{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ExercisesSolutions

ExercisesSolutions - Exercises for Tumor Dynamics Module...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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).
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern