Coupling of fluid and elastic models for biomechanical simulations of brain deformations using FEM
Introduction
The accuracy of image-guided neurosurgery generally suffers from intraoperative changes of the brain anatomy due to, e.g., tumor resection or brain shift (Hill et al., 1998). To improve upon navigation accuracy, a variety of biomechanical models have been developed (Bucholz et al., 1997, Edwards et al., 1997, Škrinjar et al., 1998, Hagemann et al., 1999b, Paulsen et al., 1999) to predict brain deformations and thus to correct the preoperative images with respect to surgery-induced effects. Other biomechanical models have been described in the context of preoperative planning for registration purposes (Davatzikos, 1996, Kyriacou and Davatzikos, 1998, Lester et al., 1998, Ferrant et al., 1999) which can be used likewise to predict brain deformations.
All these models simulate the biomechanical behavior of different anatomical structures by either spatially varying the underlying material parameter values, while assuming the same physical model for all anatomical structures (Davatzikos, 1996, Bucholz et al., 1997, Lester et al., 1998, Ferrant et al., 1999, Hagemann et al., 1999b, Paulsen et al., 1999), or by applying appropriate boundary conditions (Edwards et al., 1997, Škrinjar et al., 1998, Kyriacou and Davatzikos, 1998). An example for the latter case is the integration of the skull bone or the falx as a non-moving part (known as homogeneous Dirichlet boundary condition). However, the existing approaches generally lead to physically inadequate simulations, particularly in the case of neighbouring elastic/fluid regions. For example, in (Hagemann et al., 1999b), where the Navier equation has been used as underlying physical model, the ventricular system was modeled as a rigid object, motivated by the reported incompressibility of cerebrospinal fluid (Sahay et al., 1992, Tada et al., 1994). Using this assumption, good registration results are obtained if none of the prescribed correspondences, which drive the deformation of the model, were given in the vicinity of the ventricular system. Otherwise the model gives a poor registration result, leading to an unrealistic deformation of the fluid region as shown in Fig. 1. An approach that directly simulates the physical behavior of inhomogeneous fluids by using the Navier–Stokes equation is the model in (Lester et al., 1998), which is motivated by the homogeneous fluid model in (Christensen et al., 1996). But in both cases, it is assumed that all anatomical structures behave like fluids which is generally not the case.
In order to improve the prediction of brain deformations, we extend our biomechanical model presented in (Hagemann et al., 1999a) to cope with heterogeneous anatomical structures consisting of rigid, elastic and fluid materials while using the appropriate physical descriptions for each material, namely the Navier equation or the Stokes equation (Hagemann et al., 2000). Our approach is based on the well-established physical theory of continuum mechanics which allows to handle inhomogeneous materials. In our model, an inhomogeneous domain is divided into homogeneous regions, each simulating a different material by using the appropriate constitutive equation. The underlying necessary image segmentation for partitioning the heterogeneous domain into homogeneous regions requires at least interactive support of a semi-automated segmentation process since fully automated segmentation is not feasible yet for the general case of clinical routine images. To discretize the problem, we apply the finite element method (FEM) to each region, resulting in a set of sparse linear matrix systems. Thereafter, these matrix systems are assembled together into a single overall system of linear equations via appropriate boundary conditions, which establish a physical link between the corresponding regions. Instead of using explicit external forces, which are generally difficult to be determined from corresponding images, we use a set of given correspondences to drive the deformation of the source image. Such point-wise correspondences have to be established in a semi-automated (anatomy landmarks) or interactive (lesion landmarks) fashion. In our approach, it is ensured that the prescribed correspondences are exactly fulfilled by the resulting deformation. Experiments with synthetic as well as real tomographic images have been carried out and the results were compared against a purely linear elastic model with spatially varying material parameter values in order to assess the physical plausibility of the predicted deformations.
Section snippets
Approach
To determine the deformation of an inhomogeneous body , the body is divided into a set of homogeneous regions in accordance with the underlying anatomical structures. Different material properties are taken into account by substituting the appropriate constitutive equation, which describes the stress–strain relationship of a region , into the equilibrium equations which describe the state of equilibrium between internal and external forces. Denoting with the Eulerian stress tensor,
Experiments
Our coupled rigid/elastic/fluid model has been tested on synthetic and tomographic images to investigate the physical plausibility of the calculated deformation results. For the experiments with the synthetic images shown in Fig. 3 we model the following three different materials: rigid skull bone (black), cerebrospinal fluid (bright grey), and elastic brain tissue (dark grey). To assess the physical plausibility of our new model, we compared the results with those from our purely linear
Summary
We proposed a new biomechanical model of the human head based on the finite element method to improve upon the accuracy of image guided neurosurgery. The model uses the physical theory of continuum mechanics to explicitly simulate the deformations of coupled rigid, elastic and fluid regions. Instead of using explicit external forces, the deformation of the biomechanical model is driven by landmark correspondences. Experiments with synthetic as well as tomographic images have been carried out in
Acknowledgements
Support of Philips Research Laboratories Hamburg, project IMAGINE (IMage- and Atlas-Guided Interventions in NEurosurgery), is gratefully acknowledged. Additionally, we thank Professor Dr. med. J.M. Gilsbach and OA Dr. med. U. Spetzger of the Neurosurgical Clinic, Aachen University of Technology (RWTH) for providing us with the tomographic datasets used in the experiments. We also thank A. Pünjer for supporting us in visualizing the 3D experimental results.
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