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Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation.

Captur G, Karperien AL, Li C, Zemrak F, Tobon-Gomez C, Gao X, Bluemke DA, Elliott PM, Petersen SE, Moon JC - J Cardiovasc Magn Reson (2015)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: UCL Institute of Cardiovascular Science, University College London, Gower Street, London, WC1E 6BT, UK. gabriella.captur.11@ucl.ac.uk.

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.

No MeSH data available.


Related in: MedlinePlus

a The first 3 iterations of the Koch coastline, an exact geometrical fractal. It can be quantified by its perimeter, its AUC or its FD. With each successive iteration of the Koch coastline the original pattern is repeated at a finer level, corresponding to how with increasingly greater magnification increasingly fine detail is revealed in fractals. By traditional methods, the AUC will converge on  and the perimeter of the curve after n iterations will be  times the original perimeter (4 times more lines,  greater length per iteration), and since  perimeter will tend to infinity. These exemplify the inherent problem with traditional mathematics: it is capable of providing only scale-dependent descriptors that give limited insight into the motif’s overarching complexity. The FD of the Koch curve, on the other hand, summarises its complexity independently of scale. At every iteration (from 1 to infinity) the FD is invariant at . Biological quasi-fractals are measured by ‘sampling’ them with an imaging ‘camera’ relevant to a particular imaging modality. Different cameras have different resolutions, but in all cases increasing resolution is similar to accumulating iterations on a mathematical fractal. Natural quasi-fractals are self-similar across a finite number of scales only—a lower limit of representation is imposed by the limit of the screen (pixel resolution). For CMR cines, blurring (quite extreme in b) has the same effect as setting a lower resolution for the particular sequence, and this is equivalent to having fewer fractal iterations. With such manipulation, it can be seen that the area of the set changes little (here by 2 %), the perimeter a lot (by 43 %) and the FD less (by 8 %). This implies that high image resolution (and a fractal approach) may not add much value when attempting to measure the left ventricular volume; but image resolution (and a fractal approach) will make a considerable difference when intricate features like trabeculae are the features of interest: the perimeter length or other 1D approach will be less robust than the FD. AUC = area under the curve; d = length of segment; 1D = one-dimension/al; FD = fractal dimension; px = pixels. Other abbreviation as in Fig. 1
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Fig2: a The first 3 iterations of the Koch coastline, an exact geometrical fractal. It can be quantified by its perimeter, its AUC or its FD. With each successive iteration of the Koch coastline the original pattern is repeated at a finer level, corresponding to how with increasingly greater magnification increasingly fine detail is revealed in fractals. By traditional methods, the AUC will converge on and the perimeter of the curve after n iterations will be times the original perimeter (4 times more lines, greater length per iteration), and since perimeter will tend to infinity. These exemplify the inherent problem with traditional mathematics: it is capable of providing only scale-dependent descriptors that give limited insight into the motif’s overarching complexity. The FD of the Koch curve, on the other hand, summarises its complexity independently of scale. At every iteration (from 1 to infinity) the FD is invariant at . Biological quasi-fractals are measured by ‘sampling’ them with an imaging ‘camera’ relevant to a particular imaging modality. Different cameras have different resolutions, but in all cases increasing resolution is similar to accumulating iterations on a mathematical fractal. Natural quasi-fractals are self-similar across a finite number of scales only—a lower limit of representation is imposed by the limit of the screen (pixel resolution). For CMR cines, blurring (quite extreme in b) has the same effect as setting a lower resolution for the particular sequence, and this is equivalent to having fewer fractal iterations. With such manipulation, it can be seen that the area of the set changes little (here by 2 %), the perimeter a lot (by 43 %) and the FD less (by 8 %). This implies that high image resolution (and a fractal approach) may not add much value when attempting to measure the left ventricular volume; but image resolution (and a fractal approach) will make a considerable difference when intricate features like trabeculae are the features of interest: the perimeter length or other 1D approach will be less robust than the FD. AUC = area under the curve; d = length of segment; 1D = one-dimension/al; FD = fractal dimension; px = pixels. Other abbreviation as in Fig. 1

Mentions: 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.Fig. 2


Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation.

Captur G, Karperien AL, Li C, Zemrak F, Tobon-Gomez C, Gao X, Bluemke DA, Elliott PM, Petersen SE, Moon JC - J Cardiovasc Magn Reson (2015)

a The first 3 iterations of the Koch coastline, an exact geometrical fractal. It can be quantified by its perimeter, its AUC or its FD. With each successive iteration of the Koch coastline the original pattern is repeated at a finer level, corresponding to how with increasingly greater magnification increasingly fine detail is revealed in fractals. By traditional methods, the AUC will converge on  and the perimeter of the curve after n iterations will be  times the original perimeter (4 times more lines,  greater length per iteration), and since  perimeter will tend to infinity. These exemplify the inherent problem with traditional mathematics: it is capable of providing only scale-dependent descriptors that give limited insight into the motif’s overarching complexity. The FD of the Koch curve, on the other hand, summarises its complexity independently of scale. At every iteration (from 1 to infinity) the FD is invariant at . Biological quasi-fractals are measured by ‘sampling’ them with an imaging ‘camera’ relevant to a particular imaging modality. Different cameras have different resolutions, but in all cases increasing resolution is similar to accumulating iterations on a mathematical fractal. Natural quasi-fractals are self-similar across a finite number of scales only—a lower limit of representation is imposed by the limit of the screen (pixel resolution). For CMR cines, blurring (quite extreme in b) has the same effect as setting a lower resolution for the particular sequence, and this is equivalent to having fewer fractal iterations. With such manipulation, it can be seen that the area of the set changes little (here by 2 %), the perimeter a lot (by 43 %) and the FD less (by 8 %). This implies that high image resolution (and a fractal approach) may not add much value when attempting to measure the left ventricular volume; but image resolution (and a fractal approach) will make a considerable difference when intricate features like trabeculae are the features of interest: the perimeter length or other 1D approach will be less robust than the FD. AUC = area under the curve; d = length of segment; 1D = one-dimension/al; FD = fractal dimension; px = pixels. Other abbreviation as in Fig. 1
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Related In: Results  -  Collection

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Fig2: a The first 3 iterations of the Koch coastline, an exact geometrical fractal. It can be quantified by its perimeter, its AUC or its FD. With each successive iteration of the Koch coastline the original pattern is repeated at a finer level, corresponding to how with increasingly greater magnification increasingly fine detail is revealed in fractals. By traditional methods, the AUC will converge on and the perimeter of the curve after n iterations will be times the original perimeter (4 times more lines, greater length per iteration), and since perimeter will tend to infinity. These exemplify the inherent problem with traditional mathematics: it is capable of providing only scale-dependent descriptors that give limited insight into the motif’s overarching complexity. The FD of the Koch curve, on the other hand, summarises its complexity independently of scale. At every iteration (from 1 to infinity) the FD is invariant at . Biological quasi-fractals are measured by ‘sampling’ them with an imaging ‘camera’ relevant to a particular imaging modality. Different cameras have different resolutions, but in all cases increasing resolution is similar to accumulating iterations on a mathematical fractal. Natural quasi-fractals are self-similar across a finite number of scales only—a lower limit of representation is imposed by the limit of the screen (pixel resolution). For CMR cines, blurring (quite extreme in b) has the same effect as setting a lower resolution for the particular sequence, and this is equivalent to having fewer fractal iterations. With such manipulation, it can be seen that the area of the set changes little (here by 2 %), the perimeter a lot (by 43 %) and the FD less (by 8 %). This implies that high image resolution (and a fractal approach) may not add much value when attempting to measure the left ventricular volume; but image resolution (and a fractal approach) will make a considerable difference when intricate features like trabeculae are the features of interest: the perimeter length or other 1D approach will be less robust than the FD. AUC = area under the curve; d = length of segment; 1D = one-dimension/al; FD = fractal dimension; px = pixels. Other abbreviation as in Fig. 1
Mentions: 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.Fig. 2

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: UCL Institute of Cardiovascular Science, University College London, Gower Street, London, WC1E 6BT, UK. gabriella.captur.11@ucl.ac.uk.

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.

No MeSH data available.


Related in: MedlinePlus