The Lattice Boltzmann Method as a General Framework for Blood Flow Modelling and Simulations - Bioanalysis Advanced Materials Methods and Devices - page 158

of equal, repeated EC's. The pressure-driven axi-symmetric flow of a continuum
fluid over such a surface has been recently modelled by Pontrelli et al. [ 34 ]. It was
shown that, despite no great change in velocity profiles, there can occur significant
ESS variations between the ECs wall peaks and throats, especially in small-sized
arteries. Differently than in Sect. 10.3 , the mesoscopic particulate nature of the
blood is now addressed in the context of a bicomponent fluid model: RBC are here
deformable, neutrally buoyant liquid drops constrained by a uniform interfacial
tension and suspended in the plasma.
In addition, the endothelial surface is not only wavy in its geometry, but, at
a smaller scale, it is covered by fibrous filaments and long protein chains forming
a thin layer called the endothelial surface layer (ESL) or glycocalyx [ 44 ]. From a fluid-
dynamics point of view, the ESL can be modelled as a porous layer of constant
thickness which suits the wall undulation, through which the flow of the continuous
phase (plasma) is possible. This would alter the boundary condition of the problem,
specifically the classical no-slip condition at the vessel wall may have to be replaced to
allow for plasma penetration through the ESL. The LBmethod readily accommodates
a model of the glycocalyx itself, as it is particularly well suited to address what would
now become a multiscale model. Conceptually, the idea is to solve a two-domain
problem, whereby the bulk flow (in the lumen) is governed by the multicomponent
Navier-Stokes equations and the near-wall region by a porous-medium Brinkman
flow formulation (see below). At the mesoscale, the glycocalyx is not modelled in a
detailed form, but its effect on the flow is still properly addressed, using methods
which are amenable to coupling other, more detailed, simulations with experiments.
We develop here a two-way coupled model where the drop interface is forced by
compression of the ESL, and the effect of perturbed or compressed glycocalyx is then
communicated to the flow [33]. We assume here that the filaments are strongly
anchored in the endothelium, where they are most resistant to deformation and that
they deform preferably at their tip, that is toward the vessel lumen.
The mesoscale LB method is still used to solve the governing hydrodynamic
equations, that involves multicomponent fluid flow, off-lattice, or sub-grid, bound-
ary surfaces and a porous-layer representative of the ESL. The governing hydrody-
namic equations for flow in a porous media, with constant or variable porosity e , are
an extension of ( 10.3 )asin[ 15 ]:

r u ¼ 0 ;

@ u

@

u
e ¼

1
r rðeP

2 u

t þð

u rÞ Þþnr þ F :

(10.8)

Here, F is the total body force due to the presence of both the porous material (drag)
and other external forces:

e K u eF e

K

p

F ¼ u j u jþeH ; (10.9)