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

2. 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®

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

4. 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

5. 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

6. 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

7. 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

8. 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°

9. 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?

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

11. 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

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

13. 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