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Objective . We propose a criterion of biological effectiveness of nonuniform hypoxia-targeted dose distributions in heterogeneous hypoxic tumors based on equivalent uniform aerobic dose (EUAD). We demonstrate the utility of this criterion by applying it to the model problems in radiotherapy for tumors with different levels of oxygen enhancement ratio (OER) and different degrees of dose nonuniformity. Approach . The EUAD is defined as the uniform dose that, under well-oxygenated conditions, produces equal integrated survival of clonogenic cells in radiotherapy for heterogeneous hypoxic tumors with a non-uniform dose distribution. We define the dose nonuniformity effectiveness (DNE) in heterogeneous tumors as the ratio of the EUAD( D N ) for a non-uniform distribution D N and the reference EUAD( D U ) for the uniform dose distribution D U with equal integral tumor dose. The DNE concept is illustrated in a radiotherapy model problem for non-small cell lung cancer treated with hypoxia targeted dose escalation. A two-level cell population tumor model was used to consider the hypoxic and oxygenated tumor cells. Results . Theoretical analysis of the DNE shows that the entire region of the OER can be separated in two regions by a threshold OER th : (1) OER > OER th where DNE > 1 indicating higher effectiveness of nonuniform dose distributions and (2) OER < OER th where DNE < 1 indicating higher effectiveness of uniform dose distributions. Our simulations show that the value of the threshold OER th in radiotherapy with conventional fractionation is significant in the range of about 1.2-1.6 depending on selected radiotherapy parameters. In general, the OER th increases with reoxygenation rate, relative hypoxic volume and dose escalation factor. The threshold value of OER th is smaller of about 1.1 for hypofractionated radiotherapy. Significance . The analysis of dose distributions using the DNE shows that the uniform dose distributions may improve biological cell killing effect in heterogeneous tumors with intermediate oxygen levels compared to targeted nonuniform dose distribution., (© 2024 IOP Publishing Ltd. All rights, including for text and data mining, AI training, and similar technologies, are reserved.)

Objective. Equivalent uniform aerobic dose (EUAD) is proposed for comparison of integrated cell survival in tumors with different distributions of hypoxia and radiation dose. Approach. The EUAD assumes that for any non-uniform distributions of radiation dose and oxygen enhancement ratio (OER) within a tumor, there is a uniform distribution of radiation dose under hypothetical aerobic conditions with OER = 1 that produces equal integrated survival of clonogenic cells. This definition of EUAD has several advantages. First, the EUAD allows one to compare survival of clonogenic cells in tumors with intra-tumor and inter-tumor variation of radio sensitivity due to hypoxia because the cell survival is recomputed under the same benchmark oxygen level (OER = 1). Second, the EUAD for homogeneously oxygenated tumors is equal to the concept of equivalent uniform dose. Main results . We computed the EUAD using radiotherapy dose and the OER derived from the 18 F-Fluoromisonidazole PET ( 18 F-FMISO PET) images of hypoxia in patients with glioblastoma, the most common and aggressive type of primary malignant brain tumor. The 18 F-FMISO PET images include a distribution of SUV (Standardized Uptake Value); therefore, the SUV is converted to partial oxygen pressure (pO 2 ) and then to the OER. The prognostic value of EUAD in radiotherapy for hypoxic tumors is demonstrated using correlation between EUAD and overall survival (OS) in radiotherapy for glioblastoma. The correction to the EUAD for the absolute hypoxic volume that traceable to the tumor control probability improves the correlation with OS. Significance . While the analysis proposed in this research is based on the 18 F-FMISO PET images for glioblastoma, the EUAD is a universal radiobiological concept and is not associated with any specific cancer or any specific PET or MRI biomarker of hypoxia. Therefore, this research can be generalized to other cancers, for example stage III lung cancer, and to other hypoxia biomarkers., (© 2024 Institute of Physics and Engineering in Medicine.)

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.)

Clinical studies in the hypofractionated stereotactic body radiotherapy (SBRT) have shown a reduction in the probability of local tumor control with increasing initial tumor volume. In our earlier work, we obtained and tested an analytical dependence of the tumor control probability (TCP) on the total and hypoxic tumor volumes using conventional radiotherapy model with the linear-quadratic (LQ) cell survival. In this work, this approach is further refined and tested against clinical observations for hypofractionated radiotherapy treatment schedules. Compared to radiotherapy with conventional fractionation schedules, simulations of hypofractionated radiotherapy may require different models for cell survival and the oxygen enhancement ratio (OER). Our TCP simulations in hypofractionated radiotherapy are based on the LQ model and the universal survival curve (USC) developed for the high doses used in SBRT. The predicted trends in local control as a function of the initial tumor volume were evaluated in SBRT for non-small cell lung cancer (NSCLC). Our results show that both LQ and USC based models cannot describe the TCP reduction for larger tumor volumes observed in the clinical studies if the tumor is considered completely oxygenated. The TCP calculations are in agreement with the clinical data if the subpopulation of radio-resistant hypoxic cells is considered with the volume that increases as initial tumor volume increases. There are two conclusions which follow from our simulations. First, the extent of hypoxia is likely a primary reason of the TCP reduction with increasing the initial tumor volume in SBRT for NSCLC. Second, the LQ model can be an acceptable approximation for the TCP calculations in hypofractionated radiotherapy if the tumor response is defined primarily by the hypoxic fraction. The larger value of OER in the hypofractionated radiotherapy compared to the conventional radiotherapy effectively extends the applicability of the LQ model to larger doses., (© 2021 Institute of Physics and Engineering 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.)

The main objective of this article is to improve the stability of reconstruction algorithms for estimation of radiobiological parameters using serial tumor imaging data acquired during radiation therapy. Serial images of tumor response to radiation therapy represent a complex summation of several exponential processes as treatment induced cell inactivation, tumor growth rates, and the rate of cell loss. Accurate assessment of treatment response would require separation of these processes because they define radiobiological determinants of treatment response and, correspondingly, tumor control probability. However, the estimation of radiobiological parameters using imaging data can be considered an inverse ill-posed problem because a sum of several exponentials would produce the Fredholm integral equation of the first kind which is ill posed. Therefore, the stability of reconstruction of radiobiological parameters presents a problem even for the simplest models of tumor response. To study stability of the parameter reconstruction problem, we used a set of serial CT imaging data for head and neck cancer and a simplest case of a two-level cell population model of tumor response. Inverse reconstruction was performed using a simulated annealing algorithm to minimize a least squared objective function. Results show that the reconstructed values of cell surviving fractions and cell doubling time exhibit significant nonphysical fluctuations if no stabilization algorithms are applied. However, after applying a stabilization algorithm based on variational regularization, the reconstruction produces statistical distributions for survival fractions and doubling time that are comparable to published in vitro data. This algorithm is an advance over our previous work where only cell surviving fractions were reconstructed. We conclude that variational regularization allows for an increase in the number of free parameters in our model which enables development of more-advanced parameter reconstruction algorithms.

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.

Purpose: Cone-beam computed tomography (CBCT) is a new image-guided radiation therapy (IGRT) technique for patient alignment in radiotherapy. The CBCT x-ray volume imaging system from Elekta allows for a variety of alignment methods. The aim of this study is to assess the accuracy of soft-tissue-based automatic alignment as compared with manual alignment using intraprostatic fiducials., Methods and Materials: All patients were treated on an Elekta Synergy S linear accelerator with kilovoltage CBCT. All alignments were performed using the x-ray volume imaging system and associated software. Automatic alignment with gray-value-based registration and manual alignment to fiducial markers were performed. Transitional corrections along each axis as well as 3-dimensional vectors were compared with evaluate the accuracy of gray-value-based registration compared with fiducials., Results: The distribution of the 3-dimensional vectors between gray-value and fiducial registrations demonstrated notable differences. The mean summed vector was 0.75 cm, with a standard deviation (SD) of 0.52 cm and range from 0.04 to 2.06 cm. There was minimal difference along the lateral direction, with a mean ± SD of -0.02 cm ± 0.13 cm. However, there were large discrepancies along the superior-inferior and anterior-posterior direction alignments, with mean ± SD values of -0.55 ± 0.48 cm and -0.31 ± 0.43 cm, respectively., Conclusions: CBCT with soft-tissue-based automatic corrections is not an accurate alignment compared with manual alignment to fiducial markers for prostate IGRT. We have concluded that a daily manual alignment to fiducials is one of the most reliable methods to maintain accuracy in prostate IGRT.

The purpose of this note is to evaluate the relationship between the stochastic errors in CT numbers and the standard deviation of the computed proton beam range in radiotherapy planning. The stochastic voxel-to-voxel variation in CT numbers called 'noise,' may be due to signal registration, processing and numerical image reconstruction technique. Noise in CT images may cause a deviation in the computed proton range from the physical proton range, even assuming that the error due to CT number-stopping power calibration is removed. To obtain the probability density function (PDF) of the computed proton range, we have used the continuing slowing down approximation (CSDA) and the uncorrelated white Gaussian noise along the proton path. The model of white noise was accepted because for the slice-based fan-beam CT scanner; the power-spectrum properties apply only to the axial (x, y) domain and the noise is uncorrelated in the z domain. However, the possible influence of the noise power spectrum on the standard deviation of the range should be investigated in the future. A random number generator was utilized for noise simulation and this procedure was iteratively repeated to obtain convergence of range PDF, which approached a Gaussian distribution. We showed that the standard deviation of the range, sigma, increases linearly with the initial proton energy, computational grid size and standard deviation of the voxel values. The 95% confidence interval width of the range PDF, which is defined as 4sigma, may reach 0.6 cm for the initial proton energy of 200 MeV, computational grid 0.25 cm and 5% standard deviation of CT voxel values. Our results show that the range uncertainty due to random errors in CT numbers may be significant and comparable to the uncertainties due to calibration of CT numbers.

Purpose: To develop a fast computational radiobiologic model for quantitative analysis of tumor volume during fractionated radiotherapy. The tumor-volume model can be useful for optimizing image-guidance protocols and four-dimensional treatment simulations in proton therapy that is highly sensitive to physiologic changes., Methods: The analysis is performed using two approximations: (1) tumor volume is a linear function of total cell number and (2) tumor-cell population is separated into four subpopulations: oxygenated viable cells, oxygenated lethally damaged cells, hypoxic viable cells, and hypoxic lethally damaged cells. An exponential decay model is used for disintegration and removal of oxygenated lethally damaged cells from the tumor., Results: We tested our model on daily volumetric imaging data available for 14 head-and-neck cancer patients treated with an integrated computed tomography/linear accelerator system. A simulation based on the averaged values of radiobiologic parameters was able to describe eight cases during the entire treatment and four cases partially (50% of treatment time) with a maximum 20% error. The largest discrepancies between the model and clinical data were obtained for small tumors, which may be explained by larger errors in the manual tumor volume delineation procedure., Conclusions: Our results indicate that the change in gross tumor volume for head-and-neck cancer can be adequately described by a relatively simple radiobiologic model. In future research, we propose to study the variation of model parameters by fitting to clinical data for a cohort of patients with head-and-neck cancer and other tumors. The potential impact of other processes, like concurrent chemotherapy, on tumor volume should be evaluated.

We study the time-dependent disintegration kinetics of tumor cells that did not survive radiotherapy treatment. To evaluate the cell disintegration rate after irradiation, we studied the volume changes of solitary lung tumors after stereotactic radiotherapy. The analysis is performed using two approximations: (1) tumor volume is a linear function of the total cell number in the tumor and (2) the cell disintegration rate is governed by the exponential decay with constant risk, which is defined by the initial cell number and a half-life T(1/2). The half-life T(1/2) is determined using the least-squares fit to the clinical data on lung tumor size variation with time after stereotactic radiotherapy. We show that the tumor volume variation after stereotactic radiotherapy of solitary lung tumors can be approximated by an exponential function. A small constant component in the volume variation does not change with time; however, this component may be the residual irregular density due to radiation fibrosis and was, therefore, subtracted from the total volume variation in our computations. Using computerized fitting of the exponent function to the clinical data for selected patients, we have determined that the average half-life T(1/2) of cell disintegration is 28.2 days for squamous cell carcinoma and 72.4 days for adenocarcinoma. This model is needed for simulating the tumor volume variation during radiotherapy, which may be important for time-dependent treatment planning of proton therapy that is sensitive to density variations.

Optimization of equivalent uniform dose (EUD) in inverse planning for intensity-modulated radiation therapy (IMRT) prevents variation in radiobiological effect between different radiotherapy treatment plans, which is due to variation in the pattern of dose nonuniformity. For instance, the survival fraction of clonogens would be consistent with the prescription when the optimized EUD is equal to the prescribed EUD. One of the problems in the practical implementation of this approach is that the spatial dose distribution in EUD-based inverse planning would be underdetermined because an unlimited number of nonuniform dose distributions can be computed for a prescribed value of EUD. Together with ill-posedness of the underlying integral equation, this may significantly increase the dose nonuniformity. To optimize EUD and keep dose nonuniformity within reasonable limits, we implemented into an EUD-based objective function an additional criterion which ensures the smoothness of beam intensity functions. This approach is similar to the variational regularization technique which was previously studied for the dose-based least-squares optimization. We show that the variational regularization together with the L-curve criterion for the regularization parameter can significantly reduce dose nonuniformity in EUD-based inverse planning.

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.

The 'monoenergetic' electron loss model was derived in a previous work to account for pathlength straggling in the Fermi-Eyges pencil beam problem. In this paper, we extend this model to account for energy-loss straggling and secondary knock-on electron transport in order to adequately predict a depth dose curve. To model energy-loss straggling, we use a weighted superposition of a discrete number of monoenergetic pencil beams with different initial energies where electrons travel along the depth-energy characteristics in the continuous slowing down approximation (CSDA). The energy straggling spectrum at depth determines the weighting assigned to each monoenergetic pencil beam. Supplemented by a simple transport model for the secondary knock-on electrons, the 'energy-dependent' electron loss model predicts both lateral and depth dose distributions from the electron pencil beams in good agreement with Monte Carlo calculations and measurements. The calculation of dose distribution from a pencil beam takes 0.2 s on a Pentium III 500 MHz computer. Being computationally fast, the 'energy-dependent' electron loss model can be used for the calculation of 3D energy deposition kernels in dose optimization schemes without using precalculated or measured data.

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.