Probabilistic Analysis: Applications to Biomechanics Students: Saikat Pal, Jason Halloran, Mark Baldwin, Josh Stowe, Aaron Fields, Shounak Mitra Collaborators: Paul Rullkoetter, Anthony Petrella, Joe Langenderfer, Ben Hillberry

The Question What do I learn from probabilistic modeling that I don’t already know from deterministic modeling? Distribution of performance Assessment includes variable interaction effects Understanding of the probabilities associated with component performance –Probability of failure for a specific performance level –Minimum performance level for a specific POF Sensitivity information Two common applications Evaluation of existing components Guidance for tightening/loosening the tolerances of specific dimensions Design of future components Predict performance and identify potential issues prior to prototyping and testing

Bounding predictions of TKR performance in a knee simulator Stanmore Wear Simulator Explicit FE Model

Research Question What impact does variability in component placement and experimental setup have on the kinematic and contact mechanics results? Wear? Approach Experimental setup has inherent variability To more rigorously validate the model Scatter to setup parameters (  and  ) is introduced Distributions of results evaluated

Computational Model Explicit FE model of Stanmore simulator (Halloran, Petrella, Rullkoetter) Rigid body analysis with optimized pressure-overclosure relationship Non-linear UHMWPE material Simulated gait cycle Profiles: AP load, IE torque, flexion angle, axial force Computation time Rigid-rigid 6-8 minutes/run Rigid-deformable6-8 hours/run

Model Variables Insert_Tilt Init_Fem_FE FEax_AP Fem_IE FEax_IS IEax_ML IE Axis IEax_AP Insert_VV FE Axis Coefficient of Friction  ML Load Split  ML Spring Constant (K)

Probabilistic Approach Probabilistic Inputs Performance Measures Sensitivity Factors Probabilistic Inputs 4 translational alignments 4 angular alignments 4 experimental/setup variables Output Distributions Kinematics AP and IE position Contact pressure Wear Probabilistic Model Deterministic Inputs Deterministic Inputs Component geometry Gait profile (ISO) Material behavior

VariableDescriptionMean ValueStd.Dev. (Level A) Std.Dev. (Level B) FEax_APAP position of femoral FE axis0 mm0.25 mm0.5 mm FEax_ISIS position of femoral FE axis25.4 mm0.25 mm0.5 mm IEax_APAP position of tibial IE axis7.62 mm0.25 mm0.5 mm IEax_MLML position of tibial IE axis0 mm0.25 mm0.5 mm Init_Fem_FEInitial FE position of femoral0°0.5°1° Insert_TiltTilt of the insert0°0.5°1° Fem_IEInitial IE rotation of femoral0°0.5°1° Insert_VVInitial VV position of insert0°0.5°1°  ML ML position of spring fixation28.7 mm0.25 mm0.5 mm ML_LoadML load split (60%-40%)60%1.0% KSpring constant5.21 N/mm0.09 N/mm  Coefficient of friction Model Variables All variables assumed as normal distributions

AP Translation Model-predicted envelopes (1% to 99% confidence intervals) as a function of gait cycle Max. Range: 1.79 mm (Level A ), 3.44 mm (Level B)

IE Rotation Model-predicted envelopes (1% to 99% confidence intervals) as a function of gait cycle Max. Range: 2.17° (Level A ), 4.30° (Level B) IE

Peak Contact Pressure Model-predicted envelopes (1% to 99% confidence intervals) as a function of gait cycle Max. Range: 1.3 MPa (Level A ), 1.6 MPa (Level 40% Gait

Sensitivity Factors Normalized absolute average of sensitivity over the entire gait cycle Parameter sensitivities varied significantly throughout the gait cycle

Evaluating Measurement Uncertainty in Predicted Tibiofemoral Contact Positions using Fluoro-driven FEA

Video fluoroscopy is widely used to obtain implant kinematics in vivo Evaluate performance measures (e.g. range of motion, cam-post interaction) Uncertainty exists in spatial positioning of the implants during the model-fitting process (Dennis et al., 1998) Due to image clarity, operator experience, and differences in CAD and as-manufactured geometries Errors up to 0.5 mm and 0.5° for in-plane translations and rotations (Dennis et al., 2003) Objectives: Develop an efficient method to account for measurement uncertainty in the model-fitting process Evaluate the potential bounds of implant center-of-pressure contact estimates

Methods Probabilistic analysis based on previous fluoro-driven FE model (Pal et al., 2004) Fixed-bearing, semi-constrained, Sigma PS implant Weight-bearing knee bend from 0° to 90° Inputs: Six DOFs describing pose of each component at each flexion angle (0° to 90°, at 10° intervals) Gaussian distributions with mean based on model-fitting process In-plane DOFs: SD = 0.17 mm and 0.17° Out-of-plane DOFs: SD = 0.34 mm and 0.34° To allow both condyles to contact throughout flexion, model loading conditions were: Compressive force and in vivo kinematics (AP, IE and FE) Unconstrained in ML and VV Output: Distribution of contact location throughout flexion

Results Substantial variability in AP contact position observed Average ranges: Medial:10.9 mm (0°-30°) 5.4 mm (30°-90°) Lateral: 9.3 mm (0°-30°) 6.3 mm (30°-90°) Maximum ranges: 12.2 mm (M) and 10.7 mm (L) Uncertainty in implant position affected cam-post interaction Underscores the need for careful procedures when extracting kinematics using fluoroscopy Contact patches at 90° flexion Predicted tibiofemoral contact positions + _ Medial Lateral 1%99%

Effects of Bone Mechanical Properties on Fracture Risk Assessment

CT scans are often used to create geometry and material properties of bone Assess bone stresses Predict fracture risk Evaluate implant load transfer Significant variability present in relationships between HU and Modulus and Strength What effect does this variability have on predicted stress and risk assessment? Keller, 1994

Methods Proximal femur under stance loading (Keyak et al., 2001) Relationship Coefficient Exponent E(GPa) = a  b a (  = 1.99,  = 0.30)b (  = 3.46,  = 0.12) S(MPa) = c  d c (  = 26.9,  = 2.69)d (  = 3.05,  = 0.09) CT data Nessus Material Relations BoneMat Abaqus Stress Risk Material relation variability (Keller, 1994)

Results Average bounds (1-99%) Stress: 13.9 MPa Risk: 0.25 Potential to impact findings of bone studies Computation time < 2 hours Variability should be considered when applying lab-developed material relations to patient- specific bone models Maximum Stress FemurMeanMinMax 1R L L Fracture Risk FemurMeanMinMax 1R L L R 1L 2L

Summary Probabilistic analysis has been demonstrated as a useful computational tool in materials and biomechanics Efficient MPP-based methods make probabilistic FE analysis quite feasible Knowledge of bounds (distributions) of performance and important parameters useful in design decisions Developed framework can be “easily” applied to most computational models