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### Medical Simulation

OutlineOutline

Talk by Lisa Lyons

Surgery Simulation Requirements

- Realistic visualization of internal organs
- Organs react realistically in real time to:
- User interactions
- Environmental restrictions
- Organs react to typical surgeon’s gestures through geometric and topological modifications

Surgery Simulation Grouping

- First generation:
- Only deal with geometric nature of human anatomy
- Second generation:
- + permit physical interactions with anatomy
- Include needle-type, exploration-type, catheter installation-type simulators as well as simulators that permit training in only one task and full simulators
- Third generation:
- + consider functional nature of organs

Outline

- Physical Modeling
- Reduction of Computing Time
- Collision Detection
- Example Systems
- Results and Conclusion

Anatomical Model of the Liver

- Data set consists of about 180 slices of frozen human tissue that has been put through CT scan
- Enhance contrast
- Apply edge detection
- Semi-automatic deformable models → binary images
- Stack images to form 3D binary image [Montagnat, 1997]

Simplex Meshes

- Better than marching cubes – avoids “staircase effects
- Developed by Delingette to represent 3D objects [Delingette, 1994]
- Adaptable (figure to right)
- Working on a method to extract liver models from CT images

Force Feedback

- How physically realistic the model is correlated with how realistic force feedback is
- Model deforms with surgeon’s motion
- Contact force may be computed from deformation
- Force generated back to surgeon through mechanical actuators

Method uses linear elasticity as an approximation for tissue deformation

- Let the configuration of an elastic body be defined as Ω
- A field of volumetric and surface forces f acts on the body so it has a new configuration Ω*
- We want the displacement field u which associates the initial configuration of any particle with its final configuration
- Use FEM – Lagrange elements of type P1 [Bathe, 1996]
- Formulate the problem as a linear system
- Where [K] is the 3n by 3n stiffness matrix and n is the number of mesh vertices (more on this in a minute)

Only thing we know is endoscope position

- must use displacement not force constraints
- Given some displacements between the surgical tool and the body, we can find
- Force on end effecter
- Global deformation
- Now we use variational formulation and Lagrange multipliers to minimize
- Include constraints u = u*
- Solving for λi gives the opposite of the necessary forces to impose the displacement u*
- See Appendix A [Cotin, 1999] for full derivation

Stiffness matrix containing

3X3 “mini-matrix” of stiffness information

for each node

Matrix composed of a 3X3 identity

matrix for each constrained segment (k)

Forces required to obtain

desired state

Desired displacements

of k nodes

Linear Representation

- In theory, this behavior is only physically correct for small displacements
- Force feedback limits the range of deformations
- Feedback force on surgeon’s hand will increase as deformation increases

Quasi Non-Linear Representation

- Mix of linear representation and empirical results using a cylindrical piece of brain tissue
- [Chinsei, 1997] found that deformation depends on loading speed and is nonlinear

Outline

- Physical Modeling
- Reduction of Computing Time
- Collision Detection
- Example Systems
- Results and Conclusion

Computation Time

- Number of mesh vertices has high impact
- Makes matrices larger
- Must use speedups
- Cannot make necessary calculations in real-time

Pre-Computation Algorithm

- Specify a set of nodes to remain fixed
- Don’t have to set all three dof
- For every “free” node k and degree of freedom on the surface, emplace an “elementary” displacement constraint (δ)
- Denote this as
- Compute the displacement of every free node n in the mesh with respect to every node k
- Store as set of 3X3 tensors
- Compute elementary force at each constrained node k
- Store as 3X3 tensors

Solving The Linear System

- Must be solved 3m times where m is the total number of free nodes inside the tetrahedral mesh
- Can take anywhere from a few minutes to several hours

Linear Elasticity

- For any n where k≠ n, the relation between n and k is
- Superposition may be used to find the total displacement of a node but some modifications must be made

Use tensors of deformation found in preprocessing to generate a vector of modified constraints

where

and

From this, we can find the displacement of any node

- The force that must be applied to each node k to produce these displacements is

Quasi Non-Linear Elastic Deformations

- Computing times for a realistic looking liver model:

Outline

- Physical Modeling
- Reduction of Computing Time
- Collision Detection
- Example Systems
- Results and Conclusion

Collision Detection

- Work discussed so far uses bounding boxes with a hash table
- We know about these so lets move on to a new problem – simulating the folds of the intestines

Simulating Intestines

- Goal is simulator to allow doctors to practice a surgery that involves pulling and folding the intestines [Raghupathi L. et. al., 2003]
- Real problem here is self-collsions
- Complicated by tissue called mesentery
- Connects small intestine and blood vessels

Model

- Resting position:
- Intestines look like folded curves lying in a cylinder
- Mesentery is defined as line segments connecting folded intestine to the axis of the cylinder
- Mechanical model uses masses and springs

Collisions Between Intestines

- Model intestines like cylinders
- Find distance between their principle axes
- “Active pairs”
- Local minima satisfying certain distance threshold
- Updated every time step
- N additional random pairs of segments also generated every time step
- These are tested and thrown out if they are over the threshold or already represent a minimal pair

Mesentery Collisions

- Complexity would be too high for real-time without approximation
- Don’t consider mesentery-mesentery interactions
- Adaptive convergence
- Replace segment S1 by closest neighbor S to S2 and then replace S2 with neighbor closest to S
- When collision occurs, recursive search begins across neighbors

- Physical Modeling
- Reduction of Computing Time
- Collision Detection
- Example Systems
- Results and Conclusion

GeRTiSS

- The Generic Real Time Surgery Simulator [Monserrat et al., 2003]

Scene Generator

- Allows user to select tools and organs needed
- Systems contains modeling parameters for a variety of organs
- Mass-spring model
- Boundary element based model (BEM)

Scene Generator

- Tools:
- Loading organs
- Establishing input points for instruments
- Associating different physical properties with organs
- Establishing boundary conditions
- Linking tissues
- Adding special tissues
- Associating textures to organs

Surgery Simulator

- Takes a scene and allows user to train
- User can have interaction with organs:
- Cut
- Cauterize
- Drag
- Clip
- User can exchange instruments
- User is assessed at the end based on how many incorrect actions were taken

Results

- Use 450 MHz Pentium III with 256 MB memory
- Computational Costs:

Haptics

- For good visual image 15Hz refresh rate
- For good haptic stimulus 500 Hz refresh rate
- Use a PC cluster to solve this
- Cost of force feedback devices makes simulator 4X more expensive than without

Cataract Surgery Simulation

- Surgery aims to extract cataract and replace it with intraocular lens [Agus et al., 2006]
- Training is important
- Simulation allows:
- Flexibility
- Gradual increase in difficulty
- Exposure to rare events
- Quantification of performance

The Procedure

- Phacoemulsification: breaking hardened lens into fragments and removing them with a small sucker using the phacoemulsificator
- Create z-shaped corneal tunnel
- Capsulorhexis: removing the anterior capsule to uncover the upper surface of the crystalline

Methods

- Decoupled simulation:
- Fast subsystem for surgical instrument tracking and slower one for visual feedback
- Slow subsystem does global simulation and interaction of devices and eye
- Slow subsystem can be further broken into individual visual effects
- Force feedback is useless in this surgery
- Must use eye globe visualization
- Conjugate gradient to minimize energy constraints gives equilibrium position
- Rotate to reduce deformation

Capsulorhexis Simulation

- Use triangular mesh with a mass-spring network mapped over it
- Mass particles may be anchored, scripted or free
- Gravity, viscosity and springs contribute to acceleration
- Weak springs simulate sticking effects
- Solve ODE using semi FSAL (First Same as Last)
- Velocity found using implicit method and feedback on position is computed explicitly
- Correction routine applied after each step to correct position and velocity as required by constraints
- Tearing – breaking overextended springs

Phacoemulsification Simulation

- Lens as collection of simplices
- Tetrahedron mesh with particles placed at barycenters
- Links connecting particles maintained for rendering and determining independent particles
- Photoemulsificator modeled by eroding particles in a zone of influence
- Employ Russian roulette scheme to decide which particles to erode
- When particles are removed, simplicial mesh is updated
- Idea is to replace energies by geometric constraints and forces by distance from current position to goal
- Each connected subset of points is associated with a point cloud
- Shape matching with undeformed rest state to determine goal positions

- Physical Modeling
- Reduction of Computing Time
- Collision Detection
- Example Systems
- Results and Conclusion

Appendix A – Collision Response

- Tried penalty and constraint methods but stability of the system was reduced
- Instead alter displacement velocities to avoid penetration

Appendix A (cont.)

- Interpolating:
- Need force f’ = f so we have:
- New velocities are:
- Substituting we get:

Appendix A (cont)

- Solving for f gives:
- Condition for avoiding penetration takes radii into account:
- The force required to change the positions of the endpoints to satisfy these conditions is:

References

- Marco Agus, Enrico Gobbetti, Giovanni Pintore, Gianluigi Zanetti, and Antonio Zorcolo. Real-time Cataract Surgery Simulation for Training. In Eurographics Italian Chapter Conference. Eurographics Association, 2006.
- K.-J. Bathe, Finite Element Procedures. Prentice Hall, 1996.
- K. Chinsei and K. Miller, “Compression of Swine Brain Tissue Experiment In Vitro,” J. Mechanical Eng. Laboratory, pp. 106-115, 1997.
- S. Cotin, H. Delingette, and N. Ayache. “A Hybrid Elastic Model allowing Real-Time Cutting, Deformations and Force-Feedback for Surgery Training and Simulation.” The Visual Computer, 16(8):437-452, 2000.
- Cotin, S.; Delingette, H.; Ayache, N., "Real-time elastic deformations of soft tissues for surgery simulation," Visualization and Computer Graphics, IEEE Transactions on , vol.5, no.1, pp.62-73, Jan-Mar 1999
- H. Delingette, ”Simplex Meshes: A General Representation for 3D Shape Reconstruction,” Technical Report 2214, INRIA, Mar. 1994.
- Y.C. Fung, Biomechanics-Mechanical Properties of Living Tissues, second ed. Springer-Verlag, 1993.
- Carlos Monserrat, Oscar López, Ullrich Meier, Mariano Alcañiz Raya, M. Carmen Juan Lizandra, Vicente Grau: GeRTiSS: A Generic Multi-model Surgery Simulator. IS4TH 2003: 59-66
- J. Montagnat and H. Delingette, “Volumetric Medical Images Segmentation Using Shape Constrained Deformable Models,” Proc. First Joint Con5 CVRMed-MRCAS ’97, J. Troccaz, E. Grimson, and R. Mosges, eds. Mar. 1997.
- M. Moore and J. Wilhelms, “Collision Detection and Response for Computer Animation,” Computer Graphics (SIGGRAPH ’88), vol. 22, pp. 289-298, Aug. 1988.
- Laks Raghupathi, Laurent Grisoni, Fran?ois Faure, Damien Marchal, Marie-Paule Cani, Christophe Chaillou, "An Intestinal Surgery Simulator: Real-Time Collision Processing and Visualization," IEEE Transactions on Visualization and Computer Graphics, vol. 10, no. 6, pp. 708-718, November/December, 2004.

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