Decay data for two radioactive isotopes: optimization to find the fraction of each.

Description

Multiexponential decay curves can be fitted to data 
 on the activities, counts/min, detected by a Geiger counter or equivalent, as a 
 function of time. The data here are from 13-N-nitrogen (as NH3) contaminated with
 18-F-fluorodeoxyglucose (frF). The objective is to find the fraction of each tracer by
 fitting the data. Optimization of the model parameters, frF and bkgd, can be done
 by manual adjustment of the parameters or by using the optimizers. Because the
 data cover a three order of magnitude range, the early points make up a huge 
 proportion of any un-normalized sum of squares of the differences between the
 model solution and the data. Consequently, point weighting is critical to using
 automated optimization, e.g with Pwgt set to increase with time (type in t^2) or 
 to be the inverse of the activity (type in 1/cpm). (Changing the curve
 weighting, Cwgt, is useless since there is only one data curve to fit.).
 
 The model and data provide a simple situation in which to explore several
 optimizers and observe their routes of convergence to the best fitting solution.
 Such data can also be used to estimate the decay constants for the isotopes:
 these decay rates are all determined empirically by fitting such data curves,
 though one would prefer to have data on a chemically pure tracer. Try optimizing 
 to find the thalf for the two tracers. What is the relative accuracy of the
 estimation for each? See Notes tab below.

Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.