Tracer Kinetic Modeling

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Definition of a Tracer

A tracer is a substance that follows (“traces”) a physiologic or biochemical process. In this chapter, tracers are assumed to be radionuclides or, more commonly, small molecules or larger biomolecules (e.g., antibodies and peptides) that are labeled with radionuclides. These labeled molecules are also known as radiotracers or radiopharmaceuticals. For simplicity, we refer to them as tracers in the remainder of this discussion. Tracers can be naturally occurring substances, analogs of natural substances (i.e., substances that mimic the natural substance), or compounds that interact with specific physiologic or biochemical processes in the body. Examples include diffusible tracers for blood flow, tracers that follow important metabolic pathways in cells, and tracers that bind to specific receptors on cell surfaces. Table 21-1 lists some examples of tracers that are used in nuclear medicine and their applications.

Some specific requirements for an ideal tracer include the following:

• 1

  The behavior of the tracer should be identical or related in a known and predictable manner to that of the natural substance.

• 2

  The mass of tracer used should not alter the underlying physiologic process being studied or should be small compared with the mass of endogenous compound being traced (a typical “rule of thumb” is that the mass of tracer should be <1% of the endogenous compound).

• 3

  The specific activity of the tracer should be sufficiently high to permit imaging and blood or plasma activity assays without violating the first two requirements.

• 4

  Any isotope effect (see Chapter 3 , Section B) should be negligible or at least quantitatively predictable.

If a tracer is labeled with an element not originally present in the compound (this is often the case with radionuclides such as ^99m Tc, ¹²³ I, and ¹⁸ F), it should behave similarly to the natural substance or in a way that differs in a known manner. The strictness of this requirement depends on the process under investigation. One common use of tracers in clinical nuclear medicine is to examine gross function and distribution, including blood flow, filtration, and ventilation. Although the elements represented by radionuclides such as ^99m Tc, ⁶⁷ Ga, ¹¹¹ In, and ¹²³ I are not normally present in biologic molecules, it is possible to incorporate these radionuclides in physiologically relevant tracers that can measure simple parameters that are related to distribution, transport, and excretion.

However, these same elements are not normally present in human biochemistry (iodine is an exception when used to study thyroid metabolism). It is therefore much more difficult to mimic a biochemical reaction sequence with these radionuclides. The biochemical systems of the body are more specific than the transport processes that move or filter fluids or gases. Biochemical systems can selectively require that compounds be of one optical polarity versus the other; that compounds fit within angstroms in the cleft of an enzyme; that chemical bond angles, lengths, and strengths are appropriate; and so forth. When a compound is labeled with a foreign species, such as ^99m Tc, one cannot be sure that it will retain its natural properties and a careful examination and characterization of the compound must be undertaken. One of the advantages of radionuclides that represent elements normally involved in biochemical processes, such as ¹¹ C, ¹³ N, and ¹⁵ O, is that they generally do not alter the behavior of the labeled compound.

Analog tracers are compounds that possess many of the properties of natural compounds but with differences that change the way the analog interacts with biologic systems. In many cases, analog tracers are deliberately created to simplify the analysis of a biologic system. For example, analogs that participate through only a limited number of steps in a sequence of biologic reactions have been developed in biochemistry and pharmacology. Analogs are used to decrease the number of variables that must be measured, to increase the specificity and accuracy of the measurement, or to selectively investigate a particular step in a biochemical sequence. In other cases analog tracers are used because of the need to label the tracer with an element that is not normally present in the molecule of interest. As discussed earlier, this can lead to very significant deviations in the biologic properties (particularly in small molecules) compared with the natural compound. Correction factors based on the principles of competitive substrate or enzyme kinetics are employed in studies using analog tracers to account for differences between the analog and the natural compound. A well-known and widely used example of an analog tracer in nuclear medicine is 2-deoxy-2[ ¹⁸ F]fluoro-D-glucose (FDG) to measure glucose metabolism (see Section E.5 ).

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Definition of a Compartment

A compartment is a volume or space within which the tracer rapidly becomes uniformly distributed; that is, it contains no significant concentration gradients. In some cases, a compartment has an obvious physical interpretation, such as the intravascular blood pool, reactants and products in a chemical reaction, substances that are separated by membranes, and so forth. For other compartments, the physical interpretation may be less obvious, such as a tracer that may be metabolized or trapped by one of two different cell populations in an organ, thus defining the two populations of cells as separate compartments. Additionally, although the definition of a particular compartment may be appropriate for one tracer [e.g., the distribution of labeled red blood cells (RBCs) in the intravascular blood pool], it might not apply for a different tracer (e.g., the distribution of thallium or rubidium, which has both an intravascular and an extravascular distribution). Thus the number, interrelationship, organization, and definition of compartments in a compartmental model must be developed from knowledge of physiologic and biochemical principles.

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Distribution Volume and Partition Coefficient

A compartment may be closed or open to a tracer. A closed compartment is one from which the tracer cannot escape, whereas an open compartment is one from which it can escape to other compartments. Whether a compartment is closed or open depends on both the compartment and the tracer. Indeed, a compartment may be open to one tracer and closed to another. If a tracer is injected into a closed compartment, such as a nondiffusible tracer in the vascular system, conservation of mass requires that after the distribution of the tracer reaches equilibrium or steady-state conditions, the amount of tracer injected, A (in becquerels or other units of activity), must equal the concentration of the tracer in the compartment, C (in Bq/mL), multiplied by the distribution volume , V _d , of the compartment. Thus,

This equation is the basis for the dilution principle, which provides a convenient method for determining the distribution volume of a closed compartment, as shown by the following example.

Example 21-1

What is the distribution volume of the RBCs if 1 MBq ⁵¹ Cr-labeled RBCs is injected into the blood stream and an aliquot of blood taken after an equilibration period (10 minutes) contains 0.2 kBq/mL? Assume the hematocrit, H (fraction of the total blood volume occupied by RBCs), is 0.4.

Answer

From Equation 21-2 :

This result gives the total distribution volume, that is, total blood volume. The RBC volume is given by

More commonly, a compartment will be open; that is, the tracer will be able to escape from it. This applies, for example, to tracers that are distributed and exchanged between blood and tissue. In this case, after the tracer reaches its equilibrium distribution, * the concentration in blood will typically be different from that in the tissue ( Fig. 21-2A ). The ratio of tissue concentration C _t (Bq/g) to blood concentration C _b (Bq/mL) at equilibrium, is called the partition coefficient , λ, defined by

* Note that “equilibrium” in this case means that the concentration of the tracer in the compartments has reached a constant value with time. It does not imply equilibrium in the thermodynamic sense, that is, that there is no further transport of tracer between tissue and blood. Thus tracer equilibrium is synonymous with the term steady state (see Section B.6 ).

The equilibrium blood concentration, C _b , can be directly measured by taking blood samples. If one assumes that the concentration of tracer in tissue is the same as the concentration in blood ( Fig. 21-2B ), and applies Equation 21-2 , this leads to an apparent distribution volume in tissue given by V ₁ = A _t / C _b , in which A _t is the activity in the tissue. One also knows that A _t = C _t × V _t , in which V _t is the volume (or mass) of tissue; therefore combining these relationships and Equation 21-3 yields

Thus another interpretation of the partition coefficient is that it is the distribution volume per unit mass of tissue for a diffusible substance or tracer. This interpretation is employed in some models for estimating blood flow and perfusion, as discussed in Section E .

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Flux

Flux refers to the amount of substance that crosses a boundary or surface per unit time (e.g., mg /min or mol / min) ( Fig. 21-3 ). It also can refer to the transport of a substance between different compartments in terms of flux per unit volume or mass of tissue (e.g., mol /min /mL or mg /min /g).

Flux is a general term that can refer to a variety of processes. For example, the total mass of RBCs moving through a blood vessel per unit time is a flux. The “boundary” or “surface” in this case could be any transverse plane through the vessel. The amount of glucose moving across a cell membrane per unit time also is a flux. Fluxes therefore may either be closely related or unrelated to blood flow.

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Rate Constants

Rate constants describe the relationships between the concentrations and fluxes of a substance between two compartments. For simple first-order processes, the rate constant, k, multiplied by the amount (or concentration) of a substance in a compartment determines the flux:

For first-order processes, the units of k are (time) ^–1 . If “amount” refers to the mass of tracer in the compartment, the units of flux are mass/time (e.g., mg/min). If “amount” refers to concentration of tracer in the compartment, the units of flux are mass/time per unit of compartment volume (e.g., mg/min/mL), or mass/time per unit of compartment mass (e.g., mg/min/g). Note that, as illustrated by Figure 21-3 , different directions of transport between two compartments can be characterized by different rate constants.

A first-order rate constant also may represent the fractional rate of transport of a substance from a compartment per unit time. For example, a rate constant of 0.1 min ^–1 corresponds to a transport of 10% of the substance from the compartment per minute. The inverse of the rate constant, 1/ k , is sometimes referred to as the turnover time, or mean transit time , τ, of the tracer in the compartment (in this example, 10 minutes). Similarly, the half-time of turnover, t _1/2 , that is, the time required for the original amount of tracer in the compartment to decrease by 50% (assuming no back transfer into the compartment), is given by

Thus the fractional rate constant k is analogous to the decay constant λ for radioactive decay, whereas the mean transit time is analogous to the average lifetime of a radionuclide (see Chapter 4 , Section B.3). In first-order models, transport out of a compartment through a single pathway (without back-transport) is described by a single exponential function, e ^{– kt} , analogous to the radioactive decay factor e ^{–λ t} .

If there is more than one potential pathway for a tracer to leave a compartment, each characterized by a separate rate constant, k _i , then the turnover time of the tracer in the compartment is the inverse of the sum of all these rate constants and the half-time of turnover is

where m is the number of pathways by which the tracer can leave the compartment.

Most compartment models used in nuclear medicine are based on the assumption that first-order kinetics describe the dynamics of the system of interest. The tracer kinetics of such systems are linear. That is, doubling the input (amount or concentration) doubles the output (flux) of the system. As shown in Section E , linear first-order tracer kinetic models adequately describe many systems even when the dynamics of the natural substances are nonlinear.

A more general expression for the relationship among rate constants, fluxes, and concentrations (or masses) is

where n refers to the order of the reaction. The units of rate constants for n ^th order reactions (in terms of concentration) are [concentrations ^{(1— n )} • time ^–1 ]. Thus only first-order rate constants represent a constant fractional turnover and Equations 21-6 and 21-7 apply only to first-order processes.

Figure 21-3 illustrates a three-compartment system consisting of a blood compartment separated by a membrane barrier (e.g., capillary wall) from two sequential tissue compartments. R and P refer to chemical reactant and product, whereas the subscripts b and t refer to reactant in blood and tissue compartments, respectively. [R _b ], * [R _t ], and [P] are the blood and tissue concentrations of reactant and product, whereas the fluxes between the compartments are the first-order rate constants, k ₁ , k ₂ , k ₃ , and k ₄ , multiplied by corresponding concentrations. The thicknesses of the arrows in Figure 21-3 are proportional to the magnitude of the corresponding rate constant. In this example, the rate constants into and out of tissue are larger than the corresponding rate constants between the reactant and product compartments in tissue. Thus the majority of the reactant initially transported into the tissue space is transported back into blood without undergoing any biochemical reactions. This is a common occurrence in actual biochemical systems and introduces a reserve capacity into the system that can accommodate changes in metabolic supply and demand (e.g., by changing k ₃ ).

* The notation [X] is used to denote the concentration (usually in units of g/mL or mol/mL) of substance X.

Figure 21-4 illustrates the relationship between first-order rate constants and the relative concentrations of the substrates in a biochemical sequence. If a substrate (S) and enzyme (E) combine to form a substrate-enzyme complex (SE), which then dissociates into a product (P) with release of the enzyme, the fluxes of the first-order reaction steps are concentrations multiplied by the corresponding rate constants. If a small amount of labeled substrate is introduced into the system at time zero, the tracer will go through the reaction steps, producing concentrations of labeled S, SE, and P as shown in the graphs in Figure 21-4 . If k ₃ (the forward rate constant for the reaction converting SE to E and P) is reduced by 50% with all the other rate constants remaining unchanged, the concentrations of labeled S, SE, and P are then represented by the dotted orange lines in Figure 21-4 . Decreasing k ₃ causes a slower production of P and causes a compensatory increase in labeled S and SE.

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