**3. Results**

The weighted Spearman correlation coefficients between bone density and the predictors proposed here are shown in Table 1 for the constitutive model *AnisoH*. As stated before, three different values were used for the grey level threshold (GLT) used in Equation (4), thus, leading to three different FE models (see Figure 2). The correlation coefficients are given in the three cases for comparison.

The fact that the Spearman correlation coefficient is non-parametric makes it more appropriate to evaluate the correlation between density and the variables, as it implies no assumption on the type of relationship. It simply establishes if there is a concordance between those points having the highest density and those having the highest values of a certain predictor.

**Table 1.** Weighted Spearman correlation coefficients obtained with the constitutive model, *AnisoH*, using C3D4 elements and for different values of GLT: for the maximum variable throughout the cycle, R*M S* , and for the amplitude throughout the cycle, R*A S* . The predictors analysed are: the maximum principal stress (*σ*1), the minimum principal stress (*σ*3), the absolute maximum principal stress (AMP*σ*), the corresponding strains (*<sup>ε</sup>*1, *ε*3 and AMP*ε*), the strain energy density (SED or mechanical stimulus at the continuum level Ψ), the mechanical stimulus at the tissue level ( Ψ*t*), the maximum tensile and compressive stresses (*<sup>σ</sup>t* and *σc*, respectively), the fluctuation of stresses (*<sup>σ</sup>f* ), the von Mises and Tresca stresses, the hydrostatic stress (*<sup>σ</sup>o*) and the volumetric strain (*<sup>ε</sup>o*). Values higher than 0.9 are highlighted in boldface; negative values are in red.


It can be seen that most of the stress magnitudes are highly correlated with the density except for the peak of the minimum principal stress, *σ M* 3 , for obvious reasons, as the sign of *σ*3 (usually negative) is considered in the calculus of this peak. Therefore, *σ M* 3 usually corresponds to the lowest absolute value throughout the cycle. The amplitude, *σ<sup>A</sup>* 3 , is better correlated with the density as it usually measures the range of the compressive stress.

The strain magnitudes are poorly correlated with the density, in some cases with negative coefficients. SED (or Ψ) is only moderately correlated with the density and not correlated at all if it is corrected to account for the porosity ( Ψ*t*).

Regarding the influence of GLT, it can be seen that though the values of R are different, the same trend is observed in the three cases. In fact, if we ordered the predictors based on R, the same order would result for the three GLT.

The weighted Pearson correlation coefficients are shown in Table 2 for the constitutive model, *AnisoH*. Only some of the variables previously analysed in Table 1—those that are considered more interesting—are studied here; in particular, some of the stress variables that had a higher Spearman coefficient together with the SED at the continuum and at the tissue level for n = 1 and n = 2 (see Equation (34)). The Pearson coefficients are parametric and presuppose a certain relationship between the variables being correlated. Thus, we have tried linear, quadratic and power functions (see Appendix A for details).

Compared to the Spearman, the Pearson correlations have worsened notably as we are forcing them to fit a certain function which is probably not the most appropriate to relate the density with the predictor. Among the three types of functions tested, the quadratic is slightly better, followed by the power and the linear function. The low correlation between SED and density stands out—something that does not improve in the case of SED at the tissue level—for which even negative correlation coefficients were obtained, as in the case of the Spearman coefficients.

**Table 2.** Weighted Pearson correlation coefficients obtained with the constitutive model, *AnisoH*, using C3D4 elements and for different values of GLT: for the maximum variable throughout the cycle, R*M P* , and for the amplitude throughout the cycle, R*A P* . The predictors analysed in this case are: the absolute maximum principal stress (AMP*σ*), the maximum tensile and compressive stresses (*<sup>σ</sup>t* and *σc*, respectively), the fluctuation of stresses (*<sup>σ</sup>f* ) and the mechanical stimulus at the continuum level ( Ψ) and at the tissue level ( Ψ*t*) for two values of *n*. Values higher than 0.8 are highlighted in boldface; negative values are in red.


Tables 3 and 4 compare, respectively, the Spearman and Pearson coefficients obtained using the three constitutive models analysed in this work: *AnisoH*, *IsoH* and *IsoJ*. As indicated above, the effect of GLT was not important, and hence, only one case (GLT = 80)

was studied. It can be noted that the effect of the constitutive model is negligible on the Spearman correlations and small on the Pearson correlations, at least for the three cases tested here. The biggest difference is obtained for the power fit between the *IsoH* and *IsoJ* models, which, in turn, follow a different power correlation between the density and the Young's modulus, but is not greater than 0.03.

**Table 3.** Weighted Spearman correlation coefficients obtained for GLT = 80, constitutive models *AnisoH*, *IsoH* and *IsoJ* and using C3D4 elements: for the maximum variable throughout the cycle, R*MS* , and for the amplitude throughout the cycle, R*AS* . The predictors analysed are: the maximum principal stress (*σ*1), the minimum principal stress (*σ*3), the absolute maximum principal stress (AMP*σ*), the corresponding strains (*<sup>ε</sup>*1, *ε*3 and AMP*ε*), the strain energy density (SED or mechanical stimulus at the continuum level Ψ), the mechanical stimulus at the tissue level (Ψ*t*), the maximum tensile and compressive stresses (*<sup>σ</sup>t* and *σc*, respectively), the fluctuation of stresses (*<sup>σ</sup>f* ), the von Mises and Tresca stresses, the hydrostatic stress (*<sup>σ</sup>o*) and the volumetric strain (*<sup>ε</sup>o*). Values higher than 0.9 are highlighted in boldface; negative values are in red.


We have also analysed the spatial distribution of the correlations, in particular of the Pearson coefficients (power fit), by assessing separately the correlations for the elements of the proximal, distal and diaphyseal thirds (see Table 5). The aim of this comparison was to investigate if there are regions of the femur where the density is better to the predictors. Given the limited influence of GLT and the constitutive model, we only show the case GLT = 80 and *IsoH*. Besides, we only compare some of the variables that show a higher correlation (*<sup>σ</sup>f* , AMP*σ*) and only the coefficients for the maximum variable throughout the cycle, R*MS* . The other stress variables follow the same trend, as well as R*AS* . The comparison of the strain variables is meaningless since they are not correlated with density, as shown previously. It can be noted that the correlation coefficients are high in the diaphysis, significantly worse in the proximal and especially worse in the distal third, influenced by the simplified way the joint reaction forces were modelled. They were applied as concentrated nodal forces, as explained in Section 2.7, rather than as a load distributed over the articular surface, as it actually occurs. This simplification affects the stresses near the articular region and, therefore, the correlations. The hip joint force can be more plausibly applied as a concentrated nodal force since the pressure on that joint spans a narrower region than that on the knee joint. Probably, this makes the correlations be slightly better in the proximal third than in the distal one.

**Table 4.** Weighted Pearson correlation coefficients obtained for GLT = 80, constitutive models *AnisoH, IsoH* and *IsoJ* and using C3D4 elements: for the maximum variable throughout the cycle, R*MP* , and for the amplitude throughout the cycle, R*AP* . The predictors analysed in this case are: the absolute maximum principal stress (AMP*σ*), the maximum tensile and compressive stresses (*<sup>σ</sup>t* and *σc*, respectively), the fluctuation of stresses (*<sup>σ</sup>f* ), the mechanical stimulus at the continuum level (Ψ) and at the tissue level (Ψ*t*) for two values of *n*. Values higher than 0.8 are highlighted in boldface; negative values are in red.


**Table 5.** Weighted Pearson correlation coefficients (power fit) obtained for the maximum variable throughout the cycle, R*MS* , for GLT = 80, constitutive model *IsoH* and using C3D4 elements. The predictors analysed are: the fluctuation of stresses (*<sup>σ</sup>f* ) and the absolute maximum principal stress (AMP*σ*). Values higher than 0.8 are highlighted in boldface.


The influence of the mesh (C3D4 vs. C3D10) was analysed by comparing the correlation coefficients in Table 6; in particular, the Spearman and the power Pearson coefficients in the case GLT = 80 and using the constitutive model *IsoH*. The Pearson coefficients improved moderately with the use of quadratic elements (C3D10), but only for the good predictors, i.e., those variables that are highly correlated with density. The rest of variables, such as Ψ and the strain magnitudes (not shown), did not improve their correlations. The other types of fit (linear and quadratic) also improved with C3D10, although to a lesser extent. It is noteworthy that the Spearman coefficients were almost identical in both meshes.

**Table 6.** Influence of the FE mesh (C3D4 vs. C3D10) on the weighted Spearman and Pearson (power) correlation coefficients obtained for GLT = 80 and constitutive model *IsoH*: for the maximum variable throughout the cycle, R*MP* , and for the amplitude throughout the cycle, R*AP* . The same predictors analysed for Spearman and Pearson coefficients are compared here. Values higher than 0.9 (in Spearman) or 0.8 (in Pearson) are highlighted in boldface; negative values are in red.

