Real-time breast deformation using non-linear tissue properties

Localization of target structures in open surgical breast procedures mostly relies on localization wires that give coarse orientation hints together with a set of radiological images to convey the extent and location. Patient positioning, however, is different for image acquisition and surgery. We propose to simulate the breast deformation between these positions to track and visualize the target position. To date no sufficiently fast and accurate methods have been proposed for that purpose, which is caused by the computational expensiveness caused by the non-linear behavior of the material. In contrary, the efficient FEM-based simulation framework [GW06] employed in our work allows for an online update of material attributes, in particular the per-element elastic modulus, which affects the reaction to forces. Fast breast deformation simulation is thereby for the first time amenable to improve accuracy and confidence of breast surgeons compared to the de-facto standard techniques for localization of target structures. 1 Problem Statement and Prior Art Real-world materials exhibit non-linear reactions to stress: when a certain stress level is exceeded, they react with a non-linear change of stiffness. In contrast, linear material properties in isotropic materials following Hooke’s law are not capable to reflect this property. Our goal is to implement a simplified isotropic non-linear material law by adapting the per-element elastic modulus in every simulation step. Specifically, we adapt this method to simulate plausible breast deformations, which can be mimicked by a relatively soft base elastic modulus in combination with a stiffening of the elements under load. Other work tried to achieve similar objectives by employing more advanced non-linear material laws. However, they will not be capable of real-time simulations, though they model the physical world more convincingly. For a clinical setting, a compromise has to be found between fast calculations and sufficiently realistic material behavior which we believe we have achieved for the first time with the approach presented. Previous work tried to match prone and supine breast shapes by employing finite element analysis using non-linear material laws [RNHN08]. However, the computational complexity of this approach is high and the setup of the model difficult to be performed in time-critical clinical routine. The fastest available implementations of dynamic non-linear models [HHM11, TCO08] are based on explicit finite element approaches, which limit the largest possible time step in a dynamic simulation. Furthermore, in this case the deforINFORMATIK 2011 Informatik schafft Communities 41. Jahrestagung der Gesellschaft für Informatik , 4.-7.10.2011, Berlin www.informatik2011.de erschienen im Tagungsband der INFORMATIK 2011 Lecture Notes in Informatics, Band P192 ISBN 978-3-88579-286-4 weitere Artikel online: http://informatik2011.de/519.html mation simulation has to be followed by a non-rigid registration step to achieve the final results, which might decrease utility and certainly speed. In our work, we evaluate the performance of three stress norms controlled by two different approaches that dampen the oscillations caused by the explicit dynamic update of the perelement elastic modulus. 2 Material and Methods Standard breast MRI data was obtained from five volunteers1, once in prone (facing down) and once in supine (facing up) position. All datasets were segmented into rigid and deformable tissue using a semi-automatic tool [WFFH11], where the breast parenchyma and the fatty tissue was considered elastic, while the thorax and muscular tissue were considered as Dirichlet boundary conditions. A volumetric tetrahedral mesh was generated from a downsampled version of this data. The finite element method used to simulate the deformation of the breasts is based on the co-rotated Cauchy strain formulation [GW08]. The novel aspect of our work is the update of the per-element stiffness based on the stress it experiences in a given simulation step. We have implemented different stress tensor norms to evaluate their suitability for the task, namely one based on the eigenvalues of the stress tensor that can take into account the directionality of the principal stress, one based on the trace of the tensor, and the von Mises stress tensor norm [Bat02]. The stiffness is then adjusted per element and time step, according to the stress it encounters. Custom visualization methods allow to view the deformation along with characteristic attributes like, e.g., the principal stress directions and the von Mises norm. Since no prior work has approached the modeling of tissue stiffness with the described approach, numerous variants of the real-time stiffness update have been implemented to evaluate and compare their characteristics and performance. For better comparability, a simple and regular mesh has been set up, consisting of a ground plate and a soft box on top of it that has an adjustable size and number of finite elements (tetrahedra) and can thus be used to study microscopic effects on single elements, or macroscopic effects on a larger scale. The box has material parameters that are used for the breast model as well, i.e. a Poisson ratio (the ratio of transverse to axial strain of a stretched/compressed material; cf. Fig. 1) of 0.43 and a base stiffness of 1 kPa. In the tests, the gravity direction and strength was used to cause controlled deformations. The von Mises stress tensor norm (cf. Eqn. 1; [Bat02]) takes into account all elements of the raw stress tensor to calculate a scalar metric of the stress. It has been implemented to serve as a reference standard, since it is a well-established stress norm ||σ||Mises = √√√3 6 ∑