Simulating Cutting in Surgery Applications using Haptics and Finite Element Models C. Mendoza & C. Laugier INRIA Rhone-Alpes ZIRST 655 avenue de l’Europe 38330 Montbonnot Saint Martin FRANCE Cesar.Mendoza-Serrano, Christian.Laugier @inrialpes.fr 1. Introduction Soft tissue cutting is an important task in surgery sim- ulators. We use a separation of elements approach instead of destroying [2] or subdividing [1][5] the elements. Previ- ously we have started a new cutting approach for 2D mass- springs models: separating the elements [4]. It does not increment the number of elements (as in the subdivision approach) and maintains the same mass during simulations (against the destruction approach). Later, Nienhuys et. al. [6] has used the same idea to approach 3D cutting but they didn’t consider large displacements (required for cuts). 2 Soft Tissue Physical Model We used an explicit formulation of finite elements [2] in which, instead of solving a large matricial system of the type we solve each element independently through a local approximation and the force is obtained at each node. Geometric non linearities are taken into account by using a Green-Lagrange strain tensor, , allowing large displacements. To link strain and stress we consider that our material is isotropic and elastically linear. The total in- ternal force that a tetrahedron exerts on a node is (1) here is the volume of the tetrahedron, p the position of the nodes of the tetrahedron in the world coordinates and , the inverse barycentric matrix that links the world positions to the material coordinates. The total internal force acting on the node is obtained by summing the forces exerted by all elements that are attached to the node. Finally, we use an Euler scheme to integrate the dynamics of each node. 3. Cutting algorithm Once a collision detection has been detected between the cutting tool and the object, we follow the next steps: 3.1 Interpretation of user behavior and physical criteria First, we interpret if the user displacements on the surface of the object corresponds to a cutting attempt or not. Let be the colliding point at time between (a) (b) v (t) c v (t-1) c Cutting attempt NO cutting attempt Internal Forces Contact Surface: dA M dF Figure 1. Determining cutting attempts. the virtual tool and a facet. Let the closest vertex to , at time . Define a neighborhood (in this case a circle) around the vertex (figure 1). Thus, we consider a cutting attempt if: (1) and (2) to avoid degenerated tetrahedrons. We compute a cutting traction vector where are the normals of an infinitesimal cube, is the sharpness of the tool and is a matrix of normal and shearing stresses produced by loads caused by the tool. The object is bro- ken when the maximum eigenvalue of the matrix, , takes a value greater than the material toughness, . A cut is produced if a cutting attempt has occurred and if . Proceedings of the IEEE Virtual Reality 2003 (VR’03) 1087-8270/03 $17.00 © 2003 IEEE