Multifractal Signatures of Ageing and Dementia Development: A Multifractal Space-Filling Curve Analysis
Multifractality is an effective formalism for quantifying the nonlinear, scale-free properties of complex data. In this study, we propose a novel and efficient methodology, termed Multifractal Space-filling Curve Analysis (MFSCA), for quantifying the correlation structure of multidimensional data. Within this framework, the original multidimensional data - while preserving both local and long-range organisational properties - are projected onto a one-dimensional representation using a fractal space-filling curve. The resulting one-dimensional signal is then analysed using multifractal algorithms. We demonstrate the utility of the method using both artificially generated multifractal structures and real data. In particular, we apply MFSCA to analyse magnetic resonance imaging (MRI) data from Alzheimer patients at different stages of dementia. Based on the results, we estimate the multifractal profiles of the brain for healthy subjects of different ages as well as for dementia patients. The analysis reveals that the spatial organization of brain structures, as measured by the degree of multifractality, progressively weakens with age and the development of dementia. A transition from multifractality to monofractality is observed both in control groups, when comparing the Young Control and Elderly Control groups, and among dementia subjects of similar age but at different stages of the disease, namely early dementia and mild cognitive impairment. Thus, from the perspective of multiscaling properties, the heterogeneous characteristics of spatial brain organization deteriorate under worsening conditions, leading to a homogeneous and weakly correlated structure. These findings not only effectively capture key aspects of brain organisation, but also demonstrate that the multifractality of MRI data can serve as a marker of structural brain changes.