L.R. Ritter, Akif Ibragimov * , Jay Walton #

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

2. Contents • 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

3. Atherosclerosis 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.

4. Large Muscular Artery Arteries are thick walled, multilayered tubes. Atherosclerosis occurs in the thin innermost layer—the intima.

5. Atherosclerosis Heart Attack

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

7. Mathematical Model • The mathematical model is constructed by mass balance and consideration of the primary disease features. • Motility/diffusion • cell-chemical interaction—e.g. chemo-taxis • cell-cell interactions—e.g. foam cell formation or healthy immune function • chemical reactions—e.g. oxidation of LDL

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

9. Chemo-taxis 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.

10. Chemo-taxis 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

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

12. Mathematical Model 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)

13. Equation for Immune Cells I Self interaction (diffusion) Ii is the ith immune cell type, 1≤ i ≤ NI

14. Equation for Immune Cells I Response to chemical (chemo-taxis)

15. Equation for Immune Cells I Binding to debris (immune response)

16. Equation for Immune Cells I Subspecies interaction

17. Equation for Immune Cells I Binding to oxLDL (foam cell formation)

18. Equation for Immune Cells I Normal degradation

19. Mathematical Model We introduce a coupled nonlinear system of reaction-diffusion-convection equations characterized by chemo-taxis, Immune cells Lesion bulk Chemo-taxis agents NI immune species, ND debris species, NC chemical species

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

21. Bio-physical interpretation of parameters

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

23. Stability analysis (i) 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.

24. Stability analysis (i) Put where On an annular domain Ω with smooth boundary ∂Ω.

25. Stability analysis (i) Substitution of eigenfunctions for u, v, and w produces an algebraic system for the coefficient s. We seek nontrivial solutions u0, v0, w0.

26. Stability analysis (i) There are three eigenvalues s; 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) Y is a measure of the marginal ability of immune cells to respond to an increase in lesion debris. If Y < 0, an increase in lesion drives healthy immune function. If Y > 0 an increase in lesion results in unhealthy inflammation.

27. The role of apoptosis • Apoptotic macrophages are regularly found in atherosclerotic plaques

28. The role of apoptosis • Apoptotic macrophages are regularly found in atherosclerotic plaques • During phagocytosis, macrophages take in membrane derived as opposed to lipoprotein derived cholesterol.

29. The role of apoptosis • 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).

30. The role of apoptosis • 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).

31. Stability analysis (ii) 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.

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

33. Stability analysis (ii) • 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.

34. Stability (ii) details The equations governing perturbation variables are In an annular domain Ω On ∂Ω (Classical solutions are considered.)

35. Stability (ii) details With the various parameters defined by • Some key terms: • E, G—rates of debris removal • B—rate of apoptosis of macrophages • C—rate of foam cell formation Stabilizing: Large E, G  Strong healthy immune response Small B, C  Minimal apoptosis and foam cell formation Stabilizing: Large mu, mv, mz, Small cucv diffusion dominates over chemotaxis

36. Stability (ii) details With some effort, the following inequality is derived, and the energy functional F is defined accordingly.

37. Stability (ii) details Transition matrix for species interaction. Net mobility—diffusive motility –vs- chemo-taxis

38. The role of anti-oxidants and boundary transport • High concentrations of serum LDL is associated with high incidents of atherosclerotic lesions (Kronenberg et al. 1999).

39. The role of anti-oxidants and boundary transport • 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).

40. The role of anti-oxidants and boundary transport • 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).

41. The role of anti-oxidants and boundary transport • 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.

42. Stability analysis (iii) 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

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

44. Stability analysis (iii) 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)

45. Stability analysis (iii)

46. Stability analysis (iii) • Of particular interest: (stabilizing processes) • M & N—decay and removal rates of chemo-attractant • μ1 & μ3—motility/diffusion rates of immune cells and chemo- attractant (within the intima) • P1—a decay rate of LDL within the intima • Q3 & R3—reverse oxidation rates (due to anti-oxidant)

47. Stability analysis (iii) 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.

48. Stability analysis (iii) The stability criteria (sufficiency) consist of several inqualities: Decay and uptake of chemo-attractant by macrophages must dominate chemo-taxis both within the intima(c) and across the endothelial layer (a1 & a3).

49. Stability analysis (iii) Motility of macrophages and diffusion of chemo-attractant within the intima (m1&m3) must be significant when compared to the influx of immune cells across the endothelium (a1 & a3) and chemo-taxis within the intima (c).

50. Stability analysis (iii) If P1 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 P1 > 0. In the case of forward transport, stability requires P1 > |a4|k + oxidation rates + |Ω| where k and Ω depend on the size of the domain.