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In this paper we present a new method of solving the inverse radiation treatment planning problem. The method is based on a Maximum Likelihood Estimator with dynamically changing penalization terms. The resulting Dynamically Penalized Likelihood (DPL) algorithm achieves a dose distribution of excellent uniformity in a tumor volume and a much lower dose in regions containing sensitive volumes. A simple model of a patient and of energy deposition has been used for the initial results presented: a two-dimensional computer generated phantom and monochromatic x rays, without scattering. Three two-dimensional problems are solved with the DPL algorithm, corresponding to different size and spatial relationships between the tumor and sensitive tissue volumes. The results show that the DPL algorithm is robust and flexible; it only requires moderate computation times and leads to promising solutions, even in rather difficult problems. The results encourage the extension of the present work to more realistic therapy situations.

This paper reports on the analysis of intensity modulated radiation treatment optimization problems in the presence of non-convex feasible parameter spaces caused by the specification of dose-volume constraints for the organs-at-risk (OARs). The main aim was to determine whether the presence of those non-convex spaces affects the optimization of clinical cases in any significant way. This was done in two phases: (1) Using a carefully designed two-dimensional mathematical phantom that exhibits two controllable minima and with randomly initialized beamlet weights, we developed a methodology for exploring the nature of the convergence characteristics of quadratic cost function optimizations (deterministic or stochastic). The methodology is based on observing the statistical behaviour of the residual cost at the end of optimizations in which the stopping criterion is progressively more demanding and carrying out those optimizations to very small error changes per iteration. (2) Seven clinical cases were then analysed with dose-volume constraints that are stronger than originally used in the clinic. The clinical cases are two prostate cases differently posed, a meningioma case, two head-and-neck cases, a spleen case and a spine case. Of the 14 different sets of optimizations (with and without the specification of maximum doses allowed for the OARs), 12 fail to show any effect due to the existence of non-convex feasible spaces. The remaining two sets of optimizations show evidence of multiple minima in the solutions, but those minima are very close to each other in cost and the resulting treatment plans are practically identical, as measured by the quality of the dose-volume histograms (DVHs). We discuss the differences between fluence maps resulting from those similar treatment plans. We provide a possible reason for the observed results and conclude that, although the study is necessarily limited, the annealing characteristics of a simulated annealing method may not be justified in clinical optimization in the presence of dose-volume constraints. The results of optimizations by the Newton gradient (NG) method with a quadratic cost function are reported in detail. An adaptive simulated annealing method, optimizing the same function, and the dynamically penalized likelihood method, optimizing a log likelihood function, have also been used in the study. The results of the latter two methods have only been discussed briefly, as they yielded the same conclusions as the NG method.

Abstract-In this paper we continue the discussion of the causes for image deterioration in the Maximum Likelihood Estimator (MLE) method of tomographic image reconstruction that we initiated with the publication of a stopping rule for that iterative process. We introduce the concept of a feasible image, which is a result of a reconstruction that, if it were a radiation field, could have generated the initial projection data by the Poisson process that governs radioactive decay. From the premise that the result of a reconstruction should-be feasible, we examine the shape and characteristics of the region of feasibility in projection space. Although MLE reconstructions from computer simulated data pass through a feasibility region when started from a uniform intensity image field, as determined by our previously published stopping rule, attempts at using that rule to detect feasibility in reconstructions with real PET data failed-con;.sistently. We examine the reasons for that failure and design a more relaxed stopping rule that takes into account the fact that the prob-1 ability matrix defining a true tomographic instrument can only be known within some error margin. With the new rule, reconstructions from real data can be tested for feasibility. Results of the test and reconstructed images for the Hoffman brain phantom are shown.We conclude with a comparative examination of the current methods of dealing with MLE image deterioration and we endeavor to put the minds of current workers in the field at ease about having to stop MLE iterations when the images look acceptable.

Abstract-The work presented in this paper evaluates the statistical characteristics of regional bias and expected error in reconstructions of real PET data of human brain fluorodeoxiglucose (FDG) studies carried out by the maximum likelihood estimator (MLE) method with a robust stopping rule, and compares them with the results of filtered backprojection (FBP) reconstructions and with the method of sieves.The task that we have investigated is that of quantifying radioisotope uptake in regions-of-interest (ROI's). We first describe a robust methodology for the use of the MLE method with clinical data which contains only one adjustable parameter: the kernel size for a Gaussian filtering operation that determines h a l resolution and expected regional error. Simulation results are used to establish the fundamental characteristics of the reconstructions obtained by our methodology, corresponding to the case in which the transition matrix is perfectly known. Then, data from 72 independent human brain FDG scans from four patients are used to show that the results obtained from real data are consistent with the simulation, although the quality of the data and of the transition matrix have an effect on the final outcome. The most important results are that, for equal resolution, expected pixel-by-pixel error in the MLE and sieves reconstructions are lower in the regions of low counts than in the regions of high counts, the lowest being for the MLE. In contrast, FBP reconstructions show an expected error that is high and nearly independent of the number of counts in a region. As a consequence, the determination of radioisotope uptake in ROI's of high activity has approximately the same standard deviation in MLE, sieves, and FBP reconstructions, while the standard deviation in ROI's of low uptake is substantially lower for MLE, while sieves take an intermediate value. The use of a well-constructed Monte Carlo transition matrix improves all the results with real data in a measurable way. We conclude that our proposed MLE methodology and the method of sieves have a definite advantage over FBP. There is a tradeoff between shorter computation time, a slight bias but lower standard deviation for MLE and longer computation time, a basically unbiased estimation but higher standard deviation for sieves.

This paper presents a description of tests carried out to compare the behaviour of five algorithms in inverse radiation therapy planning: (1) The Dynamically Penalized Likelihood (DPL), an algorithm based on statistical estimation theory; (2) an accelerated version of the same algorithm: (3) a new fast adaptive simulated annealing (ASA) algorithm; (4) a conjugate gradient method; and (5) a Newton gradient method. A three-dimensional mathematical phantom and two clinical cases have been studied in detail. The phantom consisted of a U-shaped tumour with a partially enclosed 'spinal cord'. The clinical examples were a cavernous sinus meningioma and a prostate case. The algorithms have been tested in carefully selected and controlled conditions so as to ensure fairness in the assessment of results. It has been found that all five methods can yield relatively similar optimizations, except when a very demanding optimization is carried out. For the easier cases. the differences are principally in robustness, ease of use and optimization speed. In the more demanding case, there are significant differences in the resulting dose distributions. The accelerated DPL emerges as possibly the algorithm of choice for clinical practice. An appendix describes the differences in behaviour between the new ASA method and the one based on a patent by the Nomos Corporation.

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The interaction between conduction electrons with energies from 0.25 to 7.5 eV and longitudinal optical phonons in alkali halides is studied in detail by time-dependent perturbation theory. Expressions for the rate and angular distribution of scattering are obtained. The electron-transport problem is then solved with the exact quantum mechanical scattering results by a direct simulation Monte Carlo method. Probabilities of escape and average energy losses for electrons generated isotropically at a certain depth in the material, with a given initial energy, are computed for CsI, KCl, NaF, and LiF. A simple theory shows the effective mass and temperature dependence. The effect of including scattering to angles other than forward is quite apparent in the results.