On the calculation of the relative biological effectiveness of ion radiation therapy using a biological weighting function, the microdosimetric kinetic model (MKM) and subsequent corrections (non-Poisson MKM and modified MKM)

The number of clinical centers offering cancer radiation therapy with ions, mostly protons (Paganetti et al 2021) and carbon ions (Malouff et al 2020), is rapidly increasing worldwide because of the superior physical (improved dose conformity, Durante and Paganetti 2016) and biological properties (higher biological effectiveness, Tommasino and Durante 2015, Tinganelli and Durante 2020) of the treatment with respect to the X-rays of conventional radiotherapy. In this regard, it is well known that the radiation-induced damage in living organisms is strongly affected by the exposure characteristics (i.e. dose, dose rate, particle type, LET, fractionation), the system's ability to react to the stimulation, and the biological endpoint considered (Scholz 2003, McMahon and Prise 2021). Therefore, the relative biological effectiveness (RBE, IAEA 2008) of the radiation and its changes over the irradiated volume must be considered while planning a radiotherapy treatment with charged particles. Neglecting these RBE changes might result in zones of unwanted under- or over- dosage and suboptimal treatment outcomes (Wedenberg and Toma‐Dasu 2014, Peeler et al 2016, Wang et al 2020).

Because of its relevance for clinical tumor control probability (TCP) calculations (Fertil and Malaise 1985, Warkentin et al 2004, Paganetti 2017), the in vitro clonogenic survival is one of the most studied biological endpoints. Consequently, RBE calculations based on the clonogenic survival assay are used in clinics for treatment planning purposes (Kanai et al 1999, Inaniwa et al 2010, Mairani et al 2013). The clonogenic survival data (the surviving fraction S as a function of the absorbed dose D) is generally fitted with the linear-quadratic model (LQM, equation (1), Fowler 1989, McMahon 2018):

$$\begin{eqnarray}&&S=\exp (-{\alpha }D-{\beta }{D}^{2})\,,\end{eqnarray} \tag{ 1 }$$

where α and ß are exposure- and cell- specific radiosensitivity parameters representing the number of lethal events per cell per unit of dose or dose squared respectively (Jones and Dale 2018).

Although there is general consensus in literature on the ion-specific LET-dependence of α, the behavior of ß as function of the LET is topic of discussion, with different models and experiments showing contrasting results (Stewart et al 2018). Nonetheless, the linear term of the LQM for repair competent cell lines is generally characterized by an ion-specific increase with the increase LET up to 100–200 keV μm⁻¹. At higher LET values, α decreases due to overkill effects (Friedrich et al 2013). In case of repair deficient cell lines, the increase in α appears to be significantly lower than for repair competent cell lines (McMahon et al 2017). On the other hand, despite that a vanishing ß is generally observed for exposures to high LET ions, no clear trend in the LET-dependence of ß is present for energetic ions (Furusawa et al 2000, Friedrich et al 2013). This uncertainty is due to the large dispersion of the biological data (Friedrich et al 2013, Parisi et al 2021), the apparent dose-dependence of ß associated to the transition of the survival curve to a constant slope at high doses (Fowler 1989, Astrahan 2008, Park et al 2008, Matsumoto et al 2019) and possible dose-rate effects (Brenner et al 1998, Matsuya et al 2018). Nonetheless, many approaches relying on the LQM formalism were developed and used for in silico computation of the clonogenic survival (Stewart et al 2018, Bellinzona et al 2021).

It is generally accepted that predictive models dealing with systems characterized by sensitive targets in the order of micro- and nano- meters (i.e. DNA, chromatin loops) must take into account the stochastic behavior of the energy deposition at these scales. The study of the stochastic nature of the energy deposition is referred to as microdosimetry (ICRU 1983) and was developed in the 1950s to overcome the limits of average quantities such as absorbed dose and the linear energy transfer (LET) in describing the radiation effects on biological samples.

Two main approaches are used for calculating the RBE using microdosimetric distributions: the ones based on the microdosimetric kinetic model (MKM, i.e. Hawkins 1994 Hawkins 2003, Kase et al 2006, Sato and Furusawa 2012), and the biological weighting functions (BWF, i.e. Morstin et al 1989, Loncol et al 1994, Parisi et al 2020b). Microdosimetric models have the advantage of dealing with quantities such as the specific energy and the lineal energy, which are experimentally measurable (Rossi and Zaider 1996, Lindborg and Waker 2017). Alternatively, approaches based on amorphous track structure calculations could be used (Katz et al 1971), such as the clinically-implemented Local Effect Model (Scholz and Kraft 1994, Elsässer et al 2010). In such models, the clonogenic survival RBE is calculated by coupling the simulated radial dose distribution around the ion track with the experimentally-determined biological dose-response after photon exposures. Furthermore, models based on different assumptions were also proposed, such as particle-specific phenomenological RBE-versus-LET correlations (Kanai et al 1999, Jones et al 2015, Rørvik et al 2018), the repair-misrepair-fixation model (RMF, Carlson et al 2008), the giant loop binary lesion model (GLOBE, Friedrich et al 2012), the biophysical analysis of cell death and chromosome aberrations model (BIANCA, Carante et al 2018), nanodosimetric models (Villegas et al 2016, Conte et al 2018, Schneider et al 2020), the multi scale approach (MSA, Surdutovich and Solov'yov 2019), and the mechanistic DNA repair and survival model (Medras, McMahon et al 2021). Finally, it is worth mentioning that the RBE relative to endpoints other than the clonogenic survival can also be modelled. As an example, the yield of DNA damages and their clustering was computed using different approaches based on track structure codes such as KURBUC (Nikjoo et al 2016), PARTRAC (Friedland et al 2017), GEANT4-DNA (Kyriakou et al 2022), RITRACKS (Plante et al 2013), and PHITS (Matsuya et al 2020).

This work aims to explore the similarities and differences in the formalism and results of the original MKM (Hawkins 1994), the non-Poisson MKM (Hawkins 2003), the modified MKM (Kase et al 2006) and the BWF from Parisi et al (2020b). Since the BWF from Parisi et al (2020b) was developed to phenomenologically reproduce only the RBE relative to a surviving fraction of 10% (RBE_10%), this article introduces a methodology to calculate the linear term of the LQM (α) from the results of this BWF and therefore compute the RBE relative to any arbitrary surviving fractions.

In vitro data for the most used mammalian cell line (Chinese hamster lung fibroblasts, V79) were extracted from the Particle Irradiation Data Ensemble (PIDE, Friedrich et al 2013) and used to benchmark the results of the different models.

Additionally, most of the models mentioned above are based on average biophysical quantities (i.e. the dose-mean lineal energy) that are related to the biological effects (i.e. α) through linear or nonlinear relationships. The effect of pre- or post- processing the average physical quantities during the RBE calculations is also investigated.

2.1. Microdosimetry

In this article, the following definitions apply as defined in ICRU (1983, 2011).

The lineal energy y is a stochastic quantity defined as the quotient between the energy imparted ε by a single event in a given volume of matter and the mean chord length $\overline{l\,}$of that volume. The mean chord length of a volume is the mean length of randomly oriented chords through that volume. For a spherical volume, the mean chord length is equal to 2/3 of the diameter.

  

The non-stochastic dose-mean lineal energy ${\bar{y}}_{D}$ is calculated as in equation (2):

$$\begin{eqnarray}&&{\bar{y}}_{D}=\,{\int }^{}y\,d(y)\,{\rm{d}}y\,,\end{eqnarray} \tag{ 2 }$$

where d(y) is the dose probability density d(y) of the lineal energy y.

2.2. Computer simulations

2.3. Biophysical models

This section briefly describes the models used for the clonogenic survival calculations. It is worth noting that all models correlate the in vitro cell death to radiation-induced damaging events within domains located within the cell nucleus. These domains are generally modelled using simulated water spheres with a diameter of approximally 0.5–1 μm (Hawkins 1994, 2003, Kase et al 2006, Sato et al 2011, Sato and Furusawa 2012, Parisi et al 2020b). These spheres are meant to be representative of cellular structures such as giant chromatin loops or chromosome territories (Kellerer and Rossi 1978, Yokota et al 1995, Meaburn and Misteli 2007, Friedrich et al 2012). The inactivation of one of these domains is assumed to lead to cell death. Other in vitro mechanisms of cells death such as membrane rupture or intracellular communication (bystander effects) are not accounted in this study. Similarly, because of the large spread in the data and the uncomplete report of dosimetric information in most publications (Parisi et al 2021), dose rate effects are not explicitly considered.

The following models were considered in this article: the original MKM (Hawkins 1994), the non-Poisson MKM (Hawkins 1994), the modified MKM (Kase et al 2006) and the biological weighting function from Parisi et al (2020b). In all cases, the following quantities were assessed and compared with corresponding in vitro data (paragraph 2.4): RBE_α , RBE_50%, RBE_10%, RBE_1%.

The RBE for the linear term of the LQM (RBE_α ) was defined as in equation (3) as the ratio between the corresponding values after the ion (α) or the reference photon (α_ref) exposures:

$$\begin{eqnarray}&&RB{E}_{\alpha }=\,\frac{\alpha }{{\alpha }_{ref}}\,.\end{eqnarray} \tag{ 3 }$$

The RBE for the quadratic term of the LQM (RBE_ß ) was defined as in equation (4) as the ratio between the corresponding values after the ion (ß) or the reference photon (ß_ref) exposures:

$$\begin{eqnarray}&&RB{E}_{\beta }=\,\frac{\beta }{{\beta }_{ref}}\,.\end{eqnarray} \tag{ 4 }$$

Furthermore, the RBE relative to the surviving fraction S (RBE_S) was calculated using equation (5) (Parisi et al 2020b):

$$\begin{eqnarray}&&RB{E}_{S}=\,\frac{\alpha \,+\,\sqrt{{\alpha }^{2}-\,4\,\beta \,\mathrm{ln}(S)}}{{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,{\beta }_{\mathrm{ref}}\,\mathrm{ln}(S)}}\,,\end{eqnarray} \tag{ 5 }$$

where α, ß, α_ref and ß_ref are respectively the linear and quadratic terms of the LQM for the radiation under investigation and the reference photon exposure.

The expression for RBE_α (equation (3)) can be obtained as the limit of RBE_S (equation (5)) for D→0. On the other hand, the expression for RBE_ß cannot be derived from equation (5) since the limit of the RBE for D→+∞ tends to the square root of ß/ß_ref. Because negative values of ß were reported in several in vitro experiments (Friedrich et al 2013), it was decided to define RBE_ß in this study as the simple ratio between ß and ß_ref. This was done to facilitate the comparison between in silico and in vitro data, and to not be forced to reject experiments with negative ß values. The negative RBE_ß values can be due to the presence of a subpopulation of cells with increased radiation resistance (Friedrich et al 2021). Despite the definition of RBE_ß in equation (4) is quite arbitrary, the calculation of the RBE for an infinite dose as the square root of ß/ß_ref may be inaccurate since it is well-known that the in vitro clonogenic survival curves at high doses are characterized by a transition to constant slope (Fowler 1989, Astrahan 2008, Park et al 2008, Matsumoto et al 2019). In other words, since ß at high doses appears to be 0, the in vitro RBE for an infinite dose would not be equal to the square root of ß/ß_ref but to the ratio between the linear terms valid in this dose range (different from α values determined at relatively low doses).

2.3.1. Adaptation of the V79-RBE_10% BWF to α

According to the model formalism (equation (6), Parisi et al 2020b), the in vitro clonogenic survival RBE for to a surviving fraction of 10% (RBE_10%) can be calculated as the integral between d(y) and the biological weighting function ${r}_{RB{E}_{10 \% },\,BWF}\left(y\right)$

$$\begin{eqnarray}&&RB{E}_{10 \% ,\,BWF}={\int }_{{y}_{\min \,}=0.016\,\mathrm{keV}/\mu {\rm{m}}}^{{y}_{\max \,}= \tilde 4000\,\mathrm{keV}/\mu {\rm{m}}}{r}_{RB{E}_{10 \% },\,BWF}\left(y\right)\,d\left(y\right)\,{\rm{d}}y\,.\end{eqnarray} \tag{ 6 }$$

y_min is the minimum lineal energy value considered for the RBE calculations and it was set to 0.016 keV μm⁻¹ (lineal energy relative to an event of one ionization only in a 1 μm water sphere, paragraph 2.2).

The BWF is plotted in figure 1 and it is characterized by a flat profile (r(y) ∼ 1) between roughly 0.01 and 1 keV μm⁻¹, a local maximum (r(y) ∼ 4.5) around 130 keV μm⁻¹ and a subsequent decrease is due to overkill effects. The function reaches 0 around 4000 keV μm⁻¹ (y_max).

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This biological weighting function was determined through an iterative process aimed at minimizing the difference between the in silico RBE calculated using PHITS-simulated lineal energy spectra and experimental in vitro RBE_10% data points for Chinese hamster lung fibroblasts (V79 cell line) in normoxic asynchronized conditions. The lineal energy spectra were assessed in liquid water for a sphere with diameter equal to 1 μm. Using this unique function, it was possible to reproduce the non-unique LET-dependence of the V79 in vitro RBE_10% values for ions from ¹H to ²³⁸U (unrestricted LET in water range: ∼0.2–15000 keV μm⁻¹) without any adjustments of its shape. In other words, the same BWF can be unchangingly applied to calculate the RBE_10% of the V79 cell line for all particles (Parisi et al 2020b).

Recently, it was proven possible to use the V79-RBE_10% BWF for modeling in first approximation also the in vitro RBE_10% of other repair-competent asynchronized normoxic mammalian cell lines (Parisi et al 2021).

Under the assumption that the quadratic term of the LQM is radiation-independent (ß = ß_ref as in the original MKM and subsequent modifications,Hawkins 1994, Kase et al 2006, Sato and Furusawa 2012), an expression to calculate α from the results of the BWF was derived (equation (7), see Appendix):

$$\begin{eqnarray}&&\alpha \,=\,\frac{RB{E}_{10 \% }^{2}\,{\left({\alpha }_{ref}\,+\,\sqrt{{\alpha }_{ref}^{2}\,-4\,{\beta }_{ref}\,\mathrm{ln}\left(0.1\right)}\right)}^{2}+4\,{\beta }_{ref}\,\mathrm{ln}\left(0.1\right)\,}{2\,RB{E}_{10 \% }\,\left({\alpha }_{ref}\,+\,\sqrt{{\alpha }_{ref}^{2}\,-4\,{\beta }_{ref}\,\mathrm{ln}\left(0.1\right)}\,\right)}\,.\end{eqnarray} \tag{ 7 }$$

Only positive α values were accepted. The numerical values of α_ref = 0.184 Gy⁻¹ and ß_ref = 0.02 Gy⁻² were chosen for consistency with the ones used for the calculations with the MKM-based models (see paragraph 2.3.6) Nonetheless, these values are very similar to the ones obtained by averaging the corresponding ones for the V79 photon PIDE entries included in this study, namely α_ref = 0.189 Gy⁻¹ and ß_ref = 0.02 Gy⁻².

2.3.2. MKM

Following the methodology described in Hawkins (1994), the linear term of the LQM can be assessed using equation (8):

$$\begin{eqnarray}&&{\alpha }_{MKM}={\alpha }_{0}\,+\,{\beta }_{ref}\,\frac{{\bar{y}}_{D}^{\,}}{\rho \,\pi \,{r}_{d}^{2}}\,,\end{eqnarray} \tag{ 8 }$$

where α₀ is a cell-specific constant representing the initial slope of the survival curve in the limit of y → 0, ß_ref is the quadratic term of the LQM model for the reference photon exposure, ${\bar{y}}_{D}$ is the dose-mean lineal energy assessed in a domain with density ρ and radius r_d. The model parameters are listed in paragraph 2.3.6.

The quadratic term of the LQM was assumed to be radiation independent (ß = ß_ref). The same assumption holds for the other MKM-based models described in the following paragraphs.

2.3.3. Non-poisson MKM

In a later work (Hawkins 2003), the original model was modified to account for the overkill effects at high LET. An additional factor was introduced in the calculation of α (equation (9)) to consider the non-Poisson distribution of the lethal lesions among the cells:

$$\begin{eqnarray}&&{\alpha }_{non-Poisson\,MKM}\,=\left({\alpha }_{0}+{\beta }_{ref}\,\frac{{\bar{y}}_{D}^{\,}}{\rho \,\pi \,{r}_{d}^{2}}\right)\,\left\{\frac{1-\exp \left[-\left({\alpha }_{0\,}+\,{\beta }_{ref}\,\displaystyle \frac{\,{\bar{y}}_{D}^{\,}}{\rho \,\pi \,{r}_{d}^{2}}\,\right)\,\displaystyle \frac{{\bar{y}}_{D,\,n}^{\,}}{\rho \,\pi \,{R}_{n}^{2}}\,\,-\,{\beta }_{ref}\,{\left(\displaystyle \frac{{\bar{y}}_{D,\,n}^{\,}}{\rho \,\pi \,{R}_{n}^{2}}\right)}^{2}\,\right]}{\left({\alpha }_{0\,}+\,{\beta }_{ref}\,\displaystyle \frac{\,{\bar{y}}_{D}^{\,}}{\rho \,\pi \,{r}_{d}^{2}}\,\right)\,\displaystyle \frac{{\bar{y}}_{D,\,n}^{\,}}{\rho \,\pi \,{R}_{n}^{2}}\,\,+\,{\beta }_{ref}\,{\left(\displaystyle \frac{{\bar{y}}_{D,\,n}^{\,}}{\rho \,\pi \,{R}_{n}^{2}}\right)}^{2}}\right\}\,\,.\end{eqnarray} \tag{ 9 }$$

α₀ is the initial slope of the survival curve in the limit of y → 0, ß_ref is the radiation-independent quadratic term of the LQM model assessed in case of the reference photon exposure, ${\bar{y}}_{D}$ is the dose-mean lineal energy assessed in a domain (radius r_d), ${\bar{y}}_{D,\,n}^{\,}$ is the dose-mean lineal energy in the nucleus (radius R_n), and ρ is the density.

For simplicity, it is generally assumed that the dose-mean lineal energy is independent of the site diameter (${\bar{y}}_{D}^{}\sim {\bar{y}}_{D,\,n}^{\,},$ Hawkins 2003, Kase et al 2006). Consequently, all lineal energy values included in this article are relative to quantities assessed in the domain.

2.3.4. Modified MKM

An alternative way to account for the overkill effects was introduced in Kase et al (2006). In this case, the linear term of the LQM can be assessed using equation (10):

$$\begin{eqnarray}&&{\alpha }_{modified\,MKM}=\,{\alpha }_{0}\,+\,{\beta }_{ref}\,\frac{{\bar{y}}_{D\,}^{* }}{\rho \,\pi \,{r}_{d}^{2}}\,,\end{eqnarray} \tag{ 10 }$$

where α₀ is the initial slope of the survival curve in the limit of y → 0, ß_ref is the radiation-independent quadratic term of the LQM model assessed in case of the reference photon exposure, ${\bar{y}}_{D}^{* }\,$is the saturation-corrected dose-mean lineal energy assessed in a domain with density ρ and radius r_d.

The saturation-corrected dose-mean lineal energy is a phenomenological quantity introduced to reproduce the results of biological experiments (Kellerer and Rossi 1974, Sato et al 2011) and it is calculated using equation (11):

$$\begin{eqnarray}&&{\bar{y}}_{D}^{* }\,=\,{y}_{0}^{2}\,\int \frac{[1-\exp \left(-{y}^{2}/{y}_{0}^{2}\right)]}{y}\,d\left(y\right)\,dy\,,\end{eqnarray} \tag{ 11 }$$

where ${y}_{o}$ is a saturation parameter that can be assessed using equation (12) (Kase et al 2006):

$$\begin{eqnarray}&&{y}_{0}\,=\frac{\rho \,\pi \,{r}_{d}\,{R}_{n}^{2}\,}{\sqrt{\beta \,({r}_{d}^{2}\,+\,{R}_{n}^{2})}}\,.\end{eqnarray} \tag{ 12 }$$

2.3.5. Implementation of the models as weighting functions

As described in the previous paragraphs, MKMs allow the calculation of the biological effects by analyzing average biophysical quantities such as the dose-mean lineal energy (original MKM and non-Poisson MKM) or the saturation-corrected dose-mean lineal energy (modified MKM). The relationship between these quantities and the biological effect is either linear (original MKM and modified MKM) or nonlinear (non-Poisson MKM). Similarly, the RBE_10% calculated with the V79-RBE_10% BWF was related to RBE_α through equation (8). In other words, the biological effect is computed as the effect relative to a dose-weighted mean value of the microdosimetric spectrum. Since these models calculate the relevant mean quantities (i.e. ${\bar{y}}_{D},$ ${\bar{y}}_{D}^{\ast }$) before computing the resulting biological effect, this article refers to them as 'pre-processing average' models.

Alternatively, we investigate the possibility of calculating α as the dose-weighted mean of the biological effect over each bin of the microdosimetric spectrum. In other words, these 'post-processing average' models firstly calculate the α as a function of the lineal energy for an event of lineal energy y. In a second step, the mean α is calculated by performing a dose-weighted average of the computed α values in each bin.

This process can be mathematically described as in equation (13):

$$\begin{eqnarray}&&{\alpha }_{post-processing\,average}\,=\,{\int }_{{y}_{\min }}^{{y}_{\max }}\alpha \left(y\right)\,d\left(y\right)\,\,{\rm{d}}y\,,\end{eqnarray} \tag{ 13 }$$

where α(y) is the linear term of the LQM for an event of lineal energy y. y_min and y_max are the minimum and maximum lineal energy values considered for the RBE calculations. The integrations limits depend on the model used (see hereunder).

Similarly, RBE_α can be calculated with equation (14):

$$\begin{eqnarray}&&RB{E}_{\alpha ,\,post-processing\,average\,}=\,\,\frac{1}{{\alpha }_{ref}}\,{\int }_{{y}_{\min }}^{{y}_{\max }}\alpha \left(y\right)\,d\left(y\right)\,{\rm{d}}y\,=\,{\int }_{{y}_{\min }}^{{y}_{\max }}r\left(y\right)\,d\left(y\right)\,{\rm{d}}y\,,\end{eqnarray} \tag{ 14 }$$

where the weighting function ${r}_{{RBE}_{\alpha }}\left(y\right)\,$was defined as the ratio between α(y) and α_ref. Equation (14) is equivalent to the one commonly used for RBE calculations with BWFs (Loncol et al 1994, Parisi et al 2020b).

The weighting functions for the different models were derived by calculating the α value in case of lineal energy spectra composed by an event of lineal energy y. A similar approach was used in Kase et al 2006 while comparing the modified MKM and the non-Poisson MKM.

The ${r}_{{RBE}_{\alpha }}\left(y\right)\,$are listed hereunder for the different models.

r_{RBE α} (y) for the V79-RBE_10% BWF, equation ( 15 ):

$$\begin{eqnarray}&&\begin{array}{c}{r}_{RB{E}_{\alpha },\,BWF}(y)=\frac{\alpha (y)}{{\alpha }_{ref}}=\,\,\frac{{r}_{RB{E}_{10{\rm{ \% }}},\,BWF}^{2}(y)\,\,{({\alpha }_{ref}+\sqrt{{\alpha }_{ref}^{2}-4{\beta }_{ref}{\rm{l}}{\rm{n}}(0.1)})}^{2}+\,4\,{\beta }_{ref}\,{\rm{l}}{\rm{n}}(0.1)\,}{2\,{\alpha }_{ref}\,{r}_{RB{E}_{10{\rm{ \% }}},\,BWF}(y)\,\,{({\alpha }_{ref}+\sqrt{{\alpha }_{ref}^{2}-4{\beta }_{ref}{\rm{l}}{\rm{n}}(0.1)})}^{}}\,\,,\end{array}\end{eqnarray} \tag{ 15 }$$

where ${r}_{RB{E}_{10 \% },\,BWF}$ is the V79-RBE_10% BWF (figure 1, Parisi et al 2020b), y_min = 0.016 keV μm⁻¹ (lineal energy relative to an event of one ionization only in a 1 μm water sphere, paragraph 2.2), y_max = 1100 keV μm⁻¹ (see hereunder).

Since only positive values of ${r}_{RB{E}_{\alpha },\,\,BWF}\left(y\right)$ were accepted for the calculation of the RBE_α , the weighting function can be used only when the following condition is satisfied (Inequality 16):

$$\begin{eqnarray}&&\begin{array}{c}{r}_{RB{E}_{10 \% },\,BWF}^{2}(y)\,\,{\left({\alpha }_{ref}\,+\,\sqrt{{\alpha }_{ref}^{2}\,-4\,{\beta }_{ref}\,\mathrm{ln}\left(0.1\right)}\right)}^{2}\,+\,\,4\,{\beta }_{ref}\,\mathrm{ln}\left(0.1\right)\,\geqslant \,0\end{array}\,,\end{eqnarray} \tag{ 16 }$$

namely for ${r}_{RB{E}_{10 \% },\,BWF}(y)$ ≥ 0.66 and thus y ≤ ∼1100 keV μm⁻¹.

It is worth noting that the BWF function for RBE_α is equal to 0 at ∼1100 keV μm⁻¹, while the BWF for the RBE_10% (Parisi et al (2020b) reaches 0 at ∼4000 keV μm⁻¹.

Because of the different upper integration limits (y_max), it is expected that the RBE_α assessed using equations (14) and (15) will differ from the one obtained by using the α calculated with equation (7). Consequently, the RBE_α was assessed with both methodologies and compared.

r_{RBE α} (y) for the MKM, equation ( 17 ):

$$\begin{eqnarray}&&{r}_{RB{E}_{\alpha },\,MKM}\left(y\right)\,=\,\frac{\alpha (y)}{{\alpha }_{ref}}\,=\,\frac{1}{{\alpha }_{ref}}\,\left({\alpha }_{0}+\,{\beta }_{ref}\frac{y}{\rho \,\pi \,{r}_{d}^{2}}\,\,\right)\,\,,\,\end{eqnarray} \tag{ 17 }$$

y_min = 0.035 keV μm⁻¹ (lineal energy relative to an event of one ionization only in a 0.464 μm water sphere, paragraph 2.2), y_max = +∞ keV μm⁻¹.

The RBE_α values calculated with equations (14) and (17) or by using the α from equation (8) were checked to be mathematically equivalent (see Appendix).

r_{RBE α} (y) for the non-Poisson MKM, equation ( 18 ):

$$\begin{eqnarray}&&\begin{array}{c}{r}_{RB{E}_{\alpha },\,non-Poisson\,MKM}(y)=\,\frac{\alpha (y)}{{\alpha }_{ref}}=\frac{1}{{\alpha }_{ref}}\,\left({\alpha }_{0}+\,{\beta }_{ref}\frac{y}{\rho \,\pi \,{r}_{d}^{2}}\,\,\right)\left\{\frac{1-\exp \left[-\left({\alpha }_{0},+,\,,{\beta }_{ref},\,,\displaystyle \frac{\,y}{\rho \,\pi \,{r}_{d}^{2}}\,\right)\,\displaystyle \frac{y}{\rho \,\pi \,{R}_{n}^{2}}\,\,-\,{\beta }_{ref}\,{\left(\frac{y}{\rho \pi {R}_{n}^{2}}\right)}^{2}\,\right]}{\left({\alpha }_{0}+\,{\beta }_{ref}\,\displaystyle \frac{\,y}{\rho \,\pi \,{r}_{d}^{2}}\,\right)\,\displaystyle \frac{y}{\rho \,\pi \,{R}_{n}^{2}}\,\,+\,{\beta }_{ref}\,{\left(\frac{y}{\rho \pi {R}_{n}^{2}}\right)}^{2}}\right\}\,,\,\end{array}\end{eqnarray} \tag{ 18 }$$

y_min = 0.035 keV μm⁻¹ (lineal energy relative to an event of one ionization only in a 0.464 μm water sphere, paragraph 2.2), y_max = +∞ keV μm⁻¹.

Since the relationship between ${\bar{y}}_{D}$ and α is nonlinear, the RBE_α values calculated using equations (14) and (18) or using the α from equation (9) are expected to differ. Consequently, the RBE_α was assessed with both methodologies and compared.

r_{RBE α} (y) for the modified MKM, equation ( 19 ):

$$\begin{eqnarray}&&{r}_{RB{E}_{\alpha },\,modified\,MKM}\left(y\right)\,=\,\frac{\alpha (y)}{{\alpha }_{ref}}\,=\,\frac{1}{{\alpha }_{ref}}\,\left({\alpha }_{0}+\,\frac{{\beta }_{ref}}{\rho \,\pi \,{r}_{d}^{2}}\,\,\frac{{y}_{0}^{2}\,[1-\exp \left(-{y}^{2}/{y}_{0}^{2}\right)]}{y}\right)\,\,,\end{eqnarray} \tag{ 19 }$$

y_min = 0.035 keV μm⁻¹ (lineal energy relative to an event of one ionization only in a 0.464 μm water sphere, paragraph 2.2), y_max = +∞ keV μm⁻¹.

The RBE_α values calculated with equations (14) and (19) or by using the α from equation (10) were checked to be mathematically equivalent (see Appendix).

2.3.6. Model parameters

The optimal parameters for the V79 cell line were extracted from Sato et al (2011) and used for the calculations with all MKMs, namely: α₀ = 0.105 Gy⁻¹, α_ref = 0.184 Gy⁻¹, β_ref = 0.02 Gy⁻², r_d = 0.232 μm, R_n = 4.14 μm, ρ = 1 g cm⁻³, and y₀ = 133.1 keV μm⁻¹. As stated in paragraph 2.3.1, the same values of α_ref and β_ref were used also for the calculations using with the BWF.

The lineal energy spectra were assessed for spherical water targets diameter equal to 0.464 μm and 1 μm as required by the MKMs and the BWF respectively. A comparison between the PHITS-simulated lineal energy distributions for these two targets can be found in Parisi et al (2020b) for different ions and energies.

2.4.  In vitro data

2.5. Comparison between data series

In agreement with previous studies (Parisi et al 2021, 2020b), the agreement between the theoretical and the n experimental values was quantified using the average relative difference $\bar{{\rm{\Delta }}}$ (equation (20)) and the average relative absolute deviation $\bar{R}$ (equation (21)).

The first quantity $\bar{{\rm{\Delta }}}$ was used to investigate possible systematic under- or ever- estimations between the in silico and in vitro data (not possible to assess using $\bar{R}$). On the other hand, $\bar{R}$ is used as an indication of the average agreement between the data series by preventing that relative difference values of opposite sign would cancel themselves.

Finally, in vitro RBE values equal to 0 were excluded from the calculations with equations (20) and (21):

$$\begin{eqnarray}&&\bar{{\rm{\Delta }}}\,=\,\frac{1}{n}\displaystyle \sum _{i=1}^{n}\frac{RB{E}_{in\,silico}\,-\,RB{E}_{in\,vitro}}{RB{E}_{in\,vitro}}\,,\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\bar{R}=\,\frac{1}{n}\displaystyle \sum _{i=1}^{n}\frac{\left|RB{E}_{in\,silico}\,-\,RB{E}_{in\,vitro}\right|}{RB{E}_{in\,vitro}}\,.\end{eqnarray} \tag{ 21 }$$

3.1. Comparison between the different RBE_α weighting functions

The weighting functions r_{RBE α}(y) derived from the formalism of the V79-RBE_10% BWF (equation (15)), the original MKM (equation (17)), the non-Poisson MKM (equation (18)) and the modified MKM (equation (19)) are plotted in figure 2 as a function of the lineal energy in water y.

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The functions are characterized by a reasonable agreement at intermediate lineal energy values (10 < y < 100 keV μm⁻¹). However, relevant differences are present outside this range.

At lower lineal energy values (y < 10 keV μm⁻¹) the functions tend to two different values, namely 1 for the one derived from the V79-RBE_10% BWF, and ∼0.57 for the functions based on the MKM. In the first case, this is because the V79-RBE_10% BWF was constructed in Parisi et al (2020b) to be ∼1 for y < 1 keV μm⁻¹, in agreement with the corresponding in vitro RBE_10% data. On the other hand, the MKM-based r_{RBE α}(y) functions tend to the ratio between α₀ and α_ref (0.105/0.184 = ∼0.57), V79 parameters from Sato et al (2011) for y → 0.

For y > 100 keV μm⁻¹, these differences become even more marked. The r_{RBE α}(y) function based on the original MKM diverges because of its lack of corrections for the overkill effects. By contrast, the other three r_{RBE α}(y) functions show a decrease with the increase of the lineal energy after reaching a local maximum between 100 and 200 keV μm⁻¹. While the r_RBEα (y) function derived from the V79-RBE_10% BWF reaches 0 at around 1000 keV μm⁻¹, the other two functions (non-Poisson MKM and modified MKM) tend to an asymptotic value of 0 and 0.57 respectively for y → +∞.

Furthermore, a measure of the effect on RBE_α induced by an event of lineal energy value y can be obtained by multiplying the r_{RBE α} (y) functions of figure 2 by y ¹ . The resulting quantity e_RBEα (y) helps visualizing the implementation of saturation effects in the different formalisms (figure 3). It is expected that an e(y) function representing saturation effects would tend to a constant value above a certain threshold of the variable it depends on (in this case, the lineal energy y). Except for the e_{RBE α}(y) function based on the non-Poisson MKM, the other functions behave very differently from the expected saturation behavior. The e_{RBE α}(y) function for the original MKM diverges, the one for the modified MKM shows a monotonic increase after an inflection point, and the function based on the V79-RBE_10% BWF sharply decreases reaching 0 around 1000 keV μm⁻¹.

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The V79-RBE_10% BWF was determined in Parisi et al (2020b) to phenomenologically reproduce the V79 in vitro RBE_10% data. However, since the in vitro clonogenic survival curves for very high LET exposures (LET > 500 keV μm⁻¹) are characterized by a decrease in both the linear (α) and the quadratic (ß) term of the LQM with respect to the reference photon exposure, the V79-RBE_10% BWF intrinsically accounts for the decrease of both terms. In this regard, figure 4 compares the RBE_ß for the in vitro data included in this study as a function of the LET in water. Despite the large scatter in the in vitro data and an unclear trend for LET < 500 keV μm⁻¹, the RBE_ß appears to tend to 0 at higher LET values.

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However, in order to derive an expression (equation (7)) to calculate α from the results of the V79-RBE_10% BWF, a radiation independent ß was assumed (see Appendix). Consequently, the α values calculated with equation (7) are underestimated at high LET to compensate for the presence of ß = ß_ref. This causes the sharp decrease of the corresponding weighting functions in figures 2 and 3.

Furthermore, since the increasing behavior of the e_{RBE α}(y) function based on the modified MKM is caused by presence of the constant term α₀, an additional weighting function was constructed by setting α₀ = 0 in equation (19). The resulting r_RBEα (y) and e_RBEα (y) functions are also plotted in figures 2 and 3 respectively. This r_RBEα (y) is proportional to the saturation-corrected dose-mean lineal energy with no offsets, and therefore the function tends to 0 for y → 0 and for y → +∞ (figure 2). It is worth noting that this r_RBEα (y) is very similar to the one for the non-Poisson MKM for y > 200 keV μm⁻¹, i.e. after the local maximum of the functions. Finally, the corresponding e_RBEα (y) function (modified MKM with _α0 = 0) seems to follow the expected trend for the description of saturation effects (figure 3).

3.2. Benchmark against in vitro data

As a benchmark of the results obtained combining the PHITS-simulated lineal energy spectra (target = water sphere with diameter equal to 0.464 or 1.0 μm) and the different biophysical models, figures 5–8 compare the in silico calculations (RBE_α , RBE_50%, RBE_10%, RBE_1%) with corresponding in vitro V79 data from PIDE. The results are plotted as a function of the ion type and the unrestricted LET in water.

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Furthermore, figure 9 shows the average relative difference (equation (20)) and the average relative deviation (equation (21)) as a function of the ion atomic mass number for all biophysical models and RBE endpoints. To facilitate the visualization of the data, the comparison is limited to average relative difference and average relative deviation values up to 2.

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All RBE data series in figures 5, 6, 7 and 8 follow a similar trend as a function of the LET: the RBE increases with the increase of the LET up to an ion-specific local maximum located around 100–200 keV μm⁻¹. At higher LET values, the RBE decreases due to the well-known overkill effects. Furthermore, smaller RBE values are obtained for RBE values relative to smaller surviving fractions. However, relevant differences are present between the results of the different models.

In order to fully exploit the potential of cancer radiation therapy with charged particles, accurate and robust biophysical models of radiation induced effects are needed for treatment planning. Because of its relevance of TCP and NTCP, most models consider the clonogenic survival as a relevant endpoint for optimization purposes (Paganetti 2017). However, the clonogenic survival is a complex phenomenon which depends on many factors such as the radiation quality, the dose, the dose- rate, the fractionation scheme, the cellular oxygen level, possible bystander effects, and the chosen cell line (Scholz 2003, Durante et al 2021, McMahon and Prise 2021).

In this work, the effect of radiation quality and dose was systematically investigated by comparing the results of the most used microdosimetric models of clonogenic survival, namely the original MKM (Hawkins 1994), the non-Poisson MKM (Hawkins 2003) and the modified MKM (Kase et al 2006). The BWF from Parisi et al 2020b was also included for comparison and to be representative of modeling approaches based on BWFs. Clonogenic survival in vitro data for the V79 cell line (Chinese hamster lung fibroblasts) were extracted from PIDE (Friedrich et al 2013) and used to benchmark for the model calculations. All results are relative to single-dose exposures in normoxic asynchronized conditions. The V79 cell line was chosen because it is the most used mammalian cell line and represents by far the largest fraction of in vitro data in PIDE. Furthermore, model parameters for cell lines other than the V79 are not always available. The application of selected models to model the RBE of other DNA repair competent or deficient cell lines will be topic of future work.

The MKM (Hawkins 1994) and the non-Poisson MKM (Hawkins 2003) were chosen because they represent the original formulations of the model from which all subsequent MKMs were built. Furthermore, the modified MKM (Kase et al 2006) was included in the comparison because of its clinical relevance (the model has been implemented in particle therapy treatment planning systems in Japan) and its widespread application in combination with simulated or measured microdosimetric spectra (Sato et al 2009, Inaniwa et al 2010, Sato et al 2011, Kase et al 2011a, 2011b, 2013, Inaniwa et al 2015, Takada et al 2017, Debrot et al 2018, Parisi et al 2020a, Bianchi et al 2020, Tran et al 2022). The BWF from Parisi et al 2020b was chosen since it is based on to the same biological endpoint as the MKM models (clonogenic survival). By contrast, the commonly used BWF (Loncol et al 1994) is relative to a different biological endpoint, namely the early intestine tolerance assessed by in vivo crypt regeneration in mice.

Other models such as the stochastic- and double stochastic MKM (SMKM and DSMKM respectively) were proposed (Sato and Furusawa 2012) to overcome the limitations of the modified MKM (Kase et al 2006) in reproducing in vitro data relative to high LET and high dose exposures. However, the SMKM and DSMKM (Sato and Furusawa 2012) strongly differ from the models previously mentioned (Hawkins 1994, Hawkins 2003, Kase et al 2006) since they require the evaluation of dose-dependent microdosimetric distributions, which can be computationally very demanding. Despite that approximate versions of these models were developed to overcome the long computational time and memory requirements (Inaniwa and Kanematsu 2018), the resulting model formalism is based on the description of the clonogenic survival using a cubic polynomial expression in respect to dose. Furthermore, the numerical values of the optimal SMKM and DSMKM parameters (i.e. R_n) generally strongly differ (Sato and Furusawa 2012, Inaniwa and Kanematsu 2018) from the ones for the other MKMs (listed in paragraph 2.3.5). Consequently, a comparison between α and RBE values would be not straightforward. Thus, it was decided to not include the SMKM and DSMKM in the comparison.

Figure 2 shows the original implementation of the MKMs as weighting functions for the calculation of RBE_α . This representation helped to understand the differences in the processing of the microdosimetric spectra between the models. It is evident that there are relevant discrepancies between the MKMs, especially at high lineal energy values (the functions tend to different values, namely +∞, 0 or 0.57 for the original MKM, the non-Poisson MKM and the modified MKM respectively). Furthermore, the RBE_α weighting function derived from the V79-RBE_10% BWF is also included in the same plot. As discussed in paragraph 3.1, the sharp decrease of the function is due to the assumption of a radiation-independent ß. This assumption was necessary to derive equations (7) and (15) and represents a limitation of the current approach. A phenomenological BWF for RBE_α could be assessed from the in vitro data as done in Parisi et al (2020b), but it is beyond the scope of this work.

Additionally, a new quantity e(y) (figure 3) was introduced to study the implementation of saturation effects in the different models. It is expected that an e(y) function representing saturation effects would tend to a constant value for y → +∞. This appears to be the case for the non-Poisson MKM. However, things are different for the approaches based on the V79-RBE_10% BWF, the original MKM and the modified MKM. In the first case, the sharp decrease in the function is due to the inclusion of the decreasing ß in the model. The e(y) for the original MKM diverges because no overkill corrections are implemented in the model. Furthermore, after reaching an apparent saturation defined by the saturation parameter y₀ in equation (19), the effect modelled by the modified MKM increases with the increase of the lineal energy. This is conceptually in contrast with the expected saturation behavior, and it is responsible for the overestimation in the RBE_α values calculated with this model at high LET. This increase is due to the presence of the constant term α₀ in the model formalism.

To try to mitigate this overestimation, an additional biological weighting function was introduced by setting α₀ = 0 in the formalism of the modified MKM. Setting α₀ = 0 implies that pairwise combination of lesions is considered the most relevant mechanism of cell death at all dose levels (Hawkins 2005). This function (red dotted line in figure 3) seems to be very similar to the non-Poisson MKM function for y > 200 keV μm⁻¹, and it can adequately describe the saturation effects. Consequently, this version of the modified MKM with α₀ = 0 should be regarded as a high LET approximation of the non-Poisson MKM. Nonetheless, at low linear energy (y < 20 keV μm⁻¹) this function seems to be characterized by RBE_α values which are too small with respect to the experimental observations (figure 2). As an example, the dose-mean lineal energy for photons generally used as a reference in radiobiological experiments is generally between ∼2 and 4 keV μm⁻¹ (obtained with simple monoenergetic PHITS photon simulations), values in agreement with the ones reported in Kase et al (2006) and Sato et al (2011). The average RBE_α calculated by the modified MKM with α₀ = 0 ranges between approximately 0.2 and 0.4 for 2 < y < 4 keV μm⁻¹, in contrast with the values as calculated with the other two approaches (∼0.8–1.4). Therefore, an underestimation in the calculated RBE_α is expected for low lineal energy particles. This could have been mitigated by multiplying the function by a factor of ∼3. However, this would have resulted in a large overestimation of the RBE_α for y > 10 keV μm⁻¹. Consequently, it was decided to use this function without further corrections and to focus the evaluation of its accuracy at high LET.

A systematic comparison between in silico and in vitro RBE data for the V79 cell line can be found in figures 5–8. It seems from figure 5 that all models underestimate the in vitro RBE_α for hydrogen ions. The underestimation is the worst for the modified MKMwith α₀ = 0. The calculated RBE_α values are unreasonably small in case of energetic light particles whose pattern of energy deposition is similar to the one of the reference photons.

Since the original MKM does not include corrections for the overkill effects, the model fails to reproduce all in vitro data relative to exposures to ions with a LET > ∼100 keV μm⁻¹. The ion-specific results of all models are in reasonable agreement with each other and the corresponding in vitro RBE_α data for LET < 100 keV μm⁻¹.

For LET > 100 keV μm⁻¹, the in silico RBE_α results for the BWF-model, the non-Poisson MKM, the modified MKM and the modified MKM with α₀ = 0 differ because of the different implementations of overkill effects (figure 3).

Furthermore, pre- or post- processing the average values during the RBE_α calculations was found to significantly affect the results obtained with the non-Poisson MKM. In this case, the RBE_α values computed by pre-processing ${\bar{y}}_{D}$ (equation (9)) are significantly higher than the ones obtained with the post-processing approach (equation (18)) in case of ¹²C and heavier ions. The agreement with the in vitro data is significantly better for the novel post-processing average implementation of the non-Poisson MKM (equation (18)). For the methodology based on the V79-RBE_10% BWF, the RBE_α obtained by using both processing methodologies are generally equivalent. However, when the microdosimetric spectra present entries between 1100 and 4000 keV μm⁻¹ (upper integration limits for the equations (15) and (6) respectively), differences were found between the calculated RBE_α values. These differences are most evident for ⁵⁶Fe and ¹³²Xe ions at relatively low energy. No differences are present between the results of the pre- and post- processing average methods for the MKM and the modified MKM (see Appendix).

The overall best agreement with in vitro RBE_α data was found for the novel 'post- processing average' non-Poisson MKM.  

As a consequence of the similarity between the weighting functions of the non-Poisson MKM and the modified MKM with α₀=0 in the high lineal energy region (figures 2 and 3) and the similar RBE results, the modified MKM with α₀ = 0 can be considered as a simplified high-LET approximation of the non-Poisson MKM. These findings allow to provide an additional interpretation for the data included in Baiocco et al (2016). In that work, it was found possible to reproduce the energy-dependence of the neutron weighting factors by computing the normalized saturation-corrected dose-mean lineal energy as a function of the incident neutron energy. Remembering that the secondary particles induced by neutrons have generally high LET and that the α calculated using the modified MKM with α₀ = 0 is proportional to the saturation-corrected dose-mean only, the energy dependence of the neutron weighting factors (valid for low dose) can be explained as the evolution of the corresponding α (LQM term dominant at low dose) as a function of the neutron energy as calculated with the modified MKM with α₀ = 0.

On the other hand, the results of the BWF and the modified MKM are respectively under- and overestimating the in vitro RBE_α . This is particularly evident for ⁵⁶Fe and ¹³²Xe ions.

Differently from what obtained in case of RBE_α , the in silico proton RBE_50% values (figure 6) are in reasonable agreement with the corresponding in vitro data. For the other ions, the discussion of the RBE_50% results is similar to the one for RBE_α .

In case of RBE_10% and RBE_1%, all MKM models were able to reproduce reasonably well the experimental data for all ions up to a LET of approximately 200 keV μm⁻¹. At higher LET, all MKMs overestimate the results with bigger deviations observed for the heaviest ions. This is due to the fact that, while the in vitro data are characterized by a ß tending to 0 at high LET (figure 4), the MKMs assume that ß is radiation independent and equal to ß_ref. The overestimation was the worst for the modified MKM and the 'pre-processing average' non-Poisson MKM because these models tend to generally overestimate also α at high LET (figure 5).

Finally, it is worth stating that the RBE_10% values calculated using the RBE_α approach based on the V79-BWF RBE_10% differ from the ones which could be obtained by using directly the V79-BWF RBE_10% as done in Parisi et al (2020b). This happens because the V79-BWF RBE_10% was phenomenologically constructed to reproduce the in vitro RBE_10% values, i.e. accounting for both the decrease of α and ß at high LET. On the other hand, in this work the V79-BWF RBE_10% was used to derive an expression for α by assuming, as in the MKMs, a radiation independent ß = ß_ref. Therefore, the RBE_10% calculated with the RBE_α approach based on the V79-BWF RBE_10% do not fully account for the ß decrease at high LET, and thus overestimate the in vitro RBE.

This article presents a systematic comparison of the formalism and results of microdosimetric models (V79-BWF, original MKM, non-Poisson MKM, modified MKM) of clonogenic survival for a broad range of particles (¹H to ¹³²Xe) and surviving fractions (α, 50%, 10% and 1%). The results were benchmarked against corresponding the in vitro data for the most used mammalian cell line (Chinese hamster lung fibroblast, V79) in asynchronized and normoxic conditions.

The main outcomes of this study are summarized hereunder.

• (a)  
  A recently developed model (V79-RBE_10% BWF) was modified to extend its use for calculating RBE values relative to other surviving fractions.
• (b)  
  Original expressions to implement the models in the form of lineal energy weighting functions r(y) were derived and compared. Furthermore, the determination and use of an auxiliary weighting function representative of the effect of a single event e(y) helped highlighting the differences between the implementation of overkill effects in the different models.
• (c)  
  Pre- or post- processing average biophysical quantities in the RBE calculations was proven to strongly affect the results of the non-Poisson MKM.
• (d)  
  All models appear to underestimate the in vitro RBE_α for hydrogen ions.
• (e)  
  The best overall description of the in vitro RBE_α was obtained with the novel 'post-processing average' implementation of the non-Poisson MKM. On the other hand, the modified MKM and the approach based on the V79-RBE_10% BWF appear to respectively over- and under- estimate the in vitro RBE_α data for ions with a LET > ∼200 keV μm⁻¹.
• (f)  
  A high LET approximation of the 'post-processing average' non-Poisson MKM was obtained by setting α₀ = 0 in the formalism of the modified MKM. This change improved the results for LET > ∼200 keV m⁻¹. However, the results for energetic light ions seem to significantly underestimate the biological data.
• (g)  
  No model was able to correctly describe the overall trend of the RBE at high LET for surviving fractions of 10% or smaller. This is due to the common assumption of a radiation-independent quadratic term (ß) of the LQM, which is in contrast with the in vitro results.

Alessio Parisi is very thankful to Tatsuhiko Sato (Japan Atomic Energy Agency, Japan) for the fruitful discussion on the PHITS microdosimetric function.

Alessio Parisi would like to sincerely thank Thomas Friedrich (GSI Helmholtzzentrum für Schwerionenforschung, Germany) and the whole PIDE team for sharing a copy of their in vitro database.

Appendix. A.1. Derivation of equation (7)

$$\begin{eqnarray}&&\alpha \,=\,\,\frac{RB{E}_{10 \% }^{2}\,{\left({\alpha }_{ref}\,+\,\sqrt{{\alpha }_{ref}^{2}\,-4\,{\beta }_{ref}\,\mathrm{ln}\left(0.1\right)}\right)}^{2}+\,4\,{\beta }_{ref}\,\mathrm{ln}\left(0.1\right)\,}{2\,RB{E}_{10 \% }^{\,}\,{\left({\alpha }_{ref}\,+\,\sqrt{{\alpha }_{ref}^{2}\,-4\,{\beta }_{ref}\,\mathrm{ln}\left(0.1\right)}\,\right)}^{\,}}\,\,.\end{eqnarray}$$

Equation (7) was derived from equation (5):

$$\begin{eqnarray}&&RB{E}_{S}=\,\frac{\alpha \,+\,\sqrt{{\alpha }^{2}-\,4\,\beta \,\mathrm{ln}(S)}}{{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,{\beta }_{\mathrm{ref}}\,\mathrm{ln}(S)}}\,.\end{eqnarray}$$

In case of S = 10% and assuming ß = ß_ref (see article), then:

$$\begin{eqnarray*}&&RB{E}_{10 \% }=\frac{\alpha +\sqrt{{\alpha }^{2}-4\beta \mathrm{ln}(0.1)}}{{\alpha }_{\mathrm{ref}}+\sqrt{{\alpha }_{\mathrm{ref}}^{2}-4{\beta }_{\mathrm{ref}}\mathrm{ln}(0.1)}}\,,\end{eqnarray*}$$

$$\begin{eqnarray*}&&RB{E}_{10 \% }\left[{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\right]=\,\alpha \,+\,\,\sqrt{{\alpha }^{2}-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\,,\end{eqnarray*}$$

$$\begin{eqnarray*}&&RB{E}_{10 \% }\,\left[{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\right]-\alpha \,=\,\sqrt{{\alpha }^{2}-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\,,\end{eqnarray*}$$

$$\begin{eqnarray*}&&{\left\{RB{E}_{10 \% }\left[{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\right]-\alpha \right\}}^{2}=\,{\left\{\sqrt{{\alpha }^{2}-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\right\}}^{2}\,,\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{c}RB{E}_{10{\rm{ \% }}}^{2}{[{\alpha }_{{\rm{r}}{\rm{e}}{\rm{f}}}+\sqrt{{\alpha }_{{\rm{r}}{\rm{e}}{\rm{f}}}^{2}-4\beta {\rm{l}}{\rm{n}}(0.1)}]}^{2}-\,2\,\alpha \,RB{E}_{10{\rm{ \% }}}[{\alpha }_{{\rm{r}}{\rm{e}}{\rm{f}}}\,+\,\sqrt{{\alpha }_{{\rm{r}}{\rm{e}}{\rm{f}}}^{2}\,-\,4\,\beta \,{\rm{l}}{\rm{n}}(0.1)}]+\,{\alpha }^{2}={\alpha }^{2}\,-\,4\,\beta \,{\rm{l}}{\rm{n}}(0.1)\end{array}\,,\,\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{c}RB{E}_{10{\rm{ \% }}}^{2}{[{\alpha }_{\mathrm{ref}}+\sqrt{{\alpha }_{\mathrm{ref}}^{2}-4\beta \mathrm{ln}(0.1)}]}^{2}-\,2\alpha \,RB{E}_{10{\rm{ \% }}}[{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,\beta \,\mathrm{ln}(0.1)}]\,\,=\,-4\,\beta \,\mathrm{ln}(0.1)\,\end{array},\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{c}-2\,\alpha \,RB{E}_{10{\rm{ \% }}}{[{\alpha }_{{\rm{r}}{\rm{e}}{\rm{f}}}+\sqrt{{\alpha }_{{\rm{r}}{\rm{e}}{\rm{f}}}^{2}-4\beta {\rm{l}}{\rm{n}}(0.1)}]}^{}\,=-4\,\beta \,{\rm{l}}{\rm{n}}(0.1)-\,RB{E}_{10{\rm{ \% }}}^{2}\,\,{[{\alpha }_{{\rm{r}}{\rm{e}}{\rm{f}}}+\sqrt{{\alpha }_{{\rm{r}}{\rm{e}}{\rm{f}}}^{2}-4\beta {\rm{l}}{\rm{n}}(0.1)}]}^{2}\end{array}\,,\end{eqnarray*}$$

$$\begin{eqnarray*}&&\alpha {\,}^{\,}=\frac{-\,4\,\beta \,\mathrm{ln}\left(0.1\right)-\,RB{E}_{10 \% }^{2}{\left[{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\right]}^{2}}{-2RB{E}_{10 \% }\left[{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\right]}\,,\end{eqnarray*}$$

$$\begin{eqnarray*}&&\alpha =\frac{RB{E}_{10 \% }^{2}{\left[{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\right]}^{2}+\,4\,\beta \,\mathrm{ln}\left(0.1\right)\,\,}{2RB{E}_{10 \% }\left[{\alpha }_{\mathrm{ref}}\,+\,\sqrt{{\alpha }_{\mathrm{ref}}^{2}\,-\,4\,\beta \,\mathrm{ln}\left(0.1\right)}\right]}\,.\end{eqnarray*}$$

Appendix. A.2. Equivalence of the pre- and post- processing average methodologies for the MKM and the modified MKM

A.2.1. MKM

According to the original model formalism ('pre-processing average'), α can be calculated as:

$$\begin{eqnarray*}&&{\alpha }_{MKM,\,pre-processing\,average}={\alpha }_{0}+{\beta }_{ref}\,\frac{{\bar{y}}_{D}}{\rho \,\pi \,{r}_{d}^{2}}\,.\end{eqnarray*}$$

The model was also implemented in the form of a lineal energy weighting function ('post-processing average'). In this case, α is calculated as:

$$\begin{eqnarray*}&&{\alpha }_{MKM,\,post-processing\,average}\,=\,\int {\alpha }_{MKM}(y)\,d\left(y\right)\,{\rm{d}}y\,,\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&{\alpha }_{MKM}\left(y\right)\,=\,{\alpha }_{0}\,+\,{\beta }_{ref}\frac{y}{\rho \,\pi \,{r}_{d}^{2}}\,.\end{eqnarray*}$$

It follows that:

$$\begin{eqnarray*}&&\begin{array}{c}{\alpha }_{MKM,\,post-processing\,average}=\int {\alpha }_{MKM}(y)d(y)\,{\rm{d}}y\,=\,\int ({\alpha }_{0}\,+\,{\beta }_{ref}\frac{y}{\rho \,\pi \,{r}_{d}^{2}})d(y)\,{\rm{d}}y\end{array}\,=\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{c}=\int {\alpha }_{0}\,d(y)\,{\rm{d}}y\,+\,\int \left(\,\frac{{\beta }_{ref}}{\rho \,\pi \,{r}_{d}^{2}}\,y\right)d(y)\,{\rm{d}}y\,=\,{\alpha }_{0}\int d(y)\,{\rm{d}}y\,+\,\frac{{\beta }_{ref}}{\rho \,\pi \,{r}_{d}^{2}}\int y\,d(y)\,{\rm{d}}y\,.\end{array}\end{eqnarray*}$$

Since

$$\begin{eqnarray*}&&{\int }^{\,}\,d\left(y\right)\,{\rm{d}}y=1\,\,\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&{\bar{y}}_{D}=\,{\int }^{\,}y\,d\left(y\right)\,{\rm{d}}y,\end{eqnarray*}$$

then

${\alpha }_{MKM,\,post-processing\,average}\,=\,{\alpha }_{0}+{\beta }_{ref}\,\frac{{\bar{y}}_{D}^{\,}}{\rho \,\pi \,{r}_{d}^{2}}\,=\,{\alpha }_{MKM,\,pre-processing\,average}.$

A.2.2. Modified MKM

According to the original model formalism ('pre-processing average'), α can be calculated as:

$$\begin{eqnarray*}&&{\alpha }_{modified\,MKM,\,pre-proces\sin g\,average}\,=\,{\alpha }_{0}\,+\,{\beta }_{ref}\,\frac{{\overline{)y}}_{D}^{* }}{\rho \,\pi \,{r}_{d}^{2}}\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&{\bar{y}}_{D}^{* }\,=\,{y}_{0}^{2}\,\int \frac{[1-\exp \left(-{y}^{2}/{y}_{0}^{2}\right)]}{y}\,d\left(y\right)\,dy,\end{eqnarray*}$$

as described in Sato et al 2011.

The model was implemented in the form of a lineal energy weighting function ('post-processing average'). In this case, α is calculated as:

$$\begin{eqnarray*}&&\begin{array}{c}{\alpha }_{modified\,MKM,\,post-processing\,average}\,\,=\,\int {\alpha }_{modified\,MKM}(y)\,d(y)\,{\rm{d}}y,\,\end{array}\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&\begin{array}{c}{\alpha }_{modified\,MKM}(y)\,=\,\,{\alpha }_{0}\,+\,\frac{{\beta }_{ref}}{\rho \,\pi \,{r}_{d}^{2}}\,\frac{{y}_{0}^{2}\,[1-\exp (-{y}^{2}/{y}_{0}^{2})]}{y}.\end{array}\end{eqnarray*}$$

It follows that:

$$\begin{eqnarray*}&&\begin{array}{c}{\alpha }_{modified\,MKM,\,post-processing\,average}\,=\,\int {\alpha }_{modified\,MKM}(y)\,d(y)\,{\rm{d}}y\end{array}\,=\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{c}=\,\int \left\{{\alpha }_{0}+\,\frac{{\beta }_{ref}}{\rho \,\pi \,{r}_{d}^{2}}\,\,\frac{{y}_{0}^{2}\,[1-\exp (-{y}^{2}/{y}_{0}^{2})]}{y}\right\}\,d(y)\,{\rm{d}}y\,=\,{\alpha }_{0}\int d(y)\,{\rm{d}}y\,+\,\frac{{\beta }_{ref}}{\rho \,\pi \,{r}_{d}^{2}}\int \frac{{y}_{0}^{2}\,[1-\exp (-{y}^{2}/{y}_{0}^{2})]}{y}\,d(y)\,{\rm{d}}y\,.\end{array}\end{eqnarray*}$$

Since

${\int }^{\,}\,d\left(y\right)\,{\rm{d}}y\,=\,1\,\,$

and

$$\begin{eqnarray*}&&{\bar{y}}_{D}^{* }\,=\,{y}_{0}^{2}\,\int \frac{[1-\exp \left(-{y}^{2}/{y}_{0}^{2}\right)]}{y}\,d\left(y\right)\,dy\,,\end{eqnarray*}$$

then

$$\begin{eqnarray*}&&\begin{array}{c}{\alpha }_{modified\,MKM,\,post-processing\,average}=\,{\alpha }_{0}\,+\,\,{\beta }_{ref}\,\frac{{\bar{y}}_{D}^{\ast }}{\rho \,\pi \,{r}_{d}^{2}}\,\,=\,{\alpha }_{modified\,MKM,\,pre-processing\,average}\,.\end{array}\end{eqnarray*}$$