Fractal branchings: the basis of myocardial flow heterogeneities?

Characteristics of users of HIV self-testing in Kenya, outcomes, and factors associated with use: results from a population-based HIV impact assessment, 2018

Background and setting

About 20% of persons living with HIV aged 15–64 years did not know their HIV status in Kenya, by 2018. Kenya adopted HIV self-testing (HIVST) to help close this gap. We examined the sociodemographic characteristics and outcomes of self-reported users of HIVST as our primary outcome.

Methods

We used data from a 2018 population-based cross-sectional household survey in which we included self-reported sociodemographic and behavioral characteristics and HIV test results. To compare weighted proportions, we used the Rao-Scott χ-square test and Jackknife variance estimation. In addition, we used logistic regression to identify associations of sociodemographic, behavioral, and HIVST utilization.

Results

Of the 23,673 adults who reported having ever tested for HIV, 937 (4.1%) had ever self-tested for HIV. There were regional differences in HIVST, with Nyanza region having the highest prevalence (6.4%), p < 0.001. Factors independently associated with having ever self-tested for HIV were secondary education (adjusted odds ratio [aOR], 3.5 [95% (CI): 2.1–5.9]) compared to no primary education, being in the third (aOR, 1.7 [95% CI: 1.2–2.3]), fourth (aOR, 1.6 [95% CI: 1.1–2.2]), or fifth (aOR, 1.8 [95% CI: 1.2–2.7]) wealth quintiles compared to the poorest quintile and having one lifetime sexual partner (aOR, 1.8 [95% CI: 1.0–3.2]) or having ≥ 2 partners (aOR, 2.1 [95% CI: 1.2–3.7]) compared to none. Participants aged ≥ 50 years had lower odds of self-testing (aOR, 0.6 [95% CI: 0.4–1.0]) than those aged 15–19 years.

Conclusion

Kenya has made progress in rolling out HIVST. However, geographic differences and social demographic factors could influence HIVST use. Therefore, more still needs to be done to scale up the use of HIVST among various subpopulations. Using multiple access models could help ensure equity in access to HIVST. In addition, there is need to determine how HIVST use may influence behavior change towardsaccess to prevention and HIV treatment services.

Morphometric, Hemodynamic, and Multi-Omics Analyses in Heart Failure Rats with Preserved Ejection Fraction

(1) Background: There are no successive treatments for heart failure with preserved ejection fraction (HFpEF) because of complex interactions between environmental, histological, and genetic risk factors. The objective of the study is to investigate changes in cardiomyocytes and molecular networks associated with HFpEF. (2) Methods: Dahl salt-sensitive (DSS) rats developed HFpEF when fed with a high-salt (HS) diet for 7 weeks, which was confirmed by in vivo and ex vivo measurements. Shotgun proteomics, microarray, Western blot, and quantitative RT-PCR analyses were further carried out to investigate cellular and molecular mechanisms. (3) Results: Rats with HFpEF showed diastolic dysfunction, impaired systolic function, and prolonged repolarization of myocytes, owing to an increase in cell size and apoptosis of myocytes. Heatmap of multi-omics further showed significant differences between rats with HFpEF and controls. Gene Set Enrichment Analysis (GSEA) of multi-omics revealed genetic risk factors involved in cardiac muscle contraction, proteasome, B cell receptor signaling, and p53 signaling pathway. Gene Ontology (GO) analysis of multi-omics showed the inflammatory response and mitochondrial fission as top biological processes that may deteriorate myocyte stiffening. GO analysis of protein-to-protein network indicated cytoskeleton protein, cell fraction, enzyme binding, and ATP binding as the top enriched molecular functions. Western blot validated upregulated Mff and Itga9 and downregulated Map1lc3a in the HS group, which likely contributed to accumulation of aberrant mitochondria to increase ROS and elevation of myocyte stiffness, and subsequent contractile dysfunction and myocardial apoptosis. (4) Conclusions: Multi-omics analysis revealed multiple pathways associated with HFpEF. This study shows insight into molecular mechanisms for the development of HFpEF and may provide potential targets for the treatment of HFpEF.

Bioprinting Vasculature: Materials, Cells and Emergent Techniques

Despite the great advances that the tissue engineering field has experienced over the last two decades, the amount of in vitro engineered tissues that have reached a stage of clinical trial is limited. While many challenges are still to be overcome, the lack of vascularization represents a major milestone if tissues bigger than approximately 200 µm are to be transplanted. Cell survival and homeostasis is to a large extent conditioned by the oxygen and nutrient transport (as well as waste removal) by blood vessels on their proximity and spontaneous vascularization in vivo is a relatively slow process, leading all together to necrosis of implanted tissues. Thus, in vitro vascularization appears to be a requirement for the advancement of the field. One of the main approaches to this end is the formation of vascular templates that will develop in vitro together with the targeted engineered tissue. Bioprinting, a fast and reliable method for the deposition of cells and materials on a precise manner, appears as an excellent fabrication technique. In this review, we provide a comprehensive background to the fields of vascularization and bioprinting, providing details on the current strategies, cell sources, materials and outcomes of these studies.

Optimal Branching Structure of Fluidic Networks with Permeable Walls

Biological and engineering studies of Hess-Murray's law are focused on assemblies of tubes with impermeable walls. Blood vessels and airways have permeable walls to allow the exchange of fluid and other dissolved substances with tissues. Should Hess-Murray's law hold for bifurcating systems in which the walls of the vessels are permeable to fluid? This paper investigates the fluid flow in a porous-walled T-shaped assembly of vessels. Fluid flow in this branching flow structure is studied numerically to predict the configuration that provides greater access to the flow. Our findings indicate, among other results, that an asymmetric flow (i.e., breaking the symmetry of the flow distribution) may occur in this symmetrical dichotomous system. To derive expressions for the optimum branching sizes, the hydraulic resistance of the branched system is computed. Here we show the T-shaped assembly of vessels is only conforming to Hess-Murray's law optimum as long as they have impervious walls. Findings also indicate that the optimum relationship between the sizes of parent and daughter tubes depends on the wall permeability of the assembled tubes. Our results agree with analytical results obtained from a variety of sources and provide new insights into the dynamics within the assembly of vessels.

Quantitative angiography methods for bifurcation lesions: a consensus statement update from the European Bifurcation Club

Bifurcation lesions represent one of the most challenging lesion subsets in interventional cardiology. The European Bifurcation Club (EBC) is an academic consortium whose goal has been to assess and recommend the appropriate strategies to manage bifurcation lesions. The quantitative coronary angiography (QCA) methods for the evaluation of bifurcation lesions have been subject to extensive research. Single-vessel QCA has been shown to be inaccurate for the assessment of bifurcation lesion dimensions. For this reason, dedicated bifurcation software has been developed and validated. These software packages apply the principles of fractal geometry to address the "step-down" in the bifurcation and to estimate vessel diameter accurately. This consensus update provides recommendations on the QCA analysis and reporting of bifurcation lesions based on the most recent scientific evidence from in vitro and in vivo studies and delineates future advances in the field of QCA dedicated bifurcation analysis.

A mathematical model for the vessel recruitment in coronary microcirculation in the absence of active autoregulation

This paper proposes a mathematical model for vessel recruitment in the microvascular coronary network. The model is based on microvascular network units (MVNUs), where we define a MVNU as a portion of the microvascular network comprising seven generations of identical, parallel-arranged vessels (upstream arteries, large and small arterioles, capillaries, small and large venules, and downstream veins). The model implements a new mechanism to describe the variation in the number of MVNU in response to sudden variations of the local input pressure. In particular, it describes a recruitment mechanism dependent on distal pressure which operates in the coronary microcirculatory network even in maximally dilated conditions. We apply the model to interpret data from 29 patients who underwent revascularization by percutaneous coronary intervention (PCI). Treated vessels were the left anterior descending coronary artery, the left circumflex and the right coronary artery in 26, 2 and 1 patients, respectively. Following intracoronary adenosine administration, distal coronary pressure and blood flow were 48 ± 18 mmHg and 45 ± 30 ml/min before PCI, respectively, and significantly increased afterwards to 80 ± 17 mmHg and 68 ± 32 ml/min (p<0.001). The model predicts an increase in MVNU number in patients with preserved wall motion in the myocardial region which underwent PCI. On the contrary, a decrease in MVNU number is predicted by the model in patients with regional dysfunction and implies a relatively lower response of maximal flow to revascularization.

Myocardial Perfusion: Characteristics of Distal Intramyocardial Arteriolar Trees

A combination of experimental, theoretical, and imaging methodologies is used to examine the hierarchical structure and function of intramyocardial arteriolar trees in porcine hearts to provide a window onto a region of myocardial microvasculature which has been difficult to fully explore so far. A total of 66 microvascular trees from 6 isolated myocardial specimens were analyzed, with a cumulative number of 2438 arteriolar branches greater than or equal to 40 μm lumen diameter. The distribution of flow rates within each tree was derived from an assumed power law relationship for that tree between the diameter of vessel segments and flow rates that are consistent with that power law and subject to conservation of mass along hierarchical structure of the tree. The results indicate that the power law index increases at levels of arteriolar vasculature closer to the capillary level, consistent with a concomitant decrease in shear stress acting on endothelial tissue. These results resolve a long standing predicament which could not be resolved previously because of lack of data about the 3D, interconnected, arterioles. In the context of myocardial perfusion, the results indicate that the coefficient of variation of flow rate in pre-capillary distal arterioles is high, suggesting that heterogeneity of flow rate in these arterioles is not entirely random but may be due at least in part to active control.

Introduction to multifractal detrended fluctuation analysis in matlab

Fractal structures are found in biomedical time series from a wide range of physiological phenomena. The multifractal spectrum identifies the deviations in fractal structure within time periods with large and small fluctuations. The present tutorial is an introduction to multifractal detrended fluctuation analysis (MFDFA) that estimates the multifractal spectrum of biomedical time series. The tutorial presents MFDFA step-by-step in an interactive Matlab session. All Matlab tools needed are available in Introduction to MFDFA folder at the website www.ntnu.edu/inm/geri/software. MFDFA are introduced in Matlab code boxes where the reader can employ pieces of, or the entire MFDFA to example time series. After introducing MFDFA, the tutorial discusses the best practice of MFDFA in biomedical signal processing. The main aim of the tutorial is to give the reader a simple self-sustained guide to the implementation of MFDFA and interpretation of the resulting multifractal spectra.

Intraspecific scaling laws of vascular trees

A fundamental physics-based derivation of intraspecific scaling laws of vascular trees has not been previously realized. Here, we provide such a theoretical derivation for the volume-diameter and flow-length scaling laws of intraspecific vascular trees. In conjunction with the minimum energy hypothesis, this formulation also results in diameter-length, flow-diameter and flow-volume scaling laws. The intraspecific scaling predicts the volume-diameter power relation with a theoretical exponent of 3, which is validated by the experimental measurements for the three major coronary arterial trees in swine (where a least-squares fit of these measurements has exponents of 2.96, 3 and 2.98 for the left anterior descending artery, left circumflex artery and right coronary artery trees, respectively). This scaling law as well as others agrees very well with the measured morphometric data of vascular trees in various other organs and species. This study is fundamental to the understanding of morphological and haemodynamic features in a biological vascular tree and has implications for vascular disease.

A full 3-D reconstruction of the entire porcine coronary vasculature

We have previously reconstructed the entire coronary arterial tree of the porcine heart down to the first segment of capillaries. Here, we extend the vascular model through the capillary bed and the entire coronary venous system. The reconstruction was based on comprehensive morphometric data previously measured in the porcine heart. The reconstruction was formulated as a large-scale optimization process, subject to both global constraints relating to the location of the larger veins and to local constraints of measured morphological features. The venous network was partitioned into epicardial, transmural, and perfusion functional subnetworks. The epicardial portion was generated by a simulated annealing search for the optimal coverage of the area perfused by the arterial epicardial vessels. The epicardial subnetwork and coronary arterial capillary network served as boundary conditions for the reconstruction of the in-between transmural and perfusion networks, which were generated to optimize vascular homogeneity. Five sets of full coronary trees, which spanned the entire network down to the capillary level, were reconstructed. The total number of reconstructed venous segments was 17,148,946 ± 1,049,498 (n = 5), which spanned the coronary sinus (order -12) to the first segment of the venous capillary (order 0v). Combined with the reconstructed arterial network, the number of vessel segments for the entire coronary network added up to 27,307,376 ± 1,155,359 (n = 5). The reconstructed full coronary vascular network agreed with the gross anatomy of coronary networks in terms of structure, location of major vessels, and measured morphometric statistics of native coronary networks. This is the first full model of the entire coronary vasculature, which can serve as a foundation for realistic large-scale coronary flow analysis.

Mechanobiology and the microcirculation: cellular, nuclear and fluid mechanics

Endothelial cells are stimulated by shear stress throughout the vasculature and respond with changes in gene expression and by morphological reorganization. Mechanical sensors of the cell are varied and include cell surface sensors that activate intracellular chemical signaling pathways. Here, possible mechanical sensors of the cell including reorganization of the cytoskeleton and the nucleus are discussed in relation to shear flow. A mutation in the nuclear structural protein lamin A, related to Hutchinson-Gilford progeria syndrome, is reviewed specifically as the mutation results in altered nuclear structure and stiffer nuclei; animal models also suggest significantly altered vascular structure. Nuclear and cellular deformation of endothelial cells in response to shear stress provides partial understanding of possible mechanical regulation in the microcirculation. Increasing sophistication of fluid flow simulations inside the vessel is also an emerging area relevant to the microcirculation as visualization in situ is difficult. This integrated approach to study--including medicine, molecular and cell biology, biophysics and engineering--provides a unique understanding of multi-scale interactions in the microcirculation.

Accuracy of microvascular measurements obtained from micro-CT images

Early changes in branching geometry of microvasculature and its associated impact on the perfusion distribution in diseases, especially those in which different branching generations are affected differently, require the ability to analyze intact vascular trees over a wide range of scales. Micro-CT offers an excellent framework to analyze the microvascular branching geometry. Such an analysis requires methods to be developed that can accurately characterize branching properties, such as branch diameter, length, branching angle, and branch interconnectivity of the microvasculature. The purpose of this article is to report the results of a study of two human intramyocardial coronary vascular tree casts in which the accuracy of micro-CT vascular imaging and its analysis are tested against measurements made through an optical microscope (used as the "gold-standard"). Methods related to image segmentation of the vascular lumen, vessel tree centerline extraction, individual branch segment measurement, and compensating for the non-ideal modulation transfer function of micro-CT scanners are presented. The extracted centerline accurately characterized the hierarchical structure of the vascular tree casts in terms of "parent-branch" relationships which allowed each interbranch segments' dimensions to be compared to the optical measurement method. The comparison results show a close to ideal 1:1 relationship for both length and diameter measurements made by the two methods. Combining the results from both specimens, the standard deviation of the difference between measurement methods was 19 microm for the measurement of interbranch segment diameters (ranging from 12 to 769 microm), and 172 microm for the measurement of interbranch segment lengths (ranging from 14 to 3252 microm). These results suggest that our micro-CT image analysis method can be used to characterize a vascular tree's hierarchical structure, and accurately measure interbranch segment lengths and diameters.

Quantifying the genetic influence on mammalian vascular tree structure

The ubiquity of fractal vascular trees throughout the plant and animal kingdoms is postulated to be due to evolutionary advantages conferred through efficient distribution of nutrients to multicellular organisms. The implicit, and untested, assertion in this theory is that the geometry of vascular trees is heritable. Because vascular trees are constructed through the iterative use of signaling pathways modified by local factors at each step of the branching process, we sought to investigate how genetic and nongenetic influences are balanced to create vascular trees and the regional distribution of nutrients through them. We studied the spatial distribution of organ blood flow in armadillos because they have genetically identical littermates, allowing us to quantify the genetic influence. We determined that the regional distribution of blood flow is strongly correlated between littermates (r(2) = 0.56) and less correlated between unrelated animals (r(2) = 0.36). Using an ANOVA model, we estimate that 67% of the regional variability in organ blood flow is genetically controlled. We also used fractal analysis to characterize the distribution of organ blood flow and found shared patterns within the lungs and hearts of related animals, suggesting common control over the vascular development of these two organs. We conclude that the geometries of fractal vascular trees are heritable and could be selected through evolutionary pressures. Furthermore, considerable postgenetic modifications may allow vascular trees to adapt to local factors and provide a flexibility that would not be possible in a rigid system.

Scaling laws of vascular trees: of form and function

The branching pattern and vascular geometry of biological tree structure are complex. Here we show that the design of all vascular trees for which there exist morphometric data in the literature (e.g., coronary, pulmonary; vessels of various skeletal muscles, mesentery, omentum, and conjunctiva) obeys a set of scaling laws that are based on the hypothesis that the cost of construction of the tree structure and operation of fluid conduction is minimized. The laws consist of scaling relationships between 1) length and vascular volume of the tree, 2) lumen diameter and blood flow rate in each branch, and 3) diameter and length of vessel branches. The exponent of the diameter-flow rate relation is not necessarily equal to 3.0 as required by Murray's law but depends on the ratio of metabolic to viscous power dissipation of the tree of interest. The major significance of the present analysis is to show that the design of various vascular trees of different organs and species can be deduced on the basis of the minimum energy hypothesis and conservation of energy under steady-state conditions. The present study reveals the similarity of nature's scaling laws that dictate the design of various vascular trees and the underlying physical and physiological principles.

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.

Heterogeneous perfusion is a consequence of uniform shear stress in optimized arterial tree models

Using optimized computer models of arterial trees we demonstrate that flow heterogeneity is a necessary consequence of a uniform shear stress distribution. Model trees are generated and optimized under different modes of boundary conditions. In one mode flow is delivered to the tissue as homogeneously as possible. Although this primary goal can be achieved, resulting shear stresses between blood and the vessel walls show very large spread. In a second mode, models are optimized under the condition of uniform shear stress in all segments which in turn renders flow distribution heterogeneous. Both homogeneous perfusion and uniform shear stress are desirable goals in real arterial trees but each of these goals can only be approached at the expense of the other. While the present paper refers only to optimized models, we assume that this dual relation between the heterogeneities in flow and shear stress may represent a more general principle of vascular systems.

Fractal or biologically variable delivery of cardioplegic solution prevents diastolic dysfunction after cardiopulmonary bypass

OBJECTIVE

To determine whether myocardial protection is improved by restoring physiologic variability to the cardioplegia pressure signal during cardiopulmonary bypass, we compared cardiac function in pigs in the first hour after either conventional cold-blood cardioplegia (group CC) or computer-controlled biologically variable pulsatile cardioplegia (group BVC).

METHODS

Invasive monitors and sonomicrometry crystals were placed, and cardiopulmonary bypass was initiated. The aorta was crossclamped, and cold blood cardioplegic solution was infused intermittently through the aortic root with either conventional cardioplegia (n = 8) or biologically variable pulsatile cardioplegia (n = 8; mean pressure, 75 mm Hg for 85 minutes). The crossclamp was released, cardiac function was restored, and separation from cardiopulmonary bypass was completed. With stable temperature and arterial blood gases, hemodynamics and systolic and diastolic indices were compared at 15, 30, and 60 minutes after cardiopulmonary bypass.

RESULTS

Diastolic stiffness doubled from 0.027 +/- 0.016 mm Hg/mm (mean +/- SD) at baseline to 0.055 +/- 0.036 mm Hg/mm (P =.003) at 1 hour after bypass in group CC, associated with increased left ventricular end-diastolic pressure from 9 +/- 2 to 11 +/- 2 mm Hg (P =.001), mean pulmonary artery pressure from 14 +/- 2 to 20 +/- 3 mm Hg (P =.003), and serum lactate levels from 2.0 +/- 0.5 to 5.6 +/- 2.3 mmol/L (P =.008). Systolic function was not affected. In group BVC diastolic stiffness, left ventricular end-diastolic pressure, and pulmonary artery pressure values were not different from control values at any time after bypass, and serum lactate levels were significantly less than with conventional cold blood cardioplegia. Peak pressure variability with biologically variable pulsatile cardioplegia fit a power-law equation (exponent = -3.0; R(2) = 0.97), indicating fractal behavior.

CONCLUSION

Diastolic cardiac function is better preserved after cardiopulmonary bypass with biologically variable pulsatile cardioplegia and fractal perfusion. This may be attributed to enhanced microcirculatory perfusion with improved myocardial protection. A model supporting these results is presented.

Fractal branching pattern in the pial vasculature in the cat

Arborization pattern was studied in pial vascular networks by treating them as fractals. Rather than applying elaborate taxonomy assembled from measures from individual vessel segments and bifurcations arranged in their branching order, the authors' approach captured the structural details at once in high-resolution digital images processed for the skeleton of the networks. The pial networks appear random and at the same time having structural elements similar to each other when viewed at different scales--a property known as self-similarity revealed by the geometry of fractals. Fractal (capacity) dimension, Dcap, was calculated to evaluate the network's spatial complexity by the box counting method (BCM) and its variant, the extended counting method (XCM). Box counting method and XCM were subject to numerical testing on ideal fractals of known D. The authors found that precision of these fractal methods depends on the fractal character (branching, nonbranching) of the structure they evaluate. Dcaps (group mean +/- SD) for the arterial and venous pial networks in the cat (n = 6) are 1.37 +/- 0.04, 1.37 +/- 0.02 by XCM, and 1.30 +/- 0.04, 1.31 +/- 0.03 by BCM, respectively. The arterial and venous systems thus appear to be developed according to the same fractal generation rule in the cat.

A class of flow bifurcation models with lognormal distribution and fractal dispersion

We report a quantitative analysis of a simple dichotomous branching tree model for blood flow in vascular networks. Using the method of moment-generating function and geometric Brownian motion from stochastic mathematics, our analysis shows that a vascular network with asymmetric branching and random variation at each bifurcating point gives rise to an asymptotic lognormal flow distribution with a positive skewness. The model exhibits a fractal scaling in the dispersion of the regional flow in the branches. Experimentally measurable fractal dimension of the relative dispersion in regional flow is analytically calculated in terms of the asymmetry and the variance at local bifurcation; hence the model suggests a powerful method to obtain the physiological information on local flow bifurcation in terms of flow dispersion analysis. Both the fractal behavior and the lognormal distribution are intimately related to the fact that it is the logarithm of flow, rather than flow itself, which is the natural variable in the tree models. The kinetics of tracer washout is also discussed in terms of the lognormal distribution.

Intramyocardial vascular volume distribution studied by synchrotron radiation-excited X-ray fluorescence

We evaluated the vascular volume distribution with fine resolution (0.1-1.3 mg myocardial tissue) in the sagittal plane of the left ventricle by using the microsphere filling method in 21 dogs. The coronary arterial volume density in the sagittal plane did not exhibit normal distribution and was characterized by variability among the outer-to-inner layers and within the layers (+2SD/-2SD > 80 times), and the median values in the layers ranged from 4.7 to 22. 9 nl/mg myocardial tissue. The fractal analysis of vascular volume revealed a self-similar nature with a fractal dimension (D value) similar to that of flow distribution (1.20 +/- 0.05 and 1.24 +/- 0. 09 for vascular volume and flow distribution, respectively) and had a more marked variability than the flow. The correlation of the regional vascular volume between adjacent regions decreased as the distance increased. However, the correlation coefficients in the endocardial-to-epicardial direction were significantly higher than those in the anterior-to-posterior direction (P < 0.05 by paired t-test). In conclusion, we determined intramyocardial vascular volume density in the sagittal plane, and the distribution revealed considerable variability, self-similarity, and asymmetry in the correlation among the adjacent regions. These observations could be related to the characteristics of the intramural coronary vascular network.

On the design of the coronary arterial tree: a generalization of Murray's law

Murray's law has been generalized to provide morphometric relationships among various subtrees as well as between a feeding segment and the subtree it perfuses. The equivalent resistance of each subtree is empirically determined to be proportional to the cube of a subtree's cumulative arterial length (L) and inversely proportional to a subtree's arterial volume (V) raised to a power of approximately 2.6. This relationship, along with a minimization of a cost function, and a linearity assumption between flow and cumulative arterial length, provides a power law relationship between V and L. These results, in conjunction with conservation of energy, yield relationships between the diameter of a segment and the length of its distal subtree. The relationships were tested based on a complete set of anatomical data of the coronary arterial trees using two models. The first model, called the truncated tree model, is an actual reconstruction of the coronary arterial tree down to 500 microm in diameter. The second model, called the symmetric tree model, satisfies all mean anatomical data down to the capillary vessels. Our results show very good agreement between the theoretical formulation and the measured anatomical data, which may provide insight into the design of the coronary arterial tree. Furthermore, the established relationships between the various morphometric parameters of the truncated tree model may provide a basis for assessing the extent of diffuse coronary artery disease.

Fractal properties of cancellous bone of the iliac crest in vertebral crush fracture

Fractal analysis is a method for describing complex shapes, including the cancellous structure of bone. It describes the surface texture and form of individual trabecular profiles and the overall cancellous structure. Sixty-four postmenopausal women with symptoms of back pain were referred for investigation for osteoporosis. The patients were divided into two groups for comparison: vertebral crush fracture (n = 31, mean age 68.58 +/- 6.47 years), and no vertebral crush fracture (n = 33, mean age 63.36 +/- 7.21 years). Cores of cancellous bone, 3 mm in diameter, were taken from the iliac crest and sectioned. A box-counting method implemented on an image analyzer was used to measure the fractal dimension. Three fractal dimensions describing trabecular surface texture (fractal 1), trabecular shape (fractal 2), and trabecular arrangement (fractal 3) were measured, indicating that cancellous bone has sectional self-similarity. Conventional histomorphometry was also performed on the samples. The results show that fractal 2 is significantly lower in the vertebral crush fracture group than in the nonfracture group (1.15 +/- 0.10 < 1.23 +/- 0.090, p < 0.0013). The histomorphometric analysis shows that bone surface total volume (p < 0.0002), trabecular number (p < 0.0001), and osteoid surface bone surface (p < 0.028) are significantly lower in the fracture group than the nonfracture group. Eroded surface/bone surface (p < 0.056) follows this trend, whereas trabecular separation (p < 0.001) is significantly higher in the fracture group than in the nonfracture group. Fractal 1 and fractal 3 were not significantly different between study groups. The fractal dimension detects changes in the cancellous architecture and gives information about iliac bone transformation in postmenopausal women with vertebral fracture.

Morphometry of human coronary arterial trees

BACKGROUND

Some groups, including ours, have been generating arterial tree models using constrained constructive optimization (CCO). Arterial trees have been grown to arbitrary resolution without input of anatomical data. We performed this study to learn about the shortcomings that might have resulted from neglecting the anatomical data in CCO models.

METHODS

In a total of 450 segments obtained from 4 human cast hearts, the ratio ofbifurcating daughter segment radii (O < Sbif = r(2)/r(1) < 1) was examined, which corresponds to the split of the total flow of the mother segment. For any complete bifurcation, where the radii of the parent segments and the radii of daughters were known, the area expansion ratio was computed (Aexp = [r(1)2 + r(2)2]/r(parent)2).

RESULTS

The bifurcating ratio was found to be distributed in a nonnormal fashion, with a median of 0.76. The average area expansion ratio Aexp, characterizing the change of cross-sectional area of the vasculature from proximal to distal, was 0.93+/-0.26. The 'rate of branching' (d(i)/(d(0)) was defined by the segment diameter relative to the diameter of the root segment. Averaging the rate of branching over segments within each bifurcation level resulted in a decreasing function of bifurcation level.

CONCLUSIONS

This article provides new experimental data on branching geometry of coronary arteries (i.e., the trees evaluated in this study are purely delivering rather than conveying). Based on these facts, we suggest that the analytical bifurcation law in CCO might be replaced by the bifurcation rule obeyed on a stochastic basis only.

Myocardial blood flow heterogeneity in shock and small-volume resuscitation in pigs with coronary stenosis

Myocardial blood flow heterogeneity in shock and small-volume resuscitation in pigs with coronary stenosis. J. Appl. Physiol. 83(6): 1832-1841, 1997.-We analyzed the effects of shock and small-volume resuscitation in the presence of coronary stenosis on fractal dimension (D) and spatial correlation (SC) of regional myocardial perfusion. Hemorrhagic shock was induced and maintained for 1 h. Pigs were resuscitated with hypertonic saline-dextran 60 [HSDex, 10% of shed blood volume (SBV)] or normal saline (NS; 80% of SBV). Therapy was continued after 30 min with dextran (10% SBV). At baseline, D was 1.39 +/- 0.06 (mean +/- SE; HSDex group) and 1.34 +/- 0.04 (NS group). SC was 0.26 +/- 0.07 (HSDex) and 0.26 +/- 0.04 (NS). Left anterior descending coronary artery stenosis changed neither D nor SC. Shock significantly reduced D (i.e., homogenized perfusion): 1.26 +/- 0.06 (HSDex) and 1.23 +/- 0.05 (NS). SC was increased: 0.41 +/- 0.1 (HSDex) and 0.48 +/- 0.07 (NS). Fluid therapy with HSDex further decreased D to 1.22 +/- 0.05, whereas NS did not change D. SC was increased by both HSDex (0.56 +/- 0.1) and NS (0.53 +/- 0.06). At 1 h after resuscitation, SC was constant in both groups, and D was reduced only in the NS group (1.18 +/- 0.02). We conclude that hemorrhagic shock homogenized regional myocardial perfusion in coronary stenosis and that fluid therapy failed to restore this.

Fractal properties of subchondral cancellous bone in severe osteoarthritis of the hip

Primary osteoarthritis of the hip results in changes to the architecture of subchondral cancellous bone. These changes in architecture occur through the action of osteoclasts and osteoblasts in selectively removing and adding bone. The quantitative description of the bone architecture helps in understanding the etiology of primary osteoarthritis. Fractal analysis is a method for describing complex shapes, which is expressed numerically as the fractal dimension. A box counting method was used, where the perimeter of binary profiles of cancellous bone samples was measured for different box sizes. The fractal dimension was the absolute value of the slope of the straight line segments from the plot of the log number of boxes versus the log box size. Cancellous bone samples from two subchondral regions, superior and inferomedial, to the fovea were analyzed from primary severe osteoarthritic specimens taken following total hip replacement surgery (n = 19, aged 51-80 years) and autopsy controls (n = 25, aged 18-90 years). There were three straight line segments identified on the log-log plot, for each subject, indicating a fractal dimension over three different ranges of scale. The results show that in the superior region there is a highly significant difference between the groups (p < 0.0001) for fractal 1 and pivot point 2. The histomorphometry shows significant differences for bone volume/total volume, bone surface/total volume, trabecular separation, and osteoid surface/total volume between groups. In the inferomedial region fractal 1 and fractal 2 are significantly different. For the histomorphometry, trabecular thickness and eroded surface/total volume are significantly different between the groups. The pivot points, i.e., the box size at which the fractal dimension changes, were of similar magnitude to the trabecular thickness and trabecular separation. These data suggest that the fractal geometry analysis of cancellous bone identifies architectural features not easily recognized by conventional bone histomorphometry. The fractal dimension is a descriptor of bone structure which simplifies the description of a complex structure and enables changes in cancellous bone architecture, due to disease, to be identified.

Coronary microcirculation in health and disease. Summary of an NHLBI workshop

This article summarizes a 2-day workshop on the coronary microcirculation held in Bethesda, Md, in September 1994 and sponsored by the National Heart, Lung, and Blood Institute of the National Institutes of Health. The workshop explored a variety of topics pertaining to coronary microvascular physiology and pathophysiology. The latest methodologies that are being used to investigate the coronary microvasculature, including endoscopic microscopy of the intramural coronary microvasculature and micro-x-ray computerized tomography, were discussed. The most recent advances in the regulation of the coronary microcirculation-for example, myogenic and flow-dependent responses, KATP channels, and regional heterogeneity-were reported. The workshop touched on the relation of the microcirculation to clinically important conditions and offered recommendations for future research in this important area. Comparisons are made to recent advances in the peripheral circulation and current gaps in our knowledge concerning the coronary microcirculation. In recent years, research on the coronary microcirculation has made substantial advances, in part as a result of investigations in the peripheral microcirculation but also because of the application of unique methodologies. This research is providing new ways to investigate abnormalities of myocardial perfusion, an area of inquiry that until recently has been limited to examination of coronary pressure-flow relationships.

Modeling regional myocardial flows from residue functions of an intravascular indicator

The purpose of the present study was to determine the accuracy and the sources of error in estimating regional myocardial blood flow and vascular volume from experimental residue functions obtained by external imaging of an intravascular indicator. For the analysis, a spatially distributed mathematical model was used that describes transport through a multiple-pathway vascular system. Reliability of the parameter estimates was tested by using sensitivity function analysis and by analyzing "pseudodata": realistic model solutions to which random noise was added. Increased uncertainty in the estimates of flow in the pseudodata was observed when flow was near maximal physiological values, when dispersion of the vascular input was more than twice the dispersion of the microvascular system for an impulse input, and when the sampling frequency was < 2 samples/s. Estimates of regional blood volume were more reliable than estimates of flow. Failure to account for normal flow heterogeneity caused systematic underestimates of flow. To illustrate the method used for estimating regional flow, magnetic resonance imaging was used to obtain myocardial residue functions after left atrial injections of polylysine-Gd-diethylenetriaminepentaacetic acid, an intravascular contrast agent, in anesthetized chronically instrumental dogs. To test the increase in dispersion of the vascular input after central venous injections, magnetic resonance imaging data obtained in human subjects were compared with left ventricular blood pool curves obtained in dogs. It is concluded that if coronary flow is in the normal range, when the vascular input is a short bolus, and the heart is imaged at least once per cardiac cycle, then regional myocardial blood flow and vascular volume may be reliably estimated by analyzing residue functions of an intravascular indicator, providing a noninvasive approach with potential clinical application.

Heterogeneity of myocardial blood flow

Myocardial blood flow is heterogeneous, whether considered by chamber, by layers of the ventricular walls, or by microregions within layers. There is also variability of myocardial flow reserve, particularly in layers and microregions, even when the heart is arrested. The variability of flow during arrest may be associated with the resistance pathways to each region, but the variability of flows in the beating heart with vascular tone is probably due to regional differences in work and thus oxygen demand. Heterogeneity by layer may be responsible for the subendocardial ischemia that is common to many forms of heart disease. Microheterogeneity may account for the patchy necrosis that occurs with chronic ischemia.

Regional myocardial blood volume and flow: first-pass MR imaging with polylysine-Gd-DTPA

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 intensity-versus-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.

Quantification of tight junction complexity by means of fractal analysis

The concept of fractal geometry provides an elegant tool for the quantitative and objective structural description of various objects, the fractal analysis. Fractal analysis quantifies the structural complexity of objects by a characteristic singular value, the fractal dimension (FD). It can be estimated, e.g. by the box-counting method and provides a highly integrated measure in the range 1 < FD < 2 for curves extending within a plane. In this study, fractal analysis is used for the first time to evaluate the complexity of the tight junction network between adjoining cells. Bovine brain endothelial cells were cultured under various experimental conditions and the tight junctions were drawn to scale as visualized by the freeze fracture technique. These drawings were analyzed by fractal analysis, and by two other methods commonly used in this field, viz. the strand counting (SC) and complexity index (CI) methods. In contrast to the latter methods, the FD shows no directional preference and therefore no assumptions on the dynamic properties of the network's complexity are required. Thus, FD is demonstrated to provide the most sensitive, reliable and complete measure of tight junction complexity. In combination with SC and CI, additional information can be achieved concerning the directionality of the altered arrangement of tight junctional strands. Our analysis allows for the following conclusions. (1) Defined experimental influences can modify the complexity of tight junctions that are formed between endothelial cells in vitro, and (2) these structural modifications of the tight junctions are mainly due to an altered strand branching pattern.

The application of fractal geometric analysis to microscopic images

Fractal geometry is a relatively new tool for the quantitative microscopist that is a more valid way of measuring dimensions of complex irregular objects than the integer-dimensional geometries (such as Euclidean geometry). This review discusses the theory of fractal geometry using the classic examples of the Von Koch curve, the Cantor set and the Sierpinski gasket. The problems of describing the dimensions of these objects are discussed and the concept of fractal dimensionality is introduced. Methods for measuring fractal dimensions are discussed, including their implementation on microcomputer-based image analysis systems . The advantages and problems of fractal geometric analysis are discussed and current applications in the field of microscopy are reviewed.

The microvascular unit size for fractal flow heterogeneity relevant for oxygen transport

In this study we used a computer model for oxygen transport in heterogeneously perfused tissue to define the microvascular unit size of relevance to oxygen transport. Flow within this unit is presumably heterogeneous, but this internal heterogeneity is by definition of negligible importance for the oxygen tension distribution. In saline-perfused heart the linear dimension of the thus defined unit is 500 microns, in blood-perfused heart it is 100 microns.

Quantitation of the renal arterial tree by fractal analysis

To determine whether the renal arterial system has a fractal structure, the fractal dimension of renal angiograms from 52 necropsy cases was measured using an implementation of the box-counting method on an image analysis system. The method was validated using objects with known fractal dimensions. The method was accurate with errors of less than 1.5 per cent and reproducible with initial values within 1.2 per cent of the mean of ten sets of measurements (reliability coefficient 0.968, 95 per cent confidence limits 0.911-0.984). In the 36 satisfactory angiograms the mean fractal dimension was 1.61 (SD 0.06), which was significantly greater than the topological dimension of 1 (P < 0.0001), indicating that the renal arterial tree has a fractal structure. There was no significant relationship between age (P = 0.494), sex (P = 0.136), or systolic (P = 0.069) or diastolic (P = 0.990) blood pressure, but two congenitally abnormal kidneys (hypoplastic dysplasia and renal artery stenosis) had fractal dimensions at the lower end of the normal range (third percentile). Since the renal arterial tree has a fractal structure, Euclidean geometric measurements, such as area and boundary length, are invalid outside precisely defined conditions of magnification and resolution.

Trabecular bone does not have a fractal structure on light microscopic examination

The fractal dimension of the boundary of trabecular bone in 62 biopsies was measured on histological sections using a box-counting method implemented on a microcomputer image analysis system. The calculated fractal dimension had a mean value of 0.99 with a normal distribution. Since this value is not greater than the topological dimension, trabecular bone, when examined by light microscopy, does not have a fractal structure. Conventional Euclidean dimensions will continue to be the most useful measurements in bone histomorphometry.

Of the British coastline and the interest of fractals in medicine

We review a certain number of medical applications of a new non-euclidean geometry: the fractal geometry described by Mandelbrot. Examples come from anatomy, cytology, general physiology and physiopathology. Furthermore, real clinical applications are shown, in particular, in cardiology, neurology, ophtalmology, radiology and other imaging techniques. An easy reading mathematical approach is added. Some of the fractal images will certainly capture your attention and spur your interest for further applications of this new concept.

Capillary endothelial transport of uric acid in guinea pig heart

Much of the adenosine formed in the heart is degraded by endothelial enzymes to uric acid, which is exported across the coronary capillary endothelial cell membrane before renal excretion. Because previous experiments suggested that cell permeability for uric acid is either very high (similar to water) or very low, multiple indicator-dilution experiments were carried out to distinguish between the two possibilities. An intravascular reference tracer, 131I-labeled albumin, and an extracellular reference tracer, L-[3H]glucose, were injected together with [14C]uric acid as a bolus into the coronary inflow, while fractionating the venous outflow for 90 s. Recovery of injected uric acid averaged 99.0 +/- 2.9% (mean +/- SD, n = 12) that of L-glucose. Peak capillary extraction of L-glucose and uric acid averaged 0.38 +/- 0.032 and 0.42 +/- 0.035 (P less than 0.005) compared with albumin. Except at the peaks, the dilution curves for [14C]uric acid and L-[3H]glucose coincided closely, indicating that little uric acid was transported into cells. The dilution curves were analyzed using an axially distributed, multipathway, four region mathematical model, to estimate membrane permeability-surface area (PS) products. Since the endothelial cell PS for uric acid was low (0.12 +/- 0.09 ml.g-1.min-1), approximately 3% of the PS reported for adenosine, the possibility of flow-limited exchange for uric acid is ruled out. To estimate steady-state endothelial concentrations of uric acid in vivo, equations were developed describing electrochemical potential gradients for dissociated and undissociated forms of a weak acid. Despite endothelial production, intracellular concentrations that are lower than outside are expected because the negative membrane potential and lower cellular pH assist uric acid efflux.