About a generalized model of lymphoma

In this work we introduce and analyze a generalized model of precursor T-lymphoblastic lymphoma as a competition between two clonotypes of naïve T-cells, one “normal” and one tumorous. It is modeled as a continuous-time bivariate Markov process. Using an expansion of the master equation a deterministic approximation and the Fokker–Planck equation are derived. For a deterministic model we show existence and uniqueness of global solutions and positive invariance of the first quadrant of the phase space. Stability analysis of the model is performed, finding conditions guaranteeing existence of a unique, positive, steady state, which is proved to be globally stable. It is shown that expectations of fluctuations for both clonotypes tend to zero for large time. We also present numerical simulations in which two types of behavior of solutions are observed: either both clonotypes survive in the repertoire or the “normal” clonotype becomes extinct. Comparing this result with the rules of maintenance of naïve T-cell repertoire, which say that clonotypes with more specific set of receptors have longer life-span, it seems that “normal” clonotype follows them, whereas the tumorous one violates them and tends to the maximum possible expansion. The model supports the hypothesis of mutated precursor cells as an origin of cancer.