Incorporating the complex rheological behavior of blood in microvascular network simulations: Two-phase modeling and a model reduction approach

Abstract

Vessel flow resistance in microvascular network simulations is often computed using the single-phase approximation, where the no-slip boundary condition is imposed on the microvessel wall, and the "in vitro viscosity law" or "the relative hematocrit" relationships proposed by Pries et al. (1992) are utilized. However, these models have limitations due to their reliance on experimental data, which inherently contain measurement errors, thereby affecting their consistency and accuracy. To moderate these limitations, biphasic flow models can be employed, allowing for adaptable rheological descriptions of blood. Nevertheless, this approach presents challenges in terms of automation and computational cost. In order to address these challenges, we propose a monophasic steady-state model that consistently incorporates a wall-slip condition in place of the cell-free layer contribution. Model reduction is achieved by establishing an explicit relationship for the apparent slip length, which is calibrated through the regression of biphasic model results. Subsequently, simulations are conducted on microvessel network structures. The predictions of wall shear stress by the monophasic model exhibit deviations of less than 2 % from the expected values, within the range of vessel radii (10–75 μm) and discharge hematocrits (0.2–0.7). Furthermore, the computational time required is reduced by almost an order of magnitude compared to the biphasic model, rendering it suitable for multiscale biochemical simulations. The developed model closely approximates the flow resistance predictions of the biphasic model, while its modular nature allows for easy implementation of future modifications. Both models successfully reproduce experimental data by Lipowsky et al. (1978). Finally, we derive semi-analytical expressions for the apparent slip length and discuss why it is more appropriate to impose only the "relative hematocrit" relationship as a global constraint for calculating the hemodynamics in microvessels, rather than relying on the "in vitro viscosity law", which provides unreliable predictions at low values of the discharge hematocrit.