Dosimetric quantities and cell survival for spatially fractionated radiation therapy

Purpose: Spatially Fractionated Radiation Therapy (SFRT) is characterized by large differences in peak and valley doses. Preclinical and clinical studies suggest that differences in biological mechanisms lead to differential normal tissue and tumor response compared to uniform irradiation. We hypothesize that to evaluate clinical effectiveness and understand fundamental biological mechanisms, radiobiological rather than physical dose quantities should be utilized for comparisons. The aim of this work is to determine whether Equivalent Uniform Dose (EUD) is a superior predictor of cell survival than absorbed dose. Methods: Absorbed dose parameters were compared to the Equivalent Uniform Dose to assess their predictive value for the relative effectiveness of uniform and SFRT with X-rays. A Bayesian bootstrap technique was utilized to model uncertainties in the biological fit parameters for a human fibroblast (MRC5) and two human tumor cell lines (LN18 and A549). Dose uncertainties were evaluated through measurements and error modeling of SFRT profiles. A dimensionless Relative Effectiveness Factor (REF) is proposed to quantify differences between uniform and SFRT irradiation. Results: For all cell lines, cell survival after SFRT matched uniform irradiation within the estimated uncertainties at equal values of the EUD. Average and peak dose showed poor correlation with in vitro cell survival. The proposed REF factor is dose dependent and suggests enhanced cell killing for both tumor cell lines (1.14 ± .08 for LN18, 1.32 ± .13 for A549 at 8 Gy EUD) for SFRT. Normal human fibroblasts showed reduced cell killing relative to uniform irradiation (.58 ± .06 for MRC5). Synthetically generated SFRT dose profiles revealed that EUD uncertainties are dominated by valley dose uncertainties, especially at high doses. Discussion: EUD is more predictive of cell survival than average or valley dose. Valley dose is close to equal to the EUD for values > 10 Gy and has the advantage of being independent of uncertainties in biological parameters. The REF is a novel and useful metric to compare quantitative differences in SFRT and uniform irradiation. Conclusion: EUD is recommended for comparisons of SFRT and uniform irradiation. The results suggest an increase in survival of normal-human fibroblast cells and reduced survival for both tumor cell lines after SFRT relative to uniform irradiation.

Purpose: Spatially Fractionated Radiation Therapy (SFRT) is characterized by large differences in peak and valley doses. Preclinical and clinical studies suggest that differences in biological mechanisms lead to differential normal tissue and tumor response compared to uniform irradiation. We hypothesize that to evaluate clinical effectiveness and understand fundamental biological mechanisms, radiobiological rather than physical dose quantities should be utilized for comparisons. The aim of this work is to determine whether Equivalent Uniform Dose (EUD) is a superior predictor of cell survival than absorbed dose.
Methods: Absorbed dose parameters were compared to the Equivalent Uniform Dose to assess their predictive value for the relative effectiveness of uniform and SFRT with X-rays. A Bayesian bootstrap technique was utilized to model uncertainties in the biological fit parameters for a human fibroblast (MRC5) and two human tumor cell lines (LN18 and A549). Dose uncertainties were evaluated through measurements and error modeling of SFRT profiles. A dimensionless Relative Effectiveness Factor (REF) is proposed to quantify differences between uniform and SFRT irradiation.
Results: For all cell lines, cell survival after SFRT matched uniform irradiation within the estimated uncertainties at equal values of the EUD. Average and peak dose showed poor correlation with in vitro cell survival. The proposed REF factor is dose dependent and suggests enhanced cell killing for both tumor cell lines (1.14 ± .08 for LN18, 1.32 ± .13 for A549 at 8 Gy EUD) for SFRT. Normal human fibroblasts showed reduced cell killing relative to uniform irradiation (.58 ± .06 for MRC5). Synthetically generated SFRT dose profiles revealed that EUD uncertainties are dominated by valley dose uncertainties, especially at high doses.
Discussion: EUD is more predictive of cell survival than average or valley dose. Valley dose is close to equal to the EUD for values > 10 Gy and has the advantage of being independent of uncertainties in biological parameters. The REF is a novel and useful metric to compare quantitative differences in SFRT and uniform irradiation.

Introduction
Spatially fractionated radiation therapy (SFRT) is an innovative treatment concept, which seeks to exploit dosevolume effects in normal tissues without compromising tumor control. The dose-volume effect refers to the dependency of normal tissue response on the volume that is being irradiated. SFRT has been pursued using various approaches, such as microbeams, minibeams and grid therapy. With grid therapy hundreds of patients have been successfully treated and several clinical trials are underway [1][2][3][4], the development of mini-and Microbeam Radiation Therapy (MRT) is still in a preclinical phase. SFRT uses highly modulated beams with dose gradients delivered to millimeter or sub-millimeter spatial regions of tissue. Different domains and applications of SFRT exist with various radiation types, e.g. photons [5,6] and protons [7,8], geometry as well as arrangement and delivery of the beams, e.g., GRID [4] (2D pattern) and Lattice [3,9] (3D pattern) therapy. For each of the approaches reports exist in the literature. Increased normal tissue tolerance to SFRT beams compared to uniform irradiation is usually reported [10][11][12][13]. However, the exact biological mechanism for these differences are not yet understood. Similarly, published reports indicate that SFRT can potentially overcome the inherent radiation resistance of conventional low Linear Energy Transfer (LET) radiation treatments [14], presumably through dose escalation to the tumor but possibly also through enhanced tumor-cell killing mediated by inter-cellular signaling or other mechanisms.
Many laboratory studies utilize physical (absorbed) dose metrics, such as the average (or mean), peak and/or valley dose, to design studies and compare SFRT to uniform irradiation [10,12,13,15]. More recently, the shortcomings of using absorbed dose metrics for comparative pre-clinical studies of uniform irradiation and SFRT irradiation have been highlighted, and it was proposed that the Equivalent Uniform Dose (EUD) [16,17] may be useful for the design and analysis of SFRT studies with corrections for different beam parameters and the effects of relative biological effectiveness (RBE) for different particle types and energies. The most widely used formulas to compute the EUD are derived from the well-established Linear Quadratic (LQ) model for reproductive cell survival. Although the EUD may not account for all biological responses, such as systemic and bystander effects [18], it remains a useful way to compare uniform irradiation and SFRT in pre-clinical studies and ongoing efforts to document the potential benefits of SFRT compared to conventional radiation therapy approaches and to compare different types of beam modulations and low and high LET irradiation scenarios among institutions. Computing the EUD requires input of 1) the physical dose distribution and 2) fitting the LQ model parameters α and β, which describe the radiation-response of a cell-line, tissue or tumor as a function of absorbed dose (or RBE-weighted dose) in a specific tissue region or cell culture. Another factor contributing to uncertainty in the dosimetry of highly modulated SFRT beams arises because of the challenges associated with measuring dose in small, highly modulated radiation fields, which in turn requires the ability to measure dose with high spacial resolution and high dynamic range to accurately characterize peak and valley doses. Measurement uncertainties are not commonly reported [19,20] and the effects of this dosimetric uncertainty on estimates of the EUD are not well established. Similarly, the non-linear regression analysis of fits to cell survival data have large uncertainties because of uncertainties in plating efficiency, the number of replicate measurements per dose point and the limited range of doses used to characterize in vitro and in vivo radiation sensitivity. Most laboratory studies of cell survival involve a small number of dose points ( < 4-6) between 1 and 10 Gy. The small number of dose levels creates challenges for conventional statistical procedures that assume residuals from model-fits to measured data are normally distributed. The analysis of such published studies also creates a situation where numerous combinations of the LQ parameters α and β, and therefore α/ β, fit the experimental observations equally well [21,22] that is, estimates of α and β are highly correlated and degenerate.
For the experimental design of our comparative experiments between uniform irradiation (one arm) and SFRT (alternate arm), we propose the conceptual schema illustrated in Figure 1. In the first step, cells are uniformly irradiated with a range of doses, D = D 1 , D 2 , D 3 , . . ., to determine the survival fraction of cells. In the next step, we conduct a non-linear regression analysis to determine LQ parameters α U and β U . A bootstrap sampling method is used to estimate the confidence interval on pairs of LQ parameters that fit the measured data about equally well. Bootstrapping is typically used to generate a large number of samples from a single original sample when only Frontiers in Physics frontiersin.org 02 a small number of measurements is available and the distribution is unknown. Bootstrapped samples are generated from this minipopulation to approximate the population distribution. For the second arm of the study, estimates of fit parameters are used in combination with the measured SFRT beam profiles to calculate the corresponding EUD for the non-uniform beams, EUD 1 , EUD 2 , EUD 3 , . . ., that presumably results in the same number of surviving cells for an SFRT irradiation scheme as the corresponding uniform dose. This assumes that the mechanisms of cell killing and recovery in SFRT irradiations are the same as for uniform irradiation and the cellular response only depends on the locally absorbed dose, i.e., cell cultures that receive the same EUD exhibit the same overall survival response. Should the mechanisms of cell killing underlying uniform and SFRT irradiation be the same, the parameters α U , β U should be identical to α SFRT , β SFRT within the uncertainties. Should the in vitro mechanisms of cell survival differ for SFRT and uniform irradiation, we expect differences in cell survival as a function of EUD, i.e., different values of α U , β U and α SFRT , β SFRT result in different estimates of the EUD.
For the approach illustrated in Figure 1, a difference in the cell survival between the two arms would indicate some effects that cannot be explained with our current radiobiological understanding of conventional (uniform) irradiation. In order to draw such a conclusion, a careful analysis of the uncertainties in the observed cell survival and the physical dosimetry is indispensable. In this manuscript we present such an analysis of cell survival after MRT and uniform x-ray exposure of two cancer and a normal tissue (non-cancer) cell line. The approach is generally applicable to other types of beams with different spatial modulation.
The ultimate aim of this work is to determine the dosimetric quantity that best correlates with cell survival. We also aim to test the hypothesis that uniform irradiation and SFRT irradiation of several cell lines in vitro provide evidence for differences in the biological mechanisms of cell survival. We aim to highlight the FIGURE 1 Schema to test the hypothesis that uniform and SFRT irradiation are biologically equivalent for in vitro cell survival. First, cells are uniformly irradiated at different dose levels to determine α U and β U . The LQ fit parameters are then used in combination with the SFRT profile to determine the corresponding EUD. Biological equivalence is achieved if cell survival at a certain EUD leads to the same level of cell survival, given the uncertainties in dose delivery and uncertainties associated with the non-linear regression analysis of the measured plating efficiencies over a finite dose rate. This hypothesis is equivalent to testing whether or not α U ≈ α SFRT and β U ≈ β SFRT .
Frontiers in Physics frontiersin.org 03 potential impact of uncertainties of SFRT dose delivery as well as the uncertainties associated with fitting the LQ model to uniform and SFRT irradiation conditions, especially as it relates to the calculation of the EUD for SFRT. In analogy with the concept of particle RBE, we propose a similar relative effectiveness factor (REF) to quantify the impact of SFRT on tumor and normal tissue cell lines relative to the uniform irradiation of the same.

Materials and methods
Multiple approaches to deliver SFRT [23] exist. MRT is one specialized form of SFRT. Throughout the methods, results, discussion and conclusions, the abbreviations MRT and SFRT are used as most appropriate. MRT is used for points and discussion specific to the presented in vitro experiments and the abbreviation SFRT as it relates to the larger implications for spatially fractionated radiation therapy.

Clonogenic cell survival experiments
The human tumor cell lines LN18 (human glioblastoma) and A549 (human lung adenocarcinoma) were cultured in Dulbecco's Modified Eagle Medium (DMEM) high glucose (Sigma, Germany) and the normal tissue cell line MRC5 (human lung fibroblasts) was cultured in DMEM/F12 (ATCC, United States). Cells were seeded and incubated at 37°C, 5% CO 2 24 h before irradiation. Uniform irradiation was conducted at the RS225 (X-Strahl, UK) with 220 kVp photons filtered with .15 mm of copper having a dose-rate of .89 Gy/min (approximate RBE relative to Co-60 gamma-rays of 1.2 [24]). MRT irradiation of cell cultures was performed with a XenX irradiator (X-Strahl, United Kingdom) using a 225 kVp photon spectrum filtered with .15 mm copper. The photon field was collimated to an MRT field with slit widths of 50 µm separated by a ctc (center-to-center) distance of 400 µm resulting in a total field size of 20 mm × 20 mm. The peak dose-rate was 4.6 Gy/min and the valley dose-rate .112 Gy/min [25]. The dose rates were measured using Gafchromic EBT3 films. A peak-to-valley-doseratio (PVDR) of 20 was achieved by combining MRT with uniform irradiation. To circumvent the limited dynamic range of the films, separate films were used to evaluate the dose and dose rate in peak and valley regions. Subsequently, the profiles were stitched together, only including parts of the films within the calibration range to end up with one valid dose profile for the entire microbeam field [26]. After irradiation, the cells were incubated for another 30 min at 37°C and 5% CO 2 before single cell suspensions were obtained and cell numbers for each sample were counted three times using a hemocytometer. The cells were seeded in 6-well plates and after an incubation period of 2 weeks, each 6-well plate was washed with phosphate-buffered saline (PBS), fixed with methanol, and stained with crystal violet. For all 6-well plates, the number of colonies (at least 50 cells) was counted manually by means of the digital images and ultimately, both plating efficiency and survival fraction were determined. Plating efficiencies for three replicate measurements at each dose point were used for the non-linear regression analysis of LQ cell survival parameters.

LQ model and bootstrap analysis of uncertainty
The primary LQ model parameters of interest are α and β (or α and α/β). A standard approach to parameter estimation involves minimizing a positively weighted sum of errors with weights w i . This weighted sum of errors, sometimes termed a loss function or figure of merit (FOM), can be formulated in several different ways, depending on varying assumptions about the underlying probability model. The mathematical form of the loss function [27] impacts on the estimation of parameters, on confidence intervals, and on model inference. In our analyses, we used the following FOM to estimate LQ parameters, similar to [28]: Here, N is the number of data points (absorbed dose values at which survival is evaluated), S i is the ith measurement of the surviving fraction, S is the modelpredicted surviving fraction (Eq. 1) for the same exposure conditions, and α and β are the parameters in the model that are adjusted to minimize the loss function (or α and α/β). The weights are sampled for the ith dose point and replicate from an exponential distribution.
For all of the analyzed in vitro data sets, the loss function is minimized using a quasi-Newton optimization algorithm. To estimate the correlated distribution of α and β values that fit the measured data, we use a paired bootstrap method [29] with random weights [30], which is often referred to as the Bayesian bootstrap [31,32]. Estimates of the mean value of α and β are based on 10,000 individual fits to the survival data. We use random weighting for each experiment and each dose, since standard bootstrap re-sampling [29] gives, with a large probability for small data sets, a degenerate answer when data with only one or a few points are used. In effect, the estimated values of α and β (and therefore α/β) from an LQ analysis of the measurements are highly correlated for most practical doseresponse experiments, including ours.

EUD and cell survival
The EUD concept as formulated by Niemierko [33] is premised on the idea that any two dose distributions may be  S Here, v i is the fraction of cells receiving a dose D i . Taking the logarithm of both sides of Eq. 2 and solving for EUD yields Two methods are used to estimate the parameters of interest from the in vitro data. In the first method, α U and α U / β U are estimated from survival data for uniform irradiation of the cell lines ( Figure 1). Then they are applied to the calculation of cell survival and EUD for SFRT conditions. In the second method, estimates α and α/β are obtained from a fit to cell survival as a function of the EUD. The estimates of the EUD are based on values of α and α/β for uniform irradiation conditions (left-most panels in Figure 2).

Relative effectiveness factor
In analogy with particle RBE [22,35], we propose a Relative Effectiveness Factor defined as the ratio of the uniform dose D U of radiation that achieves the same level of cell survival as the EUD for an SFRT treatment, i.e., REF ≡ D U /EUD. The EUD metric is widely used for clinical comparisons of uniform and nonuniform irradiation of tumor targets and OARs [36][37][38]. In terms of the LQ cell survival model, the REF can be computed as

Simulations of dosimetric uncertainties
To investigate the impact of dosimetric uncertainties on the EUD a sensitivity analysis was carried out. MRT dose profiles were synthetically generated and parameterized to match the experimentally measured parameters: peak dose, valley dose, PVDR and beam penumbra, which is the transition area between the peak and valley and defined as the distance between 80% and 20% of the peak dose. Dosimetric uncertainties were systematically introduced to these parameters to investigate their effects on the resulting EUD. To isolate the effects of one specific parameter, all other parameters in the dose profile were kept constant and the parameter of interest was changed incrementally by ±5%, ±10% and ±20%. The effect on the EUD was analyzed for valley doses between 2 and 20 Gy while keeping the PVDR at 20.
The modeled MRT dose profiles matched the actual microbeams used for the clonogenic survival experiment described in Section 2.1 with a beam width of 50 μm, a ctc of 400 µm and a penumbra of 30 µm. For completeness, we also modeled a minibeam field consisting of 250 µm wide beams separated by a ctc of 1 mm and a penumbra of 140 µm and analyzed its effects on the EUD.

Correlation of dosimetric quantities with cell survival
The results of the LQ analysis of in vitro cell survival data for the LN18 and A549 tumor cell lines, and the MRC5 normal tissue cell line are shown in Figure 2. The left most panels show LQ fits of cell survival against average dose (D avg ), the middle panels show cell survival plotted against the valley dose (D valley ), and the right-most panel shows cell survival as a function of the EUD. The EUD values shown in Figure 2 are based on estimates of α and β from the non-linear regression analysis of measured data for uniform irradiation conditions, i.e., α U and β U . Figure 3 shows the distribution of pairs of α and β values that fit the in vitro data shown in Figure 2 about equally well, as determined by the bootstrap analysis outlined in Section 2.2. It is important to note that slightly smaller values of α can be offset by a slight increase in the value of β, and vice versa, without sacrificing the fit to the experimental observations over a finite dose range. The prediction interval for cell survival that corresponds to the pairs of LQ parameters shown in Figure 3 are illustrated by the small yellow and red circles in Figure 2. There is a clear separation between uniform irradiation and MRT in the cloud of α and β parameters for the MRC5 cell line (right panel of Figure 3), as expected from the lower panels of Figure 2. There is less separation in the cloud of α and β parameters for the LN18 and A549 cell lines, which is consistent with the data Frontiers in Physics frontiersin.org 06 shown in the upper and middle panels of Figure 2. A large separation in the parameter clouds ( Figure 3) and the plots of cell survival against EUD (right panel of Figure 3) are an indication that the biological mechanisms underlying in vitro reproductive cell survival are different for MRT and uniform irradiation.
The differences in the biological effectiveness of MRT irradiation relative to uniform irradiation for the three cell lines shown in Figures 2, 3 are depicted over a range of doses in Figure 4. The uncertainties at low doses are relatively large, especially for the LN18 cell line and therefore the results at low doses below approximately 2 Gy should be considered with a grain of salt. At 8 Gy the REF values are 1.14 ± .08, 1.32 ± .13 and .58 ± .06 for LN18, A549 and MRC5, respectively. A REF greater than unity (A549 and LN18 cells) indicates that in vitro MRT irradiation is more effective at killing these two tumor cell lines than uniform irradiation. A REF less than unity for the MRC5 cell line indicates that MRT is less effective at killing this cell line than uniform irradiation. Hence a REF < 1 implies that the EUD for MRT irradiation would have to be increased to have the same effectiveness in killing cells as uniform irradiation. In other words, sparing of cells occurs for the MRT arm at the same EUD compared to uniform irradiation.

Physical dose variation
The effects of simulated variations in certain physical dosimetric quantities in SFRT on the EUD, as described in Section 2.5, are shown in Figure 5. The upper left figure shows

Discussion
The clinically relevant parameters for the assessment and comparison of SFRT and uniform radiation therapy treatments are a matter of ongoing debate [39]. Authors frequently report physical dose quantities, such as the peak, valley and average dose, in pre-clinical and clinical studies. Of these three dose metrics, valley dose has been experimentally found most predictive of normal tissue sparing [39][40][41]. For uniform irradiation and conventional fractionation schedules (e.g., daily target doses up to 10 to possibly 20 Gy, the LQ model often suffices as a reasonable dose-response model for the endpoints of in vitro cell survival, Frontiers in Physics frontiersin.org 08 local tumor control and OAR toxicity. Clinically applied concepts, such as the equivalent dose in 2 Gy fractions (EQD2) and EUD, are derived from the LQ model and are useful concepts for the comparison of alternate treatment modalities and fractionation schedules. In this work, we investigated the potential usefulness of the LQ model and EUD concept in the analysis of in vitro cell survival for SFRT compared to uniform irradiation. While an identical EQD2 entails equal cell survival in temporally fractionated radiation therapy (e.g., different daily doses), identical EUD values imply equal levels of overall cell survival in a cell culture or tissue region of interest irradiated by spatially fractionated vs. uniform irradiation.
In Figure 2, we investigated the apparent differences in the in vitro biology of MRT and uniform irradiation for three cell lines. There are seemingly dramatic differences in the survival when average dose is used as the predictive metric for MRT relative to uniform irradiation (see left panels in Figure 2). This observation reflects the fundamental idea that cells in the peak-dose region of an SFRT field are all effectively sterilized whereas the cells in the valley region are the predominantly ones that survive. Average dose is therefore a poor predictor of SFRT outcomes because it is a simple average of the peak and valley dose, weighted for the width (area) of the peak and valley regions of an SFRT field. Average dose neglects the dramatic spatial differences in cell survival and sterilization across a SFRT field compared to uniform irradiation. In line with this observation, the middle panels in Figure 2 demonstrate that the differences in cell survival after SFRT and uniform irradiation are much more closely aligned in terms of valley dose, as expected since it is the cells in the valley region of a SFRT field are the ones most likely to survive. EUD is largely more predictive of SFRT than any physical dose parameters (right-most panels in Figure 2) because the EUD concept accounts for all cell survival, even in the penumbra region of an SFRT field, rather than the more simplistic approach that assumes only two dose levels (peak and valley dose).
As illustrated in Figure 5, the impact of the beam width and penumbra on EUD increases with decreasing valley dose and as the ctc distance changes. The impact of beam penumbra and the ctc on EUD increases as the ratio R (R ctc bw ) or the fraction dose is reduced, where bw is the beam width. For doses above 10 Gy the relative uncertainty of the EUD is below 3% for MRT (bw = 50 μm, R = 8) and below 6% for minibeams (bw = 250 μm, R = 4). As mentioned before, at larger fraction sizes there are almost no surviving cells in the peak and therefore survival of cells in the valley region of an SFRT field determines overall survival, An uncertainty in beam width or beam penumbra ΔR will lead to a relative survival uncertainty which is small for large R. Therefore it can be concluded that the dominating quantity for the biological outcome of an SFRT exposure is the valley dose. This is in line with the recent findings in a review of preclinical studies [39], where it was found that "the valley dose (rather than the peak dose) correlates best with treatment outcome." At large fraction sizes the EUD becomes approximately equal to the valley dose. Of note, the accuracy of the LQ model is often challenged for doses much larger than α/β-often on the order of 8-10 Gy for uniform irradiation. This observation implies that the LQ model might be less useful (accurate) for the analysis of SFRT irradiation than for equivalent uniform doses of radiation beyond 10-20 Gy. As illustrated in the left panels in Figure  It is intriguing to consider the possibility that tumor cell lines may respond differently to SFRT irradiation than normal cells, although additional studies comparing tumor and normal cell lines are warranted. If the mechanisms of action for cell survival were the same for SFRT and uniform irradiation, the data for cell survival vs. EUD ( Figure 2) and LQ parameter clouds ( Figure 3) would overlap with each other. The deviations seen in Figures 2,3 are consistent with the hypothesis that SFRT and uniform cell irradiation initiate mechanisms or modes of reproductive cell survival that differ to some extent. To the authors knowledge this is the first time that this difference has been quantified by using the newly introduced REF.
Accounting for the variability in plating efficiency seen in Figures 2, 3, it is reasonable to conclude that the tumor cell lines (LN18 and A549) exhibit small differences in cell survival when plotted against EUD. For human lung fibroblasts (MRC5 cell line), there seems to be considerable differences in survival for SFRT and uniform irradiation, as also reflected in the REF factor shown in Figure 4. The estimated uncertainties in the absorbed dose for the SFRT cell culture irradiations are on the order of 5%-10%. Hence, differences in survival between SFRT and conventional treatments with X-rays are smaller than the dosimetric uncertainties for the A549 and LN18 cell lines but may not be so for the MRC5 cell lines. The survival mechanisms of tumour cell lines (A549 and LN18) differ significantly from Repeat experiments are needed under the same settings to further quantity the joint experimental uncertainties associated with the SFRT dosimetry and the uncertainty in in vitro plating efficiencies. It would also be useful to extend the range of the uniform irradiation to better mimic the range of doses used in the SFRT experiments and to further map out the dose dependence of the REF. To further test and probe the concepts and results for X-ray irradiations, the experiments should be repeated with different beam types, such as protons, with consideration of the different RBE. Similar in vitro experiments for additional tumor-and normal-cell lines are needed to confirm or refute some of the reported results.
Are the biological mechanisms underlying cell survival in SFRT different from those in more uniform dose irradiation? By applying the EUD, we assume the classical mechanisms of radiation biology suffice, i.e., the survival response of one cells is independent of the survival response of other cells. Therefore, we can attribute deviations between the biological reactions towards SFRT and conventional treatments at identical EUD to differences in SFRT and uniform irradiation cell survival mechanisms that are not fully captured in the LQ model. Examples of such in vivo mechanisms include immune system responses and the effects of the tissue microenvironment, including the effects of reactive oxygen species and inflammatory signaling. However, in in vitro experiments the aforementioned factors do not, or are unlikely to, play a role. Bystander effects (inter cellular communication) or physiochemical changes in the cell culture medium are the most likely explanations for the observed differences in survival for conventional and SFRT exposures (Figures 2-4).

Conclusion
New in vitro data for three cell lines irradiated by SFRT and uniform irradiation schemes are reported for a dose range applicable to radiation therapy. The use of Bayesian bootstrap sampling to more accurately quantify the impact of uncertainties in LQ and EUD modeling of comparative studies of SFRT and uniform irradiation is useful and applicable to pre-clinical and to clinical studies of SFRT biological endpoints and outcomes. The simulation of SFRT irradiations illustrates the impact of dosimetric uncertainties, especially in the field penumbra and valley dose, which are specific to the design of the SFRT irradiation scheme.
The reported studies provide clear evidence that EUD is more predictive of in vitro cell survival for SFRT irradiations than physical dose quantities such as average or valley dose. Since EUD more fully and directly accounts for spatial variations in dose and cell survival across millimeter and sub-millimeter regions of an irradiated cell culture, it appears more predictive of SFRT cell survival than average or peak dose. As such, spatial variations in biological damage are also relevant in vivo, we recommend the use of EUD as the dosimetric metric for the comparative analysis of SFRT and uniform irradiation and discourage the use of the physical dose quantities peak dose and average dose as they can potentially be misleading and should not be used for comparisons of different types of spatial modulations and different doses. The valley dose is the most useful physical dose quantity as at higher doses the EUD approaches the valley dose. Additional in vitro and in vivo studies are needed to further test and quantify the similarities and differences in biological mechanisms arising from SFRT and uniform irradiation. The proposed concept of a relative effectiveness factor for SFRT relative to uniform irradiation is useful for inter-institutional comparisons in the same way that the concept of particle RBE is useful for comparing the effectiveness of x-rays to higher LET radiations and enables the quantification of the biological response relative to known biology.

Data availability statement
The original contributions presented in the study are available from the corresponding author.
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