Cell tracking is a subject of active research gathering great interest in medicine and biology. Positron emission tomography (PET) is well suited for tracking radio-labeled cells in vivo due to its exceptional sensitivity and whole-body capability. For validation, ground-truth data are desirable that realistically mimic the flow of cells in a clinical situation. This study develops a workflow (CeFloPS) for simulating moving radio-labeled cells in a human phantom. From the XCAT phantom, the blood vessels are reduced to nodal networks along which cells can move and distribute to organs and tissues. The movement is directed by the blood flow, which is calculated in each node using the Hagen-Poiseuille equation and Kirchhoff's laws assuming laminar flow. Organs are voxelized and movement of cells from artery entry to vein exit is generated via a biased 3D random walk. The probabilities of cells moving or remaining in tissues are derived from rate constants of tracer kinetic-based compartment modeling. PET listmode data is generated using the Monte-Carlo simulation framework GATE based on the definition of a large-body PET scanner with cell paths as moving radioactive sources and the XCAT phantom providing attenuation data. From the flow simulation of 100,000 cells, 100 sample cells were further processed by GATE and listmode data was reconstructed into images for comparison. As demonstrated by comparisons of simulated and reconstructed cell distributions, CeFloPS is capable of simulating cell behavior in whole-body PET. It achieves this simulation in a way that is anatomically and physiologically reasonable, thereby providing valuable data for the development and validation of cell tracking algorithms.

BACKGROUND: The efficient and robust statistical analysis of the shape of plant organs of different cultivars is an important investigation issue in plant breeding and enables a robust cultivar description within the breeding progress. Laserscanning is a highly accurate and high resolution technique to acquire the 3D shape of plant surfaces. The computation of a shape based principal component analysis (PCA) built on concepts from continuum mechanics has proven to be an effective tool for a qualitative and quantitative shape examination. RESULTS: The shape based PCA was used for a statistical analysis of 140 sugar beet roots of different cultivars. The calculation of the mean sugar beet root shape and the description of the main variations was possible. Furthermore, unknown and individual tap roots could be attributed to their cultivar by means of a robust classification tool based on the PCA results. CONCLUSION: The method demonstrates that it is possible to identify principal modes of root shape variations automatically and to quantify associated variances out of laserscanned 3D sugar beet tap root models. The introduced approach is not limited to the 3D shape description by laser scanning. A transfer to 3D MRI or radar data is also conceivable.

We propose a novel dynamic image reconstruction method from PET listmode data that could be particularly suited to tracking single or small numbers of cells. In contrast to conventional PET reconstruction our method combines the information from all detected events not only to reconstruct the dynamic evolution of the radionuclide distribution, but also to improve the reconstruction at each single time point by enforcing temporal consistency. This is achieved via optimal transport regularization where in principle, among all possible temporally evolving radionuclide distributions consistent with the PET measurement, the one is chosen with least kinetic motion energy. The reconstruction is found by convex optimization so that there is no dependence on the initialization of the method. We study its behaviour on simulated data of a human PET system and demonstrate its robustness even in settings with very low radioactivity. In contrast to previously reported cell tracking algorithms, our technique is oblivious to the number of tracked cells. Without any additional complexity one or multiple cells can be reconstructed, and the model automatically determines the number of particles. For instance, four radiolabelled cells moving at a velocity of 3.1 mm/s and a PET recorded count rate of 1.1 cps (for each cell) could be simultaneously tracked with a tracking accuracy of 5.3 mm inside a simulated human body.

Abstract Cubic splines in Euclidean space minimize the mean squared acceleration among all curves interpolating a given set of data points. We extend this observation to the Riemannian manifold of discrete shells in which the associated metric measures both bending and membrane distortion. Our generalization replaces the acceleration with the covariant derivative of the velocity. We introduce an effective time‐discretization for this novel paradigm for navigating shell space. Further transferring this concept to the space of triangular surface descriptors—edge lengths, dihedral angles, and triangle areas—results in a simplified interpolation method with high computational efficiency.

We generalize the notion of Bézier surfaces and surface splines to Riemannian manifolds. To this end we put forward and compare three possible alternative definitions of Bézier surfaces. We furthermore investigate how to achieve ${\mathcal{C}}^0$- and ${\mathcal{C}}^1$-continuity of Bézier surface splines. Unlike in Euclidean space and for one-dimensional Bézier splines on manifolds, ${\mathcal{C}}^1$-continuity cannot be ensured by simple conditions on the Bézier control points: it requires an adaptation of the Bézier spline evaluation scheme. Finally, we propose an algorithm to optimize the Bézier control points given a set of points to be interpolated by a Bézier surface spline. We show computational examples on the sphere, the special orthogonal group, and two Riemannian shape spaces.

Abstract We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells (modulo rigid body motions). Based on this foundation we develop a time‐ and space‐discrete geodesic calculus. In particular we show how to shoot geodesics with prescribed initial data, and we give a construction for parallel transport in shell space. This enables, for example, natural extrapolation of paths in shell space and transfer of large nonlinear deformations from one shell to another with applications in animation, geometric, and physical modeling. Finally, we examine some aspects of curvature on shell space.

Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessian reproduces the underlying Riemannian metric, and it is used to define length and energy of discrete paths in shape space. The notion of discrete geodesics defined as energy minimizing paths gives rise to a discrete logarithmic map, a variational definition of a discrete exponential map, and a time discrete parallel transport. This new concept is developed in the context of shape spaces with shapes that are described via deformations of a given reference shape, and it is applied to a particular shape space in which shapes are considered as boundary contours of physical objects consisting of viscous material. The flexibility and computational efficiency of the approach is demonstrated for topology preserving shape morphing, the representation of paths in shape space via local shape variations as path generators, shape extrapolation via discrete geodesic flow, and the transfer of geometric features.

Abstract Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the so‐called relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path.

A physically motivated approach is presented for computing a shape average of a given number of shapes. An elastic deformation is assigned to each shape. The shape average is then described as the common image under all elastic deformations of the given shapes, which minimizes the total elastic energy stored in these deformations. The underlying nonlinear elastic energy measures the local change of length, area, and volume. It is invariant under rigid body motions, and isometries are local minimizers. The model is relaxed involving a further energy which measures how well the elastic deformation image of a particular shape matches the average shape, and a suitable shape prior can be considered for the shape average. Shapes are represented via their edge sets, which also allows for an application to averaging image morphologies described via ensembles of edge sets. To make the approach computationally tractable, sharp edges are approximated via phase fields, and a corresponding variational phase field model is derived. Finite elements are applied for the spatial discretization, and a multiscale alternating minimization approach allows the efficient computation of shape averages in two and three dimensions. Various applications, e.g., averaging the shape of feet or human organs, underline the qualitative properties of the presented approach.