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.

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Gardner DG. Resolution of multi-component exponential decay curves using Fourier transforms. Ann NY Acad Sci 108: 195-203, 1963. Landaw EM and DiStefano III JJ. Multiexponential, multicompartmental, and no compartmental modeling. II. Data analysis and statistical considerations. Am J Physiol Regulatory Integrative Comp 246: R665-R677, 1984. Beard DA and Bassingthwaighte JB. Power-law kinetics of tracer washout from physiological systems. Ann Biomed Eng 26: 775-779, 1998. Brownell GL, Berman M, and Robertson JS. Nomenclature for tracer kinetics. Int J Appl Rad Isot 19: 249-262, 1968. Sheppard CW. Basic Principles of the Tracer Method. New York: Wiley, 1962.

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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

**Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants:** NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.