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Research Papers

# Adaptive Surrogate Modeling for Efficient Coupling of Musculoskeletal Control and Tissue Deformation Models

[+] Author and Article Information
Jason P. Halloran, Ahmet Erdemir

Department of Biomedical Engineering (ND-20), Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, OH 44195

Antonie J. van den Bogert1

Department of Biomedical Engineering (ND-20), Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, OH 44195bogerta@ccf.org

1

Corresponding Author.

J Biomech Eng 131(1), 011014 (Nov 26, 2008) (7 pages) doi:10.1115/1.3005333 History: Received February 29, 2008; Revised August 21, 2008; Published November 26, 2008

## Abstract

Finite element (FE) modeling and multibody dynamics have traditionally been applied separately to the domains of tissue mechanics and musculoskeletal movements, respectively. Simultaneous simulation of both domains is needed when interactions between tissue and movement are of interest, but this has remained largely impractical due to the high computational cost. Here we present a method for the concurrent simulation of tissue and movement, in which state of the art methods are used in each domain, and communication occurs via a surrogate modeling system based on locally weighted regression. The surrogate model only performs FE simulations when regression from previous results is not within a user-specified tolerance. For proof of concept and to illustrate feasibility, the methods were demonstrated on an optimization of jumping movement using a planar musculoskeletal model coupled to a FE model of the foot. To test the relative accuracy of the surrogate model outputs against those of the FE model, a single forward dynamics simulation was performed with FE calls at every integration step and compared with a corresponding simulation with the surrogate model included. Neural excitations obtained from the jump height optimization were used for this purpose and root mean square (RMS) difference between surrogate and FE model outputs (ankle force and moment, peak contact pressure and peak von Mises stress) were calculated. Optimization of the jump height required 1800 iterations of the movement simulation, each requiring thousands of time steps. The surrogate modeling system only used the FE model in 5% of time steps, i.e., a 95% reduction in computation time. Errors introduced by the surrogate model were less than $1mm$ in jump height and RMS errors of less than $2N$ in ground reaction force, $0.25Nm$ in ankle moment, and $10kPa$ in peak tissue stress. Adaptive surrogate modeling based on local regression allows efficient concurrent simulations of tissue mechanics and musculoskeletal movement.

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## Figures

Figure 1

Coupled simulation of musculoskeletal movements and tissue deformations focusing on the adaptive surrogate modeling approach. The components of q represent talus rotation and vertical position and Q are the corresponding loads. Note that when a finite element analysis is requested Q is returned back to the musculoskeletal model for movement simulations and also to the surrogate model with the corresponding q to expand the database. Data handling was performed in Matlab (Mathworks, Inc., Natick, MA) where a script was developed to link the musculoskeletal model with the FE through file input/output and with the surrogate model representation of the foot complex directly.

Figure 2

Movement of the lower extremity during jumping obtained from the simulation with maximum jump height prediction. von Mises stress distributions within the FE model of the foot are also illustrated for three time points during the simulation.

Figure 3

Jump height with respect to function call throughout the entire optimization process.

Figure 4

Percent FE analysis for each successive function call averaged over all optimization iterations. The horizontal axis represents an iteration during the optimization with 33 function calls (one initial forward dynamic simulation for function evaluation plus 32 gradient calculations). Additional function evaluations during line search were not included in this graph due to the inconsistent number of evaluations per iteration. It should be noted that in direct coupling of the musculoskeletal and FE models, FE simulations will be conducted 100% of the time.

Figure 5

Scatter plot of inputs (talus rotation and vertical ankle position) used to generate the database of FE simulations (top). Isometric view of the database for the ankle moment (middle), and vertical load (bottom) as a function of the two inputs. Database points were graded to represent steep (red/lighter) to flat (blue/darker) areas of the data set.

Figure 6

Peak pressure plot (top) and ankle vertical load (bottom) as a function of time for the optimal solution. The plot portrays close agreement between surrogate model results and those obtained from FE analysis. Toe-off occurred at approximately 0.25s and peak pressure values represent the foot-floor interaction. Contour plots for von Mises stress were included in Fig. 2 with peak values yielding a very similar behavior as peak pressure.

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