DISCRETE AND CONTINUOUS Website: www.aimSciences.org DYNAMICAL SYSTEMS Supplement 2013 pp. 825–835

FOAM CELL FORMATION IN : HDL AND REVERSE TRANSPORT

Shuai Zhang Southern Polytechnic State University Marietta, GA 30060, USA L. R. Ritter1 and A. I. Ibragimov Southern Polytechnic State University Marietta, GA 30060, USA and Texas Tech University Lubbock, TX 79409, USA

Abstract. Macrophage derived foam cells are a major constituent of the fatty deposits characterizing the disease atherosclerosis. Foam cells are formed when certain immune cells () take on oxidized low density through failed . High density lipoproteins (HDL) are known to have a number of anti-atherogenic effects. One of these stems from their abil- ity to remove excess cellular cholesterol for processing in the liver—a process called reverse cholesterol transport (RCT). HDL perform macrophage RCT by binding to forming foam cells and removing excess lipids by efflux transporters. We propose a model of foam cell formation accounting for macrophage RCT. This model is presented as a system of non-linear ordinary differential equations. Motivated by experimental observations regarding time scales for oxidation of lipids and MRCT, we impose a quasi-steady state assumption and analyze the resulting systems of equations. We focus on the existence and stability of equilibrium solutions as determined by the governing parameters with the results interpreted in terms of their potential bio-medical implications.

1. Introduction. The disease atherosclerosis, characterized by an accumulation of fatty deposits and cellular debris within the arterial wall, is understood to be a chronic inflammatory condition [3,5]. Two primary types of cholesterol carrying molecules, low density lipoproteins (LDL) and high density lipoproteins (HDL), are integral to the chemical and cellular processes characterizing this disease. These particles participate in the trafficking of the cholesterol necessary for cell metab- olism carrying lipids to the tissues throughout the body and, HDL in particular, taking excess cholesterol out of tissues for processing in the liver. While LDL per- forms the necessary function of delivering lipids for cellular use, its higher lipid to protein ratio means that these particles are subject to becoming trapped within the walls of . In an wall, an LDL molecule becomes vulnerable to oxidation by free attack. Macrophages, the primary immune cells involved in atherosclerosis, have a high affinity for oxidized LDL (oxLDL). OxLDL is recog- nized by a macrophage scavenger receptor inducing these cells to take up lipid by

2010 Mathematics Subject Classification. Primary: 92B05, 92C50; Secondary: 34D20. Key words and phrases. Atherosclerosis modeling, stability analysis, steady state. 1Author to whom correspondence should be addressed.

825 826 SHUAI ZHANG, L. R. RITTER AND A. I. IBRAGIMOV attempted phagocytosis [7, 19, 24]. Macrophages are unable to process oxLDL in this manner causing lipids to accumulate on the immune cells transforming them into a type of cellular debris called foam cells. The disruption of immune function can result in chronic inflammation, and the ongoing availability of excess lipids cou- pled with aggregation of immune cells in a region may result in accumulation of foam cells locally creating the hallmark of the disease. One focus of clinical research of atherosclerosis is the role of HDL. Of par- ticular interest is the ability of HDL to remove excess cellular cholesterol from tissue—reverse cholesterol transport [25]. Macrophage reverse cholesterol transport (MRCT) occurs when HDL molecules adhere to the surface of foam cells and ac- cept lipids from the cells by means of efflux transporters—or even through passive diffusion. This anti-atherogenic process may lead to new therapies for managing and reversing disease progression. Current clinical studies are identifying specific proteins critical to MRCT. But one important open question is how inflamma- tion, especially chronic inflammation, effects HDL ability to perform MRCT. It was thought that chronic inflammation has a debilitating effect on this HDL function, but recent in vivo studies have brought this into question [14,6, 15]. In this paper, we present an ODE model of foam cell formation and present an analysis focused on the role of HDL cholesterol through MRCT. Because the fatty streaks associated with very early lesions are present in human arteries even in early childhood [20, 21], our model allows for a nonzero concentration of foam cells and looks to characterize the “healthiness” of the system through comparison of the foam cell density to that of functional macrophages. The following section outlines a full model of foam cell formation including oxidation. We then invoke a limiting assumption based on available experimental and mathematical studies of MRCT and lipoprotein oxidation to reduce the number of equations under consideration. An analysis of long time solutions is given with a focus on parameter dependence and the potential bio-medical implications.

2. Full dynamical model. The full time dependent model consist of two cellular and three chemical species; the cellular species being macrophages denoted M and macrophage derived foam cells denoted by F . The chemicals include HDL molecules H, LDL molecules Lθ, and (free-radicals) R. The LDL are further divided into three subspecies identified by the subscript: Ln: native LDL molecules retaining some innate anti-oxidant defense, L0: LDL with no innate anti-oxidant defense with an un-oxidized lipid core, and Lox: oxidized LDL (oxLDL) following per-oxidation of the lipid core. The system governing the evolution of these species is

M˙ = φ − α1LoxM − dM M, (1)

F˙ = r1αLˆ oxM − r2hHF − dF F, (2) ˙ H = pHB − hHF − dH H, (3) ˙ Lox = kR0 RL0 − α2LoxM, (4) ˙ Ln = pLB − kRRLn + kAAoxL0 − dLLn, (5) ˙ L0 = −kR0 RL0 + kRRLn − kAAoxL0 − dLL0, (6) ˙ R = pR − kR0 RL0 − kRRLn − kH HR − ρAoxR, t > 0 (7) with some initial condition (M(0),F (0),H(0),Lox(0),Ln(0),L0(0),R(0)) given. The full model is adapted from a spatio-temporal model of atherogenesis presented FOAM CELL FORMATION IN ATHEROSCLEROSIS 827 by Ibragimov et al. in [9]2. Equations (4)–(7) are a slight modification of the lipid oxidation model given by Cobbold, Sherratt and Maxwell [2]3; the primary mod- ification is the addition of the term α2LoxM in (4) that accounts for the uptake of oxidized LDL by macrophages during attempted phagocytosis. This term, and the corresponding terms α1LoxM and r1αLˆ oxM (hereα ˆ = α1 + α2), capture the mechanism of foam cell formation in atherogenesis. The model of lipoprotein oxidation presented by Cobbold et al. in [2] and mod- ified herein is based primarily on in vitro experiments, however several terms are motivated by in vivo experimental observation—in particular, the constant source terms for HDL and LDL as well as the removal rates for these species. LDL molecules pass into the intima by transcytosis through endothelial cells or through pores between the cells. And the influx rate is determined by endothelial perme- ability as well as serum LDL level [16]. Several experimental estimates of in vivo permeability to LDL for human and animal models are summarized in [16]. A study by Schwenke and Carew shows the rate of influx of LDL into the intima during the early stage of cholesterol loading of rabbits is highly variable however this rate be- comes nearly constant after the first few days of cholesterol feeding [22]. Hence we assume that serum level and hence the source for LDL, and similarly for HDL, is constant. In contrast, the removal rates through efflux (and accounting for degra- dation as well) depend on the concentration of lipids in the tissue as opposed to constant serum levels. As such, the term dLLθ (similarly dH H for HDL) models the efflux. This is consistent with the model in [2] as motivated by in vivo observation in animal studies. The first three equations of (1)–(7) are the central focus of the present study, and these require some explanation. As the inflammatory nature of atherosclero- sis is mediated by the presence of macrophages [20, 21], the source term for this species, the term φ in (1), is of particular interest in the modeling process. Because macrophages migrate in response to chemical stimuli resulting from the presence of cellular debris (including forming foam cells) and are sensitive to the presence of oxLDL, the source φ may depend on foam cell density F as well as oxLDL con- centration Lox. A particular dependence is proposed in section3. In addition to this source, macrophages are subject to binding with oxLDL and removal (dM M) by normal cell turnover or reverse migration. The source term for foam cells comes from the binding of macrophages with oxLDL. The transfer of excess lipids to HDL, the MRCT process, appears in (2) as r2hHF . Decay of foam cells (e.g. cellular degradation) not related to MRCT is accounted for through the term dF F . (Any such contribution to foam cell growth or decay is expected to be negligible, and taking dF = 0 does not impact the results presented herein.) While the bio-physical significance of each parameter in this system can be spec- ified as seen in table1, only a small portion have known values extracted from in vitro or in vivo experimentation (see [2, 12, 18,1, 13, 16, 22]). Moreover, the primary processes—LDL modification, foam cell formation, and MRCT—appear to occur on different time scales. In vitro studies of LDL modification indicate that the process occurs over time periods of a minutes [2, 17] whereas MRCT occurs

2HDL is not explicitly included in the modeling presented in [9]. However, the process of MRCT may be incorporated into the spatio-temporal modeling framework in [9] and analyzed using the methodology in [10], and [11] 3The focus of [2] is on the lipid oxidation process. Therein the authors provide a thorough analysis of the role of HDL as a sacrificial target for free radical attack. The goal at present is to model direct effects of HDL on foam cells through MRCT. 828 SHUAI ZHANG, L. R. RITTER AND A. I. IBRAGIMOV over hours and days [26] (again in vitro). These factors motivate an analysis of the relative nature of parameters and the imposition of a quasi-steady state assumption (QSSA).

Term Bio-physical meaning Approx. Value (if known) φ Source of macrophages dM Cell death or turnover of macrophages dF Degradation or other mitigation of foam cell presence α1, α2 Rates of foam cell formation h MRCT rate by HDL r1, r2 Efficiency factors −5 −1 pLB Influx rate of native LDL from 3.84 · 10 µMs [1, 16] −4 −1 pHB Influx rate of HDL from lumen 5.8 · 10 µMs [2] pR Rate of production of free radicals −1 −1 kR Reaction rate of lipid oxidation LDL 3µM s [12] −1 −1 kH Reaction rate of lipid oxidation HDL 0.4µM s [2] −1 −1 kA Rate of transfer of electron to lipid from anti-oxidant 1.55µM s [13, 18] −5 −1 −1 kR0 Rate of per-oxidation of the lipid core of LDL 6 · 10 µM s [12] −4 −1 ∗ dH Rate of reverse transport and degradation of HDL 1.8 · 10 s [2] −5 −1 ∗ dL Rate of reverse transport and degradation of native LDL 2.4 · 10 s [2] ρ Annihilation rate of ROS by anti-oxidant Aox Anti-oxidant concentration Table 1. Parameters by bio-physical interpretation. ∗These val- ues are computed from in vivo experimental data of a rabbit model.

3. A QSSA analysis of MRCT. Motivated by preliminary in vitro evidence of different time scales for LDL modification and MRCT, we consider reduced systems of equations for which the level of oxidized LDL enters as a fixed parameter. At present we give two analyses, one for which the level of HDL likewise enters as a parameter, and a second for which the HDL level is time varying. These assumptions produce systems of two and three equations, respectively.

3.1. QSSA I: H and Lox constant. We suppose that the lipid oxidation process results in an essentially constant level of oxLDL present in the intima, and at present assume that the available level of HDL is similarly steady. Call the steady state levels of oxLDL and HDL S S Lox, and H , respectively. For future time t, we focus on equations (1) and (2) which can be restated as follows M˙ = φ(F ) − aM (8) F˙ = bM − cF, (9) S S S where a = α1Lox + dM , b = r1(α1 + α2)Lox, and c = r2hH + dF . Moreover, we introduce the functional form for the macrophage source φ ψ + φ F φ(F ) = 0 ∞ . (10) ψ + F

With φ0 < φ∞, this source term allows for increased migration in response to the lesion with a saturation effect for large F —in particular φ(F ) ∼ φ∞ as φ∞ F → ∞. For F ≈ 0, the influx is essentially linear in F , φ(F ) ≈ φ0 + ψ F . The parameter ψ is a base line level of lesion debris whose value is determined by FOAM CELL FORMATION IN ATHEROSCLEROSIS 829 the sensitivity of the immune response to changes in the lesion density when it is small—i.e. during atherogenesis. (Mathematically, it is analogous to the Michaelis constant in enzyme kinetics. In fact, (10) is a modified form of the Michaelis- Menton type source.) The parameters φ0 and φ∞ represent the level of macrophages (or which differentiate into macrophages in the tissue) entering into the intima independent of atherosclerotic processes and at the carrying capacity, respectively. For the most part, the relevant parameter values are not known—even some parameters taken from the literature shine no light on what the value of say ψ or some others would be clinically. Hence we nondimensionalize and focus on general behavior of solutions. Let us set 1 M = mψ, F = fψ, and t = τ. b Then m, f, and τ are nondimensional macrophage cell density, foam cell density, and time, respectively, and the system (8)–(9) produces dm φˆ + φˆ f = 0 ∞ − am,ˆ m(0) = m ≥ 0 (11) dτ 1 + f 0 df = m − cf,ˆ f(0) = f ≥ 0. (12) dτ 0 The nondimensional parameters are φ φ a c φˆ = 0 , φˆ = ∞ , aˆ = , andc ˆ = . 0 bψ ∞ bψ b b To avoid a proliferation of special notation, we drop the hat notation with the understanding that the following analysis involves the non-dimensional parameters (unless otherwise indicated). As mentioned, the present analysis does not necessarily equate healthiness with a zero concentration of foam cells but rather by considering foam cell concentration as it relates to overall immune related material. The precursors to clinically significant lesions contain inflammatory cells (e.g. macrophages, T lymphocytes) that may or may not include lipid droplets [21, 23] 4. Moreover, it is not necessary that small deposits of foam cells will necessarily develop into intermediate or clinically substantial lesions [23,4]. So, rather than equating the health of the system to an absence of foam cells, we propose that healthy or diseased states can be correlated f to the ratio f/m and the efficacy of MRCT with the ratio f+m —the ratio of foam cell to total immune related cellular material. An investigation of (11)–(12) reveals that both of these ratios are completely determined by the parameter c S r2hH + dF lipid efflux due to HDL c = S ∝ . r1(α1 + α2)Lox lipid uptake due to oxLDL Equations (11)–(12) have two equilibria. Only one of these is physically relevant, insomuch as m and f are nonnegative, and this equilibrium is a nodal sink. The equilibrium is given by  me  φ − ac + p(ac − φ )2 + 4acφ (me, f e) = me, where me = ∞ ∞ 0 . c 2a

4The term is frequently used to describe the earliest precursors, however the term is used inconsistently with regard to the presence of lipids. A more precise classification and morphological details can be found in [23]. 830 SHUAI ZHANG, L. R. RITTER AND A. I. IBRAGIMOV

Translating the variables m and f via u = m − me and v = f − f e, the resulting system

ψ∞v e u˙ = − au, with ψ∞ = φ∞ − acf > 0, (13) ψ1 + v e v˙ = u − cv, and ψ1 = 1 + f (14) has two steady states ψ ψ  (0, 0) and ∞ − cψ , ∞ − ψ . a 1 ac 1 e e The definition of f above ensures that ac + 2acf > φ∞. So a standard linear stability analysis readily establishes that the first equilibrium is always stable (for any choice of positive parameters a, c, φ0 and φ∞), and the second is always unstable (a saddle). Alternatively, this system of only two equations has the advantage of being amenable to phase plane analysis. To this end, we define the isoclines Im and If by (see figure1) amˆ − φˆ 1 I (m) = 0 and I (m) = m m ˆ f φ∞ − amˆ cˆ (settingm ˙ = 0 and f˙ = 0, respectively), then (me, f e) is the positive intersection of df these curves. These curves are constructed so that, for the scalar equation for dm obtained from (11)–(12) in the obvious way, the curve Im is a strong lower fence, and If is a strong upper fence on the interval [0, me]. Hence the region bounded between them

N0 = {(m, f)| 0 ≤ m ≤ me,Im(m) ≤ f ≤ If (m)} is a funnel (see [8] chapter 4) and the following result is obtained:

Theorem 3.1. If (m0, f0) ∈ N0, then the solution (m(τ), f(τ)) of (11)–(12) satis- me  fies (m, f) ∈ N0 for all τ ≥ 0 and (m, f) → me, cˆ as τ → ∞. The proposed ratio indicating the efficiency of MRCT as a control on potential lesion growth can be given in terms of the system parameters as e S f 1 r1αLˆ ox e e = = S S . (15) f + m c + 1 r2hH + r1αLˆ ox + dF c A preliminary comparison of this ratio—or rather the % change c+1 from c0 = 0 to c > 0—with the data obtained by Wang et al. in their 2004 in vitro study of the effects of specific proteins on cholesterol efflux to HDL [26] shows that the current model captures the gross observed behavior in MRCT. Additional clinical studies may allow us to isolate relevant ranges for these parameters making way for potential numerical simulations of clinical intervention or activity.

3.2. QSSA II: H varying. We may continue to hold Lox constant but assume that H is a function varying in some sense comparable to M and F . Maintaining the same assumed form for the macrophage source, our full system reduces in this case to M˙ = φ(F ) − aM (16) F˙ = bM − rhHF (17) ˙ H = pHB − hHF − dH H (18) FOAM CELL FORMATION IN ATHEROSCLEROSIS 831

Figure 1. Isoclines for the system (11)–(12). with dimensional parameters a and b defined as in the previous section. This system is slightly simplified by setting dF = 0 and ignoring degradation of foam cells not directly related to cholesterol efflux. We again nondimensionalize choosing a characteristic time using the available parameters p 1 M = ψm, F = ψf, H = HB η, and t = τ. hψ hψ The equations governing the nondimensional immune cells m, foam cells f, and HDL η, are dm φˆ + φˆ f = 0 ∞ − amˆ (19) dτ 1 + f df = ˆbm − rηfˆ (20) dτ dη = 1 − ηf − dη,ˆ (21) dτ φ φ a b rp d with φˆ = 0 , φˆ = ∞ , aˆ = , ˆb = , rˆ = HB , and dˆ= H . 0 hψ2 ∞ hψ2 hψ hψ hψ2 hψ In the interest of readability, we again drop the hat notation and proceed. This system may produce solutions, in different parameter regimes, in which both m and f are bounded and for which f becomes unbounded with m remaining finite. The latter clearly indicates a break down of the equations, at those parameter values, as a viable model of foam cell formation. In those parameter regimes for which a physically reasonable solution exists, the competing processes of inflammation and MRCT are characterized mathematically by two functions of f. Let us define the function G(f) by ar f G(f) = b . d + f 832 SHUAI ZHANG, L. R. RITTER AND A. I. IBRAGIMOV

Inflammatory response of macrophages is given by φ(f) and efflux of lipids from foam cells by G(f). Equations (19)–(21) have an equilibrium solution (me, f e, ηe) if and only if the equation φ(f) = G(f) (22) has a solution f e. (We will continue to only be interested in positive solutions.) Moreover, the linear stability of any such equilibrium state can be expressed in terms of the relationship between φ and G and their rates of change with respect to f at equilibrium. If any equilibrium state (me, f e, ηe) does exist, and we linearize about this state letting a small perturbation (u, v, w) be defined by m = me + u, f = f e + v, and η = ηe + w, then we obtain the linear system  u˙   −a φ0(f e) 0   u   v˙  =  b −rηe −rf e   v  . (23) w˙ 0 −ηe −(f e + d) w The characteristic polynomial for this matrix is P (λ) = λ3 + (f e + d + rηe + a)λ2 + (af e + ad + arηe + drηe − bφ0(f e))λ+ +darηe − (d + f e)bφ0(f e). For a solution f e of (22), 1 φ + φ f e 1 me = φ(f e) ≡ 0 ∞ , and ηe = . a a + af e d + f e Necessary and sufficient conditions for all roots of the characteristic polynomial to have negative real part are obtained by appealing to the Routh–Hurwitz criteria. For the current system, these reduce to φ0(f e) < G0(f e) and (24) (rηe + a)bφ0(f e) < (f e + d)(af e + a(d + a) + rηe(d + a)) + (a + d)rηe(rηe + a) + raηef e. (25) It is readily shown that the condition (25) holds provided 0 < G(f e) − f eφ0(f e), which is necessarily satisfied due to the profile of the curve φ(f) (it is positive and concave for f ≥ 0). The existence and stability of steady state solutions of (19)–(21) can be summarized in four cases.

Case 1. ar = bφ∞. If φ∞(1 − d) > φ0, then there is an equilibrium value of f given by f e = dφ0 . This equilibrium is linearly asymptotically stable. If φ∞(1−d)−φ0 the level of HDL is so diminished by particles exiting the intima (dH large) such that φ∞(1−d) ≤ φ0, then there is no equilibrium point and f grows without bound. The model breaks down since f → ∞ while m → φ∞/a.

Case 2. ar > bφ∞. There is one positive equilibrium value φ b + dbφ − ar + p(φ b + dbφ − ar)2 + 4dbφ (ar − bφ ) f e = 0 ∞ 0 ∞ 0 ∞ . 2(ar − bφ∞) In this case, the equilibrium is linearly asymptotically stable. Figure2 illustrates two cases of existence of equilibrium points. When a single equilibrium point does exist, the stability criterion (24) is illustrated in the figure2 a. FOAM CELL FORMATION IN ATHEROSCLEROSIS 833

Case 3. ar < bφ∞ and d is sufficiently small that both p 2 e φ0b + dbφ∞ − ar ± (φ0b + dbφ∞ − ar) + 4dbφ0(ar − bφ∞) f± = > 0. 2(ar − bφ∞) e In this case, the smaller of the two f− is such that (24) is satisfied and the larger is such that it is not. In this case, the smaller equilibrium value is linearly asymptot- ically stable, and the larger is unstable (as illustrated in2 b). And finally, Case 4. The equation G(f) = φ(f) has no solution. In this case, f grows without bound. The model breaks down much as it does in Case 1 when no equilibrium exists. Note that the function φ describes the influx of immune cells in response to the presence of foam cells—an initiated lesion. The function G has a similar profile, however it goes through the origin and its steepness for small f is proportional to pHB d . This case requires G(f) < φ(f) for all f and corresponds to a combination of (1) strong inflammation (large φ(f)), (2) low HDL influx (small pHB ), and (3) significant efflux or degradation of HDL (large d).

Figure 2. Illustration of the existence of (a) one stable (case 2) or (b) one stable and one unstable (case 3) equilibrium foam cell value f e. The first Routh-Hurwitz criterion G0(f) > φ0(f) is illustrated here as is the sufficient condition 0 < φ(f e) − f eφ0(f e) for the second Routh-Hurwitz criterion to hold. 834 SHUAI ZHANG, L. R. RITTER AND A. I. IBRAGIMOV

4. Discussion. In focusing attention on HDL and its potential to mitigate lesion growth, we have considered two related but differing systems modeling (or rather simplified cases of a single general model of) foam cell formation. The first supposes that over the time scales of interest with respect to uptake of lipids by macrophages and MRCT, the HDL level within the intima is constant. The resulting model al- ways produces a stable steady state solution for which the character of the system— with respect to being healthy or diseased—reduces to the single parameter (15). The value of this ratio could easily be computed directly when clinical or experimental values of biological parameters are available. It is not clear, but is certainly an interesting question, whether such a simple ratio can serve as a clinical marker. A collaborative study with statistical analysis is needed to approach an answer. The key parameter in this analysis is determined in large part by the concentrations of HDL and oxLDL present. However, the results of the model allowing for variable HDL concentration seem to indicate a significant interplay between HDL presence and macrophage influx aside from the specific level of oxLDL. Interestingly, the key factors determining existence and stability of physically rea- sonable solutions of (16)–(18) are φ∞, pHB and dH —as opposed to the concentration of oxLDL. Of course the model suggests that high levels of serum HDL (through pHB ) confer diminished risk of disease progression (this is commonly accepted and reflected in the popular phrase “good cholesterol”). However, the inequalities in cases 1 and 3, in particular, indicate that HDL remaining present in the intima to perform MRCT is also critical. That is, ar ≥ bφ∞ corresponds to pHB ≥ φ(∞)—i.e. HLD influx dominates the inflammatory response. If this condition is not met, a finite, stable concentration of foam cells may still be achieved provided HDL remain in the intima long enough (dH is small enough) and the rate of cholesterol efflux exceeds the rate of inflammatory response G0(f e) > φ0(f e). As mentioned at the beginning of this paper, the effects of inflammation on the ability of HDL to perform RCT are unclear and are the subject of ongoing clinical investigation (in particular, see [6, 14] and the references therein).

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