This model illustrates the variation in cell distribution from that which would be expected by the division of flow at a microvascular bifurcation.

## Description

This model represents the variation in hematocrit distribution that occurs at a microvascular bifurcation. With internal diameters larger than about 30 microns the division of red blood cells is roughly proportional to the division of blood flow at a bifurcation. Below 30 microns, the cell-free layer that is present near the cell wall begins to affect the relationship between the division of flow and cells. Pries, Ley, Claassen and Gaehtgens originally characterized this variation in their 1989 Microvascular Research paper (see References below). Drawing a parallel to the Fahraeus effect, which is the reduction of the supplied hematocrit in narrow vessels, they called this the network Fahreaus effect. Refinements have been made to the model by Pries, Secomb, Gaehtgens and Gross in 1990 and more recently by Pries and Secomb in 2005. The result of the variation in hematocrit distribution can be illustrated by this example from the JSim applet below. A 10um diameter parent vessel divides into an 8um and a 4um daughter vessel. The parent vessel has a discharge hematocrit of 0.45. If the division of flow is 80%/20% between the large and the small daughter vessel, from the graph in the JSim applet we can see that the division of cells is roughly 85%/15%.

## Equations

The hematocrits in the large and small branches in this example can be calculated as follows:

Therefore, a small variation in the division of cells at a bifurcation can lead to a wide heterogeneity in actual hematocrit values across a network.

The relationships for the phase separation effect as presented here reflect the most recent refinement made to the model by Pries and Secomb in 2005.

where FQ_{e} is the fractional erythrocyte flow in daughter branch , A is the asymmetry parameter which strongly depends on the diameter ratio of the two daughter branches, B is the nonlinearity parameter which decreases with decreasing hematocrit, FQ_{b} is the fractional blood flow into the daughter branch and X_{0} is the fractional blood flow value below which no cells enter the branch. The accompanying expressions for A, B and X_{0} are:

where is the internal diameter of daughter branch , is the internal diameter of daughter branch and H_{d} is the discharge hematocrit of the parent vessel. Finally the definition of the logit function is given below:

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

- Download JSim model MML code (text):
- Download translated SBML version of model (if available):

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Pries AR and Secomb TW. Microvascular blood viscosity in vivo and the endothelial surface layer. American Journal of Physiology 289:H2657-H2664, 2005. Pries AR, Secomb TW, Gaehtgens P and Gross JF. Blood flow in microvascular networks - Experiments and simulation Circulation Research 67:826-834, 1990. Pries AR, Ley K, Classen M and Gaehtgens P. Red cell distribution at microvascular bifurcations. Microvascular Research 38:81-101, 1989.

Please cite https://www.imagwiki.nibib.nih.gov/physiome in any publication for which this software is used and send one reprint to the address given below:

The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

**Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants:** NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.