Ionizing radiations encountered by astronauts on deep space missions produce biological damage by two main mechanisms: (1) Targeted effects (TE) due to direct traversals of cells by ionizing tracks. (2) Non-targeted effects (NTE) caused by release of signals from directly hit cells. The combination of these mechanisms generates non-linear dose response shapes, which need to be modeled quantitatively to predict health risks from space exploration. Here we used a TE+NTE model to analyze data on APC(1638N/+) mouse tumorigenesis induced by space-relevant doses of protons, 4He, 12C, 16O, 28Si or 56Fe ions, or γ rays. A customized weighted Negative Binomial distribution was used to describe the radiation type- and dose-dependent data variability. This approach allowed detailed quantification of dose-response shapes, NTE- and TE-related model parameters, and radiation quality metrics (relative biological effectiveness, RBE, and radiation effects ratio, RER, relative to γ rays) for each radiation type. Based on the modeled responses for each radiation type, we predicted the tumor yield for a Mars-mission-relevant mixture of these radiations, using the recently-developed incremental effect additivity (IEA) synergy theory. The proposed modeling approach can enhance current knowledge about quantification of space radiation quality effects, dose response shapes, and ultimately the health risks for astronauts.