*3.2. Analysis*

The linear fits between the equivalent strain of the intact and implanted femur with each stem (V1, V2 and V3) were performed for GC1 using both loading states (ISO and jogging) and materials (Ti6V4Al and Ti21S), whose metrics are summarized in Table 5. Figure 13 shows the strain shielding and the adjusted R-Square produced by each stem graphically. In addition, the *SSavg* was calculated for comparison with the *SS* obtained from the coefficient "*b*" of the regression.


**Table 5.** Results for GC1.

The designer is looking for the stem to be as close as possible to the ideal model, with zero shielding, so the implant with the lowest value should be selected. However, as explained in the previous section, the adjusted R<sup>2</sup> is a statistic that evaluates the goodness of the linear fit, so when it is closer to the unit, it is deduced that the points of the curve present a linear trend and are close to the line, which in turn validates the *SS* obtained. Therefore, there should be a compromise between the adjusted R<sup>2</sup> and the shielding. Figure 13 shows that the lowest *SS* and highest adjusted R<sup>2</sup> occur when Ti21S is used; regarding implant geometry, the V2 and V3 stems have a very similar mechanical response, with V3 being superior in the *SS* by thousandths. Then, to select which of the two implants is the indicated one, its volume was evaluated, because the prosthesis with greater volume is heavier, limits

0.00

1

0.04

0.08

0.12

Equivalent strain of the intact femur [%]

0.16

0.20

0.24

the gait and causes patient discomfort. The V2 stem has a volume of 33.25 cm<sup>3</sup> and V3, 32.368 cm<sup>3</sup> ; because V3 is lighter and has metrics similar to those of V2, it is the ideal implant for GC1. *Materials* **2022**, *15*, x FOR PEER REVIEW 20 of 32

**Figure 13.** Graphs of the and adjusted R2 for GC1. **Figure 13.** Graphs of the *SS* and adjusted R<sup>2</sup> for GC1.

The designer is looking for the stem to be as close as possible to the ideal model, with zero shielding, so the implant with the lowest value should be selected. However, as explained in the previous section, the adjusted R2 is a statistic that evaluates the goodness of the linear fit, so when it is closer to the unit, it is deduced that the points of the curve The influence of the load has not been mentioned, because evaluating either leads to the same conclusion; therefore, to further understand its effect, we plot the response of GC1 to the insertion of the selected stem (V3) when the femur is subjected to the ISO and jogging loads (Figure 14).

present a linear trend and are close to the line, which in turn validates the obtained.

0.12 **Figure 14.** Regression graph of GC1 under ISO and jogging loads.

**Figure 14.** Regression graph of GC1 under ISO and jogging loads. 0.00 0.03 0.06 0.09 0.12 0.15 0.18 Equivalent strain of the implanted femur [%] Equation y = a + b × x Graph Ti21S Ti6Al4V Constant (a) 0.01714 0.02258 Coefficient (b) 1.39913 1.4116 Adj. R-Square 0.80373 0.78037 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.04 0.08 Equivalent strain of the intact femur [%]Equivalent strain of the implanted femur [%] Equation y = a + b × x Graph Ti21S Ti6Al4V Constant (a) 0.00791 0.0126 Coefficient (b) 1.66416 1.70752 Adj. R-Square 0.70074 0.68312 Examining the range of the axes, it is perceived that jogging loads the proximal femur less in comparison to the ISO force; this is due to its mechanical nature. The femoral neck fracture is caused by high energy mechanisms such as an axial load on the femur; for this reason, ISO overloads it more. Graphically, the ISO force distributes the load better along the femur; for this reason, the points of its regression graph are more concentrated and follow a linear pattern. On the contrary, the jogging load disperses the points more and causes them not to adapt to the regression; as a result, the adjusted R<sup>2</sup> is low (Figure 13). Nevertheless, the conclusions obtained by analyzing any of the two load states do not change, i.e., whether examining the femur under ISO or jogging, the same geometry and material is selected. From this perspective, since the use of the ISO force facilitates the testing of prototypes and allows comparison of the experimental results with the finite element analysis, its use is recommended for the evaluation of femoral implants.

follow a linear pattern. On the contrary, the jogging load disperses the points more and

Examining the range of the axes, it is perceived that jogging loads the proximal femur less in comparison to the ISO force; this is due to its mechanical nature. The femoral neck

Regression graphs show the influence of the material on the femoral mechanical response. For a closer analysis of its effect, the purple-colored area of the ISO graph in Figure 14 was evaluated.

Young's modulus is related to strain shielding. Figure 15 shows that Ti6Al4V, a material with high stiffness compared to the femur, causes greater shielding, and consequently exposes the points to the bone resorption area.

**Figure 15.** Influence of the modulus of elasticity of the stem material.

In addition, due to its quadratic tendency, it alters the femoral biomechanics since it moves away from the linear behavior of the ideal stem and its adjusted R<sup>2</sup> is lower (Figure 13). In contrast, Ti21S, having lower stiffness, approaches the linear response of the stem that preserves, in a scaled form, the strain of the proximal femur anterior to the THR and maintains the points within the dead zone. In this way, it is verified that, in spite of being the same stem (V3), the selected material originates different mechanical responses; therefore, having a stiffness closer to that of the femur allows the designer to evaluate the behavior originated by the geometry and distinguish it from that caused by the mechanical properties of the material.

The regression graphs were divided by orange and purple squares enclosing the medial and lateral zones, respectively Figure 14. The medial shows a set of points that follows a negative slope and is outside of the dead zone; the stem is made of either Ti6Al4V or Ti21S. To analyze this behavior in depth, Figure 16 shows the scatter plots of the equivalent strain of the intact and implanted femur subjected to both loading states.

The plots of the equivalent strain with respect to the Z-coordinate show that the red and blue curves of the implanted femur mimic the black curve that corresponds to the strain of the intact femur, but in the medial part, from Z = −10, the curves of the implanted femur diverge due to the geometry of V3; this section originates the set of points with the negative slope mentioned above.

**Figure 16.** Scatter plots of the equivalent strain of GC1 in the medial and lateral sides under (**A**) ISO and (**B**) jogging loads. **Figure 16.** Scatter plots of the equivalent strain of GC1 in the medial and lateral sides under (**A**) ISO and (**B**) jogging loads.

The plots of the equivalent strain with respect to the Z-coordinate show that the red and blue curves of the implanted femur mimic the black curve that corresponds to the strain of the intact femur, but in the medial part, from Z = −10, the curves of the implanted femur diverge due to the geometry of V3; this section originates the set of points with the negative slope mentioned above. The scatter plots complement the results obtained from the regression graphs. These plots confirm that the material with the lower modulus of elasticity not only reduces the The scatter plots complement the results obtained from the regression graphs. These plots confirm that the material with the lower modulus of elasticity not only reduces the difference between the strain of the intact and implanted femur, but also preserves the femoral biomechanics. Furthermore, it certifies that any of the loads is useful to select the geometry and material of the stem; further proof of this is that both exhibit the alteration of the medial curve of the implanted femur from Z = −10 onwards. For this reason and due to the above advantages, to study the customized stems of GC2, the femur subjected to the ISO force was evaluated by performing the same analysis of GC1.

difference between the strain of the intact and implanted femur, but also preserves the femoral biomechanics. Furthermore, it certifies that any of the loads is useful to select the geometry and material of the stem; further proof of this is that both exhibit the alteration The metrics of the GC2 linear fits are summarized in Table 6, and Figure 17 exposes the *SS* and adjusted R<sup>2</sup> produced by each stem graphically.


**Constant ()** 0.097 0.09 0.089 0.073 0.073 0.072 **Coefficient ()** 1.213 1.052 1.043 1.26 1.082 1.079

0.176 0.049 0.041 0.206 0.076 0.073

of the medial curve of the implanted femur from Z = −10 onwards. For this reason and **Table 6.** Results for GC2.

**Figure 17.** Graphs of the and adjusted R2 for GC2. **Figure 17.** Graphs of the *SS* and adjusted R<sup>2</sup> for GC2.

Figure 17 shows that the shielding caused by the Ti21S stem is higher compared to those manufactured with Ti6Al4V, which is contradictory to the deductions obtained from the previous analysis. However, the adjusted R2 of the Ti21S stem is much higher and, because the shielding is a result of the linear fit, Ti6Al4V cannot be reliably selected as a material in this case. Regarding geometry, again, V2 and V3 have very similar metrics, with the smaller volume being the reason that the V3 stem is preferred. When the metrics do not allow correct selection of the material, the visual method is used. Figure 18 shows the regression graph generated by the chosen geometry, produced with both materials. Figure 17 shows that the shielding caused by the Ti21S stem is higher compared to those manufactured with Ti6Al4V, which is contradictory to the deductions obtained from the previous analysis. However, the adjusted R<sup>2</sup> of the Ti21S stem is much higher and, because the shielding is a result of the linear fit, Ti6Al4V cannot be reliably selected as a material in this case. Regarding geometry, again, V2 and V3 have very similar metrics, with the smaller volume being the reason that the V3 stem is preferred. When the metrics do not allow correct selection of the material, the visual method is used. Figure 18 shows the regression graph generated by the chosen geometry, produced with both materials.

the restoration of the femoral biomechanics. It is evident that Ti6Al4V locates a greater **Figure 18.** Regression graph of GC2.

tions based on the theory and the previous analysis are not contradicted by the information shown in the figure. In short, Ti21S is the ideal material for the fabrication of the customized stem. The value of the independent term () indicates whether the implant deviates from the ideal behavior and alters the load distribution along the femur, which, in this case, is The lateral area of the graph (purple box) shows the influence of material stiffness on the restoration of the femoral biomechanics. It is evident that Ti6Al4V locates a greater number of points outside the dead zone; therefore, its shielding is greater, and the deductions based on the theory and the previous analysis are not contradicted by the information shown in the figure. In short, Ti21S is the ideal material for the fabrication of the customized stem.

The lateral area of the graph (purple box) shows the influence of material stiffness on

 Ti21S Ti6Al4V Dead zone

number of points outside the dead zone; therefore, its shielding is greater, and the deduc-

The value of the independent term (*a*) indicates whether the implant deviates from the ideal behavior and alters the load distribution along the femur, which, in this case, is quantified by the equivalent strain; therefore, the farther the regression is from the center of coordinates (|*a*| > 0), the more the implanted femur strain diverges with respect to the intact one, modifying the load received by the bone and increasing the shielding. of coordinates (|| > 0), the more the implanted femur strain diverges with respect to the intact one, modifying the load received by the bone and increasing the shielding. The independent term for GC2 defines that Ti6Al4V more strongly alters the mechanical response of the femur, because the red line is more distant from the center of coordi-

quantified by the equivalent strain; therefore, the farther the regression is from the center

The independent term for GC2 defines that Ti6Al4V more strongly alters the mechanical response of the femur, because the red line is more distant from the center of coordinates. Graphically, the Ti6Al4V regression is above the Ti21S line, cutting the Y-axis at point 0.089. nates. Graphically, the Ti6Al4V regression is above the Ti21S line, cutting the Y-axis at point 0.089. Figure 18 shows, in the orange box, corresponding to the medial zone, a set of points

Figure 18 shows, in the orange box, corresponding to the medial zone, a set of points outside the dead zone and with a negative slope, and, in the purple box, corresponding to the lateral zone, a series of points moving away from the linear trend. The scatter plot of GC2 (Figure 19) supports the choice of material and allows us to identify in which specific regions the geometry of the selected stem should be optimized. In the medial region, it should be improved from Z = −15 onwards, and, in the lateral region, from Z = −30 to Z = −15 because, in these ranges, the strain of the implanted femur, with the stem made of either Ti6Al4V or Ti21S, diverges from the strain of the intact femur, with this effect resulting from the geometry of the V3. outside the dead zone and with a negative slope, and, in the purple box, corresponding to the lateral zone, a series of points moving away from the linear trend. The scatter plot of GC2 (Figure 19) supports the choice of material and allows us to identify in which specific regions the geometry of the selected stem should be optimized. In the medial region, it should be improved from Z = −15 onwards, and, in the lateral region, from Z = −30 to Z = −15 because, in these ranges, the strain of the implanted femur, with the stem made of either Ti6Al4V or Ti21S, diverges from the strain of the intact femur, with this effect resulting from the geometry of the V3.

*Materials* **2022**, *15*, x FOR PEER REVIEW 24 of 32

**Figure 19.** Scatter plots of the equivalent strain of GC2 in the medial and lateral sides. **Figure 19.** Scatter plots of the equivalent strain of GC2 in the medial and lateral sides.

Once the geometry and the material of the customized stem have been selected, it is necessary to verify whether the obtained through the proposed method is better than the ௩. For this purpose, we resort to the strain maps, which provide equivalent information to performing photoelastic tests. All the maps of the intact femur of both geometric cases distribute the color scale in the range from 0 to 0.4, while the range goes from 0 to 0.29 and from 0 to 0.21 for the maps of the implanting femur of GC1, and from 0 to 0.37 and from 0 to 0.23 for the maps of GC2 (Figure 20), both calculated from the and ௩, Once the geometry and the material of the customized stem have been selected, it is necessary to verify whether the *SS* obtained through the proposed method is better than the *SSavg*. For this purpose, we resort to the strain maps, which provide equivalent information to performing photoelastic tests. All the maps of the intact femur of both geometric cases distribute the color scale in the range from 0 to 0.4, while the range goes from 0 to 0.29 and from 0 to 0.21 for the maps of the implanting femur of GC1, and from 0 to 0.37 and from 0 to 0.23 for the maps of GC2 (Figure 20), both calculated from the *SS* and *SSavg*, respectively, a consequence of the insertion of the V3 stem made of Ti21S.

respectively, a consequence of the insertion of the V3 stem made of Ti21S. From the maps, it is verified that the *SS* adequately quantifies the strain shielding in comparison to the *SSavg*, because of the similarity between the strain maps of the intact and implanted femur when this metric is used. As the femur is mostly within the dead zone, it favors bone ingrowth, which can be supplemented with osteoconductive liners, benefiting the secondary stability, prolonging its lifespan and improving cementless fixation. Likewise, the strain maps certify that the shear planes used for postprocessing (Figure 11), which allow us to study the mechanical behavior and shielding in the lateral and medial part, are a representative sample of the mechanical response of the entire proximal femur. This plane was obtained from the Y-coordinate of the elliptical adjustment of the implantation section, and it reflects another use of the application that aids not only in design, but also in custom stem analysis and selection. The orange and purple boxes show, similar to how the traumatologist evaluates the shielding radiologically, the decrease in the color scale of the medial and lateral region, respectively, which translates into the loss of bone mass of

the femur as a natural response to the removal of the neck. This contrasts with the analysis of the scatter plots of each geometric case (Figures 16 and 19). *Materials* **2022**, *15*, x FOR PEER REVIEW 25 of 32

**Figure 20.** Equivalent strain map of the intact and implanted femur of (**A**) GC1 and (**B**) GC2. **Figure 20.** Equivalent strain map of the intact and implanted femur of (**A**) GC1 and (**B**) GC2.

From the maps, it is verified that the adequately quantifies the strain shielding in comparison to the ௩, because of the similarity between the strain maps of the intact and implanted femur when this metric is used. As the femur is mostly within the dead zone, it favors bone ingrowth, which can be supplemented with osteoconductive liners, benefiting the secondary stability, prolonging its lifespan and improving cementless fixation. Likewise, the strain maps certify that the shear planes used for postprocessing (Figure 11), which allow us to study the mechanical behavior and shielding in the lateral and medial part, are a representative sample of the mechanical response of the entire proximal femur. This plane was obtained from the Y-coordinate of the elliptical adjustment of the implantation section, and it reflects another use of the application that aids not only in design, but also in custom stem analysis and selection. The orange and purple boxes show, similar to how the traumatologist evaluates the shielding radiologically, the decrease in the color scale of the medial and lateral region, respectively, which translates into the loss of bone mass of the femur as a natural response to the removal of the neck. This contrasts with the analysis of the scatter plots of each geometric case (Figures 16 and 19). The study performed by Yan et al. [80], whose boundary conditions are similar to this research, on the shielding caused by two commercial stems—one of a conventional type and the other short calcar-loaded—concludes that the *SS* in the proximal femur caused by the conventional stem is 0.93 and that by the calcar loading stem is 0.82 approximately. Therefore, both commercial implants place the femur outside the dead zone of the bone remodeling curve, so there will be a bone resorption that, in the long-term, will cause the implant to loosen and a revision surgery will be necessary to replace it. Yamako et al. [81], using strain gauges, quantified, through the equivalent strain, that the shielding in the proximal femur caused by a conventional implant made with Ti6Al4V was 0.61, being positioned at the limit of the dead zone. The shielding resulting from the insertion of the V3 stem made of Ti21S was 0.285 and 0.073 for GC1 and GC2, respectively. Therefore, customization is beneficial in the mechanical response of the proximal femur; this is mainly due to the restoration of the parameters of the patient's anatomy (neck–shaft angle, anteversion, offset and femoral cavity) and to the selection of a material that has a modulus of elasticity close to the bone.

The study performed by Yan et al. [80], whose boundary conditions are similar to this research, on the shielding caused by two commercial stems—one of a conventional type and the other short calcar-loaded—concludes that the in the proximal femur caused by the conventional stem is 0.93 and that by the calcar loading stem is 0.82 approx-However, the precise orientation of the implant is crucial in order not to alter these parameters and consequently its biomechanics; this depends on the surgeon's expertise, but, to avoid human error in the process, technological assistance is becoming more and more common.

imately. Therefore, both commercial implants place the femur outside the dead zone of the bone remodeling curve, so there will be a bone resorption that, in the long-term, will cause the implant to loosen and a revision surgery will be necessary to replace it. Yamako et al. [81], using strain gauges, quantified, through the equivalent strain, that the shielding in the proximal femur caused by a conventional implant made with Ti6Al4V was 0.61, being positioned at the limit of the dead zone. The shielding resulting from the insertion of the V3 stem made of Ti21S was 0.285 and 0.073 for GC1 and GC2, respectively. There-Since Ti21S is an isotropic and ductile material, the Von Mises criteria were used. The V3 stem for both geometric cases, subjected to the ISO force, has an average safety factor of 6.055, which guarantees that the implant does not yield to the load. Analyzing the Von Mises stress map of V3 (Figure 21), it is observed that the area with the highest concentration of stresses is the receiving taper; this is beneficial because the stresses generate the compression of the cone walls with the articulating sphere, causing an interference fit and a cold weld between them [82].

fore, customization is beneficial in the mechanical response of the proximal femur; this is mainly due to the restoration of the parameters of the patient's anatomy (neck–shaft angle, anteversion, offset and femoral cavity) and to the selection of a material that has a modu-

*Materials* **2022**, *15*, x FOR PEER REVIEW 26 of 32

**Figure 21.** Von Mises stress map of the V3 stem. **Figure 21.** Von Mises stress map of the V3 stem.

more common.

more common.

a cold weld between them [82].

a cold weld between them [82].

To verify the implantability of the prosthesis, PLA prototypes of the V3 stem of both geometric cases were made using fused material deposition printing. With the cortical part of the osteotomy already performed, which was used in the *Implantability* (Section 2.4.3), the "round the corner" technique (Figure 6) performed by the traumatologist was imitated when inserting the stem into the canal, verifying that the implant enters normally To verify the implantability of the prosthesis, PLA prototypes of the V3 stem of both geometric cases were made using fused material deposition printing. With the cortical part of the osteotomy already performed, which was used in the *Implantability* (Section 2.4.3), the "round the corner" technique (Figure 6) performed by the traumatologist was imitated when inserting the stem into the canal, verifying that the implant enters normally (Figure 22). To verify the implantability of the prosthesis, PLA prototypes of the V3 stem of both geometric cases were made using fused material deposition printing. With the cortical part of the osteotomy already performed, which was used in the *Implantability* (Section 2.4.3), the "round the corner" technique (Figure 6) performed by the traumatologist was imitated when inserting the stem into the canal, verifying that the implant enters normally (Figure 22).

However, the precise orientation of the implant is crucial in order not to alter these parameters and consequently its biomechanics; this depends on the surgeon's expertise, but, to avoid human error in the process, technological assistance is becoming more and

However, the precise orientation of the implant is crucial in order not to alter these parameters and consequently its biomechanics; this depends on the surgeon's expertise, but, to avoid human error in the process, technological assistance is becoming more and

Since Ti21S is an isotropic and ductile material, the Von Mises criteria were used. The V3 stem for both geometric cases, subjected to the ISO force, has an average safety factor of 6.055, which guarantees that the implant does not yield to the load. Analyzing the Von Mises stress map of V3 (Figure 21), it is observed that the area with the highest concentration of stresses is the receiving taper; this is beneficial because the stresses generate the compression of the cone walls with the articulating sphere, causing an interference fit and

Since Ti21S is an isotropic and ductile material, the Von Mises criteria were used. The V3 stem for both geometric cases, subjected to the ISO force, has an average safety factor of 6.055, which guarantees that the implant does not yield to the load. Analyzing the Von Mises stress map of V3 (Figure 21), it is observed that the area with the highest concentration of stresses is the receiving taper; this is beneficial because the stresses generate the compression of the cone walls with the articulating sphere, causing an interference fit and

**Figure 22.** Imitation of the "round the corner" technique for GC1 and GC2. **Figure 22.** Imitation of the "round the corner" technique for GC1 and GC2.

### **Figure 22.** Imitation of the "round the corner" technique for GC1 and GC2. **4. Limitations and Future Proposals**
