Fractal descriptions for spatial statistics

Fractal nature of human gastrointestinal system: Exploring a new era

The morphological complexity of cells and tissues, whether normal or pathological, is characterized by two primary attributes: Irregularity and self-similarity across different scales. When an object exhibits self-similarity, its shape remains unchanged as the scales of measurement vary because any part of it resembles the whole. On the other hand, the size and geometric characteristics of an irregular object vary as the resolution increases, revealing more intricate details. Despite numerous attempts, a reliable and accurate method for quantifying the morphological features of gastrointestinal organs, tissues, cells, their dynamic changes, and pathological disorders has not yet been established. However, fractal geometry, which studies shapes and patterns that exhibit self-similarity, holds promise in providing a quantitative measure of the irregularly shaped morphologies and their underlying self-similar temporal behaviors. In this context, we explore the fractal nature of the gastrointestinal system and the potential of fractal geometry as a robust descriptor of its complex forms and functions. Additionally, we examine the practical applications of fractal geometry in clinical gastroenterology and hepatology practice.

Fractal Dimension of Tumor Microvasculature by DCE-US: Preliminary Study in Mice

Neoangiogenesis, which results in the formation of an irregular network of microvessels, plays a fundamental role in the growth of several types of cancer. Characterization of microvascular architecture has therefore gained increasing attention for cancer diagnosis, treatment monitoring and evaluation of new drugs. However, this characterization requires immunohistologic analysis of the resected tumors. Currently, dynamic contrast-enhanced ultrasound imaging (DCE-US) provides new options for minimally invasive investigation of the microvasculature by analysis of ultrasound contrast agent (UCA) transport kinetics. In this article, we propose a different method of analyzing UCA concentration that is based on the spatial distribution of blood flow. The well-known concept of Mandelbrot allows vascular networks to be interpreted as fractal objects related to the regional blood flow distribution and characterized by their fractal dimension (FD). To test this hypothesis, the fractal dimension of parametric maps reflecting blood flow, such as UCA wash-in rate and peak enhancement, was derived for areas representing different microvascular architectures. To this end, subcutaneous xenograft models of DU-145 and PC-3 prostate-cancer lines in mice, which show marked differences in microvessel density spatial distribution inside the tumor, were employed to test the ability of DCE-US FD analysis to differentiate between the two models. For validation purposes, the method was compared with immunohistologic results and UCA dispersion maps, which reflect the geometric properties of microvascular architecture. The results showed good agreement with the immunohistologic analysis, and the FD analysis of UCA wash-in rate and peak enhancement maps was able to differentiate between the two xenograft models (p < 0.05).

Estimation of noise-free variance to measure heterogeneity

Variance is a statistical parameter used to characterize heterogeneity or variability in data sets. However, measurements commonly include noise, as random errors superimposed to the actual value, which may substantially increase the variance compared to a noise-free data set. Our aim was to develop and validate a method to estimate noise-free spatial heterogeneity of pulmonary perfusion using dynamic positron emission tomography (PET) scans. On theoretical grounds, we demonstrate a linear relationship between the total variance of a data set derived from averages of n multiple measurements, and the reciprocal of n. Using multiple measurements with varying n yields estimates of the linear relationship including the noise-free variance as the constant parameter. In PET images, n is proportional to the number of registered decay events, and the variance of the image is typically normalized by the square of its mean value yielding a coefficient of variation squared (CV²). The method was evaluated with a Jaszczak phantom as reference spatial heterogeneity (CV_r²) for comparison with our estimate of noise-free or ‘true’ heterogeneity (CV_t²). We found that CV_t² was only 5.4% higher than CV_r². Additional evaluations were conducted on 38 PET scans of pulmonary perfusion using ¹³NN-saline injection. The mean CV_t² was 0.10 (range: 0.03–0.30), while the mean CV² including noise was 0.24 (range: 0.10–0.59). CV_t² was in average 41.5% of the CV² measured including noise (range: 17.8–71.2%). The reproducibility of CV_t² was evaluated using three repeated PET scans from five subjects. Individual CV_t² were within 16% of each subject's mean and paired t-tests revealed no difference among the results from the three consecutive PET scans. In conclusion, our method provides reliable noise-free estimates of CV_t² in PET scans, and may be useful for similar statistical problems in experimental data.

Forensic anthropology and human identification

This is the full summary paper of a thesis to be defended at the University of Copenhagen, May 31st, 2013

Estimation of regional myocardial mass at risk based on distal arterial lumen volume and length using 3D micro-CT images

The determination of regional myocardial mass at risk distal to a coronary occlusion provides valuable prognostic information for a patient with coronary artery disease. The coronary arterial system follows a design rule which allows for the use of arterial branch length and lumen volume to estimate regional myocardial mass at risk. Image processing techniques, such as segmentation, skeletonization and arterial network tracking, are presented for extracting anatomical details of the coronary arterial system using micro-computed tomography (micro-CT). Moreover, a method of assigning tissue voxels to their corresponding arterial branches is presented to determine the dependent myocardial region. The proposed micro-CT technique was utilized to investigate the relationship between the sum of the distal coronary arterial branch lengths and volumes to the dependent regional myocardial mass using a polymer cast of a porcine heart. The correlations of the logarithm of the total distal arterial lengths (L) to the logarithm of the regional myocardial mass (M) for the left anterior descending (LAD), left circumflex (LCX) and right coronary (RCA) arteries were log(L)=0.73log(M)+0.09 (R=0.78), log(L)=0.82log(M)+0.05 (R=0.77) and log(L)=0.85log(M)+0.05 (R=0.87), respectively. The correlation of the logarithm of the total distal arterial lumen volumes (V) to the logarithm of the regional myocardial mass for the LAD, LCX and RCA were log(V)=0.93log(M)-1.65 (R=0.81), log(V)=1.02log(M)-1.79 (R=0.78) and log(V)=1.17log(M)-2.10 (R=0.82), respectively. These morphological relations did not change appreciably for diameter truncations of 600-1400microm. The results indicate that the image processing procedures successfully extracted information from a large 3D dataset of the coronary arterial tree to provide prognostic indications in the form of arterial tree parameters and anatomical area at risk.

Fractal properties of perfusion heterogeneity in optimized arterial trees: a model study

Regional blood flows in the heart muscle are remarkably heterogeneous. It is very likely that the most important factor for this heterogeneity is the metabolic need of the tissue rather than flow dispersion by the branching network of the coronary vasculature. To model the contribution of tissue needs to the observed flow heterogeneities we use arterial trees generated on the computer by constrained constructive optimization. This method allows to prescribe terminal flows as independent boundary conditions, rather than obtaining these flows by the dispersive effects of the tree structure. We study two specific cases: equal terminal flows (model 1) and terminal flows set proportional to the volumes of Voronoi polyhedra used as a model for blood supply regions of terminal segments (model 2). Model 1 predicts, depending on the number Nterm of end-points, fractal dimensions D of perfusion heterogeneities in the range 1.20 to 1.40 and positively correlated nearest-neighbor regional flows, in good agreement with experimental data of the normal heart. Although model 2 yields reasonable terminal flows well approximated by a lognormal distribution, it fails to predict D and nearest-neighbor correlation coefficients r1 of regional flows under normal physiologic conditions: model 2 gives D = 1.69 +/- 0.02 and r1 = -0.18 +/- 0.03 (n = 5), independent of Nterm and consistent with experimental data observed under coronary stenosis and under the reduction of coronary perfusion pressure. In conclusion, flow heterogeneity can be modeled by terminal positions compatible with an existing tree structure without resorting to the flow-dispersive effects of a specific branching tree model to assign terminal flows.

Brief communication: age and fractal dimensions of human sagittal and coronal sutures

The fractal dimensions of human sagittal and coronal sutures were calculated on 31 complete skulls from the Terry Collection. The aim was to investigate whether the fractal dimension, relying on the whole sutural length, might yield a better description of age-related changes in sutural morphology, as opposed to other methods of quantification, which generally rely on more arbitrary scoring systems. However, the fractal dimension did not yield better age correlations than other previously described methods. At best, the results reflected the general observation that young adults below age 40 years display an age-related development, but that it is impossible to arrive at any precise age determinations for older adults. It seems that for some individuals, suture obliteration simply does not take place, even at an advanced age, whereas for others, suture obliteration progresses rapidly. Until a better understanding of sutural biology is reached, this will render cranial sutures only marginally useful in age determination. This does not mean, however, that investigations should not be made to elucidate more unbiased methods of sutural morphology quantification.

Minimally invasive evaluation of coronary microvascular function by electron beam computed tomography

BACKGROUND

We previously demonstrated that in vivo electron-beam computed tomography (EBCT)-based indicator-dilution methods provide an estimate of intramyocardial blood volume (BV) and perfusion (F), which relate as BV=aF+b radicalF, where a characterizes the recruitable (exchange) and b the nonrecruitable (conduit) component of the myocardial microcirculation. In the present study, we compared BV and F with intracoronary Doppler ultrasound-based coronary blood flow (CBF) as a method for detecting and quantifying differential responses of these microvascular components to vasoactive drugs in normal (control) and hypercholesterolemic (HC) pigs.

METHODS AND RESULTS

BV and F values were obtained from contrast-enhanced EBCT studies in 14 HC and 14 control pigs. BV, F, and CBF values were obtained at baseline (intracoronary infusion of saline) and after 5 minutes each of intracoronary infusion of adenosine (100 microgram. kg(-1). min(-1)) and nitroglycerin (40 microgram/min). BV and CBF reserves in response to adenosine were attenuated in HC pigs compared with controls (90+/-36% versus 127+/-42%, P<0.03, and 485+/-182% versus 688+/-160%, P<0.01, respectively). The relationship between BV and F showed consistently lower recruitable BV in HC versus control pigs. Nonrecruitable BV reserve in response to adenosine was attenuated in HC compared with controls (77+/-20% versus 135+/-28%, P<0.001). Our findings are consistent with HC-induced impairment of intramyocardial resistance vessel function.

CONCLUSIONS

EBCT technology allows minimally invasive evaluation of intramyocardial microcirculatory function and permits assessment of microvascular BV distribution in different functional components. This method may be of value in evaluating the coronary microcirculation in pathophysiological states such as hypercholesterolemia.

Evaluation of the dispersional analysis method for fractal time series

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.

Evaluating rescaled ranged analysis for time series

Rescaled range analysis is a means of characterizing a time series or a one-dimensional (1-D) spatial signal that provides simultaneously a measure of variance and of the long-term correlation or "memory," The trend-corrected method is based on the statistical self-similarity in the signal: in the standard approach one measures the ratio R/S on the range R of the sum of the deviations from the local mean divided by the standard deviation S from the mean. For fractal signals R/S is a power law function of the length tau of each segment of the set of segments into which the data set has been divided. Over a wide range of tau's the relationship is: R/S = a tau H, where kappa is a scalar and the H is the Hurst exponent. (For a 1-D signal f(t), the exponent H = 2 - D, with D being the fractal dimension.) The method has been tested extensively on fractional Brownian signals of known H to determine its accuracy, bias, and limitations. R/S tends to give biased estimates of H, too low for H > 0.72, and too high for H < 0.72. Hurst analysis without trend correction differs by finding the range R of accumulation of differences from the global mean over the total period of data accumulation, rather than from the mean over each tau. The trend-corrected method gives better estimates of H on Brownian fractal signals of known H when H > or = 0.5, that is, for signals with positive correlations between neighboring elements. Rescaled range analysis has poor convergence properties, requiring about 2,000 points for 5% accuracy and 200 for 10% accuracy. Empirical corrections to the estimates of H can be made by graphical interpolation to remove bias in the estimates. Hurst's 1951 conclusion that many natural phenomena exhibit not random but correlated time series is strongly affirmed.