Introduction

One of the main characteristics of early atherosclerosis is the accumulation of lipoproteins, such as low-density lipoprotein (LDL) within the subendothelial layer of the arterial wall.13,27 Therefore, understanding the transmural mass transport of LDL is important for gaining further insight into the development of atherosclerotic lesions.

Mass transport in arteries has been studied using both experimental and computational approaches. Many computational investigations have been concerned with the fluid phase momentum and mass transport,24–26,34–36 while some others included transport in the arterial wall.1,14,15,22,23,29–31,37 Also, since arterial wall permeability was found to be influenced by wall shear stress,4,19 a number of studies have employed shear-dependent transport properties.24–26,31 Perktold and coworkers25 investigated the effects of pulsatile flow on mass transport in an axisymmetric stenosis and found that when permeability is varied with WSS, steady flow simulations could not predict some of the features found in pulsatile flow simulations.9 However, their model omitted mass transport in the arterial wall and was only concerned about the trans-endothelial flux, hence could not provide a quantitative predication of wall concentration. There have been very few studies investigating the fluid-wall coupled mass transport under pulsatile flow conditions.30,37 Furthermore, these studies were either restricted to simple computational geometries such as a straight tube or neglected the shear-dependent trans-endothelial transport. Since atherosclerosis normally occurs in arterial branches and curved arteries, the investigation of shear-dependent fluid-wall coupled LDL transport under pulsatile flow conditions in a more complicated geometry is required.

Because the development of atherosclerosis is a long-term process, the steady flow assumption seems reasonable. However, LDL transport from the blood lumen to and through the arterial wall is coupled with pulsatile blood flow whose cycle periods are normally less than a second, indicating that the LDL transport process is influenced by two dramatically different time scales. Thus in this multi-time-scale system, the effects of blood flow pulsatility on long-term transport of LDL needs to be investigated and the validity of the assumption of steady flow needs to be examined. However, it is difficult to carry out transient simulation of flow-coupled LDL transport over a long time span. First of all, to model transient lipid transport while considering pulsatile blood flow, an extremely high temporal resolution has to be employed, which is not computationally feasible over a prolonged time period. Secondly, a transient numerical procedure requires dynamic coupling between concentration fields in the lumen and wall, further increasing the computational demand.

To circumvent these difficulties in transient simulations of LDL transmural transport, a lumen-free cyclic (LFC) and a lumen-free time-averaged (LFTA) computational procedures were developed in the present study and applied to an axisymmetric stenosis modeled within a arterial segment. Shear-dependent hydraulic conductivity of the endothelium which determines the transmural flow was incorporated to account for the effect of WSS on LDL transport.

Model Settings

Lumen-Free Procedures

In the present study, an LFC and an LFTA procedures are proposed to study the influence of transient WSS on LDL transport in the arterial wall. It is worth noting that although the proposed procedures are termed “lumen-free,” detailed hemodynamics in the fluid domain is included, but the highly convection-dominated LDL transport in the lumen is omitted and a constant blood LDL concentration is assumed in the arterial lumen. In fact, LDL transport in the arterial lumen generally has little effect on the concentration in the arterial wall. Theoretically, this is because the transport of lipid molecules, which normally have a large molecular weight, is limited by the arterial wall.2,3,5,33 Thus, unlike trans-endothelial transport of oxygen which is governed by transport efficiency in the fluid phase, passage of lipid molecules through the endothelium is determined by permeability of the membrane rather than the concentration on the lumen-side. It has also been shown numerically that the LDL concentration in the arterial wall is not sensitive to polarization of lumenal concentration when constant transport parameters are employed.22

As previously mentioned, pulsatile blood flow and LDL transport in the arterial wall are characterized by dramatically different time scales. The time scales of blood flow and mass transport of LDL are illustrated and compared in Fig. 1. It is clearly shown that the dynamics of bulk blood flow (U l ) and transmural flow (U w ) are periodic with a cycle period t p . However, LDL transport in the arterial wall (c w ) is a long-term process which finally reaches equilibrium many cardiac cycles after the perturbation is imposed. The perturbation in vivo includes, for instance, changes in local arterial geometry and changes in LDL concentration as a result of diet or lifestyle. Although concentration fluctuation is negligible within one cardiac cycle, the accumulated effect of hemodynamic factors over a longer time span could be significant. Thus a LFC procedure is proposed to investigate this accumulated effect and the influence of flow pulsatility.

Figure 1
figure 1

Time scales of the arterial mass transport system. U l denotes dynamics of bulk blood flow, U w denotes dynamics of transmural flow, and c w denotes dynamics of LDL transport in arterial wall. t p is the period of a cardiac cycle

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As shown in Fig. 2a, in the LFC procedure, the periodic bulk blood flow is simulated over five cardiac cycles using a small time step (\(\Updelta t\)) to determine instantaneous values of WSS. Based on these WSS values, shear-dependent transport properties, i.e., hydraulic conductivity of the endothelium in the present study, are calculated. Since LDL transport in the arterial wall is coupled to transmural flow, the calculated hydraulic conductivity is then used to simulate transmural flow over five cardiac cycles using the same time steps as in the pulsatile flow simulation. The resulting transmural flow field of the last cycle is then repeatedly applied to the simulation of LDL transport in the arterial wall. Thus, in the simulation of LDL transport over a long time span, the transmural flow field is used as a cyclic condition. An appropriately selected time step (\(\Updelta t'=5\Updelta t\)) is used to preserve the characteristics of transmural flow within each cardiac cycle and save computational time.

Figure 2
figure 2

Schematic views of the proposed LFC (a) and LFTA (b) procedures. U l represents the dynamics of the bulk blood flow, U w the dynamics of the transmural flow, and c w the dynamics of LDL transport in the arterial wall. L p is the shear-dependent hydraulic conductivity of the endothelium. \(\Updelta t\) is the time-step for pulsatile flow simulations. \(\Updelta t'=5\Updelta t\) is the time-step for transient LDL transport simulations. In the present study, LDL permeability is assumed to be a constant

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In addition to the LFC procedure, a LFTA procedure was also used to simulate the wall-side transport of LDL under a steady-state framework with time-averaged hydraulic conductivity calculated from pulsatile blood flow simulation using instantaneous WSS. As shown in Fig. 2b, periodic bulk blood flow is simulated over five cardiac cycles to determine the instantaneous WSS. On this basis, time-averaged shear-dependent model parameters are obtained and used in transmural momentum and LDL transport simulations.

Governing Equations

The mathematical model accounts for fluid dynamics in the arterial lumen and wall as well as LDL transport in the wall. The bulk blood flow is modeled by the Navier–Stokes equations, whereas the transmural flow that drives LDL convection in the arterial wall is modeled by Darcy’s Law. The convection-diffusion-reaction equation is employed for mass balance of LDL in the arterial wall.

Fluid Dynamics

Blood flow is assumed to be incompressible, laminar, Newtonian and hence described by the Navier–Stokes equations

$$ \rho \partial_t{\mathbf{u}}_l- \mu \nabla^2{\mathbf{u}}_l+\rho({\mathbf{u}}_l \cdot \nabla){\mathbf{u}}_l+ \nabla p_l=0 $$
(1)
$$ \nabla{\mathbf{u}}_l=0 $$
(2)

in the fluid domain, where \({{\mathbf{u}}}_l\) is blood velocity in the lumen, p l is pressure, μ is dynamic viscosity of blood, and ρ is density of blood.

The transmural flow in the arterial wall is modeled by Darcy’s Law

$$ {\mathbf{u}}_w-\nabla\cdot{\frac{\kappa}{\mu_p}}p_w=0$$
(3)
$$ \nabla {\bf u}_w=0$$
(4)

in the wall domain, where \({{\mathbf{u}}}_w\) is velocity of the transmural flow, p w is pressure in the arterial wall, μ p is viscosity of the blood plasma, and κ is the Darcian permeability coefficient of the arterial wall.

Solute Dynamics

Mass transfer in the arterial wall is coupled with transmural flow and modeled by the convection-diffusion-reaction equation as follows

$$ \partial_tc_w+\nabla\cdot(-D_w\nabla c_w+Kc_w{\mathbf{u}}_w)=r_wc_w $$
(5)

in the wall domain, where c w is the solute concentration in the arterial wall, D w is the solute diffusivity in the arterial wall, K is the solute lag coefficient, and r w is the consumption rate constant. It should be noted that in the LFTA procedure, LDL transport is modeled under a steady-state framework, in which case ∂ t c w  = 0.

Computational Geometry and Boundary Conditions

An axisymmetric stenosis with 49% area reduction was adopted. As shown in Fig. 3, the total length (z-axis) of the geometry is 25D, where D = 0.004 m is the diameter of the non-stenosed region of the artery. The length of the stenosis is 1D, leaving 4D upstream and 20D downstream of the stenosis to minimize the effects of outlet boundary conditions. Two sampling points A and B, at which LDL concentration is recorded with respect to time, are marked in the geometry.

Figure 3
figure 3

Computational geometry of a mild stenosis (49% constriction by area). Computational subdomains and dividing boundaries are denoted. The dashed line is the axis of symmetry. \(\Upomega_l\) is the fluid domain and \(\Upomega_w\) is the wall domain. Sampling points A and B in the arterial wall are marked

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The axisymmetric stenosed lumen geometry was modeled by the following cosine expression:

$$ \frac {r(z)}{R}=1-\frac {\delta}{D} 1-\cos \frac {2\pi(z-z_1-z_2)}{z_2-z_1} $$
(6)

for 4D < z < 5D, where r(z) is the radius of the artery at location z in the stenosis, R is the radius of the non-stenosed region of the artery, δ is the radius reduction at the throat of the stenosis (0.15D), z 1 = 4D is the start point of the stenosed region, and z 2 = 5D is the end point of the stenosed region. The outer wall has a radius of r = 0.575D giving a wall thickness of 0.075D at the non-stenosed region and 0.225D at the throat of the stenosis.

To solve the system of equations in the described computational domain, adequate boundary conditions need to be applied. For pulsatile flow, a sinusoidal wave form shown in Fig. 4 was used to describe the variation of averaged axial velocity at the inlet:

$$ \overline{u_{in}}(t)=U_0[1-\cos(\omega t)] $$
(7)

where U 0 = 0.24 m s−1 is the time-averaged mean inlet velocity and ω = 2π is the angular frequency with a pulse period of one second. Womersley velocity profiles corresponding to the waveform were calculated and applied at the inlet. Convective flux condition was assumed at the media-adventitia interface for LDL transport. In addition, as mentioned previously, LDL concentration on the lumenal surface was assumed to be constant.

Figure 4
figure 4

Inlet velocity waveform. U 0 is the mean inlet velocity in one cardiac cycle

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The transmural velocity (J v ) and the solute flux (J s ) at the endothelium are given by the Kedem-Katchalsky equations16

$$ J_{v}=L_{p}(\Updelta p_{end}-\sigma_{d} \Updelta\pi)$$
(8)
$$ J_{s}=P_{end}\Updelta c+(1-\sigma_{f})J_{v}\bar{c}$$
(9)

where L p is hydraulic conductivity of the endothelium, \(\Updelta c\) is the solute concentration difference across the endothelium, \(\Updelta p_{end}\) is pressure drop across the endothelium, \(\Updelta \pi\) is the oncotic pressure difference across the endothelium, σ d is the osmotic reflection coefficient, σ f is the solvent reflection coefficient, P end is the solute endothelial permeability, and \(\bar{c}\) is the mean endothelial concentration. In the present study, the oncotic pressure difference \(\Updelta \pi\) was neglected to de-couple the fluid dynamics from solute dynamics.

Numerical Details

For solution of the Navier–stokes equations, a commercial CFD code, Ansys CFX 10.0 was used and the computational domain was discretized into 91,027 nodes. The wall-side momentum and mass transfer equations were solved using a commercial finite element code, Comsol Multiphysics, Version 3.3 with 46,320 computational nodes. A mesh sensitivity test was carried out on mass transport simulations to ensure grid independence of the obtained concentration field. In the pulsatile flow simulation, the time step was \(\Updelta t=0.01\,\hbox{s}\), and in the LDL transport simulation, the time step was \(\Updelta t'=0.05\,\hbox{s}\).

The simulation parameters for fluid dynamics were chosen to approximate physiological conditions in the human coronary artery. The diameter of the non-stenosed region of the artery was D = 0.004 m with total intima-media thickness of 300 μm, dynamic viscosity μ = 0.0035 Pa s, density ρ = 1050 kg m−3, Darcian permeability of the arterial wall κ = 1 × 10−18 m2.32 Transmural pressure was chosen to be \(\Updelta p=120\)  mmHg which represents a hypertensive case in vivo. This was assumed to have a stronger convection in the wall and hence shorten the time to reach a steady state. A shear-dependent hydraulic conductivity was assumed, which allowed the endothelial hydraulic conductivity to increase with shear stress. Based on published experimental data,28 the relationship between hydraulic conductivity and shear stress was derived31

$$ L_p(|\tau_w|)=0.392\times 10^{-12}\ln(|\tau_w|+0.015)+2.7931\times 10^{-12} $$
(10)

where |τ w | is the WSS, and L p is the hydraulic conductivity of the endothelium. The LDL diffusivity in plasma D l  = 2.867 × 10−11 m2 s−1,15 LDL diffusivity in the wall D w  = 3.5 × 10−12 m2 s−1,32 the LDL solvent reflection coefficient of the endothelium σ f  = 0.997,15 LDL consumption rate in the arterial wall r w  = −6.05 × 10−4 s−1,32 and LDL solute lag coefficient K = 1.05.32 Due to the lack of detailed experimental data, a constant endothelial permeability P end  = 4.84 × 10−9 m s−1 to LDL was assumed. This was determined using an optimization approach reported separately32 based on experimental data acquired by Meyer et al.,18 who provided transmural LDL concentration distribution in the rabbit aorta. A shear-dependent endothelial permeability can be employed in the future when appropriate experimental data become available.

Numerical Results

Variations of WSS magnitude in the axial direction of the stenosis model are shown in Fig. 5. Time-averaged WSS (TAWSS) calculated using transient results in one cycle was compared with WSS calculated assuming steady flow. It was found that the location of the minimum TAWSS was further downstream in the post-stenotic region compared with the steady flow result and the magnitude of TAWSS was higher. This was because the reattachment point, where WSS became zero, moved along the arterial wall in the post-stenosis region when pulsatile flow was considered.

Figure 5
figure 5

Wall shear stress distribution in the stenosis. The solid line is WSS calculated assuming steady flow conditions. The dashed line is time-averaged WSS calculated employing the pulsatile waveform

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In the LFC simulation, the initial LDL concentration was assumed to be zero over the whole subdomain. The evolution of LDL concentration at sampling points A and B (as marked in Fig. 3) is shown in Fig. 6. It can be clearly seen that after 2 h, LDL concentration profiles at the two sampling points leveled off, indicating that the transport of LDL in the wall reached a quasi steady-state. Comparing the profiles at sampling points A (upstream of the stenosis) and B (throat of the stenosis), it was found that mass balance at point A reached the quasi steady-state faster than that at point B and the concentration at point A is higher than that at point B. This was because at the throat of the stenosis, the arterial wall is thicker and provides higher resistance to transmural LDL transport. Cross-sectional profiles of LDL concentration upstream of the stenosis at different time points are shown in Fig. 7a. It was observed that the concentration profile upstream of the stenosis developed significantly from 5 min to 30 min, moderately from 30 min to 90  min, and very little from 90 min to 150 min. Similar observation can be made at the throat of the stenosis, as shown in Fig. 7b. These results suggest that LDL concentration evolves at a higher rate within the first 30 min after the perturbation is imposed. The rate of change will then decay before equilibrium is finally reached at approximately 90 min.

Figure 6
figure 6

Variation of LDL concentration profiles at sampling points A and B with time. The solid line is the profile at sampling point A. The dashed line is the profile at sampling point B

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Figure 7
figure 7

Cross-sectional profiles of LDL concentration upstream of the stenosis (z = 3D) (a) and at the throat of the stenosis (z = 4.5D) (b) at different time points

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In Fig. 8, subendothelial LDL concentrations along the stenosis given by the LFC simulation are compared with those obtained by LFTA and steady flow simulations. It is obvious that using time-averaged hydraulic conductivity as input data for steady-state LDL transport simulation (LFTA procedure) produced very similar LDL concentration distribution to the time-dependent LDL transport simulation (LFC procedure). However, LDL concentration profiles obtained under the steady flow assumption were very different. The steady flow simulation predicted a pronounced peak in the post-stenotic region and the maximum concentration of LDL co-localized with the reattachment point. This finding was previously reported and explained by weaker local convective clearance effects of the transmural flow.31 Specifically, in the low WSS region, where hydraulic conductivity of the endothelium is low and hence endothelial resistance to transmural flow is higher, LDL particles cannot be effectively “flushed” away from the subendothelial layer. The same explanation can be applied to the wider and less marked peaks predicted by LFC and LFTA simulations. Under pulsatile flow conditions, the flow separation zone in the post-stenotic region expands and contracts during a cardiac cycle, resulting in oscillation of the reattachment point along the lumenal surface. Thus when the calculated instantaneous WSS was used to simulate transmural LDL transport, LDL distribution induced by low WSS was more diffuse than under steady flow conditions. Likewise, using the time-averaged transport properties calculated based on instantaneous WSS also tended to diffuse the LDL distribution in the post-stenotic region as shown by the LFTA simulation. The good agreement between results from the LFC and LFTA simulations implies that transmural LDL transport responds slowly to changes in WSS arising from pulsatile flow, and the transport of LDL is more likely to be influenced by the time-averaged physiological environment.33

Figure 8
figure 8

LDL concentration distribution in the sub-endothelial layer given by the LFTA simulation (solid line), LFC simulation (dashed line), and steady flow simulation (dash-dotted line)

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Discussion

Time Scale of LDL Transport

It is believed that LDL accumulation in the subendothelial layer is associated with atherogenesis, and the development of atherosclerosis is a long-term process. Factors influencing LDL accumulation in vivo are subject to continuous perturbation, ranging from local geometrical changes (e.g., thickening of arterial wall and narrowing of the lumen) to dietary changes and medication that lead to elevated or reduced circulating concentrations of LDL. Other short-term physiological responses to physical activities and diurnal variation could also perturb the transport process.

This long-term transport process is calculated using the LFC procedure in the present study and assumes no perturbation. The results show that LDL transport reaches a quasi steady-state after 2 h assuming that there is no LDL in the wall initially and a transmural pressure of 120 mmHg. This result can serve as a guideline for future experimental investigations on transmural LDL transport. However, it should be noted that the simulation represents an idealized environment without any perturbation, which may not be achievable in experiments. It is possible that LDL transport could take longer to reach quasi steady-state if minor perturbations were present. Furthermore, a reduced transmural pressure (e.g., 70 mmHg as has been used in some experiments7,18) will result in a considerably longer time for LDL transport to reach a quasi steady-state. This is because convection, driven by the transmural pressure, is the primary means of LDL transport in the arterial wall.

The Average Effect of WSS

Long-term LDL transport is coupled with fluid dynamics which is characterized by a much smaller time scale. However, most existing mathematical models assumed steady flow which excludes the effect of the pulsatile nature of blood pressure and flow. The present study was designed to evaluate this assumption. It is clearly seen that results given by the steady flow model differ considerably from those obtained from the proposed LFC procedure (see Fig. 8), and casts doubt on the validity of the steady flow assumption.

However, a complete time-dependent simulation of blood flow and transmural LDL transport is computationally very demanding. Hence, the attempt was made to identify an alternative approach that is computationally efficient and the LFTA procedure was proposed for this purpose. It has been demonstrated that there is very little difference between results given by the LFC and LFTA procedures (see Fig. 8), suggesting that the influence of transient hemodynamic conditions on LDL transport can be modeled as a time-averaged effect. Furthermore, the LFTA procedure requires significantly shorter computational time (1/30) than the LFC procedure as it avoids prolonged transient transport simulations. As an efficient tool to analyze LDL transport, the LFTA procedure can be applied to physiologically realistic geometries for subject-specific studies.

Endothelial Response to WSS

In the present study, it is assumed that endothelial transport properties respond instantaneously to WSS and that hydraulic conductivity of the endothelium changes simultaneously with WSS. Although this may not be true in vivo, there are a number of difficulties that limit the development of a mathematical model accounting for the dynamics of transport property variation.

First of all, our current understanding of the dynamic process is limited. It is known that endothelial transport properties alter as a consequence of endothelial structural and functional changes mediated by WSS.11 Since transport properties are assumed to respond linearly to the rate of structural or functional change,10 the time constant of endothelial transport variation is determined by the dynamics of structural and functional changes. Specifically, since passage of blood plasma through the endothelium mainly occurs via intercellular junctions,21 variation of hydraulic conductivity will be controlled by the dynamics of intercellular junction response to changes in WSS.8,33 Thus further studies are required to enrich the knowledge of intercellular junctional changes, which will facilitate our understanding of dynamic change in hydraulic conductivity. Secondly, the instantaneous variation of WSS provides a complicated and cyclical perturbation. Thus even if these effects are fully understood, a very high time resolution would have to be employed to capture the dynamic response of hydraulic conductivity, which would certainly require excessive computational resources.

Therefore, currently, the assumption of instantaneous response of endothelial transport properties to WSS is justifiable and has been employed in a number of studies.25,26 The present findings support speculations of a previous study,31 that under pulsatile flow conditions, the post-stenotic region would be subject to oscillatory WSS with low mean values;17,20 and that as a result, relatively low hydraulic conductivity would be found over a certain length along the endothelium,12 leading to elevated LDL concentration in the subendothelial layer in that region.

Effect of Lumen-Side LDL Transport

In the proposed LFC and LFTA procedures, lumen-side LDL transport is not taken into account and a constant LDL concentration is assumed on the lumenal surface. However, the convection of LDL to and within the endothelium could induce concentration polarization, i.e., the accumulation of LDL particles on the lumenal surface. The concentration polarization phenomenon has been observed in experiments,6 as well as in computational models.34–36

The level of concentration polarization can be influenced by the fluid shear stress by two mechanisms with opposing effects: (1) a decrease in WSS would result in less convection at the lumenal surface due to a reduction in hydraulic conductivity of the endothelium, leading to reduced LDL concentration on the lumenal surface; (2) a decrease in WSS could also lessen the removal of accumulated LDL particles from the lumenal surface, causing an elevation in LDL concentration. According to our previously published analysis,31 the first pathway seemed to be dominant since a reduction in LDL lumenal concentration was observed at the reattachment point where WSS is zero. Consequently, it is proposed that the LDL concentration polarization varies only within a small range and that the influence of fluid phase LDL transport on LDL transport is minor because the rate-limiting process is the wall-side transport,33 and LDL concentration polarization is limited (<2.5% in this case).

To validate this assumption, LDL concentration distributions in the sub-endothelial layer given by steady flow simulations with and without lumen-side transport are compared in Fig. 9. The two simulations are qualitatively very similar in terms of distribution patterns, although LDL concentration is slightly higher when lumen-side transport was considered. Even if a high level of concentration polarization (>10%) were present in low WSS regions, it would result in a more pronounced elevated LDL concentration in the subendothelial layer in accord with our findings. However, under these circumstances, a model setup that takes into account the fluid phase transport would improve the quantitative accuracy of the predictions.

Figure 9
figure 9

Comparison between LDL concentration distributions in the sub-endothelial layer given by steady flow simulations without (solid line) and with (dashed line) lumen-side transport

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Conclusion

An LFC procedure was developed to (i) study the influence of time-varying WSS on LDL transport in the subendothelial layer and (ii) examine the effect of the steady flow assumption widely employed in previous investigations. A shear-dependent hydraulic conductivity was employed to model WSS-induced LDL accumulation. The results showed that LDL transport reached a quasi steady-state after approximately 2 h (when assuming zero initial concentration and transmural pressure of 120 mmHg). Comparison showed marked differences between steady flow and pulsatile flow results, suggesting the assumption of steady flow is inadequate for the prediction of transmural LDL transport. It should be noted that this observation was made in the post-stenotic region where the flow is disturbed and complicated. The effect of pulsatile flow on LDL transport in a simple geometry without flow separation should be less prominent. An LFTA procedure, which used the values of instantaneous WSS to calculated time-averaged shear-dependent transport properties, predicted very similar results to the LFC procedure, suggesting that the influence of transient WSS on LDL transport can be satisfactorily modeled as a time-averaged effect for simulations of long-term mass transport in vivo. The LFTA procedure is a useful tool to investigate long-term transmural LDL transport, which not only greatly reduces the computational time but also adequately accounts for the influences of pulsatile flow.