Invalidity of, and alternative to, the linear quadratic model as a predictive model for postirradiation cell survival

Abstract The linear quadratic (LQ) model has been the dominant tool in preclinical radiobiological modeling of cell survival as a function of dose. However, as a second‐order polynomial approximation, it suffers from two well‐known pitfalls: nonmonotonic behavior and poor extrapolation. This study examined the raw data of 253 sets of photons and 943 sets of the ion beam from the Particle Irradiation Data Ensemble (PIDE) project to understand how often the LQ model could result in a negative β, which would give unrealistic predictions. Additionally, the predictive performance of the LQ model, the power model, and the linear model's predictive performance was studied using leave‐one‐out cross‐validation (LOOCV) and twofold cross‐validation. It was found that, when fitted to the LQ model, 7.5% of the photon and 29.8% of the ion beam dose–response data would result in negative β, compared to 0.77% and 2.0%, respectively, reported in published works. The LQ model performed poorly in LOOCV compared to the alternative power model, and performed the worst among the three models in twofold cross‐validation. The LQ model leads to unrealistic parameters, which are vastly under‐reported in published studies, and performs poorly in standard cross‐validation tests. Therefore, the LQ model is not a valid predictive dose–response model for cell survival. Alternative models need to be investigated.


| INTRODUC TI ON
Clonogenic assay of cells in vitro was first described in the 1950s 1 and has since been established as the standard technique to investigate the relationship between radiation dose and postirradiation cell survival. 2 Standard operation procedures 2,3 have been developed to ensure the reproducibility and consistency of the results.
Biological models have been developed to describe the relationship between radiation dose and cell survival. 4 The LQ model is the most frequently used model to describe the relationship between radiation dose and cell survival, and between dose and patient response in clinical applications. 5 In general, the LQ model could be derived from target theory and be viewed as the second-order polynomial approximations of the true cell survival response to dose (and the linear model being the first-order). 6 As such, there are two wellknown major pitfalls of the second-order polynomial approximation that are applicable to the problem: it does not guarantee to be a monotonic function, and it extrapolates poorly. 7 The nonmonotonic problem manifests itself when the parameter β has a negative value that leads to increased cell survival with increasing dose, according to the LQ model. 8 Fundamentally, "negative beta" data shows the cell killing decelerates with increasing dose, as opposed to accelerates with "positive beta". There is no viable explanation for negative beta within the LQ framework. One of the plausible explanations for the observed decelerating cell killing with increasing dose is that there are radioresistant subpopulations in the cell population, and the cell killing rate slows down with increasing radiation dose when only the radioresistant subpopulation survives with higher radiation dose. [9][10][11] Nevertheless, the LQ model would predict completely unrealistic cell survival with a higher dose in such scenarios. In published reports, β = 0 was often used instead, resulting in only reporting α, and it reduces the LQ model to the linear one. 12 This practice does not entirely capture the data, ignoring the radioresistant subpopulations existence. Similarly, negative β was also observed in the clinical application of the LQ model. 13,14 There is an ongoing debate regarding whether the LQ model is appropriate to model high dose per fraction effects. [15][16][17] One key issue is whether the LQ model could be extrapolated and extended to a high dose per fraction as a predictive model. Although most cell survival data are acquired with low to intermediate dose, there is an increasing need to extrapolate the data with the increasing use of hypofractionated treatment with high fractional dose.
Additionally, predicting dose response is critical for ion beam radiotherapy. One of the key parameters to characterize the ion beam compared to the photon beam is the RBE, which is defined as the ratio of photon and ion beam dose to achieve the same biological end-point, and for cell irradiation, the same cell survival level. As measuring the dose to achieve a certain cell survival directly is impossible, the quantity must be interpolated or extrapolated from the measured cell survival data using dose-response models. Therefore, the accuracy of prediction using the model is paramount for the accuracy of RBE calculation. Indeed, for ionbeam RBE models that rely on the LQ model for cell survival modeling, because the LQ model is unable to handle these data, there was not much choice but to ignore them. For example, one recent work on the proton RBE model excluded 10% of data "where β ≤ 0", 15 [18][19][20] At least one of these studies was also, inappropriately, as per the above recommendation, reported as evidence for predictive accuracy. 21,22 To properly evaluate the predictive performance of doseresponse models, 253 sets of photon irradiation and 943 sets of ion beam irradiation, which were made available through the PIDE project, 8,12 were examined. The PIDE dataset has been previously described in detail. 8,12 The models, namely the LQ model, the power model, and the linear model, with two, two, and one parameter(s), were evaluated for the goodness of fit using the coefficient of determination, and for predictive performance using was also studied. The model also has two parameters, a and b. The key difference between the LQ and the power models is that the latter guarantee to be monotonic decreasing as a function of dose, as shown in Figure 1. When β = 0 or b = 1, both models reduce to the linear model; when β > 0 or b > 1, both models would show the "shoulder"

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LI and bend down below the linear model at a high dose. When β < 0, the LQ model will predict increasing cell survival with increasing dose above the threshold point, which could be solved using the quadratic equation. However, with b < 1, the power model will bend upwards above the linear model, but cell survival remains monotonic decreasing as a function of increasing dose.

| Data processing
This study included dose-response data from 253 sets of photon and 943 sets of ion beam irradiation. The data were ensembled by the PIDE project from 115 publications (the complete list of publications is shown in Appendix S1), across 130 different cell types. Each set of data consists of 4-32 dose-cell survival pairs. Each dataset was fitted using the linear, LQ, and power models. The goodness of the fits, as quantified by the coefficient of determination, was calculated.

| Cross-validation
Variations of LOOCV and twofold CV were used in this study. The training set for the LOOCV consists of all data points in each experiment set except for the highest dose point, whereas the training set for the twofold CV consists of the first (or second) half of data points in each experiment set. The study aimed to examine the predictive performance of dose-response models at high doses. Therefore, only the highest dose points in each dataset were used as the test data. The parameters for each dose-response model were fitted for the training dataset, and cell survival at the highest dose point was predicted using each model, and compared with the raw data.

| Statistical analysis
The RMSE of the prediction using each model for the complete dataset, LOOCV, twofold CV, and photon and ion beam data, were calculated. There is no established statistical analysis regarding predictive models' performance other than comparing the residual prediction errors in CV. 17

| RE SULTS
The dose range and the number of data points of the 253 sets of photon and 943 sets of ion irradiation are summarized in Table 1.
Each dataset was fitted using the linear, LQ, and power models.
The goodness of the fits, as quantified by the coefficient of determination, is summarized in Table 2.  Subsets of the data were constructed for LOOCV and twofold CV. The dose range and the number of data points in each of these subsets were also summarized in Table 1. Each dose-response model was, again, fitted to each subset. The coefficients of determination are summarized in Table 2.

| DISCUSS ION
The first goal of the study was to understand how often the LQ  the cell killing rate slowing down, will result in β < 0 (b < 1). It will also result in a decreasing RBE with an increasing dose. It is, therefore, important to understand the physics and biological reason behind the flattened survival response with increasing doses in these curves.
The current study used a generic fitting technique and did not tune any fitting parameters or consider the experimental uncertainties. The CV process first established the baseline accuracy of the models with the complete dataset, as shown in Tables 2 and 3. The experiment uncertainty for the last data point to be predicted becomes less critical in the CV with LOOCV or the twofold CV, as when comparing different models, they would face the same experimental uncertainties built in toward the evaluation.
In the current study, three dose-response models were discussed.
The LQ and the power models have two parameters, whereas the linear models have one. As shown in Figure 1, the linear model is not suitable to model dose-response curves bending down (β > 0) or bending up (β < 0), and therefore the LQ model was developed. As shown in Table 2, both the LQ and the power model could generate good fits with complete or reduced data, but not for the linear model, which generated poor fits with the twofold CV.  survival. An alternative model, the power model, with also two parameters, was proposed and discussed.

ACK N OWLED G M ENTS
We acknowledge the PIDE project for compiling available ion beam in vitro cell irradiation data.

CO N FLI C T O F I NTE R E S T S TATE M E NT
The author have no conflict of interest.

DATA AVA I L A B I L I T Y S TAT E M E N T
All data are available through PIDE. https://www.gsi.de/work/ forsc hung/bioph ysik/forsc hungs felde r/radio biolo gical_model ling/