A phenomenological relative biological effectiveness approach for proton therapy based on an improved description of the mixed radiation field

Wedenberg et al (2013) have introduced a phenomenological model which is able to predict proton RBE based on LET, dose and tissue specific parameter ${{\left(\alpha /\beta \right)}_{\text{ph}}}$ for the reference radiation. The experimental data analyzed in Wedenberg et al (2013) included cells irradiated with near monoenergetic proton beams with well defined LET values up to around 30 keV μm⁻¹. In this work, we extended the available set of data by including tabulated values characterized by LET  >  30 keV μm⁻¹. These additional data are listed in table 1. The final database consists of a total of 31 data (24 from Wedenberg et al (2013) plus seven additional ones; see table 1) and extends the LET interval up to 37.8 keV μm⁻¹. Biological data, from proton beams with beam energies $\gtrsim$5 MeV as well as from deuteron beams at all energies, have not been included in the database to prevent a possible double counting of the biological effect of the He secondaries produced in the nuclear interactions. In fact, in the case of light target nuclei composing the cell nuclei, such as ¹²C and ¹⁶O, the lowest threshold energies¹¹ for the production of ⁴He secondaries are, respectively, about 7.9 MeV and 5.6 MeV in the case of proton irradiations and about 1.6 MeV and 0 MeV for deuteron beams.

Table 1. The list of cell lines (in the order of increasing ${{\left(\alpha /\beta \right)}_{\text{ph}}}$ of the reference radiation) added in this study to the published (Wedenberg et al 2013) database, with the calculated $\text{LE}{{\text{T}}_{\text{ph}}}$ parameters. Uncertainties are expressed as $1\sigma$ standard deviations.

Cell line	${{\left(\boldsymbol{\alpha }/\boldsymbol{\beta }\right)}_{\text{ph}}}$ [Gy]	Reference radiation	$\text{LE}{{\text{T}}_{\text{ph}}}$ [keV/$\boldsymbol{\mu }$m]	Reference
V79-753B	2.804  ±  0.319	200 kVp	1.164	Belli et al (1998)
C3H10T1/2	15.000  ±  12.751	60Co γ-rays	0.400	Bettega et al (1998)
M/10	$\infty$	60Co γ-rays	0.400	Belli et al (2000)

As in Mairani et al (2016a) and Mairani et al (2016b), we used the following formalism:

$$\begin{eqnarray}&&\text{RB}{{\text{E}}_{\alpha}}\equiv \frac{{{\alpha}_{\text{H}}}}{{{\alpha}_{\text{ph}}}}=1+\left[{{k}_{0}}+{{\left(\frac{\beta}{\alpha}\right)}_{\text{ph}}}\right]\centerdot f\left({{L}^{\ast}}\right)\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}&&{{R}_{\beta}}\equiv \frac{{{\beta}_{\text{H}}}}{{{\beta}_{\text{ph}}}}={{b}_{0}}\exp \left[-{{\left(\frac{L-{{b}_{1}}}{{{b}_{2}}}\right)}^{2}}\right]\quad \quad \text{or}\quad \quad {{R}_{\beta}}=1.\end{eqnarray} \tag{ 2 }$$

${{\alpha}_{\text{ph}}}$, ${{\alpha}_{\text{H}}}$ and ${{\beta}_{\text{ph}}}$, ${{\beta}_{\text{H}}}$ are the linear and quadratic terms for photons and for Z  =  1 particles. In accordance with the findings of Folkard (1996) for protons and deuterons, in this work protons, deuterons and tritons are assumed to have the same RBE–LET relationship.

As concerns $\text{RB}{{\text{E}}_{\alpha}}$, L^* represents the rescaled proton LET in water as suggested by Paganetti (2014) and f is the actual (rescaled) LET-dependent core of the parameterization. By following the same strategy as in Mairani et al (2016a), the ten analytical formulas for f(L^*) listed in table 2 have been studied, ranging from a simple linear expression as in Wedenberg et al (2013) to more complex ones. Following the goal of possibly refining the existing and validated (Wedenberg et al 2013) model for the $\text{RB}{{\text{E}}_{\alpha}}$ ratio, we forced the parameter k₀ to a null value in equation (1). This way, we reproduced the same (Wedenberg et al 2013) analytical parameterization when adopting the linear function $f\left({{L}^{\ast}}\right)={{k}_{1}}{{L}^{\ast}}$, except for the independent variable, here assumed to be the rescaled proton LET. For consistency, all the remaining nine models were also tested with the assumption ${{k}_{0}}\equiv 0$ on a wider LET range with respect to Wedenberg et al (2013). For the studied models, the expression of $\text{RB}{{\text{E}}_{\alpha}}$ (equation (1)) was fitted to the experimentally obtained ${{\alpha}_{\text{H}}}/{{\alpha}_{\text{ph}}}$ ratios with the ${{\chi}^{2}}$ minimization method for all the cell lines and LET values of the collected database. In contrast to the resampling approach used for He ions by Mairani et al (2016a), in this work all the ${{\alpha}_{\text{H}}}$ data were available with their experimental uncertainties, allowing for a unique definition of the minimization function (${{\chi}^{2}}$). Fit results were compared using the reduced ${{\chi}^{2}}$ values, i.e. ${{\chi}^{2}}/\nu$; $\nu =$ number of degrees of freedom.

Table 2. List of $\text{RB}{{\text{E}}_{\alpha}}$ parameterizations under investigation ($x\equiv {{L}^{\ast}}$).

	Equation
Model	$\text{RB}{{\text{E}}_{\alpha}}\left(x|{{\left(\alpha /\beta \right)}_{\text{ph}}}\right)=1+\left[{{k}_{0}}+{{\left(\frac{\beta}{\alpha}\right)}_{\text{ph}}}\right]\centerdot f(x)$
${{f}_{\text{L}}}$	$f(x)={{k}_{1}}{{x}^{{}}}$
${{f}_{\text{Q}}}$	$f(x)={{k}_{1}}{{x}^{2}}$
${{f}_{\text{LQ}}}$	$f(x)={{k}_{1}}x+{{k}_{2}}{{x}^{2}}$
${{f}_{\text{LQC}}}$	$f(x)={{k}_{1}}x+{{k}_{2}}{{x}^{2}}+{{k}_{3}}{{x}^{3}}$
${{f}_{\text{LE}}}$	$f(x)=\left({{k}_{1}}x\right)\exp \left(-{{k}_{2}}x\right)$
${{f}_{\text{QE}}}$	$f(x)=\left({{k}_{1}}{{x}^{2}}\right)\exp \left(-{{k}_{2}}x\right)$
${{f}_{\text{LQE}}}$	$f(x)=\left({{k}_{1}}x+{{k}_{2}}{{x}^{2}}\right)\exp \left(-{{k}_{3}}x\right)$
${{f}_{\text{LE}2}}$	$f(x)=\left({{k}_{1}}x\right)\exp \left(-{{k}_{2}}{{x}^{2}}\right)$
${{f}_{\text{QE}2}}$	$f(x)=\left({{k}_{1}}{{x}^{2}}\right)\exp \left(-{{k}_{2}}{{x}^{2}}\right)$
${{f}_{\text{LQE}2}}$	$f(x)=\left({{k}_{1}}x+{{k}_{2}}{{x}^{2}}\right)\exp \left(-{{k}_{3}}{{x}^{2}}\right)$

As for ${{R}_{\beta}}$, both parameterizations in equation (2) neglect the ${{\left(\alpha /\beta \right)}_{\text{ph}}}$ dependence (Friedrich et al 2013), with the second one additionally disregarding the LET dependence. The LET-dependent model has been fitted on a weighted running average of experimental ${{R}_{\beta}}$ ratios as already applied in Mairani et al (2016a) for helium ion beams.

Human adenocarcinoma cells (A549, ATCC) were grown in Dulbecco's modified Eagle medium (DMEM, ATCC) containing 10% heat-inactivated fetal bovine serum (FBS, Millipore), 2 mM glutamine and 1% penicillin–streptomycin (Gibco). Murine renal adenocarcinoma cells (RENCA, ATCC) were cultured in RPMI-1640 medium (Gibco) supplemented with 10% FBS (Gibco), non-essential amino acids (0.1 mM, Sigma), sodium pyruvate (1 mM, Sigma) and L-glutamine (2 mM, Sigma). Cells were cultured at 37 °C in 5$\%$ CO₂ atmosphere. Clonogenic assay was performed as previously described by Mairani et al (2016a). Cells were grown in 25 cm². For the reference photon irradiations, RENCA cells were irradiated using a linear accelerator (LINAC, 6 MV, Artist Siemens) at the German Cancer Research Center (DKFZ), while the irradiation of A549 cells was performed with a 320 kV x-ray irradiator (Precision X-Ray, XRAD 320). Six dose levels have been used from 1 to 10 Gy. The calculated LET_ph values were 0.204 keV μm⁻¹ and 0.981 keV μm⁻¹ for the 6 MV and 320 kV machines, respectively. The clonogenic photon survival data have been fitted with the LQ model and the best fit parameters (mean  ±  standard deviation) are reported in table 3.

Table 3. Photon parameters applied in this work.

${{\alpha}_{\text{ph}}}$ (Gy−1)	${{\beta}_{\text{ph}}}$ (Gy−2)	${{\left(\alpha /\beta \right)}_{\text{ph}}}$ (Gy)	Simulation	References
0.053  ±  0.045	0.028  ±  0.005	1.89	RENCA experiment
0.134  ±  0.025	0.021  ±  0.003	6.38	A549 experiment
0.1	0.05	2.0	SOBP9, SOBP21
0.1	0.01	10.0	SOBP9, SOBP21
0.036	0.024	1.5	Prostate case	Brenner and Hall (1999)
0.0890	0.0287	3.1	Prostate case	Terry and Denekamp (1984)
0.077	0.009	8.6	Head case	Jones and Sanghera (2007)
0.0499	0.0238	2.1	Head case	Meeks et al (2000)

Two different proton irradiation set-ups, described in section 2.3.1, have been used at the Heidelberg Ion Beam Therapy Center (HIT, (Haberer et al 2004)) for the experimental biological investigation presented in this work, studying both the dose and depth dependence in clinically relevant scenarios. Two flasks per position, per dose and per irradiation have been used. The water equivalent thickness of the flasks has been measured and taken into account in the determination of the measurement position. However, more distal positions have been avoided to reduce uncertainties from unavoidable positioning errors and possible reproducibility issues of the flask material/thickness. The clonogenic results are presented as mean  ±  standard deviation which has been used for plotting the experimental data with their error bars. Dosimetric measurements have been additionally performed using a Farmer ionization chamber (TM30010, PTW, Freiburg) in order to verify the dose values calculated by FLUKA.

For the results presented in the next section a research TPS (Krämer et al 2000) applying the HIT physical database (Parodi et al 2012) has been used, while for the results outlined in sections 2.3.2 and 2.3.3, the commercial CE (European Conformity)-labeled TPS 'syngo RT Planning' by Siemens was used to optimize the studied SOBP and patient-like cases, applying the clinical settings of the Italian National Center for Oncological Hadron Therapy (CNAO, (Rossi 2015)) and of HIT, respectively. A constant RBE value of 1.1 is assumed for protons in the commercial TPS.

Benchmarks of FLUKA predictions for proton beams have been carried out at HIT and at CNAO by comparisons with measured depth–dose distributions in water, lateral dose profiles in water and in air and patient plan verifications (Parodi et al 2012, 2013, Molinelli et al 2013, Bauer et al 2014, Tessonnier et al 2014). To perform biological calculations, we used FLUKA-based frameworks as available at HIT and at CNAO (Bauer et al 2014, Tessonnier et al 2014, Giovannini et al 2016, Mairani et al 2016b).

We simulated dose-weighted averages of the linear and quadratic terms of the mixed radiation field, ${{\bar{\alpha}}_{D}}$ and ${{\bar{\beta}}_{D}}$, calculated taking into account the biological effect of Z  =  1 particles, as described by equations (1) and (2), and of Z  =  2 particles (Mairani et al 2016b). Using the dose-weighted averages, the dose and the formalism described in Mairani et al (2010), we calculated RBE and ${{D}_{\text{RBE}}}$ values. Simulations performed in this way will be referred to as full calculations in the text. LET_D distributions have been additionally obtained by weighting the contribution of only Z  =  1 particles in order to use the LET_D values for an off-line determination of RBE and ${{D}_{\text{RBE}}}$ maps (Polster et al 2015).

2.3.1. Calculations for the experimental investigation.

2.3.2. Biological characterization of SOBP in clinically relevant scenarios.

2.3.3. Biological characterization of proton patient plans.

For the description of $\text{RB}{{\text{E}}_{\alpha}}$ as a function of the normalized LET (L^*), we have found that the minimum value of ${{\chi}^{2}}/\nu$ is given, among the parameterizations listed in table 2, by the linear one ${{f}_{\text{L}}}$. We have noticed an important variation (about 13%) of the slope parameter of ${{f}_{\text{L}}}$ as a function of the maximum LET (${{L}_{\max}}$) values considered during fitting. To achieve a more robust determination of the slope, we varied ${{L}_{\max}}$ from 30 keV μm⁻¹ to the maximum available in the experimental data, 37.8 keV μm⁻¹, and averaged the fitting results, obtaining k₁  =  0.377 Gy (keV μm⁻¹)⁻¹ The best fit parameterization, ${{f}_{\text{L}}}$, calculated with k₁  =  0.377 Gy (keV μm⁻¹)⁻¹ (solid line) and the minimum and maximum k₁ values (dashed lines), is plotted in the left-hand panel of figure 1, where plots of $\left[\text{RB}{{\text{E}}_{\alpha}}-1\right]\left(\beta /\alpha \right)_{\text{ph}}^{-1}$ (to remove the explicit dependence on the cell line) are shown as a function of L^* for ${{\beta}_{\text{ph}}}>0$. In terms of $\text{RB}{{\text{E}}_{\alpha}}$, both models introduced by Wedenberg et al (2013) and McNamara et al (2015) employed a linear $\text{RB}{{\text{E}}_{\alpha}}$–LET dependence, with recommended values of k₁  =  0.434 Gy (keV μm⁻¹)⁻¹ and k₁  =  0.356 Gy (keV μm⁻¹)⁻¹, respectively. The difference found is mainly due to the different experimental database employed and the fact that in our approach and that of McNamara et al (2015) the rescaled LET has been used while Wedenberg et al (2013) applied the pure LET concept.

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${{R}_{\beta}}$ values are shown as a function of LET in the right-hand panel of figure 1. The Gaussian fit (equation (2)), performed on the weighted running average values of ${{R}_{\beta}}$, gave the following best fit parameters: b₁  =  3.28 keV μm⁻¹ and b₂  =  27.90 keV μm⁻¹, while b₀  =  1.00 has been fixed to force the fit to converge to ${{R}_{\beta}}=1$ for low LET values where the experimental data are not available.

Comparing ${{R}_{\beta}}$ descriptions, Wedenberg et al (2013) assumed ${{R}_{\beta}}=1$, while McNamara et al (2015) applied a linear dependence of $\sqrt{{{R}_{\beta}}}$ on the rescaled LET_D and on $\sqrt{{{\left(\alpha /\beta \right)}_{\text{ph}}}}$. In our analysis, no statistically significant difference was found between a LET-dependent and a constant ${{\beta}_{\text{H}}}$ description (equation (2)) in reproducing $\text{RB}{{\text{E}}_{2~\text{Gy}-\text{ph}}}$. The Pearson correlation coefficients applying ${{f}_{\text{L}}}$ with k₁  =  0.377 Gy (keV μm⁻¹)⁻¹ in combination with ${{R}_{\beta}}=1$ or ${{R}_{\beta}}(L)$ for $\text{RB}{{\text{E}}_{2~\text{Gy}-\text{ph}}}$ vary by only 0.8%.

For the MC-based calculations presented in the next sections, for Z  =  1 particles, we applied the best fit parameterization ${{f}_{\text{L}}}$ for the linear term with k₁  =  0.377 Gy (keV μm⁻¹)⁻¹ and either ${{R}_{\beta}}(L)$ or ${{R}_{\beta}}=1$ for the quadratic term.

The first experimental configuration studied in this work is shown in the upper panel of figure 2, where FLUKA-calculated depth–dose and depth–LET_D distributions are displayed. The positions in the EC and in the middle of the SOBP, where the RENCA cell survival as a function of the proton dose has been measured, are marked with a and b, respectively. The measured reference photon clonogenic data as a function of the photon dose are shown in the lower-left panel of figure 2 together with the performed LQ fit. The resulting best-fit LQ parameters are reported in table 3. In the other two lower panels marked with (a) and (b), clonogenic data as a function of proton dose in EC and in the middle of the SOBP are compared against model predictions and the results obtained assuming an RBE of 1.1. Qualitatively comparing model outcomes and the experimental data, all the calculations presented seemed to reproduce, within the error bars, the clonogenic data. Keeping in mind the experimental uncertainties, both the full calculations and those considering biologically only Z  =  1 particles give similar levels of agreement. Assuming for both H and He particles $\beta ={{\beta}_{\text{ph}}}$ instead of a LET-dependent β term seemed to slightly improve the agreement for the high dose region (>3 Gy) for the middle of the SOBP configuration. The calculations performed assuming a constant RBE of 1.1 appeared to overestimate the survival for the middle SOBP set-up, where an experimental RBE value at 2 Gy photon dose of about 1.3 has been found.

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The second experimental set-up studied in this work is shown in the upper panel of figure 3, where FLUKA-calculated depth–dose, depth–RBE and depth–LET_D distributions are displayed together with the acquired dosimetric data. The mean and maximum absolute dose variations between MC-simulated and experimental dose values were, respectively, 1.2% and 2.6%. The absolute dose variation is calculated for each measurement point as the unsigned difference between the measured and MC doses normalized by the measured one and expressed as a percentage. In proton therapy, an accurate prediction of the dose values is important for a sound estimation of the RBE variations, which are expected to be smaller than those with higher LET beams such as helium or carbon ions (Elsässer et al 2010, Mairani et al 2016b). Looking at the survival data, a minimum in the survival has been found, as expected, in the distal part of the SOBP, mainly due to an increase of the LET of the primary beam. For the studied scenario, using a constant RBE of 1.1 seems a good approximation except in the last few millimeters, where a rise in the RBE is expected (Chaudhary et al 2014, Guan et al 2015a, Marshall et al 2016). The mean absolute survival variation between model predictions and experimental data (${{\mu}_{ \Delta S}}$) is reported in table 4 for the four different approaches used. The calculation framework developed in this work, i.e. weighting the mixed radiation field with data-driven biological models, resulted in ${{\mu}_{ \Delta S}}=2.7 \% \pm 0.5 \%$, while the clinically assumed RBE of 1.1 gives ${{\mu}_{ \Delta S}}=4.0 \% \pm 1.2 \%$. Weighting only the Z  =  1 particles slightly worsened the agreement compared to the full calculations: ${{\mu}_{ \Delta S}}=3.0 \% \pm 0.6 \%$. The best agreement has been found assuming $\beta ={{\beta}_{\text{ph}}}$: ${{\mu}_{ \Delta S}}=2.3 \% \pm 0.4 \%$. The improvement is, however, not statistically significant. Taking into account the experimental (dosimetric and biological) uncertainties, the introduced calculation scheme seems a sound candidate for predicting RBE in proton therapy. However, more experimental data in clinically relevant scenarios are needed to drive conclusions on the performance of the presented RBE phenomenological model.

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Table 4. Summary of the mean survival absolute deviation (${{\mu}_{ \Delta S}}$) between model predictions and the experimental data shown in figure 3.

RBE model	$\boldsymbol{\mu }_{{ \Delta S}}$ (%)
RBE  =  1.1	4.0  ±  1.2
Z  =  1 & Z  =  2  +  $\beta (L)$	2.7  ±  0.5
Z  =  1 & Z  =  2  +  $\beta ={{\beta}_{\text{ph}}}$	2.3  ±  0.4
only Z  =  1  +  $\beta (L)$	3.0  ±  0.6

In figure 4, MC re-calculated dose and ${{D}_{\text{RBE}}}$ distributions, assuming an RBE of 1.1, as a function of the depth in water have been plotted together with the LET_D results taking into account only Z  =  1 particles. The contribution of Z  =  1 particles to the total dose was about 98.5 (98.1)% in the EC up to about 99.9 (99.9)% in the distal part of the target (DPT) for SOBP9 (SOBP21). The contribution of Z  =  2 particles was from about 1.0 (1.2)% in EC to less than 0.1% in the DPT. The contribution of heavier secondaries (Z  >  2) in the EC was about 0.5 (0.7)%. The effect of He and heavier fragments increases as the proton beam energy increases due to an enhancement in the production of secondary particles in nuclear interactions. LET_D values for Z  =  1 particles range from about 0.85 keV μm⁻¹ and 0.70 keV μm⁻¹ in the EC to around 18 keV μm⁻¹ and 14 keV μm⁻¹ in the distal fall-off, for SOBP9 and SOBP21, respectively. The difference in the peak is mainly due to the increased mean and spread of the energy spectra, which is more pronounced for the deepest target, leading overall to lower LET_D values for SOBP21 with respect to SOBP9, as shown by Guan et al (2015b). LET_D values have been determined for dose values larger than 1.0% of the planned dose. These observations are in line with the findings of Grassberger and Paganetti (2011), using a different MC code.

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In figure 5, we studied the effect of including He fragments in the evaluation of biological quantities such as the dose-weighted mixed-radiation field ${{\bar{\alpha}}_{D}}$ (upper row) and RBE (lower row) for both SOBP9 (left panels) and SOBP21 (right panels) fields and for two representative tissues ${{\left(\alpha /\beta \right)}_{\text{ph}}}=2$ Gy and ${{\left(\alpha /\beta \right)}_{\text{ph}}}=10$ Gy, as reported in the legends. The results are presented as the percentage variation (Y) between the biological quantities under investigation calculated including or neglecting the effect of He fragments. We found that ${{Y}_{\alpha}}$ values are higher for lower ${{\left(\alpha /\beta \right)}_{\text{ph}}}$ values. This is due to the increasing trend of ${{\alpha}_{\text{He}}}$ as ${{\left(\alpha /\beta \right)}_{\text{ph}}}$ decreases for a fixed LET (Mairani et al 2016a). For both SOBP9 and SOBP21 fields, the effect of He drops on approaching to the DPT, due to a reduction in the production of higher LET He fragments and the increasing importance of Z  =  1 particles. The calculated LET_D for Z  =  2 particles ranged from about 105 keV μm⁻¹ in the EC to around 155 keV μm⁻¹ in the DPT. Comparing SOBP9 and SOBP21 results for ${{Y}_{\alpha}}$, the effect of He fragments is higher for SOBP21 due the higher relative contribution of He secondaries compared to Z  =  1 particles.

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The effect of He fragments on ${{Y}_{\beta}}$ (data not shown) assuming a variable ${{R}_{\beta}}(L)$ or a constant ${{R}_{\beta}}=1$ is negligible (less than 0.2%).

In terms of RBE variations, for the studied dose per fraction of 2 Gy (RBE), including the biological effect of He fragments increases the RBE values in the EC to about 4.3 (5.6)% and 2.8 (3.4)%, respectively, with ${{\left(\alpha /\beta \right)}_{\text{ph}}}=2$ Gy and ${{\left(\alpha /\beta \right)}_{\text{ph}}}=10$ Gy for SOBP9 (SOBP21). The effect of the ${{\left(\alpha /\beta \right)}_{\text{ph}}}$ ratio diminishes as depth increases. For the target region, the variation is within 2% independent of the tissue and the SOBP. For the studied dose per fraction, the influence on RBE of the assumption ${{R}_{\beta}}(L)$ or ${{R}_{\beta}}=1$ is negligible (solid and dashed lines in the lower row of figure 5 are indistinguishable).

In figure 6, we plotted, for the two tissues and for the two SOBPs, the RBE values as a function of depth calculated taking into account (Z  =  1 & Z  =  2) or neglecting (only Z  =  1) the contribution of helium fragments. In the case of ${{\left(\alpha /\beta \right)}_{\text{ph}}}=2.0$ Gy the RBE values are always higher than the clinically applied value of 1.1, while for ${{\left(\alpha /\beta \right)}_{\text{ph}}}=10.0$ Gy the RBE values exceed 1.1 only in the DPT. The effect as a percentage of including Z  =  2 fragments for RBE calculations has already been shown in the lower panels of figure 5.

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Figure 7 (upper panel) depicts MC-calculated ${{D}_{\text{RBE}}}$ distributions applying the biological model introduced in this work and in Mairani et al (2016b) for both Z  =  1 and Z  =  2 particles for the prostate treatment plan. ${{D}_{\text{RBE}}}$ differences between full calculations, as in panel (a), and the current clinical approach, i.e. applying a constant RBE of 1.1, are shown in panel (b). Similarly to McNamara et al (2015), the largest differences, up to about 7 Gy (RBE), have been found in the distal part of the two opposing fields due to the higher LET component (Grassberger and Paganetti 2011). In these regions RBE values of about 1.3 have been found, while the mean RBE in the PTV is about 1.18, i.e. about 7% higher than the clinical applied value of 1.1. In terms of physical dose, for dose values larger than 1% of the planned dose, the contribution of Z  =  2 and Z  >  2 particles was about 1.2% and 0.6% in the EC and about 0.6% and 0.3% in the PTV.

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Panel (c) of figure 7 shows the differences between ${{D}_{\text{RBE}}}$ values obtained with the full calculations and taking into account the biological effect of only the Z  =  1 particles using equation (1) and the LET-dependent part of equation (2). Weighting the biological effect of He secondaries increases the mean ${{D}_{\text{RBE}}}$ by about 0.9 Gy (RBE) in the PTV and by about 1.13 Gy (RBE) for the femurs in the EC. The latter means about 5% higher ${{D}_{\text{RBE}}}$.

In figure 8, we illustrate the results for the head treatment plan in terms of ${{D}_{\text{RBE}}}$ with the full calculations in panel (a) and the difference in terms of ${{D}_{\text{RBE}}}$ between full calculations and the approximation of a constant RBE of 1.1 in panel (b). In panel (c), the ${{D}_{\text{RBE}}}$ differences between the results shown in (a) and those obtained by weighting only the biological effect of Z  =  1 particles are plotted.

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${{D}_{\text{RBE}}}$ differences up to about 20 Gy (RBE) in the distal part of the field have been found, which, however, do not correspond to any critical structure. This large difference is mainly due to the low ${{\left(\alpha /\beta \right)}_{\text{ph}}}$ value assigned to the brain and the higher LET components in this region. Negative values, corresponding to RBE values lower than 1.1, have been seen in the entrance part of the PTV due the high ${{\left(\alpha /\beta \right)}_{\text{ph}}}$ value assigned. Weighting the biological effect of He secondaries increases the mean ${{D}_{\text{RBE}}}$ in the PTV by about 0.6 Gy (RBE). Variations in the EC up to 1.4 Gy (RBE) have been found.

The impact of including Z  =  2 secondaries in biological calculations for proton therapy appears to be important for low ${{\left(\alpha /\beta \right)}_{\text{ph}}}$ tissues and in the EC of the fields, especially for deep-seated tumors. The found variations of RBE and resulting ${{D}_{\text{RBE}}}$ compared to the clinically applied approach (RBE  =  1.1) and the influence of Z  =  2 secondaries on determining RBE for normal tissue complications probabilities (NTPC) are controversial, as already explained in Paganetti (2014) and in McNamara et al (2015). In fact, the published survival data used for developing the models are based on clonogenic cell survival experiments. More experimental data are needed for a better understanding of the normal tissue response to radiation and of the impact of the mixed radiation field on RBE determination for NTCP.

Additional work is needed to quantify the effect of heavier secondaries (Z  >  2) on the biological calculations in proton therapy. However, their maximum contribution to the physical dose in the EC for the cases studied in this work was about 0.5%.

The calculation approach for proton beams introduced in this study will be further experimentally verified and it will be integrated into the FLUKA MC code, allowing all the FLUKA users to perform RBE calculations with proton and He ion beams. This project goes along the line of providing the hadron therapy community with simple-to-implement data-driven approaches for light ion radiotherapy applications. Moreover, the impact of Z  =  2 and heavier secondaries on biological calculations with proton beams should be additionally investigated by means of biophysical models in order to compare their predictions and those obtained in this work.