Imaging in nuclear medicine is based on the application of radioactively labeled compounds (tracers) that are distributed throughout the body by the vascular system and take part in physiologic processes according to the properties of the tracer molecule. Each tracer is designed to assess a specific property of tissue (e.g., perfusion, metabolic turnover, or the concentration of a specific receptor). To obtain optimal images, a protocol defines how the tracer is administered and at what timepoints single images or a sequence of images is acquired. The information contained in the images represents the average tracer concentration during the image acquisition, and in most cases it cannot be directly transformed into a quantitative parameter of the target process. For instance, “perfusion-weighted” images cannot be translated directly into images representing tissue perfusion in milliliters per minute per 100 mL of tissue.
Although functionally weighted images bear valuable diagnostic information, some situations require absolute quantification, such as therapy monitoring and group comparisons. Hence there is an interest in using nuclear medicine techniques for the quantitative assessment of tissue parameters not only for the scientific evaluation of tracer characteristics but also for clinical purposes.

The transformation of the measured data into absolute metabolic parameters is based on physiologic models, often compartment models. Hereby the relevant transport and uptake parameters are mathematically modeled and a prescription derived that predicts the tracer concentration in tissue given a measured plasma concentration. Quantification of the measurement then consists of solving the following inverse problem: Which model parameters predict the measured tissue concentration most closely? It must be possible to find a model that adequately describes the processes with a small number of parameters. Otherwise, with too many parameters in the model, there is a high chance that different parameter combinations will result in approximately the same predicted tissue response, and an optimization algorithm will then have difficulty finding a definite optimal solution. Whether such a model—one that is considerably simplified with respect to the underlying physiology—still provides valuable information is a question that must be answered for each tracer individually.

Compartment modeling has the following requirements:

To describe a physiologic system, it is often helpful to decompose it into a number of interacting subsystems, called compartments. Each compartment represents a homogeneous distribution of material and may exchange material with other compartments. Note that compartments
should be understood not as a physical volume but rather as a mass of well-mixed material that behaves uniformly. Examples of frequently used compartments include these:

Compartment models are visualized by diagrams in which the rectangles symbolize compartments and the arrows represent material exchange, as illustrated in Figure 13.2. Although the mechanisms of material transport may be diverse, models that can be reasonably analyzed with standard mathematical methods assume first-order processes. As a consequence, the change of tracer concentration in one of the compartments is a linear function of the concentrations in all other compartments. Put into mathematical terms,

It is assumed that the observed system is not disturbed by the administration of the tracer. Because of the tiny amounts of tracer material applied in nuclear medicine investigations, this assumption is highly justified in most cases.

To illustrate the properties of compartment models, let us consider a tracer that is not freely diffusible and is injected intravenously as a bolus. When the tracer arrives at the heart, it is well mixed with blood and distributed by the arterial circulation. It finally arrives at the capillary bed, where exchange with the tissue can take place. Nonextracted tracer is transported back to the heart, from which a new circulation starts. This highly simplified “physiologic” model can be translated into a one-tissue compartment model, as illustrated in Figure 13.2A. It is assumed that, because of mixing, the arterial tracer concentration is the same throughout the body, so it can be measured in a peripheral artery. The tracer concentration in tissue increases by the extraction of tracer from the blood plasma, and as this is described by a first-order process, the transfer of material is proportional to the concentration in plasma C_A(t). Conversely, it is reduced by a backward transfer, which is proportional to the concentration in tissue C₁(t). Both processes compete, so the change over time of the net tracer concentration in tissue (dC₁(t)/dt) can be expressed by the following differential equation:

The two transfer coefficients, K₁ and k^″₂, have a somewhat different meaning, indicated by using capital and small letters. K₁ includes a perfusion-dependent component and has units of milliliter per minute per milliliter of tissue, whereas k^″₂ [min^-1] indicates the fraction of mass transferred per unit of time. For example, if k^″₂ equals 0.1[min^-1], the material leaves the compartment at a rate of 10% per minute. The plasma concentration C_A(t) is not modeled but is considered a measured quantity and appears as the input curve driving the system.

By solving differential equation (2), the following equation for the concentration of tracer in tissue can be derived: