As already done in the sketch of the multiscale analysis above, we will inves-tigate our models on either fast or slow timescale, e.g. the timescale of the heart beat compared to the timescale of the plaque growth. We will denote by t ∈ I := (0, T)with T > 0 the fast and by τ ∈I := (0,T)with T > 0 the slow timescale. Denoting the timescale separation parameter by 0 < ε 1, these scales will be formally related by the transformationτ=εt. In particular T =ε−1T, where the length of the slow timescale T >0 will be independent of εin our convergence results. We will not notationally distinguish between the functionst7→f(t)andτ 7→f(τ), but the scaling should be evident out of context and through the used notation for the argument. We remark that in some parts of the literature the roles oft andτ are reversed, i.e.tdenotes the slow andτ the fast time variable.
We denote by C > 0 a generic constant which is always independent of ε but may change with each occurrence. We will commonly write a ≲ b if a ≤ Cb. We write A for the closure and A◦ for the interior of a set A in some topological space and denote by 1A the associated indicator function.
The spatial dimension is denoted by d ∈ {2,3} and the Euclidean distance between a point x∈ Rd and a set A ⊂Rd by d(x, A). We use the following notations for vector calculus: We write I ∈Rd×d for the identity matrix and CofA:=detAA−Tfor the cofactor matrix of some invertible matrixA∈Rd×d. For a vector fielduwe denote by(∇u)ij=∂juiwithi, j= 1, . . . , dthe Jacobian matrix and by divu:=P
i∂iui the divergence. For a matrix (representation of a tensor) fieldAwe write(divA)i:=P
j∂jAij fori= 1, . . . , d.
InChapters 2and6, where phenomena both inside the region of free blood flow (the lumen) and inside the artery wall are examined, we will writeΩf ⊂Rd
for the fluid and Ωs⊂Rd for the structure domain. This follows the notation from [Yan+15], but note that in some reviewed models inSection 2.2fluid flow is also modelled in the wall as a porous medium. We will denote quantities defined onΩf andΩswith subscriptsf ands, respectively, e.g. writevf for the fluid velocity inΩfandvsfor the deformation velocity inΩs. The exterior unit normal vector on the boundary ofΩf andΩsis denoted bynf, respectivelyns. The abstract examples for the theory developed inChapter 3 are defined on a single domainΩ⊂Rdand no subscripts are used. In the simplified model from Chapters 4and 5 only the fluid domain is considered, but this domain depends on the plaque state q∈QforQ⊂Rn for somen∈Nand is written as Ωq = Ωf,q, i.e. omitting the subscript f. For some function f: D(f)→Y with sets X, Y and D(f)⊂X×Qwe will use the notationfq(x) :=f(x;q)if (x, q)∈D(f), where we writef(x;q)to convey thatqis often a fixed parameter.
For some generic domainΩ⊂Rd, we will denote byLp(Ω),Hs(Ω),Ws,p(Ω) for 1 ≤ p ≤ ∞ and s ≥ 0 the usual Lebesgue and Sobolev spaces, and by Lp(I, X), etc. for some Banach space X the corresponding Bochner spaces.
We will denote the scalar products on L2(Ω), L2(Ωf) and L2(Ωs) by (·,·), (·,·)f and (·,·)s respectively. For the family of (fluid) domainsΩq studied in Chapters 4and5 we write(·,·)q for theL2(Ωq)scalar product.
For Sobolev spaces a subscript0will denote the subspace with homogeneous Dirichlet boundary conditions, e.g.H01(Ω) :={u∈H1(Ω)|u= 0on∂Ω}. For Lebesgue spaces a subscript0will indicate the subspace of functions with zero mean, e.g.L20(Ω) :={p∈L2(Ω)|R
Ωpdx= 0}. The space H−1(Ω)will denote the dual of H01(Ω). Subspaces of solenoidal, i.e. divergence-free, functions will be denoted by σ, e.g. H0,σ1 (Ω) :={u ∈H01(Ω) | divu= 0}. For functions in Lp(Ω)the divergence must be understood in the sense of distributions. We will omit the spatial dimension of objects if it is clear from the context, e.g. write vf ∈H1(Ω) for the fluid velocity field instead of vf ∈H1(Ω,Rd)∼= [H1(Ω)]d. For a Banach space X we denote by Cπ(X) := {u ∈ C(R, X) | u(t+ 1) = u(t)∀t∈R}the set of continuous,1-periodic,X-valued functions.
Models for Atherosclerosis
We start with a brief summary of the disease atherosclerosis and involved structures in Section 2.1, focusing on the influence of the dynamics of blood flow (hemodynamics) and the causes of plaque growth. No prior knowledge is assumed. InSection 2.2we review mathematical models for atherosclerosis which cover different processes and stages of the disease. Three foci of this review are the influence of the wall shear stress, the realization of growth and the ad-hoc resolution of the problem of multiple timescales as discussed in the introduction. The finalSection 2.3discusses one specific model by Yang et. al.
[Yan+15] which was the motivation for this thesis. A non-dimensionalization is performed to highlight how different timescales emerge through parameter magnitudes.
2.1 The Disease Atherosclerosis
Atherosclerosis is a cardiovascular disease in which plaque builds up inside the artery’s wall which can cause a narrowing (stenosis) the region of blood flow [GJ10]. Extreme stenosis or events like a rupture of the plaque may reduce blood flow sufficiently to cause a heart attack or stroke [Her+16], which are leading causes of death in the world [GBD18]. Atherosclerosis develops over a span of decades, with initial stages of the disease found in some locations as early as in the first decade of life [Sta99]. The disease is also ubiquitous: In one study it was found in 95% of the subjects after the fourth decade of life [Sta99].
The initial stages are benign and asymptomatic, the likelihood of a progression towards later, malign stages is driven by several accepted risk factors, such as smoking, adiposity, high blood pressure, blood cholesterol, diabetes mellitus, age, sex, personal and family history [Her+16; GJ10].
Mortality due diseases caused by atherosclerosis has drastically declined since the middle of the 20th century in high-income countries [Her+16]. To-day, critically blocked arteries are either opened up with stents or bypassed [Moh+13] and progression slowed down using statins [Ped16]. But even with optimal treatment, recurrent events occur in 10% to 20% of cases in the first 12 months after acute syndromes, highlighting the need for further research [LBT16]. This research is hindered by the slow progression of the disease, diffi-culties of in-vivo monitoring and deficits of laboratory animal models [NWS08].
The description of processes involved in atherosclerosis fills volumes, so only 7
an overview can be given here. We refer to [GJ10;LBT16] for a more detailed exposition.
Review of Structures Involved in Atherosclerosis Arteries
Arteries transport blood away from the heart. They enclose the blood flow re-gion (lumen) with a wall subdivided into three enveloping layers of tissue: The innermost tunica intima, tunica media and outermost tunica adventitia. The initial stages of atherosclerosis occur in the tunica intima, but later stages can extend into tunica media and adventitia [Sta99]. Arteries differ substantially in function and composition. Arteries closest to the heart, like the aorta, are the largest and elastic to stretch in response to the blood pulse. Their walls have their own blood vessels (vasa vasorum) which enter either through the intima or adventitia [RL07]. Medium-sized arteries for blood distribution are muscular, but may still be exposed to considerable external movement, like the coronary arteries which are attached to the heart. Atherosclerosis is uncom-mon in smaller arteries [Ros99]. The intima, normally thin compared to the other layers, can thicken with age [Sta+92], either focally (eccentric thicken-ing), common near bifurcation points and at the entrances of branch vessels [Sta99], or uniformly (diffuse thickening). Regions with intimal thickening are associated to altered mechanical stress and susceptible to atherosclerosis, but the thickening itself is not considered part of the disease [Sta+92].
The tunicas are separated by elastic tissue [SK04]. The tunica intima is de-limited from the lumen by a contiguous mono-layer of cells, called endothelium, which acts as a permeability barrier [Sta+92]. The endothelium is supported by an extracellular matrix, below which lie layers of smooth muscle cells, although these may be missing in some cases and thickness of extracellular matrix and muscle cell layers varies significantly with intimal thickening [Sta+92]. The tunica media contains layers of smooth muscle cells supported by an extra-cellular matrix with elastic fibres which control the elastic behavior of large arteries [SK04]. The adventitia is a collection of smooth muscle and other cells embedded into a loose matrix containing elastic fibres [SK04].
Blood
Blood is a suspension of particles in a plasma containing, among other things, platelets, red and white blood cells. Blood is a non-Newtonian fluid with shear-thinning, viscoelastic and, albeit controversial, yield stress behaviors, see [RSK08] also for the remainder of this paragraph. These effects are mainly caused by red blood cells, which are oval biconcave disks that can deform, ag-gregate and align. Under low shear rates, they stack into a structure called rouleaux, which themselves can form complex three-dimensional networks.
These structures take seconds to minutes to form and disintegrate under high shear rates, so they can only occur in regions which experience low shear rates due to stagnation or recirculation over longer times. Due to the complexity of blood rheology, simplified models are often employed. A common, but dras-tic, simplification is the use of a Newtonian model in large and medium sized arteries, which is known to cause artifacts e.g. behind a stenosis [QVZ02b].
Lipoproteins
A function of blood of particular importance for atherosclerosis is the transport of lipids, such as cholesterol. Since lipids are hydrophobic, they are enclosed by proteins in a complex called lipoprotein. Two important characteristics of lipoproteins are their density and comprising proteins. Very low (VLDL), in-termediate (IDL) and low density lipoproteins (LDL) contain apoB-proteins, whereas high density lipoproteins (HDL) contain apoA-proteins. While this classification is already a simplification of much more heterogeneous parti-cles with complicated dynamics [Hüb+08], we focus for simplicity only on the atherogenic LDL and atheroprotective HDL [GJ10].
Progression of Atherosclerosis
There is not one single sequence of events in atherosclerosis. Using the pro-gression scheme proposed in [Sta+94; Sta+95; Sta00] one can roughly distin-guish between early and advanced stages. Early stages are benign and asymp-tomatic, whereas advanced stages can be malign with symptoms such as chest pain (angina pectoris) or sudden, possibly but not necessarily fatal, events like heart attack (myocardial infarction) or (ischemic) stroke [Sta00]. Early stages can regress to normal, advanced stages are characterized by irreversible disrup-tion of the wall’s structure and geometry with the possibility of stabilizadisrup-tion, i.e. a halting of progression, but not regression [Sta+95;Sta00].
Early Stages: Fatty Streak, Lipid Accumulation
Atherosclerosis is an inflammatory response to a dysfunction of the artery wall [LRM02]. The initial event which leads to atherosclerosis is still disputed [SK04] and some authors distinguish between initial stages based on the pres-ence or abspres-ence of intimal thickening [Vir+00]. According to the response-to-retention hypothesis the response-to-retention of LDL in the intima, and other lipoproteins with diameter<70 nm containing apoB, is the initial event in atherosclerosis [TWB07]. Regions prone to atherosclerosis differ not in the permeability but retention of lipoproteins [SC89a;SC89b], this retention predates the occurrence of immune cells [TWB07; NWS08] and is enhanced by external stimuli such as mechanical strain [NWS08]. The response-to-injury hypothesis stresses the importance of altered endothelial function, e.g. loss of nitric oxide production important for vascular homeostasis, caused by the risk factors described above through different pathways and wall shear stress [GJ10]. This causes in partic-ular monocytes, a part of the innate immune system and a subset of the white blood cells in the blood stream, to be attracted into the wall through a process of adhering to, rolling along and transmigrating through the wall [GL15]. Wall shear stress alters the expression of genes involved in this process, e.g. low shear stress increases the expression of adhesion molecule genes [Ros99].
Inside the wall, monocytes differentiate into macrophages and take up the previously retained LDL [Ros99]. The uptake of LDL in its native state is tightly regulated by the macrophage’s LDL receptor, but this regulation is cir-cumvented in atherosclerosis through the so called scavenger receptor pathway [SK04]. This pathway requires chemically modified LDL, e.g. oxidized LDL (oxLDL), and the importance of the process of chemical modification in the
intima is stressed by the oxidative modification hypothesis [SK04]. In a “pro-tective response that backfires” [SN13], macrophages can thus engorge excessive amounts of LDL and turn into foam cells, so called due to their foamy appear-ance under the microscope. Foam cells are a hallmark of early atherosclerosis and make up the fatty streak [Sta+92]. Foam cells eventually die and leave extracellular lipids in the wall, which together with cell debris form small pools in the wall [Sta00]. The occurrence of small, isolated lipid pools marks the final stage of early atherosclerosis [Sta00]. Through the process of reverse cholesterol transport, which utilizes HDL, lipids can be transported out of the wall [GJ10]
which can lead to a regression of the diseased wall back to normal [Sta00].
Advanced Stages: Plaque Formation, Stabilization, Rupture
If the accumulation of lipids outpaces the reverse transport, the small lipid pools join to a larger, confluent pool called lipid core, whose occurrence marks the first stage of advanced atherosclerosis [Sta+92]. Since lipid cores also accu-mulate debris from dying cells they are also referred to as (lipid-rich) necrotic cores. After the formation of a lipid core, smooth muscle cells migrate into the area separating the core from the lumen, forming a protective, fibrous cap [GJ10]. Together with the lipid core the cap constitutes the atheroma-tous plaque, which may cause a hardening (sclerosis) of the wall and gives the disease its name.
The further progression of the disease is largely influenced by size and com-position of the cap and core. The plaque can be unstable if it has a thin cap with thickness typically less than 100 µm1[CW14] overlying an extensive lipid core, typically 30%–50% of total plaque area [Vir+05]. Instability can also occur if the cap is structurally weakened by infiltrating macrophages, by the death of muscle cells or through the formation of microvessels [GJ10]. Stable plaque, in absence of the prior phenomena, can undergo calcification or fibra-tion, where the lipid core is slowly mineralized or the lipids removed by reverse cholesterol transport and replaced by fibrous, reparative tissue [Sta+95].
Unstable plaque may fissure or rupture. Fissuring may be seen as precursor or subtype of rupture and can cause blood to enter the plaque (intraplaque hemorrhage) [Vir+00]. A rupture exposes material from the lipid core to the blood stream, in which case platelets in the blood activate and aggregate, leading to the formation of a blood clot (thrombus) [Vir+00]. A thrombus can stay in place or travel downstream which can either lead to a fatal reduction of blood flow, manifesting e.g. as a heart attack or stroke, but can also be nonfatal and even asymptomatic [Vir+00]. In focal (eccentric) plaques, fissures and ruptures are most common in the shoulder regions where the cap is thinnest and most infiltrated by foam cells [Vir+00; Ben+14]. Multiple, nonfatal and asymptomatic thrombi can occur, which are then incorporated into the healing plaque, leading to a complicated plaque structure and composition [Vir+00].
A thrombus can also arise from a mere erosion of the most luminal part of the cap [Vir+00]. Plaque erosion is defined as occurrence of a thrombus without signs of rupture [Vir+00]. The term “erosion” is used since typically large parts of the endothelial layer are absent, a situation not observed in
1A thickness less than 65 µm is commonly used to define thin, unstable plaque, e.g. in [Vir+00], but this number has recently come under scrutiny [CW14]. Under exertion, rupture has been observed in caps with thickness up to 160 µm [CW14].
advanced plaques without thrombosis [Vir+00]. Plaque erosion occurs in about 40% of cases of sudden coronary death [Vir+00], but “the mechanisms leading to thrombus without rupture is one of the most important unresolved questions within atherosclerosis research” [Ben+14].
To quantify the previous statements, it was found in [Dav92] that 19%
of subjects with sudden coronary death had only a stenosis of > 75% cross-sectional area (or >50% diameter), while 8% had fissures and the remaining 73% had a thrombus. A rupture was the origin of the thrombus in 65% of the cases in another study [Vir+00]. Ruptured plaques are on average large [Ben+14], but the severity of stenosis leading to clinical events is still disputed [Nic+13] due to rapid changes in plaque prior to clinical events, difficulties and discrepancies in determining a reference state [FS96;Nic+13].
Mechanisms of Plaque Growth, Stenosis, Remodelling
It is necessary to distinguish growth of the plaque and the resulting stenosis, where we use the term plaque in this context also for the initial stages. Not only because plaque can grow by replacement or alteration of healthy tissue without overall increase in size, but also because arteries can compensate plaque growth through arterial remodeling.
Arterial remodelling is an adaption of vessel size over weeks to months [Dav95], controlled by the endothelium, to maintain normal blood flow charac-teristics such as wall shear stress, see [War+00] also for the remainder of this paragraph. Outward arterial remodelling can compensate plaque growth and thus postpone the development of a flow-limiting stenosis. The extend of re-modelling varies locally, hypothesized due to variability of endothelial response, local flow characteristics and wall composition. Plaques with large soft lipid core, i.e. those prone to rupture, exhibit more outward remodeling than those which are fibrous and calcified. The latter plaque type can even experience inward remodeling, which reinforces the stenosis, a phenomena which can also occur during healing from a thrombus. It is uncertain whether the association between outward remodeling and rupture-prone plaques is causal, which would make outward remodeling a “double-edged sword” [War+00]. Nonetheless, outward remodeling hides the type of plaque most vulnerable to rupture in an-giography, a common imaging technique, and may render them asymptomatic until rupture, preventing detection and treatment.
Up until the formation of a fibric cap, the size of the stenosis is mainly deter-mined by the size of the lipid core [Sta00]. Due to outward expansion by arterial remodelling, these initial stages “will not obstruct the lumen much” [Sta00] and growth is steady. In advanced stages, growth happens by lipid accumulation, smooth muscle cell and collagen increase and healing from thrombosis, while remodeling is impeded by the cap and other structures, which leads to more se-vere stenosis [Sta+95;Sta00]. The progression of stenosis in advanced plaque can be both steady or occur in sudden bursts [Yok+99], where the latter is associated with the healing of thrombi [Sta00].
Arterial remodeling necessitates a distinction between stenosis relative to the pre-disease state, which is in practise hard to determine [War+00], and stenosis relative to the remodeled artery, taking some feature as reference for a hypothetical cross-sectional area without plaque [War+00]. Further ambiguity arises since stenosis can be measured in terms of area or diameter of arteries
[War+00; Gla+87], where the latter is ill-defined for non-circular arteries or non-uniform stenosis. One seminal result about the influence of remodeling on atherosclerosis states that coronary arteries can compensate up to 40%
of stenosis, measured in terms of the area relative to the remodeled artery [Gla+87].
Biomechanics in Atherosclerosis
Although a direct correlation between sites of low and oscillatory wall shear stress and predisposition for atherosclerosis is “less robust than commonly as-sumed” [PSW13], blood dynamics influences normal endothelial function and atherosclerosis threefold: First, endothelial cells sense mechanical forces to control endothelial function [Dav95]. Second, blood flow indirectly controls the concentration of chemical substances and blood particles in the vicinity of the wall through advection and diffusion, e.g. by increasing residence times near flow stagnation points [Dav95]. Third, mechanical stress on the plaque plays a central role in plaque rupture [CW14].
Exposed to laminar flow over multiple hours, endothelial cells align and elongate in the direction of mean directional shear stress, otherwise they have an polygonal structure without preferred orientation [Dav95]. Realignment is driven by the cell’s exoskeleton and also affects surface and interior topogra-phy [Dav95]. Naturally, variations in shear stress at a sub-cellular level, where force transmission and transduction from the surface to the interior takes place,
Exposed to laminar flow over multiple hours, endothelial cells align and elongate in the direction of mean directional shear stress, otherwise they have an polygonal structure without preferred orientation [Dav95]. Realignment is driven by the cell’s exoskeleton and also affects surface and interior topogra-phy [Dav95]. Naturally, variations in shear stress at a sub-cellular level, where force transmission and transduction from the surface to the interior takes place,