Medial Techniques for Automating Finite Element Analysis
Jessica Crouch
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Motivation Deformation Modeling Aim: Model soft tissue
deformation Applications include Medical simulation, surgical
planning Tomotherapy Non-rigid registration of 3D medical
images
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Motivation Physically based deformable models Partial
differential equations (PDEs) model the deformable behavior of
materials Establish stress / strain relationship Finite element
method solves PDEs for discretized object models
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Motivation Applications of FEM in Medical Imaging Non-rigid
registration Prostate Bharatha, Hirose, et al. Brain Ferrant,
Warfield, et al. Breast Azar Motion tracking Heart wall
Papademetris, Shi, et al. Simulation Facial surgery Chabanas and
Payan Liver surgery Cotin, Delingette, Ayache Childbirth Lapeer and
Prager
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Motivation Finite Element Method (FEM) Model geometric
properties Discretize space with a mesh composed of Nodes Elements
Boundary fitted
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Motivation Finite Element Method (FEM) Model physical
properties Choose equations & coefficients that describe the
material's deformability Assemble the finite element system of
equations
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Motivation FEM for Medical Image Applications Steps include
Segmentation Mesh creation Equation and coefficient selection
Boundary condition specification Deforming forces, displacements
Solution Labor Intensive, Computationally Intensive Automate using
m-rep model framework
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Planning Image Imaging probe deforms prostate Intra-operative
image Prostate is relatively undeformed Motivation Prostate
Registration Problem
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Thesis Statement M-rep based multiscale mesh generation, M-rep
derived boundary conditions, and Multiscale solution of a finite
element system of equations are techniques that improve the
automation and efficiency of finite element analysis as it is
applied to medical imaging applications and to the prostate
brachytherapy application in particular.
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Outline Motivation & Overview FEM model construction M-reps
& image segmentation Mesh construction Finite element system of
equations Boundary conditions Solution Results for phantom prostate
image registration Conclusions & Future Work
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FEM Model Construction Automation of FEM for Imaging M-rep
models Medially based solid models Provide wealth of shape
information Global Local Facilitate segmentation, meshing, boundary
condition, and solution steps of FEM
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FEM Model Construction M-rep Models Objects are decomposed into
parts based on medial sheet branching Each branch of a medial sheet
is represented by a figure Hierarchical tree of figures is
organized by branching structure
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FEM Model Construction M-rep Models A figure consists of Single
medial sheet Functions defined on the medial sheet Radius Boundary
direction vectors Boundary displacement vectors (small) Frame
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FEM model construction M-rep Models Discrete Representation
Each figure sampled by lattice of medial atoms Lattice structure
provides (u,v) coordinate system on medial sheet
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FEM Model Construction M-rep Model Visualization Adjusting the
m-rep parameters stored in each atom affects the models
geometry
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FEM Model Construction M-rep Model Visualization A multi-figure
m-rep object consists of multiple parts, each represented by a
separate medial sheet A row of hinge atoms connects a subfigure to
its host figure
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FEM Model Construction M-rep Object Coordinate System (u,v,t, )
coordinates parameterize an m-rep model Rotating, scaling,
deforming an m- rep model changes its (u,v,t, ) (x,y,z)
mapping
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FEM Model Construction M-rep Based Image Segmentation Pablo
program Builds a new m-rep model or Adjusts an existing m-rep to
fit an object in a 3D image Optimizes atoms (medial sheet position,
radius function, boundary function, etc.) to maximize image match
expected geometry Works well with clear boundaries, still being
improved
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FEM Model Construction M-rep Segmentation Demonstration
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Outline Motivation & Overview FEM model construction M-reps
& image segmentation Mesh construction Finite element system of
equations Boundary conditions Solution Results for phantom prostate
image registration Conclusions & Future Work
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FEM Model Construction Mesh Construction Requirements Element
choices Shape Tetrahedra Hexahedra (preferred) Pyramids, wedges,
etc.
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FEM Model Construction Mesh Construction Requirements Elements
must not be overly skewed Element size should fit the Geometric
detail of an object region Solution precision needed in an object
region Meshes typically must be seamless Element face
compatibility
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FEM Model Construction Mesh Construction Top-down approach to
hexahedral mesh design Based on m-rep models Mesh generated in
m-rep object coordinate system, then mapped to world space
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Step 1: Construct a sampling grid on the (u,v) parameter plane
of the medial surface Spacing depends on object radius, and is
chosen to give elements approx. equal edge lengths in all
directions FEM Model Construction Mesh Construction: Single
Figure
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Step 2: Compute coordinates for other layers of nodes, using
illustrated meshing pattern. Result is desirable hexahedral mesh.
FEM Model Construction Mesh Construction: Single Figure
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Step 3: Optimize node locations to improve element shapes
Objective function is based on the determinant of the Jacobian of
the element shape function f( , , ) = (x,y,z) FEM Model
Construction Mesh Construction: Single Figure
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Quantitative evaluation of mesh quality Histograms of det(J)
for prostate mesh elements Left: pre-optimization Right:
post-optimization FEM Model Construction Mesh Construction: Single
Figure
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Mesh of single figure prostate m-rep model
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FEM Model Construction Mesh Construction: Single Figure 5
object male pelvis m-rep model mesh
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Mesh of space exterior to m-rep modeled objects necessary To
transmit forces between separate objects To compute a smooth
deformation field surrounding a modeled object Surrounding space
meshed with Pyramid layer on top of hexahedral elements Tetrahedra
fill remaining volume of interest generated by CUBIT FEM Model
Construction Mesh Construction: Single Figure
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Pyramid and tetrahedral elements for space external to m-rep
model
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FEM Model Construction Mesh Construction: Single Figure
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Must ensure smooth, compatible connection between host figure
and subfigure mesh elements FEM Model Construction Mesh
Construction: Multi-Figure
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Achieve compatibility by 1)Designing a host figures mesh so
that the mesh lines along its surface fit the footprint of a
subfigure 1)Designing a transition mesh pattern that fits between
the main bodies of the host and subfigure meshes FEM Model
Construction Mesh Construction: Multi-Figure
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Compute host / subfigure intersection in terms of Host figure
object coordinates Subfigure object coordinate FEM Model
Construction Mesh Construction: Multi-Figure
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Host mesh design: Fit subfigure footprint with Cartesian type
surface mesh Complete the surface mesh Interpolate interior nodes
between the surface nodes FEM Model Construction Mesh Construction:
Multi-Figure
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The mesh transition region must adjust the number of rows and
columns in the mesh pattern as well as switch between different
mesh pattern topologies. FEM Model Construction Mesh Construction:
Multi-Figure Avoid:
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Subfigure transition mesh is template based Template patterns
chosen based on the mesh patterns defined For the subfigure
footprint on the host surface Through a cross-section of the
subfigure FEM Model Construction Mesh Construction:
Multi-Figure
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Template patterns assembled in m-rep coordinate space, then
mapped to world space FEM Model Construction Mesh Construction:
Multi-Figure
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Outline Motivation & Overview FEM model construction M-reps
& image segmentation Mesh construction Finite element system of
equations Boundary conditions Solution Results for phantom prostate
image registration Conclusions & Future Work
Slide 42
Many constitutive models available Linear elastic Hyperelastic
Viscoelastic Viscous Fluid Linear elasticity chosen for prostate
registration experiment Methodology applies equally well for other
constitutive models FEM Model Construction Finite Element
Equations
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Linear elastic model Stress, , is proportional to strain, .
Linear elastic PDE: Elastic constants Youngs modulus Poissons ratio
FEM Model Construction Finite Element Equations
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Solution to the PDE is approximated on the mesh using element
interpolation functions Result is a linear system of equations The
full system of equations is singular FEM Model Construction Finite
Element Equations
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FEM Model Construction Mesh Construction: Single Figure
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Outline Motivation & Overview FEM model construction M-reps
& image segmentation Mesh construction Finite element system of
equations Boundary conditions Solution Results for phantom prostate
image registration Conclusions & Future Work
Slide 49
Boundary conditions take the form of force vectors or
displacement vectors applied to mesh nodes Displacement type
boundary conditions allow a finite element system of equations to
be reduced The displacement of at least one node must be specified
The reduced system of equations is non-singular and solvable FEM
Model Construction Boundary Conditions
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Force vectors or displacement vectors are not available
directly from images An image pair provides information about
changes in boundary shape FEM Model Construction Boundary
Conditions
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Use pair of m-rep segmentations to generate displacement type
boundary conditions M-rep correspondences are based on the shared
coordinate system of a pair of m-rep models FEM Model Construction
Boundary Conditions
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M-rep generated surface displacement vectors FEM Model
Construction Boundary Conditions
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M-rep correspondences are not necessarily physical
correspondences, so boundary condition optimization was tested
Surface correspondences were varied Potential energy of the
deformation was minimizedPotential energy FEM Model Construction
Boundary Conditions
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Optimization had a negligible effect on phantom prostate
deformation result Unoptimized m-rep generated boundary
displacements are sufficiently accurate for prostate image
registration Problems with larger deformations might benefit from
boundary condition optimization
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Outline Motivation & Overview FEM model construction M-reps
& image segmentation Mesh construction Finite element system of
equations Boundary conditions Solution Results for phantom prostate
image registration Conclusions & Future Work
Slide 57
For a 3D mesh with N nodes a 3N3N system of equations is
produced Reduced system is reduced by the number of boundary
conditions Solution options: Direct solution methods O(N 3 ) Use
iterative method with sparse matrix, get O(N 2 ) Use conjugate
gradient iterative solver for better convergence possibly as good
as O(N 9/8 ) FEM Model Construction Solution
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To improve solution accuracy, subdivide mesh elements Add nodes
at the midpoints of edges, quad faces, and hex volumes FEM Model
Construction Solution
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Subdivision with Euclidean world coordinates refines the
solution does not change the models geometric accuracy Subdivision
with m-rep object coordinates refines the solution refines the mesh
geometry FEM Model Construction Solution
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Mesh Subdivision Subdivision with m-rep object coordinates
Improved smoothness Mesh geometry more closely approximates m-rep
implied boundary with each subdivision
Multiscale 5 object pelvis mesh FEM Model Construction
Solution
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Mesh Subdivision Mesh size grows quickly with subdivision
Subdivision Improves the resolution of the model Increases solution
time Prostate mesh node and element counts for each subdivision
level:
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With iterative solution methods, an initial solution guess is
required Use coarse mesh solution to predict solution on a finer
mesh Interpolation performed in m-rep object coordinates rather
than world coordinates FEM Model Construction Solution
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Coarse-to-fine solution strategy improves solution efficiency
FEM Model Construction Solution
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Outline Motivation & Overview FEM model construction M-reps
& image segmentation Mesh construction Finite element system of
equations Boundary conditions Solution Results for phantom prostate
image registration Conclusions & Future Work
Slide 67
Planning Image Imaging probe deforms prostate Intra-operative
image Prostate is relatively undeformed Results Prostate Image
Registration Avg. seed movement: 9.4 mm Avg. movement of bottom
plane of seeds: 11.6 mm
Seed centers Blue: segmented from inflated image Green:
segmented from uninflated image, then moved by the computed
deformation
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Results Prostate Image Registration Results are averages for 75
seeds that were manually segmented in uninflated and inflated probe
images. The computed deformation was applied to uninflated seed
positions to map them into the inflated image. The difference
between mapped seed centers and seed positions identified in the
inflated image was measured. Segmentation error cannot be separated
from these error estimates
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Results Prostate Image Registration Image resolution in x and y
directions:.7mm Image resolution in z direction: 3 mm Resolution
limits segmentation accuracy, so a larger error estimate is
expected for the z direction
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Results Prostate Image Registration Registration accuracy for
the bottom plane of seeds is particularly important and is analyzed
separately
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Results Prostate Image Registration Sensitivity to segmentation
error was evaluated by perturbing the prostate model
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Results Hex / Tet mesh comparison Tet mesh constructed with
CUBIT from the surface tiles of the hex mesh
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Results Hex / Tet mesh comparison Hex mesh accuracy is better
Accuracy gap is largest in the direction with the most
deformation
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Outline Motivation & Overview FEM model construction M-reps
& image segmentation Mesh construction Finite element system of
equations Boundary conditions Solution Results for phantom prostate
image registration Conclusions & Future Work
Slide 79
Meshing Automatic hexahedral mesh generation from m-rep models
Boundary Conditions Automatic displacement boundary conditions
generated from a pair of m-rep segmentations Conclusions Summary:
Claims
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Solution Resolution adjustable with m-rep coordinate
subdivision Efficiency improvement by predicting solution on a fine
mesh based on solution from a coarser mesh Prostate Phantom Results
Seed prediction error on the order of the segmentation error /
image resolution
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Conclusions Automated Process To register images A & B by
deforming image A:
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Thesis Statement M-rep based multiscale mesh generation, M-rep
derived boundary conditions, and Multiscale solution of a finite
element system of equations are techniques that improve the
automation and efficiency of finite element analysis as it is
applied to medical imaging applications and to the prostate
brachytherapy application in particular.
Slide 83
Conclusions Future Work Now: Local subdivision Apply to other
parts of anatomy Use more sophisticated material models Long term:
Use database of deformable organ models to further automate the
creation of individualized simulations
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Acknowledgments Steve Pizer Committee members: Ed Chaney, Guido
Gerig, Sarang Joshi, Carol Lucas, and Julian Rosenman MIDAG members
MSKCC collaborators Family & friends