Pharmacokinetic Models for Dynamic Contrast-Enhanced Computed Tomography and Magnetic Resonance Imaging
Steven Sourbron
Ting-Yim Lee
The aim of this chapter is to clarify the physiologic assumptions behind the most common pharmacokinetic models and the interpretation of the parameters, and to offer guidelines on when and when not to use each model. The models and their assumptions will be represented exclusively using diagrams, without the use of mathematical formulae. The main part of the chapter will be devoted to models for vascular-interstitial exchange, which apply to the vast majority of applications discussed in this book: all cerebral diseases, all tumors with the exception of some liver tumors, myocardium, and lung. The physiologic meaning of the parameters that can be measured in such tissues is clarified in the first section. Then the four most complex models for vascular-interstitial exchange are explained. The section following discusses the important special case of the one-compartment model. This is followed by modeling strategies to deal with reduced acquisition time and temporal resolution, respectively. The chapter then discusses models for two important tissue types that cannot be modeled with the above strategies (i.e., liver see Chapter 65 and kidney see Chapter 75). Finally, we provide some concrete guidelines on how to choose a model for a given application.
Introduction
Why Do I Need a Pharmacokinetic Model?
Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) and DCE computed tomography (CT) are examples of classical indicator-dilution experiments. They aim to characterize the motion and properties of body fluids by adding an indicator (i.e., a substance that is separately detectable but does not otherwise disturb the motion of the fluids).1 DCE-MRI and DCE-CT use standard clinical contrast agents as indicators, such as gadolinium-based agents or iodine, which are known to cause signal changes in proportion to their concentration. The contrast agent is injected intravenously in the form of a short bolus, and imaging is performed dynamically to measure the concentration in the tissue of interest as a function of time (see Chapters 6 and 11). Similar tracer kinetic considerations are the foundation for quantitative arterial spin label perfusion imaging, where labeled water protons (in blood) serve as (an endogenous) tracer.
The purpose of DCE-MRI and -CT is to measure characteristic indices of tissue function or physiology by analyzing the temporal profile of the tissue concentrations.2 However, these concentration-time profiles reflect three separate effects: the manner in which the bolus is injected; the deformation of the bolus as it passes through the circulation; and the deformation of the bolus as it passes through the tissue of interest. Only the latter depends on the functional and physiologic status of the tissue. In order to separate it out, a measurement is required of the concentration-time curve in the blood of a feeding artery, known as the arterial input function (AIF) (see Chapter 26). The functional indices are then derived by analyzing the difference between AIF and tissue concentrations.
Pharmacokinetic models play a central role in this concept: they provide the link between the target of the measurement (tissue-characteristic physiologic parameters) and the data (AIF and tissue concentrations).3 Given a suitable model, values for the physiologic parameters can be derived by model fitting: one starts with an initial guess for the unknown parameters and compares the data predicted by the model to the measured data. If they agree well, this is evidence that the chosen values are correct; the model is said to “fit” the data, and the values are accepted. If they are very different, the initial values are adjusted and the process is repeated until a good fit is obtained.
In practice, this iterative optimization process is performed automatically by a computer algorithm. The goodness of fit or how well the model and associated model parameters fit the data is assessed by a metric, such as the sum of squares of the differences between measured concentrations and those predicted by the model. During the optimization process, the optimizer attempts to adjust the model parameters until the sum of squares difference reaches a minimum. There are several optimizers available, such as Downhill-Simplex, Levenberg-Marquardt, or (nonlinear) conjugate gradients, but a detailed discussion of those is outside the scope of this chapter.
Pharmacokinetic versus Signal Modeling
A pharmacokinetic model provides a link between physiologic tissue parameters and indicator concentration-time
curves, but concentrations cannot be measured directly. In reality the measured data consist of signal-time curves, and before a pharmacokinetic model can be applied, a separate analysis step is required to convert those signals into concentrations. This involves a separate model that provides the relation between signal and concentration.2 This is not a pharmacokinetic model but a modality-specific signal model that depends on the physical mechanism of signal formation. In CT, where the signals are given in Hounsfield units, this is relatively straightforward: it is generally assumed that the signal change induced by the contrast agent is directly proportional to the concentration.4 Experiments have shown that this proportionality constant is tissue independent, so it can be taken from separate phantom experiments.
curves, but concentrations cannot be measured directly. In reality the measured data consist of signal-time curves, and before a pharmacokinetic model can be applied, a separate analysis step is required to convert those signals into concentrations. This involves a separate model that provides the relation between signal and concentration.2 This is not a pharmacokinetic model but a modality-specific signal model that depends on the physical mechanism of signal formation. In CT, where the signals are given in Hounsfield units, this is relatively straightforward: it is generally assumed that the signal change induced by the contrast agent is directly proportional to the concentration.4 Experiments have shown that this proportionality constant is tissue independent, so it can be taken from separate phantom experiments.
In DCE-MRI, the relationship between signal change and concentration is typically more complex, and various models of different levels of complexity have been proposed.2 As the MRI signal is generated by the tissue water, this effectively requires a model for the exchange of water between the various tissue compartments. The most accurate methods also require a separate measurement of precontrast T1 to calibrate the dynamic signal. The problem is further complicated by the fact that the appropriate model depends on the MRI sequence and values for the sequence parameters. A particular issue of current debate is whether, and under which circumstances, there is a need for more than one compartment to model the exchange of water.5,6,7,8
The conversion of signal into concentration is the topic of another chapter and will not be further discussed here. This chapter will focus exclusively on pharmacokinetic models, assuming the conversion of signal into concentration has been performed separately.
Why are There So Many Different Pharmacokinetic Models?
Model fitting will always produce a single set of values for the unknown model parameters, but these are not necessarily correct. Three different problems can arise:
Model is too complex. The information content of DCE-MRI and -CT is limited, and strong constraints must be imposed on the model to arrive at a one-to-one relation between parameters and the data. If not enough constraints are imposed, then multiple parameter values produce a good fit, and the correct solution cannot be identified.
Model is too simple. If too many constraints are imposed, then none of the possible parameter values will produce a good fit to the data. In that case the “best” fit will not necessarily reflect the true values. In contrast to the previous case, this type of problem can be detected by comparing the data to the best fit.
Model is invalid. Even if the model has exactly the right complexity so that a single solution exists, this does not guarantee that the results are accurate. Generally different models of the same complexity fit the same data equally well. If the constraints built into the model are false, a model error arises in the measured parameters that is inherently undetectable.
The key reason why different models exist is that the optimal model depends on the tissue type and on the quality of the data (temporal resolution, acquisition time, signal-to-noise ratio, artifacts). Experience shows that in most tissues no more than four independent physiologic parameters can be measured, characterizing (a) tissue perfusion, (b) vascularity, (c) cellularity, and (d) endothelial permeability. But often this number is even < four, either because one parameter is small and falls below the detection threshold, or because the data quality is insufficient to detect it. It is critical that a model with the right level of complexity is used, and that the model error is minimized by using all available prior information.
Why Do I Need to Know All This?
Ideally, clinicians should not have to worry about the precise way the data are modeled. A better approach would be to (a) perform a pilot study using training data to identify the optimal modeling approach for a given application, and (b) embed this approach into a software tool that performs the analysis automatically. Ultimately, the field will have to move in that direction if these methods are to be applied in clinical routine, but the current reality of DCE-MRI and -CT is far removed from this ideal.
Whatever the software used, because the problem of automated model choice has not been solved in most applications, it is important that the clinical user has an insight into the scope and limitations of the pharmacokinetic models and of the possible pitfalls that may affect the confidence in the values and the interpretation of the data. Treating the software package as a black box without a clear understanding of potential limitations of the model or sources of artifacts is bound to generate trouble in a clinical environment.
A number of different problems may occur:
Early efforts toward standardisation9 have led in many areas to a dogmatic approach where the same model is always applied, without considering whether it is suitable for the given tissue and data. Some software tools follow this approach; they rely on the user not to apply the model outside its scope or to apply the necessary caution in interpreting values that may not be reliable.
More recent developments aimed to adapt and refine the modeling in response to improvements in data
quality have led to a variety of different models. Some software tools therefore offer a number of alternative models and leave the choice up to the user, who can decide on the basis of prior knowledge, experience, or visual aids such as the comparison of best fit to the data. That said, in a busy clinical environment, clinical users often do not have the time to oversee tedious postprocessing procedures and smart unsupervised analysis routines are largely desired.
Software is often application specific to allow tailor-made optimization. This may give a (false) sense of assurance that the built-in model is suitable. Caution should be exercised and one should keep in mind that it is difficult at the level of software development to foresee all possible applications or pathologies in a specific organ. This is particularly difficult when the analysis is performed on the single-pixel level, because then different areas within the same image may need to be modeled differently.
Software tools often offer a number of advanced settings or variable parameters that allow the user to optimize the output for a given type of data (e.g., interactive choice of the time window to be fitted). Results can be improved significantly by applying these options correctly, but this requires insight into the assumptions inherent in the models.
The measured parameters are always model dependent to a certain extent because of the inherent model errors. In addition, there are different conventions regarding notation and terminology for the same parameters. These differences must be understood and taken into consideration when comparing studies or individual patient outcomes against benchmarks in the literature.
Aims and Scope of the Chapter
The aim of this chapter is to clarify the physiologic assumptions behind the most common pharmacokinetic models and the interpretation of the parameters and to offer guidelines on when or when not to use which model.
The models will be represented exclusively using diagrams and the graphical conventions summarized in Figure 29.1. Equations can be derived from the diagrams by applying a fixed set of rules, but they do not add any additional physical insight and are mainly needed to build computer algorithms or to run simulations. References are inserted to direct the interested reader to more technical papers where equations and solutions can be found.
The conversion of measured signals into concentrations (see Chapters 6 and 11) is not a pharmacokinetic problem and will not be discussed here. The implication is that we need not distinguish between MRI and CT, because standard contrast agents for both modalities have a similar pharmacokinetic behavior in body tissue.10,11 The key difference between these modalities lies in the relation between signal and concentration, not in the relation between concentration and tissue structure. For the same reason, the content of this chapter is equally relevant to dynamic susceptibility contrast (DSC-MRI) (see Chapter 11). DSC-MRI and DCE-MRI are often treated as fully different topics, but because they use the same indicators, the same pharmacokinetic modeling approaches apply. Historically, a key difference is that DSC-MRI tends to be performed at high temporal resolution to capture the first pass of the bolus and measure perfusion parameters, whereas DCE-MRI is more typically associated with measurement of capillary permeability at lower temporal resolution. Because different tissue parameters were targeted, different pharmacokinetic models were required. This distinction is no longer relevant today: improvements in hardware have enabled DCE-MRI measurement at equally high temporal resolution and in most organs DCE-MRI has overtaken DSC-MRI as the method of choice for perfusion measurement as well.2 The key difference between the two methods is that DCE-MRI uses T1-weighted sequences, whereas DSC-MRI uses T2– or T2*-weighted sequences. This has implications on the relation between signal and concentration, but as radiofrequency pulses do not affect the motion of contrast agent molecules, the choice of sequence does not affect the choice of pharmacokinetic model.
 FIGURE 29.1. Graphical conventions used in the model diagrams. The spaces are color coded in red (plasma), blue (interstitium), or purple (mixed or undetermined). |
The models discussed in this chapter all assume that the AIF is known, provide a clear physiologic interpretation
for the parameters, and only depend on parameters that are measurable. This covers the most common approaches for quantifying DCE-CT and DCE-MRI. A number of alternative methods have been proposed in the literature, but these will not be discussed here:
for the parameters, and only depend on parameters that are measurable. This covers the most common approaches for quantifying DCE-CT and DCE-MRI. A number of alternative methods have been proposed in the literature, but these will not be discussed here:
Model-free analysis. Model-free analysis and pharmacokinetic modeling are both examples of deconvolution methods, but model-free analysis imposes only very weak constraints on the internal structure of the tissue. This eliminates the model error at the cost of (a) a larger numerical error, (b) a stronger dependence on the algorithm used, and (c) a reduced number of measurable parameters (see Chapters 24 and 25).
Parametric models. This is an intermediate method that is also known as parametric- or constrained deconvolution. These models are purely mathematical in nature and do not offer a clear interpretation for each model parameter. The most well-known example is the Fermi model, which for historical reasons has become a standard in myocardial DCE-MRI.12
Combined AIF tissue models. In some applications (specifically in preclinical work), it is difficult to measure an AIF, so a modeled AIF is used instead. Some authors have combined the AIF model with the tissue model to produce a larger model with more parameters.13 The AIF parameters can be fixed to population averages, or some of them can be derived from the data.
Many-parameter models. Some models have been proposed that provide a very detailed description of the tissue’s inner structure, such as the well-known multiple path, multiple tracer, indicator dilution, 4 (MMID4) region model. These have many more parameters than are actually measurable and are therefore mainly used for simulation purposes.14
Descriptive models. This type of model does not require an AIF and merely provides a parametric description of the tissue concentrations. The parameters (e.g., area-under-the-curve, time-to-peak) cannot be directly related to physiologic properties and are not absolute tissue characteristics because they depend on injection protocol and the effects of the macrocirculation.
What Can Be Measured?
For standard MRI and CT contrast agents that are extracellular in their distribution, the majority of tissue types can be pictured as shown in Figure 29.2. The contrast agent distributes over two different tissue regions: (a) the tissue blood plasma in the microvasculature, and (b) the interstitium. These two regions are often referred to as intravascular extracellular space and extravascular extracellular space (EES), respectively. Part of the contrast agent administered to the patient travels through the microvasculature (i.e., the capillary bed) and is evacuated to the venous system; the remaining portion of the contrast agent diffuses first through water pores in the vascular endothelium to the interstitium and eventually back into the vascular space via the same diffusion process. This general architecture applies to most contrast agents, but factors such as (a) the molecular size of the agent, (b) chemical properties, and (c) other factors such as protein binding may affect the rates at which these processes take place and the size of the space that is accessible to the contrast agent.
The aforementioned tissues are characterized by four independent primary parameters, which can be combined to produce a number of common derived parameters. They are listed in Table 29.1 and explained in detail below.
 FIGURE 29.2. A schematic illustration of a generic tissue with vascular-interstitial exchange of contrast agent. The interstitium is pictured in blue (volume ve), the plasma in red (volume vp), the tissue cells in white and the red blood cells in gray. Plasma flow Fp carries the contrast agent into the capillary bed, where it exchanges with the interstitium at a rate determined by PS. The contrast agent is eventually evacuated by the same flow Fp at the venous end. |
Primary Parameters: Plasma Volume, Interstitial Volume, Plasma Flow, and Permeability Surface
The four primary parameters measure the sizes of both spaces and the rate at which contrast agent enters them:
The plasma volume (vp) measures the volume of the plasma space per unit tissue volume, or the volume fraction of plasma. Typical units are milliliters per 100 mL or percentages. The plasma volume is a measure of the tissue’s vascularity: in essence, it tells how much of a voxel of interest is occupied by blood plasma. In a large blood vessel, for example, this will be 100% minus hematocrit (as we know that contrast agent is not entering blood cells). In white matter, for example, it is approximately 2% reduced by the small vessel hematocrit fraction.
The plasma flow (Fp) measures the inflow of plasma per unit of tissue volume and is usually expressed in
units of milliliters per minute per 100 mL (of tissue under observation). Whereas the plasma volume measures how much blood plasma is present in the tissue, the plasma flow measures how much plasma is delivered to the tissue per minute. Unlike contrast agent, plasma does not accumulate in tissue, so the venous outflow must be the same Fp as the arterial inflow (Fig. 29.2).
The interstitial volume (ve) measures the volume of the interstitial space per unit tissue volume. Units are the same as vp. In essence, it is the distribution volume of the contrast material outside of the blood vessels and therefore depends on the size and properties of the agent. It can only be measured when the contrast agent effectively leaks out of the vessels (i.e., ve is unmeasurable in the presence of a blood–brain barrier or when an intravascular agent is used).
The permeability (surface area) product (PS) measures the rate at which contrast agent leaks out of the tissue capillaries. More precisely, PS is the flux of contrast agent (millimole per minute) into the interstitium (across the endothelial wall), per unit tissue volume (milliliters), and per unit tissue plasma concentration (millimole per milliliter). The units of PS are typically abbreviated to 1/min. PS is in fact a composite of two different properties that are not separately measureable: the “leakiness” or “permeability” of the endothelial surface (P), and the total surface area of the perfused vessels within the voxel of observation (S). All common models assume that endothelial permeability is symmetric (i.e., transport from interstitium to plasma is characterized by the same PS as transport from plasma to the interstitium).
TABLE 29.1 AN OVERVIEW of THE MAIN PRIMARY AND DERIVED PARAMETERS FOR TISSUES WITH VASCULAR-INTERSTITIAL EXCHANGE. GIVEN ARE THE SYMBOL USED IN THE TEXT, PARAMETER NAME, UNITS, AND THE DEFINING FORMULA FOR THE DERIVED PARAMETERS. | ||||||||||||||||||||||||||||||||||||||||||||
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These four parameters are all positive and independent of one another, with the constraint that the total extracellular volume fraction ve+vp cannot be larger than 100%.
The parameters Fp and vp are typically referred to as perfusion parameters, whereas PS and ve are known as permeability parameters. An important difference is that the permeability parameters depend on the contrast agent used.15 As mentioned earlier, larger contrast agent molecules, or those that have a tendency to bind to proteins and hence assume the sizes of the proteins they bound to, generally have lower P as they do not pass through the endothelium as easily as smaller molecules can. Equivalently, a certain portion of the interstitial volume may be inaccessible to larger molecules, leading to a smaller ve measurement. Therefore, when comparing DCE measurements it is always prudent to also report the contrast agent used and consider its pharmacokinetic properties (including its ability to bind to proteins). When interpreting ve (and also PS maps), one should consider that a ve of 0 does not necessarily mean the EES is 0. If there is no contrast agent leakage (e.g., an intact blood–brain barrier or use of a macromolecular agent), then these parameters cannot be determined.
Derived Parameters: Extraction Fraction, and Volume Transfer Constant
The first class of derived parameters offers a direct measurement of contrast agent delivery to the interstitial space. These are important functional indices, because the ultimate purpose of tissue perfusion is to deliver oxygen and nutrients to the tissue cells to satisfy metabolic demands:
The extraction fraction (E) is the proportion of contrast agent that leaks into the interstitium in one pass through the tissue.16 The precise formula for E depends on the architecture of the capillary bed; for a plug-flow
capillary (discussed later in the chapter), E is related to PS and Fp via the Crone and Renkin relationship17: E = 1 − exp(− PS/Fp); the equivalent relation for a well-mixed capillary bed18 is E = PS/(PS + Fp). In most tissue types, E is <10% to 20%, meaning that <20 of 100 contrast agent molecules that enter the tissue actually extravasate. The remaining 80% to 90% of molecules passes straight through the capillary bed without leaking out.
The volume transfer constant (Ktrans) measures the “nutrient” part of the plasma flow (assuming the nutrients are of equal molecular size as the contrast agent) and is therefore a key functional index of tissue perfusion. The units of Ktrans are usually abbreviated to 1/min. Formally, Ktrans can be defined as the product of E· ×Fp (i.e., Ktrans is the part of the plasma flow that is extracted from the blood pool).9 The other part (1 − E) · Fp of the inflow passes through the tissue unused. It carries molecules that never extravasate (i.e., do not leave the blood vessel), and therefore do not have a chance to interact with the tissue cells and satisfy metabolic demands.
The key difference between PS and Ktrans is that PS measures the total flow across the capillary membrane, whereas Ktrans only measures the flow of “fresh” or “oxygenated” blood that enters straight from the arteries and has not passed through the interstitium before. Ktrans is always a function of inflow (Fp) and permeability (PS), but as a general rule the smallest of the two is the dominant factor. In the particular case where PS is much smaller than Fp (i.e., the tracer amount that diffuses into the EES can be easily replenished by the much higher plasma flow), Ktrans is equal to PS and the perfusion is said to be permeability limited. Conversely, if PS is much larger than Fp, then only a small part of the transport across the capillary membrane consists of “unused” plasma as Fp cannot provide sufficient “fresh” material; in that case Ktrans is closely approximated by Fp and the tissue perfusion is flow limited.
Derived Parameters: Mean Transit Times
A second class of derived parameters is the mean transit times (MTT), expressed in seconds or minutes; they measure the time it takes for an indicator particle on average to pass through a given tissue region. In theory, MTT can be measured directly if one measures the probability density distribution of transit times for tracer molecules, via the injection of an infinitesimally short bolus. The first moment of this probability density function is also the MTT.
As a general rule, the MTT of any region is the ratio of the volume of the region to the flow into that region (central volume theorem).1 This rule can be applied to three different tissue regions:
Plasma: The MTT of intravascular particles is Tc = vp/Fp, and measures the average time spent in the capillary bed; it is typically in the order of 2 to 5 seconds. In the DSC literature, this formula is usually written as MTT = CBV/CBF, where CBV is the cerebral blood volume and CBF is the cerebral blood flow. There is often no need to distinguish between various MTTs in DSC-MRI as the main area of application is brain tissue with an intact blood–brain barrier. In that case, only one space (plasma or blood, see discussion of conventions in this chapter) is accessible to the tracer, and only one MTT can be measured.
Interstitium: The MTT of the interstitial space is Te = ve /PS, and measures the average time the contrast agent spends in the interstitium; it is typically an order of magnitude larger than Tc, in the range of 1 to 2 minutes. This means that a contrast agent molecule, after leaking out of the blood vessels, will typically wander through the interstitium for 1 to 2 minutes before being reabsorbed into a capillary.
Total extracellular space: The total MTT for contrast agent that leaks into the interstitium is T = (vp + ve)/Fp; it measures the average time between entering the tissue at the arterial end and leaving it at the venous end.19 This is typically intermediate between Tc and Te as it presents a weighted average of the contrast agent leaking out and the part that remains intravascular. If E = 0 (i.e., no extravasation), then T = Tc.
The difference in magnitude between Tc and Te has significant implications on measurement: because the passage through the vascular bed is rapid, a high temporal resolution is required to measure the perfusion parameters. Temporal resolution can be significantly relaxed if the purpose of the measurement is to determine the permeability parameters alone.
Alternative Conventions
In some applications the parameters blood flow Fb and blood volume vb are used instead of Fp and vp. The choice is essentially a matter of convention, as one can translate between the two by inserting the hematocrit H in tissue blood: Fp = (1 − H) Fb and vp = (1 − H) Fb. A problem is that H may not be known exactly, since the hematocrit in the microvasculature may be lower than in the large vessels due to the Fahraeus effect.20,21 Often entirely ignored is the fact that H can change significantly in pathology. In this sense, the use of the parameters Fp and vp is preferable, since they are not sensitive to errors in H.
An alternative system of units is commonly used, where all quantities are normalized to the tissue mass (100 g) rather than to the tissue volume (100 mL). In this system, Fp and PS have the units milliliters per minute per 100 g, and vp and ve are in milliliters per 100 g. CT or MRI data can be converted into these units by inserting
an experimental value for the tissue density, usually 1.04 g/mL (in the brain perfusion literature often referred to as the blood–brain partition coefficient). This may lead to errors when the actual value is different; the conversion is necessary only in experiments that aim to validate MRI and CT parameters against quantitative gold standard methods such as microspheres.
an experimental value for the tissue density, usually 1.04 g/mL (in the brain perfusion literature often referred to as the blood–brain partition coefficient). This may lead to errors when the actual value is different; the conversion is necessary only in experiments that aim to validate MRI and CT parameters against quantitative gold standard methods such as microspheres.
 FIGURE 29.3. The building blocks in models for vascular-interstitial exchange. The two alternative models for the plasma space (left, red) are the well-mixed compartment (top) and the plug-flow system (bottom). The interstitium (right, blue) is either well mixed (top) or distributed (bottom). |
In a consensus paper,9 it was proposed to systematically use the notation Ktrans for the product of E × ·Fp, and ve for the interstitial volume. A notation kep was also introduced for the ratio Ktrans/ve. The convention has been of huge significance in DCE-MRI and is often adhered to in DCE-CT as well. It has enforced standardization in the terminology and notations, and in this sense has stimulated wider uptake of these methods. Nevertheless, many alternative conventions and notations are still in use.
Four-Parameter Models
The picture represented in Figure 29.2 defines the general architecture of tissues with vascular-interstitial exchange, but is not a pharmacokinetic model on itself. What is missing is more detailed information on how the contrast agent is distributed within each of the two subspaces.
This section presents the most common models for the plasma and interstitial space, then combines them into four tissue models. These are the most complex models in use today; all other models commonly used in DCE-MRI or -CT are simplified versions of these four models.
Plasma Models
For the plasma region, two fundamentally different models have been proposed (Fig. 29.3, left):
The well-mixed compartment assumes that the contrast agent distribution is approximately uniform inside the capillary bed. This arises in chaotic or randomly structured vascular beds, where arteriovenous pathways with many different lengths exist. Because different parts of the contrast agent bolus are mixed together, the bolus disperses—or broadens—when passing through a well-mixed space.
The plug-flow system assumes that contrast agent travels through the vascular bed with constant velocity and the permeability is uniform along the capillary wall. This is a good assumption for a single capillary, where the red blood cells form a series of “plugs” that force the plasma between them to pass at the same speed. Because a voxel in DCE-MRI or -CT consists of a large number of capillaries (as well as arterioles and venules), the plug-flow model is most suitable in tissues with many nearly identical capillaries.
In many ways the plug-flow system (highly structured) and well-mixed compartments (highly chaotic) are at the opposite ends of the spectrum of plausible physiologic models for the vascular bed. Realistic tissues are most certainly somewhere in between, although they may be closer to one end or the other and may move between these regimes because of pathologic processes. The choice between the two has a significant effect on the perfusion parameters, but unfortunately very limited evidence exists to decide which alternative to use for which tissue type. One study in DCE-CT in tumors has shown that both generally provide an equally good fit to the data, which means that the data themselves provide little guidance on which is most suitable.22
Interstitial Models
There are two common models for the interstitium (Fig. 29.3, right):
The well-mixed compartment assumes a uniform distribution, as in the plasma compartment. Because capillaries are densely distributed throughout the tissue, contrast agent enters the interstitium at many different locations at the same time. Hence, one would not expect large variations in concentration throughout the space, although gradients may well exist on the cellular scale.
The distributed interstitium assumes that the space consists of a large number of tiny parallel compartments, which do not exchange contrast agent with one another. Interstitial motion of contrast agent parallel to the capillary is thereby forbidden: particles can only be reabsorbed at the exact location where they are
extracted. There is no indication that such physiologic barriers exist inside the interstitium, but they simplify mathematical analysis of the model,23 and in most conditions they have little effect on the concentration-time curves.
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