1. Review Etc.

2. Modified Tumor-Immune Model

3. Dimensional Analysis Effective growth rate for tumor cells (density) 1/3 /time Carrying capacity for tumor cells density Immune-mediated death rate 1/(density*time) Tissue localization rate density/time Maximum immune cell stimulation rate 1/time Half-saturation constant for stimulation density Tumor-mediated death rate 1/(density*time) Natural death rate of immune cells 1/time

4. Parameter Values ParameterValue 180 b2 x 10 -9 1.1 x 10 -7 1.3 x 10 4 0.1245 2.0 x 10 7 3.5 x 10 -10 0.0412 The old a 1 needs to be multiplied by density 1/3 to get the new a 1. The max density is about 1 x 10 9, so new a 1 ~ 1000 old a 1.

5. Nonidemnsionalization Time is scaled relative to the tumor’s effective growth rate The immune population is scaled relative to the number of tumor cells inactivated by each immune cell per unit of rescaled time The tumor population is scaled relative to its maximum size, ie the carrying capacity

6. Nondimensional Equations

7. The Healthy Steady State Stability The healthy state appears to be unstable always!! This is different from the model we analyzed in class. To verify this we need to look at trajectories in the phase plane.

8. The Disease States Nullclines –N-nulliclines –E-nullcline

9. Graph of Nullclines for Small  The parameter values used are  = 0.7651,  = 0.1842,  = 0.0404,  = 0.5567,  = 0.0572.

10. Graph of Nullclines for Large  The parameter values used are  = 0.7651,  = 0.1842,  = 0.0404,  = 0.5567,  = 0.2859.

11. Summary of Steady States It is clear from the graph and our in class plots of the effect of moving the e-nullcline that, for this model, there will always be at least two steady states. –This is because the elimination state always exits and the e-nucline will always have a portion in the first quadrant that will insect with the n-nullcline. This is different from what we found in class, where it was possible to increase  enough to eliminate all non-healthy states –Decreasing the tissue localization rate of the effector cells results in the addition of two new disease states, representing higher tumor burdens.

12. Phase Portrait for Small  Four steady states are present. The healthy state is unstable as is the steady state representing medium tumor burden. The dormancy and immune escape states are both stable. This is similar to what we found with the model analyzed in class.

13. Phase Portrait for Large  Two steady states are present. The elimination state is again unstable and the steady state representing dormancy is stable.

14. Conclusions Using the Von Bertalanffy growth function we found that it is possible to obtain four steady states and to capture all three phases of the cancer immunoediting hypothesis. The major difference between the model with Von Bertanlanffy and the one with Logistic growth is that when using Von Bertalanffy growth, the healthy tissue state is always unstable making it impossible to achieve elimination of the tumor. –Therefore dormancy is the best possible outcome in this case.

15. Review The final exam will be comprehensive. It will cover material in the text book from Chapters 1 – 7. It will also cover material not in the text book that was provided to you in the lecture notes. Please see Exam 1 review for what to except from chapter 1- 6.2 –All of that information still applies

16. Review – Discrete Models Exam 1 –Covered some mathematical aspects of discrete equations (problem 1) Exam 2 –Will likely cover modeling aspects of discrete equations Model development, analysis, and interpretation

17. Review – Continuous Models Exam 1 –Covered some model interpretation for single-species continuous equations (problem 2) Exam 2 –This is also fair game for Exam 2 I may ask you a biological question and you need to decide what mathematics to do to answer that question

18. Review – Continuous Models Exam 1 –Covered the analysis of a system of two equations (problem 3) Exam 2 –This is also fair game for Exam 2 Nondimensionalization Produce a phase portrait Produce a bifurcation diagram

19. Review – Continuous Models Exam 1 –Covered model development Exam 2 –This is also fair game for Exam 2 The the description of a biological problem, you should be able to derive a system of equations to model it

20. Review – Specifics Chapter 6: 6.1 – 6.4 and 6.6 –Interacting Species Predator-Prey (Lab 6) Competition (Chapter 6 Assigned Reading) –SIR Models Be able to recognize, develop, analyze and interpret the results from these types of models

21. Review – Specifics Predator-Prey –Population A grows normally in the absence of population B. Population B has a significantly reduced growth rate in the absence of population A. The presence of population B significantly reduces the growth rate of population A. The presence of population A increases the growth rate of population B.

22. Review – Specifics Competition –Population A grows normally in the absence of population B. Population B grows normally in the absence of population A. The presence of population A reduces the growth rate of population B and the presence of population B reduces the growth rate of population A.

23. Review – Specifics SIR Models –Model development –Model reduction –Conditions for an epidemic –Properties of an epidemic Severity Doubling time How many get infected,...

24. Review – Specifics Chapter 7: Receptor-Ligand Binding 7.1 – 7.4 –Be able to develop, analyze and interpret the results from these types of models –Draw and interpret reaction diagrams –Derive mathematical equations Law of Mass Action –Reduce model equations Conservation QSSA

25. Review – Specifics Infectious Disease Models –Analyze and interpret models Tumor Growth –Derive, Analyze and Interpret Models