Local control of blood flow

Organ blood flow is determined by perfusion pressure and vasomotor tone in the resistance vessels of the organ. Local factors that regulate vasomotor tone include myogenic and metabolic autoregulation, flow-mediated and conducted responses, and vasoactive substances released from red blood cells. The relative importance of each of these factors varies over time, from tissue to tissue, and among vessel generations.     

Balance between myogenic,flow-dependent,and metabolic flow control in coronary arterial tree: a model study

Myogenic response, flow-dependent dilation, and direct metabolic control are important mechanisms controlling coronary flow. A model was developed to study how these control mechanisms interact at different locations in the arteriolar tree and to evaluate their contribution to autoregulatory and metabolic flow control. The model consists of 10 resistance compartments in series, each representing parallel vessel units, with their diameters determined by tone depending on either flow and pressure [flow-dependent tone reduction factor (TRF(flow)) x Tone(myo)] or directly on metabolic factors (Tone(meta)). The pressure-Tone(myo) and flow-TRF(flow) relations depend on the vessel size obtained from interpolation of data on isolated vessels. Flow-dependent dilation diminishes autoregulatory properties compared with pressure-flow lines obtained from vessels solely influenced by Tone(myo). By applying Tone(meta) to the four distal compartments, the autoregulatory properties are restored and tone is equally distributed over the compartments. Also, metabolic control and blockage of nitric oxide are simulated. We conclude that a balance is required between the flow-dependent properties upstream and the constrictive metabolic properties downstream. Myogenic response contributes significantly to flow regulation.     

Flow resistance and drag forces due to multiple adherent leukocytes in postcapillary vessels.

Computational fluid dynamics was used to model flow past multiple adherent leukocytes in postcapillary size vessels. A finite-element package was used to solve the Navier-Stokes equations for low Reynolds number flow of a Newtonian fluid past spheres adhering to the wall of a cylindrical vessel. We determined the effects of sphere number, relative geometry, and spacing on the flow resistance in the vessel and the fluid flow drag force acting to sweep the sphere off the vessel wall. The computations show that when adherent leukocytes are aligned on the same side of the vessel, the drag force on each of the interacting leukocytes is less than the drag force on an isolated adherent leukocyte and can decrease by up to 50%. The magnitude of the reduction depends on the ratio of leukocyte to blood vessel diameter and distance between adherent leukocytes. However, there is an increase in the drag force when leukocytes adhere to opposite sides of the vessel wall. The increase in resistance generated by adherent leukocytes in vessels of various sizes is calculated from the computational results. The resistance increases with decreasing vessel size and is most pronounced when leukocytes adhere to opposite sides of the vessel.     

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.     

The cost of departure from optimal radii in microvascular networks

In the Murray optimality model of branching vasculatures, the radii of vessels are related to blood viscosity, vascular metabolic rate, and blood flow rate, in such a way as to minimize the total work (hydraulic and metabolic) of the system. The model predicts that flow is proportional to the cube of a vessel radius, and that at junctions the cube of the radius of the parent vessel equals the sum of the cubes of the daughter radii. In comparing real vasculatures to the Murray model, we have previously had no expressions for evaluating the apparent energy cost for departures from the optimal junction exponent of 3. Such expressions are derived here. They show that junction exponents, from about 1.5 to large positive values, are within 5% of the energy minimum. With the new equations, observed individual junctions or entire vascular trees can be compared, energy-wise, with the Murray optimum. Junctions in the transverse arteriolar trees of cat sartorius muscle were compared to the Murray optimality model, using these new expressions. The junction exponents for these small pre-capillary vessels had a broad range, with a median value greater than the Murray optimum of 3. The exponents were restricted, however, to values requiring, at individual junctions, little increase in energy. The majority of junctions had energy costs less than 1% above the Murray minimum. For entire trees involving many junctions the departures from optimality averaged less than 10%. Thus, while the branching geometry for these microvascular trees deviates significantly from the Murray optimum in the direction of larger daughter to parent ratios, the departures are small in energy terms.     

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.     

Measurement of Streaming Potentials of Mammalian Blood Vessels, Aorta and Vena Cava, in Vivo

Attempts to measure streaming potentials in large rabbit blood vessels in vivo have been carried out. Streaming potentials, V(89), were measured by the introduction of microelectrodes through the wall of the blood vessel at separations greater than 1 cm. The outputs from these electrodes fed through calomel cells were amplified and recorded directly by using an Electronics for Medicine photorecorder (White Plains, N. Y.). "Effective streaming currents" were determined by running the output through a low impedence galvanometer while simultaneously measuring the resistance of the circuit V(8) were, therefore, calculated from two measurements and compared. Flow through vessels studied was measured using two different electromagnetic flowmeters. The results indicate that V(8) present in both aorta and vena cava are of the order of 5 to 10 mv. By using the Helmholtz-Smoluchowski equation into which flow was reintegrated, the numbers yield zeta potentials approximating 0.1 to 0.4 v in both aorta and vena cava. This number approaches the apparent upper limit for zeta (actually "interfacial potentials") potentials in biological systems. The measured "i.f." potential is considered as the interreaction of several physical and metabolic factors operating at the blood intimal interface. The polarity of the potential suggests that the interface is negative with respect to the blood flowing through the vessel. Interfacial potential and related V(8) are discussed in terms of their possible importance as a mechanism for maintaining vascular homeostasis in the living animal.     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

Network Scale Modeling of Lymph Transport and Its Effective Pumping Parameters

The lymphatic system is an open-ended network of vessels that run in parallel to the blood circulation system. These vessels are present in almost all of the tissues of the body to remove excess fluid. Similar to blood vessels, lymphatic vessels are found in branched arrangements. Due to the complexity of experiments on lymphatic networks and the difficulty to control the important functional parameters in these setups, computational modeling becomes an effective and essential means of understanding lymphatic network pumping dynamics. Here we aimed to determine the effect of pumping coordination in branched network structures on the regulation of lymph flow. Lymphatic vessel networks were created by building upon our previous lumped-parameter model of lymphangions in series. In our network model, each vessel is itself divided into multiple lymphangions by lymphatic valves that help maintain forward flow. Vessel junctions are modeled by equating the pressures and balancing mass flows. Our results demonstrated that a 1.5 s rest-period between contractions optimizes the flow rate. A time delay between contractions of lymphangions at the junction of branches provided an advantage over synchronous pumping, but additional time delays within individual vessels only increased the flow rate for adverse pressure differences greater than 10.5 cmH₂O. Additionally, we quantified the pumping capability of the system under increasing levels of steady transmural pressure and outflow pressure for different network sizes. We observed that peak flow rates normally occurred under transmural pressures between 2 to 4 cmH₂O (for multiple pressure differences and network sizes). Networks with 10 lymphangions per vessel had the highest pumping capability under a wide range of adverse pressure differences. For favorable pressure differences, pumping was more efficient with fewer lymphangions. These findings are valuable for translating experimental measurements from the single lymphangion level to tissue and organ scales.     

A circle of Willis simulation using distensible vessels and pulsatile flow

The development of a one-dimensional numerical (finite-difference) model of the arterial network surrounding the circle of Willis is described based on the full Navier-Stokes and conservation of mass equations generalized for distensible vessels. The present model assumes an elastic wall defined by a logarithmic pressure-area relation obtained from the literature. The viscous term in the momentum equation is evaluated using the slope of a Karman-Pohlhausen velocity profile at the vessel boundary. The afferent vessels (two carotids and two vertebrals) are forced with a canine physiologic pressure signature corresponding to an aortic site. The network associated with each main efferent artery of the circle is represented by a single vessel containing an appropriate amount of resistance so that the mean flow through the system is distributed in accordance with the weight of brain irrigated by each vessel as determined from a steady flow model of the same network. This resistance is placed a quarter wave-length downstream from the heart to insure proper reflection from the terminations, where the quarter wavelength is determined using the frequency corresponding to the first minimum on an input impedance-frequency diagram obtained at the heart. Computer results are given as time histories of pressure and flow at any model nodal point starting from initial conditions of null flow and constant pressure throughout the model. Variations in these pressure and flow distributions caused by the introduction of pathologic situations into the model illustrate the efficacy of the simulation and of the circle in equalizing and redistributing flows in abnormal situations.     

The pressure-flow relation for plasma in whole organ skeletal muscle and its experimental verification.

The whole-organ pressure-flow relation in resting rat skeletal muscle is examined for the flow of plasma. Due to the small size of the blood vessels in this organ, inertia and convective forces in the blood are negligible and viscous forces dominate. Direct measurements in the past have shown that skeletal muscle blood vessels are distensible. Theoretical formulations based on these measurements lead to a third order polynomial model for the pressure-flow relation. The purpose of the current study is to examine this relation experimentally in an isolated muscle organ. A high precision feedback controlled pump is used to perfuse artificial plasma into the vasodilated rat gracilis muscle. The results indicate that the pressure-flow curve in this tissue is nonlinear in the low flow region and almost linear at physiological flow rates, following closely the third order polynomial function. Vessel fixation with glutaraldehyde causes the curves to become linear at all pressures, indicating that vessel distention is the primary mechanism causing the nonlinearity. Furthermore, the resistance of the post-fixed tissue is determined by the pressure at which the fixative is perfused. At fixation pressures below 10 mmHg, the resistance is three times higher than in vessels fixed at normal physiological pressures. Dextran (229,000 Dalton) is used to obtain Newtonian perfusates at different viscosities. The pressure-flow relation is found to be linearly dependent on viscosity for all flow rates.(ABSTRACT TRUNCATED AT 250 WORDS)     

Electrical communication in branching arterial networks

Electrical communication and its role in blood flow regulation are built on an examination of charge movement in single, isolated vessels. How this process behaves in broader arterial networks remains unclear. This study examined the nature of electrical communication in arterial structures where vessel length and branching were varied. Analysis began with the deployment of an existing computational model expanded to form a variable range of vessel structures. Initial simulations revealed that focal endothelial stimulation generated electrical responses that conducted robustly along short unbranched vessels and to a lesser degree lengthened arteries or branching structures retaining a single branch point. These predictions matched functional observations from hamster mesenteric arteries and support the idea that an increased number of vascular cells attenuate conduction by augmenting electrical load. Expanding the virtual network to 31 branches revealed that electrical responses increasingly ascended from fifth- to first-order arteries when the number of stimulated distal vessels rose. This property enabled the vascular network to grade vasodilation and network perfusion as revealed through blood flow modeling. An elevation in endothelial-endothelial coupling resistance, akin to those in sepsis models, compromised this ascension of vasomotor/perfusion responses. A comparable change was not observed when the endothelium was focally disrupted to mimic disease states including atherosclerosis. In closing, this study highlights that vessel length and branching play a role in setting the conduction of electrical phenomenon along resistance arteries and within networks. It also emphasizes that modest changes in endothelial function can, under certain scenarios, impinge on network responsiveness and blood flow control.     

Microvascular blood viscosity in vivo and the endothelial surface layer

The apparent viscosity of blood in glass tubes declines with decreasing diameter (F?hraeus-Lindqvist effect) and exhibits a distinctive minimum at 6-7 microm. However, flow resistance in vivo in small vessels is substantially higher than predicted by in vitro viscosity data. The presence of a thick endothelial surface layer (ESL) has been proposed as the primary cause for this discrepancy. Here, a physical model is proposed for microvascular flow resistance as a function of vessel diameter and hematocrit in vivo; it combines in vitro blood viscosity with effects of a diameter-dependent ESL. The model was developed on the basis of flow distributions observed in three microvascular networks in the rat mesentery with 392, 546, and 383 vessel segments, for which vessel diameters, network architecture, flow velocity, and hematocrit were determined by intravital microscopy. A previously described hemodynamic simulation was used to predict the distributions of flow and hematocrit from the assumed model for effective blood viscosity. The dependence of ESL thickness on vessel diameter was estimated by minimizing deviations of predicted values for velocities, flow directions, and hematocrits from measured data. Optimal results were obtained with a layer thickness of approximately 0.8-1 microm for 10- to 40-microm-diameter vessels and declined strongly for smaller diameters, with an additional hematocrit-dependent impact on flow resistance exhibiting a maximum for approximately 10-microm-diameter vessels. These results show that flow resistance in vivo can be explained by in vitro blood viscosity and the presence of an ESL and indicate the rheologically effective thickness of the ESL in microvessels.     

Particle-fluid suspension model of blood flow through stenotic vessels with applications

The present study deals with the problem of blood flow through stenotic vessels when blood is represented by a particle-fluid suspension model, i.e. a suspension of red blood cells in plasma. The expressions for the dimensionless resistance to flow, the wall shear stress, and the shearing stress on the wall at the maximum height of the stenosis are derived. The results obtained in the analysis are discussed in brief, both qualitatively and quantitatively by comparison with other theories. It is observed that the magnitudes of the blood flow characteristics significantly increase with an increase in the red cell concentration. The importance of the decreasing vessel diameter is also pointed out. Finally, to observe the biological relevance of the analysis, the results obtained are used to compute the blood flow characteristics for normal and diseased blood using the experimental data from published literature and results are compared with those computed using the present theoretical approach.     

Plug Effect of Erythrocytes in Capillary Blood Vessels

As an idealized problem of the motion of blood in small capillary blood vessels, the low Reynolds number flow of plasma (a newtonian fluid) in a circular cylindrical tube involving a series of circular disks is studied. It is assumed in this study that the suspended disks are equally spaced along the axis of the tube, and that their centers remain on the axis of the tube and that their faces are perpendicular to the tube axis. The inertial force of the fluid due to the convective acceleration is neglected on the basis of the smallness of the Reynolds number. The solution of the problem is derived for a quasi-steady flow involving infinitesimally thin disks. The numerical calculation is carried out for a set of different combinations of the interdisk distance and the ratio of the disk radius to the tube radius. The ratio of the velocity of the disk to the average velocity of the fluid is calculated. The different rates of transport of red blood cells and of plasma in capillary blood vessels are discussed. The average pressure gradient along the axis of the tube is computed, and the dependence of the effective viscosity of the blood on the hematocrit and the diameter of the capillary vessel is discussed.     

血浆层对血管壁剪应力和剪应力梯度的影响

在细小血管中,由于血细胞明显的趋轴效应,管中的血液分为两个不同的区域,即具有血细胞的核心区和邻近管壁和血浆层。应用两相分层流模型,研究在相同的流量和管径下,当核心区中的血液分别为牛顿流体和Casson流体时,不同的血浆层厚度对细小血管壁剪应力和剪应力梯度的影响。结果表明,血浆层的存在对壁剪应力和壁剪应力梯度有较大影响,当血浆层厚度仅为血管半径的1％和3％时,壁剪应力梯度分别下降约10％和20％。     

Mathematical modeling of the response of a resistive vessel to pressure

The response of small arterial vessels to internal pressure makes an essential contribution to autoregulation in the vascular bed. It is believed that free cytosolic Ca²⁺ concentration plays a pivotal role in the regulation of smooth muscle contractility and hence of the vascular lumen. A simple mathematical model of blood flow in a resistive vessel is suggested. The model is based on the experimental data obtained for cerebral arteries, but may be used for any other resistive vessel. The model not only describes the regulation of the vascular lumen by transmural pressure but also shows realistic behavior of the vessel radius and cytosolic [Ca²⁺] at different rates of pressure change. Possible variations in the radius along the vessel due to the Bayliss effect are considered.     

Computational fluid dynamic studies of leukocyte adhesion effects on non-Newtonian blood flow through microvessels

The study of the effect of leukocyte adhesion on blood flow in small vessels is of primary interest to understand the resistance changes in venular microcirculation. Available computational fluid dynamic studies provide information on the effect of leukocyte adhesion when blood is considered as a homogeneous Newtonian fluid. In the present work we aim to understand the effect of leukocyte adhesion on the non-Newtonian Casson fluid flow of blood in small venules; the Casson model represents the effect of red blood cell aggregation. In our model the blood vessel is considered as a circular cylinder and the leukocyte is considered as a truncated spherical protrusion in the inner side of the blood vessel. The cases of single leukocyte adhesion and leukocyte pairs in positions aligned along the same side, and opposite sides of the vessel wall are considered. The Casson fluid parameters are chosen for cat blood and human blood and comparisons are made for the effects of leukocyte adhesion in both species. Numerical simulations demonstrated that for a Casson fluid with hematocrit of 0.4 and flow rate Q = 0.072 nl/s, a single leukocyte increases flow resistance by 5% in a 32 microns diameter and 100 microns long vessel. For a smaller vessel of 18 microns, the flow resistance increases by 15%.     

Network thermodynamic analysis of vasomotion in a microvascular network.

The modulation of microvascular blood flow by vasomotion in the individual vessels of a simple vascular network was simulated by means of a network thermodynamic model. The flow is driven under a pulsating pressure through two arcades of branching vasoactive arterioles into a passive resistance representing the capillary and venular beds. Each vessel was assumed to have the capability of decreasing rhythmically the local diameter over a short section by a specified fraction of the maximum value and to change the average diameter along its total length in response to alterations in intraluminal pressure. Blood was assumed to exhibit a simple linear viscous flow resistance. Alterations in flow rate and distribution through the network were determined as a function of the magnitude and frequency of vasomotion within the individual arterioles supplying blood to the microvascular bed. Specific cases are shown to illustrate how blood flow can be influenced by the patterns of vasomotion within the network.