A general algorithm for radioactive decay with branching and loss from a medium

Many areas in the field of health physics require evaluation of the change of radionuclide quantity in a medium with time. A general solution to first-order compartmental models is presented in this paper for application to systems consisting of one physical medium that contains any number of radionuclide decay chain members. The general analytical solution to the problem is first described mathematically, and then extended to four applications: (1) evaluation of the quantity of radionuclides as a function of time, (2) evaluation of the time integral of the quantity during a time period, (3) evaluation of the amount in a medium as a function of time following deposition at a constant rate, and (4) evaluation of the time integral of the amount in a medium after deposition at a constant rate for a time. The solution can be applied to any system involving physical transfers from the medium and radioactive chain decay with branching in the medium. The general solution is presented for quantities expressed in units of atoms and activity. Unlike many earlier mathematical solutions, this solution includes chain decay with branching explicitly in the equations