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 QuestionWhat 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-deformable 6-8 hours/run

0

5

10

15

20

25

30

35

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Tr

ue S

tres

s (M

Pa)

True Strain

Model Variables

Insert_Tilt

Init_Fem_FE

FEax_AP Fem_IEFEax_IS

IEax_ML

IE Axis

IEax_AP

Insert_VV

FE Axis

Coefficient of Friction

ML LoadSplit

(with medial offset)

ML

Spring Constant (K)

Probabilistic Approach

ProbabilisticInputs

Performance Measures

SensitivityFactors

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

Variable Description Mean Value Std.Dev. (Level A)

Std.Dev. (Level B)

FEax_AP AP position of femoral FE axis 0 mm 0.25 mm 0.5 mm

FEax_IS IS position of femoral FE axis 25.4 mm 0.25 mm 0.5 mm

IEax_AP AP position of tibial IE axis 7.62 mm 0.25 mm 0.5 mm

IEax_ML ML position of tibial IE axis 0 mm 0.25 mm 0.5 mm

Init_Fem_FE Initial FE position of femoral 0° 0.5° 1°

Insert_Tilt Tilt of the insert 0° 0.5° 1°

Fem_IE Initial IE rotation of femoral 0° 0.5° 1°

Insert_VV Initial VV position of insert 0° 0.5° 1°

ML ML position of spring fixation 28.7 mm 0.25 mm 0.5 mm

ML_Load ML load split (60%-40%) 60% 1.0% 1.0%

K Spring constant 5.21 N/mm 0.09 N/mm 0.09 N/mm

Coefficient of friction 0.04 0.01 0.01

Model VariablesAll variables assumed as normal distributions

AP TranslationModel-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)

-6

-4

-2

0

2

0 20 40 60 80 100% Gait Cycle

AP

Tra

nsla

tion

(mm

)

Experimental

Level A

Level B

IE RotationModel-predicted envelopes (1% to 99% confidence intervals) as a function of gait cycle

Max. Range: 2.17° (Level A ), 4.30° (Level B)

-10

-8

-6

-4

-2

0

2

4

0 20 40 60 80 100% Gait Cycle

IE R

otat

ion

(Deg

)

ExperimentalLevel ALevel B

IE

4

8

12

16

20

0 20 40 60 80 100% Gait Cycle

Con

tact

Pre

ssur

e (M

Pa)

Level A

Level B

Peak Contact PressureModel-predicted envelopes (1% to 99% confidence intervals) as a function of gait cycle

Max. Range: 1.3 MPa (Level A ), 1.6 MPa (Level B) @ 40% Gait

Sensitivity FactorsNormalized absolute average of sensitivity over the entire gait cycle

0.0

0.2

0.4

0.6

0.8

1.0F

Eax

_AP

FE

ax_I

S

IEax

_AP

IEax

_ML

Init_

Fem

_FE

Inse

rt_T

ilt

Fem

_IE

Inse

rt_V

V

ML_

Load

ΔM

L K μ

Sen

sitiv

ity

AP

IE

CP

Parameter sensitivities varied significantly throughout the gait cycle

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

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

• Video fluoroscopy is widely used to obtainimplant kinematics in vivo• Evaluate performance measures

(e.g. range of motion, cam-post interaction)

• Uncertainty exists in spatial positioning of theimplants 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 componentat 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

0 20 40 60 80 100Flexion Angle (deg.)

Lateral

-20

-10

0

10

0 20 40 60 80 100Flexion Angle (deg.)

Mean1% Bound99% Bound

Medial

A(+

)/P

(-)

Po

sitio

n (

mm

)

Contact patches at 90° flexion

Predicted tibiofemoral contact positions

+

_

Medial Lateral

Medial Lateral

1% 99%

Effects of Bone Mechanical Properties on Fracture Risk

Assessment

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)

20° Relationship Coefficient Exponent

E(GPa) = ab a ( = 1.99, = 0.30) b ( = 3.46, = 0.12)

S(MPa) = cd c ( = 26.9, = 2.69) d ( = 3.05, = 0.09)

CT dataNessus

Material Relations BoneMat

Abaqus StressRisk

Material relation variability (Keller, 1994)

-10

0

10

20

30

40

0 0.5 1 1.5 2Density [g/cm3]

Mod

ulus

[G

Pa]

0

100

200

300

400

500

Str

engt

h [M

Pa]

Modulus

Strength

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

Femur Mean Min Max

1R 111.0 104.2 117.7

1L 100.2 94.5 106.1

2L 217.1 203.2 233.5

Fracture Risk

Femur Mean Min Max

1R 0.396 0.314 0.523

1L 0.375 0.298 0.494

2L 0.658 0.521 0.870

1R

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