Calibration of the finite element model of a lumbar functional spinal unit using an optimization technique based on differential evolution

Abstract

The development of a finite element model of the lumbar spine usually involves choosing among available alternatives to decide which values should be assigned to the material properties of the different spinal structures. Furthermore, the model has to be validated so that a reasonable approximation to the mechanical response of the lumbar spine is achieved. One approach for choosing such material properties involves calibrating the model by choosing the properties that produce the best fit with the in vitro mechanical response of the lumbar spine.

This study proposes the use of an optimization method based on differential evolution to calibrate the finite element model of a functional spinal unit. Calibration was performed using reported in vitro data on the mechanical response of an intact lumbar functional unit and its successive reduced stages after the dissection of ligaments, facet joints, vertebral arch and nucleus pulposus. The loading conditions in the study were pure moments in flexion, extension, lateral bending and axial rotation. Considering all dissection stages and loading conditions, the maximum difference in vertebral rotation between the in vitro data and the model results was only 1.24°. Other model results such as facet loads and annulus fibrosus behavior also correlated well with reported data.

Introduction

The use of detailed finite element models (FEMs) of the lumbar spine to predict its mechanical response under quasi-static loading conditions is quite common in the field of spine biomechanics research [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Model development usually starts by defining the geometry of the vertebrae to which the soft tissues (disc, ligaments) are added. Then, mechanical properties are assigned to each of the spinal structures, and lastly, the model is validated by comparing the model's results to ‘in vitro’ results. One issue that must be addressed when developing the model is the great dispersion that is often found in the reported values for the material properties of the different spinal structures. In fact, the lumbar spine's biomechanical response varies greatly depending on which material properties are chosen among the different published values, something which Zander et al. [10] found for the ligaments and Yoganandan et al. [11] discovered for the elastic properties of the annulus fibrosus. To solve this problem some researchers have proposed calibrating the model by choosing the material properties of one or even several spinal components that produce the best fit between the model and reported experimental results [6], [7], [8]. Similar approaches have also been used in the development of models of other human joints [12], [13].

Problems may also arise if the validation process is carried out using a model from an intact spine, as is the case in many studies, since there is no guarantee that the model can be successfully applied to simulate situations where the spinal structure is altered due to the removal of any of its anatomical components, such as surgical stabilization techniques like fusion or artificial disc implantation. Nowadays, there are experimental studies such as those of Heuer et al. [14], [15] that detail the mechanical response of the lumbar functional spinal unit (FSU) in both an intact and anatomy-reduced state under different types of loading, thereby allowing for a more complete validation of the model. This type of data can also be used to calibrate a FEM so that it successfully simulates the contribution of the different spinal elements in its mechanical response and could therefore be used to obtain clinically relevant data for various loading cases and defect scenarios. Schmidt et al. [6], used the experimental data of Heuer et al. to calibrate a FEM of a lumbar FSU. In this study, the model was stepwise-calibrated, starting with the annulus fibrosus and successively adding subsequent spinal structures. At each stage, certain material properties of the last added structure were varied until good agreement with the experimental data was achieved. In order to improve the applicability of the methodology proposed in this study, a logical next step would be to propose a specific optimization technique which computes parameters more efficiently, while at the same time constraining the material properties of the different spinal structures in order to preserve their physiological mechanics.

This study proposes calibrating the FEM of a lumbar FSU using a verifiable evolutionary computation technique, such as differential evolution [16]. The calibration process aims to find the values of a set of parameters which define the material properties of the different structures in the model that achieve the best match between the model and the ‘in vitro’ response of the lumbar FSU. The experimental data used for model fitting was taken from the results on biomechanical response published by Heuer et al. [15]. In the calibration of the model, all of the loading cases and dissection stages covered in the study of Heuer et al. were considered. Specifically, loading cases included flexion, extension, lateral bending and axial rotation, while dissection stages started with the intact segment and progressively reduced the structure until there remained only the annulus fibrosus and vertebral bodies. In the definition of the mechanical properties of each spinal structure, care was taken to simulate the type of mechanical behavior expected for each of them. Therefore, the methodology proposed in this study attempts to obtain a model which is capable of reliably predicting the mechanical response of the lumbar FSU in its intact and anatomically altered forms under different loading conditions, while at the same time preserving the physiological mechanics of its different anatomical structures.