Fuzzy Stochastic Petri Net with Uncertain Kinetic Parameters for Modeling Tumor-Immune System

Uncertainty as inherent feature of Tumor-Immune system causes unpredictable behaviors of this complex network. Uncertainty of tumor-immune system is due to randomness in cell-cell interactions, vague, incomplete data, dynamic properties of tumor (including, e.g., extracellular ligands, mutation types, vascular status, phenotypic distribution) which are varying during time and patient-dependent properties. Fuzzy Stochastic Petri Net (FSPN) can capture this uncertainty that combine Stochastic Petri Net (SPN) with fuzzy sets. SPN model the dynamics of this complex network with regarding randomness in cell interactions and fuzzy sets consider fuzziness. FSPN of this study associate a fuzzy number instead of crisp number to kinetic parameter of SPN. Tumor-immune system of this study consider interactions of Tumor cells, Cytotoxic T lymphocytes (CTL) and Myeloid-derived suppressor cell as major component of system. CTLs are produced by immune activation of cytotoxic T cells and MDSCs augment in pathological situations such as cancer that acquire strong immunosuppressive activities. The dynamical behavior of tumor-immune system with regarding uncertain kinetic parameters is achieved by FSPN and the steady state behavior of the system with regarding fuzzy uncertain kinetic parameters is computed. The model simulates the dynamics of the cells in tumor escape and tumor elimination phases. FSPN proves that with increasing uncertainty of model parameters, the uncertainty of cell dynamics also increases. We showed that if the model kinetic parameters be a fuzzy number with a triangular membership function, the uncertainty interval of the cells is triangular in relation to the alpha-cuts.This method can be used for modeling and simulation of any biological network with uncertain information.