High Performance Computing in Surgical Simulation' An Approach Using Compact Support Radial Basis Fu

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High Performance Computing in Surgical
Simulation. An Approach Using Compact Support
Radial Basis Functions

• Mark Wachowiak, Xiaogang Wang, Aaron Fenster,
  Terry Peters
• Imaging Research Laboratories
• Robarts Research Institute
• London, ON N6A 5K8

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Soft Tissue Modeling and Surgical Simulation

• Surgical simulation for
• Pre-operative planning
• Training
• Developing new procedures

• Basic biomedical research
• Tissue properties
• Response to therapy

One of the goals of surgical simulation is to
facilitate planning of minimally invasive
procedures and to transfer the results to the
operating room. Simulation, like robotics and
tracking systems (shown above), provides great
benefits to minimally invasive surgery.
Biomechanical properties are incorporated into
surface and volumetric models generated from real
patients data/images. The resulting physical
model can then be used in pre-procedural planning
or for simulation for surgical training.
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Soft Tissue Modeling Approaches

• Geometric models
• Advantages
• Fast computation time ? real time performance can
  be obtained.
• Relatively easy to implement.
• Visually compelling.
• Disadvantages
• Do not take physical properties into account.
• May be very inaccurate for complex tissue.
• Haptic feedback is empirical.
• Examples
• Free-form deformation.
• Chain-mail.
• Sphere-filled models.

• Physical models
• Advantages
• Very realistic.
• Can be used to study tissue behaviour .
• Haptic feedback computations are straightforward.
• Disadvantages
• Very slow.
• Much pre-processing is required
• Relatively difficult to implement.
• May also be inaccurate for large topology changes
  (cutting, suturing).
• Examples
• Finite elements (widely-accepted gold
  standard).
• Mass-spring models (a little better speed).
• Hybrid geometric-physical approaches.

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Radial Basis Functions (RBFs)
Compact Support Radial Basis Functions (CSRBFs)

• A class of basis functions that can be used to
  model soft tissue deformations.
• Thin plate splines are also widely used for this
  purpose
• Many such functions exist, including
  multiquadratics and Gaussian functions.
• Can provide visual realism in surgical
  simulation, but often requires parameters to
  model biomechanical properties.
• RBF methods have an inherent parallelism.
• Very often have an undesirable global effect
  the whole tissue deforms when only a small local
  area is perturbed.

• Proposed by H. Wendland in 1995
• Unlike other RBFs or spline functions, CSRBFs
  have a local effect.
• Locality is controlled by a user-specified
  parameter.
• CSRBF matrices are positive definite, and are
  therefore guaranteed to be invertible.
• CSRBF matrices are also often sparse, and can be
  inverted with specialized methods.
• Unlike other RBF and spline functions, no
  polynomial terms are required.

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Examples of CSRBFs
Quadratic
4th order
6th order
Logarithmic/4th order
r gt 0
otherwise
Plots of CSRBFs
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CSRBFs in Soft Tissue Modeling

• Landmarks
• Placed on (virtual) surgical instruments, and on
  the (virtual) surface and in the interior of
  tissue.
• Collision detection of the tissue surface and
  surgical instrument select source landmarks.
• As the instrument deforms the tissue, the source
  landmarks (from collision detection) move toward
  the new target landmarks on the surgical tool,
  causing a deformation.

• Tissue properties
• Stiffness modeled by locality parameter (a).
• Elasticity (shape of deformation) modeled with
  smoothness of CSRBF.
• Internal tissue landmarks also help to simulate
  stiffness.

applied force
surface
Stiffness is modeled by the locality parameter
(width of deformation) and by internal landmarks.
Elasticity is modeled by shape of RBF function.
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Example 2D deformation with 4 control points and
10 internal landmarks
Colors indicate the density, measured in terms of
surface area of the material after deformation.
Red values indicate higher compression,
corresponding to higher density after deformation.
y2, a 8
y6, a 8
y2, a 12
y6, a 12
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Example 2D deformation with 4 control points and
10 internal landmarks
Colors indicate the density, measured in terms of
surface area of the material after deformation.
Red values indicate higher compression,
corresponding to higher density after deformation.
y2, a 20
y6, a 20
y2, a 40
y6, a 40
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Tissue Deformation Experiments

• The efficacy of this deformation model is
  demonstrated on data from a 3D prostate image for
  the application of needle insertion for
  implanting radioactive seeds for brachytherapy.
• This procedure is minimally invasive. A critical
  factor in the success of this procedure is
  accurate seed delivery based on a dosimetric plan
  that maximizes destruction of the cancerous
  cells, while minimizing damaging to healthy
  tissue.
• The online simulation involves
• Interactive visual simulation (30Hz)
• Haptic feedback (1 KHz)
• TCP/IP-based communication.
• Soft tissue model
• Volumetric prostate generated from a
  pre-operative 3D ultrasound volume and its
  segmented boundary.
• 8 internal landmarks.
• 14,560 3D points, including 1,200 on the surface.
  
• A simple collision detection algorithm was used
  to identify the landmark corresponding to the
  contact point between the needle tip.

• Timing experiments were run on a 4-CPU
  shared-memory 1.0GHz HP/ Compaq ES45 system.
• The following timing experiments were performed
• Four (4) CSRBF functions
• From 1 to 41 landmarks
• Three support sizes
• a 1 (sparse matrix)
• a 10
• a 20 (dense matrix)
• All timing experiments were performed on the
  prostate data, as described above.
• 100 trials were performed for each experiment.
  Mean values are reported.
• The interpolation over all 3D points was
  parallelized.
• Parallelization was performed with OpenMP.

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Timing Results
y2, a 10
y4, a 10
ylog, a 10
1 CPU
1 CPU
1 CPU
4 CPUs
4 CPUs
4 CPUs
y2, 1 and 4 CPUs
y4, 1 and 4 CPUs
ylog, 1 and 4 CPUs
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Deformation Results
Target for seed placement
Mesh representations
y4, a 6 (local)
Mesh representations
y4, a 12
With the large locality parameter (a 24), the
deformation effect was more global, and the
entire prostate base was pushed inward.
Mesh and surface representations
y6, a 24 (global)
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Discussion

• The prostate deformations were visually
  realistic, with the shape controlled by the
  locality parameter and CSRBF selection.
• The method scales well for all CSRBFs.
• As expected, the least complex CSRBF (y2) had the
  highest time performance. The most complex CSRBF
  (ylog) was the most time-complex. The
  performance of y4 was marginally better
  performance than that of y6.
• Using 4 CPUs, there is only a marginal difference
  between computation time for the sparse and dense
  matrices.
• Small glitches in the computation times for all
  experiments for 12 and 32 landmarks is most
  likely due to caching effects on the specific
  architecture used.

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Conclusions

• For applications where real-time performance is
  required, compact support radial basis functions
  provide visually compelling deformations for soft
  tissue modeling.
• These methods are also easily parallelized, and
  scale well on shared memory architectures.
• Future work includes
• Parallelizing more of the simulation, including
  CSRBF function evaluation and sparse matrix
  inversion.
• Applying RBFs and CSRBFs within a finite element
  framework, incorporating physical parameters.
  These techniques are known as mesh-free methods,
  and address many of the problems inherent in
  classical finite element approaches.
• The methods must be tested on a wide variety of
  clinical data, including non-convex and
  non-homogeneous tissue.
• Clinical validation is required.

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Acknowledgements
The authors gratefully acknowledge Ravi Gupta,
Jeff Gardiner, and Baolai Ge for technical and
scientific support, Drs. Gerard Guiraudon, Renata
Smolíková-Wachowiak, and Hualiang Zhong for
helpful discussions, and the Virtual Cardiac
Surgery Planning (VCSP) group at the Imaging
Research Laboratories, Robarts Research
Institute Martin Wierzbicki, Guy-Anne Turgeon,
Dr. Stan Szpala, and John Moore. Funding for
this project was provided by SHARCNet, NSERC
(R3146-A02), and CIHR (MT 14735, MT 11540, and
MGP 49536).