Diameter-defined Strahler system and connectivity matrix of the pulmonary arterial tree

Pulmonary vasculature development in congenital diaphragmatic hernia: a novel automated quantitative imaging analysis

PURPOSE

Impaired fetal lung vasculature determines the degree of pulmonary hypertension in the congenital diaphragmatic hernia (CDH). This study aims to demonstrate the morphometric measurements that differ in pulmonary vessels of fetuses with CDH.

METHODS

Nitrofen-induced CDH Sprague-Dawley rat fetuses were scanned with microcomputed tomography. The analysis of the pulmonary vascular tree was performed with artificial intelligence.

RESULTS

The number of segments in CDH was significantly lower than that in the control group on the left (U = 2.5, p = 0.004) and right (U = 0, p = 0.001) sides for order 1(O1), whereas there was a significant difference only on the right side for O2 and O3. The pooled element numbers in the control group obeyed Horton's law (R2 = 0.996 left and R2 = 0.811 right lungs), while the CDH group broke it. Connectivity matrices showed that the average number of elements of O1 springing from elements of O1 on the left side and the number of elements of O1 springing from elements of O3 on the right side were significantly lower in CDH samples.

CONCLUSION

According to these findings, CDH not only reduced the amount of small order elements, but also destroyed the fractal structure of the pulmonary arterial trees.

A computational growth and remodeling framework for adaptive and maladaptive pulmonary arterial hemodynamics

Hemodynamic loading is known to contribute to the development and progression of pulmonary arterial hypertension (PAH). This loading drives changes in mechanobiological stimuli that affect cellular phenotypes and lead to pulmonary vascular remodeling. Computational models have been used to simulate mechanobiological metrics of interest, such as wall shear stress, at single time points for PAH patients. However, there is a need for new approaches that simulate disease evolution to allow for prediction of long-term outcomes. In this work, we develop a framework that models the pulmonary arterial tree through adaptive and maladaptive responses to mechanical and biological perturbations. We coupled a constrained mixture theory-based growth and remodeling framework for the vessel wall with a morphometric tree representation of the pulmonary arterial vasculature. We show that non-uniform mechanical behavior is important to establish the homeostatic state of the pulmonary arterial tree, and that hemodynamic feedback is essential for simulating disease time courses. We also employed a series of maladaptive constitutive models, such as smooth muscle hyperproliferation and stiffening, to identify critical contributors to development of PAH phenotypes. Together, these simulations demonstrate an important step toward predicting changes in metrics of clinical interest for PAH patients and simulating potential treatment approaches.

A multiscale framework for defining homeostasis in distal vascular trees: applications to the pulmonary circulation

Pulmonary arteries constitute a low-pressure network of vessels, often characterized as a bifurcating tree with heterogeneous vessel mechanics. Understanding the vascular complexity and establishing homeostasis is important to study diseases such as pulmonary arterial hypertension (PAH). The onset and early progression of PAH can be traced to changes in the morphometry and structure of the distal vasculature. Coupling hemodynamics with vessel wall growth and remodeling (G&R) is crucial for understanding pathology at distal vasculature. Accordingly, the goal of this study is to provide a multiscale modeling framework that embeds the essential features of arterial wall constituents coupled with the hemodynamics within an arterial network characterized by an extension of Murray's law. This framework will be used to establish the homeostatic baseline characteristics of a pulmonary arterial tree, including important parameters such as vessel radius, wall thickness and shear stress. To define the vascular homeostasis and hemodynamics in the tree, we consider two timescales: a cardiac cycle and a longer period of vascular adaptations. An iterative homeostatic optimization, which integrates a metabolic cost function minimization, the stress equilibrium, and hemodynamics, is performed at the slow timescale. In the fast timescale, the pulsatile blood flow dynamics is described by a Womersley's deformable wall analytical solution. Illustrative examples for symmetric and asymmetric trees are presented that provide baseline characteristics for the normal pulmonary arterial vasculature. The results are compared with diverse literature data on morphometry, structure, and mechanics of pulmonary arteries. The developed framework demonstrates a potential for advanced parametric studies and future G&R and hemodynamics modeling of PAH.

Evaluation of classifications of the monopodial bronchopulmonary vasculature using clustering methods

Mammalian pulmonary arteries divide multiple times before reaching the vast capillary network of the alveoli. Morphological analyses of the arterial branches can be challenging because more proximal branches are likely biologically distinct from more peripheral parts. Thus, it is useful to group the arterial branches into groups of coherent biology. While the generational approach of dichotomous branching is straightforward, the grouping of arterial branches in the asymmetrically branching monopodial lung is less clear. Several established classification methods return highly dissimilar groupings when employed on the same organ. Here, we established a workflow allowing the quantification of grouping results for the monopodial lung and tested various methods to group the branches of the arterial tree into coherent groups. A mouse lung was imaged by synchrotron x-ray microcomputed tomography, and the arteries were digitally segmented. The arterial tree was divided into its individual segments, morphological properties were assessed from corresponding light microscopic scans, and different grouping methods were employed, such as (fractal) generation or (Strahler) order. The results were ranked by the morphological similarity within and dissimilarity between the resulting groups. Additionally, a method from the mathematical field of cluster analysis was employed for creating a reference classification. In conclusion, there were significant differences in method performance. The Strahler order was significantly superior to the generation system commonly used to classify human lung structure. Furthermore, a clustering approach indicated more precise ways to classify the monopodial lung vasculature exist.

Rigorous mathematical optimization of synthetic hepatic vascular trees

In this paper, we introduce a new framework for generating synthetic vascular trees, based on rigorous model-based mathematical optimization. Our main contribution is the reformulation of finding the optimal global tree geometry into a nonlinear optimization problem (NLP). This rigorous mathematical formulation accommodates efficient solution algorithms such as the interior point method and allows us to easily change boundary conditions and constraints applied to the tree. Moreover, it creates trifurcations in addition to bifurcations. A second contribution is the addition of an optimization stage for the tree topology. Here, we combine constrained constructive optimization (CCO) with a heuristic approach to search among possible tree topologies. We combine the NLP formulation and the topology optimization into a single algorithmic approach. Finally, we attempt the validation of our new model-based optimization framework using a detailed corrosion cast of a human liver, which allows a quantitative comparison of the synthetic tree structure with the tree structure determined experimentally down to the fifth generation. The results show that our new framework is capable of generating asymmetric synthetic trees that match the available physiological corrosion cast data better than trees generated by the standard CCO approach.

Parallel generation of extensive vascular networks with application to an archetypal human kidney model

Given the relevance of the inextricable coupling between microcirculation and physiology, and the relation to organ function and disease progression, the construction of synthetic vascular networks for mathematical modelling and computer simulation is becoming an increasingly broad field of research. Building vascular networks that mimic in vivo morphometry is feasible through algorithms such as constrained constructive optimization (CCO) and variations. Nevertheless, these methods are limited by the maximum number of vessels to be generated due to the whole network update required at each vessel addition. In this work, we propose a CCO-based approach endowed with a domain decomposition strategy to concurrently create vascular networks. The performance of this approach is evaluated by analysing the agreement with the sequentially generated networks and studying the scalability when building vascular networks up to 200 000 vascular segments. Finally, we apply our method to vascularize a highly complex geometry corresponding to the cortex of a prototypical human kidney. The technique presented in this work enables the automatic generation of extensive vascular networks, removing the limitation from previous works. Thus, we can extend vascular networks (e.g. obtained from medical images) to pre-arteriolar level, yielding patient-specific whole-organ vascular models with an unprecedented level of detail.

Stereology and three-dimensional reconstructions to analyze the pulmonary vasculature

The pulmonary vasculature consists of a large arterial and venous tree with a vast alveolar capillary network (ACN) in between. Both conducting blood vessels and the gas-exchanging capillaries are part of important human lung diseases, including bronchopulmonary dysplasia, pulmonary hypertension and chronic obstructive pulmonary disease. Morphological tools to investigate the different parts of the pulmonary vasculature quantitatively and in three dimensions are crucial for a better understanding of the contribution of the blood vessels to the pathophysiology and effects of lung diseases. In recent years, new stereological methods and imaging techniques have expanded the analytical tool box and therefore the conclusive power of morphological analyses of the pulmonary vasculature. Three of these developments are presented and discussed in this review article, namely (1) stereological quantification of the number of capillary loops, (2) serial block-face scanning electron microscopy of the ACN and (3) labeling of branching generations in light microscopic sections based on arterial tree segmentations of micro-computed tomography data sets of whole lungs. The implementation of these approaches in research work requires expertise in lung preparation, multimodal imaging at different scales, an advanced IT infrastructure and expertise in image analysis. However, they are expected to provide important data that cannot be obtained by previously existing methodology.

Geometric Features of the Pial Arteriolar Networks in Spontaneous Hypertensive Rats: A Crucial Aspect Underlying the Blood Flow Regulation

Background

Several studies indicate that hypertension causes major changes in the structure of the vessel wall by affecting the regulation of blood supply to the tissues. Recently, it has been observed that capillary blood flow is also considerably influenced by the structural arrangement of the microvascular networks that undergo rarefaction (reduction of the perfused vessel number). Therefore, this study aimed to assess the geometric arrangements of the pial arteriolar networks and the arteriolar rhythmic diameter changes in spontaneously hypertensive rats (SHRs).

Methods

Fluorescence microscopy was utilized to observe in vivo the pial microcirculation through a closed cranial window. Pial arterioles were classified according to Strahler’s method. The arteriolar rhythmic diameter changes were evaluated by a generalization short-time Fourier transform.

Result

Young SHRs showed four orders of vessels while the adult ones only three orders. The diameter, length, and branching number obeyed Horton’s law; therefore, the vessels were distributed in a fractal manner. Larger arterioles showed more asymmetrical branches than did the smaller ones in young SHRs, while in adult SHRs smaller vessels presented asymmetrical branchings. In adult SHRs, there was a significant reduction in the cross-sectional area compared with the young SHRs: this implies an increase in peripheral resistance. Young and adult age-matched normotensive rats did not show significant alterations in the geometric arteriolar arrangement with advancing age, both had four orders of arteriolar vessels, and the peripheral resistance did not change significantly. Conversely, the frequency components evaluated in arteriolar rhythmic diameter changes of young and adult SHRs showed significant differences because of a reduction in the frequency components related to endothelial activity detected in adult SHRs.

Conclusion

In conclusion, hypertension progressively causes changes in the microarchitecture of the arteriolar networks with a smaller number of vessels and consequent reduced conductivity, characteristic of rarefaction. This was accompanied by a reduction in the formation and release of independent and dependent – endothelial nitric oxide components regulating arterial vasomotion.

Hypoxic pulmonary vasoconstriction as a regulator of alveolar-capillary oxygen flux: A computational model of ventilation-perfusion matching

The relationship between regional variabilities in airflow (ventilation) and blood flow (perfusion) is a critical determinant of gas exchange efficiency in the lungs. Hypoxic pulmonary vasoconstriction is understood to be the primary active regulator of ventilation-perfusion matching, where upstream arterioles constrict to direct blood flow away from areas that have low oxygen supply. However, it is not understood how the integrated action of hypoxic pulmonary vasoconstriction affects oxygen transport at the system level. In this study we develop, and make functional predictions with a multi-scale multi-physics model of ventilation-perfusion matching governed by the mechanism of hypoxic pulmonary vasoconstriction. Our model consists of (a) morphometrically realistic 2D pulmonary vascular networks to the level of large arterioles and venules; (b) a tileable lumped-parameter model of vascular fluid and wall mechanics that accounts for the influence of alveolar pressure; (c) oxygen transport accounting for oxygen bound to hemoglobin and dissolved in plasma; and (d) a novel empirical model of hypoxic pulmonary vasoconstriction. Our model simulations predict that under the artificial test condition of a uniform ventilation distribution (1) hypoxic pulmonary vasoconstriction matches perfusion to ventilation; (2) hypoxic pulmonary vasoconstriction homogenizes regional alveolar-capillary oxygen flux; and (3) hypoxic pulmonary vasoconstriction increases whole-lobe oxygen uptake by improving ventilation-perfusion matching.

The relationship between regional ventilation (airflow) and perfusion (blood flow) is a major determinant of gas exchange efficiency. Atelactasis and pulmonary vascular occlusive diseases, such as acute pulmonary embolism, are characterized by ventilation-perfusion mismatching and decreased oxygen in the bloodstream. Despite the physiological and medical importance of ventilation-perfusion matching, there are gaps in our knowledge of the regulatory mechanisms that maintain adequate gas exchange under pathological and normal conditions. Hypoxic pulmonary vasoconstriction is understood to be the primary regulator of ventilation-perfusion matching, where upstream arterioles constrict to direct blood flow away from areas that have low oxygen supply, yet it is not understood how this mechanism affects oxygen transport at the system level. In this study we present a computational model of the ventilation-perfusion matching and hypoxic pulmonary vasoconstriction to better understand how physiological regulation at the regional level scales to affect oxygen transport at the system level. Our model simulations predict that this regulatory mechanism improves the spatial overlap of airflow and blood flow, which serves to increase the uptake of oxygen into the bloodstream. This improved understanding of ventilation-perfusion matching may offer insights into the etiology of, and therapeutic interventions for diseases characterized by ventilation-perfusion mismatching.

Fractal analysis of concurrently prepared latex rubber casts of the bronchial and vascular systems of the human lung

Fractal geometry (FG) is a branch of mathematics that instructively characterizes structural complexity. Branched structures are ubiquitous in both the physical and the biological realms. Fractility has therefore been termed nature's design. The fractal properties of the bronchial (airway) system, the pulmonary artery and the pulmonary vein of the human lung generates large respiratory surface area that is crammed in the lung. Also, it permits the inhaled air to intimately approximate the pulmonary capillary blood across a very thin blood-gas barrier through which gas exchange to occur by diffusion. Here, the bronchial (airway) and vascular systems were simultaneously cast with latex rubber. After corrosion, the bronchial and vascular system casts were physically separated and cleared to expose the branches. The morphogenetic (Weibel's) ordering method was used to categorize the branches on which the diameters and the lengths, as well as the angles of bifurcation, were measured. The fractal dimensions (DF) were determined by plotting the total branch measurements against the mean branch diameters on double logarithmic coordinates (axes). The diameter-determined DF values were 2.714 for the bronchial system, 2.882 for the pulmonary artery and 2.334 for the pulmonary vein while the respective values from lengths were 3.098, 3.916 and 4.041. The diameters yielded DF values that were consistent with the properties of fractal structures (i.e. self-similarity and space-filling). The data obtained here compellingly suggest that the design of the bronchial system, the pulmonary artery and the pulmonary vein of the human lung functionally comply with the Hess-Murray law or 'the principle of minimum work'.

Development of a globally optimised model of the cerebral arteries

The cerebral arteries are difficult to reproduce from first principles, featuring interwoven territories, and intricate layers of grey and white matter with differing metabolic demand. The aim of this study was to identify the ideal configuration of arteries required to sustain an entire brain hemisphere based on minimisation of the energy required to supply the tissue. The 3D distribution of grey and white matter within a healthy human brain was first segmented from magnetic resonance images. A novel simulated annealing algorithm was then applied to determine the optimal configuration of arteries required to supply brain tissue. The model was validated through comparison of this ideal, entirely optimised, brain vasculature with the structure and properties of real arteries. This analysis established that the human cerebral vasculature is highly optimised; closely resembling the most energy efficient arrangement of vessels. In addition to local adherence to fluid dynamical optimisation principles, the optimised vasculature reproduced expected brain perfusion territories, featuring well-defined boundaries between anterior, middle and posterior regions. This validated brain vascular model and algorithm can be used for patient-specific modelling of stroke and cerebral haemodynamics, identification of sub-optimal conditions associated with vascular disease, and optimising vascular structures for tissue engineering applications and artificial organ design.

Scaling Laws of Flow Rate, Vessel Blood Volume, Lengths, and Transit Times With Number of Capillaries

The structure-function relation is one of the oldest hypotheses in biology and medicine; i.e., form serves function and function influences form. Here, we derive and validate form-function relations for volume, length, flow, and mean transit time in vascular trees and capillary numbers of various organs and species. We define a vessel segment as a "stem" and the vascular tree supplied by the stem as a "crown." We demonstrate form-function relations between the number of capillaries in a vascular network and the crown volume, crown length, and blood flow that perfuses the network. The scaling laws predict an exponential relationship between crown volume and the number of capillaries with the power, λ, of 4/3 < λ < 3/2. It is also shown that blood flow rate and vessel lengths are proportional to the number of capillaries in the entire stem-crown systems. The integration of the scaling laws then results in a relation between transit time and crown length and volume. The scaling laws are both intra-specific (i.e., within vasculatures of various organs, including heart, lung, mesentery, skeletal muscle and eye) and inter-specific (i.e., across various species, including rats, cats, rabbits, pigs, hamsters, and humans). This study is fundamental to understanding the physiological structure and function of vascular trees to transport blood, with significant implications for organ health and disease.

Quantitative analysis of hepatic macro- and microvascular alterations during cirrhogenesis in the rat

Cirrhosis represents the end-stage of any persistent chronically active liver disease. It is characterized by the complete replacement of normal liver tissue by fibrosis, regenerative nodules, and complete fibrotic vascularized septa. The resulting angioarchitectural distortion contributes to an increasing intrahepatic vascular resistance, impeding liver perfusion and leading to portal hypertension. To date, knowledge on the dynamically evolving pathological changes of the hepatic vasculature during cirrhogenesis remains limited. More specifically, detailed anatomical data on the vascular adaptations during disease development is lacking. To address this need, we studied the 3D architecture of the hepatic vasculature during induction of cirrhogenesis in a rat model. Cirrhosis was chemically induced with thioacetamide (TAA). At predefined time points, the hepatic vasculature was fixed and visualized using a combination of vascular corrosion casting and deep tissue microscopy. Three-dimensional reconstruction and data-fitting enabled cirrhogenic features to extracted at multiple scales, portraying the impact of cirrhosis on the hepatic vasculature. At the macrolevel, we noticed that regenerative nodules severely compressed pliant venous vessels from 12 weeks of TAA intoxication onwards. Especially hepatic veins were highly affected by this compression, with collapsed vessel segments severely reducing perfusion capabilities. At the microlevel, we discovered zone-specific sinusoidal degeneration, with sinusoids located near the surface being more affected than those in the middle of a liver lobe. Our data shed light on and quantify the evolving angioarchitecture during cirrhogenesis. These findings may prove helpful for future targeted invasive interventions.

A statistical model of the penetrating arterioles and venules in the human cerebral cortex

OBJECTIVE

Models of the cerebral microvasculature are required at many different scales in order to understand the effects of microvascular topology on CBF. There are, however, no data-driven models at the arteriolar/venular scale. In this paper, we develop a data-driven algorithm based on available data to generate statistically accurate penetrating arterioles and venules.

METHODS

A novel order-based density-filling algorithm is developed based on the statistical data including bifurcating angles, LDRs, and area ratios. Three thousand simulations are presented, and the results validated against morphological data. These are combined with a previous capillary network in order to calculate full vascular network parameters.

RESULTS

Statistically accurate penetrating trees were successfully generated. All properties provided a good fit to experimental data. The k exponent had a median of 2.5 and an interquartile range of 1.75-3.7. CBF showed a standard deviation ranging from ±18% to ±34% of the mean, depending on the penetrating vessel diameter.

CONCLUSIONS

Small CBF variations indicate that the topology of the penetrating vessels plays only a small part in the large regional variations of CBF seen in the brain. These results open up the possibility of efficient oxygen and blood flow simulations at MRI voxel scales which can be directly validated against MRI data.

Universal features of dendrites through centripetal branch ordering

Dendrites form predominantly binary trees that are exquisitely embedded in the networks of the brain. While neuronal computation is known to depend on the morphology of dendrites, their underlying topological blueprint remains unknown. Here, we used a centripetal branch ordering scheme originally developed to describe river networks—the Horton-Strahler order (SO)–to examine hierarchical relationships of branching statistics in reconstructed and model dendritic trees. We report on a number of universal topological relationships with SO that are true for all binary trees and distinguish those from SO-sorted metric measures that appear to be cell type-specific. The latter are therefore potential new candidates for categorising dendritic tree structures. Interestingly, we find a faithful correlation of branch diameters with centripetal branch orders, indicating a possible functional importance of SO for dendritic morphology and growth. Also, simulated local voltage responses to synaptic inputs are strongly correlated with SO. In summary, our study identifies important SO-dependent measures in dendritic morphology that are relevant for neural function while at the same time it describes other relationships that are universal for all dendrites.

Similarly to river beds, dendritic trees of nerve cells form elaborate networks that branch out to cover extensive areas. In the 1940s, ecologist Robert E. Horton developed an ordering system for branches in river networks that was refined in the 1950s by geoscientist Arthur N. Strahler, the Horton-Strahler order (SO). Branches at the tips start with order 1 and increase their order in a systematic way when encountering new branches on the way to the root. SO relationships have recently become popular for quantifying dendritic morphologies. Various branching statistics can be studied as a function of SO. Here we describe that topological measures such as the number of branches, the branch bifurcation ratio and the size of subtrees exhibit stereotypical relations with SO in dendritic trees independently of cell type, mirroring universal features of binary trees. Other functionally more relevant features such as mean branch lengths, local diameters and simulated voltage responses to synaptic inputs directly correlate with SO in a cell type-specific manner, indicating the importance of SO for understanding dendrite growth as well as neural computation.

Architecture of the rat nephron-arterial network: analysis with micro-computed tomography

Among solid organs, the kidney's vascular network stands out, because each nephron has two distinct capillary structures in series and because tubuloglomerular feedback, one of the mechanisms responsible for blood flow autoregulation, is specific to renal tubules. Tubuloglomerular feedback and the myogenic mechanism, acting jointly, autoregulate single-nephron blood flow. Each generates a self-sustained periodic oscillation and an oscillating electrical signal that propagates upstream along arterioles. Similar electrical signals from other nephrons interact, allowing nephron synchronization. Experimental measurements show synchronization over fields of a few nephrons; simulations based on a simplified network structure that could obscure complex interactions predict more widespread synchronization. To permit more realistic simulations, we made a cast of blood vessels in a rat kidney, performed micro-computed tomography at 2.5-μm resolution, and recorded three-dimensional coordinates of arteries, afferent arterioles, and glomeruli. Nonterminal branches of arcuate arteries form treelike structures requiring two to six bifurcations to reach terminal branches at the tree tops. Terminal arterial structures were either paired branches at the tops of the arterial trees, from which 52.6% of all afferent arterioles originated, or unpaired arteries not at the tree tops, yielding the other 22.9%; the other 24.5% originated directly from nonterminal arteries. Afferent arterioles near the corticomedullary boundary were longer than those farther away, suggesting that juxtamedullary nephrons have longer afferent arterioles. The distance separating origins of pairs of afferent arterioles varied randomly. The results suggest an irregular-network tree structure with vascular nodes, where arteriolar activity and local blood pressure interact.

A multilevel framework to reconstruct anatomical 3D models of the hepatic vasculature in rat livers

The intricate (micro)vascular architecture of the liver has not yet been fully unravelled. Although current models are often idealized simplifications of the complex anatomical reality, correct morphological information is instrumental for scientific and clinical purposes. Previously, both vascular corrosion casting (VCC) and immunohistochemistry (IHC) have been separately used to study the hepatic vasculature. Nevertheless, these techniques still face a number of challenges such as dual casting in VCC and limited imaging depths for IHC. We have optimized both techniques and combined their complementary strengths to develop a framework for multilevel reconstruction of the hepatic circulation in the rat. The VCC and micro-CT scanning protocol was improved by enabling dual casting, optimizing the contrast agent concentration, and adjusting the viscosity of the resin (PU4ii). IHC was improved with an optimized clearing technique (CUBIC) that extended the imaging depth for confocal microscopy more than five-fold. Using in-house developed software (DeLiver), the vascular network - in both VCC and IHC datasets - was automatically segmented and/or morphologically analysed. Our methodological framework allows 3D reconstruction and quantification of the hepatic circulation, ranging from the major blood vessels down to the intertwined and interconnected sinusoids. We believe that the presented framework will have value beyond studies of the liver, and will facilitate a better understanding of various parenchymal organs in general, in physiological and pathological circumstances.

Development of 3D Microvascular Networks Within Gelatin Hydrogels Using Thermoresponsive Sacrificial Microfibers

A 3D microvascularized gelatin hydrogel is produced using thermoresponsive sacrificial poly(N-isopropylacrylamide) microfibers. The capillary-like microvascular network allows constant perfusion of media throughout the thick hydrogel, and significantly improves the viability of human neonatal dermal fibroblasts encapsulated within the gel at a high density.

Simulated annealing approach to vascular structure with application to the coronary arteries

Do the complex processes of angiogenesis during organism development ultimately lead to a near optimal coronary vasculature in the organs of adult mammals? We examine this hypothesis using a powerful and universal method, built on physical and physiological principles, for the determination of globally energetically optimal arterial trees. The method is based on simulated annealing, and can be used to examine arteries in hollow organs with arbitrary tissue geometries. We demonstrate that the approach can generate in silico vasculatures which closely match porcine anatomical data for the coronary arteries on all length scales, and that the optimized arterial trees improve systematically as computational time increases. The method presented here is general, and could in principle be used to examine the arteries of other organs. Potential applications include improvement of medical imaging analysis and the design of vascular trees for artificial organs.

Constructal law of vascular trees for facilitation of flow

Diverse tree structures such as blood vessels, branches of a tree and river basins exist in nature. The constructal law states that the evolution of flow structures in nature has a tendency to facilitate flow. This study suggests a theoretical basis for evaluation of flow facilitation within vascular structure from the perspective of evolution. A novel evolution parameter (Ev) is proposed to quantify the flow capacity of vascular structures. Ev is defined as the ratio of the flow conductance of an evolving structure (configuration with imperfection) to the flow conductance of structure with least imperfection. Attaining higher Ev enables the structure to expedite flow circulation with less energy dissipation. For both Newtonian and non-Newtonian fluids, the evolution parameter was developed as a function of geometrical shape factors in laminar and turbulent fully developed flows. It was found that the non-Newtonian or Newtonian behavior of fluid as well as flow behavior such as laminar or turbulent behavior affects the evolution parameter. Using measured vascular morphometric data of various organs and species, the evolution parameter was calculated. The evolution parameter of the tree structures in biological systems was found to be in the range of 0.95 to 1. The conclusion is that various organs in various species have high capacity to facilitate flow within their respective vascular structures.

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.

Hypoxia and exercise increase the transpulmonary passage of 99mTc-labeled albumin particles in humans

Intrapulmonary arteriovenous anastomoses (IPAVs) are large diameter connections that allow blood to bypass the lung capillaries and may provide a route for right-to-left embolus transmission. These anastomoses are recruited by exercise and catecholamines and hypoxia. Yet, whether IPAVs are recruited via direct, oxygen sensitive regulatory mechanisms or indirect effects secondary to redistribution pulmonary blood flow is unknown. Here, we hypothesized that the addition of exercise to hypoxic gas breathing, which increases cardiac output, would augment IPAVs recruitment in healthy humans. To test this hypothesis, we measured the transpulmonary passage of 99mTc-macroaggregated albumin particles (99mTc-MAA) in seven healthy volunteers, at rest and with exercise at 85% of volitional max, with normoxic (FIO2 = 0.21) and hypoxic (FIO2 = 0.10) gas breathing. We found increased 99mTc-MAA passage in both exercise conditions and resting hypoxia. However, contrary to our hypothesis, we found the greatest 99mTc-MAA passage with resting hypoxia. As an additional, secondary endpoint, we also noted that the transpulmonary passage of 99mTc-MAA was well-correlated with the alveolar-arterial oxygen difference (A-aDO2) during exercise. While increased cardiac output has been proposed as an important modulator of IPAVs recruitment, we provide evidence that the modulation of blood flow through these pathways is more complex and that increasing cardiac output does not necessarily increase IPAVs recruitment. As we discuss, our data suggest that the resistance downstream of IPAVs is an important determinant of their perfusion.

Radius exponent in elastic and rigid arterial models optimized by the least energy principle

It was analyzed in normal physiological arteries whether the least energy principle would suffice to account for the radius exponent x. The mammalian arterial system was modeled as two types, the elastic or the rigid, to which Bernoulli's and Hagen-Poiseuille's equations were applied, respectively. We minimized the total energy function E, which was defined as the sum of kinetic, pressure, metabolic and thermal energies, and loss of each per unit time in a single artery transporting viscous incompressible blood. Assuming a scaling exponent α between the vessel radius (r) and length (l) to be 1.0, x resulted in 2.33 in the elastic model. The rigid model provided a continuously changing x from 2.33 to 3.0, which corresponded to Uylings' and Murray's theories, respectively, through a function combining Reynolds number with a proportional coefficient of the l - r relationship. These results were expanded to an asymmetric arterial fractal tree with the blood flow preservation rule. While x in the optimal elastic model accounted for around 2.3 in proximal systemic (r >1 mm) and whole pulmonary arteries (r ≥0.004 mm), optimal x in the rigid model explained 2.7 in elastic-muscular (0.1 < r ≤1 mm) and 3.0 in peripheral resistive systemic arteries (0.004 ≤ r ≤0.1 mm), in agreement with data obtained from angiographic, cast-morphometric, and in vivo experimental studies in the literature. The least energy principle on the total energy basis provides an alternate concept of optimality relating to mammalian arterial fractal dimensions under α = 1.0.

Endothelial and smooth muscle cell ion channels in pulmonary vasoconstriction and vascular remodeling

The pulmonary circulation is a low resistance and low pressure system. Sustained pulmonary vasoconstriction and excessive vascular remodeling often occur under pathophysiological conditions such as in patients with pulmonary hypertension. Pulmonary vasoconstriction is a consequence of smooth muscle contraction. Many factors released from the endothelium contribute to regulating pulmonary vascular tone, while the extracellular matrix in the adventitia is the major determinant of vascular wall compliance. Pulmonary vascular remodeling is characterized by adventitial and medial hypertrophy due to fibroblast and smooth muscle cell proliferation, neointimal proliferation, intimal, and plexiform lesions that obliterate the lumen, muscularization of precapillary arterioles, and in situ thrombosis. A rise in cytosolic free Ca(2+) concentration ([Ca(2+)]cyt) in pulmonary artery smooth muscle cells (PASMC) is a major trigger for pulmonary vasoconstriction, while increased release of mitogenic factors, upregulation (or downregulation) of ion channels and transporters, and abnormalities in intracellular signaling cascades are key to the remodeling of the pulmonary vasculature. Changes in the expression, function, and regulation of ion channels in PASMC and pulmonary arterial endothelial cells play an important role in the regulation of vascular tone and development of vascular remodeling. This article will focus on describing the ion channels and transporters that are involved in the regulation of pulmonary vascular function and structure and illustrating the potential pathogenic role of ion channels and transporters in the development of pulmonary vascular disease.

Structure and composition of pulmonary arteries, capillaries, and veins

The pulmonary vasculature comprises three anatomic compartments connected in series: the arterial tree, an extensive capillary bed, and the venular tree. Although, in general, this vasculature is thin-walled, structure is nonetheless complex. Contributions to structure (and thus potentially to function) from cells other than endothelial and smooth muscle cells as well as those from the extracellular matrix should be considered. This review is multifaceted, bringing together information regarding (i) classification of pulmonary vessels, (ii) branching geometry in the pulmonary vascular tree, (iii) a quantitative view of structure based on morphometry of the vascular wall, (iv) the relationship of nerves, a variety of interstitial cells, matrix proteins, and striated myocytes to smooth muscle and endothelium in the vascular wall, (v) heterogeneity within cell populations and between vascular compartments, (vi) homo- and heterotypic cell-cell junctional complexes, and (vii) the relation of the pulmonary vasculature to that of airways. These issues for pulmonary vascular structure are compared, when data is available, across species from human to mouse and shrew. Data from studies utilizing vascular casting, light and electron microscopy, as well as models developed from those data, are discussed. Finally, the need for rigorous quantitative approaches to study of vascular structure in lung is highlighted.

Concepts for Liver Segment Classification: Neither Old Ones nor New Ones, but a Comprehensive One

Concepts dealing with the subdivision of the human liver into independent vascular and biliary territories are applied routinely in radiological, surgical, and gastroenterological practice. Despite Couinaud's widely used eight-segments scheme, opinions on the issue differ considerably between authors. The aim of this article is to illustrate the scientific basis for understanding and harmonizing inconsistencies between seemingly contradictory observations. Possible clinical implications are addressed.

Regional structural and biomechanical alterations of the ovine main pulmonary artery during postnatal growth

The engineering foundation for novel approaches for the repair of congenital defects that involve the main pulmonary artery (PA) must rest on an understanding of changes in the structure-function relationship that occur during postnatal maturation. In the present study, we quantified the postnatal growth patterns in structural and biomechanical behavior in the ovine PA in the juvenile and adult stages. The biaxial mechanical properties and collagen and elastin fiber architecture were studied in four regions of the PA wall, with the collagen recruitment of the medial region analyzed using a custom biaxial mechanical-multiphoton microscopy system. Circumferential residual strain was also quantified at the sinotubular junction and bifurcation locations, which delimit the PA. The PA wall demonstrated significant mechanical anisotropy, except in the posterior region where it was nearly isotropic. Overall, we observed only moderate changes in regional mechanical properties with growth. We did observe that the medial and lateral locations experience a moderate increase in anisotropy. There was an average of about 24% circumferential residual stain present at the luminal surface in the juvenile stage that decreased to 16% in the adult stage with a significant decrease at the bifurcation, implying that the PA wall remodels toward the bifurcation with growth. There were no measurable changes in collagen and elastin content of the tunica media with growth. On average, the collagen fiber recruited more rapidly with strain in the adult compared to the juvenile. Interestingly, the PA thickness remained constant with growth. When this fact is combined with the observed stable overall mechanical behavior and increase in vessel diameter with growth, a simple Laplace Law wall stress estimate suggests an increase in effective PA wall stress with postnatal maturation. This observation is contrary to the accepted theory of maintenance of homeostatic stress levels in the regulation of vascular function and suggests alternative mechanisms regulate postnatal somatic growth. Understanding the underlying mechanisms, incorporating important structural features during growth, will help to improve our understanding of congenital defects of the PA and lay the basis for functional duplication in their repair and replacement.

Airway and pulmonary vascular measurements using contrast-enhanced micro-CT in rodents

Preclinical imaging allows pulmonary researchers to study lung disease and pulmonary drug delivery noninvasively and longitudinally in small animals. However, anatomically localizing a pathology or drug deposition to a particular lung region is not easily done. Thus, a detailed knowledge of the anatomical structure of small animal lungs is necessary for understanding disease progression and in addition would facilitate the analysis of the imaging data, mapping drug deposition and relating function to structure. In this study, contrast-enhanced micro-computed tomography (CT) of the lung produced high-resolution images that allowed for the characterization of the rodent airway and pulmonary vasculature. Contrast-enhanced micro-CT was used to visualize the airways and pulmonary vasculature in Sprague-Dawley rats (200–225 g) and BALB/c mice (20–25 g) postmortem. Segmented volumes from these images were processed to yield automated measurements of the airways and pulmonary vasculature. The diameters, lengths, and branching angles of the airway, arterial, and venous trees were measured and analyzed as a function of generation number and vessel diameter to establish rules that could be applied at all levels of tree hierarchy. In the rat, airway, arterial, and venous tress were measured down to the 20th, 16th, and 14th generation, respectively. In the mouse, airway, arterial, and venous trees were measured down to the 16th, 8th, and 7th generation, respectively. This structural information, catalogued in a rodent database, will increase our understanding of lung structure and will aid in future studies of the relationship between structure and function in animal models of disease.

MDCT-based quantification of porcine pulmonary arterial morphometry and self-similarity of arterial branching geometry

The pig is frequently used as an experimental model for studies of the pulmonary circulation, yet the branching and dimensional geometry of the porcine pulmonary vasculature remains poorly defined. The purposes of this study are to improve the geometric definition of the porcine pulmonary arteries and to determine whether the arterial tree exhibits self-similarity in its branching geometry. Five animals were imaged using thin slice spiral computed tomography in the prone posture during airway inflation pressure at 25 cmH2O. The luminal diameter and distance from the inlet of the left and right pulmonary arteries were measured along the left and right main arterial pathway in each lung of each animal. A further six minor pathways were measured in a single animal. The similarity in the rate of reduction of diameter with distance of all minor pathways and the two main pathways, along with similarity in the number of branches arising along the pathways, supports self-similarity in the arterial tree. The rate of reduction in diameter with distance from the inlet was not significantly different among the five animals (P > 0.48) when normalized for main pulmonary artery diameter and total main artery pathlength, which supports intersubject similarity. Other metrics to quantify the tree geometry are strikingly similar to those from airways of other quadrupeds, with the exception of a significantly larger length to diameter ratio, which is more appropriate for the vascular tree. A simplifying self-similar model for the porcine pulmonary arteries is proposed to capture the important geometric features of the arterial tree.

Using Fractal Geometry and Universal Growth Curves as Diagnostics for Comparing Tumor Vasculature and Metabolic Rate With Healthy Tissue and for Predicting Responses to Drug Therapies

Healthy vasculature exhibits a hierarchical branching structure in which, on average, vessel radius and length change systematically with branching order. In contrast, tumor vasculature exhibits less hierarchy and more variability in its branching patterns. Although differences in vasculature have been highlighted in the literature, there has been very little quantification of these differences. Fractal analysis is a natural tool for comparing tumor and healthy vasculature, especially because it has already been used extensively to model healthy tissue. In this paper, we provide a fractal analysis of existing vascular data, and we present a new mathematical framework for predicting tumor growth trajectories by coupling: (1) the fractal geometric properties of tumor vascular networks, (2) metabolic properties of tumor cells and host vascular systems, and (3) spatial gradients in resources and metabolic states within the tumor. First, we provide a new analysis for how the mean and variation of scaling exponents for ratios of vessel radii and lengths in tumors differ from healthy tissue. Next, we use these characteristic exponents to predict metabolic rates for tumors. Finally, by combining this analysis with general growth equations based on energetics, we derive universal growth curves that enable us to compare tumor and ontogenetic growth. We also extend these growth equations to include necrotic, quiescent, and proliferative cell states and to predict novel growth dynamics that arise when tumors are treated with drugs. Taken together, this mathematical framework will help to anticipate and understand growth trajectories across tumor types and drug treatments.

Topology and hemodynamics of the cortical cerebrovascular system

The cerebrovascular system continuously delivers oxygen and energy substrates to the brain, which is one of the organs with the highest basal energy requirement in mammals. Discontinuities in the delivery lead to fatal consequences for the brain tissue. A detailed understanding of the structure of the cerebrovascular system is important for a multitude of (patho-)physiological cerebral processes and many noninvasive functional imaging methods rely on a signal that originates from the vasculature. Furthermore, neurodegenerative diseases often involve the cerebrovascular system and could contribute to neuronal loss. In this review, we focus on the cortical vascular system. In the first part, we present the current knowledge of the vascular anatomy. This is followed by a theory of topology and its application to vascular biology. We then discuss possible interactions between cerebral blood flow and vascular topology, before summarizing the existing body of the literature on quantitative cerebrovascular topology.

Rat airway morphometry measured from in situ MRI-based geometric models

Rodents have been widely used to study the environmental or therapeutic impact of inhaled particles. Knowledge of airway morphometry is essential in assessing geometric influence on aerosol deposition and in developing accurate lung models of aerosol transport. Previous morphometric studies of the rat lung performed ex situ provided high-resolution measurements (50-125 μm). However, it is unclear how the overall geometry of these casts might have differed from the natural in situ appearance. In this study, four male Wistar rat (268 ± 14 g) lungs were filled sequentially with perfluorocarbon and phosphate-buffered saline before being imaged in situ in a 7-T magnetic resonance (MR) scanner at a resolution of 0.2 × 0.2 × 0.27 mm. Airway length, diameter, gravitational, bifurcation, and rotational angles were measured for the first four airway generations from 3D geometric models built from the MR images. Minor interanimal variability [expressed by the relative standard deviation RSD (=SD/mean)] was found for length (0.18 ± 0.07), diameter (0.15 ± 0.15), and gravitational angle (0.12 ± 0.06). One rat model was extended to 16 airway generations. Organization of the airways using a diameter-defined Strahler ordering method resulted in lower interorder variability than conventional generation-based grouping for both diameter (RSD = 0.12 vs. 0.42) and length (0.16 vs. 0.67). Gravitational and rotational angles averaged 82.9 ± 37.9° and 53.6 ± 24.1°, respectively. Finally, the major daughter branch bifurcated at a smaller angle (19.3 ± 14.6°) than the minor branch (60.5 ± 19.4°). These data represent the most comprehensive set of rodent in situ measurements to date and can be used readily in computational studies of lung function and aerosol exposure.

Hypoxia recruits intrapulmonary arteriovenous pathways in intact rats but not isolated rat lungs

Intrapulmonary arteriovenous anastomoses (IPAVS) directly connect the arterial and venous circulations in the lung, bypassing the capillary network. Here, we used solid, latex microspheres and isolated rat lung and intact, spontaneously breathing rat models to test the hypothesis that IPAVS are recruited by alveolar hypoxia. We found that hypoxia recruits IPAVS in the intact rat, but not the isolated lung. IPAVS are at least 70 μm in the rat and, interestingly, appear to be recruited when the mixed venous Po(2) falls below 22 mmHg. These data provide evidence that large-diameter, direct arteriovenous connections exist in the lung and are recruitable by hypoxia in the intact animal.

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.

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.

Development of an image-based network model of retinal vasculature

The paper presents an image-based network model of retinal vasculature taking account of the 3D vascular distribution of the retina. Mouse retinas were prepared using flat-mount technique and vascular images were obtained using confocal microscopy. The vascular morphometric information obtained from confocal images was used for the model development. The network model developed directly represents the vascular geometry of all the large vessels of the arteriolar and venular trees and models the capillaries using uniformly distributed meshes. The vasculatures in different layers of the retina, namely the superficial, intermediate, and deep layer, were modeled separately in the network and were linked through connecting vessels. The branching data of the vasculatures was recorded using the method of connectivity matrix of network (the graph theory). Such an approach is able to take into account the detailed vasculature of individual retinas concerned. Using the network model developed, a circulation analysis based on Poiseuille's equation was carried out. The investigations produced predictions of spatial distribution of the pressure, flow, and wall shear stress in the entire retinal vasculature. The method developed can be used as a tool for continuous monitoring of the retinal circulation for clinical assessments as well as experimental studies.

Advanced material strategies for tissue engineering scaffolds

Tissue engineering seeks to restore the function of diseased or damaged tissues through the use of cells and biomaterial scaffolds. It is now apparent that the next generation of functional tissue replacements will require advanced material strategies to achieve many of the important requirements for long-term success. Here we provide representative examples of engineered skeletal and myocardial tissue constructs in which scaffolds were explicitly designed to match native tissue mechanical properties as well as to promote cell alignment. We discuss recent progress in microfluidic devices that can potentially serve as tissue engineering scaffolds, since mass transport via microvascular-like structures will be essential in the development of tissue engineered constructs on the length scale of native tissues. Given the rapid evolution of the field of tissue engineering, it is important to consider the use of advanced materials in light of the emerging role of genetics, growth factors, bioreactors, and other technologies.

The scaling of blood flow resistance: from a single vessel to the entire distal tree

Although the flow resistance of a single vessel segment is easy to compute, the equivalent resistance of a network of vessel segments or the entire vasculature of an organ is difficult to determine in an analytic form. Here, we propose what we believe is a novel resistance scaling law for a vascular tree (i.e., the resistance of a vessel segment scales with the equivalent resistance of the corresponding distal tree). The formulation can be written as (R(s)/R(c)) proportional, variant(L(s)/L(c)) (where R(s) and L(s) are the resistance and length of a vessel segment, respectively, and R(c) and L(c) are the equivalent resistance and total length of the corresponding distal tree, respectively), which was validated for the coronary vascular systems of the heart. The scaling law was also shown to apply to the vascular systems of the lung, mesentery, muscle, eye, and so on. The novel resistance scaling law, coupled with the 3/4-power scaling law for metabolic rates, can predict several structure-function relations of vascular trees, albeit with a different exponent. In particular, the self-similar nature of the scaling law may serve as a diagnostic tool with the help of noninvasive imaging modalities.

A scaling law of vascular volume

Vascular volume is of fundamental significance to the function of the cardiovascular system. An accurate prediction of blood volume in patients is physiologically and clinically significant. This study proposes what we believe is a novel volume scaling relation of the form: V(c)=K(v)D(s)(2/3)L(c), where V(c) and L(c) are cumulative vessel volume and length, respectively, in the tree, and D(s) is the diameter of the vessel segment. The scaling relation is validated in vascular trees of various organs including the heart, lung, mesentery, muscle, and eye of different species. Based on the minimum energy hypothesis and volume scaling relation, four structure-function scaling relations are predicted, including the diameter-length, volume-length, flow-diameter, and volume-diameter relations, with exponent values of 3/7, 1(2/7), 2(1/3), and 3, respectively. These four relations are validated in the various vascular trees, which further confirm the volume scaling relation. This scaling relation may serve as a control reference to estimate the blood volume in various organs and species. The deviation from the scaling relation may indicate hypovolemia or hypervolemia and aid diagnosis.

An automated self-similarity analysis of the pulmonary tree of the Sprague-Dawley rat

We present the results of an automated analysis of the morphometry of the pulmonary airway trees of the Sprague-Dawley rat. Our work is motivated by a need to inform lower-dimensional mathematical models to prescribe realistic boundary conditions for multiscale hybrid models of rat lung mechanics. Silicone casts were made from three age-matched, male Sprague-Dawley rats, immersed in a gel containing a contrast agent and subsequently imaged with magnetic resonance (MR). From a segmentation of this data, we extracted a connected graph, representing the airway centerline. Segment statistics (lengths and diameters) were derived from this graph. To validate this MR imaging/digital analysis method, airway segment measurements were compared with nearly 1,000 measurements collected by hand using an optical microscope from one of the rat lung casts. To evaluate the reproducibility of the MR imaging/digital analysis method, two lung casts were each imaged three times with randomized orientations in the MR bore. Diameters and lengths of randomly selected airways were compared among each of the repeated imaging datasets to estimate the variability. Finally, we analyzed the morphometry of the airway tree by assembling individual airway segments into structures that span multiple generations, which we call branches. We show that branches not segments are the fundamental repeating unit in the rat lung and develop simple mathematical relationships describing these structures for the entire lung. Our analysis shows that airway diameters and lengths have both a deterministic and stochastic character.

Geometric Characteristics of Arterial Network of Rat Pial Microcirculation

OBJECTIVE

The aim of the study was to assess the geometric characteristics of rat pial microcirculation and describe the vessel bifurcation patterns by 'connectivity matrix'.

METHODS

Male Wistar rats were used to visualize pial microcirculation by a fluorescent microscopy technique through an open cranial window, using fluorescein isothiocyanate bound to dextran (molecular weight 70 kDa). The arteriolar network was mapped by stop-frame images. Diameters and lengths of arterioles were measured with a computer-assisted method. Pial arterioles were classified according to a centripetal ordering scheme (Strahler method modified according to diameter) from the smallest order 1 to the largest order 5 arterioles in the preparation. A distinction between arteriolar segments and elements was used to express the series-parallel features of the pial arteriolar networks. A connectivity matrix was used to describe the connection of blood vessels from one order to another.

RESULTS

The arterioles were assigned 5 orders of branching by Strahler's ordering scheme, from order 1 (diameter: 16.0 +/- 2.5 microm) to order 5 (62 +/- 5.0 microm). Order 1 arterioles gave origin to capillaries, assigned order 0. The diameter, length and branching of the 5 arteriolar orders grew as a geometric sequence with the order number in accordance with Horton's law. The segments/elements ratio was the highest in order 4 and 3 arterioles, indicating the greatest asymmetry of ramifications. Finally, the branching vessels in the networks were described in details by the connectivity matrix. Fractal dimensions of arteriolar length and diameter were 1.75 and 1.78, respectively.

CONCLUSIONS

The geometric characteristics of rat pial microcirculation indicate that distribution of vessels is fractal. The connectivity matrix allowed us to describe the number of daughter vessels spreading from parent vessels. This ordering scheme may be useful to describe vessel function, according to diameter, length and branching.

Automatic segmentation of 3D micro-CT coronary vascular images

Although there are many algorithms available in the literature aimed at segmentation and model reconstruction of 3D angiographic images, many are focused on characterizing only a part of the vascular network. This study is motivated by the recent emerging prospects of whole-organ simulations in coronary hemodynamics, autoregulation and tissue oxygen delivery for which anatomically accurate vascular meshes of extended scale are highly desirable. The key requirements of a reconstruction technique for this purpose are automation of processing and sub-voxel accuracy. We have designed a vascular reconstruction algorithm which satisfies these two criteria. It combines automatic seeding and tracking of vessels with radius detection based on active contours. The method was first examined through a series of tests on synthetic data, for accuracy in reproduced topology and morphology of the network and was shown to exhibit errors of less than 0.5 voxel for centerline and radius detections, and 3 degrees for initial seed directions. The algorithm was then applied on real-world data of full rat coronary structure acquired using a micro-CT scanner at 20 microm voxel size. For this, a further validation of radius quantification was carried out against a partially rescanned portion of the network at 8 microm voxel size, which estimated less than 10% radius error in vessels larger than 2 voxels in radius.

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.