**3.3. Modeling of the biomechanical behavior of the spinal structures**

The surface models are extended to biomechanical models by adding specific biomechanical properties to all spinal structures. The biomechanical property of the translational movement of the intervertebral discs is described by a force law, which is composed of a geometry-based stiffness term and a damping term [18, 23]. Most movements of the intervertebral discs do not consist of purely translational movements but also of intervertebral rotations. If the disc is deflected from its initial position, a deflection-dependent reaction torque will be developed [24].

As already said, in the model also the anterior and posterior longitudinal ligament (ALL/PLL), the ligamentum flavum (LF), the capsular ligament (CL), the interspinous ligament (ISL), the supraspinous ligament (SSL) as well as the intertransverse ligament (ITL) are implemented. Because a ligament can be defined only between two points, broad ligament structures are realized by a bundle of several fibers. The biomechanical properties of the ligaments are represented by characteristic curves of appropriate literature [25].

In the following models, the facet joints are simulated by a 3D contact surface whose dimen‐ sions and orientation are directed to the inclinations of the facet joints of the surface model. In this way, the modeling considers the different individual dimensions and specific orientations of the facet joints in all lumbar levels. Is exclusively a detailed analysis of the load behavior of the facet joints in focus, the facet surfaces are simulated by a 3D layer of cartilage (see Section 4). It should be noted that a detailed modeling of the facet surfaces is accompanied by a large increase of computation time. Therefore, prior to the model building a specific question should be defined, in order to decide whether such a detailed facet joint modeling is needed.

A further important structure to stabilize the human body in upright position is the muscu‐ lature. According to [7], the four muscle groups, left and right musculus erector spinae and left and right rectus abdominis muscle, are modeled. The force representing the musculus erector spinae is varied so, that additional torque on the upper endplate of the vertebra L1 is no longer necessary to maintain the lumbar spine in a state of equilibrium. The model parameters used for muscles are taken from [7, 26]. These muscular structures are implemented at the moment just in the models of Chapter 4.1. A comparison of the different spine models with the corresponding muscle models is performed (see Section 4).
