Mathematical Models for Some Aspects of Blood Microcirculation (original) (raw)
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Biophysical aspects of blood flow in the microvasculature
Cardiovascular Research, 1996
The main function of the microvasculature is transport of materials. Water and solutes are carried by blood through the microvessels and exchanged, through vessel walls, with the surrounding tissues. This transport function is highly dependent on the architecture of the microvasculature and on the biophysical behavior of blood flowing through it. For example, the hydrodynamic resistance of a microvascular network, which determines the overall blood flow for a given perfusion pressure, depends on the number, size and arrangement of microvessels, the passive and active mechanisms governing their diameters, and on the apparent viscosity of blood flowing in them. Suspended elements in blood, especially red blood cells, strongly influence the apparent viscosity, which varies with several factors, including vessel diameter, hematocrit and blood flow velocity. The distribution of blood flows and red cell fluxes within a network, which influences the spatial pattern of mass transport, is determined by the mechanics of red cell motion in individual diverging bifurcations. Here, our current understanding of the biophysical processes governing blood flow in the microvasculature is reviewed, and some directions for future research are indicated.
Hemodynamics: Macroscopic and Microscopic Modeling
International Journal of Innovation in Science and Mathematics, 2022
This article presents a work of modeling blood circulation. More specifically, we are interested in macrocirculation and microcirculation. These two circulations depend on the size of the blood vessels.
Mechanics and computational simulation of blood flow in microvessels
Medical Engineering & Physics, 2011
Blood is a concentrated suspension of red blood cells (RBCs). Motion and deformation of RBCs can be analyzed based on knowledge of their mechanical characteristics. Axisymmetric models for single-file motion of RBCs in capillaries yield predictions of apparent viscosity in good agreement with experimental results for diameters up to about 8 micron. Two-dimensional simulations, in which each RBC is represented as a set of interconnected viscoelastic elements, predict that offcentre RBCs in an 8-micron channel take asymmetric shapes and drift toward the centre-line. Predicted trajectories agree with observations in microvessels of the rat mesentery. An isolated RBC initially positioned near the wall of a 20-micron channel is deformed into an asymmetric shape, migrates away from the wall, and then enters a complex tumbling motion with continuous shape change. Realistic simulation of multiple interacting RBCs in microvessels remains as a major challenge.
Blood exhibits a heterogeneous nature of hematocrit, velocity, and effective viscosity in microcapillaries. Microvascular bifurcations have a significant influence on the distribution of the blood cells and blood flow behavior. This paper presents a simulation study performed on the two-dimensional motions and deformation of multiple red blood cells in microvessels with diverging and converging bifurcations. Fluid dynamics and membrane mechanics were incorporated. Effects of cell shape, hematocrit, and deformability of the cell membrane on rheological behavior of the red blood cells and the hemodynamics have been investigated. It was shown that the blood entering the daughter branch with a higher flow rate tended to receive disproportionally more cells. The results also demonstrate that red blood cells in microvessels experienced lateral migration in the parent channel and blunted velocity profiles in both straight section and daughter branches, and this effect was influenced by the shape and the initial position of the cells, the hematocrit, and the membrane deformability. In addition, a cell free region around the tip of the confluence was observed. The simulation results are qualitatively consistent with existing experimental findings. This study may provide fundamental knowledge for a better understanding of hemodynamic behavior of micro-scale blood flow. Microvascular system is a network of microvessels connected by irregular short bifurcated segments. The local blood flow behavior at the vicinity of the apexes of the diverging and converging bifurcation significantly affects the hemodynamics in microcirculation, such as the heterogeneous distribution of red blood cells throughout the microvasculature, the effective viscosity, and blood flow velocity 1. The direct influence of this nonuniform distribution is on the microvascular oxygen transportation. Therefore, detailed investigation and deeper understanding of rheological property of red blood cells in micro-bifurcations are extremely important. Blood flow in microcirculation can be affected by many factors. One of the most crucial factors is hematocrit, which is the volumetric percentage of red blood cells in whole blood, and it ranges from 37% to 52% for adults. Due to the Fahraeus-Lindqvist effect, the tube hematocrit in smaller branches can be as low as 10–20%, much lower than the hematocrit in the main vessels. The variation in hematocrit may cause fluctuation in blood effective viscosity which is a primary measure of blood flow resistance and a crucial factor of friction against the vessel walls. Thus, an increase in the blood viscosity will lead to decreasing oxygen transported to tissues and organs and will cause problems with blood circulation. Another important factor that affects the microscopic blood flow is the deformability of the red blood cell. Under normal conditions, red blood cells deform into various shapes to compensate the flow resistance. The cell membrane can be significantly altered under some disease conditions. For example, the malaria infected red blood cells could be much more rigid in comparison with healthy ones 2,3. The ability that red blood cells possess to deform themselves influences the aggregation of the cells, the cell free layer, as well as the velocity of the cells to transit the vessels. The mechanical property of the red blood cell membrane has been modeled and the effects on blood flow have been extensively studied 4–6. Due to the complexity of the microvascular system, a detailed in vivo study of blood flow in the microcircula-tion is a challenging task. Nowadays, in vitro data 7,8 utilize idealised geometries and simplified dynamics for the
Blood pressure distribution in microvascular networks
2014
Abstract: Blood rheology is complex and nonlinear. The effective viscosity variations are important due to red blood cells packing inside capillaries, the socalled FåhræusLindquist effect, whilst concomitantly phase segregation appears in bifurcations. We have performed direct numerical simulations of different nonlinear rheological models of the blood on realistic threedimensional microvascular networks. These simulations point out two significant results. First, various rheological models lead to very similar pressure distributions over the whole range of physiologically relevant hematocrits. Secondly, different models for phase segregation lead to very distinct hematocrit distributions in the microvascular network. Moreover, for all the investigated rheological models, the hematocrit distribution very weakly affects the pressure distribution, when prescribing uniform pressure boundary conditions.
Flow-dependent rheological properties of blood in capillaries
Microvascular Research, 1987
Velocity-dependent flow of human red blood cells in capillaries with inside diameters of 4 to 8 pm is described theoretically. Cells are assumed to flow in single file, with axisymmetric shapes. Plasma flow in the gaps between cells and vessel walls is described by lubrication theory. The model takes into account the elastic properties of red cell membrane, including its responses to shear and bending. Cell shape is computed numerically as a function of tube diameter and cell velocity over the range 0.001 to 10 cm/set. Relative apparent viscosity and dynamic hematocrit reduction (Fahraeus effect) are also computed. Since effects of interactions between cells are neglected, the Fahraeus effect is independent of hematocrit, while viscosity varies linearly with hematocrit. At moderate or high cell velocities, about 0.1 cm/set or more, cell shapes and rheological parameters approach flow-independent limits. At lower velocities, cells broaden as a result of membrane shear and bending resistance and approach the walls more closely. Consequently, apparent viscosity increases with decreasing flow rate. Predicted values are in agreement with in vitro experimental determinations. Flow cessation is not predicted to occur in uniform tubes at positive driving pressures. Elastic deformational energies associated with red cell shapes are computed, leading to estimates of the pressure difference required to drive red cells past typical irregularities in capillary lumen cross sections. The hindrance to flow resulting from such structural irregularities represents a potential rheological mechanism for cessation of capillary flow at very low driving pressures.
Microvascular Research, 2011
We compare the predictive capability of two mathematical models for red blood cells (RBCs) focusing on blood flow in capillaries and arterioles. Both RBC models as well as their corresponding blood flows are based on the dissipative particle dynamics (DPD) method, a coarsegrained molecular dynamics approach. The first model employs a multiscale description of the RBC (MS-RBC), with its membrane represented by hundreds or even thousands of DPD-particles connected by springs into a triangular network in combination with out-of-plane elastic bending resistance. Extra dissipation within the network accounts for membrane viscosity, while the characteristic biconcave RBC shape is achieved by imposition of constraints for constant membrane area and constant cell volume. The second model is based on a low-dimensional description (LD-RBC) constructed as a closed torus-like ring of only 10 large DPD colloidal particles. They are connected into a ring by worm-like chain (WLC) springs combined with bending resistance. The LD-RBC model can be fitted to represent the entire range of nonlinear elastic deformations as measured by optical-tweezers for healthy and for infected RBCs in malaria. MS-RBCs suspensions model the dynamics and rheology of blood flow accurately for any vessel size but this approach is computationally expensive for vessel diameters above 100 microns. Surprisingly, the much more economical suspensions of LD-RBCs also capture the blood flow dynamics and rheology accurately except for small-size vessels comparable to RBC diameter. In particular, the LD-RBC suspensions are shown to properly capture the experimental data for the apparent viscosity of blood and its cell-free layer (CFL) in tube flow. Taken together, these findings suggest a hierarchical approach in modeling blood flow in the arterial tree, whereby the MS-RBC model should be employed for capillaries and arterioles below 100 microns, the LD-RBC model for arterioles, and the continuum description for arteries.
Microcirculatory model relating geometrical variation to changes in pressure and flow rate
Annals of Biomedical Engineering, 1981
A primary first step in defining the functional capabilities of a given microcirculatory bed is to describe the distribution of intravascular pressure and flow throughout the network using an idealized geometrical model By applying an approach developed by the geomorphological sciences, changes in vessel lengths, numbers, and diameters were systematically related to order number by applying three mathematical relationships. From measurements of a number of vascular modules, generalized relationships were obtained which applied to the microcirculation as a whole. An idealized model was developed by the computergeneration of node-node connections based on their probabilities of occurrence, and by generation of lengths and diameters, based on equations relating diameter and length to branching order number. Networks of this type were readily analyzable in terms of pressure-flow characteristics, using linear network analysis techniques. The current study focused on two specific microvascular beds:
Red blood cell mechanics and capillary blood rheology
Cell Biophysics, 1991
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.