In the following, the compartment models are illustrated using a tracer that is injected intravenously as a bolus and is not freely diffusible. Please refer to the consensus paper of Innis et al [56] for the details of the nomenclature and the interpretation of the models in the context of brain receptor tracers.
When the tracer arrives in the heart chambers, 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. Some fraction of the tracer is extracted into tissue and metabolized, the rest is transported back to the heart, from where a new circulation starts.
This simplified “physiologic” model can be translated into a 1tissue compartment model (1TCM):
Because of mixing it is assumed that, the arterial tracer concentration is the same throughout the body, so that it can be measured in a peripheral artery. The tracer concentration in tissue C_{1}(t) increases by the extraction of tracer from the arterial blood plasma. A usual condition for a quantitative analysis is, that only the unchanged tracer, called the authentic tracer or parent, can enter tissue, whereas labeled metabolites which may also circulate cannot. Because extraction is described by a firstorder process, the transfer of material is proportional to the parent concentration C_{P}(t). Parallel to the uptake, tracer in tissue is reduced by a backwards transfer (or washout), which is proportional to the concentration in tissue. Both processes compete, so that the change over time of the net tracer concentration in tissue (dC_{1}(t)/dt) can be expressed by the following differential equation
The two transfer coefficients, K_{1} and k_{2}, have a somewhat different meaning, indicated by using capital and small letters. K_{1} includes a perfusiondependent component and has units of milliliter per minute per milliliter tissue, whereas k_{2} [min^{–1}] indicates the fraction of mass transferred per unit time. For example, if k_{2} equals 0.1[min^{–1}], material leaves the tissue compartment at a rate of 10% per minute. The parent concentration C_{P}(t) is not a compartment, but considered as a measured quantity and appears as the input curve driving the system.
The differential equation of the 1tissue compartment model can be analytically solved (integrated), yielding the following equation for the concentration of tracer in tissue:
Hence the time course of the tissue concentration essentially results from a convolution Ä of the input curve with a decaying exponential. K_{1} acts as a scaling factor of the concentration course, whereas the washout coefficient k_{2} has an impact on its form.
More complex compartment models distinguish different forms of tracer in tissue, which is usually interpreted as follows. After entering a cell, the tracer is available for binding in a free form at a concentration C_{1}(t). Free tracer can directly be bound to its target molecule, C_{2}(t), but it also may bind to some cell components that are not known in detail, C_{3}(t). Considering these pathways, a 3tissue compartment model (3TCM) with six transfer coefficients is established, whereby C_{2} and C_{3} communicate only by C_{1}.
The system of differential equations can be derived in analogy to the 1tissue compartment model, but is much more complex. The equations describing this model are given by
This system contains six unknown parameters (K_{1} to k_{6}) that are difficult to assess experimentally. For practical purposes, the system is therefore most often reduced to a 2tissue compartment model (2TCM) by treating free and nonspecifically bound tracer as a single compartment C_{1 }(nondisplaceable compartment).
This simplification is justified if the exchange free « nonspecific binding is significantly faster than the exchange free « specific binding. Otherwise specific and nonspecific binding cannot be distinguished, and the resulting transfer coefficients cannot be attributed to specific binding alone. The simplified twotissue compartment model is given by
By using a mathematical method called Laplace transformation, analytic solutions can be derived for linear multicompartment models, but the mathematical formulas are complicated expressions already for two compartments. However, they demonstrate that generally K_{1} represents a scaling factor like in the solution given for the onetissue compartment.
Model Simplification
While complex models are a more realistic description of the involved processes, they include many parameters. PET data from dynamic measurements has a limited statistical quality, and represents the summed contributions from all compartments. When trying to estimate a large number of model parameters from such data, the variance of the resulting parameters tends to be too high for reliable interpretation. Therefore, only relatively simple models with 2 to 4 (at most 6) parameters are feasible in practice.
When the 3tissue compartment model is simplified to a 2tissue compartment model, and further to a 1tissue compartment model, the meaning of the transfer coefficients changes (except for K_{1}). The “lumped” parameters of the simplified models can be expressed by the parameters of the complex models as shown in the tables below. Note that k_{2 }and k_{3} have a different meaning in the different models, as well as C_{1}.
Parameters in 1TCM 
Expressed by parameters of the 2TCM 
Expressed by parameters of the 3TCM 
K_{1} 
K_{1} 
K_{1} 
k_{2} 
Parameters in 2TCM 
Expressed by parameters of the 3TCM 
K_{1} 
K_{1} 
k_{2} 

k_{3} 

k_{4} 
k_{4} 
The term 1/(1+k_{5}/k_{6}) represents the free fraction of tracer in the nondisplaceable tissue compartment which is available for transfer back to blood or for specific binding, and is often called f_{2}.
A good summary on the configuration and interpretation of linear models for receptor tracers is given by Koeppe et al. [15]. Starting from the 3tissue model it is nicely shown how the parameters of simplified models are related to the true rate constants. PKIN can take these relations into account when switching from a more complex model to a simpler one. This behavior can be switched on by the Model Conversion check on the Extras panel.
Relation of K1 with Perfusion
As mentioned earlier, K_{1} is related to tissue perfusion F. With a capillary model, the relation
K_{1} = E F
can be derived. E represents the unidirectional firstpass extraction fraction (i.e., the fraction of tracer that penetrates the capillary wall and is extracted into the tissue during the first pass of tracer through the capillary). The reverse transport is not regarded in the calculation of E. Renkin [40] and C. Crone [41] calculated the extraction as
E = 1 – e^{–PS/F}
P denotes permeability, and S, the surface of the endothelium by which the capillary exchanges with the tissue. The relation demonstrates that the extraction depends on the properties of the endothelium with respect to the tracer (PS), as well as the perfusion F. The higher the perfusion, the smaller the extraction, because the average time spent close to the capillary wall decreases. However, total mass transport in general still increases with flow, as the reduced extraction is more than compensated by the higher abundance of tracer (F in the term EF). For tracers with very high permeability, the extraction is virtually independent of perfusion and E approaches 1 (such as for ^{15}Olabeled water). In this case, K_{1} equals perfusion.
Volumes of Distribution for Receptor Tracers
For receptor tracers, it is important that the extraction from plasma to tissue is sufficient, such that the kinetics is not limited by the extraction step. Additionally, the relation of the transfer coefficients for specific and nonspecific binding determines whether these different types of binding can be distinguished.
Naturally, the main interest is in specific binding. Besides the individual rate constants k_{3} and k_{4}, which are often difficult to estimate precisely, the distribution volume of specific binding V_{S} is a measure which is often used to quantify specific binding. V_{S} is defined by the ratio of specific binding concentration to total parent at equilibrium (which may not occur during the experiment). However, it can also be calculated from the rate constants of the 3tissue compartment model as follows.
V_{s} = K_{1}/k_{2}*k_{3}/k_{4} 3Tissue Compartment Model
Under the assumption of a rapid equilibration between the free and nonspecific compartment the same expression with the rate constants from the 2tissue compartment model is often used as an approximation of V_{S}.
If the assumption of rapid equilibration is not justified one must resort to considering the entire tracer concentration in tissue as an indicator of specific binding. The measure used in these cases is called the "total distribution volume" V_{T}. In general it is defined as the ratio of the tracer concentration in tissue to total parent in plasma at equilibrium. A V_{T} equal to 20 hence means that the tracer is being concentrated in tissue by 20:1. V_{T} can be easily computed from the rate constants of the compartments model.
V_{T} = K_{1}/k_{2} 1Tissue Compartment Model
V_{T} = K_{1}/k_{2} (1+k_{3}/k_{4}) 2Tissue Compartment Model
K_{1}/k_{2} can be interpreted as the distribution volume of the first tissue compartment C_{1}. In receptor tracer experiments, this compartment is usually called the nondisplaceable compartment, and K_{1}/k_{2} the distribution volume V_{ND} of nondisplaceable uptake,
V_{ND} = K_{1}/k_{2} 2Tissue Compartment Model
This naming indicates that the concentration of free and nonspecifically bound tracer in tissue is not reduced when a nonlabeled ligand is brought into the tissue, competing with binding to the same receptor as the tracer. The rational for this assumption is that the potential for nonspecific binding to various cell components is practically unlimited as compared to the number of specific binding sites.
Rate Constants and Binding Potential for Receptor Tracers
k_{2} represents the fractional rate constant for the washout of free tracer from tissue to plasma. It it is a composite rate constant and is related to flow in the same way as K_{1}. Consequently it is often reasonable to assume that the ratio K_{1}/k_{2} is insensitive to flow and regionally constant. This property can be exploited as a restriction in a model fit, hereby reducing the number of fitted parameters.
k_{3} is regarded as the rate of association of the ligand with the specific binding sites. It is a function of the concentrations of the free radioligand, the available binding sites and intrinsic second order association rate constants. Consequently it is a pseudo first order rate constant incorporating the concentration of available binding sites.
k_{4} is considered as the dissociation rate constant of the receptorligand complex; often it is assumed to be invariant, what is still a matter of debate.
The binding potential (BP) quantifies the equilibrium concentration of specific binding as a ratio to some other reference concentration. Particularly, the ratio at equilibrium of specifically bound radioligand to that of nondisplaceable radioligand in tissue is referring to as BP_{ND} (unitless) and equals the ratio k_{3}/k_{4}.