Technical noteEfficacy and efficiency of multivariate linear regression for rapid prediction of femoral strain fields during 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 ( = 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.
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