Anthony J Petrella, PhD

Statistical Shape Modeling &Probabilistic Methods

20° q

Practical Challenges in Prob Analysis

Quality deterministic model, validated

How to estimate input distributions

Correlated input variables

Complex systems with long solution times require more efficient alternatives to Monte Carlo

Implementation Matlab, Excel – simple problems Commercial FE modules (Abaqus, ANSYS, PAM-CRASH) NESSUS – dedicated prob code, integrates with model

Validation

Anatomical variation?

Practical Challenges

Quality deterministic model, validated

How to estimate input distributions

Correlated input variables

Complex systems with long solution times require more efficient alternatives to Monte Carlo

Implementation Matlab, Excel – simple problems Commercial FE modules (Abaqus, ANSYS, PAM-CRASH) NESSUS – dedicated prob code, integrates with model

Validation

Anatomical variation?

What is the best way to parameterize anatomical shape so that we can easily do Prob simulation to explore effects of

anatomical variation?

Parameterizing Anatomy

Approximate with Primitives

Statistical Shape Modeling (SSM)

(Laville et al., 2009)

Red: m + 1*sBlue: m – 1*s

Lumbar (Huls et al., 2010)

Hip (Barratt et al., 2008)

Knee (Fitzpatrick et al., 2007)

KS Huls, AJ Petrella, PhDColorado School of MinesGolden, Colorado USA

A Agarwala, MDPanorama Orthopaedics & Spine CenterGolden, Colorado USA

ICCB 2009, Bertinoro, Italy

September 16-18, 2009

Modeling Anatomic Variability for Application in Probabilistic Simulation of Lumbar Spine Biomechanics

SSM Background

1

… (N = 8)

Training set

2 Morph template mesh →

…

unique geometryidentical topology

SSM Background

3

4 Assemble data matrix →

Least squares fit → remove variations intranslation & rotation

retain variations insize and shape

(Spoor and Veldpaus, 1980)

SSM Background

5 Principal Component Analysis = eigenanalysison covariance matrix of data, D

eigenvectors (cj ) = fundamental shape modes

eigenvalues = variance of a each shapemode across specimens

→

where the bj coefficients are the “principal components”of specimen P

SSM Background

6 New virtual specimens instantiated from SSM

→

Coefficients bj are assumed normally distributed

PDF for each bj randomly sampled to instantiate any number of virtual specimens

…

bj

SSM in Orthopaedics & Biomechanics

2D kinematic measures for functional evaluation of cervical spine (McCane et al., 2006)

3D pelvis and femur anatomy for computer-navigated total hip arthroplasty (Barratt et al., 2008)

3D model of hemi-pelvis for use in computer-navigated THA (Meller and Kalender, 2004)

Lumbar vertebral bodies (Lorenz and Krahnstöver, 2000)

Previous work focused only on individual bones

Relative position, alignment, and conformity of articulating surfaces not considered

Objectives

Develop SSM for lumbar spine, focusing initially on the L3-L4 functional spinal unit (FSU)

Determine if virtual specimens instantiated from the SSM are biomechanically viable

use finite element (FE) modeling to demonstrate normal facet articulation

Methods: Lumbar FSU

Lumbar geometry L1-L5 extracted from 8 CT data sets using Mimics software (Materialise, Inc.)

2 Male/6 Female, 54 ± 16 yrs

Quadrilateral FE mesh created for L3 and morphed to L4 using HyperMesh software (Altair, Inc.)

Methods: Independent SSM for L3 & L4

SSM created for L3 bodies

Independent SSM created for L4 bodies

L3 and L4 bodies independently instantiated and combined to form L3-L4 functional spinal units

We refer to these models as:L3+L4 pairs

L3

L4

Methods: Unified SSM for L3-L4

For each of the 8 specimens…

Least squares fit of L3 to L4to remove non-physiological alignment created in scanner

Unified SSM then created for L3-L4

Virtual specimens instantiated as FSU

We refer to these models as:L3-L4 FSU

Methods: Leave-one-out Validation

Assess ability of SSM to represent the shape of an unknown specimen

Randomly selected L3 specimen removed from training data set of 8 CT scans

SSM recalculated with only seven specimens

Non-linear least squares optimization scheme was used to fit the SSM to the “left-out” specimen

Methods: FE Model

Virtual specimens created with L3+L4 SSM

Specimens also created fromSSM of the L3-L4 FSU model

ABAQUS (Simulia, Inc.) model constructed for virtual specimens Ligaments: non-linear springs Facet cartilage: linear elastic Annulus: hyper-elastic matrix, linear fibers Nucleus: incompressible fluid cavity

L4 fixed, L3 loaded with 10 N·m right axial torque

Results: Fundamental Modes of Shape

First five PC’s captured 95% of variance in data PC1: scaling PC2: shape and angulation of facet joints PC3: variations in the transverse processes Higher modes were not visually obvious

Red: m + 1*sBlue: m – 1*s

Results: Leave-one-out Validation

Maximum Euclidian distance error: 5.6 mm

Mean error: 1.9 mm

Red: “left out” specimenBlue: SSM fit

Results: Lumbar Model Instantiation

Appearance of virtual L3+L4 pairs

Appearance of virtual FSU specimens

FSU-1 FSU-2 FSU-4FSU-3

L3+L4-1 L3+L4-2 L3+L4-4L3+L4-3

Results: FE Model

Facet contact area (p = 0.33)Natural: 158 ± 43 mm2

FSU specimens: 120 ± 59 mm2

Average contact pressure (p = 0.55)Natural: 0.79 ± 0.16 MPaFSU specimens: 0.88 ± 0.22 Mpa

No FE for L3+L4 specimens dueto facet interaction

Natural

FSU

L3+L4

Conclusions

SSM reasonable for spine, 95% of variance captured by just 5 variables

Leave-one-out validation Errors similar to pelvis (Meller and Kalender, 2004)

Maybe acceptable for CAOS, but not for biomechanics More specimens needed in training set

SSM instantiation L3+L4 facet articulation not viable FSU similar to natural facet interaction Lumbar SSM must include all bodies to ensure reasonable

inter-body articulation

SSM Summary

Why? Continuous parameterization of anatomical shape

The basic steps…1. Collect data for training set

2. Morph to each specimen → identical topology, inter-specimen correspondence

3. Register specimens to common reference

4. Assemble data matrix

5. Eigenanalysis on COV matrix

6. Instantiate new specimens

1. Collect data for training set

Usually done with medical images

CT common for bone geometry

MR common for soft tissue, but bone extraction protocols do exist

Commercial software typically used Mimics (Materialise) Simpleware

Result is usually STLgeometry

2. Morph template mesh to each specimen

Essential for statistical analysis because morph creates Identical topology Inter-specimen correspondence – every vertex / landmark is

at the same anatomical location for every specimen

Beyond the scope of our discussion, but… Coherent Point Drift: may be used for rigid or non-rigid reg.

https://sites.google.com/site/myronenko/research/cpd Non-rigid surface registration: see class website for PDF Other morphing studies: Grassi et al., 2011; Sigal et al.,

2008; Viceconti et al., 1998

3. Register specimens to common reference

SSM quantifies variation in vertex coordinates across all specimens

Must remove variation that does not pertain to shape

Rigid registration of point sets with different topology Coherent Point Drift – stated previously Iterative Closest Point (ICP) – well known

From step 2: we have identical topology Analytical registration method: Spoor & Veldpaus, 1980

based on calculus of variations, see class website for PDF

4. Assemble data matrix

Arrange each specimen in a column vector with x, y, z coordinates “stacked” as shown

Normalize to the mean specimen shape

5. Eigenanalysis on COV (C) matrix of D

Eigenanalysis equivalent to Principal Component Analysis (PCA)

Eigenanalysis of original system is tedious…

Rank = max # linearly independent col vectors

Rank of C is driven by n (specimens), not by m (data points)

Solving the alternate problem is much faster…

e.g., 104 × 104

e.g., 10 × 10

5. Eigenanalysis on COV (C) matrix of D

6. Instantiate new specimens

Assume bi normally distributed

New specimens may be created simply with

Femur SSM Exercise

Exercise is based on a 2D femur meshed with triangular elements

See if you can build SSM and instantiate it

Mode 1: Blue(-), Red(+) Mode 2: Blue(-), Red(+) Mode 3: Blue(-), Red(+) Mode 4: Blue(-), Red(+)Unregistered Raw Data Registered Data