Including oxygen enhancement ratio in ion beam treatment planning: model implementation and experimental verification

The first dependence to be mapped is obviously the one on oxygen concentration. For that, we decided to extend for every LET level the dependence at asymptotically low LET, i.e. in the case of photon radiation. For this case in the corresponding photon studies (Sovik et al 2007a, 2007b) the parameterization provided by Alper and Howard-Flanders (1956) (AHF) is universally used. They, on the basis of several experiments, describe the relative radiosensitivity RR at a given concentration of oxygen pO₂, i.e., the ratio of the sensitivities respectively in that condition, RS(pO₂), and in total absence of oxygen, RS(0), with the formula

$$\begin{equation} RR(p\mathrm{O}_2)=\frac{RS(p\mathrm{O}_2)}{RS(0)}=\frac{M p\mathrm{O}_2+ K}{K+p\mathrm{O}_2} \end{equation} \tag{ 2 }$$

where M is a maximum effect (generally found for x-rays, for different cell lines, to be M ∼ 3) and K the concentration corresponding to a half maximum sensitization, i.e. (M − 1)/2. Then, considering that RS(pO₂)∝1/D(pO₂), from (1) the OER may be expressed with a slight approximation in a similar way

$$\begin{equation} \mathrm{OER}(p\mathrm{O}_2)\approx M/RR(p\mathrm{O}_2)=\frac{Mb+p\mathrm{O}_2}{b+p\mathrm{O}_2} \end{equation} \tag{ 3 }$$

with b = K/M

It is worth here to point out that some authors, especially in recent literature (e.g. Stewart et al (2011)), prefer to adopt for this quantity the term 'Hypoxia reduction factor' (HRF). This gives emphasis to the fact that the varying quantity is the pressure (and thus the required isoeffective dose) of the hypoxic phase with respect to the fixed dose required in full oxygenation. On the contrary, the term OER is there left to its historical conception and thus to a ratio where the varying dose is in the normoxic phase, while the reference is taken as the dose in total anoxic (pO₂ = 0) conditions. To avoid confusion we stress that the quantity which we will deal in this paper will be always the one defined in (1).

In principle all OER experimental data are reported as a function of the survival level S or of the dose D. In fact, according to the linear quadratic model (LQ)

$$\begin{equation} -\ln S=\alpha D+\beta D^{2} \end{equation} \tag{ 4 }$$

with α, β general parameters, the OER dependence should be:

$$\begin{equation} \fl \mathrm{OER}(S)=\frac{D(p\mathrm{O}_2;S)}{D_{a}(S)}=\frac{\sqrt{\alpha ^2(p\mathrm{O}_2)-4\beta (p\mathrm{O}_2)\ln S}-\alpha (p\mathrm{O}_2)}{\sqrt{\alpha _{a}^{2}-4\beta _{a}\ln S}-\alpha _{a}}\cdot \frac{\beta _{a}}{\beta (p\mathrm{O}_2)}. \end{equation} \tag{ 5 }$$

This means that for a given pO₂ one cannot, in general, neglect the dependence in S. Alternatively, the same dependence may be expressed on the dose D. It is object of debate for several years if this S dependence should play a role or not, since it is known, for example, that the sensitization effect on β is different from that on α and even some studies report a remarkable dependence (Skarsgard and Harrison 1991), especially at low dose, for x-rays. A very recent paper also attempts to systematize these data and to estimate a small correction factor for neglecting this dose dependence in clinical cases (Wenzl and Wilkens 2011b). Nevertheless it was clear from most of the literature that if the effect should be accounted for, this will be not as relevant as for example it happens markedly for RBE, and that it even reduces drastically, until vanishing, for increasing LET values. Thus, in a first approximation, a dose modifying factor $\overline{\mathrm{OER}}$, at any survival level, can be assumed. E.g. in Carlson et al (2006) it was shown that even if an independent fit of the LQ parameters for an hypoxic and a normoxic curve will lead in general to a dose dependent OER value, it is possible to perform a different fit with the additional constraint of imposing such an average, constant dose modifying factor, with an almost negligible deviation in the resulting curves compared to the experimental uncertainties. The value of this factor is normally very close to the OER(10%). This allows us to safely neglect this effect especially for ion beams, at least for the initial input tables for the code; considering, still, the possibility to reintroduce it in refined tables, as mentioned at the start of this section, or to correct the outcome with the estimations provided in Wenzl and Wilkens (2011b).

As for the LET dependence, we decided to map explicitly its dose averaged value, where the dose averaged LET is defined as

$$\begin{equation} \overline{\mathrm{LET}}=\langle\mathrm{LET}\rangle_D=\frac{\sum _i \mathrm{LET}_i \, D_i}{\sum _i D_i}=\frac{\sum _i \mathrm{LET}_i^2 \, F_i}{\sum _i \mathrm{LET}_i \, F_i} \end{equation} \tag{ 6 }$$

with D_i dose and F_i fluence of the radiation component i. It was shown that dose averaged LET is a better descriptive quantity of the radiobiological effect of an ion producing a mixed radiation field (ICRP 2003), as compared e.g., to a track average, and in this case it is the more natural indicator of radiation quality.

Firstly, after the asymptotically low LET, the other asymptotic condition was considered, i.e. pO₂ = 0. This condition (pure anoxia) is not really interesting in clinical practice, since it is mostly related to already dying cells, while the hypoxic fraction at intermediate levels of oxygen is most dangerous. Nevertheless it is useful to describe the maximum effect, and it is also the condition where most experiments are available. For our purposes, then, we used the large Japanese dataset (Furusawa et al 2000), with several OER measurements for several ions and two exemplary tissues (V79 and HSG cells). There it was shown a very slight dependence on the ion type (more pronounced for the lighter ones). In fact, results for ions as different as carbon and neon lead to a difference which is sensible but small as compared to the experimental spread of the data. Moreover, the tissue dependence has also been shown to be relatively small, since the two, quite different, sampled cell lines present very similar OER profiles. This fact was found also in previous works, e.g. in Freyer et al (1991), in the systematic review of Wenzl and Wilkens (2011b) and also in recent measurements from ourselves (Tinganelli et al 2013). There, cell lines even with remarkable differences in sensitivities showed a quite similar OER. This allows us to take a table extracted from V79 measurements as a primer also for modeling different cell types. Certainly, in case of future availability of extensive data for a specific cell line, more accurate tables for the latter can be generated and imported in the program in the same way. Then the OER data at pO₂ = 0, for 10% survival in V79 cells, as a function of LET were fitted with a form similar to the AHF one but where, in order to reproduce the steeper fall-off toward higher LET, we introduced an additional parameter γ:

$$\begin{equation} \mathrm{OER}_0(\overline{\mathrm{LET}}) = \frac{M_0a+\overline{\mathrm{LET}}^{\gamma }}{\overline{\mathrm{LET}}^{\gamma }+a} \end{equation} \tag{ 7 }$$

with fitting parameters found to be a = 8.27 · 10⁵(keV μm⁻¹)^γ and γ = 3.0; and M₀ a maximum value at zero pressure set to 3.0.

A confirmation of the very small OER dependence on the dose and then of the validity of an approximation employing a dose modifying factor, arises from the analysis of these data and specifically comparing the fit obtained on the 10% survival data with the other data reported in that collection (Furusawa et al 2000): in figure 1 the survival dependence is extracted through inserting the reported LQ parameters in (5), and it is especially visible how, at increasing LET, the assumption of the dose modifying factor becomes even more safe, as also reported in Wenzl and Wilkens (2011b).

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Other intensive experiments of heavy ions with hypoxia were reported by the Berkeley group (Blakely et al 1979) and Rijswyk/London (Barendsen et al 1966) but since data reported are versus track averaged LET and not dose averaged, a comparison is not straightforward; and for the same reason in the Japanese report mentioned above no comparison is performed with these older data.

Finally the extension in the full two-dimensional plane ($p\mathrm{O}_2,\overline{\mathrm{LET}}$) was obtained combining the two asymptotic relations by assuming a similar pO₂ dependence at any $\overline{\mathrm{LET}}$, refining an approach we introduced in Scifoni et al (2011), and thus substituting to the M in (3) a maximum value dependent on $\overline{\mathrm{LET}}$, coming from (7):

$$\begin{equation} \mathrm{OER}(\overline{\mathrm{LET}},p\mathrm{O}_2) = \frac{b(a M_0+\overline{\mathrm{LET}}^\gamma )/(a+\overline{\mathrm{LET}}^\gamma )+p\mathrm{O}_{2}}{b+p\mathrm{O}_{2}} \end{equation} \tag{ 8 }$$

with M₀ = 3.0 and b = 0.25%.

The complete surface is shown in figure 2, with the parameter values introduced above. Independent measurements performed at GSI, with the techniques described in following section 4, confirmed the behavior of the fully anoxic curve, and additionally gave some insights of the partial hypoxic conditions with a set of measurements at pO₂ = 0.5%, for a single LET value, which resulted, after averaging on the three performed experiments, in very good agreement with the corresponding computed curve. These measurements were performed on CHO-K1 cells, a tissue type similar but not identical to the Japanese one, thus confirming the above mentioned very low tissue dependence. The complete original experimental data, partially reported in Tinganelli et al (2011) and Durante et al (2012), will be published separately (Tinganelli et al 2013), while the extracted OER values are summarized in figure 2 and superimposed to our independent parameterization. There, the numbers close to the points correspond to the number of experiments performed; and when only a single experiment was possible, mean and standard deviation are derived from the colonies counts. These data also confirmed the almost negligible dependence on ion type, for a discrete range of Z, once the dose average LET is considered, as also found in Furusawa et al (2000).

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The complete lack of experimental data (except the single one reported) at intermediate values of pO₂ and LET, impose a serious uncertainty to the present as well as to any other model, especially since these are the most relevant conditions for treatment planning. In fact, the typical $\overline{\mathrm{LET}}$ values in an ETI with carbon ion are between 70 and 130 keV μm⁻¹, exactly in the region where the slope of the OER surface is maximum and the most typical values for hypoxic tumors are for a pO₂ between 0.15 and 2%. We want to point out that the single measurement we reported at 100 keV and pO₂ = 0.5% is the first one in this region, which should be definitely explored more extensively.

Results accessible from this new TRiP98 extension, which we call TRiP-OER, are of a different kind. The quantity immediately retrievable from this model and from the $\overline{\rm {LET}}$ resulting at any point of an irradiated target is the $\overline{\rm {OER}}$ as a function of ion penetration depth. This analysis is shown for several pO₂ values and for two different primary ions on figure 5. There an extended target of 4 cm centered at a depth of 8.3 cm was planned in order to get an RBE-weighted dose of 6.5 Gy(RBE) in a normoxic target for carbon. The effect of stopping particles, dominating at the distal end of the target volume, in suppressing the radioresistance effect is clearly evident, especially for the more drastic (lower) pO₂ conditions. But it is also evident how this LET 'counter-effect' plays a role, especially for carbon ions, only in a relatively small region of the irradiated extended target volume.

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This allows one to draw some estimates of an expectable clinical advantage of the use of different ions as the ones available, e.g., in the Heidelberg HIT facility, where the oxygen beam was recently characterized (Kurz et al 2012). In fact, while one may appreciate the overall effect of a carbon ion, it is remarkable how an oxygen ion irradiation, with a prescribed dose chosen in order to maintain the same dose in the entrance channel, may drastically further reduce the effect in the intermediate pO₂ values corresponding to relevant clinical hypoxia.

For a more detailed comparative analysis see tables 1 and 2. In table 1 a similar situation as the one plotted in figure 5 is shown, for different extensions of the tumor target (and then of the region covered with the extended irradiation) along the penetration depth dimension z. The effect of the two ions, delivering the same dose to the entrance channel are explored at the proximal end and in the central point of the extended volume. From the relative reduction ($R_{O/C}=(\overline{\mathrm{OER}}_C-\overline{\mathrm{OER}}_O)/\overline{\mathrm{OER}}_C$) it is quite visible how the effect of the higher LET ion is especially advantageous in small tumors, highly hypoxic, reaching a value of 30%. In table 2 the same analysis is done for tumors of the same extension (4 cm along z) placed at different depth, i.e. where the isocenter is placed respectively at 6.3, 8.3 (as in figure 5) and 10.3 cm in water equivalent path length (WEPL) from the tissue entrance. In this case a clear trend as a function of depth seems not evident, and the improvement appears to be almost constant for more deep seated tumors as compared to the ones closer to the surface. A similar comparative analysis for protons and carbon ions, but just for a single irradiation configuration, is reported in Wenzl and Wilkens (2011a). In the case of oxygen and carbon this analysis is less sensitive to error in the OER versus LET parameterization, since as shown, the different ions behave very similarly in the experiments.

Table 1. $\overline{\mathrm{OER}}$ for carbon and oxygen and relative reduction (R_O/C) for a tumor of different sizes (extension along z), centered at a depth of 8.3 cm WEPL, computed at different positions within the extended volume. Data for 4 cm size correspond to figure 5.

	Tumor size (cm)	pO2	0%	0.15%	0.5%
Target center	2.0	$\overline{\mathrm{OER}}_C$	2.1	1.7	1.4
		$\overline{\mathrm{OER}}_O$	1.5	1.3	1.2
		RO/C(%)	29	23	15
	4.0	$\overline{\mathrm{OER}}_C$	2.5	1.9	1.5
		$\overline{\mathrm{OER}}_O$	1.9	1.5	1.3
		RO/C(%)	26	21	13
	6.0	$\overline{\mathrm{OER}}_C$	2.6	2.0	1.6
		$\overline{\mathrm{OER}}_O$	2.1	1.7	1.4
		RO/C(%)	21	17	12
Proximal end	2.0	$\overline{\mathrm{OER}}_C$	2.5	1.9	1.5
		$\overline{\mathrm{OER}}_O$	1.9	1.5	1.3
		RO/C(%)	25	22	17
	4.0	$\overline{\mathrm{OER}}_C$	2.7	2.1	1.6
		$\overline{\mathrm{OER}}_O$	2.3	1.8	1.4
		RO/C(%)	17	15	11
	6.0	$\overline{\mathrm{OER}}_C$	2.8	2.2	1.6
		$\overline{\mathrm{OER}}_O$	2.6	2.0	1.5
		RO/C(%)	10	9	6

Table 2. $\overline{\mathrm{OER}}$ for carbon and oxygen and relative reduction (R_O/C) for a 4 cm WEPL extended tumor centered at different depths, computed at different positions within the extended volume. Data for 8.3 cm depth correspond to figure 5.

	Tumor depth (cm)	pO2	0%	0.15%	0.5%
Target center	6.3	$\overline{\mathrm{OER}}_C$	2.5	2.0	1.5
		$\overline{\mathrm{OER}}_O$	1.8	1.5	1.3
		RO/C(%)	28	23	16
	8.3	$\overline{\mathrm{OER}}_C$	2.5	1.9	1.5
		$\overline{\mathrm{OER}}_O$	1.9	1.5	1.3
		RO/C(%)	26	21	13
	10.3	$\overline{\mathrm{OER}}_C$	2.6	2.0	1.5
		$\overline{\mathrm{OER}}_O$	1.8	1.5	1.3
		RO/C(%)	28	23	16
Proximal end	6.3	$\overline{\mathrm{OER}}_C$	2.8	2.1	1.6
		$\overline{\mathrm{OER}}_O$	2.2	1.8	1.4
		RO/C(%)	20	17	12
	8.3	$\overline{\mathrm{OER}}_C$	2.7	2.1	1.6
		$\overline{\mathrm{OER}}_O$	2.3	1.8	1.4
		RO/C(%)	17	15	11
	10.3	$\overline{\mathrm{OER}}_C$	2.8	2.1	1.6
		$\overline{\mathrm{OER}}_O$	2.2	1.8	1.4
		RO/C(%)	19	16	12

This kind of analysis may help to lay the basis for the conditions which drive the choice for a multi-ion treatment; a point which will be further developed in tight connection with the TRiP multi-ion development (Krämer et al 2012a, 2012b). The latter recent extension, in fact, allows to perform inverse planning with the use of different beam modalities and then to optimize at once, e.g., two beams of different ions, selecting accordingly the proper fluences for any beam at any raster point. The full application of this multiple ion feature to the present hypoxia problem goes beyond the scope of this paper, where we focus on carbon beams, but will be exploited in a forthcoming one.

Then, using the survival calculations implemented as explained above, it is possible to perform forward planning calculations yielding the corresponding survival level, assuming an homogeneous target at different pO₂ states (figure 6). One can now appreciate in terms of actual survival level along the irradiated extended volume, how the combined role of oxygen concentration and dose averaged LET drastically affect the biological outcome, as shown for an exemplary tissue of CHO-K1 cells (α_x = 0.22 Gy⁻¹, β_x = 0.02 Gy⁻²), very similar, as mentioned, to the ones used in the parameterization.

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A refined tissue model can also be considered, as shown in figure 7, where the oxygenation level is spatially inhomogeneous (as in the interesting clinical cases) and the code is shown to correctly deal with changes of pO₂ values from one voxel to the other.

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After exploiting the capability of TRiP-OER to predict the survival including the OER, in 3D along an ETI, attempts of dose compensation and inverse planning have been performed. In the first version, the biological effect accounting for OER is computed directly in each iteration step, while the mechanism of the iterative solution is not altered from the previous version (Gemmel et al 2008, Horcicka 2012, Horcicka et al 2012).

Results are plotted in figure 8. As is visible, in the case of a homogeneous hypoxic tissue (left panels), dose compensation is able to restore the required survival level. Also in the case of a composed tissue, even if with slight fluctuations, the dose compensation in the tissue is successful, but the price to pay is a larger damage in the entrance channel which eventually needs to be prevented, but cannot at this point. The effect of the correction is clearly visible in the resulting absorbed (or physical) dose, which, as evident in the composed tissue case, may consequently also have discontinuous shapes in correspondence to abrupt changes in tissue sensitivity. Another technical point that is arising from this data is that as long as a strong sensitivity variation is along the z axis, it is technically not problematic to adjust the dose delivery accordingly, but when it occurs within the same isoenergetic slice, an abrupt jump in fluence might be required to the ion scanning system that is not always feasible. This point might require the design of specific irradiation geometries in order to minimize the gradient in the x, y directions. A two-dimensional view of the survival profile in a similar, slightly more complex, model tissue, shows clearly the uniformity of the dose coverage (figure 9).

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The normal tissue can be much better spared adopting an additional field and performing a MFO, i.e. optimizing at the same time the contributions from both fields to the overall effective dose. This means that the vector $\vec{N}$ in (9) is containing as many components as all the raster spots of all the fields, and the minimization of the χ² function is performed on this full space at once. This method has been already shown to be convenient in RBE-weighted dose optimization, especially of complex target geometries (Gemmel et al 2008). In this specific case, where hypoxia is additionally included, considering the OER dependence on LET it is especially advantageous to use several fields, since the differently oxygenated regions can be irradiated using the best components of the two fields, as also suggested in Bassler et al (2010). In the example of figure 10, where two opposed fields are simultaneously optimized, it is evident how the overall survival has a more convenient profile, moreover, the contributions from different fields to the absorbed dose, show how the modified optimization algorithm selects the beams accordingly.

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In a first beamtime, a tumor tissue with three different concentrations of oxygen has been simulated. A 4 cm water equivalent length target centered at 8.1 cm from the beam entrance was planned, divided in a first region (from the proximal end) of 13 mm of normoxia, a second layer of 13 mm of hypoxia (pO₂ = 0.5%), and a last one of complete anoxia (pO₂ = 0%), 14 mm, a phantom corresponding to that of figure 8, right. Three chambers, each one containing three rings and entirely gased at the corresponding concentration, were then placed in a corresponding position after accounting for the path length conversion factors. The oxygenation condition has been checked to be maintained during the time of irradiation, after performed tests on the concentration changes after interruption of the gasing procedure. Additionally, two normoxic single chambers were placed in the entrance channel as a further control (figure 11). The plan was prepared in order to have a 10% survival in the target in a normoxic case, but was later recomputed according to the average photon parameters obtained from all the measurements performed in our group in the last three years on these cells on film support: α_x = 0.171 Gy⁻¹, β_x = 0.02 Gy⁻², D_t = 17.5 Gy (Tinganelli et al 2013). In order to assure uniformity in the dose coverage, the target was chosen as a broad parallelepiped, with Δx = 6 cm and Δy = 10 cm. A single beam was used, with energies optimized by the program to cover the requested range (E_min = 173.63 MeV/u, E_max = 224.81 MeV/u). For having a reasonable fast irradiation we imposed a set of energies corresponding to depth steps of 3 mm, corresponding to the size of the ripple filter employed. For the lateral extension the raster spots were imposed at steps of 2 mm in both directions (x, y) and for each of them beam size was 5.4 mm FWHM.

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After the irradiation, one of the biofilms sealing the ring is removed (the one opposed to the cell layer) and cells are trypsinized for 5 min at 37 °C and then counted and reseeded in tissue culture flasks according to the expected survival and the plating efficiency. After one week the colonies are counted and the survival level is computed. Details on the standard procedure for plating/counting are available in Tinganelli (2012).

The results (figure 12), where the error bars derived from the colonies count are also reported, show an overall qualitative agreement between measured survival and the TPS predictions and suggested refined insights for a second experiment. In fact, apart for an overall scaling factor, which may be induced to a collective increased radioresistance of all the cells, the most puzzling fact was the impossibility to detect any kind of LET effect at the distal end of the peak. This could be due to range uncertainties which displaced the cell layers to lower LET regions. The two sources of these uncertainties are tiny leaks of medium, which may occur and slightly reduce the effective depth, and deformation of the rings with accumulation of the medium toward the bottom due to gravity, inducing a non uniform exposure of different compartments of the target cell layers. Such uncertainties, in the present case, accumulate, especially in the last of the three triple chambers placed in a row along the beam.

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The following series of beamtimes was focused on the distal part of the irradiated extended target, which has been analyzed in total anoxic conditions; and, in order to check the consistency, an identical configuration with normoxic cells was irradiated too. No additional chambers were placed upstream the beam, in order to reduce the range uncertainties. Moreover, in order to minimize the time of vertical placing of the rings, and then the gravity-induced deformations mentioned above, the chambers were kept with the rings parallel to the ground until positioning for irradiation.

Three triple chambers were prepared in normoxic conditions and another three at pO₂ = 0. In addition, two unirradiated controls, for each oxygenation condition, were prepared and subjected to the same procedure as the others, except irradiation. In order to increase the number of sampling points within the target volume, the three triple chambers, for every oxygenation state were placed on a conveyor belt perpendicular to the beam, with respectively 0, 1 and 2 slices of material (poly-methyl-methacrylate) inducing a shift of 1.255 mm WEPL each (figure 13). In this way the last 8 mm of the irradiated extended target volume, where the LET effect is supposed to be larger, have been sampled uniformly and fairly densely. The beam features were the same of the previous experiment, a part for a slightly deeper centering of the target (8.3 cm), for convenience of the setup construction, and a reduction of the lateral extension of the field (Δx = Δy = 4 cm), to spare irradiation time.

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This experiment was repeated in its complete form two times. Two additional measurements were performed of the single anoxic curve. On one of these occasions, additionally, a survival curve at intermediate oxygen concentration (pO₂ = 0.5%) and a fixed LET value (100 MeV mm⁻¹), obtained from an irradiation on a small (1 cm) extended volume, was performed to test the introduced model for generating the OER tables in a crucial region, confirming the point plotted in figure 2.

The average of the results obtained in all the beamtimes (four anoxic and two normoxic measurements) are shown in figure 14, where they are plotted together with the bar of errors resulting from the standard error of all the corresponding independent experiments. In this case, as visible, the resulting data showed a much better agreement with the predictions, as compared to the pilot experiment. The agreement is anyhow not fully satisfactory, and progressively worse, at the last 4–5 mm of the extended target, and still the pure effect of the LET in suppressing the OER seems not to be visible. This is, however, understandable when looking to the corresponding deviation in the normoxic curve. In this region (the extreme distal part of an extended target volume) as already commented in Friedrich et al (2010, 2012), the uncertainty of the photon parameters (α_x, β_x) which are input in the advanced RBE modeling (LEM IV) used in TRiP98, will induce a larger uncertainty in RBE. This is, in particular, the case for in vitro cell lines which might underly temporal variations: significant variations, indeed, for anticorrelated LQ variables for multiple measurements are observed. And this is the case for the employed CHO cells, which we choosed for the convenient behaviour in the hypoxic chambers. This 'distal end' effect becomes even more pronounced if associated to slight range uncertainties, of the kind we mentioned above, which still may occur in a non-rigid setup as the one we used. In this case, a slight progressive overestimation of the RBE it is quite evident in this last region, returning a smoother curved profile, instead of the predicted steep gradient as requested in the optimization. In usual experiments, this deviation does not play a crucial role, but in this specific case where we are interested exactly in the very last 5 mm of the target, the overestimation of the RBE, and then an underestimation of the required physical dose, is enough to compensate the OER-suppressing effect of the high LET occurring at those depths and resulting in an overall almost flat survival profile. This deviation is quite typical for RBE tables produced with LEM IV, which slightly tends to overestimate the RBE at high LET, inversely of what happened, e.g., in the LEM I case. The impact of this effect on treatment planning, specifically on the distal ends of ETI, and especially in comparison with tables produced with LEM I was extensively analyzed and commented in a very recent publication (Grün et al 2012). There it was also shown that any deviation is strongly reduced by the application of two opposed fields, and then, in practical clinical situations. It is anyhow also worth mentioning that TRiP98 biological optimization is not bound to a specific LEM version. In fact, as recalled in the previous section, the RBE model provides the initial intrinsic RBE_α(E,Z) tables for any energy and particle type, which after importing are combined for each voxel by different internal models.

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Nevertheless, since the above effect is affecting in the same way, the normoxic and anoxic data, a confirmation of the above mentioned argument and a better evaluation of the goodness of the present OER prediction is visible in the following figure 15. There, the RBE dependence is disentangled from the OER one, by taking the ratio of the logarithm of the survival data, collected in the same occasion (only the two anoxic measurements performed in combination with normoxic controls were taken), and considering that at high LET, i.e. for depth z in the distal end, it is valid

$$\begin{equation} \overline{\mathrm{OER}}(z)=\frac{\alpha _{ox}(z)}{\alpha _{\rm anox}(z)}\approx \frac{\ln S_{ox}(z)}{\ln S_{\rm anox}(z)}. \end{equation} \tag{ 14 }$$

In this way also the impact of range uncertainties is substantially leveled, considering a similar effect in the anoxic and the normoxic setup. So it is possible to compare the resulting $\overline{\mathrm{OER}}$ along the irradiated volume, in that region, resulting from the code predictions and from the set of experiments over reported. In this case the agreement is more remarkable and the trend in reducing the $\overline{\mathrm{OER}}$ is evident also from the experimental data. The oscillations of the computed curves, here and in previous figures, are a consequence of the energy resolution employed in the active scanning (corresponding to steps of 3 mm in depth, as specified above in the beam features). The similar plot of figure 5, with smoother profile, was instead optimized employing more beam energies (corresponding to steps of 1 mm in depth).

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From the obtained results it is possible to conclude that the effect of LET in suppressing the $\overline{\mathrm{OER}}$ in an extended carbon ion irradiation is detectable, through biological dosimetry, even if the region where this effect plays a role is quite small and present a large uncertainty in RBE assessment. Moreover, the effect is in agreement with the predictions of the enhanced TRiP98 code presented here. This possibility would open the way to optimize the treatment even with the only use of carbon beams, e.g. simply concentrating that high LET component in the more hypoxic region either with the more sophisticated optimization approaches accessible through TRiP98. As mentioned above, anyhow, for a more remarkable and spatially extended effect, the combined use of different ions would be of larger efficiency, as indicated in table 1. In the latter case, also absolute survival measurements are expected to be less affected by RBE uncertainties, as also already verified in the case of multiple fields of carbon beams (Grün et al 2012).