Thanks dear Elman,

You are right! The behavior of the ODE models depends on their parameters! My model is not ODE, actually! I just need to know about ODE models to validate my results. In fact, I want to compare my results with ODE or DDE models of this area.

How can I know which ODE models or which parameters are right while all of them are published in scientific journals?

I agree with Elman.  Most systems in the body are nonlinear and many (e.g. macrophages) function on the edge of criticality between order and chaos.  It is the most effective way to both be responsive to multiple needs (stimuli), yet robust enough not to be perturbed by everything that comes along.  Since the possibilities of this type of system, which functions far from thermodynamic equilibrium, also allow for bifurcations, it is not strange that you are getting the kinds of results that you are getting

I agree with the above discussions. In addition, it should be noted that ODE models for cancer exhibit different behaviours because they are inherently different and might have different assumptions in their derivations. You could also have different models with different assumptions yielding similar quantitative behaviours. So you need to look carefully at the assumptions of the models.

Hi Negin,

First of all, and concerning the reference you are using. Either the authors are mistaken or the title of the paper. And the reason is that I am one of the co-authors of the paper with that title.

A Validated Mathematical Model of Tumor Growth Including Tumor–Host Interaction, Cell-Mediated Immune Response and Chemotherapy by Álvaro G. López, Jesús M. Seoane and Miguel A. F. Sanjuán. Bulletin of Mathematical Biology 76, 2884-2906 (2014) that one can download from any one of the following  links

http://bit.ly/1QksmO3

http://bit.ly/1Lqm2ES

The ODE model indeed is nonlinear and strongly depends on the parameters. There is another slightly different ODE model

http://bit.ly/1R6DG0h

with transient chaotic solutions that were controlled by using the partial control method of chaos.

Certainly, there are many models using  using ODE, PDE, DDE, CA, etc...I suggest you to start understanding one type, perhaps ODE, since the methods and techniques are very different.

Miguel AF Sanjuan

 Negin, defining and making a model should depend on the question that you want to answer. A quick search of ODE models of cancer will give you all sorts of models from very simple to very complex containing vasculature, hypoxia, necrosis, stem cells, immune cells, normal tissue, cells with different molecular mechanisms, sensitive and resistant cells, etc. Not all models will be relevant to your study. Perhaps some will give you insight on yours, or maybe give you a starting point for you to create your own based on your specific question. You have to figure out which ones are useful, and which assumptions you need to make to abstract the system in a way to make it relevant to the biological problem that you are addressing.  

Dear Miguel,

I did a mistake about the name of the paper. I've corrected it now. Please, accept my apologies!

Thanks everyone,

Actually, I want to know which behavior tumor cells would show through interactions with the effector cytotoxic T cells (CTLs)  (i.e., an exponentially decrease, an exponentially increase, an oscillatory decrease, an oscillatory increase, or some thing else). Do you think I can fit my model with Gompertz curve?

Hi Negin, I believe that the answer you could be looking for may be found in the above comments. You shall not get a straight answer to say (eg. tumor-immune interactions are oscillatory). Modeling tumor-immune interactions would depend on the assumptions one makes (eg on the tumor growth stage, the geometry, local kinetics ,  and parameters chosen). In simple terms: all the above scenarios from which you want to choose the exact behavior of tumor-immune interactions are possible depending on the model assumptions, the model used, and/or the parameter values, which is rather realistic.

I think the Gompertz curve may only be used to fit tumor growth, without considering the immune response.

I agree with Joseph Malinzi. It might depend on the chosen parameters to obtain a different behavior of course, and certainly you can use a logistic or Gompertz growth term in your model equation, but as Dr. Malinzi says, this has nothing to do with the interaction with the T cells.

it's depends on parameter. you can see it by looking at simple spring mass system with damping. based on values of damping, and internal frequency, you have different behavior. In your system, parameters depend on properties of medium, cell and other involved processes. Usually in biological processes, models are simply rate model. In absence of any interaction any species of system simply growth or damp, but interaction between these processes, produce non-trivial behaviors like oscillations, as you mentioned in question.