3D Visualization of Radiation Treatment Plans

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3D Visualization of Radiation Treatment Plans

• Jenny Sager
• Friday, October 23, 2009
• 348 AM

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Organization

• Spent the summer learning
• LISP
• SLIK (a button package for LISP)
• CLX
• OpenGL
• PRISM
• Parallel Contour Reconstruction

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Organization

• PRISM overview
• The Tiling Problem
• Terminology
• Basic approaches
• Implementation in PRISM
• Limitations and future improvements

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PRISM

• PRISM - a radiation treatment planning system
  for designing plans for treating cancer patients.
• An satisfactory plan delivers enough radiation
  to a tumor without unacceptable damage to the
  surrounding tissues

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PRISM Contour Editor
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PRISM Typical Session
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PRISM

• It is difficult for a person to visualize complex
  2D data as 3D objects
• Currently PRISM has the following visualizations
• Parallel xy plane contours
• Beams eye view (from the radiation beams point
  of view)
• 2D planes
• Coronal (xz plane) from the top
• Saggital (yz plane) from the side
• Transverse (xy plane) from the front
• But PRISM does not have a 3D room view!

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Meyers Terminology

• Reconstruction of parallel contours can be broken
  into 4 sub-problems
• Correspondence problem
• Tiling problem
• Branching problem
• Surface-fitting problem
• (Meyers, Skinner, and Sloan Surfaces from
  Contours 1992)

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Correspondence Problem

• Which contours should be connected together by
  the surface?
• The question arises whenever there are multiple
  contours in a section
• Solution arrange contours into groups organized
  by which objects they represent
• Difficult to make automatic without making
  assumptions because of the ambiguities
• In PRISM this task is done by a human in the
  contour editor

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Tiling Problem

• How should the contours be connected?
• More about this topic later

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Branching Problem

• What do we do when there are branches in the
  surface?
• One solution is user intervention
• In PRISM, contours do not branch or intersect by
  system definition

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Surface-fitting Problem

• Once a triangulated mesh is found, how should a
  smooth surface be made that approximates or
  interpolates the vertices of the mesh and
  maintains the same topology?

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Basic Approaches - Volume (Grid) Based Approach

• Volume (Grid) Based - assumes that data is
  available as a 3D grid or lattice
• Voxel Technique (spatial equivalent of a pixel)
• Marching Cubes (Lorensen and Cline 1987)
• Surface Normals (Hoehne and Bernstein 1986 Levoy
  1988)
• Geometrically Deformed Models
• Miller et al. 1991

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Basic Approaches - Surface Based Approach

• Surface Based assumes that the data define a
  set of closed contours and attempts to
  reconstruct a sequence of surfaces, one between
  each pair of adjacent contours
• Growing Cylinder
• Soroka 1981 Generalized cylinder/elliptical
  cylinder (Correspondence problem)
• MST/Min Path
• Keppel 1975 (Tiling Problem)
• Fuchs, Kedem, Uselton 1977 (Tiling Problem)
• Meyers 1992, 1993 (Tiling Problem)

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Basic Approaches - Volume (Contour) Based

• Volume (Contour) Based assumes that data is a
  set of closed contours (same form as surface
  based) but attempts to construct the volume from
  the contours instead
• Delaunay triangulation
• Boissonnat (1986) and Geiger (1993) (Branching
  Problem)
• Capable of handling some branching structures
  without user intervention

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Fuchs, Kedem, and Uselton Algorithm, 1977

• Choose to implement the algorithm of Fuchs et al.
  in PRISM because
• PRISM contours do not branch or intersect
• Fuchs requires on 1-to-1 relationship between
  contours (no branching, or complicated shape
  variations
• Fuchs Algorithm is fairly simple and efficient
• O (n2 log n), where n is total number of vertices
  on the contours bounding the triangles
• Meyers describes several additional improvements
  in efficiency that can be made

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Fuchs, Kedem, and Uselton Algorithm, 1977

• Construct a graph shaped as a torus so that
• Points on Contour A are the Rows
• Points on Contour B are the Columns
• The intersection of a row and a column on the
  toriod graph will be called a node

Node
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Fuchs, Kedem, and Uselton Algorithm, 1977

• Thus a Torus node (i, j) represents the line
  connecting (Pt i of Contour A) to (Pt j of
  Contour B) , called a span

Node (i, j)
Span
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Fuchs, Kedem, and Uselton Algorithm, 1977

• A Single Triangle
• Toroidal Graph 2 nodes and an arc
• Contours 2 spans and 1 contour segment

Span
Contour Segment
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Fuchs, Kedem, and Uselton Algorithm, 1977

• Horizontal Triangle Fan

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Fuchs, Kedem, and Uselton Algorithm, 1977

• Vertical Triangle Fan

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Fuchs, Kedem, and Uselton Algorithm, 1977

• Triangle Strip

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Fuchs, Kedem, and Uselton Algorithm, 1977

• Find the minimum cost Eulerian Trail (a closed
  path in which every arc of the graph occurs
  exactly once) by
• Find each Eulerian Trail that starts at node (i,
  0) for i 0 to ( of vertices in Contour A 1)
• The minimum of these paths is the graphs
  minimum path

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Fuchs, Kedem, and Uselton Algorithm, 1977

• Found a min path using Dijkstras algorithm
• Divide and Conquer
• Use the previously found cycles to limit the
  nodes to search for next
• A cycle starting at node (i, 0) is upper bounded
  by the cycle starting at (i - m) and lower
  bounded by the cycle at (i n) where m, n are
  positive integers
• (sharing nodes with boundary cycles is ok)

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A Few Metric Functions

• How should the edges be weighed?
• Volume (Keppel)
• Difficult to calculate the contribution of a
  single tile to total volume
• Minimum Surface Area (Fuchs et. al) -
• Easy to compute (Area of Triangle ½ bh)
• Match direction
• Matching the directions of points from their
  respective contours center of mass
• Problematic for concave objects

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Implementation in PRISM
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Implementation in PRISM
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Implementation in PRISM
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Implementation in PRISM

• Orthographic/perspective option
• Wireframe/Translucent/Opaque option
• Navigational buttons translate, rotate
• Coronal, Sagittal, and Transverse pre-set buttons
• Scaling
• Ruler (imported from its implementation in other
  views in PRISM)
• Cut Plane that cuts off where z lt user-input
• Lighting (partially implemented)

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Limitations of Fuchs et al.

• Tiling Problem
• The minimal area constraint will result in a
  surface being constructed containing two cones,
  joined at a line, rather than a cylinder
• Calculations are slow

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Future Improvements

• Solution to skewed cylinder problem (Sloan et
  al.)
• Normalize the two cross-sections so that their
  centers lie on an axis perpendicular to the plane
  of section
• Solve the tiling problem to find lines joining
  points on the contours
• Then use this correspondence to construct a
  surface from the original contours

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Future Improvements

• Improvement in Speed (Meyers)
• Claims to have modification to Fuchs et al. that
  appears to require O (n) space and O ( n log n)
  time
• The original Fuchs et al. algorithm had O (n2)
  space and O ( n2 log n) time

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Future Improvements

• Things to add to PRISM
• Option to show a specific coronal CT scan cutting
  though at the users chosen cut plane
• Cut plane is implemented
• Texture mapping of the CT scan is NOT implemented
  yet
• Ability to render/draw beams
• Smooth surface (surface-fitting problem)
• Support for color index mode

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Additional Problems

• Gitlin, ORourke, and Subramanian proved that
  (1993)
• It is not always possible to find an
  interpolating polyhedron between two polygons
  that lie in parallel planes
• If the shape of of the contours differs
  sufficiently, no simple polyhedron can be
  constructed
• However, in practice adjacent contours are
  usually fairly similar in shape
• Exceptions exist

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Additional Problems

• A Collection of triangles intersect properly if
  any given pair of triangles are either
• Disjoint
• Have one vertex in common
• Have two vertices and consequently, the entire
  edge joining them
• In this definition, adding extra vertices is not
  allowed
• (def. From Giblin)

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Additional Problems

• Example of Improper Triangles

It is necessary to add vertices to make the
triangles not share interior points
These triangles share interior points
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Conclusion

• Questions? Comments?