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The purpose of this present work is to provide a tool to better understand mechanically related pathologies of the lumbar unit and the spinal structure by providing spinal cord deformations in different loading cases. In fact, spinal cord injury (SCI) resulting from a traumatic movement leades to a deformation of the neural and vascular structure of the spinal cord. And since the magnitude of the spinal cord stress is correlated with the pressure of the vertebral elements, stresses will be computed on all theses components. Physical properties of the vertebrae, various ligaments, the discs, and the spinal cord are described under simple loading as compression, and combined loading, flexion and lateral bending to evaluate the pressure undergone by different components of the lumbar unit. A nonlinear three-dimensional finite element method is used as a numerical tool to perform all the computations. This study provides accurate results for the localisation and the magnitude of maximum equivalent stress and shear stress on the lumbar unit and especially for the spinal cord. These results showed that stresses are more important when a compression of 500 N is combined with a flexion and a lateral bending. In particular, shear stresses are maximum for the spinal cord and the four intervertebral discs for the case of a flexion of 3.8 N.m and a lateral bending of 6.5 N.m.

Spinal cord injury (SCI) is usually a consequence of a traumatic movement resulting in a deformation of the neural and vascular structure of the spinal cord exceeding their structural and physiological limits. As a matter of fact, the neurological damage is due to different loading combination. The identification of the parameters, leading to neurological deficit during walking (Neutrogenic claudication) can help to understand and prevent all these phenomena. Based on clinical observations, the physiological changes of soft tissues throughout life is another factor to be taken into account. They could reduce the available space for the neural structures (spinal cord, medullar cone and roots) [

In general, the increase of stenosis during walking and prolonged standing position leads to neutrogenic claudication and in common to sphincter disorder. In some cases, these degenerative lesions exist whereas the clinical phenomena did not appear during rest.

In medical revues, the spinal component displacement (mainly the intervertebral disc) have been often related to neurological deficits or complications after lumbar spine manipulations. These observations have been made in [

From biomechanical point of view, many authors provided some answer to how different components of the spinal column interact with each other under different loading situations but none of these works examined the spinal cord behavior in the lumbar unit. One example of these works is the one presented in [

Finally, the biomechnical study of cervical flexion myelopathy throughout a finite element model presented in [

A numerical biomechanical model will then be presented in this article to describe the mechanical behavior of a complete functional human spinal unit; this anatomical model consists in five lumbar vertebrae, four intervertebral discs, the physiological ligaments, capsular articular parts, and spinal cord (terminal cone).

The biomechanical study of this model has been established to study the influence of the stresses on all the elements of this model and to explain advanced state of neurological deficit signs.

Three dimensional finite element model is built up using the computed tomography (CT) of the L1-L5 lumbar unit including mechanical properties designed for the five vertebrae, the four intervertebral discs, the ligaments (anterior, posterior, flavum, interspinous and supraspinous), articular and capsular parts, and spinal cord (cauda equina).

The numerical modelling based on a finite element methods (FEM) simplifies the structure whether it is anatomical or not by reflecting its mechanical properties. This method requires specification of the geometry of the modeled structure, the loads and pressure applied to that^{ }structure, and the elastic properties of the components. The^{ }geometry is subdivided into small regions (elements) and the^{ }differential equations governing the deformation of solids are^{ }numerically solved. Computed quantities include^{ }local deformations in response to the applied loads, as well^{ }as the corresponding stresses.

Three steps are needed to accomplish these tasks:

1) Definition of the geometry of the column constitutive parts;

2) Establishement of the laws which govern the behavior of every part of the lumbar spinal unit;

3) Evaluation of the model by performing a series of numerical computations.

Concerning the first step, the total obtained meshed model representing the lumbar is given in

The discs have been added to fit between two vertebrae as well as the totality of the physiological ligaments. Moreover, as it is suggested in most described physiological cases, the spinal cord is added to fit into the medullar canal which is between the first and the second

lumbar vertebra. The annulus fibers and the cartilaginous plates are respectively shown in the same figure. Details of the meshing of vertebrae are shown in

The appropriate meshing for the disc is shown in

The second step consists in modeling the vertebrae as an elastic orthotropic structure with Young’s Modulii and Poisson’s ratios obtained through the mineral bone density. The maximun and the minimum values of the the young modulii (E_{x}, E_{y}, E_{z}) taken in the three directions (x, y, z) are summeraized in the _{xy}, G_{yz}, G_{xz}) and

the three Poisson ratios (v_{xy}, v_{xz}, v_{zx}). For the intervertebral disc which is divided into the annulus and pulposus part, Young’s Modulii and Poisson’s ratios were taken in the litterature [

The third step is performed through different loadings using the finite element method, FEM. The boundary

conditions are specified in

The boundary conditions for the different cases use a fixed bottom of the lumbar human unit (in the three directions) and the appropriate load for each presented case.

For the case of compression, a pressure is assigned to the top surface of the geometry to ensure the homogeinity of the applied load. The pressure is computed by dividing each load by the area located on the first lumbar L1.

The equivalent von Mises stress given in the following equation is used to evaluate the maximum value. This value is taken as a limit to the ultimate elastic stress for most structural computation, Equation (1).

where, and are the principal stresses.

The results will be given for the whole structure. Results obtained for the bony structure (vertebrea) and the intervetbral disc are presented here after followed by the maximun (critical values) of stresses obtained for the spinal cord.

The disc is also split into six different parts as done for each vertebra. As known, the disc is composed of nucleus pulposus and annulus fibrosis. In the normal healthy disc, the pulposus hydrated core exerts a hydrostatic pressure (pressure intradiscal (IDP) on fibers of the annulus fibrosis [

^{th} and the 5^{th} vertebra (L45). The values of indicate that the nucleus pulpous can absorb some compressive loads and that the maximum values are obtained for the outer posterior and anterior points, which are located in the annulus region as pointed out in [