A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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• What are some applications of the Keller-Segel model? In the next chapter, I will focus on the last question which will include some examples of the previous two. 35
Chapter 4 Applications of the Keller-Segel Model The model as first derived by Keller and Segel was done so in the context of a specific example of chemotaxis—i.e. aggregation of slime mold [11]. In this final chapter, we will explore two other examples of problems for which the Keller-Segel model has been used to describe the evolution of a system in which organisms (in these cases human cells) respond to chemicals in their environment. 4.1 Angiogenesis 4.1.1 What is Angiogenesis? When a tumor first appears in the human body it is in a stage that would be termed avascular meaning that it does not have its own blood supply but rather depends on diffusion of oxygen and other nutrients from nearby tissues for its survival 36
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: or in two or more dimensions 34 ∂
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 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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