Elsevier

Medical Engineering & Physics

Volume 63, January 2019, Pages 88-92
Medical Engineering & Physics

Technical note
Efficacy and efficiency of multivariate linear regression for rapid prediction of femoral strain fields during activity

https://doi.org/10.1016/j.medengphy.2018.12.001Get rights and content

Highlights

  • Full femoral strain was calculated using a multi-linear regression (MLR) model.

  • A similar behaviour observed between different activities.

  • The peak difference with corresponding finite-element (FE) calculations was 8.6% of the maximum computed micro-strain.

  • The MLR-based surrogate model was faster than the finite-element analysis for solving 209 frames or more.

  • The MLR's solution time for 1000 frames of motion was approximately 21% of the time required for the corresponding FE model.

Abstract

Multivariate Linear Regression-based (MLR) surrogate models were explored to reduce the computational cost of predicting femoral strains during normal activity in comparison with finite element analysis. The musculoskeletal model of one individual, the finite-element model of the right femur, and experimental force and motion data for normal walking, fast walking, stair ascent, stair descent, and rising from a chair were obtained from a previous study. Equivalent Von Mises strain was calculated for 1000 frames uniformly distributed across activities. MLR surrogate models were generated using training sets of 50, 100, 200 and 300 samples. The finite-element and MLR analyses were compared using linear regression. The Root Mean Square Error (RMSE) and the 95th percentile of the strain error distribution were used as indicators of average and peak error. The MLR model trained using 200 samples (RMSE < 108 µε; peak error < 228 µε) was used as a reference. The finite-element method required 66 s per frame on a standard desktop computer. The MLR model required 0.1 s per frame plus 1848 s of training time. RMSE ranged from 1.2% to 1.3% while peak error ranged from 2.2% to 3.6% of the maximum micro-strain (5020 µε). Performance within an activity was lower during early and late stance, with RMSE of 4.1% and peak error of 8.6% of the maximum computed micro-strain. These results show that MLR surrogate models may be used to rapidly and accurately estimate strain fields in long bones during daily physical activity.

Introduction

Quantifying femoral strain distribution is important for studying bone adaptation [1], [2], [3], diagnosing individuals most at risk of femoral fracture [4], [5], [6], and optimizing the biomechanical behaviour of implantable devices [7], [8]. Over the last few decades, finite-element analysis has been used extensively to quantify the entire femoral strain field [9], [10], [11], and there is growing interest in using this method to characterise strain distributions in multiple individuals [12], [13] and across multiple trials and tasks [14]. In addition, there is need to investigate the influence of the musculoskeletal (MS) modelling process on femoral strain predictions, by performing probabilistic analyses to account for uncertainties in the MS model inputj parameters [14], [15], [16] and examining alternative muscle recruitment strategies [17]. Unfortunately, the computational cost of performing such analyses can be prohibitive, thus new methods are needed to accurately and rapidly estimate the femoral strain field to enable large-scale studies of 100′s–1000′s of simulations to be performed.

Surrogate models represent a viable solution in that they can be trained using finite element calculations of femoral strain for a limited number of training sets and then used to rapidly provide femoral strain estimates for an arbitrary frame of motion or an entire activity. A variety of surrogate models have been used by the biomechanics community including Multivariate Linear Regression [18], [19], Bayesian modelling [20], Artificial Neural Networks [18], [21], [22], Random Forest [23] and Kriging [24–26], either for linear problems, (e.g., assessment of femoral neck fracture during a single load case [18]) or for non-linear problems (e.g., modelling the contact between bone and implant [19]). Most studies predict a single scalar outcome, such as joint moments and muscle forces [27]; contact forces and contact pressure [[21], [25], [26], [28], [29], [30]; femoral neck strain and fracture load [18]; implant micro-movement and stress shielding [20], [31]. Multivariate Linear Regression has been used for predicting femoral neck strain [32], fracture load [18] and the micro-movement at the bone-implant interface [19]. However, the error and computational advantage of MLR over finite-element models remains unclear for the calculation of strain over the femoral volume and across normal activities of daily living.

The aim of this work was to explore the use of MLR for predicting femoral strain fields for a range of activities of daily living. Muscle forces, joint reaction forces and femoral strain were calculated for a single individual performing five tasks using a previously developed musculoskeletal and finite-element model [16]. A MLR surrogate model was trained using femoral strain, muscle forces, and joint reaction forces for a limited number of randomly selected frames of motion and then used to estimate femoral strain for multiple motor tasks. Model performance was assessed by comparing MLR estimates of the femoral strain field to corresponding results obtained from finite-element calculations.

Section snippets

Data

A full-body musculoskeletal model, finite-element model of the femur of the dominant leg, marker trajectories, and ground reaction forces for a single healthy participant (female, 68-year-old, 53 kg weight, 157 cm height) were obtained from a previous study [16]. All experimental and computational methods are described in detail by Martelli et al. [16] and Dorn et al. [33], respectively. Briefly, marker trajectories and ground reaction forces were recorded for five trials of each of the

Results

The trial-by-trial comparison showed that the coefficient of determination and slope were close to unity for the training datasets greater than 50 (RTrial2 = 0.84 − 0.94; slopeTrial=0.97−0.99). The prediction error of the surrogate model was a function of the size of the training set. Increasing the size of the training set from 100 to 200 frames reduced the average RMSE across trials from 132 µε to 108 µε, while a relatively small decrease in RMSE to 107 µε was obtained by increasing the

Discussion

Finite element analysis has been used extensively in orthopaedic biomechanics research [37], but there are a number of barriers involved in the translation of FE modelling to the clinic. One problem is that predicting the full femoral strain for multiple tasks using a coupled FE-musculoskeletal modelling approach is computationally expensive. The current study represents a first step in overcoming this barrier, by demonstrating that reliable estimates of strain distributions may be obtained

Conclusions

A Multivariate Linear Regression model was successfully developed for a single individual and used to rapidly predict the full femoral strain field for a range of activities of daily living. The MLR model was able to predict the femoral strain field for each studied activity within an error comparable to the intrinsic error in finite-element models based on clinical CT images and was computationally advantageous when 209 loading cases or more were analysed. Hence, MLR enables large statistical

Competing interests

There are no conflicts of interest associated with the work performed in this study.

Funding

This work was supported by the Australian Government Research Training Program Scholarship (AGRTPS); and the Australian Research Council (Grant no. DP180103146).

Ethical approval

Not required.

References (39)

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