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Transforming MRI volumes with Space-Filling Curves reveals Multifractal Signatures of Ageing and Dementia Progression

This study introduces Multifractal Space-filling Curve Analysis (MFSCA) to demonstrate that the spatial organization of brain structures, as measured by multifractality in MRI data, progressively deteriorates from a complex, heterogeneous state to a simpler, homogeneous one with advancing age and the progression of dementia.

Original authors: Marta Lotka, Jacek Grela, Zbigniew Drogosz, Jeremi K Ochab, Paweł Oświęcimka

Published 2026-07-30

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CC BY 4.0

Original authors: Marta Lotka, Jacek Grela, Zbigniew Drogosz, Jeremi K Ochab, Paweł Oświęcimka

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine your brain not as a static, smooth lump of gray matter, but as a bustling, chaotic city. In this city, every neuron is a building, and the connections between them are the roads, bridges, and power lines. For a long time, scientists have tried to map this city using standard tools, looking for big, obvious changes like a building collapsing or a road disappearing. But what if the real story isn't about the buildings falling down, but about the rhythm of the traffic? What if the city is still standing, but the traffic patterns have become too simple, too predictable, or too chaotic in a bad way? This is the world of fractals. Think of a fractal like a snowflake or a fern leaf: a pattern that repeats itself no matter how much you zoom in. A "monofractal" is like a perfect, repeating wallpaper pattern—very orderly, but a bit boring. A "multifractal" is like a wild, stormy ocean: it has patterns within patterns, with some areas calm and others raging, creating a rich, complex, and self-similar texture. Scientists believe that a healthy brain is a "multifractal" masterpiece, full of this rich, complex organization. As we age or develop diseases like dementia, this complex rhythm might fade, turning our brain's "city" into something simpler and less connected. Understanding this shift could help us spot brain diseases earlier and better than ever before.

This paper introduces a clever new way to listen to that brain rhythm, called Multifractal Space-Filling Curve Analysis (MFSCA). The researchers faced a tricky problem: brain scans (MRI) are 3D objects, like a giant block of cheese, but the math needed to measure this "complexity" usually works best on simple, one-dimensional lines, like a string of beads. To solve this, the team used a mathematical trick called a space-filling curve (specifically, a Hilbert curve). Imagine taking that giant 3D block of cheese and unrolling it into a single, long, winding snake without ever breaking the cheese or losing the order of the ingredients. This snake preserves the local neighborhood of every piece of cheese while turning the whole 3D volume into a 1D line. Once the brain scan is "unrolled" into this line, the team applied a sophisticated math test (Multifractal Detrended Fluctuation Analysis) to measure how complex the signal is.

The team tested this method first on computer-generated patterns. They created artificial "brain-like" structures with known levels of complexity and then tried to "scramble" them to remove the complex, non-linear parts while keeping the simple, linear parts. Their method worked perfectly: it could tell the difference between the rich, complex patterns and the scrambled, simple ones, proving it was actually measuring the "wildness" of the data and not just the basic order.

When they applied this to real MRI scans from the OASIS-1 database, the results were striking. They looked at healthy people of different ages (young, middle-aged, and elderly) and patients with different stages of cognitive decline (from very mild impairment to mild dementia). The study found that as people get older, the "richness" of their brain's signal naturally fades. The complex, multifractal patterns slowly turn into simpler, weaker patterns, almost like a vibrant symphony turning into a single, repetitive drumbeat. This "simplification" happens even more drastically in people with dementia.

Specifically, the researchers found that the Hurst exponent (a number that measures how connected the signal is over long distances) drops significantly in older adults and even lower in dementia patients. In the most impaired groups, the brain's signal looked much more like "white noise"—random and unconnected—than the organized, complex signal seen in younger, healthy brains. The study suggests that the brain's spatial organization is deteriorating, moving from a state of high complexity (multifractality) to a state of low complexity (monofractality). This transition happens both as part of normal aging and accelerates with the progression of dementia.

Interestingly, the method didn't just show that the brain gets "simpler"; it showed where this happens. The changes weren't uniform across the whole brain; they were concentrated in specific regions, particularly around the ventricles (fluid-filled spaces in the brain) and in the back of the brain. The team also used computer models to see if these complex numbers could help tell different groups apart. They found that these "multifractal fingerprints" were very good at distinguishing between young, middle-aged, and elderly people, and even better at separating healthy elderly people from those with early or mild dementia.

The authors are careful to note that while their method is powerful, it has limits. The data they used was a snapshot in time (cross-sectional), so they can't say for sure that the brain changed in a specific person over time, only that older people look different from younger people right now. They also couldn't account for every possible factor, like how much a patient moved during the scan or their genetic makeup. However, the results strongly suggest that the "complexity" of the brain's structure is a vital marker of health. By turning a 3D brain scan into a 1D line and measuring its fractal rhythm, this new method offers a fresh, transparent, and mathematically grounded way to see the invisible signs of aging and dementia, potentially helping doctors spot trouble before it becomes too late.

Technical Summary: Transforming MRI volumes with Space-Filling Curves reveals Multifractal Signatures of Ageing and Dementia Progression

Problem Statement

The detection and monitoring of Alzheimer's Disease (AD) and normal brain ageing currently face significant challenges. While biomarkers like amyloid PET and CSF assays exist, they are invasive, costly, or limited in scalability. Structural MRI is widely available but often lacks robust, routine quantification methods in clinical practice. Existing machine learning (ML) approaches for neuroimaging often suffer from a "black box" nature, lacking interpretability, and struggle with generalization due to data scarcity and diversity. Furthermore, traditional fractal analysis in neuroscience has largely been restricted to the calculation of the Fractal Dimension (FD), which captures linear dependencies but fails to quantify the complex, scale-free, and nonlinear correlations inherent in multidimensional brain data. There is a critical need for a transparent, interpretable methodology capable of analyzing multidimensional neuroimaging data to capture both linear and nonlinear organizational properties.

Methodology: Multifractal Space-Filling Curve Analysis (MFSCA)

The authors propose Multifractal Space-Filling Curve Analysis (MFSCA), a two-stage methodology designed to quantify the correlation structure of multidimensional data by transforming it into a one-dimensional time series while preserving local and long-range organizational properties.

Dimensionality Reduction via Space-Filling Curves (SFC):
- Multidimensional MRI volumes (specifically T1-weighted slices) are converted into one-dimensional signals.
- The Hilbert curve is utilized as the space-filling mechanism. This curve is chosen for its Hölder continuity, which ensures that the distance between points along the curve ( $|x-y|$ ) relates to the Euclidean distance in the embedded space ( $\|f(x)-f(y)\|$ ), thereby preserving the local spatial structure of the voxel intensities.
- The voxel intensities along the Hilbert curve traversal order form a 1D time series for each slice.
Multifractal Analysis via MFDFA:
- The resulting 1D signals are analyzed using Multifractal Detrended Fluctuation Analysis (MFDFA).
- This process calculates the generalized Hurst exponent $h(q)$ and the multifractal spectrum $f(\alpha)$ .
- Key metrics extracted for each slice include:
  - Hurst Exponent ( $H$ ): The value of $h(2)$ , indicating long-range correlations.
  - Spectrum Width ( $\Delta\alpha$ ): The difference between maximum and minimum singularity strengths ( $\alpha_{max} - \alpha_{min}$ ), quantifying the degree of multifractality and nonlinearity. A wider spectrum indicates stronger nonlinear correlations.
  - Spectrum Asymmetry ( $A$ ): Indicates whether the correlation structure is dominated by small or large fluctuations.
Statistical Framework:
- The study utilizes data from the OASIS-1 database, organizing subjects into five cohorts: Young Control (YC), Middle-Aged Control (MAC), Elderly Control (EC), Early Mild Cognitive Impairment (EMCI), and Mild Cognitive Impairment (MCI).
- A pooled hierarchical generalized additive model (GAM) is employed to analyze the per-slice profiles of $H$ , $\Delta\alpha$ , and $A$ . This model adjusts for demographic covariates (age, sex, education, eTIV) and accounts for within-subject dependence along the slice axis using an AR1 residual structure.
- Linear Discriminant Analysis (LDA) combined with Principal Component Analysis (PCA) is used to assess feature importance and cohort separability, reducing the high-dimensional feature space (approx. 777 features) to identify discriminatory metrics.

Key Contributions

Novel Methodology: The introduction of MFSCA as a generalizable framework for applying multifractal analysis to multidimensional data, overcoming the limitations of standard 2D/3D multifractal methods.
Validation on Synthetic Data: The method was validated using deterministic two-dimensional multifractal cascades. The analysis confirmed that MFSCA accurately recovers theoretical multifractal spectra and distinguishes between data with preserved linear correlations (monofractal surrogates) and those with nonlinear structures.
Comprehensive Profiling: The generation of "brain profiles" mapping multifractal metrics across anatomical axes (x, y, z) for different age and disease cohorts, allowing for spatial localization of changes.

Results

Synthetic Data Validation: MFSCA successfully recovered the theoretical multifractal spectrum width and maximum location for deterministic cascades. When applied to Fourier-surrogate data (where nonlinear correlations were removed), the method correctly identified a narrowing of the spectrum, confirming its sensitivity to nonlinear organization.
Ageing Effects (Healthy Subjects):
- Hurst Index ( $H$ ): A pervasive reduction in $H$ was observed with increasing age (comparing age 25 to 75), particularly in regions surrounding the lateral ventricles. This suggests a weakening of long-range correlations and a shift toward white-noise-like behavior in older brains.
- Multifractal Spectrum ( $\Delta\alpha$ ): The spectrum width generally decreased with age in localized regions, indicating a transition from multifractality toward monofractality (loss of heterogeneous scaling). However, the changes were spatially structured and direction-dependent, with some regions showing an increase in width.
Dementia Progression (EC vs. EMCI vs. MCI):
- Hurst Index: The MCI group exhibited significantly lower $H$ values compared to Elderly Controls (EC) and EMCI, particularly in central and posterior regions. The EMCI group was largely indistinguishable from controls, except in small segments.
- Spectrum Width: MCI subjects showed a significantly narrower spectrum width compared to EC and EMCI in specific regions, reinforcing the finding of reduced multifractal complexity.
- Transition to Monofractality: The study observed a transition from multifractality to monofractality in both the ageing process (Young vs. Elderly controls) and disease progression (EMCI vs. MCI). This indicates a deterioration of the heterogeneous spatial organization of brain structures into a more homogeneous, weakly correlated state.
Feature Importance: Dimensionality reduction (t-SNE and LDA) revealed that multifractal features (specifically asymmetry and spectrum width) contain sufficient information to discriminate between cohorts, including separating groups of similar age (e.g., Elderly vs. EMCI).

Significance and Claims

The paper claims that the spatial organization of brain structures, as measured by the degree of multifractality, progressively weakens with both age and the development of dementia. The primary significance of the work lies in demonstrating that:

Multifractality as a Marker: The multifractality of MRI data serves as a sensitive marker for structural brain changes, capturing key aspects of brain organization that linear methods or simple fractal dimensions might miss.
Interpretability: Unlike "black box" ML models, MFSCA provides a theoretically grounded, transparent framework where the physical meaning of the metrics (correlation strength, nonlinearity) is clear.
Methodological Utility: The proposed MFSCA framework effectively bridges the gap between multidimensional neuroimaging data and advanced multifractal analysis, offering a robust tool for investigating complex systems in neuroscience.

The authors remain modest regarding clinical deployment, noting limitations such as the cross-sectional nature of the OASIS-1 dataset (preventing distinction between birth cohort and ageing effects), the lack of APOE $\epsilon$ 4 genotype data, and potential motion artifacts in older subjects. They position the findings as a demonstration of the method's utility and a step toward developing simpler, routine, and more robust MRI analysis methods for AD prognosis.