2.4. Parameter Estimation

Some parameters of our model, such as the decay/death rates of immune cells and cytokines, were taken from available research (more details in Appendix B.1), while others were estimated. We follow the common approach from mathematical biological models to use assumptions on the steady state values of the system to derive those unknown parameters [102,103]. In particular, we make the assumption that after a tumor reaches a very large size, the immune variation within the tumor microenvironment is minuscule, and we denote this state as the steady state of our system.

Different immune patterns of tumors, such as high or low levels of helper and cytotoxic T cells in one group versus another group, indicate that the activation rates of different T cell sub-types from naive T cells vary from one group of tumors to another group. Hence, many parameters of the model, such as the activation rates of T cell sub-types, depend on the tumor immune profile, and therefore we estimated the parameters separately for each cluster.

We assumed the samples with a large number of cancer cells were at the steady state. For each cluster, we used the 85th percentile of cancer abundance as the cutoff, and calculated the steady state values for the cluster by averaging the values from samples that had more cancer cells than this cutoff. Table 2 shows the steady state values of every cluster.

Steady-state abundance of cells and cytokines.

Our assumption above asserts that the rate of change of our model’s variables is 0 at the steady state, or equivalently dXdt=0 at the steady state. With the additional assumptions in Appendix B.1, as well as knowing the steady state values of our model’s variables, we can derive parameter values for each cluster using the fsolve function from the SciPy package in Python. The parameter values for each cluster are given in Table A1.