*4.2.2. Finite element analysis of PU-foam based synthetic pelvis*

The Sawbone pelvis used in the experiment was CT scanned (Philips Brilliance 64, Philips). Each CT image was manually segmented and a finite element model of the hemi pelvis was generated using the previously validated procedure (Shim et al., 2007, Shim et al., 2008)(Figure 11). Our model is both geometrically and materially non-linear. Geometric nonlinearity was achieved by using finite elasticity governing equations rather than linear elasticity approximations (Shim et al., 2008). Material non-linearity was incorporated in a similar manner as in (Keyak, 2001). In our model, the material behaviour of the synthetic PU-foam based pelvis was divided into two regions – 1) an elastic region with a modulus E; 2) perfectly plastic regions with a plastic strain εAB (Figure 11) according to the material behaviour of polyurethane materials(Thompson et al., 2003). The value for εAB was obtained from the specifications provided by the manufacturer (Pacific Research Laboratories, INC, Washington, WA, USA).

Use of Polyurethane Foam in Orthopaedic Biomechanical Experimentation and Simulation 185

 

*Gauss point von Mises stress (from FE simulation)* (2)

*Change in input parameter* (3)

1 (1)

   

The contact between the femoral head and the acetabulum was modeled as frictional contact (μ = 0.3) and the boundary conditions used in FE simulation were the same as the experiment. The nodes on the superior region of the iliac crest were fixed and two different loading conditions used in the experiment were used as the boundary condition. As in the experiment, the standing and seating positions were differentiated by varying the angle α defined in Figure

The failure behaviour was characterized using the distortion energy (DE) theory of failure (Keyak et al., 1997). The DE theory is a simplified form of the Hoffman failure theory which was proposed for brittle fracture of orthotropic materials (Hoffman, 1967). Assuming

> 

In this equation, σi is the principal stresses and St is the tensile strength and Sc is the compressive strength. If St and Sc are equal, Equation (1) becomes the distortion energy (DE) theory of failure, which was used in our study as a failure criterion. Since we used Gauss points in assigning material properties, we calculated a factor of safety (FOS) for every Gauss point (Equation 2) and if the FOS value was predicted to be less than one the Gauss point was regarded as in failure (Keyak et al., 1997). The fracture load and location were

*Gauss point strength (from CT and material property) FOS=*

Sensitivity analysis was performed to find out which material parameters affect the strength of the pelvic bone most. The parameters of interest for sensitivity analysis were: 1) cortical thickness; 2) cortical modulus; 3) trabecular modulus. Since we used synthetic bones made of polyurethane foam, the three corresponding material parameters for the Sawbone FE model were 1) solid polyurethane foam thickness; 2) solid polyurethane modulus; 3) cellular polyurethane modulus. The values for the parameters were varied to see their effects on the predicted fracture load. The following equation (Equation 3) was used to measure the

*Change in predicted fracture load <sup>S</sup>*

recorded for each loading condition and compared with the experimental results.

10. A vertically directed force was exerted on the femoral head mesh until failure.

isotropy, the fracture condition is reduced to the following (Lotz et al., 1991)

 

1 2 *t c*

> 1 1 *t c*

*4.2.3. Finite element model sensitivity analysis* 

% %

sensitivity of the chosen parameters.

where

123

456

*CCC*

*S S*

*S S*

*CCC*

222 1 2 3 2 3 1 3 1 2 41 52 63 *C C C CCC*

**Figure 11.** FE model and it nonlinear material behaviour. The polyurethane foam material behavior was represented by an elastic region with modulus E until stress S, followed by a perfectly plastic region with plastic strain εAB.

Two materials were incorporated in our model – 1) solid polyurethane foam that mimicked the cortical bone property; 2) cellular rigid polyurethane foam for cancellous bone. The material properties for the two polyurethane foams are given in Table 2.


**Table 2.** Material properties of solid and cellular polyurethane foam

The previously developed algorithm that determines cortical thickness was used to distinguish between solid polyurethane and cellular polyurethane foam regions from CT scans. We used Gauss points inside the mesh to assign material properties and those points placed in the solid region was given the solid polyurethane foam material properties while those in the cellular region was assigned with the cellular polyurethane foam material property(Shim et al., 2008). This allowed our model to have location dependent cortical thickness.

The contact between the femoral head and the acetabulum was modeled as frictional contact (μ = 0.3) and the boundary conditions used in FE simulation were the same as the experiment. The nodes on the superior region of the iliac crest were fixed and two different loading conditions used in the experiment were used as the boundary condition. As in the experiment, the standing and seating positions were differentiated by varying the angle α defined in Figure 10. A vertically directed force was exerted on the femoral head mesh until failure.

The failure behaviour was characterized using the distortion energy (DE) theory of failure (Keyak et al., 1997). The DE theory is a simplified form of the Hoffman failure theory which was proposed for brittle fracture of orthotropic materials (Hoffman, 1967). Assuming isotropy, the fracture condition is reduced to the following (Lotz et al., 1991)

$$\left[\mathbf{C}\_{1}\left[\boldsymbol{\sigma}\_{2}-\boldsymbol{\sigma}\_{3}\right]^{2}+\mathbf{C}\_{2}\left[\boldsymbol{\sigma}\_{3}-\boldsymbol{\sigma}\_{1}\right]^{2}+\mathbf{C}\_{3}\left[\boldsymbol{\sigma}\_{1}-\boldsymbol{\sigma}\_{2}\right]^{2}+\mathbf{C}\_{4}\boldsymbol{\sigma}\_{1}+\mathbf{C}\_{5}\boldsymbol{\sigma}\_{2}+\mathbf{C}\_{6}\boldsymbol{\sigma}\_{3}=1\right]\tag{1}$$

where

184 Polyurethane

*4.2.2. Finite element analysis of PU-foam based synthetic pelvis* 

Laboratories, INC, Washington, WA, USA).

region with plastic strain εAB.

The Sawbone pelvis used in the experiment was CT scanned (Philips Brilliance 64, Philips). Each CT image was manually segmented and a finite element model of the hemi pelvis was generated using the previously validated procedure (Shim et al., 2007, Shim et al., 2008)(Figure 11). Our model is both geometrically and materially non-linear. Geometric nonlinearity was achieved by using finite elasticity governing equations rather than linear elasticity approximations (Shim et al., 2008). Material non-linearity was incorporated in a similar manner as in (Keyak, 2001). In our model, the material behaviour of the synthetic PU-foam based pelvis was divided into two regions – 1) an elastic region with a modulus E; 2) perfectly plastic regions with a plastic strain εAB (Figure 11) according to the material behaviour of polyurethane materials(Thompson et al., 2003). The value for εAB was obtained from the specifications provided by the manufacturer (Pacific Research

**Figure 11.** FE model and it nonlinear material behaviour. The polyurethane foam material behavior was represented by an elastic region with modulus E until stress S, followed by a perfectly plastic

Two materials were incorporated in our model – 1) solid polyurethane foam that mimicked the cortical bone property; 2) cellular rigid polyurethane foam for cancellous bone. The

The previously developed algorithm that determines cortical thickness was used to distinguish between solid polyurethane and cellular polyurethane foam regions from CT scans. We used Gauss points inside the mesh to assign material properties and those points placed in the solid region was given the solid polyurethane foam material properties while those in the cellular region was assigned with the cellular polyurethane foam material property(Shim et al., 2008).

Solid polyurethane foam 0.32 8.8 260 Cellular polyurethane foam 0.16 2.3 23

Density (g/cc) Strength (MPa) Modulus (MPa)

material properties for the two polyurethane foams are given in Table 2.

**Table 2.** Material properties of solid and cellular polyurethane foam

This allowed our model to have location dependent cortical thickness.

$$\mathbf{C}\_1 = \mathbf{C}\_2 = \mathbf{C}\_3 = \frac{1}{2\mathbf{S}\_t\mathbf{S}\_c}$$

$$\mathbf{C}\_4 = \mathbf{C}\_5 = \mathbf{C}\_6 = \frac{1}{\mathbf{S}\_t} - \frac{1}{\mathbf{S}\_c}$$

In this equation, σi is the principal stresses and St is the tensile strength and Sc is the compressive strength. If St and Sc are equal, Equation (1) becomes the distortion energy (DE) theory of failure, which was used in our study as a failure criterion. Since we used Gauss points in assigning material properties, we calculated a factor of safety (FOS) for every Gauss point (Equation 2) and if the FOS value was predicted to be less than one the Gauss point was regarded as in failure (Keyak et al., 1997). The fracture load and location were recorded for each loading condition and compared with the experimental results.

$$FOS = \frac{\text{Gauss point strength (from CT and material property)}}{\text{Gauss point von Mises stress (from FE simulation)}} \tag{2}$$

#### *4.2.3. Finite element model sensitivity analysis*

Sensitivity analysis was performed to find out which material parameters affect the strength of the pelvic bone most. The parameters of interest for sensitivity analysis were: 1) cortical thickness; 2) cortical modulus; 3) trabecular modulus. Since we used synthetic bones made of polyurethane foam, the three corresponding material parameters for the Sawbone FE model were 1) solid polyurethane foam thickness; 2) solid polyurethane modulus; 3) cellular polyurethane modulus. The values for the parameters were varied to see their effects on the predicted fracture load. The following equation (Equation 3) was used to measure the sensitivity of the chosen parameters.

$$S = \frac{\text{\%} \cdot \text{Change in predicted fracture load}}{\text{\%} \cdot \text{Change in input parameter}} \tag{3}$$

The range of simulated variation in the input parameters is given in Table 3. Since we used Gauss points in assigning material properties, the number of Gauss points in the transverse direction was varied from 4 to 6, which had the equivalent effect of varying the solid polyurethane thickness by -40% to +50% (Shim et al., 2008). As for the modulus values, the values were varied by ±25%, which was the next available value in the manufacturer's specification for material properties (Table 3). Multiple FE simulations were run with these values and the change in the predicted fracture load was recorded.

Use of Polyurethane Foam in Orthopaedic Biomechanical Experimentation and Simulation 187

**Predicted fracture** 

0.6 3200 0.395 1.4 4500 0.461

153 2200 0.858 400 5000 0.874

12.4 3200 0.128 47.5 3500 0.028

**load Sensitivity** 

**Figure 12.** Fracture location (a) and load (b) predictions from FE models

**Amount of variations in input parameters** 

*4.2.5. Feasibility of the use of synthetic PU-based bone in validating FE fracture* 

FE models have been extensively used in predicting fracture load. Our approach to fracture mechanics was based on the work by Keyak and co-workers (Keyak et al., 1997, Korn et al.,

**Type of input parameters** 

**Solid polyurethane foam thickness** 

**Solid polyurethane foam Modulus** 

**Cellular polyurethane foam Modulus** 

*predictions* 

**Table 4.** Results of sensitivity analysis

Once the fracture prediction was done, the sensitivity analysis was performed. Three input parameters (solid polyurethane thickness and modulus, cellular polyurethane modulus) were varied to examine the effect of their variation on the predicted fracture load. Among the three input parameters solid polyurethane foam modulus had the greatest impact on the resulting fracture load. Solid polyurethane foam thickness also had some effect on the predicted fracture load, but the sensitivity of this parameter was not as high as the modulus. Cellular polyurethane foam modulus, on the other hand, did not have any significant impact on the predicted fracture load as can be seen in Table 4. Since the pelvis has a sandwich structure where the outer cortical shell bears most of the load, our results indicate that this structural characteristic is also preserved even when the pelvis undergoes fracture.

(a) Fracture location prediction results (b) Fracture load prediction results


**Table 3.** Sensitivity analysis of polyurethane foam thickness and modulus

#### *4.2.4. Fracture experiment with PU-foam pelvis and corresponding FE model predictions*

The fracture behaviour of Sawbone pelves was linear elastic fracture of brittle material (Figure 12 (a)), which coincides with other studies involving fractures of polyurethane (Mcintyre and Anderston, 1979)

The fracture loads from the mechanical experiments are given in Figure 12 (b). The standing case has a slightly higher fracture load (mean 3400N) than the seating case (mean 2600N). Our FE model predicted the fracture load for both cases with a good accuracy as the predicted values are within the standard deviation of the experimental values for both case (Figure 12). The predicted fracture loads from the FE model are 3200N and 2300N for standing and seating cases respectively.

The predicted and actual fracture locations were consistent for both experiments and FE simulations and the fractures occurred mainly in the posterior region of the acetabulum. Different fracture patterns were obtained from two different loading conditions. For the fall from standing experiment, the fracture pattern resembled posterior column fracture according to the Letournel's classification (Letournel, 1980) while the dashboard experiment produced posterior wall fractures. Since the main cause of posterior wall fracture is car accidents (Spagnolo et al., 2009), our experimental set-up was able to capture the main features present in this fracture. The fracture locations predicted by the FE model were similar to the actual fracture patterns from the experiment. For the fall from heights fracture case, the failed Gauss points were concentrated at the region that extends from the dome of the acetabulum to the posterior superior region and then to the posterior column of the pelvis. This resembled the posterior column fracture that was observed from the experiment. For the seating case, on the other hand, the failed Gauss points were more or less limited in the posterior wall region of the acetabular rim, resembling the posterior wall fracture (Figure 12 (a)).

**Figure 12.** Fracture location (a) and load (b) predictions from FE models

Once the fracture prediction was done, the sensitivity analysis was performed. Three input parameters (solid polyurethane thickness and modulus, cellular polyurethane modulus) were varied to examine the effect of their variation on the predicted fracture load. Among the three input parameters solid polyurethane foam modulus had the greatest impact on the resulting fracture load. Solid polyurethane foam thickness also had some effect on the predicted fracture load, but the sensitivity of this parameter was not as high as the modulus. Cellular polyurethane foam modulus, on the other hand, did not have any significant impact on the predicted fracture load as can be seen in Table 4. Since the pelvis has a sandwich structure where the outer cortical shell bears most of the load, our results indicate that this structural characteristic is also preserved even when the pelvis undergoes fracture.


**Table 4.** Results of sensitivity analysis

186 Polyurethane

Solid polyurethane foam

Solid polyurethane foam

Cellular polyurethane foam

(Mcintyre and Anderston, 1979)

standing and seating cases respectively.

fracture (Figure 12 (a)).

thickness

The range of simulated variation in the input parameters is given in Table 3. Since we used Gauss points in assigning material properties, the number of Gauss points in the transverse direction was varied from 4 to 6, which had the equivalent effect of varying the solid polyurethane thickness by -40% to +50% (Shim et al., 2008). As for the modulus values, the values were varied by ±25%, which was the next available value in the manufacturer's specification for material properties (Table 3). Multiple FE simulations were run with these

Simulated of variation

obtained from CT scans

*4.2.4. Fracture experiment with PU-foam pelvis and corresponding FE model predictions* 

The fracture behaviour of Sawbone pelves was linear elastic fracture of brittle material (Figure 12 (a)), which coincides with other studies involving fractures of polyurethane

The fracture loads from the mechanical experiments are given in Figure 12 (b). The standing case has a slightly higher fracture load (mean 3400N) than the seating case (mean 2600N). Our FE model predicted the fracture load for both cases with a good accuracy as the predicted values are within the standard deviation of the experimental values for both case (Figure 12). The predicted fracture loads from the FE model are 3200N and 2300N for

The predicted and actual fracture locations were consistent for both experiments and FE simulations and the fractures occurred mainly in the posterior region of the acetabulum. Different fracture patterns were obtained from two different loading conditions. For the fall from standing experiment, the fracture pattern resembled posterior column fracture according to the Letournel's classification (Letournel, 1980) while the dashboard experiment produced posterior wall fractures. Since the main cause of posterior wall fracture is car accidents (Spagnolo et al., 2009), our experimental set-up was able to capture the main features present in this fracture. The fracture locations predicted by the FE model were similar to the actual fracture patterns from the experiment. For the fall from heights fracture case, the failed Gauss points were concentrated at the region that extends from the dome of the acetabulum to the posterior superior region and then to the posterior column of the pelvis. This resembled the posterior column fracture that was observed from the experiment. For the seating case, on the other hand, the failed Gauss points were more or less limited in the posterior wall region of the acetabular rim, resembling the posterior wall

modulus ±25 % from the original density value of 0.32g/cc

modulus ±25% from the original density 0.16 g/cc

**Table 3.** Sensitivity analysis of polyurethane foam thickness and modulus


values and the change in the predicted fracture load was recorded.

#### *4.2.5. Feasibility of the use of synthetic PU-based bone in validating FE fracture predictions*

FE models have been extensively used in predicting fracture load. Our approach to fracture mechanics was based on the work by Keyak and co-workers (Keyak et al., 1997, Korn et al.,

2001) which used the DE theory of fracture as well as material non-linearity. However, our approach differs from their work in that we simulated acetabular fractures not fractures of the proximal femur which are generally more complicated than fractures of the proximal femur. Moreover, rather than applying force directly to the bone of interest as done in majority of the FE fracture studies, we employed a contact mechanics approach where the force was applied to the acetabulum via the femoral head. Another novel approach of our study is that we incorporated geometric non-linearity to the model by using full finite elasticity governing equations, which has been found to enhance the fracture prediction capabilities of FE models (Stˆlken and Kinney, 2003).

Use of Polyurethane Foam in Orthopaedic Biomechanical Experimentation and Simulation 189

(Parker and Copeland, 1997) for the treatment of minimally displaced acetabular fractures without comminution or free fragment in the joint. However the biomechanical stability of percutaneous fixation has not been studied thoroughly, especially in terms of interfragmentary movement. In particular, the stability of percutaneous fixation in acetabular fractures has not been compared with the more conventional ORIF involving a plate with screws. There have been previous biomechanical studies that compared different types of stabilization in posterior wall fractures (Goulet et al., 1994, Zoys et al., 1999). But the main focus of such studies was to compare the strength of several types of osteosynthesis. However it is interfragmentary movement that exerts major influences on the primary stability and fracture healing (Klein et al., 2003, Wehner et al., 2010). As discussed in Section 3, PU-foam based synthetic bones have been used extensively in testing stability of various osteosynthesis techniques. Therefore we have further developed our FE model capable of prediction acetabular fractures to simulate stability in osteosynthesis. Specifically, we have developed a fast and efficient way of predicting the interfragmentary movement in percutaneous fixation of posterior wall fractures of the acetabulum and validated with a matching biomechanical experiment using PU-foam

Seven synthetic pelves (Full Male Pelvis 1301-1, Pacific Research Laboratories Inc) were loaded until failure with the loading condition that resembled seating fracture[10], creating posterior wall fractures[11]. The fractures were then reduced and fixed with two fixation methods –with two screws (3.5mm Titan Screws, Synthes) and then with a 10-12 hole plates (3.5mm Titan Reconstruction- or LCDC-Plates, Synthes) by an experienced surgeon (JB)

(a) Screw fixation (b) Plate fixation

based synthetic pelves.

*4.3.1. Mechanical experiment with PU-foam based synthetic pelvis* 

(Figure 13 A and B). The maximum remaining crack was 0.7 mm.

**Figure 13.** Two fixation methods performed on the fractured acetabulum

However the most notable feature of our approach is the use of PU-based synthetic bone in validating FE fracture predictions. At present, it is not known whether our model can predict human bone fractures with the same degree of accuracy as the synthetic bones. Therefore caution is required when interpreting the data. However we are confident that our result will translate into human bones due to the following reasons. Firstly our experimental results with PU-foam pelves showed similar results as other human cadaver results as the fracture patterns generated in seating and standing cases correspond well with clinical results. Moreover the sensitivity analysis revealed that our model behaves in a similar manner as the cadaver bones despite the apparent difference in absolute magnitudes in modulus values between PU-foams and bones.

In fact, PU-foam based synthetic bone served our purpose of model validation very well due to their uniformity and consistency (Nabavi et al., 2009). As such, the ASTM standard states that it is "an ideal material for comparative testing" of various orthopaedic devices (American Society for Testing and Materials, 2008b). Although the fracture load is expected to be different from the fracture load of human pelvis, the material behaviour is expected to be comparable to human bones, both of which exhibit brittle fracture (Schileo et al., 2008, Thompson et al., 2003). Therefore the model's ability to predict fracture load and location of the synthetic bone can be regarded as a positive indication that it will also be applicable to human cases. Therefore we continued to use this approach in developing and validating FE model predictions for fracture stability with PU-foam based synthetic bones, which will be described in the next section(Shim et al., 2011).
