The paper describes a mathematical model of blood ow in capillaries with accounting for the endothelial surface layer (ESL). The in uence of ESL is modeled by a boundary layer with zero ow velocity. Numerical simulations for di erent levels of the discharge hematocrit are conducted using the nite element method. The reliability of the results obtained is veri ed using known experimental data.
Modeling blood circulation in human brain requires understanding the dynamics of blood ow both in the entire vascular network and in an individual vessel. Assuming the blood ow in a vessel as a moving laminar Newtonian uid, we describe it by the Poiseuille's law [16]:
Q =
D4
p:
(
The dynamic viscosity is a material property of the liquid, which re ects the internal resistance to shearing motions. This law is a reasonable approximation for the ow in large blood vessels. As the diameter of the blood vessel decreases, the behavior of the blood ow increasingly deviates from the Poiseuille's law. For small vessels, e.g. for capillaries, the blood cannot be considered as continuous uid with a xed viscosity. Instead, it is regarded as plasma with suspended red blood cells (RBCs), also called erythrocytes. Other elements of blood, as for example, white blood cells and platelets have negligible e ects on the blood ow due to their tiny volume fractions. Thus, the blood in a microvessel can be considered as a two-phase liquid consisting of plasma and erythrocytes [2, 3], wherein the RBCs phase is modeled by a high viscosity substance. Numerical realization of the two-phase model of blood ow in capillaries can be carried out using the nite element method (see, e.g., [3]).
The RBCs have a tendency to migrate away from the vessel walls to the centerline, which causes the formation of cell-free regions near the vessel walls. Since the velocity increases towards the vessel centerline, the velocities of RBCs are higher than the average blood velocity. This means that there is a di erence between the tube hematocrit (the volume fraction of RBCs in the vessel at a given time instant) and the discharge hematocrit (the volume fraction of RBCs collected at the end of the tube at a given time period); the latter is higher than the tube hematocrit. This phenomenon is called Fahr us e ect [14]. The discharge hematocrit level has a signi cant impact on the parameters of blood ow in microvessels [14{16].
As noted above, Poiseuille's law does not apply to microvessels. However, to
estimate the resistance to blood ow in microvessels, it is possible, on the base
of (
D4
p
Q
:
(
By calculating the blood ow Q for a given pressure drop p using the nite
element method, we can nd the apparent viscosity by formula (
Note that the microcirculatory hemodynamic parameters (e.g., on the apparent viscosity and ow resistance) are signi cantly in uenced by the endothelial surface layer (ESL). The e ect of ESL on blood circulation, as well as its characteristics are discussed in [12{14, 16]. Following [13], the term \Endothelial surface layer" (ESL) is used for a boundary layer in which the plasma motion is signi cantly retarded. In particular, the ESL includes the glycocalyx layer. A. Copley studied the endothelium-plasma interface and developed a concept in which an immobile layer of plasma at the vessel wall is present [4, 5]. In the present paper, the in uence of ESL on the apparent viscosity is investigated by means of the boundary layer with zero velocity.
Construction of a mathematical model of blood ow in microvessels accounting for the in uence of ESL allows us to determine adequate vessels resistances in the brain capillary network. It is important to calculate the cerebral pressure distribution, for example, using the algorithm proposed in [2]. In particular, it can be applied to nd most dangerous pressure gradients to estimate the risk of bleeding in the germinal matrix of preterm infants. 2
Experimental Observations of the Apparent Viscosity The dependence of the relative apparent viscosity rel (the ratio of apparent viscosity to plasma viscosity) on the discharge hematocrit and vessel diameter both in vitro and in vivo is provided by T. Secomb and A. Pries [15]. The in vitro data corresponding to blood ow in a glass tube are represented by the following equation: where and rel = 1 + ( 0:45 1) (1 (1
HD)C 0:45)C In these equations, D denotes the diameter of the vessel (glass tube) in m, and HD is the discharge hematocrit. The parameter 0:45 is the viscosity for HD = 0:45 which is a typical hematocrit for humans.
The plots of the relative viscosity obtained by the formula (
Similar measurements in vivo are di cult due to technicalities of measuring the pressure drop in capillaries. Therefore, it was assumed that the reduction of the viscosity with increasing diameter of living vessels was similar to that in glass tubes. However, the estimates of the apparent viscosity obtained by Lipowsky et al. [7, 8] for blood ow in microvessels were much higher than expected from the in vitro data. An alternative parametric description that is consistent with the observed behavior in vivo was found by Pries et al. [11]: rel = 1 + ( 0:45 1) (1 (1
HD)C
0:45)C
and C remains the same as in equation (
The plots of the relative viscosity obtained by the formula (
The di erence between the in vivo and in vitro data can be explained by the
presence of the endothelial surface layer on the inner surface of blood vessels,
which has a signi cant e ect on the blood ow.
(
4
3
3.5
Fig. 2. Dependence in vivo of the relative viscosity ( rel) on the vessel diameter (D) for
di erent values of the discharge hematocrit: HD = 0:1 (solid line), HD = 0:3 (dashed
line), and HD = 0:5 (dot-dashed line). The data represented by (
As it was proposed in [2, 3], the RBCs and blood plasma are considered as one ow with two di erent viscosities (much larger viscosity for red blood cells): the viscosity of blood plasma is assumed to be 1 = 0:001 Pa s, whereas the viscosity of RBCs is set to be 2 = 0:1 Pa s to make RBCs e ectively rigid. Moreover, it is assumed that the ow is steady-state, without transition e ects. Therefore, the model is described by the steady state Stokes equation with space variable viscosity.
Assume that the ow is axisymmetric, that is all variables depend only on the radial and longitudinal coordinates, r and z. Let ur and uz be the radial and longitudinal ow velocities, respectively, and p the pressure. Therefore, it is possible to reduce the problem to two dimensions (see Fig. 3). Here, the radius r0 determines the boundary of the sequence of RBCs and hence rc r0 is the thickness of the plasma gap between the RBCs and the vessel wall.
Denote = (0; rc) (0; L). The model is mathematically formulated in [17] in a weak form, which allows us to use spatially discontinuous viscosity functions. With x1 = r, x2 = z, u1 = ur, u2 = uz, u = (u1; u2)T , p(r; 0) = p0, p(r; L) = 0, the weak formulation reads in cylindrical coordinates as follows: Z
2 x1 2 (x1; x2) X Dij (u)Dij (v) + u1v1 dx
Z i;j=1 x1 p div(v)dx =
x1 p0 v2 dx; Z 0 x2 1 "
Z where
Z x1pq dx x1 div(u) q dx = 0; " = 10 6; uj 2 = 0; vj 2 = 0; Dij (u) =
Functions v = (v1; v2)T , and q are the test ones. The viscosity distribution (x1; x2) is equal to 0.001 Pa s in the plasma part and 0.1 Pa s in the RBCs part.
Thus, the RBCs are modeled as uid with the high viscosity to make them
e ectively rigid. The model (
Z
2 x1 2 (x1; x2) X Dij (u)Dij (v) + ux1v21 + rpv dx = 0;
1 "
Z x1 pq
div(u)q dx = 0;
pj 0 = p0; pj 1 = 0; uj 2 = 0; vj 2 = 0;
rp =
:
(
To set the value of the pressure drop at a capillary with the length L, rst, we estimate the pressure drop in the capillary network (pressure di erence between inlets and outlets). For the estimation, the cerebral ow rate and total resistance of the capillary network are required. According to [18], the cerebral blood ow rate Q = 600 ml/min is a realistic value for an adult brain. Moreover, a cerebrovascular network model from [10] yields the total resistance RT of the capillary system to be equal to 0:1Pa s=mm3, which gives the pressure drop to be equal to 1000 Pa (computed as Q RT ). Following [10], where the parallel topology for capillaries with the length of 600 m and radius of 2.8 m is utilized, we consider in further modeling that the pressure drop of 1000 Pa corresponds to the capillary length of 600 m. Therefore, for the capillary length equals to L, the pressure drop is 5L=3 Pa. In the numerical modeling, the capillary length is chosen in the range from 50 to 150 m in accordance with the level of the hematocrit.
When conducting computer simulations, along with specifying the radius of
the vessel, it is required to determine the radius r0 of the core zone lled with
RBCs and the linear dimensions of RBCs. To specify r0, we use the following
approximation [2]:
r0 = 0:3 m + 0:8rc:
(
Comparison of the results of the nite element modeling (conducted by using FreeFEM++ package [6]) with in vitro data shows a signi cant di erence (see Fig. 4). This is because the velocity no-slip condition, uj 2 = 0, is not suitable for modeling the blood ow in glass tubes. More adequate is using the velocity slip condition: u1j 2 = 0 and @u2=@x1 + u2j 2 = 0. By appropriate choosing the parameter in numerical modeling, we can provide a closer approximation of the in vitro data.
4 3 3.5 4.5 5 5.5 6 6.5 Vessel diameter, μm 7 7.5 8
To take into account the in uence of ESL in modeling the blood ow in microvessels, we assume zero ow velocity in some neighborhood of the vessel wall, that is u = 0 for r < r rc (r > r0). The presence of the sublayer of ESL with zero ow velocity is mentioned in particular in [14], where the longitudinal velocity pro le measured in a rat venule is presented. Note that a similar e ect regarding zero velocity in a neighborhood of the surface somewhat akin to ESL is also observed [1] in modeling of elasto-optical biosensors.
To t the results of nite elements modeling to experimental data described
by (
The computed values of the relative viscosity and in vivo data calculated
with (
It is worth to note that taking into account the in uence of ESL through the layer with zero ow velocity is characterized by a signi cant increase of the relative viscosity and, respectively, by a proportional increase of the microvessel resistance. As a consequence, it leads to a proportional decrease in the longitudinal velocity. The results of a numerical experiment for the vessel diameter of 6 m and HD = 0:3 are shown in Fig. 6.
22 20 18 16 5.5 6 6.5 Vessel diameter, μm 7 7.5 8
In the example considered, accounting for the ESL leads to the 3.9-fold increase of the viscosity and, respectively, to the 3.9-fold decrease of the blood ow. 4
The mathematical model of blood ow in capillaries containing the endothelial surface layer was proposed. The ESL in uence is described by the presence of the boundary layer with zero ow velocity. The reliability of the results obtained has been veri ed for di erent values of the discharge hematocrit and vessel diameter using experimental data from the literature.
Further e orts of the authors will be aimed at applying this approach to calculate the characteristics of the cerebral capillary network with the subsequent calculation of the blood ow and pressure drop distributions in the germinal matrix of preterm infants.