# 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)
In particular, parameter \(\rho\) (Lean/Fat ratio) and fractal features, parameter \(\beta\) (Lacunarity) and parameters \(\mu\) (Succolarity) were used to quantify the fatty infiltration of lean muscle area, which are also potentially usable as indices of muscle contractile function. Table 3 summarizes the results related to the three new indices proposed to quantify fatty infiltration. The results from the fractal analysis of right paraspinal muscles show that lacunarity parameter \(\beta\), related to fatty infiltration, increases with age and physio-pathological status. In particular, osteoporosis and age-matched control groups are statistically significantly different (\(p\) = 0.012), with \(\beta\) values higher in osteoporosis than in 'healthy' aging. These results are consistent with fatty mass levels found higher in osteoporosis than in age-matched controls in spite of comparable lean mass levels. Figure 5 shows the age-related distribution of fatty infiltration expressed as lacunarity (parameter \(\beta\)) in the whole sample and in osteoporosis compared to the age-matched control group. Lacunarity parameter \(\alpha\) from our hyperbola model function-based lacunarity texture analysis, by correlating with fractal dimension, provides an estimate of the complexity of the lean muscle structure. Lacunarity parameter \(\alpha\) values (Table 3) are higher in the young group and statistically significantly differ from the others (\(p\) = 0.007, 0.017, 0.032 respectively vs. pre-menopause, menopause, and osteoporosis groups). Osteoporosis and age-matched groups show similar values of parameter \(\alpha\) (\(p\) = 0.420); nevertheless, they can be separated by lacunarity parameter \(\beta\) (\(p\) = 0.012). Figure 4: Composition of paraspinal muscles in aging and osteoporosis. (**a**) Lean mass (black) and fatty mass (gray) of paraspinal muscles, represented as mean values \(\pm\) SEM (standard error of mean) of right (R), left (L), and both (R + L) muscle groups, show comparable values between R and L groups; (**b**) Lean/Fat ratio of paraspinal muscles represented as mean value of right (R), left (L), and whole (R + L) muscle area. Figure 6 compares the fractal features used to measure lean muscle fatty infiltration (lacunarity parameter \(\beta\) and succolarity parameter \(\mu\) for the four directions considered) in the groups of different age and physio-pathological status. Lacunarity can discriminate between osteoporosis and age-matched control groups (\(p\) = 0.012), while succolarity does not statistically change between aging and disease. It is worth noting that the groups showing similar lacunarity values (young, pre-menopause, and menopause) can be further discriminated by succolarity that shows increasing values of parameters \(\mu\) from young to pre-menopause to menopause. In particular, the young group statistically significantly differs from both pre-menopause and menopause groups when the succolarity \(\mu_{\uparrow}\), \(\mu_{\rightarrow}\), and \(\mu_{\leftarrow}\) values are considered (respectively, \(p\) values equal to 0.045, 0.035, and 0.049 in young vs. pre-menopause; \(p\) values equal to 0.043, 0.014, and 0.049 in young vs. menopause groups).