Two finite-element methods have been developed for mathematical modeling of the time-dependent behaviors of deformable objects and, more specifically, the mechanical responses of soft tissues and organs in contact with surgical tools. These methods may afford the computational efficiency needed to satisfy the requirement to obtain computational results in real time for simulating surgical procedures as described in "Simulation System for Training in Laparoscopic Surgery" (NPO-21192) on page 31 in this issue of NASA Tech Briefs.

Simulation of the behavior of soft tissue in real time is a challenging problem because of the complexity of soft-tissue mechanics. The responses of soft tissues are characterized by nonlinearities and by spatial inhomogeneities and rate and time dependences of material properties. Finite-element methods seem promising for integrating these characteristics of tissues into computational models of organs, but they demand much central- processing- unit (CPU) time and memory, and the demand increases with the number of nodes and degrees of freedom in a given finite-element model. Hence, as finite-element models become more realistic, it becomes more difficult to compute solutions in real time.

In both of the present methods, one uses approximate mathematical models — trading some accuracy for computational efficiency and thereby increasing the feasibility of attaining real-time update rates. The first of these methods is based on modal analysis. In this method, one reduces the number of differential equations by selecting only the most significant vibration modes of an object (typically, a suitable number of the lowest-frequency modes) for computing deformations of the object in response to applied forces.

The second method involves the use of the spectral Lanczos decomposition to obtain explicit solutions of the finite-element equations that describe the dynamics of the deformations. The explicit solutions are used to generate an "impedance map" of the object: this involves the precomputation of displacement fields (in effect, a look-up table), each field being the response to a unit load along each nodal degree of freedom. Thereafter, the deformation of an object is computed as a superposition of the individual responses of the nodes. In computing the response of a given node, one uses the responses of only those neighboring nodes that lie within an arbitrary radius of influence. This method is suitable for a linear (but not for a nonlinear) finite-element model of tissue.

This work was done by Cagatay Basdogan of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com/tsp under the Information Sciences category.

This software is available for commercial licensing. Please contact Don Hart of the California Institute of Technology at (818) 393-3425. Refer to NPO-21190.