Arterial Branching within the Confines of Fractal L-System Formalism

Zamir, M.

doi:10.1085/jgp.118.3.267

Cited by 88 publications

(65 citation statements)

References 24 publications

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“…The symbols produced are then interpreted graphically to visualize a geometry encoded by the output L-system string (representing the branching system) (18). A branching L-system is characteristically fractal, with self-similar elements visible at decreasing size scales (38). Parametric L-systems allow branching parameter values (such as branching angles and growth rates) to vary between branches (for example, of different orders or ages), enabling realistic representation of biological structures with nonuniform branching patterns (18).…”

Section: Methodsmentioning

confidence: 99%

Fractal branching organizations of Ediacaran rangeomorph fronds reveal a lost Proterozoic body plan

Cuthill

Morris

2014

Proc. Natl. Acad. Sci. U.S.A.

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The branching morphology of Ediacaran rangeomorph fronds has no exact counterpart in other complex macroorganisms. As such, these fossils pose major questions as to growth patterns, functional morphology, modes of feeding, and adaptive optimality. Here, using parametric Lindenmayer systems, a formal model of rangeomorph morphologies reveals a fractal body plan characterized by self-similar, axial, apical, alternate branching. Consequent morphological reconstruction for 11 taxa demonstrates an adaptive radiation based on 3D space-filling strategies. The fractal body plan of rangeomorphs is shown to maximize surface area, consistent with diffusive nutrient uptake from the water column (osmotrophy). The enigmas of rangeomorph morphology, evolution, and extinction are resolved by the realization that they were adaptively optimized for unique ecological and geochemical conditions in the late Proterozoic. Changes in ocean conditions associated with the Cambrian explosion sealed their fate.paleobiology | paleontology I n parallel with large-scale geochemical transitions associated with ocean oxygenation (1-3), the Ediacaran Period (635-541 Ma) records a major diversification of multicellular eukaryotes. Rangeomorph fronds (575-541 Ma) dominated early Ediacaran biotas (4) and have a characteristic branching morphology, distinct from any known Phanerozoic organism (5). Although the fronds are often preserved as flattened impressions, exceptional moldic fossils preserve details of the 3D branching structure to a resolution of 30 μm (6). Qualitative classifications for rangeomorph branching patterns have been proposed (7,8), but no quantitative model has previously been formulated. Because branching is repeated over decreasing size scales (with up to four observed orders of branching), rangeomorph fronds have been informally described as self-similar and fractal (4-6). Although this has potential implications for the functional optimality of their morphologies (5, 9), the extent to which they are formally fractal and self-similar (10, 11) has not previously been tested. Furthermore, until now, evolutionary transitions in branching patterns have not been characterized within any quantitative framework.Rangeomorphs inhabited shallow to abyssal marine environments (1,8,(12)(13)(14), evidently precluding photosynthesis for most taxa (12). Preservational features, including bending and overfolding (4, 15), suggest that rangeomorphs were soft-bodied. No evidence exists for either motility or active feeding (such as musculature, filter feeding organs, or a mouth). Consequently, rangeomorphs have been reconstructed as sessile, feeding on organic carbon by diffusion (or possibly endocytosis) through the body surface (3,5,12,16), with a large surface area to volume ratio aiding nutrient uptake (5, 16). The adaptive potential of their different branching morphologies has, however, never been quantified.Here, using parametric Lindenmayer systems (L-systems) (17, 18), we present a unified model to describe the branching structure of Ed...

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Section: Methodsmentioning

confidence: 99%

Fractal branching organizations of Ediacaran rangeomorph fronds reveal a lost Proterozoic body plan

Cuthill

Morris

2014

Proc. Natl. Acad. Sci. U.S.A.

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“…This may be due to a combination of normal biological scatter [5] [20], the value of the power law exponent k [4], experimental measurement errors, or intrinsic vascular variability [21]. Some of the variation may relate to pathology in those subjects with diabetic retinopathy.…”

Section: A Theoretical Predictionsmentioning

confidence: 99%

Manual measurement of retinal bifurcation features

Al-Diri

Hunter

Steel³

et al. 2010

2010 Annual International Conference of the IEEE Engineering in Medicine and Biology

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Abstract-This paper introduces a new computerized tool for accurate manual measurement of features of retinal bifurcation geometry, designed for use in investigating correlations between measurement features and clinical conditions. The tool uses user-placed rectangles to measure the vessel width, and lines placed along vessel center lines to measure the angles. An analysis is presented of measurements taken from 435 bifurcations. These are compared with theoretical predictions based on optimality principles presented in the literature. The new tool shows better agreement with the theoretical predictions than a simpler manual method published in the literature, but there remains a significant discrepancy between current theory and measured geometry.

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“…This variability may be due to normal biological scatter [4], the values of the power law exponent k [6], experimental measurement errors, the scatter in physiological data [12] or the presence of a multifractal pattern which suggests that the bulk of the scatter is because of vascular design variability [13]. The noticed variations could also be attributed to the fact that some of the used images were of patients with variable grades of diabetic retinopathy, thus their geometrical features could deviate more from the theoretical optimum due to pathological alterations of the applied forces on the retinal vessels.…”

Section: Discussionmentioning

confidence: 99%

Automated Measurements of Retinal Bifurcations

Al-Diri

Hunter

2009

IFMBE Proceedings

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Abstract-This paper presents an analysis of the bifurcations of retinal vessels. The angles and relative diameters of blood vessels in 230 bifurcations were measured using a new automated procedure, and used to calculate the values of several features with known theoretical properties. The measurements are compared with predictions from theoretical models, and with manual measurements. The automated measurements agree with the theoretical prediction measurements with slightly different bias. The automated method can measure a large number of retinal bifurcations very rapidly, and may be useful in correlating bifurcation geometry with clinical conditions.

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Arterial Branching within the Confines of Fractal L-System Formalism

Cited by 88 publications

References 24 publications

Fractal branching organizations of Ediacaran rangeomorph fronds reveal a lost Proterozoic body plan

Fractal branching organizations of Ediacaran rangeomorph fronds reveal a lost Proterozoic body plan

Manual measurement of retinal bifurcation features

Automated Measurements of Retinal Bifurcations

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