The evolution of practical radiobiological modelling

Designing combination therapies for cancer treatment: application of a mathematical framework combining CAR T-cell immunotherapy and targeted radionuclide therapy

Introduction

Cancer combination treatments involving immunotherapies with targeted radiation therapy are at the forefront of treating cancers. However, dosing and scheduling of these therapies pose a challenge. Mathematical models provide a unique way of optimizing these therapies.

Methods

Using a preclinical model of multiple myeloma as an example, we demonstrate the capability of a mathematical model to combine these therapies to achieve maximum response, defined as delay in tumor growth. Data from mice studies with targeted radionuclide therapy (TRT) and chimeric antigen receptor (CAR)-T cell monotherapies and combinations with different intervals between them was used to calibrate mathematical model parameters. The dependence of progression-free survival (PFS), overall survival (OS), and the time to minimum tumor burden on dosing and scheduling was evaluated. Different dosing and scheduling schemes were evaluated to maximize the PFS and optimize timings of TRT and CAR-T cell therapies.

Results

Therapy intervals that were too close or too far apart are shown to be detrimental to the therapeutic efficacy, as TRT too close to CAR-T cell therapy results in radiation related CAR-T cell killing while the therapies being too far apart result in tumor regrowth, negatively impacting tumor control and survival. We show that splitting a dose of TRT or CAR-T cells when administered in combination is advantageous only if the first therapy delivered can produce a significant benefit as a monotherapy.

Discussion

Mathematical models are crucial tools for optimizing the delivery of cancer combination therapy regimens with application along the lines of achieving cure, maximizing survival or minimizing toxicity.

A novel, multi-active emollient for the prevention of acute radiation dermatitis in breast cancer patients: a randomized clinical trial

PURPOSE

To investigate the efficacy of a novel, multi-active emollient in preventing and managing acute radiation dermatitis (ARD) in breast cancer patients undergoing moderate hypofractionated (HF) radiotherapy (RT) compared to standard of care.

METHODSA

A monocentric, open-label, randomized clinical trial (RCT) with breast cancer patients receiving moderate HF (dose: 40.05-55.86 Gy, fractions: 15-21) was conducted between January 2022 and May 2023. The experimental group received the novel emollient, while the control group received the standard skin care. Patients applied the skin care products twice daily during the complete RT course. The primary outcome was the severity of ARD at the final RT session measured by the modified Radiation Therapy Oncology Group (RTOG) criteria. Secondary outcomes included patient symptoms, quality of life (QoL), and treatment satisfaction.

RESULTS

A total of 100 patients with 50 patients per group were enrolled. In the control group, 50% of the patients developed RTOG grade 1 ARD and 48% grade 2 or higher, while in the experimental group, the severity of ARD was significantly lower with 82% grade 1 and 16% grade 2 ARD (P = .013, χ2-test). The frequency and severity of xerosis were significantly lower in the experimental compared to the control group (Ps ≤ .036, Mann Whiney U test). The impact of ARD on the QoL was low, and treatment satisfaction was high in both groups, with no significant difference.

CONCLUSION

This RCT shows that the novel, multi-active emollient significantly reduced the ARD RTOG grade. Research in a more diverse patient population is warranted.

TRIAL REGISTRATION

ClinicalTrials.gov: NCT04929808 (11/06/2021).

Re-irradiation for recurrent/progressive pediatric brain tumors: from radiobiology to clinical outcomes

INTRODUCTION

Brain tumors are the most common solid tumors in children. Neurosurgical excision, radiotherapy, and/or chemotherapy represent the standard of care in most histopathological types of pediatric central nervous system (CNS) tumors. Even though the successful cure rate is reasonable, some patients may develop recurrence locally or within the neuroaxis.

AREA COVERED

The management of these recurrences is not easy; however, significant advances in neurosurgery, radiation techniques, radiobiology, and the introduction of newer biological therapies, have improved the results of their salvage treatment. In many cases, salvage re-irradiation is feasible and has achieved encouraging results. The results of re-irradiation depend upon several factors. These factors include tumor type, extent of the second surgery, tumor volume, location of the recurrence, time that elapses between the initial treatment, the combination with other treatment agents, relapse, and the initial response to radiotherapy.

EXPERT OPINION

Reviewing the radiobiological basis and clinical outcome of pediatric brain re-irradiation revealed that re-irradiation is safe, feasible, and indicated for recurrent/progressive different tumor types such as; ependymoma, medulloblastoma, diffuse intrinsic pontine glioma (DIPG) and glioblastoma. It is now considered part of the treatment armamentarium for these patients. The challenges and clinical results in treating recurrent pediatric brain tumors were highly documented.

The influence of hypoxia on LET and RBE relationships with implications for ultra-high dose rates and FLASH modelling

Objective. To investigate relationships between linear energy transfer (LET), fluence rates, changes in radiosensitivity and the oxygen enhancement ratio (OER) in different ion beams and extend these concepts to ultra-high dose rate (UHDR) or FLASH effects. Approach. LET values providing maximum relative biological effect (RBE), designated as LETU, are found for neon, carbon and helium beams. Proton experiments show reduced RBEs with depth in scattered (divergent) beams, but not with scanned beams, suggesting that instantaneous fluence rates (related to track separation distances) can modify RBE, all other RBE-determining factors being equal. Micro-volumetric energy transfer per μm3 (mVET) is defined by LET × fluence. High fluence rates will increase mVET rates, with proportional shifts of LETU to lower values due to more rapid energy transfer. From the relationship between LETU and OER at conventional dose rates, OER reductions in UHDR/FLASH exposures can be estimated and biological effective dose analysis of experimental lung and skin reactions becomes feasible. Main results. The Furusawa et al data show that hypoxic LETU values exceed their oxic counterparts. OER reduces from around 3–1.25 at LETU, although the relative radiosensitivities of the oxic and hypoxic α parameters (the OER(α)) exceed those of the standard OER values. Increased fluence rates are predicted to reduce LETU and OER. Large FLASH single doses will minimise RBE increments due to the β parameter reducing by a factor of 0.5–0.25 consistent with oxygen depletion, causing radioresistance. Similar results will occur for photons. Tissue α/β ratios increase by around 10 in FLASH conditions, agreeing with derived ion-beam dose rate equations. Significance. Increasing dose rates enhance local energy deposition rate per unit volume, probably causing oxygen depletion and radioresistance in pre-existing hypoxic sites during UHDR/FLASH exposures. The modelled equations provide testable hypotheses for further dose rate investigations in photon, proton and ion beams.

On the calculation of the relative biological effectiveness of ion radiation therapy using a biological weighting function, the microdosimetric kinetic model (MKM) and subsequent corrections (non-Poisson MKM and modified MKM)

Objective. To investigate similarities and differences in the formalism, processing, and the results of relative biological effectiveness (RBE) calculations with a biological weighting function (BWF), the microdosimetric kinetic model (MKM) and subsequent modifications (non-Poisson MKM, modified MKM). This includes: (a) the extension of the V79-RBE10% BWF to model the RBE for other clonogenic survival levels; (b) a novel implementation of MKMs as weighting functions; (c) a benchmark against Chinese Hamster lung fibroblast (V79) in vitro data; (d) a study on the effect of pre- or post- processing the average biophysical quantities used for the RBE calculations; (e) a possible modification of the modified MKM parameters to improve the model accuracy at high linear energy transfer (LET). Methodology. Lineal energy spectra were simulated for two spherical targets (diameter = 0.464 or 1.0 μm) using PHITS for 1H, 4He, 12C, 20Ne, 40Ar, 56Fe and 132Xe ions. The results of the in silico calculations were compared with published in vitro data. Main results. All models appear to underestimate the RBE α of hydrogen ions. All MKMs generally overestimate the RBE50%, RBE10% and RBE1% for ions with an LET greater than ∼200 keV μm−1. This overestimation is greater for small surviving fractions and is likely due to the assumption of a radiation-independent quadratic term of clonogenic survival (ß). The overall RBE trends seem to be best described by the novel ‘post-processing average’ implementation of the non-Poisson MKM. In case of calculations with the non-Poisson MKM, pre- or post- processing the average biophysical quantities affects the computed RBE values significantly. Significance. This study presents a systematic analysis of the formalism and results of widely used microdosimetric models of clonogenic survival for ions relevant for cancer particle therapy and space radiation protection. Points for improvements were highlighted and will contribute to the development of upgraded biophysical models.

Variability of α/β ratios for prostate cancer with the fractionation schedule: caution against using the linear-quadratic model for hypofractionated radiotherapy

BACKGROUND

Prostate cancer (PCa) is known to be suitable for hypofractionated radiotherapy due to the very low α/β ratio (about 1.5-3 Gy). However, several randomized controlled trials have not shown the superiority of hypofractionated radiotherapy over conventionally fractionated radiotherapy. Besides, in vivo and in vitro experimental results show that the linear-quadratic (LQ) model may not be appropriate for hypofractionated radiotherapy, and we guess it may be due to the influence of fractionation schedules on the α/β ratio. Therefore, this study attempted to estimate the α/β ratio in different fractionation schedules and evaluate the applicability of the LQ model in hypofractionated radiotherapy.

METHODS

The maximum likelihood principle in mathematical statistics was used to fit the parameters: α and β values in the tumor control probability (TCP) formula derived from the LQ model. In addition, the fitting results were substituted into the original TCP formula to calculate 5-year biochemical relapse-free survival for further verification.

RESULTS

Information necessary for fitting could be extracted from a total of 23,281 PCa patients. A total of 16,442 PCa patients were grouped according to fractionation schedules. We found that, for patients who received conventionally fractionated radiotherapy, moderately hypofractionated radiotherapy, and stereotactic body radiotherapy, the average α/β ratios were 1.78 Gy (95% CI 1.59-1.98), 3.46 Gy (95% CI 3.27-3.65), and 4.24 Gy (95% CI 4.10-4.39), respectively. Hence, the calculated α/β ratios for PCa tended to become higher when the dose per fraction increased. Among all PCa patients, 14,641 could be grouped according to the risks of PCa in patients receiving radiotherapy with different fractionation schedules. The results showed that as the risk increased, the k (natural logarithm of an effective target cell number) and α values decreased, indicating that the number of effective target cells decreased and the radioresistance increased.

CONCLUSIONS

The LQ model appeared to be inappropriate for high doses per fraction owing to α/β ratios tending to become higher when the dose per fraction increased. Therefore, to convert the conventionally fractionated radiation doses to equivalent high doses per fraction using the standard LQ model, a higher α/β ratio should be used for calculation.

A Multi-Compartment Model of Glioma Response to Fractionated Radiation Therapy Parameterized via Time-Resolved Microscopy Data

Purpose

Conventional radiobiology models, including the linear-quadratic model, do not explicitly account for the temporal effects of radiation, thereby making it difficult to make time-resolved predictions of tumor response to fractionated radiation. To overcome this limitation, we propose and validate an experimental-computational approach that predicts the changes in cell number over time in response to fractionated radiation.

Methods

We irradiated 9L and C6 glioma cells with six different fractionation schemes yielding a total dose of either 16 Gy or 20 Gy, and then observed their response via time-resolved microscopy. Phase-contrast images and Cytotox Red images (to label dead cells) were collected every 4 to 6 hours up to 330 hours post-radiation. Using 75% of the total data (i.e., 262 9L curves and 211 C6 curves), we calibrated a two-species model describing proliferative and senescent cells. We then applied the calibrated parameters to a validation dataset (the remaining 25% of the data, i.e., 91 9L curves and 74 C6 curves) to predict radiation response. Model predictions were compared to the microscopy measurements using the Pearson correlation coefficient (PCC) and the concordance correlation coefficient (CCC).

Results

For the 9L cells, we observed PCCs and CCCs between the model predictions and validation data of (mean ± standard error) 0.96 ± 0.007 and 0.88 ± 0.013, respectively, across all fractionation schemes. For the C6 cells, we observed PCCs and CCCs between model predictions and the validation data were 0.89 ± 0.008 and 0.75 ± 0.017, respectively, across all fractionation schemes.

Conclusion

By proposing a time-resolved mathematical model of fractionated radiation response that can be experimentally verified in vitro, this study is the first to establish a framework for quantitative characterization and prediction of the dynamic radiobiological response of 9L and C6 gliomas to fractionated radiotherapy.

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.

Clinical and practical considerations in the design of appropriate compensation schedules following treatment interruptions

Compensatory dose calculations to mitigate the deleterious effect of unscheduled treatment interruptions remain important. They may be increasingly required during and after epidemics, as with the present Covid-19 virus. The information presented to those involved in the actual dose estimations is often limited, thereby increasing the likelihood of confusion, further time delays and possibly incorrect decisions. This article sets out what aspects need to be considered by the Clinical Oncologist (or Radiation Oncologist), and the reasons why, in order to provide greater clarity. The key issues are: (a) the biological nature of the tumour (and hence its repopulation potential), (b) patient age and pre-existing medical risk factors that influence radiation tolerance, the use of chemotherapy, surgery etc, (c) the acceptable dose limits of the relevant normal tissues at risk and (d) consideration of the possibility of further field size adjustments, a change in treatment plan or acceptance of a greater role for alternative forms of radiation treatment (e.g. brachytherapy, electron boosts, etc.) or reliance on radical surgery. Only then can a compensatory schedule be more safely estimated.

Clinical Radiobiology of Fast Neutron Therapy: What Was Learnt?

Neutron therapy was developed from neutron radiobiology experiments, and had identified a higher cell kill per unit dose and an accompanying reduction in oxygen dependency. But experts such as Hal Gray were sceptical about clinical applications, for good reasons. Gray knew that the increase in relative biological effectiveness (RBE) with dose fall-off could produce marked clinical limitations. After many years of research, this treatment did not produce the expected gains in tumour control relative to normal tissue toxicity, as predicted by Gray. More detailed reasons for this are discussed in this paper. Neutrons do not have Bragg peaks and so did not selectively spare many tissues from radiation exposure; the constant neutron RBE tumour prescription values did not represent the probable higher RBE values in late-reacting tissues with low α/β values; the inevitable increase in RBE as dose falls along a beam would also contribute to greater toxicity than in a similar megavoltage photon beam. Some tissues such as the central nervous system white matter had the highest RBEs partly because of the higher percentage hydrogen content in lipid-containing molecules. All the above factors contributed to disappointing clinical results found in a series of randomised controlled studies at many treatment centres, although at the time they were performed, neutron therapy was in a catch-up phase with photon-based treatments. Their findings are summarised along with their technical aspects and fractionation choices. Better understanding of fast neutron experiments and therapy has been gained through relatively simple mathematical models—using the biological effective dose concept and incorporating the RBEmax and RBEmin parameters (the limits of RBE at low and high dose, respectively—as shown in the Appendix). The RBE itself can then vary between these limits according to the dose per fraction used. These approaches provide useful insights into the problems that can occur in proton and ion beam therapy and how they may be optimised. This is because neutron ionisations in living tissues are mainly caused by recoil protons of energy proportional to the neutron energy: these are close to the proton energies that occur close to the Bragg peak region. To some extent, neutron RBE studies contain the highest RBE ranges found within proton and ion beams near Bragg peaks. In retrospect, neutrons were a useful radiobiological tool that has continued to inform the scientific and clinical community about the essential radiobiological principles of all forms of high linear energy transfer therapy. Neutron radiobiology and its implications should be taught on training courses and studied closely by clinicians, physicists, and biologists engaged in particle beam therapies.

The Impact of Insulin-Like Growth Factor Index and Biologically Effective Dose on Outcomes After Stereotactic Radiosurgery for Acromegaly: Cohort Study

BACKGROUND

Stereotactic radiosurgery (SRS) is a safe and effective treatment for acromegaly.

OBJECTIVE

To improve understanding of clinical and dosimetric factors predicting biochemical remission.

METHODS

A single-institution cohort study of nonsyndromic, radiation-naïve patients with growth hormone-producing pituitary adenomas (GHA) having single-fraction SRS between 1990 and 2017. Exclusions were treatment with pituitary suppressive medications at the time of SRS, or <24 mo of follow-up. The primary outcome was biochemical remission-defined as normalization of insulin-like growth factor-1 index (IGF-1i) off suppression. Biochemical remission was assessed using Cox proportional hazards. Prior studies reporting IGF-1i were assessed via systematic literature review and meta-analysis using random-effect modeling.

RESULTS

A total of 102 patients met study criteria. Of these, 46 patients (45%) were female. The median age was 49 yr (interquartile range [IQR] = 37-59), and the median follow-up was 63 mo (IQR = 29-100). The median pre-SRS IGF-1i was 1.66 (IQR = 1.37-3.22). The median margin dose was 25 Gy (IQR = 21-25); the median estimated biologically effective dose (BED) was 169.49 Gy (IQR = 124.95-196.00). Biochemical remission was achieved in 58 patients (57%), whereas 22 patients (22%) had medication-controlled disease. Pre-SRS IGF-1i ≥ 2.25 was the strongest predictor of treatment failure, with an unadjusted hazard ratio (HR) of 0.51 (95% CI = 0.26-0.91, P = .02). Number of isocenters, margin dose, and BED predicted remission on univariate analysis, but after adjusting for sex and baseline IGF-1i, only BED remained significant-and was independently associated with outcome in continuous (HR = 1.01, 95% CI = 1.00-1.01, P = .02) and binary models (HR = 2.27, 95% CI = 1.39-5.22, P = .002). A total of 24 patients (29%) developed new post-SRS hypopituitarism. Pooled HR for biochemical remission given subthreshold IGF-1i was 2.25 (95% CI = 1.33-3.16, P < .0001).

CONCLUSION

IGF-1i is a reliable predictor of biochemical remission after SRS. BED appears to predict biochemical outcome more reliably than radiation dose, but confirmatory study is needed.

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.

Use of radiobiology in medical jurisprudence, with particular reference to delays in diagnosis and therapeutic onset

OBJECTIVE

This paper considers aspects of radiobiology and cell and tissue kinetics applicable to legal disputations concerned with diagnostic and treatment onset delays.

METHODS

Various models for tumour volume changes with time are reviewed for estimating volume ranges at earlier times, using ranges of kinetic parameters. Statistical cure probability methods, using Poisson statistics with allowances for parameter heterogeneity, are also described to estimate the significance of treatment delays, as well as biological effective dose (BED) estimations of radiation effectiveness.

RESULTS

The use of growth curves, based on parameters in the literature but with extended ranges, can identify a window of earlier times when such tumour volumes would be amenable to a cure based on the literature for curability with stage (and dimensions). Also, where tumour dimensions are not available in a post-operative setting, higher cure probabilities can be achieved if treatment had been given at earlier times.

CONCLUSION

The use of radiobiological modelling can provide useful insights, with quantitative assessments of probable prior conditions and future outcomes, and thus be of assistance to a Court in deciding the most correct judgement.

ADVANCES IN KNOWLEDGE

This study collates prior knowledge about aspects of radiobiology that can be useful in the accumulation of sufficient proof within medicolegal claims involving diagnostic and treatment days.

The many faces of mathematical modelling in oncology

The application of modelling to solve problems in biology and medicine, and specifically in oncology and radiation therapy, is increasingly established and holds big promise. We provide an overview of the basic concepts of the field and its current state, along with new tools available and future directions for research. We will outline radiobiology models, examples of other anticancer therapy models, multiscale modelling, and we will discuss mechanistic and phenomenological approaches to modelling.

Modelling the influence of treatment time on the biological effectiveness of single radiosurgery treatments: derivation of "protective" dose modification factors

OBJECTIVE

To provide simpler models for adjusting total dose to compensate for significant variations in central nervous system radiosurgical treatment times, which vary and will influence treatment bioeffectiveness. At present, no allowance is made for time variations. A framework of simpler equations would allow radiosurgical outcomes to be analysed with respect to treatment time, and a system for dose adjustments between radioisotope and linac-based techniques with different treatment durations.

METHODS

The standard biological effective dose (BED) equations for fractionated and protracted radiations have been combined, using biexponential DNA repair kinetics, to provide the following equation:BED=x.nd(1+(ndk-dk)f(μ1T)+dkf(μ1t))+(1-x). nd(1+(ndk-dk)f(μ2T)+dkf(μ2t))for "n" isocentres (or subfractions), each treated to a variable dose "d" in time "t", the overall time-being, T, µ1, µ2, are fast and slow repair rate coefficients, with partition factors of x and (1-x), respectively and k is the alpha/beta ratio, with f(μT) being the function that summates sublethal damage repair. Thus, repair during the period of irradiation and in the time interval between each isocentre can be taken into account. Simpler monoexponential and linear models are also used.

RESULTS

The results obtained using simpler models are compared with those obtained using more complex retrospective Gamma Knife BED treatment planning by Millar et al. (2015) in a group of 23 patients on a 13 Gy physical isodose surface. The above equation provides a BED value around 3% above their minimum values, 4% below their average value and 10% below their maximum BED values. Changes in isocentre numbers used, due to treatment plan complexity, can influence total treatment time, producing variations in the BED-time data: instead of a unique curve for each "n" value, in aggregate form the data (ranging from around 20 to 140 min treatment times) can be fitted by monoexponential time functions and further approximated to a linear function for more rapid estimations. Worked examples show how dose can then be tailored to the expected treatment times in order to obtain isoeffective treatments for central nervous system tissues.

CONCLUSION

The models allow better analysis of radiosurgical treatment time data and guidance to the choice of dose to match the overall time. Although this study is based on Gamma Knife treatments, in principle the methods will also apply to any radiosurgical technique, so that dose-time compensations can be made between differing techniques.

ADVANCES IN KNOWLEDGE

The new BED equation-based framework is relevant to analyse and optimise radiosurgical treatments.