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Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation
Journal of Cardiovascular Magnetic Resonance volume 17, Article number: 80 (2015)
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.
Fractals-irregularity and complexity in nature
The earliest formal references to fractal geometry were made by Leibniz  in the mid-1600s. Centuries later, the first fractal prototype was abstractly introduced (only in passing) by German mathematician Georg Cantor in 1883. But the word ‘fractal’ did not come into existence until at least one century later. Inspired by the Latin fractus, meaning “broken”, the term was first coined in 1975 by Benoit Mandelbrot  to describe complex patterns that were self-similar across infinite scales. A fractal object is defined as a rough, fragmented, or detailed geometric shape that can be subdivided into parts, each of which is a reduced copy or approximate copy of the whole, where their self-similarity may be exact, quasi, or statistical.
Theoretical mathematical fractals are indeed infinitely self-similar. We can generate limited practical graphical representations of them by repeating a pattern at different scales in a recursive or iterative loop or by recursion of algebraic equations. Algebraic fractals typically require thousands or millions of iterations before their fractal nature is realised, and thus are usually visualised using computer software. Not surprisingly, widespread appreciation of fractal complexity developed only after the advent of the computer in the 1980s and thanks to Mandelbrot’s work .
Natural quasi fractal objects, unlike theoretical fractals but much like graphical representations of fractals, are scale invariant across only a limited range of scales. We are surrounded by natural objects that iterate, branch or spiral, spanning a wide range of scales. Some large-scale examples in the physical world include recursing coastlines, branching tree and river networks, and spiralling galaxies (Fig. 1a) and hurricanes. Some small-scale examples in biology include the spirals of a nautilus and whorls of a seashell (Fig. 1b). Small-scale examples in the human body include the lattices of cancellous bone (Fig. 1c), neuronal dendrites, tumor growth patterns, and specifically for the cardiovascular system, branching vascular networks (Fig. 1d), endocardial trabeculae, and the quasi-fractal ordering of collagen and fibrosis in the diseased myocardium as seen by micro-histology .
In cardiovascular magnetic resonance (CMR), much of what we see, report, measure and compute in everyday clinical practice also has some quasi-fractal property and is amenable to description and quantification by fractal mathematics, generating an index of their space-filling. To date however, much more emphasis on Fourier analysis and processing of CMR data has existed. Fractal analysis of magnitude images is a more recent application—although more than 100 [3–6] publications indexed in PubMed have described fractal analysis in magnetic resonance imaging of the brain, only 4 publications exist for CMR [7–10]. Summing up this biological complexity in medical images is clinically important, to guide treatment decisions and improve disease diagnosis, but attempting to do so using traditional mathematics (perimeter estimates or area under the curve) is unsatisfactory—it will tend to either oversimplify the motif’s detail and/or vary with the iteration being interrogated (Fig. 2). In general, the fractal approach is ideal for measuring complicated image details that are beyond simple calliper measurement, and permits results from different scanners to be meaningfully compared.
By summarising some of the fundamental principles underpinning the science of deterministic fractals, and by pointing to existing tools and approaches, this paper invites CMR scientists to experiment with fractal analysis as a means of developing an alternative breed of quantitative cardiac imaging tools.
How to measure
Geometrically a fractal would exist in between our more familiar topological dimensions (D T ): between the 1st and 2nd D T , or between the 2nd and 3rd, etc. An understanding of the concept of fractal dimensionality begins therefore with at least some understanding of D T and Euclidean dimensionality (D E ) (Fig. 3). Euclidean space refers to an object’s embedding space and encompasses dimensions that we define using Cartesian coordinates (real numbers e.g., x, y and z). Figure 3 explains why some objects will have D T = D E , while others will have D T < D E . Unlike the topological and Euclidean dimensions, the fractal dimension (FD) measures the detailed self-similarity of fractals—the space-filling capacity of a set of points embedded in space or its complexity. It is related to D E and D T by Eq 1:
These definitions also apply to fractal analysis in CMR. The heart itself exists in three-dimensional (3D) space, but diagnostic images provide 2D data a large part of the time, from which we extract patterns. The pattern of a drawn endocardial contour, for example on a left ventricular short axis CMR cine slice, appears more complicated than a simple curved line so its FD will be > 1. Because it partly but not completely ‘fills’ 2D space however its FD will be < 2. Therefore the range of possible FD s for a quasi-fractal object like the endocardial contour extracted from a CMR sequence will be consistently a value between 1 and 2.
The mathematical details of a fractal analysis are generally taken care of by software, but this is typically preceded by some medical image preparation. It may be necessary to generate the needed image format (e.g., grayscale, binary or red-green-blue (RGB) data type) or to remove image complexity unrelated to the feature to be measured. For example, a short-axis cine slice may carry signal originating from the myocardium, blood-myocardial boundary, blood pool, and surrounding tissues, all of which are measurable, either separately or together. To be able to measure the quasi-fractal properties of an endocardial contour (the blood-myocardial boundary) some image transformation would be needed in order to extract its relevant pattern, in particular its binary outline. In a segmented image, derived according to a fixed thresholding rule, the meaning of each single pixel is reduced to the binary logic of existence (pixel present/foreground) and nonexistence (pixel absent/background). Typically, the FD of a binary filled object (e.g., the binary mask of the blood pool) is greater than that of its binary outlined counterpart (e.g., the edge image of the endocardial contour), and the FD of such binary images (whether filled or outlined) will be generally greater than the equivalent FD  of the original grayscale object (Fig. 4) .
Assuming the preprocessing approaches used (threshold, subtract background, dilate, trace, find edges, binarise or skeletonize, either automatically or manually) are appropriate for the type of image [13, 14], then it is reasonable to expect that the FD of the resultant region of interest (ROI) will closely approximate the real FD of the aspect of the physical object or process being investigated, at least over a range of image resolutions, and that it will encode potentially valuable biological information.
Once the ROI is extracted, the FD can be calculated using many analysis methods (Table 1). Each will compute a different type of FD but fundamentally they all measure the same property of the ROI—they are all meters of complexity. Even for a single method (e.g., box-counting) multiple algorithmic variants may exist (box-counting may use either a conventional, overlapping, folded or symmetric surface scanning approach ). The conventional procedure for box-counting (Fig. 5) rests on simple arbitrary scaling and can be applied to structures lacking strictly self-similar patterns. It works by systematically laying a series of grids of boxes of decreasing calibre onto the ROI and counting (at each level) the number of boxes that overlies pixel detail. The FD is derived from the slope of the logarithmic regression line graphing the relationship of box count and scale. The number of data points used to generate these log-log plots is related to the number of measuring steps. Theoretically, given a priori knowledge of the scaling rules, a mathematical fractal would generate data points that lie along a perfect straight line. The point of practical analysis, however, is to find the scaling rule in the first place. For anisotropic biological objects (like left ventricular endocardial contours) as well as for precisely generated fractal images analysed without knowledge of the scaling rule, the data points do not generally lie on a straight line, reflecting sampling limitations as well as limited self-similarity , thus the slope is estimated from the regression line for the log-log plot. The choice of image preparation routine and the details of the method used to gather the data for fractal analysis are important as they can either increase or decrease the correlation coefficient of the double logarithmic plot (more linear or more sigmoid fit respectively).
The FD is not the only tool available in fractal geometry—others such as lacunarity also exist that provide a different layer of information relating more to the texture of objects . Lacunarity (λ) mesures the size distribution of gaps (lacunae) in an image, providing a measure of heterogeneity . It is the counterpart to the FD but the two are non-identical (Fig. 6). If an image has few, small, and regular gaps and is translationally and rotationally invariant, it will have low λ; if it has many large and irregular gaps with notable translational and rotational variance, it will have high λ. The translational invariance (spatial heterogeneity ) that is measured by lacunarity implies that: 1) λ is highly scale-dependent, meaning an image that appears highly heterogenous at low scale may appear much more homogenous at large scale producing two very different values of λ; and 2) λ (like the related box-counting fractal analysis) may be used to study non-fractal objects. λ and the FD are usually used complementarily, but for some biomedical applications lacunarity may be preferred (e.g., quantification of trabecular bone by MR  where the widely varying pattern of emptiness between spicules is the feature of interest, Fig. 1c), and in others the FD is preferred (e.g., endocardial contours with large central emptiness and edge detail, Fig. 5).
Previous use of fractal analysis in medicine
Fractal geometry has already found effective research application in the medical imaging field across several modalities (such as plain radiography, retinal photography, ultrasonography , computed tomography, MR and nuclear MR ). It has been used to study a wide variety of processes: the complex geometries of biological cell types ; tumor growth patterns ; gene expression ; retinopathy ; cellular differentiation in space and time ; bone and dental matrix composition [24, 28]; brain matter changes  etc. Fractal methods are popular and convenient because they lend themselves to automated computer-assisted image processing providing a precise and quantitative metric. Robust measurement of biological complexity in the medical imaging field is clinically important and worth pursuing because fractal indices have been shown to permit early diagnosis of disease (in osteoporosis ), predict likelihood of malignancy (in mediastinal nodes imaged by endobronchial ultrasound ), predict outcome (of lacunar strokes on the basis of retinal vessel complexity ) and measure treatment response (to radiochemotherapy in malignant head and neck tumors ).
Utility in CMR
Pertinent to CMR, and for certain applications (e.g., myocardial trabecular quantification), there are clear advantages in using the FD: because it is less susceptible to magnification, it works on different CMR sequences, with different voxel sizes acquired on different platforms; because it is independent of the size of the ROI, it works for small as well as large hearts. There are also potential limitations. For cine imaging, loss of image detail is a particular concern due to partial volume effects at the blood-myocardial boundary in the relatively extended through-plane voxel dimension. Left ventricular cine stacks may be prone to variable spatial resolution but we have previously shown how FD is robust to small changes in slice thickness (6 mm vs. 7 mm vs. 8 mm ). Future work should explore whether the higher spatial resolution of computerised tomography provides more suitable image data for fractal analysis than does CMR, especially with respect to vascular trees and probably also myocardial trabeculae provided blood-myocardial contrast is sufficient.
Experimenting with fractal analysis of images in the CMR domain, typically involves the in-house development of scripts written for a specific programming environment (e.g., MATLAB, ImageJ , Insight Toolkit [ITK] , etc.). It may be possible to repurpose already available tools in the form of commercial and open-source fractal plugins and codes . Examples include: Fractalyse (ThèMA, F), Whinrhizo (Regent Instruments Inc.), Image Pro Plus (Media Cybernetics), FDSURFFT (in MATLAB) and Fraclac for ImageJ . Our group started off with Fraclac and then moved to an in-house MATLAB implementation. We applied fractal analysis to CMR cine data for trabecular quantification. In left ventricular noncompaction (n = 30) compared to healthy volunteers (n = 75) fractal analysis (Fig. 7) revealed FD elevation in the apical half of the left ventricle  (1.392 ± 0.010 versus 1.235 ± 0.004). When we studied patients at our centre with hypertrophic cardiomyopathy (n = 107), fractal analysis showed abnormally increased apical FD not only in overt disease, but also in sarcomere gene mutation carriers without left ventricular hypertrophy (G + LVH-, 1.249 ± 0.07) compared to controls (1.199 ± 0.05) . In a multi-centre setting high FD was further shown to predict hypertrophic cardiomyopathy sarcomere gene mutation carriage in G + LVH- (n = 73) . Applied to 2547 participants in the population-representative MESA study, fractal analysis was able to provide ethnically-appropriate normal reference ranges for left ventricular endocardial complexity .
Whether to measure endocardial complexity or any other imaging feature of interest, all novel CMR fractal tests will invariably need to satisfy the usual STAndards for the Reporting of Diagnostic accuracy studies (STARD) . To become useful clinical tools, they will need to pass the 15 developmental “check-points” . Table 2 underscores how two efforts in this field are still some way off from clinical utility (e.g., the further developed of the two is at step 11—development of normal reference values).
Nonetheless, on the broader frontier, there is reason for optimism with regard to developing useful CMR applications in the future. We think that potential, as yet untested applications could include such things as textural analysis to quantify scar in late gadolinium enhancement images; spatiotemporal analysis to track cardiac motion of cine objects; stochastic fractal models  to study nonperiodic fluctuations in physiological parameters in MR flow data; and fractal analysis in general to aid pattern recognition in pixel-wise parametric mapping.
Although the description of modern fractal analysis by Mandelbrot occurred more than 40 years ago and in spite of clinical practice bringing us face to face with multifarious fractal features daily, the CMR community is only beginning to evaluate potential applications of fractal analysis to cardiac imaging. This review reminds us of the accessibility of fractal mathematics and methods and aspires to attract more cardiac imagers to the library of efficient fractal analysis tools available, as well as invite them to innovate. A deeper fractal exploration of the human heart by CMR has the ability to teach us new facts relating to cardiac function, haemodynamics and tissue characterisation. With additional validation, software tools based on fractal analysis may ultimately prove to have clinical utility in the field of CMR.
Cardiovascular magnetic resonance
- D T :
- D E :
- FD :
Sarcomere gene mutation carriers without left ventricular hypertrophy
- λ :
Region of interest
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JCM: Higher Education Funding Council for England and the UK National Institute for Health Research, University College London, Biomedical Research Centre; GC: NIHR BRC University College London. DAB: Intramural research program, National Institutes of Health.
Academic collaboration with Circle CVI.
The authors declare that they have no competing interests.
All authors have contributed significantly to the submitted work: JCM and GC wrote the article. ALK, CTG, XG, PB, FZ, SEP, DAB and CL provided expert advice and critical review of the manuscript. All authors read and approved the final manuscript.
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Captur, G., Karperien, A.L., Li, C. et al. Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation. J Cardiovasc Magn Reson 17, 80 (2015). https://doi.org/10.1186/s12968-015-0179-0
- Cardiovascular magnetic resonance
- Image processing