Once inside the arterial wall, the LDL and monocytes were characterised by means of their transport and chemical interactions (Figure 2.10). A monolayer approach was used, considering the arterial intima and media with IEL and EEL as forming one single layer. The adventitial layer was considered as the limiting boundary of the monolayer and neglected from the model, as it does not play an active role in the initial stage of atherosclerosis formation.
Figure 2.10: Details of Figure 2.1 showing the model specifications for the arterial wall.
2.1.3.1 LDL Transport
The transport of LDL inside the arterial intima-media layer has been modelled in the direction normal to the arterial lumen via a convection-diffusion-reaction equation[83]:
dcw
dt =−uw· ∇cw+ Dw∆cw− rwcw (2.24) The transmural velocity uw is calculated using Darcys law:
uw= k
µp∇p (2.25)
where µp is the viscosity of plasma. The transport properties of the arterial wall with respect to LDL are described by the arterial wall Darcys permeabil-ity k and the diffusion coefficient Dw. The last term of Equation (2.24) is the degradation of the LDL particles, with rw as the reaction coefficient.
The LDL fluxes (Equation (2.6) and Equation (2.7)) were imposed normal to the endothelium and they provided the sub-endothelial layer boundary value for cw,end (Equation (2.24)); a LDL value of ccw,adv
w,end = 0.005 as from experimental findings from Meyer et al.[90] was used at the boundary corresponding to the adventitia (cw,adv) (Figure 2.11). The portion of reacting LDL (rwcw) is the quantity prone to degradation and it represents the source term for the LDL oxidation model which will be described in the next section.
Figure 2.11: Macromolecule transport through the artery wall and endothelium;
where c is the macromolecule concentration, Γend and Γadv are the boundary conditions at the endothelium and adventitia layer respectively.
2.1.3.2 LDL Oxidation
LDL oxidation is believed to be one of the triggering factors of atherosclerosis formation. As described in section 1.1.2.1 (LDL and its role in atherosclerosis), once inside the arterial wall the LDL will be exposed to oxidation by free radi-cals[22]. Its antioxidant defence, represented by the vitamin E contained in its outermost layer, will be depleted during the interaction between LDL and free radicals, leaving the LDL lipid core prone to oxidation.
On average six molecules of vitamin E are associated with an LDL molecule[11];
following the model developed by Cobbold et al.[22] each interaction with a free radical would deplete one of these vitamin E (Figure 1.5).
LDLn+ R−−−→ LDLk n−1 (2.26) where Ln represents the LDL particle with n vitamin E, R is the free radical and ke is the kinetic constant for vitamin E oxidation. When the vitamin E is totally depleted, LDL will become prone to oxidation.
LDL0+ R −−−→ LDLk0 ox (2.27)
L0 is the vitamin E depleted LDL, Lox is the oxidised LDL and k0 is the LDL oxidation kinetic constant. The kinetic constant for oxidation of LDL k0 is of a much lower rate than ke, as the cholesterol core of LDL has a lower reactivity respect to the vitamin E[21].
As free radicals were assumed to be always available, their concentration vari-ation in time was neglected and a fixed value was adopted. The chemical reactions of the oxidation cascade are considered to follow a first order reaction kinetics, the concentration of the reaction product depends solely on the concentration of the reactant.
Following the law of mass action, the concentration in time of the different LDL species in the oxidation cascade was:
dL5 The native LDL particle with 6 Vitamin E (L6) is represented here by the quantity of LDL prone to degradation (rwcw) coming from Equation (2.24).R represents the free radicals, modelled here as a fixed quantity of 1 µM. The last equation of the chemical cascade (Equation (2.34)) models the concentration of
of oxidised LDL by interaction with plasma macrophages (Mw).
Oxidised LDL is noxious to the body and its presence triggers an immune re-sponse[12]. Cytokines are secreted by the endothelial cells, leading to recruitment and migration of plasma monocytes into the arterial wall.
2.1.3.3 Monocytes Transport
Monocyte migration into the arterial wall is the body’s immune response to ox-idised LDL (Section 1.1.2.1 – From Monocytes to Foam Cells – The immune response to oxidised LDL). Once inside the arterial wall the assumption is made that all monocytes will differentiate into macrophages, neglecting the time needed for their differentiation. Their transport has been imposed in the normal direction to the arterial lumen and modelled with the following diffusion-reaction equation (Figure 2.10):
dMw
dt = Dm∆Mw− kmLoxMw (2.35) As a consequence of the activation of the immune system, monocytes were free to travel through the endothelial layer. Following the same principle, macrophages are also relatively free to travel inside the arterial wall, and show a high diffusion coefficient (Dm) (Table 2.1). The reaction term represents macrophages becoming foam cells after taking up oxLDL, with the kinetic constant km (Table 2.1) show-ing the rate of this reaction. The transport of monocytes-macrophages does not present an advection term. This term should represent the chemical force given by chemotaxis[102], and is in this case ignored; with the chemotaxis action rep-resented merely by the activation of cytokines regulating the influx of monocytes in the arterial wall. The sub-endothelial interface boundary condition:
Mw,end = K5Lox (2.36)
shows a monocytes source proportional to the quantity of oxLDL (Lox) by a factor K5 (Table 2.1), representing the cytokines secreted by the endothelial cells in response to the oxLDL produced by the chemical cascade (Equations (2.28) to (2.34)). At the media-adventitia interface a constant concentration value as for the LDL model was used, MMw,adv
w,end = 0.005 (Figure 2.11).
The product of the interaction between oxLDL and macrophages is the for-mation of foam cells.
2.1.3.4 Foam cell formation and accumulations
Once a macrophage is filled up with oxLDL, it will become a lipid laden macrophage, also known as a foam cell. The interaction between oxLDL and macrophages has been modelled as a first order kinetic reaction:
LDLox+ Mw −−−−→ Fkm w (2.37) giving the following change in foam cells concentration with time:
dFw
dt = kmLoxMw (2.38)
Fw is the quantity of foam cells in the arterial wall and km is the kinetic constant for foam cells formation. Foam cells do not have any transport term in their equation as they are not “supposed” to move inside the arterial wall. Once created, they are unable to migrate out of the arterial wall and constrained in term of space, where they eventually form a stratified aggregation (Figure 2.12).
Foam cell stratification is the basis of the fatty streak formation.
Foam cell
Figure 2.12: Electron micrograph of sub-endothelial fatty streak with two layers of foam cells (image bar = 1 µm). Lipid droplets can be visualised inside the foam cells. The figure on the right shows the close hexagonal packing technique used to approximate the foam cells stratification in the modelled arterial wall.
Electron micrograph image from Faggiotto et al.[6].
To resemble the fatty streak structure, shown in Figure 2.12, foam cells were assumed to have a constant volume and spherical shape. Their quite regular pattern of stratification was modelled as following close hexagonal packing (Fig-ure 2.12). The volume occupied by the foam cells would be:
VF = Fw,n· Fw,V
0.7450 (2.39)
with Fw,n as the number of foam cells given by:
Fw,n = Fw· NA (2.40)
where Fw is the foam cell concentration, NAis Avogadro’s number, and Fw,V is the volume of a spherical foam cell with a radius of 20 µm[6]. VF is the volume that the foam cells stratification formed in the model would occupy. At the early stage of formation this stratification of foam cells would be easily accommodated in the arterial wall portion considered, which has a volume (V ) of:
V = Atot· IMT · s0 (2.41)
with IM T being the intima-media thickness and Atot the considered portion
of endothelium. The space available for the accommodation of new material without significant structural modification in the arterial wall is represented by the scaling factor s0. When the volume represented by the accumulated foam cells is larger than V , the portion of arterial wall considered will start swelling:
∆h = VF − V Atot
(2.42) with ∆h as the intima-media thickness (IMT) growth in the normal direction, leading to the initial stage of atherosclerotic plaque formation, the fatty streak.