Regional myocardial flow heterogeneity explained with fractal networks

Spatial architecture of biliary tree in mammals: Fractal and Euclidean geometric features

The study of the fractal architecture of various organs and structures expanded the possibilities for determining the ranges of their functioning and structural arrangement, which, as a result, was reflected in the development of new approaches to diagnostics and therapeutic impacts. The architecture of the excretory duct systems, similar to the hemo- and lymph- circulatory beds and the bronchial tree, is considered fractal. At the same time, information about hitherto unknown structures of the biliary tree continues to appear in the literature. We aimed to study the features of the spatial geometry of the biliary tree and assess the significance of both its fractal and Euclidean characteristics for the development of approaches that facilitate comprehensive description of intrahepatic biliary tract architecture. We investigated the architecture of the biliary trees of six men, seven male canines, and seven male Wistar rats using the corrosion casting method. Corrosion casts were prepared by injecting solidifying latexes into the bile ducts. The preparations were studied using a light stereomicroscope and a scanning electron microscope. Biliary tree branching is of various types. In addition, the correlation between variations in the caliber and length of the bile ducts and their order is not significant. Therefore, the biliary tree should not be considered as a classical fractal and it consists of the main modules, represented by the network of the bile canaliculi (first nonfractal module) and a biliary tree with a fractal branching (second module) that drains the bile canaliculi mesh and the additional modules represented by the mucosal biliary glands (in mammals with the gallbladder) or the periportal biliary plexus (in mammals without a gallbladder) and the aberrant biliary ducts. Such a configuration of the biliary bed should optimally ensure the smooth implementation of the physiological function of the liver, as well as its adaptation to different pathologies accompanied by biliary hypertension. It also might be considered in the diagnosis and assessment of ductular reaction, biliary regeneration, and/or carcinogenesis.

Development of Novel Fractal Method for Characterizing the Distribution of Blood Flow in Multi-Scale Vascular Tree

Blood perfusion is an important index for the function of the cardiovascular system and it can be indicated by the blood flow distribution in the vascular tree. As the blood flow in a vascular tree varies in a large range of scales and fractal analysis owns the ability to describe multi-scale properties, it is reasonable to apply fractal analysis to depict the blood flow distribution. The objective of this study is to establish fractal methods for analyzing the blood flow distribution which can be applied to real vascular trees. For this purpose, the modified methods in fractal geometry were applied and a special strategy was raised to make sure that these methods are applicable to an arbitrary vascular tree. The validation of the proposed methods on real arterial trees verified the ability of the produced parameters (fractal dimension and multifractal spectrum) in distinguishing the blood flow distribution under different physiological states. Furthermore, the physiological significance of the fractal parameters was investigated in two situations. For the first situation, the vascular tree was set as a perfect binary tree and the blood flow distribution was adjusted by the split ratio. As the split ratio of the vascular tree decreases, the fractal dimension decreases and the multifractal spectrum expands. The results indicate that both fractal parameters can quantify the degree of blood flow heterogeneity. While for the second situation, artificial vascular trees with different structures were constructed and the hemodynamics in these vascular trees was simulated. The results suggest that both the vascular structure and the blood flow distribution affect the fractal parameters for blood flow. The fractal dimension declares the integrated information about the heterogeneity of vascular structure and blood flow distribution. In contrast, the multifractal spectrum identifies the heterogeneity features in blood flow distribution or vascular structure by its width and height. The results verified that the proposed methods are capable of depicting the multi-scale features of the blood flow distribution in the vascular tree and further are potential for investigating vascular physiology.

A quasi-3D model of the whole lung: airway extension to the tracheobronchial limit using the constrained constructive optimization and alveolar modeling, using a sac-trumpet model

Existing computational models used for simulating the flow and species transport in the human airways are zero-dimensional (0D) compartmental, three-dimensional (3D) computational fluid dynamics (CFD), or the recently developed quasi-3D (Q3D) models. Unlike compartmental models, the full CFD and Q3D models are physiologically and anatomically consistent in the mouth and the upper airways, since the starting point of these models is the mouth-lung surface geometry, typically created from computed tomography (CT) scans. However, the current resolution of CT scans limits the airway detection between the 3rd-4th and 7th-9th generations. Consequently, CFD and the Q3D models developed using these scans are generally limited to these generations. In this study, we developed a method to extend the conducting airways from the end of the truncated Q3D lung to the tracheobronchial (TB) limit. We grew the lung generations within the closed lung lobes using the modified constrained constructive optimization, creating an aerodynamically optimized network aiming to produce equal pressure at the distal ends of the terminal segments. This resulted in a TB volume and lateral area of ∼165 cc and ∼2000 cm², respectively. We created a "sac-trumpet" model at each of the TB outlets to represent the alveoli. The volumes of the airways and the individual alveolar generations match the anatomical values by design: with the functional residual capacity at 2611 cc. Lateral surface areas were scaled to match the physiological values. These generated Q3D whole lung models can be efficiently used for conducting multiple breathing cycles of drug transport and deposition simulations.

Impact of baseline calibration on semiquantitative assessment of myocardial perfusion reserve by adenosine stress MRI

In this study, we sought to investigate the impact of baseline calibration, which is used in quantitative cardiac MRI perfusion analysis to correct for surface coil inhomogeneity and noise, on myocardial perfusion reserve index (MPRI) and its contribution to previously reported paradoxical low MPRI < 1.0 in patients with unobstructed coronary arteries. Semiquantitative perfusion analysis was performed in 20 patients with unobstructed coronary arteries undergoing stress/rest perfusion CMR and in ten patients undergoing paired rest perfusion CMR. The following baseline calibration settings were compared: (1) baseline division, (2) baseline subtraction and (3) no baseline calibration. In uncalibrated analysis, we observed ~ 20% segmental dispersion of signal intensity (SI)-over-time curves. Both baseline subtraction and baseline division reduced relative dispersion of t0-SI (p < 0.001), but only baseline division corrected for dispersion of peak-SI and maximum upslope also (p < 0.001). In the assessment of perfusion indices, however, baseline division resulted in paradoxical low MPRI (1.01 ± 0.23 vs. 1.63 ± 0.38, p < 0.001) and rest perfusion index (RPI 0.54 ± 0.07 vs. 0.94 ± 0.12, p < 0.001), respectively. This was due to a reversed ratio of blood-pool and myocardial baseline-SI before the second perfusion study caused by circulating contrast agent from the first injection. In conclusion, baseline division reliably corrects for inhomogeneity of the surface coil sensitivity profile facilitating comparisons of regional myocardial perfusion during hyperemia or at rest. However, in the assessment of MPRI, baseline division can lead to paradoxical low results (even MPRI < 1.0 in patients with unobstructed coronary arteries) potentially mimicking severely impaired perfusion reserve. Thus, in the assessment of MPRI we propose to waive baseline calibration.

Fractal regional myocardial blood flows pattern according to metabolism, not vascular anatomy

Regional myocardial blood flows are markedly heterogeneous. Fractal analysis shows strong near-neighbor correlation. In experiments to distinguish control by vascular anatomy vs. local vasomotion, coronary flows were increased in open-chest dogs by stimulating myocardial metabolism (catecholamines + atropine) with and without adenosine. During control states mean left ventricular (LV) myocardial blood flows (microspheres) were 0.5-1 ml·g(-1)·min(-1) and increased to 2-3 ml·g(-1)·min(-1) with catecholamine infusion and to ∼4 ml·g(-1)·min(-1) with adenosine (Ado). Flow heterogeneity was similar in all states: relative dispersion (RD = SD/mean) was ∼25%, using LV pieces 0.1-0.2% of total. During catecholamine infusion local flows increased in proportion to the mean flows in 45% of the LV, "tracking" closely (increased proportionately to mean flow), while ∼40% trended toward the mean. Near-neighbor regional flows remained strongly spatially correlated, with fractal dimension D near 1.2 (Hurst coefficient 0.8). The spatial patterns remain similar at varied levels of metabolic stimulation inferring metabolic dominance. In contrast, adenosine vasodilation increased flows eightfold times control while destroying correlation with the control state. The Ado-induced spatial patterns differed from control but were self-consistent, inferring that with full vasodilation the relaxed arterial anatomy dominates the distribution. We conclude that vascular anatomy governs flow distributions during adenosine vasodilation but that metabolic vasoregulation dominates in normal physiological states.

Microsphere skimming in the porcine coronary arteries: Implications for flow quantification

Particle skimming is a phenomenon where particles suspended in fluid flowing through vessels distribute disproportionately to bulk fluid volume at junctions. Microspheres are considered a gold standard of intra-organ perfusion measurements and are used widely in studies of flow distribution and quantification. It has previously been hypothesised that skimming at arterial junctions is responsible for a systematic over-estimation of myocardial perfusion from microspheres at the subendocardium. Our objective is to integrate coronary arterial structure and microsphere distribution, imaged at high resolution, to test the hypothesis of microsphere skimming in a porcine left coronary arterial (LCA) network. A detailed network was reconstructed from cryomicrotome imaging data and a Poiseuille flow model was used to simulate flow. A statistical approach using Clopper-Pearson confidence intervals was applied to determine the prevalence of skimming at bifurcations in the LCA. Results reveal that microsphere skimming is most prevalent at bifurcations in the larger coronary arteries, namely the epicardial and transmural arteries. Bifurcations at which skimming was identified have significantly more asymmetric branching parameters. This finding suggests that when using thin transmural segments to quantify flow from microspheres, a skimming-related deposition bias may result in underestimation of perfusion in the subepicardium, and overestimation in the subendocardium.

Algorithmically generated rodent hepatic vascular trees in arbitrary detail

Physiologically realistic geometric models of the vasculature in the liver are indispensable for modelling hepatic blood flow, the main connection between the liver and the organism. Current in vivo imaging techniques do not provide sufficiently detailed vascular trees for many simulation applications, so it is necessary to use algorithmic refinement methods. The method of Constrained Constructive Optimization (CCO) (Schreiner et al., 2006) is well suited for this purpose. Its results after calibration have been previously compared to experimentally acquired human vascular trees (Schwen and Preusser, 2012). The goal of this paper is to extend this calibration to the case of rodents (mice and rats), the most commonly used animal models in liver research. Based on in vivo and ex vivo micro-CT scans of rodent livers and their vasculature, we performed an analysis of various geometric features of the vascular trees. Starting from pruned versions of the original vascular trees, we applied the CCO procedure and compared these algorithmic results to the original vascular trees using a suitable similarity measure. The calibration of the postprocessing improved the algorithmic results compared to those obtained using standard CCO. In terms of angular features, the average similarity increased from 0.27 to 0.61, improving the total similarity from 0.28 to 0.40. Finally, we applied the calibrated algorithm to refine measured vascular trees to the (higher) level of detail desired for specific applications. Having successfully adapted the CCO algorithm to the rodent model organism, the resulting individual-specific refined hepatic vascular trees can now be used for advanced modeling involving, e.g., detailed blood flow simulations.

Physiology of endothelin in producing myocardial perfusion heterogeneity: a mechanistic study using darusentan and positron emission tomography

BACKGROUND

Heterogeneity of resting perfusion may be due in part to up-regulation of coronary vasoconstriction via endothelin (ET) type A receptors, as homogeneity increases during subsequent vasodilatory hyperemia. Therefore, we conducted a mechanistic study using an ET receptor antagonist to determine if it could alter the homogeneity of myocardial perfusion.

METHODS

Included subjects demonstrated a low myocardial perfusion homogeneity index (HI) compared to normal volunteers. Four serial cardiac positron emission tomography Rb-82 scans were performed 2 weeks apart. Before the middle two scans, subjects were randomized to receive either darusentan first then placebo or visa versa. Absolute flow and coronary flow reserve were quantified for each study. Rest flow was adjusted for the pressure-rate product (PRP).

RESULTS

We screened 37 subjects and randomized 20 who satisfied entry criteria. Rest HI increased significantly while taking darusentan (0.39 ± 0.10 vs 0.33 ± 0.04 on placebo, P = .030, compared to a normal range of 0.52 ± 0.10) without an increase in the PRP (6,859 ± 1,503 vs 6,976 ± 1,092, P = .79), leading to a higher adjusted flow at rest (0.69 ± 0.18 cc/minute/g at 7,000 PRP vs 0.59 ± 0.07 with placebo).

CONCLUSIONS

Antagonism of the type A ET receptor increases homogeneity of resting myocardial perfusion. The mechanism appears to be increased absolute rest flow without an increase in either the PRP or myocardial perfusion during hyperemia. Our translational results are consistent with one mechanism for the observed heterogeneity of myocardial perfusion in humans.

Heterogeneity and matching of ventilation and perfusion within anatomical lung units in rats

Prior studies exploring the spatial distributions of ventilation and perfusion have partitioned the lung into discrete regions not constrained by anatomical boundaries and may blur regional differences in perfusion and ventilation. To characterize the anatomical heterogeneity of regional ventilation and perfusion, we administered fluorescent microspheres to mark regional ventilation and perfusion in five Sprague-Dawley rats and then using highly automated computer algorithms, partitioned the lungs into regions defined by anatomical structures identified in the images. The anatomical regions ranged in size from the near-acinar to the lobar level. Ventilation and perfusion were well correlated at the smallest anatomical level. Perfusion and ventilation heterogeneity were relatively less in rats compared to data previously published in larger animals. The more uniform distributions may be due to a smaller gravitational gradient and/or the fewer number of generations in the distribution trees before reaching the level of gas exchange, making regional matching of ventilation and perfusion less extensive in small animals.

Random stride intervals with memory

The stride interval in normal human gait is not strictly constant, butfluctuates from step to step in a random manner. Herein we show thatcontrary to the traditional assumption of uncorrelated random errors,these fluctuations have long-time memory. However, rather than being amonofractal process as found earlier, there exists a multiplicative timescale that characterizes the process in addition to the fractal dimension.Further, these long-time correlations are interpreted in terms of anallometric control process.

3D Imaging of vascular networks for biophysical modeling of perfusion distribution within the heart

One of the main determinants of perfusion distribution within an organ is the structure of its vascular network. Past studies were based on angiography or corrosion casting and lacked quantitative three dimensional, 3D, representation. Based on branching rules and other properties derived from such imaging, 3D vascular tree models were generated which were rather useful for generating and testing hypotheses on perfusion distribution in organs. Progress in advanced computational models for prediction of perfusion distribution has raised the need for more realistic representations of vascular trees with higher resolution. This paper presents an overview of the different methods developed over time for imaging and modeling the structure of vascular networks and perfusion distribution, with a focus on the heart. The strengths and limitations of these different techniques are discussed. Episcopic fluorescent imaging using a cryomicrotome is presently being developed in different laboratories. This technique is discussed in more detail, since it provides high-resolution 3D structural information that is important for the development and validation of biophysical models but also for studying the adaptations of vascular networks to diseases. An added advantage of this method being is the ability to measure local tissue perfusion. Clinically, indices for patient-specific coronary stenosis evaluation derived from vascular networks have been proposed and high-resolution noninvasive methods for perfusion distribution are in development. All these techniques depend on a proper representation of the relevant vascular network structures.

Modeling to link regional myocardial work, metabolism and blood flows

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, the 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.

Analysis and algorithmic generation of hepatic vascular systems

A proper geometric model of the vascular systems in the liver is crucial for modeling blood flow, the connection between the organ and the rest of the organism. In vivo imaging does not provide sufficient details, so an algorithmic concept for extending measured vascular tree data is needed such that geometrically realistic structures can be generated. We develop a quantification of similarity in terms of different geometric features. This involves topological Strahler ordering of the vascular trees, statistical testing, and averaging. Invariant features are identified in human clinical in vivo CT scans. Results of the existing "Constrained Constructive Optimization" algorithm are compared to real vascular tree data. To improve bifurcation angles in the algorithmic results, we implement a postprocessing step calibrated to the measured features. This framework is finally applied to generate realistic additional details in a patient-specific hepatic vascular tree data set.

Computational analyses of intravascular tracer washout reveal altered capillary-level flow distributions in obese Zucker rats

Intravascular tracer washout data obtained from gastrocnemius muscle of lean Zucker rats (LZRs) and obese Zucker rats (OZRs) were analysed to investigate flow distributions in the OZR, a model of non-atherosclerotic peripheral vascular disease. A computer model used to simulate the network washout curves was developed based on experimentally observed relative dispersions in large vessels and asymmetrical flow distributions at bifurcations in dichotomous microvascular networks. The model results of simulations were compared to experimental washout data of (125)I-labelled albumin, an intravascular tracer, to uncover flow distributions on the arterial-network and capillary levels. The lean and obese Zucker rats demonstrated distinct capillary-level flow distributions, with higher dispersion and significantly more low-flow capillaries in the OZRs than in the LZRs. Targeted pharmacological treatments against identified sites of vascular dysfunction in OZRs (adrenoreceptor blockade with phentolamine, antioxidant treatment with Tempol and thromboxane receptor antagonism with SQ-29548) were shown to improve the capillary-level flow distributions in treated OZRs toward distributions determined in control LZRs. Combination therapy with multiple pharmacological interventions resulted in a greater degree of recovery. This study demonstrates that the enhanced perfusion heterogeneity at arteriole bifurcations is a potential mechanism underlying perfusion-demand mismatching in OZRs, and suggests that amelioration of this dysfunction must involve a multi-faceted interventional approach.

Indicator dilution models for the quantification of microvascular blood flow with bolus administration of ultrasound contrast agents

Indicator dilution methods have a long history in the quantification of both macro- and microvascular blood flow in many clinical applications. Various models have been employed in the past to isolate the primary pass of an indicator after an intravenous bolus injection. The use of indicator dilution techniques allows for the estimation of hemodynamic parameters of a tumor or organ and thus may lead to useful diagnostic and therapy monitoring information. In this paper, we review and discuss the properties of the lognormal function, the gamma variate function, the diffusion with drift models, and the lagged normal function, which have been used to model indicator dilution curves in different fields of medicine. We fit these models to contrast-enhanced ultrasound time-intensity curves from liver metastases and the ovine corpora lutea. We evaluate the models' performance on the image data and compare their predictions for hemodynamic-related parameters such as the area under the curve, the mean transit time, the full-width at half-maximum, the time to the peak intensity, and wash-in time. The models that best fit the experimental data are the lognormal function and the diffusion with drift.

Biophysical model of the spatial heterogeneity of myocardial flow

The blood flow in the myocardium has significant spatial heterogeneity. The objective of this study was to develop a biophysical model based on detailed anatomical data to determine the heterogeneity of regional myocardial flow during diastole. The model predictions were compared with experimental measurements in a diastolic porcine heart in the absence of vessel tone using nonradioactive fluorescent microsphere measurements. The results from the model and experimental measurements showed good agreement. The relative flow dispersion in the arrested, vasodilated heart was found to be 44% and 48% numerically and experimentally, respectively. Furthermore, the flow dispersion was found to have fractal characteristics with fractal dimensions (D) of 1.25 and 1.27 predicted by the model and validated by the experiments, respectively. This validated three-dimensional model of normal diastolic heart will play an important role in elucidating the spatial heterogeneity of coronary blood flow, and serve as a foundation for understanding the interplay between cardiac mechanics and coronary hemodynamics.

Origins of heterogeneity in tissue perfusion and metabolism

AIMS

In the heart and other tissues, perfusion and metabolic activity are heterogeneous and spatially correlated. The goal of this work is to investigate the causes of this behaviour. Theoretical simulations are used to examine the effects on flow distribution and oxygen levels in terminal vascular beds of inherent irregularity in network structure, considering structural adaptation of vessel diameters to haemodynamic and metabolic stimuli, and adaptation of oxygen demand to local oxygen availability.

METHODS AND RESULTS

A mathematical model based on experimentally observed microvascular network structures (rat mesentery and m. sartorius) is used to simulate blood flow, oxygen transport, and adaptation of vessel diameters and tissue oxygen demand. Inherent geometric heterogeneities of vascular networks cause heterogeneity of blood flow and oxygen levels that cannot be eliminated by increasing metabolic sensitivity of diameter adaptation. Adaptation of oxygen demand to differences in oxygen availability causes increased oxygen extraction, implying improved functional capacity, and establishes a correlation between local oxygen demand and flow rate, as observed experimentally. Such a correlation is not predicted if the heterogeneity of oxygen demand is assumed to be an intrinsic tissue property.

CONCLUSION

A central mechanism generating heterogeneous perfusion is the inevitable structural heterogeneity of terminal vascular beds, which cannot be fully compensated by structural adaptation of vessel diameters. Heterogeneity of metabolism may result from adaptation of tissue function to the heterogeneous oxygen availability. These findings are of interest for the understanding of tissue function, including the heart, and of results obtained by corresponding imaging approaches.

Nitric oxide and prostaglandins influence local skeletal muscle blood flow during exercise in humans: coupling between local substrate uptake and blood flow

Synergic action of nitric oxide (NO) and prostaglandins (PG) in the regulation of muscle blood flow during exercise has been demonstrated. In the present study, we investigated whether these vasodilators also regulate local blood flow, flow heterogeneity, and glucose uptake within the exercising skeletal muscle. Skeletal muscle blood flow was measured in seven healthy young men using near-infrared spectroscopy and indocyanine green and muscle glucose uptake using positron emission tomography and 2-fluoro-2-deoxy-d-[18F]glucose without and with local blockade of NO and PG at rest and during one-legged dynamic knee-extension exercise. Local blockade was produced by infusing nitro-l-arginine methyl ester and indomethacin directly in the muscle via a microdialysis catheter. Blood flow and glucose uptake were measured in the region of blockade and in two additional regions of vastus lateralis muscle 1 and 4 cm away from the infusion of blockers. Local blockade during exercise at 25 and 40 watts significantly decreased blood flow in the infusion region and in the region 1 cm away from the site of infusion but not in the region 4 cm away. During exercise, muscle glucose uptake did not show any regional differences in response to blockade. These results show that NO and PG synergistically contribute to the local regulation of blood flow in skeletal muscle independently of muscle glucose uptake in healthy young men. Thus these vasodilators can play a role in regulating microvascular blood flow in localized regions of vastus lateralis muscle but do not influence regional glucose uptake. The findings suggest that local substrate uptake in skeletal muscle can be regulated independently of regional changes in blood flow.

Comparing microsphere deposition and flow modeling in 3D vascular trees

Blood perfusion in organs has been shown to be heterogeneous in a number of cases. At the same time, a number of models of vascular structure and flow have been proposed that also generate heterogeneous perfusion. Although a relationship between local perfusion and vascular structure has to exist, no model has yet been validated as an accurate description of this relationship. A study of perfusion and three-dimensional (3D) arterial structure in individual rat kidneys is presented, which allows comparison between local measurements of perfusion and model-based predictions. High-resolution computed tomography is used to obtain images of both deposited microspheres and of an arterial cast in the same organ. Microsphere deposition is used as an estimate of local perfusion. A 3D cylindrical pipe model of the arterial tree is generated based on an image of the arterial cast. Results of a flow model are compared with local microsphere deposition. High correlation (r(2) > 0.94) was observed between measured and modeled flows through the vascular tree segments. However, the relative dispersion of the microsphere perfusion measurement was two- to threefold higher than perfusion heterogeneity calculated in the flow model. Also, there was no correlation in the residual deviations between the methods. This study illustrates the importance of comparing models of local perfusion with in vivo measurements of perfusion in the same biologically realistic vascular tree.

Bifurcation asymmetry of the porcine coronary vasculature and its implications on coronary flow heterogeneity

The branching pattern of the coronary arteries and veins is asymmetric, i.e., many small vessels branch off of a large trunk such that the two daughter vessels at a bifurcation are of unequal diameters and lengths. One important implication of the geometric vascular asymmetry is the dispersion of blood flow at a bifurcation, which leads to large spatial heterogeneity of myocardial blood flow. To document the asymmetric branching pattern of the coronary vessels, we computed an asymmetry ratio for the diameters and lengths of all vessels, defined as the ratio of the daughter diameters and lengths, respectively. Previous data from silicone elastomer cast of the entire coronary vasculature including arteries, arterioles, venules, and veins were analyzed. Data on smaller vessels were obtained from histological specimens by optical sectioning, whereas data on larger vessels were obtained from vascular casts. Asymmetry ratios for vascular areas, volumes, resistances, and flows of the various daughter vessels were computed from the asymmetry ratios of diameters and lengths for every order of mother vessel. The results show that the largest orders of arterial and venous vessels are most asymmetric and the degree of asymmetry decreases toward the smaller vessels. Furthermore, the diameter asymmetry at a bifurcation is significantly larger for the coronary veins (1.7–6.8 for sinus veins) than the corresponding arteries (1.5–5.8 for left anterior descending coronary artery) for orders 2–10, respectively. The reported diameter asymmetry at a bifurcation leads to significant heterogeneity of blood flow at a bifurcation. Hence, the present data quantify the dispersion of blood flow at a bifurcation and are essential for understanding flow heterogeneity in the coronary circulation.

High-resolution imaging reveals a limit in spatial resolution of blood flow measurements by microspheres

Density of 15-μm microspheres after left atrial application is the standard measure of regional perfusion. In the heart, substantial differences in microsphere density are seen at spatial resolutions <5 ml, implying perfusion heterogeneity. Microsphere deposition imaging permits a superior evaluation of the distribution pattern. Therefore, fluorescent microspheres (FMS) were applied, FMS deposition in the canine heart was imaged by epifluorescence microscopy in vitro, and the patterns were observed compared with MR images of iron oxide microspheres (IMS) obtained in vivo and in vitro. FMS deposition in myocardial slices revealed the following: 1) a nonrandom distribution, with sequentially applied FMS of different color stacked within the same vessel, 2) general FMS clustering, and 3) rather large areas devoid of FMS ( n = 3). This pattern was also seen in reconstructed three-dimensional images (<1 nl resolution) of FMS distribution ( n = 4). Surprisingly, the deposition pattern of sequentially applied FMS remained virtually identical over 3 days. Augmenting flow by intracoronary adenosine (>2 μM) enhanced local microsphere density, but did not alter the deposition pattern ( n = 3). The nonrandom, temporally stable pattern was quantitatively confirmed by a three-dimensional intermicrosphere distance analysis of sequentially applied FMS. T2-weighted short-axis MR images (2-μl resolution) of IMS revealed similar patterns in vivo and in vitro ( n = 6), as seen with FMS. The observed temporally stable microsphere patterns are not consistent with the notion that microsphere deposition is solely governed by blood flow. We propose that at high spatial resolution (<2 μl) structural aspects of the vascular network dominate microsphere distribution, resulting in the organized patterns observed.

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.

Flow and pressure distributions in vascular networks consisting of distensible vessels

We examine the influence of vessel distensibility on the fraction of the total network flow passing through each vessel of a model vascular network. An exact computational methodology is developed yielding an analytic proof. For a class of structurally heterogeneous asymmetric vascular networks, if all the individual vessels share a common distensibility relation when the total network flow is changed, this methodology proves that each vessel will continue to receive the same fraction of the total network flow. This constant flow partitioning occurs despite a redistribution of pressures, which may result in a decrease in the diameter of one and an increase in the diameter of the other of two vessels having a common diameter at a common pressure. This theoretical observation, taken along with published experimental observations on pulmonary vessel distensibilities, suggests that vessel diameter-independent distensibility in the pulmonary vasculature may be an evolutionary adaptation for preserving the spatial distribution of pulmonary blood flow in the face of large variations in cardiac output.

Branching tree model with fractal vascular resistance explains fractal perfusion heterogeneity

Perfusion heterogeneities in organs such as the heart obey a power law as a function of scale, a behavior termed "fractal." An explanation of why vascular systems produce such a specific perfusion pattern is still lacking. An intuitive branching tree model is presented that reveals how this behavior can be generated as a consequence of scale-independent branching asymmetry and fractal vessel resistance. Comparison of computer simulations to experimental data from the sheep heart shows that the values of the two free model parameters are realistic. Branching asymmetry within the model is defined by the relative tissue volume being fed by each branch. Vessel ordering for fractal analysis of morphology based on fed or drained tissue volumes is preferable to the commonly used Strahler system, which is shown to depend on branching asymmetry. Recently, noninvasive imaging techniques such as PET and MRI have been used to measure perfusion heterogeneity. The model allows a physiological interpretation of the measured fractal parameters, which could in turn be used to characterize vascular morphology and function.

Muscle fractal vascular branching pattern and microvascular perfusion heterogeneity in endurance-trained and untrained men

Less heterogeneous skeletal muscle perfusion has recently been reported in endurance-trained compared to untrained men at macrovascular level. The causes of this difference in perfusion heterogeneity are unknown as is whether the same difference is observed in microvasculature. We hypothesised that the difference could be caused by changes in muscle vascular branching pattern. Perfusion was measured in resting and exercising muscle in 14 endurance-trained and seven untrained men using [(15)O]water and positron emission tomography. Fractal dimension (D) of perfusion distribution was calculated as a measure of fractal characteristics of muscle vascular branching pattern. Perfusion heterogeneity in microvascular units (1 mm(3) samples) was estimated using the measured heterogeneity in voxels of positron emission tomography (PET) images (relative dispersion, RD = S.D./mean) and corresponding D values. D was similar between the groups (exercising muscle 1.11 +/- 0.07 and 1.14 +/- 0.06, resting muscle 1.12 +/- 0.06 and 1.14 +/- 0.03, trained and untrained, respectively). Trained men had lower perfusion (151 +/- 44 vs. 218 +/- 87 ml min(-1) kg(-1), P < 0.05) and macrovascular perfusion heterogeneity (relative dispersion 21 +/- 5 vs. 25 +/- 5 %, P < 0.05) in exercising muscle than untrained men. Furthermore, estimated perfusion heterogeneity in microvascular units in exercising muscle was also lower in trained men (33 +/- 7 vs.48 +/- 19 %, P < 0.05). These results show that fractal vascular branching pattern is similar in endurance-trained and untrained men but perfusion is less heterogeneous at both the macro- and the microvascular level in endurance-trained men. Thus, changes in fractal branching pattern do not explain the differences in perfusion heterogeneity between endurance-trained and untrained men.

The mechanical and metabolic basis of myocardial blood flow heterogeneity

Precise measurements of regional myocardial blood flow heterogeneity had to be developed before one could seek causation for the heterogeneity. Deposition techniques (particles or molecular microspheres) are the most precise, but imaging techniques have begun to provide high enough resolution to allow in vivo studies. Assigning causation has been difficult. There is no apparent association with the regional concentrations of energy-related enzymes or substrates, but these are measures of status, not of metabolism. There is statistical correlation between flow and regional substrate uptake and utilization. Attribution of regional flow variation to vascular anatomy or to vasomotor control appears not to be causative on a long-term basis. The closest relationships appear to be with mechanical function, but one cannot say for sure whether this is related to ATP hydrolysis at the crossbridge or associated metabolic reactions such as calcium uptake by the sarcoplasmic reticulum.

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

The theory of exponential dispersion models was applied to construct a stochastic model for heterogeneities in regional organ blood flow as inferred from the deposition of labeled microspheres. The requirements that the dispersion model be additive (or reproductive), scale invariant, and represent a compound Poisson distribution, implied that the relative dispersion (RD = standard deviation/mean) of blood flow should exhibit self-similar scaling in macroscopic tissue samples of masses m and m(ref) such that RD(m) = RD(m(ref)). (m/m(ref))(1-D), where D was a constant. Under these circumstances this empirical relationship was a consequence of a compound Poisson-gamma distribution that represented macroscopic blood flow. The model also predicted that blood flow, at the microcirculatory level, should also be heterogeneous but obey a gamma distribution-a prediction supported by observation.

Pulmonary blood flow remains fractal down to the level of gas exchange

The spatial distribution of pulmonary blood flow is increasingly heterogeneous as progressively smaller lung regions are examined. To determine the extent of perfusion heterogeneity at the level of gas exchange, we studied blood flow distributions in rat lungs by using an imaging cryomicrotome. Approximately 150,000 fluorescent 15-microm-diameter microspheres were injected into tail veins of five awake rats. The rats were heavily anesthetized; the lungs were removed, filled with an optimal cutting tissue compound, and frozen; and the spatial location of every microsphere was determined. The data were mathematically dissected with the use of an unbiased random sampling method. The coefficients of variation of microsphere distributions were determined at varying sampling volumes. Perfusion heterogeneity increased linearly on a log-log plot of coefficient of variation vs. volume, down to the smallest sampling size of 0.53 mm(3). The average fractal dimension, a scale-independent measure of perfusion distribution, was 1.2. This value is similar to that of other larger species such as dogs, pigs, and horses. Pulmonary perfusion heterogeneity increases continuously and remains fractal down to the acinar level. Despite the large degree of perfusion heterogeneity at the acinar level, gases are efficiently exchanged.

Effects of primary resuscitation from shock on distribution of myocardial blood flow

Hemorrhagic shock alters heterogeneity of regional myocardial perfusion (RMP) in the presence of critical coronary stenosis in pigs. Conventional resuscitation has failed to reverse these effects. We hypothesized that improvement of the resuscitation regime would lead to restoration of RMP heterogeneity. Diaspirin-cross-linked hemoglobin (10 g/dl; DCLHb) and human serum albumin (8.0 g/dl; HSA) were used. After baseline, a branch of the left coronary artery was stenosed; thereafter, hemorrhagic shock was induced. Resuscitation was performed with either DCLHb or HSA. At baseline, the fractcal dimension (D) of subendocardial myocardium was 1.31 +/- 0.083 (HSA) and 1.35 +/- 0.106 (DCLHb) (mean +/- SD). Coronary stenosis increased subendocardial D slightly but consistently only in the DCLHb group (1.39 +/- 0.104; P < 0.05). Shock reduced subendocardial D: 1.21 +/- 0.093 (HSA; P = 0.10), 1.25 +/- 0.092 (DCLHb; P < 0.05). Administration of DCLHb increased subendocardial D in 7 of 10 animals (1.31 +/- 0.097; P = 0.066). HSA was ineffective in this respect. DCLHb infusion restored arterial pressure and increased cardiac index (CI) to 80% of baseline values. Administration of HSA left animals hypotensive (69 mmHg) and increased CI to 122% of the average baseline value. Shock-induced disturbances of the distribution of RMP were improved by administration of DCLHb but not by HSA.

The 400 microsphere per piece "rule" does not apply to all blood flow studies

Microsphere experiments are useful in measuring regional organ perfusion as well as heterogeneity of blood flow within organs and correlation of perfusion between organ pieces at different time points. A 400 microspheres/piece "rule" is often used in planning experiments or to determine whether experiments are valid. This rule is based on the statement that 400 microspheres must lodge in a region for 95% confidence that the observed flow in the region is within 10% of the true flow. The 400 microspheres precision rule, however, only applies to measurements of perfusion to a single region or organ piece. Examples, simulations, and an animal experiment were carried out to show that good precision for measurements of heterogeneity and correlation can be obtained from many experiments with <400 microspheres/piece. Furthermore, methods were developed and tested for correcting the observed heterogeneity and correlation to remove the Poisson "noise" due to discrete microsphere measurements. The animal experiment shows adjusted values of heterogeneity and correlation that are in close agreement for measurements made with many or few microspheres/piece. Simulations demonstrate that the adjusted values are accurate for a variety of experiments with far fewer than 400 microspheres/piece. Thus the 400 microspheres rule does not apply to many experiments. A "rule of thumb" is that experiments with a total of at least 15,000 microspheres, for all pieces combined, are very likely to yield accurate estimates of heterogeneity. Experiments with a total of at least 25,000 microspheres are very likely to yield accurate estimates of correlation coefficients.

Fractal fluctuations in cardiac time series

Human heart rate, controlled by complex feedback mechanisms, is a vital index of systematic circulation. However, it has been shown that beat-to-beat values of heart rate fluctuate continually over a wide range of time scales. Herein we use the relative dispersion, the ratio of the standard deviation to the mean, to show, by systematically aggregating the data, that the correlation in the beat-to-beat cardiac time series is a modulated inverse power law. This scaling property indicates the existence of long-time memory in the underlying cardiac control process and supports the conclusion that heart rate variability is a temporal fractal. We argue that the cardiac control system has allometric properties that enable it to respond to a dynamical environment through scaling.

Heterogeneities in myocardial flow and metabolism: exacerbation with abnormal excitation

Because regional myocardial blood flows are remarkably heterogeneous-with a 6- to 10-fold range of flows in normal hearts-and because the spatial profiles of the flows are stable over long periods and over a range of conditions, the relation between flows and other physiologic functions has been explored. Local fatty acid uptake and oxygen consumption are almost linearly related to the flows. Coronary network structure and hydrodynamic resistances give suitable flow heterogeneity but are thought to be a response to local needs rather than being causative. Presumably the cause is the need for adenosine triphosphate (ATP) synthesis locally, and therefore flows, substrate delivery, and oxygen utilization are driven primarily by local rates of ATP hydrolysis, mainly by contractile proteins. This hypothesis is by no means fully tested. Data on pacing dog hearts from different sites, on patients with left bundle branch block, and on unloading transplanted rat hearts, all point in the same direction: unloading ventricular muscle leads to diminished flow and exaggeratedly diminished glucose uptake. The mechanism is likely to be that discovered by Taegtmeyer and colleagues, namely, the expression of fetal genes in regions where the muscle is unloaded and particular metabolic enzymes and transporters are downregulated.

The Cardiome Project. An integrated view of cardiac metabolism and regional mechanical function

The goal, to develop a functioning three-dimensional computational model of the excitation, metabolism and contraction of the heart within three years, is one of the beginnings for the Cardiome Project. Our first stage will not be likely to provide highly accurate prediction of physiological behavior in general, but will be focussed so that it is adequate for at least three specific purposes: response to regional flow reduction, response to heart rate changes, and response to increased metabolic drive. We would like to make the model visualizable by three-dimensional viewing, with cross-sectional and transparency viewing approaches, illustrate the fiber directions, the arteries, the deformation with contraction and images of regional functions such as oxygen consumption, preejection strain, or lactate concentration. The display techniques developed by Hunter et al. and by McCulloch et al. would be excellent for such demonstration and teaching purposes, and should be attractive enough for public display. The Physiome Project is underway now, with growing government and private support. Now we are going from the era of molecular biology, led by the Genome Project, into a new era of integrative biology. The goal is to understand biology so deeply and so broadly that predictions about interventions can be made. Methods of experimentation and of diagnosis are critical to acquiring the data, and therefore in making the prediction, and so all aspects of our Society's efforts and interests are relevant to undertaking this grand challenge.

The dynamic regulation of myocardial oxidative phosphorylation: analysis of the response time of oxygen consumption

Although usually steady-state fluxes and metabolite levels are assessed for the study of metabolic regulation, much can be learned from studying the transient response during quick changes of an input to the system. To this end we study the transient response of O2 consumption in the heart during steps in heart rate. The time course is characterized by the mean response time of O2 consumption which is the first statistical moment of the impulse response function of the system (for mono-exponential responses equal to the time constant). The time course of O2 uptake during quick changes is measured with O2 electrodes in the arterial perfusate and venous effluent of the heart, but the venous signal is delayed with respect to O2 consumption in the mitochondria due to O2 diffusion and vascular transport. We correct for this transport delay by using the mass balance of O2, with all terms (e.g. O2 consumption and vascular O2 transport) taken as function of time. Integration of this mass balance over the duration of the response yields a relation between the mean transit time for O2 and changes in cardiac O2 content. Experimental data on the response times of venous [O2] during step changes in arterial [O2] or in perfusion flow are used to calculate the transport time between mitochondria and the venous O2 electrode. By subtracting the transport time from the response time measured in the venous outflow the mean response time of mitochondrial O2 consumption (tmito) to the step in heart rate is obtained. In isolated rabbit heart we found that tmito to heart rate steps is 4-12 s at 37 degrees C. This means that oxidative phosphorylation responds to changing ATP hydrolysis with some delay, so that the phosphocreatine levels in the heart must be decreased, at least in the early stages after an increase in cardiac ATP hydrolysis. Changes in ADP and inorganic phosphate (Pi) thus play a role in regulating the dynamic adaptation of oxidative phosphorylation, although most steady state NMR measurements in the heart had suggested that ADP and Pi do not change. Indeed, we found with 31P-NMR spectroscopy that phosphocreatine (PCr) and Pi change in the first seconds after a quick change in ATP hydrolysis, but remarkably they do this significantly faster (time constant approximately 2.5 s) than mitochondrial O2 consumption (time constant 12 s). Although it is quite likely that other factors besides ADP and Pi regulate cardiac oxidative phosphorylation, a fascinating alternative explanation is that the first changes in PCr measured with NMR spectroscopy took exclusively place in or near the myofibrils, and that a metabolic wave must then travel with some delay to the mitochondria to stimulate oxidative phosphorylation. The tmito slows with falling temperature, intracellular acidosis, and sometimes also during reperfusion following ischemia and with decreased mitochondrial aerobic capacity. In conclusion, the study of the dynamic adaptation of cardiac oxidative phosphorylation to demand using the mean response time of cardiac mitochondrial O2 consumption is a very valuable tool to investigate the regulation of cardiac mitochondrial energy metabolism in health and disease.

Blood flows and metabolic components of the cardiome

This is a plan for the first stage of The Cardiome Project. The cardiome is the representation, in quantitative, testable form, of the functioning of the normal heart and its responses to intervention. The goal is to integrate the efforts of many years into a comprehensive understandable scheme. Past efforts have spanned the fields of transport within blood vessels, the distributions of regional coronary blood flows, permeation processes through capillary and cell walls, mediated cell membrane transport, extra- and intracellular diffusion, cardiac electrophysiology, the uptake and metabolism of the prime substrates (fatty acid and glucose), the metabolism of the purine nucleosides and nucleotides (mainly adenosine and ATP), the regulation of the ionic currents and of excitation-contraction coupling and finally the regulation of contraction. The central theme is to define the coronary flows and metabolic components of a computer model that will become a part of a three-dimensional heart with appropriate fibre shortening and volume ejection. The components are: (a) coronary flow distributions with appropriate heterogeneity, (b) metabolism of the substrates for energy production, (c) ATP, PCr and energy metabolism and (d) calcium metabolism as it relates to excitation-contraction coupling. The modeling should provide: (1) appropriate responses to regional ischemia induced by constriction of a coronary artery, including tissue contractility loss and aneurysmal dilation of the ischemic region; (2) physiological responses to rate changes such as treppe and changes in metabolic demand and (3) changes in local metabolic needs secondary to changes in the site of pacing stimulation and shortening inactivation or stretch activation of contraction.

Limited bifurcation asymmetry in coronary arterial tree models generated by constrained constructive optimization

Models of coronary arterial trees are generated by the algorithm of constrained constructive optimization (CCO). In a given perfusion area a binary branching network of straight cylindrical tubes is generated by successively adding terminal segments to the growing structure. In each step the site of connection is chosen according to an optimization target function (total intravascular volume), and in any stage of development the tree fulfills physiologic boundary conditions (constraints involving pressures, flows and bifurcation rules). CCO generates structures which in many aspects resemble real coronary arterial trees, except for very asymmetric bifurcations, occurring when a large branch gives off a tiny terminal segment. In the present work we evaluate an additional constraint within CCO, namely imposing a limit on the asymmetry of bifurcations during the construction process. Model trees are grown with different limits imposed, and the effects on structure are studied both phenomenologically and via statistical descriptors. As the limit to asymmetry is tightened, blood is conveyed to the perfusion sites via detours rather than directly and the comparison with measured data shows that the structure to change from a conveying to a delivering type of function. Simultaneously total intravascular volume, surface and sum of segments' lengths increase. It is shown why and how local bifurcation asymmetry is able to determine the global structure of the optimized arterial tree model. Surprisingly, the pressure profile from inlet to terminals, being a functional characteristic, remains unaffected.

A fractal approach to heterogeneous drug distribution: calcium pharmacokinetics

PURPOSE

To point out the importance of heterogeneity in drug distribution processes and develop a noncompartmental approach for the description of the distribution of drug in the body.

METHODS

A dichotomous branching network of vessels for the arterial tree connected to a similar venous network was used to describe the heterogeneity of blood flow in the successive generations of the networks. The relevant kinetics of drug distribution in the well perfused and the deep tissues was considered to take place under well stirred (homogeneous) and understirred (heterogeneous) conditions, respectively.

RESULTS

A "homogeneous model" with classical kinetics (which is mathematically equivalent with the one-compartment model) was developed for these drugs which are confined to well perfused ("well stirred") spaces. A "heterogeneous model" was proposed for the drugs reaching understirred spaces using a decreasing with time rate coefficient (fractal kinetics) to model the diffusion of drug under heterogeneous conditions. The analysis of the model equations revealed that the homogeneous model can be considered as a special case of the heterogeneous model. Concentration-time plots of multiexponential type were generated using the heterogeneous model equation. The empirically used power functions of time for the analysis of calcium clearance curves, were found to be similar to the equation adhering to the heterogeneous model. Fittings comparable to multiexponential models were obtained when the heterogeneous model equation with only one adjustable parameter was applied to six sets of long period calcium data.

CONCLUSIONS

The heterogeneous processes of drug distribution in the body can obey the principles of fractal kinetics. Calcium clearance curves were analysed with the heterogeneous model. The validity of multicompartmental models which are based on the concept of homogeneity to describe drug distribution should be reconsidered.

Fractal 15O-labeled water washout from the heart

To characterize the washout of water from the heart, we used a flow-limited (not diffusion- or permeability-limited) marker for blood-tissue exchange, namely, tracer-labeled water. Experiments were performed by injecting 15O-labeled water into the inflow to isolated blood-perfused rabbit hearts and by recording the tracer content in the heart and in the outflow simultaneously for up to 5 minutes. The data exhibit a particular combination of power law forms: (1) The downslopes of the residue and outflow curves were both power law functions, with the residue diminishing as t-alpha and the outflow as t-alpha-1, where alpha is interpreted to be the dimensionless exponent of a fractal power law relation characterizing the self-similarity inherent in each curve. (2) The fractional escape rate, given by the outflow curve divided by the residue curve, diminished almost exactly as t-1. In 18 sets of curves, alpha averaged 2.21 +/- 0.27. These results lead to an improved method for extrapolating the downslopes of indicator dilution curves to estimate their areas and therefore the blood flows. The evidence also points strongly to the conclusions that myocardial water washout is a fractal process and that stirred tank models are inappropriate for the heart.

Autoradiographic assessment of blood flow heterogeneity in the hamster heart

OBJECTIVE

Provide regional flow measurement in the hearts of small mammals using a new, higher-resolution technique based on the deposition of a molecular marker.

METHODS

We determined the instantaneous extraction and retention of the "molecular microsphere" radiolabeled desmethylimipramine in retrogradely perfused hamster hearts. In a separate series of experiments, autoradiography was used to measure regional myocardial deposition densities in hamster hearts of about 0.5 g with spatial area resolution of 16 x 16 microns.

RESULTS

Radiolabeled desmethylimipramine is almost 100% extracted during a single transcapillary passage and is retained in the tissue for many minutes. Autoradiographic images demonstrated a spatial flow heterogeneity with standard deviations of 31 +/- 4% of the mean flow (N = 5) in 16 x 16 x 20-micronm3 voxels. This is equivalent to the projections made using fractal relationships from cruder observations obtained with microspheres in the hearts of baboons, sheep, and rabbits.

CONCLUSIONS

Autoradiography using a molecular deposition marker provides quantitative information on myocardial flow heterogeneities with resolution at the size of cardiac myocytes. Because the regions resolved are smaller than the volume of regions supplied by single arterioles, the results must slightly exaggerate the true heterogeneity of regional flows.

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.

FRACTAL VASCULAR GROWTH PATTERNS

Flow distributions in the heart and lung are heterogeneous but not at all random. The apparent degree of heterogeneity increases as one reduces the size of observable elements; the fact that the dispersion of flows shows a logarithmic relation to element size says that the system is statistically fractal. The fractal characterization is a statement that the system is nonrandom and that it shows correlation. The close near neighbor correlation has as the corollary of long tailing or falloff in correlation with distance, so called spatial persistence. Correlation can be expected because flow is delivered via a branching vascular system, and so it appears that the structure of the vasculature itself contributes. Since it is also practical and efficient for growth to occur via recursive rules, such as branch, grow, and repeat the branching and growing, it appears that fractals may be useful in understanding the ontological aspects of growth of tissues and organs, thereby minimizing the requirements for genetic material.

Applications of fractal analysis to physiology

This review describes approaches to the analysis of fractal properties of physiological observations. Fractals are useful to describe the natural irregularity of physiological systems because their irregularity is not truly random and can be demonstrated to have spatial or temporal correlation. The concepts of fractal analysis are introduced from intuitive, visual, and mathematical perspectives. The regional heterogeneities of pulmonary and myocardial flows are discussed as applications of spatial fractal analysis, and methods for estimating a fractal dimension from physiological data are presented. Although the methods used for fractal analyses of physiological data are still under development and will require additional validation, they appear to have great potential for the study of physiology at scales of resolution ranging from the microcirculation to the intact organism.

Fractal descriptions for spatial statistics

Measures of spatial statistics have been available for estimating means, calculating or assessing differences, estimating nearest neighbor distances, and such, but have not provided a general approach to describing variances. Because measures of heterogeneity depend upon choosing a particular element size in the domain, estimates of apparent heterogeneity are larger with high-resolution observations than with low-resolution data. Many descriptors might be used to describe the relationships between apparent heterogeneity and the size of the observed spatial elements, but we have found that fractal relationships provide concise and precise descriptions of many types of data over large ranges of element sizes. Perhaps more importantly, the fractal approaches give additional insight, such as measures of spatial correlation, and often suggest ways of approaching the underlying basis of the heterogeneity.