Modelling a radionuclide generator in Excel

We’ve been teaching our first year (second year now) residents about nuclear medicine physics, and one of the topics we’ve covered is producing radionuclides for medical use. So I’m trying to whip up a spreadsheet that will model the activity of a typical Mo-Tc radionuclide generator to show off transient equilibrium and what happens when the generator is eluted.
Modelling the Mo/Tc activity in the generator is easy. What I’m finding a little more difficult is including the effects of generator elution where some of the Tc activity is removed from the generator. I’m thinking if I can reformulate the transient equilibrium equation as a recursive equation that looks something like A(t+dt) = f(A(t)) then I can get it to work. Spreadsheets are good at dealing with recursive equations. Should be simple.

So let’s start with the Bateman equation (need to learn some MathML). For a 2 radionuclide (parent/daughter) setup, it looks like

A_d(t) = A_p(0)(λ_d/(λ_d-λ_p))(exp(-λ_pt)-exp(-λ_dt)) + A_d(0)exp(-λ_dt)

Now, it’s a fairly simple exercise to show that when transient equilibrium is established, the recursive equation has the form

A_d(t+1) = A_d(t)exp(-λ_p)

But, when the generator is eluted, transient equilibrium no longer exists, and we need to go back to the Bateman equation to determine the daughter activity.
So, supposing that at time t=0, we have no initial daughter activity. Our equation looks like

A_d(0) = A_p(0)(λ_d/(λ_d-λ_p))

And at time t=1, we have

A_d(1) = A_p(0)(λ_d/(λ_d-λ_p))(exp(-λ_p)-exp(-λ_d))

At time t=2,

A_d(1) = A_p(0)(λ_d/(λ_d-λ_p))(exp(-2λ_p)-exp(-2λ_d))

Already we can see that the term containing the difference of exponentials

exp(-λ_p)-exp(-λ_d)

is going to cause a lot of grief. A recursive Bateman equation may not be possible. I may have to come up with another way to do my spreadsheet.