Concurrent Prediction of Muscle Force and Cartilage Stress during Movement

Trent M. Guess, PhD University of Missouri-Kansas City, Kansas City, MO

RPI-NSF Workshop on Multiscale Modeling of Complex Data

September 13, 2011

MUSCULOSKELETAL BIOMECHANICS

RESEARCH LAB

Introduction n Knowledge of joint and tissue level loading during

movement can benefit n Ligament injury and repair n Tissue engineered cartilage and menisci n Degenerative joint disease

n Chondromalacia n Osteoarthritis (OA)

Full-depth OA 12 weeks after meniscal release procedure

Canine meniscal release model of OA

caudal horn attachments

Introduction n Primary computational tools of musculoskeletal

biomechanics n Multibody dynamics (rigid body dynamics) n Finite Element Method

Multibody Dynamics

n Musculoskeletal Modeling and Movement Simulation n Body level n Computationally efficient n Optimization techniques to

estimate muscle forces/activations nMuscle Redundancy

McLean S G, Su A and van den Bogert A J. 2003. Development and validation of a 3-D model to predict knee joint loading during dynamic movement. J Biomech Eng, 125:864-874.

Multibody Dynamics

McLean S G, Su A and van den Bogert A J. 2003. Development and validation of a 3-D model to predict knee joint loading during dynamic movement. J Biomech Eng, 125:864-874.

Simple Hinge

n Musculoskeletal Modeling and Movement Simulation n Rigid body

nNo deformable tissue

n Simplified representation of joints

Finite Element Method n Tissue level or joint level n Deformable (stress, strain) n Computationally intensive (development intensive)

n Simplified loading n Assumed or measured boundary conditions

nNo movement simulation

Weiss J A, Gardiner J C, Ellis B J, Lujan T J and Phatak N S. 2005. Three-dimensional finite element modeling of ligaments: technical aspects. Med Eng Phys, 27:845-861.

Musculoskeletal Interdependency n Interdependency of muscle force and tissue response

justifies a concurrent modeling approach

Muscle forces affect tissue loading

Tissue loading and joint motion affect muscle

forces

Body Joint Tissue

Multibody Data Driven Surrogate

n Goal n Concurrent Simluation – Body to Tissue Level

nMuscle Force n Anatomical Joints n Tissue (cartilage)

Compare Kinematics

Method (body to joint)

MRI

Cadaver Knee (55 yr, female, left

knee, 170 cm, 72 kg)

Dynamic Simulator

Gait Measurements

Validated Knee Model

Musculoskeletal Model

Hip Angle, Quad Force

Ankle Loads

Hip Position, Hip Load

Multibody Anatomical Knee Validation

Optotrak 3020 3D motion tracking system

Kansas Knee Simulator, University of Kansas, Lawrence

Loading is known Bone kinematics directly measured

5 second Walk Cycle

Compare Kinematics

Translation (mm) Orientation (deg) x y z 1 2 3

5.6 4.4 2.9 1.8 1.0 7.1

Kinematic Envelope of Motion (Tibia relative to Femur)

Derive zero-load length lo

Walk Cycle RMS Error (Tibia relative to Femur )

Femur Coord Sys

Tibia Coord Sys

X Y

Z

X

Z

Ligaments

Blenkevoort et al. 1991, Wismans et al. 1980

Multibody Cartilage n Macro

n Divides cartilage geometries into discrete elements n 4 mm x 4 mm cross-section n 72 discrete elements medial n 61 discrete element lateral

n Attaches each to the tibia n Defines a compliant contact

w/ femur cartilage

Multibody Cartilage n Advantage

n Prediction of contact pressure n Input to tissue surrogate

models n Greater control of contact

models n thickness, non-homogeneity, etc.

Multibody Menisci Meniscus geometry

from MRI

Macro divides geometries and connects w/ 6x6 stiffness matrices

Stiffness matrix parameters optimized to minimize position error of an identically loaded FE model

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Objective Function: Displacement Error

Optimization

N/mm N-mm/(rad)

Multibody Menisci

100 N 100 N

ddd &)(exp BkFc +=

Multibody Menisci

Experiment

Model

Meniscus

Deformable Contacts

ddd &)(exp BkFc +=

Femur with Cartilage

Patella with Cartilage

Tibia

Cartilage Medial Meniscus

Lateral Meniscus

𝐹𝑐 = 𝑘𝛿𝑒𝑒𝑒 + 𝐵(𝛿)�̇�

Gait Measurements

n Motion (7 camera Vicon Mx-T40) n Plug-in Gait marker set

n Surface EMG (Delsys Myomonitor IV) n 8 muscles, left leg

n Ground Reaction Forces (AMTI OR-6)

Extension 16° 0.25 s

Flexion 87° 1.63 s

Musculoskeletal Model

LifeMOD (LifeModeler, Inc., San Clemente, CA)

Bushing

Spherical joints

Mirrored

Musculoskeletal Model 1.) Motion Driven Inverse Dynamic simulations to position knee

2.) Add Muscles (45 per leg) and quad attachments on patella

3.) Run Inverse Dynamics To “train” muscles

4.) Run Forward Dynamics muscle driven simulation

Results

Results

Knee Loading Studies n With and without menisci

n Significant increase in peak tibia cartilage contact pressure without menisci

n Anterior Cruciate Ligament (ACL) transection (full and partial) n Lateral meniscus becomes “wedged” between the tibia and

femur for full and partial ACL transection n Small shift in peak contact pressure location

n Both findings are consistent with experimental measurements and observations n (Scarvell, Smith et al. 2005; Scarvell, Smith et al. 2005; Van de Velde, Bingham et al. 2009)

n Patello-femoral contact pressure n Magnitude of peak patella cartilage contact pressure does not

significantly change with altered vasti recruitment n Location of peak patella cartilage contact pressure does change

with altered vasti recruitment

Post

Ant

Intact Partial Full

Medial Plateau Extension

Post

Ant

Medial Plateau Flexion

Partial and Full Anterior Cruciate Ligament Transection

Post

Ant

Intact Partial Full

Lateral Plateau Extension

Post

Ant

Lateral Plateau Flexion

Partial and Full Anterior Cruciate Ligament Transection

Extension Flexion

Intact

Partial

Full

Location of Peak Contact Pressure

Predicted and Measured Ground Reaction Forces

Patello-femoral Cartilage

<------Lateral --------------------------------Medial------>

Custom deformable contact

Elastic foundation model for large deformations (Blakevoort et. al. 1991, Bei and Fregly 2004)

Patello-femoral Cartilage

Uniform Vasti

Low VM

Low VL

Patello-femoral Cartilage

Uniform Vasti Low VM Low VL

11.6 MPa 85°

1.78 sec 11.1 MPa

84°

1.82 sec 11.6 MPa

87°

1.70 sec

121mm2 +56mm2 = 178mm2 139mm2 +43mm2 = 182mm2 175mm2 +50mm2 = 225mm2

<------Lateral ---------Medial------>

*

<------Lateral ---------Medial------>

*

<------Lateral ---------Medial------>

*

Patello-femoral Cartilage

Low VM <------Lateral -----------Medial------> <------Lateral -----------Medial------> <------Lateral -----------Medial------>

Uniform Vasti Low VL

Contact Pressure at 80° flexion

Future Work

- Patient specific, in vivo knee models - Geometries from MRI, ligament properties from in vivo laxity testing

- Modeling more activities (e.g. walking)

Tissue Models

n Multibody musculoskeletal models n Rigid body dynamics

nNo deformation n no stress and strain

n Limited to contact and interface loading n e.g. contact pressures, reaction forces

n Loading at tissue interfaces is not sufficient n Osteoarthritis n Mechanotransduction n Tissue engineering

d Femur Fc

FR

ddd &)(exp BkFc +=

Cartilage Surrogate

Cartilage bone interface reaction forces

Cartilage Surrogate

𝐹𝑐 = 𝑘𝛿𝑒𝑒𝑒 + 𝐵(𝛿)�̇�

𝐹𝑐

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐹𝑅𝐹𝑅𝑅𝐹

𝑇𝑅 𝑇𝑅𝐹𝐹𝑇𝑅 𝑀𝑅𝑀𝑅𝑀

n Goals n Identify and develop data-driven “surrogate” models

(black box nonlinear system identification) to predict von Mises stress from multibody reaction forces

n Develop a fast but accurate model with low computational footprint: nNeural nets are good candidates: besides their universal

function approximation properties, they can run in near “real time” post-training, even when representing a complex system.

Cartilage Surrogate

Cartilage Surrogate

n Multibody n Tune contact parameters

n Match FE displacement n Geometry, boundary

conditions n Outputs – rigid body

reaction forces

n Finite Element Analysis n Linear elastic cartilage

model n Geometry, boundary

conditions n Outputs – element von

Mises stress

Surrogate Training

Cartilage Surrogate

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐹𝑅𝐹𝑅𝑅𝐹

Von Mises Stress

Data Driven Surrogate

Cartilage Surrogate

n Trial 1 n Surrogate predicts von Mises stress on

center (blue) element n 15 inputs, 1 output n Advantage

n Could be subject independent nSimpler surrogate

n Disadvantage n Cartilage curvature n Cartilage thickness nNon-homogeneity nMulti-body cartilage segments are

independent n No reaction forces if not in contact with indenter n FE elements are interconnected

Cartilage Surrogate

Multibody Reaction Forces FE Stress

Cartilage Surrogate n Trial 2

n Surrogate predicts von Mises stress on all elements n For 20 x 20 grid, 1200 inputs, 400 outputs

Trial 2 Approach: Feed Forward Neural Networks

n NN setup: 20x20 grid of input elements (reaction forces), 20x20 grid of output elements (von Mises stress), one hidden layer with 10 neurons (hyperbolic tangent activation functions, 1200 inputs), and 400 linear output nodes.

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h2

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input grid

output grid

Neural Net Training n Division of dataset (random sampling): 70% for calibrating

neural net parameters (training), 15% for iterative validation of training results, and 15% for blind test.

n Assumed that the system is memoryless (i.e. model outputs do not depend on past inputs).

n Scaled conjugate gradient for gradient descent solution to the neural net model parameters (goal: minimize the MSE on training set without increasing the MSE on validation set) n Other gradient decent MSE minimizers, such as Levenberg

Marquardt, can be too taxing on the memory and CPU resources, or they could be fast and light but not accurate enough (e.g. resilient backprop method)

Neural Net Performance (Trial 2) n MSE plots for training, validation, and test datasets (each

epoch is one pass or iteration through the whole dataset during gradient decent solution):

Neural Net Performance (2) • Regression fit plots

for training, validation, and test subsetsà

• A good fit is represented by a data-target scatter plot that is tightly packed around the 45o line, also indicated by a correlation coefficient (R) that is close to 1.

• The neural net has successfully modeled not only the training but also the unseen test data.

Cartilage Surrogate n Trial 3

n Surrogate predicts Von misses stress on all elements n For 20 x 20 grid, 1200 inputs, 400 outputs n Curved Tibia surface

n Trial 4 n Tibia and femur cartilage geometries

Acknowledgements n National Science Foundation

n CMMI-1039524 n CBET-0821459 n CMS-0506297

n National Institutes of Health n NIAMS RAR061698A

n Missouri Life Sciences Research Board n 09-1078

n Researchers n Reza Derakhshani, PhD n Antonis Stylianou, PhD n Yunkai Lu, PhD n Mohammad Kia, MS n Gavin Paiva, MS n Katherine Bloemker, MS n Sampath Bhashyam, BS

Thank You