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Experimental studies reporting murine Harderian gland (HG) tumourigenesis have been a NASA concern for many years. Studies used particle accelerators to produce beams that, on beam entry, consist of a single isotope also present in the galactic cosmic ray (GCR) spectrum. In this paper synergy theory is described, potentially applicable to corresponding mixed-field experiments, in progress, planned, or hypothetical. The "obvious" simple effect additivity (SEA) approach of comparing an observed mixture dose-effect relationship (DER) to the sum of the components' DERs is known from other fields of biology to be unreliable when the components' DERs are highly curvilinear. Such curvilinearity may be present at low fluxes such as those used in the one-ion HG experiments due to non-targeted ('bystander') effects, in which case a replacement for SEA synergy theory is needed. This paper comprises in silico modeling of published experimental data using a recently introduced, arguably optimal, replacement for SEA: incremental effect additivity (IEA). Customized open-source software is used. IEA is based on computer numerical integration of non-linear ordinary differential equations. To illustrate IEA synergy theory, possible rapidly-sequential-beam mixture experiments are discussed, including tight 95% confidence intervals calculated by Monte-Carlo sampling from variance-covariance matrices. The importance of having matched one-ion and mixed-beam experiments is emphasized. Arguments are presented against NASA overemphasizing accelerator experiments with mixed beams whose dosing protocols are standardized rather than being adjustable to take biological variability into account. It is currently unknown whether mixed GCR beams sometimes have statistically significant synergy for the carcinogenesis endpoint. Synergy would increase risks for prolonged astronaut voyages in interplanetary space.

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