Microcirculatory model relating geometrical variation to changes in pressure and flow rate

A constrained constructive optimization model of branching arteriolar networks in rat skeletal muscle

Blood flow regulation within the microvasculature reflects a complex interaction of regulatory mechanisms and varies spatially and temporally according to conditions such as metabolism, growth, injury, and disease. Understanding the role of microvascular flow distributions across conditions is of interest to investigators spanning multiple disciplines; however, data collection within networks can be labor-intensive and challenging due to limited resolution. To overcome these experimental challenges, computational network models that can accurately simulate vascular behavior are highly beneficial. Constrained constructive optimization (CCO) is a commonly used algorithm for vascular simulation, particularly well known for its adaptability toward vascular modeling across tissues. The present work demonstrates an implementation of CCO aimed to simulate a branching arteriolar microvasculature in healthy skeletal muscle, validated against literature including comprehensive rat gluteus maximus vasculature datasets, and reviews a list of user-specified adjustable model parameters to understand how their variability affects the simulated networks. Network geometric properties, including mean element diameters, lengths, and numbers of bifurcations per order, Horton's law ratios, and fractal dimension, demonstrate good validation once model parameters are adjusted to experimental data. This model successfully demonstrates hemodynamic properties such as Murray's law and the network Fahraeus effect. Application of centrifugal and Strahler ordering schemes results in divergent descriptions of identical simulated networks. This work introduces a novel CCO-based model focused on generating branching skeletal muscle microvascular arteriolar networks based on adjustable model parameters, thus making it a valuable tool for investigations into skeletal muscle microvascular structure and tissue perfusion.NEW & NOTEWORTHY The present work introduces a CCO-based algorithm for generating branching arteriolar networks, with adjustable model parameters to enable modeling in varying skeletal muscle tissues. The geometric and hemodynamic parameters of the generated networks have been comprehensively validated using experimental data collected previously in-house and from literature. This is one of few validated CCO-based models to specialize in skeletal muscle microvasculature and acts as a beneficial tool for investigating the microvasculature for hypothesis testing and validation.

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.

Supporting measurements or more averages? How to quantify cerebral blood flow most reliably in 5 minutes by arterial spin labeling

Purpose

To determine whether sacrificing part of the scan time of pseudo‐continuous arterial spin labeling (PCASL) for measurement of the labeling efficiency and blood T1 is beneficial in terms of CBF quantification reliability.

Methods

In a simulation framework, 5‐minute scan protocols with different scan time divisions between PCASL data acquisition and supporting measurements were evaluated in terms of CBF estimation variability across both noise and ground truth parameter realizations taken from the general population distribution. The entire simulation experiment was repeated for a single‐post‐labeling delay (PLD), multi‐PLD, and free‐lunch time‐encoded (te‐FL) PCASL acquisition strategy. Furthermore, a real data study was designed for preliminary validation.

Results

For the considered population statistics, measuring the labeling efficiency and the blood T1 proved beneficial in terms of CBF estimation variability for any distribution of the 5‐minute scan time compared to only acquiring ASL data. Compared to single‐PLD PCASL without support measurements as recommended in the consensus statement, a 26%, 33%, and 42% reduction in relative CBF estimation variability was found for optimal combinations of supporting measurements with single‐PLD, free‐lunch, and multi‐PLD PCASL data acquisition, respectively. The benefit of taking the individual variation of blood T1 into account was also demonstrated in the real data experiment.

Conclusions

Spending time to measure the labeling efficiency and the blood T1 instead of acquiring more averages of the PCASL data proves to be advisable for robust CBF quantification in the general population.

Methods to label, image, and analyze the complex structural architectures of microvascular networks

Microvascular networks play key roles in oxygen transport and nutrient delivery to meet the varied and dynamic metabolic needs of different tissues throughout the body, and their spatial architectures of interconnected blood vessel segments are highly complex. Moreover, functional adaptations of the microcirculation enabled by structural adaptations in microvascular network architecture are required for development, wound healing, and often invoked in disease conditions, including the top eight causes of death in the Unites States. Effective characterization of microvascular network architectures is not only limited by the available techniques to visualize microvessels but also reliant on the available quantitative metrics that accurately delineate between spatial patterns in altered networks. In this review, we survey models used for studying the microvasculature, methods to label and image microvessels, and the metrics and software packages used to quantify microvascular networks. These programs have provided researchers with invaluable tools, yet we estimate that they have collectively attained low adoption rates, possibly due to limitations with basic validation, segmentation performance, and nonstandard sets of quantification metrics. To address these existing constraints, we discuss opportunities to improve effectiveness, rigor, and reproducibility of microvascular network quantification to better serve the current and future needs of microvascular research.

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.

Cerebral oxygenation and optimal vascular brain organization

The cerebral vascular network has evolved in such a way so as to minimize transport time and energy expenditure. This is accomplished by a subtle combination of the optimal arrangement of arteries, arterioles and capillaries and the transport mechanisms of convection and diffusion. Elucidating the interaction between cerebral vascular architectonics and the latter physical mechanisms can catalyse progress in treating cerebral pathologies such as stroke, brain tumours, dementia and targeted drug delivery. Here, we show that brain microvascular organization is predicated on commensurate intracapillary oxygen convection and parenchymal diffusion times. Cross-species grey matter results for the rat, cat, rabbit and human reveal very good correlation between the cerebral capillary and tissue mean axial oxygen convective and diffusion time intervals. These findings agree with the constructal principle.

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.

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 model for capillary vasculature of retina

The paper presents a method of development of a detailed network model to represent retinal capillary vasculature. The capillary model is a circular mesh consisting of concentric rings with an increasing diameter. Each of the rings has uniformly distributed bifurcation nodes to represent capillary vessels. The model is customized using the data that has been measured from confocal microscopic images of a mouse retina. The capillary model developed can be connected to networks of larger vessels of the vasculature such as arterial and venous networks to form a complete model of the retinal network. A method to automate such interface connections between capillary and other vascular networks using connecting vessels (i.e., pre-capillary and post-capillary) is also presented in the paper. Such a detailed image-based capillary model together with the arterial and venular networks can be used for various circulation simulations to obtain accurate information on hemodynamic quantities such as the spatial distribution of pressure and flow in the vasculature for both physiological and pathological conditions. The method presented for the development of the capillary model can also be adopted for vasculatures of other organs.

Intravascular pillars and pruning in the extraembryonic vessels of chick embryos

To investigate the local mechanical forces associated with intravascular pillars and vessel pruning, we studied the conducting vessels in the extraembryonic circulation of the chick embryo. During the development days 13-17, intravascular pillars and blood flow parameters were identified using fluorescent vascular tracers and digital time-series video reconstructions. The geometry of selected vessels was confirmed by corrosion casting and scanning electron microscopy. Computational simulations of pruning vessels suggested that serial pillars form along pre-existing velocity streamlines; blood pressure demonstrated no obvious spatial relationship with the intravascular pillars. Modeling a Reynolds number of 0.03 produced 4 pillars at approximately 20-μm intervals matching the observed periodicity. In contrast, a Reynolds number of 0.06 produced only 2 pillars at approximately 63-μm intervals. Our modeling data indicated that the combination of wall shear stress and gradient of shear predicted the location, direction, and periodicity of developing pillars.

Blood flow in microvascular networks: a study in nonlinear biology

Plasma skimming and the Fahraeus-Lindqvist effect are well-known phenomena in blood rheology. By combining these peculiarities of blood flow in the microcirculation with simple topological models of microvascular networks, we have uncovered interesting nonlinear behavior regarding blood flow in networks. Nonlinearity manifests itself in the existence of multiple steady states. This is due to the nonlinear dependence of viscosity on blood cell concentration. Nonlinearity also appears in the form of spontaneous oscillations in limit cycles. These limit cycles arise from the fact that the physics of blood flow can be modeled in terms of state dependent delay equations with multiple interacting delay times. In this paper we extend our previous work on blood flow in a simple two node network and begin to explore how topological complexity influences the dynamics of network blood flow. In addition we present initial evidence that the nonlinear phenomena predicted by our model are observed experimentally.

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.

Microvascular Rheology and Hemodynamics

The goal of elucidating the biophysical and physiological basis of pressure-flow relations in the microcirculation has been a recurring theme since the first observations of capillary blood flow in living tissues. At the birth of the Microcirculatory Society, seminal observations on the heterogeneous distribution of blood cells in the microvasculature and the rheological properties of blood in small bore tubes raised many questions on the viscous properties of blood flow in the microcirculation that captured the attention of the Society's membership. It is now recognized that blood viscosity in small bore tubes may fall dramatically as shear rates are increased, and increase (dramatically with elevations in hematocrit. These relationships are strongly affected by blood cell deformability and concentration, red cell aggregation, and white cell interactions with the red cells anti endothelium. Increasing strength of red cell aggregation may result in sequestration of clumps of red cells with either reductions or increases in microvascular hematocrit dependent upon network topography. During red cell aggregation, resistance to flow may thus decrease with hematocrit reduction or increase due to redistribution of red cells. Blood cell adhesion to the microvessel wall may initiate flow reductions, as, for example, in the case of red cell adhesion to the endothelium in sickle cell disease, or leukocyte adhesion in inflammation. The endothelial glycocalyx has been shown to result from a balance of the biosynthesis of new glycans, and the enzymatic or shear-dependent alterations in its composition. Flow-dependent reductions in the endothelial surface layer may thus affect the resistance to flow and/or the adhesion of red cells and/or leukocytes to the endothelium. Thus, future studies aimed at the molecular rheology of the endothelial surface layer may provide new insights into determinants of the resistance to flow.

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.

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.

Loss of M5 muscarinic acetylcholine receptors leads to cerebrovascular and neuronal abnormalities and cognitive deficits in mice

The M5 muscarinic acetylcholine receptor (M5R) has been shown to play a crucial role in mediating acetylcholine-dependent dilation of cerebral blood vessels. We show that male M5R-/- mice displayed constitutive constriction of cerebral arteries using magnetic resonance angiography in vivo. Male M5R-/- mice exhibited a significantly reduced cerebral blood flow (CBF) in the cerebral cortex, hippocampus, basal ganglia, and thalamus. Cortical and hippocampal pyramidal neurons from M5R-/- mice showed neuronal atrophy. Hippocampus-dependent spatial and nonspatial memory was also impaired in M5R-/- mice. In M5R-/- mice, CA3 pyramidal cells displayed a significantly attenuated frequency of the spontaneous postsynaptic current and long-term potentiation was significantly impaired at the mossy fiber-CA3 synapse. Our findings suggest that impaired M5R signaling may play a role in the pathophysiology of cerebrovascular deficits. The M5 receptor may represent an attractive novel therapeutic target to ameliorate memory deficits caused by impaired cerebrovascular function.

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.

First phase microcirculatory reaction to mechanical skin irritation: a three layer model of a compliant vascular tree

Mechanical skin irritation creates vasodilatation in the line of a stroke and in the surrounding tissue. To obtain further insight on underlying physiological mechanisms we developed a model of the vascular network comprised of three layers, where the first and the last one have a tree structure. They represent the arterial and the venous system, respectively. Both are connected via an intermediate zone representing the core of the microcirculation, which is described by means of a compliant compartment model. Irritation induces change in compliance of vessels situated at the entrance of the intermediate zone. Thus the model describes flow and pressure behavior due to mechanical skin irritation.

A graph theory analysis of renal glomerular microvascular networks

A graph theory model and its invariants are used to compare previously published renal glomerular networks of six adult rats, one adult uremic rat, and one newborn rat. Invariants calculated include order, size, cycle rank, eccentricity, root distance, planarity, and vertex degree distribution. These invariants enabled the differentiation of six normal adult glomerular microvascular networks from that of the uremic glomerulus and from that of the normal newborn glomerulus. These invariants might then be used to differentiate between normal and pathological vascular networks. Also proposed are graph theory invariants that might be used to develop a quantitative model for angiogenesis.

Effect of local shell conductance on the vascularisation of the chicken chorioallantoic membrane

The vascularisation of the chorioallantoic membrane (CAM) of avian embryos is influenced by environmental oxygen partial pressure (P(O(2))) on a global level: incubation at high P(O(2)) reduces the density of pre- and post-capillary vessels of the CAM and decelerates the thinning of the blood-gas barrier, and vice versa. This study investigates the effects of local P(O(2)) on vascular development during the formative period of days ten to fifteen, by making half of the egg hypoxic and the other half hyperoxic. The densities of arterioles, venules and capillaries were reduced under the hypoxic side, compared to untreated eggs, but not significantly changed on the hyperoxic side. Harmonic mean thickness of the tissue barrier and total CAM blood volume were not affected by the treatments. Vascular development of the CAM was therefore only partly influenced by local P(O(2)).

Effects of vitamin E and sesamin on hypertension and cerebral thrombogenesis in stroke-prone spontaneously hypertensive rats

The preventive effects of sesamin, a lignan from sesame oil, and vitamin E on hypertension and thrombosis were examined using stroke-prone spontaneously hypertensive rats (SHRSP). At 5 weeks of age the animals were separated into four groups: (i) a control group; (ii) a vitamin E group, which was given a 1,000 mg alpha-tocopherol/kg diet; (iii) a sesamin group, given a 1,000 mg sesamin/kg diet; and (iv) a vitamin E plus sesamin group, given a 1,000 mg alpha-tocopherol plus 1,000 mg sesamin/kg diet for 5 weeks from 5 to 10 weeks of age. Resting blood pressure was measured by the tail-cuff method once weekly. A closed cranial window was created and platelet-rich thrombi were induced in vivo using a helium-neon laser technique. The number of laser pulses required for formation of an occlusive thrombus was used as an index of thrombotic tendency. In control rats, systolic blood pressure and the amount of urinary 8-hydroxy-2'-deoxyguanosine (8-OHdG) became significantly elevated with age. However, the elevation in blood pressure and 8-OHdG were significantly suppressed in rats administrated vitamin E, sesamin, or vitamin E plus sesamin. At 10 weeks, the number of laser pulses required to induce an occlusive thrombus in arterioles of the control group was significantly lower than in the other groups (p<0.05). These results indicate that chronic ingestion of vitamin E and sesamin attenuated each of elevation in blood pressure, oxidative stress and thrombotic tendency, suggesting that these treatments might be beneficial in the prevention of hypertension and stroke.

Effects of thrombin and of the phospholipase C inhibitor, D609, on the vascularity of the chick chorioallantoic membrane

Microvascular corrosion casting was used to assess the effects of thrombin and D609, a phospholipase C inhibitor, on the vascularity of the chick embryo chorioallantoic membrane (CAM). Discs containing vehicle, thrombin or D609 were placed on the CAM of fertilized white Leghorn eggs on Day 9 of gestation and vascularity was assessed on Day 11. Thrombin caused significant increases in the numbers (43%), diameters (5%) and lengths (17%), of both pre- and postcapillaries (first-order vessels by centripetal ordering). Conversely, D609 caused a decrease in the numbers (27%), lengths (12%) and diameters (8%) of first-order vessels. D609 decreased the total vascular volume of first- to third-order vessels by 32%, whereas thrombin increased vascular volume by 27%. Additionally, thrombin increased capillary plexus density by 6%, whereas D609 decreased capillary plexus density by 3%. These findings provide a quantitative assessment of changing vascularity in the chick CAM--a model assay system in the development of pro- and antiangiogenic agents.

Hindlimb unweighting alters endothelium-dependent vasodilation and ecNOS expression in soleus arterioles

The purpose of this study was to test the hypothesis that endothelium-dependent dilation is impaired in soleus resistance arteries from hindlimb-unweighted (HLU) rats. Male Sprague-Dawley rats (300-350 g) were exposed to HLU (n = 14) or weight-bearing control (Con, n = 14) conditions for 14 days. After the 14-day treatment period, soleus first-order (1A) arterioles were isolated and cannulated with micropipettes to assess vasodilator responses to an endothelium-dependent dilator, ACh (10(-9)-10(-4) M), and an endothelium-independent dilator, sodium nitroprusside (SNP, 10(-9)-10(-4) M). Arterioles from HLU rats were smaller than Con arterioles (maximal passive diameter = 140 +/- 4 and 121 +/- 4 microm in Con and HLU, respectively) but developed similar spontaneous myogenic tone (43 +/- 3 and 45 +/- 3% in Con and HLU, respectively). Arteries from Con and HLU rats dilated in response to increasing doses of ACh, but dilation was impaired in arterioles from HLU rats (P = 0.03), as was maximal dilation to ACh (85 +/- 4 and 65 +/- 4% possible dilation in Con and HLU, respectively). Inhibition of nitric oxide (NO) synthase (NOS) with N(omega)-nitro-L-arginine (300 microM) reduced ACh dilation by approximately 40% in arterioles from Con rats and eliminated dilation in arterioles from HLU rats. The cyclooxygenase inhibitor indomethacin (50 microM) did not significantly alter dilation to ACh in either group. Treatment with N(omega)-nitro-L-arginine + indomethacin eliminated all ACh dilation in Con and HLU rats. Dilation to sodium nitroprusside was not different between groups (P = 0.98). To determine whether HLU decreased expression of endothelial cell NOS (ecNOS), mRNA and protein levels were measured in single arterioles with RT-PCR and immunoblot analysis. The ecNOS mRNA and protein expression was significantly lower in arterioles from HLU rats than in Con arterioles (20 and 65%, respectively). Collectively, these data indicate that HLU impairs ACh dilation in soleus 1A arterioles, in part because of alterations in the NO pathway.

Effect of restricted water exchange on cerebral blood flow values calculated with arterial spin tagging: a theoretical investigation

Arterial spin tagging techniques originally used the one-compartment Kety model to describe the dynamics of tagged water in the brain. The work presented here develops a more realistic model that includes the contribution of tagged water in the capillary bed and accounts for the finite time required for water to diffuse across the blood-brain barrier. The new model was used to evaluate potential errors in cerebral blood flow values calculated using the one-compartment Kety model. The results predict that if the one-compartment Kety model is used to analyze arterial spin tagging data the observed grey matter cerebral blood flow values should be relatively insensitive to restricted diffusion of water across the capillary bed. For instance, the observed grey matter cerebral blood flow should closely approximate the true cerebral blood flow and not the product of the extraction fraction and the cerebral blood flow. This prediction is in agreement with recent experimental arterial spin tagging results.

Heat transfer in a microvascular network: the effect of heart rate on heating and cooling in reptiles (Pogona barbata and Varanus varius)

Thermally-induced changes in heart rate and blood flow in reptiles are believed to be of selective advantage by allowing animal to exert some control over rates of heating and cooling. This notion has become one of the principal paradigms in reptilian thermal physiology. However, the functional significance of changes in heart rate is unclear, because the effect of heart rate and blood flow on total animal heat transfer is not known. I used heat transfer theory to determine the importance of heat transfer by blood flow relative to conduction. I validated theoretical predictions by comparing them with field data from two species of lizard, bearded dragons (Pogona barbata) and lace monitors (Varanus varius). Heart rates measured in free-ranging lizards in the field were significantly higher during heating than during cooling, and heart rates decreased with body mass. Convective heat transfer by blood flow increased with heart rate. Rates of heat transfer by both blood flow and conduction decreased with mass, but the mass scaling exponents were different. Hence, rate of conductive heat transfer decreased more rapidly with increasing mass than did heat transfer by blood flow, so that the relative importance of blood flow in total animal heat transfer increased with mass. The functional significance of changes in heart rate and, hence, rates of heat transfer, in response to heating and cooling in lizards was quantified. For example, by increasing heart rate when entering a heating environment in the morning, and decreasing heart rate when the environment cools in the evening a Pogona can spend up to 44 min longer per day with body temperature within its preferred range. It was concluded that changes in heart rate in response to heating and cooling confer a selective advantage at least on reptiles of mass similar to that of the study animals (0. 21-5.6 kg).

Effects of voluntary exercise and L-arginine on thrombogenesis and microcirculation in stroke-prone spontaneously hypertensive rats

1. The preventive effects of exercise and L-arginine intake on hypertension and thrombosis in stroke-prone spontaneously hypertensive rats (SHRSP) were studied. 2. Stroke-prone spontaneously hypertensive rats were divided into three groups: (i) the control, sedentary group; (ii) the exercise group, which was allowed to run voluntarily on running wheels; and (iii) the L-arginine intake group, which was given 2.25% L-arginine solution for 8 weeks from 4 to 12 weeks of age. In the control group, one rat died from stroke and symptoms of stroke were observed in the remaining animals. Similar symptoms were recorded in one rat of the exercise group, but not in the L-arginine group. 3. Blood pressure increased in the control group and this increase was suppressed significantly in the exercise and L-arginine groups. Thrombotic potential in cerebral vessels was the lowest at 4 weeks in all groups and was increased significantly at 12 weeks in the control group, but not in the exercise and L-arginine groups. Plasma concentrations of NO2/NO3 were lower in all animals at 12 weeks compared with those at 4 weeks. This reduction was significantly less marked in the L-arginine group. Cerebral arterioles in control rats at 12 weeks of age were significantly smaller in diameter than those at 4 weeks and these changes were less pronounced in the exercise and L-arginine groups. 4. The results provide clear evidence for the beneficial effects of L-arginine intake and voluntary exercise in mechanisms related to hypertension, thrombosis and stroke.

Longitudinal position matrix of the pig coronary vasculature and its hemodynamic implications

Hemodynamic analysis of coronary blood flow must be based on a statistically valid geometric model of the coronary vasculature. We have previously developed a diameter-defined Strahler model for the arterial and venous trees and a network model for the capillaries. A full set of data describing the geometric properties of the porcine coronary vasculature was given. The order number, diameter, length, connectivity matrix [m,n] (CM), and parallel-series features were measured for all orders of vessels of the right coronary artery (RCA), left anterior descending artery (LAD), left circumflex artery (LCX), and coronary venous system. The purpose of the present study is to present another feature of the branching pattern of the coronary vasculature: the longitudinal position matrix [m,n] (LPM), whose component in row m and column n is the fractional longitudinal position of the branch point on vessels of order n at which vessels of order m branch off (m < or = n). The LPM of the pig RCA, LAD and LCX arterial trees, as well as the coronary sinusal and thebesian venous trees, are presented. The hemodynamic implications of the LPM are illustrated by comparing two kinds of circuits: one, the CM + LPM model, simulates the mean data on the morphology (diameters, lengths, and numbers), CM, and LPM of vessels, whereas the other, the CM model, simulates the mean data on the morphology and CM without considering the LPM. We found that the LPM affects the hemodynamics of coronary blood flow especially with regard to the nonuniformity or dispersion of flow distribution.

Analysis of pig's coronary arterial blood flow with detailed anatomical data

Blood flow to perfuse the muscle cells of the heart is distributed by the capillary blood vessels via the coronary arterial tree. Because the branching pattern and vascular geometry of the coronary vessels in the ventricles and atria are nonuniform, the flow in all of the coronary capillary blood vessels is not the same. This nonuniformity of perfusion has obvious physiological meaning, and must depend on the anatomy and branching pattern of the arterial tree. In this study, the statistical distribution of blood pressure, blood flow, and blood volume in all branches of the coronary arterial tree is determined based on the anatomical branching pattern of the coronary arterial tree and the statistical data on the lengths and diameters of the blood vessels. Spatial nonuniformity of the flow field is represented by dispersions of various quantities (SD/mean) that are determined as functions of the order numbers of the blood vessels. In the determination, we used a new, complete set of statistical data on the branching pattern and vascular geometry of the coronary arterial trees. We wrote hemodynamic equations for flow in every vessel and every node of a circuit, and solved them numerically. The results of two circuits are compared: one asymmetric model satisfies all anatomical data (including the mean connectivity matrix) and the other, a symmetric model, satisfies all mean anatomical data except the connectivity matrix. It was found that the mean longitudinal pressure drop profile as functions of the vessel order numbers are similar in both models, but the asymmetric model yields interesting dispersion profiles of blood pressure and blood flow. Mathematical modeling of the anatomy and hemodynamics is illustrated with discussions on its accuracy.

On the pressure and flow-rate distributions in tree-like and arterial-venous networks

A solution algorithm yielding the pressure and flow-rate distributions for steady flow in an arbitrary, tree-like network is provided. Given the tree topology, the conductance of each segment and the pressure distribution at the boundary nodes, the solution is obtained from a simple recursion based on perfect Gauss elimination. An iterative solution method using this algorithm is suggested to solve for the pressure and flow-rate distributions in an arbitrary diverging-converging (arterial-venous) network consisting of two tree-like networks which are connected to each other at the capillary nodes. A number of special solutions for tree-like networks are obtained for which the general algorithm is either simplified or can be replaced by closed form solutions of the pressure and flow-rate distributions. These special solutions can also be obtained for each tree of diverging-converging networks having particular topologies and conductance distributions. Sample numerical results are provided.

A quantitative interpretation of IVIM measurements of vascular perfusion in the rat brain

Pulsed gradient spin echo (PGSE) sequences have been used to measure the signal loss of 19F in perfluorinated hydrocarbon blood substitutes moving within the vasculature of the rat brain in the experimental conditions of the study. The signal loss is not characterized by a single apparent pseudodiffusion coefficient. A simple vascular network model based on self-similarity has been used to calculate the shape of the signal loss. Excellent agreement with the experiment has been obtained showing that the IVIM measurements are sensitive to flow over a wide range of vessel diameters and flow rates. This model of vascular structure may serve well for other MR measurements that are sensitive to perfusion.

Semi-automatic segmentation of vascular network images using a rotating structuring element (ROSE) with mathematical morphology and dual feature thresholding

A method for measuring the spatial concentration of specific categories of vessels in a vascular network consisting of vessels of several diameters, lengths, and orientations is demonstrated. It is shown that a combination of the mathematical morphology operation, opening, with a linear rotating structuring element (ROSE) and dual feature thresholding can semi-automatically segment categories of vessels in a vascular network. Capillaries and larger vessels (arterioles and venules) are segmented here in order to assess their spatial concentrations. The ROSE algorithm generates the initial segmentation, and dual feature thresholding provides a means of eliminating the nonedge artifact pixels. The subsequent gray-scale histogram of only the edge pixels yields the correct segmentation threshold value. This image processing strategy is demonstrated on micrographs of vascular casts. By adjusting the structuring element and rotation angles, it could be applied to other network structures where a segmentation by network component categories is advantageous, but where the objects can have any orientation.

Branching patterns in the porcine coronary arterial tree. Estimation of flow heterogeneity

The aim of this study is to quantify the porcine coronary arterial branching pattern and to use this quantification for the interpretation of flow heterogeneity. Two casts of the coronary arterial tree were made at diastolic arrest and maximal dilation. The relation between length and diameter of arterial segments was quantified, as well as the area expansion ratio and diameter symmetry of vascular nodes. These relations were used to construct computer models of the coronary arterial tree, covering diameters between 10 and 500 microns. Topology of these simulated trees was analyzed using Strahler ordering: Bifurcation ratio, diameter ratio, and length ratio were constant along orders 2-8 and equal to 3.30, 1.51, and 1.63, respectively. In each order, the number of segments per Strahler vessel was almost geometrically distributed. For the lowest orders, these predictions were confirmed by direct observations. From the network model, local pressure and flow were also predicted: Pressure fell from 90 to 32 mm Hg at the 10-microns level. The coefficient of variation (CV) of flow in individual segments was dependent on the number of perfused terminal segments (Nt) according to the fractal relation CV(Nt) approximately Nt(1-D), where D is the fractal dimension (1.20). CV of flow in 1-g tissue units was predicted to be 18%. This study shows that the structure of the coronary arterial bed is an important determinant of the fractal nature of local flow heterogeneity.

Microvascular architecture in rat soleus and extensor digitorum longus muscles

Microvascular architecture was investigated in the slow-twitch soleus (SOL) and fast-twitch extensor digitorum longus (EDL) muscles. Rats (n = 5) were anesthetized and papaverine was infused into a carotid artery cannula to induce vasodilation. Microfil casting compound was then infused at an inflation pressure (caudal artery) of 100 mm Hg. Bilateral SOL and EDL muscles were excised 24-72 hr postcasting, dehydrated in ethanol, and cleared in methyl salicylate. Branch frequencies (BR) and segment lengths (SL) of intramuscular arterioles and venules were quantified along primary (1 degree), secondary (2 degrees), and tertiary (3 degrees) order microvessels using microscopy. In both muscles, BR decreased with increasing vessel order. Regional differences in network organization were observed within the EDL muscle. SL of 1 degrees arterioles was 47% shorter in the SOL muscle indicating more compact microvascular networks compared to the EDL muscle. These findings provide a structural basis for reported differences in blood flow between the SOL and EDL muscles at rest and during exercise.

A three-dimensional analysis of plasma skimming at microvascular bifurcations

This paper analyzes an important underlying mechanism for the discharge hematocrit reduction observed in microvessels, which refers to the plasma skimming from the cell-free layer near the parent tube wall in the presence of a side branch. The three-dimensional theory recently developed by the authors (Yan et al., 1991, J. Fluid Mech., in press) for treating the simple shear flow past a side branch tube in a plane wall with suction is first summarized and then extended to treat T bifurcations from parent vessels with an upstream Poiseuille flow. For unequal vessel bifurcations, a fundamental new dimensionless group, Q = 1/8(qb/qp)(Rp/Rb)3, is derived whose value determines the shape of the upstream capture tube of the plasma phase, when the partitioning qb/qp of the flow into the side branch and the ratio Rp/Rb of the radii of the parent and side branch vessels are varied. Closed form expressions are then presented for the three-dimensional fluid capture tube shape upstream of the bifurcation which are valid when Q greater than 1 or Q less than 0.2. Based on this theory and its modification for an upstream Poiseuille velocity profile, the separating surface shape, the critical minimum fractional flux for incipient cell capture, and the discharge hematocrit defect and its dependence on the flow rate are predicted. It is shown, furthermore, that for flows typical of the microcirculation, a single dimensionless number, P = 3 pi Q(Rb/gamma 2), with gamma being the cell-free layer thickness, can be defined whose value determines the discharge hematocrit defect that arises from plasma skimming. The minimum critical flow rate for any red cells to enter the side branch is then given by the criterion P = 1. Although this theory does not account for the cell screening effect arising from the hydrodynamic interaction between the cells and the tube walls, it leads to predictions which exhibit the same trends as the experimental observations and is able to explain the results of several seemingly contradictory microvascular experiments that have puzzled investigators in recent years.

Dye studies on flow through branching tubes

This study was aimed at identifying the shape of the separating surface at the junction of two vessels and determining the shift in concentration profiles due to streamline bending at the junction. These data are useful in modeling phase separation during plasma skimming in serial microvascular bifurcations. It was hypothesized that the separating surface shape should be a function of the ratio of the branch diameters and the fractional flow split at the junction. Streamlines bend as blood flows through a junction thereby disturbing the concentration profile of several blood components downstream from the branch point. Model experiments have been conducted to test these ideas. Scaled-up dye studies have shown that the separating surface is indeed a function of the branch diameter ratio. When all branches have the same diameter the separating surface is virtually flat. When the side branch diameter is half that of the parent branch, the surface is curved, bulging away from the opening of the side branch. At Reynold's numbers greater than 20, when vortices form in the daughter branches, the shape of the separating surface is complicated and may even be discontinuous. Other dye experiments have been used to illustrate the magnitude of streamline bending. The data are then used to estimate the shift in concentration profiles due to flow through junctions. A technique for mapping upstream profiles to their corresponding downstream location has been developed. This mapping technique provides the necessary initial condition for the equation which describes the dispersion of red blood cells as they flow between the junctions.

Mapping of the microcirculation in the chick chorioallantoic membrane during normal angiogenesis

The microcirculation within the chorioallantoic membrane (CAM) of the chick is particularly well suited for in vivo observation and has been used extensively as an assay to detect angiogenic activity. Although progressive chronological expansion of the CAM capillary network occurs normally during embryogenesis, descriptions of the branching patterns of CAM pre- and postcapillary microvessels during embryonic development have not been recorded. In the present study chick embryos were incubated, using an established shell-less culture technique, and observed in vivo at Days 6, 10, and 14 of embryonic development. Morphometric analyses of photomicrographs of CAM microvessels were based upon the centripetal ordering method of microvascular mapping of the first three orders of pre- and postcapillary microvessels with the capillaries serving as the initial point of reference. For both pre- and postcapillary vessels, the number of first-order vessels exceeded the number of second-order vessels which, in turn, outnumbered third-order vessels during each observation period. First- and second-order vessels progressively increased in number from Day 6 to Day 14; however, the number of third-order vessels remained essentially constant during this period. Further, the number of precapillary vessels was greater than postcapillary vessels in their respective orders at Days 6 and 10; however, by Day 14 the numbers were comparable. Average diameters and lengths of the third-order vessels were greater than the second-order vessels which, in turn, were greater than the first-order vessels in both the pre- and postcapillary compartments. Further, mean lengths of each of the three vessel orders in both compartments decreased progressively and by Day 14 were significantly less than at Day 6. Average diameters of each vessel order, on the other hand, remained unchanged from Day 6 to Day 14. Finally, intercapillary distances, based on measurements from fluorescent micrographs obtained after microinjections of fluorescein isothiocyanate (FITC)-dextran, were substantially less at Day 10 and 14 than at Day 6. Based on these morphometric data, the endothelial precursor responsible for continuous neoformation of first- and second-order microvessels during embryogenesis remains uncertain. Whether existing first-, second-, or third-order vessel endothelia serve as this precursor or histodifferentiation of existing capillaries enables continuous expansion of the first- and second-order microvessels remains to be tested.

Carbon dioxide exchange across the walls of arterioles: implication for the location of the medullary chemoreceptors

The location of the medullary chemoreceptors is not conclusively established. The original experiments, which were believed to suggest a shallow surface location in the ventrolateral medulla, have been questioned because substances, particularly CO2, applied on the surface of the medulla could diffuse into small arterioles. Because the whole tissue blood flow is supplied by surface arterioles, they could transport substances from the surface into the tissue to the respiratory centers. We studied simple transport equations describing movement of CO2 in arterioles bathed by rapidly flowing cerebrospinal fluid (CSF) and arterioles in tissue perfused by capillaries. Substantial exchange of CO2 could occur across the arteriole wall for all expected sizes of vessels when the partial pressure of CO2 at the outside wall was determined by CSF. When an arteriole is surrounded by tissue, only vessels with inside diameters (ID) less than or equal to 50 micron will exchange substantial amounts of CO2 but the smallest arterioles may be nearly in equilibrium with the tissue. The CO2 gradient in tissue around the arteriole will extend approximately 1 mm. Our simple theoretical description of CO2 transport in arterioles predicts substantial exchange in precapillary vessels. CO2 picked up by the smallest surface arterioles when the medulla is perfused at a high rate with CSF will not stay in the blood past the putative depth of the chemoreceptors. In arterioles greater than 30 micron, however, the CO2 could be carried to the respiratory centers.

Analysis of vascular pattern and dimensions in arteriolar networks of the retractor muscle in young hamsters

A quantitative analysis of the distribution of microvascular blood flow and oxygen delivery requires a detailed description of the vascular network geometry. The distributions of lengths and diameters were determined in terminal arteriolar networks of the cheek pouch retractor muscle of young (34 +/- 2 days) hamsters. We compared the Strahler centripetal vessel ordering scheme, which assigns lowest order to the capillaries and proceeds upstream toward the larger vessels, with the centrifugal ordering scheme, which begins with the input arteriole and proceeds downstream toward the capillaries. The terminal networks of the retractor muscle typically contain 2 to 4 Strahler orders and 2 to 6 centrifugal orders. The coefficients of variation of diameter and length are smaller for Strahler ordering than for centrifugal ordering. In addition, for Strahler ordering, we found that the sequence of number of vessels obeyed Horton's law. We have compared three different methods of calculating the bifurcation, diameter, and length ratios. As an alternative method for analyzing network topology, we also studied the distribution of the number of segments on each pathway from the inlet of a network to a capillary. The information obtained from this analysis is useful for the mathematical modeling of flow in the microvascular network.

Topological structure of rat mesenteric microvessel networks

Microvascular lengths, diameters, and flow directions were determined in all vessel segments (n = 1303) between bifurcations in three complete rat mesenteric microvessel networks (25 mm2 each) using intravital video- and photomicroscopy. The classification of vessel segments as arteriolar, venular, or av-segments (all segments connecting the arteriolar to the venular tree) was based on purely topological criteria. The topological structure of the networks was analyzed using the Horton-Strahler technique and a new generation scheme. Generation numbers were assigned to the vessel segments on the basis of the number of upstream (in the arteriolar tree) and downstream (in the venular tree) bifurcations. The mean generation number of the av-segments, a characteristic parameter of the generation scheme, reflects the topological structure of the network more accurately than Horton's branching ratio Rb. Both the arteriolar and venular tree of the mesenteric networks were found to be dichotomous branching structures which were neither strictly symmetric nor strictly asymmetric. The topological information obtained was compared to network models generated by different random branching algorithms. The result of this comparison suggests that the network structure changes at a certain generation level. Distal to this generation level, the mesenteric networks resemble a model network generated by random branching at any segment, while the proximal portion is similar to a model allowing random branching at terminal segments only.

Power dissipation as a measure of peripheral resistance in vascular networks

Peripheral resistance was examined in the microcirculation of the rat cremaster muscle using a network-conserved parameter, power dissipation. Previous studies of peripheral resistance used network-sensitive parameters, and their interpretation is limited by tacit assumptions about the structure of the peripheral vasculature. Power dissipation is directly linked to the resistive process, providing a measure of resistance based on the actual hemodynamics of the network. The dissipation parameter was quantified with the usual vascular parameters of velocity and vessel segment length; 991 segment lengths were measured in 12 normotensive Wistar-Kyoto rats and 16 spontaneously hypertensive rats. Arterial power dissipation was significantly elevated over a wide range of vessel segments; blood flow ranged from 0.08 to 80 nl/sec. Since the largest vessels showed the greatest power dissipation, the organ resistance elevation seen in hypertension in the cremaster apparently is mediated by the larger vessels in the high flow range. Vessel segment length and number of dissipative vessels were unchanged. The increase in power dissipation was due to a network-averaged reduction in mean vessel diameter. Power dissipation also increased significantly in the fastest flowing venous microvessels (greater than 25 nl/sec), also due to a reduction in vessel segment diameter.

Effect of dispersion of vessel diameters and lengths in stochastic networks. I. Modeling of microcirculatory flow

A microvascular network model is proposed with random arrangement and random dimensions of vessels. In addition to stochasticity of the topological characteristics of the model networks, as previously introduced by Fenton and Zweifach (1981, Ann. Biomed. Eng., 9, 303-321), the vessel diameters and lengths are treated as random variables following certain probability distributions for each vascular order. Flow and pressure distributions are calculated for each network configuration assuming a linear relationship between the blood flow rate and pressure drop for each vascular segment. The mean, coefficient of variation, skewness, kurtosis, and histograms of the hemodynamic variables are computed using an ensemble of random networks. The results indicate that dispersion of vessel diameters and lengths may significantly affect the distributions of microvascular variables such as capillary flow and pressure, and the flow distribution at bifurcations. It is shown that the dispersion of vessel diameters causes a decrease of total flow whereas the dispersion of lengths causes its increase.

The microvasculature in skeletal muscle. I. Arteriolar network in rat spinotrapezius muscle

A quantitative analysis of blood flow dynamics in skeletal muscle requires a detailed picture of the microvascular network. This report presents an analysis of the arteriolar network structure in the spinotrapezius muscle of the rat. The microvasculature is visualized by injection of a carbon suspension and recorded in the form of photomicrographs with a complete reconstruction of the microvasculature on transparent overlays. The spinotrapezius muscle has several major feeding arterioles which supply blood into an extensive meshwork of interconnecting or arcading arterioles spanning the entire muscle. The connections from the arcade arterioles to the capillaries are provided by transverse arterioles, which branch from the arcades at regular intervals. Each transverse arteriole forms a single asymmetric dichotomous tree and within each muscle there is a wide range in the size of transverse arterioles. A new branching schema is proposed to describe the arteriolar network. A set of network parameters is derived and typical values of these parameters in the spinotrapezius muscle of the rat are provided.