Medial Techniques for Automating Finite Element Analysis Jessica Crouch
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- Slide 1
- Medial Techniques for Automating Finite Element Analysis
Jessica Crouch
- Slide 2
- Motivation Deformation Modeling Aim: Model soft tissue
deformation Applications include Medical simulation, surgical
planning Tomotherapy Non-rigid registration of 3D medical
images
- Slide 3
- 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
- Slide 4
- 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
- Slide 5
- Motivation Finite Element Method (FEM) Model geometric
properties Discretize space with a mesh composed of Nodes Elements
Boundary fitted
- Slide 6
- Motivation Finite Element Method (FEM) Model physical
properties Choose equations & coefficients that describe the
material's deformability Assemble the finite element system of
equations
- Slide 7
- 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
- Slide 8
- Planning Image Imaging probe deforms prostate Intra-operative
image Prostate is relatively undeformed Motivation Prostate
Registration Problem
- Slide 9
- 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 10
- 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 11
- 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
- Slide 12
- 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
- Slide 13
- 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
- Slide 14
- 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
- Slide 15
- FEM Model Construction M-rep Model Visualization Adjusting the
m-rep parameters stored in each atom affects the models
geometry
- Slide 16
- 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
- Slide 17
- 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
- Slide 18
- 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
- Slide 19
- FEM Model Construction M-rep Segmentation Demonstration
- Slide 20
- 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 21
- FEM Model Construction Mesh Construction Requirements Element
choices Shape Tetrahedra Hexahedra (preferred) Pyramids, wedges,
etc.
- Slide 22
- 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
- Slide 23
- 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
- Slide 24
- 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
- Slide 25
- 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
- Slide 26
- 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
- Slide 27
- Slide 28
- 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
- Slide 29
- Mesh of single figure prostate m-rep model
- Slide 30
- FEM Model Construction Mesh Construction: Single Figure 5
object male pelvis m-rep model mesh
- Slide 31
- 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
- Slide 32
- Pyramid and tetrahedral elements for space external to m-rep
model
- Slide 33
- FEM Model Construction Mesh Construction: Single Figure
- Slide 34
- Must ensure smooth, compatible connection between host figure
and subfigure mesh elements FEM Model Construction Mesh
Construction: Multi-Figure
- Slide 35
- 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
- Slide 36
- Compute host / subfigure intersection in terms of Host figure
object coordinates Subfigure object coordinate FEM Model
Construction Mesh Construction: Multi-Figure
- Slide 37
- 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
- Slide 38
- 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:
- Slide 39
- 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
- Slide 40
- Template patterns assembled in m-rep coordinate space, then
mapped to world space FEM Model Construction Mesh Construction:
Multi-Figure
- Slide 41
- 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
- Slide 43
- Linear elastic model Stress, , is proportional to strain, .
Linear elastic PDE: Elastic constants Youngs modulus Poissons ratio
FEM Model Construction Finite Element Equations
- Slide 44
- Slide 45
- Slide 46
- 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
- Slide 47
- FEM Model Construction Mesh Construction: Single Figure
- Slide 48
- 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
- Slide 50
- 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
- Slide 51
- 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
- Slide 52
- M-rep generated surface displacement vectors FEM Model
Construction Boundary Conditions
- Slide 53
- 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
- Slide 54
- Slide 55
- 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
- Slide 56
- 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
- Slide 58
- To improve solution accuracy, subdivide mesh elements Add nodes
at the midpoints of edges, quad faces, and hex volumes FEM Model
Construction Solution
- Slide 59
- 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
- Slide 60
- Mesh Subdivision Subdivision with m-rep object coordinates
Improved smoothness Mesh geometry more closely approximates m-rep
implied boundary with each subdivision
- Slide 61
- Mesh Subdivision Subdivided prostate mesh 3 levels
- Slide 62
- Multiscale 5 object pelvis mesh FEM Model Construction
Solution
- Slide 63
- 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:
- Slide 64
- 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
- Slide 65
- Coarse-to-fine solution strategy improves solution efficiency
FEM Model Construction Solution
- Slide 66
- 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
- Slide 68
- Results Prostate Image Registration
- Slide 69
- Slide 70
- Red: Inflated image Gray: Computed deformation applied to
uninflated image Results Prostate Image Registration
- Slide 71
- Seed centers Blue: segmented from inflated image Green:
segmented from uninflated image, then moved by the computed
deformation
- Slide 72
- 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
- Slide 73
- 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
- Slide 74
- Results Prostate Image Registration Registration accuracy for
the bottom plane of seeds is particularly important and is analyzed
separately
- Slide 75
- Results Prostate Image Registration Sensitivity to segmentation
error was evaluated by perturbing the prostate model
- Slide 76
- Results Hex / Tet mesh comparison Tet mesh constructed with
CUBIT from the surface tiles of the hex mesh
- Slide 77
- Results Hex / Tet mesh comparison Hex mesh accuracy is better
Accuracy gap is largest in the direction with the most
deformation
- Slide 78
- 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
- Slide 80
- 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
- Slide 81
- Conclusions Automated Process To register images A & B by
deforming image A:
- Slide 82
- 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
- Slide 84
- Acknowledgments Steve Pizer Committee members: Ed Chaney, Guido
Gerig, Sarang Joshi, Carol Lucas, and Julian Rosenman MIDAG members
MSKCC collaborators Family & friends