Multiscale modeling of lymphatic drainage from tissues using homogenization theory

Lymphatic capillary drainage of interstitial fluid under both steady-state and inflammatory conditions is important for tissue fluid balance, cancer metastasis, and immunity. Lymphatic drainage function is critically coupled to the fluid mechanical properties of the interstitium, yet this coupling is poorly understood. Here we sought to effectively model the lymphatic-interstitial fluid coupling and ask why the lymphatic capillary network often appears with roughly a hexagonal architecture. We use homogenization method, which allows tissue-scale lymph flow to be integrated with the microstructural details of the lymphatic capillaries, thus gaining insight into the functionality of lymphatic anatomy. We first describe flow in lymphatic capillaries using the Navier-Stokes equations and flow through the interstitium using Darcy's law. We then use multiscale homogenization to derive macroscale equations describing lymphatic drainage, with the mouse tail skin as a basis. We find that the limiting resistance for fluid drainage is that from the interstitium into the capillaries rather than within the capillaries. We also find that between hexagonal, square, and parallel tube configurations of lymphatic capillary networks, the hexagonal structure is the most efficient architecture for coupled interstitial and capillary fluid transport; that is, it clears the most interstitial fluid for a given network density and baseline interstitial fluid pressure. Thus, using homogenization theory, one can assess how vessel microstructure influences the macroscale fluid drainage by the lymphatics and demonstrate why the hexagonal network of dermal lymphatic capillaries is optimal for interstitial tissue fluid clearance.     

Analysis of coupled intra- and extraluminal flows for single and multiple capillaries

Matched asymptotic expansions are used to study a model of the coupled fluid flow in the capillaries and tissue of the microcirculation. These capillaries are long, narrow cylindrical tubes embedded in a uniform tissue space. The capillary, or intraluminal, flow is assumed to be that of an incompressible Navier-Stokes fluid wherein colloids are represented as dilute solute; the extraluminal flow in the tissue is according to Darcy's law. Central to this fluid exchange is the boundary condition on the fluid radial velocity at the semipermeable wall of the capillary. This boundary condition, involving the local hydrostatic and colloidal osmotic pressures in both the capillary and the tissue, together with the radial gradient of the tissue hydrostatic pressure, couples the intra- and extraluminal flow fields. With this model we investigate the relationship between transport properties, hydrostatic pressures, and flow exchange for a single capillary, and describe the fluid transport in the tissue space produced by an array of such capillaries.     

The flow of sickle blood in glass capillaries: Fundamentals and potential applications

We have characterized the imbibed horizontal flow of sickle blood into 100-μm-diameter glass capillaries. We find that blood containing sickled cells typically traverses the capillaries between three and four times as slowly as oxygenated cells from the same patient for all genotypes tested, including SS, AS, SC and Sβ⁺ thalassemia blood. Blood from SS patients treated with hydroxyurea has a viscosity intermediate between the SS and AA values. Blood containing cells that are not rigidified, such as normal red cells or oxygenated sickle cells, follows a simple Lucas-Washburn flow throughout the length of the 3-cm capillary. By fitting the flexible-cell data to the Lucas-Washburn model, a viscosity can be derived that is in good agreement with previous measurements over a range of volume fractions and is obtained using an apparatus that is far more complex. Deoxygenation sickles and thus rigidifies the cells, and their flow begins as Lucas-Washburn, albeit with higher viscosity than flexible cells. However, the flow further slows as a dense mass of cells forms behind the meniscus and increases in length as flow progresses. By assuming that the dense mass of cells exerts a frictional force proportional to its length, we derive an equation that is formally equivalent to vertical imbibition, even though the flow is horizontal, and this equation reproduces the observed behavior well. We present a simple theory using activity coefficients that accounts for this viscosity and its variation without adjustable parameters. In the course of control experiments, we have found that deoxygenation increases the flexibility of normal human red cells, an observation only recently published for mouse cells and previously unreported for human erythrocytes. Together, these studies form the foundation for an inexpensive and rapid point-of-care device to diagnose sickle cell disease or to determine blood viscosity in resource-challenged settings.     

Dynamics of blood flow: modeling of Fåhraeus and Fåhraeus–Lindqvist effects using a shear-induced red blood cell migration model

Blood flow in micro capillaries of diameter approximately 15–500 μm is accompanied with a lower tube hematocrit level and lower apparent viscosity as the diameter decreases. These effects are termed the Fåhraeus and Fåhraeus–Lindqvist effects, respectively. Both effects are linked to axial accumulation of red blood cells. In the present investigation, we extend previous works using a shear-induced model for the migration of red blood cells and adopt a model for blood viscosity that accounts for the suspending medium viscosity and local hematocrit level. For fully developed hematocrit profiles (i.e., independent of axial location), the diffusion fluxes due to particle collision frequency and viscosity gradients are of equal magnitude and opposite directions. The ratio of the diffusion coefficients for the two fluxes affects both the Fåhraeus and Fåhraeus–Lindqvist effects and is found related to the capillary diameter and discharge hematocrit using a well-known data-fit correlation for apparent blood viscosity. The velocity and hematocrit profiles were determined numerically as functions of radial coordinate, tube diameter, and discharge hematocrit. The velocity profile determined numerically is consistent with the derived analytical expression and the results are in good agreement with published numerical results and experimental data for hematocrit ratio and hematocrit and velocity profiles.     

Pressure regulation in the microcirculation.

The results of direct pressure measurements are described which demonstrate that pressures in a certain fraction of mesenteric capillaries remain remarkably constant during large changes in systemic pressure. The results of isogravimetric studies, reported in the literature, are also described which indicate that this phenomenon may also occur in the intestine. The question is raised whether capillary pressures may therefore be regulated. Pressures recorded from mesenteric arterioles and capillaries are shown which indicate that maintenance of a constant capillary pressure is primarily the consequence of the vascular architecture peculiar to this tissue, and is merely a secondary reflection of mechanisms associated with flow regulation. The results of direct pressure measurements recorded in the microcirculation of intestinal muscle are also shown. These data indicate that capillary pressures in innervated, denervated, and xylocaine-treated intestinal muscle change in direct proportion to variations in arterial pressure. It is concluded that capillary pressures in the intestinal muscle layers are therefore not regulated, so that the observation that capillary pressures may be maintained is probably a phenomenon unique to the mesentery. Pressures recorded from capillaries in the mucosal villi are also shown and compared to capillary pressures measured in the microvasculature of mesentery and intestinal muscle. When systemic pressure was normal (107 +/- 10 mm Hg), capillary pressure in the mesentery averaged 30 to 33 mm Hg; capillary pressures in the intestinal muscle averaged 22 to 24 mm Hg; and capillary pressures in the mucosal villi averaged 13 to 15 mm Hg. These data suggest that mesenteric capillaries are primarily a filtering network; intestinal muscle capillaries are normally in fluid balance; whereas at rest mucosal capillaries are primarily absorptive. These pressures, recorded from the three major regions of the rat intestine, were used to calculate a weighted average for the whole organ. The calculated value, based on assumed values for relative capillary densities, was 17 mm Hg. This result compares favorably with data from whole organ, isogravimetric studies, and may clarify some of the apparent discrepancies between previous isogravimetric and servopressure studies.     

Theoretical and experimental study of the time dependent flow of red blood cell suspension through narrow pores

The Hemorheometer has been adapted to allow the recording of the flow rate during the filtration process. For newtonian fluids, the flow rate variation versus time through the pores is well approximated by Poiseuille's law. For dilute red blood cell suspensions, the same analysis can be applied by introducing the concept of "apparent filtration viscosity" which is higher than the usual viscosity measured by Couette viscometry. The apparent filtration viscosity parameter is related to the deformations undergone by red blood cells as they pass through the narrow pores. Apparent filtration viscosity can be used to obtain a precise determination of the erythrocyte deformability. Measurements performed, for a given blood sample, with pores of different diameters (5 microns, 8 microns and 12 microns) show that the error on the value of apparent filtration viscosity is less than 3%. As a result, the sensitivity of the filtration method allows to discriminate among normal blood samples. High concentrations of erythrocytes or leucocytes are found to modify the apparent filtration viscosity. These factors are apparent in the recorded filtration curves. Their effects on filtration measurements can be easily estimated.     

Dynamical clustering of red blood cells in capillary vessels

We have modeled the dynamics of a 3-D system consisting of red blood cells (RBCs), plasma and capillary walls using a discrete-particle approach. The blood cells and capillary walls are composed of a mesh of particles interacting with harmonic forces between nearest neighbors. We employ classical mechanics to mimic the elastic properties of RBCs with a biconcave disk composed of a mesh of spring-like particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, translational and angular momenta. Realistic behavior of blood cells is modeled by considering RBCs and plasma flowing through capillaries of various shapes. Three types of vessels are employed: a pipe with a choking point, a curved vessel and bifurcating capillaries. There is a strong tendency to produce RBC clusters in capillaries. The choking points and other irregularities in geometry influence both the flow and RBC shapes, considerably increasing the clotting effect. We also discuss other clotting factors coming from the physical properties of blood, such as the viscosity of the plasma and the elasticity of the RBCs. Modeling has been carried out with adequate resolution by using 1 to 10 million particles. Discrete particle simulations open a new pathway for modeling the dynamics of complex, viscoelastic fluids at the microscale, where both liquid and solid phases are treated with discrete particles. Figure A snapshot from fluid particle simulation of RBCs flowing along a curved capillary. The red color corresponds to the highest velocity. We can observe aggregation of RBCs at places with the most stagnant plasma flow.     

Microcontinuum model for pulsatile blood flow through a stenosed tube

The effects of polar nature of blood and pulsatility on flow through a stenosed tube have been analysed by assuming blood as a micropolar fluid. Linearized solutions of basic equations are obtained through consecutive applications of finite Hankel and Laplace transforms. The analytical expressions for axial and particle angular velocities, wall shear stress, resistance to flow and apparent viscosity have been obtained. The axial velocity profiles for Newtonian and micropolar fluids have been compared. The interesting observation of this analysis is velocity, in certain parts of cycle, for micropolar fluid is higher than Newtonain fluid. Variation of apparent viscosity eta a with tube radius shows both inverse Fahraeus-Lindqvist and Fahraeus-Lindqvist effects. Finally, the resistance to flow and wall shear stress for normal and diseased blood have been computed and compared.     

A mechanism for erythrocyte-mediated elevation of apparent viscosity by leukocytes in vivo without adhesion to the endothelium

In spite of the relatively small number of leukocytes in the circulation, they have a significant influence on the perfusion of such organs as skeletal muscle or kidney. However, the underlying mechanisms are incompletely understood. In the current study a combined in vivo and computational approach is presented in which the interaction of individual freely flowing leukocytes with erythrocytes and its effect on apparent blood viscosity are explored. The skeletal muscle microcirculation was perfused with different cell suspensions with and without leukocytes or erythrocytes. We examined a three-dimensional numerical model of low Reynolds number flow in a capillary with a train of erythrocytes (small spheres) in off-axis positions and single larger leukocytes in axisymmetric positions. The results indicate that in order to match the slower axial velocity of leukocytes in capillaries, erythrocytes need to position themselves into an off-axis position in the capillary. In such off-axis positions at constant mean capillary velocity, erythrocyte axial velocity matches on average the axial velocity of the leukocytes, but the apparent viscosity is elevated, in agreement with the whole organ perfusion observations. Thus, leukocytes influence the whole organ resistance in skeletal muscle to a significant degree only in the presence of erythrocytes.     

Red blood cell mechanics and capillary blood rheology.

Blood contains a high vol fraction of erythrocytes (red blood cells), which strongly influence its flow properties. Much is known about the mechanical properties of red cells, providing a basis for understanding and predicting the rheological behavior of blood in terms of the behavior of individual red cells. This review describes quantitative theoretical models that relate red cell mechanics to flow properties of blood in capillaries. Red cells often flow in single file in capillaries, and rheological parameters can then be estimated by analyzing the motion and deformation of an individual red cell and the surrounding plasma in a capillary. The analysis may be simplified by using lubrication theory to approximate the plasma flow in the narrow gaps between the cells and the vessels walls. If red cell shapes are assumed to be axisymmetric, apparent viscosities are predicted that agree with determinations in glass capillaries. Red cells flowing in microvessels typically assume nonaxisymmetric shapes, with cyclic "tank-treading" motion of the membrane around the interior. Several analyses have been carried out that take these effects into account. These analyses indicate that nonaxisymmetry and tank-treading do not significantly influence the flow resistance in single-file or two-file flow.     

Red blood cell mechanics and capillary blood rheology

Blood contains a high vol fraction of erythrocytes (red blood cells), which strongly influence its flow properties. Much is known about the mechanical properties of red cells, providing a basis for understanding and predicting the rheological behavior of blood in terms of the behavior of individual red cells. This review describes quantitative theoretical models that relate red cell mechanics to flow properties of blood in capillaries. Red cells often flow in single file in capillaries, and rheological parameters can then be estimated by analyzing the motion and deformation of an individual red cell and the surrounding plasma in a capillary. The analysis may be simplified by using lubrication theory to approximate the plasma flow in the narrow gaps between the cells and the vessel walls. If red cell shapes are assumed to be axisymmetric, apparent viscosities are predicted that agree with determinations in glass capillaries. Red cells flowing in microvessels typically assume nonaxisymmetric shapes, with cyclic “tank-treading” motion of the membrane around the interior. Several analyses have been carried out that take these effects into account. These analyses indicate that nonaxisymmetry and tank-treading do not significantly influence the flow resistance in single-file or two-file flow.     

Motion of nonaxisymmetric red blood cells in cylindrical capillaries

We analyze theoretically the single-file flow of asymmetric red blood cells along cylindrical capillaries. Red cells in narrow capillaries are typically nonaxisymmetric, with the cell membrane moving continuously around the cell. In our analysis, cell shape and streamlines of membrane motion are prescribed. Lubrication theory is used to compute velocities and pressures in the fluid surrounding the cell. Conditions of zero lift, zero torque, zero drag, and energy conservation in the cell are imposed. Predicted tank-treading frequency, cell inclination and transverse displacement are small. Cell asymmetry and tank-treading are found to have little effect on the apparent viscosity of blood in capillaries with diameters up to 7 microns.     

Characterization of the capillary network in skeletal muscles from 3D data

In this review we present immunohistochemical methods for visualization of capillaries and muscle fibres in thick muscle sections. Special attention is paid to the procedures that preserve good morphology. Applying confocal microscopy and virtual 3D stereological grids, or tracing of capillaries in virtual reality, length of capillaries within a muscle volume or length of capillaries adjacent to a muscle fibre per fibre length, fibre surface area or fibre volume can be evaluated by an unbiased approach. Moreover, 3D models of capillaries and muscle fibres can be produced. Comparison of the developed methods with counting capillary profiles from 2D sections is discussed and the reader is warned that counting capillary profiles from 2D sections can underestimate the capillary length by as much as 75 percent. Application of the described 3D methodology is illustrated by the anatomical remodelling of capillarity during acute denervation and early reinnervation in the rat soleus and extensor digitorum longus muscles.     

Red Blood Cells: Centerpiece in the Evolution of the Vertebrate Circulatory System

All vertebrates except cold-water ice fish transport oxygenvia hemoglobin packaged in red blood cells (RBCs). VertebrateRBCs vary in size by thirtyfold. Differences in RBC size havebeen known for over a century, but the functional significanceof RBC size remains unknown. One hypothesis is that large RBCsare a primitive character. Agnathans have larger RBCs than domammals. However, the largest RBCs are found in urodele amphibianswhich is inconsistent with the hypothesis that large RBCs areprimitive. Another possibility is that small RBCs increase bloodoxygen transport capacity. Blood hemoglobin concentration ([Hb])and mean RBC hemoglobin concentration (MCHC) increase from Agnathato birds and mammals. However, the changes in [Hb] and MCHCdo not parallel changes in RBC size. In addition, RBC size doesnot affect blood viscosity. Thus, there is no clear link betweenRBC size and oxygen transport capacity. We hypothesize thatRBC size attends changes in capillary diameter. This hypothesisis based on the following observations. First, RBC width averages25% larger than capillary diameter which insures cell deformationduring capillary flow. Functionally, RBC deformation minimizesdiffusion limitations to gas exchange. Second, smaller capillariesare associated with increased potential for diffusive gas exchange.However, smaller capillaries result in higher resistances toblood flow which requires higher blood pressures. We proposethat the large capillary diameters and large RBCs in urodelesreflect the evolutionary development of a pulmonary vascularsupply. The large capillaries reduced systemic vascular resistancesenabling a single ventricular heart to supply blood to two vascularcircuits, systemic and pulmonary, without developing high pressureson the pulmonary side. The large RBCs preserved diffusive gasexchange efficiency in the large capillaries.     

In vivo measurement of leukocyte viscosity during capillary plugging.

Leukocyte plugging of capillaries in vivo was measured in rat spinotrapezius muscle. The plug durations, leukocyte and capillary dimensions, and arteriolar pressure at the plug sites were applied to the mechanical model of Needham and Hochmuth (1990) to estimate the leukocyte viscosities. The viscosity distribution of 389 cells was lognormal with a median value of 232 Poise. 3.1 percent of the cells were apparently activated and displayed viscosities greater than 3000 Poise. The median viscosity suggests that inactivated leukocytes have a minimal effect on blood flow, but that leukocyte activation may result in significant increases in microvascular flow resistance.     

Alginate plasma expander maintains perfusion and plasma viscosity during extreme hemodilution

Extreme hemodilution was performed in the hamster chamber window model using 6% Dextran 70, lowering systemic hematocrit by 60%. Animals were subsequently divided into three groups and hemodiluted to a hematocrit of 11% using 6% Dextran 70, 6% Dextran 500, and a 4% Dextran 70 + 0.7% alginate solution (n = 6 each group). Final plasma viscosities were 1.4 +/- 0.2, 2.2 +/- 0.1, and 2.7 +/- 0.2 cp, respectively, (P < 0.05, high viscosity vs. low viscosity). Blood viscosities were 2.1 +/- 0.2, 2.9 +/- 0.4, and 3.9 +/- 0.3 cp, respectively. The lowest blood and plasma viscosity group had a significantly lower functional capillary density, 37 +/- 16%, whereas the two high-viscosity solutions were 71 +/- 15% and 76 +/- 12% (P < 0.05, high viscosity vs. low viscosity), respectively. Arteriolar and venular flow in the Dextran 500 and alginate groups was higher than baseline (i.e., normal nontreated animals), whereas the low-viscosity group showed a reduction in flow. These microvascular changes were paralleled by changes in base excess, which was negative for the Dextran 70 group and positive for the other groups. However, tissue Po(2) was uniformly low for all groups (average of 1.4 mmHg). Calculation of tissue oxygen consumption in the window chamber based on the microvascular data, flow, and intravascular Po(2) showed that only the alginate + Dextran 70 solution-exchanged animals returned to baseline oxygen consumption, whereas the other groups were lower than baseline (P < 0.05). These results show that hemodilution performed with high-viscosity plasma expanders yields systemic arterial pressures and functional capillary densities that are significantly higher (P < 0.05) than those obtained with 6% Dextran 70, a fluid whose viscosity is similar to that of plasma. A condition for obtaining these results is that the oncotic pressure of the plasma expander be titrated to near normal, so that autotransfusion of fluid from the tissue into the vascular compartment does not reduce the effects of increasing plasma viscosity and increased shear stress on the microvascular wall.     

Numerical Modeling of Interstitial Fluid Flow Coupled with Blood Flow through a Remodeled Solid Tumor Microvascular Network

Modeling of interstitial fluid flow involves processes such as fluid diffusion, convective transport in extracellular matrix, and extravasation from blood vessels. To date, majority of microvascular flow modeling has been done at different levels and scales mostly on simple tumor shapes with their capillaries. However, with our proposed numerical model, more complex and realistic tumor shapes and capillary networks can be studied. Both blood flow through a capillary network, which is induced by a solid tumor, and fluid flow in tumor’s surrounding tissue are formulated. First, governing equations of angiogenesis are implemented to specify the different domains for the network and interstitium. Then, governing equations for flow modeling are introduced for different domains. The conservation laws for mass and momentum (including continuity equation, Darcy’s law for tissue, and simplified Navier–Stokes equation for blood flow through capillaries) are used for simulating interstitial and intravascular flows and Starling’s law is used for closing this system of equations and coupling the intravascular and extravascular flows. This is the first study of flow modeling in solid tumors to naturalistically couple intravascular and extravascular flow through a network. This network is generated by sprouting angiogenesis and consisting of one parent vessel connected to the network while taking into account the non-continuous behavior of blood, adaptability of capillary diameter to hemodynamics and metabolic stimuli, non-Newtonian blood flow, and phase separation of blood flow in capillary bifurcation. The incorporation of the outlined components beyond the previous models provides a more realistic prediction of interstitial fluid flow pattern in solid tumors and surrounding tissues. Results predict higher interstitial pressure, almost two times, for realistic model compared to the simplified model.     

Insulin binding and vesicular ingestion in capillary endothelium

Capillary endothelium can actively regulate vascular permeability of various serum proteins. Hormones such as insulin must interact with this capillary barrier in order to reach their respective target tissues. We have studied the binding and subsequent internalization of 125I-insulin in both native (freshly isolated) and primary cultured capillary endothelium derived from rat epididymal fat pads. Insulin association with the endothelium, internalization and degradation differed between freshly isolated and primary cultured capillaries. Specific binding in freshly isolated and cultured capillaries was temperature dependent, and was competitively inhibited in the presence of unlabelled insulin. Primary cultures of capillaries grown to confluence did not exhibit specific binding of insulin. Despite the lack of specific receptors for insulin, cultured cells vesicularly internalized insulin. Greater than 50% of the total associated insulin was not degraded by cultured endothelium. Morphological examinations using ferritin labelled insulin localized insulin associated to the capillary endothelial cell membrane and sequestered within pinocytotic vesicles. Incubation of freshly isolated capillaries with insulin stimulated the fluid phase endocytosis of 14C-sucrose; however, insulin had no effect on fluid phase endocytosis in cultured capillaries. These results indicate that capillary endothelium, isolated from rat epididymal fat, exhibit specific receptors for insulin. Binding of insulin to the capillary membrane is followed by internalization into cytoplasmic vesicles and partial degradation.     

Pulmonary capillaries are recruited during pulsatile flow.

Capillaries recruit when pulmonary arterial pressure rises. The duration of increased pressure imposed in such experiments is usually on the order of minutes, although recent work shows that the recruitment response can occur in <4 s. In the present study, we investigate whether the brief pressure rise during cardiac systole can also cause recruitment and whether the recruitment is maintained during diastole. To study these basic aspects of pulmonary capillary hemodynamics, isolated dog lungs were pump perfused alternately by steady flow and pulsatile flow with the mean arterial and left atrial pressures held constant. Several direct measurements of capillary recruitment were made with videomicroscopy. The total number and total length of perfused capillaries increased significantly during pulsatile flow by 94 and 105%, respectively. Of the newly recruited capillaries, 92% were perfused by red blood cells throughout the pulsatile cycle. These data provide the first direct account of how the pulmonary capillaries respond to pulsatile flow by showing that capillaries are recruited during the systolic pulse and that, once open, the capillaries remain open throughout the pulsatile cycle.     

Frequency bandwidth limitation of external pulse electric fields in cylindrical micro-channel electrophoresis with analyte velocity modulation

In capillary electrophoresis, effective optical signal quality improvement is obtained when high frequency (>100 Hz) external pulse fields modulate analyte velocities with synchronous lock-in detection. However, the pulse frequency is constrained under a critical value corresponding to the time required for the bulk viscous flow, which arises due to viscous momentum diffusion from the electro-osmotic slip in the Debye layer, to reach steady-state. By solving the momentum diffusion equation for transient bulk flow in the micro-channel, we show that this set-in time to steady-state and hence, the upper limit for the pulse frequency is dependent on the characteristic diffusion length scale and therefore the channel geometry; for cylindrical capillaries, the set-in time is approximately one half of that for rectangular slot channels. From our estimation of the set-in time and hence the upper frequency modulation limit, we propose that the half width of planar channels does not exceed 100 microm and that the radii of cylindrical channels be limited to 140 microm such that there is a finite working bandwidth range above 100 Hz and below the upper limit in order for flicker noise to be effectively suppressed.