*3.1. Results for the Stage of Tightening the Screw Simulation*

In the beginning of the study, the initial step of screw pretension was analysed, and the optimal value for the screw force was chosen. Since there was no specific value for the pretension force, it was decided to investigate this point in more detail. For the final assembly, the calculations were carried out with several values of the pretension force from 100 N up to 1500 N, and the force values were considered incrementally.

Figure 7 demonstrates the dependence between the maximum stress in each of the bolts and the applied tension force. The typical stress distribution in the screws and implant itself caused by the screw preload force of 1000 N is shown in Figure 8.

**3. Results** 

preload values.

the screw force values of 500 and 1000 N.

*3.1. Results for the Stage of Tightening the Screw Simulation* 

In this study, the problem of assessing the screw force effect on overall stress distribution is solved for the four values of pretension load equal to 100, 500, 1000, and 1500 N, and reliable results were obtained. Based on these results, a detailed analysis of the structural strength in the case of slow walking is presented below with emphasis for

In the beginning of the study, the initial step of screw pretension was analysed, and the optimal value for the screw force was chosen. Since there was no specific value for the pretension force, it was decided to investigate this point in more detail. For the final assembly, the calculations were carried out with several values of the pretension force

Figure 7 demonstrates the dependence between the maximum stress in each of the bolts and the applied tension force. The typical stress distribution in the screws and

from 100 N up to 1500 N, and the force values were considered incrementally.

implant itself caused by the screw preload force of 1000 N is shown in Figure 8.

**Figure 7.** Dependence of the maximum von Mises stress in the screws as a function of the applied **Figure 7.** Dependence of the maximum von Mises stress in the screws as a function of the applied preload values.

**Figure 8.** Equivalent von Mises stress (MPa) distribution caused by a pretension load of 1000 N: (**a**) in the screw system; (**b**) in the implant. **Figure 8.** Equivalent von Mises stress (MPa) distribution caused by a pretension load of 1000 N: (**a**) in the screw system; (**b**) in the implant.

Particular attention should be given to the stresses in the pelvic bones because the destruction of the bone might be initiated due to the high value of the screw force. Figure 9 represents comparative graphs of the maximum stress values arising in the spongy (a) and cortical (b) layers in the first row of the finite elements of holes 1, 2, and 3 in the spongy tissue and holes 5, 6, and 7 in the cortical tissue.

Figure 10 shows the typical stress distribution around the holes in the resected pelvic bone after surgery.

The comparison was carried out separately for the spongy and cortical layers because screws with numbers 1, 2, 3, and 4 interacted only with the spongy layer. However, screws with numbers 5, 6, and 7 crossed the cortical layer. The screw with number 4 entered the spongy tissue but was located very close to the cortical layer.

As shown in Figures 8 and 9, the maximum von Mises stress values that occurred in screw 7 and holes 4 and 7 in the case of a pretension force of 1500 N were close to the ultimate stresses according to Table 2. A significant change in the stresses took place near the holes, as shown in Figure 10, while the remaining volume of the bone was almost free from the stresses.

The analysis proved that a pretension force of close to or higher than 1500 N may lead to local bone fracture, and further analysis of the walking simulation made no sense. The optimal screw force was expected to be between 500 N and 1000 N, since it did not cause any bone destruction but provided sufficient contact between the bone and implant. Therefore, the following strength analysis of the "bone–endoprosthesis" system was carried out for a walking condition assuming screw forces equal to 500 N and 1000 N.

Particular attention should be given to the stresses in the pelvic bones because the destruction of the bone might be initiated due to the high value of the screw force. Figure 9 represents comparative graphs of the maximum stress values arising in the spongy (a) and cortical (b) layers in the first row of the finite elements of holes 1, 2, and 3 in the

Figure 10 shows the typical stress distribution around the holes in the resected pelvic

spongy tissue and holes 5, 6, and 7 in the cortical tissue.

bone after surgery.

**Figure 9.** Comparative graphs of the maximum von Mises stresses occurring in the reconstructed pelvis bone on the hole boundaries: (**a**) the spongy tissue; (**b**) the cortical tissue. **Figure 9.** Comparative graphs of the maximum von Mises stresses occurring in the reconstructed pelvis bone on the hole boundaries: (**a**) the spongy tissue; (**b**) the cortical tissue. screws with numbers 1, 2, 3, and 4 interacted only with the spongy layer. However, screws with numbers 5, 6, and 7 crossed the cortical layer. The screw with number 4 entered the spongy tissue but was located very close to the cortical layer.

**Figure 10.** Equivalent von Mises stress (MPa) distribution in the bone tissue caused by a pretension load of 1000 N: (**a**) the upper part of the resected pelvic bone; (**b**) the lower part of the resected pelvic bone. The images on the left show the stresses in the cortical tissue, and the images on the right show the stresses in the spongy tissue. **Figure 10.** Equivalent von Mises stress (MPa) distribution in the bone tissue caused by a pretension load of 1000 N: (**a**) the upper part of the resected pelvic bone; (**b**) the lower part of the resected pelvic bone. The images on the left show the stresses in the cortical tissue, and the images on the right show the stresses in the spongy tissue.

### *3.2. Results of the Walking Cycle Simulation* the holes, as shown in Figure 10, while the remaining volume of the bone was almost free from the stresses.

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The detailed walking FE simulation was carried out for the prepared model with screw pretension values equal to 500 N and 1000 N as follows from the previous section. The analysis proved that a pretension force of close to or higher than 1500 N may lead to local bone fracture, and further analysis of the walking simulation made no sense.

As shown in Figures 8 and 9, the maximum von Mises stress values that occurred in screw 7 and holes 4 and 7 in the case of a pretension force of 1500 N were close to the ultimate stresses according to Table 2. A significant change in the stresses took place near

First, Figure 11 shows the von Mises stresses in the screws preloaded with forces of 500 and 1000 N, while the hip joint reaction force altered according to the graph in Figure 5 that simulates the gait cycle. The optimal screw force was expected to be between 500 N and 1000 N, since it did not cause any bone destruction but provided sufficient contact between the bone and implant. Therefore, the following strength analysis of the "bone–endoprosthesis" system was

In addition, for the initial stage of screw pretension, the stress state in the titanium parts of the endoprosthesis in the walking condition was also considered. Particular attention was given to the load case corresponding to 17% of the gait cycle because this phase is the phase of the maximum reaction force applied to the centre of the left joint, where the implant was placed. However, obviously, extremely high stresses in the screws occurred at 45% of the gait cycle (Figure 11). carried out for a walking condition assuming screw forces equal to 500 N and 1000 N. *3.2. Results of the Walking Cycle Simulation*  The detailed walking FE simulation was carried out for the prepared model with screw pretension values equal to 500 N and 1000 N as follows from the previous section.

Similarly, the peak von Mises stresses for the considered walking phases were expected on the hole edges (Figure 12). The highest stresses in the implant took place at 45% of the gait cycle in the case of a 1000 N preload but at 17% and 65% for a 500 N preload. First, Figure 11 shows the von Mises stresses in the screws preloaded with forces of 500 and 1000 N, while the hip joint reaction force altered according to the graph in Figure 5 that simulates the gait cycle.

(**b**)

**Figure 11.** Maximum von Mises stress in each of the screws with pretension in the walking cycle phases: (**a**) pretension force equal to 500 N; (**b**) pretension force equal to 1000 N. **Figure 11.** Maximum von Mises stress in each of the screws with pretension in the walking cycle phases: (**a**) pretension force equal to 500 N; (**b**) pretension force equal to 1000 N.

preload.

In addition, for the initial stage of screw pretension, the stress state in the titanium parts of the endoprosthesis in the walking condition was also considered. Particular attention was given to the load case corresponding to 17% of the gait cycle because this phase is the phase of the maximum reaction force applied to the centre of the left joint, where the implant was placed. However, obviously, extremely high stresses in the screws

Similarly, the peak von Mises stresses for the considered walking phases were expected on the hole edges (Figure 12). The highest stresses in the implant took place at 45% of the gait cycle in the case of a 1000 N preload but at 17% and 65% for a 500 N

The general view of the typical stress distribution in the endoprosthesis is shown in

Figure 13 for a preload of 1000 N and gait cycle phase equal to 45%.

occurred at 45% of the gait cycle (Figure 11).

(**b**)

**Figure 12.** Maximum von Mises stress in the implant on the boundaries of the holes in the walking cycle phases: (**a**) pretension force equal to 500 N; (**b**) pretension force equal to 1000 N. **Figure 12.** Maximum von Mises stress in the implant on the boundaries of the holes in the walking cycle phases: (**a**) pretension force equal to 500 N; (**b**) pretension force equal to 1000 N.

The general view of the typical stress distribution in the endoprosthesis is shown in Figure 13 for a preload of 1000 N and gait cycle phase equal to 45%.

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system; (**b**) implant body.

**Figure 13.** Equivalent von Mises stress (MPa) distribution in the endoprosthesis titanium parts caused by the pretension load of 1000 N and joint reaction forces at 45% of the gait cycle: (**a**) screw **Figure 13.** Equivalent von Mises stress (MPa) distribution in the endoprosthesis titanium parts caused by the pretension load of 1000 N and joint reaction forces at 45% of the gait cycle: (**a**) screw system; (**b**) implant body.

N.

*Materials* **2021**, *14*, x FOR PEER REVIEW 18 of 24

for the screws at every load point in the walking cycle are shown in Figure 14.

The total stress distribution that occurred in the pelvic bones for the typical gait phase

Finally, the maximum von Mises stresses evaluated on the boundaries of the holes for the screws at every load point in the walking cycle are shown in Figure 14. was 45%, and the screw force of 1000 N is presented in Figure 15.

**Figure 14.** Maximum von Mises stress in the bone tissue on the boundaries of the holes шт in the walking cycle phases: (**a**) the pretension force equal to 500 N; (**b**) the pretension force equal to 1000 **Figure 14.** Maximum von Mises stress in the bone tissue on the boundaries of the holes in the walking cycle phases: (**a**) the pretension force equal to 500 N; (**b**) the pretension force equal to 1000 N.

The total stress distribution that occurred in the pelvic bones for the typical gait phase was 45%, and the screw force of 1000 N is presented in Figure 15.

**Figure 15.** Equivalent von Mises stress (MPa) distribution in the bone tissue caused by the pretension load of 1000 N and joint reaction forces at 45% of the gait cycle: (**a**) the upper part of the resected pelvic bone; (**b**) the lower part of the resected pelvic bone. The images on the left show stresses in the cortical tissue, and the images on the right show stresses in the spongy tissue. **Figure 15.** Equivalent von Mises stress (MPa) distribution in the bone tissue caused by the pretension load of 1000 N and joint reaction forces at 45% of the gait cycle: (**a**) the upper part of the resected pelvic bone; (**b**) the lower part of the resected pelvic bone. The images on the left show stresses in the cortical tissue, and the images on the right show stresses in the spongy tissue.

### **4. Discussion**  For a more convenient assessment and comparison of the obtained results, a **4. Discussion**

summary of the maximum von Mises stresses (MPa) occurring in the endoprosthesis parts and bone tissue are presented in Table 3 for the two considered values for the screw pretension loads: 500 and 1000 N. For a more convenient assessment and comparison of the obtained results, a summary of the maximum von Mises stresses (MPa) occurring in the endoprosthesis parts and bone tissue are presented in Table 3 for the two considered values for the screw pretension loads: 500 and 1000 N.


pretension loads equal to 500 N and 1000 N. **Maximum Von Mises Stresses (MPa) and Their Location Table 3.** Summary of the maximum von Mises stresses occurring in the endoprosthesis parts and bone tissue for screw pretension loads equal to 500 N and 1000 N.

**Table 3.** Summary of the maximum von Mises stresses occurring in the endoprosthesis parts and bone tissue for screw

The first step in the assessment of the long-term strength and reliability of the biomechanical "bone–endoprosthesis" system is to evaluate the mechanical strength of the implant and its fixation system. According to Figure 7, the stresses occurring in the screws vary mostly linearly depending on the applied pretension force. This finding allows us to obtain the required force in a relatively simple way. The high stresses are distributed quite locally. The highest stress level takes place in the areas of contact between the screws and the screw holes in the implant (Figure 8) and the bone tissue (Figure 10).

However, it should be mentioned that the real stresses in the screws differ significantly from the tensile stress in the metal rod loaded with a similar tensile force, which is calculated as the ratio between the force and the cross-sectional area. For example, for screws 5, 6, and 7 with a diameter of 4.5 mm loaded with a longitudinal force of 1000 N, the sectional stresses are equal to 63 MPa, whereas the total equivalent stresses considering the contact interaction reach values from 117 to 212 MPa. This fact emphasizes the importance of considering the contact interaction of the bodies of the "bone–endoprosthesis" system both for the first stage of screw pretension and for the subsequent walking step.

According to Table 3, the screws and implant at pretension values of 500 N and 1000 N have a safety factor of more than 4.0 based on data in the literature [3] (Table 2). The Mises equivalent contact stresses on the edges of the screw holes in the implant are close to the total stresses in the corresponding screws. However, the stresses in the implant are slightly lower. The difference approximately equals the pretension stresses in the screws.

Analysis of the static strength of the screws allows us to assume that there should be no destruction of the titanium components of the system when the patient is walking. However, this assumption becomes debatable in further analysis of the structural strength under periodic loads occurring in the hip joint during normal human activity.

In the case of walking, the development of fatigue damage is highly possible on the edges of the implant holes for both considered options for screw tightening (Figures 12 and 13). The maximum equivalent stresses exceed the lower bound of the fatigue limit for 3D printed Ti-6Al-4V, which is 200 MPa according to the obtained material data (Table 2). At a pretension force of 500 N, the stresses exceed the limit only in the region of hole 1, whereas at a force of 1000 N, the limit is exceeded for all holes except hole 2. However, most likely, the risk of fracture due to fatigue effects may decrease with successful osseointegration and should not have any significant effect thereafter. Regardless, the use of high-quality titanium powders and the application of advanced manufacturing technologies for producing implants [22] should have a high priority in the planning stage for such surgeries.

In the fixation system, screws 1 and 4–7, which fix the implant to the upper part of the pelvic bone, are mostly affected by the periodic loads that occur during walking. In screws 2 and 3, the stresses do not exceed 80 MPa since these screws do not carry any significant external load. However, these screws cannot be excluded from the system because they connect the implant and the lower part of the pelvic bone, reconstructing the pelvic ring. In all screws, the maximum stresses did not reach the lower bound of the fatigue limit for Ti-6Al-4V according to the obtained material data (Table 2). The highest stress value equals 263 MPa and occurs in screw 7.

The analysis of the stresses in the bone tissue of the analysed biomechanical system requires special attention. As mentioned above, at the stage of screw tightening, the equivalent von Mises stresses on the edges of the screw holes in the cortical and spongy tissues (Figure 9) for tightening forces of 1000 N and above approach the lower bound of the strength limit of the corresponding tissue (Table 2) but do not reach the critical values. The cortical tissue of the upper pelvic bone has a safety margin against the strength limit at the preloading stage for all the considered values of the pretension forces. The spongy tissue also has a significant safety margin in the case of the pretension of 500 N. In the case of 1000 N, the stresses are close to the limit but do not exceed it. The remaining regions of the bone remain mostly unstressed, which is quite reasonable due to the chosen detailed setup of the problem.

In the cortical layer of the lower part of the pelvis, maximum stresses are situated in the pubic joint. These stresses are caused by the rigid connection of the left and right pelvic bones in this area. In other areas, at all stages of the simulation, stresses do not exceed 20 MPa, and there is a safety margin of at least 4.0. A similar situation occurs in the area of the rigid connection between the upper part of the pelvic bone and the sacrum. The increased stresses in these areas might be neglected because they are caused by artificial rigid constraints and occur due to the absence of the soft cartilage layer in the model. Furthermore, these areas are not subjected to surgical intervention, and their physiological condition remains unchanged compared with the healthy biomechanical system of the pelvis, where there are no high mechanical stresses that exceed the strength limits during normal human activities [30].

During the walking phase, areas with extremely high stresses, which exceed the allowed limits, appear both in the cortical and spongy tissues in the upper part of the resected pelvic bone and for both pretension forces considered (Figures 14 and 15). According to Table 3, the most dangerous regions where local destruction might be expected are the edges of screw holes 1 and 4 (spongy layer, equivalent stress of 4–5 MPa) and holes 5 and 7 (cortical layer, equivalent stress of 160–170 MPa). The limits are significantly exceeded in the walking cycle phases of 17% and 45%. The stress near hole 6 equals 95 MPa and exceeds the lower limit of the allowable stress range (80 MPa). This fact confirms the high risk of bone damage.

The stress values obtained in the "bone–endoprosthesis" system during the walking simulation are slightly higher than the stress values reported previously [28], where a similar approach was used. The reported peak stresses in the implant were approximately 105 MPa during the walking phase, and the stresses in the screws and pelvic bones were approximately 50 MPa under the same loading condition. Therefore, the authors [28] expected that the fatigue limit could be reached only in the case of more severe loading scenarios, such as stair climbing. The difference in the results can be explained by the difference in the implant design and the differences in the approaches of finite element modelling of the behaviour of the bone tissue.

Thus, according to the performed analysis, slight destruction of the bone might be expected in local areas near the screw holes in cases of walking. This destruction can affect the stability of the implant fixation to the upper bone. However, bone tissue is capable of regeneration when it is loaded with an external mechanical field, in particular, with periodic loading of a sufficient level. Therefore, in the case of a moderate dynamic compressive load, the process of regeneration could be initiated in the area that might be initially damaged [31].

The current study particularly focuses on the screw tightening process and on the analysis of the values of the forces applied to the screws. This issue is usually neglected in studies, and the number of related publications is very limited [2–4]. Nevertheless, this point is quite important for proper patient surgery planning. The degree of fixation between the implant and the bones directly depends on the value of the screw pretension force. Additionally, fixation affects the reliability of the reconstructed pelvis in terms of cyclic loading and fatigue effects. If the screws are tightened loosely, then the process of bone regeneration might take much more time. If the screws are tightened excessively, local destruction will occur in the bones near the holes. This fact can lead to more serious consequences over time. Considering previous results [2], it should be pointed out that the screw pretension force equals 3000 N. However, according to the current study, increasing the screw tightening force from 1000 to 1500 N may lead to local bone destruction. Comparison of the obtained results with the charts provided [2] shows the compliance of stresses for tightening forces up to 500 N. The results from previous research [2] also confirm the linear growth of stresses in the screws.

The shape of the endoprosthesis is customized, which causes additional problems for the stress-strain state analysis of the structure. It should be mentioned that the methods of designing individual implants differ in complexity from the methods of developing serial endoprostheses [1]. Additionally, the design of the individual endoprosthesis can change during the modelling process. However, the design of serial implants is usually as efficient as possible, as opposed to individual implants, which are designed and produced only once for a specific patient. Using the finite element method, it is possible to predict the areas of stress concentrations and to choose the optimal number and parameters of the screws. In the current study, there were several screws that did not carry much load. This finding indicates that they may be removed from the structure. The authors presume that if the stress concentration does not occur in the screws, such action should not affect the general performance and reliability of the structure. A previous study [32] confirms this statement, saying that such screws accumulate excess stresses and should be removed.

When the current stress-strain state results are compared to results from similar studies [1,2,10], it should be considered that the forces applied to the structure are not universal and vary depending on the patient. For the stress-strain state comparison [2], the stress concentration arises around holes, and the stress values match the results of current research. The same trend takes place in other studies [11], especially for cup-shaped structures.
