Probabilistic Methods: Theory and Application to Human Anatomy Anthony J Petrella, PhD Colorado School of Mines

Why Do Prob? Quantify risk and reliability Reduce over-conservatism in design Identify critical variables and failure mechanisms Minimize variation sensitivity – more robust design Minimize physical testing – use testing to validate models Analysis can examine wider range of variables, scenarios Quantify reliability of high-consequence of failure systems Probability-based performance optimization Copyright SwRI® Copyright SwRI® Copyright SwRI®

Why Prob in Biomechanics? Implant & device design Recent trends…subject-specific simulation

Why Prob in Biomechanics? Simulation driving interventions Many parameters difficult to measure in vivo Understand uncertainty within subject-specific predictions Deterministic model → population Star Trek – advanced diagnostics and automated/optimized intervention

How to Do Prob Probabilistic simulation is not new Monte Carlo methods perhaps best-known Named after famous Casino in Monaco Monte Carlo was a code name coined in the 1940’s at Los Alamos National Lab

How to Do Prob with Monte Carlo Outcome Probabilities & Sensitivities Model Input Uncertainties Validated Deterministic Model Tissue Properties Probability Performance Metric External Loads Response and Failure Prediction Device Placement Sensitivity Factors

Monte Carlo Example: Recumbent Cycling 2D model with rigid links No co-contraction, no dependence on length or velocity Global origin at hip, ankle coincident with pedal axis

Monte Carlo Example: Recumbent Cycling Inverse dynamic solution performed, resultant force and moment at knee computed Muscle forces and knee contact force estimated Focus on: FAP at 240°

Monte Carlo Example: Recumbent Cycling quad tub_x tub_y ham_x tibia femur Probabilistic variables Assume all parameters normally distributed with a COV = s/m = 0.033 How does uncertainty in moment arms affect FAP?

Monte Carlo Results Results represent distribution of FAP at 240° Other angles → upper and lower bounding curves

Monte Carlo Results Probabilistic sensitivity factors for the i’th input Sm (or s) > 0 negative correlation variability increased (or decreased in converse scenario) Sm (or s) < 0 positive correlation

Monte Carlo Results Sensitivity to changes in input mean values Sm > 0 → negative correlation quad tub_y Sm < 0 → positive correlation

Monte Carlo Results Sensitivity to changes in input standard deviations Increased variation in inputs increases variation in FAP Factors that drive down uncertainty in outcome metric? variability increased