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Introduction to Compartment Models

A physiologic system is often described by decomposition into a number of interacting subsystems, called compartments. Compartments should not be understood as a physical volume, but rather as a mass of well-mixed, homogeneous material that behaves uniformly. Each compartment may exchange material with other compartments. Examples of frequently used compartments are

Compartment models are visualized by diagrams wherein the rectangles symbolize compartments and the arrows represent material exchange, as illustrated for the different models below. Although the mechanisms of material transport between compartments may differ, models that can be reasonably analyzed with standard mathematical methods assume first-order processes. As a consequence, the change of tracer concentration in each compartment is a linear function of the concentrations in the other compartments. Because of the tiny amounts of tracer material applied in nuclear medicine it is usually assumed that the observed system is not disturbed by the tracer.

In the following, the properties and terminology of compartment models are illustrated using a tracer that is injected intravenously as a bolus and is not freely diffusible. 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 1-tissue compartment model (1TCM):

It is assumed that, because of mixing, 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 C1(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, but not labeled metabolites which may also circulate. Because extraction is described by a first-order process, the transfer of material is proportional to the parent concentration CP(t). Parallel to the uptake, tracer in tissue is reduced by a backwards transfer, which is proportional to the concentration in tissue. Both processes compete, so that 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 k2, 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 tissue, whereas k2 [min–1] indicates the fraction of mass transferred per unit time. For example, if k2 equals 0.1[min–1], material leaves the tissue compartment at a rate of 10% per minute. The parent concentration CP(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 1-tissue compartment model can be analytically solved, yielding the following equation for the concentration of tracer in tissue:

Hence the time course of the tissue concentration essentially results from a convolution of the input curve with a decaying exponential. K1 acts as a scaling factor of the concentration course, whereas k2 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 C1(t). Free tracer can directly be bound to its target molecule, C2(t), but it also may bind to some cell components that are not known in detail, C3(t). Considering these pathways, a 3-tissue compartment model (3TCM) with six transfer coefficients is established, whereby C2 and C3 communicate only by C1.

The system of differential equations can be derived in analogy to the 1-tissue compartment model, but is much more complex. The equations describing this model are given by

This system contains six unknown parameters (K1 to k6) that are difficult to assess experimentally. For practical purposes, the system is therefore reduced to a 2-tissue compartment model (2TCM) by treating free and non-specifically bound tracer as a single compartment C1 (non-displaceable 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 two-tissue compartment model is given by

By using a mathematical method called Laplace transformation, analytic solutions can be derived for linear multi-compartment models, but the mathematical formulas are complicated expressions already for two compartments. However, they demonstrate that K1 represents a scaling factor like in the solution given for the one-tissue 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 3-tissue compartment model is simplified to a 2-tissue compartment model, and further to a 1-tissue compartment model, the meaning of the transfer coefficients changes (except for K1). 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 k2 and k3 have a different meaning in the different models, as well as C1.

Parameters in 1TCM

Expressed by parameters of the 2TCM

Expressed by parameters of the 3TCM

K1

K1

K1

k2

Parameters in 2TCM

Expressed by parameters of the 3TCM

K1

K1

k2

k3

k4

k4

The term 1/(1+k5/k6) represents the free fraction of tracer in the non-displaceable tissue compartment which is available for transfer back to blood or for specific binding, and is often called f2.

A good summary on the configuration and interpretation of linear models for receptor tracers is given by Koeppe et al. [15]. Starting from the 3-tissue 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.

Relation of K1 with Perfusion

As mentioned earlier, K1 is related to tissue perfusion F. With a capillary model, the relation

K1 = E F

can be derived. E represents the unidirectional first-pass 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. In this case, K1 equals perfusion.

General Assumptions

The following sections sections provide a specification of the compartment models available in PKIN. The following is generally assumed:

Exceptions to these rules are specified in the description of the individual models.