As an alternative to motion compensation approaches on the acquisition level, or complementary to them, motion correction may be performed on the post-processing level [21]. Technically, the major difficulty in DCE-MRI compared to similar problems in medical imaging is the changing signal intensities during bolus passage. The challenge for a (semi) automatic motion correction technique is to distinguish these changes from those due to motion, and the development of robust techniques remains an open problem. For an ROI-based analysis, the most straightforward approach is to redraw or modify the ROI for every individual dynamic acquisitions. The process is tedious and time consuming, difficult to automatize [22], and is not suitable for a pixel analysis. An alternative approach is based on co-registration techniques, which aim to match motion-affected images to a reference image by a rigid or nonrigid deformation of the image [23–25]. Co-registration is attractive in theory, as it fully removes motion effects and reconstructs the data that would be measured in the absence of motion. However, it is computationally challenging and usually requires expert intervention especially in those body organs where the displacement is different depending on the organ position.

Different strategies can be applied to extract quantitative parameters from perfusion studies. The simplest approach uses curve descriptor like contrast arrival time (time when the contrast arrive to a pixel), time to peak (time for the maximum signal intensity), or area under the curve (AUC, integral of the contrast concentration time curve for a specific time interval definition).

On the other hand, once the contrast concentration is obtained from each dynamic acquisition, the tracer kinetic theory provides a relation between those quantities and thus forms the basis for determining the tissue status from the measured concentrations in the tissue C(t) and the feeding artery C _A(t). This theory establishes a relation between both quantities that follows:
where F _p is the flow of the blood plasma in the tissue, and R(t) is normally called residue function and represents the fraction left in the tissue at time t of a dose injected at time t = 0, and ⊗  represent the convolution operator. The residual function R(t) is the relative amount of contrast in the VOI in an idealized perfusion experiment, where a unit area bolus is instantaneously injected and subsequently washed out by the perfusion.

The blood volume (BV) can be estimated as:
where the k _H represents a correction factor that takes into account the extracellular behavior of the contrast.

Following the central volume theorem, the passage of contrast can also be measured with DSC-MRI through the MTT, expressed as MTT = V/F _p which gives a measure of the washout time of the blood in the volume of interest. In the case of cerebral perfusion, F _p is known as CBF and the contrast volume is known as CBV (Fig. 16.3).

One step further in this theory is the pharmacokinetic modeling of the residue function in terms of the hemodynamic parameters. The fundamental building block of any tracer-kinetic model is the compartment, often defined as a space where the tracer is well mixed or evenly distributed over the space at all times. Following this theory, the residue function can be described as a combination of different compartments:
where n represents the number of compartments.

Following this compartmental theory, cerebral perfusion can be seen as a single compartment model where the residual function can be expressed as:

In most of the tissues other than brain, standard MRI tracers distribute over two different spaces: the blood plasma P and the extravascular, extracellular space E. The two-compartment exchange model [26–28] is defined by the assumption that (1) P and E are compartments; (2) E does not exchange tracer directly with the environment; (3) the clearances for the outlets connecting P and E are equal; and (4) the clearance for the outlet of P to the environment equals the plasma flow.

There are four different situations where a tissue with this structure can be effectively reduced to a one-compartment model: (1) and (2) when one of the spaces has a negligible volume, (3) if the tracer extravasates slowly, so that, the concentration in the extravascular space is negligible within the acquisition time, and (4) if the tracer extravasates rapidly, so the system behaves as a single well mixed space. In each of these regimes, the residue function becomes mono-exponential, but the precise interpretation of the parameters is different. Hence, it is recommended to use an abstract notation K _trans and k _ep for the model parameters of a one-compartment model:

A one-compartment model with these notations is often referred to as a Tofts model [29] or as a modified Tofts model when a term is added to account for the tracer in the vasculature [30]. If prior information regarding the state of the tissue is available, then K _trans and k _ep can be interpreted in more concrete terms. However, to avoid errors of misinterpretation [31, 32], it is prudent to leave the interpretation open [33, 34].

In those cases where the vascular volume cannot be neglected, it is required to include the plasma contribution in the model with a resulting total contrast concentration equal to:
where v _p represents the volume of plasma inside the voxel and is normally called plasma volume, v _e represents the extravascular-extracellular volume inside the voxel, K _trans represents the transfer rate of the blood from the vessels to the extravascular-extracellular space, and k _ep the transfer rate of the blood from the extravascular-extracellular space to the vessel. A graphical interpretation of these parameters can be found in Fig. 16.4 and the results of the analysis of a highly vascularized tumor.

For an accurate perfusion estimation, it is necessary to obtain the arterial input function (AIF) of a contrast agent C _A(t); the time-concentration curve in plasma has long presented a problem in MR DCE-MRI and DSC-MRI studies. A poor estimation of the AIF due to insufficient time sampling or signal saturation effects produces substantially increase bias in the estimated hemodynamic and permeability maps [28]. To overcome this limitations, some authors have proposed to use a simplified functional form for the AIF, which is assumed valid for all individuals [35, 36]. However, others have shown that using a simplified standard AIF can lead to large systematic errors in kinetic tracer model output parameters, such as the volume transfer constant K _trans and the fractional blood volume [37, 38]. Because interindividual variations in heart rate and kidney function have the potential to lead to differences in the AIF form, the use of an accurately measured AIF for each subject is often the preferred approach, even though it is only attainable in a minority of studies [35, 39]. It is often not possible to perform a reliable AIF measurement due to experimental constraints or due to the lack of a suitable artery in the field of view. Hence, recent published work [40] of a clinical DCE-MRI study has suggested the use of an assumed form of AIF that is sufficiently similar to the true AIF to allow estimation of model parameters. Parker et al. [40] demonstrated that in the presence of measurement errors in an experimentally acquired AIF, there are significant benefits to be gained in the reproducibility of compartmental model parameters by using an assumed form. However, there have been limited studies conducted, so far, to quantify the effects of using an assumed form of AIF in experimental models or in differentiating the effects of varying assumed AIF forms, including population averages.

Among MRI methods, arterial spin labeling (ASL) techniques have shown their potential for tissue perfusion quantification. The complete noninvasiveness and nonionizing nature of this technique makes ASL a very interesting method for studying perfusion in healthy individuals or patients who require repetitive follow-ups. Furthermore, the use of any radioactive tracers or exogenous contrast agents, which are necessary in most conventional techniques, may be restricted in patients with particular conditions, such as kidney failure, or in pediatric populations. Finally, ASL-based methods are useful for functional studies and evaluations of new therapies, in which physiological changes due to the pharmacological stimuli must be monitored over time. Due to movement artifacts, most of the clinical experience of ASL has been limited to the brain (Fig. 16.5), although recently it has been proven useful in body organ such as kidney.

The common goal of all existing ASL techniques is to produce a flow-sensitized image (also known as a labeled image) and a control image in which the static tissue signals are identical. This is usually performed by inverting or saturating the water protons in the blood supplying the imaged organ. After a delay between labeling and image acquisition, called the inversion delay (TI), the labeled blood spins reach the capillaries, where they exchange with tissue water, giving rise to the perfusion signal. The subtraction of the label from the control yields a difference signal that directly reflects local perfusion, since the signal from stationary tissue is completely eliminated.

There exist two main classes of ASL techniques: continuous ASL (CASL) and pulsed ASL (PASL). In CASL, the supplying blood is continuously labeled below the imaging slab, until the tissue magnetization reaches a steady state. The PASL approach labels a thick slab of arterial blood at a single instance in time, and the imaging is performed after a time long enough to allow the spatially labeled blood to reach the tissue and exchange at the region of interest. Both methods need a control experiment in order to visualize and quantify the perfusion (Fig. 16.6).

Diffusion-weighted imaging (DWI) is an MRI technique sensible to microscopic mobility of water and is classically referred as Brownian motion. This mobility is due to thermal agitation and is highly influenced by the cellular environment of water. In oncologic imaging, DWI has been linked to lesion aggressiveness and proposed to evaluate tumor response, although the biophysical basis for this is incompletely understood. In biologic tissues, these motions include molecular diffusion of water and microcirculation of blood in the capillary network (perfusion). Molecular diffusion refers to physical properties of tissue that allow tissues to be characterized. The sensitivity to water movement is controlled by a sequence parameter called b value. This acquisition parameter controls the water dephasing due to the movement; when this parameter increases, the signal from water molecules with a higher movement disappears, remaining only the signal from those water molecules with restricted movement. In this scenario microcirculation of blood or perfusion can also be considered an incoherent motion due to the pseudorandom organization of the capillary network at the voxel level that can be imaged using low b values during the diffusion acquisition.

IVIM imaging is a method initially developed by Le Bihan et al. [52] to quantitatively assess the microscopic translational motions that occur in each image voxel at MRI. Le Bihan et al. [52, 53] demonstrated that both pure molecular diffusion and microcirculation, or blood perfusion, can be distinguished by using IVIM analysis of diffusion signal decay. For this, it is necessary to obtain multiple b values to encompass both low b values (<200 s/mm²) and high b values (>200 s/mm²). The low b values are used to study the fast water movement associated to the blood flowing through the capillary network in a pseudorandom manner. Conversely, for b values over 200 s/mm², the signal information from the capillary network is completely dephased obtaining only signal from the extravascular space (Fig. 16.7).