SENSOP: A derivative-free solver for nonlinear least squares with sensitivity scaling

Chan, I. S.; Goldstein, Allen H.; Bassingthwaighte, James B.

doi:10.1007/bf02368642

Cited by 53 publications

(34 citation statements)

References 19 publications

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“…In the classic case from Richardson (22), using calipers to measure the length of a coastline, he found that over a wide range, the relationship between the logarithm of the apparent measured length and the logarithm of the caliper length was linear, and slope gave a measure of the fractal dimension. Likewise, the variance in regional flows within the tissue of an organ was found by Bassingthwaighte (1) to exhibit a linear relationship between the log of the variance, or the standard deviation, SD, and the log of the size, m, of the observed unit: (8) from which the fractal dimension D is obtained from the slope of the log-log regression: (9) where m is the element size used to calculate SD, and m 0 is the arbitrarily chosen reference size, where "size" is the number of points aggregated together in a one-dimensional series, or a length over which an average is obtained, or the mass of a tissue sample in which a concentration is measured. The fractal dimension D = 2 − H, where H is the Hurst coefficient; we slide back and forth using D and H, but the advantage of H is that it gives a direct indication of the degree of smoothness or correlation which has the same meaning for signals of any Euclidean dimension, E. In general, H = E + 1 − D. A Hurst coefficient of 0.5 indicates a random noncorrelated signal (otherwise known as white noise) whether the signal is one-, two-, or three-dimensional.…”

Section: Dispersional Analysismentioning

confidence: 91%

“…In our introductory paper, an application of dispersional analysis to the velocities of red blood cells in capillaries did give a straight-line relationship between log(variance) and log(time), but the data set used was short, and therefore this result did not really give an indication that the method was valid (1). A later analysis by Schepers et al (23) using four different techniques for examining time series as fractals showed that the method worked apparently well on signals of 8,192 elements or more: the test signals were fractal signals generated by the spectral synthesis method outlined by Saupe, in Peitgen and Saupe (21), the same signal generation method used for the present study.…”

Section: Introductionmentioning

confidence: 99%

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Evaluation of the dispersional analysis method for fractal time series

1995

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Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is <0.5 for rough anticorrelated series, >0.5 for positively correlated series, and =0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 − D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (>0.8) and low H (<0.5), and biases are greater for short time series than for long; and (iii) the addition of Gaussian noise (H = 0.5) degrades the signals: for those with negative correlation (H < 0.5) the degradation is great, the noise has only mild degrading effects on signals with H > 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H.Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.

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Section: Dispersional Analysismentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Evaluation of the dispersional analysis method for fractal time series

1995

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“…The simulated data, with or without added noise, were fitted using SENSOP, a derivativefree solver with sensitivity scaling for nonlinear least-squares equations (13). The amplitude of the data covered a large range, usually about four orders of magnitude.…”

Section: Methodsmentioning

confidence: 99%

Modeling blood flow heterogeneity

1996

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It has been known for some time that regional blood flows within an organ are not uniform. Useful measures of heterogeneity of regional blood flows are the standard deviation and coefficient of variation or relative dispersion of the probability density function (PDF) of regional flows obtained from the regional concentrations of tracers that are deposited in proportion to blood flow. When a mathematical model is used to analyze dilution curves after tracer solute administration, for many solutes it is important to account for flow heterogeneity and the wide range of transit times through multiple pathways in parallel. Failure to do so leads to bias in the estimates of volumes of distribution and membrane conductances. Since in practice the number of paths used should be relatively small, the analysis is sensitive to the choice of the individual elements used to approximate the distribution of flows or transit times. Presented here is a method for modeling heterogeneous flow through an organ using a scheme that covers both the high flow and long transit time extremes of the flow distribution. With this method, numerical experiments are performed to determine the errors made in estimating parameters when flow heterogeneity is ignored, in both the absence and presence of noise. The magnitude of the errors in the estimates depends upon the system parameters, the amount of flow heterogeneity present, and whether the shape of the input function is known. In some cases, some parameters may be estimated to within 10% when heterogeneity is ignored (homogeneous model), but errors of 15-20% may result, even when the level of heterogeneity is modest. In repeated trials in the presence of 5% noise, the mean of the estimates was always closer to the true value with the heterogeneous model than when heterogeneity was ignored, but the distributions of the estimates from the homogeneous and heterogeneous models overlapped for some parameters when outflow dilution curves were analyzed. The separation between the distributions was further reduced when tissue content curves were analyzed. It is concluded that multipath models accounting for flow heterogeneity are a vehicle for assessing the effects of flow heterogeneity under the conditions applicable to specific laboratory protocols, that efforts should be made to assess the actual level of flow heterogeneity in the organ being studied, and that the errors in parameter estimates are generally smaller when the input function is known rather than estimated by deconvolution.

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“…Both approaches give equivalent results if the vascular dispersion is set appropriately (ie, approximately 18%) (23). To fit model solutions to the MR tissue contentversus-time curves, a nonlinear least-squares optimizer routine, based on sensitivity functions (24), was used to simultaneously adjust the values of flow and volume for large vessels and microvessels.…”

Section: Mr Blood Flow Estimatesmentioning

confidence: 99%

Regional myocardial blood volume and flow: First‐pass MR imaging with polylysine‐Gd‐DTPA

Wilke

Kroll

Merkle

et al. 1995

Magnetic Resonance Imaging

132

104

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The authors investigated the utility of an intravascular magnetic resonance (MR) contrast agent, poly-L-lysine-gadolinium diethylenetriaminepentaacetic acid (DTPA), for differentiating acutely ischemic from normally perfused myocardium with first-pass MR imaging. Hypoperfused regions, identified with microspheres, on the first-pass images displayed significantly decreased signal intensities compared with normally perfused myocardium (P < .0007). Estimates of regional myocardial blood content, obtained by measuring the ratio of areas under the signal intensityversus-time curves in tissue regions and the left ventricular chamber, averaged 0.12 mL/g ± 0.04 (n = 35), compared with a value of 0.11 mL/g ± 0.05 measured with radiolabeled albumin in the same tissue regions. To obtain MR estimates of regional myocardial blood flow, in situ calibration curves were used to transform first-pass intensity-time curves into content-time curves for analysis with a multiple-pathway, axially distributed model. Flow estimates, obtained by automated parameter optimization, averaged 1.2 mL/min/g ± 0.5 [n = 29), compared with 1.3 mL/min/g ± 0.3 obtained with tracer microspheres in the same tissue specimens at the same time. The results represent a combination of T1-weighted first-pass imaging, intravascular relaxation agents, and a spatially distributed perfusion model to obtain absolute regional myocardial blood flow and volume. Index termsContrast agent, blood pool; Contrast enhancement; Coronary vessels, diseases, 54.76; Heart, flow dynamics; Heart, MR, 51.12143; Model, mathematical; Myocardium, blood supply, 511.12143; Myocardium, MR, 511.12143 NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author ManuscriptA growing body of evidence supports the feasibility of evaluating myocardial perfusion with magnetic resonance (MR) imaging with contrast agents. A variety of intravascular and extracellular MR contrast agents have been studied for delineation of infarcted myocardium (1-3) and for identification of acutely ischemic and reperfused cardiac muscle (4,5). The first-pass MR imaging technique with the extracellular contrast agent gadolinium diethylenetriaminepentaacetic acid (DTPA) has been used to assess myocardial perfusion in patients at rest (6,7) and during pharmacologic stress (8,9). Although experimental (10) and patient studies have demonstrated the suitability of extracellular contrast agents for detecting hypoperfused myocardium, such agents are inherently disadvantageous for flow quantification because they are not confined to a single volume compartment within the tissue. Consequently, first-pass kinetics of an extracellular agent are determined not only by blood flow but by the intravascular and interstitial distribution volumes and by permeability across the capillary wall. This exchange between the vascular and interstitial compartments greatly complicates quantification of tissue blood flow unless it is either very fast or slow compared with flow rates (11).The objective of the present study was to...

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SENSOP: A derivative-free solver for nonlinear least squares with sensitivity scaling

Cited by 53 publications

References 19 publications

Evaluation of the dispersional analysis method for fractal time series

Evaluation of the dispersional analysis method for fractal time series

Modeling blood flow heterogeneity

Regional myocardial blood volume and flow: First‐pass MR imaging with polylysine‐Gd‐DTPA

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