The response surface method-genetic algorithm for identification of the lumbar intervertebral disc material parameters

Highlights

• An algorithm was proposed to identify material of lumbar intervertebral disk.

• The algorithm combined response surface and genetic algorithm.

• The accuracy and efficiency of parameter identification improved considerably.

• Compared with the normal disk, permeability decreased for the enucleated disk.

• Elastic modulus decreased and Poisson's ratio increased for the enucleated disk.

Abstract

Long-term compressive load on the lumbar intervertebral disc (IVD) might lead to lumbar IVD herniation. Exploring the material parameters of normal and degenerative enucleated IVDs is the basis for studying their mechanical behavior. According to the inverse analysis principle of the parameter estimation, an optimization method was proposed to identify the parameters of the porous material of the lumbar IVD based on finite element inverse analysis. The poroelastic finite element models were established in line with the compression creep experiment. The material parameters were combined by Box–Behnken design (BBD), and the response surface (RS) models were constructed using a quadratic polynomial with cross terms and optimized by genetic algorithm (GA). The results showed that the simulation result of the best material parameter combination had a good agreement with the experiment. Compared with the normal lumbar IVD, the elastic modulus and permeability decreased, and Poisson's ratio increased for the enucleated disk, resulting in a significant difference in mechanical properties. The algorithm used in this study can reduce the parameter identification error compared with only the RS method, and decrease the number of finite element simulations compared with only the GA.

Introduction

Long-term compression load on the lumbar IVD can easily give rise to the degeneration and protrusion of the IVD; the resulting compression of the spinal nerve might cause low back pain, which is clinically called lumbar IVD herniation (or LDH for lumbar disk herniation) [[1], [2], [3], [4]]. This disease severely affects patients' lives and occupations. Studies have shown that it gradually exhibits a trend toward the younger population and has become a high-incidence medical condition worldwide [5]. The resection of the nucleus pulposus (NP) has become one of the main clinical treatment modalities for this condition in recent years. By removing the degenerating tissue, the lumbar IVD is decompressed, and the condition can be improved [6]. Since the mechanical state and material properties of the lumbar IVD change after surgery [7], studying the material's parameters and characteristics can predict the mechanical behavior of the degenerative lumbar IVD. The process can provide accurate material models for finite element simulation [8,9], reveal its mechanical mechanism, and provide a theoretical basis for clinical treatment.

Some mechanical properties of biological materials, including their permeability, are difficult to measure directly or indirectly by experimental methods. Therefore, the material properties can be identified by using the inverse problem principle of parameter estimation. Identification methods include direct and indirect methods. The direct method derives the constitutive equation from the mechanical model and the input load, fits the experimental data, and obtains the material parameters through the inverse analysis [[10], [11], [12]]. However, this method cannot solve the problem of complex boundary conditions, and the accuracy of the biological simulation is low. The indirect method is a trial algorithm. The error between the simulation and experiment results is within the acceptable range, and the predicted values of material parameters are considered to be close to the actual value by repeatedly adjusting the combination of material parameters. The indirect method is suitable for the analysis of nonlinear inverse problems under various complicated boundary conditions; however, it exhibits low calculation efficiency as a disadvantage. A series of experimental design methods and intelligent optimization algorithms have been derived to improve search efficiency and accuracy, including the RS method, the GA, and the BP (backpropagation) neural network algorithm.

Concerning material identification methods, Wagnac et al. [13] calibrated the hyperelastic material properties of IVD under fast dynamic compressive loads by using experiments and finite element models. Malandrino et al. [14] carried out a sensitivity study of IVD material parameters in a finite element model by using a statistical factorial analysis approach and reported that permeability and stiffness strongly affected the response. Nikkhoo et al. [15] implemented compression creep experiments on pig lumbar IVDs, performed an inverse analysis using the finite element method, and obtained the material parameters of intact and degenerated discs by an RS method. Chen et al. [16] established the finite element model and constructed an RS model through animal tissue experiments to obtain rib material parameters by using inverse analysis. Hua et al. [17] proposed an inverse method combined with experiments, finite element simulation, and optimization algorithms to identify the material parameters of the skin of a dummy's head and knee. Delalleau et al. [18] proposed a procedure to characterize the skin's mechanical properties by using random inverse recognition by minimizing the function between the experiment and the finite element model. Newell et al. [19] used the inverse finite element algorithm to characterize the material properties of human lumbar IVDs combined with experiments and found that IVD stiffness and strain rate had a log-linear relationship. Currently, the material parameters of lumbar IVDs in the literature are quite varied. There are few studies on the material parameters of lumbar IVD after nucleus pulposus removal. Therefore, according to the indirect method, the algorithm was proposed combined with the experiment, a finite element method, RS method, and GA in this study. The algorithm was implemented to identify the material parameters of normal and enucleated lumbar IVD. The material parameters and mechanical properties were compared.