A stochastic model for the self-similar heterogeneity of regional organ blood flow

Kendal, Wayne S.

doi:10.1073/pnas.021347898

Cited by 17 publications

(13 citation statements)

References 16 publications

(28 reference statements)

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“…None of these population models could account for the nonecological manifestation of Taylor's law, as seen with cancer metastasis [30], regional organ blood flow [8], and the genomic distributions of SNPs [9] and genes [10]. Fronczak and Fronczak attempted to provide a more general explanation for Taylor's law on the basis of a thermodynamic model that employed fluctuation dissipation in the presence of an external physical field [12].…”

Section: Discussionmentioning

confidence: 99%

“…Taylor, who first observed this effect with viruses, protozoa, insects, mollusks, vertebrates, and plants [1,2], explained it in terms of intraspecies behavior [3]; others have postulated demographic mechanisms [4], stochastic variations in reproductive rates [5], and interspecies interactions [6]. Taylor's power law also manifests within nonecological systems such as HIV epidemiology [7], regional organ blood flow [8], the genomic distributions of single nucleotide polymorphisms (SNPs) [9], and genes [10], as well as within physical and econometric systems where it has been called fluctuation scaling [11,12]. When applied to sequential data, with expanding enumerative bins, Taylor's law also implies long-range correlations and 1/f noise [13].…”

Section: Introductionmentioning

confidence: 91%

“…The correspondence between the PG distribution and the GUE/GOE eigenvalue fluctuations, as well as its correspondence within other biological systems that have exhibited Taylor's law [8][9][10]29,30], might be dismissed as an artifact of curve fitting of little consequence were it not for the mathematical role that the Tweedie models have as foci of statistical convergence: For EDMs E D (μ,σ 2 ) with unit variance functions of the form V (μ) ∼ μ p , Jørgensen et al have proven that as μ → 0 or μ → ∞ then, within the constant factor c, c −1 E D (cμ,σ 2 c 2−p ) will converge to the form of a Tweedie model as either c ↓ 0 or c → ∞ [14]. Since the variance functions for many probability distributions approximate the form V (μ) ∝ μ p , for small or large values of μ, the Tweedie EDMs act as the foci of convergence for a wide variety of data [14].…”

Section: Statistical Convergence Explains Both Taylor's Law and 1/mentioning

confidence: 99%

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Tweedie convergence: A mathematical basis for Taylor's power law,1/fnoise, and multifractality

Kendal

Jørgensen

2011

Phys. Rev. E

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Plants and animals of a given species tend to cluster within their habitats in accordance with a power function between their mean density and the variance. This relationship, Taylor's power law, has been variously explained by ecologists in terms of animal behavior, interspecies interactions, demographic effects, etc., all without consensus. Taylor's law also manifests within a wide range of other biological and physical processes, sometimes being referred to as fluctuation scaling and attributed to effects of the second law of thermodynamics. 1/f noise refers to power spectra that have an approximately inverse dependence on frequency. Like Taylor's law these spectra manifest from a wide range of biological and physical processes, without general agreement as to cause. One contemporary paradigm for 1/f noise has been based on the physics of self-organized criticality. We show here that Taylor's law (when derived from sequential data using the method of expanding bins) implies 1/f noise, and that both phenomena can be explained by a central limit-like effect that establishes the class of Tweedie exponential dispersion models as foci for this convergence. These Tweedie models are probabilistic models characterized by closure under additive and reproductive convolution as well as under scale transformation, and consequently manifest a variance to mean power function. We provide examples of Taylor's law, 1/f noise, and multifractality within the eigenvalue deviations of the Gaussian unitary and orthogonal ensembles, and show that these deviations conform to the Tweedie compound Poisson distribution. The Tweedie convergence theorem provides a unified mathematical explanation for the origin of Taylor's law and 1/f noise applicable to a wide range of biological, physical, and mathematical processes, as well as to multifractality.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 91%

Section: Statistical Convergence Explains Both Taylor's Law and 1/mentioning

confidence: 99%

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Tweedie convergence: A mathematical basis for Taylor's power law,1/fnoise, and multifractality

Kendal

Jørgensen

2011

Phys. Rev. E

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“…The next question is “What gives rise to the fractal characteristics of the flow distributions?” 72 [Kendal 2001]. A probable cause is the nature of the vascular branching; van Beek 20,119 (Bassingthwaighte and vanBeek 1988, 1989) found that a dichotomous branching system with a small degree of asymmetry in flows in successive branches sufficed to give the observed flow heterogeneities and fractal dimensions.…”

Section: Section II the Fractal Nature Of Regional Myocardial Blood mentioning

confidence: 99%

Modeling to Link Regional Myocardial Work, Metabolism and Blood Flows

et al. 2012

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Given the mono-functional, highly coordinated processes of cardiac excitation and contraction, the observations that regional myocardial blood flows, rMBF, are broadly heterogeneous has provoked much attention, but a clear explanation has not emerged. In isolated and in vivo heart studies the total coronary flow is found to be proportional to the rate-pressure product (systolic mean blood pressure times heart rate), a measure of external cardiac work. The same relationship might be expected on a local basis: more work requires more flow. The validity of this expectation has never been demonstrated experimentally. In this article we review the concepts linking cellular excitation and contractile work to cellular energetics and ATP demand, substrate utilization, oxygen demand, vasoregulation, and local blood flow. Mathematical models of these processes are now rather well developed. We propose that the construction of an integrated model encompassing the biophysics, biochemistry and physiology of cardiomyocyte contraction, then combined with a detailed three-dimensional structuring of the fiber bundle and sheet arrangements of the heart as a whole will frame an hypothesis that can be quantitatively evaluated to settle the prime issue: Does local work drive local flow in a predictable fashion that explains the heterogeneity? While in one sense one can feel content that work drives flow is irrefutable, there are no cardiac contractile models that demonstrate the required heterogeneity in local strain-stress-work; quite the contrary, cardiac contraction models have tended toward trying to show that work should be uniform. The object of this review is to argue that uniformity of work does not occur, and is impossible in any case, and that further experimentation and analysis are necessary to test the hypothesis.

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“…4 is a gamma distribution is not altogether surprizing given the prevalence of this distribution in other biological systems generally [24][25][26], and the analysis of models of vascular networks [27][28][29] in particular. While the gamma distribution is consistent with branching governed by processes with exponential (Table I. waiting times, the 3-month adjustment phase could mean that other processes are operative.…”

mentioning

confidence: 99%