When a contrast agent is injected and enters a vascular system with a single inlet and a single outlet as a bolus as shown in Figure 4.1, the concentration-time curve measured at the exit of the vascular system is dispersed as the molecules in contrast agent may have different transit times through the system. Furthermore, there is a
delay time before they can be detected at the exit, because a specific time is necessary for traversing the system.

Figure 4.1A shows an ideal bolus input at time 0. Mathematically, this is represented by Dirac delta function defined by

Figure 4.1B shows the corresponding tissue response function (R(t)), which shows the fraction of contrast agent remaining in the vascular system and is referred to as the “impulse residue (retention) function.” Figure 4.1C shows the distribution of transit times of contrast agent at the venous outlet through the vascular system, which is denoted by h(t) and is referred to as the “impulse response function.”

When we analyze the kinetic behavior of contrast agent in the vascular system (Fig. 4.1), a relationship between the inflow and outflow is based on the fundamental assumption that the system is both linear and stationary. This means that the response to an influx is assumed to be proportional to the injected dose and that the distribution of transit times through the system is constant in time.⁵

Mathematical solutions are represented for simple models under the condition of an ideal bolus, that is, the tracer appears for one capillary transit without recirculation. Actually, however, the input to the tissue is represented by a continuous concentration-time curve. In other words, the tissue response is a sum of the individual responses scaled in magnitude and shifted in time to match each bolus input, which is mathematically represented by convolution integral.

Figure 4.2 illustrates the concept of convolution integral. Figure 4.2A shows an ideal bolus input of contrast agent injected at time 0, that is, δ(t) defined by Eq. (5), and Figure 4.2B shows the corresponding impulse residue (retention) function (R(t)). Figure 4.2C shows an ideal bolus input injected at time t, that is, δ(t — τ), and Figure 4.2D shows the corresponding tissue response to this bolus input, which is represented by R(t — τ). When the arterial
input function (AIF) is not bolus (ideal) but continuous (real) as shown in Figure 4.2E, it can be interpreted as a series of bolus inputs with an area of C_a(τ)dτ (black square in Fig. 4.2E). In this case, the tissue response to the continuous input function can be interpreted as the sum of the responses to each bolus input given by C_a(τ)R(t-τ)dτ (Fig. 4.2F). Mathematically, this can be expressed as

where the symbol ⊗ denotes convolution integral and τ is a dummy integration variable. When R(t) = 0 for t < 0, the following equation holds

because the area of the convolution of two functions is equal to the product of the areas of the functions. If all tracers that enter the system leave it eventually, the principle of conservation of mass, that is, , holds (see Eq. (16)). Substituting this relation into Eq. (7) yields

The idea of a system’s impulse response is commonly used in systems theory.⁶ When we consider a hypothetical tracer experiment with an ideal bolus input of tracer at the arterial inlet of the vascular system (Fig. 4.1A), the injected tracer would not appear all at once at the venous outlet⁶ and the transit times of tracer (contrast agent) through the vascular system are dispersed, as previously described. The distribution of transit times of tracer (contrast agent) at the venous outlet through the vascular system is described by the impulse response function (h(t)) (Fig. 4.1C). The relationship between h(t) and R(t) can be given by⁶

Differentiating Eq. (9) yields the alternative relation

Because h(t) is the probability density function of transit times of tracer through the vascular system, the first moment of h(t) gives the mean transit time (MTT) of the tracer, that is,

Substituting Eq. (10) into Eq. (11) and using the formula for integration by parts yield

When a contrast agent is injected and enters a vascular system as shown in Figure 4.1, the concentration of contrast agent at the venous outlet of the system cannot be measured using DCE-CT, but only the average concentration within the distribution space of contrast agent in the tissue can be measured. Thus, we consider the vascular system consisting of the tissue region including the distribution space of contrast agent as illustrated in Figure 4.3. If the administered tracer enters and leaves the system with blood flow F as illustrated in Figure 4.3, then the temporal change of the tracer mass (concentration × volume) inside the tissue distribution space at time t is described by Fick principle as

where C_d(t), C_a(t), and C_v(t) are the concentrations of the tracer in the tissue distribution space, arterial blood, and venous blood at time t, respectively, and M_d(t) is the mass of tracer inside the tissue distribution space with a volume of V_d, that is, M_d(t) = C_d(t)V_d. If we use the concentration of contrast agent in whole blood (C_b(t)) for C_a(t), F represents whole blood flow. If the concentration of contrast agent in arterial plasma (C_p(t)) is used for C_a(t), F represents plasma flow and C_p(t) is given by C_p(t) = C_b(t)/(1 — Hct), where Hct denotes hematocrit.

The extraction fraction (E) is defined by dividing both sides of Eq. (13) by the influx (FC_a(t)) as

E is the fraction of the inflowing tracer that does not leave the system via outflow. Note that E is usually given in the form of E = (C_a(∝) — C_v(∝))/C_a(∝). The complement of E is also sometimes used, that is, T = 1 — E. Correspondingly, T is defined as the fraction of the inflowing tracer that leaves the system via outflow and is called the “throughput” or “transmission.”

Integrating both sides of Eq. (13) from t = 0 to ∝ with an assumption of C_d(0) = 0 yields

Equation (15) is known as “Kety-Schmidt equation.”⁷ When C_d(0) = 0 and C_d(∝) = 0, that is, all tracers entering the tissue leave it eventually, we obtain the following relation from Eq. (13):

Equation (16) represents the principle of conservation of mass.

It should be noted that blood flow is the amount of blood flowing through an organ, tissue, or vessel at a given time, whereas
perfusion is the flow per given volume or mass of tissue. Thus, a large organ could have greater blood flow but less perfusion.

The concentration of the tracer in venous blood (C_v(t)) in Eq. (13) can be given by convolution integral between C_a(t) and h(t) as C_v(t) = C_a(t) ⊗ h(t). Substituting this relation into Eq. (13) and integrating both sides of Eq. (13) yield²

where

R_d(t) is referred to as the “tissue response function.” When we integrate both sides of Eq. (17) and use the principle of conservation of mass in a similar manner to that in Eq. (8), we obtain . Furthermore, integrating both sides of Eq. (18) and

using this relation yield . Therefore, from Eq. (12), we obtain the MTT of tracer in the tissue distribution space (MTT_d) as

Equation (19) is the basic equation of “central volume principle,” stating that the MTT of tracer in the tissue distribution space can be given by the quotient of distribution volume and blood flow, and it is equal to the reciprocal of perfusion.

The MTT of tracer in the tissue (MTT_t) is given by replacing V_d by V_t as²

As illustrated in Figure 4.4, the following relation holds:

Thus, we obtain

Equation (22) implies that the initial height of the tissue response function is equal to F/V_d, that is, perfusion.

A tracer kinetic model is a mathematical description of the transport processes of the tracer within the tissue.⁶ Recently, various methods have been proposed for the quantification of kinetic parameters including perfusion.⁸,⁹ These methods can be roughly classified into two groups; one is a model-independent method and the other is a model-dependent method.

The model-independent method yields kinetic parameters for a tissue region described as a “black box” and thus does not provide insight into its internal structure.² For quantification of perfusion using a model-independent method, deconvolution between the measured tissue residue curve and the AIF is needed to obtain the tissue response function, from which the perfusion can be quantified based on the central volume principle (Eq. (19)).¹⁰ Thus, the model-independent method is also called “deconvolution analysis.”

For a model-dependent method, compartment models are usually employed as tracer kinetic models, and kinetic parameters are quantified using compartment analysis based on the assumption that tissues can be kinetically represented by a set of interacting subcompartments that represent distinguishable states of an administered tracer.²

The kinetic models employed in compartment analysis can be classified as lumped parameter or distributed parameter models. The compartments are used to describe the physical distribution spaces of the tracer in the tissue. Conventionally, these compartments are assumed to be well mixed, that is, tracer movement is sufficiently fast and distributes evenly throughout the compartments, such that tracer concentration is only a function of time, but not space. Thus, conventional compartment models are called lumped parameter models. On the other hand, a distributed parameter model attempts to account for tracer concentration gradient within the compartments, such that concentration can be a function of both time and space.⁶

In contrast to the model-independent method, the modeldependent method generally requires specific knowledge about the vascular and compartmental architecture, the volumes of the compartments comprising the model, and the tracer-dependent permeability of compartment barriers. The assumptions in these models are generally valid under normal conditions, but for pathologic conditions, it is often difficult to independently verify which model parameters changed significantly, in particular when the model involves a large number of parameters or if the sensitivity ranges for the different parameters overlap significantly.¹¹

Generally, model-independent deconvolution analysis has been applied to the data obtained by intravascular tracers.¹² Recently, the accuracy of model-independent deconvolution analysis when applied to the data obtained by extravascular contrast agents, that is, the effect of the leakage of contrast agent from the intravascular space to the extravascular space, has been studied in Ref. ¹³

To investigate the effect of the leakage of contrast agent from the intravascular space to the extravascular space, it is necessary to use a model in which the concentration of contrast agent within the tissue is treated as a function of both time and position, that is, distributed parameter models.⁹ The distributed parameter models describe tracer concentration in the vascular compartment as a function of both time and position along the capillary and are generally believed to be more realistic than the lumped parameter models. One of the simple distributed parameter models is the tissue homogeneity model, which was initially developed by Johnson and Wilson.¹⁴ This model is based on both convective and diffusive transport of contrast agent in capillaries represented together as
a single cylinder and is composed of the intravascular space surrounded by the extravascular space. The two spaces are separated by a permeable barrier. St. Lawrence and Lee¹⁵ derived an approximate closed-form solution to the tissue homogeneity model using the adiabatic approximation. The details of the adiabatic approximation to the tissue homogeneity model will be described afterwards.