Linear-quadratic-linear Models Research Articles

Obtaining organ-specific radiobiological parameters from clinical data for radiation therapy planning of head and neck cancers

Objective. Different radiation therapy (RT) strategies, e.g. conventional fractionation RT (CFRT), hypofractionation RT (HFRT), stereotactic body RT (SBRT), adaptive RT, and re-irradiation are often used to treat head and neck (HN) cancers. Combining and/or comparing these strategies requires calculating biological effective dose (BED). The purpose of this study is to develop a practical process to estimate organ-specific radiobiologic model parameters that may be used for BED calculations in individualized RT planning for HN cancers. Approach. Clinical dose constraint data for CFRT, HFRT and SBRT for 5 organs at risk (OARs) namely spinal cord, brainstem, brachial plexus, optic pathway, and esophagus obtained from literature were analyzed. These clinical data correspond to a particular endpoint. The linear-quadratic (LQ) and linear-quadratic-linear (LQ-L) models were used to fit these clinical data and extract relevant model parameters (alpha/beta ratio, gamma/alpha, d T and BED) from the iso-effective curve. The dose constraints in terms of equivalent physical dose in 2 Gy-fraction (EQD2) were calculated using the obtained parameters. Main results. The LQ-L and LQ models fitted clinical data well from the CFRT to SBRT with the LQ-L representing a better fit for most of the OARs. The alpha/beta values for LQ-L (LQ) were found to be 2.72 (2.11) Gy, 0.55 (0.30) Gy, 2.82 (2.90) Gy, 6.57 (3.86) Gy, 5.38 (4.71) Gy, and the dose constraint EQD2 were 55.91 (54.90) Gy, 57.35 (56.79) Gy, 57.54 (56.35) Gy, 60.13 (59.72) Gy and 65.66 (64.50) Gy for spinal cord, optic pathway, brainstem, brachial plexus, and esophagus, respectively. Additional two LQ-L parameters d T were 5.24 Gy, 5.09 Gy, 7.00 Gy, 5.23 Gy, and 6.16 Gy, and gamma/alpha were 7.91, 34.02, 8.67, 5.62 and 4.95.Significance. A practical process was developed to extract organ-specific radiobiological model parameters from clinical data. The obtained parameters can be used for biologically based radiation planning such as calculating dose constraints of different fractionation regimens.

Radiobiological Meta-Analysis of the Response of Prostate Cancer to Different Fractionations: Evaluation of the Linear-Quadratic Response at Large Doses and the Effect of Risk and ADT.

The purpose of this work was to investigate the response of prostate cancer to different radiotherapy schedules, including hypofractionation, to evaluate potential departures from the linear-quadratic (LQ) response, to obtain the best-fitting parameters for low-(LR), intermediate-(IR), and high-risk (HR) prostate cancer and to investigate the effect of ADT on the radiobiological response. We constructed a dataset of the dose-response containing 87 entries/16,536 patients (35/5181 LR, 32/8146 IR, 20/3209 HR), with doses per fraction ranging from 1.8 to 10 Gy. These data were fit to tumour control probability models based on the LQ model, linear-quadratic-linear (LQL) model, and a modification of the LQ (LQmod) model accounting for increasing radiosensitivity at large doses. Fits were performed with the maximum likelihood expectation methodology, and the Akaike information criterion (AIC) was used to compare the models. The AIC showed that the LQ model was superior to the LQL and LQmod models for all risks, except for IR, where the LQL model outperformed the other models. The analysis showed a low α/β for all risks: 2.0 Gy for LR (95% confidence interval: 1.7-2.3), 3.4 Gy for IR (3.0-4.0), and 2.8 Gy for HR (1.4-4.2). The best fits did not show proliferation for LR and showed moderate proliferation for IR/HR. The addition of ADT was consistent with a suppression of proliferation. In conclusion, the LQ model described the response of prostate cancer better than the alternative models. Only for IR, the LQL model outperformed the LQ model, pointing out a possible saturation of radiation damage with increasing dose. This study confirmed a low α/β for all risks.

Modeling the impact of tissue oxygen profiles and oxygen depletion parameter uncertainties on biological response and therapeutic benefit of FLASH.

Ultra-high dose rate (FLASH) radiation has been reported to efficiently suppress tumor growth while sparing normal tissue; however, the mechanism of the differential tissue sparing effect is still not known. Oxygen has long been known to profoundly impact radiobiological responses, and radiolytic oxygen depletion has been considered to be a possible cause or contributor to the FLASH phenomenon. This work investigates the impact of tissue pO2 profiles, oxygen depletion per unit dose (g), and the oxygen concentration yielding half-maximum radiosensitization (the average of its maximum value and one) (k) in tumor and normal tissue. We developed a model that considers the dependent relationship between oxygen depletion and change of radiosensitivity by FLASH irradiation. The model assumed that FLASH irradiation depletes intracellular oxygen more rapidly than it diffuses into the cell from the extracellular environment. Cell survival was calculated based on the linear quadratic-linear model and the radiosensitivity related parameters were adjusted in 1Gy increments of the administered dose. The model reproduced published experimental data that were obtained with different cell lines and oxygen concentrations, and was used to analyze the impact of parameter uncertainties on the radiobiological responses. This study expands the oxygen depletion analysis of FLASH to normal human tissue and tumor based on clinically determined aggregate and individual patient pO2 profiles. The results show that the pO2 profile is the most essential factor that affects biological response and analyses based on the median pO2 rather than the full pO2 profile can be unreliable and misleading. Additionally, the presence of a small fraction of cells on the threshold of radiobiologic hypoxia substantially alters biological response due to FLASH oxygen depletion. We found that an increment in the k value is generally more protective of tumor than normal tissue due to a higher frequency of lower pO2 values in tumors. Variation in the g value affects the dose at which oxygen depletion impacts response, but does not alter the dose-dependent response trends, if the g value is identical in both tumor and normal tissue. The therapeutic efficacy of FLASH oxygen depletion is likely patient and tissue-dependent. For breast cancer, FLASH is beneficial in a minority of cases; however, in a subset of well oxygenated tumors, a therapeutic gain may be realized due to induced normal tissue hypoxia.

Extracting Organ-Specific Radiobiological Model Parameters of Abdominal Organs from Clinical Outcome Data for Biological Radiation Treatment Planning

<h3>Purpose/Objective(s)</h3> It is desirable to consider organ-specific treatment response in a variety of biological radiotherapy (RT) planning scenarios, e.g., adaptive RT with varying daily dose, reirradiation, multi-modality RT, varying fractionation. This study aims to extract organ-specific radiobiological model parameters for selected abdominal organs and to demonstrate their applications in determining OAR dose constraints using biologically effective dose (BED) and equivalent dose in 2Gy fractions (EQD2). <h3>Materials/Methods</h3> Available clinical studies from the literature reporting dose volume constraints for abdominal organs, e.g., duodenum, stomach, small bowel, and rectum, along with the compiled data in available sources such as TG101, QUANTEC, HyTEC, national and institutional trial protocols were analyzed. The data ranging from conventional fractionation RT (CFRT) to stereotactic body RT (SBRT) were analyzed to determine the maximum point doses (MPD) corresponding to no more than 5% grade ≥ 3 toxicity. The endpoints were selected according to toxicity scales (e.g., CTCAE (v3 or v4), RTOG). Iso-effect fittings (e.g., MPD vs. fractional dose) on the data collected for each organ were performed using both the linear quadratic (LQ) and the linear-quadratic-linear (LQL) models to extract the organ-specific radiobiological parameters. The dose constraint MPD in terms of BED and EQD2 for each OAR was calculated using the obtained organ-specific parameters. The use of the obtained MPD constraints was demonstrated in reirradiation planning. <h3>Results</h3> More than 50 studies were screened to find an average of 10 to 20 relevant studies for each OAR. The LQL and LQ models fit the clinical data from CFRT to SBRT reasonably well, although the LQL model showed a better fit compared to the LQ model. The table below shows the obtained LQ and LQL model parameters and the MPD constraints in terms of BED and EQD2 for selected abdominal organs, where χ²/df is the reduced chi-square. Using the data in the table, the MPD in the small bowel for a vaginal tumor case re-irradiated with 4 × 5Gy after one year from the initial RT of 10 × 3Gy was estimated as 50.2 Gy EQD2, below the constraint of 53.1 Gy. <h3>Conclusion</h3> We developed a robust method to successfully extract abdominal organ-specific radiobiological parameters from clinical data. These parameters can be used to convert OAR dose constraints from different RT fractionations/techniques for biologically optimized RT planning.

Tumor Control Probability After Radiosurgery of Brain Metastases With and Without Retreatment

To develop and compare tumor-control probability (TCP) models for single-fraction stereotactic radiosurgery (SRS) for brain metastasis (BMs) with and without retreatment. We developed three different schemas to model TCP of BMs treated with linear accelerator-based SRS. Dose to 99% of each planning target volume (PTV D99) and 6-month local control were fit using linear-quadratic-linear (LQ-L) models based on equivalent-dose conversions in 2 Gy (EQD2). The M1 schema had separate LQ-L TCP models for initial dose (M1-initial) and retreatment dose (M1-retreat), and the M2 schema had an LQ-L model using the sum of 50% of the initial SRS dose plus the retreatment SRS dose. The M1-initial and M1-retreat schema modeled local control after first SRS to 48 lesions (patients=22) and second SRS to 46 lesions (patients=21). The M0 schema included a whole data set of 349 lesions (patients=136) receiving first SRS (no retreatment and M1-initial). LQ-L models fitted the data well (χ2=0.059-0.525 and P=0.999-1.000). For M0 and M1-retreat, the fitted models EQD250 and γ50 parameters were similar. The LQ-L fitted EQD250 was ∼8.0 Gy for M0 and M1-retreat, ∼24 Gy for M1-initial, and ∼19 Gy for M2. The model fitted γ50 was 0.1 Gy for M0, M1-retreat, and M2 and 0.5 for M1-initial. For the PTV D99 of 10 and 20 Gy, the steepest to shallowest dose-response or largest change in TCP, that is, TCP20Gy - TCP10Gy, was observed in M1-initial (0.49) and M2 (0.17). M0 and M1-retreat showed a similar change in TCP of 0.21. The model-fitted parameters predicted the recurrent BMs required a higher threshold dose and had a steeper dose-response for first SRS versus second SRS and M0. Alternatively, the recurrent BMs required ∼2 Gy higher predicted PTV D99 dose for first SRS to achieve the same TCP of 0.75 compared with second SRS and M0. Further investigations on larger patient cohorts are needed for validating our findings in predictive modeling of recurrent BMs.

A Practical Workflow to Calculate 3D BED Composite Plan Using Organ-Specific Radiobiological Parameters

Biological effective dose (BED) distribution calculation is desirable for combining multiple treatments based on image sets at different times and/or with different fractionations, e.g., re-irradiation. In addition, the composite BED plan needs to be converted back to a physical dose plan in a common fractionation for plan evaluation and/or comparison. The purpose of this work is to develop a practical workflow to composite 3D BED plan and to convert it back to a physical dose plan.The workflow was developed using contouring software. The organ-specific radiobiological parameters (e.g., α/β, γ/α and dt) for different organs at risk (OAR) were obtained by fitting available clinical dose volume constraint data using a linear-quadratic-linear (LQ-L) model and were stored in the software. The BEDs in each voxel for two separate treatments were calculated using the LQ-L model for each OAR. By contour-based deformable image registration (DIR) algorithm in the software, the two images were registered to obtain a 3D composite BED. Using the rearranged LQ-L model in the software extension, the physical dose was calculated for a given fractionation scheme voxel by voxel from the 3D composite BED.The contouring software workflow along with the organ-specific BED parameters was implemented and successfully tested with two head and neck cancer re-irradiation cases. Case I was initially treated with 36 Gy in 6 fractions in 2017 and was re-irradiated with 35 Gy in 5 fractions in 2020. The case II had the first treatment of 70 Gy in 35 fractions in 2008 and the second treatment with 40 Gy in 5 fractions in 2020. The composite BED distributions calculated by the contouring software workflow were converted back to physical dose distributions with 2 Gy per fraction. The total physical doses received by the OARs after second treatments were all within acceptable clinical dose limits, consistent with the fact that no radiation related toxicity was reported for both patients. The obtained organ-specific parameters and the calculation results for these two cases are tabulated below.A practical workflow along with organ-specific radiobiological parameters was developed to generate BED composite plan and to convert it back into physical dose distribution. This study provides an additional tool to evaluate overall dose for each OAR in re-irradiation or iso-toxic adaptive radiation therapy.

Evaluation of indirect damage and damage saturation effects in dose–response curves of hypofractionated radiotherapy of early-stage NSCLC and brain metastases

Background and purposeTo investigate the possible contribution of indirect damage and damage saturation to tumour control obtained with SBRT/SRS treatments for early-stage NSCLC and brain metastases. Methods and materialsWe have constructed a dataset of early-stage NSCLC and brain metastases dose–response. These data were fitted to models based on the linear-quadratic (LQ), the linear-quadratic-linear (LQL), and phenomenological modifications of the LQ-model to account for indirect cell damage. We use the Akaike-Information-Criterion formalism to compare performance, and studied the stability of the results with changes in fitting parameters and perturbations on dose/TCP values. ResultsIn NSCLC, a modified LQ-model with a beta-term increasing with dose yields the best-fits for α/β = 10 Gy. Only the inclusion of very fast accelerated proliferation or low α/β values can eliminate such superiority. In brain, the LQL model yields the best-fits, and the ranking is not affected by variations of fitting parameters or dose/TCP perturbations. ConclusionsFor α/β = 10 Gy, a modified LQ-model with a beta-term increasing with dose provides better fits to NSCLC dose–response curves. For brain metastases, the LQL provides the best fit. This might be interpreted as a hint of indirect damage in NSCLC, and damage saturation in brain metastases. The results for NSCLC are strongly dependent on the value of α/β and may require further investigation, while those for brain seem to be clearly significant. Our results can assist in the design of improved radiotherapy for NSCLC and brain metastases, aiming at avoiding over/under-treatment.

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.

Establishment of a New Three-Dimensional Dose Evaluation Method Considering Variable Relative Biological Effectiveness and Dose Fractionation in Proton Therapy Combined with High-Dose-Rate Brachytherapy.

Purpose:The purpose of this study is to evaluate the influence of variable relative biological effectiveness (RBE) of proton beam and dose fractionation has on dose distribution and to establish a new three-dimensional dose evaluation method for proton therapy combined with high-dose-rate (HDR) brachytherapy.Materials and Methods:To evaluate the influence of variable RBE and dose fractionation on dose distribution in proton beam therapy, the depth-dose distribution of proton therapy was compared with clinical dose, RBE-weighted dose, and equivalent dose in 2 Gy fractions using a linear-quadratic-linear model (EQD2LQL). The clinical dose was calculated by multiplying the physical dose by RBE of 1.1. The RBE-weighted dose is a biological dose that takes into account RBE variation calculated by microdosimetric kinetic model implemented in Monte Carlo code. The EQD2LQL is a biological dose that makes the RBE-weighted dose equivalent to 2 Gy using a linear-quadratic-linear (LQL) model. Finally, we evaluated the three-dimensional dose by taking into account RBE variation and LQL model for proton therapy combined with HDR brachytherapy.Results:The RBE-weighted dose increased at the distal of the spread-out Bragg peak (SOBP). With the difference in the dose fractionation taken into account, the EQD2LQL at the distal of the SOBP increased more than the RBE-weighted dose. In proton therapy combined with HDR brachytherapy, a divergence of 103% or more was observed between the conventional dose estimation method and the dose estimation method we propose.Conclusions:Our dose evaluation method can evaluate the EQD2LQL considering RBE changes in the dose distribution.

Isobio software: biological dose distribution and biological dose volume histogram from physical dose conversion using linear-quadratic-linear model.

PurposeTo develop an in-house software program that is able to calculate and generate the biological dose distribution and biological dose volume histogram by physical dose conversion using the linear-quadratic-linear (LQL) model.Material and methodsThe Isobio software was developed using MATLAB version 2014b to calculate and generate the biological dose distribution and biological dose volume histograms. The physical dose from each voxel in treatment planning was extracted through Computational Environment for Radiotherapy Research (CERR), and the accuracy was verified by the differentiation between the dose volume histogram from CERR and the treatment planning system. An equivalent dose in 2 Gy fraction (EQD2) was calculated using biological effective dose (BED) based on the LQL model. The software calculation and the manual calculation were compared for EQD2 verification with pair t-test statistical analysis using IBM SPSS Statistics version 22 (64-bit).ResultsTwo and three-dimensional biological dose distribution and biological dose volume histogram were displayed correctly by the Isobio software. Different physical doses were found between CERR and treatment planning system (TPS) in Oncentra, with 3.33% in high-risk clinical target volume (HR-CTV) determined by D90%, 0.56% in the bladder, 1.74% in the rectum when determined by D2cc, and less than 1% in Pinnacle. The difference in the EQD2 between the software calculation and the manual calculation was not significantly different with 0.00% at p-values 0.820, 0.095, and 0.593 for external beam radiation therapy (EBRT) and 0.240, 0.320, and 0.849 for brachytherapy (BT) in HR-CTV, bladder, and rectum, respectively.ConclusionsThe Isobio software is a feasible tool to generate the biological dose distribution and biological dose volume histogram for treatment plan evaluation in both EBRT and BT.

Challenges in radiobiological modeling: can we decide between LQ and LQ-L models based on reviewed clinical NSCLC treatment outcome data?

AimTo study the dose-response of stage I non-small-cell lung cancer (NSCLC) in terms of long-term local tumor control (LC) after conventional and hypofractionated photon radiotherapy, modeled with the linear-quadratic (LQ) and linear-quadratic-linear (LQ-L) approaches and to estimate the clinical α/β ratio within the LQ frame.Material and methodsWe identified studies of curative radiotherapy as single treatment through MedLine search reporting 3-year LC as primary outcome of interest. Logistic models coupled with the biologically effective dose (BED) at isocenter and PTV edge according to both the LQ and LQ-L models with α/β = 10 Gy were fitted. Additionally, α/β was estimated from direct LQ fits.ResultsThirty one studies were included reporting outcome of 2319 patients. The LQ-L fit yielded a significant value of 11.0 ± 5.2 Gy for the dose threshold (Dt) for BED10 at the isocenter. The LQ and LQ-L fits did not differ substantially. Concerning the estimation of α/β, the value obtained from the direct LQ fit for the complete fractionation range was 3.9 [68 % CI: 2.2–9.0] Gy (p > 0.05).ConclusionBoth LQ and LQ-L fits can model local tumor control after conventionally and hypofractionated irradiation and are robust methods for predicting clinical effects. The observed dose-effect for local control in NSCLC is weaker at high doses due to data dispersion. For BED10 values of 100–150 Gy in ≥3 fractions, the differences in isoeffects predicted by both models can be neglected.Electronic supplementary materialThe online version of this article (doi:10.1186/s13014-016-0643-5) contains supplementary material, which is available to authorized users.

Strategy for stochastic dose-rate induced enhanced elimination of malignant tumour without dose escalation.

The efficacy of radiation therapy, a primary modality of cancer treatment, depends in general upon the total radiation dose administered to the tumour during the course of therapy. Nevertheless, the delivered radiation also irradiates normal tissues and dose escalation procedure often increases the elimination of normal tissue as well. In this article, we have developed theoretical frameworks under the premise of linear-quadratic-linear (LQL) model using stochastic differential equation and Jensen's inequality for exploring the possibility of attending to the two therapeutic performance objectives in contraposition-increasing the elimination of prostate tumour cells and enhancing the relative sparing of normal tissue in fractionated radiation therapy, within a prescribed limit of total radiation dose. Our study predicts that stochastic temporal modulation in radiation dose-rate appreciably enhances prostate tumour cell elimination, without needing dose escalation in radiation therapy. However, constant higher dose-rate can also enhance the elimination of tumour cells. In this context, we have shown that the sparing of normal tissue with stochastic dose-rate is considerably more than the sparing of normal tissue with the equivalent constant higher dose-rate. Further, by contrasting the stochastic dose-rate effects under LQL and linear-quadratic (LQ) models, we have also shown that the LQ model over-estimates stochastic dose-rate effect in tumour and under-estimates the stochastic dose-rate effect in normal tissue. Our study indicates the possibility of utilizing stochastic modulation of radiation dose-rate for designing enhanced radiation therapy protocol for cancer.

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 Pade 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 χ (2)-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.

A modified hypoxia-based TCP model to investigate the clinical outcome of stereotactic hypofractionated regimes for early stage non-small-cell lung cancer (NSCLC)

Stereotactic body radiotherapy (SBRT) has been applied to lung tumors at different stages and sizes with good local tumor control (LC) rates. The linear quadratic model (LQM), in its basic formulation, does not seem to be appropriate to describe the response to radiotherapy for clinical trials, based on a few fractions. Thus, the main aim of this work was to develop a model, which takes into account the hypoxic cells and their reoxygenation. A parameter named B has been introduced in a modified tumor control probability (TCP) from LQM and linear-quadratic-linear model (LQLM), and represents the fraction of hypoxic cells that survive and become oxygenated after each irradiation. Based on published trials evaluating LC at 3 yr (LC3), values of B were obtained by maximum likelihood minimization between predicted TCP and clinical LC3. Two oxygen enhancement ratio (OER) parameter sets (1 and 2) from literature have been adopted to calculate the B-factors. Initial hypoxic cell fractions (η(h)) from 0.05 to 0.50 were assumed. Log-likelihood (L) and Akaike information criterion (AIC) were determined in an independent clinical validation dataset. The B-values of modified TCPs spanned the whole interval from 0 to 1, depending on the fractionation scheme (number of fractions and dose/fraction), showing a maximum (close to 1) at doses/fraction of 8-12 Gy. The B-values calculated using the OER parameter set 1 exhibited a smoother falloff than set 2. An analytical expression was derived to describe the B-value's dependence on the fractionation scheme. The R(2)-adjusted values varied from 0.63 to 0.67 for LQ models and OER set 1 and from 0.75 to 0.78 for LQ model and OER set 2. Lower values of R(2)-adjusted were found for LQLM and both OER sets. L and AIC, calculated using a fraction of η(h) = 0.15 and the B-value from the authors analytical expression were higher than for other η(h)-values, irrespective of model or OER set. The authors model allows to predict the clinical outcome associated with SBRT treatment, taking into account both direct killing and indirect vasculature or stromal damage.

TH-A-BRB-02: Mechanistic Linear-Quadratic-Linear (LQL) Model for Large Doses Per Fraction

The Linear-Quadratic (LQ) model has been successfully used for at least two decades to quantify radiation effects at standard (1.2-3Gy) doses. The validity of the standard LQ model for larger doses per fraction has been called into question by many authors, in part due to the continuous bending of the cell survival curve at large doses, contrary to results from in-vitro experiments. In recent years, Stereotactic Body Radiation Therapy (SBRT) treatments with large fraction doses have become more common, making the modeling for large doses per fraction of significant clinical relevance. Many authors have proposed extensions of the LQ model to address the shape of the cell survival curve at larger doses by arbitrary adding a phenomenological parameter to the LQ model. In this work, we discuss a Linear-Quadratic-Linear (LQL) model with a mechanistic formulation. This LQL model has one additional parameter with respect to the standard LQ model and can be derived by mechanistic arguments. The additional parameter makes the survival curve linear at high doses and has been estimated using in-vivo and in-vitro data. The advantage of a mechanistic model is that it allows for modeling of radiation effects for any given dose- rate and fractionation scheme. As an example, we present the LQL prediction for split dose experiments and low-dose-rate (LDR) prostate brachytherapy. The LQL model is equivalent to other mechanistic models like the Lethal-Potentially-Lethal model (LPL) but it retains the well-known parameters from the standard LQ model. As a clinical application, using estimated values of the additional parameter, we investigate the impact on the biologically effective dose (BED) for clinical SBRT treatment schedules of various cancer sites. Learning Objectives: 1. Understand the limitations of the standard LQ model for large doses per fraction. 2. Understand the mechanistic formulation of the proposed LQL. 3. Understand how the LQL can be used to calculate the BED for SBRT radiation treatment schedules and compare them with standard fractionation.

TH‐C‐220‐11: Interplay of Radiobiological Parameters in BED Calculations for a Linear‐Quadratic‐Linear Model

Purpose: In the framework of stereotactic body radiation therapy (SBRT), dose per fraction larger than 2Gy have become more common, challenging the applicability of the linear‐quadratic (LQ) model. Several authors have proposed ways to modify the LQ model to make it linear at large doses per fraction. Qualitatively, the modification implies a reduction of biological effect with respect of the standard LQ model for large doses. The purpose of this work is to quantitatively study the effect of the survival curve linearization at high doses for BED calculation relevant to prostate and lung SBRT. Method: We use a previously published linear‐quadratic‐linear model (LQL) to calculate BEDLQL and the equivalent dose in 2Gy fractions EQD2 for several clinically used fractionation schemes of prostate and lung cancer. The LQL model adds a parameter delta that makes the survival curve linear at high doses and is estimated to be 0.05–0.2Gy‐1. We compare BED and EQD2 for the LQ and LQL models. Results: For prostate cancer if alpha/beta=1.5–3Gy the difference in EQD2 between the LQ and LQL is 5 to 10% for cases with delta=0.2Gy‐1 and moderate dose per fraction d=3–4.3Gy. For extreme hypofractionation the effect is more pronounced with differences of up to 35% in EQD2. For the lung cancer SBRT, the BEDLQL for doses of 10‐12Gy is 3–25% smaller than BEDLQ with alpha/beta=10Gy while even bigger differences ranging from 10 to 40% are possible for normal tissues alpha/beta=3Gy. Conclusions: There is a clear interplay of the alpha/beta ratio and the parameter delta in terms of BED and EQD2 calculations for different fractionations. The impact of the parameter delta is significantly higher for low alpha/beta ratios. This is particularly important for the case of prostate cancer where a large value of delta (0.2Gy‐1 or so) could bring EQD2 below acceptable therapeutic.

TU-EE-A2-01: A Linear-Quadratic-Linear Formulation to Model Radiation Dose-Response

Purpose: Recent technological advances enable radiotherapy to be delivered in a highly conformal manner almost anywhere in the body. This has renewed interest in hypofractionation wherein the tumor is delivered a few fractions of large dose/fraction. Extrapolating clinical experience with conventional fractionations to fractions of high dose is important when designing hypofractionated regimens. Method and Materials: The concept of biologically effective dose (BED) based on the linear-quadratic (LQ) formulation is useful for intercomparing conventional fractionations but is suspect at high dose because the LQ curve bends continuously on the log-linear plot. A linear-quadratic-linear (LQ-L) formulation which better fits the final exponential response of experimental dose-response studies at high dose is described. This new formulation requires only one new term, the dose DT at which the LQ curve transitions to a linear tail. LQ-L is applied to published dose-response curves and the clinical implications of LQ-L are examined across a wide range of fractionations. Results: For fractions of high dose, the LQ formulation underestimates the dose per fraction required to maintain equivalency with conventional regimens. The LQ-L model fits a wide variety of experimental survival data over a wide range of dose. When , the line tangent to the LQ curve at DT intersects the e−αD and e−βD2 curves at dose α/β and also closely fits the linear response in the high dose region of many in vitro studies. Conclusion: For fractions of high dose LQ-L gives better estimates of BED than LQ because LQ-L better fits experimental dose-response in the high dose region. This is particularly important when planning hypofractionated regimens for reactions with low α/β such as prostate cancer or late sequelae because for these reactions falls within the contemplated range of hypofractional doses.