# Anthony J Petrella, PhD Statistical Shape Modeling & Probabilistic Methods 

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Anthony J Petrella, PhD Statistical Shape Modeling & Probabilistic Methods 

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:  + 1*  Blue:  – 1*  Lumbar (Huls et al., 2010) Hip (Barratt et al., 2008) Knee (Fitzpatrick et al., 2007)

KS Huls, AJ Petrella, PhD Colorado School of Mines Golden, Colorado USA A Agarwala, MD Panorama Orthopaedics & Spine Center Golden, 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 geometry identical topology

SSM Background 3 4 Assemble data matrix → Least squares fit → remove variations in translation & rotation retain variations in size and shape (Spoor and Veldpaus, 1980)

SSM Background 5 Principal Component Analysis = eigenanalysis on covariance matrix of data, D eigenvectors ( c j ) = fundamental shape modes eigenvalues = variance of a each shape mode across specimens → where the b j coefficients are the “principal components” of specimen P

SSM Background 6 New virtual specimens instantiated from SSM → Coefficients b j are assumed normally distributed PDF for each b j randomly sampled to instantiate any number of virtual specimens … bjbj

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 L4 to 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 from SSM 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:  + 1*  Blue:  – 1* 

Results: Leave-one-out Validation  Maximum Euclidian distance error: 5.6 mm  Mean error: 1.9 mm Red: “left out” specimen Blue: SSM fit

Results: Lumbar Model Instantiation  Appearance of virtual L3+L4 pairs  Appearance of virtual FSU specimens FSU-1FSU-2FSU-4FSU-3 L3+L4-1L3+L4-2L3+L4-4L3+L4-3

Results: FE Model  Facet contact area (p = 0.33) Natural: 158 ± 43 mm 2 FSU specimens: 120 ± 59 mm 2  Average contact pressure (p = 0.55) Natural: 0.79 ± 0.16 MPa FSU specimens: 0.88 ± 0.22 Mpa  No FE for L3+L4 specimens due to 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 STL geometry

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., 10 4 × 10 4 e.g., 10 × 10

5. Eigenanalysis on COV (C) matrix of D

6. Instantiate new specimens  Assume b i 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

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