# Clonogenic Survival RBE Calculations in Carbon Ion Therapy: The Importance of the Absolute Values of α and β in the Photon Dose-Response Curve and a Strategy to Mitigate Their Anticorrelation

^{*}

## Abstract

**:**

^{12}C ions of different LET. Furthermore, we introduce a theory-based strategy to potentially mitigate the anticorrelation between α and β during the fit of the photon dose-response biological data.

## 1. Introduction

_{10%}of two human cell lines calculated with the MCF MKM could be significantly different if dissimilar published sets of LQM terms (obtained by different authors fitting the same photon survival curve) were used as input for the calculations. It must be noted that these different LQM sets (α and β for the photon reference radiation, named α

_{ref}and β

_{ref}here and in the following) could lead to markedly dissimilar α

_{ref}/β

_{ref}ratios for the same in vitro survival curve. For instance, α

_{ref}/β

_{ref}values of 3.5 and 6.9 Gy were obtained for photon-exposed human brain glioblastoma cells (A-172 cell line) [34]. Since the α

_{ref}/β

_{ref}ratio is used as a relevant parameter for RBE calculations with models other than the MCF MKM (i.e., proton phenomenological models [35]) and for fractionation studies [36], it is very important to assess it accurately.

_{ref}/β

_{ref}respectively equal 3.5 Gy and 6.9 Gy [34]).

_{ref}/β

_{ref}equal to 3.5 Gy and 6.9 Gy, respectively.

_{0}, namely α in the limit of LET → 0. Negative values α

_{0}were reported in the case of cell lines with a low α

_{ref}/β

_{ref}[24,39]. However, the physical and biological meaning of these negative values of α

_{0}(which might lead to the computation of unreasonable negative RBE values) is unclear, if any. It is possible that the anticorrelation between α

_{ref}and β

_{ref}is one of the causes behind the computation of these negative α

_{0}values.

_{ref}/β

_{ref}after photon irradiation can be used as an indication of the radiosensitivity of the cell line for ion irradiations, with higher RBE values expected for cell lines with a lower α

_{ref}/β

_{ref}[35]. Similarly, in the case of carbon ions, higher RBE values were observed at low doses for cell lines with a lower α

_{ref}/β

_{ref}[6,23]. However, at higher doses (i.e., for a surviving fraction of 10% or 50%), the RBE values appear to not significantly depend on the α

_{ref}/β

_{ref}[6]. In addition, despite the ion RBE values calculated with the clinically implemented LEM I [8] for a fixed α

_{ref}/β

_{ref}were reported to be nearly independent of the absolute values of α

_{ref}and β

_{ref}[40], the knowledge of the absolute values of α

_{ref}and β

_{ref}appears to be necessary [41] for

^{12}C ion RBE calculations with the latest version of the LEM (LEM IV [30]) and the modified MKM [9].

- (1)
- Study the effect of α
_{ref}/β_{ref}on the RBE of^{12}C ions calculated with the MCF MKM. - (2)
- Investigate the effect of the absolute value of α
_{ref}and β_{ref}on the RBE of^{12}C ions computed using the MCF MKM in the case of two clinically relevant values of α_{ref}/β_{ref}(2 and 10 Gy). - (3)
- Explore the possibility of mitigating the anticorrelation between α
_{ref}and β_{ref}by means of an MKM-based LQM fit of published in vitro photon survival curves. The RBE values predicted with the MCF MKM in combination with the α_{ref}and β_{ref}obtained with this MKM-based LQM fit are compared with the RBE values computed using the published α_{ref}and β_{ref}values obtained by other authors fitting the same survival curve. A benchmark of the RBE against corresponding in vitro data for irradiations with^{12}C ions is also included.

## 2. Materials and Methods

#### 2.1. Relative Biological Effectiveness

_{ref}) and the radiation under investigation (D) (Equation (3)).

_{S}) using Equation (4) [28].

_{ref}, and β

_{ref}are the linear and quadratic terms of the LQM for the radiation under investigation and the reference photon exposure, respectively.

#### 2.2. Mayo Clinic Florida Microdosimetric Kinetic Model (MCF MKM)

#### 2.2.1. Historical Background

_{0}is the cell-specific value of α in the limit of y → 0 where y is the lineal energy, β

_{ref}is the quadratic term of the LQM model for the reference photon exposure, ${\overline{y}}_{D}^{}$ is the dose-mean lineal energy assessed in a domain with density ρ and radius r

_{d}.

_{ref}and β

_{ref}are determined with Equation (1) by fitting the clonogenic survival curve after photon exposure, the two MKM parameters (α

_{0}and r

_{d}) are generally assessed by fitting the results of ion-exposed in vitro experiments with Equation (5) [43].

_{0}in the modified MKM [9]. Nonetheless, it is worth mentioning that the results of these newer MKMs [25,27] were found to be equivalent to that of the original MKM [25] for LET < ~30 keV/µm [46], where Equations (5) and (7) are still valid.

#### 2.2.2. MCF MKM Formalism

_{0}and β

_{0}are the LQM terms in the limit of y → 0, ${R}_{n}$ is the mean radius of the cell nucleus, r

_{d}is the mean radius of the subnuclear domains, and ρ is the density (=1 g/cm

^{3}).

#### 2.2.3. MCF MKM Parameters

_{n}and r

_{d}) are assessed a priori without using clonogenic survival in vitro data [23,24]. As first choice, R

_{n}can be determined by means of morphometric measurements for (unattached) cells presenting a spherical nucleus. However, this information is not always available. If the cross-sectional area A of the cell nucleus is reported, an estimate of the mean radius of the cell nucleus can be obtained with Equation (11) [23,24].

_{n}can be obtained using an empirical correlation between R

_{n}and the mean DNA content of an asynchronized population Γ [Gbp] (Equation (12) [23,24])

_{n}is the number of chromosomes in a normal set (23 for humans, 21 for rats, 11 for Chinese hamsters, and 20 for mice [47]).

_{d}is computed with Equation (15) [23,24]

_{n}and r

_{d}are known, and the correction factor c(y) (Equation (10)) is ~1 for the reference photon exposure [23], then

#### 2.3. Effect of α_{ref}/β_{ref} on the Calculated RBE for ^{12}C Ions

_{ref}/β

_{ref}for a model assumption of a fixed α

_{ref}/β

_{ref}, we carried out calculations with the MCF MKM for a synthetic cell line with R

_{n}= 4.7 µm and r

_{d}= 0.29 µm. These parameters represent the mean of the corresponding values obtained for the cell lines used in previous works [23,24,34], where R

_{n}ranged between 4.0 and 5.7 µm and r

_{d}ranged between 0.26 and 0.32 µm.

_{ref}was fixed to 0.2 Gy

^{−1}while α

_{ref}/β

_{ref}was set to 100 Gy, 50 Gy, 10 Gy, 5 Gy, 1.4 Gy (α

_{ref}/β

_{ref}for which α

_{0}~0, see Section 2.5), 1 Gy, 0.5 Gy, and 0.1 Gy. The values of 0.1 and 100 Gy were chosen as the lower and upper bounds of α

_{ref}/β

_{ref}for most of the published in vitro data [33,50], as shown in Figure 1. The value of α

_{ref}= 0.2 Gy

^{−1}was chosen as intermediate value for the cell lines analyzed in [23,24,34]. The effect of varying α

_{ref}for a fixed α

_{ref}/β

_{ref}was evaluated in a second step, as described in Section 2.4.

^{12}C ions with energy between 1 and 1000 MeV/n: α/α

_{ref}, β/β

_{ref}, RBE

_{50%}, RBE

_{10%}, and RBE

_{1%}. The ratio α/α

_{ref}is known as low dose RBE (RBE

_{α}) and can be derived from Equation (4) in the limit of S → 1. The reference radiation chosen for the RBE calculations was 6 MV X-rays (${\overline{y}}_{D,ref}^{}$~2.3 keV/μm [23]).

#### 2.4. Effect of the Absolute Values of α_{ref} and β_{ref} on the Calculated RBE for ^{12}C Ions

_{ref}and β

_{ref}for two fixed values of α

_{ref}/β

_{ref}(2 and 10 Gy). These two α

_{ref}/β

_{ref}values were chosen because of their clinical relevance and for consistency with previous studies [35,51,52]. Once fixed the α

_{ref}/β

_{ref}ratio, α

_{ref}was allowed to assume the following values: 2 Gy

^{−1}, 1 Gy

^{−1}, 0.5 Gy

^{−1}, 0.2 Gy

^{−1}, 0.1 Gy

^{−1}, 0.05 Gy

^{−1}, 0.02 Gy

^{−1}, and 0.01 Gy

^{−1}. The values of 2 and 0.1 Gy

^{−1}were chosen as the lower and upper bounds of α

_{ref}for most of the published in vitro data [33,50] (see Figure 1). The biological endpoints, the reference radiation, and the numerical values of R

_{n}, r

_{d}, and ${\overline{y}}_{D,ref}^{}$ were the same of Section 2.3.

**Figure 1.**α

_{ref}/β

_{ref}plotted as a function of α

_{ref}for the asynchronized in vitro data from PIDE 3.2 [53]. The two data series represent the results of the LQM fit performed by the authors of the original publication (full diamonds) and by the PIDE team (open diamonds). To improve the readability of the plot, two in vitro data points were excluded: α

_{ref}= 1.73 Gy

^{−1}, α

_{ref}/β

_{ref}= −211 Gy; and α

_{ref}= 1.04 Gy

^{−1}, α

_{ref}/β

_{ref}= 3059 Gy.

#### 2.5. MKM-Based Fit of the Photon Survival Curve

_{ref}and β

_{ref}. Since no physical nor biological constraints are generally applied during the LQM fits, null and negative values of both α

_{ref}and β

_{ref}were reported in literature [33]. This could immediately result in negative or infinite RBE values at low doses (RBE

_{α}).

_{0}are calculated with Equation (16). Therefore, if the dose-mean lineal energy for the radiation under investigation $({\overline{y}}_{D}^{}$) is lower than that for the reference radiation ${\overline{y}}_{D,ref}^{}$, a negative α value is computed and thus a negative RBE

_{α}.

_{0}≥ 0.

_{d}= 0.29 µm, ${\overline{y}}_{D,ref}^{}$ = 2.3 keV/µm), this constraint is equal to α

_{ref}/β

_{ref}≥ ~1.4 Gy.

#### 2.6. Effect of Different Fits of the Same Photon Survival Curve on the Calculated RBE for ^{12}C Ions

_{ref}and β

_{ref}) on the RBE calculated with the MCF MKM for

^{12}C ions. PIDE is a database of in vitro clonogenic survival data for human and rodent cell lines exposed to photons and ions up to

^{238}U. The following information is reported for each entry in PIDE: cell line details (name, human or rodent, tumor or healthy, synchronized or asynchronized), ion exposure details (ion energy, ion LET, monoenergetic or spread-out Bragg peak (SOBP)), the reference photons used (i.e., 200 kVp X-rays or

^{60}Co γ-rays), and the results of the LQM fits (α, β, α

_{ref}, and β

_{ref}). When available, two sets of LQM terms are included for each entry in PIDE: the results of the LQM fits performed by the authors of the original publication and the results of LQM fits performed by the PIDE team. As performed in previous studies [23,24,46], we initially discarded entries for synchronized cells, irradiations along SOBPs, and exposures to ions with energy lower than 1 MeV/n. In the second step, we included only PIDE entries with non-negative values of β for the ion exposure. Negative β could be due to the presence of a more radioresistant cell subpopulation [53].

_{0}, we selected the cell lines with most entries (for a specific ion) for which α

_{ref}was equal to 0 in one of the two LQM sets (i.e., fit by the original author of the publication or by the PIDE team) and greater than 0 in the other photon LQM set: human astrocytoma cells (U-251MG cell line) and human mammary epithelial cells (M/10 cell line). In both cases, these two cell lines were irradiated with

^{12}C ions. It must be stated that the RBE for the M/10 and the U-251MG cell lines was computed in previous work with the MCF MKM [24] using only the photon LQM terms with α

_{ref}≠ 0. These published RBE results were compared with the novel RBE calculations using the MCF MKM in combination with the photon LQM terms with α

_{ref}= 0 and the new LQM terms obtained fitting the in vitro photon survival curve with the MKM-based approach of Section 2.7. The a priori assessed model parameters (r

_{d}and R

_{n}) for both cell lines were extracted from [24] and listed in Table 1 together with the model parameters derived analyzing the different LQM fits of the photon in vitro survival curve (α

_{ref}, β

_{ref}, α

_{0}, β

_{0}). Additionally, in vitro data for human brain glioblastoma cells (A-172 cell line) irradiated with

^{12}C ions were also included since marked differences were observed between the photon LQM terms (α

_{ref}and β

_{ref}) obtained by the original authors of the publication and by the PIDE team [34]. Different RBE

_{10%}values were computed when these two sets of photon-exposed LQM terms were used in combination with the MCF MKM [34]. In this work, we use these computed RBE

_{10%}s as a benchmark for the novel RBE calculation based on the α

_{ref}and β

_{ref}assessed with the MKM-based fit of Section 2.5. Furthermore, we extend the analysis to other RBE levels (RBE

_{α}, RBE

_{50%}, and RBE

_{1%}). The reference photon radiation was

^{137}Cs γ-rays for the A-172 and U-251MG cell lines from [55], while

^{137}Cs or

^{60}Co γ-rays (not specified) for the M/10 cell line data from [56].

#### 2.7. Computer Simulations

^{12}C ion beams with energy between 1 and 1000 MeV/n.

_{2}O

_{3}:C optically stimulated luminescent detectors [66], and an optical-fiber-based BaFBr:Eu detector [67]. The unrestricted linear energy transfer (LET) in water was calculated with the PHITS [T-LET] tally. All results of this study are plotted as a function of a generic “unrestricted LET in water” because of the lacking details and the contrasting approaches used in the LET assessment between the authors of the different studies presenting the in vitro data. A previous study reported a good agreement between the PHITS-calculated LET values and corresponding literature results for

^{12}C ions [68]. More details on the simulations can be found in [23,24] and in the PHITS manual (https://phits.jaea.go.jp/manual/manualE-phits.pdf, accessed on 24 January 2023).

## 3. Results

#### 3.1. Effect of α_{ref}/β_{ref} on the Calculated RBE for ^{12}C Ions

_{ref}= 0.2 Gy

^{−1}and varying α

_{ref}/β

_{ref}between 0.1 and 100 Gy. A decrease in α

_{ref}/β

_{ref}causes a larger curvature in the simulated survival curve and smaller surviving fractions at the same absorbed dose. These synthetic survival curves were used to determine the MCF MKM parameters α

_{0}and β

_{0}using Equations (16) and (17). As described in the methodology, R

_{n}and r

_{d}were fixed at 4.7 and 0.29 µm. Since the biological weighting function α(y) (Equation (8)) is the most important factor in the RBE calculations with the MCF MKM, Figure 2B compares its shape as a function of the lineal energy for the different α

_{ref}/β

_{ref}included in this study (0.1–100 Gy). The chosen reference radiation was 6 MV X-rays. Therefore, for any value of α

_{ref}/β

_{ref}, α

_{ref}is equal to 0.2 Gy

^{−1}at 2.3 keV/µm. For y < 2.3 keV/µm, lower α values were calculated in case of smaller α

_{ref}/β

_{ref}, with negative values computed for α

_{ref}/β

_{ref}< 1.4 Gy. For y > 2.3 keV/µm, the opposite is observed: higher α values were calculated in the case of smaller α

_{ref}/β

_{ref}. It must be noted that the maximum of α(y) is shifted to higher linear energy values with the increase in α

_{ref}/β

_{ref}. Nonetheless, all α(y) functions tend to 0 when y → +∞.

_{ref}/β

_{ref}on the RBE

_{α}= α/α

_{ref}, β/β

_{ref}, RBE

_{50%}, RBE

_{10%}, and RBE

_{1%}computed with the MCF MKM as a function of the LET of

^{12}C ions. The results are plotted as a function of the LET between 10 and 1000 keV/µm for better visualization of the data. Strong differences were found between the low dose RBE (RBE

_{α}, Figure 3A) values computed using the different α

_{ref}/β

_{ref}. Both the maximum RBE

_{α}values (i.e., ~45 for α

_{ref}/β

_{ref}= 0.1 Gy and ~2.5 for α

_{ref}/β

_{ref}= 100 Gy) and the LET at which this maximum RBE is located (i.e., ~30 keV/µm for α

_{ref}/β

_{ref}= 0.1 Gy and ~500 keV/µm for α

_{ref}/β

_{ref}= 100 Gy) were strongly affected by the α

_{ref}/β

_{ref}chosen for the calculations. The general trend consists of a shift of the RBE

_{α}maximum to lower values and higher LET with the increase in α

_{ref}/β

_{ref}. Except for α

_{ref}/β

_{ref}= 0.1 Gy, the maximum value of β/β

_{ref}(=1) for carbon ion appears to not be affected by α

_{ref}/β

_{ref}, as shown in Figure 3B. On the other hand, the decrease in β/β

_{ref}with the increase in the LET is sharper for lower α

_{ref}/β

_{ref}values. As an example, β/β

_{ref}= 0.5 is reached at ~30 keV/µm for α

_{ref}/β

_{ref}= 0.1 Gy, while at ~450 keV/µm for α

_{ref}/β

_{ref}= 100 Gy). In the case of RBE

_{50%}, RBE

_{10%}, and RBE

_{1%}(Figure 3C–E), the shift in the position of the RBE maximum as a function of the LET follows that of RBE

_{α}. However, the variation in the maximum RBE values is less pronounced than that for RBE

_{α}. As mentioned before, the maximum value of RBE

_{α}varied between 2.5 (α

_{ref}/β

_{ref}= 100 Gy) and 45 (α

_{ref}/β

_{ref}= 0.1 Gy) as a function of α

_{ref}/β

_{ref}. By contrast, the maximum value of RBE varied between 2.4 and 7.2 for RBE

_{50%}, between 2.2 and 4.3 for RBE

_{10%}, and between 2.1 and 3.2 for RBE

_{1%}.

#### 3.2. Effect of the Absolute Values of α_{ref} and β_{ref} on the Calculated RBE for ^{12}C Ions

_{ref}/β

_{ref}on the MCF MKM results for

^{12}C ions is shown in Figure 4. Two values of α

_{ref}/β

_{ref}were analyzed (2 and 10 Gy), while α

_{ref}was varied between 2 and 0.01 Gy

^{−1}. The RBE values for α

_{ref}/β

_{ref}= 10 Gy (dashed lines in Figure 4) are systematically lower than that for α

_{ref}/β

_{ref}= 2 Gy (solid lines in Figure 3). Since the effect of the absolute values of α

_{ref}and β

_{ref}is similar for both α

_{ref}/β

_{ref}values, the discussion below is limited to the results for α

_{ref}/β

_{ref}= 2 Gy. As can be seen in Figure 4A, RBE

_{α}is strongly affected by the absolute values of α

_{ref}and β

_{ref}. The maximum of RBE

_{α}is higher for smaller values of α

_{ref}. Furthermore, the maximum of RBE

_{α}is shifted at higher LET values for smaller values of α

_{ref}. As an example for α

_{ref}/β

_{ref}= 2 Gy, the maximum RBE

_{α}was ~3.8 at ~40 keV/µm for α

_{ref}= 2 Gy

^{−1}, while ~63 at ~500 keV/µm for α

_{ref}= 0.01 Gy

^{−1}. The LET dependence of β/β

_{ref}was also found to be affected by the absolute value of α

_{ref}: a sharper decrease in β/β

_{ref}as a function of the LET was computed for higher α

_{ref}values. In this regard, β/β

_{ref}= 0.5 was obtained in the case of carbon ions with an LET of ~350 keV/µm for α

_{ref}= 0.01 Gy, while at ~35 keV/µm for α

_{ref}= 2 Gy. In the case of RBE

_{50%}, RBE

_{10%}, and RBE

_{1%}(Figure 4C–E), the change in the position of the RBE maximum as a function of the LET is the same as that for RBE

_{α}. For α

_{ref}/β

_{ref}= 2 Gy, the maximum value of RBE varied between 3.8 and 63 for RBE

_{α}, between 3.3 and 9.9 for RBE

_{50%}, between 2.7 and 5.7 for RBE

_{10%}, and between 2.3 and 4.1 for RBE

_{1%}. For α

_{ref}/β

_{ref}= 10 Gy, the maximum value of RBE varied between 2.1 and 29 for RBE

_{α}, between 2.1 and 9.2 for RBE

_{50%}, between 1.9 and 5.5 for RBE

_{10%}, and between 1.8 and 4.1 for RBE

_{1%}.

#### 3.3. Effect of Different Fits of the Same Photon Survival Curve on the Calculated RBE for ^{12}C Ions

_{ref}and β

_{ref}) were used as an input to the MCF MKM to calculate the LET dependence of α, β, RBE

_{50%}, RBE

_{10%}, and RBE

_{1%}in case of exposures to monoenergetic carbon ions (panels C, D, E, F, and G of Figure 5, Figure 6 and Figure 7). It was decided not to present the results in the form of RBE

_{α}= α/α

_{ref}since infinite RBE values would be computed for the M/10 and U-251MG cell lines in case of the PIDE fits with α

_{ref}= 0. The results of these in silico calculations (panels C, D, E, F, and G of Figure 5, Figure 6 and Figure 7) are compared with the in vitro results presented by authors of the original articles (black diamonds) [55,56] and by the subsequent reanalysis performed by the PIDE team (open red diamonds) [53] on the same raw data. All the in vitro results are reported without error bars since the PIDE database, including the results of the analyses by original authors and by the PIDE team, currently does not provide uncertainty intervals for the clonogenic survival data. As previously discussed [24], this is likely due to the fact that most published biological papers do not include a systematic uncertainty analysis nor include sufficient information to perform a retrospective uncertainty study.

_{ref}and β

_{ref}from the PIDE team are used in combination with the MCF MKM, the model appears to overestimate the in vitro data.

_{50%}and RBE

_{10%}results are not included in Figure 6E,F. Except for the carbon-irradiated data point at ~300 keV/µm (there is a minor underestimation), the MCF MKM results agree well with the in vitro data (Figure 6C,G) when the photon LQM terms from the original article [56] and the MKM-based fit are used as input to the MCF MKM calculations. By contrast, the in silico calculations based on the photon LQM terms obtained by the PIDE team (dashed red lines in Figure 6C,G) overestimate the corresponding in vitro results.

## 4. Discussion

_{n}) and the subnuclear domains (r

_{d}) and the numerical values of the LQM terms in the limit of zero lineal energy (α

_{0}and β

_{0}). While the first two parameters (R

_{n}and r

_{d}) can be assessed by means of morphometric measurements of the cell nucleus and estimates of the mean DNA content, α

_{0}and β

_{0}are derived from the in vitro LQM terms of the photon dose-response of the cell line under investigation (α

_{ref}and β

_{ref}) [23,24]. Since the two LQM terms are anticorrelated during the LQM fit of the photon dose response [33], different sets of α

_{ref}and β

_{ref}might be obtained from the same in vitro survival curve. This affects the results of the following calculations for photons and the computation of the RBE of ion therapy treatments.

_{ref}and β

_{ref}(and therefore a different α

_{ref}/β

_{ref}) are derived from the same survival curve. In the BED calculations, the ratio α

_{ref}/β

_{ref}is used as an indication of the sensitivity of the cell line. Similarly, calculations of the sublethal lesion repair during beam interruption or a change in the radiation dose rate [69,70] can be affected by the different LQM fits. This happens because these MKM-based repair models [69,70] rely on the application of a temporal correction factor to β, while α is left unchanged. The anticorrelation between the two LQM terms could therefore result in different sets of LQM terms which lead to a different estimation of the temporal dependence (i.e., fractionation, dose rate, beam interruption) of the radiation-induced effects.

_{ref}and β

_{ref}are used as input to the RBE calculations (i.e., via Equations (16) and (17) in the case of the MCF MKM), a different estimation of their values could result in a different α

_{ref}/β

_{ref}. A recent analysis of published in vitro clonogenic survival data suggests that the low dose RBE (RBE

_{α}) of carbon ions strongly depends on α

_{ref}/β

_{ref}, with higher RBE

_{α}values found for cells with lower α

_{ref}/β

_{ref}[6]. By contrast, this relationship between α

_{ref}/β

_{ref}and RBE was not visible at higher doses (i.e., RBE

_{10%}) [6]. One could argue that the need for grouping the in vitro data in two broad categories (radioresistant if α

_{ref}/β

_{ref}< 4 Gy and radiosensitive if α

_{ref}/β

_{ref}> 4 Gy [6]) and the relatively large uncertainty in the biological data [68] could be the reason behind this lack of observable correlation between RBE and α

_{ref}/β

_{ref}at high dose. In order to explore the effect of α

_{ref}/β

_{ref}on the RBE of carbon ions, we computed the RBE with the MCF MKM as a function of the ion LET and of the surviving fraction for α

_{ref}/β

_{ref}between 0.1 and 100 Gy. As summarized in Figure 3, α

_{ref}/β

_{ref}appears to affect both the maximum value of the RBE and its position when plotted as a function of the carbon ion LET. Though the differences in the maximum values of the RBE are very large at low doses (RBE

_{α}) between the different α

_{ref}/β

_{ref}data series, the values of the RBE maxima are closer at higher doses. When the comparison is restricted to the α

_{ref}/β

_{ref}range encompassing most of the in vitro data for

^{12}C ions (1 Gy < α

_{ref}/β

_{ref}< 10 Gy, Figure S1 of Supplementary Materials), our in silico results with the MCF MKM agree well with what previously observed on the in vitro data [6]. As an example, the maximum RBE

_{α}was strongly affected by α

_{ref}/β

_{ref}: 16 for α

_{ref}/β

_{ref}= 1 and 6 for α

_{ref}/β

_{ref}= 10 Gy. On the other hand, the maximum RBE

_{10%}values were much closer: 4.2 for α

_{ref}/β

_{ref}= 1 and 3.6 for α

_{ref}/β

_{ref}= 10.

_{ref}/β

_{ref}is not sufficient to uniquely characterize the RBE of carbon ions. As shown in Figure 4 for two clinically relevant values of α

_{ref}/β

_{ref}(2 and 10 Gy), the absolute values of α

_{ref}and β

_{ref}significantly affect the RBE computed with the MCF MKM in the case of carbon ions and likely also for other ions. In the framework of implementing a variable RBE in proton therapy practice, many phenomenological RBE-vs-LET models were developed. A numerical comparison between the results of these models revealed large differences between the results of these LET-based proton RBE models [35]. As concluded in a subsequent review of these phenomenological models [71], these differences could be attributed to the significant disagreement between the role of α

_{ref}/β

_{ref}on the computed RBE among the different models. Further studies with the MCF MKM are warranted to investigate if this disagreement on the role of α

_{ref}/β

_{ref}could be due to the negligence of the effect of absolute values of α

_{ref}and β

_{ref}in the RBE calculations for protons.

_{ref}and β

_{ref}during the LQM fit and the uncertainty in the clonogenic assay at high doses (i.e., few residual colonies) could lead to the computation of very low α

_{ref}/β

_{ref}(even 0 in some cases, see Table 1) and therefore negative α

_{0}values with Equation (16). As a result, the MCF MKM α(y) functions computed with Equation (8) could assume negative values, as shown in Figure 2B. This might lead to the calculation of unreasonable negative RBE

_{α}values for sparsely ionizing radiation. Though α

_{0}= 0 denotes that all lethal lesions arise from the combination of sublethal lesions [72], the biological meaning of negative α

_{0}values is, at best, unclear.

_{0}values, to avoid the calculation of negative RBE values for ions whose dose-mean lineal energy is smaller than that of the reference photons, and to prevent the calculations of infinite RBE

_{α}for cell lines whose LQM fit of the reference photon exposure resulted in α

_{ref}= 0 (Table 1). When the photon LQM terms (α

_{ref}and β

_{ref}) obtained with this MKM-based fit were used as input to the MCF MKM calculations, the predicted cell response to carbons (solid blue lines in panels C to G of Figure 5, Figure 6 and Figure 7) were generally found to be in reasonable agreement with the in vitro data. The results obtained using the other two LQM fits of the photon survival curve (short-dashed red lines and dashed black lines in panels C to G of Figure 5, Figure 6 and Figure 7) were found to describe the in vitro data in a comparable or worse way (i.e., some large deviations were observed especially in case of the photon fits with α

_{ref}= 0).

_{0}values are computed using the published values of α

_{ref}and β

_{ref}, as shown in Table S1 of the Supplementary Materials.

_{ref}and β

_{ref}and their ratio α

_{ref}/β

_{ref}appears to be necessary for an accurate calculation of the RBE in carbon ion therapy. Care must be taken in fitting the reference in vitro photon survival curve. It is suggested to pay particular attention to the relatively-low dose range since α

_{0}and β

_{0}represent the LQM parameters in the limit of very sparsely ionizing radiation and low dose. However, at low doses (i.e., below ~1–2 Gy), phenomena such as low-dose hypersensitivity [74], likely due to intercellular signaling (bystander effects) [75], could further complicate the fitting process. Furthermore, it should be remembered that LQM fits over different dose ranges could produce different sets of LQM terms [76]. This is particularly evident at relatively high doses where the in vitro clonogenic survival curves transition to a fixed slope [77]. The MKM-based fit of the in vitro photon survival curve appears to be a useful tool to determine α

_{0}and β

_{0}for the MCF MKM calculations and to check the physical and biological meaning of the mathematical values of α

_{ref}and β

_{ref}.

## Supplementary Materials

_{ref}/β

_{ref}plotted as a function of α

_{ref}for

^{12}C ions only. Figure S2: Proton-irradiated human glioblastoma cells (U-87 cell line): comparison between MCF MKM results and in vitro data. Table S1: MCF MKM parameters used in the calculations for the U-87 cell line.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

α | linear term of the linear-quadratic model of clonogenic survival |

α_{0} | α in the limit of the lineal energy y → 0 |

α_{ref} | α for the reference photon exposure |

β | quadratic term of the linear-quadratic model of clonogenic survival |

β_{0} | β in the limit of the lineal energy y → 0 |

β_{ref} | β for the reference photon exposure |

A-172 cell line | human glioblastoma cells |

BED | biologically effective dose [37] |

DNA | deoxyribonucleic acid |

LEM | local effect model [8] |

LEM IV | fourth version of the local effect model [30] |

LET | linear energy transfer |

LQM | linear-quadratic model of clonogenic survival [32] |

M/10 cell line | human mammary epithelial cells |

MCF | Mayo Clinic Florida, Jacksonville, Florida, United States of America |

MCF MKM | Mayo Clinic Florida microdosimetric kinetic model [23] |

MKM | microdosimetric kinetic model [25] |

modified MKM | modified microdosimetric kinetic model [9] |

PHITS | Particle and Heavy Ion Transport code System [57] |

PIDE | Particle Irradiation Data Ensemble [53] |

RBE | relative biological effectiveness |

RBE_{S} | in vitro clonogenic cell survival RBE for the surviving fraction S (Equation (4)) |

r_{d} | mean radius of the subnuclear domains |

R_{n} | mean radius of the cell nucleus |

U-87 cell line | human glioblastoma cells |

U-251MG cell line | human astrocytoma cells |

y | lineal energy |

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**Figure 2.**(

**A**) Survival curves calculated for α

_{ref}= 0.2 Gy

^{−1}and α

_{ref}/β

_{ref}ranging between 0.1 and 100 Gy. (

**B**) Effect of the α

_{ref}/β

_{ref}ratio on the MCF MKM biological weighting function for the linear term of the LQM (α(y), Equation (8)) computed for a synthetic cell line with mean characteristics (R

_{n}= 4.7 µm and r

_{d}= 0.29 µm) and α

_{ref}= 0.2 Gy

^{−1}.

**Figure 3.**Effect of the α

_{ref}/β

_{ref}ratio on the RBE

_{α}= α/α

_{ref}(

**A**), β/β

_{ref}(

**B**), RBE

_{50%}(

**C**), RBE

_{10%}(

**D**), and RBE

_{1%}(

**E**) of

^{12}C ions as calculated with the MCF MKM for a synthetic cell line with mean characteristics (R

_{n}= 4.7 µm and r

_{d}= 0.29 µm) and α

_{ref}= 0.2 Gy

^{−1}.

**Figure 4.**Effect of the absolute values of α

_{ref}and β

_{ref}on the RBE

_{α}= α/α

_{ref}(

**A**), β/β

_{ref}(

**B**), RBE

_{50%}(

**C**), RBE

_{10%}(

**D**), and RBE

_{1%}(

**E**) of

^{12}C ions as calculated with the MCF MKM for a synthetic cell line with mean characteristics (R

_{n}= 4.7 µm and r

_{d}= 0.29 µm) and α

_{ref}/β

_{ref}= 2 and 10 Gy.

**Figure 5.**(

**A**) Clonogenic survival of human glioblastoma cells (A-172 cell line) exposed to photons: comparison between the in vitro data (black diamonds) and the LQM fits performed by the authors of the original article (short-dashed black line), by the PIDE team (dashed red line), and the novel MKM-based fit (solid blue line). (

**B**) Detailed view of the survival data in linear-linear scale. α (

**C**), β (

**D**), RBE

_{50%}(

**E**), RBE

_{10%}(

**F**), and RBE

_{1%}(

**G**) as a function of the LET of carbon ions as experimentally determined by the authors of the original article (black diamonds) and by PIDE team (open red diamonds) in comparison with MCF MKM predictions based on the α

_{ref}and β

_{ref}obtained by authors of the original article (black short-dashed line), by the PIDE team (dashed red line), and the novel MKM-based fit (solid blue line).

**Figure 6.**(

**A**) Clonogenic survival of human mammary epithelial cells (M/10 cell line) exposed to photons: comparison between the in vitro data (black diamonds) and the LQM fits performed by the authors of the original article (short-dashed black line), by the PIDE team (dashed red line), and the novel MKM-based fit (solid blue line). (

**B**) Detailed view of the survival data in linear-linear scale. α (

**C**), β (

**D**), RBE

_{50%}(

**E**), RBE

_{10%}(

**F**), and RBE

_{1%}(

**G**) as a function of the LET of carbon ions as experimentally determined by the authors of the original article (black diamonds) and by PIDE team (open red diamonds) in comparison with MCF MKM predictions based on the α

_{ref}and β

_{ref}obtained by authors of the original article (black short-dashed line), by the PIDE team (dashed red line), and the novel MKM-based fit (solid blue line).

**Figure 7.**(

**A**) Clonogenic survival of human astrocytoma cells (U-251MG cell line) exposed to photons: comparison between the in vitro data (black diamonds) and the LQM fits performed by the authors of the original article (short-dashed black line), by the PIDE team (dashed red line), and the novel MKM-based fit (solid blue line). (

**B**) Detailed view of the survival data in linear-linear scale. α (

**C**), β (

**D**), RBE

_{50%}(

**E**), RBE

_{10%}(

**F**), and RBE

_{1%}(

**G**) as a function of the LET of carbon ions as experimentally determined by the authors of the original article (black diamonds) and by PIDE team (open red diamonds) in comparison with MCF MKM predictions based on the α

_{ref}and β

_{ref}obtained by authors of the original article (black short-dashed line), by the PIDE team (dashed red line), and the novel MKM-based fit (solid blue line).

**Table 1.**MCF MKM parameters used in the calculations to benchmark the MKM-based LQM fit of the photon in vitro survival curve.

Cell Line | R_{n}[µm] | r_{d}[µm] | LQM Fit | α_{ref}[Gy ^{−1}] | β_{ref} = β_{0}[Gy ^{−2}] | α_{ref}/β_{ref}[Gy] | α_{0}[Gy ^{−1}] |
---|---|---|---|---|---|---|---|

A-172 | 5.7 [34] | 0.3 [34] | original article [55] | 0.405 | 0.0588 | 6.89 | 0.328 |

PIDE team [53] | 0.310 | 0.0888 | 3.49 | 0.194 | |||

MKM-based | 0.428 | 0.0544 | 7.87 | 0.357 | |||

M/10 | 4.7 [24] | 0.29 [24] | original article [56] | 0.300 | 0.0680 | 4.41 | 0.205 |

PIDE team [53] | 0 | 0.1120 | 0 | −0.156 | |||

MKM-based | 0.288 | 0.0785 | 3.67 | 0.178 | |||

U-251MG | 4.9 [24] | 0.29 [24] | original article [55] | 0.031 | 0.0551 | 0.56 | −0.046 |

PIDE team [53] | 0 | 0.0609 | 0 | −0.085 | |||

MKM-based | 0.0635 | 0.0455 | 1.39 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Parisi, A.; Beltran, C.J.; Furutani, K.M.
Clonogenic Survival RBE Calculations in Carbon Ion Therapy: The Importance of the Absolute Values of *α* and *β* in the Photon Dose-Response Curve and a Strategy to Mitigate Their Anticorrelation. *Quantum Beam Sci.* **2023**, *7*, 3.
https://doi.org/10.3390/qubs7010003

**AMA Style**

Parisi A, Beltran CJ, Furutani KM.
Clonogenic Survival RBE Calculations in Carbon Ion Therapy: The Importance of the Absolute Values of *α* and *β* in the Photon Dose-Response Curve and a Strategy to Mitigate Their Anticorrelation. *Quantum Beam Science*. 2023; 7(1):3.
https://doi.org/10.3390/qubs7010003

**Chicago/Turabian Style**

Parisi, Alessio, Chris J. Beltran, and Keith M. Furutani.
2023. "Clonogenic Survival RBE Calculations in Carbon Ion Therapy: The Importance of the Absolute Values of *α* and *β* in the Photon Dose-Response Curve and a Strategy to Mitigate Their Anticorrelation" *Quantum Beam Science* 7, no. 1: 3.
https://doi.org/10.3390/qubs7010003