The Lattice Boltzmann Method as a General Framework for Blood Flow Modelling and Simulations - Bioanalysis Advanced Materials Methods and Devices - page 160

e ( x , y ) that tends to 1 in the lumen region and gradually reduces, as we enter the

glycocalyx region, where it approaches a minimum value, e G . This porosity

transition is modelled through the increasing smooth function:

1 e G

2

eð x Þ¼e G þ ½ ðxð s l ÞÞ ;

1 tanh (10.12)

where l is themeanESL thickness and the parameter 1 determines the distributionof

(i.e., the effective standard deviation of) protein chain lengths, while s ( x )denotes
distancemeasured normally to the endothelial boundary. Note that

x

e G e ( x ) 1and

that, for H [see ( 10.9 )], and ( 10.8 )-( 10.9 ) reduce to the multi-

component LB Navier-Stokes equations for free multicomponent fluid flows, and the
described procedure reduces to the standard LB method for two-component, incom-
pressible fluid. On the other hand, an additional, fictitious, repulsive body force density
acts on the drop interface which enters the ESL region, impinging on the lumen. This
force distribution is so designed that its accumulation produces an effective Hookean
force acting at the center of the local volume. Specifically, the erythrocyte is subjected
to a surface force distribution, effective in the ESL only, which is directed everywhere
in the drop-surface normal direction. This force device effectively models the
glycocalyx as a continuum of elastic springs, with modulus E, gradually decaying
from a maximum value, and E G (in the ESL) to 0 (towards the bulk):

e ! 1, we have F !

E G
2

E ð x Þ¼ ½ 1 tanh ðxð s l ÞÞ ; (10.13)

where all notations are given in correspondence to ( 10.12 ). It is important to note
that the above force acts solely on the red fluid (drop) and not upon the plasma.
Hence, the relative density of the material which comprises the drop may be
modelled by appropriate choice of the spring constant E G in the above equation.
A number of simulations have been carried out in the case of an axi-symmetric
channel having the same corrugation repeated along the length. Its size (of order of
m m) is slightly larger than a single RBC flowing through it, driven by a constant

pressure gradient with periodic conditions. At such fine scale, for accuracy
purposes, the off-lattice non-slip endothelial surface uses continuous bounce-back
conditions [ 4 ]. The ESL structure has been modelled as a porous layer of constant
thickness over the undulated wall. As one may expect, the average velocity of the
drop is slower in the presence of the glycocalyx, which constitutes a hindrance for
the lumen flow. Also, the mean deformation of the drop is more pronounced with
the glycocalyx force (Fig. 10.4 ). Hence, when the drop is in the ESL influence
region, it is subjected to the elastic force which squeezes and lifts it away from the
boundary while making its shape more elongated. Considering the action of the
glycocalyx as a sensor of mechanical forces, it is worth computing the shear stress
at the ESL/lumen boundary. Our results evidence, in the latter case with ESL, a
reduction of the shearing stress either at the wall (due to the plasma only) and at the
ESL top (due to the particulate fluid) [ 33 ]. It is seen that ESL is more likely to
protect the endothelial cells from ESS fluctuations associated with particle transits.