# 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)
Given this premise and less strictly mathematically speaking, we show that fractal analysis can characterize muscle wasting better than classic methods. In particular, we found that lacunarity, as a tool to estimate the fatty infiltration of paraspinal muscles, is able to distinguish between aging and age-dependent disease (osteoporosis). In fact, fatty infiltration, as estimated by lacunarity parameter \(\beta\), from our hyperbola-based method [34; 36], increases with aging and it is statistically higher in osteoporotic patients than in healthy individuals with similar age. It is worth noting that, in spite of the age-related increasing trend of lacunarity parameter \(\beta\), lacunarity analysis fails to separate the other age/physio-pathological groups: young, pre-menopause, and menopause. However, succolarity analysis is able to discriminate among these three groups showing a similar lacunarity; parameter \(\mu\), used to quantify succinolarity of paraspinal muscle in the four directions, is statistically significantly different among the three groups considered. These results are consistent with the use of succolarity to better characterize fractals showing the same lacunarity [39]. Figure 6: Fractal features of lean muscle area. Lacunarity and succolarity as indices of muscle fatty infiltration: lacunarity (black) discriminates between osteoporotic patients and age-matched controls but not among the other age and physio-pathological groups; succolarity (gray scale from dark to light gray for the four directions, \(\uparrow\), \(\rightarrow\), \(\downarrow\), \(\leftarrow\), respectively) shows increasing values from young to pre-menopause to menopause, the three age/physio-pathological groups showing similar lacunarity values. Data are represented as mean \(\pm\) SEM. The need to consider new approaches to characterize muscle wasting with aging and pathology also stems from the lack of consistency of results on muscle composition based on classic measurements [44; 45]. Our results on paraspinal muscle composition by classic measurements 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; 46]. The high degree of inter-individual variability observed, however, suggests introducing an alternative index able to correlate changes, positive and/or negative, occurring at the levels of both lean and fatty muscle mass. In fact, both inter- and intra-lean muscle infiltration of fatty tissue contribute to the reduction in contractile muscle responsible for altered muscle strength. Mechanical consequences on muscle function due to fat infiltration are not completely clear yet; however, it has been hypothesized that fat physically limits the strength of the muscle by interfering with its natural design [47]. It is known that mechanical stimulation induced by adequate training can favor muscle function preservation by reducing the accumulation of fat and adipose cells in the skeletal muscle [48]. In this study, we propose parameter \(\rho\), an index representative of the ratio between lean and fatty mass in the whole muscle area (TotCSA) and in the muscle contractile area (LeanCSA), with the latter being used to obtain an index to provide an estimate of fatty infiltration of contractile muscle from classic measurements and, therefore, being better comparable with the new proposed fractal measurements. The Lean/Fat ratio results from TotCSA show that parameter \(\rho\) is lower in the osteoporosis group than in the age-matched control one. This is consistent with a higher fatty mass in osteoporotic patients than in the healthy aging control group in spite of similar lean muscle mass in the two groups. The results on parameter \(\rho\) from LeanCSA, as an estimate of fatty infiltration of contractile muscle, confirm higher amounts of fatty tissue in LeanCSA of osteoporotic patients than that of age-matched controls, showing similar amounts of lean tissue.