A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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which is intended to account for both chemotaxis in response to protease c and haptotaxis in response to fibronectin f. This is based on an ad hoc postulate that the mechanics obey a power law (τ1(c) = c ˜γ1 for example) which is then mollified (that is τi is bounded away from both zero and infinity). The problem is scaled with initial conditions η(x, 0) = 1, c(x, 0) = 0, f(x, 0) = 1, and v(x, 0) = 15κ100(1 − cos(2πx)) 100 . The value of κ100 = 0.933 × 10−30 is chosen so that � 1 v(x, 0) dx = 1. This choice 0 of initial condition on the TAFs corresponds essentially to a unit pulse 6 of TAF at the center of the lattice at the onset of the numerical experiment. The remaining parameter values used are given in table 4.1. Table 4.1: Parameter Values [16] D = 3.5e − 5 α1 = 0.001 α2 = 1.0 γ1 = 1.2 β1 = 1.0 β2 = 0.001 γ2 = 1.2 λ1 = 73.0 ν1 = 0.007 β = 0.222 λ2 = 19.0 ν2 = 1.28 Figure 4.2 shows the decay of fibronectin in an interval x ∈ (0.44, 0.56) about which the TAFs are originally concentrated. The width of the channel corresponds to about 6 microns which is in the range of diameters of a typical capillary. Figures 4.3 and 4.4 show the corresponding bimodal concentration of endothelial cells at the ends of this interval which is indicative of the tube structure growth of a new capillary. The remaining figures, 4.5 and 4.6, show the phenomenon of rapid uptake of TAFs by the endothelial cells—the TAF is nearly completely depleted within 6 The sequence Fm(x) = 15κm(1−cos(2πx)) m tends to the Dirac function δ(x−1/2) as m → ∞ in the sense of distributions. 45
1/40 th of an hour, and during this same short period the density of protease attains a near steady profile. For an analysis of the system with respect to changes in some of the parameter values please see [16]. 4.2 Atherogenesis A more recent application of the Keller-Segel model of chemotaxis is the current study of the disease atherosclerosis. Before discussing the modeling approach, let me familiarize the reader with the this disease. 4.2.1 What is Atherosclerosis? Atherosclerosis is a chronic disease involving the accumulation of lipid laden cells in the arterial wall. The precursors of the disease can begin in humans during early childhood with the appearance of fatty streaks which are purely inflamma- tory lesions [23]. These early lesions, which consist primarily of monocyte 7 derived macrophages 8 , T-lymphocytes 9 , and lipids 10 , may be found throughout the arterial 7 A large white blood cell with finely granulated chromatin dispersed throughout the nucleus that is formed in the bone marrow, enters the blood, and migrates into the connective tissue where it differentiates into a macrophage. 8 A phagocytic tissue cell of the mononuclear phagocyte system that may be fixed or freely motile, is derived from a monocyte, and functions in the protection of the body against infection and noxious substances 9 Any of several lymphocytes (as a helper T cell) that differentiate in the thymus, possess highly specific cell-surface antigen receptors, and include some that control the initiation or suppression of cell-mediated and humoral immunity. 10 Any of various substances that are soluble in nonpolar organic solvents (as chloroform and ether), that with proteins and carbohydrates constitute the principal structural components of 46
Page 1 and 2: 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: arrive at the system of interest
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 and 62: to be characterized by a change in
Page 63 and 64: eaction with a free radical R at a
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) =

A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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