ACCOUNTING FOR OVERDISPERSION OF LETHAL LESIONS IN THE LINEAR QUADRATIC MODEL IMPROVES PERFORMANCE AT BOTH HIGH AND LOW RADIATION DOSES

Abstract

Purpose

The linear-quadratic (LQ) model represents a simple and robust approximation for many mechanistically-motivated models of radiation effects. We believe its tendency to overestimate cell killing at high doses derives from the usual assumption that radiogenic lesions are distributed according to Poisson statistics.

Materials and methods

In that context, we investigated the effects of overdispersed lesion distributions, such as might occur from considerations of microdosimetric energy deposition patterns, differences in DNA damage complexities and repair pathways, and/or heterogeneity of cell responses to radiation. Such overdispersion has the potential to reduce dose response curvature at high doses, while still retaining LQ dose dependence in terms of the number of mean lethal lesions per cell. Here we analyze several irradiated mammalian cell and yeast survival data sets, using the LQ model with Poisson errors, two LQ model variants with customized negative binomial (NB) error distributions, and the Padé-linear-quadratic and Two-component models. We compare the performances of all models on each data set by information-theoretic analysis, and assess the ability of each to predict survival at high doses, based on fits to low/intermediate doses.

Results

Changing the error distribution, while keeping the LQ dose dependence for the mean, enables the NB LQ model variants to outperform the standard LQ model, often providing better fits to experimental data than alternative models.

Conclusions

The NB error distribution approach maintains the core mechanistic assumptions of the LQ formalism, while providing superior estimates of cell survival following high doses used in radiotherapy. Importantly, it could also be useful in improving the predictions of low dose/dose rate effects that are of major concern to the field of radiation protection.

Introduction

Many types of mechanistically-motivated models of radiation effects can be approximated by the linear-quadratic (LQ) formalism (Brenner et al. 1998). This classic model has a long history of use in radiation biology and oncology because it provides robust descriptions/predictions for radiation-induced cell death and chromosomal aberration dose responses (Iliakis et al. 2004; Lea 1946; Catcheside et al. 1943), and provides interpretations of these phenomena in terms of single-track vs. multiple-track effects, using a very small number of adjustable parameters.

The standard combination of the LQ dose dependence for the mean number of lethal radiation-induced lesions/cell with a Poisson error distribution predicts a continuously-curving cell survival dose response. On many data sets, this predicted dose response shape is not optimal, for example because it tends to underestimate cell survival at high doses/low surviving fractions (Garcia et al. 2006), such as those used in modern stereotactic radiotherapy. To mitigate these issues, several researchers advocated the use of more complex (or ad hoc) radiobiological models (reviewed in (Andisheh et al. 2013)). They used different functional forms for the dose dependence of the mean number of lethal lesions/cell, but generally still relied on the Poisson distribution to describe variations around the mean.

We explored an alternative approach, where the mean is described by the LQ function, but the error distribution is changed from Poisson to a more flexible one (Shuryak et al. 2017). The history and rationale for the use of the Poisson in describing the random distribution of lethal lesions is nearly as old radiation biology itself. Its roots can be traced to the introduction of formal target theory initially used to describe the inactivation of viruses and bacteria, in which the survival to x rays decreased with dose as a simple exponential via “single-hit” kinetics (Lea 1955). The practice carried over to more complex models of cellular inactivation that were required to model the curvilinear or “shouldered” responses typical of eukaryotic cells (Atwood et al. 1949). Later, this included biophysical models of radiation action that stressed the importance of track structure and lesion interaction, such as the Theory of Dual Radiation Action (Kellerer et al. 1972), from which the ubiquitous linear-quadratic formalism derives.

Cytogenetics, coupled with subsequent advances in cell culture techniques, made it possible to observe likely-lethal events (e.g., dicentric chromosomes) on a cell-by-cell basis, thereby allowing a direct measure of the distribution of lesions among cells. In general, experiments conducted using x- or gamma rays served to substantiate the continued use of the Poisson. However, not all such experiments supported its use (Shuryak et al. 2017). In particular, as investigators gained access to more exotic sources of ionizing radiation it became clear that the distribution of chromosome aberrations from high LET radiations was decidedly overdispersed, meaning the variance exceeded the mean (Edwards et al. 1979; Virsik et al. 1981). From our prospective, the Poisson can be viewed as a simplified approximation to the binomial distribution (Elkind et al. 1967), and as such, the latter is inherently better suited to deal with more complex distributions.

According to the Poisson distribution, the data variance is equal to the mean, whereas other distributions like negative binomial (NB) allow variances greater than the mean – this phenomenon is frequently called “overdispersion”. Overdispersion of radiation damage is often associated with densely ionizing radiations like α-particles and neutrons, notably through cytogenetic studies (Virsik et al. 1981; Lloyd et al. 1983). However, we provided and discussed evidence for overdispersion of lethal chromosomal aberrations after sparsely ionizing radiation (gamma rays and x-rays) in human cells (lymphocytes and fibroblasts) (Shuryak et al. 2017). Plausible reasons for this phenomenon include the microdosimetric distribution of radiation energy deposition (Hanin et al. 2010), diversity of DNA double-strand break (DSB) complexities and repair pathways (Nikitaki et al. 2016; Schipler et al. 2016), variety of lethal chromosomal aberration types (McKenna et al. 2019; Cornforth et al. 2018), and heterogeneity in cellular responses to DNA damage, e.g. due to differences in cell cycle stage and epigenetics (Segerman et al. 2016; Shuryak et al. 2015; Ciccarese et al. 2017).

Our analysis showed that changing the error distribution around the mean from Poisson to NB, while keeping the same dose response for the mean (the LQ function), causes the dose response for clonogenic cell survival (i.e. for the fraction of cells with zero lethal lesions) at high radiation doses to approach log-linear (i.e. “straightening”, rather than continuously curving) behavior (Shuryak et al. 2017). Therefore, we hypothesized that LQ model predictions can potentially become more consistent with those experimental data sets that suggest “straightening” of radiation dose responses at high doses by modifying the error distribution, rather than by using an alternative model for the mean number of lethal lesions per cell.

In addition, better model fits over a wide dose range could be useful for more accurately predicting the effects of low doses/dose rates by improving the accuracy of estimating the linear parameter (α) of the LQ model. On data sets where the standard LQ model with Poisson errors does not perform well because it cannot adequately explain the shape of the data such as a pronounced “shoulder” region at low doses followed by dose response “straightening” at high doses, the α parameter estimate can be unreliable. Risk assessments for low doses/dose rates based on such α estimates would be unreliable as well. In contrast, a more “flexible” LQ model variant such as one with an NB error distribution could fit the data better over a wide dose range and generate a more realistic α estimate and corresponding low dose/dose rate risk predictions. This approach would be potentially useful in radiation protection.

Here we test these hypotheses by analyzing several published cell survival data sets for mammalian cells and yeast exposed to acute photon or α-particle doses. On each data set, we use the standard LQ model with a Poisson error distribution, two LQ model variants with customized NB error distributions, and two alternative models capable of dose response “straightening”: the Padé linear quadratic (PLQ) and Two Components (2C) (Andisheh et al. 2013). The performances of these different models are compared by Akaike information criterion with sample size correction (AICc) (Burnham et al. 2014; Wagenmakers et al. 2004). We also use root mean squared error (RMSE) to assess the ability of each model to predict survival data at high doses, based on fits to low/intermediate doses. The results of this detailed model comparison provide useful information on how changing the error distribution for radiation-induced lethal lesions from Poisson to NB, without changing the LQ dose dependence for the mean, alters the performance of the LQ model in describing survival curve data and making predictions at high doses and at low doses/dose rates.

Methods

Data sets

For this analysis we chose the very detailed data published by Garcia et al. (Garcia et al. 2006) on four mammalian cell lines – human prostate carcinoma (DU145 and CP3), human glioblastoma (U373MG), and Chinese hamster fibroblasts (CHOAA8) – exposed acutely to 250 kVp x-rays in 0.5 Gy increments from 0 up to 10.5 or even 16 Gy, depending on cell line. As an additional example of eukaryotic but non-mammalian cells, we chose the large survival data sets on yeast (Saccharomyces cerevisiae) exposed to acute doses of x-rays and α-particles, published by Bertsche et al (Bertsche 1978). These data sets were selected because they thoroughly cover high, medium and low surviving fractions, allowing informative analyses of survival curve shapes.

The data points and error bars were estimated from published graphs by GetData Graph Digitizer software (http://getdata-graph-digitizer.com/). When no error bars were reported, we assumed them to be 0.05 on the log₁₀ scale. The resulting data sets are shown in supplementary materials.

Radiobiological models

According to the LQ model, the mean number of lethal lesions per cell (μ) is described by the following function of radiation dose D, where α and β are the model parameters:

$$\mu =\alpha \times D+\beta \times D^2$$ (1)

The Poisson distribution provides the following formula for the probability of k lethal lesions in a cell, when the mean number of such lesions per cell is μ:

$$p_{\mathit{\text{pois}}}(k)={\mu }^k\times \text{exp}[-\mu ]/k!$$ (2)

The predicted clonogenic cell surviving fraction (SF) is defined as p_Pois(k=0), which is:

$$S{F}_{\mathit{\text{pois}}}=\text{exp}[-\mu ]$$ (3)

Therefore, the cell surviving fraction based on the classic LQ model with a Poisson error distribution can be calculated by substituting Eq. (1) into Eq. (3):

$$S{F}_{\mathit{\text{pois}}}=\text{exp}\left[-\alpha \times D-\beta \times D^2\right]$$ (4)

As an alternative, the negative binomial (NB) distribution provides the following formula for the probability of k lethal lesions in a cell, when the mean number of such lesions per cell is μ, Γ is the gamma function, and r is an additional model parameter:

$$p_{NB}(k)={(1/[1+\mu \times r])}^{\left(\frac{1}{r}\right)}\times {(\mu /[\mu +1/r])}^k\times \frac{\Gamma \left(k+\frac{1}{r}\right)}{\left[k!\times \Gamma \left(\frac{1}{r}\right)\right]}$$ (5)

Here μ is still assumed to be described by the LQ function of dose (Eq. 1). Parameter r quantifies the magnitude of “overdispersion” relative to the Poisson distribution (Shuryak et al. 2017). Small r values (close to zero) generate behaviors similar to the Poisson distribution, whereas larger r values produce distributions with larger “tails”.

The cell surviving fraction based on the LQ model with an NB error distribution can be calculated by substituting k=0 into Eq. (5). The solution is as follows, where μ is still assumed to be described by Eq. (1):

$$S{F}_{NB}={\left(\frac{1}{[1+\mu \times r]}\right)}^{\left(\frac{1}{r}\right)}$$ (6)

As an additional alternative, we modified the NB distribution so that the “overdispersion parameter” r is not constant, but depends on the mean μ. As a plausible possibility, overdispersion can be negligible at low μ values, but increase up to a certain limit at high μ values. This example of a modified NB distribution (MNB) is mathematically represented by the following equation, which does not introduce any additional adjustable parameters compared to Eq. (5):

$$p_{MNB}(k)={(1/[1+\mu \times X])}^{\left(\frac{1}{x}\right)}\times {(\mu /[\mu +1/X])}^k\times \frac{\Gamma \left(k+\frac{1}{x}\right)}{\left[k!\times \Gamma \left(\frac{1}{x}\right)\right]},X=r\times (1-\text{exp}[-\mu ])$$ (7)

The cell surviving fraction based on the LQ model with an MNB error distribution can be calculated by substituting k=0 into Eq. (7). The solution is as follows, where μ is still assumed to be described by Eq. (1):

$$S{F}_{MNB}={\left(\frac{1}{[1+\mu \times X]}\right)}^{\left(\frac{1}{X}\right)}={\left(\frac{1}{[1+\mu \times r\times (1-\text{exp}[-\mu ])]}\right)}^{\left(\frac{1}{r\times (1-\text{exp}[-\mu ])}\right)}$$ (8)

The NB and MNB error distributions alter the shapes of cell survival curves, compared with the Poisson distribution, even though the dose dependence of μ remains unchanged (Eq. 1). The differences from the standard LQ model with a Poisson error distribution are controlled by an extra adjustable parameter (in addition to α and β), called r in the NB and MNB models.

For comparison, we used two additional cell survival models from the literature, which also have the same number of adjustable parameters as the NB and MNB formalisms, but have different mathematical structures and motivations. The Padé Linear Quadratic (PLQ) model (Andisheh et al. 2013) modifies the standard LQ model for cell survival by introducing a “curvature-reducing” term 1+γ×D, where γ is the extra adjustable parameter:

$$S{F}_{PLQ}=\text{exp}\left[-\alpha \times D-\beta \times D^2\right]/(1+\gamma \times D)$$ (9)

This model behaves in a standard LQ manner at low doses, but approaches exponential (rather than Gaussian) behavior at high doses.

The Two Components (2C) model (Bender et al. 1962) is not structurally based on the LQ function, and provides a smooth dose-dependent transition between an “initial slope” and a “terminal slope” for the natural logarithm of cell surviving fraction. This model is described by the following equation with adjustable parameters α₁, α_n, and n:

$$S{F}_{2C}=\text{exp}\left[-{\alpha }_1\times D\right]\times \left[1-{\left(1-\text{exp}\left[-{\alpha }_n\times D\right]\right)}^n\right]$$ (10)

Here the “initial slope” is α₁ and the “terminal slope” is α₁+α_n.

Model fitting and parameter estimation

We used maximum likelihood estimation to find best-fit parameters for each model on each data set, assuming that the statistical uncertainties are normally distributed on a logarithmic scale. We maximized the following log likelihood function LL_M (Banks et al. 2017), where N is the number of data points, σ is the statistical uncertainty parameter (which was considered an extra adjustable parameter in each model), ln[SF_O,i] is the logarithm of the observed cell surviving fraction for the i-th data point, ln[SF_M,i] is the corresponding cell surviving fraction predicted by the M-th model (i.e. by one of the tested models), and w_i is the “weight” for the i-th data point (estimated from data error bars, defined so that the variance = [σ×w_i]²):

$$L{L}_M=-\text{N}\times \left(\frac{\text{ln}[2\times \pi ]}{2}-\text{ln}[\sigma ]\right)-{\sum }_{i=1}^N\text{ln}\left[w_i\right]-\frac{1}{2\times {\sigma }^2}\times {\sum }_{i=1}^N{\left(\text{ln}\left[S{F}_{O,i}\right]-\text{ln}\left[S{F}_{M,i}\right]\right)}^2/w_i^2$$ (11)

The log likelihood maximization was performed by the sequential quadratic programming (SQP) algorithm in Maple 2018® software (Lawrence et al. 2001). The following procedure was used to increase the probability of finding the global, rather than a local, log likelihood maximum: (1) Initial parameter values for each model on each data set were identified manually by substituting various values and keeping those which generated a model curve that passed, by visual inspection, through most of the data. (2) One thousand parameter values were randomly selected from the parameter space in the vicinity of the initial values. (3) The model was fitted using each of these starting parameter combinations, and the combination which produced the highest log likelihood was retained as the best-fit parameter set. During the optimization procedure, the parameters were restricted to positive values to maintain biological plausibility. Uncertainties (95% confidence intervals, CI) for each parameter were estimated using profile likelihood (Pek et al. 2015).

In addition to using the model equations 1–9 to represent the Poisson, NB, MNB, and PLQ formalisms, we also fitted the same models, but with α/β instead of β as the quadratic dose response parameter. This reformulation did not change the models conceptually, but allowed CI estimation by profile likelihood for the α/β ratio, which is an important quantity in LQ-based models (but not in the structurally distinct 2C model).

Information theoretic model selection

The performances of different models fitted to the same data set were compared using the Akaike information criterion with sample size correction (AICc) (Wagenmakers et al. 2004; Burnham et al. 2014). AICc for the M-th model (AICc_M) is calculated below, where Λ_M is the number of adjustable parameters and LL_M is the maximized log-likelihood value from Eq. (11):

$$AIC{c}_M=-2\times L{L}_M+2\times {\Lambda }_M+2\times {\Lambda }_M\times \left({\Lambda }_M+1\right)/\left(N-{\Lambda }_M-1\right)$$ (12)

The relative likelihood of the M-th model is called the evidence ratio (ER_M). It is calculated as follows, where AICc_min is the lowest AICc value generated by the set of compared models:

$$E{R}_M=\text{exp}\left[-\frac{\Delta AIC{c}_M}{2}\right],\Delta AIC{c}_M=AIC{c}_M-AIC{c}_{min}$$ (13)

The evidence ratio for M-th model, divided by the sum of the evidence ratios for all models, is the Akaike weight, W_M, described by the following equation:

$$W_M=E{R}_M/{\sum }_{M}E{R}_M$$ (14)

The Akaike weight represents the relative likelihood of a model divided by the sum of these values across all models. This normalized relative likelihood is useful in model selection.

Extrapolation of models from low/intermediate to high doses

The ability of a model to predict data beyond the dose range to which the model was fitted is a measure of the model’s practical usefulness. We assessed this by fitting each model to 2/3 of the dose range (i.e. from zero dose up to 2/3 of the maximum dose in the analyzed data set), and calculating model predictions on the remaining 1/3 of the dose range (i.e. the highest doses in the data set). We calculated root mean squared error (RMSE) between the predictions of each model and the observed data points over the highest 1/3 of the dose range, and divided the results by the minimum RMSE value for all models. The resulting RMSE ratio was defined as 1.0 for the best possible performing model (the one with the smallest RMSE), and was >1 for the other models. Larger RMSE ratios indicate poorer performance for a given model, when extrapolated from low/intermediate doses to high doses.

We also tested the power of extrapolation for each model independently of the others by comparing two distinct RMSE values, both calculated over remaining 1/3 of the dose range but based on two different fits. (a) One using the best-fit parameters over the full dose range and (b) the other based on the best-fit parameters over the 2/3 range. The ratio (b-a)/a between the two RMSE values (obtained with the same model) quantifies the effect of the missing information when only a portion of the data is used to extrapolate outside the range.

The cutoff of 2/3 of the dose range for model fitting was selected by trying several plausible values. If a smaller data fraction (e.g. ½) was used, extrapolations from all models were poor because there were too few data points used in the fit to capture the dose response shape adequately. If a larger data fraction (e.g. ¾) was used for the fits, too few data points remained available for RMSE estimates. Therefore, 2/3 was a compromise value which was sufficient to discriminate between model performances. It was held constant for all models, regardless of model structure or parameter values, to provide a fair RMSE comparison across models.

Results

Information theoretic comparison of model performances

The classic LQ model (labeled as the Poisson model here) did a generally poorer job in describing data, compared to the other tested models (Table 1). This conclusion is supported by visual examination of model fits (Figs. 1–3). The sole exception was for the CHOAA8 cell line: on this data set, the Poisson model described the data as well as any of the other models (Fig. 2), but was favored by information theoretic analysis simply because it has the smallest number of adjustable parameters (Table 1).

Table 1.

Information theoretic comparison of model performances by Akaike weights on different data sets. The sum of Akaike weights for all models on each data set is 1. Larger weights indicate stronger support for a given model, taking into account the number of adjustable parameters and data set size.

Cell type	DU145	CP3	U373MG	CHOAA8	Yeast	Yeast
Radiation type	x-rays (250 kVp)				x-rays (70 kVp)	241Am α-particles
Poisson model	2.96×10−3	6.52×10−5	8.02×10−5	0.532	2.02×10−16	1.03×10−11
NB model	0.253	0.468	0.184	0.130	5.81×10−6	0.389
MNB model	0.319	0.427	0.230	0.136	2.53×10−6	0.610
PLQ model	0.334	0.102	0.324	0.129	7.91×10−10	2.45×10−4
2C model	0.0909	2.89×10−3	0.262	0.0728	~1.00	1.30×10−3

Figure 1.

Model fits to DU145 and CP3 cell data. In this and the following figures, all models were fitted to the data at all tested doses for each cell type (left panels). Separate plots that magnify the low dose (middle panels) and high dose regions (right panels) are shown for improved visualization of model behaviors. Error bars represent standard deviations.

Figure 3.

Model fits to data for yeast irradiated with x-rays or α-particles.

Figure 2.

Model fits to U373MG and CHOAA8 cell data.

In general, the reasons for the Poisson model’s poor performance included underestimation of the data at low doses and at high doses, with overestimation at intermediate doses (Figs. 1–2). In other words, the Poisson model was not flexible enough to describe the shapes of the dose responses in most of the analyzed data sets, except for CHOAA8 cell data. The four alternative models (NB, MNB, PLQ and 2C), which included an extra adjustable parameter, tended to attain much higher Akaike weights (Table 1) and visually fit the data better (Figs. 1–3), compared with the Poisson model. In short, the added flexibility of these models in describing survival dose responses overcame the AICc penalty for an extra adjustable parameter, resulting in higher Akaike weights relative to the Poisson model (Table 1).

Among these alternative models, our proposed NB and MNB formalisms achieved the highest Akaike weights on 3 out of 6 data sets: DU145, CP3, and yeast α–particle (Table 1). The PLQ model slightly outperformed them on U373MG data, the 2C model strongly outperformed them on yeast x-rays data, and the Poisson model dominated on CHOAA8 data (Table 1). In general, the NB, MNB, PLQ and 2C models had comparable success rates over the analyzed data sets (Table 1). These results suggest that survival dose response data tend to be better described by any model that has an extra adjustable parameter, compared with the Poisson model. The main important distinctions between these models are the differences in assumptions and interpretations for the predicted dose response shape, rather than fit quality.

Model parameter values

The best-fit parameters and their uncertainties for each model are listed in Table 2. The 2C model is structurally distinct from the others because it is not based on the LQ formalism. The Poisson, NB, MNB and PLQ models, however, share some fundamental common assumptions, allowing the best-fit linear and quadratic dose dependence terms (α and β parameters) to be compared. These parameters differed considerably between models (Table 1). For example, the α/β ratio is an important quantity in LQ-based models because it determines the magnitude of dose response “curvature” and predicts the sensitivity of survival dose responses to dose fractionation, such as in cancer radiotherapy applications. Best-fit α/β ratio values for the NB, MNB and PLQ models tended to be smaller, compared with values from the Poisson model (Table 3). This difference may have substantial clinical implications for cancer radiotherapy in situations where the NB, MNB and/or PLQ models strongly outperform the Poisson model in fitting the data, e.g. for the DU145, CP3 and U373MG cancer cell lines (Table 1).

Table 2.

Best-fit model parameter values on each analyzed data set.

Cell type		DU145a	CP3a	U373MGa	CHOAA8a	Yeastb	Yeastb
Radiation type		x-rays (250 kVp)				x-rays (70 kVp)	241Am α-particles
Poisson model parameters	α (dose−1)	0.22	0.14	0.16	0.18	1.24	9.10
	95% CIs	0.20	0.10	0.14	0.17	0.79	7.85
		0.24	0.18	0.18	0.20	1.70	10.38
	β (dose−2)	0.012	0.044	0.020	0.018	2.68	5.59
	95% CIs	0.010	0.040	0.017	0.017	2.25	3.23
		0.014	0.047	0.023	0.020	3.11	8.21
NB model parameters	α (dose−1)	0.16	0.00	0.08	0.18	0.00	2.05
	95% CIs	0.12	0.00	0.04	0.16	0.00	0.42
		0.20	0.05	0.12	0.20	0.11	3.65
	β (dose−2)	0.028	0.075	0.047	0.019	5.80	52.67
	95% CIs	0.019	0.064	0.036	0.017	5.37	42.14
		0.037	0.079	0.060	0.023	6.27	64.26
	r	0.126	0.069	0.241	0.000	0.148	0.242
	95% CIs	0.064	0.049	0.155	0.000	0.124	0.204
		0.183	0.083	0.326	0.034	0.173	0.284
MNB model parameters	α (dose−1)	0.15	0.00	0.08	0.18	0.00	1.76
	95% CIs	0.10	0.00	0.04	0.16	0.00	0.19
		0.19	0.05	0.11	0.20	0.11	3.30
	β (dose−2)	0.029	0.074	0.045	0.019	5.70	53.3
	95% CIs	0.020	0.064	0.035	0.017	5.28	43.3
		0.038	0.079	0.057	0.024	6.15	64.5
	r	0.130	0.069	0.220	0.000	0.143	0.244
	95% CIs	0.067	0.049	0.141	0.000	0.120	0.207
		0.186	0.082	0.300	0.035	0.168	0.283
PLQ model parameters	α (dose−1)	0.13	0.00	0.01	0.18	0.00	0.00
	95% CIs	0.03	0.00	0.00	0.15	0.00	0.00
		0.19	0.05	0.09	0.20	0.21	2.32
	β (dose−2)	0.057	0.090	0.108	0.019	7.63	176.8
	95% CIs	0.029	0.075	0.062	0.017	6.52	121.1
		0.109	0.100	0.130	0.029	8.93	238.7
	γ (dose−1)	0.10	0.05	0.21	0.00	0.86	12.6
	95% CIs	0.04	0.03	0.10	0.00	0.63	8.2
		0.22	0.07	0.27	0.02	1.17	17.7
2C model parameters	α1 (dose−1)	0.18	0.19	0.08	0.27	0.00	0.00
	95% CIs	0.10	0.10	0.00	0.25	0.00	0.00
		0.24	0.26	0.15	0.29	0.39	2.89
	αn (dose−1)	0.29	0.76	0.37	0.40	6.13	13.7
	95% CIs	0.26	0.70	0.33	0.36	5.78	11.0
		0.36	0.83	0.42	0.45	6.27	14.3
	n	3.97	28.4	3.04	26.4	6.82	2.20
	95% CIs	2.84	18.8	2.38	14.3	5.99	1.88
		6.04	45.4	4.14	50.6	8.05	2.66

^{a =}data sets with doses in Gy,

^{b =}data sets with doses in kGy.

Table 3.

Best-fit α/β ratio values for the Poisson, NB, MNB and PLQ models on each analyzed data set. The 2C model is not included here because it is structurally distinct and does not include an α/β ratio concept.

Cell type		DU145a	CP3a	U373MGa	CHOAA8a	Yeastb	Yeastb
Radiation type		x-rays (250 kVp)				x-rays (70 kVp)	241Am α-particles
Poisson model parameters	α (dose−1)	0.22	0.14	0.16	0.18	1.24	9.10
	95% CIs	0.20	0.10	0.14	0.17	0.79	7.85
		0.24	0.18	0.18	0.20	1.70	10.38
	α/β (dose)	18.59	3.22	8.04	10.04	0.46	1.63
	95% CIs	14.87	2.28	6.14	8.37	0.29	0.86
		22.87	4.89	10.53	12.05	0.84	2.75
NB model parameters	α (dose−1)	0.16	0.00	0.08	0.18	0.00	2.05
	95% CIs	0.12	0.00	0.04	0.16	0.00	0.42
		0.20	0.05	0.12	0.20	0.11	3.65
	α/β (dose)	5.70	0.00	1.68	9.84	0.00	0.04
	95% CIs	3.15	0.00	0.88	8.20	0.00	0.02
		10.26	0.01	4.31	14.17	0.00	0.23
MNB model parameters	α (dose−1)	0.15	0.00	0.08	0.18	0.00	1.76
	95% CIs	0.10	0.00	0.04	0.16	0.00	0.19
		0.19	0.05	0.11	0.20	0.11	3.30
	α/β (dose)	5.16	0.00	1.66	9.93	0.00	0.03
	95% CIs	2.77	0.00	0.86	8.28	0.00	0.01
		9.70	0.01	4.44	14.60	0.00	0.37
PLQ model parameters	α (dose−1)	0.13	0.00	0.01	0.18	0.00	0.00
	95% CIs	0.03	0.00	0.00	0.15	0.00	0.00
		0.19	0.05	0.09	0.20	0.21	2.32
	α/β (dose)	2.18	0.00	0.11	9.68	0.00	0.00
	95% CIs	0.73	0.00	0.01	8.07	0.00	0.00
		16.63	0.01	1.55	17.52	0.00	0.00

^{a =}data sets with doses in Gy,

^{b =}data sets with doses in kGy.

The best-fit α parameter values which, in the context of the LQ formalism, predict the dose response at low doses/dose rates, also tended to be smaller for the NB, MNB and PLQ models, compared with values from the Poisson model (Table 3). The lower α estimates (i.e. low dose “slopes”) from the non-Poisson models tended to be visibly closer to the data, than the Poisson model estimates (Figs. 1–3). However, in some of the analyzed data sets (e.g. DU145, CP3, CHOAA8) the dose responses at low doses were complex, presumably due to poorly understood phenomena such as low dose hypersensitivity and induced radioresistance, which are not explicitly modeled by either of the formalisms considered here.

For the yeast α-particle data, a low dose rate experiment with the same cell strain and culture conditions was performed and the α parameter for those data was estimated (Bertsche 1978). This allowed a form of validation of our model fits by comparing best-fit α parameter values from each model with the independently estimated value at low dose rate, which was 2.1 ± 0.3 kGy⁻¹ (Bertsche 1978). Clearly, the NB and MNB model α parameter estimates of 2.05 and 1.76 kGy⁻¹, respectively (Table 3), were very close to the independently estimated value from low dose rate studies, within the uncertainties. In contrast, the Poisson model predicted a much larger (overestimated) α value of 9.10 kGy⁻¹, whereas the PLQ model predicted an underestimated value of 0 (Table 3). On this data set, the NB and MNB models also achieved the highest Akaike weights, whereas the other models were essentially rejected (Table 1).

These results suggest that the added flexibility of the NB and MNB models compared with the Poisson formalism, while keeping the fundamental LQ model assumptions, can allow better predictions of the survival dose response at low doses/dose rates, based on model fits to data at high dose rates. As mentioned previously, this may be useful in radiation protection, where low dose/dose rate effects are important for estimating risks from occupational radiation exposure (e.g. nuclear industry workers, medical professionals, pilots, astronauts) and some accidental exposure scenarios (e.g. exposure to radioactive fallout from nuclear power plant accidents).

Extrapolation from low to high doses

Extrapolation of model fits obtained on low/intermediate doses to high doses showed that the Poisson model performed well on 2 out of 6 data sets (DU145 and CHOAA8 cells), but worse on the remaining 4 (Table 4, Fig. 4). Consequently, predictions of survival at high doses based on model fits at lower doses can be made more reliable by using non-Poisson models instead of the Poisson model. Such predictions can have important clinical implications for planning high dose stereotactic cancer radiotherapy or radiosurgery. However, no single model was optimal for all the tested data sets, which probably reflects the well-established difficulty of attempting an extrapolation.

Table 4.

Comparisons of root mean squared error (RMSE) values for different models on each analyzed data set. Comparison of different models: RMSE values calculated over the highest 1/3 of the dose range, using predictions from models fitted to the lower 2/3 of the dose range (i.e. low/intermediate doses), were compared across different models. A ratio value of 1.0 indicates the best-fitting model (the one with smallest RMSE) on the given data set, and larger values indicate worse performance. Values <1.05 are shown in bold font to indicate the best-performing model(s). Comparison within each model: RMSE values calculated for each model the highest 1/3 of the dose range but based on two different fits. (a) One using the best-fit parameters over the full dose range and (b) the other based on the best-fit parameters over the 2/3 dose range. The ratio (b-a)/a between the two RMSE (obtained with the same model) quantifies the effect of the missing information. The lowest value is shown in bold font to indicate the best-performing model.

Cell type	DU145	CP3	U373MG	CHOAA8	Yeast	Yeast
Radiation type	x-rays (250 kVp)				x-rays (70 kVp)	241Am α-particles
Comparison of different models
Poisson model	1.00	4.07	2.13	1.00	1.02	7.46
NB model	2.31	1.16	2.23	1.01	1.01	3.08
MNB model	2.04	1.00	1.82	1.00	1.00	2.61
PLQ model	1.70	1.35	1.00	1.01	1.02	1.25
2C model	1.76	3.18	1.27	1.25	1.22	1.00
Comparison within each model
Poisson model	0.62	2.21	1.20	0.02	3.53	4.09
NB model	5.05	0.36	3.92	0.02	8.51	3.77
MNB model	4.27	0.16	2.78	0.02	8.33	3.04
PLQ model	3.22	0.43	0.96	0.02	6.41	0.25
2C model	3.73	2.89	1.60	0.15	10.87	0.04

Figure 4.

Extrapolation of models fitted to data at low/medium doses (lower 2/3 of the dose range in each data set) to high doses (higher 1/3 of the dose range).

Discussion and Conclusions

Our analysis of diverse cell survival data shows that the standard LQ model with Poisson errors does not have enough flexibility to describe data sets with a small initial slope (i.e. a strongly curved or shouldered dose response at low doses) combined with dose response “straightening” at high doses. In contrast, the alternative models, which have an additional parameter, do have sufficient flexibility for such situations. The added flexibility can translate into better descriptions of available survival dose response data, and better predictions of the dose responses at low doses/dose rates, as well as at high doses.

Importantly, the meaning and interpretation for the extra adjustable parameter differ between models. This situation stresses the need to take into account the mechanistic basis for each evaluated model. The NB and MNB model variants preserve the main assumptions of the LQ formalism, whereas some other dose response models (e.g. 2C) do not. In the NB and MNB models, the extra parameter (r) represents “overdispersion” and determines the relationship between mean and variance for lethal lesions.

Likely contributors to overdispersion of radiogenic lethal lesions include spatial clustering of microdosimetric energy deposition events, and heterogeneity of the cell population’s responses to radiation that result from cell cycle effects, epigenetics, and double strand break repair pathway choice (Oliveira et al. 2016). In principle, such heterogeneity could be accounted for mathematically, by assuming the α parameter in the LQ model is not identical for all irradiated cells, but follows a gamma distribution, whereas the α/β ratio is held constant for all cells (Shuryak et al. 2015). Because a continuous mixture of Poisson distributions where the Poisson mean is gamma- distributed produces an NB distribution (Zhou et al. 2015), such an assumption applied to the α parameter would yield an NB distribution of lethal lesions per cell. More complicated situations— where both the α and β parameters vary between cells, but are correlated to each other—would also result in survival curves that are similar in shape to those generated using the proposed NB and MNB models. Therefore, the LQ model with an NB error distribution is a reasonable approximation for the effects of heterogeneity in cellular radiosensitivity and/or the complex patterns of microdosimetric energy deposition by ionizing tracks.

In addition to improving LQ model fits over a wide dose range, using the NB error distribution instead of the Poisson distribution tends to improve α parameter estimates and consequent predictions of responses at low doses/dose rates, which have implications for radiation protection. For example, people can be exposed to low dose and/or low dose rate radiation in occupational settings (e.g. pilots, astronauts, some medical personnel) and radioactive contamination from nuclear power plant accidents or malicious events. Indirectly, more accurate α parameter estimates for sparsely ionizing radiations (e.g. gamma rays) could be useful for improving the estimation of relative biological effectiveness (RBE) for densely ionizing radiations. This is so because RBE at low doses approaches the ratio of α parameters for the studied densely ionizing radiation and the reference sparsely ionizing radiation. Therefore, adjustments of the error distribution within the LQ model framework can potentially improve radiobiological modeling and risk estimation in many situations, including high dose cancer radiotherapy and low dose/dose rate radiation protection.