Fractal topology of gene promoter networks at phase transitions

Novel Fractal-Based Sub-RPE Compartment OCT Radiomics Biomarkers Are Associated With Subfoveal Geographic Atrophy in Dry AMD

OBJECTIVE

The purpose of this study was to quantitatively characterize the shape of the sub-retinal pigment epithelium (sub-RPE, i.e., space bounded by RPE and Bruch's membrane) compartment on SD-OCT using fractal dimension (FD) features and evaluate their impact on risk of subfoveal geographic atrophy (sfGA) progression.

METHODS

This was an IRB-approved retrospective study of 137 subjects with dry age-related macular degeneration (AMD) with subfoveal GA. Based on sfGA status at year five, eyes were categorized as "Progressors" and "Non-progressors". FD analysis allows quantification of the degree of shape complexity and architectural disorder associated with a structure. To characterize the structural irregularities along the sub-RPE surface between the two groups of patients, a total of 15 shape descriptors of FD were extracted from the sub-RPE compartment of baseline OCT scans. The top four features were identified using minimum Redundancy maximum Relevance (mRmR) feature selection method and evaluated with Random Forest (RF) classifier using three-fold cross validation from the training set (N = 90). Classifier performance was subsequently validated on the independent test set (N = 47).

RESULTS

Using the top four FD features, a RF classifier yielded an AUC of 0.85 on the independent test set. Mean fractal entropy (p-value = 4.8e-05) was identified as the most significant biomarker; higher values of entropy being associated with greater shape disorder and risk for sfGA progression.

CONCLUSIONS

FD assessment holds promise for identifying high-risk eyes for GA progression.

SIGNIFICANCE

With further validation, FD features could be potentially used for clinical trial enrichment and assessments for therapeutic response in dry AMD patients.

Prime numbers and random walks in a square grid

In recent years, computer simulations have played a fundamental role in unveiling some of the most intriguing features of prime numbers. In this paper, we define an algorithm for a deterministic walk through a two-dimensional grid, which we refer to as a prime walk. The walk is constructed from a sequence of steps dictated by and dependent on the sequence of the last digits of the primes. Despite the apparent randomness of this generating sequence, the resulting structure-in both two and three dimensions-created by the algorithm presents remarkable properties and regularities in its pattern, which we proceed to analyze in detail.

Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation

Many of the structures and parameters that are detected, measured and reported in cardiovascular magnetic resonance (CMR) have at least some properties that are fractal, meaning complex and self-similar at different scales. To date however, there has been little use of fractal geometry in CMR; by comparison, many more applications of fractal analysis have been published in MR imaging of the brain.This review explains the fundamental principles of fractal geometry, places the fractal dimension into a meaningful context within the realms of Euclidean and topological space, and defines its role in digital image processing. It summarises the basic mathematics, highlights strengths and potential limitations of its application to biomedical imaging, shows key current examples and suggests a simple route for its successful clinical implementation by the CMR community.By simplifying some of the more abstract concepts of deterministic fractals, this review invites CMR scientists (clinicians, technologists, physicists) to experiment with fractal analysis as a means of developing the next generation of intelligent quantitative cardiac imaging tools.

Dynamics and processing in finite self-similar networks

A common feature of biological networks is the geometrical property of self-similarity. Molecular regulatory networks through to circulatory systems, nervous systems, social systems and ecological trophic networks show self-similar connectivity at multiple scales. We analyse the relationship between topology and signalling in contrasting classes of such topologies. We find that networks differ in their ability to contain or propagate signals between arbitrary nodes in a network depending on whether they possess branching or loop-like features. Networks also differ in how they respond to noise, such that one allows for greater integration at high noise, and this performance is reversed at low noise. Surprisingly, small-world topologies, with diameters logarithmic in system size, have slower dynamical time scales, and may be less integrated (more modular) than networks with longer path lengths. All of these phenomena are essentially mesoscopic, vanishing in the infinite limit but producing strong effects at sizes and time scales relevant to biology.