I assume that the cancer patient needs to receive a certain amount of radiation for each treatment  call it P particles. The time required for each treatment is then equal to the number of particles required to be emitted divided by the rate of particle emission:
T_treatment= P/R
The rate of particle emission slows down as the radioactive source ages:
R(t) = R_0 x (1/2)^(t/t_hl)
where R_0 is the initial rate of particle emission, and t_hl is the half life of the material. In this case you have t=2 years, t_hl = 2 years, so:
R(2 years) = R_0 x (1/2)^1 = R_0/2.
The time initially required for treatment at t=0 years is
T_treatment = P/R_0 = 10 minutes.
The time required for treatment at t=2 years is:
T_treatment = P/(R_0/2) = 2 P/R_0 = 2 x 10 minutes = 20 minutes.
