2.5.4. Boundary Conditions

Solórzano et al. [18] studied the mechanical behavior of the proximal femur against the nine loads proposed by Bergmann et al. [74], including the ISO force [75], widely used to test femoral stems. They concluded that the representative loads that increase the risk of fracture are ISO and jogging; consequently, both were used to evaluate the differences between the biomechanics of the intact and implanted femur. The jogging load for the intact femur is only composed by the contact forces (*FX*, *F<sup>Y</sup>* and *FZ*); however, when the stem is implanted, the moments that stress the fixation in the acetabulum appear; hence, the implanted femur is subjected to contact forces and frictional moments (*MX*, *M<sup>Y</sup>* and *MZ*). This load depends on the body weight of each patient; nevertheless, in this study, the same load state (shown in Table 3) was used for GC1 and GC2 of our recently explained procedure [18], since, being equal, the boundary conditions allowed us to evaluate and compare the influence of the femoral morphology in the stem design.


**Table 3.** Standardized loads for the intact and implanted femur.

To apply the load on the intact femur, the body was first placed in a frontal position and rotated (90-MAS)◦ clockwise with respect to the Y-axis, and then it was rotated at an angle equal to the patient's anteversion clockwise with respect to the Z-axis, to finally place the load on the cortical femoral nodes that make up the acetabular region—the region located from the beginning to the middle of the femoral head. Since the prostheses were designed using the geometric parameters of the patient, in the implanted femur, the load was applied on the flat part of the receiving taper, which happened to be the same position as in the intact femur—since the cone was designed considering the middle of the femoral head (sphere), the mechanical angle and the anteversion. Therefore, no torque was produced by displacement of the forces, allowing a fair comparison between the intact and implanted femur. In both situations, the movement of the femur was limited through the fixed constraint in the flat part of the cortical and trabecular bone (Figure 10).

**Figure 10.** Boundary conditions for the intact and implanted femur. **Figure 10.** Boundary conditions for the intact and implanted femur.

Once the meshes were generated, the material and the boundary conditions for each of the bodies were defined, and the simulations were carried out in NX®—first for the intact femur of both geometric cases (GC1 and GC2) with the two defined load states (jogging and ISO load), and then for the implanted femur for both geometric cases with the load states and using the two materials (Ti6Al4V ELI and Ti21S) in each of the 3 stems (V1, V2 and V3). Once the results of each simulation were obtained, they were processed to extract the information useful in the evaluation of the . Once the meshes were generated, the material and the boundary conditions for each of the bodies were defined, and the simulations were carried out in NX®—first for the intact femur of both geometric cases (GC1 and GC2) with the two defined load states (jogging and ISO load), and then for the implanted femur for both geometric cases with the load states and using the two materials (Ti6Al4V ELI and Ti21S) in each of the 3 stems (V1, V2 and V3). Once the results of each simulation were obtained, they were processed to extract the information useful in the evaluation of the *SS*.

the femoral head (sphere), the mechanical angle and the anteversion. Therefore, no torque was produced by displacement of the forces, allowing a fair comparison between the intact and implanted femur. In both situations, the movement of the femur was limited through the fixed constraint in the flat part of the cortical and trabecular bone (Figure 10).

### 2.5.5. Postprocessing 2.5.5. Postprocessing

According to Wolff's law [76], the adaptation of the bone to the mechanical stimulus causes the remodeling process. However, according to the definition of the mechanostat [77], bone adapts towards a target strain; hence, osteocytes sense this stimulus and send biochemical signals that activate cellular action to remodel bone. In vitro or in vivo studies even use strain gauge rosettes to quantify the strain of the femur and study the relationship between in vivo loading and bone adaptation. Strain data recorded in extensometer studies are usually summarized in terms of principal strains. Therefore, it is necessary to represent the multiaxial strain state as an equivalent metric, i.e., to reduce the complicated and directionally specific strain state to a scalar quantity that is independent of direction. This metric is called "equivalent strain"; it was first introduced by Mikić and Carter [78] with the aim of incorporating strain gauge data in the context of bone adaptation models. Turner et al. [79], through clinical testing of patients with femoral prostheses, evaluated changes in their bone density and found that it adequately modeled bone remodeling. It is easy to interpret, direction-invariant and a positive scalar, because, mathematically, it is the norm of the strain tensor (): According to Wolff's law [76], the adaptation of the bone to the mechanical stimulus causes the remodeling process. However, according to the definition of the mechanostat [77], bone adapts towards a target strain; hence, osteocytes sense this stimulus and send biochemical signals that activate cellular action to remodel bone. In vitro or in vivo studies even use strain gauge rosettes to quantify the strain of the femur and study the relationship between in vivo loading and bone adaptation. Strain data recorded in extensometer studies are usually summarized in terms of principal strains. Therefore, it is necessary to represent the multiaxial strain state as an equivalent metric, i.e., to reduce the complicated and directionally specific strain state to a scalar quantity that is independent of direction. This metric is called "equivalent strain"; it was first introduced by Miki´c and Carter [78] with the aim of incorporating strain gauge data in the context of bone adaptation models. Turner et al. [79], through clinical testing of patients with femoral prostheses, evaluated changes in their bone density and found that it adequately modeled bone remodeling. It is easy to interpret, direction-invariant and a positive scalar, because, mathematically, it is the norm of the strain tensor (*εij*):

$$
\varepsilon\_{ij} = \begin{bmatrix}
\varepsilon\_x & \varepsilon\_{xy} & \varepsilon\_{xz} \\
\varepsilon\_{xy} & \varepsilon\_y & \varepsilon\_{yz} \\
\varepsilon\_{xz} & \varepsilon\_{yz} & \varepsilon\_z
\end{bmatrix} = \begin{bmatrix}
\varepsilon\_1 & 0 & 0 \\
0 & \varepsilon\_2 & 0 \\
0 & 0 & \varepsilon\_3
\end{bmatrix} \tag{22}
$$

To perform postprocessing, the proximal femur was cut longitudinally using a plane coincident with the Y-coordinate of the elliptical adjustment performed for the implantation section (I) of each geometric case (Figure 11A). Mimicking the position of the strain gauges and the orthopedist's analysis of bone density using medical imaging, shielding and bone remodeling over the outer medial (M) and lateral (L) sides of the proximal femur To perform postprocessing, the proximal femur was cut longitudinally using a plane coincident with the Y-coordinate of the elliptical adjustment performed for the implantation section (I) of each geometric case (Figure 11A). Mimicking the position of the strain gauges and the orthopedist's analysis of bone density using medical imaging, shielding and bone remodeling over the outer medial (M) and lateral (L) sides of the proximal femur were evaluated using the equivalent strain, in the region bounded by section I and VI (Figure 11B), since these regions undergo more bone resorption in the proximal femur.

*Materials* **2022**, *15*, x FOR PEER REVIEW 17 of 32

**Figure 11.** (**A**) Elliptical adjustment of the implantation section. (**B**) Medial and lateral side of the proximal femur. **Figure 11.** (**A**) Elliptical adjustment of the implantation section. (**B**) Medial and lateral side of the proximal femur. **3. Results and Discussion** 

were evaluated using the equivalent strain, in the region bounded by section I and VI (Figure 11B), since these regions undergo more bone resorption in the proximal femur.

were evaluated using the equivalent strain, in the region bounded by section I and VI (Figure 11B), since these regions undergo more bone resorption in the proximal femur.

### **3. Results and Discussion 3. Results and Discussion** *3.1. Remodeling Curve and Regression Graph*

### *3.1. Remodeling Curve and Regression Graph 3.1. Remodeling Curve and Regression Graph* The bone adapts towards a target strain, and, if this is greater than desired, the bone

The bone adapts towards a target strain, and, if this is greater than desired, the bone mass increases, and if it is less, it decreases. The dead zone is defined as the zone where bone resorption and bone formation are in equilibrium. All these characteristics are summarized graphically in the bone remodeling curve (Figure 12), which relates the variations in apparent density to the mechanical stimulus. Likewise, bone density is directly proportional to stiffness and strength and inversely proportional to its ductility; it is understood that an increase or decrease in density causes undesirable mechanical performance. The bone adapts towards a target strain, and, if this is greater than desired, the bone mass increases, and if it is less, it decreases. The dead zone is defined as the zone where bone resorption and bone formation are in equilibrium. All these characteristics are summarized graphically in the bone remodeling curve (Figure 12), which relates the variations in apparent density to the mechanical stimulus. Likewise, bone density is directly proportional to stiffness and strength and inversely proportional to its ductility; it is understood that an increase or decrease in density causes undesirable mechanical performance. mass increases, and if it is less, it decreases. The dead zone is defined as the zone where bone resorption and bone formation are in equilibrium. All these characteristics are summarized graphically in the bone remodeling curve (Figure 12), which relates the variations in apparent density to the mechanical stimulus. Likewise, bone density is directly proportional to stiffness and strength and inversely proportional to its ductility; it is understood that an increase or decrease in density causes undesirable mechanical performance.

**Figure 12.** Remodeling curve and regression graph. **Figure 12.** Remodeling curve and regression graph. **Figure 12.** Remodeling curve and regression graph.

Iatrogenic remodeling is related to the bone changes caused by the implant; this type of remodeling should be avoided by the designer and the orthopedist since it contributes to implant loosening and periprosthetic fractures and complicates revision surgeries. Therefore, the ideal stem is one that does not change the femoral biomechanics, does not cause iatrogenic bone remodeling and integrates perfectly through bone ingrowth. However, each stem leads to a specific change in the mechanical response of the femur. Consequently, the designer wants the implant to keep the femur within the dead zone and not cause an excessive increase or decrease in its density. To analyze bone adaptation, the equivalent strain of the mesh element before (̅௧) and after (̅) stem insertion was obtained; then, the bone remodeling curve was defined, where is ̅௧, and in order to establish the dead zone, the "*s*" value was necessary, which, according to the study by Iatrogenic remodeling is related to the bone changes caused by the implant; this type of remodeling should be avoided by the designer and the orthopedist since it contributes to implant loosening and periprosthetic fractures and complicates revision surgeries. Therefore, the ideal stem is one that does not change the femoral biomechanics, does not cause iatrogenic bone remodeling and integrates perfectly through bone ingrowth. However, each stem leads to a specific change in the mechanical response of the femur. Consequently, the designer wants the implant to keep the femur within the dead zone and not cause an excessive increase or decrease in its density. To analyze bone adaptation, the equivalent strain of the mesh element before (̅௧) and after (̅) stem insertion was obtained; then, the bone remodeling curve was defined, where is ̅௧, and in order to establish the dead zone, the "*s*" value was necessary, which, according to the study by Iatrogenic remodeling is related to the bone changes caused by the implant; this type of remodeling should be avoided by the designer and the orthopedist since it contributes to implant loosening and periprosthetic fractures and complicates revision surgeries. Therefore, the ideal stem is one that does not change the femoral biomechanics, does not cause iatrogenic bone remodeling and integrates perfectly through bone ingrowth. However, each stem leads to a specific change in the mechanical response of the femur. Consequently, the designer wants the implant to keep the femur within the dead zone and not cause an excessive increase or decrease in its density. To analyze bone adaptation, the equivalent strain of the mesh element before (*εint*) and after (*εimp*) stem insertion was obtained; then, the bone remodeling curve was defined, where *Sre f* is *εint*, and in order to establish the dead zone, the "*s*" value was necessary, which, according to the study by Turner et al. [79], is 0.6. Once the parameters were established, the *εimp* was located on the abscissae to determine whether it was inside or outside of the dead zone.

Despite defining whether or not the femoral region under study is in the dead zone, many designers analyze a region of the femur by averaging the mechanical stimulus of the mesh elements before (*εint*, *avg*) and after (*εimp*, *avg*) surgery, and calculate the respective strain shielding (*SSavg*). However, the mean of the mechanical stimulus may not represent the loading pattern caused by the stem; consequently, the designer may reach erroneous conclusions using only the average parameters (Table 4).


**Table 4.** Errors caused by average equivalent strains.

Consequently, using concepts related to calculus and statistics, a method was found not only analytically but also graphically to evaluate strain shielding, bone remodeling and femoral biomechanics. This consists of transferring the information from the remodeling curve to a regression graph in an equivalent strain plane of the intact implanted femur, as shown in Figure 12. To obtain the regression graph, an assumption is made: the equivalent strain of the intact femur is dependent on the strain of the implanted femur (*εint* = *f εimp* ). This may seem contradictory; however, this assumption is very useful because, if a linear regression is performed between the values of the elemental strain before and after the insertion, the result is:

$$
\mathbb{E}\_{\rm int} = a + b\mathbb{E}\_{\rm imp} \tag{23}
$$

From this equation, it is possible to obtain the particular designs of femoral stems. For example, the ideal stem, defined as that which fully restores the femoral biomechanics, whose shielding is zero, describes its behavior through Equation (23) when *a* = 0 and *b* = 1. A stem that preserves the femoral biomechanics will be one whose fit results in the Equation (23) with |*a*| ∼= 0; this means that the strain prior to THR is equal to that after but multiplied by a factor "*b*", and the trend in the mechanical response of the femur is maintained in a scaled manner. For this reason, the shielding of this type of implant is:

$$SS = \frac{\overline{\varepsilon}\_{int} - \overline{\varepsilon}\_{imp}}{\overline{\varepsilon}\_{int}} = 1 - \frac{\overline{\varepsilon}\_{imp}}{\overline{\varepsilon}\_{int}} = 1 - \frac{1}{b} \tag{24}$$

The stem altering biomechanics is defined by Equation (23) for values of "*a*" and "*b*" ∈ R, and as a result, the shielding is:

$$\text{SS} = 1 - \frac{\overline{\varepsilon}\_{imp}}{\overline{\varepsilon}\_{int}} = \frac{b-1}{b} + \frac{a}{b\overline{\varepsilon}\_{int}}\tag{25}$$

High values of *εint* result in a strain shielding equal to Equation (24). Therefore, this expression was used to approximate the shielding caused by stems that do not restore the femoral biomechanics. In the regression graph (Figure 12), the ideal stem is represented by the dashed black line, the stem that restores biomechanics by the blue line and the stem that modifies the mechanical response by the red line. We defined the types of stems from the assumption *εint* = *f εimp* , and it is necessary to bound the dead zone within the graph:

$$
\overline{\varepsilon}\_{imp} = (1+s)\overline{\varepsilon}\_{\rm int} \Leftrightarrow \overline{\varepsilon}\_{imp} = 1.6\overline{\varepsilon}\_{\rm int} \Leftrightarrow \overline{\varepsilon}\_{\rm int} = 0.625\overline{\varepsilon}\_{\rm imp} \tag{26}
$$

$$
\mathfrak{E}\_{imp} = (1 - s)\mathfrak{E}\_{int} \Leftrightarrow \mathfrak{E}\_{imp} = 0.4\mathfrak{E}\_{int} \Leftrightarrow \mathfrak{E}\_{int} = 2.5\mathfrak{E}\_{imp} \tag{27}
$$

These lines limit the dead zone, which corresponds to the gray area in Figure 12. From this zone, another two are defined: the purple and green indicate the loss and increase of bone mass, respectively. The adjusted R-Square was used to evaluate how good the linear fit of the equivalent strain of the elements is. The definition of this statistical metric is the proportion of the variance of the dependent variable that can be explained by the independent variable or how well the linear fit is able to model the dependent variable from the independent variable, so it is an indirect measure of how dispersed the points are around the fit line.

The results obtained from the simulations were used for the analysis, with the equivalent strains of the elements of both regions, lateral and medial. The linear adjustment was performed to obtain the "*a*" and "*b*" coefficients, the adjusted R-Square and the *SS*, and to evaluate the response of each femur to the implantation of the customized stems and the influence of the material from which it is made. Then, using scatter plots by regions, areas of the femoral stem that can be optimized in a following work to mitigate shielding were visualized. Finally, equivalent strain maps extracted from NX® were obtained for the intact and implanted femur with the selected stem and material to verify if the analysis in the lateral and medial zone is representative for both femurs.
