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Background: Brachytherapy for ocular melanoma is based on the application of eye plaques with different spatial dose nonuniformity, time-dependent dose rates and relative biological effectiveness (RBE)., Purpose: We propose a parameter called the equivalent uniform RBE-weighted dose (EUD RBE ) that can be used for quantitative characterization of integrated cell survival in radiotherapy modalities with the variable RBE, dose nonuniformity and dose rate. The EUD RBE is applied to brachytherapy with 125 I eye plaques designed by the Collaborative Ocular Melanoma Study (COMS)., Methods: The EUD RBE is defined as the uniform dose distribution with RBE = 1 that causes equal cell survival for a given nonuniform dose distribution with the variable RBE > 1. The EUD RBE can be used for comparison of cell survival for nonuniform dose distributions with different RBE, because they are compared to the reference dose with RBE = 1. The EUD RBE is applied to brachytherapy with 125 I COMS eye plaques that are characterized by a steep dose gradient in tumor base-apex direction, protracted irradiation during time intervals of 3-8 days, and variable dose-rate dependent RBE with a maximum of about 1.4. The simulations are based on dose of 85 Gy prescribed to the farthest intraocular extent of the tumor (tumor apex). To compute the EUD RBE in eye plaque brachytherapy and correct for protracted irradiation, the distributions of physical dose have been converted to non-uniform distributions of biologically effective dose (BED) to include the biological effects of sublethal cellular repair, Our radiobiological analysis considers the combined effects of different time-dependent dose rates, spatial dose non-uniformity, dose fractionation and different RBE and can be used to derive optimized dose regimens brachytherapy., Results: Our simulations show that the EUD RBE increases with the prescription depths and the maximum increase may achieve 6% for the tumor height of 12 mm. This effect stems from a steep dose gradient within the tumor that increases with the prescription depth. The simulations also show that the EUD RBE increase may achieve 12% with increasing the dose rate when implant duration decreases. The combined effect of dose nonuniformity and dose rate may change the EUD RBE up to 18% for the same dose prescription of 85 Gy to tumor apex. The absolute dose range of 48-61 Gy (RBE) for the EUD RBE computed using 4 or 5 fractions is comparable to the dose prescriptions used in stereotactic body radiation therapy (SBRT) with megavoltage X-rays (RBE = 1) for different cancers. The tumor control probabilities in SBRT and eye plaque brachytherapy are very similar at the level of 80% or higher that support the hypothesis that the selected approximations for the EUD RBE are valid., Conclusions: The computed range of the EUD RBE in 125 I COMS eye plaque brachytherapy suggests that the selected models and hypotheses are acceptable. The EUD RBE can be useful for analysis of treatment outcomes and comparison of different dose regimens in eye plaque brachytherapy., (© 2024 American Association of Physicists in Medicine.)

Purpose: Clinical studies in radiation therapy with conventional fractionation show a reduction in the tumor control probability (TCP) with an increase in the total and hypoxic tumor volumes. The main objective of this article is to derive an analytical relationship between the TCP and the hypoxic and total tumor volumes. This relationship is applied to clinical data on the TCP reduction with increasing total tumor volume and, also, dose escalation to target tumor hypoxia., Methods: The TCP equation derived from the Poisson probability distribution predicts that both (a) an increase in the number of tumor clonogens and (b) an increase in the average cell surviving fraction are the factors contributing to the loss of local control. Using asymptotic mathematical properties of the TCP formula and the linear quadratic (LQ) cell survival model with two levels of hypoxic and oxygenated cells, we separated the TCP dependence on the total and hypoxic tumor volumes. The predicted trends in the local control as a function of total and hypoxic tumor volumes were evaluated in radiotherapy model problems with conventional dose fractionation for head and neck and non-small cell lung cancers. Tumor-specific parameters in the LQ model and the density of clonogens in the TCP model were taken from published data on predictive assays and the plating efficiency measurements, respectively., Results: Our simulations show that, at the dose levels used in conventional radiation therapy for head and neck and non-small cell lung cancers, the TCP dependence on the total tumor volume is negligible for completely oxygenated tumors. However, the presented results demonstrate that tumor hypoxia introduces a significant volume effect into estimates of the TCP. The extent of tumor hypoxia is a plausible mechanism to explain the TCP reduction with increasing total tumor volume observed in clinical studies. To achieve the same level of tumor control in a hypoxic tumor region relative to well oxygenated tumor regions, the delivered dose should, in principle, be escalated by a factor equal to the oxygen enhancement ratio (OER). The theoretically required hypoxia-targeted dose escalation could be as large as 100% because it has been estimated that hypoxic tumor regions may have an OER = 2 for conventional fractionation. However, our results indicate that clinically acceptable values of the TCP would require much lower hypoxia-targeted dose escalation (<50%) when the effects of total and hypoxic tumor volumes are taken into account., Conclusions: The reported studies and models suggest that the effect of total tumor volume on the TCP is negligible for oxygenated head and neck and non-small cell lung tumors treated with conventional fractionation. According to our simulations, the volume effects in the TCP observed in clinical studies are defined primarily by the hypoxic volume. This information can be useful for the analysis of treatment outcomes and the dose escalation to target tumor hypoxia., (© 2020 American Association of Physicists in Medicine.)

Purpose: Dose painting techniques based on dose escalation to hypoxic regions are usually governed by the spatial distribution of Oxygen Enhancement Ratio (OER) derived from PET or MRI images. The goal of this article is to show that the volume of the hypoxic region is also a parameter which may affect the radiobiological effectiveness of hypoxia-targeted dose escalation., Methods: Our analysis is performed using the equation for Tumor Control Probability (TCP) derived from Poisson statistics. A tumor response model based on two levels of oxygenated and hypoxic cells with the survival curves described by the Linear Quadratic (LQ) model was used in the calculations. The model allows for analytical solutions which can be used for additional validation of our numerical simulations. The model was used to compute the average tumor cell survival that defines TCP. Dependence of average tumor cell survival on the relative hypoxic volume and dose escalation was computed for the conventional 30 × 2 Gy and hypofractionated 4 × 12 Gy dose regimens in a model problem with radiobiological parameters for nonsmall cell lung cancer., Results: We show that equal average tumor cell survival and TCP can be achieved with smaller hypoxia-targeted dose escalation for smaller relative hypoxic volumes if the dose escalation does not exceed 50% of dose in the oxygenated region. This effect is explained by the dependence of average tumor cell survival and TCP mostly on the hypoxic tumor volume in this dose escalation range. Assuming a linear relationship between the number of clonogens and the hypoxic volume, smaller hypoxic volumes have smaller number of clonogens, therefore, smaller doses are needed to eradicate them., Conclusions: The theoretical analysis performed in this article predicts that tumors with smaller relative hypoxic volumes may require smaller hypoxia-targeted dose escalation to maintain equal TCP. The results of this research can be used for dose de-escalation strategies in dose painting techniques which may reduce dose to normal tissue and normal tissue complications without deterioration of TCP., (© 2018 American Association of Physicists in Medicine.)

Purpose: The goal of this article is to compute the cell survival during fractionated radiotherapy with non-uniform hypoxia-targeted dose distribution relative to cell survival for a uniform dose distribution with equal integral tumor dose. The analysis is performed for different parameters of radiotherapy with conventional and hypofractionated dose regimens., Methods: Our analysis is done using a parsimonious tumor response model that describes the major components of tumor response to radiotherapy such as radiosensitivity, cell proliferation, and hypoxia using as few variables as possible. Two levels of oxygenated and hypoxic cells with the survival curves described by the linear quadratic (LQ) model are implemented in the model. The model allows for analytical solutions for relative cell survival in a single fraction simulation which can be used for additional validation of our numerical simulations. The relative cell survival was computed for conventional and hypofractionated dose regimens in a model problem with radiobiological parameters for the non-small-cell lung cancer. Sensitivity of cell survival to different parameters of radiotherapy such as the relative volume of hypoxic fraction, boost dose ratio, re-oxygenation rate, and delivery errors was investigated., Results: Our computational and analytical results show that relative cell survival for non-uniform and uniform dose distributions is larger than 1.0 during the first few fractions of radiotherapy with conventional fractionation. This indicates that the uniform dose distribution produces a higher cell killing effect for the equal integral dose. This may stem from domination of linear contribution to the cell killing effect seen in the dose range of 1-2 Gy and a steeper slope of survival curve in the oxygenated tumor region. This effect can only happen if the distribution of clonogens is nearly uniform; therefore, after the first few fractions, the non-uniform dose distributions outperform the uniform dose distribution and relative cell survival becomes less than 1.0. However, re-oxygenation and delivery errors can also reduce the effectiveness of hypoxia-targeted non-uniform dose distributions toward the end of treatment. For hypofractionated radiotherapy with fractional dose >3 Gy, the relative cell survival was always below 1.0, which means the non-uniform dose distributions produced higher cell killing effect than the uniform dose distribution during the entire treatment., Conclusion: The data obtained in this work suggest that non-uniform hypoxia-targeted dose distributions are less effective at cell killing than uniform dose distributions at the beginning of radiotherapy with conventional fractionation. However; non-uniform dose distributions can outperform uniform dose distribution by the end of the treatment if the effects of re-oxygenation and delivery errors are reduced. In hypofractionated radiotherapy, non-uniform hypoxia-targeted dose distributions are more efficient than uniform dose distributions during the entire treatment., (© 2017 American Association of Physicists in Medicine.)

Purpose: In our previous work, the authors showed that a distribution of cell surviving fractions S2 in a heterogeneous group of patients could be derived from tumor-volume variation curves during radiotherapy for head and neck cancer. In this research study, the authors show that this algorithm can be applied to other tumors, specifically in nonsmall cell lung cancer. This new application includes larger patient volumes and includes comparison of data sets obtained at independent institutions., Methods: Our analysis was based on two data sets of tumor-volume variation curves for heterogeneous groups of 17 patients treated for nonsmall cell lung cancer with conventional dose fractionation. The data sets were obtained previously at two independent institutions by using megavoltage computed tomography. Statistical distributions of cell surviving fractions S2 and clearance half-lives of lethally damaged cells T(1/2) have been reconstructed in each patient group by using a version of the two-level cell population model of tumor response and a simulated annealing algorithm. The reconstructed statistical distributions of the cell surviving fractions have been compared to the distributions measured using predictive assays in vitro., Results: Nonsmall cell lung cancer presents certain difficulties for modeling surviving fractions using tumor-volume variation curves because of relatively large fractional hypoxic volume, low gradient of tumor-volume response, and possible uncertainties due to breathing motion. Despite these difficulties, cell surviving fractions S2 for nonsmall cell lung cancer derived from tumor-volume variation measured at different institutions have similar probability density functions (PDFs) with mean values of 0.30 and 0.43 and standard deviations of 0.13 and 0.18, respectively. The PDFs for cell surviving fractions S2 reconstructed from tumor volume variation agree with the PDF measured in vitro., Conclusions: The data obtained in this work, when taken together with the data obtained previously for head and neck cancer, suggests that the cell surviving fractions S2 can be reconstructed from the tumor volume variation curves measured during radiotherapy with conventional fractionation. The proposed method can be used for treatment evaluation and adaptation.

Purpose: The main goal of this paper is to reconstruct a distribution of cell survival fractions from tumor-volume variation for a heterogeneous group of head and neck cancer patients and compare this distribution to the data from predictive assays., Methods: To characterize the tumor-volume variation during radiation therapy treatment, the authors use a two-level tumor-volume model of cell population that separates the entire tumor cell population into two subpopulations of viable cells and lethally damaged cells. This parameterized radiobiological model is integrated with a least squares objective function and a simulated annealing optimization algorithm to describe time-dependent tumor-volume variation rates in individual patients. Several constraints have been used in the optimization problem because tumor-volume variation during radiotherapy is described by a sum of exponentials; therefore, the problem of accurately fitting a model to measured data is ill-posed. The model was applied to measured tumor-volume variation curves from a clinical study on tumor-volume variation during radiotherapy for 14 head and neck cancer patients in which an integrated CT/linear particle accelerator (LINAC) system was used for tumor-volume measurements., Results: The two-level cell population tumor-volume modeling is capable of describing tumor-volume variation throughout the entire treatment for 11 of the 14 patients. For three patients, the tumor-volume variation was described only during the initial part of treatment, a fact that may be related to the neglected hypoxia in the two-level approximation. The predicted probability density distribution for the survival fractions agrees with the data obtained using in vitro studies with predictive assays. The mean value 0.35 of survival fraction obtained in this study is larger than the value 0.32 from in vitro studies, which could be expected because of greater repair in vivo. The mean half-life obtained in this study for the head-and-neck squamous cell carcinoma (SCC) is equal to 3.8 mean potential doubling times, which agrees with 4.0 mean potential doubling times obtained previously for lung SCC., Conclusions: The distribution of cell survival fractions obtained in this study support the hypothesis that the tumor-volume variation during radiotherapy treatment for head and neck cancer can be described by the two-level cell population tumor-volume model. This model can be used for in vivo evaluation of patient-specific radiobiological parameters that are needed for tumor-control probability evaluation.

Electron spectral reconstruction of medical accelerators from measured depth doses is a practical method for providing the input initial phase space distribution at the patient surface that is required by Monte Carlo treatment planning systems. The posed inverse problem of spectral reconstruction is ill conditioned and this may lead to nonphysical oscillations in the reconstructed spectra. Use of a variational method of solution with a regularization technique removes the oscillations but tends to smooth the sharp (deltalike) energy peak that is a common feature in electron spectra generated by medical accelerators. Because the sharp peak contains a large percentage of the electrons in the spectrum, an accurate estimate of the peak width, height and position is critical to the success of the technique for spectrum reconstruction with regularization. We propose use of an adaptive regularization term as a special form of the general Tichonov regularization function. The variational method with the adaptive regularization term is applied to reconstruct electron spectra for the 6, 9, and 18 MeV electron beams of a Varian Clinac 2100C accelerator and proves to be a very simple, effective and accurate approach. Results using this variational method with adaptive regularization almost perfectly reconstruct electron spectra from depth dose distributions.

We attempt to select an optimal value of regularization parameter in the optimization problems for intensity-modulated radiotherapy which are solved using a variational regularization technique. We apply to inverse treatment planning the L-curve method which was developed to determine the regularization parameter in the discrete ill-posed problems. The L-curve method is based on finding the regularization parameter which minimizes the residual norm which is a measure of accuracy of fit and the solution norm which is a measure of smoothness of solution. The main idea of the L-curve method is to plot the smoothing norm as a function of the residual norm for all values of the regularization parameter. This characteristic curve has an L-shaped dependence and the optimal value of regularization parameter can be found at the "corner" of the L-curve. We plot the L-curves for the optimization problems which simulate prostate radiotherapy cancer treatment with intensity-modulated beams. Different numerical methods are applied to calculate the point of maximum curvature of the L-curves which is a criterion to locate the corner. We show that the point of maximum curvature can be located in a most robust way using a formula derived from the singular value decomposition analysis.

The performance of a variational regularization technique to improve robustness of inverse treatment planning for intensity modulated radiotherapy is analyzed and tested. Inverse treatment planning is based on the numerical solutions to the Fredholm integral equation of the first kind which is ill-posed. Therefore, a fundamental problem with inverse treatment planning is that it may exhibit instabilities manifested in nonphysical oscillations in the beam intensity functions. To control the instabilities, we consider a variational regularization technique which can be applied for the methods which minimize a quadratic objective function. In this technique, the quadratic objective function is modified by adding of a stabilizing functional that allows for arbitrary order regularization. An optimal form of stabilizing functional is selected which allows for both regularization and good approximation of beam intensity functions. The regularized optimization algorithm is shown, by comparison for a typical case of a head-and-neck cancer treatment, to be significantly more accurate and robust than the standard approach, particularly for the smaller beamlet sizes.

Techniques for reconstruction of electron spectra from the depth-dose curves used to date have ignored the angular distribution of incident electrons scattered at large angles. The approximation adopted is to employ a database of monoenergetic depth-dose curves generated for normal incidence of electrons at the surface. This approximation is acceptable for direct electrons with small angular spread. However, electrons scattered from the treatment head and collimating system may have large average angles of incidence which affects the depth-dose distribution significantly at shallow depths by increasing energy deposition close to the surface. We show that ignoring the electron incident angular distribution leads to systematic errors in the low energy region of reconstructed electron spectra. We propose a simple 1-D model to correct for these systematic errors using only electron angular distribution at the central beam axis. This model provides reconstructed spectra in excellent agreement with Monte Carlo simulation in the low energy region.

Reconstruction of electron spectra of medical accelerators from measured depth dose distributions is an attractive tool for commissioning of a Monte Carlo treatment planning system. However, the reconstruction method is an inverse radiation transport problem which is poorly conditioned, in the sense it may become unstable due to small perturbations in the input data. Predicting the sharp (delta-like) peak in the electron spectrum provides an additional challenge for the numerical reconstruction technique. To improve efficiency and robustness of the reconstruction technique, we developed an algorithm based on a separation of the electron spectrum into singular and regular components. We approximate the singular peak of the spectrum by a narrow weighted Gaussian function. The parameters of this Gaussian function are sought using only the fall-off and toe regions of the depth-dose curve. Analytical representation of the spectral peak by a Gaussian has benefit since only one weight and the mean and variance must be derived from the depth-dose curve instead of multiple spectra weights. The regular part of the spectrum is reconstructed from the residual depth-dose distribution using a variational method combined with a regularization technique to avoid the nonphysical oscillations. The effectiveness of the method is demonstrated by comparing predictions to "benchmark" spectra and depth-dose distributions from Monte Carlo simulation of medical accelerators.

Inverse planning for intensity- and energy-modulated radiotherapy (IEMRT) with proton beams involves the selection of (i) the relative importance factors to control the relative importance of the target and sensitive structures, (ii) an appropriate energy resolution to achieve an acceptable depth modulation, (iii) an appropriate beamlet width to modulate the beam laterally, and (iv) a sufficient number of beams and their orientations. In this article we investigate the influence of these variables on the optimized dose distribution of a simulated prostate cancer IEMRT treatment. Good dose conformation for this prostate case was achieved using a constellation of I factors for the target, rectum, bladder, and normal tissues of 500, 50, 15, and 1, respectively. It was found that for an active beam delivery system, the energy resolution should be selected on the basis of the incident beams' energy spread (sigmaE) and the appropriate energy resolution varied from 1 MeV at sigmaE = 0.0 to 5 MeV at sigmaE= 2.0 MeV. For a passive beam delivery system the value of the appropriate depth resolution for inverse planning may not be critical as long as the value chosen is at least equal to one-half the FWHM of the primary beam Bragg peak. Results indicate that the dose grid element dimension should be equal to or no less than 70% of the beamlet width. For this prostate case, we found that a maximum of three to four beam ports is required since there was no significant advantage to using a larger number of beams. However for a small number (< or = 4) of beams the selection of beam orientations, while having only a minor effect on target coverage, strongly influenced the sensitive structure sparing and normal tissue integral dose.

A transport algorithm called the proton loss (PL) model is developed for proton pencil beams of therapeutic energies. The PL model takes into account inelastic nuclear reactions, pathlength straggling, and energy-loss straggling and predicts the 3D dose distribution from a proton pencil beam. In proton beams, the multiple scattering and ionizational energy loss processes approach their diffusional limit where scattering and energy loss probability densities become Gaussian. Therefore we chose to derive the PL model from the Fermi-Eyges diffusional multiple scattering theory and the Gaussian theory of energy straggling. We first introduce a generalization of the Fermi-Eyges equation for proton pencil beams, labeled the proton loss (PL) transport equation. This new equation includes terms that model inelastic nuclear reactions as a depth-dependent absorption and pathlength straggling as a quasi-absorption. Then energy straggling is taken into account by using a weighted superposition of a discrete number of elementary pencil beams. These elementary pencil beams have different initial energies and lose energy according to the CSDA, thus they have different ranges of penetration. A final solution for the proton beam transport is obtained as a linear combination of elementary pencil beam solutions with weights defined by the Gaussian evolution of the proton energy spectrum with depth. A numerical comparison of the dose distribution predictions of the PL model with measurements and PTRAN Monte Carlo simulations indicates the model is both computational fast and accurate.