**2.7 Motion segment response to small loads**

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

The determination of the ligament attachment points is carried out on the basis

The ligament's characteristic is modeled by the load displacement curves [13, 22, 23].

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

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

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.

attaches all the posterior tips of the spinous processes in between [21].

of the vertebral geometry and is checked by an expert.

initial lever arm that could lead to unintentional torques.

**2.5 Load case configuration**

*Recent Advances in Numerical Simulations*

**2.6 Model validation**

(**Table 3**).

**Model C7**

Current FSU model

Hueston et al. [23]

Tan et al. [25]

Yoganandan et al. [20]

Pooni et al. [26]

**Table 3.**

**84**

**EPWu [mm]**

**C7 EPDu [mm]**

*Comparison of the vertebra C7 and C6 anthropometry.*

**C6 EPWl [mm]**

**C6 EPDl [mm]**

19.0 15.1 19.5 15.7 220.8 316.3

**C7 EPAu [mm2 ]**

*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.*

**C6 EPAl [mm2 ]**

21.7 16.9 19.2 15.4 288.0 232.2 20.45 16.15 0.0068 260.1

19.0 15.1 19.5 15.7 220.8 316.3 268.6

**C6-C7 DW [mm]**

**C6-C7 DD [mm]**

**C6-C7 DH [mm]**

0.005– 0.0075

**C6-C7 DA [mm<sup>2</sup> ]**

> 168– 502

200– 502

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 evaluation of the models deformation.

#### **2.8 Motion segment response to large loads**

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 intervertebral discs in the current model.

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

#### **Figure 7.**

*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.*

In order to identify the effect of both stiffness and damping variations on the current FSU model, one-way sensitivity analysis is performed. After every simulation run the corresponding changes in the intervertebral disc pressure in the C6-C7 segment are reported as a difference between initial and current disc pressure

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

The first series of simulations consider the variation of the stiffness term *c*, which is expressed in Eq. (1), however the damping value *d* is held constant by 50000 *Ns=m*. For the sake of simplicity, the same value *ci* is assigned to *cx*,*cy*,*cz* in every *i*-th simulation. Starting from the initial value of 500000 *N=m* in each experiment repetition the stiffness parameter is increased and decreased by a

It can be seen in **Figure 9**, that the variation in the stiffness term results in the

The second part of the experiments aims the analysis of the system sensitivity with respect to alternations of the damping constant *d* (see Eq. (1)). In order to simplify the experiment execution, the same value *di* is assigned to *dx*, *dy*, *dz* in each *i*-th simulation run. The stiffness term is set to be constant, i.e. *ci* ¼ 500000 *N=m* for all *N* trials. The damping parameter *di* is increased and decreased by a fraction

The impact of the damping parameter changes on the disc pressure is presented in **Figure 10**. Similarly to the results obtained in Section 3.2, an obvious influence of the damping term on the disc pressure can be observed, where the linear changes of

tude of the change is not symmetrical for decreasing and increasing values of the

*Decreasing (left side) and increasing (right side) the stiffness term c by factor f impact the intradiscal pressure.*

. However, the magni-

linear change of the disc pressure, which points out to the linear relationship between these two variables. However, the change plot indicates the opposite linear relationship between the stiffness and the disc pressure, where increasing of the stiffness causes decreasing of the pressure. Note, that the course of the disc pressure changes is symmetrical, i.e. the minimum and maximum changes in the pressure value are of the same magnitude. According to the revealed results the maximal absolute pressure change is reported to be 7*:*<sup>0935249</sup> � <sup>10</sup>�<sup>5</sup> *MPa* or 0.02% of the

*δPi* ¼ *Pi* � *Pinit*, where the initial pressure value is 0*:*301315 *MPa*.

**3.2 Simulation results of stiffness term alternation**

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

**3.3 Simulation results of damping term alternation**

*f* ∈f g 0*:*1, 0*:*2, 0*:*3, 0*:*4, 0*:*5 of its initial value *dinit* ¼ 50000 *Ns=m*.

*di* are reflected in the linear changes of the disc pressure *pi*

*The pressure change at the initial point is* 0 *and is marked green.*

fraction *f* ∈ f g 0*:*1, 0*:*2, 0*:*3, 0*:*4, 0*:*5 .

initial value.

**Figure 9.**

**87**

#### **Figure 8.**

*Comparison of the intervertebral disc displacement with experimental results [20], FE model [24] and MBS model [14]. The results of the currant FSU model are presented in blue bars, die orange bar represent the results of [24], the yellow one of [14] and the green one of [20].*

applied loads. The response of model motion segments C5–C6 to the applied translational load of 500 N is shown.

A review with the focus on soft tissue structural responses with an emphasis on finite element mathematical models is done by [20]. Biomechanical data of intervertebral disc under compression test are provided for the FSU C6-C7. Under a load of 673 N the intervertebral disc between vertebrae C6 and C7 is deformed with 1.7 mm.

The comparison of the intervertebral disc pressure under the loads of 80 N and 200 N are shown in **Figure 7**.

**Figure 8** compares the effect on the intervertebral disc deformation of the loads 80 N, 500 N and 673 N. The intradiscal pressure almost agrees with the published pressure at the loads of 80 N and 673 N. The current model has about one-third lower disc displacement than the comparison model [14]. One reason for this can be, that under certain circumstances the muscles, that are not taken into account in the current FSU model, are accompanied by a lack of muscle tension, which leads to the less compression of the intervertebral discs.

### **3. Effects of stiffness and damping variations**

#### **3.1 Method**

In the biomechanical modeling the quality of the simulation can be considered valid only when the model and the input parameters are accurate and robust [28]. To examine the robustness of the modeled system a method called sensitivity analysis is the first choice. Generally speaking, *sensitivity analysis is collection of approaches, that determine, quantify and analyze the impact of the input parameters on the model outcome* [29]. The sensitivity analysis can also identify those components of the model that might need additional studies to be performed. Further, in the model optimization the sensitivity analysis can be used to refine the values of the critical parameters as well as to simplify or ignore those factors, which do not show any impact on the model response [30].

One of the simplest and effective techniques used to determine the level of the sensitivity or insensitivity of the model outputs to the plausible variation of one particular parameter is *one-way sensitivity analysis* [31, 32].

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

In order to identify the effect of both stiffness and damping variations on the current FSU model, one-way sensitivity analysis is performed. After every simulation run the corresponding changes in the intervertebral disc pressure in the C6-C7 segment are reported as a difference between initial and current disc pressure *δPi* ¼ *Pi* � *Pinit*, where the initial pressure value is 0*:*301315 *MPa*.

#### **3.2 Simulation results of stiffness term alternation**

The first series of simulations consider the variation of the stiffness term *c*, which is expressed in Eq. (1), however the damping value *d* is held constant by 50000 *Ns=m*. For the sake of simplicity, the same value *ci* is assigned to *cx*,*cy*,*cz* in every *i*-th simulation. Starting from the initial value of 500000 *N=m* in each experiment repetition the stiffness parameter is increased and decreased by a fraction *f* ∈ f g 0*:*1, 0*:*2, 0*:*3, 0*:*4, 0*:*5 .

It can be seen in **Figure 9**, that the variation in the stiffness term results in the linear change of the disc pressure, which points out to the linear relationship between these two variables. However, the change plot indicates the opposite linear relationship between the stiffness and the disc pressure, where increasing of the stiffness causes decreasing of the pressure. Note, that the course of the disc pressure changes is symmetrical, i.e. the minimum and maximum changes in the pressure value are of the same magnitude. According to the revealed results the maximal absolute pressure change is reported to be 7*:*<sup>0935249</sup> � <sup>10</sup>�<sup>5</sup> *MPa* or 0.02% of the initial value.

#### **3.3 Simulation results of damping term alternation**

The second part of the experiments aims the analysis of the system sensitivity with respect to alternations of the damping constant *d* (see Eq. (1)). In order to simplify the experiment execution, the same value *di* is assigned to *dx*, *dy*, *dz* in each *i*-th simulation run. The stiffness term is set to be constant, i.e. *ci* ¼ 500000 *N=m* for all *N* trials. The damping parameter *di* is increased and decreased by a fraction *f* ∈f g 0*:*1, 0*:*2, 0*:*3, 0*:*4, 0*:*5 of its initial value *dinit* ¼ 50000 *Ns=m*.

The impact of the damping parameter changes on the disc pressure is presented in **Figure 10**. Similarly to the results obtained in Section 3.2, an obvious influence of the damping term on the disc pressure can be observed, where the linear changes of *di* are reflected in the linear changes of the disc pressure *pi* . However, the magnitude of the change is not symmetrical for decreasing and increasing values of the

#### **Figure 9.**

*Decreasing (left side) and increasing (right side) the stiffness term c by factor f impact the intradiscal pressure. The pressure change at the initial point is* 0 *and is marked green.*

applied loads. The response of model motion segments C5–C6 to the applied trans-

*Comparison of the intervertebral disc displacement with experimental results [20], FE model [24] and MBS model [14]. The results of the currant FSU model are presented in blue bars, die orange bar represent the results*

finite element mathematical models is done by [20]. Biomechanical data of

A review with the focus on soft tissue structural responses with an emphasis on

intervertebral disc under compression test are provided for the FSU C6-C7. Under a load of 673 N the intervertebral disc between vertebrae C6 and C7 is deformed with

The comparison of the intervertebral disc pressure under the loads of 80 N and

**Figure 8** compares the effect on the intervertebral disc deformation of the loads 80 N, 500 N and 673 N. The intradiscal pressure almost agrees with the published pressure at the loads of 80 N and 673 N. The current model has about one-third lower disc displacement than the comparison model [14]. One reason for this can be, that under certain circumstances the muscles, that are not taken into account in the current FSU model, are accompanied by a lack of muscle tension, which leads to

In the biomechanical modeling the quality of the simulation can be considered valid only when the model and the input parameters are accurate and robust [28]. To examine the robustness of the modeled system a method called sensitivity analysis is the first choice. Generally speaking, *sensitivity analysis is collection of approaches, that determine, quantify and analyze the impact of the input parameters on the model outcome* [29]. The sensitivity analysis can also identify those components of the model that might need additional studies to be performed. Further, in the model optimization the sensitivity analysis can be used to refine the values of the critical parameters as well as to simplify or ignore those factors, which do not show

One of the simplest and effective techniques used to determine the level of the sensitivity or insensitivity of the model outputs to the plausible variation of one

lational load of 500 N is shown.

*of [24], the yellow one of [14] and the green one of [20].*

*Recent Advances in Numerical Simulations*

200 N are shown in **Figure 7**.

the less compression of the intervertebral discs.

any impact on the model response [30].

particular parameter is *one-way sensitivity analysis* [31, 32].

**3. Effects of stiffness and damping variations**

1.7 mm.

**Figure 8.**

**3.1 Method**

**86**

**Figure 10.**

*Decreasing (left side) and increasing (right side) the damping value by f effects the disc pressure in a linear manner. The disc pressure at the initial point is* 0*:*301315*MPa, the change of pressure at initial point is 0 (marked green).*

damping factor. Moreover, the smaller damping values result in higher changes of the intradiscal pressure. The maximal pressure change is found to be 0*:*00593 *MPa* for *d* ¼ 25000 *Ns=m*, which is approximately 1.991% of the initial pressure value. In comparison, for *d* ¼ 75000 *Ns=m* the change is �0*:*00226 *MPa* and 0.6% respectively.

In **Figure 11** the disc pressure changes affected by percentage decreasing of both model factors *d* and *c* following the one-way sensitivity analysis approach are depicted. It can be seen, that the same alternation of the damping term causes approximately two orders higher magnitude of the disc pressure. To examine the hypothesis, that the damping parameter has much stronger influence on the system, the calculation of a further sensitivity metric called **sensitivity coefficient** is elaborated [33]. In our particular setting the sensitivity coefficient *sv* is defined to be an average quotient of the disc pressure change *pi* to the *i*-th change in the parameter value *vi*:

$$s\_v = \frac{1}{N} \sum \frac{\delta p\_i}{\delta v\_i},\tag{2}$$

Determined coefficients for the stiffness *sc* and damping term *sd* are shown in **Table 4**. The obtained results support the above statement, that the behavior of the current model is approximately 12.59 times more sensitive to the damping term

**Sensitivity coefficient value** �1.145 � <sup>10</sup>�<sup>5</sup> �9.093�<sup>5</sup>

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

**s***<sup>c</sup>* **s***<sup>d</sup>*

10�<sup>7</sup>

**4. Impact of different load cases and intervertebral disc areas on**

One of the main functional task of the intervertebral disc is transmitting the compressive loads through the spine [34]. Therefore, it is important to study the sensitivity of the input parameters as well as mechanical responses of the model considering multiple loading cases. For this experiment, the acting of various external loads *l* ∈*L*, where *L* ¼ f g 100*N*, 200*N*, … , 800*N* on the upper endplate of the C6 vertebra (see **Figure 3**) is simulated. Such high forces are selected in order to

The disc pressure responded by the current model is reported in **Figure 12**. It can be seen, that the stiffness alternations among the load cases do not lead to significant change in the disc pressure. An unusual pattern is observed in each particular load situation, where the stiffness variation causes the linear growth in the disc pressure followed by piece-wise non-linear regions. Please note, that this disc

*Maximal intradiscal pressure for C6-C7 segment calculated for multiple compressive loads and different stiffness value. The initial stiffness term is decreased and increased by a factor up to 50% of its initial value*

*c* ¼ 500000 *N=s. The damping term is help constant d* ¼ 50000 *Nm=s.*

**4.1 Impact of different loads on intervertebral disc pressure**

*Sensitivity coefficient determined for parameters c and d using Eq. (2).*

investigate the model behavior under different boundary conditions.

than to the stiffness parameter.

**Table 4.**

**Figure 12.**

**89**

**intervertebral disc pressure**

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

where *N* is a total number of trials and *δvi* is the *i*-th change in the observing parameter, *vi* ∈ *ci* f g , *di* .

#### **Figure 11.**

*Comparison of the maximal disc pressure changes [MPa] given with respect to the variability of the input parameters c and d. The stiffness (marked red) and damping (marked blue) terms are decreased by the factor of 10–50%.*

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


**Table 4.**

damping factor. Moreover, the smaller damping values result in higher changes of the intradiscal pressure. The maximal pressure change is found to be 0*:*00593 *MPa* for *d* ¼ 25000 *Ns=m*, which is approximately 1.991% of the initial pressure value. In comparison, for *d* ¼ 75000 *Ns=m* the change is �0*:*00226 *MPa* and 0.6%

*Decreasing (left side) and increasing (right side) the damping value by f effects the disc pressure in a linear manner. The disc pressure at the initial point is* 0*:*301315*MPa, the change of pressure at initial point is 0*

In **Figure 11** the disc pressure changes affected by percentage decreasing of both

X*δpi δvi*

where *N* is a total number of trials and *δvi* is the *i*-th change in the observing

*Comparison of the maximal disc pressure changes [MPa] given with respect to the variability of the input parameters c and d. The stiffness (marked red) and damping (marked blue) terms are decreased by the factor*

, (2)

model factors *d* and *c* following the one-way sensitivity analysis approach are depicted. It can be seen, that the same alternation of the damping term causes approximately two orders higher magnitude of the disc pressure. To examine the hypothesis, that the damping parameter has much stronger influence on the system, the calculation of a further sensitivity metric called **sensitivity coefficient** is elaborated [33]. In our particular setting the sensitivity coefficient *sv* is defined to be an average quotient of the disc pressure change *pi* to the *i*-th change in the parameter

> *sv* <sup>¼</sup> <sup>1</sup> *N*

respectively.

**Figure 10.**

*(marked green).*

*Recent Advances in Numerical Simulations*

value *vi*:

**Figure 11.**

*of 10–50%.*

**88**

parameter, *vi* ∈ *ci* f g , *di* .

*Sensitivity coefficient determined for parameters c and d using Eq. (2).*

Determined coefficients for the stiffness *sc* and damping term *sd* are shown in **Table 4**. The obtained results support the above statement, that the behavior of the current model is approximately 12.59 times more sensitive to the damping term than to the stiffness parameter.
