4.1.1. Effects of different standardized cage sizes

variables "height loss," "steophyte formation," and "diffuse sclerosis." According to Wilke et al. [9], each of these three variables first has to be graded individually on lateral and posteroanterior radiographs. Finally, the so-called overall degree of degeneration is assigned on a four-

Recent approaches try to estimate biomechanical properties of humans by tracking motions both in color image sequences as in distance measurements. Such data are available from devices that were designed for consumer games, but have also been used experimentally in

Biomechanical modeling is an established method to simulate physics and physiology of a human body. Depending on the scientific question, it is possible to create whole body models for humans [11, 12] or parts of the body like the human heart [13, 14] or the spine [15, 16]. A distinction can be made between the multibody simulation (MBS) modeling and the finite element (FE) modeling. Depending on the scientific question, either the MBS or the FE simulation method can be used. For analyzing highly sophisticated problems, the FE modeling is the appropriate modeling method. The system is divided into a finite number of small geometric elements, called the finite elements. At the connection point, the so-called nodes, boundary and transition conditions are defined in accordance to specific material laws [17]. If the biomechanical behavior of high dynamic movements or larger parts or the entire of the human body is the focus, the MB simulation is the suitable method. A further possibility is to combine MBS and FE to ensure a higher degree of fine specific structure modeling. Due to such hybrid models (e.g. MBS-coupled with FE models), short computing times are guaranteed. The rapid availability of results enables a future usability in medical routine for spinal operation planning. Further detailed explanations of the basics of simulations can be taken out

The possibility of using the biomechanical simulation in the area of spinal surgery is diverse. Preoperatively, the effects of mono- or multisegmental spinal fusion on adjacent segments can be analyzed. In addition, an optimized positioning of the inserted implant can be demonstrated by taking into account the reconstruction of the sagittal balance or an adequate stress distribution. One can also compare the effects of minimally invasive surgical methods to

Although most manufacturers of implants offer different sizes of implants, a full-area contact of the implant with the vertebral body is not always ensured. An insufficient anchorage can lead to local stress peaks at the contact points. With the help of computer-assisted simulations, such stress peaks can be analyzed. To ensure the best possible anchoring, the effects of different contact surface designs of the implant can be determined. Thus, the simulation can contribute to a development of patient-specific shaped implant surfaces, which ensure a

surgical procedures with high degree of resections of spinal structures.

point scale from 0 (no degeneration) to 3 (severe degeneration).

medical applications, for example, in Ref. [38].

52 Innovations in Spinal Deformities and Postural Disorders

4.1. Application examples implant design

of [18].

4. Biomechanical modeling and simulation

This example is intended to show the effects of the cage size on the biomechanics of the lumbar spine. For this, a biomechanical simulation a model of a person was created, which considers gender (male), age (35 years), weight (75 kg, body mass index (BMI) 22), and body height (1.85 m), including detailed lumbar spine structures (Figure 2). The lumbar spine model includes the biomechanical properties of the intervertebral discs, the facet joints, the ligamentous structures and the muscle groups left, right m. erector spinae, left and right m. rectus abdominus according to [21]. The exact model configuration as well as the validation can be found in Refs. [18–20].

To investigate the effects of fusing implants with five different sizes, the size of this optimally fitted implant is varied about 2.5 and 5%. In this context, an optimal fit is the planar resting of the cage on the endplates of the corresponding vertebral body. It should be noted that the entire cage base area is not in contact with the vertebral endplate, because it is a standardized implant without considering the patient-specific vertebral endplate morphology. The load situation, which is simulated, is the upright position under load of the body weight and a fused functional spinal unit (FSU) L4-L3. The weight force solves the kinematics of the MBS model and the motion equations, which form a system of coupled differential equations, are integrated for each simulation time step. This means that this weight force causes small movements in the spinal structures, and they are brought out of their equilibrium state. The reaction forces of the individual spinal structures build up until a new equilibrium state is reached. The following results refer to this new equilibrium state.

The basic implant size is chosen so that the implant fills the entire space between the corresponding vertebral bodies, and thus has contact with the endplates of the vertebral bodies. This basis cage is named in the Figures 3–5 as "optimal fit." The cage is increased or reduced by a certain percentage and is shown in Figures 3–5 as follows: "plus 2.5%" and "plus 5%" for the cages, which are enlarged 2.5 and 5%, respectively, and "minus 2.5%" and "minus 5%" for the size reduction of 2.5 and 5%, respectively (Figure 3).

When the basic cage is implemented in the FSU L3-L4, the other FSUs will undergo the least load compared to a smaller or larger cage size.

In the simulation cage "optimal fit," the lowest FSU sac-L5 is loaded the highest of all the FSU (Figure 4). The smallest stress is recorded in the FSU L5-L4. When a larger or smaller implant is selected, the load in all FSUs increases sharply. The load of the different FSUs hardly differ in height, whether the choice of a larger (plus 2.5% or plus 5%) or smaller (minus 2.5% or minus 5%) cage. The difference is marginal when comparing the simulations of the simulation cases "plus 2.5%," "plus 5%," "minus 2.5%," and "minus 5%" within the FSUs. The results show how

Figure 2. Simulation model of a person.

Figure 3. Vertical force intervertebral discs.

Figure 4. Loads of the facet joints.

Figure 2. Simulation model of a person.

54 Innovations in Spinal Deformities and Postural Disorders

important the correct choice of cage size could be, so that the intervertebral discs are not loaded more heavily.

On the other hand, if considering the loads of the facet joints, the choice of the implant size has a small influence on their load height. However, the facets of the FSU L3-L2 are much more heavily loaded than any other. This is due to the alignment of the facet surface. In the case of the FSUs L3- L2, the corresponding sagittal superior facet angles are relatively large so that the value of the force component of the acting external force, which is almost perpendicular to this surface, is high. Furthermore this FSU rotates in dorsal direction, so this boosts also the load situation. What's more, the facets joints of the FSU L4-L3 are not loaded in the cage when choosing the cage size "optimal fit." The reason is that the spinal alignment is modified by the body weight, so that the vertebral bodies above the implemented cage FSU situated move in such a position that the lower endplate of FSU is in contact with the cage. As a result, the cranial facet joint surfaces of L4 and the caudal facet joint surfaces of L3 come directly and strongly into contact, and are therefore correspondingly highly loaded. The uppers facet joint L2-L1 are loaded in none of the simulation cases. The reason could be that the alignment of the facet joints is nearly parallel to the direction of movement, and so the facet surfaces "slide" through each other.

In general, the direction of rotation of the FSU is determined by the acting torque resulting from the lever arm and the acting weight force. Thereby, a force whose line of action runs vertically in front of the axis of rotation produces a positive torque, and a force whose line of action runs dorsally behind the axis of rotation produces a negative torque. A positive torque results in flexion movement, and a negative torque results in an extension movement of the affected vertebral body segments. From this model configuration or rather specific spinal alignment, the rotations seen in Figure 5 are obtained. It should be noted, however, that these results are only valid for this model configuration and cannot be transferred to other patients. Already in the case of a changed spinal alignment, completely different results can occur [22]. But this example shows the effects of choosing a non-optimal fitting cage and the significance of the appropriate choice of the right cage size.

#### 4.1.2. Cervical vertebral replacement

The superior surface of a cervical vertebral body is shaped like a tub. On its sides, it has small branches which are called uncinate processes. These margins build the uncovertebral joints (Figure 6) [23]. Thereby, the angle of the uncovered joints of the different vertebral bodies is not of the same magnitude, but increases significantly from C5 to C7 [24].

Due to the special anatomical conditions, the implantation of a vertebral body replacement implant or its baseplates cannot be optimally brought into contact with the corresponding superior or inferior anchor vertebra (Figure 7).

Figure 5. Rotation of the intervertebral discs.

Computational Simulation as an Innovative Approach in Personalized Medicine http://dx.doi.org/10.5772/intechopen.68835 57

Figure 6. Illustration of the uncovertebral joint inclination.

the vertebral bodies above the implemented cage FSU situated move in such a position that the lower endplate of FSU is in contact with the cage. As a result, the cranial facet joint surfaces of L4 and the caudal facet joint surfaces of L3 come directly and strongly into contact, and are therefore correspondingly highly loaded. The uppers facet joint L2-L1 are loaded in none of the simulation cases. The reason could be that the alignment of the facet joints is nearly parallel to the direction

In general, the direction of rotation of the FSU is determined by the acting torque resulting from the lever arm and the acting weight force. Thereby, a force whose line of action runs vertically in front of the axis of rotation produces a positive torque, and a force whose line of action runs dorsally behind the axis of rotation produces a negative torque. A positive torque results in flexion movement, and a negative torque results in an extension movement of the affected vertebral body segments. From this model configuration or rather specific spinal alignment, the rotations seen in Figure 5 are obtained. It should be noted, however, that these results are only valid for this model configuration and cannot be transferred to other patients. Already in the case of a changed spinal alignment, completely different results can occur [22]. But this example shows the effects of choosing a non-optimal fitting cage and the significance

The superior surface of a cervical vertebral body is shaped like a tub. On its sides, it has small branches which are called uncinate processes. These margins build the uncovertebral joints (Figure 6) [23]. Thereby, the angle of the uncovered joints of the different vertebral bodies is not

Due to the special anatomical conditions, the implantation of a vertebral body replacement implant or its baseplates cannot be optimally brought into contact with the corresponding

of movement, and so the facet surfaces "slide" through each other.

of the same magnitude, but increases significantly from C5 to C7 [24].

of the appropriate choice of the right cage size.

56 Innovations in Spinal Deformities and Postural Disorders

superior or inferior anchor vertebra (Figure 7).

Figure 5. Rotation of the intervertebral discs.

4.1.2. Cervical vertebral replacement

In order to guarantee planar rest of the implant, the implant base should either be relatively narrow, so that it preferably rests only on the endplate surface of the corresponding vertebral body or be a "negative replica" of the patient-specific vertebral body surface, including consideration of the uncovered joint angle. A small implant baseplate has the disadvantage that the stress on the vertebral body is thereby increased by the reduced contact surface. A baseplate adapted to the superior vertebral surface could result in a much larger area of contact, and therefore a more balanced load distribution is achieved. Because this is a recently launched research project, the following examples are not intended to be final versions, but merely should represent the possibilities of a future implementation of biomechanical simulation in a process chain. The main focus will be to demonstrate the model creation and not to present validated results. The multibody simulation model is therefore a prototype. It should also be noted that we focus on the MBS modeling because this type of simulation is a much faster calculation method than the finite element method. In addition, we aim to implement fine-structured parts, such as the spine, into a whole-body model in order to simulate the dynamic situation of everyday life and thus to determine the stresses. Depending on the question, it is also possible to create a hybrid model of MBS and FE parts. A more detailed explanation can be found in Ref. [18] and in Section 5.

#### 4.1.2.1. Basic model description

The MBS prototype model consists of the vertebral bodies C3-C6, where an extractable vertebral body replacement implant is implemented between the anchor vertebrae C3 and C6 (Figure 8). The intervertebral body surfaces C4 and C5 are adapted accordingly to the real operative procedure laminectomy. The anatomy of vertebral bodies C6 and C3 is retained. Because of the prevailing anatomical conditions of the uncovertebral joint and the relating lateral margins, a complete contact of the implant baseplates with the vertebral endplate of the anchor vertebra cannot be realized. As a result of the typically slightly corrugated form of the

Figure 7. X-ray image of an implemented vertebral body replacement implant.

anchor vertebra, the parts of the implant base located medial have no direct contact with the anchor vertebra in the unload state. The cervical vertebral curvature corresponds to an average spine curvature and is 28 degrees [25]. The biomechanical properties of the ligaments and facet joints are taken from literature [26, 27].

#### 4.1.2.2. Realization of the surface contact

The modeling of the contact between the vertebral body and the implant is realized by means of a special three-dimensional contact force element. The contacting surfaces, the baseplate of the implant, and the superior or inferior vertebral surface of the C3 and C6 are tessellated in such a way that the surfaces of the objects are composed of equally large polygons. In addition, Computational Simulation as an Innovative Approach in Personalized Medicine http://dx.doi.org/10.5772/intechopen.68835 59

Figure 8. MBS model C6-C3 including vertebral body replacement implant.

anchor vertebra, the parts of the implant base located medial have no direct contact with the anchor vertebra in the unload state. The cervical vertebral curvature corresponds to an average spine curvature and is 28 degrees [25]. The biomechanical properties of the ligaments and facet

The modeling of the contact between the vertebral body and the implant is realized by means of a special three-dimensional contact force element. The contacting surfaces, the baseplate of the implant, and the superior or inferior vertebral surface of the C3 and C6 are tessellated in such a way that the surfaces of the objects are composed of equally large polygons. In addition,

joints are taken from literature [26, 27].

58 Innovations in Spinal Deformities and Postural Disorders

Figure 7. X-ray image of an implemented vertebral body replacement implant.

4.1.2.2. Realization of the surface contact

the baseplate was dissected into smaller subunits to allow a more detailed analysis of the contact behavior (Figure 9).

For each of these polygons, a contact force is determined which is essentially oriented according to Hippmann [28] on the boundary layer model, and combined with a half-space approximation and contact force elements. So the contact force Fk is composed of a normal force Fnk and a tangential force Ftk. The normal force Fnk is composed as follows:

$$F\_{nk} = \begin{cases} F\_{ck} + F\_{dk} : F\_{ck} + F\_{dk} > 0 \\ 0 : F\_{ck} + F\_{dk} \not\le 0 \end{cases} \tag{6}$$

Annotation: in the following the subscript E and F stand for the corresponding bodies E and F. For the case Fck þ Fdk > 0, the equation is composed of a stiffness term,

Figure 9. Illustration of the tessellated polygon meshes of the caudal implant base and explanation of terms.

$$F\_{ck} = \mathfrak{c}\_l \* A\_k \* \mathfrak{u}\_{nk} \tag{7}$$

cl : stiffness of the contact element; Ak : total area of the contact element; unk : penetration of the contact element.

which results from

$$\mathbf{c}\_{l} = \frac{\mathbf{c}\_{lE} \* \mathbf{c}\_{lF}}{\mathbf{c}\_{lE} + \mathbf{c}\_{lF}} \tag{8}$$

with

$$c\_{\rm IE} = \frac{K\_E}{b\_E} = \frac{1 - \nu\_E}{(1 + \nu\_E) + (1 - 2\nu\_E)} \ast \frac{E\_E}{b\_E} \tag{9}$$

for clF analog and a damping term

$$F\_{dk} = \begin{cases} \, \_{d}d\_{l} \* A\_{k} \* \upsilon\_{nk} : \mathfrak{u}\_{nk} \geq \mathfrak{u}\_{d} \\\, d\_{l} \* A\_{k} \* \upsilon\_{nk} \* \frac{\mathfrak{u}\_{nk}}{\mathfrak{u}\_{d}} : \mathfrak{u}\_{nk} < \mathfrak{u}\_{d} \end{cases} \tag{10}$$

where

$$
\upsilon\_{nk} = \mathfrak{n}\_k \* \upsilon\_k \tag{11}
$$

stands for the relative speed projected in the normal direction

$$
\sigma\_k = \sigma\_{M\_\ast M\_\circ} + \omega\_{M\_\ast M\_\circ} \times r\_{M\_\circ \mathbb{C}\_k} \tag{12}
$$

of both contact bodies at the position ck of the contact element.

According to Hippmann [28], the parameter ud can be used to define a penetration depth up to which the damping force acts linear. This makes it possible to avoid unrealistic forces during rapid contact processes. Further input parameters are the layer depth b, the E-modulus, and Poisson's ratio ν of each surface and the damping constant d. Because the vertebral body replacement implant consists of a titanium alloy, the corresponding material property for E and ν has been entered into the model. The E-modulus and Poisson's ratio ν for the superior and inferior vertebral body surfaces are taken from [29, 30]. The damping is 10% of the E-modulus of the vertebral body replacement implant. The resulting total force of the contact surface is determined by summing the acting forces of all contact elements. Because the vertebral body replacement implant is actually fastened to the vertebral body by means of a screw, this fixation has been realized by a force element that realizes spring and damper forces and moments between bodies in multiple axis direction. These parameters (c ¼ 108 N/m, d ¼ c \* 0.1 Ns/m) are determined by means of sensitivity analysis.

The tangential force Ftk is calculated as follows [28] and is determined in dependence of the tangential relative velocity

$$
\boldsymbol{\upsilon}\_{\boldsymbol{k}} = \boldsymbol{\upsilon}\_{\boldsymbol{k}} - \boldsymbol{\upsilon}\_{\boldsymbol{n}\boldsymbol{k}} \ast \boldsymbol{n}\_{\boldsymbol{k}} \tag{13}
$$

$$\left| \boldsymbol{\upsilon}\_{\boldsymbol{k}} = \left| \boldsymbol{\upsilon}\_{\boldsymbol{k}} \right| \tag{14}$$

and the normal force Fnk of the contact element

Fck ¼ cl � Ak � unk ð7Þ

� EE bE

: unk < ud

υnk ¼ nk � vk ð11Þ

vk ¼ vMeMf þ ωMeMf � rMf Ck ð12Þ

ð8Þ

ð9Þ

ð10Þ

cl : stiffness of the contact element; Ak : total area of the contact element; unk : penetration of

Figure 9. Illustration of the tessellated polygon meshes of the caudal implant base and explanation of terms.

cl <sup>¼</sup> clE � clF clE þ clF

<sup>¼</sup> <sup>1</sup> � <sup>ν</sup><sup>E</sup>

dl � Ak � <sup>υ</sup>nk � unk

ð1 þ νEÞþð1 � 2νEÞ

dl � Ak � υnk : unk ≥ ud

ud

clE <sup>¼</sup> KE bE

Fdk ¼

stands for the relative speed projected in the normal direction

of both contact bodies at the position ck of the contact element.

8 < :

the contact element. which results from

for clF analog and a damping term

60 Innovations in Spinal Deformities and Postural Disorders

with

where

$$F\_{tk} = \left\{ \begin{array}{c} \mu \* F\_{nk} : \ \upsilon\_{tk} \ge \upsilon\_{\varepsilon} \\ \mu \* F\_{nk} \* \frac{\upsilon\_{tk}}{\upsilon\_{\varepsilon}} \* \left(2 - \frac{\upsilon\_{tk}}{\upsilon\_{\varepsilon}}\right) : \ \upsilon\_{tk} < \upsilon\_{\varepsilon} \end{array} \right. \tag{15}$$

To avoid a set-valued static friction, the frictional force is disabled when the sliding speed falls below a given small value vε.

The total force Fk and the torque Mk of the single contact element are now:

$$F\_k = F\_{nk} \* n\_k + F\_{tk} \* \frac{\mathbf{v}\_{tk}}{\mathbf{v}\_{tk}} \tag{16}$$

$$M\_k = r\_{M\_f \mathbb{C}\_k} \times F\_k \tag{17}$$

Finally, the forces and torques of all contact elements are summed to the resulting total force screw:

$$F\_E^{M\_\uparrow} = \sum\_k F\_k \tag{18}$$

$$M\_{\rm E} = \sum\_{k} M\_{k} \tag{19}$$

A detailed description of the determination of the contact force and further information can be found in [28].

#### 4.1.2.3. Simulation results

The external force that causes the kinematics of the model is 50 N, which corresponds to the average weight force of the corresponding upper body segments and is taken from [31]. In general, it is possible to analyze both the kinematic and kinetic parameters of the modeled spinal structures in this model configuration, such as the ligaments or facet joints, as well as the contact behavior "vertebral body surface-implant plate." In the following, the contact behavior is analyzed and the parameters "total contact patch area," "weighted penetration," "maximum penetration," "maximum contact pressure," and "weighted contact pressure" are discussed.

The surfaces of the subunits of the caudal implant base surface, which comes into contact with the superior anchor vertebral surface, are of different size (Table 1). The sinister regions are more in contact than the dexter regions.

Both the dorsally central (dm) and dorsally dexterous region (dr) of the implant baseplate have no contact with the vertebral body C6. The percentage total contact surface is 52%.

The average penetrations and the maximum penetrations of the subunits are shown in Figure 10. The right front (vr) and central front (vm) subunits penetrate the superior vertebral body surfaces on average most strongly. The weighted penetration of the left middle (ml) and right middle (mr) subunits is half as high. Due to the missing contact in the subunits "dm"and "dr," there is no penetration.

The maximum penetration behaves in an analogous manner. The difference between the "weighted penetration" and the "maximum penetration" of the ventral right "vr" and the left dorsal "ld" subunit stands out. The maximum penetration is 2.5 times and 2 times higher than the weighted penetration. In the remaining subunits, the maximum penetration is more than 1.5 times higher than the weighted penetration. We concluded that within the different subunits (SU) the penetration depth of the specific areas of this subunit (SASU) can be very different in some cases.


Table 1. Percentage contact of the subunits of the implant base with the vertebral endplate.

Looking at the contact pressure in Figure 11, the right ventral (rv) subunit of the implant baseplate is much more heavily loaded than the other subunits. The maximum contact pressure in this subunit of the implant base is 6 times stronger and the weighted contact pressure is 5 times stronger. Comparing the maximum and the weighted contact pressure of the right ventral (rv) subunit, the subunit "rv" certain specific areas of subunit are loaded up to 2.5 times more than others.

Figure 10. Penetration of the contact surfaces.

4.1.2.3. Simulation results

there is no penetration.

different in some cases.

vr 15.0 vm 9.8 vl 1.3 ml 15.7 dl 1.2 dm 0.0 dr 0.0 mr 8.9

more in contact than the dexter regions.

62 Innovations in Spinal Deformities and Postural Disorders

discussed.

The external force that causes the kinematics of the model is 50 N, which corresponds to the average weight force of the corresponding upper body segments and is taken from [31]. In general, it is possible to analyze both the kinematic and kinetic parameters of the modeled spinal structures in this model configuration, such as the ligaments or facet joints, as well as the contact behavior "vertebral body surface-implant plate." In the following, the contact behavior is analyzed and the parameters "total contact patch area," "weighted penetration," "maximum penetration," "maximum contact pressure," and "weighted contact pressure" are

The surfaces of the subunits of the caudal implant base surface, which comes into contact with the superior anchor vertebral surface, are of different size (Table 1). The sinister regions are

Both the dorsally central (dm) and dorsally dexterous region (dr) of the implant baseplate have

The average penetrations and the maximum penetrations of the subunits are shown in Figure 10. The right front (vr) and central front (vm) subunits penetrate the superior vertebral body surfaces on average most strongly. The weighted penetration of the left middle (ml) and right middle (mr) subunits is half as high. Due to the missing contact in the subunits "dm"and "dr,"

The maximum penetration behaves in an analogous manner. The difference between the "weighted penetration" and the "maximum penetration" of the ventral right "vr" and the left dorsal "ld" subunit stands out. The maximum penetration is 2.5 times and 2 times higher than the weighted penetration. In the remaining subunits, the maximum penetration is more than 1.5 times higher than the weighted penetration. We concluded that within the different subunits (SU) the penetration depth of the specific areas of this subunit (SASU) can be very

Percentage contact patch area (%)

Table 1. Percentage contact of the subunits of the implant base with the vertebral endplate.

no contact with the vertebral body C6. The percentage total contact surface is 52%.

Figure 11. Contact pressure of the different subunits.
