Equivalent uniform aerobic dose in radiotherapy for hypoxic tumors

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 the18F-Fluoromisonidazole PET (18F-FMISO PET) images of hypoxia in patients with glioblastoma, the most common and aggressive type of primary malignant brain tumor. The18F-FMISO PET images include a distribution of SUV (Standardized Uptake Value); therefore, the SUV is converted to partial oxygen pressure (pO2) 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 the18F-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.

Equivalent uniform RBE-weighted dose in eye plaque brachytherapy

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 (EUDRBE) that can be used for quantitative characterization of integrated cell survival in radiotherapy modalities with the variable RBE, dose nonuniformity and dose rate. The EUDRBE is applied to brachytherapy with 125I eye plaques designed by the Collaborative Ocular Melanoma Study (COMS).

METHODS

The EUDRBE 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 EUDRBE 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 EUDRBE is applied to brachytherapy with 125I 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 EUDRBE 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 EUDRBE 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 EUDRBE 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 EUDRBE 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 EUDRBE 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 EUDRBE are valid.

CONCLUSIONS

The computed range of the EUDRBE in 125I COMS eye plaque brachytherapy suggests that the selected models and hypotheses are acceptable. The EUDRBE can be useful for analysis of treatment outcomes and comparison of different dose regimens in eye plaque brachytherapy.

Tumor control probability in hypofractionated radiotherapy as a function of total and hypoxic tumor volumes

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.

Theoretical effectiveness of cell survival in fractionated radiotherapy with hypoxia-targeted dose escalation

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.

Assessment of interpatient heterogeneity in tumor radiosensitivity for nonsmall cell lung cancer using tumor-volume variation data

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.

The influence of CT image noise on proton range calculation in radiotherapy planning

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.

Tumor-volume simulation during radiotherapy for head-and-neck cancer using a four-level cell population model

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.

Time-dependent cell disintegration kinetics in lung tumors after irradiation

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 using the L-curve criterion

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.

Reconstruction of electron spectra from depth doses with adaptive regularization

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.

L-curve analysis of radiotherapy optimization problems

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.

Regularization of inverse planning for intensity-modulated radiotherapy

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.

Angular correction in reconstruction of electron spectra from depth dose distributions

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 using singular component decomposition

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.

Intensity and energy modulated radiotherapy with proton beams: variables affecting optimal prostate plan

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 energy-dependent electron loss model for pencil beam dose kernels

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.

Proton loss model for therapeutic beam dose calculations

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.