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ExercisesSolutions

# ExercisesSolutions - Exercises for Tumor Dynamics Module...

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Exercises for Tumor Dynamics Module * L.G. de Pillis and A.E. Radunskaya September 22, 2004 Exercises for Equation Development Module 1. Purpose: To interpret model equations biologically and to go through the preliminary steps of qualitative analysis. This model is derived from the paper by Panetta ([Pan96]). It is developed further in the Projects. Exercise: Another simple model of tumor/host interaction describes the growth of two populations, each growing according to a logistic law and competing with each other for resources. In this model, we lump together all non-tumor cells which are at the tumor site, including normal tissue as well as immune cells. We do not assume a constant source of immune cells. Let X ( t ) denote the normal cell population at time t , (including immune cells), and let Y ( t ) denote the tumor cell population at time t . The system of differential equations which describes the model is: dX dt = a 1 X (1 - b 1 X ) - c 1 XY dY dt = a 2 Y (1 - b 2 Y ) - c 2 XY (a) What is the biological interpretation of each of the parameters a 1 , a 2 , b 1 , b 2 , c 1 , and c 2 ? Are they all necessarily positive or negative? (b) Describe hypothetical experiments which would allow the determination of these parameters. (c) Determine the nullclines of this system. Use these nullclines to sketch a few representative phase portraits. Find and label all of the equilibria. (d) What condition must the parameters satisfy in order that the tumor-free equilibrium be stable? Solution: (a) The biological interpretation of the parameters is as follows: a 1 , a 2 : Growth rates of the normal and tumor cells b 1 , b 2 : Carrying capacity of the normal and tumor cells c 1 , c 2 : Competition rate parameters of the normal and tumor cells They are all necessarily positive since the specified system of equations has the required negative signs to account for decrease in numbers wherever necessary. (b) The growth rate could be determined by examining a fixed number of cells (normal and tumor cells in different dishes) with an infinite nutrient supply, while the carrying capacity could be determined by by a similar experiment in which the nutrient supply was limited. The competition rate parameters may be determined using an assay procedure. This involves setting up different ratios of tumor cells to normal cells, wherein the normal cells take in a fixed amount * This work was supported in part by a grant from the W.M. Keck Foundation 1

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of chromium. Varying amounts of tumor cells will kill proportional amounts of normal cells and release chromium, which is then measured using centrifugation and other processes. Thus the parameters c 1 and c 2 may be determined. (c) The four cases that result depend on parameters are shown in Figure 1 (adapted from Boyce and DiPrima, Sixth Edition).
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