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Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and contains a variety of theorems which provide sharp relationships between this bound and notions of {\em{width}}. Some inequalities leverage geometric quantities such as boundary mean curvature, while others involve topological conditions in the form of linking requirements or homological constraints. In several of these results open and incomplete manifolds are studied, one of which partially addresses a conjecture of Gromov in this setting. The majority of results are accompanied by rigidity statements which isolate various model geometries -- both complete and incomplete -- including a new characterization of round lens spaces, and other models that have not appeared elsewhere. As a byproduct, we additionally give new and quantitative proofs of several classical comparison statements such as Bonnet-Myers' and Frankel's Theorem, as well as a version of Llarull's Theorem and a notable fact concerning asymptotically flat manifolds. The results that we present vary significantly in character, however a common theme is present in that the lead role in each proof is played by \emph{spacetime harmonic functions}, which are solutions to a certain elliptic equation originally designed to study mass in mathematical general relativity., Minor technical improvements

The Theorem of Bonnet--Myers implies that manifolds with topology $M^{n-1} \times \mathbb{S}^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $\mathbb{T}^n$ does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so called $m$-intermediate curvature), and use stable weighted slicings to show that for $n \leq 7$ the manifolds $N^n = M^{n-m} \times \mathbb{T}^m$ do not admit a metric of positive $m$-intermediate curvature., All comments welcome!

The AneuX morphology database is an open-access, multi-centric database containing 3D geometries of 750 intracranial aneurysms curated in the context of the AneuX project (2015-2020). The database combines data from three different projects (AneuX, @neurIST and Aneurisk) standardized using a single processing pipeline.

Effective visualizations were evaluated to reveal relevant health patterns from multi-sensor real-time wearable devices that recorded vital signs from patients admitted to hospital with COVID-19. Furthermore, specific challenges associated with wearable health data visualizations, such as fluctuating data quality resulting from compliance problems, time needed to charge the device and technical problems are described. As a primary use case, we examined the detection and communication of relevant health patterns visible in the vital signs acquired by the technology. Customized heat maps and bar charts were used to specifically highlight medically relevant patterns in vital signs. A survey of two medical doctors, one clinical project manager and seven health data science researchers was conducted to evaluate the visualization methods. From a dataset of 84 hospitalized COVID-19 patients, we extracted one typical COVID-19 patient history and based on the visualizations showcased the health history of two noteworthy patients. The visualizations were shown to be effective, simple and intuitive in deducing the health status of patients. For clinical staff who are time-constrained and responsible for numerous patients, such visualization methods can be an effective tool to enable continuous acquisition and monitoring of patients' health statuses even remotely., 17 pages, 9 figures, conference

Effective visualizations were evaluated to reveal relevant health patterns from multi-sensor real-time wearable devices that recorded vital signs from patients admitted to hospital with COVID-19. Furthermore, specific challenges associated with wearable health data visualizations, such as fluctuating data quality resulting from compliance problems, time needed to charge the device and technical problems are described. As a primary use case, we examined the detection and communication of relevant health patterns visible in the vital signs acquired by the technology. Customized heat maps and bar charts were used to specifically highlight medically relevant patterns in vital signs. A survey of two medical doctors, one clinical project manager and seven health data science researchers was conducted to evaluate the visualization methods. From a dataset of 84 hospitalized COVID-19 patients, we extracted one typical COVID-19 patient history and based on the visualizations showcased the health history of two noteworthy patients. The visualizations were shown to be effective, simple and intuitive in deducing the health status of patients. For clinical staff who are time-constrained and responsible for numerous patients, such visualization methods can be an effective tool to enable continuous acquisition and monitoring of patients' health statuses even remotely.

Proceedings: https://www.compbiomed.net/2021/cmbe-proceedings.htm, The typical characteristics of intracranial aneurysms vary for different anatomic locations. Here, we study the location-dependent variability of aneurysm shape, and propose means to model and visualize this variability. We further elaborate to which extent the configuration of the cerebral vasculature could affect the outcome of an aneurysmal lesion.

Proceedings: https://www.compbiomed.net/2021/cmbe-proceedings.htm Intracranial aneurysms (IAs) are a dynamically evolving disease with severe or fatal consequences of IA rupture. Survival analysis methods were used to gain insights into the influence of routinely acquired clinical risk factors onto IA rupture. Patient data consisting of 788 records from the Geneva university hospitals was analyzed. For this censored data, Kaplan-Meier analysis together with Cox regression using proportional hazards were employed to determine the influence of the risk factors. Smoking males with large IAs in high-risk locations had the highest risk for early IA rupture. Patients who were aware of hypertension (irrespective of treatment), stopped smoking and with a positive family history had a reduced rupture risk.

Proceedings: https://www.compbiomed.net/2021/cmbe-proceedings.htm, Intracranial aneurysm (IA) shape is proposed to be a predicting factor of rupture. In this study, using 3D-angiographies, surgical and histological images, we ranked 11 IAs according to different characteristics (homogeneity, aspect and thickness), and correlations between the different ranking systems were investigated. We showed positive correlations between IA morphology (normalized total Gaussian curvature, GLN) and wall aspect ranking, and between GLN and histology ranking. Correlations between increased GLN, inhomogeneity of IA wall aspect and thickness were shown. This exploratory study supports the GLN in its ability to quantify IA shape and to be used as an IA wall feature predictor.

Proceedings: https://www.compbiomed.net/2021/cmbe-proceedings.htm Various intracranial aneurysm (IA) rupture risk factors are used for risk assessment, but little is understood about their complex interactions leading to IA rupture. In this study, Bayesian networks (BN) were learned on data of nine risk factors from 790 patients and compared to standard descriptive and regression analyses. The results from standard methods agreed with previous studies and could be extended with BNs by uncovering additional associations in the data. The graphical representation of the IA rupture risk factors in BNs facilitated the result understanding and provided new hypotheses on the risk factor interdependencies.

Intracranial aneurysms (IAs) are usually asymptomatic with a low risk of rupture, but consequences of aneurysmal subarachnoid hemorrhage (aSAH) are severe. Identifying IAs at risk of rupture has important clinical and socio-economic consequences. The goal of this study was to assess the effect of patient and IA characteristics on the likelihood of IA being diagnosed incidentally versus ruptured. Patients were recruited at 21 international centers. Seven phenotypic patient characteristics and three IA characteristics were recorded. The analyzed cohort included 7992 patients. Multivariate analysis demonstrated that: (1) IA location is the strongest factor associated with IA rupture status at diagnosis; (2) Risk factor awareness (hypertension, smoking) increases the likelihood of being diagnosed with unruptured IA; (3) Patients with ruptured IAs in high-risk locations tend to be older, and their IAs are smaller; (4) Smokers with ruptured IAs tend to be younger, and their IAs are larger; (5) Female patients with ruptured IAs tend to be older, and their IAs are smaller; (6) IA size and age at rupture correlate. The assessment of associations regarding patient and IA characteristics with IA rupture allows us to refine IA disease models and provide data to develop risk instruments for clinicians to support personalized decision-making. The @neurIST project was supported by the 6th framework program of the European Commission (FP6-IST-2004-027703). Geneva data collection was part of the AneuX project supported by the Swiss SystemsX.ch initiative (PB), evaluated by the Swiss National Science Foundation, and which also funded the SyBIT project (web applications for data exploration). YMR has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (PRYSM, grant agreement No. 852173). MKB and YMR were supported by the Netherlands Cardiovascular Research Initiative: An initiative with support of the Dutch Heart Foundation, CVON2015-08 ERASE. DW was supported by NIH Funding. RR was supported by the French Regional Council of Pays-de-la-Loire (VaCaRMe program) and the Agence Nationale de la Recherche (ANR-15-CE17-0008-01 to G.L). HD and RB were supported by the French Ministry of Health (Clinical trial NCT02848495 to HD), the Genavie Foundation, the Société Française de Radiologie and the Société Française de Neuroradiologie. JJC and ECG were supported in part by Spain’s Ministry of Health (Instituto de Salud Carlos III Fondo de Investigaciones sanitarias P19/00011 and by “RICORS-ICTUS RD21/0006/0021). GAR was supported by the Canadian Institutes of Health Research. MN was supported by the Helsinki University Central Hospital EVO grant TYH2018316. The GOSH study was funded by the Stroke Association. Funders were not involved in the study design, in the analysis of the data or in the interpretation of the results.

Proceedings: https://www.compbiomed.net/2021/cmbe-proceedings.htm, Although the shape of intracranial aneurysms and the geometry of the surrounding vasculature are commonly taken into account by clinicians when assessing and treating aneurysms, it remains dif- ficult to quantify shape and develop clinical guidelines or tools that accommodate aneurysm shape. Here, we present new evidence that aneurysm shape is a meaningful proxy for disease status, the re- sults of a benchmark analysis comparing novel and established measurement methods for their ability to discriminate between ruptured and unruptured aneurysms, and how these findings can be trans- lated into clinics. We conclude with a plea for multi-centric data collections and present our own contributions to it.

An author correction to this article published in December 2020 is available at https://doi.org/10.1038/s41588-020-00760-4. Rupture of an intracranial aneurysm leads to subarachnoid hemorrhage, a severe type of stroke. To discover new risk loci and the genetic architecture of intracranial aneurysms, we performed a cross-ancestry, genome-wide association study in 10,754 cases and 306,882 controls of European and East Asian ancestry. We discovered 17 risk loci, 11 of which are new. We reveal a polygenic architecture and explain over half of the disease heritability. We show a high genetic correlation between ruptured and unruptured intracranial aneurysms. We also find a suggestive role for endothelial cells by using gene mapping and heritability enrichment. Drug-target enrichment shows pleiotropy between intracranial aneurysms and antiepileptic and sex hormone drugs, providing insights into intracranial aneurysm pathophysiology. Finally, genetic risks for smoking and high blood pressure, the two main clinical risk factors, play important roles in intracranial aneurysm risk, and drive most of the genetic correlation between intracranial aneurysms and other cerebrovascular traits.

Let $X$ be a closed $3$-manifold, $\mathcal{M}_{R>0}$ the space of metrics on $X$ with positive scalar curvature, and $\text{Diff}(X)$ the group of diffeomorphisms of $X$. Marques proves the fundamental result that $\mathcal{M}_{R>0}/ \text{Diff}(X)$ is path connected. Using this and the theorem of Cerf in differential topology, Marques shows that the space of asymptotically flat metrics with nonnegative scalar curvature on $\mathbb{R}^3$ is path connected. Based on Carlotto-Li's generalization of Marques' result to the case of compact manifold with boundary, we show that the space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on $\mathbb{R}^3\backslash B^3$ is path connected. The differential topology part of Marques' argument no longer yields the desired result for $\mathbb{R}^3\setminus B^3$, but we bypass this issue by finding a more elementary proof. We also include a path-connectedness result for the space of black hole initial data sets, which can be thought of as a necessary condition for the Final State Conjecture., 14 pages, 1 figure

Intracranial aneurysms are focal deformations of larger cerebral arteries that occur in 2-5% of the population. Although they remain quiescent most of the time, aneurysms may rupture at a rate of about 1% per year, leading to subarachnoid hemorrhage with potentially catastrophic effects on the patient. It is exceedingly challenging to predict the clinical fate of intracranial aneurysms. Currently, physicians associate shape irregularity with vessel wall instability. However, there is no consensus on which shape features reliably predict aneurysm rupture. Here we present two approaches that aim to eliminate the subjectivity of rater assessment. In a first approach, we have implemented a semi-automated classification pipeline to predict the rupture status using morphometric parameters. These parameters were computed from 3D geometries of intracranial aneurysms obtained from 3D rotational angiographies. The main objective of this first study was to identify morphometric parameters that efficiently encode the disease status and understand how well morphology predicts disease status in general. In the second study, we followed a psychometric approach to better understand how human raters assess aneurysm morphology. We acquired rating data from 39 clinical experts and informed laypersons on perceived irregularity and the presence of 5 different morphological attributes (presence of a rough surface, of blebs, lobules, asymmetry and a complex parent vasculature). We related this data to clinically relevant parameters using regression analysis and binary classification. Our investigations confirmed that aneurysm morphology provides significant information about the disease. For example, a logistic regression model based on perceived irregularity alone is able to discriminate relatively well ruptured from unruptured aneurysms (AUC=0.81±0.04). Extending that model by aneurysm location increased the AUC significantly to 0.87±0.08, suggesting that morphology varies with location. Morphometric parameters well correlating to perceived irregularity (e.g. non-sphericity NSI, or total Gaussian curvature GLN) likewise predict the disease status well, but to a lesser extent. More specific parameters are required that are able to encode relevant morphological structure such as blebs/lobules or asymmetry.

Clinical data science is an emerging discipline, owing to recent developments in the acquisition, storage and processing of large amounts of clinical data. The increasing wealth of data, however, demands for interdisciplinary collaborations, which imposes new challenges. The need for properly dealing with selection biases and establishing balanced databases becomes a key issue to be addressed in the field of digital health. Pre-existing beliefs about the disease sometimes are badly supported by evidence. Wrong assumptions or selection biases, however, will skew the subsequent analyses and mislead the interpretation of results. Subjective or approximate assessments by clinicians, just like missing/censored information about the patients, the data acquisition process or the pathology under examination make data scientific approaches introduce uncertainty about the data to be processed. All this requires robust approaches and very good domain knowledge to avoid false predictions. To successfully reach the clinically relevant statements calls for transparent methods and efficient tools for creating and communicating insights to practitioners. This talk will present some of the experiences made throughout our research on intracranial aneurysms and discuss how we manage these challenges. It will touch the detection of visual biomarkers with machine learning, present an approach to quantifying rather vaguely defined subjective estimations and demonstrate how visualization helps to detect clinical pathways in high dimensional diagnostic data.

The shape of intracranial aneurysms is used by clinicians for risk assessment. It has been suggested that irregular shape is linked to degenerate wall conditions and rupture. With the goal to better characterize shape irregularity, we acquired rating data of 27 participants (9 clinical experts, 18 laypersons) that assessed 134 aneurysms for perceived irregularity. 9 raters additionally examined the presence of 5 morphological attributes. Alongside with quantitative metrics for morphology, we used this rating data to model perceived irregularity. Preliminary results suggest that a combination of quantitative and qualitative shape assessment may characterize perceived irregularity the best.

An ever increasing amount of medical data is collected and used for scientific and clinical purposes. To benefit from the abundance of data, however, one has to deal with several challenges. The diversity of data sources, the variability seen in the biological systems and the biases and distortions inherent in the acquired data request for robust and flexible data processing pipelines. Here, we illustrate some of these challenges at the hand of our research on intracranial aneurysms and share insights how to deal with these challenges. We present the AneuX AneurysmDataBase. It stores data acquired at multiple clinical centers, supports heterogeneous data (clinical data, imaging data, genetic data, morphological and histological data, etc.) and is aimed for use in both scientific and industrial contexts. We further present five scientific studies that demonstrate the usage of the AneurysmDataBase. In the first application, we evaluated the PHASES score, a recent scoring scheme to guide the clinicians whether to treat an unruptured intracranial aneurysm. We further examined and improved existing morphological descriptors with the goal to associate aneurysm shape with its disease status. In a third study, we quantified the qualitative rating of aneurysm shape by humans. A fourth study aims at inferring information about the disease directly from imaging data by means of convolutional neural nets. Finally, we sketch how to query aneurysms with similar anatomical and morphological properties from a database. With our work, we demonstrate how clinical data sharing can be used for quantitative analyses of aneurysm properties and for the development of diagnostic and prognostic tools.

Introduction: Current methods of medical imaging provide key information for the clinical assessment of intracranial aneurysms. To date, broadly accepted quantitative criteria to compare aneurysm morphology are yet lacking. In this study, we relate established shape descriptors to expert assessments of aneurysm irregularity in view of establishing clinically meaningful descriptors. Method: To address morphology in an isolated manner, from 3D angiographies, we produced replicas of aneurysms and adjacent segments of the parent vascular tree, extracted computed geometry indices and measures based on curvature, surface writhe number and geometry moments. Independently, experts evaluated the morphology of these aneurysms in a 3D viewer by inspecting their shapes and rating the degree of irregularity on a 9-point rating scale. Besides these 3D views, no further information was given to the raters. We then examined the univariate and multivariate correlations between the aggregated ratings and the quantitative descriptions of the 3D models. Results: Preliminary results are based on 134 aneurysm models and 15 raters, who all assessed the perceived irregularity of the aneurysm shape. Univariate analysis shows that curvature metrics reproduce the rater assessment of shape irregularity the best (Spearman correlation ρ=0.86, p

Recent studies have found supporting evidence that the shape of an intracranial aneurysm can be used as a proxy for disease status. Although the shape, as seen in 3D imaging data, already plays a role in the clinical assessment of aneurysms today, tools to quantify and compare aneurysm morphology in a generic, standardized way are still lacking. Here, we present a machine learning approach based on a broad spectrum of shape descriptors to predict the aneurysm rupture status. Results are based on a dataset consisting of over 400 segmented aneurysm models. We extended our analysis by including human ratings of aneurysm shape. A correlation analysis of these ratings with quantifiable morphological parameters allowed us to identify shape descriptors mimicking the human assessment. Preliminary results based on 134 geometric aneurysm models and 15 assessments of human raters show that human assessment of irregular shape correlates well with curvature metrics, spread of the writhe number distribution and non-sphericity index.

Aneurysm wall changes its morphology and mechanical properties under a sequential wall remodeling dysfunction including de-differentiation and loss of endothelial cells and smooth muscle cells. To date, many studies have been performed CFD analysis on aneurysmal flow and revealed that flow-induced stress are partly correlated with histopathological events on vessel wall such as initiation, growth, and rupture. Nowadays, vessel wall enhancement is efficiently visualized using "black-blood"-T1-W high-resolution MRI sequences, and it has been demonstrated to be strongly associated with rupture in cases diagnosed with multiple aneurysms. Enhancement on aneurysm wall is observed as a result of deposition or penetration of contrast medium. This consequentially suggests that the enhanced area changes wall permeability and can be thought as an expression of wall degradation. Therefore, we systematically collected over the past three years, information about lesion stability based on MRI for better understanding of wall degeneration, aneurysm morphology, and flow-induced stress. This presentation exhibits selected aneurysm cases with wall enhancement. CFD results show a frequent collocation of wall enhancement with low WSS condition. However, results also investigate a case without visible wall enhancement, even though aneurysm wall experiences lower WSS than enhanced wall regions in same patient. The cases we present suggest that local flow condition can explain wall enhancement where active remodeling occurs, however, there are still limitations on prediction of enhancement by flow field alone.

The rupture of intracranial aneurysms is a potentially deadly event. Aneurysm detection is based on imaging techniques, amongst which 3D angiograms deliver the highest quality. Currently, neuroradiologists assess such images only in a qualitative manner, even though it remains unclear how morphological properties relate to the disease status quantitatively. In an effort to identify suitable descriptors that capture the irregularity of aneurysms, we relate a range of shape descriptors to expert assessments of aneurysm morphology. In a first step, we extracted 3D models of aneurysms from 3D angiographies and calculated geometry indices and moment invariants describing size and shape of aneurysms and surrounding arteries. In a second step, we compared these descriptors to human assessments of irregularity of the aneurysm dome. Preliminary results are presented based on 134 aneurysm models and 15 raters (5 clinicians, 10 informed laymen). Univariate correlation revealed that curvature-based metrics predict most accurately the human assessment of irregularity (rank correlation: ?=0.86). This study was performed within the scope of the AneuX project, funded by SystemsX.ch, and received support by SNSF NCCR Kidney.CH.