Unsteady LDL Transport through Patient-Specific Multi-Layer Left Coronary Artery

Aims: The objective of the study is to study the transport and distribution of Low Density Lipoprotein (LDL) within patient-specific multi-layer arterial wall model under unsteady flow using computational fluid dynamic analysis.

Methods: A Left Coronary Artery (LCA) patient-specific model was incorporated. Both flow-mass transport equations in lumen as well as flow-mass transport equations within the patient-specific multi-layer arterial wall are numerically analyzed.

Results: The lumen-side LDL concentration preferably occurs at the concave geometry parts denoting concentration polarization. The Average Wall Shear Stress (AWSS) is not the only factor that can determine the lumen-side concentration of LDL. Increased time-averaged luminal concentration develops mainly in the proximal rather than to distal segment flow parts. The LDL concentration at the endothelium/intima interface is substantially lower, almost 90 times, than its value at lumen/endothelium interface. The concentration drop across the intima layer is negligible, whereas the concentration reduction across the Internal Elastic Layer (IEL) is remarkable. LDL concentration values at the IEL/media interface are one order of magnitude smaller to ones occurring at the intima layer.

Conclusion: The transportation of LDL through the multi-layer arterial wall is affected by the flow pattern itself, the arterial wall thickness and the physical values of the layers.

C: concentration (gr/l);$$\stackrel{-}{\complement }$$ : average concentration gr/l; D: diffusivity (m2/s); D’ : diffusivity per unit length (m/s); Δp: pressure drop (N/m2); J: flux (m/s); K: permeability (m2); K’: solute lag; L: hydraulic conductivity (m2 s/gr); L”: thickness (m); R: radial location from the centerline (m); p: pressure (N/m2); r: constant consumption rate; u: velocity (m/s)

l: lumen; lag: coefficient; o: inlet; p: plasma; s: solute; v: volume; eff: effective property; w: wall; in: instantaneous.

end: endothelium; in: intima; iel: internal elastic layer; m: media.

μ: molecular viscosity (Ρa s); σd: osmotic reflection coefficient of the endothelium; σf: solvent reflection coefficient; ρ: density (Kg/m3); ε: porosit

Acquisition of anatomy data

The LCA patient specific model was derived by the IVUS Angio tool which has been previously validated [22]. The basis for generating 3D geometry is the creation of numerous closed 3D curves using Rhinoceros 4.0 (Figure 1A & 1B). The distance between two successive curves is 0.5 mm. The length of the acquired patient-specific segment is 6.2 cm with lumen average diameter of 3.0 mm and average wall thickness of 0.24 mm. The acquired surfaces: endothelium, Internal Elastic Layer (IEL) and adventitia as well as the acquired layers: lumen, intima and media were reconstructed using ANSYS Design Modeler software.

Model construction

The reconstruction of each layer constitutes a difficult task. The acquired LCA model has variable wall thickness. Specifically, the intima layer is comprised of constant wall thickness with 0.012 mm in order to avoid intimal hyperplasia while the media layer has variable wall thickness. The endothelium and IEL were assumed to be semi-permeable biological membranes due to the extremely low wall thickness. The constructed model was imported into appropriate software and the boundary zones were accordingly specified. A mesh sensitive study was performed to solve the fundamental flow-mass transport equations in the lumen. The mesh independence criterion was mainly based in the WSS convergence criterion satisfaction (Figure 2A & 2B). The final grid size was comprised of 586204 nodes, 491164 cells and 1567980 faces for lumen, 753732 nodes, 501984 cells and 1757448 faces for intima layer and 237742 nodes, 1061714 cells and 2245134 faces for media layer.

Flow and mass transport equations

The applied numerical code solves the governing unsteady 3D Navier-Stokes equations for lumen flow and the mass transport equation. The LCA is treated as being of non-elastic, stationary and rigid material. For calculation simplicity, an assumption has been made about the transport properties of the endothelium. This assumption states that solute endothelial permeability coefficient is not shear flow dependent. The intima and the patient-specific media layer are considered to be porous media.

Flow equations

Lumen flow: In their generality the mass flow equation for the lumen is,

$$\frac{\partial \rho }{\partial t}+\nabla •\left(\rho {\overrightarrow{u}}_l\right)=0\text{ (1)}$$

ρ (kg/m3) is the blood density; t (s) the time and $${\overrightarrow{\text{u}}}_{\text{l}}$$ (m/s) is the lumen velocity vector. The conservation of momentum is,

$$\frac{\partial }{\partial \text{}t}\left(\rho {\overrightarrow{u}}_l\right)+\nabla •\left(\rho {\overrightarrow{u}}_l{\overrightarrow{u}}_l\right)+\nabla p_l=\nabla •(\overline{\tau })+\rho \overrightarrow{g}\text{ (2)}$$

pl (N/m2 or Pa) is the blood static pressure within lumen; ¯τ (N/m2) the shear stress tensor and ρ $$\overrightarrow{g}$$ (N/m3) the gravitational body force. The shear stress tensor ¯τ is,