Efficient finite element methods for deformable bodies in medical applications.

Simulation techniques for deformable bodies are of major relevance for a broad range of medical applications. In recent decades, a lot of work has been performed to improve simulation methods, allowing interactivity or even real time. However, this work often focused on applications such as computer games or virtual environments, where physical accuracy is not a primary goal. The goal of this report is to give an overview of efficient physics-based techniques for deformable objects, focusing on finite element methods, and to discuss the applicability of these techniques in medical scenarios. As a result, we focus on techniques that are amenable to simulating highly resolved meshes, which for instance can be generated from computed tomography (CT) or magnetic resonance (MR) images, and we review the so-called corotated finite element method that has shown a high potential in recent years. Specifically, we will capture in detail the related work in this field and demonstrate the current state of the art in efficient deformable bodies simulations.

Stress tensor field visualization for implant planning in orthopedics.

We demonstrate the application of advanced 3D visualization techniques to determine the optimal implant design and position in hip joint replacement planning. Our methods take as input the physiological stress distribution inside a patient's bone under load and the stress distribution inside this bone under the same load after a simulated replacement surgery. The visualization aims at showing principal stress directions and magnitudes, as well as differences in both distributions. By visualizing changes of normal and shear stresses with respect to the principal stress directions of the physiological state, a comparative analysis of the physiological stress distribution and the stress distribution with implant is provided, and the implant parameters that most closely replicate the physiological stress state in order to avoid stress shielding can be determined. Our method combines volume rendering for the visualization of stress magnitudes with the tracing of short line segments for the visualization of stress directions. To improve depth perception, transparent, shaded, and antialiased lines are rendered in correct visibility order, and they are attenuated by the volume rendering. We use a focus+context approach to visually guide the user to relevant regions in the data, and to support a detailed stress analysis in these regions while preserving spatial context information. Since all of our techniques have been realized on the GPU, they can immediately react to changes in the simulated stress tensor field and thus provide an effective means for optimal implant selection and positioning in a computational steering environment.

Solving the Fluid Pressure Poisson Equation Using Multigrid-Evaluation and Improvements.

In many numerical simulations of fluids governed by the incompressible Navier-Stokes equations, the pressure Poisson equation needs to be solved to enforce mass conservation. Multigrid solvers show excellent convergence in simple scenarios, yet they can converge slowly in domains where physically separated regions are combined at coarser scales. Moreover, existing multigrid solvers are tailored to specific discretizations of the pressure Poisson equation, and they cannot easily be adapted to other discretizations. In this paper we analyze the convergence properties of existing multigrid solvers for the pressure Poisson equation in different simulation domains, and we show how to further improve the multigrid convergence rate by using a graph-based extension to determine the coarse grid hierarchy. The proposed multigrid solver is generic in that it can be applied to different kinds of discretizations of the pressure Poisson equation, by using solely the specification of the simulation domain and pre-assembled computational stencils. We analyze the proposed solver in combination with finite difference and finite volume discretizations of the pressure Poisson equation. Our evaluations show that, despite the common assumption, multigrid schemes can exploit their potential even in the most complicated simulation scenarios, yet this behavior is obtained at the price of higher memory consumption.

A System for High-Resolution Topology Optimization.

A key requirement in 3D fabrication is to generate objects with individual exterior shapes and their interior being optimized to application-specific force constraints and low material consumption. Accomplishing this task is challenging on desktop computers, due to the extreme model resolutions that are required to accurately predict the physical shape properties, requiring memory and computational capacities going beyond what is currently available. Moreover, fabrication-specific constraints need to be considered to enable printability. To address these challenges, we present a scalable system for generating 3D objects using topology optimization, which allows to efficiently evolve the topology of high-resolution solids towards printable and light-weight-high-resistance structures. To achieve this, the system is equipped with a high-performance GPU solver which can efficiently handle models comprising several millions of elements. A minimum thickness constraint is built into the optimization process to automatically enforce printability of the resulting shapes. We further shed light on the question how to incorporate geometric shape constraints, such as symmetry and pattern repetition, in the optimization process. We analyze the performance of the system and demonstrate its potential by a variety of different shapes such as interior structures within closed surfaces, exposed support structures, and surface models.

Real-time haptic cutting of high-resolution soft tissues.

We present our systematic efforts in advancing the computational performance of physically accurate soft tissue cutting simulation, which is at the core of surgery simulators in general. We demonstrate a real-time performance of 15 simulation frames per second for haptic soft tissue cutting of a deformable body at an effective resolution of 170,000 finite elements. This is achieved by the following innovative components: (1) a linked octree discretization of the deformable body, which allows for fast and robust topological modifications of the simulation domain, (2) a composite finite element formulation, which thoroughly reduces the number of simulation degrees of freedom and thus enables to carefully balance simulation performance and accuracy, (3) a highly efficient geometric multigrid solver for solving the linear systems of equations arising from implicit time integration, (4) an efficient collision detection algorithm that effectively exploits the composition structure, and (5) a stable haptic rendering algorithm for computing the feedback forces. Considering that our method increases the finite element resolution for physically accurate real-time soft tissue cutting simulation by an order of magnitude, our technique has a high potential to significantly advance the realism of surgery simulators.

Multi-Charts for Comparative 3D Ensemble Visualization.

A comparative visualization of multiple volume data sets is challenging due to the inherent occlusion effects, yet it is important to effectively reveal uncertainties, correlations and reliable trends in 3D ensemble fields. In this paper we present bidirectional linking of multi-charts and volume visualization as a means to analyze visually 3D scalar ensemble fields at the data level. Multi-charts are an extension of conventional bar and line charts: They linearize the 3D data points along a space-filling curve and draw them as multiple charts in the same plot area. The bar charts encode statistical information on ensemble members, such as histograms and probability densities, and line charts are overlayed to allow comparing members against the ensemble. Alternative linearizations based on histogram similarities or ensemble variation allow clustering of spatial locations depending on data distribution. Multi-charts organize the data at multiple scales to quickly provide overviews and enable users to select regions exhibiting interesting behavior interactively. They are further put into a spatial context by allowing the user to brush or query value intervals and specific distributions, and to simultaneously visualize the corresponding spatial points via volume rendering. By providing a picking mechanism in 3D and instantly highlighting the corresponding data points in the chart, the user can go back and forth between the abstract and the 3D view to focus the analysis.

Large-Scale Liquid Simulation on Adaptive Hexahedral Grids.

Regular grids are attractive for numerical fluid simulations because they give rise to efficient computational kernels. However, for simulating high resolution effects in complicated domains they are only of limited suitability due to memory constraints. In this paper we present a method for liquid simulation on an adaptive octree grid using a hexahedral finite element discretization, which reduces memory requirements by coarsening the elements in the interior of the liquid body. To impose free surface boundary conditions with second order accuracy, we incorporate a particular class of Nitsche methods enforcing the Dirichlet boundary conditions for the pressure in a variational sense. We then show how to construct a multigrid hierarchy from the adaptive octree grid, so that a time efficient geometric multigrid solver can be used. To improve solver convergence, we propose a special treatment of liquid boundaries via composite finite elements at coarser scales. We demonstrate the effectiveness of our method for liquid simulations that would require hundreds of millions of simulation elements in a non-adaptive regime.

Distance visualization for interactive 3D implant planning.

An instant and quantitative assessment of spatial distances between two objects plays an important role in interactive applications such as virtual model assembly, medical operation planning, or computational steering. While some research has been done on the development of distance-based measures between two objects, only very few attempts have been reported to visualize such measures in interactive scenarios. In this paper we present two different approaches for this purpose, and we investigate the effectiveness of these approaches for intuitive 3D implant positioning in a medical operation planning system. The first approach uses cylindrical glyphs to depict distances, which smoothly adapt their shape and color to changing distances when the objects are moved. This approach computes distances directly on the polygonal object representations by means of ray/triangle mesh intersection. The second approach introduces a set of slices as additional geometric structures, and uses color coding on surfaces to indicate distances. This approach obtains distances from a precomputed distance field of each object. The major findings of the performed user study indicate that a visualization that can facilitate an instant and quantitative analysis of distances between two objects in interactive 3D scenarios is demanding, yet can be achieved by including additional monocular cues into the visualization.