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TUTORIAL: Principles of Tracer Modeling
Mathematical Modeling
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Single Compartment Model
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Suppose we inject a bolus of some inert substance into a volume, where it undergoes instantaneous mixing with the liquid in the volume. If we could observe the concentration of the substance in the compartmental volume continuously over time, then the curve would look something like the graph on the image above.
Discrete Sampling
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If we could observe the concentration of the substance in the compartmental volume only at discrete time points (e.g., by drawing small samples with a needle), then we might obtain a plot similar to that shown above.
Plasma Curve
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Now suppose we inject a bolus of an inert substance into the human body. Initially, it would exist only in the plasma - the liquid of the circulatory system. Because of this, a plot of the initial concentration values over time would appear similar to injection into a one-compartment system.
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If the tracer could not cross the capillary walls into the tissues of the body, the rest of the kinetic curve would show a constant concentration. An example of what one might observe in arterial plasma at later time points is shown above.
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However, assuming that the tracer could cross the capillary walls into the tissues of the body, the rest of the kinetic curve would show a drop in concentration after that initial peak. An example of what one might observe in arterial plasma at later time points is shown above.
Two Compartment Model
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In its simplest form, the human body could be treated as a plasma compartment and a tissue compartment. Simulating a bolus injection into the plasma, we see that the PET tracer concentration would initially peak in the plasma compartment and later peak in the tissue compartment. The downward arrow from the plasma compartment to the tissue compartment represents the transfer of tracer from the former to the latter. Note that the instantaneous mixing assumption in any compartmental-model is only approximately satisfied in a living system.
Plasma and Tissue Curves
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If we could observe these two compartments (plasma and tissue) continuously over time, then we might obtain kinetic concentration curves such as those shown above.
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With PET, following a bolus injection of a positron-emitter-labeled tracer into the plasma, we might obtain the measurements plotted above. The plasma radioactivity concentration measurements would come from well-counted blood samples, and the tissue measurements would come from regional analysis of a sequence of PET images obtained over scan intervals of various lengths.
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