**2.4 Ligament modeling**

In reality the intervertebral disc is not only deformed by loads, but also bent by

Through the facets, adjacent vertebrae are connected via a thin layer of cartilage.

The stiffness coefficients are taken from [18] and the damping values is defined

as 10% of the stiffness term. The damping coefficient is used to obtain a better attenuation of the maximum linear and angular accelerations of the head [19].

*Representation of the facet width and height, which builds the basis area for the facet contact simulation.*

In the model the facet cartilage layers are approximated by an unilateral contact spring-damper element, whose contact area is determined by the facet geometry. The contact area is a rectangular region, which represents the facet width and height. With an additional dimension the cartilage layer of the facet joint is simulated. The cartilage layer thickness of the lower cervical spine bases on [16] and is determined to be 0.00045 m for the superior layer and 0.00049 m for the inferior. The parameterization of the geometry, positioning and orientation of the 3D facet contact area is determined with respect to the C7 upper facet surface. The modeled facet contact surface is assumed to be an average facet width and facet height of the superior facet surface C7 and the inferior facet surface C6. The average model geometry results in a facet width (FW) of 0.0094 m and a facet height (FH) of 0.009 m. Comparison of the approximate facet area (FCA) of the current model with FCA = 0.000085 m2 with the average facet area superior C7 and inferior C6 reported in [17], a discrepancy of FCA = 0.000089 m2 (**Figure 5**) can be observed. This information is given at this point in order to show the extent to which the model assumption differs from the

external torques. Depending on the action direction of the external torque the intervertebral disc performs a flexion and extension movement, an axial rotation or a lateral flexion. To counteract this rotations, the intervertebral disc develops a counter-torque. This non-linear disc torque is defined by two-dimensional functions that describe the relationship between the disc torque and the relative angle. A specific input function is assigned to the torques acting around three axes of

rotation x, y and z. The applied input function bases on [15].

experimental measurements with regard to the geometry.

**2.3 Facet joint modeling**

*Recent Advances in Numerical Simulations*

**Figure 5.**

**82**

The spinal ligaments provide stability to the motion segments allowing motion within physiological limits. Ligaments are uniaxial structures that resist only tensile or distractive forces becoming slack in compression [14, 20].

In the FSU model the following ligaments are incorporated: anterior and posterior longitudinal ligament (ALL and PLL), flava ligament (FL), interspinous ligament (ISL), nuchal ligament (NL) and the left and right capsular ligaments (CL) (**Figure 6**). Ligaments, which have a broad structure, are represented by several fiber bundles. For instant, ALL and PLL are composed of a right, left and middle ligament structure. CL is approximated by four individual ligament structures that attach to the top, bottom, left and right surfaces of the articular processes. The ISL

**Figure 6.** *Representation of the ligament attachment points.*

extends over the entire edge of the spinous process and is therefore modeled using three bundles of ligaments. The LF attaches to the proximal edge of the lamina and is represented by six ligament fibers. The NL is an extension of the SSL which extends from the external occipital protuberance to the spinous process of C7 and attaches all the posterior tips of the spinous processes in between [21].

**2.7 Motion segment response to small loads**

*DOI: http://dx.doi.org/10.5772/intechopen.98211*

evaluation of the models deformation.

**2.8 Motion segment response to large loads**

intervertebral discs in the current model.

**Figure 7.**

**85**

A validated intact FE model of the C4-C5-C6 cervical spine to simulate progressive disc degeneration at the C5-C6 level is presented by [24]. The intact and three degenerated cervical spine models are exercised under the compression load of 80 N. The results of the intact spine model are used to compare the intervertebral disc pressure between vertebrae C5-C6 in the current FSU model. The motion segments were subjected to a small static compression load of 80 N in z-direction. While in the current model the resulting displacement of the intervertebral disc is measured, in the FSU model the overall force displacement response of C4 with respect to C6 is determined. Therefore, the comparison can only be taken as a rough

*Parameter Dependencies of a Biomechanical Cervical Spine FSU - The Process of Finding…*

In the second stage of validation, the FSU model is subjected to larger loads of 200 N, 500 N and 673 N to determine its intervertebral disc pressure and disc deformation. The load of 200 N is chosen to represent the combined effects of head weight and muscle tension [27]. The human cervical disc pressure using a pressure transducer, side-mounted in a 0.9 mm diameter needle is investigated by [27]. Forty-six cadaverous cervical motion segments aged 48–90 years are subjected to a compressing load of 200 N for 2 s. Due to the lack of data available for high load cases, these data are used to analyze the characteristics of the intervertebral discs. The deformation value under a certain load is only provided for the specific healthy disc segment C7-T1. These results are used to compare the characteristics of the

A MBS model of human head and neck C7-T1 is presented by [14]. The MBS model comprise soft tissues, i.e. muscles, ligaments, intervertebral discs and supported through facet joints. Also eighteen muscle groups and 69 individual muscle segments of the head and neck are included in the model. For load–

displacement testing, each motion segment is mounted so that the inferior vertebra is rigidly fixed whereas the superior vertebra is free to move in response to the

*Response of model motion segments to applied compressive loads of 80 N and 200 N. the orange bar shows the intervertebral disc pressure for the corresponding motion segment as reported by [24] and the yellow one as*

*reported by [27]. The results of the current FSU model is highlighted in blue.*

The determination of the ligament attachment points is carried out on the basis of the vertebral geometry and is checked by an expert.

The ligament's characteristic is modeled by the load displacement curves [13, 22, 23]. When a ligament is stretched, it develops a force that is specific to the ligament in question. It acts against the direction of the stretch with no resistance in compression.

### **2.5 Load case configuration**

In order to analyze the reaction of the spinal structures to a load, an external force of 80 N is applied to the endplate of the vertebra C6. This loading case is chosen because the cervical spine is permanently loaded by the weight of the head [24]. To prevent additional torques, the y-coordinates of the external load markers have the same position as the y-coordinates of the disc joint, so that there is no initial lever arm that could lead to unintentional torques.

#### **2.6 Model validation**

An important step in the simulation process is the model validation, with which the simulation results are checked for correctness. The correctness of the FSU is proven by comparing the intervertebral disc pressure and disc deformation to existing published data. After researching the literature, it turned out that there is only a limited possibility of validation data that exactly depicts the simulation scenario we have modeled at the moment. In general, there is the difficulty that the own model configuration does not necessarily exactly match to that of other researchers, since different specific research questions have to be answered. In order to get the response of the FSU model to different loads, the FSU is exposed to small and large external loads. The disc pressure and deformation are compared (**Table 3**).


*The width (W), depth (D) and area (A) of upper (u) and lower (l) endplates (EP) are presented of different studies. Further, the disc width (DW), the disc depth (DD), the disc area (DA) and the disc height (DH) is presented. The idea is to present the various possible measures to be able to assess the model parameters of the current model.*

#### **Table 3.**

*Comparison of the vertebra C7 and C6 anthropometry.*

*Parameter Dependencies of a Biomechanical Cervical Spine FSU - The Process of Finding… DOI: http://dx.doi.org/10.5772/intechopen.98211*
