In the September 1986 edition of the Journal of Nuclear
Medicine Technology an article appeared on "*An
Integration Technique for Rapid Estimation of Activity of Waste
in Storage*". One of the problems hospitals face is
disposing of radioactive wastes. An increasingly popular method
is to allow radionuclides with short half-lives to decay at the
hospital until they can be disposed of as nonradioactive waste
(after about 10 half-lives of the radioactive nuclide have
elapsed.). This is a cheaper method of disposal and reduces the
volume of low-level radioactive waste sent to shallow-land burial
sites.

However, radioactive wastes held in storage contribute to the
overall activity which is in the user’s possession. Since
there are limits on the amounts of various radioactive nuclides
that can be possessed, a **method for determining the
activity of radoactive wastes held in storage** is needed.
The following method overestimates the activity in storage which
results in a conservative restriction on activity on hand in the
laboratory.

In a facility where the **usage rate of a radionuclide is
well defined and relatively constant**, the total
activity, *A*, of this radionuclide held in storage at any
time, *t*, can calculated. The rate of change in activity
as a function of time, *dA*/*dt*, is expressed as
the difference between the rate at which waste is placed in
storage and the rate of decay:

where:

*R*is usage rate of the radionuclide (its units are activity/time),

*f*is the fraction of the used activity in the facility which is transferred to storage (no units)

*λ*is the decay constant of the radionuclide (its units are 1/time)

- Determine an equation for the activity
*A*in terms of the time*t*if initially*A*= 0. - The activity accrued in storage increases as
a function of time. However, as time becomes large, the activity
approaches an equilibrium value
*Ae*. Find this equilibrium value*Ae*that*A*approaches.

- This
**differential equation**can be solved using the method of separation of variables. Isolate*A*on the left and*t*on the right side of the equation, and integrate:To carry out the integration, make the change of variable

*u*=*f R*-*λ**A*. Calculate its differential:These substitutions give:

Both sides can now be integrated to give:

Substituting back for

*u*gives:We are interested in the case where the activity in storage grows, ie. in the case that

*f R*>*λ**A*. Therefore we can drop the absolute value symbol. Next isolate the logarithm and put the equation in exponential form:Substituting the initial condition that the activity

*A*= 0 at time*t*= 0 into this equation implies the following condition on the integration constant*C*:Using this to replace the integration constant in the equation gives:

Solving this for

*A*finally gives the desired equation for the activity*A*as a function of time*t*: - To find the equilibrium value
*Ae*that*A*approaches, use the fact that:This means that the equilibrium activity is given by:

The graph below shows the approach to equilibrium: