Generalizations of Ripley's K-function with Application to Space Curves

The intensity function and Ripley's K-function have been used extensively in the literature to describe the first and second moment structure of spatial point sets. This has many applications including describing the statistical structure of synaptic vesicles. Some attempts have been made to extend Ripley's K-function to curve pieces. Such an extension can be used to describe the statistical structure of muscle fibers and brain fiber tracks. In this paper, we take a computational perspective and construct new and very general variants of Ripley's K-function for curves pieces, surface patches etc. We discuss the method from [Chiu, Stoyan, Kendall, & Mecke 2013] and compare it with our generalizations theoretically, and we give examples demonstrating the difference in their ability to separate sets of curve pieces.Comment: 9 pages & 8 figure

Atlas-Based Reduced Models of Blood Flows for Fast Patient-Specific Simulations

The original publication is available at www.springerlink.com.International audienceModel-based interpretation of the complex clinical data now available (shape, motion, flow) can provide quantitative information for diagnosis as well as predictions. However such models can be extremely time consuming, which does not always fit with the clinical time constraints. The aim of this work is to propose a model reduction technique to perform faster patient-specific simulations with prior knowledge built from simulations on an average anatomy. Rather than simulating a full fluid problem on individual patients, we create a representative `template' of the artery shape. A full flow simulation is carried out only on this template, and a reduced model is built from the results. Then this reduced model can be transported to the individual geometries, allowing faster computational analysis. % Here we propose a preliminary validation of this idea. A well-posed framework based on currents representation of shapes is used to create an unbiased template of the pulmonary artery for 4 patients with Tetralogy of Fallot. Then, a reduced computational fluid dynamics model is built on this template. Finally, we demonstrate that this reduced model can represent a specific patient simulation

Topology preserving atlas construction from shape data without correspondence using sparse parameters

pre-printStatistical analysis of shapes, performed by constructing an atlas composed of an average model of shapes within a population and associated deformation maps, is a fundamental aspect of medical imaging studies. Usual methods for constructing a shape atlas require point correspondences across subjects, which are difficult in practice. By contrast, methods based on currents do not require correspondence. However, existing atlas construction methods using currents suffer from two limitations. First, the template current is not in the form of a topologically correct mesh, which makes direct analysis on shapes difficult. Second, the deformations are parametrized by vectors at the same location as the normals of the template current which often provides a parametrization that is more dense than required. In this paper, we propose a novel method for constructing shape atlases using currents where topology of the template is preserved and deformation parameters are optimized independently of the shape parameters. We use an L1-type prior that enables us to adaptively compute sparse and low dimensional parameterization of deformations.We show an application of our method for comparing anatomical shapes of patients with Down's syndrome and healthy controls, where the sparse parametrization of diffeomorphisms decreases the parameter dimension by one order of magnitude

The Science Behind the Springs: Using Biomechanics and Finite Element Modeling to Predict Outcomes in Spring-Assisted Sagittal Synostosis Surgery

Spring-assisted surgery for the correction of scaphocephaly has gained popularity over the past 2 decades. Our unit utilizes standardized torsional springs with a central helix for spring-assisted surgery. This design allows a high degree of accuracy and reproducibility of the force vectors and force distance curves. In this manuscript, we expand on the biomechanical testing and properties of these springs. Standardization of design has enabled us to study the springs on bench and in vivo and a comprehensive repository of calvarial remodeling and spring dynamics has been acquired and analyzed. Finite element modeling is a technique utilized to predict the outcomes of spring-assisted surgery. We have found this to be a useful tool, in planning our surgical strategy and improving outcomes. This technique has also contributed significantly to the process of informed consent preoperatively. In this article, we expand on our spring design and dynamics as well as the finite element modeling used to predict and improve outcomes. In our unit, this practice has led to a significant improvement in patient outcomes and parental satisfaction and we hope to make our techniques available to a wider audience

Currents and finite elements as tools for shape space

The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper we study a general representation of shapes that is based on linear spaces and is suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the $H^{-s}$ norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples

Understanding the influence of surgical parameters on craniofacial surgery outcomes: a computational study

Sagittal craniosynostosis (SC) is a congenital condition whereby the newborn skull develops abnormally owing to the premature ossification of the sagittal suture. Spring-assisted cranioplasty (SAC) is a minimally invasive surgical technique to treat SC, where metallic distractors are used to reshape the newborn’s head. Although safe and effective, SAC outcomes remain uncertain owing to the limited understanding of skull−distractor interaction and the limited information provided by the analysis of single surgical cases. In this work, an SC population-averaged skull model was created and used to simulate spring insertion by means of the finite-element analysis using a previously developed modelling framework. Surgical parameters were varied to assess the effect of osteotomy and spring positioning, as well as distractor combinations, on the final skull dimensions. Simulation trends were compared with retrospective measurements from clinical imaging (X-ray and three-dimensional photogrammetry scans). It was found that the on-table post-implantation head shape change is more sensitive to spring stiffness than to the other surgical parameters. However, the overall end-of-treatment head shape is more sensitive to spring positioning and osteotomy size parameters. The results of this work suggest that SAC surgical planning should be performed in view of long-term results, rather than immediate on-table reshaping outcomes

Investigating Cardiac Motion Patters Using Synthetic High-Resolution 3D Cardiovascular Magnetic Resonance Images and Statistical Shape Analysis

Diagnosis of ventricular dysfunction in congenital heart disease is more and more based on medical imaging, which allows investigation of abnormal cardiac morphology and correlated abnormal function. Although analysis of 2D images represents the clinical standard, novel tools performing automatic processing of 3D images are becoming available, providing more detailed and comprehensive information than simple 2D morphometry. Among these, statistical shape analysis (SSA) allows a consistent and quantitative description of a population of complex shapes, as a way to detect novel biomarkers, ultimately improving diagnosis and pathology understanding. The aim of this study is to describe the implementation of a SSA method for the investigation of 3D left ventricular shape and motion patterns and to test it on a small sample of 4 congenital repaired aortic stenosis patients and 4 age-matched healthy volunteers to demonstrate its potential. The advantage of this method is the capability of analyzing subject-specific motion patterns separately from the individual morphology, visually and quantitatively, as a way to identify functional abnormalities related to both dynamics and shape. Specifically, we combined 3D, high-resolution whole heart data with 2D, temporal information provided by cine cardiovascular magnetic resonance images, and we used an SSA approach to analyze 3D motion per se. Preliminary results of this pilot study showed that using this method, some differences in end-diastolic and end-systolic ventricular shapes could be captured, but it was not possible to clearly separate the two cohorts based on shape information alone. However, further analyses on ventricular motion allowed to qualitatively identify differences between the two populations. Moreover, by describing shape and motion with a small number of principal components, this method offers a fully automated process to obtain visually intuitive and numerical information on cardiac shape and motion, which could be, once validated on a larger sample size, easily integrated into the clinical workflow. To conclude, in this preliminary work, we have implemented state-of-the-art automatic segmentation and SSA methods, and we have shown how they could improve our understanding of ventricular kinetics by visually and potentially quantitatively highlighting aspects that are usually not picked up by traditional approaches