3.3.1. LDL

Once the LDL molecules cross the endothelial barrier, they can suffer both convection and diffusion, because of their small sizes. When an LDL molecule enters the arterial wall, it is oxidised. Therefore, the reactive term of LDL molecules represents their oxidation.

$$\frac{\partial \mathbb{C}\_{LDL,w}}{\partial t} + \nabla \cdot (-D\_{\mathbb{C}\_{LDL,w}} \nabla \mathbb{C}\_{LDL,w}) + K\_{\text{lag}} \cdot u\_w \cdot \nabla \mathbb{C}\_{LDL,w} = -d\_{LDL} \mathbb{C}\_{LDL,w} \tag{22}$$

*dLDL* is the degradation ratio of LDL in the arterial wall and *CLDL*,*<sup>w</sup>* is its concentration at each time.

It is necessary to define as a boundary condition the LDL concentration in the adventitia ( *CLDL*,*adv*) [49]. The Kedem–Katchalsky equation is used to determine the LDL flow across the endothelium [11]:

$$J\_{S,LDL} = \mathbb{C}\_{LDL,l} \cdot LDL\_{dep} \cdot P\_{app} \tag{23}$$

where *CLDL*,*<sup>l</sup>* is the LDL concentration in blood and *LDLdep* is the amount of LDL molecules that are deposited from the lumen to the arterial wall. Finally, *Papp* is the apparent permeability of the arterial wall. LDL can flow from the lumen to the arterial wall through normal and leaky junctions and vesicular pathways, so the apparent permeability of the arterial wall is composed of three types of permeabilities [50,51]:

$$P\_{app} = P\_{app,lj} + P\_{\mu\eta\eta\tilde{p}\zeta\eta\tilde{i}}{}^{\prime} + P\_{app,\nu\eta\nu} \tag{24}$$

 with *Papp*,*nj*, *Papp*,*lj*, and *Papp*,*vp* being the permeabilities of normal and leaky junctions and vesicular pathways, respectively.

The transport of molecules through the endothelium is dependent on the size of the particles. For the case of LDL, which has a radius of 11 nm [50]), transport across normal junctions is not possible due to their small size. Therefore, LDL transport through the endothelium can only occur through leaky junctions and vesicular pathways [52].

In addition, LDL transport through vesicular pathways is 10% of the flux through leaky junctions [11]:

$$P\_{app,vp} = 0.1 \cdot P\_{app,lj} \tag{25}$$

The apparent permeability of leaky junctions can be defined as:

$$P\_{app,l\dot{l}} = P\_{l\dot{j}} Z\_{l\dot{l}} + f\_{v,l\dot{j}} \cdot (1 - \sigma\_{f,l\dot{j}}) \, \tag{26}$$

where *Plj*, *Zlj*, and *σf* ,*lj* are the diffusive permeabilities of the leaky junctions, factors of reduction of the LDL concentration gradient at the inlet of the flow and the solvent-drag coefficient of leaky junctions. So, the LDL flux across the endothelium can be written as:

$$J\_{S,LDL} = 1.1 \cdot C\_{LDL,l} \cdot LDL\_{dep} \cdot \left( P\_{l\bar{j}} Z\_{l\bar{j}} + I\_{v,l\bar{j}} (1 - \sigma\_{f,l\bar{j}}) \right) \tag{27}$$

The diffusive permeability of leaky junctions is defined as:

$$P\_{l\dot{j}} = \frac{A\_p}{S} \chi P\_{sl\dot{j}\prime} \tag{28}$$

where *χ* is the difference between the total area of endothelial cells and the area of cells separated by leaky junctions:

$$
\chi = 1 - \mathfrak{a}\_{lj},
\tag{29}
$$

with *<sup>α</sup>lj* being the ratio between the radius of an LDL molecule (*am*) and the half-width of a leaky junction (*wl*):

$$\alpha\_{I\bar{j}} = \frac{a\_m}{w\_I} \tag{30}$$

*Pslj* is the permeability of a single leaky junction, and can be determined by knowing the LDL diffusion coefficient in a leaky junction (*Dlj*) and the length of a leaky junction (*llj*):

$$P\_{slj} = \frac{D\_{lj}}{l\_{lj}}\tag{31}$$

LDL diffusion coefficient in a leaky junction is related to the LDL diffusion coefficient by [11]:

$$\frac{D\_{lj}}{D\_l} = F\left(a\_{l\dot{j}}\right) = 1 - 1.004a\_{l\dot{j}} + 0.418a\_{l\dot{j}}^3 - 0.16a\_{l\dot{j}}^5\tag{32}$$

On the other hand, *Zlj* depends on a modified Péclet number:

$$Z\_{l\circ} = \frac{P e\_{l\circ}}{e^{(P e\_{l\circ})} - 1} \tag{33}$$

This modified Péclet number can be defined as:

$$Pe\_{lj} = \frac{Iv\_{l}lj \cdot (1 - \sigma\_{f,lj})}{P\_{lj}}\tag{34}$$

Finally, the solvent-drag coefficient of leaky junctions is given by [11]:

$$\sigma\_{f,lj} = 1 - \frac{2}{3} a\_{lj}^2 (1 - a\_{lj}) \cdot F(a\_{lj}) - (1 - a\_{lj}) \left( \frac{2}{3} + \frac{2a\_{lj}}{3} - \frac{7a\_{lj}^2}{12} \right) \tag{35}$$

3.3.2. Oxidised LDL

Once LDL is oxidised, it is considered to not experience convection [4]. However, due to its similar size to LDL, oxidised LDL shows diffusion in the arterial wall. Once LDL enters the arterial wall, it becomes oxidised, so one of its reactive terms refers to this oxidation. On the other hand, oxidised LDL is phagocytosed by macrophages, which corresponds to its second reactive term.

$$\frac{\partial \mathbb{C}\_{\text{oxLDL,w}}}{\partial t} + \nabla \cdot (-D\_{\text{CoxLDL,w}} \nabla \mathbb{C}\_{\text{oxLDL,w}}) = d\_{\text{LDL}} \mathbb{C}\_{\text{LDL,w}} - LDL\_{\text{oxL}} \mathbb{C}\_{\text{oxLDL}\_w} \mathbb{C}\_{M,w} \tag{36}$$

where *CoxLDLw* and *CM*,*<sup>w</sup>* are the oxidised LDL and macrophage concentrations in the arterial wall. In addition, *LDLox*,*<sup>r</sup>* is a ratio of the quantity of oxidised LDL that a macrophage can ingest.
