Technical aspects of MR perfusion

European Journal of Radiology 76 (2010) 304–313

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Technical aspects of MR perfusion Steven Sourbron ∗ Division of Medical Physics, University of Leeds, Worsley Building, Clarendon Way, LS2 9JT Leeds, UK

a r t i c l e

i n f o

Article history: Received 17 February 2010 Accepted 23 February 2010 Keywords: Perfusion Permeability Dynamic contrast-enhanced MRI Tracer kinetics Signal analysis

a b s t r a c t The most common methods for measuring perfusion with MRI are arterial spin labelling (ASL), dynamic susceptibility contrast (DSC-MRI), and T1 -weighted dynamic contrast enhancement (DCE-MRI). This review focuses on the latter approach, which is by far the most common in the body and produces measures of capillary permeability as well. The aim is to present a concise but complete overview of the technical issues involved in DCE-MRI data acquisition and analysis. For details the reader is referred to the references. The presentation of the topic is essentially generic and focuses on technical aspects that are common to all DCE-MRI measurements. For organ-specific problems and illustrations, we refer to the other papers in this issue. In Section 1 “Theory” the basic quantities are defined, and the physical mechanisms are presented that provide a relation between the hemodynamic parameters and the DCE-MRI signal. Section 2 “Data acquisition” discusses the issues involved in the design of an optimal measurement protocol. Section 3 “Data analysis” summarizes the steps that need to be taken to determine the hemodynamic parameters from the measured data. © 2010 Elsevier Ireland Ltd. All rights reserved.

1. Introduction The term perfusion refers to the delivery of arterial blood to the capillary bed in biological tissue. The most common methods to characterize perfusion with MRI are arterial spin-labelling (ASL), or dynamic contrast-enhanced MRI (DCE-MRI) [1,2]. ASL uses the tissue water as an endogenous tracer. The magnetization of blood flowing into the tissue is manipulated by excitation pulses, and the changes in tissue magnetization caused by the deposit of labelled blood are measured. ASL has been applied extensively in the brain, but applications in the body are currently limited [3–5]. DCE-MRI uses an intravenous injection of a bolus of paramagnetic tracer, and a dynamic imaging technique which measures the signal changes induced by the tracer in the tissue as a function of time. First DCE-MRI experiments date back to the mid 1980s [6,7], but true perfusion weighting was only obtained when tracer injection and data sampling were both performed in a time scale of seconds [8,9]. Since this is the time scale of typical tracer transit times through the capillary bed, the tracer is essentially intravascular during the first pass of the bolus, providing pure perfusion weighting. DCE-MRI can be performed with T1 - or T2∗ -weighted imaging. For the remainder of this text we follow the convention to

∗ Tel.: +44 0 113 343 6063. E-mail address: [email protected]. 0720-048X/$ – see front matter © 2010 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.ejrad.2010.02.017

designate T2∗ -weighted techniques with DSC-MRI (dynamic susceptibility contrast-enhanced MRI) and reserve the acronym DCE-MRI for T1 -weighted techniques only. The observation that T2∗ -effects are significantly stronger for intravascular tracers [8] led to the wide-spread use of DSC-MRI for perfusion MRI [10–13]. However, the increasing list of quantification issues in DSC-MRI [14] has raised the interest in perfusion imaging with DCE-MRI in recent years. One major problem with DSC-MRI is the loss of T2∗ -weighting when the tracer extravasates in the interstitial space [15,16]. T1 -weighting is not affected by extravasation, so that additional information on vessel permeability can be derived from DCE-MRI by analysing the slow component of the signal [17,18]. Moreover, technological advances leading to shorter echo times and higher field strengths have improved the DCE-MRI data quality. In the brain, DCE-MRI signal change remains weaker than with DSC-MRI, but the difference is less relevant in more highly vascularized tissue such as tumors [19] or most abdominal organs. Currently, DSC-MRI remains the standard approach for perfusion MRI in the brain [20–22], but this position is increasingly challenged in recent years [23–27]. DCE-MRI remains the method of choice for permeability MRI, and has replaced DSC-MRI as the standard approach for perfusion MRI outside the brain [28–35]. In order to reduce the scope, the discussion in this paper is restricted to DCE-MRI. In Section 1 , the basic principles of the physical mechanisms involved in DCE-MRI signal generation are presented. Section 2 discusses the various issues involved in designing an optimal measurement protocol. Section 3 presents the steps

S. Sourbron / European Journal of Radiology 76 (2010) 304–313

that need to be taken to determine the hemodynamic parameters from the measured data. 2. Theory The main perfusion parameters are the tissue plasma flow (ml/100 g/min), which measures the volume of plasma (ml) flowing through the capillaries of a given amount of tissue (100 g) per unit of time (min); and the tissue plasma volume (ml/100 g), which measures the volume of plasma in the capillary bed of 100 g of tissue. Other authors use blood flow and -volume as target parameters [36], but all formulae can easily be translated between both conventions by scaling with the hematocrit. For the sake of transparency the plasma flow and -volume will be used consistently throughout this text. The main permeability parameters are the extravascular, extracellular volume (ml/100 g), and the permeability-surface area product PS (ml/100 g/min), which measures the volume of plasma (ml) flowing across the capillary wall of 100 g tissue per unit of time (min). In specific organs, hemodynamic parameters can be defined that do not apply in other tissues. Important examples are the arterial fraction of the plasma flow in the liver [37,31], or the glomerular filtration rate in the kidney [38,34]. In this section, the physical mechanisms are described that link those hemodynamic parameters to the measured DCE-MRI signal. This involves two different levels, each governed by entirely different physical principles: (i) tracer-kinetic theory relates the hemodynamic parameters to the concentration–time curves in the tissue; (ii) MRI signal theory relates those concentrations to changes in MR signals. 2.1. Tracer kinetics

The transit time of a tracer particle through the tissue is defined as the time elapsed between entering and leaving the tissue. Since typically different paths through the tissue are available to the tracer, the tissue is characterized by a distribution of transit times. It can be shown that the area under the residue function equals the mean transit time (MTT) of the tracer in the tissue [40]:



∞

MTT =

dt  R(t  )

(3)

0

Broadly speaking, MTT is in the order of seconds in the capillary bed, and in the order of minutes for the extravascular space. However, in particular tissue types the difference between both may be smaller [35]. The systemic mean transit time theorem [40] applied to an extracellular tracer, leads to a relation between the extracellular volume (ECV), the MTT of the tracer (3) and the flow of plasma through the inlet: ECV = FP · MTT

(4)

The ECV in this relation equals the plasma volume for an intravascular tracer, and the total extracellular volume (intra- and extravascular) otherwise. Compartment models. Tracer-kinetic models provide a parametric representation 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. More generally, a compartment is a space where the outflux Jo (t) (mol/min) of tracer through any arbitrary outlet o is directly proportional to the average concentration C(t) (mol/ml) inside the space [42]: Jo (t) = Fo C(t)

The precise form of the tracer concentration–time curves C(t) in the tissue are fully determined by the hemodynamic parameters, and the concentration CA (t) in the blood plasma of an arterial vessel feeding the tissue: the arterial input function (AIF). Tracerkinetic theory provides a relation between those quantities, and thus forms the basis for determining the tissue status from the measured concentrations C(t) and CA (t) [39–42]. Linear and stationary tissues. The theory of linear and stationary systems provides the general framework for any tracer-kinetic analysis. It is valid as soon as the response of the tissue to an injection of tracer at any given time is proportional to the injected dose (linearity) and independent on the time of injection (stationarity). For such tissues, C(t) and CA (t) are related by convolution with a residue function R(t):

305

(5)

The proportionality constant Fo (ml/min) is referred to as the clearance or transfer constant of the outlet o. In the particular situation where the tracer is carried through the outlet by convection, Fo is the flow through the outlet [40]. In a tissue that is modelled by n different interacting compartments, a set of n equations can be built by expressing conservation of tracer mass in each compartment, and using the definition of a compartment (5). The solution of such systems always produces a multi-exponential residue function [41]: FP R(t) =

n 

Fi e−t/Ti

(6)

i=1

Here FP is the tissue plasma flow, and R(t) is the fraction left in the tissue at time t of a dose injected at time t = 0. The residue function is a tissue characteristic that fully defines the kinetics of a particular tracer. It is always a positive, decreasing function which satisfies R(0) = 1. Eq. (1) is valid only if the AIF is measured directly at the inlet to the tissue. In practice this cannot be realized exactly, and CA (t) is usually measured at a more upstream location. The bolus then may shift or disperse during the transit to the tissue, a process which can be formalized by convolution with the distribution HA (t) of arterial transit times [43]:

This implies that at most 2n parameters are measurable with an n-compartment system, which means that additional constraints must be imposed when the number of free parameters in the model is larger. Unfortunately, analytical solutions for the parameters Fi , Ti are only available for n ≤ 3 [42]. Two-compartment models. In most tissue types, standard MR tracers distribute over two different spaces: the blood plasma P and the extravascular, extracellular space E. The two-compartment exchange model [44,45,27] is defined by the assumption that (i) P and E are compartments; (ii) E does not exchange tracer directly with the environment; (iii) the clearances for the outlets connecting P and E are equal; (iv) the clearance for the outlet of P to the environment equals the plasma flow. Applying tracer mass conservation and definition (5) to P and E leads to:

C = FP CA ⊗ HA ⊗ R

VP CP = PS (CE − CP ) + FP (CA − CP )

C = FP CA ⊗ R

(1)

(2)

By definition, HA (t) is a positive function with unit area. In the special case HA (t) = ı(t) (i.e. in the absence of delay or dispersion), Eq. (2) reduces to (1).

VE CE = PS (CP − CE )

(7)

The solution is a biexponential (6) with four parameters that are fully defined by the flows FP , PS and the volumes VP , VE [16].

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There are four different situations where a tissue with this structure effectively reduces to a one-compartment model: (i) and (ii) when one of the spaces has a negligible volume, (iii) if the tracer extravasates slowly, so that the concentration in the extravascular space is negligible within the acquisition time, and (iv) if the tracer extravasates rapidly, so the system behaves as a single wellmixed space. In each of these regimes, the residue function becomes mono-exponential, but the precise interpretation of the parameters is different [46]. Hence it is recommended to use an abstract notation K trans and kep for the model parameters of a one-compartment model: C(t) = K trans e−tkep ⊗ CA (t)

(8)

A one-compartment model with these notations is often referred to as a Tofts model [46], or as a modified Tofts model when a term VP CA (t) is added to account for the tracer in the vasculature [47]. If prior information regarding the state of the tissue is available, then K trans and kep can be interpreted in more concrete terms. However, to avoid errors of misinterpretation [48,49], it is prudent to leave the interpretation open [50,51]. Alternative models. More intricate tracer distributions can always be constructed by combining more compartments [52], though in more complex models not all parameters may be measurable. An alternative to the compartment models is the distributed-parameter models, which allow for a gradient in the concentration by fixing the flow patterns inside the tissue [53–56]. They produce the same parameters as the simpler compartment models, but it is currently unclear whether they lead to an improved accuracy [57]. Models such as the two-compartment exchange model (7) are generic, but particular organs may need particular solutions. A typical example is the liver, where tracer enters through arterial and venous inlets [37,58]. Another example is the kidney, where processes of filtration require a different model architecture than typical tissue types, and where reabsorption leads to interpretation issues in some of the parameters [59,60,52]. 2.2. DCE-MRI signal DCE-MRI signal theory provides a relation between the measured signal–time courses S(t), and the concentration C(t). Physically, this involves two issues. The relation between C and T1 is determined by the physical interactions between the tracer particles and the tissue. The relation between S and T1 is dependent on the details of the MR sequence used. Relaxation. T1 is shortened by the short-range interactions between the contrast agent particles and the proton spins. The details of the interactions may be quite complicated, but experiments show that the relaxation rate R1 = 1/T1 is linearly related to C [61] in the range 0–5 mM, which corresponds to the concentrations encountered in a standard DCE-MRI measurement [62]: R1 = R10 + r1 C

(9)

Here R10 is the precontrast relaxation rate. The proportionality constant r1 is the T1 -relaxivity of the tracer, defined as the change in relaxation rate per unit of concentration. Relaxivities in living tissue are difficult to determine by measurement, but in vitro results suggest that for standard tracers they are largely independent of tissue type [63]. Some tracers, however, are designed to have a certain amount of protein binding to enhance their relaxivity and/or modify their pharmacokinetics. As a result, their relaxivity is dependent on protein concentration. Since this differs between tissue compartments, the relaxivity of such protein-bound tracers is dependent on tissue type [64]. Signal. The precise relation between R1 and the signal S depends on the chosen measurement sequence, but virtually all existing

sequences used in DCE-MRI take the following general form: S = ˝ · c · M0 · e

−TE R∗

2

· sin ˛ · mz (R1 )

(10)

In DCE-MRI, all factors apart from mz (R1 ) are regarded as constant in time: the global calibration constant ˝, the coil sensitivity c, the equilibrium magnetization M0 , the flip angle ˛, and the exponential T2∗ -weighting. However, due to variations in coil sensitivity, B1 -fields or proton density, these constants may be dependent on voxel position. The (normalized) longitudinal magnetization mz is a function of R1 and the sequence parameters, but its precise analytical form is dependent on the design of the sequence. For a spoiled gradientecho sequence with repetition time TR , it has the following form in the steady state: mz (R1 ) =

1 − e−TR R1 1 − cos ˛ e−TR R1

(11)

Whatever the precise analytical form, mz is always an increasing function of R1 . Hence the T1 - and T2∗ -effects of the tracer are competing: at higher concentrations, shortening of T2∗ will reduce the signal, whereas shortening of T1 will increase the signal. The signal model acquires an additional dependence on the blood flow when measurements are performed in ROIs containing larger arteries with fast-flowing blood [65–67]. The effect comes about because blood flowing into the slice or slab has a different magnetization than the static spins, which have experienced all previous excitation pulses. The magnetization of the tissue then becomes a superposition of static and flowing contributions. Water exchange. An implicit assumption in ((9) and (11)) is that longitudinal relaxation is mono-exponential. This is accurate in unenhanced and weakly enhanced tissue types [68], because the exchange of water between tissue compartments (e.g. intra- and extracellular spaces) is rapid compared to the differences in R1 -values. However, when the tracer is compartmentalized, large R1 -differences arise between compartments that contain different concentrations of tracer. In that case, water exchange may no longer be sufficiently rapid to achieve the fast-exchange limit [69,70], and longitudinal relaxation becomes multi-exponential. During a typical DCE-MRI experiment, large concentration differences arise (i) between the blood plasma P and the extravascular, extracellular space E (during the first pass, when most of the tracer is in the blood), and (ii) between E and the intracellular space C (in the later phases, when the tracer has extravasated). Simulations and data have suggested that the signal may leave the fast-exchange regime in both these phases [71,70]. In that case, the longitudinal magnetization (10) becomes a function of the relaxation rates R1P , R1E , R1C of each space, and of the water-exchange rates FPE , FEC between them [72]: mz (R1 ) → mz (R1P , FPE , R1E , FEC , R1C )

(12)

The dependence on concentration can be found by applying (9) to R1P and R1E separately. It remains an issue of debate whether limited intraextravascular or cellular-interstitial water-exchange plays an important role in DCE-MRI. Older results [71,70] cannot necessarily be translated directly to more recent data, since the dependence of mz on water exchange rates (12) depends strongly on the measurement sequence, and on the sequence parameters [73]. Recent evidence suggests that, with appropriate sequence optimization, the effect is negligible [74]. 3. Data acquisition Setting up a DCE-MRI measurement involves the choice and optimization of an MRI sequence, the choice of a contrast agent and

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injection protocol, and of appropriate hardware. The optimal setup depends to some extent on the tissue under investigation, and the clinical constraints in terms of coverage and spatial resolution, but also on the objective of the measurement. 3.1. Imaging Sequence choice. Most sequences applied in DCE-MRI are either 2D or 3D gradient-echo sequences. Earliest applications [75–77,24,37,78] used a 2D sequence with a limited number of slices to achieve the high temporal resolutions required for perfusion MRI. However, since the introduction of parallel imaging and acceleration schemes for dynamic imaging [79], fast 3D acquisition has become feasible and is gradually replacing 2D approaches in current practice [53,80,58,81]. The essential difference between 2D and 3D methods is that 2D acquisitions always contain a non-selective inversion- or saturation prepulse. In combination with a low flip angle, this serves to minimize inflow effects in the larger vessels. In a 3D sequence, inflow effects are minimized by positioning the slab so that the major artery travels a large enough distance through the slab [53]. Inflowing spins then always reach a steady-state before they exit the slab, and an AIF can be acquired near the point where the vessel leaves the slab. When an AIF is not required, these inflow-issues are of less significance. For absolute quantification of DCE-MRI, a precontrast mapping of T1 is often performed (see below). In the 2D strategy this is usually achieved by varying the preparation delay [44,82], in 3D by varying the flip angle [53,35]. These calibration measurements are run immediately before the DCE-MRI sequence. An alternative measurement approach is the use of multi-echo sequences, which allow for a measurement of T1 at every individual time point by fitting the mono-exponential T2∗ decay [83–85]. They have the advantage that any T2∗ effects are fully eliminated, and produce an additional T2∗ measurement, allowing combined DCE-DSC-MRI [86]. Multi-echo sequences form a promising path for future development [16], but their use is currently limited. Sampling strategy. Since slower sampling allows for increased spatial resolution, coverage or SNR, it is important to identify the longest admissible sampling interval [81]. Unfortunately, there is very little consensus in the literature on this point. A necessary requirement is that all signal–time curves are adequately sampled, in particular the rapid signal changes in the AIF. Additionally, it appears plausible that the sampling interval must be shorter than the typical time scales of the processes to be measured [87–89]. For tissues where tracer transit times in the capillary bed are in the 3–5 s range, a sampling interval <2 s is usually recommended for the measurement of perfusion [90,91]. Since the transit times in the extravascular space are of the order of minutes, studies that target the permeability alone may use a lower temporal resolution [92]. Equivalently, it may be assumed that the total acquisition time should be longer than the time scale of the processes to be measured. For the measurement of perfusion, an acquisition of 1 min is usually sufficient [93,89], but permeability measurements require longer acquisition times. Most current studies use no more than 5 min, but significantly longer acquisitions may be required in slowly enhancing tissues [17,92,35]. Sequence optimization. In DCE-MRI, the echo time TE is chosen near its minimal value in order to minimize T2∗ -weighting, i.e. in the range 1–2 ms with current scanners. The repetition time TR is usually minimized as well in the interest of temporal resolution (2–4 ms). For a 2D sequence, the same holds for the delay between preparation pulse and centre of k-space, often called inversion time. The field of view depends on the body part under examination, but matrix size and number of slices can be maximized within the con-

307

straints set by the temporal resolution. Parallel imaging techniques to speed up the acquisition are increasingly common. The optimal choice for the flip angle reflects a compromise of competing effects [94]. Higher flip angles increase the dynamic range (the maximum concentration that is measureable), but reduce the sensitivity (the slope of the signal as a function of concentration). High flip angles also increase the specific absorption rate, which may impose a more fundamental limitation. In a 2D sequence with a preparation pulse, higher flip angles amplify the inflow effects in the AIF [78], but in a 3D sequence they minimize inflow effects. Most authors use a value in the intermediate range 30–60◦ [95], but a value of 90◦ has been recommended as well [94]. In prepared 2D sequences, either a low flip angle (< 15◦ ) must be used to minimize inflow effects; or the data must be triggered prospectively [94] or retrospectively [96] to diastole, where flow is minimal. The conventional measure SNR does not reflect the data quality in DCE-MRI, and should be replaced by a contrast-to-noise ratio (CNR) that compares the signal change S − S0 to the noise level. One possible definition for CNR is the ratio of maximum signal change to the baseline noise (standard deviation of the precontrast signal). For a given measurement protocol, CNR depends on tissue type, and the required CNR depends on other factors such as the temporal resolution. CNR is improved at higher field strength, so 3 T scanners are generally recommended over 1.5 T for DCE-MRI [97,98,26]. Motion minimization. A major issue in abdominal perfusion is the problem of breathing motion. On the acquisition level, there are essentially two possible strategies to minimize the effect: either the data are acquired in breath hold [99,80,100], or a form of triggering is applied to ensure that data are always acquired at the same point in the breathing cycle [89]. Both methods have distinctive limitations. Since a breath hold for the entire duration of the acquisition is not always feasible, multiple breath holds are often necessary. This is challenging from the perspective of practicality and patient comfort, runs the risk of mismatches in position between different breath holds, and creates gaps in the time curves which may affect quantification. The main problem with triggering approaches is that this may reduce the temporal resolution to below the requirements for an accurate perfusion measurement. An alternative strategy is to perform data acquisition in free breathing, and remove motion effects on the post-processing level if necessary [76,37,101,102]. 3.2. Injection protocol All standard MRI contrast agents have been used for the measurement of perfusion, but the use of protein-bound tracers must be avoided for permeability measurements. Due to the sensitivity of relaxivity (9) on protein concentration, errors arise in the permeability parameters when such a tracer leaks into a space with a different protein content [60,27]. In addition, since bound and unbound particles have different pharmacokinetics, a precise tracer-kinetic analysis can only be performed if the bound fraction is known [103]. Data acquisition should be started some time (10–20 s) before the bolus is injected, so that a sufficient amount of precontrast data is acquired. A typical injection protocol in DCE-MRI uses a standard dose (0.1 mmol/kg body weight for standard tracers) and injects the tracer in 5–10 s [87]. However, significantly lower doses are common, particularly in highly perfused tissue types [76,24,104]. The bolus is usually flushed with 20–30 ml of saline at the same injection rate. At peak concentrations beyond the dynamic range of the sequence, the signal reaches a saturation regime where it is no longer sensitive to changes in the concentration. Even with optimal sequence optimization, signal saturation cannot always be avoided

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in the larger arteries due to the high concentrations in the first pass. In that case, the injection protocol must be modified to reduce the peak concentration. This can be achieved by reducing the dose, extending the injection time, or both. However, in weakly perfused tissues, these procedures may reduce the CNR in the tissue to a point where it is no longer adequate. In order to reconcile these constraints, it has been proposed to reconstruct the arterial input function from a measurement with a small prebolus—typically one-tenth of a normal dose [105,99]. A limitation of this prebolus approach is that it requires extra postprocessing and twice the acquisition time, which is not desirable in clinical practice for permeability measurements of several minutes. A compromise might be to split up a full dose in two equal injections [106], which combines the advantage of a full dose with a reduction in peak concentration. 4. Data analysis Generally speaking, data can be analysed by visual assessment, descriptive parameters, or quantitative parameters [29]. A visual assessment of images at various time points is the most intuitive approach [9], but it does not produce a quantifiable index, it does not separate perfusion and permeability, and produces no information on the rate of tracer uptake or washout. Descriptive parameters are indices that characterize the shape and structure of the curves, such as the time to peak enhancement, bolus arrival time, maximum upslope, maximum downslope, area under the curve, or maximum enhancement [107–113]. Deriving descriptive parameters is straightforward, but the link to physiology is not always clear, and they are only reproducible when an identical measurement protocol is used. A quantitative analysis aims to directly measure physiological parameters such as tissue blood flow, blood volume, interstitial volume or permeability-surface area [114]. From a measurement perspective, the main complication for a quantification is the need to accurately measure the concentration in the lumen of a major feeding artery. Also, additional post-processing steps are required: MRI signal analysis to calculate or approximate the tracer concentrations from the MRI signals, and tracer-kinetic analysis to determine quantitative parameters from the concentrations. 4.1. Preprocessing ROI or voxel. Quantitative or descriptive parameters can be calculated on the level of voxels, or of regions of interest (ROI). For an ROI analysis, a region is outlined manually [37,115] or by some (semi)automatic segmentation procedure [116–118], and the signal–time curves of all voxels in the ROI are averaged to produce one single curve. The post-processing protocol is then applied to this curve. For a voxel-based analysis, a curve is extracted for each voxel. The post-processing protocol is applied to each voxelcurve individually [78,119], producing an image for each calculated parameter. The main advantage of a voxel-based analysis is that it produces information on the heterogeneity of perfusion and/or permeability within the organ or tissue [120–122]. Summary parameters (mean, standard deviation, . . .) for a lesion or an anatomical structure can always be derived by defining an ROI on one of the parametric maps [123,60]. A ROI analysis produces more accurate average values since the CNR of the signal–time curves is improved by the averaging over the ROI. A hybrid approach is to perform a voxel-based analysis first, define ROIs on the parametric maps, and repeat the analysis on ROI-level. In this case, it may be sufficient to perform a simple and robust analysis on the pixel level, and a complete quantification on the ROI level only [124].

Motion compensation. As an alternative to motion compensation approaches on the acquisition level (see above), or complementary to them, motion correction may be performed on the postprocessing level [125]. 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. The process is tedious and time-consuming, difficult to automatize [29], 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 non-rigid deformation of the image [126–131]. 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. 4.2. DCE-MRI signal analysis The aim of DCE-MRI signal analysis is to calculate or approximate the longitudinal relaxation rates R1 (t) from the measured signals S(t), then the concentrations C(t) from R1 (t). We restrict the discussion here to methods that assume fast water exchange (10). The major difficulty with incorporating limited water exchange effects is that it introduces unknown parameters in the model (12). Corrections have been proposed using approximations and experimental values for the water exchange rates [71]. But such values may be inaccurate, and recent evidence suggests that the assumption of fast water-exchange is accurate with current measurement sequences [74]. Relaxation rates. The signal Eq. (10) forms the basis for any method aiming to derive R1 (t) from S(t). A number of approaches can be found in the literature, with different trade-offs between accuracy and complexity. All methods assume that T2∗ -weighting is negligible. This is mostly justified by the use of typically small echo times (∼1 ms), but in the presence of highly concentrated tracer, such as in arterial blood or urine, a certain amount of T2∗ -shine through cannot be avoided. If this cannot be minimized by dose reduction or sequence optimization, it can only be fully eliminated by the use of a multiecho sequence. If T2∗ -changes are negligible, the amplitude of mz (R1 ) in (10) can be treated as a constant and eliminated by scaling out the baseline signal S0 . Solving for R1 leads to the basic formula for signal analysis in DCE-MRI: R1 = m−1 z



mz (R10 )

S S0



(13)

The formula shows that an additional measurement of precontrast relaxation rate R10 is required [44,132,80]. T1 -measurement does however require additional measurement time and postprocessing, and in abdominal organs might require an additional breath hold and/or co-registration step [80]. Two approximate strategies have been proposed that avoid the need for a precontrast T1 -measurement. For an ROI based analysis, a literature value for T1 may be used. This approach is often applied to determine the concentration in arterial blood, where accurate T1 -measurement in vivo is more difficult due to flow effects. However, T1 -values in tissues are variable and may change in pathology. An alternative is to use a receiver/transmit coil with a maximally uniform coil sensitivity, so that the amplitude of mz (R1 ) in (10) is independent of voxel position. It can then be determined by a ref-

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erence measurement in a tissue type with a known T1 such as fat [76], or in an item with a known T1 placed in the field of view during imaging [78]. A second potential complication with (13) is the dependence of mz on the flip angle, since the exact value may be unknown due to B1 -inhomogeneities or imperfect slice profiles. One possible solution is to insert a second precontrast calibration sequence to measure the flip angle [133,134]. An alternative solution is to impose the additional assumption that mz is proportional to R1 . In that case, the value of the flip angle no longer plays a role in (13): R1 = R10

S S0

(14)

The linearity assumption is valid at small enough concentrations, and its validity may be improved by sequence optimization [94]. However, it is often violated in blood and highly vascularized tissues, where concentrations may enter the non-linear regime during the first pass [135]. Concentrations. Eq. (9) shows that the tracer concentration can be derived directly from the change in relaxation rate induced by the tracer: C=

R1 − R10 r1

(15)

Absolute values can be quantified if the relaxivity r1 is known. However, if the relaxivity is independent of tissue type [63], it scales out in a quantitative analysis (1). In that case the outcome of the measurement is independent of the value chosen for r1 . For a quantitative analysis, an accurate measurement of the concentration in the blood of an arterial feeder is required (1). If the AIF is measured far from the tissue, the transit time distribution of the arteries contaminates the impulse response (2) and causes significant dispersion errors that are difficult to correct [43,136]. Hence dispersion errors should be minimized by measuring the AIF close to the tissue of interest. If only small arteries are available, partial-volume errors may arise by contributions of vessel wall or surrounding tissue [137]. However, in contrast to dispersion errors, they can be corrected in a straightforward manner by a reference measurement in a large vein or artery [16,137]. An alternative strategy involves a separate measurement of cardiac output [138]. Since MR tracers are extracellular, the AIF is defined in (1) as the concentration in the plasma of the artery. Since a measurement will only produce the values for whole blood, the measured curve must be divided by a factor (1 − H), where H is the patients’ hematocrit. If the value is not known from laboratory data, a standard value of H = 0.45 is often used. 4.3. Tracer-kinetic analysis The second step in a quantitative data analysis is to apply the principles from tracer-kinetic theory (Section 2.1) to derive the hemodynamic parameters from the concentration–time curves. We distinguish between (i) direct methods, which produce some of these parameters without explicitly determining the residue function (1), and (ii) deconvolution methods, which calculate the full residue function from (1). The group of deconvolution methods can be classified into model-free, parametric, and model-based methods. Direct methods. Integration of (1) and using (3) and (4) produces a useful formula to calculate the extracellular volume: ECV =

∞   dt C(t )  0∞ 0

dt  CA (t  )

(16)

A disadvantage of such relations is that concentration–time curves must return to zero within the acquisition window to allow calcu-

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lation of the areas [39]. In DCE-MRI this not the case, since half-lifes of tracers in the blood are significantly longer. Hence such methods are usually accompanied by a preprocessing step that extracts the first pass of the bolus [36]. However, this is difficult to justify in tissue types with rapid tracer uptake, where the first pass is not clearly differentiated. A direct method for calculating plasma flow is obtained by considering only the time points shortly after the bolus arrival in the tissue. If the time since arrival is sufficiently short, no tracer has yet left the tissue, so that R = 1 in (1):



C(t) = FP

t

dt  CA (t  )

(17)

0

FP can be determined directly from this equation by linear fitting [139], or from the maximum of the derivative of both sides [76]: FP =

max(C  ) max(CA )

(18)

The limitation of this approach is that the relations are – at most – valid only in the first few seconds after bolus arrival. Hence only very few data points can be used, and the slope (18) may not maximize in this range. Model-free deconvolution. Model-free deconvolution methods do not impose any constraints on the form of the residue function or the structure of the tissue [140,141,78]. They produce a measurement of the impulse response FP R(t) directly from the data CA (t) and C(t) (1). Since R(0) = 1 the plasma flow FP can then be found as the initial value FP R(0), or, since R(t) is decreasing, as the maximum of FP R(t). In practice the latter approach is preferred, since it remains exact when the AIF is shifted [142], and is approximately valid in the presence of a small amount of bolus dispersion (2). The extracellular volume ECV is found by integration of FP R(t) (4) and (3), a method which is insensitive to dispersion effects since HA (t) has unit area (2). Finally, the MTT of the tracer can be found from the ratio of ECV and FP (4). The first model-free deconvolution methods proposed in DCEMRI were based on the Fourier transform [36], but since a calculation of the Fourier transform involves integration, they suffer from the same limitation as (16). Most current methods are based on a discretization of (1) which rewrites the formula as a matrix equation [140,142]: C = t FP CA R

(19)

A naive solution of (19) by multiplying both sides with the matrix inverse C−1 A produces unphysical solutions with high-frequency components of unlimited amplitude. A physical solution can be obtained by the method of truncated singular value decomposition [140,143]. Briefly, all singular values of CA below a certain threshold  are set to zero, and (19) is solved in the least-squares sense by multiplication with the pseudo-inverse of the truncated matrix CA (). The appearance of large and unphysical high-frequency components is a common feature of all model-free methods, and results from the inherent ill-posedness of the deconvolution problem [143]. Truncating the singular values is an example of a regularization method, i.e. an approach to resolve the ill-posedness. The cutoff value  here is the regularization parameter, which determines exactly how much regularization is applied. Since calculated values for the plasma flow depend very strongly on the precise value of the  [144], and the optimal value depends on the properties of the data, a regularization method must always be complemented with an unbiased criterion to select a value for the regularization parameter [145,146]. Parametric deconvolution. Parametric methods represent the next level of generality in the sense that they do not make any

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explicit physiological assumptions, but they do assume that the residue function has some known analytical form [115,147]. The function depends on a number of free parameters, the value of which is determined by minimizing the difference between both sides of (1). One example of a parametrization is the Fermi model, which has been used extensively for analysis of DCE-MRI data in the heart [75]: R(t) =

1+ˇ 1 + ˇ e˛t

(20)

Alternative parameterizations proposed are gamma-variate functions [148] or polynomial representations [12,26]. Compared to model-free methods, parametric methods may improve the accuracy in plasma flow estimates when the residue function has the required shape—particularly when the data quality is poor. On the other hand, a systematic error may arise when the actual residue function has a different functional structure. Parametric methods usually produce a number of additional parameters apart from blood flow and MTT (e.g. ˛ and ˇ in the Fermi model), but these cannot be interpreted due to the lack of an underlying physiological model. Model-based deconvolution. Tracer-kinetic models provide a well-defined relation between a parametric representation of the residue function (6) and physiological parameters. Model-based approaches therefore offer the possibility to measure other additional hemodynamic parameters such as the volume of different tissue compartments, or the exchange rates between them (7). A typical problem is that multiple models may be appropriate for a given tissue type. Since the results depend on the chosen model [149–151], it is important that the most appropriate model is selected. For particular applications, prior knowledge and experience may be available to eliminate all but one of the possible models. In general, however, the natural variability in the tissue types of interest, and in the quality of the data, may be too large to reduce the number of possible models to one. A general rule of thumb for model selection is that the most appropriate model is the simplest that provides a good fit to the data. For instance, the two-compartment exchange model (7) has more free parameters than the Tofts model (8) and will therefore always provide the better fit. However, if the tissue is in one of the boundary regimes discussed, the full biexponential structure of the residue function cannot be resolved from the data, and the simpler mono-exponential Tofts model must be used. The twocompartment exchange model would in that case produce arbitrary values for the two redundant parameters. From a practical perspective, therefore, model selection is reduced to assessing whether additional free parameters significantly improve the fit to the data. For an analysis on ROI level, the accuracy of the fit can be evaluated by an expert observer. Apart from the practical limitations of requiring expert intervention, this introduces a level of subjectivity in the results. Automatic methods for model selection have been proposed [152], the most popular being the F-test and the Akaike criterion [153,154]. However, model selection remains inherently ambiguous when two models provide a nearly equal fit. In this sense it is useful to observe that the choice of a model is not only determined by the state of the tissue, but also by the quality of the data [87,88,155]. In particular, the injection protocol, temporal resolution, acquisition time and noise level play an important role. For instance, if the injection rate is too slow, intra- and extravascular spaces are in constant equilibrium, and the mono-exponential Tofts model must be applied (8). Conversely, a more rapid injection rate may create strong intra- and extravascular concentration differences in the first pass, so that only a full biexponential model (7) provides a good fit. This implies that ambi-

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