Data-driven RBE parameterization for helium ion beams

Radiotherapy with light ions is spreading worldwide due to the favorable physical characteristics of light ion beams allowing highly conformal dose distributions in the tumor region while sparing adjacent normal tissues (Schardt et al 2010). Previously, clinical trials with different ion species of protons, helium ions, carbon ions, neon ions and argon ions have been performed at Lawrence Berkley Laboratory (LBL) and more than 2000 patients have been treated with helium ions (Castro 1995). Nowadays, proton and carbon ions are used in clinical practice, and there is an increase in interest for future clinical applications of helium beams. For example, at the Heidelberg Ion Beam Therapy Center (HIT) (Haberer et al 2004), helium and oxygen ion beams are already available for research purposes. Helium ions, in comparison to protons, have smaller lateral scattering (Ströbele et al 2012). Moreover, when compared to carbon ions, they have a reduced fragmentation tail. In order to support the future clinical application of helium ions and exploit their potential advantages, treatment planning calculations of both dose and RBE-weighted dose distributions are necessary. In proton therapy, a constant RBE of 1.1 is usually applied clinically as recommended by ICRU (2007), although there are experimental evidences that RBE varies with dose, linear energy transfer (LET), cell line and chosen endpoint (Paganetti 2014). For carbon ion beams, three clinical models are employed: the Local Effect Model (LEM) (Krämer and Scholz 2000), the Kanai model (Kanai et al 1999) and, more recently, the microdosimetric-kinetic-model (MKM) (Kase et al 2008). Currently, biophysical models which describe the biological effect of He beams such as the MKM, the LEM (Elsässer et al 2010), and the repair-misrepair-fixation (RMF) model (Frese et al 2012) are available. However, differently from protons (Wilkens and Oelfke 2004, Tilly et al 2005, Carabe-Fernandez et al 2007, Wedenberg et al 2013), no easy-to-use phenomenological models for RBE predictions have been introduced. Therefore, in this work, we developed a data-driven and easy-to-implement model, which is able to predict the $\text{RBE}$ for helium ions, in a clinically relevant range of doses, based on LET and the tissue specific parameter ${{(\alpha /\beta )}_{\text{ph}}}$ of photons—the ratio of the linear-quadratic (LQ) parameters from photon exposure—used to characterize the radiosensitivity of a cell type (Sinclair 1966). Phenomenological analytic expressions were investigated to describe how the $\text{RB}{{\text{E}}_{\alpha}}={{\alpha}_{\text{He}}}/{{\alpha}_{\text{ph}}}$ ratio varies with LET and were then tested against statistical goodness-of-fit methods. Due to the large uncertainties affecting the β parameter (Friedrich et al 2013), the ${{\text{R}}_{\beta}}={{\beta}_{\text{He}}}/{{\beta}_{\text{ph}}}$ ratio parameterization has been instead determined by fitting the computed value for the weighted running average as a function of LET.

In addition, due to the lack of experimental data in the literature, an experimental benchmark of the introduced formalism at low LET has been performed and is presented.

2.1. Database generation

In order to study and determine the radiobiological properties of helium beams, we collected in vitro data of cell survival available in literature. A database for helium and other ions can be found in Friedrich et al (2013) (Particle Irradiation Data Ensemble—PIDE). However, we reanalyzed the data in order to fully handle the fitting results taking into account the parameter uncertainties, if available. We examined all the publications reported therein and excluded the ones for which it was not possible to extract the LQ parameters of the response both to helium ions and to the reference radiation (${{\alpha}_{\text{He}}}$, ${{\beta}_{\text{He}}}$, ${{\alpha}_{\text{ph}}}$ and ${{\beta}_{\text{ph}}}$), either in tables or retrievable by fitting data from the reported figures of survival curves.

When photons and helium LQ parameters were reported as numbers, they were directly taken. If not, whenever possible, they were calculated from other related published quantities, e.g. from RBE and the reference radiation dose for a certain survival level. When only survival curves were provided, experimental points and their error bars were acquired and digitized. The LQ parameters were obtained by fitting the data with an in-house tool based on the CERN ROOT framework (http://root.cern.ch) and the MINUIT minimization package (Brun and Rademakers 1997). This software allows to fit data-sets with or without errors by minimizing, respectively, a weighted or unweighted sum of squared differences computed between the experimental survival value and the chosen model. Since all data reported in the published tables were derived from the LQ model, we decided to exclude from our work those ones (always from photon irradiation) for which the suggested best fit model was different, in order to build up a consistent analysis based on the LQ formalism only. Those include either the linear quadratic cubic (LQC) or the linear quadratic-linear (LQ-L) models (Joiner and van der Kogel 2009) (this is the case for Hall et al (1972), for the HeLaS3 cell line in Goodhead et al (1992) and for the L5178Y cell line in Eguchi-Kasai et al (1996)). We also rejected data from survival curves where the experimental points, mainly at low doses, were indistinguishable or difficult to determine (this is the case for Takahashi et al (2000), for the irs1 and M10 cell lines in Eguchi-Kasai et al (1996) and for the V79 cell line in Jenner et al (1993)).

Helium ion beams, which the available literature refers to, were produced by accelerators or originated from the α-decay of radionuclides (e.g. ²³⁸Pu and ²¹¹At). All papers described the radiation quality of the helium beams in terms of LET. It should be noticed that an intrinsic uncertainty affected the collected LET values. In fact, LET was not always unequivocally calculated: some papers used the dose-averaged LET, others the track-averaged LET or the volume-averaged LET, but most of the time the method of LET calculation was not even specified. Moreover, in some experiments, before reaching the cells, the helium beam passed through passive elements, thus degrading its energy and producing secondary particles.

As reference radiations, γ-rays from ⁶⁰Co decay or x-rays produced by x-ray machines (voltage ranging from 80 to 250 kVp) were usually employed. In order to estimate the effectiveness of different photon beams, a RBE could be defined:

$$\begin{eqnarray}&&\text{RB}{{\text{E}}_{\text{ph}}}=\frac{{{D}_{{{}^{60}}\text{Co}}}}{{{D}_{\text{ph}}}},\end{eqnarray} \tag{ 1 }$$

where ${{D}_{\text{ph}}}$ is the dose of x-rays at a certain kVp and ${{D}_{{{}^{60}}\text{Co}}}$ is the dose of ⁶⁰Co γ-rays to induce the same biological effect. For example, 200 kVp x-rays have, on average, an $\text{RB}{{\text{E}}_{\text{ph}}}$ of 1.15 relative to ⁶⁰Co (Sinclair 1962). The choice of using ⁶⁰Co as a reference, among the photon beams, relates to its known clinical relevance. In calculating helium RBE values, the effectiveness of the different photon radiations should be considered. However, as the $\text{RB}{{\text{E}}_{\text{ph}}}$ for a certain endpoint is usually not available, correcting the RBE values deduced from these experiments is usually not possible. Normalizing the helium LET is more advisable, as recommended in Paganetti (2014) and references cited therein:

$$\begin{eqnarray}&&\text{LE}{{\text{T}}^{\ast}}\equiv \text{LET}-\text{LE}{{\text{T}}_{\text{ph}}}+\text{LE}{{\text{T}}_{{{}^{60}}\text{Co}}}.\end{eqnarray} \tag{ 2 }$$

The list of $\text{LE}{{\text{T}}_{\text{ph}}}$ values, calculated as suggested in Paganetti (2014) and applied in this work, is reported in table 1. Around 20 cell lines, ranging from radioresistant to radiosensitive cells, were examined. The V79 cell line was the mostly investigated in the collected publications. Only data from experiments performed with asynchronous cells were selected, excluding those that resulted from the irradiation of cells synchronized in a certain phase of the cell cycle. Asynchronous cells are indeed more representative of an in vivo scenario related to the clinical practice.

Table 1. The list of cell lines used in this study with the calculated radiobiological parameters. When present, uncertainties are expressed as $1\sigma$ standard deviations. In case of asymmetric errors, the mean error is reported.

Cell line	${{(\alpha /\beta )}_{\text{ph}}}$ [Gy]	Reference radiations	$\text{LE}{{\text{T}}_{\mathbf{ph}}}$ [keV $\mu$m−1]	Reference
LS174T	55.437	60Co γ-rays	0.400
U343MG	4.598	60Co γ-rays	0.400	(Carlsson et al 1995)
V79-379A	3.765	60Co γ-rays	0.400
T1	$10.000\pm 0.705$	200–250 kVp	1.117	(Carlson et al 2008)
V79	$2.206\pm 0.735$	220 kVp	1.127	(Chapman et al 1977)
V79-379A	$4.941\pm 3.980$	100 kVp	1.443	(Claesson et al 2011)
HF19	$\infty$	250 kVp	1.075	(Cox and Masson 1979)
irs2	7.971	200 kVp	1.164	(Eguchi-Kasai et al 1996)
V79	$2.708\pm 0.489$	240 kVp	1.092	(Folkard 1996)
HSG	5.089	200 kVp	1.164	(Furusawa et al 2000)
V79	9.200	200 kVp	1.164
C3H10T1/2	$6.586\pm 1.278$	250 kVp	1.075	(Goodhead et al 1992)
HeLa	$21.560\pm 1.885$	250 kVp	1.075
AG01522	$\infty$	60Co γ-rays	0.400	(Hamada et al 2006)
CHOK1	6.897	250 kVp	1.075
irs1	$\infty$	250 kVp	1.075
irs2	8.577	250 kVp	1.075
irs3	10.075	250 kVp	1.075	(Hill et al 2004)
V79	3.700	250 kVp	1.075
xrs5	$\infty$	250 kVp	1.075
C3H10T1/2	2.172	250 kVp	1.075	(Miller et al 1995)
C3H10T1/2	$9.000\pm 1.803$	80 kVp	1.549	(Napolitano et al 1992)
V79	$4.074\pm 0.869$	250 kVp	1.075	(Prise et al 1990)
CHO10B	$7.667\pm 0.852$	60Co γ-rays	0.400
HS23	$7.222\pm 0.728$	60Co γ-rays	0.400
C3H10T1/2	$20.000\pm 2.088$	60Co γ-rays	0.400	(Raju et al 1991)
V79	$9.333\pm 0.741$	60Co γ-rays	0.400
AG1522	$4.886\pm 0.797$	60Co γ-rays	0.400
B16	$2.837\pm 2.198$	60Co γ-rays	0.400
DU145	$7.875\pm 2.491$	60Co γ-rays	0.400
HTh7	$5.173\pm 1.263$	60Co γ-rays	0.400
IGR	$1.929\pm 1.825$	60Co γ-rays	0.400	(Stenerlöw et al 1995)
LS174T	$\infty$	60Co γ-rays	0.400
U343MG	$0.910\pm 0.919$	60Co γ-rays	0.400
V79	$6.483\pm 0.793$	60Co γ-rays	0.400
V79-4	5.521	60Co γ-rays	0.400	(Thacker et al 1979)

All the cell lines with their experimental parameters used in this study are presented in table 1.

2.2. The $\text{RBE}$ model

Within the LQ framework, one can straightforwardly derive a dependency of RBE on the respective photon and helium ion LQ-parameters (see also Carabe-Fernandez et al (2007) and Wedenberg et al (2013)). Considering that helium absorbed dose D and photon dose ${{D}_{\text{ph}}}$ are isoeffective if:

$$\begin{eqnarray}&&{{\alpha}_{\text{He}}}D+{{\beta}_{\text{He}}}{{D}^{2}}={{\alpha}_{\text{ph}}}{{D}_{\text{ph}}}+{{\beta}_{\text{ph}}}D_{\text{ph}}^{2},\end{eqnarray}$$

by rearranging the expression and solving a second-degree equation for the positive root of ${{D}_{\text{ph}}}$, as in Wedenberg et al (2013), and by dividing the solution by D, one can obtain:

$$\begin{eqnarray}&&\text{RBE}\left({{\alpha}_{\text{He}}},{{\beta}_{\text{He}}},{{\alpha}_{\text{ph}}},{{\beta}_{\text{ph}}},D\right)=-\frac{1}{2D}{{\left(\frac{\alpha}{\beta}\right)}_{\text{ph}}}+\frac{1}{D}\sqrt{\frac{1}{4}\left(\frac{\alpha}{\beta}\right)_{\text{ph}}^{2}+\left(\frac{{{\alpha}_{\text{He}}}}{{{\alpha}_{\text{ph}}}}\right){{\left(\frac{\alpha}{\beta}\right)}_{\text{ph}}}D+\left(\frac{{{\beta}_{\text{He}}}}{{{\beta}_{\text{ph}}}}\right){{D}^{2}}},\end{eqnarray} \tag{ 3 }$$

where $\text{RBE}={{D}_{\text{ph}}}/D$ by definition. Then, we define two ratios as:

$$\begin{eqnarray}&&\text{RB}{{\text{E}}_{\alpha}}\equiv \frac{{{\alpha}_{\text{He}}}}{{{\alpha}_{\text{ph}}}},\end{eqnarray} \tag{ 4 }$$

defined for example in Friedrich et al (2013), and

$$\begin{eqnarray}&&{{\text{R}}_{\beta}}\equiv \frac{{{\beta}_{\text{He}}}}{{{\beta}_{\text{ph}}}}.\end{eqnarray} \tag{ 5 }$$

Using equations (4) and (5), equation (3) can be rearranged as follows:

$$\begin{eqnarray}&&\text{RBE}\left({{\left(\frac{\alpha}{\beta}\right)}_{\text{ph}}},D,\text{RB}{{\text{E}}_{\alpha}},{{\text{R}}_{\beta}}\right)=-\frac{1}{2D}{{\left(\frac{\alpha}{\beta}\right)}_{\text{ph}}}+\frac{1}{D}\sqrt{\frac{1}{4}\left(\frac{\alpha}{\beta}\right)_{\text{ph}}^{2}+\text{RB}{{\text{E}}_{\alpha}}{{\left(\frac{\alpha}{\beta}\right)}_{\text{ph}}}D+{{\text{R}}_{\beta}}{{D}^{2}}},\end{eqnarray} \tag{ 6 }$$

to better highlight the dependence of RBE on photon parameter (${{(\alpha /\beta )}_{\text{ph}}}$) and helium ion parameters ($\text{RB}{{\text{E}}_{\alpha}}$, ${{\text{R}}_{\beta}}$ and D).

2.2.1. $\text{RB}{{\text{E}}_{\alpha}}$ parameterization

The investigated $\text{RB}{{\text{E}}_{\alpha}}$ parameterizations were developed with the aim to capture basic features of the $\text{RB}{{\text{E}}_{\alpha}}$ using a minimum of assumptions supported by experimental data and previously published papers.

Data collected from the available publications (table 1) have shown how $\text{RB}{{\text{E}}_{\alpha}}$ for helium ions increases with increasing LET approximately up to a maximum value $\tilde{L}$ of about 120–150 keV μm⁻¹, after which it starts decreasing (see also Friedrich et al (2013)). We started with the assumption that ${{\alpha}_{\text{He}}}$ approaches ${{\alpha}_{\text{ph}}}$ for decreasing LET values, as already done for protons in a previous work by Wedenberg et al (2013). Referring to the same work, we also assumed that the initial slope was affected by the cell line via the inverse relationship with ${{(\alpha /\beta )}_{\text{ph}}}$, so that the $\text{RB}{{\text{E}}_{\alpha}}$ of cell lines with high ${{(\alpha /\beta )}_{\text{ph}}}$ was less dependent on LET (L), namely:

$$\begin{eqnarray}&&\text{RB}{{\text{E}}_{\alpha}}\left(L\,|{{(\alpha /\beta )}_{\text{ph}}}\right)=1+\left[{{k}_{0}}+{{\left(\frac{\beta}{\alpha}\right)}_{\text{ph}}}\right]\centerdot f(L).\end{eqnarray} \tag{ 7 }$$

The assumed and fixed dependency on ${{(\alpha /\beta )}_{\text{ph}}}$ can be further justified by looking at the $\text{RB}{{\text{E}}_{\alpha}}$ behavior as a function of ${{(\beta /\alpha )}_{\text{ph}}}$ for different groups of LET ranges. For this purpose, figure 1 shows how the cell line parameters affect the $\text{RB}{{\text{E}}_{\alpha}}$ ratio: $\text{RB}{{\text{E}}_{\alpha}}$ generally increases with decreasing ${{(\alpha /\beta )}_{\text{ph}}}$ and with increasing LET, however, at high ${{(\alpha /\beta )}_{\text{ph}}}$ (or equivalently, low ${{(\beta /\alpha )}_{\text{ph}}}$), the change in $\text{RB}{{\text{E}}_{\alpha}}$ due to different LETs is very limited. For low LETs, the ${{(\alpha /\beta )}_{\text{ph}}}$-induced variation of $\text{RB}{{\text{E}}_{\alpha}}$ is also small, as indicated by the lower slope of the line referring to the 20–30 keV μm⁻¹ LET interval. Figure 1 somewhat illustrates also the spread and uncertainty of the experimental data available.

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The multiplicative function f(L) should act on $\text{RB}{{\text{E}}_{\alpha}}$ to possibly satisfy the empirical properties suggested by experimental findings:

$$\begin{eqnarray}&&\text{RB}{{\text{E}}_{\alpha}}(L)\geqslant 0\quad \forall L\lesssim {{L}_{\text{max}}}\quad ;\quad \text{RB}{{\text{E}}_{\alpha}}\left(\tilde{L}\right)\approx \text{RBE}_{\alpha}^{\text{max}},\end{eqnarray} \tag{ 8 }$$

where ${{L}_{\text{max}}}$ is the maximum LET of He beams in water, i.e.  ∼200 keV μm⁻¹ and $\text{RBE}_{\alpha}^{\text{max}}$ is the maximum $\text{RB}{{\text{E}}_{\alpha}}$ for a given cell line.

In order not to include any bias in the estimation of the fitting parameters, no a-priori boundary limits for their variation were specified and the requirements for the $\text{RB}{{\text{E}}_{\alpha}}$ function were only verified a-posteriori. The first free parameter—k₀—which plays a common role for all the tested models, corrects for ${{\beta}_{\text{ph}}}=0$ cell lines, where $\text{RB}{{\text{E}}_{\alpha}}\geqslant 1$ for ${{\beta}_{\text{ph}}}=0$. This assumption is confirmed by experimental evidence (see section 3).

Differently from Wedenberg et al (2013), who limited their analysis with proton beams to LET values up to about 30 keV μm⁻¹, where the maximum of the $\text{RB}{{\text{E}}_{\alpha}}$ is located, we used several fitting functions, via f(L), with the aim of predicting $\text{RB}{{\text{E}}_{\alpha}}$ for the entire relevant LET range (up to about 200 keV μm⁻¹). A summary of the studied parameterizations is reported in table 2. We decided to choose fitting functions of increasing complexity and number of parameters, starting from a simple linear model (${{f}_{\text{L}}}$) to a more elaborated one (${{f}_{\text{LQE}2}}$) with the aim of improving the description of the experimental data until no relevant improvements could be found (F-statistic analysis in section 3). We started with purely polynomial expressions and mixed afterwards polynomial and exponential terms. Mixtures of polynomial and $\exp \left(-{{L}^{2}}\right)$ terms have been also tested for evaluating their capability in fitting the $\text{RB}{{\text{E}}_{\alpha}}$ values in the fall-off region after the maximum.

Table 2. List of $\text{RB}{{\text{E}}_{\alpha}}$ parameterizations under investigation.

	Equation
Model	$\text{RB}{{\text{E}}_{\alpha}}\left(L\,|{{(\alpha /\beta )}_{\text{ph}}}\right)=1+\left[{{k}_{0}}+{{\left(\frac{\beta}{\alpha}\right)}_{\text{ph}}}\right]\centerdot f(L)$
${{f}_{\text{L}}}$	f(L)  =  k1 L
${{f}_{\text{Q}}}$	$f(L)={{k}_{1}}{{L}^{2}}$
${{f}_{\text{LQ}}}$	$f(L)={{k}_{1}}L+{{k}_{2}}{{L}^{2}}$
${{f}_{\text{LQC}}}$	$f(L)={{k}_{1}}L+{{k}_{2}}{{L}^{2}}+{{k}_{3}}{{L}^{3}}$
${{f}_{\text{LE}}}$	$f(L)=\left({{k}_{1}}L\right)\exp \left(-{{k}_{2}}L\right)$
${{f}_{\text{QE}}}$	$f(L)=\left({{k}_{1}}{{L}^{2}}\right)\exp \left(-{{k}_{2}}L\right)$
${{f}_{\text{LQE}}}$	$f(L)=\left({{k}_{1}}L+{{k}_{2}}{{L}^{2}}\right)\exp \left(-{{k}_{3}}L\right)$
${{f}_{\text{LE}2}}$	$f(L)=\left({{k}_{1}}L\right)\exp \left(-{{k}_{2}}{{L}^{2}}\right)$
${{f}_{\text{QE}2}}$	$f(L)=\left({{k}_{1}}{{L}^{2}}\right)\exp \left(-{{k}_{2}}{{L}^{2}}\right)$
${{f}_{\text{LQE}2}}$	$f(L)=\left({{k}_{1}}L+{{k}_{2}}{{L}^{2}}\right)\exp \left(-{{k}_{3}}{{L}^{2}}\right)$

In order to link experimental $\text{RB}{{\text{E}}_{\alpha}}$ data with models, we set-up a procedure based on the jackknife (JK) resampling technique (Efron 1979, Cameron and Trivedi 2005), which is considered to be a robust procedure for the estimation of statistical population parameters. The main advantage of JK is that it eliminates the need to make strong assumptions about the underlying probability density functions governing a given statistical system. However, it requires that the observed data are a representative sample of the population. This allows an estimation of the population parameters by analysing the sample itself. In particular, the JK technique can be a useful tool for identifying outliers and biases in statistical estimates. Given the variability of collected radiobiological data, it is of interest to examine the influence that one observation can have on the overall outcome. As required by the JK technique, the implicit assumption has been made that observations are independent of each other.

In order to deal with the highly inhomogeneous data set, where the specific tissue parameters ${{\alpha}_{{}}}$ and ${{\beta}_{{}}}$, as well as LET, are mixed up either having experimental uncertainties or not, we adopted a two-step procedure where the JK technique was coupled to a data resampling technique which took into account the data uncertainties, if available. In the first step of the adopted procedure, both the independent ($\text{LE}{{\text{T}}^{\ast}}$) and the dependent ($\text{RB}{{\text{E}}_{\alpha}}$) variable vectors were replicated N times, each entry being replaced by the original value plus a fluctuation given by its uncertainty, if available (otherwise, the value without uncertainties was kept). A Gaussian dispersion was assumed around the reported mean value, so that a normal (or split-normal for asymmetric errors) distribution with standard deviation equal to the given data uncertainty was used to randomly sample the new point, eventually re-sample when negative (unphysical) LET and/or $\text{RB}{{\text{E}}_{\alpha}}$ values were obtained ('sampling inefficiency' η in the following). This allowed to randomly work with equally weighted squared residuals when minimizing the sum of squared residuals (SSR) between the observed and the predicted values.

The actual data-set, on which the minimization procedure was applied, was determined by the JK iterative process for the best estimation of the fit parameters. From the whole sample, i.e. the 'uncertainty-free' data-set discussed above, each element was dropped out and the parameters of interest were estimated from that smaller sample. This estimation is known as partial estimate or JK replication. The control variable ${{\Delta }^{2}}$ was then computed as the sum of the quadratic differences between the resulting model fits (the partial estimates) and the excluded data points. Through the computation of ${{\Delta }^{2}}$, each model is actually used to predict the left-out observation, which has the role of a new observation in this context.

The JK iterative algorithm returns a distribution of best estimates for each parameter of the specific model under study. To retrieve a single value, the sample mean was computed as final best estimator.

The CERN ROOT framework and the MINUIT minimization package were used to build the core part of our C++ based programs. In order to speed-up the simulation time, a pre-fitter based on MINUIT was implemented to work on the whole unmodified database. At this initial stage, no range constraints for the free parameters were set and the resulting best estimates and their $1\sigma$ standard deviations were passed as input for the initialization of the parameters in our macros, which were instead bounded within $\pm 5\sigma$ to quicken the fit convergence.

The final best estimates of the fit parameters derived from an overall statistics of 101 k minimizations (N  =  1000 random data samplings  ×  JK over 101 experimental data).

Fit results were compared using the root mean squared error ($\text{RMSE}\equiv \sqrt{\text{SSR}/\nu}$, $\nu =$ number of degrees of freedom), the JK control variable ${{\Delta }^{2}}$ and then also tested against the requirements in equation (8).

2.2.2. The ${{\text{R}}_{\beta}}$ parameterization.

To estimate a trend component for the ${{\text{R}}_{\beta}}-\text{LET}(L)$ relation, we applied a filter to the data, similarly to Friedrich et al (2013). Due to the large degree of dispersion shown by the ${{\text{R}}_{\beta}}$ ratio when plotted against LET (see figure 4), a low-pass filter could be chosen to remove fluctuating components, yielding an estimate of the slow-moving trend of collected data. A Gaussian filter has been adopted for the computation of a weighted running average of ${{\text{R}}_{\beta}}$ ratios. Weights of the Gaussian filter are proportional to ordinates of a normal probability density function with a specified standard deviation sigma chosen as suggested in Friedrich et al (2013). The smoothed ${{\text{R}}_{\beta}}$ curve has been then fitted minimizing the unweighted sum of squared differences computed between the calculated running mean values and the following parameterization:

$$\begin{eqnarray}&&{{\text{R}}_{\beta}}(L)={{b}_{0}}\exp \left[-{{\left(\frac{L-{{b}_{1}}}{{{b}_{2}}}\right)}^{2}}\right].\end{eqnarray} \tag{ 9 }$$

Unlike equation (7), the dependence of ${{\text{R}}_{\beta}}$ on ${{(\alpha /\beta )}_{\text{ph}}}$ has been neglected as a first approximation as supported by experimental evidence (Friedrich et al 2013). LET corrections, i.e. taking into account the LET of the reference radiation, have been neglected as well at this stage, due to the large uncertainties affecting the ${{\beta}_{\text{He}}}$ component.

2.3. Experimental investigation

In figure 2, comparisons between experimental data, as listed in table 1, (points with error bars) and fits (solid lines) for the 10 parameterizations under study are reported. Data and fitting results are depicted in terms of $\left(\text{RB}{{\text{E}}_{\alpha}}-1\right){{\left[{{k}_{0}}+{{(\beta /\alpha )}_{\text{ph}}}\right]}^{-1}}$ to remove the explicit dependence on the cell line. The ${{f}_{\text{L}}}$ and the ${{f}_{\text{Q}}}$ functions are, as expected, not able to match the experimental LET dependency, as shown in the upper panels of figure 2, while the other functions reproduce qualitatively the main features of the data.

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The obtained results, together with the RMSE and the square root of the JK control variable ${{\Delta }^{2}}$, are given in table 3. A major discrimination between the 10 models can be obtained via the RMSE, which basically gives, in the same units of the fitted function, the final value of the SSR minimization procedure adjusted for the degrees of freedom. In this metric, the best model for the description of $\text{RB}{{\text{E}}_{\alpha}}$ as a function of the normalized LET ($\text{LE}{{\text{T}}^{\ast}}$) is given by the quadratic-exponential parameterization ${{f}_{\text{QE}}}$, for which the minimum value of the RMSE is registered (1.665).

Table 3. Summary of the fit results obtained with the JK technique coupled with a data resampling technique for N  =  1000 random samplings.

Model	RMSE	$\sqrt{{{\Delta }^{2}}}$	k0	k1	k2	k3
${{f}_{\text{L}}}$	2.343	31.4	1.62498E-01	1.59047E-01	—	—
${{f}_{\text{Q}}}$	3.696	49.2	2.14500E-01	8.53959E-04	—	—
${{f}_{\text{LQ}}}$	1.723	25.0	1.42057E-01	2.91783E-01	−9.52500E-04	—
${{f}_{\text{LQC}}}$	1.702	31.7	1.30576E-01	1.82335E-01	9.53070E-04	−7.04393E-06
${{f}_{\text{LE}}}$	1.827	25.7	1.53848E-01	2.96541E-01	4.90821E-03	—
${{f}_{\text{QE}}}$	1.665	24.9	1.36938E-01	9.73154E-03	1.51998E-02	—
${{f}_{\text{LQE}}}$	1.681	41.4	1.28962E-01	2.36454E-01	−9.50865E-04	−4.12185E-03
${{f}_{\text{LE}2}}$	1.686	24.7	1.39642E-01	2.55597E-01	2.35585E-05	—
${{f}_{\text{QE}2}}$	1.737	26.1	1.17047E-01	4.74970E-03	6.66283E-05	—
${{f}_{\text{LQE}2}}$	1.686	25.7	1.31564E-01	1.24090E-01	2.15385E-03	4.44092E-05

Note: For models names and related parameters, refer to table 2. A sampling inefficiency $\eta =0.01$% was registered. The RMSE and the square root of the JK control variable ${{\Delta }^{2}}$ are reported, together with the parameter values. For the polynomial part of the equations, coefficients are expressed as [Gy][keV μm⁻¹]⁻ⁿ, with n  =  1, 2, 3 for terms of first, second and third degree in the LET variable, respectively. At the exponent, coefficients are given as [keV μm⁻¹]^−m, with m  =  1, 2 for terms of first and second degree in the LET variable. k₀ is in [Gy]⁻¹.

The value assumed by the JK control variable ${{\Delta }^{2}}$ is a good indicator to quantify a model's performance in predicting out-of-sample data (new observations). In this sense, ${{f}_{\text{QE}2}}$, ${{f}_{\text{L}}}$, ${{f}_{\text{LQC}}}$, ${{f}_{\text{LQE}}}$ and ${{f}_{\text{Q}}}$ may be classified as moderately to highly unstable predictors of $\text{RB}{{\text{E}}_{\alpha}}$, with a relative deviation from the minimum $\sqrt{{{\Delta }^{2}}}$—belonging to ${{f}_{\text{LE}2}}$—ranging from 5.8% for ${{f}_{\text{QE}2}}$ up to 99.2% for ${{f}_{\text{Q}}}$. For the remaining functions, this relative residual is well contained within 5%. In particular it is around 4% for both ${{f}_{\text{LE}}}$ and ${{f}_{\text{LQE}2}}$, while a 1.0% difference is registered for both ${{f}_{\text{LQ}}}$ and ${{f}_{\text{QE}}}$, a further indication that the latter is a good candidate as model for $\text{RB}{{\text{E}}_{\alpha}}$. The caveat here may be comparing goodness-of-fit as merely quantified by RMSE, given that a variety of models were investigated in order to reproduce at best what is empirically observed, without a rigorous theoretical formalism about the physical/biological processes behind. Therefore one may argue that this analysis may result to be too simplistic, especially because functions chosen as models for the parameterization of $\text{RB}{{\text{E}}_{\alpha}}$ differ by additive and/or multiplicative terms, being in most of the cases effectively nested versions one of each other, as the complexity of the equation increases. Fits based on more complicated equations (i.e. with more variables) will tend to have a lower SSR, simply because they have more inflection points and/or a reduced number of degrees of freedom. This is however not strictly the case here, since there are more complicated models than ${{f}_{\text{QE}}}$ which can be rejected by means of the RMSE statistic. However, as a precaution, one can arbitrarily neglect differences within 3% with respect to the minimum RMSE, thus remaining with a subset of eligible accurate models other than ${{f}_{\text{QE}}}$, namely $\left\{\,{{f}_{\text{LQE}}};\,{{f}_{\text{LE}2}};\,{{f}_{\text{LQE}2}};\,{{f}_{\text{LQC}}}\right\}$. A 3% threshold may be considered as an acceptable deviation, given the purely data-driven procedure. However, once differences in RMSE are leveled out, a direct comparison between nested models only can be performed, if the more complicated one better fits data according to the considered statistics. An F-test was therefore conducted to decide which model to accept, avoiding any arbitrary interpretation and quantifying whether the decrease in SSR is worth the cost of the additional variables (loss of degrees of freedom). For the purpose the F-statistic was defined as:

$$\begin{eqnarray}&&F=\frac{\left(\text{SS}{{\text{R}}_{s}}-\text{SS}{{\text{R}}_{c}}\right)/\left({{\nu}_{s}}-{{\nu}_{c}}\right)}{\text{SS}{{\text{R}}_{c}}/{{\nu}_{c}}},\end{eqnarray} \tag{ 10 }$$

where labels s and c are referred to the simpler and more complicated models under comparison, respectively. It was used to determine a p-value on a $F\left({{\nu}_{s}}-{{\nu}_{c}},{{\nu}_{c}}\right)$-distribution after fixing a significance level at 1% and the null hypothesis that the simpler model is the better one. The most trivial case is given by the $\left\{\,{{f}_{\text{QE}}}\,;{{f}_{\text{LQE}}}\right\}$ pair, since having $\text{SS}{{\text{R}}_{s}}<\text{SS}{{\text{R}}_{c}}$ ensures that the simpler ${{f}_{\text{QE}}}$ function is the best choice. For the second pair, $\left\{\,{{f}_{\text{LE}2}}\,;{{f}_{\text{LQE}2}}\right\}$, the more complicated model is not significantly better than the simpler one. The remaining ${{f}_{\text{LQC}}}$ model is affected by a relatively high value of the control variable ${{\Delta }^{2}}$. However, one of its nested models, ${{f}_{\text{LQ}}}$, which is above our exclusion threshold on the RMSE deviation by only 0.5%, substantially lowers the latter parameter. For the pair $\left\{\,{{f}_{\text{LQ}}}\,;{{f}_{\text{LQC}}}\right\}$, the F-test supports the hypothesis that the more complicated model does not describe significantly better the data. Therefore, with the choice of ${{f}_{\text{LQ}}}$, at the expenses of a slightly worse RMSE (3.5% above the minimum one, instead of 2.2%), one improves the capability of predicting new observables. The remaining 2 models $\left\{\,{{f}_{\text{QE}}};{{f}_{\text{LE}2}}\right\}$ share the same number of free parameters (3) and a further comparison cannot be made at this level. Additional experimental data are needed in order to understand the capabilities of the listed models in predicting data not included in the database. All the presented parameterizations fulfill the constraints stated in equation (8).

The resulting best fit parameterization, ${{f}_{\text{QE}}}$, has been used in figure 3 where $\text{RB}{{\text{E}}_{\alpha}}$ is plotted as a function of $\text{LET}_{{}}^{\ast}$ using the values reported in table 3. For display purposes, the mean value of each selected ${{(\alpha /\beta )}_{\text{ph}}}$ interval in figure 3 was used in the model equation. Taking into account this unavoidable approximation, the model still reproduces the experimental data satisfactorily for the four representative tissues characterized by low, medium, high ${{(\alpha /\beta )}_{\text{ph}}}$ ratios and ${{\beta}_{\text{ph}}}=0$, additionally confirming the chosen parameterization for $\text{RB}{{\text{E}}_{\alpha}}$.

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The biophysical models, available in literature, assume or predict different behaviors of the ${{\beta}_{\text{He}}}$ as a function of LET. In the RMF model, ${{\text{R}}_{\beta}}$ is assumed to increase as the LET increases (Frese et al 2012), while in the MKM (Kase et al 2008), it is assumed to be one. In the LEM, ${{\beta}_{\text{He}}}$ decreases with increasing LET in the single hit-based version. An improved version of LEM is able to reproduce better the experimental data (see Friedrich et al (2013) and references presented therein).

In this work, we have fitted the running average of ${{\text{R}}_{\beta}}$ as a function of LET (see figure 4) without additional efforts in finding a best fit parameterization as performed for $\text{RB}{{\text{E}}_{\alpha}}$, due to the large fluctuations of experimental data. Fitting the weighted running averages with equation (9), produced the following best fit parameters: b₀  =  2.51, b₁  =  65.18 keV μm⁻¹ and b₂  =  54.80 keV μm⁻¹. Comparing our running averages and the ones shown in Friedrich et al (2013), we have found differences mainly at low/intermediate LET values (<30 keV μm⁻¹), probably due the actual implementation of the running mean calculations and the different experimental database as already explained in section 2.1. However, it should be noted that a limited number of experimental data is available at low LET values (<15 keV μm⁻¹), which however could be important for predicting RBE values in the entrance channel of patient plans.

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We performed an experiment at HIT irradiating A549 cells in the plateau region of a 56.4 MeV u⁻¹ helium beam, as an initial investigation for low LET values. In figure 5, we have compared A549 clonogenic survival data for photons (open circles with error bars) and He (points with error bars) with LQ fits of the photon data (dotted line) and of the He data (solid line). The best fit LQ parameters for He were ${{\alpha}_{\text{He}}}=0.241\pm 0.066$ Gy⁻¹ and ${{\beta}_{\text{He}}}=0.026\pm 0.015$ Gy⁻², which agree within the experimental uncertainties with the values calculated applying equation (7), choosing ${{f}_{\text{QE}}}$ and equation (9): ${{\alpha}_{\text{He}}}=0.205$ Gy⁻¹ and ${{\beta}_{\text{He}}}=0.027$ Gy⁻². The resulting model-based cell survival as a function of dose, applying the LQ formalism, is depicted with a dashed line in figure 5.

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To test the validity of our statistical analysis and the robustness of the chosen parameterizations for $\text{RB}{{\text{E}}_{\alpha}}$ and ${{\text{R}}_{\beta}}$, cell survival data sets described by LQC and LQ-L models by our ROOT–MINUIT based software were forced to follow a LQ-like trend, and then included in the analyzed database. The same analysis as described in section 2.2.1 was performed on this enlarged database and the obtained results were fully consistent with previous outcomes. With respect to the standard database, the ${{f}_{\text{QE}}}$-based $\text{RB}{{\text{E}}_{\alpha}}$ and ${{\text{R}}_{\beta}}$ parameterizations showed, on average, a 3.3% and a  −1.4% variation of the best fit parameters respectively, thus confirming the robustness of the previous analysis.

In figure 6, we compared calculated RBE values versus experimental RBE values for all the analyzed data at 2 Gy photon dose ($\text{RB}{{\text{E}}_{2~\text{Gy}-\text{ph}}}$, open circles) and at 10% survival level ($\text{RB}{{\text{E}}_{10}}$, points) assuming ${{\text{R}}_{\beta}}=1$ (left panel) or ${{\text{R}}_{\beta}}(L)$ as in equation (9) (right panel). The Pearson's correlation coefficients are respectively: 0.851/0.853 for $\text{RB}{{\text{E}}_{2~\text{Gy}-\text{ph}}}$ and 0.824/0.836 for $\text{RB}{{\text{E}}_{10}}$. Calculated p-values  <10⁻⁵ confirm that the results are significant at 1% level, proving the capability of our approach to reproduce experimental data. From the comparison of RBE prediction by assuming ${{\text{R}}_{\beta}}=1$ or ${{\text{R}}_{\beta}}(L)$, we did not find any statistically significant difference between the two approaches.

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For completeness, the same analysis was performed for all the introduced models. The resulting correlation coefficients are reported in table 4. They were all found to reproduce the $\text{RB}{{\text{E}}_{2~\text{Gy}-\text{ph}}}$ and the $\text{RB}{{\text{E}}_{10}}$ values, with a strong positive correlation between calculated and experimental RBE values. The Pearson's correlation coefficients ranged from 0.473 ($\text{RB}{{\text{E}}_{10}}$ with ${{\text{R}}_{\beta}}=1$ and ${{f}_{\text{Q}}}$-based $\text{RB}{{\text{E}}_{\alpha}}$) to 0.859 ($\text{RB}{{\text{E}}_{2\text{Gy}-\text{ph}}}$ with ${{\text{R}}_{\beta}}(L)$ and ${{f}_{\text{QE}2}}$-based $\text{RB}{{\text{E}}_{\alpha}}$). Results were significant at p  <  0.01 and no statistically significant difference was found between the two ${{\text{R}}_{\beta}}$ descriptions.

Table 4. Summary of the Pearson's correlation coefficients applying the ten $\text{RB}{{\text{E}}_{\alpha}}$ parameterizations in combination with ${{\text{R}}_{\beta}}=1$ or ${{\text{R}}_{\beta}}(L)$ for $\text{RB}{{\text{E}}_{2~\text{Gy}-\text{ph}}}$ and $\text{RB}{{\text{E}}_{10}}$.

Model	$\text{RB}{{\text{E}}_{2~\text{Gy}-\text{ph}}}$ ${{\text{R}}_{\beta}}=1$	$\text{RB}{{\text{E}}_{2~\text{Gy}-\text{ph}}}$ ${{\text{R}}_{\beta}}(L)$	$\text{RB}{{\text{E}}_{10}}$ ${{\text{R}}_{\beta}}=1$	$\text{RB}{{\text{E}}_{10}}$ ${{\text{R}}_{\beta}}(L)$
${{f}_{\text{L}}}$	0.715	0.722	0.658	0.685
${{f}_{\text{Q}}}$	0.520	0.532	0.473	0.508
${{f}_{\text{LQ}}}$	0.843	0.844	0.791	0.805
${{f}_{\text{LQC}}}$	0.841	0.845	0.794	0.813
${{f}_{\text{LE}}}$	0.822	0.824	0.765	0.781
${{f}_{\text{QE}}}$	0.851	0.853	0.824	0.836
${{f}_{\text{LQE}}}$	0.839	0.842	0.789	0.807
${{f}_{\text{LE}2}}$	0.847	0.848	0.800	0.814
${{f}_{\text{QE}2}}$	0.856	0.859	0.832	0.845
${{f}_{\text{LQE}2}}$	0.845	0.848	0.805	0.822

The developed RBE formula (equation (6)) depends only on dose, ${{(\alpha /\beta )}_{\text{ph}}}$, $\text{RB}{{\text{E}}_{\alpha}}$ and ${{\text{R}}_{\beta}}$. The best fit parameterization of $\text{RB}{{\text{E}}_{\alpha}}$ and ${{\text{R}}_{\beta}}$ as a function of $\text{LE}{{\text{T}}^{\ast}}$ and ${{(\alpha /\beta )}_{\text{ph}}}$ and of LET, respectively, reproduce satisfactorily the analyzed experimental data as shown both qualitatively in the figures 3, 5 and 6 and quantitatively by Pearson's coefficients.

The parameterization introduced in this work predicts α and β values for He ions in the track segment condition, i.e. for ions at a given energy/LET. In therapeutic scenarios this condition is never met, due to energy loss processes and nuclear interactions. The mixed radiation field is composed mainly of primary He beam and secondary charged fragments with charges (Z) equal to 1 or 2 (Ströbele et al 2012). Our formula can be applied for predicting the biological response for Z  =  2 while for Z  =  1, the Wedenberg et al. parameterization can be used as a sound starting point. Hence, to compute the biological effect on-the-fly for such mixed radiation fields, the common approach based on the dose-weighted averages ${{\bar{\alpha}}_{j}}$ and ${{\bar{\beta}}_{j}}$ can be applied (Zaider and Rossi 1980, Kanai et al 1997) as already described for example in Mairani et al (2013):

$$\begin{eqnarray}&&{{\bar{\alpha}}_{j}}=\frac{{\mathop \sum}_{i}\,\Delta {{d}_{i,j}}\centerdot {{\alpha}_{i,j}}}{{\mathop \sum}_{i}\,\Delta {{d}_{i,j}}}\quad \text{and}\quad \sqrt{{{{\bar{\beta}}}_{j}}}=\frac{{\mathop \sum}_{i}\,\Delta {{d}_{i,j}}\centerdot \sqrt{{{\beta}_{i,j}}}}{{\mathop \sum}_{i}\,\Delta {{d}_{i,j}}}~,\end{eqnarray} \tag{ 11 }$$

where $\Delta {{d}_{i,j}}$ is the dose from the ith charged particle (with Z  =  1, 2) with associated ${{\alpha}_{i,j}}$ and ${{\beta}_{i,j}}$ in voxel j and i runs over all particles depositing dose in voxel j. RBE and RBE-weighted dose values can be determined for each voxel of the patient knowing the absorbed dose and the dose-weighted averages ${{\bar{\alpha}}_{j}}$ and ${{\bar{\beta}}_{j}}$ (see for example Mairani et al (2010)).

In this work, a RBE parameterization for helium beams has been presented. It is able to describe the experimental in vitro data available in literature and the data from clonogenic survival experiment performed at HIT. A model for $\text{RB}{{\text{E}}_{\alpha}}$ has been introduced that depends only on ${{(\alpha /\beta )}_{\text{ph}}}$ and on $\text{LE}{{\text{T}}^{\ast}}$ values. A simpler formulation for ${{\text{R}}_{\beta}}(L)$ has been studied due to the large uncertainties affecting the β parameters. However, based on the analyzed data, there is not a statistically significant gain in using ${{\text{R}}_{\beta}}(L)$ instead of a constant ${{\text{R}}_{\beta}}=1$. Treatment planning studies with helium beams will be carried out relying on the biological model explained in this study.

T Tessonnier acknowledges support from the German Research Foundation DFG (KFO214). We acknowledge support from the German Krebshilfe (Deutsche Krebshilfe, Max Eder 108876).