The Hug-Kellerer equation as the universal cell survival curve

Formulation of Time-Dependent Cell Survival with Saturable Repairability of Radiation Damage

This study aims to provide a model that compounds historically proposed ideas regarding cell survival irradiated with X rays or particles. The parameters used in this model have simple meanings and are closely related to cell death-related phenomena. The model is adaptable to a wide range of doses and dose rates and thus can consistently explain previously published cell survival data. The formulas of the model were derived by using five basic ideas: 1. "Poisson's law"; 2. "DNA affected damage"; 3. "repair"; 4. "clustered affected damage"; and 5. "saturation of reparability". The concept of affected damage is close to but not the same as the effect caused by the double-strand break (DSB). The parameters used in the formula are related to seven phenomena: 1. "linear coefficient of radiation dose"; 2. "probability of making affected damage"; 3. "cell-specific repairability", 4. "irreparable damage by adjacent affected damage"; 5. "recovery of temporally changed repairability"; 6. "recovery of simple damage which will make the affected damage"; 7. "cell division". By using the second parameter, this model includes cases where a single hit results in repairable-lethal and double-hit results in repairable-lethal. The fitting performance of the model for the experimental data was evaluated based on the Akaike information criterion, and practical results were obtained for the published experiments irradiated with a wide range of doses (up to several 10 Gy) and dose rates (0.17 Gy/h to 55.8 Gy/h). The direct association of parameters with cell death-related phenomena has made it possible to systematically fit survival data of different cell types and different radiation types by using crossover parameters.

[Limits of dose constraint definition for organs at risk specific to stereotactic radiotherapy]

Stereotactic radiotherapy is a very hypofractionated radiotherapy (>7.5Gy per fraction), and therefore is more likely to induce late toxicities than conventional normofractionated irradiations. The present study examines four frequent and potentially serious late toxicities: brain radionecrosis, radiation pneumonitis, radiation myelitis, and radiation-induced pelvic toxicities. The critical review focuses on the toxicity scales, the definition of the dose constrained volume, the dosimetric parameters, and the non-dosimetric risk factors. The most commonly used toxicity scales remain: RTOG/EORTC or common terminology criteria for adverse events (CTCAE). The definition of organ-at-risk volume requiring protection is often controversial, which limits the comparability of studies and the possibility of accurate dose constraints. Nevertheless, for the brain, whatever the indication (arteriovenous malformation, benign tumor, metastasis of solid tumors...), the association between the volume of brain receiving 12Gy (V12Gy) and the risk of cerebral radionecrosis is well established for both single and multi-fraction stereotactic irradiation. For the lung, the average dose received by both lungs and the V20 seem to correlate well with the risk of radiation-induced pneumonitis. For the spinal cord, the maximum dose is the most consensual parameter. Clinical trial protocols are useful for nonconsensual dose constraints. Non-dosimetric risk factors should be considered when validating the treatment plan.

The role of the spatially fractionated radiation therapy in the management of advanced bulky tumors

Spatially fractionated radiation therapy (SFRT) refers to the delivery of a single large dose of radiation within the target volume in a heterogeneous pattern using either a custom GRID block, multileaf collimators, and virtual methods such as helical tomotherapy or synchrotron-based microbeams. The potential impact of this technique on the regression of bulky deep-seated tumors that do not respond well to conventional radiotherapy has been remarkable. To date, a large number of patients have been treated using the SFRT techniques. However, there are yet many technical and medical challenges that have limited their routine use to a handful of clinics, most commonly for palliative intent. There is also a poor understanding of the biological mechanisms underlying the clinical efficacy of this approach. In this article, the methods of SFRT delivery together with its potential biological mechanisms are presented. Furthermore, technical challenges and clinical achievements along with the radiobiological models used to evaluate the efficacy and safety of SFRT are highlighted.

Improved Asymptotic Expansions in High- and Low-Dose Ranges for Generalized Multi-Hit Model of Radiation-Induced Cell Survival

By considering an upper bound on the number of radiation-induced potential lethal damages that can be repaired in a cell, we have proposed the generalized multi-hit (GMH) model with a closed-form solution, which can better fit various radiation-induced cell survival curves. Recent analysis shows that the asymptotic expansions that we gave before can be used to approximate the generalized single-hit single-target (GSHST) model rather than the GMH model. To illustrate the asymptotic trends of radiation-induced cell survival curves, in this study, we improve the asymptotic expansions of the GMH model in low- and high-dose ranges based on the limit formula of the incomplete gamma function in the corresponding dose ranges. When the upper limit of the number of radiation-induced potential lethal damages is one, the improved expansions of the GMH model can be reduced to the previous expansions of the GSHST model, and the improved asymptotic expansions of the GMH model also indicate that the GMH model has the generalized linear-quadratic-linear (LQL) feature. The numerical simulations indicate that the improved asymptotic expansions in high- and low-dose ranges agree well with the non-linear fitting of the GMH model in six kinds of cell lines under the corresponding dose ranges. In addition, we analyze the relative errors of the improved expansions of the GMH model in high- and low-dose ranges to demonstrate the accuracy and effectiveness of the improved expansions. Based on the error analysis, we further give the reasonable ranges of radiation dose applicable to the improved asymptotic expansions of the GMH model.

Generalized Multi-Hit Model of Radiation-Induced Cell Survival with a Closed-Form Solution: An Alternative Method for Determining Isoeffect Doses in Practical Radiotherapy

The standard linear-quadratic (LQ) model is currently the preferred model for describing the ionizing radiation-induced cell survival curves and tissue responses. And the LQ model is also widely used to calculate isoeffect doses for comparing different fractionated schemes in clinical radiotherapy. Despite its ubiquity, because the actual dose-response curve may appear linear at high doses in the semilogarithmic plot, the application of the LQ model is greatly challenged in the high-dose region, while the dose employed in stereotactic body radiotherapy (SBRT) is often in this area. Alternatively, the biophysical models of radiation-induced effects with a linear-quadratic-linear (LQL) characteristic can well fit the dose-survival curve of cells in vitro. However, most of these LQL models are phenomenological and have not fully considered the biophysical mechanism of radiation-induced damage and repair, and the fitting quality decreases in some high-dose ranges. In this work, to provide an alternative model to describe the cell survival curves in high-dose ranges and predict the biologically effective dose (BED) for SBRT, we propose a novel generalized multi-hit model with a closed-form solution by considering an upper bound on the number of lethal damages induced by radiation that can be repaired in a cell. This model has a clear biophysical basis and a simple expression, and also has the LQL characteristic under low- and high-dose approximate conditions. The experimental data fitting indicated that compared to the standard LQ model and our previously generalized target model, the current model can better fit the radiation-induced cell survival curves in the high-dose ranges (P < 0.05). The current model parameters and parameter ratios were determined from the fits in different kinds of cell lines irradiated with various dose rates and linear energy transfer (LET), which indicates that the model parameters significantly depend on the dose rate and LET. Based on the current model, we derived two equivalence formulae for the BED calculations in the low- and high-dose ranges, and then calculated the BED for the clinical data of SBRT from 17 selected studies. The correlation analysis showed that there were significant linear correlations between the BED at isocenter and planning target volume (PTV) edge calculated by this model and the LQ model (R > 0.86, P < 0.001). In conclusion, the generalized multi-hit model proposed in this work can be used as an alternative tool to handle in vitro radiation-induced cell survival curves in high-dose ranges, and calculate the in vivo BED for comparing the dose fractionation schemes in clinical radiotherapy.

Grid Block Design Based on Monte Carlo Simulated Dosimetry, the Linear Quadratic and Hug-Kellerer Radiobiological Models

Purpose

The clinical efficacy of Grid therapy has been examined by several investigators. In this project, the hole diameter and hole spacing in Grid blocks were examined to determine the optimum parameters that give a therapeutic advantage.

Methods

The evaluations were performed using Monte Carlo (MC) simulation and commonly used radiobiological models. The Geant4 MC code was used to simulate the dose distributions for 25 different Grid blocks with different hole diameters and center-to-center spacing. The therapeutic parameters of these blocks, namely, the therapeutic ratio (TR) and geometrical sparing factor (GSF) were calculated using two different radiobiological models, including the linear quadratic and Hug-Kellerer models. In addition, the ratio of the open to blocked area (ROTBA) is also used as a geometrical parameter for each block design. Comparisons of the TR, GSF, and ROTBA for all of the blocks were used to derive the parameters for an optimum Grid block with the maximum TR, minimum GSF, and optimal ROTBA. A sample of the optimum Grid block was fabricated at our institution. Dosimetric characteristics of this Grid block were measured using an ionization chamber in water phantom, Gafchromic film, and thermoluminescent dosimeters in Solid Water™ phantom materials.

Results

The results of these investigations indicated that Grid blocks with hole diameters between 1.00 and 1.25 cm and spacing of 1.7 or 1.8 cm have optimal therapeutic parameters (TR > 1.3 and GSF~0.90). The measured dosimetric characteristics of the optimum Grid blocks including dose profiles, percentage depth dose, dose output factor (cGy/MU), and valley-to-peak ratio were in good agreement (±5%) with the simulated data.

Conclusion

In summary, using MC-based dosimetry, two radiobiological models, and previously published clinical data, we have introduced a method to design a Grid block with optimum therapeutic response. The simulated data were reproduced by experimental data.

The Impact of the Geometrical Structure of the DNA on Parameters of the Track-Event Theory for Radiation Induced Cell Kill

Background and Purpose

When fractionation schemes for hypofractionation and stereotactic body radiotherapy are considered, a reliable cell survival model at high dose is needed for calculating doses of similar biological effectiveness. An alternative to the LQ-model is the track-event theory which is based on the probabilities for one- and two two-track events. A one-track-event (OTE) is always represented by at least two simultaneous double strand breaks. A two-track-event (TTE) results in one double strand break. Therefore at least two two-track-events on the same or different chromosomes are necessary to produce an event which leads to cell sterilization. It is obvious that the probabilities of OTEs and TTEs must somehow depend on the geometrical structure of the chromatin. In terms of the track-event theory the ratio ε of the probabilities of OTEs and TTEs includes the geometrical dependence and is obtained in this work by simple Monte Carlo simulations.

Materials and Methods

For this work it was assumed that the anchors of loop forming chromatin are most sensitive to radiation induced cell deaths. Therefore two adjacent tetranucleosomes representing the loop anchors were digitized. The probability ratio ε of OTEs and TTEs was factorized into a radiation quality dependent part and a geometrical part: ε₌ε_{ion ∙}ε_geo. ε_geo was obtained for two situations, by applying Monte Carlo simulation for DNA on the tetranucleosomes itself and for linker DNA. Low energy electrons were represented by randomly distributed ionizations and high energy electrons by ionizations which were simulated on rays. ε_ion was determined for electrons by using results from nanodosimetric measurements. The calculated ε was compared to the ε obtained from fits of the track event model to 42 sets of experimental human cell survival data.

Results

When the two tetranucleosomes are in direct contact and the hits are randomly distributed ε_geo and ε are 0.12 and 0.85, respectively. When the hits are simulated on rays ε_geo and ε are 0.10 and 0.71. For the linker-DNA ε_geo and ε for randomly distributed hits are 0.010 and 0.073, and for hits on rays 0.0058 and 0.041, respectively. The calculated ε fits the experimentally obtained ε = 0.64±0.32 best for hits on the tetranucleosome when they are close to each other both, for high and low energy electrons.

Conclusions

The parameter ε_geo of the track event model was obtained by pure geometrical considerations of the chromatin structure and is 0.095 ± 0.022. It can be used as a fixed parameter in the track-event theory.

Is grid therapy useful for all tumors and every grid block design?

Grid therapy is a treatment technique that has been introduced for patients with advanced bulky tumors. The purpose of this study is to investigate the effect of the radiation sensitivity of the tumors and the design of the grid blocks on the clinical response of grid therapy. The Monte Carlo simulation technique is used to determine the dose distribution through a grid block that was used for a Varian 2100C linear accelerator. From the simulated dose profiles, the therapeutic ratio (TR) and the equivalent uniform dose (EUD) for different types of tumors with respect to their radiation sensitivities were calculated. These calculations were performed using the linear quadratic (LQ) and the Hug-Kellerer (H-K) models. The results of these calculations have been validated by comparison with the clinical responses of 232 patients from different publications, who were treated with grid therapy. These published results for different tumor types were used to examine the correlation between tumor radiosensitivity and the clinical response of grid therapy. Moreover, the influence of grid design on their clinical responses was investigated by using Monte Carlo simulations of grid blocks with different hole diameters and different center-to-center spacing. The results of the theoretical models and clinical data indicated higher clinical responses for the grid therapy on the patients with more radioresistant tumors. The differences between TR values for radioresistant cells and radiosensitive cells at 20 Gy and 10 Gy doses were up to 50% and 30%, respectively. Interestingly, the differences between the TR values with LQ model and H-K model were less than 4%. Moreover, the results from the Monte Carlo studies showed that grid blocks with a hole diameters of 1.0 cm and 1.25 cm may lead to about 19% higher TR relative to the grids with hole diameters smaller than 1.0 cm or larger than 1.25 cm (with 95% confidence interval). In sum-mary, the results of this study indicate that grid therapy is more effective for tumors with radioresistant characteristics than radiosensitive tumors.

On the analysis of clonogenic survival data: Statistical alternatives to the linear-quadratic model

Background

The most frequently used method to quantitatively describe the response to ionizing irradiation in terms of clonogenic survival is the linear-quadratic (LQ) model. In the LQ model, the logarithm of the surviving fraction is regressed linearly on the radiation dose by means of a second-degree polynomial. The ratio of the estimated parameters for the linear and quadratic term, respectively, represents the dose at which both terms have the same weight in the abrogation of clonogenic survival. This ratio is known as the α/β ratio. However, there are plausible scenarios in which the α/β ratio fails to sufficiently reflect differences between dose-response curves, for example when curves with similar α/β ratio but different overall steepness are being compared. In such situations, the interpretation of the LQ model is severely limited.

Methods

Colony formation assays were performed in order to measure the clonogenic survival of nine human pancreatic cancer cell lines and immortalized human pancreatic ductal epithelial cells upon irradiation at 0-10 Gy. The resulting dataset was subjected to LQ regression and non-linear log-logistic regression. Dimensionality reduction of the data was performed by cluster analysis and principal component analysis.

Results

Both the LQ model and the non-linear log-logistic regression model resulted in accurate approximations of the observed dose-response relationships in the dataset of clonogenic survival. However, in contrast to the LQ model the non-linear regression model allowed the discrimination of curves with different overall steepness but similar α/β ratio and revealed an improved goodness-of-fit. Additionally, the estimated parameters in the non-linear model exhibit a more direct interpretation than the α/β ratio. Dimensionality reduction of clonogenic survival data by means of cluster analysis was shown to be a useful tool for classifying radioresistant and sensitive cell lines. More quantitatively, principal component analysis allowed the extraction of scores of radioresistance, which displayed significant correlations with the estimated parameters of the regression models.

Conclusions

Undoubtedly, LQ regression is a robust method for the analysis of clonogenic survival data. Nevertheless, alternative approaches including non-linear regression and multivariate techniques such as cluster analysis and principal component analysis represent versatile tools for the extraction of parameters and/or scores of the cellular response towards ionizing irradiation with a more intuitive biological interpretation. The latter are highly informative for correlation analyses with other types of data, including functional genomics data that are increasingly beinggenerated.

Track-event theory of cell survival with second-order repair

When fractionation schemes for hypofractionation and stereotactic body radiotherapy are considered, a reliable cell survival model at high dose is needed for calculating doses of similar biological effectiveness. In this work, a simple model for cell survival which is valid also at high dose is developed from Poisson statistics. It is assumed that a cell is killed by an event that is defined by two double-strand breaks on the same or different chromosomes. Two different mechanisms can produce events. A one-track event is always represented by two simultaneous double-strand breaks. A two-track event results in one double-strand break. Therefore, at least two two-track events on the same or different chromosomes are necessary to produce an event. It is assumed that two double-strand breaks can be repaired with a certain repair probability. Both the one-track events and the two-track events are statistically independent. From the stochastic nature of cell killing which is described by the Poisson distribution, the cell survival probability was derived. The model was fitted to experimental data. It was shown that a solution based on Poisson statistics exists for cell survival. It exhibits exponential cell survival at high dose and a finite gradient of cell survival at vanishing dose, which is in agreement with experimental cell studies. The model fits the experimental data as well as the LQ model and is based on two free parameters. It was shown that cell survival can be described with a simple analytical formula on the basis of Poisson statistics. This solution represents in the limit of large dose the typical exponential behavior and predicts cell survival as well as the LQ model.

A Comparative Analysis of Radiobiological Models for Cell Surviving Fractions at High Doses

For many years the linear-quadratic (LQ) model has been widely used to describe the effects of total dose and dose per fraction at low-to-intermediate doses in conventional fractionated radiotherapy. Recent advances in stereotactic radiosurgery (SRS) and stereotactic radiotherapy (SRT) have increased the interest in finding a reliable cell survival model, which will be accurate at high doses, as well. Different models have been proposed for improving descriptions of high dose survival responses, such as the Universal Survival Curve (USC), the Kavanagh-Newman (KN) and several generalizations of the LQ model, e.g. the Linear-Quadratic-Linear (LQL) model and the Padé Linear Quadratic (PLQ) model. The purpose of the present study is to compare a number of models in order to find the best option(s) which could successfully be used as a fractionation correction method in SRT. In this work, six independent experimental data sets were used: CHOAA8 (Chinese hamster fibroblast), H460 (non-small cell lung cancer, NSLC), NCI-H841 (small cell lung cancer, SCLC), CP3 and DU145 (human prostate carcinoma cell lines) and U1690 (SCLC). By detailed comparisons with these measurements, the performance of nine different radiobiological models was examined for the entire dose range, including high doses beyond the shoulder of the survival curves. Using the computed and measured cell surviving fractions, comparison of the goodness-of-fit for all the models was performed by means of the reduced χ2-test with a 95% confidence interval. The obtained results indicate that models with dose-independent final slopes and extrapolation numbers generally represent better choices for SRT. This is especially important at high doses where the final slope and extrapolation numbers are presently found to play a major role. The PLQ, USC and LQL models have the least number of shortcomings at all doses. The extrapolation numbers and final slopes of these models do not depend on dose. Their asymptotes for the cell surviving fractions are exponentials at low as well as high doses, and this is in agreement with the behaviour of the corresponding experimental data. This is an important improvement over the LQ model which predicts a Gaussian at high doses. Overall and for the highlighted reasons, it was concluded that the PLQ, USC and LQL models are theoretically well-founded. They could prove useful compared to the other proposed radiobiological models in clinical applications for obtaining uniformly accurate cell surviving fractions encountered in stereotactic high-dose radiotherapy as well as at medium and low doses.