# Fractal Features of Muscle to Quantify Fatty Infiltration in Aging and Pathology
###### Abstract
Numerous and different techniques have been proposed to assess muscle fatty infiltration; nevertheless, there is a lack of methods for its quantitative estimate [25; 26; 27; 28; 29]. The aim of this study was to set up a computerized method for quantitative characterization of fatty infiltration of the contractile muscle. Considering that the term 'fatty infiltration' has been using with a non-univocal meaning [30], it has to be stressed that the term 'fatty infiltration' in this study identifies inter- and intra-non-contractile tissue (mainly fat and connective tissue) that substitutes myofibrillar tissue portions within the lean muscle area (contractile muscle). Image texture analysis is a key approach to understand the relationships between the microstructures of the tissues under consideration and their properties. Different approaches can be used to extract image texture features [31] and the statistical approach based on the co-occurrence matrix [32] is probably the most common approach. Paradigms such as complexity, chaos, and fractality, introduced in the field of biomedicine and gerontology in particular, are sources of new approaches and tools to their study. The proposed method, set up considering the complexity of living beings and the fractality of many structures and functions of humans, can be seen as a model-based approach; in fact, it aims to estimate the fractal features of LeanCSA (lean muscle cross-sectional area) in MR images of paraspinal muscles. In particular, the hyperbola-based method was used for lacunarity texture analysis; it provides three parameters (\(\alpha\), \(\beta\), \(\gamma\)), for which \(\alpha\) correlates with the fractal dimension and \(\beta\) quantifies the lacunarity of the set, while \(\gamma\) represents the translation term on x axis [33; 34; 35; 36]. [PERSON] introduced the term 'lacunarity' (from the Latin _lacuna_, which means 'gap or hole') to characterize fractal objects with the same fractal dimension but differently appearing [37]. Later, lacunarity analysis was also introduced as a more general texture analysis method to describe complex patterns with or without fractal properties [38]. Succolarity, the other fractal-derived feature, was also considered to complete the characterization of muscle fatty infiltration. Succolarity (from the Latin _sucoclare_\(<sub+\)_colare_, which means 'to filter, strain, purify' and is associated with _perolare_, which means 'to strain or filter through, to percolate') was introduced by [PERSON] [39] to discriminate fractals with the same lacunarity. Succolarity quantifies the capacity of a hypothetic fluid to percolate in a given direction such that it can estimate the connectivity and the intercommunication in the structure of a set [40]. Parameter \(\mu\) for the four main directions was introduced to quantify succolarity by using a revised version of the method proposed to compute succolarity [41]. While fractal dimension and lacunarity are widely used, succolarity was not considered for a long time after its introduction in fractal geometry. More recently, the concept of succolarity has been revisited and quantification has been also proposed [42]. In the following, with the misuse of the terminology, we refer to such parameters as fractal features for the sake of brevity as they were born to better characterize fractal objects even if, later, they were also introduced as more general texture analyses to characterize complex objects with or without fractal properties. In this study, fractal-derived texture analysis was applied to lumbar paraspinal muscle MRI axial images acquired by the multislice spin-echo technique from subjects of different age and physio-pathological status to verify the potential of fractal parameters as new indices of muscle wasting in aging (sarcopeniaa) and age-related pathology (osteoporosis). Classic indices of muscle mass composition were also considered, together with a new one, namely the Lean/Fat ratio.