Observations on the accuracy of photometric techniques used to measure some in vivo microvascular blood flow parameters.

OBJECTIVE: The accuracy of optical methods used to measure in vivo microvascular blood flow parameters is investigated using measurements made in all vessels of microvascular networks of the rat mesentery. METHODS: The principle of mass conservation was applied to in vivo blood flow rate and discharge hematocrit data, which were determined by photometric methods. One of the several implied assumptions of most interpretations of in vivo optical data is that the vessels are circular in cross-section: to see the impact of vessel lumen shape on one of these measurements, the average velocity of blood flowing through a D-shaped glass capillary tube was measured by the dual-slit method. RESULTS: For in vivo data, significant imbalance exists in a large number of bifurcations, and the correlation between the blood flow imbalance and the red cell flux imbalance is very small (r2 = 0.39), indicating multiple sources of error. Furthermore, the measured discharge hematocrits were consistent with the observed flow directions at bifurcations in only 39% to 46% of the bifurcations in a network. The imbalance at these bifurcations is not simply caused by the inaccuracy of measurements in only a few microvessels that join such bifurcations, i.e., the inaccuracies are evenly distributed among the vessels. The results of the in vitro study of blood velocity measurement in D-shaped tubes indicates that the ratio of dual-slit velocity to the actual average blood velocity is sensitive to the shape of the vessel lumen, and is a function of blood flow rate, hematocrit, vessel lumen shape, and orientation. CONCLUSIONS: Significant inaccuracies exist in the flow and hematocrit data obtained by current methods of interpretation of in vivo photometric measurements. These inaccuracies must be considered when making vessel to-vessel comparisons, or vessel-by-vessel comparisons between in vivo observations and model predictions, even though the inaccuracies are greatly reduced when comparing averaged data.

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.

Diameter variability and microvascular flow resistance.

Microvessels are known to exhibit irregular shapes, deviating substantially from an idealized cylindrical tube geometry. Such irregularities must be taken into account in calculating microvascular flow resistance and may add to the observation that flow resistance in living microvessels in vivo is about twice that predicted on the basis of tube flow studies in vitro. The present study was aimed at providing a comprehensive database describing the apparent diameter variability for all segments of a complete microvascular network in the rat mesentery and assessing the impact of this variability on segmental flow resistance and the pressure drop across the network. Diameters were estimated by intravital microscopy at axial intervals of 20 microns along the 546 vessel segments of a mesenteric microvessel network, resulting in 6,319 separate diameter measurements. The amplitude of diameter variations in individual vessel segments decreased from approximately 15% of the mean vessel diameter in the smallest segments (approximately 5 microns diam) to approximately 5% in the largest segments (approximately 60 microns diam). Segmental hindrance was estimated to be 10-23% higher than calculated from arithmetic mean diameter, depending on the model used to estimate the hydrodynamically effective segment diameter. The overall pressure drop across the network calculated using a mathematical flow simulation was increased by 7-13.5%. This increase in flow resistance can explain approximately 10% of the observed discrepancy between flow resistance in vivo and in vitro.

Model studies of leukocyte-endothelium-blood interactions. II. Hemodynamic impact of leukocytes adherent to the wall of post-capillary vessels.

Computational fluid dynamics (CFD) and large scale model experiments were used to analyze the hemodynamic impact of leukocytes adherent to the wall of post-capillary venules. Using a large scale model and, with the aid of a finite element package, solving the Navier Stokes equations for low Reynolds number flow in a cylinder past an adherent sphere, we have developed a dimensionless correlation which permits the estimation of the pressure drop across an adherent leukocyte in an in vivo vessel. This relationship is: f.Re = exp[2.877+4.630 (d/D)4] where f is the Fanning friction factor, Re is the Reynolds number and d/D is the leukocyte to vessel diameter ratio. The friction factor is proportional to the pressure drop across the leukocyte, and does not significantly increase until d/D is greater than 0.5, and then increases rapidly with increasing d/D. Computations indicate that the length of the disturbed flow region generated by an adherent leukocyte increases with decreasing vessel size. The average wall stress in the disturbed flow region remains constant, and equal to the wall stress in the undisturbed region for d/D less than approximately 0.5. For d/D greater than 0.5, the average wall stress in the disturbed flow region increases rapidly with increasing d/D. There is an even larger increase, up to five times greater than the average disturbed stress, in the peak wall stress in the disturbed flow region. This indicates that significant wall stress gradients can be generated by an adherent leukocyte in post-capillary size vessels.

Additional pressure drop at a bifurcation due to the passage of flexible disks in a large scale model.

The flow of red blood cells (RBC) through a microvascular capillary bifurcation was modeled in a large scale system in which rigid circular tubes and bifurcations (diameter = .95 cm) simulated capillaries and capillary bifurcations, flexible disks (undeformed diameter = 0.75 cm) simulated RBC and glycerol simulated plasma. At low Reynolds numbers (0.01 to 0.1), pressure drop was measured in the tubes upstream and downstream from the bifurcation as well as across the bifurcation itself, for various flow splits at the bifurcation while the inflow in the upstream tube was held constant. Pressure gradient across the bifurcation is taken to be the average of the upstream and downstream pressure gradients if the additional pressure drop at the bifurcation due to the partitioning of flow and disks is negligible. For the case of glycerol alone, the ratio of pressure gradient (G) at the bifurcation to the one at the upstream region was always greater than expected and reached 1.14 when the flow in the side branch was zero. With introduction of flexible disks into the system, G at the bifurcation was as much as 10 times the G at the upstream region as disks came in contact with, or close to, the dividing line of the bifurcation and paused momentarily before they entered one or the other side of the bifurcation. The largest G was for even flow split at the bifurcation and the smallest G was for the case where the flow in the side branch was smallest. Therefore, for the range of tube hematocrits (0-30 percent) and flow splits tested here, a significant additional pressure drop at the bifurcation is observed.

Fluctuations in microvascular blood flow parameters caused by hemodynamic mechanisms.

We have developed a mathematical model of microvascular network blood flow in which the nonlinear flow properties of blood and the nonuniform axial distribution of red blood cells in each vessel, as well as disproportionate cell partitioning at bifurcations, are all accounted for. The movements of red blood cells in the network are tracked; hence, the model is able to simulate temporal variations in local flow parameters in the network due to hemodynamic mechanisms. The model was applied to four rat mesenteric networks for which the topology, boundary conditions, blood velocity, and discharge hematocrit (Hctd) had been measured for each branch. Temporal variations in Hctd and blood velocity after simulation convergence were predicted. In some cases of the three vessels connected to a node, Hctd of one vessel fluctuates in a simple periodic form, Hctd of the second one oscillates in a more complex periodic form, whereas the Hctd of the third one does not oscillate at all. These variations were obtained with constant flow boundary conditions and, therefore, are due to hemodynamic factors alone. The temporal variations in flow parameters predicted by the model simulations are caused by hemorheological mechanisms and would be superimposed on variations caused by other mechanisms (e.g., vasomotion). The frequencies of the predicted fluctuations in blood velocity are in qualitative agreement with observed in vivo variations in dual-slit velocity in the arterioles of the cremaster muscle of anesthetized Golden hamster.

In vivo polymerization of sickle-cell hemoglobin: a theoretical study.

Several studies on the gelation and oxygenation state of sickle red blood cells have been done under conditions of equilibrium. The kinetics of sickle hemoglobin (HbS) polymerization have also been studied extensively in fully deoxygenated HbS solutions. The issue of the relevance of these investigations to the physiological in vivo situation has not been addressed. Here, we use a theoretical model to compare theoretical equilibrium predictions of HbS polymer concentration and cellular oxygen content, previously validated against equilibrium data, with the corresponding values under physiologic oxygen unloading conditions. We also use the model to simulate polymerization in almost completely deoxygenated sickle erythrocytes, validate the theoretical polymerization curves against published data, and compare them with the corresponding curves from the dynamic oxygen unloading analyses. Our model shows that equilibrium predictions severely overestimate intracellular polymer concentrations and underestimate cellular oxygen content, during the unloading of oxygen. Also, the delay times to significant polymerization in the physiologic situation are substantially longer than the corresponding values measured in completely deoxygenated HbS solutions. These results indicate that in vivo HbS polymerization is strongly influenced by the rate of oxygen desaturation. Equilibrium estimates of intracellular polymer content, or polymerization kinetic data from fully deoxygenated solutions, could be misleading and should be used in the proper perspective.

Effect of diameter variability along a microvessel segment on pressure drop.

In most microcirculatory studies, the diameter of a blood vessel is characterized by a single, average value, obtained from a projected two-dimensional microscopic view of the vessel. Such diameter values are often used to calculate microvascular hemodynamic variables (e.g., pressure drop along vessels). The validity, and some possible consequences, of the assumption that vessel diameter is constant along vessels were examined for (a) single capillaries of the hamster cremaster muscle and (b) a network model of blood flow in a rat mesenteric network. Vessel diameter was measured in cremaster microvessels of anesthetized golden hamsters (Nembutal, 70 mg/kg ip) at regular intervals along the lengths of individual vessels. The standard deviation of the diameter measurements (ranging from 0.2 to 4.0 microns) in each vessel increased with the average diameter (ranging from 4 to 27 microns, 132 vessels, 12 animals). The coefficient of variation (CV = SD/Mean) was close to 0.1 for vessels larger than 15 microns and up to 0.4 for smaller vessels. In individual isolated vessels, the ratio of calculated pressure drop using the actual diameter measurements (delta P) to pressure drop assuming an average diameter (delta Po) was between 1.05 and 3.0 (at constant volume flow rate); this ratio correlated significantly with the CV for that vessel. Using an iterative mathematical model of network blood flow in a mesenteric network, we investigated the effects on single-vessel delta P/delta P0 of using a single average estimate of diameter for each microvessel segment, compared to the inclusion of variations in each vessel diameter expected from the single-vessel data. The results from the model indicated that single vessel delta P/delta Po values obtained with expected diameter variations in each vessel varied between 0.01 and 100. This implies that the axial variations in vessel diameter that exist in vivo may affect the calculations of vessel pressure drop both in single vessels and in microvascular networks.

Perceived vessel lumen and cell-blood velocity ratio: impact on in vivo blood flow rate determination.

Microscopic images of blood flow through individual capillary segments and divergent capillary bifurcations in cremaster muscle of golden hamster were videotaped, and the data from the videotapes (projected vessel image width and individual red cell velocities) were used in two ways to evaluate blood flow rate. The first method assumes 1) that the vessel lumen is circular with a diameter equal to the projected image width and 2) that the blood average velocity is proportional to the average red cell velocity. The second method makes neither of these assumptions but relies only on the principle of mass conservation. It is demonstrated that the two assumptions of the first method can lead to significant errors in hemodynamic relationships deduced from in vivo data. Although the second method cannot independently give absolute values of average velocity at one vessel location, it can give absolute values of the ratio of blood flow rates through two (or more) vascular lumina.

Decreased hydrodynamic resistance in the two-phase flow of blood through small vertical tubes at low flow rates.

The aggregation of red blood cells in blood flowing through small tubes at very low shear rates leads to the two-phase flow of an inner core of rouleaux surrounded by a cell-depleted peripheral layer. The formation of this layer is known to be accompanied by a decrease in hydrodynamic resistance to flow. To quantitate this effect, we measured the pressure gradient, flow rate, and the radius of the red blood cell core in suspensions flowing through tubes of 172-microns radius at mean linear flow rates (U) from 50 to 0.15 tube diameters.sec-1. Washed red blood cells were suspended in 1.5% buffered dextran 110 at hematocrits of 34-52%. Using syringe pumps, blood flowed from a stirred reservoir through a vertical 12-cm length of tube in either the upward or downward direction. The pressure drop was measured with transducers. Mean values in distributions in the core radius were obtained by analyzing cine films of flow taken through a microscope with flow in the upward direction, measuring the core radius at five equally spaced axial positions of the tube in each of 100 frames. At 34% and 46% hematocrit, the hydrodynamic resistance increased as U decreased from 50 sec-1, reaching a maximum at U-2 sec-1. It then decreased to a minimum at U less than 0.5 sec-1 as the red blood cell core formed in the tube, and the mean core radius/tube radius ratio decreased from 0.98 to 0.74 with marked axial fluctuations at the lower U. At higher hematocrits, both the increase and decrease in hydrodynamic resistance were greater. In a red blood cell albumin-saline suspension, where there is no aggregation of red blood cells and no two-phase flow, hydrodynamic resistance increases linearly with decreasing U. The experimental results were compared with the predictions of a two-phase steady-flow model, assuming axisymmetric flow of a core surrounded by cell-free suspending medium. Two models were considered, one in which the core is solid, the other in which the rheological properties of the suspension in the core are given by the Quemada equation. The effects of sedimentation of the core resulting in a zero net flow pressure gradient were taken into account. Provided that an experimentally extrapolated value for the zero pressure gradient was used, the Quemada-fluid model gave good agreement with the experimentally observed core radius as a function of U and hematocrit.

Dynamics of oxygen unloading from sickle erythrocytes.

The objective of this work is to theoretically model oxygen unloading in sickle red cells. This has been done by combining into a single model diffusive transport mechanisms, which have been well-studied for normal red cells, and the hemoglobin polymerization process, which has been previously been studied for deoxyhemoglobin-S solutions and sickle cells in near-equilibrium situations. The resulting model equations allow us to study the important processes of oxygen delivery and polymerization simultaneously. The equations have been solved numerically by a finite-difference technique. The oxygen unloading curve for sickle erythrocytes is biphasic in nature. The rate of unloading depends in a complicated way on (a) the kinetics of hemoglobin S polymerization, (b) the kinetics of hemoglobin deoxygenation, and (c) the diffusive transport of both free oxygen and oxy-hemoglobin. These processes interact. For example, the hemoglobin S polymer interferes with the transport of both free oxygen and unpolymerized oxy-hemoglobin, and this is accounted for in the model by diffusivities which depend on the polymer and solution hemoglobin concentration. Other parameters which influence the interaction of these processes are the concentration of 2,3-diphosphoglycerate and total hemoglobin concentration. By comparing our model predictions for oxygen unloading with simpler predictions based on equilibrium oxygen affinities, we conclude that the relative rate of oxygen unloading of cells with different physical properties cannot be correctly predicted from the equilibrium affinities. To describe the unloading process, a kinetic calculation of the sort we give here is required.

Robin Fåhraeus: evolution of his concepts in cardiovascular physiology.

We give an account of the work of Robin Fåhraeus over the years 1917-1938, his contribution to our understanding of blood rheology, and its relevance to circulatory physiology. Fåhraeus published few original papers on this subject, yet he clearly understood the phenomena occurring in the tube flow of mammalian blood. 1) The concentration of cells in a tube less than 0.3 mm in diameter differs from that in the larger feed tube or reservoir, the Fåhraeus effect. This is due to a difference in the mean velocity of cells and plasma in the smaller vessel associated with a nonuniform distribution of the cells. 2) In tubes less than 0.3 mm in diameter, the resistance to blood flow decreases with decreasing tube diameter, the Fåhraeus-Lindqvist effect. We define and generalize the two effects and describe how red cell aggregation at low shear rates affects cell vessel concentration and resistance to flow. The fluid mechanical principles underlying blood cell lateral migration in tube flow and its application to Fåhraeus' work are discussed. Experimental data on the Fåhraeus and Fåhraeus-Lindqvist effects are given for red cells, white cells, and platelets. Finally, the extension of the classical Fåhraeus effect to microcirculatory beds, the Fåhraeus Network effect, is described. One of the explanations for the observed, very low average capillary hematocrits is that the low values are due to a combination of the repeated phase separation of red cells and plasma at capillary bifurcations (network effect) and the single-vessel Fåhraeus effect.

Perturbation of red blood cell flow in small tubes by white blood cells.

The flow of blood in the microcirculation is facilitated by the dynamic reduction in viscosity (Fahraeus-Lindquist effect) resulting from the axial flow of deforming erythrocytes (RBCs) and from the decrease in the ratio of cell to vessel diameter. RBC velocity exceeds that of average fluid velocity; however the slower moving white blood cells (WBC) perturb flow velocity and the ratio of cell to vessel diameter by obstructing red cell flow through formation of "trains" of red cells collecting behind the white cell. This effect of white cells was studied quantitatively in a model in vitro tubes less than 10 microns in diameter with the demonstration that flow resistance increases linearly with white cell numbers up to 1,000 WBC/mm3 at tube hematocrit of 17.7%. The increase in resistance exceeds the flow resistance of WBC and appears to relate directly to train formation. A mechanical model of train formation developed to predict WBC influence in flow resistance over the range of WBC studied reasonably fits observed WBC effects.

Radial distribution of white cells during blood flow in small tubes.

The radial distribution of white blood cells (WBC) in blood flowing through glass tubes (i.d. 69 micron) was studied as a function of wall shear stress (range 0.1-2.5 Pa) and suspending medium (plasma, buffered saline, high-molecular-weight dextran solution). It was found that, irrespective of the choice of suspending medium, the highest leukocyte flux at high shear stresses was found in the tube center. WBC redistribution was seen upon lowering the shear stresses: A significant shift of WBC flux toward the marginal fluid layers occurred at the expense of the axial region. After replacement of plasma by other media the flow-dependent redistribution of WBCs was qualitatively unaffected. However, suspension of cells in dextran solution (inducing strong red cell aggregation) resulted in enhanced WBC margination, while in saline (no red cell aggregation) axial accumulation was accentuated. The results support the concept of size-dependent radial distribution of particles in flow of mixed suspensions. If applied to the living microcirculation, the data serve to explain WBC margination in microvessels (the first step in the series of events leading to emigration) in terms of a hydrodynamic phenomenon resulting from red cell/white cell interaction. The pronounced flow dependence of WBC margination results primarily from the effect of shear on red cell aggregation which leads to an alteration of the effective particle size distribution in the flowing blood.

Analysis of the effects of measured white blood cell entrance times on hemodynamics in a computer model of a microvascular bed.

To quantify the interdependence of capillary leukocyte plugging and microvascular hemodynamics, experimental measurements were made of the time required for lymphocytes and granulocytes to enter a micropipette. Using standard micropipette deformation techniques, entrance times were found to be a function of both cell diameter and pipette diameter, with no significant dependence on aspiration pressure over the differential pressure range of 200-400 Pa. Experimental results were combined with a computer network model to describe changes in red cell distribution and flow rate resulting from the delayed entrance of leukocytes (WBC) into capillaries. The network model is based on geometrical measurements from the capillary bed of a hamster cremaster muscle (Sarelius et al. 1981) and utilizes previous work describing: 1. preferential cell distribution at a bifurcation, 2. increased apparent viscosity due to the presence of red and white cells, and 3. increased velocities of red and white cells relative to blood. Red and white cell positions within the network were computed at discrete time increments, and WBC plugging was simulated by a temporary cessation of flow in vessels of smaller diameter than the white cell. In contrast with previous studies, the increased viscosity due to the presence of WBCs was found to have an insignificant effect on overall network flow rate. Instead, a major flow reduction occurs only when capillaries are plugged by the white cells. At normal physiological concentrations (1,000 RBC/WBC), time-averaged overall network flow is reduced by 4.4%, based on averaged experimentally measured entrance times, and up to 14.8% if maximal entrance times are used.(ABSTRACT TRUNCATED AT 250 WORDS)

Oxygen delivery from red cells.

This paper deals with the theoretical analysis of the unloading of oxygen from a red cell. A scale analysis of the governing transport equations shows that the solutions have a boundary layer structure near the red-cell membrane. The boundary layer is a region of chemical nonequilibrium, and it owes its existence to the fact that the kinetic time scales are shorter than the diffusion time scales in the red cell. The presence of the boundary layer allows an analytical solution to be obtained by the method of matched asymptotic expansions. A very useful result from the analysis is a simple, lumped-parameter description of the oxygen delivery from a red cell. The accuracy of the lumped-parameter description has been verified by comparing its predictions with results obtained by numerical integration of the full equations for a one-dimensional slab. As an application, we calculate minimum oxygen unloading times for red cells.

Nonuniform red cell distribution in 20 to 100 micrometers bifurcations.

Several potentially important parameters influencing the disproportionate distribution of red cell flux and blood flow at a bifurcation are examined. These include parent vessel hematocrit, vessel diameter, suspending medium, cell distensibility, parent vessel blood flow rate, and local geometry. Measurements were performed on 20 to 100 micrometers bifurcations, fabricated such that all vessels of a given bifurcations have the same diameter. Suspensions of human red cells, hardened red cells, and mixtures of each in albumin-saline, Dextran 75, or plasma were flowed through the bifurcations and determinations of flow rates and discharge hematocrits were made for each of the channels. For the 20-micrometers channels, hematocrits were found using videophotometric techniques, and for the larger channels, hematocrits were measured directly from the exit streams. Flow rates for both were measured by meniscus travel downstream in small-diameter glass tubes. Within the limits of the present experiments, three of the variables proved to be of major importance: feed hematocrit, tube diameter, and flow-rate distribution. It was clearly demonstrated that red cell flux varies nonlinearly with fractional flow rate. Critical flow rates, at which all or none of the cells entered one of the branches, were found to vary with diameter and hematocrit as has been reported in other studies. The data were analyzed with a theoretical model which assumes that the parent vessel contains a core of uniformly distributed red cells surrounded by a marginal gap of suspending medium; in the parent vessel lumen, the flows to the two daughter branches were assumed to be separated by a chord. The marginal gap widths and tube hematocrits deduced from the data with this model are of reasonable magnitudes.

Influence of wall surface on the flow of blood through endothelial-lined glass tubes.

It has been proposed that the presence of endothelial cells lining small vessels decreases resistance to microcirculatory blood flow. Results of previous investigations have been inconclusive when fibrin-coated glass tubes were used to approximate the endothelial layer. Estimates of the decrease in apparent viscosity for blood flow in these tubes have ranged from near-zero to 50% when compared to flow in unlined glass tubes. The present study was devised to determine the effect of endothelial cell layer on blood flow in vessels under in vitro conditions in which pressure--flow relationships can be monitored precisely. Monolayers of human endothelial cells were grown over the major portion of the inner surface of glass tubes (I.D. = 1130 microns, length = 7.5 cm), and differential pressure and flow in the tubes were measured for plasma and suspensions of erythrocytes in plasma. Controls of phosphate buffer solution and culture medium were used to calculate inner diameters for both tube types. Results show statistically insignificant differences between apparent viscosities calculated from pressure-flow data in unlined and endothelial cell-lined tubes. These data indicate that the presence of endothelial lining does not have a marked influence on apparent viscosity of either blood or plasma when flowing through tubes of this diameter.