Finitary Abstract Elementary Classes

The research in model theory has extended from the study of elementary classes to non-elementary classes, i.e. to classes which are not completely axiomatizable in elementary logic. The main theme has been the attempt to generalize tools from elementary stability theory to cover more applications arising in other branches of mathematics. In this doctoral thesis we introduce finitary abstract elementary classes, a non-elementary framework of model theory. These classes are a special case of abstract elementary classes (AEC), introduced by Saharon Shelah in the 1980's. We have collected a set of properties for classes of structures, which enable us to develop a 'geometric' approach to stability theory, including an independence calculus, in a very general framework. The thesis studies AEC's with amalgamation, joint embedding, arbitrarily large models, countable Löwenheim-Skolem number and finite character. The novel idea is the property of finite character, which enables the use of a notion of a weak type instead of the usual Galois type. Notions of simplicity, superstability, Lascar strong type, primary model and U-rank are inroduced for finitary classes. A categoricity transfer result is proved for simple, tame finitary classes: categoricity in any uncountable cardinal transfers upwards and to all cardinals above the Hanf number. Unlike the previous categoricity transfer results of equal generality the theorem does not assume the categoricity cardinal being a successor. The thesis consists of three independent papers. All three papers are joint work with Tapani Hyttinen.

Accuracy of absorbed dose in external photon beam radiotherapy : what level is sufficient and how to approach it?

Radiation therapy (RT) plays currently significant role in curative treatments of several cancers. External beam RT is carried out mostly by using megavoltage beams of linear accelerators. Tumor eradication and normal tissue complications correlate to dose absorbed in tissues. Normally this dependence is steep and it is crucial that actual dose within patient accurately correspond to the planned dose. All factors in a RT procedure contain uncertainties requiring strict quality assurance. From hospital physicist´s point of a view, technical quality control (QC), dose calculations and methods for verification of correct treatment location are the most important subjects. Most important factor in technical QC is the verification that radiation production of an accelerator, called output, is within narrow acceptable limits. The output measurements are carried out according to a locally chosen dosimetric QC program defining measurement time interval and action levels. Dose calculation algorithms need to be configured for the accelerators by using measured beam data. The uncertainty of such data sets limits for best achievable calculation accuracy. All these dosimetric measurements require good experience, are workful, take up resources needed for treatments and are prone to several random and systematic sources of errors. Appropriate verification of treatment location is more important in intensity modulated radiation therapy (IMRT) than in conventional RT. This is due to steep dose gradients produced within or close to healthy tissues locating only a few millimetres from the targeted volume. The thesis was concentrated in investigation of the quality of dosimetric measurements, the efficacy of dosimetric QC programs, the verification of measured beam data and the effect of positional errors on the dose received by the major salivary glands in head and neck IMRT. A method was developed for the estimation of the effect of the use of different dosimetric QC programs on the overall uncertainty of dose. Data were provided to facilitate the choice of a sufficient QC program. The method takes into account local output stability and reproducibility of the dosimetric QC measurements. A method based on the model fitting of the results of the QC measurements was proposed for the estimation of both of these factors. The reduction of random measurement errors and optimization of QC procedure were also investigated. A method and suggestions were presented for these purposes. The accuracy of beam data was evaluated in Finnish RT centres. Sufficient accuracy level was estimated for the beam data. A method based on the use of reference beam data was developed for the QC of beam data. Dosimetric and geometric accuracy requirements were evaluated for head and neck IMRT when function of the major salivary glands is intended to be spared. These criteria are based on the dose response obtained for the glands. Random measurement errors could be reduced enabling lowering of action levels and prolongation of measurement time interval from 1 month to even 6 months simultaneously maintaining dose accuracy. The combined effect of the proposed methods, suggestions and criteria was found to facilitate the avoidance of maximal dose errors of up to even about 8 %. In addition, their use may make the strictest recommended overall dose accuracy level of 3 % (1SD) achievable.