Radiotracers provide a means for investigation of biochemical or physiologic processes without altering the normal functions of the biologic system. Each radiotracer must be targeted to provide a physiologic parameter of interest, such as blood flow, glucose metabolism, oxygen utilization, protein synthesis, and receptor or binding site density, etc. in the body. The concentration of the radiotracer introduced into the biologic system is assumed to be negligible so that it does not perturb the natural process of the system. Otherwise, the measurement does not represent the process we want to measure but the effect induced by the introduction of the radiotracer. External measurable data is the time course of total tissue activity concentration obtained from the PET images, and the time course of blood (or plasma) activity concentration (i.e. the input function of the compartment model), obtained from peripheral blood sampling. These curves are described as time-activity curves (TACs), where the term "activity" refers to concentration of the radiotracer rather than the tissue (or blood) activity. Yet, the measured time course of tracer uptake and delivery does not directly provide quantitative information about the biologic and physiologic processes but the kinetic information of the radiotracer. Mathematical modeling of the measured tracer kinetics is thus required to transform the kinetic information into physiologically meaningful information, i.e. the physiologic parameters of interest. This can be accomplished through the use of an analysis technique commonly referred to as compartmental or tracer kinetic modeling.

Mathematical modeling of biologic processes and systems is well established and a wide variety of models have been developed [61]. Although nonlinear models should be used to study biological systems which are commonly nonlinear, linear compartmental models have properties which make them attractive for radiotracer experiments with PET and SPECT [62]. A given system can be described by a compartment model, which consists of a finite number of interconnected compartments (or pools), each of which is assumed to behave as a distinct component of the biologic system with well-mixed and homogeneous concentration [63]. An example is shown in Fig. 2.9 for [18F]fluorodeoxyglucose, which is the primary radiopharmaceutical used in PET to assess glucose metabolism. A compartment can be a physical space, such as plasma or tissue, or a chemical entity, where tracer may exist in different forms

Figure 2.9: The three-compartment model for transport and metabolism of [18F]fluorodeoxyglucose (FDG).

Figure 2.9: The three-compartment model for transport and metabolism of [18F]fluorodeoxyglucose (FDG).

(FDG and its phosphorylated form FDG-6-PO4). The compartments of a tracer kinetic model are linked by a set of parameters called rate constants, k,, which represent the rates at which the radiotracer in one compartment is transported to the connected compartments. More precisely, these rate constant parameters represent specific physiologic or biochemical processes (e.g. flow or transport across physical spaces, or rates of transformation from one chemical form to the other in a chemical entity) within the biologic system. For the FDG model as shown in Fig. 2.9, the three compartments represent (from left) vascular space for FDG, tissue space for free FDG, and tissue space for FDG-6-phosphate (FDG-6-P). The rate constants describe the movement of FDG between compartments: k1 and k2 for the forward and backward transport of FDG across the blood-brain barrier, k3 for the phosphorylation of FDG to FDG-6-P, and k4 for the dephos-phorylation of FDG-6-P back to FDG.

The aim of modeling is to interpret the fate of the administered radiotracer quantitatively in terms of the standard parameters in the compartmental model. In conjunction with knowledge of the transport and metabolism of the radiotracer, it is possible to relate the rate constants to physiologic parameters of interest. Figure 2.10 summarizes the key steps in physiologic parameter estimation

Acquired arterial blood samples (input function)

Plasma/Blood TAC

Acquired arterial blood samples (input function)

Plasma/Blood TAC

Compartmental model fitting

Weighted integration

Graphical techniques

Figure 2.10: (Color slide) Quantitative physiological parameter estimation with PET includes radiotracer administration, data acquisition with a PET scanner, measurement of tracer plasma concentration, a suitable mathematical model and a parameter estimation method to estimate the physiological parameter of interest.

Acquired PET images (output function)

Compartmental model fitting

Weighted integration

Physiological parameters

Graphical techniques

Spectral analysis

Physiological parameters e.g. rCBF, rCMRGIc, Vd, BP, etc.

Figure 2.10: (Color slide) Quantitative physiological parameter estimation with PET includes radiotracer administration, data acquisition with a PET scanner, measurement of tracer plasma concentration, a suitable mathematical model and a parameter estimation method to estimate the physiological parameter of interest.

in a quantitative PET study. After radiopharmaceutical administration, PET data is acquired at a predefined sampling schedule and individual voxel values in the reconstructed images represent the localized radiotracer time-activity concentration in the body upon correction for some degrading factors (e.g. attenuation and scatter) and cross-calibration. A vector formed by extracting a voxel curve from the sequence of images corresponds to a tissue TAC, which represents the response of the local tissue as a function of time after the tracer administration. Alternatively, the tissue TAC can be obtained by manual delineation of region of interest (ROI) on the reconstructed PET images. Plasma tracer concentration is typically measured by means of arterial blood sampling. A mathematical model is applied to the tissue and the plasma tracer concentration to estimate the physiological parameters of interest. Kinetic modeling approaches based on the framework of tracer kinetic modeling could be applied to estimate the physiologic parameters. The same analysis procedures can also be applied to dynamic SPECT without loss of generality, although the challenges tend to be much greater in SPECT.

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