Pre-implant magnetic resonance and transrectal ultrasound imaging in high-dose-rate prostate brachytherapy : comparison of prostate volumes, craniocaudal extents, and contours

Purpose: The purpose of this study was to compare the prostate contours drawn by two radiation oncologists and one radiologist on magnetic resonance (MR) and transrectal ultrasound (TRUS) images. TRUS intra- and inter-fraction variability as well as TRUS vs. MR inter-modality and inter-operator variability were studied. Material and methods: Thirty patients affected by localized prostate cancer and treated with interstitial high-dose-rate (HDR) prostate brachytherapy at the National Cancer Institute in Milan were included in this study. Twenty-five patients received an exclusive two-fraction (14 Gy/fraction) treatment, while the other 5 received a single 14 Gy fraction as a boost after external beam radiotherapy. The prostate was contoured on TRUS images acquired before (virtual US) and after (real US) needle implant by two radiation oncologists, whereas on MR prostate was independently contoured by the same radiation oncologists (MR1, MR2) and by a dedicated radiologist (MR3). Absolute differences of prostate volumes ( 06V) and craniocaudal extents ( 06dz) were evaluated. The Dice's coefficient (DC) was calculated to quantify spatial overlap between MR contours. Results: Significant difference was found between Vvirtual and Vlive (p < 0.001) for the first treatment fractions and between VMR1 and VMR2 (p = 0.043). Significant difference between cranio-caudal extents was found between dzvirtual and dzlive (p < 0.033) for the first treatment fractions, between dzvirtual of the first treatment fractions and dzMR1 (p < 0.001) and between dzMR1 and dzMR3 (p < 0.01). Oedema might be responsible for some of the changes in US volumes. Average DC values resulting from the comparison MR1 vs. MR2, MR1 vs. MR3 and MR2 vs. MR3 were 0.95 \ub1 0.04 (range, 0.82-0.99), 0.87 \ub1 0.04 (range, 0.73-0.91) and 0.87 \ub1 0.04 (range, 0.72-0.91), respectively. Conclusions: Our results demonstrate the importance of a multiprofessional approach to TRUS-guided HDR prostate brachytherapy. Specific training in MR and US prostate imaging is recommended for centers that are unfamiliar with HDR prostate brachytherapy

On the spectral Adam property for circulant graphs

We investigate a certain condition for isomorphism between circulant graphs (known as the Ádám property) in a stronger form by referring to isospectrality rather than to isomorphism of graphs. We describe a wide class of graphs for which the Ádám conjecture holds. We apply these results to establish an asymptotic formula for the number of non-isomorphic circulant graphs and connected circulant graphs. Circulant graphs arise in many applications including telecommunication networks, VLSI design and distributed computation and have been extensively studied in the literature. In the important case of double loops (particular circulant graphs of degree 4) we give a complete classification of all possible isospectral graphs. Our method is based on studying the graph spectra with the aid of some deep results of algebraic number theory.21 page(s

Coincidences in the values of the Euler and Carmichael functions

http://www.math.missouri.edu/~bbanks/papers/index.htmlThe Euler function has long been regarded as one of the most basic of the arithmetic functions. More recently, partly driven by the rise in importance of computational number theory, the Carmichael function has drawn an ever-increasing amount of attention. A large number of results have been obtained, both about the growth rate and about various arithmetical properties of the values of these two functions; see for example [2, 3, 5-7, 10-18, 20, 22, 23] and the references therein

Contributions to zero-sum problems

A prototype of zero-sum theorems, the well-known theorem of Erdős, Ginzburg and Ziv says that for any positive integer n, any sequence a1,a2,…,a2n-1 of 2n-1 integers has a subsequence of n elements whose sum is 0 modulo n. Appropriate generalizations of the question, especially that for (Z/pZ)d, generated a lot of research and still have challenging open questions. Here we propose a new generalization of the Erdős–Ginzburg–Ziv theorem and prove it in some basic cases

Coincidences in the values of the Euler and Carmichael functions

The well-known Carmichael conjecture concerns the set of positive integers (presumed empty) which occur as a value of the Euler function at one positive integer but at no others. We study the analogous problem for the Carmichael function. We also study the (far more numerous) sets of integers which occur in the image of one of these functions but not of the other, as well as those which occur in the image of both