Pulsatile pressure and flow in the skeletal muscle microcirculation

Although blood flow in the microcirculation of the rat skeletal muscle has negligible inertia forces with very low Reynolds number and Womersley parameter, time-dependent pressure and flow variations can be observed. Such phenomena include, for example, arterial flow overshoot following a step arterial pressure, a gradual arterial pressure reduction for a step flow, or hysteresis between pressure and flow when a pulsatile pressure is applied. Arterial and venous flows do not follow the same time course during such transients. A theoretical analysis is presented for these phenomena using a microvessel with distensible viscoelastic walls and purely viscous flow subject to time variant arterial pressures. The results indicate that the vessel distensibility plays an important role in such time-dependent microvascular flow and the effects are of central physiological importance during normal muscle perfusion. In-vivo whole organ pressure-flow data in the dilated rat gracilis muscle agree in the time course with the theoretical predictions. Hemodynamic impedances of the skeletal muscle microcirculation are investigated for small arterial and venous pressure amplitudes superimposed on an initial steady flow and pressure drop along the vessel.     

Blood flow regulation in the cerebral microvasculature with an arcadal network: a numerical simulation

Blood flow regulation in the cerebral microvasculature with an arcadal network was investigated using a numerical simulation. A mathematical model for blood flow in the arcadal network, based on in vivo data of cat cerebral microvasculature and flow velocity was developed. The network model consists of 45 vessel segments and 25 branching points. To simulate microvascular response to blood flow, non-reactive (solid), cerebral arteriole-like, or skeletal muscle arteriole-like responses to wall shear stress were taken into account. Numerical calculation was carried out in the flow condition where the inlet (arterial) pressure was changed from 60 to 120 mmHg. Flow-rate in each efferent vessel and the mean flow-rate over all efferent vessels were evaluated for assessment of blood supply to the local area of cerebral tissue. The simulation demonstrated the wall shear stress-induced vasodilation in the arcadal network worked to maintain the blood flow at a constant level with pressure variable in a wide range. It is suggested that an individual microvessel (segment) should join in the regulatory process of flow, interacting with other microvessels (cooperative regulation).     

Pulsatile flow in a constricted channel.

A nonuniform channel is used as a simple model of a constricted arterial vessel. Flow patterns have been calculated for pulsatile flow with both sinusoidal and nonsinusoidal flow rates for a range of Reynolds number, Re, and Strouhal number, St. The results show that even for relatively low frequency flows a strong vortex wave will be generated with a complex wall shear stress distribution and peak values much greater than those found in steady or unsteady parallel flow. The vortex wave increases in strength with increasing Re and St, with its total length and wavelength independent of Re but inversely proportional to St. The form of the imposed flow rate is found to have an important effect on the flow and the shear stress distribution.     

Pressure and flow in the umbilical cord

A fluid dynamic study of blood flow within the umbilical vessels of the human maternal-fetal circulatory system is considered. It is found that the umbilical coiling index (UCI) is unable to distinguish between cords of significantly varying pressure and flow characteristics, which are typically determined by the vessel curvature, torsion and length. Larger scale geometric non-uniformities superposed over the inherent coiling, including cords exhibiting width and/or local UCI variations as well as loose true knots, typically produce a small effect on the total pressure drop. Crucially, this implies that a helical geometry of mean coiling may be used to determine the steady vessel pressure drop through a more complex cord. The presence of vessel constriction, however, drastically increases the steady pressure drop and alters the flow profile. For pulsatile-flow within the arteries, the steady pressure approximates the time-averaged value with high accuracy over a wide range of cords. Furthermore, the relative peak systolic pressure measured over the period is virtually constant and approximately 25% below the equivalent straight-pipe value for a large range of non-straight vessels. Interestingly, this suggests that the presence of vessel helicity dampens extreme pressures within the arterial cycle and may provide another possible evolutionary benefit to the coiled structure of the cord.     

Pulsed magnetohydrodynamic blood flow in a rigid vessel under physiological pressure gradient

Blood flow in a steady magnetic field has been of great interest over recent years. Many researchers have examined the effects of magnetic fields on velocity profiles and arterial pressure, and major studies have focused on steady or sinusoidal flows. In this paper, we present a solution for pulsed magnetohydrodynamic blood flow with a somewhat realistic physiological pressure wave obtained using a Windkessel lumped model. A pressure gradient is derived along a rigid vessel placed at the output of a compliant module which receives the ventricle outflow. Then, velocity profile and flow rate expressions are derived in the rigid vessel in the presence of a steady transverse magnetic field. As expected, results showed flow retardation and flattening. The adaptability of our solution approach allowed a comparison with previously addressed flow cases and calculations presented a good coherence with those well established solutions.     

Distally based free flap using retrograde arterial flow

In lower extremity injuries, the major arteries are frequently damaged and the remaining artery may be located in the traumatized zone. Recently, vascular anastomoses to recipient vessels distal to the zone of injuries have been advocated in the lower limb reconstruction and flaps by using retrograde arterial flow are well established in island flaps. Twelve patients with soft-tissue defects below the knee underwent lower extremity reconstruction with distally based free flaps by using retrograde arterial flow. For assessment of retrograde flow, the intraoperative retrograde arterial pressure was quantitated and compared with the systolic blood pressure taken at the same time (n = 6). In this study, suitable candidates had been chosen with pulsatile retrograde flow from the distal cut end of the vessel because patients without pulsatile flow had recorded diastolic and systolic retrograde arterial pressures that were very low. Four different free flaps were used. All flaps were successful. Retrograde flow anastomosis could not interrupt the major blood vessels, which were essential for survival of the distal limb, the compromise of fracture or wound healing might be prevented. In cases in which arteriography demonstrates significant vascular flow interruption within the zone of injury, retrograde arterial anastomosis may also be a candidate.     

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.     

A theory of blood flow in skeletal muscle

A theoretical analysis of blood flow in the microcirculation of skeletal muscle is provided. The flow in the microvessels of this organ is quasi steady and has a very low Reynolds number. The blood is non-Newtonian and the blood vessels are distensible with viscoelastic properties. A formulation of the problem is provided using a viscoelastic model for the vessel wall which was recently derived from measurements in the rat spinotrapezius muscle (Skalak and Schmid-Sch?nbein, 1986b). Closed form solutions are derived for several physiologically important cases, such as perfusion at steady state, transient and oscillatory flows. The results show that resting skeletal muscle has, over a wide range of perfusion pressures an almost linear pressure-flow curve. At low flow it exhibits nonlinearities. Vessel distensibility and the non-Newtonian properties of blood both have a strong influence on the shape of the pressure-flow curve. During oscillatory flow the muscle exhibits hysteresis. The theoretical results are in qualitative agreement with experimental observations.     

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.     

Theoretical comparison of wall-derived and erythrocyte-derived mechanisms for metabolic flow regulation in heterogeneous microvascular networks

The objective of this study is to compare the effectiveness of metabolic signals derived from erythrocytes and derived from the vessel wall for regulating blood flow in heterogeneous microvascular networks. A theoretical model is used to simulate blood flow, mass transport, and vascular responses. The model accounts for myogenic, shear-dependent, and metabolic flow regulation. Metabolic signals are assumed to be propagated upstream along vessel walls via a conducted response. Arteriolar tone is assumed to depend on the conducted metabolic signal as well as local wall shear stress and wall tension, and arteriolar diameters are calculated based on vascular smooth muscle mechanics. The model shows that under certain conditions metabolic regulation based on wall-derived signals can be more effective in matching perfusion to local oxygen demand relative to regulation based on erythrocyte-derived signals, resulting in higher extraction and lower oxygen deficit. The lower effectiveness of the erythrocyte-derived signal is shown to result in part from the unequal partition of hematocrit at diverging bifurcations, such that low-flow vessels tend to receive a reduced hematocrit and thereby experience a reduced erythrocyte-derived metabolic signal. The model simulations predict that metabolic signals independent of erythrocytes may play an important role in local metabolic regulation of vascular tone and flow distribution in heterogeneous microvessel networks.     

Particle deposition in arteries ex vivo: effects of pressure,flow, and waveform

The goal of this study was to quantify the effect of hemodynamic pressure, flow and waveform perturbations on the deposition of protein-sized particles in porcine carotid arteries ex vivo. An ex vivo perfusion system was used to control the pressure and flow environment for excised arterial tissue. Confocal laser microscopy images revealed that 200 nm particles were deposited intimally and that more spheres were evident along vessels perfused under oscillatory waveform conditions than all others. Under all pressure, flow and waveform conditions, particles were excluded from the media and adventitia of the vessel wall. The steady flow data support the use of Darcy's Law with pressure-dependent hydraulic permeability to model arterial tissue.     

Effects of bradykinin on the hemodynamics of tumor and granulating normal tissue microvasculature.

Bradykinin (BK) is an important endogenous mediator of microvascular flow modulation. Since the structure of the microcirculation is very different in tumor tissues than in normal tissues, bradykinin may elicit different responses in tumors. This study was designed to test the hypothesis that local administration of bradykinin increases blood flow preferentially in normal tissue relative to adjacent tumor tissue, resulting in a "vascular steal" phenomenon. Microvessel diameters (D), velocities (Vc), length densities, shear rates, and intermittent flow frequencies were measured every 10 min before, during, and after 40 min exposure to BK in rats with dorsal flap window chambers 9 days after tumor implantation. Separate studies were made of normal vessels outside the tumor margin, the hypervascular tumor periphery, and the tumor center. Bradykinin was administered with a suffusion medium flowing over the tissue at 1-2 ml/min with a BK concentration of 1.6 x 10(7) M. Administration of BK created five distinct changes in normal and tumor vessel function that varied over time, but coincidentally reached a maximum effect after 20 min exposure to BK. In normal vessels, increased Vc and D led to increased flow, which reached a peak 20 min after onset of suffusion with BK. In contrast, in centrally located tumor vessels, decreased D and Vc were observed in most vessels during the initial 10-20 min of suffusion. In addition, there was a significant increase in intermittent flow frequency in tumor central vessels, which peaked after 20 min of suffusion with BK. These five separate observations that coincided at 20 min of suffusion are consistent with a "vascular steal" phenomenon. The increase in normal microvessel D and Vc at 20 min suggests that BK causes vasodilation in arterioles. The coincident decrease in tumor microvessel D and Vc suggests that tumor feeding vessels are less able to respond to BK by vasodilating. The concomitant increase in intermittent flow frequency in tumor vessels suggests that a reduction in pressure drop occurred after 20 min exposure to BK, which is also consistent with "vascular steal." Since BK is also known to increase vascular permeability, it is possible that increases in interstitial fluid pressure brought on by exposure to BK contributed to the observed reduction in tumor blood flow. In normal vessels, reduced D and Vc, relative to peak values, were noted after 40 min suffusion with BK. Adherence of leukocytes to the vessel walls was prominent and microthrombi were also observed during this period. No evidence of such adhesion was seen in tumor vessels, although microthrombi were observed.(ABSTRACT TRUNCATED AT 400 WORDS)     

Study of human cerebral blood flow by corrosion casts of the carotid system

20 corrosion casts of human anterior cerebral arteries and middle cerebral arteries were analyzed. In 588 vessel sections in between 2 bifurcations, the relation of vessel length and vessel diameter was investigated. Measurements were taken starting from the arterial trunk down to vessels with a diameter of 0.4 mm. The mean values of the lengths show that both the anterior cerebral artery and the middle cerebral artery can be grouped into 5 diameter ranges with significant different lengths. With decreasing diameter there are short vessels in the groups 1, 3 and 5 and long vessels in the groups 2 and 4. However, the long branches in the groups 2 and 4 of the middle cerebral artery are longer and thinner than in the anterior cerebral artery. Due to the higher pressure loss in thin and long vessels, blood pressure drop occurs earlier in the branches of the middle cerebral artery than in those of the anterior cerebral artery.     

A constrained mixture model for developing mouse aorta

Mechanical stresses influence the structure and function of adult and developing blood vessels. When these stresses are perturbed, the vessel wall remodels to return the stresses to homeostatic levels. Constrained mixture models have been used to predict remodeling of adult vessels in response to step changes in blood pressure, axial length and blood flow, but have not yet been applied to developing vessels. Models of developing blood vessels are complicated by continuous and simultaneous changes in the mechanical forces. Understanding developmental growth and remodeling is important for treating human diseases and designing tissue-engineered blood vessels. This study presents a constrained mixture model for postnatal development of mouse aorta with multiple step increases in pressure, length and flow. The baseline model assumes that smooth muscle cells (SMCs) in the vessel wall immediately constrict or dilate the inner radius after a perturbation to maintain the shear stress and then remodel the wall thickness to maintain the circumferential stress. The elastin, collagen and SMCs have homeostatic stretch ratios and passive material constants that do not change with developmental age. The baseline model does not predict previously published experimental data. To approximate the experimental data, it must be assumed that the SMCs dilate a constant amount, regardless of the step change in mechanical forces. It must also be assumed that the homeostatic stretch ratios and passive material constants change with age. With these alterations, the model approximates experimental data on the mechanical properties and dimensions of aorta from 3- to 30-day-old mice.     

Influence of arterial wall compliance on the pressure drop across coronary artery stenoses under hyperemic flow condition

Hemodynamic endpoints such as flow and pressure drop are often measured during angioplasty procedures to determine the functional severity of a coronary artery stenosis. There is a lack of knowledge regarding the influence of compliance of the arterial wall-stenosis on the pressure drop under hyperemic flows across coronary lesions. This study evaluates the influence in flow and pressure drop caused by variation in arterial-stenosis compliance for a wide range of stenosis severities. The flow and pressure drop were evaluated for three different severities of stenosis and tested for limiting scenarios of compliant models. The Mooney-Rivlin model defined the non-linear material properties of the arterial wall and the plaque regions. The non-Newtonian Carreau model was used to model the blood flow viscosity. The fluid (blood)-structure (arterial wall) interaction equations were solved numerically using the finite element method. Irrespective of the stenosis severity, the compliant models produced a lower pressure drop than the rigid artery due to compliance of the plaque region. A wide variation in the pressure drop was observed between different compliant models for significant (90% area occlusion) stenosis with 41.0, 32.1, and 29.8 mmHg for the rigid artery, compliant artery with calcified plaque, and compliant artery with smooth muscle cell proliferation, respectively. When compared with the rigid artery for significant stenosis the pressure drop decreased by 27.7% and 37.6% for the calcified plaque and for the smooth muscle cell proliferation case, respectively. These significant variations in pressure drop for the higher stenosis may lead to misinterpretation and misdiagnosis of the stenosis severity.     

Blood flow in the capillary bed

From arteries to veins, the blood has to go through the ‘capillary’ blood vessels. These blood vessels are so small that often their diameter is smaller than that of the red blood cells. Intimate interactions occur, therefore, between the blood cells and the blood vessels.

A general survey of recent works on capillary blood flow is given in this article. Some details are presented for two problems: the problem of deformation of the flexible red blood cells, their motion in the capillary blood vessels, and the pressure drop due to the red cell blood vessel interaction; and the problem of the flow of plasma ‘bolus’ between neighboring red cells.

The solution supplies many details about the microcirculation phenomenon. Taken together, a method is offered for the calculation of pressure drop in the capillary as a function of various physical parameters: the red cell volume per unit blood volume, (hematocrit), the ratio of the cell diameter to the blood vessel diameter, the ratio of the length of the blood vessel to its length, the volume of individual red cells, and a parameter relating the cell membrane elasticity, plasma viscosity and the cell velocity.     

A framework for incorporating 3D hyperelastic vascular wall models in 1D blood flow simulations

We present a novel framework for investigating the role of vascular structure on arterial haemodynamics in large vessels, with a special focus on the human common carotid artery (CCA). The analysis is carried out by adopting a three-dimensional (3D) derived, fibre-reinforced, hyperelastic structural model, which is coupled with an axisymmetric, reduced order model describing blood flow. The vessel transmural pressure and lumen area are related via a Holzapfel–Ogden type of law, and the residual stresses along the thickness and length of the vessel are also accounted for. After a structural characterization of the adopted hyperelastic model, we investigate the link underlying the vascular wall response and blood-flow dynamics by comparing the proposed framework results against a popular tube law. The comparison shows that the behaviour of the model can be captured by the simpler linear surrogate only if a representative value of compliance is applied. Sobol’s multi-variable sensitivity analysis is then carried out in order to identify the extent to which the structural parameters have an impact on the CCA haemodynamics. In this case, the local pulse wave velocity (PWV) is used as index for representing the arterial transmission capacity of blood pressure waveforms. The sensitivity analysis suggests that some geometrical factors, such as the stress-free inner radius and opening angle, play a major role on the system’s haemodynamics. Subsequently, we quantified the differences in haemodynamic variables obtained from different virtual CCAs, tube laws and flow conditions. Although each artery presents a distinct vascular response, the differences obtained across different flow regimes are not significant. As expected, the linear tube law is unable to accurately capture all the haemodynamic features characterizing the current model. The findings from the sensitivity analysis are further confirmed by investigating the axial stretching effect on the CCA fluid dynamics. This factor does not seem to alter the pressure and flow waveforms. On the contrary, it is shown that, for an axially stretched vessel, the vascular wall exhibits an attenuation in absolute distension and an increase in circumferential stress, corroborating the findings of previous studies. This analysis shows that the new model offers a good balance between computational complexity and physics captured, making it an ideal framework for studies aiming to investigate the profound link between vascular mechanobiology and blood flow.

     

Distribution of blood flow during exercise after blood volume expansion in swine

To study the distribution of blood flow after blood volume expansion, seven miniature swine ran at high speed (17.6-20 km/h, estimated to require 115% of maximal O2 uptake) on a motor-driven treadmill on two occasions: once during normovolemia and once after an acute 15% blood volume expansion (homologous whole blood). O2 uptake, cardiac output, heart rate, mean arterial pressure, and distribution of blood flow (with radiolabeled microspheres) were measured at the same time during each of the exercise bouts. Maximal heart rate was identical between conditions (mean 266); mean arterial pressure was elevated during the hypovolemic exercise (149 +/- 5 vs. 137 +/- 6 mmHg). Although cardiac output was higher and arterial O2 saturation was maintained during the hypervolemic condition (10.5 +/- 0.7 vs. 9.3 +/- 0.6 l/min), O2 uptake was not different (1.74 +/- 0.08 vs. 1.74 +/- 0.09 l/min). Mean blood flows to cardiac (+12.9%), locomotory (+9.8%), and respiratory (+7.5%) muscles were all elevated during hypervolemic exercise, while visceral and brain blood flows were unchanged. Calculated resistances to flow in skeletal and cardiac muscle were not different between conditions. Under the experimental conditions of this study, O2 uptake in the miniature swine was limited at the level of the muscles during hypervolemic exercise. The results also indicate that neither intrinsic contractile properties of the heart nor coronary blood flow limits myocardial performance during normovolemic exercise, because both the pumping capacity of the heart and the coronary blood flow were elevated in the hypervolemic condition.     

Human muscle length-dependent changes in blood flow

Although a multitude of factors that influence skeletal muscle blood flow have been extensively investigated, the influence of muscle length on limb blood flow has received little attention. Thus the purpose of this investigation was to determine if cyclic changes in muscle length influence resting blood flow. Nine healthy men (28 ± 4 yr of age) underwent a passive knee extension protocol during which the subjects' knee joint was passively extended and flexed through 100-180° knee joint angle at a rate of 1 cycle per 30 s. Femoral blood flow, cardiac output (CO), heart rate (HR), stroke volume (SV), and mean arterial pressure (MAP) were continuously recorded during the entire protocol. These measurements revealed that slow passive changes in knee joint angle did not have a significant influence on HR, SV, MAP, or CO; however, net femoral blood flow demonstrated a curvilinear increase with knee joint angle (r(2) = 0.98) such that blood flow increased by ～90% (125 ml/min) across the 80° range of motion. This net change in blood flow was due to a constant antegrade blood flow across knee joint angle and negative relationship between retrograde blood flow and knee joint angle (r(2) = 0.98). Thus, despite the absence of central hemodynamic changes and local metabolic factors, blood flow to the leg was altered by changes in muscle length. Therefore, when designing research protocols, researchers need to be cognizant of the fact that joint angle, and ultimately muscle length, influence limb blood flow.