Comments on “Formalization and Theoretical Analysis of the Local Effect Model” by Beuve (Radiat. Res. 172, 394–402, 2009)

With great interest we have read the article by Beuve (1) concerning the theoretical analysis of the Local Effect Model (LEM), which we developed and have continuously improved over the last decade (2–4). Unfortunately, we feel that most of the criticism of the model assumptions addressed in the article by Beuve is due to his serious misconception of the general ideas of the amorphous track structure approach and that it requires clarification. An essential point addressed in the paper concerns the assumption of a homogeneous distribution of dose in the case of photon radiation and the related assumption of a continuous radial dose distribution in the case of particle tracks. In both cases, the term ‘‘local dose’’ in our model refers to the expected value of the dose deposition on the nanometer scale rather than the actual dose distribution. In that sense, the local dose deposition at a given radius represents the average over a large number of individual tracks. Similarly, for photons, the local dose is interpreted as the expected value of the dose deposition for a large number of trials with a given distribution of impinging photons. Whereas this aspect is correctly addressed in the Introduction of the paper, in his own contribution in the section Local Dose Simulation, the author switches to the stochastic view of energy depositions and thus spuriously leaves the framework of the LEM. Although Beuve also uses the term local dose, he gives it a different meaning from the definition used in amorphous track structure models. We are aware that for individual tracks and individual secondary electrons the actual energy depositions obey a broad stochastic distribution. However, it is a key feature of the amorphous track structure models that they express the biological effects in terms of the expected value of the energy deposition instead of the actual values, because only in this framework is the reference to the photon dose–response curves feasible. Reference to the photon dose–response curve also implies averaging over a large population, assuming the same intrinsic sensitivity of all cells in a first approximation. Only for such a large population is the expected value of the dose uniquely related to the expected value of the response. The amorphous track structure approach therefore is not intended to predict the response of individual cells. For this case, not only would the granularity of the dose deposition as indicated in Fig. 3 of the article by Beuve have to be exploited but also the granularity of the target, i.e. the DNA, since it covers only a few percent of the nuclear volume. One essential approximation introduced in amorphous track models is that for the same expected value of the local dose the underlying stochastic energy distributions are similar for photons and ions, respectively (5). In that sense, Fig. 3 of the article of Beuve can be interpreted even more as being supportive of this general assumption of amorphous track models rather than contradictory. The analysis presented by Beuve highlights even more the difficulty of extracting useful biological information from stochastic distributions as shown in his Fig. 3. Here we would have been grateful to the author for presenting alternative approaches, which would allow making use of the photon dose–response curve. Furthermore, also in the analysis of the origin of the ß term in the dose–response curve, the apparent inconsistencies are due to the inappropriate mixing of the amorphous track and stochastic track approach. Figure 5 is thus highly misleading, because it tries to construct an inconsistency based on a single representation of tracks. However, local dose as an expected value can be interpreted only in a probabilistic way, and thus low local doses are just reflecting low probabilities for interaction. Interaction effects are implicitly taken into account in the framework of the LEM by making reference to the photon dose–response curve but without making any assumption about the mechanisms leading to these interaction effects or about their range. Since we derive the density of lethal events from the photon dose–response curve (for simplicity we neglect the threshold parameter Dt), we obtain