  The recent advances in cancer immunotherapy boosted the development of
tumor-immune system models aiming to provide mechanistic understanding and
indicate more efficient treatment regimes. However, the complexity of such
models, their multi-scale dynamics and their overparameterized character
renders them inaccessible for wide utilization. In this work, the dynamics of a
fundamental model formulating the interactions of tumor cells with natural
killer cells, CD8$^+$ T cells and circulating lymphocytes is examined. It is
first shown that the long-term evolution of the system towards high-tumor or
tumor-free equilibria is determined by the dynamics of an initial
\emph{explosive stage} of tumor progression. Focusing on this stage, the
algorithmic Computational Singular Perturbation methodology is employed to
identify the underlying mechanisms confining the system's evolution towards the
equilibrium and the governing slow dynamics along them. It is shown that these
insights are preserved along different tumor-immune system and
patient-dependent realizations. Utilizing the obtained mechanistic
understanding, a novel reduced model is constructed in an algorithmic fashion,
which accurately predicts the dynamics of the system during the explosive stage
and includes half of the parameters of the detailed model. This present
analysis demonstrates the potential of algorithmic asymptotic analysis to
simplify the complex, overeparameterized and multi-scale nature of cancer
immunology models and to indicate the interactions and cell types to target for
more effective treatment development.
