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

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

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 investigate the model behavior under different boundary conditions.

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

#### **Figure 12.**

*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.*

behavior is not detected in the simulations applying the load of 80 N (see Sections 3.2 and 3.3). In details, the non-linear pressure change is detected withing the following ranges of the stiffness term: *<sup>c</sup>*<sup>∈</sup> <sup>2</sup>*:*<sup>5</sup> � <sup>10</sup>6, 3*:*<sup>0</sup> � <sup>10</sup><sup>6</sup> as well as for *<sup>c</sup>*<sup>∈</sup> <sup>4</sup>*:*<sup>0</sup> � <sup>10</sup>6, 4*:*<sup>5</sup> � <sup>10</sup><sup>6</sup> and 6*:*<sup>5</sup> � <sup>10</sup>6, 7*:*<sup>0</sup> � <sup>10</sup><sup>6</sup> .

**Figure 13** illustrates the results of the simulations where the applied loads *l*∈ *L* and the damping factor *d* are varied simultaneously. The perspective view of this diagram is slightly different in order to emphasize the regions where the nonlinearity in the disc pressure occurs. The dark spots in the plot indicate the jumps in the disc pressure value over the loads, where the step-wise patterns show the nonlinear responses of the current model for the following cases: 3*:*<sup>0</sup> � <sup>10</sup><sup>4</sup> *Ns=<sup>m</sup>* for the applied force of 200 *<sup>N</sup>*, 3*:*<sup>5</sup> � <sup>10</sup><sup>4</sup> *Ns=<sup>m</sup>* for 100 *<sup>N</sup>* load, another peaks are observed for the exerting force of 500 *<sup>N</sup>* at damping value of 5*:*<sup>0</sup> � <sup>10</sup><sup>4</sup> *Ns=m*.

### **4.2 Impact of different intervertebral disc areas on disc pressure**

The size of the disc area is presented in the literature with different values. This leads to the question how different disc areas influence the disc pressure. In order to investigate this effect, approximated intervertebral disc areas from the literature [11, 20, 25, 26] are used as examples. Values of the FSU C6-C7 disc area with minimum of 168 mm<sup>2</sup> and maximum of 502 *mm*<sup>2</sup> are published in [20]. Estimated disc areas of 180 mm<sup>2</sup> , 230 mm2 and 295 mm<sup>2</sup> are published by [26] and represented as mean values from 3 specimens of their cervical spine. The EPAu of C7 and the EPAl of C6 is specified in [11, 25]. The mean values of EPAu of C7 and EPAl of C6 with 284 mm<sup>2</sup> and 269 mm<sup>2</sup> respectively are taken to approximate the area of the corresponding intervertebral disc.

intervertebral disc area on the intradiscal pressure are considered under the load

*Representation of the relationship between disc area size and intradiscal pressure. The pressure is determined for eight disc areas of different sizes and an external load of 80 N. the blue points in the plot are the data points connected by a best-fit straight line. The disc areas are based on literature data. The assignment of the data points with their specific intervertebral disc areas to the corresponding literature is as follows (starting from the top left side): Data point 1 [20], data point 2 [26], data point 3 [26], data point 4 (current FSU model), data*

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

*<sup>x</sup>*,*<sup>y</sup>*. Pearson correlation coefficient is determined for data pairs

*<sup>i</sup>*¼<sup>1</sup>ð Þ *xi* � *<sup>x</sup>*

The calculated R2 value of 1 (see **Figure 14**) shows a perfect correlation of the variables. Based on this correlation, the disc pressure can be determined by means of the third degree polynomial for given disc surface areas. The resulted polynomial is:

This method offers the possibility of comparing and checking the disc pressure calculated in the simulation model with the one determined by the polynomial.

This study should be seen as a first approach to analyze the cervical spine's sensibility to different influencing factors. The focus is on the analysis of the effects of various stiffness and damping parameters and disc area on the intradiscal

where *<sup>N</sup>* is sample size, *xi*, *yi* are the individual sample points, *<sup>x</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>i</sup>*¼<sup>1</sup>ð Þ *xi* � *<sup>x</sup> yi* � *<sup>y</sup>* � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup>P*<sup>N</sup>*

*<sup>i</sup>*¼<sup>1</sup> *yi* � *<sup>y</sup>* � �<sup>2</sup> <sup>q</sup> , (3)

*N* P*<sup>N</sup>*

*<sup>x</sup>*<sup>2</sup> <sup>þ</sup> <sup>5</sup>*:*<sup>89</sup> � <sup>10</sup><sup>3</sup> <sup>þ</sup> <sup>1</sup>*:*05*:* (4)

*<sup>i</sup>*¼<sup>1</sup>*xi* the

P*<sup>N</sup>*

P*<sup>N</sup>*

sample mean for *x*, which is calculated for *y* analogically and *i*∈ *N*.

*rx*,*<sup>y</sup>* ¼ �1*:*<sup>16</sup> � <sup>10</sup><sup>10</sup>*x*<sup>3</sup> <sup>þ</sup> <sup>1</sup>*:*<sup>39</sup> � <sup>10</sup><sup>7</sup>

In **Figure 14** it can be clearly seen that the size of the disc area has a direct effect on the disc pressure. The course can be approximated by a 3-degree polynomial. In order to assess the goodness of the polynomial fit, the coefficient of determination *R*<sup>2</sup> is calculated. *R*<sup>2</sup> is defined to be the square of Pearson correlation coefficient *rx*,*<sup>y</sup>*,

case of 80 N.

**Figure 14.**

i.e. *<sup>R</sup>*<sup>2</sup> <sup>¼</sup> *<sup>r</sup>*<sup>2</sup>

*x*1, *y*<sup>1</sup>

**5. Conclusions**

**91**

� �, … , *xn*, *yn*<sup>Þ</sup> � � � as follows:

*rx*,*<sup>y</sup>* ¼

*point 5 [25], data point 6 [11], data point 7 [26], data point 8 [20].*

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

All disc area values listed above serve as input for the analysis of the relationship between intervertebral disc pressure and disc area. The effects of different

#### **Figure 13.**

*Maximal intradiscal pressure calculated for C6-C7 segment. Different compressive loads and values of the damping term are simultaneously changed, however the stiffness factor is set constant to c* ¼ 500000 *N=s. The simulation results reveal the areas (dark spot regions in the plot) with a non-linear changes of the maximal disc pressure.*

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

#### **Figure 14.**

behavior is not detected in the simulations applying the load of 80 N (see Sections 3.2 and 3.3). In details, the non-linear pressure change is detected withing the following ranges of the stiffness term: *<sup>c</sup>*<sup>∈</sup> <sup>2</sup>*:*<sup>5</sup> � <sup>10</sup>6, 3*:*<sup>0</sup> � <sup>10</sup><sup>6</sup> as well as for

**Figure 13** illustrates the results of the simulations where the applied loads *l*∈ *L* and the damping factor *d* are varied simultaneously. The perspective view of this diagram is slightly different in order to emphasize the regions where the nonlinearity in the disc pressure occurs. The dark spots in the plot indicate the jumps in the disc pressure value over the loads, where the step-wise patterns show the nonlinear responses of the current model for the following cases: 3*:*<sup>0</sup> � <sup>10</sup><sup>4</sup> *Ns=<sup>m</sup>* for the applied force of 200 *<sup>N</sup>*, 3*:*<sup>5</sup> � <sup>10</sup><sup>4</sup> *Ns=<sup>m</sup>* for 100 *<sup>N</sup>* load, another peaks are observed

The size of the disc area is presented in the literature with different values. This leads to the question how different disc areas influence the disc pressure. In order to investigate this effect, approximated intervertebral disc areas from the literature [11, 20, 25, 26] are used as examples. Values of the FSU C6-C7 disc area with minimum of 168 mm<sup>2</sup> and maximum of 502 *mm*<sup>2</sup> are published in [20]. Estimated

represented as mean values from 3 specimens of their cervical spine. The EPAu of C7 and the EPAl of C6 is specified in [11, 25]. The mean values of EPAu of C7 and EPAl of C6 with 284 mm<sup>2</sup> and 269 mm<sup>2</sup> respectively are taken to approximate the

between intervertebral disc pressure and disc area. The effects of different

*Maximal intradiscal pressure calculated for C6-C7 segment. Different compressive loads and values of the damping term are simultaneously changed, however the stiffness factor is set constant to c* ¼ 500000 *N=s. The simulation results reveal the areas (dark spot regions in the plot) with a non-linear changes of the maximal disc*

All disc area values listed above serve as input for the analysis of the relationship

, 230 mm2 and 295 mm<sup>2</sup> are published by [26] and

for the exerting force of 500 *<sup>N</sup>* at damping value of 5*:*<sup>0</sup> � <sup>10</sup><sup>4</sup> *Ns=m*.

**4.2 Impact of different intervertebral disc areas on disc pressure**

*<sup>c</sup>*<sup>∈</sup> <sup>4</sup>*:*<sup>0</sup> � <sup>10</sup>6, 4*:*<sup>5</sup> � <sup>10</sup><sup>6</sup> and 6*:*<sup>5</sup> � <sup>10</sup>6, 7*:*<sup>0</sup> � <sup>10</sup><sup>6</sup> .

*Recent Advances in Numerical Simulations*

disc areas of 180 mm<sup>2</sup>

**Figure 13.**

*pressure.*

**90**

area of the corresponding intervertebral disc.

*Representation of the relationship between disc area size and intradiscal pressure. The pressure is determined for eight disc areas of different sizes and an external load of 80 N. the blue points in the plot are the data points connected by a best-fit straight line. The disc areas are based on literature data. The assignment of the data points with their specific intervertebral disc areas to the corresponding literature is as follows (starting from the top left side): Data point 1 [20], data point 2 [26], data point 3 [26], data point 4 (current FSU model), data point 5 [25], data point 6 [11], data point 7 [26], data point 8 [20].*

intervertebral disc area on the intradiscal pressure are considered under the load case of 80 N.

In **Figure 14** it can be clearly seen that the size of the disc area has a direct effect on the disc pressure. The course can be approximated by a 3-degree polynomial. In order to assess the goodness of the polynomial fit, the coefficient of determination *R*<sup>2</sup> is calculated. *R*<sup>2</sup> is defined to be the square of Pearson correlation coefficient *rx*,*<sup>y</sup>*, i.e. *<sup>R</sup>*<sup>2</sup> <sup>¼</sup> *<sup>r</sup>*<sup>2</sup> *<sup>x</sup>*,*<sup>y</sup>*. Pearson correlation coefficient is determined for data pairs *x*1, *y*<sup>1</sup> � �, … , *xn*, *yn*<sup>Þ</sup> � � � as follows:

$$r\_{\mathbf{x},\mathbf{y}} = \frac{\sum\_{i=1}^{N} (\mathbf{x}\_i - \overline{\mathbf{x}}) \left( y\_i - \overline{\mathbf{y}} \right)}{\sqrt{\sum\_{i=1}^{N} (\mathbf{x}\_i - \overline{\mathbf{x}})^2 \sum\_{i=1}^{N} (y\_i - \overline{\mathbf{y}})^2}},\tag{3}$$

where *<sup>N</sup>* is sample size, *xi*, *yi* are the individual sample points, *<sup>x</sup>* <sup>¼</sup> <sup>1</sup> *N* P*<sup>N</sup> <sup>i</sup>*¼<sup>1</sup>*xi* the sample mean for *x*, which is calculated for *y* analogically and *i*∈ *N*.

The calculated R2 value of 1 (see **Figure 14**) shows a perfect correlation of the variables. Based on this correlation, the disc pressure can be determined by means of the third degree polynomial for given disc surface areas. The resulted polynomial is:

$$r\_{x,y} = -1.16 \times 10^{10} x^3 + 1.39 \times 10^7 x^2 + 5.89 \times 10^3 + 1.05. \tag{4}$$

This method offers the possibility of comparing and checking the disc pressure calculated in the simulation model with the one determined by the polynomial.

### **5. Conclusions**

This study should be seen as a first approach to analyze the cervical spine's sensibility to different influencing factors. The focus is on the analysis of the effects of various stiffness and damping parameters and disc area on the intradiscal

pressure of the FSU C6-C7 in order to indicate the model weaknesses and optimize the model design.

**Conflict of interest**

MBS Multi-Body Simulation FSU Functional Spinal Unit CoG Center of Gravity CR Center of Rotation MI Moment of Inertia

FL Flava Ligament ISL Interspinous Ligament NL Nuchal Ligament CL Capsular Ligament SPL Spinous Process Length

FC Facet DC Disc EP Endplate A Area W Width H Height D Depth

r right u upper

**Author details**

Sabine Bauer<sup>1</sup>

Germany

**93**

ALL Anterior Longitudinal Ligament PLL Posterior Longitudinal Ligament

l left or lower (depending on the context)

\*† and Ivanna Kramer1,2†

\*Address all correspondence to: bauer@uni-koblenz.de

Koblenz-Landau, Koblenz, Germany

† These authors contributed equally.

provided the original work is properly cited.

1 Institute for Medical Engineering and Information Processing, University of

2 Institute for Computer Visualistics, University of Koblenz-Landau, Koblenz,

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Abbreviations**

The authors declare no conflict of interest.

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

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

In the first part of this study an *one-way* sensitivity analysis is performed in order to indicate, whether one of the given input parameter, namely stiffness or damping term, has a dominant influence on the model behavior. The experimental results show, that both parameters exhibit an identical impact on the disc pressure. However the variations of the damping term indicate a slightly stronger effect on the intradiscal pressure measurements, which is reflected in relatively higher value of the calculated sensitivity coefficient. When applying compressive loads from 100 N up to 800 N on the FSU model and varying the analyzing parameters a not foreseeable response pattern in the disc pressure is explored. Simultaneous change of the load and the corresponding parameter values results in a non-linear outcome regarding the intradiscal pressure, which is not detected in the simulations that consider the exerting external force of 80 N.

Further, it could be shown that the correlation between disc area and disc pressure can be approximated by a third-degree polynomial. This allows a further possibility for model validation of the simulated intervertebral disc pressure. For this purpose, the simulation result can be compared with the intervertebral disc pressure calculated by the polynomial with a known disc surface area.

An essential point to be considered in the next step is the implementation of the musculature. This is not taken into account in this model. It is still unclear what influence other cervical parameters, e.g. the facet joints, ligaments or muscles have and how these affect the overall mechanic when changed. Therefore, following this investigation, the effect of model parameters of others spinal structures, such as facet alignment and size, on the load on the intervertebral discs will be evaluated. Further, it must be questioned critically whether these results can be transferred to a model with a larger spinal column section. In order to discuss this question, in a further step not only an FSU should be considered, but the sensitivity of model parameters in a model that contains an entire spinal column section should be analyzed.

In case when additional elements are integrated into the model and the number of input factors grows, another broadly used method called multivariate sensitivity analysis can be applied in order to investigate the model response affected by the simultaneous variations of the underlying parameters. This procedure can help to optimize the model structure by finding the variables, that primarily impact the model outcomes. Moreover, using the sensitivity analysis methods the values of the principal parameters can be determined so that realistic simulation of model behavior is possible.

The experimental design of the presented sensitivity analysis follows the recommendations found in the literature. In the future work, the boundary conditions of the experiments should be extended. For instance, the range of the stiffness value might be increased up to 8*:*<sup>3</sup> <sup>10</sup><sup>6</sup> as it was used in the model proposed in [14]. Then the response of the current FSU model can be compared with the outputs of the referenced model.

## **Acknowledgements**

We like to thank Prof. Dietrich Paulus, Institute of Computer Visualistics, University Koblenz-Landau for the fruitful discussion and Dr. Francis Kilian, Head of the Clinic for Spinal Surgery, Head of the Spinal Center Catholic Clinic Koblenz-Montabaur and PD Dr. Roland Jacob, specialist in ear, nose and throat medicine for guidance on medical and anatomical questions.

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