We consider multi-target models for use in analyzing data of the dose-response relationship. The target sizes we are concerned with here are both homogeneous, as assumed in the classical model, and heterogeneous, as simplified using geometric progression. We apply two models for establishing the multi-target models: a Poisson regression model constructed by assuming that the response variable Y follows Poisson distribution, and a gamma-frailty model as a Poisson mixture model derived by adding random common risks having a gamma distribution. Applying these models to experimental data relating the effects of miso fermentation-stages on the survival rate of cells of intestinal crypts of mice exposed to radiation yielded the result that there were substantial frailties associated with all miso fermentation-stages. Short-term and medium-term fermented miso provided similar effects, whereas long-term fermentation had the lowest-relative risk value, indicating a significant protection of the crypts against exposure effects. A gamma-frailty model based on heterogeneous target size was more suitably applied when there were at least 3 dead stem cells having 10 target genes.