Background: Due to imprecise/missing data used for parameterization of ordinary differential equations (ODEs), model parameters are uncertain. Uncertainty of parameters has hindered the application of ODEs that require accurate parameters. Methods: We extended an available ODE model of tumor-immune system interactions via fuzzy logic to illustrate the fuzzification procedure of an ODE model. The fuzzy ODE (FODE) model assigns a fuzzy number to the parameters, to capture parametric uncertainty. We used the FODE model to predict tumor and immune cell dynamics and to assess the efficacy of 5-fluorouracil (5-FU) chemotherapy. Result: FODE model investigates how parametric uncertainty affects the uncertainty band of cell dynamics in the presence and absence of 5-FU treatment. In silico experiments revealed that the frequent 5-FU injection created a beneficial tumor microenvironment that exerted detrimental effects on tumor cells by enhancing the infiltration of CD8+ T cells, and natural killer cells, and decreasing that of myeloid-derived suppressor cells. The global sensitivity analysis was proved model robustness against random perturbation to parameters. Conclusion: ODE models with fuzzy uncertain kinetic parameters cope with insufficient/imprecise experimental data in the field of mathematical oncology and can predict cell dynamics uncertainty band.

Abstract Background: Ordinary differential equation (ODE) models widely have been used in mathematical oncology to capture dynamics of tumor and immune cells and evaluate the efficacy of treatments. However, for dynamic models of tumor-immune system (TIS), some parameters are uncertain due to inaccurate, missing or incomplete data, which has hindered the application of ODEs that require accurate parameters. Methods: We extended an available ODE model of TIS interactions via fuzzy logic to illustrate the fuzzification procedure of an ODE model. Fuzzy ODE (FODE) models, in comparison with the stochastic differential equation (SDE) models, assigns a fuzzy number instead of a random number (from a specific probability density function) to the parameters, to capture parametric uncertainty. We used FODE model to predict tumor and immune cells dynamics and assess the efficacy of 5-FU. The present model is configurable for 5-FU chemotherapy injection timing and propose testable hypothesis in vitro/ in vivo experiments. Result: FODE model was used to explore the uncertainty of cells dynamics resulting from parametric uncertainty in presence and absence of 5-FU therapy. In silico experiments revealed that the frequent 5-FU injection created a beneficial tumor microenvironment that exerted detrimental effects on tumor cells by enhancing the infiltration of CD8+ T cells, and NK cells, and decreasing that of myeloid-derived suppressor (MDSC) cells. We investigate the effect of perturbation on model parameters on dynamics of cells through global sensitivity analysis (GSA) and compute correlation between model parameters and cell dynamics. Conclusion: ODE models with fuzzy uncertain kinetic parameters cope with insufficient experimental data in the field of mathematical oncology and can predict cells dynamics uncertainty band. In silico assessment of treatments considering parameter uncertainty and investigating the effect of the drugs on movement of cells dynamics uncertainty band may be more appropriate than in crisp setting.