# Fractal Features of Muscle to Quantify Fatty Infiltration in Aging and Pathology
## 2 Materials and Methods
### Image Processing and Analysis
#### 2.2.2 Computation of Index \(\rho\) (Lean/Fat Ratio)
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Subjects** & **Young** & **Pre-Menopause** & **Menopause** & **Age-Matched** & **Osteoporosis** & \(p\) **Value 1** \\ \hline Lean mass 2 & 8586 \(\pm\) 1778 & 7889 \(\pm\) 1002 & 7277 \(\pm\) 1638 & 6581 \(\pm\) 1024 & 6890 \(\pm\) 1402 & 0.286 \\ Fatty mass 2 & 2234 \(\pm\) 756 & 2444 \(\pm\) 732 & 2894 \(\pm\) 936 & 2813 \(\pm\) 1041 & 3627 \(\pm\) 1114 & 0.046 \\ Lean/Fat ratio \(\rho\) & 4.21 \(\pm\) 1.33 & 3.57 \(\pm\) 0.68 & 2.70 \(\pm\) 0.76 & 2.61 \(\pm\) 0.89 & 2.08 \(\pm\) 0.67 & 0.057 \\ \hline Lacunarity \(\alpha\) & 0.900 \(\pm\) 0.403 & 0.572 \(\pm\) 0.350 & 0.639 \(\pm\) 0.303 & 0.672 \(\pm\) 0.316 & 0.639 \(\pm\) 0.420 & 0.420 \\ Lacunarity \(\beta\) & 0.062 \(\pm\) 0.039 & 0.113 \(\pm\) 0.040 & 0.101 \(\pm\) 0.044 & 0.092 \(\pm\) 0.041 & 0.148 \(\pm\) 0.062 & 0.012 \\ \hline Succolarity \(\mu_{\uparrow}\) & 0.203 \(\pm\) 0.075 & 0.243 \(\pm\) 0.047 & 0.252 \(\pm\) 0.070 & 0.253 \(\pm\) 0.084 & 0.264 \(\pm\) 0.077 & 0.371 \\ Succolarity \(\mu_{\rightarrow}\) & 0.240 \(\pm\) 0.085 & 0.288 \(\pm\) 0.050 & 0.308 \(\pm\) 0.070 & 0.298 \(\pm\) 0.077 & 0.298 \(\pm\) 0.080 & 0.499 \\ Succolarity \(\mu_{\downarrow}\) & 0.243 \(\pm\) 0.096 & 0.275 \(\pm\) 0.042 & 0.276 \(\pm\) 0.072 & 0.270 \(\pm\) 0.086 & 0.283 \(\pm\) 0.086 & 0.361 \\ Succolarity \(\mu_{\leftarrow}\) & 0.218 \(\pm\) 0.080 & 0.258 \(\pm\) 0.042 & 0.267 \(\pm\) 0.071 & 0.266 \(\pm\) 0.084 & 0.263 \(\pm\) 0.079 & 0.470 \\ \hline \hline \end{tabular} Results are mean \(\pm\) standard deviation; 1 \(p\) values calculated by one-tail \(t\)-test for \(p\leq 0.05\) to compare differences between osteoporosis and age-matched control groups; 2 lean and fatty mass related to lean muscle area (LeanCSA); \(\alpha\) and \(\beta\): fractal parameters from our hyperbola model method where \(\alpha\) correlates with fractal dimension and \(\beta\) measures lacunarity, \(\gamma\): the third coefficient from our hyperbola model method representing the translation term on x axis assumed a mean value \(\pm\) standard deviation in the whole sample equal to 0.9818 \(\pm\) 0.0370; \(\mu_{\downarrow}\), \(\mu_{\leftarrow}\), \(\mu_{\uparrow}\), \(\mu_{\rightarrow}\), \(\mu_{\rightarrow}\) are the succolarity indices for the directions considered.

\end{table} 
Table 3: Fatty infiltration of paraspinal lean muscle mass. Figure 5: Lacunarity of lean muscle area to estimate muscle fatty infiltration. (**a**) Age-related increasing trend of lacunarity parameter \(\beta\) in the whole sample; (**b**) age-related distribution of lacunarity parameter \(\beta\) in osteoporoptic patient (black circle) vs. age-matched controls (gray circle) shows similar increasing trend but with values higher in osteoporoptic patients than in age-matched controls. ## 4 Discussion

In this study, we present a new method to quantify muscle fatty infiltration, an important index of muscle wasting usable in the diagnosis of sarcopenia. The method was set up taking into consideration the complexity of human beings and the pseudo-fractality of many biomedical structures and functions. It is worth noting that the mathematics of fractals, introduced to describe fractals in nature, has been developed on ideal fractals, characterized by self-similarity, and invariance of scale and fractal dimension was introduced to characterize a fractal. When we deal with natural objects, we have to face the problem of statistical similarity in a limited range of scale. Lacunarity and succolarity were introduced to better characterize fractal objects with the same fractal dimension but with a very different appearance. Lacunarity and succolarity analyses were then introduced for a better featuring of fractals. These fractal analyses were later introduced with a more general application as they represent texture analyses usable to describe complex patterns with fractal, multifractal, or no fractal properties, thus overcoming the limits of the fractal dimension applied to natural objects.