Various approaches have been proposed to modelling deformable objects in virtual reality systems, many of them driven by the desire to develop surgical simulators, including simple deformable body models and various enhancements to the Finite Element (FE) method.

The requirements of neurosurgical applications are discussed, and the BE method is discussed in relation to previous approaches.. An initial demonstration is presented to demonstrate the use of the BE method to provide haptic feedback, i.e. force feedback typically via small robot-like actuators notably the Phantom system produced by Sensable Technologies [1].

The focus of the paper is on the BE method [2], which so far has received only limited attention for the modelling of deformable objects within virtual reality, e.g. the work of James and Pai [3] and Monserrat et al [4]; both teams identify surgical simulation as a primary or possible application. None of this published work has as yet considered the issues of geometric non-linearity caused by large deformations, the issue of contact between flexible bodies, the enclosure of the object in a tough membrane, or the cutting of a solid, all of which are important considerations in surgical simulation. These issues would therefore appear to be the critical ones that need to be addressed before BE-based surgical simulation can become a reality.

Viewed from the modeller's (rather than the surgeon's) perspective, a simulation of the surgery process on an organ such as the brain based upon the BE method must be capable of representing the following features, listed approximately in order of difficulty:

1. Indentation of the tissue with a sharp object such as a scalpel or forceps prior to cutting, with haptic and visual feedback at a suitable refresh rate (say 25 - 50 Hz) for the whole body and haptic feedback at (say) 1000 Hz.
2. Self weight of the tissue, i.e. so that it naturally deforms under its own weight.
3. Retraction of the cut tissue using a shaped and approximately rigid implement having a finite surface.
4. Cutting of the tissue, either using a scalpel or (in the case of neurosurgery) using a forceps-like device, thus creating new surfaces resulting in additional deformations due to the change in the structural behaviour of the tissue under existing loads.
The contact of tissue with itself and with its surroundings.

For simulation of indenting or prodding an object, the conventional BE formulation is used to model the object. However, since a virtual reality simulation of large 3D problems would require a real-time solution of a very large fully-populated solution matrix, a pre-solution BE approach, based on the superposition principle, is implemented. Triangular elements with constant shape functions provide the fastest solution and the simplest meshing option with adequate accuracy and are compatible with the data requirements for the visual and haptic simulation.

Prior to the real-time simulation, a look-up table or a set of pre-solutions can be created by imposing a hypothetical unit displacement or unit traction on all surface nodes in turn, solving the system of equations and storing the solutions.

Practical surgery involves the creation of new surfaces as cutting progresses. This can be modelled as the creation of a crack in a deformable object (based upon fracture mechanics techniques) and exploiting the "dual" BE approach to permit the formation of initially-coincident crack faces [5]. An efficient algorithm based on the inverse of the initial system matrix can be developed by partitioning. Since each incremental step of cutting only adds a few surface elements, the matrix extensions are small, and the resulting computations can be performed at real-time speed.

1

http://www.sensable.com

2

Becker, A.A. "The Boundary Element Method in Engineering", McGraw-Hill, London, 1992.

3

James, D.L. and Pai, D.K., "ArtDefo: Accurate real time deformable objects", Proc SIGGRAPH99, 65-72, 1999.

4

Monserrat, C., Meier, U., Alcaniz, M., Chinesta, F and Juan, M.C., "A new approach for the real-time simulation of tissue deformations in surgery simulation", Comp Meth Prog in Biomedicine, 64, 77-85, 2001. doi:10.1016/S0169-2607(00)00093-6

5

A. Portela, "Dual Boundary Element Analysis of Crack Growth", In C.A. Brebbia, J.J. Connor (Editors) Topics in Engineering - Vol. 14. Computational Mechanics Publications, Southampton, 1993.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)