A Multiphase Model of Growth Factor-Regulated Atherosclerotic Cap Formation
Michael G. Watson, Helen M. Byrne, Charlie Macaskill, Mary R. Myerscough
AA Multiphase Model of Growth Factor-RegulatedAtherosclerotic Cap Formation
Michael G. Watson, Helen M. Byrne, Charlie Macaskill, Mary R. MyerscoughAugust 9, 2019
Abstract
Atherosclerosis is characterised by the growth of fatty plaques in the inner (intimal)layer of the artery wall. In mature plaques, vascular smooth muscle cells (SMCs) arerecruited from the adjacent medial layer to deposit a cap of fibrous collagen over thefatty plaque core. The fibrous cap isolates the thrombogenic content of the plaquefrom the bloodstream and prevents the formation of blood clots that cause myocardialinfarction or stroke. Despite the important protective role of the cap, the mechanismsthat regulate cap formation and maintenance are not well understood. It remainsunclear why certain caps become stable, while others become vulnerable to rupture.We develop a multiphase PDE model with non-standard boundary conditions toinvestigate collagen cap formation by SMCs in response to growth factor signals fromthe endothelium. Diffusible platelet-derived growth factor (PDGF) stimulates SMCmigration, proliferation and collagen degradation, while diffusible transforming growthfactor (TGF)- β stimulates SMC collagen synthesis and inhibits collagen degradation.The model SMCs respond haptotactically to gradients in the collagen phase and havereduced rates of migration and proliferation in dense collagenous tissue. The model,which is parameterised using a range of in vivo and in vitro experimental data, re-produces several observations from studies of plaque growth in atherosclerosis-pronemice. Numerical simulations and model analysis demonstrate that a stable cap can beformed by a relatively small SMC population and emphasise the critical role of TGF- β in effective cap formation and maintenance. These findings provide unique insightinto the cellular and biochemical mechanisms that may lead to plaque destabilisationand rupture. This work represents an important step towards the development of acomprehensive in silico plaque. a r X i v : . [ q - b i o . CB ] A ug Introduction
Cardiovascular diseases are the leading global cause of mortality (World Health Organization,2017). Atherosclerosis, the growth of fat-filled plaques in the walls of arteries, is the primarycause of most cardiovascular disease-related deaths. Unstable atherosclerotic plaques canrupture, and subsequent blood clot formation may occlude blood flow and lead to myocar-dial infarction or stroke (Hansson and Libby, 2006). Smooth muscle cells (SMCs) play animportant role in the prevention of these grave clinical outcomes by forming a stabilising capof fibrous tissue over the lipid-rich core in developing plaques (Alexander and Owens, 2012).To understand how plaques become unstable and dangerous, it is important to understandthe behaviour of plaque SMCs and the corresponding mechanisms of fibrous cap synthesisor degradation. Despite extensive experimental investigation, the in vivo dynamics of capformation remain poorly understood.The absence of a dynamic understanding of cap formation and maintenance can be largelyattributed to the slow rate of plaque development. In humans, atherosclerosis can begin inchildhood and plaques may take several decades to progress towards a dangerous, unstablestate (Lusis, 2000). Even in the apolipoprotein-E (ApoE) deficient mouse (the most commonanimal model for experimental atherosclerosis studies), plaques require a period of severalmonths to grow to an advanced stage. Consequently, observations of plaque tissue resectedfrom experimental mice are typically made at distinct (or even single) time points. Mathe-matical modelling provides a powerful tool to study the dynamic mechanisms that underliethese patchy experimental observations. In this paper, we use the multiphase framework de-veloped in Watson et al. (2018) and build a comprehensive new model to study the formationof the protective fibrous cap by atherosclerotic plaque SMCs.Atherosclerotic plaques develop in the narrow intimal layer of the artery wall (Figure 1).The intima is located between the endothelium (a thin sheet of endothelial cells that lines thevessel lumen) and the media, which houses several striated layers of quiescent (contractile)SMCs. The intima and the media are separated by a dense tissue membrane known as theinternal elastic lamina (IEL). The outermost layer of the artery wall beyond the media iscalled the adventitia. Plaque formation is initiated by blood-borne low-density lipoproteins(LDL), which accumulate in the intima and become oxidised or modified in different ways.The presence of modified LDL triggers the recruitment of immune cells (monocytes) fromthe bloodstream by a process of transendothelial migration. These monocytes differentiateinto macrophages, which can consume the lipid and remove it from the intima by migrationto the adventitial lymphatics (Moore et al., 2013). However, when lipid-filled macrophages(known as foam cells) die within the plaque, they release their lipid content and other cellulardebris. This reinforces the immune response and can lead to a cycle of chronic inflammationthat promotes plaque growth and, ultimately, the development of necrotic tissue.2 umenEndotheliumIntimaInternalElasticLaminaMedia
Figure 1: Schematic diagram of a cross-section through the inner artery wall (layer widthsnot to scale and outer adventitial layer not shown). Atherosclerotic plaques develop in the intima , which is separated from the blood flow in the lumen by a thin layer of cells calledthe endothelium . Deeper in the wall, the intima is separated from the media by a membranecalled the internal elastic lamina .At an intermediate stage of plaque formation, medial SMCs adopt an active (synthetic)state and migrate through the IEL into the intima. SMC activation is believed to be initiatedby an injured endothelium, which may suffer mechanical disruption as the intimal layer ex-pands to accommodate the infiltration of lipid and immune cells (Faggiotto and Ross, 1984).Once inside the plaque, SMCs replace the existing intimal extracellular matrix (ECM) witha dense matrix of fibrillar collagens (Adiguzel et al., 2009). SMCs are subject to an abun-dance of signalling cues in the plaque, but two chemicals known to be particularly critical tocap formation are platelet-derived growth factor (PDGF) and transforming growth factor- β (TGF- β ). PDGF plays an important role as a SMC mitogen and chemoattractant (Ruther-ford et al., 1997; Sano et al., 2001), while TGF- β enhances plaque stability by stimulatingSMC collagen production (Mallat et al., 2001; Lutgens et al., 2002). PDGF and TGF- β can be produced by a variety of plaque cells, but circulating platelets that adhere to sitesof endothelial injury are believed to be a significant source of both growth factors (Ross,1999; Toma and McCaffrey, 2012). TGF- β is secreted as a latent complex and the activecomponent of TGF- β must be liberated before it can bind to target cells (Singh and Ramji,2006). Liberation of active TGF- β typically requires chemical or mechanical stimulation,and TGF- β derived from platelets is believed to be activated by shear forces in the blood-stream (Ahamed et al., 2008). A schematic diagram of the cap formation process is providedin Figure 2. 3 UMEN INTIMA MEDIA endothelium collagenousECM internal elastic laminaadherent platelets contractile SMCssynthetic SMCs
PDGF, TGF- β Gradients
Figure 2: Schematic diagram of the key processes in atherosclerotic cap formation. Lipopro-tein accumulation in the intima triggers an immune response that leads to intimal growthand endothelial disruption. Diffusible growth factors PDGF and TGF- β are released fromsites of injury by endothelial cells and adherent platelets. Contractile SMCs in the media arestimulated to adopt a synthetic state and respond chemotactically to PDGF by migratingthrough the internal elastic lamina. Once inside the plaque, synthetic SMCs remodel theexisting ECM and are stimulated by TGF- β to deposit a dense cap of collagenous tissueadjacent to the endothelium.Recent advances in cell lineage tracing techniques have begun to reveal new insightsinto the behaviour of plaque SMCs. Chappell et al. (2016) and Jacobsen et al. (2017)have performed studies using so-called Confetti transgenic mice, where individual medialSMCs can be labelled by inducing the unique expression of one of four possible fluorescentproteins in each cell. Both of these studies have identified that SMC populations in advancedplaques consist of several large monochromatic patches, indicating that plaque SMCs areoligoclonal and derived from only a limited number of medial progenitor cells. These resultsare significant because they highlight, in the ApoE mouse at least, that proliferation makesa much larger contribution to SMC accumulation in plaques than previously thought.These studies provide an exciting new window into the cellular mechanisms of fibrous capformation, and have significant potential to advance current understanding of atherosclerosisprogression. However, much is still unknown about the factors that underlie the formationand maintenance of the collagenous cap and, in particular, the reasons why certain capsremain thick and stable whilst others become thin and vulnerable to rupture. Several possiblemodes of long-term cap degradation have been proposed in the literature. These include anincrease in SMC death — possibly due to cellular senescence (Wang et al., 2015) — or4n increase in collagen destruction due to either elevated matrix metalloproteinase (MMP)production by inflammatory cells (Hansson et al., 2015) or reduced SMC sensitivity to TGF- β signalling (Chen et al., 2007; Vengrenyuk et al., 2015). The model that we develop inthis paper addresses these gaps in biological understanding by establishing a framework tosimulate cap formation dynamics and assess the deleterious impact of a variety of ECMdegradation mechanisms.We have recently published a foundational two-phase PDE model that studied the migra-tion and proliferation of media-derived SMCs in response to an endothelium-derived source ofdiffusible PDGF (Watson et al., 2018). The model domain was taken to be a one-dimensionalcross-section through the diseased intima and the flux of SMCs and PDGF into the plaquewere captured by a set of non-standard boundary conditions. The model did not explicitlyconsider the synthesis of a fibrous cap, but the approach did provide several interesting in-sights into the mechanisms that regulate SMC migration to the cap region. In particular, themodel predicted that SMC recruitment from the media is likely to be a rate-limiting factorfor SMC accumulation in plaques — an observation that is supported by the recent lineagetracing studies of Chappell et al. (2016) and Jacobsen et al. (2017). In the current paper,we build upon this earlier model by incorporating a profile of diffusible, endothelium-derivedTGF- β and an explicit representation of collagenous ECM remodelling. This allows us toperform a detailed investigation of growth factor-stimulated fibrous cap formation by plaqueSMCs. Note that the new model is more than just an elementary extension of the earlierapproach. We assume that the SMCs and the nascent ECM are coupled by a nonlinear me-chanical feedback that allows us to investigate the influence of factors including haptotacticSMC migration, which may play an important role in the cap formation process (Nelsonet al., 1996; Lopes et al., 2013).Interest in mathematical modelling of the cell activity in atherosclerosis has grown overrecent years. The majority of studies published to date have focussed on modelling theinflammatory response that characterises the early stages of plaque development (El Khatibet al., 2007; Pappalardo et al., 2008; Cohen et al., 2014; Chalmers et al., 2015), whereseveral groups have coupled their models to detailed representations of blood flow and/orintimal growth (Bulelzai and Dubbeldam, 2012; Filipovic et al., 2013; Islam and Johnston,2016; Yang et al., 2016; Bhui and Hayenga, 2017). In contrast, only a handful of modelshave been proposed to study events in the later stages of plaque progression. McKay et al.(2004) performed foundational work in this area by proposing a PDE model that includedboth plaque SMC recruitment and subsequent collagen deposition. The only other model toconsider collagen synthesis by invading SMCs is that of Cilla et al. (2014), who developed acomprehensive 3D model of blood flow, transmural transport and plaque growth. Neither ofthese modelling approaches, however, provide an adequate description of the cap formationprocess. Several other models of plaque progression (Poston and Poston, 2007; Friedman5nd Hao, 2015) — including studies of PDGF-induced intimal thickening (Fok, 2012) andintraplaque haemorrhage (Guo et al., 2018) — have considered SMCs independently of theircollagen-synthesising activity. Interested readers are referred to Parton et al. (2016) fora comprehensive review of mathematical and computational approaches to atherosclerosismodelling.Beyond atherosclerosis, a variety of mathematical models have been developed to studythe response of vascular SMCs to other sources of endothelial injury. One area that has at-tracted significant research interest is in-stent restenosis — the rapid recurrence of a narrowedlumen after surgical deployment of an artery-widening stent. Mechanical stresses imposedby the stent can locally denude the endothelium and elicit an intense healing response thatinvolves rapid proliferation of medial SMCs and significant neointima formation. Discreteand continuous models of SMC behaviour during in-stent restenosis have been developedby several research groups (Lally and Prendergast, 2006; Evans et al., 2008; Zahedmaneshet al., 2014; Tahir et al., 2015). Researchers have also developed models of the vascularSMC response to surgical interventions such as vein grafting (Budu-Grajdeanu et al., 2008;Garbey et al., 2017) and blood filter insertion (Nicolas et al., 2015). The tissue repair carriedout by artery wall SMCs in response to vascular injury also shares several similarities withthe process of dermal wound healing. In dermal wounds, fibroblasts take the role of theSMC and migrate to the wound site to regenerate the damaged collagenous tissue. Woundfibroblasts are known to be stimulated by growth factors including PDGF and TGF- β , andthe corresponding implications for healing have been studied in a variety of modelling frame-works (Olsen et al., 1995; Cobbold and Sherratt, 2000; Haugh, 2006; McDougall et al., 2006;Cumming et al., 2010; Menon et al., 2012).The model of cap formation that we present in this paper is inspired by the multiphasetheory developed in Byrne and Owen (2004), Lemon et al. (2006) and Astanin and Preziosi(2008). Multiphase models have been widely developed to study articular cartilage (reviewedin Klika et al. (2016)), tumour growth (Preziosi and Tosin, 2009; Hubbard and Byrne, 2013)and tissue engineering applications (O’Dea et al., 2013; Pearson et al., 2014), but our previousstudy (Watson et al., 2018) was the first application of the approach in atherosclerosis.Other existing studies of plaque development have predominantly utilised reaction-diffusionequations to model the spatio-temporal evolution of cells and tissues in the plaque (McKayet al., 2004; El Khatib et al., 2007; Filipovic et al., 2013; Cilla et al., 2014; Chalmers et al.,2015, 2017). However, multiphase models can provide a more detailed representation ofplaque formation dynamics because they provide a natural framework to account for volumeexclusion and mechanical interactions between individual plaque constituents. We continueto develop our multiphase approach in this study because we firmly believe that both volumeexclusion and mechanical effects can play a significant role in the development of plaquespatial structure. 6n the next section, we derive the model equations and introduce the model parameterisa-tion. Where possible, we base our assumptions and our parameter selections on observationsof plaque growth in the ApoE mouse, which is the most well-characterised animal model of in vivo atherosclerosis in the experimental literature (Getz and Reardon, 2012). We presentthe model results in Section 3, where we report some interesting analysis before performinga series of numerical simulations. We use these results to investigate the dynamic cellular,biochemical and mechanical mechanisms that regulate cap formation and aim to identify thefactors that are most important to the development and maintenance of plaque stability.Finally, we conclude with a discussion of the outcomes of the study in the context of bothcomputational and experimental atherosclerosis research. The plaque tissue in the model is represented as a mixture of three distinct phases: a cellularphase that comprises matrix-synthesising SMCs, an ECM phase that comprises collagen-richfibrous tissue, and a generic phase that is assumed to comprise the remaining plaque con-stituents (foam cells, extracellular lipids, interstitial fluid, etc.). We derive mass and mo-mentum balance equations for each phase, and the equations are closed by imposing suitableconstitutive assumptions and conditions for the transfer of mass and momentum betweeneach pair of phases. Equations for the concentrations of diffusible PDGF and (active) TGF- β are also included in the model. We assume that PDGF promotes a migroproliferative SMCphenotype, while TGF- β promotes a matrigenic SMC phenotype (Alexander and Owens,2012). These distinct SMC phenotypes are not explicitly represented in the model. Instead,we assume that the SMC phase contains a continuum of different phenotypes whose aver-age behaviour is regulated by the local growth factor concentrations. We assume that bothPDGF and TGF- β reside exclusively in the generic tissue phase, and that they exert influ-ence on the SMC and ECM phases via stress and/or mass transfer terms that are functionsof their respective concentrations.We model the above system on a one-dimensional Cartesian domain that represents across-section of diseased arterial intima far from the edges of the plaque. We assume that thedomain is bounded by the endothelium at x = 0 and by the IEL at x = L . The dependentvariables are therefore functions of time t (cid:62) x ∈ [0 , L ]. We denote the volumefractions of the SMC, ECM and generic tissue phases by m ( x, t ), ρ ( x, t ) and w ( x, t ), andtheir corresponding intraphase stresses by τ m ( x, t ), τ ρ ( x, t ) and τ w ( x, t ), respectively. Weassume that the ECM phase takes the form of a rigid scaffold and the ECM phase thereforehas zero velocity for all x and t . We denote the (non-zero) velocities of the SMC and generictissue phases by v m ( x, t ) and v w ( x, t ). The interstitial fluid pressure is denoted by p ( x, t ),and the concentrations of PDGF and TGF- β in the generic tissue phase are denoted by7 ( x, t ) and T ( x, t ), respectively. In this section we use the principles of mass and momentum conservation to derive equationsthat describe the SMC, ECM and generic tissue phase dynamics in response to diffusiblePDGF and TGF- β in the plaque. Assuming that all three phases share the same constant density, the mass balance equationsfor m , ρ and w can be expressed as follows: ∂m∂t + ∂∂x ( v m m ) = S m , (1) ∂ρ∂t = S ρ , (2) ∂w∂t + ∂∂x ( v w w ) = − ( S m + S ρ ) . (3)Here, the functions S m and S ρ represent the net rates of SMC and ECM production. Wehave assumed in equation (3) that there is no local source or sink of material and thatmass simply transfers from one phase to another as SMCs proliferate or die, and as ECMis synthesised or degraded. We further assume that there are no voids in the plaque tissue,and the three volume-occupying phases therefore satisfy the condition: m + ρ + w = 1 . (4) Neglecting inertial effects and assuming that no external forces act on the system, the mo-mentum balance equations for m , ρ and w reduce to a balance between the intraphase and8nterphase forces that act on each phase: ∂∂x ( τ m m ) = F ρm + F wm , (5) ∂∂x ( τ ρ ρ ) = F mρ + F wρ , (6) ∂∂x ( τ w w ) = F mw + F ρw , (7)where F ij (= − F ji ) denotes the force exerted by phase j on phase i .Following Lemon et al. (2006), we assume that the F ij terms in equations (5)–(7) arecomprised of interphase pressure and interphase drag components: F mρ = ( p + ψ ) ρ ∂m∂x − ( p + ψ ) m ∂ρ∂x − k mρ mρv m , (8) F mw = pw ∂m∂x − pm ∂w∂x + k mw mw ( v w − v m ) , (9) F ρw = pw ∂ρ∂x − pρ ∂w∂x + k ρw ρwv w . (10)The first term on the right hand side of each of equations (8)–(10) describes the interfacialforce exerted by phase j on phase i . This force is assumed to be proportional to the interphasepressure between the phases and to the degree of contact of phase j with phase i . The forceis assumed to act in the direction of increasing interfacial contact and, on the macroscale,this contributes a further term proportional to the gradient in phase i . The second termon the right hand side of each equation contributes a corresponding reaction force, which isexerted by phase i on phase j . Note that we have included an additional (contact-dependent)interphase pressure contribution ψ ≡ ψ ( m, ρ ) in the equation for F mρ . This additionalpressure is included to account for traction forces that are generated as the SMCs translocatethrough the ECM. We assume that traction forces between the other pairs of phases in thesystem are negligible in comparison, so the equations for F mw and F ρw include only thecontact-independent pressure p in their interphase pressure terms. The final term on theright hand side of each equation represents the interphase drag, which we assume to beproportional to the relative velocities of phases i and j with constant of proportionality k ij (= k ji ). We further assume that the drag depends linearly on the volume fraction of eachphase i and j so that no drag can occur in the absence of either phase.The generic tissue phase contains several active constituents such as foam cells andmacrophages, but we shall assume here that it behaves as an inert isotropic fluid. We9ake this assumption in order to focus on how the SMC phase interacts with other keyfactors in the plaque environment. We shall assume that the SMC phase displays morecomplex behaviour than the generic tissue phase and, in particular, that SMCs alter theirmotility in response to both the local PDGF concentration and the local ECM volume frac-tion. Neglecting viscous effects, and combining the approaches of Byrne and Owen (2004)and Lemon et al. (2006), we adopt the following models for the intraphase stress terms inequations (5)–(7): τ m = − [ p + ρψ ( m, ρ ) + Λ( P ) ] , (11) τ ρ = − [ p + mψ ( m, ρ ) ] , (12) τ w = − p. (13)In equation (11), the extra pressures Λ( P ) and ρψ ( m, ρ ) describe the respective influences ofthe PDGF concentration and the ECM volume fraction on the pressure in the SMC phase.The ECM-derived extra pressure term is proportional to the interphase pressure contribution ψ and is weighted according to the degree of contact between the SMC and ECM phases.We assume that the SMC phase has a complementary influence on the pressure in the ECMphase, and this is captured by the extra pressure term mψ ( m, ρ ) in equation (12). We definethe functions Λ and ψ to be (Byrne and Owen, 2004; Lemon et al., 2006):Λ( P ) = χ P κP ) n P , (14) ψ ( m, ρ ) = − χ ρ + δm n ρ (1 − m − ρ ) n ρ , (15)where χ P , κ , n P , χ ρ , δ and n ρ are positive parameters. Equation (14) is deliberately chosento be a decreasing function of P as this ensures that chemotactic SMC migration up PDGFgradients will provide stress relief in the SMC phase. The first term in equation (15), whichensures that ψ will be negative for sufficiently small m , reflects an assumption that the SMCshave an affinity for the collagenous ECM. Note that this adhesion mechanism introduces thepotential for haptotactic SMC migration in the presence of ECM gradients. The second termin equation (15) contributes a net repulsion on the SMC phase that increases significantlyas m or ρ become sufficiently large. 10 .1.3 Mass Transfer Terms We assume that the SMC phase source term S m in equation (1) includes both a linear deathterm and a proliferation term that requires the uptake of material from the generic tissuephase. We assume that the net SMC proliferation rate is proportional to the sum of aconstant baseline mitosis rate r m and an additional rate that depends on the local PDGFconcentration. PDGF is a well-known mitogen for SMCs, and in vitro evidence suggests thatSMC proliferation can be significantly upregulated with saturating kinetics in the presenceof increasing PDGF concentrations (Munro et al., 1994). Combining these considerations,we define: S m = r m mw (cid:20) A m Pc m + P (cid:21) − β m m, (16)where A m quantifies the maximum possible PDGF-stimulated increase in the baseline SMCmitosis rate, c m represents the PDGF concentration at which the half-maximal increase inthe baseline SMC mitosis rate occurs, and β m denotes the rate of SMC death. We remarkhere that, while normal vascular SMCs are known to be growth inhibited by TGF- β , in vitro experiments have shown that atherosclerotic plaque-derived SMCs retain their proliferativecapacity in the presence of TGF- β (McCaffrey et al., 1995).We define the ECM phase source term S ρ in equation (2) by assuming that the collagenousmatrix is continuously remodelled by the cells that occupy the plaque. Based on a variety ofexperimental observations, we assume that both PDGF and TGF- β are important mediatorsof this remodelling process. We propose the following form for the ECM phase source term,where we assume that ECM is synthesised by SMCs but is also degraded by both SMCs andplaque immune cells: S ρ = r s mw (cid:20) A s Tc s + T (cid:21) − r d mρ (cid:20) A d P ( c d + P ) (1 + γ d T ) (cid:21) − β ρ ρw (cid:20) ε γ ρ T γ ρ T (cid:21) . (17)The first term in equation (17) represents the net rate of ECM synthesis by the SMCphase, which we assume to be proportional to the sum of a constant baseline ECM produc-tion rate r s and an additional rate that depends on the local TGF- β concentration. The roleof TGF- β as a stimulant for SMC collagen production in plaques has been clearly demon-strated by in vivo studies (Mallat et al., 2001; Lutgens et al., 2002) and, following in vitro observations (Kubota et al., 2003), we assume that the rate of ECM synthesis increases withsaturating kinetics in response to increasing T . The parameter A s quantifies the maximumpossible TGF- β -stimulated increase in the baseline ECM synthesis rate, while c s denotes theTGF- β concentration at which the half-maximal increase in the baseline ECM synthesis rateoccurs. Note that the ECM synthesis rate is also proportional to w because we assume thatECM synthesis by SMCs again requires the uptake of material from the generic tissue phase.11he second and third terms in equation (17) represent the respective net rates of ECMdegradation by plaque SMCs and by plaque immune cells, both of which are known to pro-duce MMPs. We assume that the net rate of ECM degradation by SMCs is proportionalto the sum of a constant baseline degradation rate r d and an additional rate that dependson the local concentration of both growth factors. In vitro studies indicate that SMCs up-regulate their production of MMP-2 and MMP-9 in response to stimulation with PDGF,but also that co-stimulation with TGF- β can inhibit this response (Borrelli et al., 2006;Risinger et al., 2010). We represent this behaviour with a simple functional form where A d quantifies the maximum possible PDGF-stimulated increase in the baseline ECM degrada-tion rate, c d denotes the PDGF concentration at which the half-maximal PDGF-stimulatedincrease in the baseline ECM degradation rate occurs and γ d quantifies the extent of inhi-bition of PDGF-stimulated ECM degradation by TGF- β . Experimental observations alsoindicate that TGF- β can protect the plaque ECM from degradation by MMP-9-producingmacrophages (Vaday et al., 2001; Ogawa et al., 2004). Assuming that macrophages andother inflammatory cell types comprise a certain fraction of the generic tissue phase, wecapture this behaviour in the final term by assuming that the net rate of immune cell ECMdegradation is proportional to both w and a decreasing function of T . The parameter β ρ denotes the maximum possible rate of immune cell ECM degradation, which we assume tobe related to the extent of inflammation in the plaque tissue. The dimensionless parameter ε (cid:54) β -inhibited immune cell ECM degration(i.e. εβ ρ ), while γ ρ quantifies the rate at which immune cell ECM degradation decreases withincreasing T . In this section we derive reaction-diffusion equations that describe how the concentrationsof diffusible PDGF and TGF- β evolve in the plaque tissue. In addition to the equationspresented below, the model also includes a source term for each growth factor at the en-dothelium ( x = 0) and sink term for each growth factor at the media ( x = L ). Theseadditional terms are given by boundary conditions and will be discussed in Section 2.4. We assume that the characteristic timescales of PDGF and TGF- β diffusion are significantlyshorter than the characteristic timescales of SMC migration and ECM remodelling. Wetherefore neglect advective growth factor transport in the generic tissue phase and assume12he following quasi-steady state mass balance equations for P and T :0 = D P ∂∂x (cid:20) w ∂P∂x (cid:21) + S P , (18)0 = D T ∂∂x (cid:20) w ∂T∂x (cid:21) + S T . (19)Here, D P and D T are constant chemical diffusion coefficients, while S P and S T denote localsink terms for PDGF and TGF- β , respectively. Note that, following Astanin and Preziosi(2008), we have assumed that the diffusive flux of both chemicals is modulated by the factor w . This implies that the capacity for growth factor transport through the intimal tissue willreduce as SMCs and ECM accumulate in the plaque. We assume that both growth factors are taken up by SMCs in the plaque, and also thatboth growth factors undergo natural decay. These assumptions lead to the sink terms: S P = − η P mwP − β P wP, (20) S T = − η T mwT − β T wT, (21)where η P , η T , β P and β T denote the rates of PDGF uptake by SMCs, TGF- β uptake bySMCs, PDGF decay and TGF- β decay, respectively. Note that we include a factor of w ineach term to ensure that neither uptake nor decay of either growth factor can occur in theabsence of the generic tissue phase. In this section, we show how the three-phase model developed in Section 2.1 can be reducedto a system of two coupled nonlinear equations for the SMC and ECM phases. In all thatfollows, we shall assume that the volume fractions of all three phases m , ρ and w are strictlypositive for all x and t .Using equation (4) to replace all occurrences of w with the equivalent expression 1 − m − ρ ,the sum of the mass balance equations (1)–(3) reduces to: ∂∂x (cid:104) v m m + v w (1 − m − ρ ) (cid:105) = 0 . (22)Assuming a zero-flux condition for the SMC phase at the endothelium (i.e. v m = v w = 0 at13 = 0), integration of equation (22) with respect to x gives the following expression for thevelocity of the generic tissue phase: v w = − v m (cid:18) m − m − ρ (cid:19) . (23)Summing the momentum balance equations (5)–(7), and substituting in the expres-sions (11)–(13) gives the following equation for the tissue pressure gradient: ∂p∂x = − ∂∂x (cid:104) m (Λ + 2 ρψ ) (cid:105) . (24)A second expression for the tissue pressure gradient is obtained by substituting the expres-sions (9), (10) and (13) into equation (7) (note that equation (8) becomes redundant at thisjuncture). Cancelling terms, we obtain: ∂p∂x = k mw mv m − v w ( k mw m + k wρ ρ ) . (25)Equating the right hand sides of equations (24) and (25), and then using equation (23) toeliminate v w , we derive the following expression for v m in terms of m , ρ and P : v m = − (cid:20) − m − ρk mw m (1 − ρ ) + k wρ mρ (cid:21) ∂∂x (cid:104) m (Λ + 2 ρψ ) (cid:105) . (26)In the interest of simplicity, we shall assume that the drag between the generic tissue phaseand the SMC phase and between the generic tissue phase and the ECM phase is uniform(i.e. k mw = k wρ = k ). Making this simplification and substituting equation (26) backinto equation (1), we arrive at the final mass balance relationship for the SMC phase in theintima: ∂m∂t = 1 k ∂∂x (cid:20) (1 − m − ρ ) ∂∂x (cid:104) m (Λ + 2 ρψ ) (cid:105) (cid:21) + S m ( m, ρ, P ) . (27)Our reduced model therefore comprises equation (27) alongside the following expressions for14 MCs CollagenousECM Other Cells& Tissues
PDGFActiveTGF- β chemotaxismitosisdegradationsynthesis haptotaxisinhibition ofmovement/mitosis degradation byimmune cells SMC Phase 𝑚𝑚 𝑥𝑥 , 𝑡𝑡 CollagenousECM Phase 𝜌𝜌 𝑥𝑥 , 𝑡𝑡 GenericTissue Phase 𝑤𝑤 𝑥𝑥 , 𝑡𝑡 PDGF
𝑃𝑃 𝑥𝑥 , 𝑡𝑡 TGF- β 𝑇𝑇 𝑥𝑥 , 𝑡𝑡 Figure 3: Schematic diagram of the primary interactions that regulate fibrous cap formationin the model. The volume-occupying phases are shown in red and the volumeless chemicalgrowth factors are shown in blue. Labelled (solid) lines indicate interactions that act toincrease (arrows) or decrease (bars) the SMC and ECM volume fractions in the cap region.Unlabelled (dashed) lines denote the stimulatory (arrows) and inhibitory (bars) effects ofthe growth factors on the synthesis and degradation of the ECM phase. ρ , P and T : ∂ρ∂t = S ρ ( m, ρ, P, T ) , (28)0 = D P ∂∂x (cid:20) (1 − m − ρ ) ∂P∂x (cid:21) + S P ( m, ρ, P ) , (29)0 = D T ∂∂x (cid:20) (1 − m − ρ ) ∂T∂x (cid:21) + S T ( m, ρ, T ) . (30)A schematic diagram that summarises the key model interactions involved in fibrous capformation is provided in Figure 3. 15 .4 Boundary and Initial Conditions In this section we define the boundary conditions required to solve equations (27), (29)and (30), and the initial conditions required to solve equations (27) and (28). Note thatthe assumptions that we make are largely consistent with those made previously in Watsonet al. (2018).Recall that in the model simplification in Section 2.3 we have already assumed a zero-fluxcondition for SMCs at the endothelium. We therefore have the following boundary conditionfor the SMC phase: 1 k (1 − m − ρ ) ∂∂x (cid:104) m (Λ + 2 ρψ ) (cid:105) = 0 at x = 0= ⇒ ∂∂x (cid:104) m (Λ + 2 ρψ ) (cid:105) = 0 at x = 0 . (31)Note that, in practice, this boundary condition stipulates that any flux of SMCs towardsthe endothelium at x = 0 must be identically balanced by an equivalent flux of SMCs in theopposite direction.SMC invasion of the plaque requires the activation of quiescent SMCs in the media bychemical signalling, and subsequent production of progeny that can negotiate the porousIEL (Chappell et al., 2016; Jacobsen et al., 2017). The IEL provides a physical barrier tocell movement, so it is likely that a sustained and directed migratory response is requiredfor synthetic SMCs to enter the plaque. We therefore assume that chemotaxis is the domi-nant mechanism of SMC migration across the IEL, and that any contribution from passiveSMC diffusion or from ECM-mediated haptotaxis is negligible. We introduce the parameter m M < k (1 − m − ρ ) ∂∂x (cid:104) m (Λ + 2 ρψ ) (cid:105) = 1 k (1 − m − ρ ) d Λ dP m M ∂P∂x at x = L = ⇒ ∂∂x (cid:104) m (Λ + 2 ρψ ) (cid:105) = d Λ dP m M ∂P∂x at x = L. (32)There are several points to note regarding the format of this boundary condition. First, themodel assumes that no SMCs will enter the intima in the absence of a PDGF gradient atthe medial boundary. Second, the condition does not stipulate continuity of m across theboundary (i.e. in general, m (cid:54) = m M at x = L ). Finally, since we are assuming a closed system,the boundary condition implies that any flux of SMCs into the intima must be balanced by16n equivalent efflux from the generic tissue phase into the media. Such an efflux of tissueconstituents such as macrophages and foam cells may not be entirely physically realistic,but we resolve that our modelling assumptions remain reasonable provided that the totalinflux of plaque SMCs remains relatively small. An extended model that explicitly capturesintimal growth could, of course, provide a more comprehensive treatment of the problem.However, we postpone modelling of domain growth for future work because we believe thatthe current approach provides significant insight into the cap formation process without theadded complexity of tracking a moving boundary.Stimulated endothelial cells and adherent platelets at the lesion site are believed to bethe dominant sources of PDGF in the plaque (Funayama et al., 1998), and we model thisinflux of endothelium-derived PDGF with the following boundary condition: D P (1 − m − ρ ) ∂P∂x = − α P (1 − m − ρ ) at x = 0= ⇒ ∂P∂x = − α P D P at x = 0 , (33)where α P quantifies the rate of PDGF flux into the intima, and the term (1 − m − ρ ) isincluded for consistency with the modulated PDGF transport inside the model domain. Notethe corresponding implication that the absolute rate of PDGF influx into the plaque willreduce as the SMC and ECM phases accumulate at the luminal boundary. This introduces animplicit self-regulation mechanism in the model whereby SMCs can effectively downregulatethe signal that recruits them to the cap region.The boundary condition (32) stipulates that SMC migration into the plaque from themedia requires a negative PDGF gradient at the medial boundary. We therefore treat theIEL as a permeable membrane and assume that PDGF can diffuse into the media from theintima according to the following boundary condition: D P (1 − m − ρ ) ∂P∂x = σ P ( P M − P ) (1 − m − ρ ) at x = L = ⇒ ∂P∂x = σ P D P ( P M − P ) at x = L. (34)Here, σ P denotes the permeability of the IEL to PDGF, P M represents the PDGF concen-tration in the media, and the term (1 − m − ρ ) is again included for consistency with themodulated chemical transport inside the domain.The boundary conditions that we impose for TGF- β take an identical format to thosedefined for PDGF. Like PDGF, TGF- β is also released by degranulating platelets and acti-17ated endothelial cells at the lesion site (Toma and McCaffrey, 2012). We therefore definethe following condition at the endothelium: D T (1 − m − ρ ) ∂T∂x = − α T (1 − m − ρ ) at x = 0= ⇒ ∂T∂x = − α T D T at x = 0 , (35)where α T quantifies the rate of TGF- β flux into the intima. At the medial boundary, weassume: D T (1 − m − ρ ) ∂T∂x = σ T ( T M − T ) (1 − m − ρ ) at x = L = ⇒ ∂T∂x = σ T D T ( T M − T ) at x = L, (36)where σ T denotes the permeability of the IEL to TGF- β , and T M represents the TGF- β concentration in the media. Note that the decision to allow TGF- β to diffuse out of thedomain is made simply for consistency with the boundary condition (34) for PDGF. Thepresence of an explicit TGF- β gradient at the medial boundary is not required to initiatecap formation in the model.Finally, we must define initial profiles for the SMC and collagenous ECM phases in theplaque. Observations of plaque development in mice indicate that the SMC population inthe intima is negligible prior to cap formation (Bennett et al., 2016). The model equationscannot support a domain that is entirely devoid of SMCs (see the singularity that arises inequation (26), for example), so we initiate the plaque with a small and uniform initial SMCvolume fraction: m ( x,
0) = m i , where 0 < m i (cid:28) . (37)We also assume a small and uniform initial profile for the collagenous ECM phase in theplaque: ρ ( x,
0) = ρ i , where 0 < ρ i (cid:28) . (38)This initial collagen is assumed to represent a fraction of the ECM that exists in theintima prior to SMC invasion. The remaining constituents of the initial plaque ECM, whosequantities we do not explicitly track, are assumed to reside in the generic tissue phase andare notionally replaced by the deposition of new collagen over the course of a simulation.18 .5 Model Non-Dimensionalisation Using tildes to denote dimensionless quantities, we rescale space x , time t , PDGF concen-tration P and TGF- β concentration T as follows (note that the SMC volume fraction m andthe ECM volume fraction ρ do not require rescaling): (cid:101) x = xL , (cid:101) t = tt , (cid:101) P = PP , (cid:101) T = TT , (cid:101) m = m, (cid:101) ρ = ρ, where t , P and T represent characteristic time, PDGF concentration and TGF- β concen-tration values, respectively. The model parameters can now be non-dimensionalised in thefollowing way: (cid:102) χ P = χ P t kL , (cid:101) κ = κP , (cid:102) χ ρ = 2 χ ρ t kL , (cid:101) δ = 2 δt kL , (cid:102) r m = r m t , (cid:102) c m = c m P , (cid:102) β m = β m t , (cid:101) r s = r s t , (cid:101) c s = c s T , (cid:101) r d = r d t , (cid:101) c d = c d P , (cid:101) γ d = γ d T , (cid:101) β ρ = β ρ t , (cid:101) γ ρ = γ ρ T , (cid:102) η P = η P L D P , (cid:102) β P = β P L D P , (cid:102) η T = η T L D T , (cid:102) β T = β T L D T , (cid:102) α P = α P LD P P , (cid:102) α T = α T LD T T , (cid:102) σ P = σ P LD P , (cid:102) σ T = σ T LD T , (cid:102) P M = P M P , (cid:102) T M = T M T . ∂m∂t = ∂∂x (cid:20) (1 − m − ρ ) ∂∂x (cid:104) m (Λ + ρψ ) (cid:105) (cid:21) + r m m (1 − m − ρ ) (cid:20) A m Pc m + P (cid:21) − β m m, (39) ∂ρ∂t = r s m (1 − m − ρ ) (cid:20) A s Tc s + T (cid:21) − r d mρ (cid:20) A d P ( c d + P ) (1 + γ d T ) (cid:21) − β ρ ρ (1 − m − ρ ) (cid:20) ε γ ρ T γ ρ T (cid:21) , (40) ∂∂x (cid:20) (1 − m − ρ ) ∂P∂x (cid:21) = η P mP (1 − m − ρ ) + β P P (1 − m − ρ ) , (41) ∂∂x (cid:20) (1 − m − ρ ) ∂T∂x (cid:21) = η T mT (1 − m − ρ ) + β T T (1 − m − ρ ) , (42) ∂∂x (cid:104) m (Λ + ρψ ) (cid:105) = 0 at x = 0 , ∂∂x (cid:104) m (Λ + ρψ ) (cid:105) = d Λ dP m M ∂P∂x at x = L, (43) ∂P∂x = − α P at x = 0 , ∂P∂x = σ P ( P M − P ) at x = L, (44) ∂T∂x = − α T at x = 0 , ∂T∂x = σ T ( T M − T ) at x = L, (45) m ( x,
0) = m i , (46) ρ ( x,
0) = ρ i , (47)wherein Λ( P ) = χ P κP ) n P and ψ ( m, ρ ) = − χ ρ + δm n ρ (1 − m − ρ ) n ρ . A summary of the base case parameter values is provided in Table 1. Our parameter selec-tions have mostly been informed by data and observations from relevant in vitro and in vivo experimental studies. When appropriate data could not be found, we have chosen values20hat ensure biologically realistic results. Several important parameters (e.g. A m , c m , χ P , κ , n P ) have been assigned values that are dimensionally the same (or very similar) to those inour earlier model of cap formation. We provided detailed justifications for these parametervalues in Watson et al. (2018), so below we focus on explaining the parameter selections thatare unique to the current study. In vivo studies in the ApoE-deficient mouse have shown that plaque fibrous caps typi-cally form over the course of several months (Kozaki et al., 2002). We therefore assume acharacteristic timescale of approximately 1 month and set t = 2 . × s. For the domainlength, we set L = 120 µ m. This value is based on data from Reifenberg et al. (2012), whomeasured intimal thickness in the diseased aortic arches of mice around the time of initialplaque SMC infiltration. For PDGF and TGF- β , we assume the characteristic concentrations P = 10 ng ml − and T = 1 ng ml − . These values reflect reported concentrations of PDGFin platelets (15–50 ng ml − ; Huang et al. (1988)) and TGF- β in blood plasma (2–12 ng ml − ;Wakefield et al. (1995)).In a detailed model of the fibroblast response to PDGF in dermal wound healing, Haugh(2006) estimated PDGF diffusion coefficient and decay rate values that suggest a PDGF dif-fusion distance of approximately 300 µ m. Based on these estimates, we set our dimensionlessPDGF decay rate to be β P = 0 .
2. As TGF- β has an almost identical molecular weight toPDGF, we expect that TGF- β will diffuse at a similar rate to PDGF in the plaque tissue.However, active TGF- β has been reported to decay around two orders of magnitude fasterthan PDGF (Wakefield et al., 1990), and we therefore set β T = 20. The other parametersthat determine the plaque growth factor profiles are more difficult to estimate. We assume,for consistency, that σ P = σ T and we choose corresponding values for α P and α T that givereasonable growth factor concentration ranges in the intima. The response of the model tochanges in the values of α P and α T will be an important focus of our numerical simulationstudies in Section 3.2. Note that we also set P M = T M = 0 because neither growth factor isknown to have a prominent source in the medial tissue.ECM remodelling in the model is governed by a range of concentration-dependent stim-ulatory and inhibitory effects of PDGF and TGF- β . The parameter values in these rela-tionships have all been informed by data from in vitro experiments. Kubota et al. (2003)used SMCs derived from human atherosclerotic plaques to show that TGF- β can increasebaseline collagen synthesis up to 2-fold, with a half-maximal response occurring for a TGF- β concentration in the interval 0.2–0.5 ng ml − . We therefore set A s = 1 and c s = 0 .
35. Valuesfor ε and γ ρ are based on results from Vaday et al. (2001), who studied the impact of TGF- β on tumour necrosis factor (TNF)- α -induced expression of MMP-9 in monocytes. Note thatTNF- α is known to be an important inflammatory cytokine in atherosclerosis progression(Urschel and Cicha, 2015). Vaday et al. (2001) showed that, in the presence of 1 ng ml − TNF- α , MMP-9 activity decreased with increasing TGF- β concentration and was reduced by21 arameter Description Dimensionless Referencevalue n P Exponent in SMC phase extra pressure function Λ 1.8 Schachter (1997) κ Reciprocal of reference PDGF concentration in SMC phase extrapressure function Λ 5.5 Schachter (1997) χ P Baseline SMC phase motility coefficient 1.75 Cai et al. (2007) n ρ Exponent in SMC phase extra pressure function ψ δ SMC phase repulsion coefficient 0.45 χ ρ SMC phase affinity for ECM phase 0.3 r m Baseline rate of SMC phase proliferation 0.25 Breton et al. (1986)Chappell et al. (2016) A m Maximal factor of PDGF-stimulated increase in rate of SMC pro-liferation 14 Munro et al. (1994) c m PDGF concentration for half-maximal increase in rate of SMCproliferation 1.5 Munro et al. (1994) β m Rate of SMC phase loss 0.6 Lutgens et al. (1999) r s Baseline rate of ECM phase synthesis by SMCs 1.8 Reifenberg et al. (2012) A s Maximal factor of TGF- β -stimulated increase in rate of ECM syn-thesis 1 Kubota et al. (2003) c s TGF- β concentration for half-maximal increase in rate of ECMsynthesis 0.3 Kubota et al. (2003) r d Baseline rate of ECM phase degradation by SMCs 1.5 A d Maximal factor of PDGF-stimulated increase in rate of ECMdegradation by SMCs 4 Borrelli et al. (2006) c d PDGF concentration for half-maximal increase in rate of ECMdegradation by SMCs 2.5 Borrelli et al. (2006) γ d Reciprocal of reference TGF- β concentration for inhibition ofPDGF-stimulated ECM degradation by SMCs 0.5 Borrelli et al. (2006) β ρ Baseline rate of ECM phase degradation by immune cells 0.75 ε Maximal factor of TGF- β -stimulated decrease in rate of ECMdegradation by immune cells 0.25 Vaday et al. (2001) γ ρ Reciprocal of reference TGF- β concentration for inhibition ofECM degradation by immune cells 10 Vaday et al. (2001) η P Rate of PDGF uptake by SMC phase 2.5 β P Rate of PDGF decay 0.2 Haugh (2006) η T Rate of TGF- β uptake by SMC phase 2.5 β T Rate of TGF- β decay 20 Wakefield et al. (1990) m M Volume fraction of synthetic SMCs in media 0.01 α P Rate of PDGF influx from endothelium 0.7 σ P Permeability of IEL to PDGF 4 P M PDGF concentration in media 0 α T Rate of TGF- β influx from endothelium 2.5 σ T Permeability of IEL to TGF- β T M TGF- β concentration in media 0 m i Initial SMC volume fraction in intima 10 − Bennett et al. (2016) ρ i Initial ECM volume fraction in intima 0.02 Reifenberg et al. (2012)
Table 1: Base case parameter values. The final column reports any references that havebeen used to calculate individual parameter values. Values that do not have references havebeen chosen to ensure biologically reasonable results. Unless otherwise stated, all reportedsimulations use these values. 22round 50–75% for TGF- β concentrations in the range 0.1–1 ng ml − . We therefore estimate ε = 0 .
25 and γ ρ = 10. The parameter values that quantify the competing effects of PDGFand TGF- β on ECM remodelling by SMCs have been calculated using data from Borrelliet al. (2006). Borrelli et al. (2006) showed that SMCs demonstrate very similar qualitativepatterns of MMP-2 and MMP-9 release in the presence of varying concentrations of PDGFand TGF- β . We therefore base our quantitative estimates on the averaged responses of thesetwo MMPs to PDGF and TGF- β . In the absence of TGF- β , the data suggest an average2.5-fold increase in basal MMP release at 20 ng ml − PDGF and an average 4-fold increaseat 50 ng ml − PDGF. When 5 ng ml − TGF- β is introduced at 50 ng ml − PDGF, the releaseof MMPs is cut, on average, by around 50%. Based on these observations, we select A d = 4, c d = 2 . γ d = 0 . r s , r d and β ρ ) must be determined separately. It is difficult to ascertain typicalvalues for these parameters during in vivo plaque progression, so we base our parameterselections on the following considerations. First, we assume that, as invasive plaque SMCsundertake a process of continual ECM remodelling, the values of r s and r d should be of asimilar order. Furthermore, we assume that the value of r s must be sufficiently large toallow plaque collagen to accumulate on a timescale similar to that reported in Reifenberget al. (2012). Second, we assume that the value of β ρ should be smaller than the value of r d . We make this assumption because, while inflammatory cells such as macrophages maydegrade the plaque ECM more aggressively than SMCs, these cells make up only a fractionof the content of the generic tissue phase. Based on these assumptions, we estimate r s = 1 . r d = 1 . β ρ = 0 . r m = 0 .
02. This value was estimated based on previous experimental observations thatsuggested a proliferative index of around 1% for plaque SMCs (i.e. 1% of plaque SMCsdisplaying evidence of mitotic activity at a given point in time). However, as discussed inSection 1, recent SMC lineage tracing studies have indicated that plaque SMC proliferationmay be more prominent than previously appreciated. For example, Chappell et al. (2016)have recently reported a plaque SMC proliferative index of approximately 4%. Furthermore,this value was determined at an advanced stage of plaque progression, where a significantSMC population had already assembled. Hence, accounting for likely effects such as cell-cell and cell-ECM contact inhibition, we anticipate that the plaque SMC proliferative indexcould be even larger in early cap formation. We therefore assume a significant increase inour previous estimate and set r m = 0 .
25. To our knowledge, no recent lineage tracing studyhas reported an equivalent calculation for the SMC apoptotic index. Hence, for the SMCdeath rate in the model, we use the apoptotic index of 1% that was calculated by Lutgens23t al. (1999) and set β m = 0 .
6. This value corresponds to a typical SMC apoptosis time ofapproximately 8 hours.Finally, we briefly comment on the rationale for our parameterisation of the SMC affinityfor the ECM phase χ ρ . Expansion of the flux term in equation (39) shows that χ ρ representsan effective haptotaxis coefficient for SMC movement up gradients in the collagen-rich ECM.The extent to which haptotactic SMC migration contributes to in vivo cap formation is notwell known. We therefore make a conservative initial estimate of χ ρ = 0 . χ ρ also arises due to practical considerations. Numerical simulations indicatethat the model can become ill-posed if the value of χ ρ is chosen to be too large. This issueappears to be related to the onset of negative SMC diffusion, which is an occurrence thatwe envisage to be unlikely during in vivo plaque growth. We therefore avoid such parameterregimes in the simulations that are presented in this paper. In Section 4, we will discuss thisissue in greater detail and propose amendments that can be made to the model to reduce oreliminate the ill-posed region of the parameter space.
Before presenting numerical solutions of the model system (39)–(47), we begin this section byperforming a simplified steady state analysis that provides useful insight into the relationshipbetween the SMC phase and the collagenous ECM phase in the model plaque.
The mathematical model derived in Section 2 provides a significant challenge for analyticalstudies. Indeed, given the highly nonlinear nature of the model equations, it seems veryunlikely that closed-form expressions could be derived for any spatially inhomogeneous steadystate solutions. Worthwhile progress can be made, however, if we exploit the absence of aflux term in the ECM equation and investigate the steady state ECM phase solution at afixed spatial position for fixed values of the other model variables. This analysis, whichwe will use to interpret the numerical results in Section 3.2, allows us to understand whya particular combination of steady state values for P , T and m correspond to a particularsteady state value for ρ at a given location in the plaque.Consider a fixed point inside the model domain, and let us assume that, at that point, thePDGF and TGF- β concentrations take the fixed (steady state) values P ∗ and T ∗ . Assumingthat the SMC volume fraction has the fixed (steady state) value m ∗ at the same point,24quation (40) gives the following expression for the steady state ECM volume fraction ρ ∗ : R s m ∗ (1 − m ∗ − ρ ∗ ) − R d m ∗ ρ ∗ − B ρ ρ ∗ (1 − m ∗ − ρ ∗ ) = 0 , (48)where ρ ∗ + m ∗ <
1. Note that, in the above expression, we have introduced the positiveconstants R s ( T ∗ ) = r s (cid:104) A s T ∗ c s + T ∗ (cid:105) , R d ( P ∗ , T ∗ ) = r d (cid:104) A d P ∗ ( c d + P ∗ )(1+ γ d T ∗ ) (cid:105) , and B ρ ( T ∗ ) = β ρ (cid:104) ε γ ρ T ∗ γ ρ T ∗ (cid:105) . These constants represent the (local) steady state rates of ECM synthesis bySMCs, ECM degradation by SMCs, and ECM degradation by immune cells, respectively.Rearranging equation (48) gives the following quadratic in ρ ∗ : ρ ∗ + (cid:20) m ∗ (cid:18) − R s B ρ − R d B ρ (cid:19) − (cid:21) ρ ∗ + R s B ρ m ∗ (1 − m ∗ ) = 0 . (49)Setting µ ( T ∗ ) = R s B ρ and λ ( P ∗ , T ∗ ) = R d B ρ , this quadratic gives two possible solutions ρ ∗± forthe steady state ECM volume fraction in terms of m ∗ : ρ ∗± ( m ∗ ) = 12 (cid:34) − m ∗ (1 − µ − λ ) ± (cid:114)(cid:104) − m ∗ (1 − µ − λ ) (cid:105) − µm ∗ (1 − m ∗ ) (cid:35) , (50)where 0 < m ∗ <
1. It is trivial to show that the discriminant of equation (50) is strictlypositive and, hence, that the two solutions are real-valued, positive and distinct. However, foreither solution to be physically meaningful, it must satisfy the condition 0 < ρ ∗± < − m ∗ .It can therefore be shown that ρ ∗− gives the only admissable steady state solution and,furthermore, that this steady state is always stable. From here, we drop the subscript on ρ ∗− and use ρ ∗ to refer to the negative root in equation (50).For fixed values of µ and λ , plots of ρ ∗ ( m ∗ ) for m ∈ (0 ,
1) demonstrate a biphasicbehaviour where ρ ∗ initially increases from zero with increasing m ∗ but then diminishes backtowards zero as m ∗ approaches one. The region of decreasing ρ ∗ is attributable to the factthat as m ∗ increases, ECM synthesis is increasingly inhibited by the reduced availability ofgeneric tissue material. An interesting consequence of this biphasic relationship between ρ ∗ and m ∗ is that, for each pair of growth factor concentrations P ∗ and T ∗ (or parameter values µ and λ ), there exists a SMC volume fraction (cid:98) m ∗ that produces a maximum steady stateECM volume fraction (cid:98) ρ ∗ . Therefore, in what follows, we shall derive expressions for both (cid:98) m ∗ and (cid:98) ρ ∗ , and use these expressions to explore the conditions that lead to optimal ECMdeposition at a given point in the plaque.We begin by looking for stationary points of the function ρ ∗ ( m ∗ ), whose first derivative25s given by the following: dρ ∗ dm ∗ = 12 µ + λ − µ − λ − m ∗ (cid:104) (1 − µ − λ ) + 4 µ (cid:105)(cid:114)(cid:104) − m ∗ (1 − µ − λ ) (cid:105) − µm ∗ (1 − m ∗ ) . (51)Before proceeding, we remark that the second derivative of ρ ∗ ( m ∗ ) is strictly negative and,hence, that any stationary point found from equation (51) will indeed be a maximum. Settingequation (51) equal to zero, rearranging terms and squaring leads to the following quadraticequation in (cid:98) m ∗ : (cid:104) (1 − µ − λ ) + 4 µ (cid:105) (cid:98) m ∗ − µ − λ ) (cid:98) m ∗ + 1 − λ = 0 , (52)which has corresponding solutions: (cid:98) m ∗± = 1 + µ − λ ± √ λ (1 − µ − λ )(1 − µ − λ ) + 4 µ . (53)The physically meaningful solution, which must satisfy the condition 0 < (cid:98) m ∗± <
1, is againgiven by the negative root. Selecting the negative root and dropping the subscript, it ispossible to show that (cid:98) m ∗ can be written in the reduced form: (cid:98) m ∗ ( µ, λ ) = 1 + √ λ µ + λ + 2 √ λ , (54)which, upon substitution into the negative root in equation (50), gives the correspondingform of (cid:98) ρ ∗ : (cid:98) ρ ∗ ( µ, λ ) = µ µ + λ + 2 √ λ . (55)The dependence of (cid:98) ρ ∗ and (cid:98) m ∗ on µ and λ can be inspected visually by plotting 2Dheatmaps of the right hand sides of equations (54) and (55). However, these plots havelimited utility for the interpretation of numerical simulation results because, at steady statein a given simulation, the values of µ and λ at each location in the plaque are not immediatelyobvious. This is because µ and λ depend not only on the local growth factor concentrations P ∗ and T ∗ , but also on the ten parameters ( r s , A s , c s , r d , A d , c d , γ d , β ρ , ε , γ ρ ) that appear inthe definitions of R s , R d and B ρ . Therefore, to support the interpretation of the numericalresults that will be presented in Section 3.2, we instead plot heatmaps of (cid:98) ρ ∗ and (cid:98) m ∗ asfunctions of P ∗ and T ∗ where the above parameters have been assigned their base case26 a) (b) Figure 4: Heatmaps that show how (a) the maximum steady state ECM volume fraction (cid:98) ρ ∗ and (b) the corresponding SMC volume fraction (cid:98) m ∗ at a fixed point in the plaque vary withthe local growth factor concentrations P ∗ and T ∗ for ( P ∗ , T ∗ ) ∈ [0 , × [0 , (cid:98) ρ ∗ and (cid:98) m ∗ ( r s , A s , c s , r d , A d , c d , γ d , β ρ , ε , γ ρ )have been assigned their base case values (see Table 1). Explicit expressions for (cid:98) ρ ∗ ( P ∗ , T ∗ )and (cid:98) m ∗ ( P ∗ , T ∗ ) are not provided but can be inferred from the expressions in equations (54)and (55), and the definitions in the text after equations (48) and (49).values (Figure 4). Note that expressions for (cid:98) ρ ∗ ( P ∗ , T ∗ ) and (cid:98) m ∗ ( P ∗ , T ∗ ) can be writtendown explicitly (see the relevant definitions above), but we omit these here for brevity.Figures 4a and 4b present respective plots of (cid:98) ρ ∗ ( P ∗ , T ∗ ) and (cid:98) m ∗ ( P ∗ , T ∗ ) for ( P ∗ , T ∗ ) ∈ [0 , × [0 , P ∗ and T ∗ reflect the typical ranges of dimensionless growthfactor concentrations that will be considered in our numerical simulations in Section 3.2.Recall that, for each parameter combination ( P ∗ , T ∗ ), these heatmaps indicate the maximumsteady state ECM volume fraction that could be reached at a fixed point in the plaque(Figure 4a), and the local SMC volume fraction that is required to achieve this maximum(Figure 4b). For SMC volume fractions that are either above or below those plotted for each( P ∗ , T ∗ ) in Figure 4b, the corresponding steady state ECM volume fractions will be smaller than those plotted in Figure 4a. Intuitively, (cid:98) ρ ∗ takes its largest values when T ∗ is largeand P ∗ is small (i.e. high net rate of ECM synthesis and low net rate of ECM degradation),and its smallest values when T ∗ is small and P ∗ is large (i.e. low net rate of ECM synthesisand high net rate of ECM degradation). The corresponding behaviour of (cid:98) m ∗ , on the otherhand, is less intuitive. Figure 4b indicates that, over the majority of the plotted ( P ∗ , T ∗ )parameter space, the maximum ECM volume fraction can be generated by a relatively small27olume fraction (0.1–0.2) of SMCs.Figure 4a indicates that, while both growth factors can have a significant impact on plaqueECM levels, (cid:98) ρ ∗ is much more sensitive to T ∗ than to P ∗ over the relevant concentration range.This is because TGF- β plays a role not just in synthesising the collagenous ECM, but alsoin preventing its degradation. To quantify the relative sensitivity to T ∗ and P ∗ , we notethat, in general, an increase in T ∗ from zero to one causes (cid:98) ρ ∗ to increase by approximately0.2, while a similar change in P ∗ causes (cid:98) ρ ∗ to decrease by approximately 0.1. Of course, it isalso important to consider the SMC levels that are required to attain these maximum ECMvolume fractions at different plaque growth factor levels. For T ∗ close to zero, Figure 4bindicates that an SMC volume fraction of around 25–30% is required to maximise the steadystate ECM volume fraction at around 20–30%. However, for T ∗ close to one, a maximumsteady state ECM volume fraction of 40–50% can be generated by an SMC volume fractionof less than 15%. The model therefore suggests that, in the presence of favourable TGF- β levels, it is possible to synthesise around twice as much collagenous ECM with about half asmany tissue-synthesising SMCs.The above results provide a useful tool with which to interpret the final outcome of capformation in a given model simulation. However, it is also important to emphasise that thissimplified analysis does not allow us to readily predict cap formation efficacy in a particularcase. We have focussed on a fixed spatial position in the plaque and have disregardeddynamic interactions between the various model variables, including the role of PDGF inSMC recruitment and the mechanical feedback between the SMCs and the ECM that theydeposit. In the next section, we investigate the importance of these factors, and others, byperforming numerical simulations with the full system of model equations. We begin this section by presenting results from a simulation with the base case param-eter values. This is followed by a sensitivity analysis that investigates the impact of keyparameters on cap formation. All numerical solutions have been obtained by discretisingequations (39)–(45) in space and/or time using a central differencing scheme for P and T ,the trapezoidal rule for ρ , and the Crank-Nicolson method for m . The resulting systems ofnonlinear equations were solved sequentially for each variable at each timestep by iteration. We initiate the base case simulation by prescribing small and spatially uniform volumefractions for m and ρ in the plaque. The initial profiles are presented in Figure 5 alongsideapproximate initial solutions for the plaque growth factor concentrations P and T . By usingthe fact that 1 − m − ρ = const and m ≈ t = 0, the P and T profiles represent exact28 Figure 5: Initial conditions for each of the model variables in the base case simulation. ThePDGF concentration is shown in blue, the TGF- β concentration in green, the ECM volumefraction in magenta, and the SMC volume fraction in red.solutions of simplified forms of equations (41) and (42) (i.e. ∂ G∂x = β G G , for G = P, T ),subject to the corresponding boundary conditions (44) and (45). An important feature tonote about the initial growth factor profiles is the distinct difference between the distributionsof endothelium-derived PDGF and TGF- β . The PDGF profile, which is diffusion-dominated( β P (cid:28) β , which isdegradation-dominated ( β T (cid:29) β is more strongly localised to the endothelium and, of course, to the desired region ofcollagenous cap formation.The time evolution of the model variables in the base case simulation is presented inFigure 6 (note that the time points for the ECM phase plots in Figure 6b do not correspondexactly to those for the plots of the other variables). The results demonstrate significantaccumulations of SMCs (Figure 6a) and ECM in the cap region over the course of thesimulation and a concurrent reduction in the overall concentrations of PDGF (Figure 6c)and TGF- β (Figure 6d). The reduction in plaque PDGF and TGF- β levels is partly due touptake by the invading SMCs, but can mostly be attributed to reduced chemical diffusionthrough the increasingly populated cap region. Note that the PDGF profile undergoes a moredramatic change than the TGF- β profile as a consequence of this modulated diffusivity.Figure 6a shows that the media-derived model SMCs respond to the diffusible PDGF sig-nal by migrating towards the endothelium and proliferating rapidly over the first 2 months ofsimulation time. Due to the ongoing increase in ECM deposition and concurrent reductionin PDGF levels, the SMCs divide more slowly after this time point and the plaque SMCpopulation peaks at around 4 months. Interestingly, Figure 6b demonstrates that cap for-29 (a) (b) (c) (d) Figure 6: (a) SMC volume fraction, (b) ECM volume fraction, (c) PDGF concentrationand (d) TGF- β concentration profiles in the plaque at several times during the base casesimulation. Arrows indicate the direction of sequential time points, which correspond to t = 1 . , . , . t = 0 . , . , . t = 2 . t = 8(solid lines). The simulation is close to steady state at this time point, and the only notablechange from the results at the latest time points in Figure 5 is a small reduction in the SMCvolume fraction throughout the domain. This reduction in SMC numbers appears to be dueto the delayed accumulation of ECM, which causes increased inhibition of SMC migrationand SMC proliferation at a late stage of cap formation. At t = 8, the total volume fractions ofSMCs and collagenous ECM in the plaque are approximately 8.6% and 21.4%, respectively.These values are consistent with observations from several studies of plaque growth in theApoE knockout mouse (Mallat et al., 2001; Clarke et al., 2006; Reifenberg et al., 2012).In Figure 7, we have also plotted the values of (cid:98) m ∗ and (cid:98) ρ ∗ at each spatial position inthe plaque (dotted lines). These plots, which have been constructed using equations (54)and (55), represent the maximum ECM volume fraction ( (cid:98) ρ ∗ ) that could be attained at eachposition x , and the corresponding SMC volume fraction ( (cid:98) m ∗ ) that would be required toachieve this maximum. For each x , the values of (cid:98) m ∗ and (cid:98) ρ ∗ are given by a unique pair ofvalues µ and λ , which depend on the local PDGF and TGF- β concentration values P ( x, T ( x, (cid:98) m ∗ and (cid:98) ρ ∗ curves show that, in the majority of the plaque tissue ( x (cid:38) . m ( x,
8) is below the level required to maximiseECM deposition. For x (cid:46) .
08, the extent of ECM deposition is again sub-optimal, but,in this case, the reduced ECM volume fraction ρ ( x,
8) is the result of over -recruitmentof SMCs. Interestingly, Figure 7 shows that the simulated ECM volume fraction remainsrelatively close to maximal levels throughout the region proximal to the endothelium (i.e. x (cid:46) . (cid:98) m ∗ for x ≈ (cid:98) m ∗ for x ≈ . near-maximal ECM de-position can be attained for a wide range of SMC volume fractions near the endothe-lium in the base case simulation. For a series of spatial positions in the model plaque( x = 0 , . , . , . , . , Figure 7: Base case simulation results that show approximate steady state profiles (solidlines) for the PDGF concentration (blue), TGF- β concentration (green), ECM volume frac-tion (magenta) and SMC volume fraction (red). Dotted lines represent the analytically-derived maximum steady state ECM volume fraction (cid:98) ρ ∗ (pink) and corresponding steadystate SMC volume fraction (cid:98) m ∗ (red) at each x . The (cid:98) m ∗ and (cid:98) ρ ∗ values are functions of thelocal PDGF and TGF- β concentrations and have been calculated at each x by equations (54)and (55), respectively. All plots taken at time t = 8.ECM volume fraction ρ ∗ ( m ∗ ) (negative root in equation (50); solid lines). Each individual ρ ∗ ( m ∗ ) curve again uses the corresponding local growth factor concentrations P ( x,
8) and T ( x, m, ρ ) values at each x at t = 8 (open circles), and the maximum ( (cid:98) m ∗ , (cid:98) ρ ∗ ) of each individual ρ ∗ ( m ∗ ) curve (asterisks). Note, firstly, that the simulated ( m, ρ ) data points are all stronglyaligned with their corresponding ρ ∗ ( m ∗ ) curves. This confirms that, at t = 8, the base casesimulation is indeed very close to steady state. Each individual ρ ∗ ( m ∗ ) curve in Figure 8 hasa unique shape and a unique maximum value, but all of the plots share similar qualitativefeatures. In particular, each curve is relatively flat for a wide range of m ∗ values either sideof its maximum at (cid:98) m ∗ . Therefore, when the simulated steady state value of m lies insidethe shallow region of the ρ ∗ ( m ∗ ) curve — as is the case near the endothelium in the basecase simulation — the simulated steady state value of ρ will differ only slightly from itstheoretical maximum. At x = 0 in the base case simulation, for example, the analyticalresults give (cid:98) ρ ∗ ≈ .
415 and corresponding (cid:98) m ∗ ≈ . x = 0 for t = 8 is much larger ( m ≈ . (cid:98) m ∗ , but the corresponding simulatedECM volume fraction ( ρ ≈ . (cid:98) ρ ∗ . Indeed, Figure 8 indicates that,at x = 0, any simulated steady state m in the range [0 . , . ρ > . Figure 8: Plot that compares near-steady state m and ρ values for several values of x inthe base case simulation with the corresponding analytical steady state results derived inSection 3.1. Solid lines represent the expressions ρ ∗ ( m ∗ ) (negative root in equation (50))at each of the spatial positions x = 0 (red), x = 0 . x = 0 . x = 0 . x = 0 .
45 (orange) and x = 1 (purple). Each ρ ∗ ( m ∗ ) curve is a function of thelocal PDGF and TGF- β concentrations. Individual data points represent the near-steadystate ( m, ρ ) values from the base case simulation (open circles) and the maximum ( (cid:98) m ∗ , (cid:98) ρ ∗ )of each individual ρ ∗ ( m ∗ ) curve (asterisks) at each x . All curves and data points taken attime t = 8.generated in the base case simulation would be robust to a relatively large reduction in capregion SMC numbers. The base case simulation results demonstrate how SMCs can accumulate in the plaque andsynthesise a collagenous cap in response to diffusible signals from endothelium-derived PDGFand TGF- β . In this section, we perform additional sensitivity analyses to investigate how thevalues of certain parameters determine the extent of collagen deposition in the cap region.In particular, we investigate both the impact of SMC adhesion to the nascent collagenousmatrix and the role of the relative rates of growth factor influx into the plaque. SMC Affinity for CollagenIn vitro studies have demonstrated that vascular SMCs can elicit a strong haptotactic re-sponse to types I, IV and VIII collagen (Nelson et al., 1996; Hou et al., 2000). The extent ofany haptotactic contribution to SMC migration during in vivo cap formation is unclear, but,33ince plaque ECM is known to contain both collagen I and collagen VIII (Adiguzel et al.,2009), it seems plausible that SMC haptotaxis may play an important role. We investigatethis possibility below by examining the sensitivity of the model to the parameter χ ρ , whichquantifies the affinity of the plaque SMCs for the collagenous ECM phase.We find that setting χ ρ = 0 produces results that are only marginally different to thosefrom the base case simulation. This highlights that our choice of χ ρ = 0 . χ ρ = 0 .
8, we find that the solutiondynamics are practically identical to the base case simulation for the first 5–6 weeks of sim-ulation time. Beyond this point, however, the emerging ECM phase exerts and increasinginfluence on the SMC behaviour and the model solution for larger χ ρ diverges from the basecase simulation dynamics. Figure 9 compares the approximate steady state SMC and ECMprofiles from the base case simulation and from the case with χ ρ = 0 .
8. As would be ex-pected, the increase in χ ρ results in increased recruitment of SMCs to the region proximalto the endothelium. More surprising, however, is that the overall plaque SMC content isreduced from 8.6% to 7.1%, and that there is no overall increase in ECM deposition in thecap region. The reduction in SMC numbers appears to be due to an increased squeeze on thePDGF influx from the endothelium, which reduces both SMC recruitment from the mediaand SMC mitosis inside the intima. The lack of increase in cap ECM deposition, on theother hand, can be attributed to the fact that the SMC volume fraction near the endothe-lium has increased well beyond the level required to maximise ECM synthesis (c.f. Figure 4).Of course, a potential benefit of the increased haptotaxis simulated here is that less ECM isdeposited beyond the cap region (due to the drop in SMC numbers), which may contributeto a reduction in the overall hardening of the diseased arterial tissue. Growth Factor Influx
A key target of this study has been to understand how plaque SMC behaviour and fibrouscap formation are regulated by the growth factors PDGF and TGF- β . Since it is difficultto quantify the concentrations of these chemicals in a given in vivo plaque, we perform arange of sensitivity analyses below to investigate how the relative rates of PDGF and TGF- β influx from the endothelium can contribute to the density, and therefore the stability, of thecollagenous cap. We first examine the consequences of varying the growth factor influx rates α T and α P independently, before considering the impact of varying them simultaneously.To facilitate the comparison of cap formation dynamics in the simulations that we performin this section, we introduce a simple metric to measure the SMC and ECM levels proximalto the endothelium at a given point in time. Specifically, we define the quantity V i ( t ; X ),which denotes the average volume fraction of phase i ( i = m, ρ ) in the interval x ∈ [0 , X ] at34 Figure 9: Approximate steady state SMC volume fraction (red lines) and ECM volumefraction (magenta lines) profiles in the plaque for the base case simulation (solid lines) anda simulation with increased SMC affinity for the ECM phase ( χ ρ = 0 .
8; dashed lines). Allplots taken at time t = 8.time t : V i ( t ; X ) = 1 X (cid:90) X i ( x, t ) d x, (56)where 0 < X (cid:28)
1. In all that follows, we choose (arbitrarily) to set X = 0 . x ∈ [0 , .
2] represents what we shall refer to as the cap region. Weremark that we have calculated V i values for several different choices of the cap region width,but (within reasonable bounds) the precise choice of X does not alter our qualitative findingsin any of the below sensitivity studies.Figure 10 presents the approximate steady state SMC and ECM profiles for a simulationwith no influx of TGF- β (i.e. α T = 0). This scenario mimics the experimental set-upsused by Mallat et al. (2001) and Lutgens et al. (2002), both of whom studied the impactof blocking TGF- β signalling during plaque growth in the ApoE knockout mouse. Theresults show that the absence of TGF- β has reduced the total collagen deposition in theplaque by approximately 15%. This is a smaller reduction in collagen deposition than mayhave been expected, but it is clear that collagen levels have been rescued to some degreeby a corresponding increase in SMC numbers. Indeed, this increase in SMC numbers isdirectly attributable to the reduced rate of collagen deposition because contact inhibition ofSMC proliferation remains lower, and PDGF influx from the endothelium remains higher,throughout the simulation. Having said that, the simulation results still demonstrate thatthe absence of TGF- β can decimate the fibrous cap. Figure 10 shows that, near steady state,the average ECM volume fraction in the cap region V ρ ( t ; 0 .
2) has decreased by approximately35
Figure 10: Approximate steady state SMC volume fraction (red lines) and ECM volumefraction (magenta lines) profiles in the plaque for the base case simulation (solid lines) anda simulation with no influx of TGF- β ( α T = 0; dashed lines). All plots taken at time t = 8.40% from 0.381 to 0.235. This result is consistent with the findings of Mallat et al. (2001)and Lutgens et al. (2002), who report that inhibition of TGF- β signalling can lead to anunstable plaque phenotype.For PDGF, we cannot simulate a case where chemical signalling is blocked entirely be-cause cap formation in the model is reliant upon PDGF-stimulated recruitment of SMCs fromthe media. The model does, however, elicit interesting dynamics over a range of non-zerovalues of α P , and we therefore consider the impact of both an increase and a decrease in thisparameter. Figure 11 compares the initial PDGF profile from the base case simulation withthe initial PDGF profiles for α P = 0 . α P = 1 .
1. Note that increasing the value of α P not only increases the PDGF gradient but also increases the PDGF concentration through-out the plaque tissue. Figure 12 presents the corresponding SMC and ECM profiles for eachvalue of α P after 8 months of simulation time. The SMC profiles show a large disparity intheir volume fractions, which is particularly pronounced near the endothelium (Figure 12a).This reflects the significant differences in the relative rates of PDGF-stimulated chemotaxisand proliferation in each case. The ECM profiles, on the other hand, show relatively littlevariation in their volume fractions, but do still demonstrate some interesting features (Fig-ure 12b). For example, the case with larger α P produces a slightly lower cap collagen contentcompared to the base case simulation despite recruiting over 60% more SMCs to the capregion. This result can mostly be attributed to the fact that the cap SMC volume fractionhas significantly exceeded the level required to attain optimum ECM synthesis. However,note that the elevated plaque PDGF levels also contribute to an enhanced rate of ECMdegradation by increasing MMP production by the invading SMCs. In contrast, in the sim-36 Figure 11: Initial PDGF concentration profiles in the plaque for the base case simulation(solid line), a simulation with a smaller rate of PDGF influx from the endothelium ( α P = 0 . α P = 1 .
1; dashed line).ulation with smaller α P , the sparse SMC population makes a remarkably strong attempt atcap synthesis. In this case, the collagen content in the cap region is only 20% less than thatin the base case simulation despite a cap SMC content that has been depleted by over 60%.Of course, if we examine these SMC and ECM profiles at only one time point, we getno real indication of the overall simulation dynamics. Hence, in Figure 13, we also presentthe average cap SMC volume fraction V m ( t ; 0 .
2) and the average cap ECM volume fraction V ρ ( t ; 0 .
2) as functions of time t . For α P = 1 .
1, Figure 13a shows a slow initial rate of capSMC recruitment, followed by a period of rapid population growth that saturates after 3–4months. However, despite a cap SMC population that significantly exceeds the base caselevels from 2 months onwards, the temporal accumulation of cap ECM shows a very similarpattern to the base case simulation and fails to exceed the base case value of V ρ at anytime (Figure 13b). For α P = 0 .
3, however, we observe completely different cap formationdynamics. The reduced PDGF levels in this case inhibit both the rate and the extent of capSMC recruitment (Figure 13a), which, in turn, creates a significant lag in the rate of capECM accumulation (Figure 13b). Hence, although these SMCs clearly have the capacity togenerate large quantities of ECM, their limited numbers ensure that this process takes anextended period of time.We conclude this study by briefly considering the consequences of simultaneously chang-ing the rates of influx of both PDGF and TGF- β . Figure 14 presents bar charts of V ρ ( t ; 0 . V m ( t ; 0 .
2) at time t = 8 for simulations with a range of values of α T and α P . Specifically,we report results for each unique pair of values from α T = 0, 1.25, 2.5, 3.75, 5 and α P =37 (a) (b) Figure 12: Approximate steady state (a) SMC volume fraction and (b) ECM volume fractionprofiles in the plaque for the base case simulation (solid lines), a simulation with a smallerrate of PDGF influx from the endothelium ( α P = 0 .
3; dot-dash lines) and a simulation witha larger rate of PDGF influx from the endothelium ( α P = 1 .
1; dashed lines). All plots takenat time t = 8. (a) (b) Figure 13: Plots that show how the average cap region volume fractions of (a) SMCs V m ( t ; 0 .
2) and (b) ECM V ρ ( t ; 0 .
2) vary with time for the base case simulation (solid lines),a simulation with a smaller rate of PDGF influx from the endothelium ( α P = 0 .
3; dot-dashlines) and a simulation with a larger rate of PDGF influx from the endothelium ( α P = 1 . .100.1 0.95 0.2 3.75 0.70.3 2.5 0.50.4 1.25 0.30 (a) (b) Figure 14: Charts that show how the rates of PDGF influx α P and TGF- β influx α T influencethe average volume fractions of (a) SMCs V m ( t ; 0 .
2) and (b) ECM V ρ ( t ; 0 .
2) in the cap regionafter 8 months of simulation time. Individual simulations were performed for each uniquepair of values from the lists α P = 0.3, 0.5, 0.7, 0.9, 1.1 and α T = 0, 1.25, 2.5, 3.75, 5. Thecentral bar on each chart represents the outcome of the base case simulation.0.3, 0.5, 0.7, 0.9, 1.1. Note that results from the base case simulation are represented by thecentral bar on each chart. Figure 14a, which plots the V m value for each simulation, showstwo clear trends: (1) increasing V m with increasing α P (mainly due to increased SMC prolif-eration) and (2) decreasing V m with increasing α T (due to the inhibitory effects of increasedECM deposition). Interestingly, the plot for V ρ indicates a relatively large region of the( α T , α P ) parameter space where the model predicts a robust cap formation response (Fig-ure 14b). This region corresponds to parameter regimes with large TGF- β levels, moderatePDGF levels and, consequently, moderate volume fractions of cap SMCs. Mature atherosclerotic plaques contain a lipid-rich core of necrotic material and release of thismaterial into the circulation can cause myocardial infarction or stroke. Fibrous cap formationprovides protection against these outcomes by stabilising the plaque and sequestering thedangerous plaque content from the bloodstream. However, the mechanisms that regulate capformation, and the factors that may subsequently lead the cap to fail, are poorly understood.In this paper, we have developed a multiphase model to investigate collagenous cap synthesisby growth factor-stimulated SMCs. The model includes representations of the growth factors39DGF and TGF- β , and we have used the model to study the co-operation (and competition)between these diffusible chemicals in the recruitment of vascular SMCs to remodel the plaqueECM. The model that we have developed in this study is designed to investigate cap formationin the ApoE-deficient mouse. We have chosen to focus on this transgenic mouse modelof atherosclerosis because there exists an extensive experimental literature that we haveused to inform and validate our modelling assumptions. It would, of course, be possible toextrapolate our model to human atherosclerosis by solving the equations on a domain ofsignificantly greater dimensional width (e.g. 1 mm). However, the behaviour of vascularSMCs in human plaques remains poorly understood (Bennett et al., 2016). Unlike mice,healthy human arteries contain a resident population of intimal SMCs and the extent ofmedial SMC recruitment to human plaques has yet to be established.
The research presented in this paper has some similarities and several key differences tothe earlier approach of Watson et al. (2018). In Watson et al. (2018), we established anovel modelling framework with non-standard boundary conditions to allow us to studythe chemotactic recruitment of plaque SMCs from the media in response to an influx ofendothelium-derived PDGF. We did not explicitly model the deposition of a fibrous cap.Instead, we assumed that the plaque ECM profile would evolve to more-or-less reflect theplaque SMC profile and, hence, that an increase in SMC recruitment to the cap region wouldgenerally improve cap stability. The current work uses the modelling framework of Watsonet al. (2018) as a foundation, but builds a substantial new model that focusses on ECMsynthesis and degradation by plaque cells in response to an influx of both PDGF and TGF- β from the endothelium. The new model also includes haptotactic SMC migration in responseto dynamically varying gradients in the ECM and, hence, to the best of our knowledge, thispaper presents the first attempt to include both chemotactic and haptotactic cell movementin a multiphase model. The results that we have reported have demonstrated that theplaque SMC profile does not necessarily provide a reliable indication of the correspondingECM profile, nor of the overall likelihood of fibrous cap stability. This is in contrast to theassumptions that we made previously in the simpler model of Watson et al. (2018).By retaining the modelling framework from our previous study, we have also retainedsome of the limitations of the earlier approach. In particular, the model remains on afixed domain and does not explicitly account for the lipoprotein influx and the associated40nflammatory response that may contribute to continued plaque growth during the processof cap formation. The current work can therefore be interpreted as a model of cap formationin a plaque that otherwise exists in a dynamic equilibrium (i.e. where any influx of LDLand immune cells from the bloodstream is exactly balanced by an efflux of lipid-loadedfoam cells to the adventitial lymphatics). Of course, even with this interpretation of themodel, it is possible to argue that we should incorporate domain growth to account forthe accumulation of SMCs and collagenous ECM in the plaque. In the numerical resultspresented in Section 3.2, for example, the total fraction of the plaque occupied by SMCsand ECM increases over the course of a simulation by between 17–35% depending on thechoice of parameters. In the absence of domain growth, we assume that this increase inthe plaque SMC and ECM levels is balanced by an equivalent reduction in the volumefraction of the generic tissue phase. This reduction in generic tissue phase material can beinterpreted as a loss of the initial non-collagenous ECM, which is degraded and subsequentlyrecycled, and as a loss of interstitial fluid, which is squeezed out of the plaque or absorbedby SMCs to produce new material. We note, however, that this interpretation becomes lessreasonable if the total SMC and ECM volume fractions in the plaque become large becausesignificant quantities of other generic tissue phase constituents (e.g. immune cells, lipids) arealso effectively lost from the tissue during cap formation. Despite these limitations of theapproach, we believe that the fixed domain model provides a useful and computationallystraightforward approximation to the problem of modelling atherosclerotic cap growth. Infuture studies, we will develop a comprehensive moving boundary approach to explore howdynamic expansion of the intima by immune cell and SMC activity can influence the efficacyof cap formation. One of the key benefits of employing a fixed domain is that the model remains amenableto mathematical analysis. In Section 3.1, we used the absence of a flux term in the ECMphase equation to study the factors that determine the steady state ECM volume fractionat a given position in the plaque. Specifically, we showed, for fixed concentrations of thegrowth factors PDGF and TGF- β , that the steady state ECM volume fraction ρ ∗ has abiphasic dependence on the steady state SMC volume fraction m ∗ , and, correspondingly,that there exists a steady state SMC volume fraction (cid:98) m ∗ that gives a maximum steadystate ECM volume fraction (cid:98) ρ ∗ . Moreover, we derived simple expressions for (cid:98) m ∗ and (cid:98) ρ ∗ interms of the dimensionless parameter groupings µ and λ , where µ represents the ratio of thenet rate of ECM synthesis by SMCs to the net rate of ECM degradation by immune cellsand λ represents the ratio of the net rate of ECM degradation by SMCs to the net rate ofECM degradation by immune cells. The most significant finding of this analysis is that, for41arameter values informed by a range of in vitro and in vivo studies, the model predicts thatthe maximum ECM volume fraction can generally be achieved by a relatively small volumefraction of SMCs (e.g. <
20% even at moderate TGF- β concentrations). Interestingly, forseveral of the numerical simulations presented in Section 3.2, the model predicts steadystate SMC volume fractions proximal to the endothelium that are above the level requiredto maximise ECM deposition. Consequently, the steady state ECM levels in the cap regionin these simulations are typically smaller than their maximum possible level. With regard toplaque stability, this outcome initially appears to be undesirable. However, note that a slightexcess in SMC numbers above the optimum level may be beneficial because it affords theplaque greater robustness to destabilisation by mechanisms such as SMC loss or diminishedTGF- β signalling. When the typical steady state SMC volume fraction in the cap regionis below the level required for maximum ECM deposition, then an equivalent drop in SMCnumbers or loss of TGF- β signalling would have far more pronounced consequences for plaquestability. The base case simulation results reproduced several observations from experimental studiesof plaque growth in the ApoE-deficient mouse. For example, temporal changes in the modelplaque SMC and collagenous ECM levels were consistent with the qualitative pattern re-ported in Reifenberg et al. (2012). The plaque SMC content accumulated rapidly over thefirst 2 months of simulation time and maintained a relatively stable level thereafter. Theplaque ECM content, on the other hand, accumulated more slowly and eventually reacheda stable level after 4–6 months. The total SMC content and the total ECM content in themodel plaque after 4–8 months of simulation time were in good quantitative agreement withthe values reported in Reifenberg et al. (2012) and in other experimental studies (Mallatet al., 2001; Clarke et al., 2006). An interesting aspect of the base case simulation wasthat the total plaque SMC content reached a peak of around 9.2% after 4 months of sim-ulation time and then dropped to around 8.6% as the plaque ECM content continued toincrease. This is a relatively subtle effect, but for simulations where the overall plaque ECMcontent reaches higher levels (e.g. sensitivities with larger r s , smaller r d or smaller β ρ ), weobserve a more pronounced suppression of SMC numbers in the latter stages of cap forma-tion (results not shown). These findings are consistent with observations made by Fukumotoet al. (2004), who studied plaque growth in ApoE mice that were resistant to degradation oftype I collagen. Fukumoto et al. (2004) found that, relative to standard ApoE mice, thesecompound-mutant mice had significantly more collagen and significantly fewer SMCs in theirplaques. Moreover, the plaques that were resistant to collagen degradation also showed ev-idence of reduced SMC proliferation and increased SMC death. These observations suggest42hat excessive accumulation of collagen in plaques can tip the balance towards a net re-duction in plaque SMC levels. The model presented here makes a similar prediction andsuggests that the loss of SMCs from collagen-rich plaques can be explained by an increasein ECM-mediated contact inhibition of SMC proliferation and a decrease in the capacity forPDGF transport in the dense intimal tissue.A consistent finding throughout this modelling study is that the growth factor TGF- β plays a critical role in the ability of plaque SMCs to deposit a stable, collagen-rich fibrouscap. The analytical results in Section 3.1, for example, indicated that the presence of TGF- β can enable significantly fewer SMCs to deposit significantly more ECM than would otherwisebe possible. These analytical results have been further supported by numerical simulations.Compared to the base case, a sensitivity simulation with no TGF- β ( α T = 0) showed anoverall decline in steady state plaque ECM levels, which included a substantial reductionin ECM deposition in the cap region. These observations are qualitatively consistent withthose from equivalent experimental studies in ApoE mice (Mallat et al., 2001; Lutgens et al.,2002). However, we also note some inconsistencies between the in vivo and in silico results.For example, the model predicted a 15% decrease in plaque ECM content coupled to a 55%increase in plaque SMC content, whereas both of the above experimental studies showed a50% decrease in plaque collagen and no overall change in plaque SMC content after TGF- β blockade. The reasons for these differences are not clear, but the experimental results alsoshowed that inhibition of TGF- β signalling led to increased inflammatory cell content andlarger lipid cores in the murine plaques. The SMC response in these in vivo plaques maytherefore have been blunted by increased competition for space or by increased SMC death inthe noxious and highly inflammatory plaque environment. The absence of these confoundingfactors in the model could potentially explain why the plaque SMC and ECM content areover-predicted in the current in silico approach.In order for TGF- β to play its role in cap formation, PDGF is first required to stimulateSMC recruitment to the cap region. PDGF may contribute both positively and negatively tocap formation. Experimental evidence suggests that PDGF can increase SMC production ofmatrix-degrading MMPs, in addition to stimulating SMC migration and SMC mitosis. Thesefactors were included in the model, and sensitivity simulations with different rates of PDGFinflux from the endothelium α P showed interesting results. For α P values in the range 0.3–1.1, simulations showed that SMC recruitment to the cap region increased significantly withincreasing α P . ECM levels in the cap region close to steady state, on the other hand, wererelatively insensitive to α P and showed a biphasic response, with maximal ECM depositionat the base case value ( α P = 0 . α P have been shown to be independent of the corresponding rateof TGF- β influx (for α T in the range 0–5).The reported simulation with α P = 0 . α T at its base case value) is interesting43ecause it demonstrates that even a small amount of SMCs can generate a substantialamount of ECM. From a total plaque SMC volume fraction of only 4.0% (vs. 8.6% in thebase case simulation), the total plaque ECM volume fraction reaches 15.0% (vs. 21.4% inthe base case simulation). This result is qualitatively consistent with observations made byClarke et al. (2006) in a study that examined the impact of diphtheria toxin (DT)-inducedSMC apoptosis in the plaques of ApoE mice. Plaque SMC content in DT treated micewas reduced more than 4-fold relative to untreated mice (2.5% vs. 10.2%), but the plaquecollagen content was correspondingly reduced by just over 2-fold (8.0% vs. 18.1%). Of course,despite the relatively small drop in the long-term plaque collagen content, the simulationwith α P = 0 . in vivo fibrous cap formation is entirely consistent with themodelling results presented in this paper. SMCs are well-known to respond haptotactically to gradients in collagenous substrates andwe have included this phenomenon in the model by assuming that the SMC phase has anaffinity for the ECM phase (quantified by the parameter χ ρ ). Unlike SMC chemotaxis,haptotactic SMC migration is not critical to the formation and maintenance of a cap in themodel. We therefore assumed a relatively low base case value for χ ρ , and later performed asensitivity study to investigate the impact of an increase in this value. The sensitivity studyemphasised the importance of SMC chemotaxis on the initiation of cap formation because,even with a relatively large χ ρ value, it took over a month of simulation time for the nascentECM profile to have a discernible influence on the SMC phase. In the long term, the increasein SMC haptotaxis led to steeper profiles in both the SMC and ECM phases in the plaquebut no overall increase in ECM deposition in the cap region. It would be interesting toperform a more comprehensive investigation of the impact of the relative contributions ofSMC haptotaxis and SMC chemtotaxis on fibrous cap formation by simultaneously varyingthe values of χ ρ and χ P (or α P ). An in-depth investigation of the role of SMC haptotaxiswas not the primary intention of the current work, but the model presented herein creates44he opportunity for a more focussed future study.One reason why a thorough study of the role of SMC haptotaxis is challenging is thatthe model equations can become ill-posed if the value assigned to χ ρ is too large. To un-derstand why this is the case, consider the form of the effective SMC diffusion coefficient inequation (39). This can be shown to be proportional to:Λ + ρ (cid:18) ψ + m ∂ψ∂m (cid:19) = χ P κP ) n P − ρχ ρ + δρm n ρ (cid:2) (1 + n ρ ) (1 − ρ ) − m (cid:3) (1 − m − ρ ) n ρ +1 . (57)Note that the third term on the right hand side of this expression is strictly positive because n ρ > m < − ρ . Recall that we haveassumed zero flux for SMCs at the endothelium (equation (31)). This boundary conditionstipulates that, at x = 0, any SMC flux towards the endothelium must be exactly balancedby an equivalent SMC flux in the opposite direction. In practice, the chemotactic and hap-totactic SMC fluxes almost always act towards the endothelium and this creates a negativegradient in the SMC phase at x = 0. Hence, to satisfy the zero flux boundary condition at x = 0, the diffusive SMC flux must always be positive and act down the SMC gradient awayfrom the endothelium. If the diffusive SMC flux becomes negative at any stage, the problemimmediately becomes ill-posed. Inspection of equation (57) indicates that negative SMCdiffusion can occur if the second term on the right hand side is dominant. This can occur ifSMC adhesion to the collagenous ECM is sufficiently strong to counteract the other mech-anisms of cell spreading. Note that this scenario is particularly likely to occur if P is large(first term on the right hand side of equation (57) negligible) and/or if m is small (third termon the right hand side of equation (57) negligible). In general, it is difficult to remove thisproblem of ill-posedness entirely, but here we propose several possible strategies to at leastreduce the region of parameter space where the problem occurs. First, we could regularisethe governing equations by including viscous effects in equation (11) for the SMC phase. Sec-ond, as done by Byrne and Owen (2004), we could introduce a constant background SMCdiffusion by including an additional constant SMC phase pressure in the function Λ. Third,we could assume that the SMC phase affinity for the ECM phase decreases with increasingPDGF concentration (i.e. χ ρ ≡ χ ρ ( P )). In this latter case, the first and second terms on theright hand side of equation (57) would become complementary and, since we also assumethat SMC proliferation increases with increasing P , this would represent the introductionof a “go-or-grow” mechanism in the SMC phase. Note, however, that the inclusion of a P dependency in χ ρ would also introduce a new term into the effective SMC chemotaxiscoefficient. This term would resemble an ECM phase-dependent chemorepulsion. Of course,the downside of these potential solutions to the problem of ill-posedness is that each solutionwould introduce additional model parameters whose values may be difficult to determine in45ractice. So, what conclusions can we draw from the model about factors that have been reportedto contribute to plaque instability? The model results suggest that, under conditions whereinitial SMC recruitment is strong and plaque TGF- β levels remain favourable, the fibrous capshould be robust to a relatively significant loss of SMC numbers. However, if as suggestedby Wang et al. (2015) that long-term loss of SMCs can occur due to replicative senescence,the model would predict rapid degradation of the cap by immune cell activity as the SMCpopulation falls to critically low levels. In conditions where the SMC numbers and the TGF- β concentration in the plaque remain at moderate levels, the current model suggests thatan increase in ECM degradation by immune cells alone is unlikely to destabilise the plaque.In results that we have not reported in this paper, a sensitivity simulation with a 2-foldincrease in the baseline rate of ECM degradation by immune cells ( β ρ = 1 .
5) predicts only arelatively small change in the ECM volume fraction in the cap region (32% vs. 38% in thebase case simulation). This makes sense because, based on our modelling assumptions, themajority of the plaque immune cell content resides deeper in the plaque and beneath the capregion. Of course, the current model is unable to predict the consequences of immune cellECM degradation at the plaque shoulders, where rupture events are commonly reported tooccur (Bennett et al., 2016). Investigation of this phenomenon will require models for capformation to be developed in two or three spatial dimensions.The most critical factor in maintenance of plaque stability, as predicted by the currentmodel, is the continued presence of plaque TGF- β and, more importantly, the continuedcapacity for stimulation of SMCs (and immune cells) by this growth factor. Our analysisand our numerical simulations have indicated that TGF- β can play a critical role not onlyin the formation of a stable cap region, but also in ensuring that cap formation is efficientand does not require excessive recruitment of SMCs. Interestingly, recent reports from in vivo and in vitro experiments have shown that cholesterol loading in several cell types,including vascular SMCs, can significantly attenuate cellular responsiveness to TGF- β (Chenet al., 2007; Vengrenyuk et al., 2015). Given the extent of cholesterol accumulation in theintimal tissue during atherosclerotic plaque growth, it is therefore plausible that modifiedLDL consumption by SMCs (and by macrophages) indirectly contributes to the breakdown ofplaque ECM by inhibiting the protective actions of TGF- β . Investigation of this hypothesisedcap destabilisation mechanism is another important target for future modelling studies.46 Conclusions
In this paper, we have developed a novel three-phase model of fibrous cap formation inthe atherosclerosis-prone mouse. The model investigates the roles of endothelium-derivedPDGF and TGF- β in the regulation of collagen cap deposition by synthetic vascular SMCs,which migrate both chemotactically and haptotactically in the arterial intima. We haveparameterised the model using data from a wide range of in vitro and in vivo studies andour numerical simulations reproduce a number of experimental observations. Our resultsprovide some interesting insights into potential mechanisms of plaque instability and long-term cap degradation. The model presented in this paper can be extended in several newdirections, and this work therefore represents an important step in the development of adynamic and mechanistic understanding of atherosclerotic plaque formation. Acknowledgements
MGW, CM and MRM acknowledge funding from an Australian Research Council DiscoveryGrant (DP160104685).
References
E. Adiguzel, P. J. Ahmad, C. Franco, and M. P. Bendeck. Collagens in the progression andcomplications of atherosclerosis.
Vasc. Med. , 14:73–89, 2009.J. Ahamed, N. Burg, K. Yoshinaga, C. A. Janczak, D. B. Rifkin, and B. S. Coller. In vitroand in vivo evidence for shear-induced activation of latent transforming growth factor- β Blood , 112:3650–3660, 2008.M. R. Alexander and G. K. Owens. Epigenetic control of smooth muscle cell differentiationand phenotypic switching in vascular development and disease.
Annu. Rev. Physiol. , 74:13–40, 2012.S. Astanin and L. Preziosi. Multiphase models of tumour growth. In A. Angelis, M. A. J.Chaplain, and N. Bellomo, editors,
Selected Topics in Cancer Modeling: Genesis, Evolu-tion, Immune Competition, and Therapy. Modeling and Simulation in Science, Engineer-ing and Technology , pages 223–253. Birkh¨auser, Boston, 2008.M. R. Bennett, S. Sinha, and G. K. Owens. Vascular smooth muscle cells in atherosclerosis.
Circ. Res. , 118:692–702, 2016. 47. Bhui and H. N. Hayenga. An agent-based model of leukocyte transendothelial migrationduring atherogenesis.
PLoS Comput. Biol. , 13:e1005523, 2017.V. Borrelli, L. di Marzo, P. Sapienza, M. Colasanti, E. Moroni, and A. Cavallaro. Roleof platelet-derived growth factor and transforming growth factor β in the regulation ofmetalloproteinase expressions. Surgery , 140:454–463, 2006.M. Breton, E. Berrou, M.-C. Brahimi-Horn, E. Deudon, and J. Picard. Synthesis of sulfatedproteoglycans throughout the cell cycle in smooth muscle cells from pig aorta.
Exp. CellRes. , 166:416–426, 1986.P. Budu-Grajdeanu, R. C. Schugart, A. Friedman, C. Valentine, A. K. Agarwal, and B. H.Rovin. A mathematical model of venous neointimal hyperplasia formation.
Theor. Biol.Med. Model. , 5:2, 2008.M. A. K. Bulelzai and J. L. A. Dubbeldam. Long time evolution of atherosclerotic plaques.
J. Theor. Biol. , 297:1–10, 2012.H. M. Byrne and M. R. Owen. A new interpretation of the Keller-Segel model based onmultiphase modelling.
J. Math. Biol. , 49:604–626, 2004.A. Q. Cai, K. A. Landman, and B. D. Hughes. Multi-scale modeling of a wound healing cellmigration assay.
J. Theor. Biol. , 245:576–594, 2007.A. D. Chalmers, A. Cohen, C. A. Bursill, and M. R. Myerscough. Bifurcation and dynamicsin a mathematical model of early atherosclerosis.
J. Math. Biol. , 71:1451–1480, 2015.A. D. Chalmers, C. A. Bursill, and M. R. Myerscough. Nonlinear dynamics of earlyatherosclerotic plaque formation may determine the efficacy of high density lipoproteins(HDL) in plaque regression.
PLoS ONE , 12:e0187674, 2017.J. Chappell, J. L. Harman, V. M. Narasimhan, H. Yu, K. Foote, B. D. Simons, M. R.Bennett, and H. F. Jørgensen. Extensive proliferation of a subset of differentiated, yetplastic, medial vascular smooth muscle cells contributes to neointimal formation in mouseinjury and atherosclerosis models.
Circ. Res. , 119:1313–1323, 2016.C.-L. Chen, I.-H. Liu, S. J. Fliesler, X. Han, S. S. Huang, and J.S. Huang. Cholesterolsuppresses cellular TGF- β responsiveness: implications in atherogenesis. J. Cell Sci. , 120:3509–3521, 2007.M. Cilla, E. Pena, and M. A. Martinez. Mathematical modelling of atheroma plaque forma-tion and development in coronary arteries.
J. Royal Soc. Interface , 11:20130866, 2014.48. C. H. Clarke, N. Figg, J. J. Maguire, A. P. Davenport, M. Goddard, T. D. Littlewood,and M. R. Bennett. Apoptosis of vascular smooth muscle cells induces features of plaquevulnerability in atherosclerosis.
Nat. Med. , 12:1075–1080, 2006.C. A. Cobbold and J. A. Sherratt. Mathematical modelling of nitric oxide activity in woundhealing can explain keloid and hypertrophic scarring.
J. Theor. Biol. , 204:257–288, 2000.A. Cohen, M. R. Myerscough, and R. S. Thompson. Athero-protective effects of high densitylipoproteins (HDL): an ODE model of the early stages of atherosclerosis.
Bull. Math. Biol. ,76:1117–1142, 2014.B. D. Cumming, D. L. S. McElwain, and Z. Upton. A mathematical model of wound healingand subsequent scarring.
J. R. Soc. Interface , 7:19–34, 2010.N. El Khatib, S. Genieys, and V. Volpert. Atherosclerosis initiation modeled as an inflam-matory process.
Math. Model. Nat. Phenom. , 2:126–141, 2007.D. J. W. Evans, P. V. Lawford, J. Gunn, D. Walker, D. R. Hose, R. H. Smallwood,B. Chopard, M. Krafczyk, J. Bernsdorf, and A. Hoekstra. The application of multiscalemodelling to the process of development and prevention of stenosis in a stented coronaryartery.
Phil. Trans. R. Soc. A , 366:3343–3360, 2008.A. Faggiotto and R. Ross. Studies of hypercholesterolemia in the nonhuman primate II.Fatty streak conversion to fibrous plaque.
Arteriosclerosis , 4:341–356, 1984.N. Filipovic, Z. Teng, M. Radovic, I. Saveljic, D. Fotiadis, and O. Parodi. Computer simu-lation of three-dimensional plaque formation and progression in the carotid artery.
Med.Biol. Eng. Comput. , 51:607–616, 2013.P. Fok. Mathematical model of intimal thickening in atherosclerosis: vessel stenosis as a freeboundary problem.
J. Theor. Biol. , 314:23–33, 2012.A. Friedman and W. R. Hao. A mathematical model of atherosclerosis with reverse choles-terol transport and associated risk factors.
Bull. Math. Biol. , 77:758–781, 2015.Y. Fukumoto, J. Deguchi, P. Libby, E. Rabkin-Aikawa, Y. Sakata, M. T. Chin, C. C. Hill,P. R. Lawler, N. Varo, F. J. Schoen, S. M. Krane, and M. Aikawa. Genetically determinedresistance to collagenase action augments interstitial collagen accumulation in atheroscle-rotic plaques.
Circulation , 110:1953–1959, 2004.49. Funayama, U. Ikeda, M. Takahashi, Y. Sakata, S.-I. Kitagawa, Y.-I. Takahashi, J.-I.Masuyama, Y. Furukawa, Y. Miura, S. Kano, M. Matsuda, and K. Shimada. Humanmonocyte-endothelial cell interaction induces platelet-derived growth factor expression.
Cardiovasc. Res. , 37:216–224, 1998.M. Garbey, S. Casarin, and S. A. Berceli. Vascular adaptation: pattern formation and crossvalidation between an agent based model and a dynamical system.
J. Theor. Biol. , 429:149–163, 2017.G. S. Getz and C. A. Reardon. Animal models of atherosclerosis.
Arterioscler. Thromb.Vasc. Biol. , 32:1104–1115, 2012.M. Guo, Y. Cai, X. Yao, and Z. Li. Mathematical modeling of atherosclerotic plaque desta-bilization: role of neovascularization and intraplaque hemorrhage.
J. Theor. Biol. , 450:53–65, 2018.G. K. Hansson and P. Libby. The immune response in atherosclerosis: a double-edged sword.
Nat. Immunol. , 6:508–519, 2006.G. K. Hansson, P. Libby, and I. Tabas. Inflammation and plaque vulnerability.
J. Intern.Med. , 278:483–493, 2015.J. M. Haugh. Deterministic model of dermal wound invasion incorporating receptor-mediatedsignal transduction and spatial gradient sensing.
Biophys. J. , 90:2297–2308, 2006.G. Hou, D. Mulholland, M. A. Gronska, and Bendeck M. P. Type VIII collagen stimulatessmooth muscle cell migration and matrix metalloproteinase synthesis after arterial injury.
Am. J. Pathol. , 156:467–476, 2000.J. S. Huang, T. J. Olsen, and S. S. Huang. The role of growth factors in tissue repair I.Platelet-derived growth factor. In R. A. F. Clark and P. M. Henson, editors,
The molecularand cellular biology of wound repair , pages 243–251. Plenum, New York, 1988.M. E. Hubbard and H. M. Byrne. Multiphase modelling of vascular tumour growth in twospatial dimensions.
J. Theor. Biol. , 316:70–89, 2013.M. H. Islam and P. R. Johnston. A mathematical model for atherosclerotic plaque formationand arterial wall remodelling.
ANZIAM J. , 57:C320–C345, 2016.K. Jacobsen, M. B. Lund, J. Shim, S. Gunnersen, E.-M. F¨uchtbauer, M. Kjolby, L. Car-ramolino, and J. F. Bentzon. Diverse cellular architecture of atherosclerotic plaque derivesfrom clonal expansion of a few medial SMCs.
JCI Insight , 2:e95890, 2017.50. Klika, E. A. Gaffney, Y.-C. Chen, and C. P. Brown. An overview of multiphase cartilagemechanical modelling and its role in understanding function and pathology.
J. Mech.Behav. Biomed. , 62:139–157, 2016.K. Kozaki, W. E. Kaminski, J. Tang, S. Hollenbach, P. Lindahl, C. Sullivan, J.-C. Yu,K. Abe, P. J. Martin, R. Ross, C. Betsholtz, N. A. Giese, and E. W. Raines. Blockadeof platelet-derived growth factor or its receptors transiently delays but does not preventfibrous cap formation in ApoE null mice.
Am. J. Pathol. , 161:1395–1407, 2002.K. Kubota, J. Okazaki, O. Louie, K. C. Kent, and B. Liu. TGF- β stimulates collagen (I)in vascular smooth muscle cells via a short element in the proximal collagen promoter. J.Surg. Res. , 109:43–50, 2003.C. Lally and P. Prendergast. Simulation of in-stent restenosis for the design of cardiovascularstents. In A. Holzapfel and R. W. Ogden, editors,
Mechanics of Biological Tissue , pages255–267. Springer, Berlin, 2006.G. Lemon, J. R. King, H. M. Byrne, O. E. Jensen, and K. M. Shakesheff. Mathematicalmodelling of engineered tissue growth using a multiphase porous flow mixture theory.
J.Math. Biol. , 52:571–594, 2006.J. Lopes, E. Adiguzel, S. Gu, S.-L. Liu, G. Hou, S. Heximer, R. K. Assoian, and M. P.Bendeck. Type VIII collagen mediates vessel wall remodeling after arterial injury andfibrous cap formation in atherosclerosis.
Am. J. Pathol. , 182:2241–2253, 2013.A. J. Lusis. Atherosclerosis.
Nature , 407:233–241, 2000.E. Lutgens, E. D. de Muinck, P. J. E. H. M. Kitslaar, J. H. M. Tordoir, H. J. J. Wellens, andM. J. A. P. Daemen. Biphasic pattern of cell turnover characterises the progression fromfatty streaks to ruptured human atherosclerotic plaques.
Cardiovasc. Res. , 41:473–479,1999.E. Lutgens, M. Gijbels, M. Smook, P. Heeringa, P. Gotwals, V. E. Koteliansky, and M. J.A. P. Daemen. Transforming growth factor- β mediates balance between inflammation andfibrosis during plaque progression. Arterioscler. Thromb. Vasc. Biol. , 22:975–982, 2002.Z. Mallat, A. Corbaz, A. Scoazec, P. Graber, S. Alouani, B. Esposito, Y. Humbert,Y. Chvatchko, and A. Tedgui. Interleukin-18/Interleukin-18 binding protein signalingmodulates atherosclerotic lesion development and stability.
Circ. Res. , 89:e41–e45, 2001.51. A. McCaffrey, S. Consigli, B. Du, D. J. Falcone, T. A. Sanborn, A. M. Spokojny, andH. L. Bush Jr. Decreased type II/type I TGF- β receptor ratio in cells derived fromhuman atherosclerotic lesions. Conversion from an antiproliferative to profibrotic responseto TGF- β J. Clin. Invest. , 96:2667–2675, 1995.S. McDougall, J. Dallon, J. Sherratt, and P. Maini. Fibroblast migration and collagen de-position during dermal wound healing: mathematical modelling and clinical implications.
Phil. Trans. R. Soc. A , 364:1385–1405, 2006.C. McKay, S. McKee, N. Mottram, T. Mulholland, S. Wilson, S. Kennedy, andR. Wadsworth. Towards a model of atherosclerosis. Technical report, University of Strath-clyde, 2004.S. N. Menon, J. A. Flegg, S. W. McCue, R. C. Schugart, R. A. Dawson, and D. L. S.McElwain. Modelling the interaction of keratinocytes and fibroblasts during normal andabnormal wound healing processes.
Proc. R. Soc. B , 279:3329–3338, 2012.K. Moore, F. Sheedy, and E. Fisher. Macrophages in atherosclerosis: a dynamic balance.
Nat. Rev. Immunol. , 13:709–721, 2013.E. Munro, M. Patel, P. Chan, L. Betteridge, K. Gallagher, M. Schachter, J. Wolfe, andP. Server. Effect of calcium channel blockers on the growth of human vascular smoothmuscle cells derived from saphenous vein and vascular graft stenosis.
J. Cardiovasc. Phar-macol. , 23:779–784, 1994.P. R. Nelson, S. Yamamura, and K. C. Kent. Extracellular matrix proteins are potentagonists of human smooth muscle cell migration.
J. Vasc. Surg. , 24:25–32, 1996.M. Nicolas, E. Pe˜na, M. Malv`e, and M. A. Mart´ınez. Mathematical modeling of the fibrosisprocess in the implantation of inferior vena cava filters.
J. Theor. Biol. , 387:228–240, 2015.R. D. O’Dea, J. M. Osborne, A. J. El Haj, H. M. Byrne, and S. L. Waters. The interplaybetween tissue growth and scaffold degradation in engineered tissue constructs.
J. Math.Biol. , 67:1199–1225, 2013.K. Ogawa, F. Chen, C. Kuang, and Y. Chen. Suppression of matrix metalloproteinase-9transcription by transforming growth factor- β is mediated by a nuclear factor- κ B site.
Biochem. J. , 381:413–422, 2004.L. Olsen, J. A. Sherratt, and P. K. Maini. A mechanochemical model for adult dermal woundcontraction and the permanence of the contracted tissue displacement profile.
J. Theor.Biol. , 177:113–128, 1995. 52. Pappalardo, S. Musumeci, and S. Motta. Modeling immune system control of atherogen-esis.
Bioinformatics , 24:1715–1721, 2008.A. Parton, V. McGilligan, M. O’Kane, F. R. Baldrick, and S. Watterson. Computationalmodelling of atherosclerosis.
Brief. Bioinform. , 17:562–575, 2016.N. C. Pearson, R. J. Shipley, S. L. Waters, and J. M. Oliver. Multiphase modelling ofthe influence of fluid flow and chemical concentration on tissue growth in a hollow fibremembrane bioreactor.
Math. Med. Biol. , 31:393–430, 2014.R. N. Poston and D. R. M. Poston. Typical atherosclerotic plaque morphology producedin silico by an atherogenesis model based on self-perpetuating propagating macrophagerecruitment.
Math. Model. Nat. Phenom. , 2:142–149, 2007.L. Preziosi and A. Tosin. Multiphase modelling of tumour growth and extracellular matrixinteraction: mathematical tools and applications.
J. Math. Biol. , 58:625–656, 2009.K. Reifenberg, F. Cheng, C. Orning, J. Crain, I. K¨upper, E. Wiese, M. Protschka, M. Bless-ing, K. J. Lackner, and M. Torzewski. Overexpression of TGF- β PLoS ONE , 7:e40990, 2012.G. M. Risinger, D. L. Updike, E.C. Bullen, J. J. Tomasek, and E. W. Howard. TGF- β suppresses the upregulation of MMP-2 by vascular smooth muscle cells in response toPDGF-BB. Am. J. Physiol. Cell Physiol. , 298:C191–C201, 2010.R. Ross. Atherosclerosis – an inflammatory disease.
N. Engl. J. Med. , 340:115–126, 1999.C. Rutherford, W. Martin, M. Carrier, E. E. ˚Angg˚ard, and G. A. A. Ferns. Endogenouslyelicited antibodies to platelet derived growth factor-BB and platelet cystolic protein inhibitaortic lesion development in the cholesterol-fed rabbit.
Int. J. Exp. Path. , 78:21–32, 1997.H. Sano, T. Sudo, M. Yokode, T. Murayama, H. Kataoka, N. Takakura, S. Nishikawa, S.-I.Nishikawa, and T. Kita. Functional blockade of platelet-derived growth factor receptor- β but not of receptor- α prevents vascular smooth muscle cell accumulation in fibrous caplesions in apolipoprotein E-deficient mice. Circulation , 103:2955–2960, 2001.M. Schachter. Vascular smooth muscle cell migration, atherosclerosis, and calcium channelblockers.
Int. J. Cardiol. , 62:S85–S90, 1997.N. N. Singh and D. P. Ramji. The role of transforming growth factor- β in atherosclerosis. J. R. Soc. Interface , 17:487–499, 2006. 53. Tahir, I. Niculescu, C. Bona-Casas, R. M. H. Merks, and A. G. Hoekstra. An in silicostudy on the role of smooth muscle cell migration in neointimal formation after coronarystenting.
Cytokine Growth Factor Rev. , 12:20150358, 2015.I. Toma and T. A. McCaffrey. Transforming growth factor- β and atherosclerosis: interwovenatherogenic and atheroprotective aspects. Cell Tissue Res. , 347:155–175, 2012.K. Urschel and I. Cicha. TNF- α in the cardiovascular system: from physiology to therapy. Int. J. Interferon Cytokine Mediat. Res. , 7:9–25, 2015.G. G. Vaday, H. Schor, M. A. Rahat, N. Lahat, and O. Lider. Transforming growth factor- β suppresses tumor necrosis factor α -induced matrix metalloproteinase-9 expression inmonocytes. J. Leukoc. Biol. , 69:613–621, 2001.Y. Vengrenyuk, H. Nishi, X. Long, M. Ouimet, N. Savji, F. O. Martinez, C. P. Cassella,K. J. Moore, S. A. Ramsey, J. M. Miano, and E. A. Fisher. Cholesterol loading repro-grams the microRNA-143/145-myocardin axis to convert aortic smooth muscle cells to adysfunctional macrophage-like phenotype.
Arterioscler. Thromb. Vasc. Biol. , 35:535–546,2015.L. M. Wakefield, T. S. Winokur, R. S. Hollands, K. Christopherson, A. D. Levinson, andM.B. Sporn. Recombinant latent transforming growth factor β β
1, and a different tissue distribution.
J. Clin. Invest. , 86:1976–1984, 1990.L. M. Wakefield, J. J. Letterio, T. Chen, D. Danielpour, R. S. Allison, L. H. Pai, A. M.Denicoff, M. H. Noone, K. H. Cowan, J. A. O’Shaughnessy, and M.B. Sporn. Transform-ing growth factor- β Clin. Cancer Res. , 1:129–136, 1995.J. Wang, A. K. Uryga, J. Reinhold, N. Figg, L. Baker, A. Finigan, K. Gray, S. Kumar,M. Clarke, and M. Bennett. Vascular smooth muscle cell senescence promotes atheroscle-rosis and features of plaque vulnerability.
Circulation , 132:1909–1919, 2015.M. G. Watson, H. M. Byrne, C. Macaskill, and M. R. Myerscough. A two-phase model ofearly fibrous cap formation in atherosclerosis.
J. Theor. Biol. , 456:123–136, 2018.World Health Organization. Cardiovascular diseases fact sheet. , May 2017. Ac-cessed April 2019. 54. Yang, W. J¨ager, M. Neuss-Radu, and T. Richter. Mathematical modeling and simulationof the evolution of plaques in blood vessels.
J. Math. Biol. , 72:973–996, 2016.H. Zahedmanesh, H. Van Oosterwyck, and C. Lally. A multi-scale mechanobiological modelof in-stent restenosis: deciphering the role of matrix metalloproteinase and extracellularmatrix changes.