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L.R. Ritter, Akif Ibragimov *, Jay Walton # *-Texas Tech University, #-Texas A & M University April 18, 2011 Mathematical Modeling of Atherogenesis

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The disease process Building the mathematical model of atherosclerosis (Keller-Segel model of chemo-taxis) Atherogenesis viewed as an instability Simple stability analysis (i) Stability analysis (ii): apoptosis/differing roles of macrophages Stability analysis (iii): boundary transport and anti-oxidant

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Atherosclerosis is a vascular disease characterized by the build up of lipid laden cells in the walls of large muscular arteries. Low density lipoproteins can become trapped in the artery wall where they become oxidatively modified. This triggers an immune response that can become corrupted as immune cells become lipid laden. An inflammatory feedback loop mediated by chemical signaling may be instigated leading to runaway inflammation and disease progression. Cardiovascular disease continues to be the primary cause of adult mortality through out the United States, Europe, and much of Asia.

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Arteries are thick walled, multilayered tubes. Atherosclerosis occurs in the thin innermost layerthe intima.

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Heart Attack

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injury or leakiness of the endothelial layer Macrophage Endothelial cells or immune cells send chemical signal Macrophage reads signal and migrates to intervene Ox-LDL Oxidized LDL attracts macrophage that attempt to consume the particles: foam cells form Foam cell Macrophage Inflammatory spiral may result

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The mathematical model is constructed by mass balance and consideration of the primary disease features. Motility/diffusion cell-chemical interactione.g. chemo-taxis cell-cell interactionse.g. foam cell formation or healthy immune function chemical reactionse.g. oxidation of LDL

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We consider several generalized species: I : Immune cells, primarily macrophages D : Debrisi.e. the bulk of the lesion C : chemo-taxis inducing agents (chemical signals) L : Low density lipoproteins in a native or partially oxygenated state L ox : LDL molecules after full peroxidation of the lipid core R : Reactive oxygen species (free-radicals) In addition to many parameters including: A ox : Anti-oxidant species L B : Serum LDL concentration (and several reaction rates and additional parameters)

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che·mo·tax·is (kē'mōtāk'sĭs,kěm'ō) n. The characteristic movement or orientation of an organism or cell along a chemical concentration gradient either toward or away from the chemical stimulus. The American Heritage® Dictionary of the English Language, Fourth Edition.

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Keller-Segel Model of Chemo-taxis: For motile species U influenced by the chemical species V: The flux field for U = flux due to diffusion (reaction to U) + flux due to chemo-taxis (reaction to V) measure of motility chemo-tactic sensitivity function

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The point-wise equation for U: where Q is an appropriately defined reaction term.

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In general, each variable I, D, C, and L, is vector of several components representing different phenotypes, chemical agents, or in the case of LDL, levels of oxidation. So that the system consists of a large number of distinct equations. Several studies indicate for example that macrophages play many different roles in the disease process: see The role of macrophages in atherosclerosis Libby, Peter; Clinton, Steven K. (1993), Multifunctional roles of macrophages in the development and progression of atherosclerosis in humans and experimental animals K. Takahashi, Motohiro Takeya, Naomi Sakashita (2002) Pivotal Advance: Macrophages become resistant to cholesterol-induced death after phagocytosis of apoptotic cells Cui et al (2007)

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Self interaction (diffusion) I i is the i th immune cell type, 1 i N I

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Response to chemical (chemo- taxis)

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Binding to debris (immune response)

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Subspecies interaction

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Binding to oxLDL (foam cell formation)

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Normal degradation

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We introduce a coupled nonlinear system of reaction-diffusion- convection equations characterized by chemo-taxis, Immune cells Lesion bulk Chemo-taxis agents N I immune species, N D debris species, N C chemical species

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and linear oxidation of LDL with reverse oxidation* and diffusion. * This is a model adapted from Cobbold, Sherratt and Maxwell, 2002 Number of native LDL subspecies

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The disease is characterized by local accumulation a cells. We propose that atherogenesis the onset of lesion growthcan be viewed as due to an instability in a uniform configuration of cells and chemical species.

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A greatly simplified analysis of a single immune cell type, single debris type and single chemical agent after full oxidation of present LDL: In the interior of an annular domain (or annular cylinder) with homogeneous Neumann B.C. Introduction of small perturbations u, v, and w.

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Put where On an annular domain with smooth boundary.

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Substitution of eigenfunctions for u, v, and w produces an algebraic system for the coefficient We seek nontrivial solutions u 0, v 0, w 0.

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There are three eigenvalues ; two are necessarily real and negative, and one is real and can be positive or negative depending on parameter values. For a given level of hostility (i.e. concentration of oxLDL) is a measure of the marginal ability of immune cells to respond to an increase in lesion debris. If < 0, an increase in lesion drives healthy immune function. If > 0 an increase in lesion results in unhealthy inflammation.

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Apoptotic macrophages are regularly found in atherosclerotic plaques

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During phagocytosis, macrophages take in membrane derived as opposed to lipoprotein derived cholesterol.

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Apoptotic macrophages are regularly found in atherosclerotic plaques. During phagocytosis, macrophages take in membrane derived as opposed to lipoprotein derived cholesterol. This has been seen to induce a survival response in macrophages (Cui et al 2007).

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Apoptotic macrophages are regularly found in atherosclerotic plaques. During phagocytosis, macrophages take in membrane derived as opposed to lipoprotein derived cholesterol. This has been seen to induce a survival response in macrophages (Cui et al 2007). Apoptotic macrophages may also lead to plaque rupture in later stages of the disease (Tabas 2004).

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We consider two competing macrophage species, one healthy and one corrupt. A similar eigenvalue perturbation is not feasible. Again imposing a zero flux condition on the boundary of the domain.

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We again introduce perturbation variables u, v, w, and z, but consider another approach to the stability analysis.

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We can state a stability theorem giving a specific set of requirements on the relative size of the system parameters. The significant requirements include Certain domination of diffusive motility of immune cells over chemo-taxis, An inequality relating healthy immune response to foam cell formation, Certain domination of diffusion of the chemo-attractant over its production in response to the lesion.

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The equations governing perturbation variables are (Classical solutions are considered.) In an annular domain On

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With the various parameters defined by Stabilizing: Large E, G Strong healthy immune response Small B, C Minimal apoptosis and foam cell formation Stabilizing: Large u, v, z, Small u v diffusion dominates over chemotaxis Some key terms: o E, Grates of debris removal o Brate of apoptosis of macrophages o Crate of foam cell formation

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With some effort, the following inequality is derived, and the energy functional F is defined accordingly.

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Transition matrix for species interaction. Net mobilitydiffusive motility –vs- chemo-taxis

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High concentrations of serum LDL is associated with high incidents of atherosclerotic lesions (Kronenberg et al. 1999).

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Transport of LDL molecules and immune cells across the endothelial layer are key to atherogenesis (Ross 1999).

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High concentrations of serum LDL is associated with high incidents of atherosclerotic lesions (Kronenberg et al. 1999). Transport of LDL molecules and immune cells across the endothelial layer are key to atherogenesis (Ross 1999). Anti-oxidant levels can mitigate the disease onset and progression (Howard and Kritchevsky 1997).

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High concentrations of serum LDL is associated with high incidents of atherosclerotic lesions (Kronenberg et al. 1999). Transport of LDL molecules and immune cells across the endothelial layer are key to atherogenesis (Ross 1999). Anti-oxidant levels can mitigate the disease onset and progression (Howard and Kritchevsky (1997). The present energy-estimate approach can be extended to include additional species and boundary transport.

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We seek to capture the full role of LDL oxidation, anti-oxidant presence and boundary transport. To include these, we begin with a system of one of each species

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and allow for movement of immune cells, chemo-attractant, and LDL molecule across the endothelial layer. subendothelial intima (domain of equations) Endothelial layer internal elastic lamina forward & reverse transport from blood

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We again consider a perturbation off of a constant equilibrium solution The linearized equations are considered with third type boundary conditions for those variables representing immune cells u, chemo- attractant w, and native LDL z. forward transport (from blood to tissue) reverse transport (from tissue to blood)

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Of particular interest: (stabilizing processes) o M & Ndecay and removal rates of chemo-attractant o μ 1 & μ 3 motility/diffusion rates of immune cells and chemo- attractant (within the intima) o P 1 a decay rate of LDL within the intima o Q 3 & R 3 reverse oxidation rates (due to anti-oxidant)

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Construction of appropriate functionals requires treatment of the boundary terms. The following inequalities are useful and result in explicit inclusion of the geometry of the domain.

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The stability criteria (sufficiency) consist of several inqualities: Decay and uptake of chemo-attractant by macrophages must dominate chemo-taxis both within the intima and across the endothelial layer

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Motility of macrophages and diffusion of chemo-attractant within the intima ( 1 3 ) must be significant when compared to the influx of immune cells across the endothelium ( ) and chemo-taxis within the intima ().

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If P 1 is the rate of LDL decay within the intima Then in the case of reverse transport of LDL (from the intima back to the blood), stability requires P 1 > 0. In the case of forward transport, stability requires P 1 > |4| + oxidation rates + || where and depend on the size of the domain.

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Anti-oxidant reactions must occur at rates superior to the rates of the LDLfree radical reaction (R 1 ) and the peroxidation rates of the lipid core (Q 2 & Q 5 ).

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We impose a pair of conditions:

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And define a pair of functionals:

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A complete generalization for an unspecified number of each species is currently under construction. Such analysis will give a base line set of inequalities relating various parameters that can eventually be compared with bio- medical data.

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R. Ross, Cell biology of atherosclerosis, Annu. Rev. Physiol., 57 (1995) 791804 R. Ross, Atherosclerosis: An inflammatory disease, N. Eng. J. of Med., 340(2) (1999) 115126 C.A. Cobbold, J.A. Sherratt, and S.J.R. Maxwell, Lipoprotein oxidation and its significance for atherosclerosis: a mathematical approach. Bull. Math. Biol., 64 (2002), 6595 E.F.Keller and L.A. Segel, Model for chemotaxis. J. Theor. Biol., 30 (1971), 235248

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Stability Analysis of a Reaction-Diffusion System Modeling Atherogenesis, With A.I. Ibragimov, and J.R. Walton, SIAM J. Appl. Math. 70 (2010), pp. 2150-2185 Stability analysis of a model of atherogenesis: An energy estimate approach II, With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, J. of Comp. and Math. Meth. in Med., Vol.11(1), (2010) pp. 67-88 Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis, With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, Discrete and Continuous Dynamical Systems, Supplement (2009), pp. 630-639 Stability analysis of a model of atherogenesis: An energy estimate approach With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, J. of Comp. and Math. Meth. in Med., Vol.9(2),(2008) pp. 121-142 A dynamic model of atherogenesis as an inflammatory response, With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, DCDIS A Supplement, Advances in Dynamical Systems, Vol.14(S2) (2007), pp. 185--189 A Mathematical Model of Atherogenesis as an Inflammatory Response, With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, Math. Med. Biol., 22 (2005), pp. 305--333

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