A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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of cholesterol is contained within the latter two particles. The lipoprotein structure consists of a lipid core containing cholesterol esters and triglicerides, and a coat that is composed of regulatory surface proteins, unesterified cholesterol, phospho- lipids and a variety of other minor components that may include molecules and proteins associated with antioxidant defenses. LDL particles transport cholesterol that is needed for various cellular functions such as cell membrane formation and hormone synthesis. About 60% to 70% of the total body cholesterol is contained in the LDL particles. HDL particles account for most of the remaining cholesterol. The function of the HDL particles appears to be involved with the return of excess lipids from tissues to the liver for subsequent processing (a process referred to as reverse transport). Many studies have unequivocably shown that elevated blood levels of LDL cholesterol confer a higher risk of developing CVD. Although LDL particles are not found in atherosclerotic plaques, oxidatively modified LDL particles are. A mathematical model of the in vitro oxidation process of LDL is described by Cobbold, Sherratt and Maxwell [2]. As described above, the LDL particle contains on its surface a number of antioxidant defenses including α-tocopherol (vitamin E). In the plasma where the concentration of free radicals is low and other antioxidant particles are present, LDL usually remain in its native, unoxidized form. As the LDL particle migrates into the intima, it may expend all of its innate defenses against oxidation. According to the model in [2], each contact with a free radical depletes the LDL particle of one of its vitamin E molecules. Once these are exhausted, the next contact with a free radical initiates peroxidation of the lipid core. In [2], the authors consider a typical LDL particle with six α-tocopherol molecules on its surface exposed to free radical attack. Letting L6(t) denote the concentration of LDL particles with six antioxidant defenses at time t, the authors assume a linear 57
eaction with a free radical R at a rate ke (L6 + R ke −→ L5), to obtain the variable L5(t), the concentration of LDL particles with five antioxidant defenses remaining. In this fashion, they obtain a system of nine ordinary differential equations governing the concentration of free radicals R(t), the concentration of LDL particles with i antioxidant defenses Li(t), i = 0..6, and the concentration of oxidized LDL Lox(t). The time rate of change of each species of LDL is given by the linear reactions, and the change in radical concentration is increased by a production term (that is estimated since it is not measurable) and decreased by reaction with LDL species. In the model of atherogenesis proposed in this paper, we adopt Cobbold Sherratt and Maxwell’s model of LDL oxidation. We modify their equations only by adding spatial dependence in the form of a diffusion term. LDL in both its native and oxidized form are only slightly mobile within the arterial tissue. We account for this by a small diffusion coefficient. This process is obligatory in the formation of the atherosclerotic plaque. In 1977, Goldstein et al. [5] discovered that certain immune cells, in particular macrophages, have a high affinity for oxidatively-modified LDL but not native LDL. This results in trapping of cholesterol within the arterial wall. Macrophages engorged with lipids are referred to as foam cells. Unable to perform their normal duty of degrading debris, these lipid-laden cells accumulate and signal other immune cells to the site instigating plaque growth. Corruption of the normal immune process Circulating immune cells in the blood, such as monocytes and T-lymphocytes, migrate into tissues in response to chemical signals secreted by various cells including endothelial cells and other immune cells. Under normal healthy conditions, these immune cells aid in the degradation of apoptic cells as well as the removal of foreign 58
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A SHORT COURSE IN THE MODELING OF C
Page 3 and 4:
Contents List of Figures iii 1 Intr
Page 5 and 6:
List of Figures 1.1 Life Cycle of D
Page 7 and 8:
mobile species.” Referring back t
Page 9 and 10:
this new form, as a plasmoid, the i
Page 11 and 12: the macroscopic (see Patlak [20]),
Page 13 and 14: η(x, t) The concentration of acras
Page 15 and 16: The only way for this result to hol
Page 17 and 18: Equation (2.2) is the identifying e
Page 19 and 20: 2.2 The Keller-Segel Model of Chemo
Page 21 and 22: Figure 2.1: The experimental config
Page 23 and 24: , b ′ → 0 and s → s0 as z →
Page 25 and 26: 2.4 Some conclusions In this chapte
Page 27 and 28: Chapter 3 Modeling: The Microscopic
Page 29 and 30: master equation of our pdf we need
Page 31 and 32: This equation assumes that our walk
Page 33 and 34: A number of observations can be mad
Page 35 and 36: The notable characteristic of this
Page 37 and 38: efore, the corresponding continuous
Page 39 and 40: or in two or more dimensions 34 ∂
Page 41 and 42: Chapter 4 Applications of the Kelle
Page 43 and 44: (SMCs). The ECs are the main cellul
Page 45 and 46: the review [17] and the references
Page 47 and 48: The first two assumptions immediate
Page 49 and 50: arrive at the system of interest
Page 51 and 52: 1/40 th of an hour, and during this
Page 53 and 54: Figure 4.3: Time evolution of the e
Page 55 and 56: Figure 4.5: Evolution of the growth
Page 57 and 58: tree. More advanced atherosclerotic
Page 59 and 60: three distinctive layers-the intima
Page 61: to be characterized by a change in
Page 65 and 66: Lesion progression Smooth muscle ce
Page 67 and 68: Modeling Assumptions We begin by id
Page 69 and 70: gradient. For the immune cells and
Page 71 and 72: Both production and consumption app
Page 73 and 74: assume that the threshold values of
Page 75 and 76: The system (4.22)-(4.24) was solved
Page 77 and 78: Bibliography [1] E. De Angelis and
Page 79 and 80: [15] I. R. Lapidus, ”Pseudochemot
Page 81 and 82: Appendix A Linear Chemical Reaction
Page 83 and 84: proportional to A”. So that there
Page 85 and 86: Appendix B Bachmann-Landau Order Sy
Page 87: 3. Let h(τ) = e −τ and p(τ) =
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A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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