Volume flow estimation in the precapillary mesenteric microvasculature in vivo and the principle of constant pressure gradient

Volume flow was estimated from axial erythrocyte velocity measurements in 30 mesenteric microvessels of 6 rabbits and was compared to Murray's law predictions. The diameters of capillaries and precapillary arterioles ranged between 5.6 and 12 microm. The significant pulsating flow component existing in these microvessels was taken into account by measuring instantaneous axial blood velocity throughout the course of a cardiac cycle and then averaging over the period. In addition, the effect of the velocity profile variation with diameter was taken into account, for the first time, by using a profile factor function. According to Murray's law, the relation between blood volume flow and diameter is governed by a 'cube' law. Curve fitting to volume flow and diameter data pairs for rabbits, showed a dependence of volume flow on diameter raised to the 4th power (with a correlation coefficient equal to 0.97). The above result supports the hypothesis that, in the precapillary part of microvasculature, the principle of constant longitudinal pressure gradient rather than the principle of minimum work may be valid.     

Pulsatile blood flow, shear force, energy dissipation and Murray's Law

Background  

Murray's Law states that, when a parent blood vessel branches into daughter vessels, the cube of the radius of the parent vessel is equal to the sum of the cubes of the radii of daughter blood vessels. Murray derived this law by defining a cost function that is the sum of the energy cost of the blood in a vessel and the energy cost of pumping blood through the vessel. The cost is minimized when vessel radii are consistent with Murray's Law. This law has also been derived from the hypothesis that the shear force of moving blood on the inner walls of vessels is constant throughout the vascular system. However, this derivation, like Murray's earlier derivation, is based on the assumption of constant blood flow.     

Investigating Murray's law in the chick embryo.

According to the optimization principle known as Murray's law, the blood vessel geometry at a bifurcation satisfies the relation alpha = (D3(1) + D3(2))/D3(0) = 1, where D0, D1, and D2 are the diameters of the parent and two daughter vessels, respectively. Previous investigations have shown that mature blood vessels adhere to this law fairly closely. The purpose of this study was to test Murray's law in the developing extraembryonic blood vessels of 2-4 day-old chick embryos. Vessel diameters were measured manually using image analysis software. The measurements for the group of all vessels at all studied stages (n = 449) gave alpha = 1.01+/-0.34 (mean +/- SD), and the value of alpha is similar at all stages. These results indicate that Murray's law holds in the chick embryo, even before medial smooth muscle becomes functional, suggesting that blood vessels follow the same basic morphogenetic rules throughout life.     

An optimization principle for vascular radius including the effects of smooth muscle tone.

An optimization principle is proposed for the regulation of vascular morphology. This principle, which extends Murray's law, is based on the hypothesis that blood vessel diameter is controlled by a mechanism that minimizes the total energy required to drive the blood flow, to maintain the blood supply, and to support smooth muscle tone. A theoretical analysis reveals that the proposed principle predicts that the optimum shear stress on the vessel wall due to blood flow increases with blood pressure. This result agrees qualitatively with published findings that the fluid shear stress in veins is significantly smaller than it is in arteries.     

The evaluation of Murray's law in Psilotum nudum (Psilotaceae), an analogue of ancestral vascular plants

Previous work has shown that the xylem of seed plants follows Murray's law when conduits do not provide structural support to the plant. Here, compliance with Murray's law was tested in the stem photosynthesizer Psilotum nudum, a seedless vascular plant. Psilotum nudum was chosen because the central stele does not provide structural support, which means that Murray's law is applicable, and because its simple shoot structure resembles the earliest vascular plants. Murray's law predicts that the sum of the conduit radii cubed (Σr(3)) should decrease in direct proportion with the volume flow rate (Q) to maximize the hydraulic conductance per unit vascular investment. Agreement with Murray's law was assessed by estimating the transpiration rate distal to a cross-section, which should determine Q under steady state conditions, and comparing that with the Σr(3) of that cross-section. As predicted, regressions between the Σr(3) of the cross-section and Q resulted in a linear relationship with a y-intercept that was not different from zero. Two more rigorous statistical tests were also unable to reject Murray's law. Psilotum nudum plants also increased their conductance per investment by having more conduits distally than proximally, which is more efficient hydraulically than equal or declining conduit numbers distally.     

Remodeling of conduit arteries in hypertension and flow-overload obeys a minimum energy principle

Arterial remodeling is an important process in physiology and pathophysiology. Based on an energy minimization method, Murray's law predicts the optimal inner radius. Application of Darcy's law in the wall results in an optimal outer radius. The average wall stress is computed by the Laplace's law. Using these formulas, a large porcine coronary artery in hypertension is studied. The results reveal how wall thickness and average circumferential stress change after increasing blood pressure and volume flow rate. The theoretical predictions are in good qualitative agreement with experimental observations. The advantage and limitation of the current approach are discussed.     

The influence of boundary conditions on wall shear stress distribution in patients specific coronary trees

Patient specific geometrical data on human coronary arteries can be reliably obtained multislice computer tomography (MSCT) imaging. MSCT cannot provide hemodynamic variables, and the outflow through the side branches must be estimated. The impact of two different models to determine flow through the side branches on the wall shear stress (WSS) distribution in patient specific geometries is evaluated. Murray's law predicts that the flow ratio through the side branches scales with the ratio of the diameter of the side branches to the third power. The empirical model is based on flow measurements performed by Doriot et al. (2000) in angiographically normal coronary arteries. The fit based on these measurements showed that the flow ratio through the side branches can best be described with a power of 2.27. The experimental data imply that Murray's law underestimates the flow through the side branches. We applied the two models to study the WSS distribution in 6 coronary artery trees. Under steady flow conditions, the average WSS between the side branches differed significantly for the two models: the average WSS was 8% higher for Murray's law and the relative difference ranged from -5% to +27%. These differences scale with the difference in flow rate. Near the bifurcations, the differences in WSS were more pronounced: the size of the low WSS regions was significantly larger when applying the empirical model (13%), ranging from -12% to +68%. Predicting outflow based on Murray's law underestimates the flow through the side branches. Especially near side branches, the regions where atherosclerotic plaques preferentially develop, the differences are significant and application of Murray's law underestimates the size of the low WSS region.     

Extension of Murray's law using a non-Newtonian model of blood flow

Background  

So far, none of the existing methods on Murray's law deal with the non-Newtonian behavior of blood flow although the non-Newtonian approach for blood flow modelling looks more accurate.     

Biomimetic design of artificial micro-vasculatures for tissue engineering

Over the last decade, highly innovative micro-fabrication techniques have been developed that are set to revolutionise the biomedical industry. Fabrication processes, such as photolithography, wet and dry etching, moulding, embossing and lamination, have been developed for a range of biocompatible and biodegradable polymeric materials. One area where these fabrication techniques could play a significant role is in the development of artificial micro-vasculatures for the creation of tissue samples for drug screening and clinical applications. Despite the enormous technological advances in the field of tissue engineering, one of the major challenges is the creation of miniaturised fluid distribution networks to transport nutrients and waste products, in order to sustain the viability of the culture. In recent years, there has been considerable interest in the development of microfluidic manifolds that mimic the hierarchical vascular and parenchymal networks found in nature. This article provides an overview of microfluidic tissue constructs, and also reviews the hydrodynamic scaling laws that underpin the fluid mechanics of vascular systems. It shows how Murray's law, which governs the optimum ratio between the diameters of the parent and daughter branches in biological networks, can be used to design the microfluidic channels in artificial vasculatures. It is shown that it is possible to introduce precise control over the shear stress or residence time in a hierarchical network, in order to aid cell adhesion and enhance the diffusion of nutrients and waste products. Finally, the paper describes the hydrodynamic extensions that are necessary in order to apply Murray's law to the rectangular channels that are often employed in artificial micro-vasculatures.     

The impact of simplified boundary conditions and aortic arch inclusion on CFD simulations in the mouse aorta: a comparison with mouse-specific reference data

Computational fluid dynamics (CFD) simulations allow for calculation of a detailed flow field in the mouse aorta and can thus be used to investigate a potential link between local hemodynamics and disease development. To perform these simulations in a murine setting, one often needs to make assumptions (e.g. when mouse-specific boundary conditions are not available), but many of these assumptions have not been validated due to a lack of reference data. In this study, we present such a reference data set by combining high-frequency ultrasound and contrast-enhanced micro-CT to measure (in vivo) the time-dependent volumetric flow waveforms in the complete aorta (including seven major side branches) of 10 male ApoE -/- deficient mice on a C57Bl/6 background. In order to assess the influence of some assumptions that are commonly applied in literature, four different CFD simulations were set up for each animal: (i) imposing the measured volumetric flow waveforms, (ii) imposing the average flow fractions over all 10 animals, presented as a reference data set, (iii) imposing flow fractions calculated by Murray's law, and (iv) restricting the geometrical model to the abdominal aorta (imposing measured flows). We found that - even if there is sometimes significant variation in the flow fractions going to a particular branch - the influence of using average flow fractions on the CFD simulations is limited and often restricted to the side branches. On the other hand, Murray's law underestimates the fraction going to the brachiocephalic trunk and strongly overestimates the fraction going to the distal aorta, influencing the outcome of the CFD results significantly. Changing the exponential factor in Murray's law equation from 3 to 2 (as suggested by several authors in literature) yields results that correspond much better to those obtained imposing the average flow fractions. Restricting the geometrical model to the abdominal aorta did not influence the outcome of the CFD simulations. In conclusion, the presented reference dataset can be used to impose boundary conditions in the mouse aorta in future studies, keeping in mind that they represent a subsample of the total population, i.e., relatively old, non-diseased, male C57Bl/6 ApoE -/- mice.     

Analysis of blood flow in the entire coronary arterial tree

A hemodynamic analysis of coronary blood flow must be based on the measured branching pattern and vascular geometry of the coronary vasculature. We recently developed a computer reconstruction of the entire coronary arterial tree of the porcine heart based on previously measured morphometric data. In the present study, we carried out an analysis of blood flow distribution through a network of millions of vessels that includes the entire coronary arterial tree down to the first capillary branch. The pressure and flow are computed throughout the coronary arterial tree based on conservation of mass and momentum and appropriate pressure boundary conditions. We found a power law relationship between the diameter and flow of each vessel branch. The exponent is approximately 2.2, which deviates from Murray's prediction of 3.0. Furthermore, we found the total arterial equivalent resistance to be 0.93, 0.77, and 1.28 mmHg.ml(-1).s(-1).g(-1) for the right coronary artery, left anterior descending coronary artery, and left circumflex artery, respectively. The significance of the present study is that it yields a predictive model that incorporates some of the factors controlling coronary blood flow. The model of normal hearts will serve as a physiological reference state. Pathological states can then be studied in relation to changes in model parameters that alter coronary perfusion.     

Quantitative relationship between the octanol/water partition coefficient and the diffusion limitation of the exchange between adipose and blood

Background  

The goal of physiologically based pharmacokinetics (PBPK) is to predict drug kinetics from an understanding of the organ/blood exchange. The standard approach is to assume that the organ is "flow limited" which means that the venous blood leaving the organ equilibrates with the well-stirred tissue compartment. Although this assumption is valid for most solutes, it has been shown to be incorrect for several very highly fat soluble compounds which appear to be "diffusion limited". This paper describes the physical basis of this adipose diffusion limitation and its quantitative dependence on the blood/water (K_bld-wat) and octanol/water (K_ow) partition coefficient.     

Which diameter and angle rule provides optimal flow patterns in a coronary bifurcation?

The branching angle and diameter ratio in epicardial coronary artery bifurcations are two important determinants of atherogenesis. Murray's cubed diameter law and bifurcation angle have been assumed to yield optimal flows through a bifurcation. In contrast, we have recently shown a 7/3 diameter law (HK diameter model), based on minimum energy hypothesis in an entire tree structure. Here, we derive a bifurcation angle rule corresponding to the HK diameter model and critically evaluate the streamline flow through HK and Murray-type bifurcations. The bifurcations from coronary casts were found to obey the HK diameter model and angle rule much more than Murray's model. A finite element model was used to investigate flow patterns for coronary artery bifurcations of various types. The inlet velocity and pressure boundary conditions were measured by ComboWire. Y-bifurcation of Murray type decreased wall shear stress-WSS (10%-40%) and created an increased oscillatory shear index-OSI in atherosclerosis-prone regions as compared with HK-type bifurcations. The HK-type bifurcations were found to have more optimal flow patterns (i.e., higher WSS and lower OSI) than Murray-type bifurcations which have been traditionally believed to be optimized. This study has implications for changes in bifurcation angles and diameters in percutaneous coronary intervention.     

Vascular metabolic dissipation in Murray's law

The metabolic dissipation in Murray's minimum energy hypothesis includes only the blood metabolism. The metabolic dissipation of the vascular tree, however, should also include the metabolism of passive and active components of the vessel wall. In this study, we extend the metabolic dissipation to include blood metabolism, as well as passive and active components of the vessel wall. The analysis is extended to the entire vascular arterial tree rather than a single vessel as in Murray's formulation. The calculations are based on experimentally measured morphological data of coronary artery network and the longitudinal distribution of blood pressure along the tree. Whereas the model includes multiple dissipation sources, the total metabolic consumption of a complex vascular tree is found to remain approximately proportional to the cumulative arterial volume of the unit. This implies that the previously described scaling relations for the various morphological features (volume, length, diameter, and flow) remain unchanged under the generalized condition of metabolic requirements of blood and blood vessel wall.     

Comparison and definition of spleen and lymph node: a phylogenetic analysis

The hitherto largely unsolved problem with a biological definition of spleen versus lymph node seems possible to solve from a phylogenetic point of view. Thus, it is suggested that the spleen be defined as a hemopoietic organ which is able to filter blood with sinusoids. In contradistinction, a lymph node is defined as a hemopoietic organ which is able to filter lymph with sinusoids. Comparative anatomical studies show that the spleen appears as a condensation of the lymphomyeloid complex in the spiral fold of the gut in cyclostomes. The spiral fold spleen vanishes with the bony fishes, while in cartilaginous fishes a similar spleen appears in the dorsal mesentery. The dorsal spleen remains in a retroperitoneal position in higher vertebrates and is regarded as a specialized blood vessel compartment closely connected with the blood stream. In "higher" vertebrates the spleen is a stagnated organ because splenic functions are gradually transferred to other sites. The bone marrow takes over the erythro-, thrombo- and granulocytopoiesis while the lymph nodes take over the lymphocytopoiesis. This transfer of the splenic functions is first seen in anurans and seems to be a marvelous adaptation to life on land where the need for local defence against a large number of antigens is necessary before spread of the antigens to central parts of the body. In higher vertebrates, the great number of lymph nodes at peripheral positions, derived from the lymphatic vessels, are able to do so. It is demonstrated that the definitions of spleen and lymph nodes as hemopoietic organs which by their sinusoids are able to filter blood and lymph, respectively, are not only of semantic interest but also useful in regard the immunohematological system as an entity.     

A new fundamental bioheat equation for muscle tissue--part II: Temperature of SAV vessels

In this study, a new theoretical framework was developed to investigate temperature variations along countercurrent SAV blood vessels from 300 to 1000 microm diameter in skeletal muscle. Vessels of this size lie outside the range of validity of the Weinbaum-Jiji bioheat equation and, heretofore, have been treated using discrete numerical methods. A new tissue cylinder surrounding these vessel pairs is defined based on vascular anatomy, Murray's law, and the assumption of uniform perfusion. The thermal interaction between the blood vessel pair and surrounding tissue is investigated for two vascular branching patterns, pure branching and pure perfusion. It is shown that temperature variations along these large vessel pairs strongly depend on the branching pattern and the local blood perfusion rate. The arterial supply temperature in different vessel generations was evaluated to estimate the arterial inlet temperature in the modified perfusion source term for the s vessels in Part I of this study. In addition, results from the current research enable one to explore the relative contribution of the SAV vessels and the s vessels to the overall thermal equilibration between blood and tissue.     

Rhythm assimilation in the isolated canine heart under isovolumic conditions

Isolated canine heart has an expressed ability for autoregulation of bioelectrical and contractile functions irrespective of the neurohumoral factors influence on the work of the heart, and Frank-Starling law. Under the change of stimulation frequency, the autoregulation of heart functions is carried out as rhythm assimilation at organ (cell) level. The heart has a higher ability to bioelectrical rhythm assimilation rather than the mechanical rhythm assimilation. Incomplete rhythm assimilation is characterised by the alternation of contractions. The "Everything or nothing" law has no applicability to the work of the heart.     

On connecting large vessels to small. The meaning of Murray's law

A large part of the branching vasculature of the mammalian circulatory and respiratory systems obeys Murray's law, which states that the cube of the radius of a parent vessel equals the sum of the cubes of the radii of the daughters. Where this law is obeyed, a functional relationship exists between vessel radius and volumetric flow, average linear velocity of flow, velocity profile, vessel-wall shear stress, Reynolds number, and pressure gradient in individual vessels. In homogeneous, full-flow sets of vessels, a relation is also established between vessel radius and the conductance, resistance, and cross- sectional area of a full-flow set.     

孤雌生殖长角血蜱的哈氏器超微结构与发育

为阐明长角血蜱Haemaphysalis longicornis孤雌生殖种群的哈氏器结构及发育特征, 用扫描电镜对其各虫期哈氏器进行了观察, 分析了血餐对哈氏器发育的影响。结果表明： 该种群幼蜱、 若蜱和成蜱哈氏器形态结构基本相同, 均由前窝和后囊构成。幼蜱前窝感毛6根, 位于同一基盘; 若蜱和成蜱哈氏器相似, 前窝感毛7根, 其中1根孔毛位于外侧基盘, 另6根感毛位于内侧基盘。各虫期饱血后哈氏器大小均比饥饿状态下显著增大（P<0.05）。幼蜱前窝与后囊面积比值与若蜱相比无显著差异（P>0.05）, 若蜱前窝与后囊面积比值与成蜱相比差异显著（P<0.05）。各虫期哈氏器均在发育, 且血餐对哈氏器发育有重要影响。幼蜱至若蜱期哈氏器前窝与后囊的发育速度相似, 若蜱至成蜱期哈氏器前窝发育快于后囊。本研究结果在一定程度上揭示了孤雌生殖长角血蜱的哈氏器发育规律。     

Podocytes in the axial organ of echinoderms

The axial organ of two sea urchin genera, Echinometra and Eucidaris , has been analysed with the electron microscope. It is mainly built up by an extracoelomic tissue rich in various leucocytic cells and blood vessels lacking an endothelium. The axial sinus (axocoel) is located on the interior of the axial organ, and evaginations extend into this loosely constructed tissue. The epithelium of the axial sinus and its extensions are characterized by epitheliomuscular cells and abundant podocytes, which overlie the blood vessels. Since podocytes are the well known structural basis for ultrafiltration, it is postulated that this is a function of the axial organ at least in echinoids and asteroids.