to hear about the publication by Friedrich et al. (2012a) that actually presents a model which is for the description of sparsely ionizing radiation conceptually the same as the one we presented in Mairani et al. (2013a). The paper by Friedrich et al. (2012a) was published in Radiation Research at the same time of re-submitting our manuscript to International Journal of Radiation Biology. In fact, looking at the Radiation Research paper, it was published a month after and it was available online 10 days before our re-submission to International Journal of Radiation Biology, i.e., 10 months after our first draft was initially submitted to another journal in January 2012. We are happy that our results for sparsely ionizing radiations are in line with the work by Friedrich et al. (2012a) and we interpret this as an affirmation of the proposed modelling approach. We would like to emphasize that in addition to presenting and evaluating the basic modelling approach various additional results are presented in our paper. This includes the development of the model for the description of sparsely ionizing radiation at different oxygen levels. Moreover, we have performed ‘initial comparisons with experimental data’, to see if the approach holds for higher LET radiations, and we have studied the impact of the giant-loop variation for better reproducing the cell survival with sparsely ionizing radiation for two cell lines at high doses, where the linear-quadratic model typically fails.

Despite agreeing with our modelling strategy Friedrich et al. (2014) had difficulties to reproduce some of our results.

We do not understand why Friedrich et al. claim that our Figure 1 should intriguingly suggest an equivalence between the α and β values from Suzuki et al. (2000) and Combs et al. (2009) and the values derived from the new model. Differences in the LQ parameters which Friedrich et al. noted between the linear quadratic (LQ) parameters published in the original experimental papers (Suzuki et al. 2000, Combs et al. 2009) and the K_iDSB and K_cDSB values we reported in Table I and Figure 5 of Mairani et al. (2013a) can be mostly explained by the fact that we factored in the non-uniform uncertainties of the experimental data points in our fits. Details of the fitting procedure were not discussed in our original publication as they are, as we will show, not of principal concern for demonstrating the validity of the proposed model, and particularly in view of the initial modelling stage and rough accuracy which we claimed in our publication for high LET radiation. However, under the given circumstances we will make up for this omission and we will discuss in the following differences obtained due to differing fitting procedures and reason for a certain fitting approach.

Figure 1. Comparison of experimental survival data (Combs et al. 2009) for X-rays (points with uncertainty bars), for ¹²C ions at 103 keV/μm (empty circles with error bars) and for ¹²C ions at 170 keV/μm (squares with error bars) with model calculations (lines) for the LN229 cell line.

As we have reported in Mairani et al. (2013a), we chose to perform our own fit to the experimental data for both the LQ model and our approach: ‘when fitting with the two models the data from Suzuki et al. (2000)’. Firstly, this choice was motivated by the fact that by fitting both models using a unique approach, we could assure that we are focusing on differences in our model and the LQ model, rather than differences in different fitting approaches, as exemplified by the results of Friedrich et al. (2014). Secondly, Suzuki et al. (2000) and Combs et al. (2009) reported only that the data are fitted by the least squares method to a linear-quadratic equation without giving further details. We believe that if we want to properly estimate the biological effect of high LET radiation in future one should take into account the often substantial uncertainties in the experimental data points when fitting sparsely ionizing radiation. It should be also kept in mind that substantial uncertainties exist for single experimental cell survival points (up to several tens of percents) and propagate to the model parameters obtained by the fits. For the concrete case fitting experimental data while considering the uncertainties of the individual data points is complicated by the fact that the experimental uncertainties are not reported for all data points. Hence, it is not obvious how to weight these points when fitting. Consequently, there are certainly different ways to perform fits and we started with an educated guess. By using a fitting approach which reproduces the LQ parameters obtained by Suzuki et al. (2000) and Combs et al. (2009) when fitting their data with the LQ model, we obtain when fitting with our model and computing the derived LQ parameters (see Equation 7 of Mairani et al. [2013a]), as expected, ratios for α (R_α) and β (R_β) very close to one (mean ± standard deviation): R_α = 1.00 ± 0.03 and R_β = 1.00 ± 0.00.

We re-analyzed the results of our fitting procedure applying three changes. In our original paper, there was a typo in Table I for the K_iDSB parameter and its error for the LC-1-sq cell line. The correct value is 14E-3 ± 1E-3. Secondly, due to an accidental omission we have not included a data point at 5 Gy and 6 Gy in the analysis for the U87MG and LN229 cell lines, respectively. While the point at 5 Gy for the U87MG cell line does not have a large impact on the fitting result, the point at 6 Gy for the LN229 cell line influences the outcomes of the fit and for this reason we do not include it in the analysis here and we present a new fit of the LN229 cell line in Figure 1. We thank Friedrich et al. for pointing out this mistake. Moreover, we have not included the U-251MG(KO) cell line in the analysis due to the different fitting constraints used in Mairani et al. (2013a) (positive K_iDSB and K_cDSB as well as positive α and β values) and in Suzuki et al. (2000) (eventually negative α value). When comparing the corresponding χ² for our model using the K_iDSB and K_cDSB parameters obtained from the fit as reported in Table I of Mairani et al. (2013a) and the K_iDSB and K_cDSB obtained, as suggested from Friedrich et al. (2014), starting from the α and β values of Suzuki et al. (2000) and Combs et al. (2009) we obtain R_α = 1.00 ± 0.29, R_β = 1.11 ± 0.28 and Rχ² = 0.83 ± 0.65. This underlines that the fits we presented taking into account data uncertainties have on average a similar quality compared to the α and β values published by Suzuki et al. (2000) and Combs et al. (2009).

Hence, we believe that rather than having ‘substantial inconsistencies’ in our approach we could show that we merely employed a different fitting approach compared to the one used in the original experimental publications which factors in relative uncertainties of individual data points. Differences are however not of relevance for demonstrating the performances of the novel model in comparison with the LQ model since equivalent conclusions can be drawn, as shown, when employing a fitting approach which uses an equal weighting of the data points.

Friedrich et al. commented also on the supposed inconsistencies between the results shown in Table I and in Figure 9 and Figure 10 of our publication. We cannot reproduce their problems. Revising and recalculating the results using the K_iDSB and K_cDSB of Table I and using N_iDSB and N_cDSB values shown in Figure 3 of our article we obtain the same findings as reported in the published figures, apart from the typo in Table I of Mairani et al. (2013a) as reported already in the previous section. The likely explanation is that the differences are due to the different modeling of the DSB distribution with high LET radiation. Friedrich et al. claim that ‘a more detailed description of the modeling procedures would be required’. However, their original implementations of the Local Effect Model (LEM) are not openly available. As a consequence several groups have followed the publications by the GSI Helmholtzzentrum für Schwerionenforschung GmbH in order to implement their own version of the LEM in its different versions. However, as usually the case, the actual implementation of a model may be slightly different from group to group (e.g., Russo 2011, Wiklund 2012). This is specifically the case for the computation of the DSB distribution in the LEM-IV approach, since not all details of the original model implementation are given in Elsässer et al. (2010) and Friedrich et al. (2012b). Hence, we wonder that Friedrich et al. (2014) assume for their comparisons with our results that our implementation for determining the DSB distribution and all inherent assumptions to be exactly the same. This is certainly not the case. For example, we have found that a SSB-to-SSB distance forming a DSB conversion of 15 bp produces good results for us, compared to the larger parameter of 25 bp used by their implementation within the LEM-IV approach. The values we obtained for the enhancement factor for the DSB production are in line with the results of Russo (2011) and references therein. In addition also our approach for treating the voxelization of the nucleus in giant loops might slightly differ bringing 5–10% variation for N_iDSB and N_cDSB for the studied LET values. Although the presentation of an extensive benchmark of the model for high LET radiation was beyond the purpose of the work, we want to stress here that we performed several checks of our implementation of the LEM-IV model with the same experimental data also used in Elsässer et al. (2010) and obtained comparable results with similar agreement, but applying different dose threshold values for the description of the linear-quadratic-linear response to sparsely ionizing radiation (Mairani et al. 2013b). Hence, as we have explained in our original paper, we have followed the approach described in Elsässer et al. (2010) and Friedrich et al. (2012b) but the actual implementation certainly varies slightly.

In conclusion, we thank Friedrich et al. for their careful reading and re-analysis of our paper. However, we cannot agree on the conclusions contained in their comment. We did not find any inconsistencies in the results shown in Mairani et al. (2013a) except for a typo in Table I for the K_iDSB parameter and its uncertainty for the LC-1-sq cell line and for the accidental omission of the experimental data points at 5 and 6 Gy for U87-MG and LN229 cell lines, respectively. We believe that our original conclusions that the ‘high LET extension yields a reasonable agreement with the data in aerobic condition’, having in mind that the ‘capability of the model for reproducing cell survival data of ¹²C ions should be further improved with regard to its potential use for hadron therapy’ of our original publication (Mairani et al. 2013a) remain valid. All the differences shown in Friedrich et al. (2014) could be due to different fitting procedures and model implementations for high LET radiation. Finally, we agree with Friedrich et al., that a ‘thorough check of the implementation’, and we would like to add here: also of the modelling and fitting approach and its approximations, should be done prior to its application in hadron therapy. However, we feel that it is a common practice to propose a model and to perform in the next steps additional benchmarks (e.g., including other ions, LET values and cell lines), improvements and extensions – thereby obtaining a more thorough description of biological effects. As a matter of fact, a similar path has been taken over the last two decades for the LEM resulting in various updated and refined versions of the model.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.