Abstract
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.
Highlights
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
Left ventricular cine stacks may be prone to variable spatial resolution but we have previously shown how fractal dimension (FD) is robust to small changes in slice thickness (6 mm vs. 7 mm vs. 8 mm [10])
When we studied patients at our centre with hypertrophic cardiomyopathy (n = 107), fractal analysis showed abnormally increased apical FD in overt disease, and in sarcomere gene mutation carriers without left ventricular hypertrophy (G + LVH, 1.249 ± 0.07) compared to controls (1.199 ± 0.05) [9]
Summary
Achieved – method first implemented in Java [8] and in MATLAB [9] to improve computational efficiency; many segmentation algorithms tested before choosing a region-based level-set function [40]. Achieved – fractal dimension used as an index of pulmonary perfusion heterogeneity; image preparation included a coil inhomogeneity correction. 2. Comparison with gold-standard or tissue biopsy (animal models and human biopsy material). Achieved – validated against episcopic mouse embryo Part achieved – validated using 3 MR reference datasets and using synthetically constructed phantoms applied to each patient’s chest phantoms with well-known FD: 1) regular geometrical objects (plane, cube surface, sphere surface) and 2) ideal monofractal signals (4th, 5th and 6th iteration of the Sierpinski carpet or 9th, 10th and 11th iteration of the Sierpinski gasket)
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