Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 4235937, 10 pages

https://doi.org/10.1155/2019/4235937

## Accounting for Tube Hematocrit in Modeling of Blood Flow in Cerebral Capillary Networks

Correspondence should be addressed to Andrey E. Kovtanyuk; ed.mut.am@uynatvok

Received 26 November 2018; Revised 3 February 2019; Accepted 28 July 2019; Published 18 August 2019

Academic Editor: Michele Migliore

Copyright © 2019 Nikolai D. Botkin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim of this paper consists in the derivation of an analytic formula for the hydraulic resistance of capillaries, taking into account the tube hematocrit level. The consistency of the derived formula is verified using Finite Element simulations. Such an effective formula allows for assigning resistances, depending on the hematocrit level, to the edges of networks modeling biological capillary systems, which extends our earlier models of blood flow through large capillary networks. Numerical simulations conducted for large capillary networks with random topologies demonstrate the importance of accounting for the hematocrit level for obtaining consistent results.

#### 1. Introduction

Simulation of blood circulation in large capillary networks is a challenging task. Realistic modeling of microvessel structures should take into account not only sophisticated topologies of blood vessel networks but also correct hydraulic resistance of microvessels. The latter is characterized by the apparent blood viscosity which depends on the vessel diameter as well as the discharge and tube hematocrit. The discharge hematocrit is the volume fraction of the red blood cells (RBCs) in the blood delivered by the flow in the vessel. The tube hematocrit is the volume fraction of RBCs that are inside the vessel at a given time instant. The discharge hematocrit is larger than the tube one because the velocity profile in the radial direction is nonuniform; namely, the RBCs velocity is higher than the mean bulk flow speed, which is called the Fåhraeus effect [1, 2]. The velocity profile in the radial direction is affected by the presence of the endothelial surface layer (ESL) [1].

The importance of accounting for the hematocrit level in blood flow simulations attracts attention of many researches. Animal models are utilized, for example, to measure and analyze the distributions of cell velocity and cell flux in the capillary network for different values of systemic hematocrit [3]. By using fluorescent microscopic analysis of rat cerebral capillary networks, the influence of hematocrit on mean RBC capillary velocity and mean arterial pressure can be assessed [4]. Moreover, the effect of hematocrit can be investigated in artificial microvascular branching networks [5]. A combination of an animal model with an effective iterative algorithm allows for finding the distribution of discharge hematocrit and blood flow velocity in a cerebrocortical microvascular network [6]. This approach takes into account the heterogeneity of blood flow and partitioning of red cells at bifurcations.

The current paper is related to modeling of computer-generated blood microvessel networks with vessel diameters less than 10 *μ*m. This is motivated by our previous work on simulating cerebral blood flow of preterm infants [7]. In such thin blood vessels, RBCs move in single file. Due to their ability to deform, RBCs can pass through vessels down to 2.7 *μ*m in size, which is less than their diameter, without damaging their membrane [8]. The hematocrit level and the shape of single erythrocytes during their motion in a capillary with diameter less than 8 *μ*m depend on RBC velocity [9].

Two approaches to mathematical modeling of RBC transport through microvessels can be mentioned.

The first one is based on continuum models [7, 10], where erythrocytes are considered as a homogeneous substance, and the RBC motion is described as a two-phase blood flow; namely, the erythrocyte homogeneous substance moves in the central core of the vessel, whereas the plasma fraction moves in a cell-free layer formation near the wall. Reasonable assumptions allow for deriving an explicit formula for the hydraulic resistance of a single capillary [7], which is very important for the simulation of blood flow in large capillary networks. A model derived in [10] studies the relations between the tube hematocrit level, vessel diameter, and apparent viscosity.

The second approach relies on discrete models [11, 12], where the effect of each erythrocyte is taken into account. The capillary blood flow is considered as single-file flow of red cells in blood plasma playing the role of lubrication and filling gaps between the erythrocytes. Such an ansatz [11] is used to develop a model resulting in an efficient algorithm for computing the pressure and flow field as well as the hematocrit distribution in simplified capillary networks. This approach shows a strong influence of single-file arrangement of RBCs on flow behavior. A coupled model describing the delivery of oxygen to tissue cells is considered [12] at the scale allowing to take into account the size and shape of individual RBCs as well as their deformation. The proposed approach [11, 12] takes into account the level of hematocrit in simulations of blood flow.

In the present paper, the approaches described in [7, 11] are combined to obtain an analytical formula for the computation of blood flow resistance in microvessels. This formula accounts for gaps between RBCs and, therefore, reflects the dependence on tube hematocrit. The tuning and validation of this formula are performed using hydrodynamical computations based on the representation of the cell-plasma mixture as a fluid with two different viscosities (much larger viscosity for blood cells). A very good consistency of the analytically computed values with the numerical results is obtained. An example of finding the pressure distribution in a relatively large capillary network (the germinal matrix or the whole brain), accounting for the level of tube hematocrit, is presented. The ability to account for the hematocrit level significantly enhances the algorithm proposed in [7] for finding the pressure distribution in the germinal matrix. Numerical experiments show a significant influence of the hematocrit level on the pressure distribution.

#### 2. Continuous Model of Red Cell Transport

In [7], the transport of red cells in capillaries was modeled as a continuum flow with spatially variable viscosity. A high viscosity was assigned to the central part of the capillary, RBC substance, whereas the layer between the RBC substance and capillary wall was assigned with a small viscosity typical for blood plasma (Figure 1.)