Low-density lipoprotein transport through an arterial wall under hyperthermia and hypertension conditions – An analytical solution
Introduction
The formation of an atherosclerotic plaque inside the wall of an artery is a very dangerous phenomenon for the human health. Computational fluid dynamic has a very important role in the prediction of such phenomena (Abraham et al., 2008), also for other similar phenomena such as aneurysms (Naughton et al., 2014), in which modeling fluid-structure interactions is a primary task (Chung and Vafai, 2012, Sun et al., 2015). If this plaque breaks, a thrombous in the blood vessel occurs, causing vascular obstruction. The thrombous is the cause of many cardiovascular diseases, like strokes and heart attack. The growth of this atherosclerotic plaque is caused by the infiltration of Low-Density Lipoprotein (LDL) through the wall. The LDLs derive from triglycerides hydrolysis of Intermediate-Density Lipoproteins (IDLs) which are not metabolized by the liver. When LDL infiltrates in the intima layer through the endothelium, it can oxidize, attracting monocytes. These monocytes absorb ox-LDL, forming foam cells. Smooth Muscle Cells (SMC) are then prone to move from the tunica media to the intima, promoting the growth of the foam cells that are going to form the plaque. The process of the plaque formation is characterized by the growth of a fibrous connective tissue layer, namely the fibrous cap, that has an important role in calcifying atherosclerotic plaques. The plaque formation can be either stable or unstable. When the plaque is stable, the fibrous cap is thick and solid. In this case, the risk consists in the occlusion of the artery. However, because the growth of the plaque is relative slow, a collateral circulation can occur. On the other hand, an unstable plaque is more dangerous, because it can break, forming a thrombous. In that case, there is no collateral circulation that compensates the problem. When the artery becomes occluded, the reduction of blood flow could cause several fatal problems. For example, an ischemic stroke occurs when blood perfusion of the brain is diminished by a thrombous or by a vascular lumen restriction caused by a growing plaque. Among the various techniques used to counteract such phenomena, an example of emerging therapy for the treatment of LDL is the LDL-apheresis (McGowan, 2013), that reminds of dialysis, in which modeling mass transfer through the vascular access has a primary role. A generic mass transfer model for vascular access was presented by Chelikani et al. (2011). The study of the mechanisms that regulate LDL accumulation through the wall is then extremely important, due to its primary role in the growth of the atherosclerotic plaque.
Following Iasiello et al. (2015), two methods are used to investigate LDL transport through an arterial wall: the first is based on experiments on animals (Meyer et al., 1996, Xie et al., 2013), and the second on predictions. Predictive models can also be distinguished in models based on realistic arteries (Kenjereš and de Loor, 2014) and on idealized arteries (Prosi et al., 2005, Yang and Vafai, 2006, Ai and Vafai, 2006) Depending on how much in detail the wall is represented, predictive methods can also be divided into three categories (Prosi et al., 2005): wall-free models, in which the arterial wall is represented by means of a boundary condition (Wada and Karino, 2000), fluid-wall models, that are also used for other similar transport problems such as the penetration of liquid medication through an arterial wall (Abraham et al., 2013), in which an homogeneous layer represents the arterial wall (Stangeby and Ethier, 2002), and multi-layer model, in which the heterogeneity of the wall layers is taken into account (Yang and Vafai, 2006, Ai and Vafai, 2006).
Most of the predictive models are based on numerical simulations. Because of the complexity of the problem, only few analytical solutions for comparisons with numerical models based on the multi-layer model were obtained during the years. The first analytical solution for the LDL transport through a straight artery was developed by Yang and Vafai (2008). They obtained flow fields and LDL concentration profiles along the wall radius, considering a simplified one-dimensional case. They used several assumptions to obtain the analytical solution, however, comparisons with literature data showed that their results were quite accurate. A comprehensive solution was presented by Khakpour and Vafai (2008) and Wang and Vafai (2013), using the method of matched asymptotic expansions in conjunction with Laplace transformation to calculate fluid flow fields and LDL distributions. Their results matched the literature data. Effects due to the insertion of a stent were analytically analyzed by Wang and Vafai (2013), showing how stent compactness affects transport phenomena. A model for the transport of therapeutic drugs through a pressurized balloon, was also proposed by Stark et al. (2013). Curvature effects of an artery were exhaustively analytically analyzed by Wang and Vafai (2015). In their study they demonstrate that low curvature ratio increases concentration polarization, i.e. the accumulation of solute on a membrane surface (Colton et al., 1972), at the lumen/endothelium interface.
When hyperthermia occurs in the human body, temperature gradients are generated. This hyperthermia can occur naturally or artificially, for example for the treatment of some diseases like arrhythmias or cancer (Soares et al., 2012, Roesch and Mueller-Huebenthal, 2015, Hernàndez et al., 2015). Predictions of heat transfer during hyperthermia treatments were carried out by Mahjoob and Vafai (2009), Alamiri et al. (2014) and by Wang et al. (2015). Hyperthermia can be induced also with laser angioplasty, that is a technique used to remove plaques. The mass transfer is influenced by temperature gradients by means of Ludwig-Soret effect (Ludwig, 1856, Soret, 1879). When temperature gradients are applied to a solution, the solute tends to move from the hot to the cold zone of the solution (Platten, 2006). The counter part for the effects of mass transfer on temperature is the Dufour effect (Ingle and Horne, 1973). Numerical studies for the LDL transport through the arterial wall under hyperthermia load, applied either from the exterior or the interior of the artery, were recently performed by Chung and Vafai (2014) for a straight artery, and by Iasiello et al. (2015) for a stenosed artery. There are no analytical results in literature for the comparisons of the cited numerical models describing or predicting hyperthermia effects on LDL transport through an arterial wall.
An analytical solution is established here for the LDL transport through the arterial wall, under hyperthermia conditions. Mass, momentum and energy dimensionless governing equations are solved for both external and internal hyperthermia loads. Comparisons with numerical and analytical results from literature are presented. Effects of hyperthermia, set of thermophysical properties and media/adventitia boundary condition, are also discussed.
Section snippets
Geometry of an artery: the multi-layer model
The sketch of the arterial wall is shown in Fig. 1a. The space in which the blood flows is called the lumen. The lumen is in direct contact with a layer of cells elongated in the blood flow direction, namely the endothelium, that is a semi-selective barrier which has a primary role into the mass exchanges between the lumen and the rest of the wall. This layer is covered by the glycolcalyx, that is a coating of molecules rich in carbohydrates. After the endothelium, the tunica intima is a layer
Dimensionless governing equations and boundary conditions
Governing Eqs. (7), (8), (9), (10) are further simplified by observing the following attributes established in earlier literature:
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filtration velocity in the axial direction is much smaller than filtration velocity in the radial direction. This means that LDL transport can be approximated as independent of axial direction (Yang and Vafai, 2006, Yang and Vafai, 2008);
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heat transfer is independent of axial direction (Chung and Vafai, 2014, Iasiello et al., 2015);
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effect of curvature is negligible,
Hypertension effects under hyperthermia conditions
Hypertension effects under hyperthermia conditions are investigated here. Hypertension conditions are imposed by varying transmural pressure, such as =70, 120 and 160 mmHg, with = 40 K, for values of 0, 0.005 and 0.01. Results are reported in Fig. 7 for external hyperthermia load, and in Fig. 8 for internal hyperthermia load.
Results on external heating show that hypertension generally increases LDL concentration in each layer. This effect is also amplified by hyperthermia. Indeed, the case
Conclusions
An analytical solution has been presented for the problem of LDL transport through an arterial wall under hyperthermia conditions. Results are in very good agreement with previous numerical and analytical studies from literature for isothermal case, and also with numerical results when hyperthermia is considered. It is shown that hyperthermia generally increases LDL concentration through an arterial wall, and hypertension combined with hyperthermia further augments this LDL accumulation.
Conflict of Interest
There is no conflict of interest. This manuscript has not been submitted to anywhere else.
Acknowledgments
The stay of Marcello Iasiello at the University of California, Riverside was financially supported by UniNA and Compagnia di San Paolo, in the frame of Programme STAR.
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