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Edited by: Anatoly Dritschilo, Georgetown University, United States

Reviewed by: Dalong Pang, Georgetown University, United States; Vinay Sharma, University of the Witwatersrand, South Africa; Timothy John Jorgensen, Georgetown University, United States

Specialty section: This article was submitted to Radiation Oncology, a section of the journal Frontiers in Oncology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Recent technological advances allow precise radiation delivery to tumor targets. As opposed to more conventional radiotherapy—where multiple small fractions are given—in some cases, the preferred course of treatment may involve only a few (or even one) large dose(s) per fraction. Under these conditions, the choice of appropriate radiobiological model complicates the tasks of predicting radiotherapy outcomes and designing new treatment regimens. The most commonly used model for this purpose is the venerable linear-quadratic (LQ) formalism as it applies to cell survival. However, predictions based on the LQ model are frequently at odds with data following very high acute doses. In particular, although the LQ predicts a continuously bending dose–response relationship for the logarithm of cell survival, empirical evidence over the high-dose region suggests that the survival response is instead log-linear with dose. Here, we show that the distribution of lethal chromosomal lesions among individual human cells (lymphocytes and fibroblasts) exposed to gamma rays and X rays is somewhat overdispersed, compared with the Poisson distribution. Further, we show that such overdispersion affects the predicted dose response for cell survival (the fraction of cells with zero lethal lesions). This causes the dose response to approximate log-linear behavior at high doses, even when the mean number of lethal lesions per cell is well fitted by the continuously curving LQ model. Accounting for overdispersion of lethal lesions provides a novel, mechanistically based explanation for the observed shapes of cell survival dose responses that, in principle, may offer a tractable and clinically useful approach for modeling the effects of high doses per fraction.

At traditional doses per fraction (e.g., <8 Gy) used for cancer radiotherapy, the linear-quadratic (LQ) model of cell killing by radiation continues to be successfully used. This simple formalism has a mechanistic justification in terms of lethal lesions produced by single ionizing tracks (intratrack action) versus those produced by multiple independent tracks (intertrack action). A notable feature of the LQ model, as it is usually applied, is that it presupposes that the number of lethal lesions per cell is Poisson distributed. For low LET radiations, the LQ dose response for the mean number of lethal lesions per cell exhibits a characteristic upward curvature throughout the entire dose range. With a few possible exceptions (

In fact, cell survival data often produce dose responses that approach a constant slope on a logarithmic scale (“terminal exponential”) (

A common feature of these approaches is that they focus on the

For the current studies, we take advantage of the well-established relationship between cytogenetic damage and cell killing (

Although the Poisson distribution is a reasonable approximation for lethal lesion data, there are reasons to believe that it may not be the best choice for therapeutically relevant doses of sparsely ionizing radiation (e.g., gamma rays and X rays with energies typically used in radiation oncology). For example, the microdosimetric distribution of radiation energy deposition is not optimally approximated by the Poisson distribution (

Overdispersion of radiation-induced lethal lesion yields is not merely of theoretical interest, but can be clinically important for cancer radiotherapy. This is because overdispersion alters the relationship between the mean number of lethal lesions per cell and survival (the probability of a cell having zero lethal lesions). It follows that even if the

Here, we used data on clonogenically lethal chromosomal aberrations in human cells (lymphocytes and fibroblasts) exposed to gamma rays and X rays to search for overdispersion and to quantify clinically relevant effects that overdispersion may have on survival curve shape. Our objectives are to show the following: (1) There is indeed cytogenetic evidence for overdispersion of lethal lesions per cell. (2) The estimated overdispersion is sufficient to produce effectively log-linear behavior of the cell surviving fraction at high doses, even if the mean number of lethal lesions per cell is generated from the continuously curving LQ model. We also argue that accounting for overdispersion may therefore have clinically relevant implications for explaining and predicting radiotherapy effects at high doses per fraction.

We quantified the numbers of clonogenically lethal chromosomal lesions in human lymphocytes exposed to 4 Gy of gamma rays (

For the lymphocytes data, blood was drawn from two healthy male volunteers following procedures approved by the UTMB Institutional Review Board. Written informed consent was obtained from the participants of this study. Cells were cultured in 25-cm^{2} tissue culture flasks containing 5 ml RPMI 1640 medium supplemented with 15% fetal bovine serum plus penicillin and streptomycin. To this was added 0.1 ml phytohemagglutinin and 0.4 ml whole blood. Immediately thereafter, cells were irradiated using a JL Shepherd Mark 69-1 Irradiator with 4 Gy of ^{137}Cs gamma rays at a dose rate of 1.3 Gy/min. Irradiation cells were incubated for 48 h before harvest with 0.1 µg/ml Colcemid, present during the final 3 h. Cells were then fixed and spread onto slides following standard cytogenetic procedures. Slides were processed for mFISH hybridization using SpectraVision 24-color probe cocktail (Vysis). Images were captured using a Zeiss Axiophot epifluorescence microscope equipped with a black and white CCD camera controlled by Power Gene image analysis software (Applied Imaging, Inc.). Karyotypes were constructed from these images, and exchanges were scored. These were assigned mPAINT descriptors as described previously (

For the fibroblast data, low-passage AG1522 normal human fibroblasts were obtained from the NIA cell repository. To prepare samples for metaphase chromosome analysis, two 6-h Colcemid (0.1 µg/ml) collection intervals were used that encompassed the time range spanning the peak of the first-division mitotic index. The first collection interval was between 30 and 36 h after subculture and the last between 36 and 42 h. Following the addition of 0.075 M KC1, cell suspensions were fixed onto glass microscope slides using standard cytogenetic methods to include staining in Sorensen’s-buffered Giemsa. Barring the infrequent occurrence of tricentric chromosomes, Giemsa staining does not directly allow for the identification of complex exchange aberrations. It is, nevertheless, capable of indirectly detecting many types of complex exchanges as pseudosimple exchanges (

We modeled the mean yield

Here, α represents the contribution of lesions resulting from the same ionizing track, and β represents the contribution from different tracks. For simplicity, we ignore the small probability of lethal lesions in cells under background conditions (at

The Poisson distribution is often used to model the distribution of lethal lesions per cell. It predicts the probability _{Pois}(

A fundamental property of the Poisson distribution is that its variance is equal to the mean. The negative binomial (NB) distribution is frequently used as a more flexible alternative that allows the variance to be larger than the mean. Here, we employed a customized NB distribution parametrization described in the Appendix (Eq. ^{2}. Consequently, if

This specific formulation was chosen merely as a convenient example of an overdispersed distribution that is sufficiently general to represent a variety of mechanisms for overdispersion (e.g., the effects of microdosimetric energy deposition heterogeneity, DSB complexity, and repair pathway differences). Other distributions targeted to more specific mechanisms have also been used: e.g., the compound Poisson (Neyman) distribution, which accounts for stochasticity of the number of ionizing track traversals per cell and the number of chromosomal aberrations per track (

The clonogenic cell surviving fraction _{NB} = _{NB}(0) and _{Pois} = _{Pois}(0). The solutions are as follows:

We used maximum likelihood estimation to fit the Poisson and NB distributions to the data on gamma-ray irradiated lymphocytes. This popular approach involves finding parameter values that maximize the likelihood of making the observations given the parameters. For the Poisson distribution, the only adjustable parameter is the mean, _{Pois}) values across all analyzed cells. The log-likelihood function is described in the Appendix (Eq. ^{®} software.

To fit the data on fibroblasts, for which the full distribution of lethal lesions per cell was not available (

Ninety-five percent confidence intervals (CIs) for the adjustable parameters in each analyzed distribution were estimated by profile likelihood.

Relative performances of different probability distributions fitted to the same data were assessed by the Akaike information criterion with sample size correction (AICc) (

Biologically effective dose is a convenient and frequently used metric for comparing the predicted potency of radiotherapy protocols with different dose fractionation schemes. Assuming complete radiation damage repair between dose fractions, we can calculate the BED for the _{M}_{frac} is the number of dose fractions and _{M}

Explicit BED solutions for the Poisson and NB distributions are below, where

Summary statistics suggested that the distribution of lethal chromosomal lesions per cell in human lymphocytes exposed to 4 Gy of gamma rays was somewhat “overdispersed”: the mean was 1.96 and the variance was 2.51, so the ratio of variance/mean was ~1.3, rather than 1.0 as in the Poisson distribution. The upper 95% CI for skewness based on 10,000 bootstrap samples from the observed distribution was 2.16-fold greater than expected for the Poisson distribution.

Evidence for overdispersion was also found when we compared fits to the data from: (a) single Poisson distribution, (b) double Poisson distributions, and (c) the NB distribution. As shown graphically in Figure

Best fits of different probability distributions to data on lethal lesions in lymphocytes exposed to 4 Gy of gamma rays. Gray bars, data; blue curve, single Poisson distribution; green curves, double Poisson distributions; brown curve, sum of double Poisson distributions; red curve, NB distribution.

Therefore, the NB and double Poisson distributions produced better fits than the single Poisson, particularly for the “upper tail” region of the data (Figure

Evidence for overdispersion of lethal lesions was also found at relevant doses in another cell type: human fibroblasts (

These results from lymphocytes and fibroblasts provide support for our contention that the distribution of lethal chromosomal lesions per cell at high doses of sparsely ionizing radiation (e.g., gamma rays) may not be optimally described by the Poisson distribution. Alternative approaches such as the NB distribution, which allow the variance to be larger than the mean, allow better fits to such data.

Importantly, when the Poisson and NB distributions have the same mean number of lethal lesions per cell, the probability of a cell to contain zero lethal lesions predicted by the NB distribution is larger than that predicted by the Poisson distribution. This effect increases rapidly with increasing mean number of lethal lesions per cell and with increasing overdispersion (NB parameter

Ratio of cell surviving fractions (probabilities to have zero lethal lesions) predicted by the negative binomial (NB) distribution, relative to those predicted by the Poisson distribution, as function of the average number of lethal lesions per cell. This ratio is defined as _{NB}/_{Pois} from Eq.

From a theoretical perspective, binary misrepair serves as the cornerstone concept of the LQ formalism and the theory of dual radiation action (TDRA), upon which its mathematical underpinnings rest (^{2}) components of the dose–response relationship, respectively. As mentioned, the use of mean values for lethal lesions in the LQ model tends to produce fits that bend continuously downward and away from actual survival data at high doses. As shown, consideration of the distribution of lesions among cells mitigates this tendency, producing fits that approach a terminal log-linear survival response (Figure

Effects of overdispersion of lethal lesions on cell survival curve shapes. The mean number of lethal lesions per cell in all curves was calculated using the linear-quadratic model with α = 0.15 Gy^{−1} and α/β = 10 Gy

The LQ model can be contrasted to more sophisticated and complex radiobiological models that emphasize biological repair kinetics, as opposed to spatial/temporal aspects of damage with respect to track structure. We take the view that the intratrack and intertrack concepts embodied in the LQ model serve to “set the stage” for subsequent repair/misrepair processes following initial radiogenic damage. For the purposes of predicting responses of cells or tissues exposed to acute radiation doses, it is debatable whether a detailed consideration of such repair processes is necessary.

There is no question that the concept of binary misrepair evolved from early radiation cytogenetic theory (

From a clinical perspective, even modest differences in distribution shapes for lethal lesions per cell can have a potentially important effect on cell survival curve shapes at high radiation doses used for stereotactic radiotherapy (Figure

The differences in cell survival predicted by using an overdispersed distribution instead of the Poisson distribution can be important not only for single-dose radiotherapy but also for fractionated stereotactic treatments. For example, moderate overdispersion of lethal lesions per cell modeled by the NB distribution substantially modifies BED estimates for radiotherapy regimens that use three to five dose fractions of ≥5 Gy/fraction (Figure

Effects of overdispersion of lethal lesions on biologically effective dose (BED) estimates for fractionated radiotherapy. Blue curves, Poisson distribution; red curves, NB distribution with the overdispersion parameter

Our results do not contradict other explanations for cell survival curve “straightening” at high doses because an overdispersed error distribution and non-LQ dose dependence for the mean number of lethal lesions per cell are not mutually exclusive. Therefore, we do not argue specifically in favor of the LQ formalism and/or of the NB distribution. Our point is that even a continuously curving dose–response formalism coupled with an overdispersed error distribution is capable of accounting for survival curve “straightening” at high doses without invoking complicated or

All authors: study design, data analysis, and manuscript writing.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewers TJ and DP and the handling editor declared their shared affiliation.

The authors express their appreciation to Dr. Dudley Goodhead for valuable insight regarding radiobiological models. This work supported by the following grants from the National Aeronautics and Space Administration (NASA): NNX15AG74G (MC) and NNX14AC76G (BL) and from the National Institute of Allergy and Infectious Diseases (NIAID): U19AI067773 (IS).

The customized negative binomial (NB) distribution, where _{NB}(

The Poisson log-likelihood can be written as:

The NB log-likelihood can be written as:

Binomial approximations for the Poisson (BLL_{Pois}) and NB (BLL_{NB}) distributions, which were used to analyze fibroblast data, are described as follows:

Here, BLL represents the Binomial log-likelihood, _{obs} is the observed fraction of cells with zero lethal lesions, and _{obs} is the observed mean number of lethal lesions per cell.

Akaike information criterion with sample size correction (AICc) for the _{M}_{M}_{M}_{tot} is the total number of analyzed cells for all radiation doses in the data set:

The distribution with the lowest AICc value is considered to be best supported among those considered. The relative likelihood of the _{M}

Here, AICc_{min} is the lowest AICc value generated by the set of distributions being compared.

The evidence ratio for the tested distribution divided by the sum of the evidence ratios for all distributions being compared is the Akaike weight, _{M}