Evaluation of a surrogate contact model of TKA

Abstract

INTRODUCTION: Simultaneous prediction of body-level dynamics and detailed joint mechanics in the frame of musculoskeletal (MS) modeling represents still a highly computationally demanding task. Marra et al. (2014) recently presented and validated a MS model capable of concurrent prediction of muscle forces, knee ligament forces, tibiofemoral (TF) and patellofemoral (PF) contact forces in a MS model of Total Knee Arthroplasty (TKA) [1]. Simulation time for one complete gait cycle was in the order of 3 hours, and the iterative process that solved the equilibrium in the knee joint was thought to be the main source of overhead. Surrogate modeling techniques were suggested [2]. In this study, we develop a surrogate contact model of TKA to decrease the simulation time in the MS simulation. We hypothesize that the algorithm that allows the muscle fibers to wrap around the bones constitutes another source of overhead in the MS model. Therefore, we will also evaluate the performances of the surrogate model with and without muscle wrapping. METHODS: The original tibial component from our TKA model [1] was split in a medial and lateral hemi-part and fixed to the ground, whereas the femoral part was left with 6 degrees of freedom (DOF). The contacting pairs exchanged three forces and three moments, which were assumed functions of the relative pose only. Translations (X, Y, Z) were defined relative to the tibial component frame and rotations of the femoral component (RotX, RotY, RotZ) were described with Cardan angles, using the z-y-x rotation sequence. Similarly to Lin et al (2010) [3] we identified two sensitive directions, Y and RotX and, therefore, we defined a sample point as composed by four pose parameters and the two loads in the sensitive directions: X, Fy, Z, Tx, RotY, RotZ. Reference load-pose data were obtained from four simulations of gait, squat, chair-rise, and right-turn trials using the original contact model. The design space was populated using the Hammersley quasi-random sequence and adopting a multi-domain approach, as proposed by Eskinazi and Fregly (2015) [2]. One domain consisted of 20 data points per each frame of the four dynamic simulations, spanning the boundaries of ± 1 standard deviation from the time-varying reference envelopes. A second domain of 2500 points was generated in the principal component space of the reference load-pose data of each dynamic simulation, with boundaries enlarged by 50%. A third domain of 1000 data points represented one-side-contact situations, in which Tx was bounded to ± 4 Nm. In total, 36000 data points were sampled in the three different domains. Data points were evaluated using the original contact model (Fig. 1) by repeated Force-Dependent Kinematics (FDK) analyses. Data points which did not lead to equilibrium were discarded. The remaining 27620 points were randomly subdivided into a training (70%) and testing (30%) group. Three separate Feed-Forward Artificial Neural Networks (FFANN), consisting of four inner layers of 20 hyperbolic tangent sigmoid neurons each, were configured within the Neural Network Toolbox in MATLAB 8.1 (The MathWorks Inc., Natick, MA, 2013). The first network was trained to learn the relationships between the four –two medial and two lateral– sensitive loads (output) and the six pose parameters (input). Two other networks –one medial and one lateral– were trained separately to learn the relationships between the remaining loads of each side (output) and all the pose parameters plus the two sensitive loads from each side (input). We used the popular Levenberg-Marquardt training algorithm in conjunction with Bayesian regularization to avoid over-fitting. Stopping criterion was a training time of two hours for each network. The trained networks were translated to custom C++ DLL functions for successive inclusion in our MS model. The surrogate contact model replaced the original contact model and one gait trial was simulated with 4 different combination of the following model settings: original versus surrogate contact model, wrapping enabled versus disabled. RESULTS: The contact sampling model required 238 hours to evaluate the 36000 data points. Predicted tibiofemoral compressive forces under all simulated cases are shown in Fig. 2. A comparison with experimental measurements (eTibia line) is also shown. Surrogate model predictions showed a very good agreement with the original model counterparts. Fig. 3 summarizes the computation times: simulations took the longest when muscle wrapping was enabled and the benefits of using the surrogate model became evident only when the wrapping algorithm was switched off, leading to a 6x speed-up. Simulation time with the original contact model decreased by a factor of 8 by switching off the wrapping algorithm. DISCUSSION: The use of FFANN-based surrogate contact model, in place of the original rigid contact model, could substantially reduce the simulation time of a full gait cycle down to 3 minutes, when the wrapping algorithm was turned off. Such improvement could not be achieved when using the wrapping algorithm. This enlightens another important source of overhead in MS modeling –the muscle wrapping algorithm– which unexpectedly was found to dominate the simulation time. At each FDK iteration, the wrapping algorithm needs to be solved as well, introducing overhead. If the wrapping algorithm is slower than the contact algorithm, then the computation time of each step will be dominated by the former, leaving only a small fraction to be gained from the latter. SIGNIFICANCE: We showed that surrogate contact model could reduce the simulation time in a MS model of TKA down to a level which allows parametric studies and/or optimization to be feasible. We also discovered that the muscle wrapping algorithm constituted an unexpectedly large source of overhead during dynamic simulations. These represent new and important findings for the MS modeling community. REFERENCES: [1] M. A. Marra, V. Vanheule, R. Fluit, B. H. F. J. M. Koopman, J. Rasmussen, N. J. J. Verdonschot, and M. S. Andersen, “A Subject-Specific Musculoskeletal Modeling Framework to Predict in Vivo Mechanics of Total Knee Arthroplasty.,” J. Biomech. Eng., Nov. 2014. [2] I. Eskinazi and B. J. Fregly, “Surrogate modeling of deformable joint contact using artificial neural networks.,” Med. Eng. Phys., Jul. 2015. [3] Y.-C. Lin, R. T. Haftka, N. V Queipo, and B. J. Fregly, “Surrogate articular contact models for computationally efficient multibody dynamic simulations.,” Med. Eng. Phys., vol. 32, no. 6, pp. 584–94, Jul. 2010. ACKNOWLEDGEMENTS: This study was conducted within the ERC ‘BioMechTools’ project, funded by the European Research Council.

Full text

Introduction
•Since the calculation of joint contact forces is often carried
out using expensive finite-element or elastic-foundation
models, concurrent simulation of body-level dynamics and
detailed joint mechanics is computationally demanding.
•Simulation time for a single activity of daily living may reach
several hours, as shown in a recent Total Knee Arthroplasty
(TKA) musculoskeletal (MS) model [1].
•To speed up the computation, surrogate modeling techniques
have been proposed to replace the original contact model
(OCM) with a faster surrogate model (SCM)[2,3].
•Overhead may also arise from the computation of muscle and
ligament lines of action over obstacles, which require the
solution of a contact problem. Simple wrapping conditions
can be solved both analytically and numerically.
Objective
We developed and tested a surrogate contact model of TKA and
we assessed its performance during gait simulation using both
numerical and analytical wrapping algorithm.
Materials and Methods
•Sampling. 135.000 sample points were randomly generated
using a multi-domain approach [3]. The OCM (Fig. 1) was
created in the AnyBody Modeling System (AnyBody
Technology A/S, Aalborg, Denmark) and used to calculate the
TF loads resulting from the TF pose for each sample.
Additionally, 20.000 samples were evaluated for testing.
•Training. Feed-forward artificial neural networks (FFANN)
were trained until convergence to learn the implicit relations
between TF loads and pose (Fig. 2) [2,3].
•Gait simulation. A gait trial from a publicly available dataset
[4] was simulated using the OCM, the SCM, numerical and
analytical wrapping algorithm¹. Simulation times were noted.
M.A. Marra¹, M.S. Andersen², H.F.J.M. Koopman³, D. Janssen¹, N. Verdonschot¹,³
¹Orthopaedic Research Lab, Radboud University Medical Center, Nijmegen, The Netherlands, ²Department of Mechanical and
Manufacturing Engineering, Aalborg University, Aalborg East, Denmark, ³Department of Biomechanical Engineering, University of
Twente, Enschede, The Netherlands
Results
Surrogate model accuracy
a b
Gait simulation
Discussion and Conclusion
•Approximately 213 hours were necessary on an Intel® Core™
i5-4570 quad-core computer with 16 gigabytes of RAM for
the creation of the surrogate model. This time was paid up
front and could be reduced using parallel-computing.
•There were no substantial differences in predicted versus
experimental TF forces during a gait simulation using either
contact models and wrapping algorithms (Fig. 4).
•The SCM provided the largest acceleration in conjunction
with the analytical wrapping algorithm (Fig. 4). The latter is
preferable over the more general numerical algorithm when
computation time is a concern.
Conclusion
When used together with an analytical wrapping algorithm, our
surrogate contact model could reduce simulation time by 67%.
Figure 1. The original contact model used to
evaluate sample points by repeated static analyses.
The TF pose is defined by the relative position
between the femur (blue frame) and tibial (red
frame) component.
Figure 2. 2-stage FFANN used to learn the relations between TF pose (input) and
TF loads (output). In stage I (left half) MedFy, MedTx, LatFy, LatTx were fit as
functions of TF pose. In stage II (right half) the remaining TF loads were fit as
functions of the TF pose and the TF loads of stage I. HL: hidden layer, W: network
weight, b: network bias.
Figure 4. Left: proximo-distal component of tibiofemoral force predictions during
gait. Right: simulation times and the musculoskeletal model used.
Legend: eTibia: experimental TF force; NumWrp, numerical wrapping; AnlWrp,
anlytical wrapping; OCM, original contact model; SCM, surrogate contact model.
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Figure 3. Accuracy of the surrogate model on a testing dataset of ca. 20.000
sample points. (a) Regression plot of output versus target loads and (b) root-
mean-square errors of predicted medial and lateral forces and moments.
[1] Marra et al., “A Subject-Specific Musculoskeletal Modeling Framework to Predict in Vivo Mechanics of Total Knee
Arthroplasty”, J Biomech Eng. 2015 Feb 1;137(2):020904; [2] Eskinazi and Fregly, “Surrogate modeling of deformable joint
contact using artificial neural networks.”, Med Eng Phys. 2015 Sep;37(9):885-91; [3] Lin et al., “Surrogate articular contact
models for computationally efficient multibody dynamic simulations.”, Med Eng Phys. 2010 Jul;32(6):584-94; [4] Fregly et
al., “Grand challenge competition to predict in vivo knee loads.”, J Orthop Res. 2012 Apr;30(4):503-13
¹ The analytical wrapping algorithm was made available to us by AnyBody Technology A/S in a prototype version of the
AnyBody Modeling System for the solution of a cylindrical wrapping case.
Evaluation of a surrogate contact model of TKA.
Marco Marra, MSc
Marco.Marra@radboudumc.nl
Orthopaedic Research Laboratory, Radboud umc
P.O. Box 9101, 6500 HB Nijmegen, The Netherlands
The research leading to these results has received funding from the
European Research Council under the European Union's Seventh
Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 323091
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