# 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)
From functions \(S_{d}\), \(d\in\{\downarrow,\leftarrow,\uparrow,\rightarrow\}\), the succolarity indices \(\mu_{\downarrow}\), \(\mu_{\leftarrow}\), \(\mu_{\uparrow}\), \(\mu_{\rightarrow}\) are computed as follows:

\[\mu_{d}=max\{S_{d}(b),\ \ \ b\in[b_{min},\,b_{max}]\},\ \ \ \ \ \ d\in\{\downarrow, \leftarrow,\uparrow,\rightarrow\}. \tag{8}\]

## 3 Results

### Mass Composition of Paraspinal Muscles

The results on paraspinal muscle composition confirm an age-related decreasing trend for lean mass and an increasing trend for fatty mass. These results are consistent with those reported in the literature for different muscles [19; 20; 21; 22; 23; 24]. The Lean/Fat ratio results of TotCSA show an age-related decreasing trend. In osteoporosis, the Lean/Fat ratio lower than that of the age-matched control group reflects the higher fatty mass in disease than in aging as both show similar amounts of lean mass. Table 2 summarizes classic measurement results on paraspinal muscle composition (area in pixel, percentage, and ratio) expressed as mean \(\pm\) SD of the average of the right and left paraspinal muscle groups. Figure 3 shows that osteoporotic subjects, when compared with age-matched controls, have an opposite trend of Lean/Fat ratio that is in contradiction with muscle composition, expressed as mean \(\pm\) SD (Table 2). In fact, despite similar lean mass values, osteoporotic subjects show a higher amount of fatty mass when compared with age-matched controls (Figure 4). This would have to account for a more marked decreasing trend of Lean/Fat ratio. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Subjects** & **TotCSA** & **LeanCSA \({}^{*}\)** & **FatCSA \({}^{\ddagger}\)** & **Lean/Fat Ratio** \\ \hline All & 11,114 \(\pm\) 1858 & 8507 \(\pm\) 2608 (77\%) \({}^{*}\) & 1925 \(\pm\) 1153 (23\%) & 4.40 \(\pm\) 3.53 \\ Young & 11,222 \(\pm\) 1939 & 9364 \(\pm\) 1929 (84\%) & 1858 \(\pm\) 1015 (16\%) & 7.17 \(\pm\) 5.73 \\ Pre-menopause & 10,652 \(\pm\) 1686 & 8481 \(\pm\) 1105 (80\%) & 2171 \(\pm\) 838 (20\%) & 4.34 \(\pm\) 1.35 \\ Menopause & 11,062 \(\pm\) 2043 & 8332 \(\pm\) 1609 (75\%) & 2730 \(\pm\) 1226 (25\%) & 3.50 \(\pm\) 1.54 \\ Age-matched \({}^{1}\) & 10,356 \(\pm\) 1215 & 7714 \(\pm\) 1364 (75\%) & 2642 \(\pm\) 1018 (25\%) & 3.51 \(\pm\) 1.87 \\ Osteoporosis \({}^{2}\) & 11,517 \(\pm\) 1865 & 7837 \(\pm\) 1449 (68\%) & 3680 \(\pm\) 1315 (32\%) & 2.52 \(\pm\) 1.34 \\ \hline \hline \end{tabular} Values are mean \(\pm\) standard deviation expressed in pixel; \({}^{*}\) percentage values within round brackets are % of LeanCSA referred to TotCSA; \({}^{*}\) percentage values within round brackets are % of FatCSA referred to TotCSA; CSA: cross-sectional area; \({}^{1}\) age-matched control group for comparison with Osteoporosis group; \({}^{2}\) osteoporosis: bone mineral density T-score \(\leq-2.5\) as defined by the World Health Organization.

\end{table} 
Table 2: Paraspinal muscle composition by classic estimates. Figure 3: Lean/Fat ratio of the whole muscle mass area in aging and disease. (**a**) Age-related decreasing trend of Lean/Fat ratio in the whole sample analyzed; (**b**) comparison of Lean/Fat ratio between osteoporotic patients (black) and age-matched control group (gray) showing an increasing trend (dashed line) for osteoporotic group vs. a decreasing trend (dotted line) for age-matched controls. ### Fractal Features of Paraspinal Muscle

To overcome this discrepancy, another index, lacunarity parameter \(\beta\), was introduced to analyze the fatty infiltration of lean mass. Lacunarity analysis, in fact, provides a holistic estimate of muscle composition comprehensive of changes occurring at both lean and fatty muscle mass. In addition, to better characterize the fatty infiltration of the muscle, succolarity was also considered. As comparable results of muscle composition were obtained for both right and left paraspinal muscles (Figure 4), additional analyses were performed only on the right paraspinal muscle group.