TIsolns

Review Etc.: 

Review Etc.

Modified Tumor-Immune Model: 

Modified Tumor-Immune Model

Dimensional Analysis: 

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

Parameter Values: 

Parameter Values The old a1 needs to be multiplied by density1/3 to get the new a1. The max density is about 1 x 109, so new a1 ~ 1000 old a1.

Nonidemnsionalization: 

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

Nondimensional Equations: 

Nondimensional Equations

The Healthy Steady State: 

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.

The Disease States: 

The Disease States Nullclines N-nulliclines E-nullcline

Graph of Nullclines for Small : 

Graph of Nullclines for Small  The parameter values used are m = 0.7651, d = 0.1842, g = 0.0404, a = 0.5567, s = 0.0572.

Graph of Nullclines for Large : 

Graph of Nullclines for Large  The parameter values used are m = 0.7651, d = 0.1842, g = 0.0404, a = 0.5567, s = 0.2859.

Summary of Steady States: 

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.

Phase Portrait for Small  : 

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.

Phase Portrait for Large : 

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

Conclusions: 

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.

Another Mystery: 

Another Mystery Why would a tumor first treated with chemotherapy first grow, then shrink? How could we use modeling to address this?

Interacting Species Models: 

Interacting Species Models When two or more species interact, their effect on each other usually falls into one of three categories. 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.

Interacting Species Models: 

Interacting Species Models When two or more species interact, their effect on each usually fall into one of three categories. 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.

Interacting Species Models: 

Interacting Species Models When two or more species interact, their effect on each usually fall into one of three categories. Mutualism or Symbiosis 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 increases the growth rate of population B and the presence of population B increases the growth rate of population A.

Interacting Species Models: 

Interacting Species Models What type of interaction is this? Tumor-Immune Model

Review: 

Review Discrete Equations - Chapters 1 - 3 Linear Single Species Homogeneous (Bacteria growth), Inhomogeneous (Breathing model) Order Number of previous generations need to determine a future generation Finding solutions xn = Cn Behavior of solutions - determine by the eigenvalues,  Increasing, decreasing, oscillating

Review: 

Review Discrete Equations Linear Systems (Tumor Growth) - 1.3, 1.6, 1.8, 1.9 Increase the order any system of two or more linear, first order discrete equations can be written as a single higher order equation Solutions Characteristic equations - eigenvalues Behavior of Solutions Dominant eigenvalue

Review: 

Review Discrete Equations Nonlinear, Single Species - Chapter 2 Single species (Discrete Logistic) Steady states - analytically/graphically Stability - analytically/graphically Cobweb Diagrams |f’(xe)| < 1 for stability Don’t worry about 2 point cycles Chaos Look at 2.1, 2.2, 2.5, 3.1

Review: 

Review Discrete Equations Nonlinear Systems - Chapters 2.7, 2.8, 3.2 Systems (Host-Parasitoid) - 3.2 - 3.4 Steady states Analytically Stability

Review: 

Review Bifurcations What are they? Bifurcation diagrams What are they, why are they useful? Generate them Read them and interpret them

Review: 

Review Continuous Models - Chapters 4 - 6.2 Single Species (Logistic Equation and Spruce Budworm) - Lectures 5 and 6 Model Development Given a description of a biological problem, be able to derive a mathematical model Nondimensionalization (Lecture 5 Notes) Be able to do it Be able to express why its useful Be able to interpret the scales Steady states (Lecture 5 Notes ) Analytically Graphically Stability (Lecture Notes 5) Analytically Graphically Don’t worry about hysteresis!

Review: 

Review Continuous Models - Chapters 4 - 6.2 Systems of Nonlinear ODEs (The Chemostat, Tumor-Immune Interactions, Predator-Prey) - Lectures 7 and 8, Lab 5, 4.2 - 6.2 Model Development Given a description of a biological problem, be able to derive a mathematical model of interacting species Nondimensionalization (Lecture 5 Notes, 4.5) Be able to do it Be able to express why its useful Be able to interpret the scales Steady states (Lecture 7 and 8 Notes, 4.6) Analytically Graphically (5.5) Nullclines Stability (Lecture Notes 7 and 8, 4.7, 4.9) Analytically RE( Graphically Phase portraits (Chapter 5, Lectures 7 and 8)

Review: 

Review In general Model Development Model Analysis Model Interpretation