Abstract
Patient specific biomedical applications of the Finite Element Method are generally more expensive and time consuming compared with standard CAD based ones, common in structural engineering. They usually require segmentation and geometry creation procedures which are hard to automatize or other techniques which directly convert each pixel into a finite element hence increasing the computational resources required.
We present a method for the fast and automatic creation of h-adapted patient-specific FE models from medical images by using a hierarchical structure of nested Cartesian grids (image-based cgFEM). The method is oriented to the simulation of hard tissue such as bone and provides a useful tool for lightening the burden of segmentation, geometry creation and meshing during the pre-processing stage. Our goal is to show the potential of the method through a first implementation for linear elasticity which uses simplified relationships between material properties and pixel values taken from the literature.
This method allows numerical models to be easily obtained with a reasonable number of degrees of freedom, skipping the segmentation and geometry creation stages, with the exception of the surfaces used for the imposition of the boundary conditions. The volume boundaries, implicitly described in the image, do not require an explicit geometrical representation, but, nonetheless, are captured by the mesh refinement process.
The resulting numerical models are characterized by nonconforming meshes, in which the continuity of the solution is enforced using Multi-Point Constraints. Each element of the mesh will contain a number of pixels whose values describe the material heterogeneity. This is taken into account at the numerical integration stage. This provides the stiffness matrix that describes a homogenized behaviour over the element.
In the process, mesh adaptivity prevents excessive homogenization and properly captures the boundaries implicitly described in the image. This is carried out by limiting the pixel value variability inside each element by splitting if the variability exceeds a prescribed value. The results of this process is a h-adapted Finite Element mesh able to model the heterogeneity represented in the medical image using a reduced number of degrees of freedom to minimize the computational cost.
The boundary conditions are enforced on geometrical entities, such as curves and surfaces, which are independent from the mesh, only have to be defined locally and do not have any particular requirement. The Dirichet boundary condition is imposed by using the Lagrange multiplier technique.
The process of generating the FE model, which usually requires a relevant amount of time, is, in contrast, fast in image-based cgFEM due to the efficiency of the hierarchical structure of Cartesian grids, despite the fact that the code is completely implemented in Matlab.
