y la irrupción de las TIC Desde el estallido social del 18 de octubre de

The endothelium is usually only characterised in multi-layer models, where the different strata of the artery are modelled separately. An example of endothelium modelling in a simple model such the free-wall model is presented by Wada et al.[66]. In this free-wall model, the transport of macromolecules in the lumen is considered in both the axial and the radial direction, with the arterial endothelium as a boundary condition. At the blood-endothelium boundary, the steady state amount of LDL passing through the endothelium is modelled as the difference between the LDL that is carried to the wall and the LDL that diffuses back to the mainstream

where D is the LDL diffusivity and the total flux of LDL to the endothelium

can be expressed as:

Kcw = nνwcw+ kcw (1.9)

with cw as the wall macromolecule concentration, n as the fraction of LDL carried through the vessel wall by filtration flow, ν_w as the filtration velocity of water through the arterial wall, and K as the endothelial uptake of LDL through both vesicular transport and receptor-mediated uptake (endocytosis).

A more complete description of the endothelial interface between the arte-rial wall and lumen was proposed using fluid-wall models and multi-layer mod-els[84, 88]. The endothelium is modelled as a porous membrane where the macro-molecule (solute) penetrates together with the macromacro-molecule carrier (solvent).

The transport of a solute and solvent through a membrane can be modelled with the Kedem-Katchalsky equations[60, 69, 83, 89]:

J_v = L_p(∆p_end− σend∆Π) (1.10)

J_s = P_end∆c + (1− σf)J_vc_end (1.11) Where J_v is the volume flux (solvent flux) and J_s is the solute flux through the membrane (the endothelium), L_p and P_end are the hydraulic conductivity and permeability of the endothelium, ∆p is the pressure difference and ∆c is the solute concentration difference across the endothelium, c_end is the mean solute concentration inside the endothelium, and ∆Π is the osmotic pressure with σ_end as the osmotic reflection coefficient. The net filtration pressure in Equation (1.10) is given by the term ∆p_endσ_end∆Π, and it stands for the driving force of volume flux penetration. In vitro findings show that the LDL concentration in the arterial wall increases under elevated transmural pressure[90, 91]. In Equation (1.11) the term (1−σf) represents the frictional reflection coefficient (or Staverman filtration coefficient); i.e. the extent to which the particles of LDL (the solute) are reflected by the endothelial membrane. The osmotic effect ∆Π is usually neglected when calculating the solvent flux as it is far below the hydraulic pressure gradient

and blood pressure on the distribution of albumin in the vascular wall, modelling the endothelium with the help of Kedem-Katchalsky equations. Their model showed that the main resistance to the macromolecule flux to the arterial wall is represented by the endothelium.

The properties of the endothelium have been shown in vitro to be linked to the local haemodynamics[90, 94, 95], as discussed in section 2.1.3. Sun et al.[88]

developed an initial approach to model haemodynamic effects on the endothe-lium transport properties using a shear-dependent hydraulic conductivity (L_p) to calculate the volume flux through it (Equation (1.11)).

Factors that affect the permeability of endothelial cells are still not fully un-derstood and they constitute an active field of research[41, 95]. Findings from Levesque et al. [32] show that in areas of flow recirculation the endothelial cells have a higher permeability to macromolecules. Some hypotheses attribute the change in endothelial permeability to the relaxation of the endothelial tissue leading to an increase in the endothelial clefts gap size[96]. Other experiments show that in areas of decreased blood flow, endothelial mitosis is stimulated and accelerated. These experiments suggest a new hypothesis that considers mitotic cells as “leaky cells” (meaning that the clefts surrounding these cells would no longer be the normal endothelial tight clefts) (Figure 1.9)[97, 98].

In order to account for this heterogeneous behaviour of the endothelium the three pores model[99] was created. The three pores model considers three path-ways through which LDL can penetrate inside the endothelium (Figure 1.9):

Normal cell pathway Endothelial cells take up and metabolise plasma LDL macromolecules (endocytosis), no passage of LDL is allowed through in-tercellular clefts as the tight junctions block the passage of solutes with a radius greater than 2 nm (the average LDL radius is 11 nm).[83]

Vesicular pathway LDL macromolecules travel through the endothelial layer via vesicles (transcytosis) and are discharged in the arterial wall.

Leaky cell pathway Mitotic endothelial cells present leaky junction (the in-tercellular protein strands forming the tight junctions are broken down), these are 40 nm wide clefts that allow macromolecules such as LDL to pass through them.

Olgac et al.[83] linked the quantity of mitotic cells to local WSS values, with an inverse relationship between WSS and number of leaky cells sites. Modified versions of the Kedem-Katchalsky equations (Equations (2.8) and (2.9)) were used to include the three pores pathways [99] into their transport model of the endothelium.

Figure 1.9: Diagram representing endothelium as described by the three pores model. Perpendicular cross section view showing two normal endothelial cells and a leaky endothelial cell. The three macromolecule transport pathways: tight junctions, vesicles and leaky junctions are shown.

Further extensions to this model included the heterogeneous endothelial trans-port properties and the glycocalyx layer[84]. Another approach to the Kedem-Katchalsky equations is to model the solvent flux through the endothelial clefts and the leaky junctions of the endothelium using the steady state Navier-stokes and continuity equations (Equations (1.5) and (1.6)) with the velocity u as the solvent velocity through the leaky clefts or junctions:

−∇plj+ µ∇²u_lj = ρ(u_lj· ∇)ulj (1.12) where p_lj is the pressure in the corresponding leaky cleft or junction, and the material density and viscosity ρ and µ are taken for blood plasma.

The flux of solute (LDL) through the endothelial leaky junctions was coupled to the solvent flux with a convection-diffusion equation[87, 100] in a steady state environment:

∇ · (−Dlj∇clj) + u_lj· ∇clj = 0 (1.13)

diffusion coefficient of LDL through the leaky junction.

To account for the semi-permeable endothelial coating represented by the glycocalyx, Brinkman’s equation for a porous medium was used:

µ

K_plu_l =−∇pl+ µ

ϵ_l∇²u_l (1.14)

∇ · ul = 0 (1.15)

where ϵ_land K_plare the porosity and hydraulic permeability of the glycocalyx, p_l is the pressure and u_l is the superficial velocity vector of the glycocalyx.

Brinkman’s equation (Equation (1.14)) was used to link the pressure drop with the velocity across the porous media[91]:

ul =−k

µ_l∇pl (1.16)

Brinkman’s equation (Equation (1.14)) allows a heterogeneous material with more than one pore size to be modelled, by adding a viscous term to Darcys Law (Equation (1.16)) (in order to account for the presence of a solid boundary, the glycocalyx).

Yang et al. [100] modelled transendothelial macromolecule transport using an extended version of the Brinkman’s equation (accounting also for the temperature of the membrane) coupled with a convection-diffusion equation (Equation (1.13)) in a volume-averaged stationary form[91, 100]. The convection-diffusion equa-tion was here improved by including the Staverman filtraequa-tion and the osmotic reflection coefficient, to model the membrane pores behaviour in relation to the macromolecule penetrating them.