Mathematical analysis of the life-span shortening in mice induced by radiation

At Institute for Environmental Sciences, Japan, they are conducting experiments to continuously irradiate mice with low dose-rate radiation. In one experiment, they sacrifice a certain number of mice every 100 days and count the number of cancers in the body. They found that, in the irradiated group, the occurrence of cancer happens earlier and the average number of cancer is larger.In the other experiment, they keep the mice until the natural death and they found that the life span is shortened with the increase in the dose rate.
We analyzed the data with a mathematical model. Let F_C(t) be the probability that a mouse has cancers at time t and P_D(t, s) be the probability that a mouse dies by a time t due to a cancer that occurred at time s. Then the survival probability at time t, F_S(t), is given as F_S(t)=1 - ∫dF_C(s)/ds P_D(t, s) ds. In our analysis, F_C(t) is given by the cancer occurrence data. Then by assuming a function form of P_D(t, s), we calculate F_S(t) to compare with the survival data. We found that we can reproduce the survival data by using a simple function for P_D(t, s). We obtained a parameter which expresses the interval between the occurrence of cancer and the death.