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Blood Flow in Microvascular Networks

Axel R. Pries, Timothy W. Secomb

10.1002/cphy.cp020401

Source: Supplement 9: Handbook of Physiology, The Cardiovascular System, Microcirculation

Originally published: 2008

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Introduction
2 Flow in Single Microvessels
2.1 Flow Resistance, Poiseuille's Law, and Effective Viscosity
2.2 Fåhraeus Effect
2.3 Fåhraeus‐Lindqvist Effect
2.4 Aggregation and Sedimentation
2.5 Endothelial Surface Layer
2.6 Effective Blood Viscosity in vivo
3 Microvascular Networks
3.1 General Features
3.2 Topology
3.3 Topological Growth Models
3.4 Segment Lengths and Diameters
3.5 Heterogeneity and Correlations
3.6 Flow and Transit Time
3.7 Pressure and Wall Stresses
3.8 Phase Separation in Bifurcations
3.9 Network Fåhraeus Effect
4 Relationship of Network Structure and Flow to Physiological Functions
4.1 Transport Functions
4.2 Inflammatory and Immune Functions
4.3 Regulation of Blood Flow
4.4 Structural Adaptation
5 Conclusions
Figure 1. Figure 1.

Blood flow through microvessels in the rat mesentery with inner diameters 7, 12, and 16μm (top to bottom). Flow is from left to right.
Figure 2. Figure 2.

Human erythrocytes during flow through a glass tube with inner diameter 7 μm. The discharge hematocrits are 0.2, 0.34, 0.41, 0.52, and 0.65 (top to bottom). Flow is from left to right.
Figure 3. Figure 3.

Human erythrocytes during flow through glass tubes with inner diameters 4, 7, and 17μm (top to bottom). Flow is from left to right.
Figure 4. Figure 4.

(A) Radial hematocrit profiles in arterioles of the rat mesentery determined by microdensitometry 21. The absolute dimension of the cell‐depleted layer at the vessel wall is similar for the smaller and the larger arteriole. However, the impact of the cell‐depleted layer on tube hematocrit and on flow resistance is larger in smaller vessels. (B) Profiles obtained by microparticle velocimetry in a rat cremaster venule (diameter ∼40.5 μm) before (continuous lines) and after (dashed lines) hemodilution 22. Left panel: flow velocity. Right panel: relative viscosity, which correlates to the local hematocrit at a given radial position. The vertical dark and light‐shaded regions indicate the width of the endothelial surface layer before (0.82μm) and after (0.62 μm) hemodilution (see Section 5). With the exception of this layer, the velocity profiles are similar for microvessels and similar‐sized glass tubes.
Figure 5. Figure 5.

Relation between tube hematocrit (HT) and discharge hematocrit (HD) as determined in glass tubes of different luminal diameter 23 Data for three different levels of are given together with the respective parametric fits according to eq. 13. (See page 1 in colour section at the back of the book)
Figure 6. Figure 6.

(A) Relative apparent viscosity of blood in long, straight glass tubes at a discharge hematocrit of 0.45. Experimental data for human blood (C4, H10, S17, Y26, X62, O69, T78 V80, G84, A145, N136, E153, J201, K200, U235, P252, Z179) are given together with the results of the parametric fit defined In eqs (15–17) (scaling to other species can be attempted by scaling the diameter with the cubic root of the ratio of the respective mean red cell volumes). (B) Experimental data and approximations for different levels of discharge hematocrit. Modified after 47. (See page 1 in colour section at the back of the book)
Figure 7. Figure 7.

Dependence of relative apparent viscosity on tube diameter. Comparison of experimental data (represented by the parametric description of eqs 15–17) and predictions of theoretical models. Modified after (48].
Figure 8. Figure 8.

Photographic images of blood flowing in vertical (left) and horizontal (right) glass tubes of diameter 60 μm. Following a sudden reduction of flow rate, a central core of aggregated red blood cells is formed, surrounded by enlarged plasma spaces near the tube walls. In the horizontal tube, this aggregated core eventually approaches the lower side of the tube as a result of sedimentation.

Reproduced by permission from 55
Figure 9. Figure 9.

Effective thickness of the endothelial surface layer (ESL), as a function of microvessel luminal diameter. Estimates were obtained by minimizing the differences between observed and predicted flow velocities in microvascular networks of the rat mesentery 73. Values in the dark area to the upper left are excluded by the constraint that red blood cells must be able to pass through the vessel when deformed to a minimal diameter of 2.8 μm. The vertically hatched area indicates the estimated physical thickness of the layer. The optimization indicated that the layer exerts an additional hematocrit‐dependent increase in flow resistance with a peak for vessel diameters of about 10μm. For different levels of discharge hematocrit (HD), this increase is given as equivalent increase in the layer thickness (lines).
Figure 10. Figure 10.

Components of the endothelial glycocalyx. (A) Schematic representation of a glycoprotein, two major types of proteoglycans belonging to the syndecan and glypican families, and hyaluronan (hyaluronic acid) on the endothelial surface. The carbohydrate side chains of glycoproteins (e.g., selectins. integrins, members of the immunoglobulin superfamily) are short and branched, while proteoglycans are characterized by long unbranched side chains. Hyaluronan may be produced by endothelial cells 89 or adsorbed from the plasma [90) to endothelial surface receptors 91,92. (B) Chemical composition of a typical heparan sulfate proteoglycan and of hyaluronan (gly: glycin, ala: alanin, ser: serine, Xyl; xylose, Gal: galactose, GlcA; glucuronis acid, IdoA: iduronic acid. IodA 2S: 2‐O‐sulfated iduronic acid, GlcNAc: N‐acetylglucosamine, GlcNAc 6s: 6‐O‐sulfated N‐acetylglucosamine. GlcNS 3S: 3‐0‐sulfated N‐sulfated glucosamine, GlcNS 38 6S: 3‐0‐ and 6‐O‐sulfated N‐sulfated glucosamine). The saccharide sequence shown represents the specific binding site for antithrombin III (ATIII), and is thus crucial for the anticoagulatory properties of the glycocalyx. (See page 2 in colour section at the back of the book)
Figure 11. Figure 11.

Concept for the composition of the endothelial surface layer. The glycocalyx is the thin (50–100 nm) domain adjacent to the endothelial surface which is constituted by glycoproteins and proteoglycans bound directly to the plasma membrane. The main part of the endothelial surface layer (∼0.5 μm) consists of a complex array of soluble plasma components possibly Including a variety of proteins, solubilized glycosaminoglycans, and hyaluronan. This layer is in a dynamic equilibrium with the flowing plasma and stabilized by osmotic forces due to low solid fraction in the range of ∼0.001 76. The surface layer may be degraded by mechanisms targeting the glycocalyx proper (e.g., enzymes, inflammatory mediators) or by changing the plasma composition (e.g., by infusion of artificial plasma replacement fluids). Molecules and components of the layer are not drawn to scale. (See page 2 in colour section at the back of the book)
Figure 12. Figure 12.

Apparent viscosity in vivo derived from an analysis of blood flow in microvascular networks (eqs 19, 20) compared with experimental measurements 95,98. Modified after 99.
Figure 13. Figure 13.

Microvascular networks in different tissues as visualized by scanning electron microscopy. Scale bars indicate a length of 40 μm. These show a sample of the wide variety of microvascular patterns found in the body. Different network structures are seen in heart muscle, skin, brain, lung, liver, spleen, lymph node, bone marrow, etc. (Micrographs courteously provided by Valentin Djonov).
Figure 14. Figure 14.

Microvascular network in the rat mesentery with a total area of 31 mm 55. The main feeding arteriole and draining venule enter at the left part of the lower boundary.

Adapted from Pries et al. 116
Figure 15. Figure 15.

Idealized (and unrealistic) representation of a microvascular network as a set of vessels of different categories coupled in series, in which the parallel vessels within each category exhibit identical properties.
Figure 16. Figure 16.

Topological description of trees by the Horton‐Strahler and generation nomenclatures. The Horton‐Strahler approach starts at the capillary (or terminal) level and proceed centripetally. The order is increased if two segments of equal order join at a bifurcation. The generation (centrifugal) scheme starts from the most central vessel considered and proceeds to the capillary level, increasing the generation by one at every branch point.
Figure 17. Figure 17.

Distributions of capillary generation numbers (top) and the fraction of capillaries relative to all segments of a given generation level (bottom) for an arterial and venous vessel tree with 180 capillary segments. The experimental data are compared with predictions of the random terminal branching (RTB) and the random segment branching (RSB) growth models.
Figure 18. Figure 18.

Data of Figure 17 compared with predictions of an area‐restricted sprouting growth model and an area‐restricted diffusion growth model (diffusion‐limited aggregation).
Figure 19. Figure 19.

Distributions of length and diameter for arteriolar, capillary, and venular segments in microvascular networks in the rat mesentery (total 2720 segments). For each vessel category, the mean value and the standard deviation are given.
Figure 20. Figure 20.

Variation of diameter, length, relative intravascular pressure and longitudinal relative pressure gradient of arterial, capillary, and venular vessel segments with vessel generation. The relative pressure scale was set to 1 for the pressure in the main feeding arterioles and to 0 in the main draining venules. For each parameter and class, mean values were determined for all segments fed by the main arteriolar inputs in each of seven microvascular networks In the rat mesentery. These data were then averaged across networks. Modified after 103.
Figure 21. Figure 21.

Distributions of parameters for 361 arterio‐venous flow pathways through a microvascular network with 913 segments. Data are based on experimental measurements combined with mathematical simulations of blood flow. The skewness given in addition to mean values and standard deviations characterizes the asymmetry of the distribution.
Figure 22. Figure 22.

Correlations between blood flow through 361 individual pathways through a microvascular network in the rat mesentery (913 segments) and pathway length (A) and transit time (B).
Figure 23. Figure 23.

Frequency distribution of intravascular pressure levels for all vessel segments of seven microvascular networks (top, modified after [10.3]) and variation of wall shear stress with intravascular pressure (middle) and with vessel diameter (bottom) for arterial, capillary, and venular vessel segments of six microvascular networks (modified after 179).
Figure 24. Figure 24.

Relationship between log of circumferential wall stress and log of vessel diameter. Data from experimental studies 190,195,196,197,198,199,200,201,202,203,204,205 have been analyzed using the relation between microvascular pressure and diameter derived with a mathematical flow model 179. The r2 value of the linear regression to all data points shown is 0.85 (modified after 193,194).
Figure 25. Figure 25.

Mesenteric microvascular network with an area of about 35 mm2 and 546 vessel segments, color coded for wall shear stress (left) and circumferential wall stress (right). The main inflow arteriole and outflow venule are marked. Circumferential stress is higher in larger vessels, while shear stress is high in the arteriolar and low venous portion of the network. (See page 3 in colour section at the back of the book)
Figure 26. Figure 26.

Microvascular bifurcation in the rat mesentery (flow from top to bottom).
Figure 27. Figure 27.

Parameters of red cell distribution at microvascular bifurcations. Upper panels; schematic drawing of a microvascular bifurcation (left). Data for a single bifurcation: hematocrit relative to that in the feeding vessel (middle) and fractional erythrocyte flow (right) in the daughter branches vs. the fractional blood flow in the respective branch. Experimental data, the fits obtained with eq. 21 and the respective parameters are given. Lower panels: parameters of the logit fit (A, B, XO) obtained for 65 bifurcations in the mesentery plotted vs. the relevant combinations of independent variables. The respective linear regression lines are shown with their 95% confidence intervals. (Adapted from Pries et al. 223).
Figure 28. Figure 28.

Reduction of discharge hematocrit in microvascular networks in the rat mesentery by the network Fåhraeus effect under control conditions (n = 5, systemic hematocrit 0.48–0.51) and after hemodilution (n = 6, systemic hematocrit 0.29–0.35). Shown is the mean discharge hematocrit (weighted by the cross‐sectional area of the segments) in all vessels segments of a complete flow cross section divided by the systemic hematocrit. A complete flow cross section consists of all arteriolar and capillary segments of the respective generation number plus all capillaries with lower generation numbers. The final flow cross section includes all capillaries. By this definition, each flow cross section carries the complete inflow of the network. Standard errors correspond to the variation among networks (modified after 116).
Figure 29. Figure 29.

Variation of disccharge hematocrit averaged for arteriolar and capillary segments for a microvascular network with 913 segments. The hematocrit was normalized with respect to the discharge hematocrit in the main arteriole feeding the network.


Figure 1.

Blood flow through microvessels in the rat mesentery with inner diameters 7, 12, and 16μm (top to bottom). Flow is from left to right.



Figure 2.

Human erythrocytes during flow through a glass tube with inner diameter 7 μm. The discharge hematocrits are 0.2, 0.34, 0.41, 0.52, and 0.65 (top to bottom). Flow is from left to right.



Figure 3.

Human erythrocytes during flow through glass tubes with inner diameters 4, 7, and 17μm (top to bottom). Flow is from left to right.



Figure 4.

(A) Radial hematocrit profiles in arterioles of the rat mesentery determined by microdensitometry 21. The absolute dimension of the cell‐depleted layer at the vessel wall is similar for the smaller and the larger arteriole. However, the impact of the cell‐depleted layer on tube hematocrit and on flow resistance is larger in smaller vessels. (B) Profiles obtained by microparticle velocimetry in a rat cremaster venule (diameter ∼40.5 μm) before (continuous lines) and after (dashed lines) hemodilution 22. Left panel: flow velocity. Right panel: relative viscosity, which correlates to the local hematocrit at a given radial position. The vertical dark and light‐shaded regions indicate the width of the endothelial surface layer before (0.82μm) and after (0.62 μm) hemodilution (see Section 5). With the exception of this layer, the velocity profiles are similar for microvessels and similar‐sized glass tubes.



Figure 5.

Relation between tube hematocrit (HT) and discharge hematocrit (HD) as determined in glass tubes of different luminal diameter 23 Data for three different levels of are given together with the respective parametric fits according to eq. 13. (See page 1 in colour section at the back of the book)



Figure 6.

(A) Relative apparent viscosity of blood in long, straight glass tubes at a discharge hematocrit of 0.45. Experimental data for human blood (C4, H10, S17, Y26, X62, O69, T78 V80, G84, A145, N136, E153, J201, K200, U235, P252, Z179) are given together with the results of the parametric fit defined In eqs (15–17) (scaling to other species can be attempted by scaling the diameter with the cubic root of the ratio of the respective mean red cell volumes). (B) Experimental data and approximations for different levels of discharge hematocrit. Modified after 47. (See page 1 in colour section at the back of the book)



Figure 7.

Dependence of relative apparent viscosity on tube diameter. Comparison of experimental data (represented by the parametric description of eqs 15–17) and predictions of theoretical models. Modified after (48].



Figure 8.

Photographic images of blood flowing in vertical (left) and horizontal (right) glass tubes of diameter 60 μm. Following a sudden reduction of flow rate, a central core of aggregated red blood cells is formed, surrounded by enlarged plasma spaces near the tube walls. In the horizontal tube, this aggregated core eventually approaches the lower side of the tube as a result of sedimentation.

Reproduced by permission from 55


Figure 9.

Effective thickness of the endothelial surface layer (ESL), as a function of microvessel luminal diameter. Estimates were obtained by minimizing the differences between observed and predicted flow velocities in microvascular networks of the rat mesentery 73. Values in the dark area to the upper left are excluded by the constraint that red blood cells must be able to pass through the vessel when deformed to a minimal diameter of 2.8 μm. The vertically hatched area indicates the estimated physical thickness of the layer. The optimization indicated that the layer exerts an additional hematocrit‐dependent increase in flow resistance with a peak for vessel diameters of about 10μm. For different levels of discharge hematocrit (HD), this increase is given as equivalent increase in the layer thickness (lines).



Figure 10.

Components of the endothelial glycocalyx. (A) Schematic representation of a glycoprotein, two major types of proteoglycans belonging to the syndecan and glypican families, and hyaluronan (hyaluronic acid) on the endothelial surface. The carbohydrate side chains of glycoproteins (e.g., selectins. integrins, members of the immunoglobulin superfamily) are short and branched, while proteoglycans are characterized by long unbranched side chains. Hyaluronan may be produced by endothelial cells 89 or adsorbed from the plasma [90) to endothelial surface receptors 91,92. (B) Chemical composition of a typical heparan sulfate proteoglycan and of hyaluronan (gly: glycin, ala: alanin, ser: serine, Xyl; xylose, Gal: galactose, GlcA; glucuronis acid, IdoA: iduronic acid. IodA 2S: 2‐O‐sulfated iduronic acid, GlcNAc: N‐acetylglucosamine, GlcNAc 6s: 6‐O‐sulfated N‐acetylglucosamine. GlcNS 3S: 3‐0‐sulfated N‐sulfated glucosamine, GlcNS 38 6S: 3‐0‐ and 6‐O‐sulfated N‐sulfated glucosamine). The saccharide sequence shown represents the specific binding site for antithrombin III (ATIII), and is thus crucial for the anticoagulatory properties of the glycocalyx. (See page 2 in colour section at the back of the book)



Figure 11.

Concept for the composition of the endothelial surface layer. The glycocalyx is the thin (50–100 nm) domain adjacent to the endothelial surface which is constituted by glycoproteins and proteoglycans bound directly to the plasma membrane. The main part of the endothelial surface layer (∼0.5 μm) consists of a complex array of soluble plasma components possibly Including a variety of proteins, solubilized glycosaminoglycans, and hyaluronan. This layer is in a dynamic equilibrium with the flowing plasma and stabilized by osmotic forces due to low solid fraction in the range of ∼0.001 76. The surface layer may be degraded by mechanisms targeting the glycocalyx proper (e.g., enzymes, inflammatory mediators) or by changing the plasma composition (e.g., by infusion of artificial plasma replacement fluids). Molecules and components of the layer are not drawn to scale. (See page 2 in colour section at the back of the book)



Figure 12.

Apparent viscosity in vivo derived from an analysis of blood flow in microvascular networks (eqs 19, 20) compared with experimental measurements 95,98. Modified after 99.



Figure 13.

Microvascular networks in different tissues as visualized by scanning electron microscopy. Scale bars indicate a length of 40 μm. These show a sample of the wide variety of microvascular patterns found in the body. Different network structures are seen in heart muscle, skin, brain, lung, liver, spleen, lymph node, bone marrow, etc. (Micrographs courteously provided by Valentin Djonov).



Figure 14.

Microvascular network in the rat mesentery with a total area of 31 mm 55. The main feeding arteriole and draining venule enter at the left part of the lower boundary.

Adapted from Pries et al. 116


Figure 15.

Idealized (and unrealistic) representation of a microvascular network as a set of vessels of different categories coupled in series, in which the parallel vessels within each category exhibit identical properties.



Figure 16.

Topological description of trees by the Horton‐Strahler and generation nomenclatures. The Horton‐Strahler approach starts at the capillary (or terminal) level and proceed centripetally. The order is increased if two segments of equal order join at a bifurcation. The generation (centrifugal) scheme starts from the most central vessel considered and proceeds to the capillary level, increasing the generation by one at every branch point.



Figure 17.

Distributions of capillary generation numbers (top) and the fraction of capillaries relative to all segments of a given generation level (bottom) for an arterial and venous vessel tree with 180 capillary segments. The experimental data are compared with predictions of the random terminal branching (RTB) and the random segment branching (RSB) growth models.



Figure 18.

Data of Figure 17 compared with predictions of an area‐restricted sprouting growth model and an area‐restricted diffusion growth model (diffusion‐limited aggregation).



Figure 19.

Distributions of length and diameter for arteriolar, capillary, and venular segments in microvascular networks in the rat mesentery (total 2720 segments). For each vessel category, the mean value and the standard deviation are given.



Figure 20.

Variation of diameter, length, relative intravascular pressure and longitudinal relative pressure gradient of arterial, capillary, and venular vessel segments with vessel generation. The relative pressure scale was set to 1 for the pressure in the main feeding arterioles and to 0 in the main draining venules. For each parameter and class, mean values were determined for all segments fed by the main arteriolar inputs in each of seven microvascular networks In the rat mesentery. These data were then averaged across networks. Modified after 103.



Figure 21.

Distributions of parameters for 361 arterio‐venous flow pathways through a microvascular network with 913 segments. Data are based on experimental measurements combined with mathematical simulations of blood flow. The skewness given in addition to mean values and standard deviations characterizes the asymmetry of the distribution.



Figure 22.

Correlations between blood flow through 361 individual pathways through a microvascular network in the rat mesentery (913 segments) and pathway length (A) and transit time (B).



Figure 23.

Frequency distribution of intravascular pressure levels for all vessel segments of seven microvascular networks (top, modified after [10.3]) and variation of wall shear stress with intravascular pressure (middle) and with vessel diameter (bottom) for arterial, capillary, and venular vessel segments of six microvascular networks (modified after 179).



Figure 24.

Relationship between log of circumferential wall stress and log of vessel diameter. Data from experimental studies 190,195,196,197,198,199,200,201,202,203,204,205 have been analyzed using the relation between microvascular pressure and diameter derived with a mathematical flow model 179. The r2 value of the linear regression to all data points shown is 0.85 (modified after 193,194).



Figure 25.

Mesenteric microvascular network with an area of about 35 mm2 and 546 vessel segments, color coded for wall shear stress (left) and circumferential wall stress (right). The main inflow arteriole and outflow venule are marked. Circumferential stress is higher in larger vessels, while shear stress is high in the arteriolar and low venous portion of the network. (See page 3 in colour section at the back of the book)



Figure 26.

Microvascular bifurcation in the rat mesentery (flow from top to bottom).



Figure 27.

Parameters of red cell distribution at microvascular bifurcations. Upper panels; schematic drawing of a microvascular bifurcation (left). Data for a single bifurcation: hematocrit relative to that in the feeding vessel (middle) and fractional erythrocyte flow (right) in the daughter branches vs. the fractional blood flow in the respective branch. Experimental data, the fits obtained with eq. 21 and the respective parameters are given. Lower panels: parameters of the logit fit (A, B, XO) obtained for 65 bifurcations in the mesentery plotted vs. the relevant combinations of independent variables. The respective linear regression lines are shown with their 95% confidence intervals. (Adapted from Pries et al. 223).



Figure 28.

Reduction of discharge hematocrit in microvascular networks in the rat mesentery by the network Fåhraeus effect under control conditions (n = 5, systemic hematocrit 0.48–0.51) and after hemodilution (n = 6, systemic hematocrit 0.29–0.35). Shown is the mean discharge hematocrit (weighted by the cross‐sectional area of the segments) in all vessels segments of a complete flow cross section divided by the systemic hematocrit. A complete flow cross section consists of all arteriolar and capillary segments of the respective generation number plus all capillaries with lower generation numbers. The final flow cross section includes all capillaries. By this definition, each flow cross section carries the complete inflow of the network. Standard errors correspond to the variation among networks (modified after 116).



Figure 29.

Variation of disccharge hematocrit averaged for arteriolar and capillary segments for a microvascular network with 913 segments. The hematocrit was normalized with respect to the discharge hematocrit in the main arteriole feeding the network.

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Article Title: Blood Flow in Microvascular Networks
 

How to Cite

Axel R. Pries, Timothy W. Secomb. Blood Flow in Microvascular Networks. Compr Physiol 2011, Supplement 9: Handbook of Physiology, The Cardiovascular System, Microcirculation: 3-36. First published in print 2008. doi: 10.1002/cphy.cp020401