Monte Carlo evaluation of target dose coverage in lung stereotactic body radiation therapy with flattening filter-free beams.

Aim: Previous studies showed that replacing conventional flattened beams (FF) with flattening filter-free (FFF) beams improves the therapeutic ratio in lung stereotactic body radiation therapy (SBRT), but these findings could have been impacted by dose calculation uncertainties caused by the heterogeneity of the thoracic anatomy and by respiratory motion, which were particularly high for target coverage. In this study, we minimized such uncertainties by calculating doses using high-spatial-resolution Monte Carlo and four-dimensional computed tomography (4DCT) images. We aimed to evaluate more reliably the benefits of using FFF beams for lung SBRT. Materials and methods: For a cohort of 15 patients with early stage lung cancer that we investigated in a previous treatment planning study, we recalculated dose distributions with Monte Carlo using 4DCT images. This included fifteen FF and fifteen FFF treatment plans. Results: Compared to Monte Carlo, the treatment planning system (TPS) over-predicted doses in low-dose regions of the planning target volume. For most patients, replacing FF beams with FFF beams improved target coverage, tumor control, and uncomplicated tumor control probabilities. Conclusions: Monte Carlo tends to reveal deficiencies in target coverage compared to coverage predicted by the TPS. Our data support previously reported benefits of using FFF beams for lung SBRT.

Using FFF Beams to Improve the Therapeutic Ratio of Lung SBRT.

Aim: To investigate the extent to which lung stereotactic body radiotherapy (SBRT) treatment plans can be improved by replacing conventional flattening filter (FF) beams with flattening filter-free (FFF) beams. Material and Methods: We selected 15 patients who had received SBRT with conventional 6-MV photon beams for early-stage lung cancer. We imported the patients' treatment plans into the Eclipse 13.6 treatment planning system, in which we configured the AAA dose calculation model using representative beam data for a TrueBeam accelerator operated in 6-MV FFF mode. We then created new treatment plans by replacing the conventional FF beams in the original plans with FFF beams. Results: The FFF plans had better target coverage than the original FF plans did. For the planning target volume, FFF plans significantly improved the D98, D95, D90, homogeneity index, and uncomplicated tumor control probability. In most cases, the doses to organs at risk were lower in FFF plans. FFF plans significantly reduced the mean lung dose, V10, V20, V30, and normal tissue complication probability for the total lung and improved the dosimetric indices for the ipsilateral lung. For most patients, FFF beams achieved lower maximum doses to the esophagus, heart, and the spinal cord; and a lower chest wall V30. Findings: Compared with FF beams, FFF beams achieved lower doses to organs at risk, especially the lung, without compromising tumor coverage; in fact, FFF beams improved coverage in most cases. Thus, replacing FF beams with FFF beams can achieve a better therapeutic ratio.

Impact of Intra-Fractional Motion on Dose Distributions in Lung IMRT.

AIM: To investigate the impact of intra-fractional motion on dose distribution in patients treated with intensity-modulated radiation therapy (IMRT) for lung cancer. MATERIALS AND METHODS: Twenty patients who had undergone IMRT for non-small cell lung cancer were selected for this retrospective study. For each patient, a four-dimensional computed tomography (CT) image set was acquired and clinical treatment plans were developed using the average CT. Dose distributions were then re-calculated for each of the 10 phases of respiratory cycle and combined using deformable image registration to produce cumulative dose distributions that were compared with the clinical treatment plans. RESULTS: Intra-fractional motion reduced planning target volume (PTV) coverage in all patients. The median reduction of PTV volume covered by the prescription isodose was 3.4%; D98 was reduced by 3.1 Gy. Changes in the mean lung dose were within ±0.7 Gy. V20 for the lung increased in most patients; the median increase was 1.6%. The dose to the spinal cord was unaffected by intra-fractional motion. The dose to the heart was slightly reduced in most patients. The median reduction in the mean heart dose was 0.22 Gy, and V30 was reduced by 2.5%.The maximum dose to the esophagus was also reduced in most patients, by 0.74 Gy, whereas V50 did not change significantly. The median number of points in which dose differences exceeded 3%/3 mm was 6.2%. FINDINGS: Intra-fractional anatomical changes reduce PTV coverage compared to the coverage predicted by clinical treatment planning systems that use the average CT for dose calculation. Doses to organs at risk were mostly over-predicted.

Accumulation of sublethal radiation damage and its effect on cell survival.

Objective.Determine the extent of sublethal radiation damage (SRD) in a cell population that received a given dose of radiation and the impact of this damage on cell survival.Approach.We developed a novel formalism to account for accumulation of SRD with increasing dose. It is based on a very general formula for cell survival that correctly predicts the basic properties of cell survival curves, such as the transition from the linear-quadratic to a linear dependence at high doses. Using this formalism we analyzed extensive experimental data for photons, protons and heavy ions to evaluate model parameters, quantify the extent of SRD and its impact on cell survival.Main results.Significant accumulation of SRD begins at doses below 1 Gy. As dose increases, so does the number of damaged cells and the amount of SRD in individual cells. SRD buildup in a cell increases the likelihood of complex irrepairable damage. For this reason, during a dose fraction delivery, each dose increment makes cells more radiosensitive. This gradual radosensitization is evidenced by the increasing slope of survival curves observed experimentally. It continues until the fraction is delivered, unless radiosensitivity reaches its maximum first. The maximum radiosensitivity is achieved when SRD accumulated in most cells is the maximum damage they can repair. After this maximum is reached, the slope of a survival curve, logarithm of survival versus dose, becomes constant, dose independent. The survival curve becomes a straight line, as experimental data at high doses show. These processes are random. They cause large cell-to-cell variability in the extent of damage and radiosensitivity of individual cells.Significance.SRD is in effect a radiosensitizer and its accumulation is a significant factor affecting cell survival, especially at high doses. We developed a novel formalism to study this phenomena and reported pertinent data for several particle types.

On calculation of the average linear energy transfer for radiobiological modelling.

Applying the concept of linear energy transfer (LET) to modeling of biological effects of charged particles usually involves calculation of the average LET. To calculate this, the energy distribution of particles is characterized by either the source spectrum or fluence spectrum. Also, the average can be frequency-or dose-weighted. This makes four methods of calculating the average LET, each producing a different number. The purpose of this note is to describe which of these four methods is best suited for radiobiological modelling. We focused on data for photons (x-rays and gamma radiation) because in this case differences in the four averaging methods are most pronounced. However, our conclusions are equally applicable to photons and hadrons. We based our arguments on recently emerged Monte Carlo data that fully account for transport of electrons down to very low energies comparable to the ionization potential of water. We concluded that the frequency average LET calculated using the fluence spectrum has better predictive power than does that calculated using any of the other three options. This optimal method is not new but is different from those currently dominating research in this area.

A simple model for calculating relative biological effectiveness of X-rays and gamma radiation in cell survival.

OBJECTIVES: The relative biological effectiveness (RBE) of X-rays and γ radiation increases substantially with decreasing beam energy. This trend affects the efficacy of medical applications of this type of radiation. This study was designed to develop a model based on a survey of experimental data that can reliably predict this trend. METHODS: In our model, parameters α and ß of a cell survival curve are simple functions of the frequency-average linear energy transfer (LF) of delta electrons. The choice of these functions was guided by a microdosimetry-based model. We calculated LF by using an innovative algorithm in which LF is associated with only those electrons that reach a sensitive-to-radiation volume (SV) within the cell. We determined model parameters by fitting the model to 139 measured (α,ß) pairs. RESULTS: We tested nine versions of the model. The best agreement was achieved with [Formula: see text] and ß being linear functions of [Formula: see text] .The estimated SV diameter was 0.1-1 µm. We also found that α, ß, and the α/ß ratio increased with increasing [Formula: see text] . CONCLUSIONS: By combining an innovative method for calculating [Formula: see text] with a microdosimetric model, we developed a model that is consistent with extensive experimental data involving photon energies from 0.27 keV to 1.25 MeV. ADVANCES IN KNOWLEDGE: We have developed a photon RBE model applicable to an energy range from ultra-soft X-rays to megaelectron volt γ radiation, including high-dose levels where the RBE cannot be calculated as the ratio of α values. In this model, the ionization density represented by [Formula: see text] determines the RBE for a given photon spectrum.

AAPM Task Group 329: Reference dose specification for dose calculations: Dose-to-water or dose-to-muscle?

Linac calibration is done in water, but patients are comprised primarily of soft tissue. Conceptually, and specified in NRG/RTOG trials, dose should be reported as dose-to-muscle to describe the dose to the patient. Historically, the dose-to-water of the linac calibration was often converted to dose-to-muscle for patient calculations through manual application of a 0.99 dose-to-water to dose-to-muscle correction factor, applied during the linac clinical reference calibration. However, many current treatment planning system (TPS) dose calculation algorithms approximately provide dose-to-muscle (tissue), making application of a manual scaling unnecessary. There is little guidance on when application of a scaling factor is appropriate, resulting in highly inconsistent application of this scaling by the community. In this report we provide guidance on the steps necessary to go from the linac absorbed dose-to-water calibration to dose-to-muscle in patient, for various commercial TPS algorithms. If the TPS does not account for the difference between dose-to-water and dose-to-muscle, then TPS reference dose scaling is warranted. We have tabulated the major vendors' TPS in terms of whether they approximate dose-to-muscle or calculate dose-to-water and recommend the correction factor required to report dose-to-muscle directly from the TPS algorithm. Physicists should use this report to determine the applicable correction required for specifying the reference dose in their TPS to achieve this goal and should remain attentive to possible changes to their dose calculation algorithm in the future.

Treatment vault shielding for a flattening filter-free medical linear accelerator.

The requirements for shielding a treatment vault with a Varian Clinac 2100 medical linear accelerator operated both with and without the flattening filter were assessed. Basic shielding parameters, such as primary beam tenth-value layers (TVLs), patient scatter fractions, and wall scatter fractions, were calculated using Monte Carlo simulations of 6, 10 and 18 MV beams. Relative integral target current requirements were determined from treatment planning studies of several disease sites with, and without, the flattening filter. The flattened beam shielding data were compared to data published in NCRP Report No. 151, and the unflattened beam shielding data were presented relative to the NCRP data. Finally, the shielding requirements for a typical treatment vault were determined for a single-energy (6 MV) linac and a dual-energy (6 MV/18 MV) linac. With the exception of large-angle patient scatter fractions and wall scatter fractions, the vault shielding parameters were reduced when the flattening filter was removed. Much of this reduction was consistent with the reduced average energy of the FFF beams. Primary beam TVLs were reduced by 12%, on average, and small-angle scatter fractions were reduced by up to 30%. Head leakage was markedly reduced because less integral target current was required to deliver the target dose. For the treatment vault examined in the current study, removal of the flattening filter reduced the required thickness of the primary and secondary barriers by 10-20%, corresponding to 18 m(3) less concrete to shield the single-energy linac and 36 m(3) less concrete to shield the dual-energy linac. Thus, a shielding advantage was found when the linac was operated without the flattening filter. This translates into a reduction in occupational exposure and/or the cost and space of shielding.

Stereotactic radiotherapy for lung cancer using a flattening filter free Clinac.

The objective of this study was to assess the feasibility of stereotactic radiotherapy for early stage lung cancer using photon beams from a Varian Clinac accelerator operated without a flattening filter. Treatment plans were generated for 10 lung cancer patients with isolated lesions less than 3 cm in diameter. For each patient, two plans were generated, one with and one without the flattening filter. Plans were generated with Eclipse 8.0 (Varian Medical Systems) commissioned with beam data measured on a Clinac 21EX (Varian Medical Systems) operated with and without the flattening filter. Removal of the flattening filter increased the dose rate. The median beam-on time per field was reduced from 25 sec (with the filter) to 11 sec (without the filter), increasing the feasibility of breath-hold treatments and the efficiency of gated treatments. Differences in a dose heterogeneity index for the planning target volume between plans with flattened and unflattened beams were statistically insignificant. Differences in mean doses to organs at risk were small, typically about 10 cGy over the entire treatment. The study concludes that radiotherapy with unflattened beams is feasible and requires substantially less beam-on time, facilitating breath-hold and gating techniques.

Systematic microdosimetric data for protons of therapeutic energies calculated with Geant4-DNA.

The purpose of this study was to generate physical data needed for microdosimetry-based models of proton RBE. Our focus was on the frequency and dose average lineal energies, y  F and y  D . We report data for proton energies from 0.1 to 100 MeV, for spherical volumes 2-103 nm in diameter. These data were calculated using Geant4-DNA Monte Carlo software. The physics implemented in Geant4-DNA has been extensively tested for this type of calculations but data on y  F and y  D for protons generated with this code have been very limited. An innovative aspect of our study is that we introduced a straightforward procedure for calculation of y  F and y  D for polyenergetic beams and presented the data in a format that simplifies these calculations. We compared our data with previous studies that used different Monte Carlo codes and with experimental data.

Feasibility of a multigroup deterministic solution method for three-dimensional radiotherapy dose calculations.

PURPOSE: To investigate the potential of a novel deterministic solver, Attila, for external photon beam radiotherapy dose calculations. METHODS AND MATERIALS: Two hypothetical cases for prostate and head-and-neck cancer photon beam treatment plans were calculated using Attila and EGSnrc Monte Carlo simulations. Open beams were modeled as isotropic photon point sources collimated to specified field sizes. The sources had a realistic energy spectrum calculated by Monte Carlo for a Varian Clinac 2100 operated in a 6-MV photon mode. The Attila computational grids consisted of 106,000 elements, or 424,000 spatial degrees of freedom, for the prostate case, and 123,000 tetrahedral elements, or 492,000 spatial degrees of freedom, for the head-and-neck cases. RESULTS: For both cases, results demonstrate excellent agreement between Attila and EGSnrc in all areas, including the build-up regions, near heterogeneities, and at the beam penumbra. Dose agreement for 99% of the voxels was within the 3% (relative point-wise difference) or 3-mm distance-to-agreement criterion. Localized differences between the Attila and EGSnrc results were observed at bone and soft-tissue interfaces and are attributable to the effect of voxel material homogenization in calculating dose-to-medium in EGSnrc. For both cases, Attila calculation times were <20 central processing unit minutes on a single 2.2-GHz AMD Opteron processor. CONCLUSIONS: The methods in Attila have the potential to be the basis for an efficient dose engine for patient-specific treatment planning, providing accuracy similar to that obtained by Monte Carlo.

Energy spectra, sources, and shielding considerations for neutrons generated by a flattening filter-free Clinac.

Neutron production is an unwanted result of high-energy radiation therapy and results in secondary exposure of patients and radiation therapists to radiation. Recent studies have shown that delivering therapy using a standard medical accelerator with the flattening filter removed may reduce neutron fluence by nearly 70% over the course of prostate intensity-modulated radiation therapy (IMRT). In the current study, the 197Au Bonner sphere technique was used to compare the neutron spectrum produced when the filter is present and when it is absent. In addition, the following was calculated: (1) the neutron-shielding parameters of source strength and ambient dose equivalent (H0) and (2) using the Monte Carlo technique, the sources of neutron production in the accelerator head. It was found that the neutron spectrum was nearly constant, regardless of the presence of the flattening filter; however, the total fluence and ambient dose equivalent over the course of prostate IMRT were more than 70% lower when the filter was removed. Similarly, shielding parameters were lower when the filter was removed. Finally, the primary collimator and jaws accounted for the majority of neutron production, both with and without the flattening filter; however, with the flattening filter removed, the upper jaw accounted for much more neutron production relative to when the filter was present. Ultimately, removal of the flattening filter may offer several clinical advantages, including a reduction in the dose from neutrons to the patient and to radiation personnel.

Monte Carlo investigation of collimator scatter of proton-therapy beams produced using the passive scattering method.

As a proton-therapy beam passes through the field-limiting aperture, some of the protons are scattered off the edges of the collimator. The edge-scattered protons can degrade the dose distribution in a patient or phantom, and these effects are difficult to model with analytical methods such as those available in treatment planning systems. The objective of this work was to quantify the dosimetric impact of edge-scattered protons for a representative variety of clinical treatment beams. The dosimetric impact was assessed using Monte Carlo simulations of proton beams from a contemporary treatment facility. The properties of the proton beams were varied, including the penetration range (6.4-28.5 cm), width of the spread-out Bragg peak (SOBP; 2-16 cm), field size (3 x 3 cm(2) to 15 x 15 cm(2)) and air gap, i.e. the distance between the collimator and the phantom (8-48 cm). The simulations revealed that the dosimetric impact of edge-scattered protons increased strongly with increasing range (dose increased by 6-20% with respect to the dose at the center of the spread-out Bragg peak), decreased strongly with increasing field size (dose changed by 2-20%), increased moderately with increasing air gap (dose increased by 2-6%) and increased weakly with increasing SOBP width (dose change <4%). In all cases examined, the effects were largest at shallow depths. We concluded that the dose deposited by edge-scattered protons can distort the dose proximal to the target with varying contributions due to the proton range, treatment field size, collimator position and thickness, and width of the SOBP. Our findings also suggest that accurate predictions of dose per monitor-unit calculations may require taking into account the dose from protons scattered from the edge of the patient-specific collimator, particularly for fields of small lateral size and deep depths.

Average stopping powers for electron and photon sources for radiobiological modeling and microdosimetric applications.

This study concerns calculation of the average electronic stopping power for photon and electron sources. It addresses two problems that have not yet been fully resolved. The first is defining the electron spectrum used for averaging in a way that is most suitable for radiobiological modeling. We define it as the spectrum of electrons entering the sensitive to radiation volume (SV) within the cell nucleus, at the moment they enter the SV. For this spectrum we derive a formula that combines linearly the fluence spectrum and the source spectrum. The latter is the distribution of initial energies of electrons produced by a source. Previous studies used either the fluence or source spectra, but not both, thereby neglecting a part of the complete spectrum. Our derived formula reduces to these two prior methods in the case of high and low energy sources, respectively. The second problem is extending electron spectra to low energies. Previous studies used an energy cut-off on the order of 1 keV. However, as we show, even for high energy sources, such as 60Co, electrons with energies below 1 keV contribute about 30% to the dose. In this study all the spectra were calculated with Geant4-DNA code and a cut-off energy of only 11 eV. We present formulas for calculating frequency- and dose-average stopping powers, numerical results for several important electron and photon sources, and tables with all the data needed to use our formulas for arbitrary electron and photon sources producing electrons with initial energies up to ∼1 MeV.

Radiotherapy of lung cancers: FFF beams improve dose coverage at tumor periphery compromised by electronic disequilibrium.

The purpose of this work was to investigate radiotherapy underdosing at the periphery of lung tumors, and differences in dose for treatments delivered with flattening filter-free (FFF) beams and with conventional flattened (FF) beams. The true differences between these delivery approaches, as assessed with Monte Carlo simulations, were compared to the apparent differences seen with clinical treatment planning algorithms AAA and Acuros XB. Dose was calculated in a phantom comprised of a chest wall, lung parenchyma, and a spherical tumor (tested diameters: 1, 3, and 5 cm). Three lung densities were considered: 0.26, 0.2, and 0.1 g cm-3, representing normal lung, lung at full inspiration, and emphysematous lung, respectively. The dose was normalized to 50 Gy to the tumor center and delivered with 7 coplanar, unmodulated 6 MV FFF or FF beams. Monte Carlo calculations used EGSnrc and phase space files for the TrueBeam accelerator provided by Varian Medical Systems. Voxel sizes were 0.5 mm for the 1 cm tumor and 1 mm for the larger tumors. AAA and Acuros XB dose calculations were performed in Eclipse™ with a 2.5 mm dose grid, the resolution normally used clinically. Monte Carlo dose distributions showed that traditional FF beams underdosed the periphery of the tumor by up to ~2 Gy as compared to FFF beams; the latter provided a more uniform dose throughout the tumor. In all cases, the underdosed region was a spherical shell about 5 mm thick around the tumor and extending into the tumor by 2-3 mm. The effect was most pronounced for smaller tumors and lower lung densities. The underdosing observed with conventional FF beams was not captured by the clinical treatment planning systems. We concluded that FFF beams mitigate dose loss at tumor periphery and current clinical practice fails to capture tumor periphery underdosing and possible ways to mitigate it.

Reduced neutron production through use of a flattening-filter-free accelerator.

PURPOSE: To measure and compare neutron fluence around an accelerator operating at 18 MV, both with the flattening filter present (FF mode) and absent (flattening-filter-free [FFF] mode). METHODS AND MATERIALS: The neutron fluence was measured at several locations in the patient plane using gold foil activation in neutron moderators. Differences in neutron fluence between the FF and FFF mode were assessed in three frameworks: (1) measured per monitor unit of machine-on time, (2) determined per dose on the central axis, and (3) determined for a complete course of prostate intensity-modulated radiotherapy. RESULTS: Neutron fluence per monitor unit was approximately 20% lower when the accelerator was operated in the FFF mode than when it was in FF mode. The total amount of neutron fluence that would be obtained during the entire course of prostate intensity-modulate radiotherapy was 69% lower when the accelerator was operated in the FFF mode than when it was in the FF mode. This reduction in neutron fluence would correspond to a drastic reduction in the neutron dose equivalent received by the patient as a byproduct of high-energy radiotherapy. It would also correspond to a reduction in activation within the treatment vault and subsequent exposure to radiation therapists. CONCLUSION: When feasible, operating the accelerator without a FF will benefit both patients and radiation therapists by reducing the number of unwanted neutrons and resultant exposure. This reduces the risk of negative effects from such exposure (e.g., second cancers).

Treatment-planning study of prostate cancer intensity-modulated radiotherapy with a Varian Clinac operated without a flattening filter.

PURPOSE: To assess the feasibility of intensity-modulated radiotherapy for prostate cancer using photon beams from an accelerator operated without a flattening filter; and to determine potential benefits and drawbacks of using unflattened beams for this type of treatment. METHODS AND MATERIALS: Intensity-modulated radiotherapy plans were generated for 10 patients with early-stage prostate cancer. For each patient, four plans were generated: with and without the flattening filter, at 6 and 18 MV. The prescription dose was 75.6 Gy to 98% of the planning target volume. The number of beams, their orientations, and optimization constraints were the same for all plans. Plans were generated with Eclipse 8.0 (Varian Medical Systems). RESULTS: All the plans developed with unflattened beams were clinically acceptable. In terms of patient dose distributions, plans with unflattened beams were similar to the corresponding plans with flattened beams. Plans with unflattened beams required fewer monitor units (MUs) per plan: on average, by a factor of 2.0 at 6 MV and 2.6 at 18 MV, assuming that removal of the flattening filter was not followed by recalibration of MUs. CONCLUSIONS: Clinically acceptable intensity-modulated radiotherapy plans for prostate cancer can be developed with unflattened beams at both 6 and 18 MV. Dosimetrically, flattened and unflattened beams generated similar treatment plans. The plans with unflattened beams required substantially fewer MUs. The reduction in the number of MUs indicates corresponding reduction in beam-on time and in the amount of radiation outside the target.

A Monte Carlo model for out-of-field dose calculation from high-energy photon therapy.

As cancer therapy becomes more efficacious and patients survive longer, the potential for late effects increases, including effects induced by radiation dose delivered away from the treatment site. This out-of-field radiation is of particular concern with high-energy radiotherapy, as neutrons are produced in the accelerator head. We recently developed an accurate Monte Carlo model of a Varian 2100 accelerator using MCNPX for calculating the dose away from the treatment field resulting from low-energy therapy. In this study, we expanded and validated our Monte Carlo model for high-energy (18 MV) photon therapy, including both photons and neutrons. Simulated out-of-field photon doses were compared with measurements made with thermoluminescent dosimeters in an acrylic phantom up to 55 cm from the central axis. Simulated neutron fluences and energy spectra were compared with measurements using moderated gold foil activation in moderators and data from the literature. The average local difference between the calculated and measured photon dose was 17%, including doses as low as 0.01% of the central axis dose. The out-of-field photon dose varied substantially with field size and distance from the edge of the field but varied little with depth in the phantom, except at depths shallower than 3 cm, where the dose sharply increased. On average, the difference between the simulated and measured neutron fluences was 19% and good agreement was observed with the neutron spectra. The neutron dose equivalent varied little with field size or distance from the central axis but decreased with depth in the phantom. Neutrons were the dominant component of the out-of-field dose equivalent for shallow depths and large distances from the edge of the treatment field. This Monte Carlo model is useful to both physicists and clinicians when evaluating out-of-field doses and associated potential risks.

Radial dose distributions from carbon ions of therapeutic energies calculated with Geant4-DNA.

We report on radial dose distributions [Formula: see text] for carbon ions calculated with Geant4-DNA code. These distributions characterize ion tracks on a nanoscale and are important for understanding the biological effects of ion beams. We present data for carbon ion beams in the energy range from 20 to 400 MeV u-1. To approximate the Monte Carlo results, we developed a simple formula that combines the well-known inverse square distance dependence with a factor correcting [Formula: see text] for small [Formula: see text]. The proposed formula can be used to calculate [Formula: see text] for any energy within the above range and for distances [Formula: see text] from 1 nm to 2 µm with a maximum error not exceeding 14%. This range of distances corresponds to a dose range of over seven orders of magnitude. Differences between our results and those of previously published analytical models are discussed.

A new formalism for modelling parameters α and ß of the linear-quadratic model of cell survival for hadron therapy.

We propose a new formalism for calculating parameters α and ß of the linear-quadratic model of cell survival. This formalism, primarily intended for calculating relative biological effectiveness (RBE) for treatment planning in hadron therapy, is based on a recently proposed microdosimetric revision of the single-target multi-hit model. The main advantage of our formalism is that it reliably produces α and ß that have correct general properties with respect to their dependence on physical properties of the beam, including the asymptotic behavior for very low and high linear energy transfer (LET) beams. For example, in the case of monoenergetic beams, our formalism predicts that, as a function of LET, (a) α has a maximum and (b) the α/ß ratio increases monotonically with increasing LET. No prior models reviewed in this study predict both properties (a) and (b) correctly, and therefore, these prior models are valid only within a limited LET range. We first present our formalism in a general form, for polyenergetic beams. A significant new result in this general case is that parameter ß is represented as an average over the joint distribution of energies E 1 and E 2 of two particles in the beam. This result is consistent with the role of the quadratic term in the linear-quadratic model. It accounts for the two-track mechanism of cell kill, in which two particles, one after another, damage the same site in the cell nucleus. We then present simplified versions of the formalism, and discuss predicted properties of α and ß. Finally, to demonstrate consistency of our formalism with experimental data, we apply it to fit two sets of experimental data: (1) α for heavy ions, covering a broad range of LETs, and (2) ß for protons. In both cases, good agreement is achieved.