Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.Comment: 76 pages, 8 figures, final version, one section on open problems added, and references expanded. Appears in "Advances in Noncommutative Geometry - on the occasion of Alain Connes' 70th birthday

Characterization of active and infiltrative tumorous subregions from normal tissue in brain gliomas using multiparametric MRI

Background: Targeted localized biopsies and treatments for diffuse gliomas rely on accurate identification of tissue subregions, for which current MRI techniques lack specificity. Purpose: To explore the complementary and competitive roles of a variety of conventional and quantitative MRI methods for distinguishing subregions of brain gliomas. Study Type: Prospective. Population: Fifty‐one tissue specimens were collected using image‐guided localized biopsy surgery from 10 patients with newly diagnosed gliomas. Field Strength/Sequence: Conventional and quantitative MR images consisting of pre‐ and postcontrast T1w, T2w, T2‐FLAIR, T2‐relaxometry, DWI, DTI, IVIM, and DSC‐MRI were acquired preoperatively at 3T. Assessment: Biopsy specimens were histopathologically attributed to glioma tissue subregion categories of active tumor (AT), infiltrative edema (IE), and normal tissue (NT) subregions. For each tissue sample, a feature vector comprising 15 MRI‐based parameters was derived from preoperative images and assessed by a machine learning algorithm to determine the best multiparametric feature combination for characterizing the tissue subregions. Statistical Tests: For discrimination of AT, IE, and NT subregions, a one‐way analysis of variance (ANOVA) test and for pairwise tissue subregion differentiation, Tukey honest significant difference, and Games‐Howell tests were applied (P < 0.05). Cross‐validated feature selection and classification methods were implemented for identification of accurate multiparametric MRI parameter combination. Results: After exclusion of 17 tissue specimens, 34 samples (AT = 6, IE = 20, and NT = 8) were considered for analysis. Highest accuracies and statistically significant differences for discrimination of IE from NT and AT from NT were observed for diffusion‐based parameters (AUCs >90%), and the perfusion‐derived parameter as the most accurate feature in distinguishing IE from AT. A combination of “CBV, MD, T2_ISO, FLAIR” parameters showed high diagnostic performance for identification of the three subregions (AUC ∼90%). Data Conclusion: Integration of a few quantitative along with conventional MRI parameters may provide a potential multiparametric imaging biomarker for predicting the histopathologically proven glioma tissue subregions