Shear stress regulation in cylindrical arteries through medial growth and nitric oxide release.

The mechanisms employed by blood vessels in order to adapt to their hemodynamic environment are important for our general understanding of disease and development. Changes in arterial geometry are generally induced by two effects: vasodilation and/or constriction; and growth and remodeling ("G &R"). The first can occur over short periods of a few minutes, while the second usually occurs over timescales of weeks or months. The free radical Nitric oxide (NO) is one of the few biological signaling molecules that is gaseous. When smooth muscle cells internalize NO, they lengthen and ultimately induce a relaxation of the artery. Platelet-Derived Growth Factor (PDGF) is a growth factor released by smooth muscle cells and platelets that regulates cell growth and division. In this paper we present a single-layered, axisymmetric hyperelastic model for a deforming, growing artery in which the opening angle is regulated by NO and growth is induced by PDGF. Our model describes vasodilation and G &R in a long cylindrical artery regulated by a steady-state Poiseuille flow. The transport of NO released by the endothelium is governed by a diffusion equation with a shear-stress dependent flux boundary condition. Arterial opening angle is assumed to be a Hill function of the wall-averaged NO concentration. We find that both growth and NO help to regulate shear stress with respect to the flow rate, but regulation through growth occurs only at large times. In contrast, regulation through NO is immediate but can only occur as long as the opening angle is able to continually decrease as a function of flow rate. Our model is calibrated using experimental data from ligated, control, and anastomosed carotid arteries of adult and weanling rabbits. Our results generate shear stress/flow rate and lumen radius/flow rate curves that agree with experimental data from control and NO-inhibited rabbit carotid arteries.

A new approach to calculating fiber fields in 2D vessel cross sections using conformal maps.

An arterial vessel has three layers, namely, the intima, the media and the adventitia. Each of these layers is modeled to have two families of strain-stiffening collagen fibers that are transversely helical. In an unloaded configuration, these fibers are coiled up. In the case of a pressurized lumen, these fibers stretch and start to resist further outward expansion. As the fibers elongate, they stiffen, affecting the mechanical response. Having a mathematical model of vessel expansion is crucial in cardiovascular applications such as predicting stenosis and simulating hemodynamics. Thus, to study the mechanics of the vessel wall under loading, it is important to calculate the fiber configurations in the unloaded configuration. The aim of this paper is to introduce a new technique of using conformal maps to numerically calculate the fiber field in a general arterial cross-section. The technique relies on finding a rational approximation of the conformal map. First, points on the physical cross section are mapped to points on a reference annulus using a rational approximation of the forward conformal map. Next, we find the angular unit vectors at the mapped points, and finally a rational approximation of the inverse conformal map is used to map the angular unit vectors back to vectors on the physical cross section. We have used MATLAB software packages to achieve these goals.

Medial Arterial Calcification: JACC State-of-the-Art Review.

Medial arterial calcification (MAC) is a chronic systemic vascular disorder distinct from atherosclerosis that is frequently but not always associated with diabetes mellitus, chronic kidney disease, and aging. MAC is also a part of more complex phenotypes in numerous less common diseases. The hallmarks of MAC include disseminated and progressive precipitation of calcium phosphate within the medial layer, a prolonged and clinically silent course, and compromise of hemodynamics associated with chronic limb-threatening ischemia. MAC increases the risk of complications during vascular interventions and mitigates their outcomes. With the exception of rare monogenetic defects affecting adenosine triphosphate metabolism, MAC pathogenesis remains unknown, and causal therapy is not available. Implementation of genetics and omics-based approaches in research recognizing the critical importance of calcium phosphate thermodynamics holds promise to unravel MAC molecular pathogenesis and to provide guidance for therapy. The current state of knowledge concerning MAC is reviewed, and future perspectives are outlined.

Simple model of atherosclerosis in cylindrical arteries: impact of anisotropic growth on Glagov remodeling.

In 1987, Seymour Glagov observed that arteries went through a two-stage remodeling process as a result of plaque growth: first, a compensatory phase where the lumen area remains approximately constant and second, an encroachment phase where the lumen area decreases over time. In this paper, we investigate the effect of growth anisotropy on Glagov remodeling in five different cases: pure radial, pure circumferential, pure axial, isotropic and general anisotropic growth where the elements of the growth tensor are chosen to minimize the total energy. We suggest that the nature of anisotropy is inclined towards the growth direction that requires the least amount of energy. Our framework is the theory of morphoelasticity on an axisymmetric arterial domain. For each case, we explore their specific effect on the Glagov curves. For the latter two cases, we also provide the changes in collagen fiber orientation and length in the intima, media and adventitia. In addition, we compare the total energy produced by growth in radial, circumferential and axial direction and deduce that using a radially dominant anisotropic growth leads to lower strain energy than isotropic growth.

An integrated approach to simulating the vulnerable atherosclerotic plaque.

Analyses of individual atherosclerotic plaques are mostly descriptive, relying, for example, on histological classification by spectral analysis of ultrasound waves or staining and observing particular cellular components. Such passive methods have proved useful for characterizing the structure and vulnerability of plaques but have little quantitative predictive power. Our aim is to introduce and discuss a computational framework to provide insight to clinicians and help them visualize internal plaque dynamics. We use partial differential equations (PDEs) with macrophages, necrotic cells, oxidized lipids, oxygen concentration, and platelet-derived growth factor (PDGF) as primary variables coupled to a biomechanical model to describe vessel growth. The model is deterministic, providing mechanical, morphological, and histological characteristics of an atherosclerotic vessel at any desired future time point. We use our model to create computer-generated animations of a plaque evolution that are in qualitative agreement with published serial ultrasound images and hypothesize possible atherogenic mechanisms. A systems biology model consisting of five differential equations is able to capture the morphology of necrotic cores residing within vulnerable atherosclerotic plaque. In the context of the model, the distribution of oxidized low-density lipoprotein (Ox-LDL) particles, endothelial inflammation, plaque oxygenation (via the presence of vasa vasora), and intimal oxygenation are four important factors that drive changes in core morphology.NEW & NOTEWORTHY In this article, we propose a quantitative framework to describe the evolution of atherosclerotic plaque. We use partial differential equations (PDEs) with macrophages, necrotic cells, oxidized lipids, oxygen concentration, and PDGF as primary variables coupled to a biomechanical model to describe vessel growth. A feature of our method is that it outputs color-coded vessel sections corresponding to regions of the plaque that are necrotic and fibrous, qualitatively similar to images generated by enhanced intravascular ultrasound.

Media sclerosis drives and localizes atherosclerosis in peripheral arteries.

Media sclerosis (MS) and peripheral artery disease (PAD) may coincide, particularly in type 2 diabetics (T2D) and in patients with chronic kidney disease (CKD). In contrast to non-diabetics, in T2D PAD is more severe and more distal. Although MS is suspected to play a role, the underlying pathophysiological reasons for the differences still remain elusive today. We tested the hypothesis that MS is a promoter of atherosclerosis as it occurs in T2D with PAD by interfering with arterial remodeling using an in-silico simulation. We confirmed that MS aggravates PAD by promoting negative remodeling. We found that the effect is more pronounced in smaller distal arteries compared to larger proximal ones. Our results suggest that the degree of this divergence depends on the ratio between the thickness of the intima relative to the thickness of the media/adventitia of the individually affected arteries.

A biochemical and mechanical model of injury-induced intimal thickening.

In this paper, we investigate an axisymmetric model of intimal thickening using hyperelasticity theory. Our model describes the growth of the arterial intima due to cell proliferation which, in turn, is driven by the release of a cytokine such as platelet-derived growth factor (PDGF). With the growth rate tied to both local stress and the local concentration of PDGF, we derive a quadruple free boundary problem with different regions of the vessel wall characterized by different homeostatic stress. We compare our model predictions to rabbit and rodent models of atherosclerosis and find that in order to achieve the growth rates reported in the experiments, growth must be mainly cytokine induced rather than stress induced. Our model is also able to reproduce Glagov remodelling where, as a vessel becomes more diseased, the lumen expands before rapidly contracting.

Multi-Layer Mechanical Model of Glagov Remodeling in Coronary Arteries: Differences between In-Vivo and Ex-Vivo Measurements.

When blood vessels undergo remodeling because of the buildup of atherosclerotic plaque, it is thought that they first undergo compensatory or outward remodeling, followed by inward remodeling: the lumen area stays roughly constant or increases slightly and then decreases rapidly. The second phase of remodeling is supposed to start after the plaque burden exceeds about 40%. These changes in the vessel were first observed by S. Glagov who examined cross-sections of coronary arteries at different stages of the disease. In this paper, we use a mathematical model based on growth and elasticity theory to verify the main aspects of Glagov's result. However, both our model and curve-fitting to the data suggest that the critical stenosis is around 20% rather than 40%. Our model and data from the PROSPECT trial also show that Glagov remodeling is qualitatively different depending on whether measurements are taken ex-vivo or in-vivo. Our results suggest that the first outward phase of "Glagov remodeling" is largely absent for in-vivo measurements: that is, the lumen area always decreases as plaque builds up. We advocate that care must be taken when infering how in-vivo vessels remodel from ex-vivo data.

Bayesian Uncertainty Quantification for Bond Energies and Mobilities Using Path Integral Analysis.

Dynamic single-molecule force spectroscopy is often used to distort bonds. The resulting responses, in the form of rupture forces, work applied, and trajectories of displacements, are used to reconstruct bond potentials. Such approaches often rely on simple parameterizations of one-dimensional bond potentials, assumptions on equilibrium starting states, and/or large amounts of trajectory data. Parametric approaches typically fail at inferring complicated bond potentials with multiple minima, while piecewise estimation may not guarantee smooth results with the appropriate behavior at large distances. Existing techniques, particularly those based on work theorems, also do not address spatial variations in the diffusivity that may arise from spatially inhomogeneous coupling to other degrees of freedom in the macromolecule. To address these challenges, we develop a comprehensive empirical Bayesian approach that incorporates data and regularization terms directly into a path integral. All experimental and statistical parameters in our method are estimated directly from the data. Upon testing our method on simulated data, our regularized approach requires less data and allows simultaneous inference of both complex bond potentials and diffusivity profiles. Crucially, we show that the accuracy of the reconstructed bond potential is sensitive to the spatially varying diffusivity and accurate reconstruction can be expected only when both are simultaneously inferred. Moreover, after providing a means for self-consistently choosing regularization parameters from data, we derive posterior probability distributions, allowing for uncertainty quantification.

Simulation of a pulsatile non-Newtonian flow past a stenosed 2D artery with atherosclerosis.

Atherosclerotic plaque can cause severe stenosis in the artery lumen. Blood flow through a substantially narrowed artery may have different flow characteristics and produce different forces acting on the plaque surface and artery wall. The disturbed flow and force fields in the lumen may have serious implications on vascular endothelial cells, smooth muscle cells, and circulating blood cells. In this work a simplified model is used to simulate a pulsatile non-Newtonian blood flow past a stenosed artery caused by atherosclerotic plaques of different severity. The focus is on a systematic parameter study of the effects of plaque size/geometry, flow Reynolds number, shear-rate dependent viscosity and flow pulsatility on the fluid wall shear stress and its gradient, fluid wall normal stress, and flow shear rate. The computational results obtained from this idealized model may shed light on the flow and force characteristics of more realistic blood flow through an atherosclerotic vessel.

Mathematical model of intimal thickening in atherosclerosis: vessel stenosis as a free boundary problem.

Atherosclerosis is an inflammatory disease of the artery characterized by an expansion of the intimal region. Intimal thickening is usually attributed to the migration of smooth muscle cells (SMCs) from the surrounding media and proliferation of SMCs already present in the intima. Intimal expansion can give rise to dangerous events such as stenosis (leading to stroke) or plaque rupture (leading to myocardial infarction). In this paper we propose and study a mathematical model of intimal thickening, posed as a free boundary problem. Intimal thickening is driven by damage to the endothelium, resulting in the release of cytokines and migration of SMCs. By coupling a boundary value problem for cytokine concentration to an evolution law for the intimal area, we reduce the problem to a single nonlinear differential equation for the luminal radius. We analyze the steady states, perform a bifurcation analysis and compare model solutions to data from rabbits whose iliac arteries are subject to a balloon pullback injury. In order to obtain a favorable fit, we find that migrating SMCs must enter the intima very slowly compared to cells in dermal wounds. This cell behavior is indicative of a weak inflammatory response which is consistent with atherosclerosis being a chronic inflammatory disease.

Growth of necrotic cores in atherosclerotic plaque.

Plaques are fatty deposits that grow mainly in arteries and develop as a result of a chronic inflammatory response. Plaques are characterized as 'vulnerable' when they have large internal regions of necrosis and are heavily infiltrated by macrophages. The particular composition of a vulnerable plaque renders it susceptible to rupture, which releases thrombogenic agents into the bloodstream and can result in myocardial infarction. In this paper, we propose a mathematical model to predict the development of a plaque's necrotic core. By solving coupled reaction-diffusion equations for macrophages and dead cells, we focus on the joint effects of hypoxic cell death and chemoattraction to oxidized low-density lipoprotein (Ox-LDL), a molecule that is strongly linked to atherosclerosis. We do not model the mechanical properties of the plaque, its growth or rupture. Our model predicts cores that have approximately the right size and shape when compared to ultrasound images. Because our model is linear and autonomous, normal mode analysis and subsequent calculation of the smallest eigenvalue allow us to compute the times taken for the necrotic core to form. We find that the spatial distribution of Ox-LDL within the plaque determines not only the placement and size of cores, but their time of formation. Although plaques are biochemically complex, our study shows that certain aspects of their composition can be predicted and are, in fact, governed by simple physical models.

Accelerated search kinetics mediated by redox reactions of DNA repair enzymes.

A charge transport (CT) mechanism has been proposed in several articles to explain the localization of base excision repair (BER) enzymes to lesions on DNA. The CT mechanism relies on redox reactions of iron-sulfur cofactors that modify the enzyme's binding affinity. These redox reactions are mediated by the DNA strand and involve the exchange of electrons between BER enzymes along DNA. We propose a mathematical model that incorporates enzyme binding/unbinding, electron transport, and enzyme diffusion along DNA. Analysis of our model within a range of parameter values suggests that the redox reactions can increase desorption of BER enzymes not already bound to lesions, allowing the enzymes to be recycled--thus accelerating the overall search process. This acceleration mechanism is most effective when enzyme copy numbers and enzyme diffusivity along the DNA are small. Under such conditions, we find that CT BER enzymes find their targets more quickly than simple passive enzymes that simply attach to the DNA without desorbing.

Charge-transport-mediated recruitment of DNA repair enzymes.

Damaged or mismatched bases in DNA can be repaired by base excision repair enzymes (BER) that replace the defective base. Although the detailed molecular structures of many BER enzymes are known, how they colocalize to lesions remains unclear. One hypothesis involves charge transport (CT) along DNA [Yavin et al., Proc. Natl. Acad. Sci. U.S.A. 102, 3546 (2005)]. In this CT mechanism, electrons are released by recently adsorbed BER enzymes and travel along the DNA. The electrons can scatter (by heterogeneities along the DNA) back to the enzyme, destabilizing and knocking it off the DNA, or they can be absorbed by nearby lesions and guanine radicals. We develop a stochastic model to describe the electron dynamics and compute probabilities of electron capture by guanine radicals and repair enzymes. We also calculate first passage times of electron return and ensemble average these results over guanine radical distributions. Our statistical results provide the rules that enable us to perform implicit-electron Monte Carlo simulations of repair enzyme binding and redistribution near lesions. When lesions are electron absorbing, we show that the CT mechanism suppresses wasteful buildup of enzymes along intact portions of the DNA, maximizing enzyme concentration near lesions.

Dynamic boundaries in asymmetric exclusion processes.

We investigate the dynamics of a one-dimensional asymmetric exclusion process with Langmuir kinetics and a fluctuating wall. At the left-hand boundary, particles are injected onto the lattice; from there, the particles hop to the right. Along the lattice, particles can adsorb or desorb, and the right-hand boundary is defined by a wall particle. The confining wall particle has intrinsic forward and backward hopping, a net leftward drift, and cannot desorb. Performing Monte Carlo simulations and using a moving-frame finite segment approach coupled to mean field theory, we find the parameter regimes in which the wall acquires a steady-state position. In other regimes, the wall will either drift to the left and fall off the lattice at the injection site, or drift indefinitely to the right. Our results are discussed in the context of nonequilibrium phases of the system, fluctuating boundary layers, and particle densities in the laboratory frame versus the frame of the fluctuating wall.

Continuum theory of nanostructure decay via a microscale condition.

The morphological relaxation of faceted crystal surfaces is studied via a continuum approach. Our formulation includes (i) an evolution equation for the surface slope that describes step line tension, g1, and step repulsion energy, g3; and (ii) a condition at the facet edge (a free boundary) that accounts for discrete effects via the collapse times, t(n), of top steps. For initial cones and t(n) approximately t(n)4, we use t(g) from step simulations and predict self-similar slopes in agreement with simulations for any g = g3/g1 > 0. We show that for g >> 1, (i) the theory simplifies to an equilibrium-thermodynamics model; (ii) the slope profiles reduce to a universal curve; and (iii) the facet radius scales as g(-3/4).