Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

Removal of Limited Nodal Disease in Patients Undergoing Radical Prostatectomy: Long-Term Results Confirm a Chance for Cure

Abstract PURPOSE: In 2003 we reported on the outcomes of 88 patients with node positive disease who underwent radical prostatectomy and pelvic lymph node dissection (median 21 nodes) between 1989 and 1999. Patients with limited nodal disease appeared to have a good chance of long-term survival, even without immediate adjuvant therapy (androgen deprivation therapy and/or radiotherapy). In this study we update the followup in these patients and verify the reported projected probability of survival. MATERIALS AND METHODS: The projected 10-year cancer specific survival probability after the initially reported followup of 3.2 years was 60% for these patients with node positive disease. The outcome has been updated after a median followup of 15.6 years. RESULTS: Of the 39 patients with 1 positive node 7 (18%) remained biochemically relapse-free, 11 (28%) showed biochemical relapse only and 21 (54%) experienced clinical progression. Of these 39 patients 22 (57%) never required deferred androgen deprivation therapy and 12 (31%) died of prostate cancer. All patients with 2 (20) or more than 2 (29) positive nodes experienced biochemical relapse and only 5 (10%) of these 49 experienced no clinical progression. Of these 49 patients 39 (80%) received deferred androgen deprivation therapy. CONCLUSIONS: Biochemical relapse is likely in patients with limited nodal disease after radical prostatectomy and pelvic lymph node dissection, but for 47% of patients this does not imply death from prostate cancer. Patients with 1 positive node have a good (75%) 10-year cancer specific survival probability and a 20% chance of remaining biochemical relapse-free even without immediate adjuvant therapy.

Differentiating atrioventricular nodal reentrant tachycardia from tachycardia via concealed accessory pathway.

Studies analyzing the diagnostic value of 12-lead electrocardiographic criteria differentiating slow-fast atrioventricular nodal reentrant tachycardia (AVNRT) from atrioventricular reentrant tachycardia (AVRT) due to concealed accessory pathway have shown inconsistent results. In 97 patients (50 with AVNRT, 47 with AVRT) 12-lead electrocardiograms (ECGs) were recorded during sinus rhythm and tachycardia (QRS <120 ms). The ECGs were blinded for diagnosis and patient and analyzed independently by 2 electrophysiologists. The studied criteria differentiating AVNRT from AVRT included pseudo-r'/S, the presence of a retrograde P wave, RP interval, ST-segment depression >/=2 mm with the number and location of the affected leads, QRS amplitude, and cycle length alternans.

Left-septal ablation of the fast pathway in AV nodal reentrant tachycardia refractory to right septal ablation.

In more than 95% of patients with atrioventricular nodal reentrant tachycardia (AVNRT), curative treatment can be achieved with selective ablation of the slow pathway in the right-sided septum. We report a patient with typical AVNRT who had failed attempts to perform conventional right septal ablation of the slow as well as of the fast pathway and finally underwent successful ablation of the fast pathway on the left side of the interatrial septum using a transseptal approach.

Loss of TRAIL-receptors is a recurrent feature in pancreatic cancer and determines the prognosis of patients with no nodal metastasis after surgery

INTRODUCTION Agonistic antibodies targeting TRAIL-receptors 1 and 2 (TRAIL-R1 and TRAIL-R2) are being developed as a novel therapeutic approach in cancer therapy including pancreatic cancer. However, the cellular distribution of these receptors in primary pancreatic cancer samples has not been sufficiently investigated and no study has yet addressed the issue of their prognostic significance in this tumor entity. AIMS AND METHODS Applying tissue microarray (TMA) analysis, we performed an immunohistochemical assessment of TRAIL-receptors in surgical samples from 84 consecutive patients affected by pancreatic adenocarcinoma and in 26 additional selected specimens from patients with no lymph nodes metastasis at the time of surgery. The prognostic significance of membrane staining and staining intensity for TRAIL-receptors was evaluated. RESULTS The fraction of pancreatic cancer samples with positive membrane staining for TRAIL-R1 and TRAIL-R2 was lower than that of cells from surrounding non-tumor tissues (TRAIL-R1: p<0.001, TRAIL-R2: p = 0.006). In addition, subgroup analyses showed that loss of membrane staining for TRAIL-R2 was associated with poorer prognosis in patients without nodal metastases (multivariate Cox regression analysis, Hazard Ratio: 0.44 [95% confidence interval: 0.22-0.87]; p = 0.019). In contrast, analysis of decoy receptors TRAIL-R3 and -R4 in tumor samples showed an exclusively cytoplasmatic staining pattern and no prognostic relevance. CONCLUSION This is a first report on the prognostic significance of TRAIL-receptors expression in pancreatic cancer showing that TRAIL-R2 might represent a prognostic marker for patients with early stage disease. In addition, our data suggest that loss of membrane-bound TRAIL-receptors could represent a molecular mechanism for therapeutic failure upon administration of TRAIL-receptors-targeting antibodies in pancreatic cancer. This hypothesis should be evaluated in future clinical trials.

Characterization of granite matrix porosity and pore-space geometry by in situ and laboratory methods

Most available studies of interconnected matrix porosity of crystalline rocks are based on laboratory investigations; that is, work on samples that have undergone stress relaxation and were affected by drilling and sample preparation. The extrapolation of the results to in situ conditions is therefore associated with considerable uncertainty, and this was the motivation to conduct the ‘in situ Connected Porosity’ experiment at the Grimsel Test Site (Central Swiss Alps). An acrylic resin doped with fluorescent agents was used to impregnate the microporous granitic matrix in situ around an injection borehole, and samples were obtained by overcoring. The 3-D structure of the porespace, represented by microcracks, was studied by U-stage fluorescence microscopy. Petrophysical methods, including the determination of porosity, permeability and P -wave velocity, were also applied. Investigations were conducted both on samples that were impregnated in situ and on non-impregnated samples, so that natural features could be distinguished from artefacts. The investigated deformed granites display complex microcrack populations representing a polyphase deformation at varying conditions. The crack population is dominated by open cleavage cracks in mica and grain boundary cracks. The porosity of non-impregnated samples lies slightly above 1 per cent, which is 2–2.5 times higher than the in situ porosity obtained for impregnated samples. Measurements of seismic velocities (Vp ) on spherical rock samples as a function of confining pressure, spatial direction and water saturation for both non-impregnated and impregnated samples provide further constraints on the distinction between natural and induced crack types. The main conclusions are that (1) an interconnected network of microcracks exists in the whole granitic matrix, irrespective of the distance to ductile and brittle shear zones, and (2) conventional laboratory methods overestimate the matrix porosity. Calculations of contaminant transport through fractured media often rely on matrix diffusion as a retardation mechanism.

CD8/CD45RO T-cell infiltration in endoscopic biopsies of colorectal cancer predicts nodal metastasis and survival

BACKGROUND AND AIMS: Reliable prognostic markers based on biopsy specimens of colorectal cancer (CRC) are currently missing. We hypothesize that assessment of T-cell infiltration in biopsies of CRC may predict patient survival and TNM-stage before surgery. METHODS: Pre-operative biopsies and matched resection specimens from 130 CRC patients treated from 2002-2011 were included in this study. Whole tissue sections of biopsy material and primary tumors were immunostained for pancytokeratin and CD8 or CD45RO. Stromal (s) and intraepithelial (i) T-cell infiltrates were analyzed for prediction of patient survival as well as clinical and pathological TNM-stage of the primary tumor. RESULTS: CD8 T-cell infiltration in the preoperative biopsy was significantly associated with favorable overall survival (CD8i p = 0.0026; CD8s p = 0.0053) in patients with primary CRC independently of TNM-stage and postoperative therapy (HR [CD8i] = 0.55 (95% CI: 0.36-0.82), p = 0.0038; HR [CD8s] = 0.72 (95% CI: 0.57-0.9), p = 0.0049). High numbers of CD8i in the biopsy predicted earlier pT-stage (p < 0.0001) as well as absence of nodal metastasis (p = 0.0015), tumor deposits (p = 0.0117), lymphatic (p = 0.008) and venous invasion (p = 0.0433) in the primary tumor. Infiltration by CD45ROs cells was independently associated with longer survival (HR = 0.76 (95% CI: 0.61-0.96), p = 0.0231) and predicted absence of venous invasion (p = 0.0025). CD8 counts were positively correlated between biopsies and the primary tumor (r = 0.42; p < 0.0001) and were reproducible between observers (ICC [CD8i] = 0.95, ICC [CD8s] = 0.75). For CD45RO, reproducibility was poor to moderate (ICC [CD45i] = 0.16, ICC [CD45s] = 0.49) and correlation with immune infiltration in the primary tumor was fair and non-significant (r[CD45s] = 0.16; p = 0.2864). For both markers, no significant relationship was observed with radiographic T-stage, N-stage or M-stage, indicating that assessment of T-cells in biopsy material can add additional information to clinical staging in the pre-operative setting. CONCLUSIONS: T-cell infiltration in pre-operative biopsy specimens of CRC is an independent favorable prognostic factor and strongly correlates with absence of nodal metastasis in the resection specimen. Quantification of CD8i is highly reproducible and allows superior prediction of clinicopathological features as compared to CD45RO. The assessment of CD8i infiltration in biopsies is recommended for prospective investigation.

Mechanical constraints imposed by 3D cellular geometry and arrangement modulate growth patterns in the Arabidopsis embryo

Morphogenesis occurs in 3D space over time and is guided by coordinated gene expression programs. Here we use postembryonic development in Arabidopsis plants to investigate the genetic control of growth. We demonstrate that gene expression driving the production of the growth-stimulating hormone gibberellic acid and downstream growth factors is first induced within the radicle tip of the embryo. The center of cell expansion is, however, spatially displaced from the center of gene expression. Because the rapidly growing cells have very different geometry from that of those at the tip, we hypothesized that mechanical factors may contribute to this growth displacement. To this end we developed 3D finite-element method models of growing custom-designed digital embryos at cellular resolution. We used this framework to conceptualize how cell size, shape, and topology influence tissue growth and to explore the interplay of geometrical and genetic inputs into growth distribution. Our simulations showed that mechanical constraints are sufficient to explain the disconnect between the experimentally observed spatiotemporal patterns of gene expression and early postembryonic growth. The center of cell expansion is the position where genetic and mechanical facilitators of growth converge. We have thus uncovered a mechanism whereby 3D cellular geometry helps direct where genetically specified growth takes place.

IMRT with ¹⁸FDG-PET\CT based simultaneous integrated boost for treatment of nodal positive cervical cancer

BACKGROUND To evaluate toxicity and outcome of intensity modulated radiotherapy (IMRT) with simultaneous integrated boost (SIB) to the positive lymph nodes in patients with loco-regional advanced cervical cancer (LRACC). METHODS The study population comprised ten patients with 18FDG-PET\CT positive lymph nodes (LNs), who underwent chemoradiation with IMRT and SIB. A dose of 50.4 Gy, in daily fractions of 1.8 Gy, was delivered to primary tumor and draining LNs. Primary tumor received an additional external beam boost to a total dose of 55.8 Gy. A SIB of 62 Gy, in daily fractions of 2 Gy, was delivered to the 18FDG-PET\CT positive LNs. Finally, a high dose rate brachytherapy (HDRB) boost (15 - 18 Gy) was administered to the primary tumor. The primary goal of this study was to evaluate acute and early late toxicity and loco-regional control. RESULTS The median number of irradiated LNs per patient was 3 (range: 1-6) with a median middle nodal SIB-volume of 26.10 cm3 (range, 11.9-82.50 cm3). Median follow-up was 20 months (range, 12 to 30 months). Acute and late grade 3 toxicity was observed in 1 patient. Three of the patients developed a recurrence, one in the form of a local tumor relapse, one had a paraaortic LN metastasis outside the treated volume and the last one developed a distant metastasis. CONCLUSION IMRT with SIB in the region of 18FDG-PET positive lymph nodes appears to be an effective therapy with acceptable toxicity and might be useful in the treatment of patients with locally advanced cervical cancer.

How does the ascending aorta geometry change when it dissects?

OBJECTIVES The purpose of this study is to delineate changes in aortic geometry and diameter due to dissection. BACKGROUND Aortic diameter is the major criterion for elective ascending aortic replacement for dilated ascending aortas to prevent aortic dissection. However, recommendations are made on the basis of clinical experience and observation of diameters of previously dissected aortas. METHODS Six tertiary centers on 2 continents reviewed their acute aortic dissection type A databases, which contained 1,821 patients. Included were all non-Marfan patients with nonbicuspid aortic valves who had undergone computed tomography angiography <2 years before and within 12 h after aortic dissection onset. Aortic geometry before and after dissection onset were compared. RESULTS Altogether, 63 patients were included (27 spontaneous and 36 retrograde dissections, median age 68 [57; 77] years; 54% were men). In all but 1 patient, maximum ascending aortic diameter was <55 mm before aortic dissection onset. The largest increase in diameter and volume induced by the dissection were observed in the ascending aorta (40.1 [36.6; 45.3] mm vs. 52.9 [46.1; 58.6] mm, +12.8 mm; p < 0.001; 124.0 [90.8; 162.5] cm(3) vs. 171.0 [147.0; 197.0] cm(3), +47 cm(3); p < 0.001). Mean aortic arch diameter increased from 39.8 (30.5; 42.6) mm to 46.4 (42.0; 51.6) mm (+6.6 mm; p < 0.001) and descending thoracic aorta diameter from 31.2 (27.0; 33.3) mm to 34.9 (30.9; 39.5) mm (+3.7 mm; p < 0.001). Changes in thoracic aorta geometry were similar for spontaneous and retrograde etiology. CONCLUSIONS Geometry of the thoracic aorta is affected by aortic dissection, leading to an increase in diameter that is most pronounced in the ascending aorta. Both spontaneous and retrograde dissection result in similar aortic geometry changes.

Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

Geometry and kinematics of the Roisan-Cignana Shear Zone, and the orogenic evolution of the Dent Blanche Tectonic System (Western Alps)

The Dent Blanche Tectonic System (DBTS) is a composite thrust sheet derived from the previously thinned passive Adriatic continental margin. A kilometric high-strain zone, the Roisan-Cignana Shear Zone (RCSZ) defines the major tectonic boundary within the DBTS and separates it into two subunits, the Dent Blanche s.s. nappe to the northwest and the Mont Mary nappe to the southeast. Within this shear zone, tectonic slices of Mesozoic and pre-Alpine meta-sediments became amalgamated with continental basement rocks of the Adriatic margin. The occurrence of high pressure assemblages along the contact between these tectonic slices indicates that the amalgamation occurred prior to or during the subduction process, at an early stage of the Alpine orogenic cycle. Detailed mapping, petrographic and structural analysis show that the Roisan-Cignana Shear Zone results from several superimposed Alpine structural and metamorphic stages. Subduction of the continental fragments is recorded by blueschist-facies deformation, whereas the Alpine collision is reflected by a greenschist facies overprint associated with the development of large-scale open folds. The postnappe evolution comprises the development of low-angle brittle faults, followed by large-scale folding (Vanzone phase) and finally brittle extensional faults. The RCSZ shows that fragments of continental crust had been torn off the passive continental margin prior to continental collision, thus recording the entire history of the orogenic cycle. The role of preceding Permo-Triassic lithospheric thinning, Jurassic rifting, and ablative subduction processes in controlling the removal of crustal fragments from the reactivated passive continental margin is discussed. Results of this study constrain the temporal sequence of the tectono-metamorphic processes involved in the assembly of the DBTS, but they also show limits on the interpretation. In particular it remains difficult to judge to what extent precollisional rifting at the Adriatic continental margin preconditioned the efficiency of convergent processes, i.e. accretion, subduction, and orogenic exhumation.