A3 : Laboratory for Human and Machine Haptics, Department of Mechanical Engineering

Contact information

A1 : Kim

A2 : Jung

A3 : Laboratory for Human and Machine Haptics, Department of Mechanical Engineering,

A5 : 77. Mass. Ave. Massachusetts Institute of Technology, Cambridge, MA 01239

A6 : A Hybrid Modeling Scheme for Soft Tissue Simulation in Virtual Reality Based Medical Trainers

J. Kim1, S. De2, M. A. Srinivasan1

The TouchLab

1Dept. of Mechanical Engineering and Research Laboratory for Electronics

Massachusetts Institute of Technology, Cambridge, MA 02139C2 : All authors

Jung Kim, M.S., Dept. of Mechanical Engineering, Massachusetts Institute of Technology

Suvranu De, Sc.D., Dept. of Mechanical, Aerospace and Nuclear Engineering and the Faculty of Information Technology, Rensselaer Polytechnic Institute

M.A. Srinivasan, Ph.D., Dept. of Mechanical Engineering, Massachusetts Institute of Technology

2 Dept. of Mechanical, Aerospace and Nuclear Engineering,

Rensselaer Polytechnic Institute, Troy, NY12180

Abstract

This paper describes a hybrid modelinghybrid-modeling scheme of soft tissue deformation for a virtual reality based medical simulator. By using two different physically based models, it satisfies two competitive requirements of real time performance and accuracy. A local point collocation-based method of finite spheres is coupled with a global boundary element technique to capture local features of the interaction (e.g., nonlinearities of the soft tissue) without sacrificing global accuracy. The technique is demonstrated using realistic examples.

Keywords: Soft tissue simulation, Physically based model, Medical simulation, Force feedback

C3: Background / Proble

1.Introductionm:

To display realistic deformations and reaction forces in a virtual reality based medical simulator (see figure 1), accurate modeling of the soft tissue is necessary. However, simulation of soft tissue behavior in real time is challenging. In multimodal surgical simulation the visual loop must be updated at 30 Hz for real time graphics while a much higher update rate of around 1kHz is required for stable display force interactions [1].. This imposes severe restrictions on the complexity of the models that can be rendered in real time. Therefore developing an efficient computational model without sacrificing fidelity is one of the major issues in surgical simulation.

Various techniques can be found in literature for the simulation and display of deformable objects [2]. Among them physicallythem physically based approaches, like the finite element method, which model the underlying physics of deformable objects have drawn attention of researchers primarily due to the fact that tool-tissue interactions may be modeled with accuracy and robustness. Bro-Nielson [3] and Cotin at el. [4] developed a simulation system to utilize three dimensional solid finite element models (FEM) based on linear elasticity. In these systems, real time performance was achieved by the use of condensation, and precomputation of the stiffness matrix governing tissue behaviors. James and Pai [5]modeled real-time quasi-static deformations using the boundary element method (BEM) and achieved real time performance for haptic interaction using low rank updates of the stiffness matrix, although they did not apply their technique to surgical simulation.

The physically based techniques found in literature are inherently computationally expensive and real time implementation is not possible without extensive precomputations or gross simplifications. It is well known that soft tissue behavior is highly nonlinear, especially in the vicinity of the surgical tooltip where the deformations are large, and surgerical procedures almost always involve cutting.

In section 2 we describe a hybrid modeling approach to achieve real time performance in surgical simulation without sacrificing computational accuracy. In section 3 we discuss numerical implementation and some software issues and demonstrate a realistic example. C4: Method:

2. Hybrid modeling strategy

We had presented a localized point collocation-based method of finite spheres technique in [6] where only the vicinity of the tooltip was discretized using a novel meshfree technique. The rest of the domain was, however, assumed to be rigid. This technique is quite appealing when the size of the organ is rather large and surgical cutting is not performed. The major advantage of this technique is that it is not limited to linear tissue behavior and real time performance may be obtained without using any precomputations.

However, when the size of the organ is small and surgical cutting needs to be performed then the assumption of zero deformation outside the “region of influence” of the tool is not quite accurate. We have therefore developed a novel hybrid modeling paradigm which allows us to incorporate local nonlinearities as well as local changes in mesh topology by coupling the localized point collocation-based method of finite spheres with a linear global model. This is a multi-rate simulation approach and the global model is updated at a lower rate than the localized point collocation model (see Figure 2).

2.1 The global model

We use an integral formulation of the equations of linear elastostatics posed on the surface of the computational domain, discretized using the boundary element method (BEM)[6][1ref of BEM book]which. The integral equation formulation reduces the dimensionality of the problem by one and only the surface, instead of the volume, is discretized. The same triangular surface mesh used for rendering the geometry of the organ can be used as the boundary element mesh. The main problem associated with the integral equation formulation is, however, that dense global stiffness matrices are generated.

The displacement () and traction () vectors at a point x in the domain or on the boundary may be represented by their three Cartesian components

(1)

Using piece-wise constant elements (i.e., the displacements and tractions are assumed to be constant over each element), the BEM equations of linear elastostatics with ‘E’ elements is given by [6]

(2)

where iis the surface of the ith element, and are the ‘fundamental solutions’(see [6]). The coefficient ‘c’ depends on the smoothness of the boundaries and can be found in literature (for a Lipschitz boundary, c = 0.5). Satisfying this equation at the centroids of each of the elements and incorporating the boundary conditions, we obtain the following system of linear algebraic equations:

(3)

where Y is a vector of length ‘N’ and contains the unknown deformations and tractions at the centroids of the boundary elements. A is ‘N’ by ‘N’ dense. F is the known right hand side vector. The solution of this system is symbolically represented as,

(4)

Usually, computing the inverse of A is an O( N3 )process therefore it is expedient to precompute it. A structural reanalysis technique using the Sherman-Morrison-Woodbury formula is used (a similar technique has been described in ref. [5]) to incorporate the change in boundary conditions. We note here that the global model is linear and it is not possible to achieve real time haptic updates without precomputation.

2.2 The local model

To enhance the accuracy in the vicinity of the tool-tissue interaction region, we couple the global boundary element model with A structural reanalysis technique is used to compute fast updates.. the localized point collocation based method of finite spheres [7][2ref of MMVR2000 paper]. This technique uses a sprinkled set of nodal points around the tooltip to perform discretization. An “influence zone” is associated with each nodal point. The approximation uh of a variable u (e.g. displacement), using ‘M’ spheres, may be written as

(5)

where is the nodal unknown at node J. The nodal shape function at node J is generated using a moving least squares technique (see [7]). We satisfy the equations of motion only at the nodal points to obtain the discretized set of equations

(6)

where K is the stiffness matrix and f is the vector containing nodal loads. A point to note is that the formulation is very general and can account for nonlinearities. K represents the tangent stiffness matrix if an incremental analysis is performed for nonlinear problems. The number of nodal points in the local model is much smaller than the number of nodal points in the global model and therefore only a small set of equations have to be solved at every time step.

As an example of our hybrid modeling approach, we simulate the palpation of a kidney model

3. Real time computatio. Figure 3 shows the deformation of a human kidney pulled by a virtual laparoscopic tool. It shows realistic deformations, which reflects underlying soft tissue mechanics in real time.

4. 3. Conclusion

C5: Results:

In this paper, we propose a hybrid modeling approach to compute the deformations deformable organs in multimodal medical simulations. We used two complementary physically-based modeling schemes. While the global model is linear and can be updated rapidly using a structural reanalysis technique, the local model can handle nonlinearities in tissue behavior quite effectively. We are able to model complicated behavior of biological tissues in real time by using this approach. We demonstrate the effectiveness of the method using realistic examples.

C6: Concluding remarks:

Reference

[1]M. A. Srinivasan and C. Basdogan, “Haptics in Virtual Environments: Taxonomy, Research Status, and Challenges.,” Computer & Graphics, vol. 21, pp. 393-404, 1997.

[2]C. Basdogan, C. Ho, and M. A. Srinivasan, “Virtual Environments for Medical Training: Graphical and Haptic Simulation of Laparoscopic Common Bile Duct Exploration,” IEEE/ASME Transactions on Mechatronics, vol. 6, pp. 269-285, 2001.

[3]M. Bro-Nielsen, “Finite Element Modeling in Surgery Simulation,” Proceeding of IEEE, vol. 86, pp. 490-503, 1998.

[4]S. Cotin, H. Delingette, and N. Ayache, “Real-time elastic deformations of soft tissue for surgery simulation,” IEEE Trans. On Visualization and computer graphics, vol. 5, pp. 62-73, 1999.

[5]D. James and D. K. Pai, “ArtDefo, Accurate Real Time Deformable Objects,” presented at Computer Graphics (ACM SIGGRAPH 99 Conference Proceedings), 1999.

[6]C. A. Brebbia, J.C.F.Telles, and L. C. Wrobel, Boundary Element Technique: Theory and Applications in Engineering. New York: Springer-Verlag, 1984.

[7]S. De, J. Kim, and M. A. Srinivasan, “A Meshless Numerical Technique for Physically Based Real Time Medical Simulations,” presented at Proceeding of MMVR 2001, 2001.

Figure 1

The components of a real time medical simulation system.

Figure 2

Schematic of the modeling strategy for real time rendering of deformable objects

Figure 3

The deformations of a kidney model using the hybrid modeling scheme.