A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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∂n1 ∂t ∂n2 ∂t � = ∇ · µ1∇n1 − χ 0 11 � = ∇ · µ2∇n2 − χ21 n1 c1 n2 c1 n1 � ∇c1 − χ 0 13 ∇c3 − d1n1 c3 ∇c1 + δ n2 � ∇n1 − d2n2 n1 65 (4.16) (4.17) ∂n3 ∂t = ∇ · (µ3∇n3) + d1n1 + d2n2 − f2(c3)n1n3 (4.18) ∂c1 ∂t = ∇ · (ν1∇c1) − α1n1c1 − α2n2c1 + f1(n3)n3 (4.19) ∂c2 ∂t = ∇ · (ν2∇c2) − kRc2 (4.20) ∂c3 ∂t = ∇ · (ν3∇c3) + kRc2 (4.21) The motility coefficients µ1 and µ2 can depend on the concentrations of the chemical species as well as position. The diffusion coefficients µ3, ν1, ν2, and ν3 can in gen- eral depend on position. The parameters d1 and d2 that appear in equations (4.16), (4.17), and (4.18) are the death (or corruption) rates of immune cells and SMCs, respectively. For simplicity, we have not explicitly denoted any functional depen- dence of these parameters on any variables. However, we could easily allow them to depend on other variables in the system. For example, d1 could depend on c3 if it is observed that the rate of conversion of macrophages into foam cells varies significantly with oxLDL concentration. The consumption of debris, or growth of debris in the disease case, is inherent in the term f2(c3) which can change signs. If the concentration of oxLDL is small, say below some threshold value, then f2 > 0 which corresponds to healthy immune function. In this case, the concentration of debris is decreased as immune cells degrade dying cells and keep the tissue free of foreign agents. Once the level of oxLDL exceeds the threshold level, the function f2 can become negative. This term in equation (4.18) is then a growth term reflecting the cessation of normal immune response and the progression of plaque formation.
Both production and consumption appear explicitly in the equation for the chemo-attractant. The two consumption terms, one corresponding to each of immune cells and SMCs, appear because we assume that upon responding to a chemical signal a cell indicates its response by consuming or otherwise neutralizing a chemical particle. The production of chemo-attractant is in response to the concentration of debris, although the exact mechanism of this production is not specified in equation (4.19). The last two equations reflect the conversion of native LDL into oxidized LDL. The parameter R is the concentration of free-radical. We have taken it to be a constant, however, it could be considered a time dependent variable as is the case in [2]. Native LDL is converted into oxidized LDL at a constant rate k per unit of radicals and per unit of native LDL, where we simplify this conversion to occur in a single chemical step. Accounting for a six (or more) step conversion process imposes no additional difficulty and is omitted here so as not to obscure the more salient features of the current model. Boundary and initial conditions To complete the description of the model and in order to obtain a well-posed mathematical problem it remains to specify the domain of interest and to impose boundary and initial conditions on each of the species. In the results that appear in the following sections herein, we consider Ω ⊂ R 2 as a thin annulus of inner radius r1 and outer radius r2. Let Γ1 denote the inner boundary and Γ2 the outer boundary of Ω. Then we consider each of the dependent variables to depend on the polar-cylindrical coordinates (r, θ) so that r = r1 and r = r2 on Γ1 and Γ2, respectively. Moreover, we let Ω represent the intima so that Γ1 corresponds to the 66
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A SHORT COURSE IN THE MODELING OF C
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Contents List of Figures iii 1 Intr
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List of Figures 1.1 Life Cycle of D
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mobile species.” Referring back t
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this new form, as a plasmoid, the i
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the macroscopic (see Patlak [20]),
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η(x, t) The concentration of acras
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The only way for this result to hol
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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 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: gradient. For the immune cells and
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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