# InVivoViscosity

Model number
0157

This model represents changes of blood viscosity due to the Fahraeus-Lindqvist effect as observed by Pries, Secomb et al. for flow in the rat mesentery.

## Description

This blood viscosity model represents the in vivo viscosity relationship, developed by
Axel Pries, Tim Secomb et al., of the Fahraeus-Lindqvist effect. The development by
Pries et al. is the latest in a series of refinements of the initial observation made
by Fahraeus, that the hematocrit of blood flowing through a narrow tube differs from the
hematocrit of the blood supply at the inlet.

Fahraeus's observation led him to distinguish between tube and discharge hematocrit and,
more to the point of this model, led to his conclusion, with Lindqvist, that effective
viscosity of the blood in narrow tubes must also vary with the diameter of the tube. Fahraeus
and Lindqvist noticed that in tubes greater than 1 mm in diameter the bulk viscosity values
for blood were acceptable for matching blood flow. On the other hand, as the tubes became
smaller the effective viscosity was reduced. The effective viscosity is defined here as the
viscosity calculated from Poiseuille's equation for a given pressure drop, tube inside
diameter and tube length. Quantification and mathematical description of this effect, first in
narrow glass tubes and later in vivo, followed.

The work by Pries, Secomb et al., represented in this model, shows that the effect in actual
vessels in vivo is not as significant as the effect in narrow glass tubes, but must be
accounted for in order to accurately represent flow in the microvasculature.

## Equations

The expression used here is for the effective viscosity of blood relative to the viscosity of plasma, and therefore the expression is unitless.

where HD is the discharge hematocrit, D is the diameter of the vessel lumen (in m), *0.45 is the relative effective viscosity at 45% hematocrit and C is the shape of the dependence of the viscosity on discharge hemtocrit. The expression for *0.45 is given by:

and C is given by:

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.

References
Pries AR, Secomb TW, Gebner T, Sperandio MB, Gross JF and Gaehtgens P;
Resitance to blood flow in microvessels in vivo. Circulation Research 75:904-915, 1994.

Fahraeus R, Lindqvist T; The viscosity of the blood in narrow capillary tubes.
American Journal of Physiology 96:562-568, 1931.

Key terms
Fahraeus-Lindqvist effect
in vivo
blood viscosity
microvasculature
discharge hematocrit
vessel diameter
Cardiovascular System
Microcirculation
blood flow
Acknowledgements

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.