Keywords: computer tomography, finite element method, geometry reconstruction, Delaunay triangulation, mesh optimization, material properties of cancellous bone, apparent density, Hounsfield units.

The finite element method has now become the most widely used numerical tool for use in the solution of partial differential equations on regions of arbitrary shape. This is especially true in the field of computational biomechanics, where the regions of interest represent very complicated structures, usually inner organs. One of the most common problems in computational biomechanics is stress analysis of a new type of an implant and particularly assessment of its influence on the stress state in the respective tissues. It is therefore necessary to construct detailed mathematical model of selected part of the human skeletal system. As a convenient source of data for the model creation, Computer Tomography (CT), Magnetic Resonance Imaging (MRI) and Ultrasound (US) techniques are used.

The reconstruction process starts with tissue segmentation, that is, selection of the tissue of interest in the whole stack of medical images. Several techniques are presented in the paper (intensity based segmentation, edge detection techniques, region growing techniques). Traditional intensity based segmentation is suitable for images of high quality where every tissue of interest is well separated from each other. Edge detection belongs to boundary based methods that work by detecting discontinuities in the gray level values of the pixels. Most of the edge detection methods are based on finding places of abrupt change of intensity between them and neighbours. These techniques make use of standard mathematical operators with a small spatial extent to detect whether there is a local edge in the image. The standard operators, such as Gradient, Laplacian, Sobel, Roberts, Prewitt or Kirsch, are approximate mathematical gradient operators or operators using a template matching technique. The Canny operator, used in the work for cortical bone tissue segmentation, works as a multi-stage operator. Segmentation using active contours (region growing) is another possibility to identify regions of interest within an image based on an attempt to match the image to a parametric edge model. These methods are very robust in case of images of poor quality (ultrasound pictures) or finding an object that does not create big differences in the pixel intensity but is of known shape (e.g. tumor detection).

After the tissue segmentation, the surface of the organ must be detected. Three different detection categories are available: (i) contour-based methods: these methods consist in finding corresponding contours in each adjacent slice and connecting consistently oriented contours (usually represented by splines) by triangle meshes, (ii) table-lookup methods: these methods use lookup-tables to decide how to connect data from adjacent slices to create polygonal surface without holes. The most commonly used method is the Marching Cubes Algorithm [1], (iii) adaptive methods: this group of surface-generation methods uses different approaches to adapt a surface to the object structure.

The main objective of the whole reconstruction process is to create a volumetric representation of organs to be used in finite element simulations. It is therefore necessary to create 3-D mesh of the organ. To describe the organ as a set of finite elements, it is necessary to fill the surface of the organ with volumetric elements. Because the Marching Cubes algorithm generates surface elements as triangles, the volumetric elements have tetrahedral shape. For the discretization of the volume Delaunay triangulation in 3-D is used. The Delaunay triangulation [2] is one of the methods for subdivision of an area into triangles or in 3-D volume into tetrahedrons. The result is a set of tetrahedra filling the entire 3-D domain. Delaunay triangulation used for the discretization of the volume does not guarantee elements with good aspect ratios. The resulting mesh can contain elements of bad shape that can slow down the solution or produce undesirable errors. The Delaunay triangulation must be therefore followed by mesh optimization to avoid sliver elements (elements with very small spatial angle between two sides) as well as other elements of bad shape. Three methods are used to improve the quality of existing tetrahedral mesh are covered: (i) mesh smoothing, (ii) nodal points insertion and deletion and (iii) local re-meshing.

Acknowledgment

The research has been sponsored by research plan of Academy of Sciences of the Czech Republic (grant No. AV 0Z 2071913) and research fund of the Ministry of Education, Youth and Sports of the Czech Republic (grant No. MSM 212600025).

1

W.E. Lorensen and H.E. Cline. "Marching Cubes: A High Resolution 3D Surface Construction Algorithm", Computer Graphics, 21(2):155-168, July 1987. doi:10.1145/37402.37422

2

Tsung-Pao Fang and Les A. Piegl. "Delaunay Triangulation in Three Dimensions", IEEE Computer Graphics and Applications, 15(5), September 1995. doi:10.1109/38.403829

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