Diffusion and perfusion MRI: basic physics

European Journal of Radiology 38 (2001) 19 – 27 www.elsevier.nl/locate/ejrad

Diffusion and perfusion MRI: basic physics R. Luypaert *, S. Boujraf, S. Sourbron, M. Osteaux MR-Centre, Academisch Ziekenhuis, Vrije Uni6ersiteı´t Brussels, Laarbeeklaan 101, 1090 Brussels, Belgium Received 3 January 2001; received in revised form 9 January 2001; accepted 10 January 2001

Abstract Diffusion and perfusion MR imaging are now being used increasingly in neuro-vascular clinical applications. While diffusion weighted magnetic resonance imaging exploits the translational mobility of water molecules to obtain information on the microscopic behaviour of the tissues (presence of macromolecules, presence and permeability of membranes, equilibrium intracellular–extracellular water,…), perfusion weighted imaging makes use of endogenous and exogenous tracers for monitoring their hemodynamic status. The combination of both techniques is extremely promising for the early detection and assessment of stroke, for tumor characterisation and for the evaluation of neurodegenerative diseases. This article provides a brief review of the basic physics principles underlying the methodologies followed. © 2001 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Diffusion MRI; Perfusion MRI; Magnetic resonance imaging; Apparent diffusion coefficient; Hemodynamics

1. Introduction Over the past 20 years, MR imaging has become a powerful tool for the evaluation of the anatomic characteristics of various organs. More recently, a number of techniques have been introduced that allow additional evaluation of functional parameters. Diffusion and perfusion imaging are typical examples that have gained considerable clinical acceptance in neuro-vascular imaging. MR diffusion weighted imaging (DWI) uses the signal loss associated with the random thermal motion of water molecules in the presence of magnetic field gradients to derive a parameter (the so-called apparent diffusion coefficient) that directly reflects the translational mobility of the water molecules in the tissues. Applications of this technique in the context of neuro-vascular imaging include the early detection and assessment of stroke, tumor characterisation, evaluation of multiple sclerosis. MR perfusion weighted imaging (PWI) refers to methods that make use of the effect of endogenous or exogenous tracers on the MR images for deriving various hemodynamic quantities such as cerebral blood volume, cerebral blood flow and mean transit time. Potential applications include the identification of tissue * Corresponding author. Fax: +32-2-4775362.

at risk after acute stroke, assessment of tumors, evaluation of neurodegenerative conditions. Although DWI and PWI on their own can answer a number of questions, the information they provide is to a large extent complementary and their combination, for instance in application to stroke, appears to be extremely promising. The purpose of this article is to provide a basic understanding of the methodology underlying both imaging techniques and their application in the neuro-vascular clinical environment. 2. Diffusion weighted imaging

2.1. Background While the signal attenuation caused by molecular diffusion in the presence of magnetic field gradients was recognized in MR spectroscopy as early as 1954 by Carr and Purcell [1], the pulsed gradient technique developed by Stejskal and Tanner in 1965 [2] forms the basis of today’s diffusion weighted imaging methods. Interest in the potential medical usefulness of the technique was further stimulated by the 1990 discovery by Moseley and co-workers that the apparent diffusion coefficient of cat brain decreased by up to 50% within 30 min after the onset of focal ischemia, while the conventional MR images remained normal [3].

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& & TE



2

t

Molecular diffusion is the result of brownian motion, the constant random walk of the individual molecules in a fluid due to thermal agitation. Although the mean displacement of the molecules remains zero, as time goes by, there is a non-zero probability of finding an individual molecule at a distance from its point of origin. In fact, the root-mean-square displacement can be shown to increase in proportion to the square root of time, the constant of proportionality being a diffusion constant D characterising the fluid studied. At 25°, for instance, the diffusion coefficient of pure water is about 2.2×10 − 3 mm2/s. Soft tissues tend to behave like aqueous protein solutions and, due to the reduced mobility of the water molecules, the corresponding diffusion coefficient is generally smaller than that of pure water. In many tissues, boundaries with various degrees of permeability hinder the free diffusion of water, further decreasing the diffusion coefficient. Applying the brownian motion model in these circumstances leads to an ‘apparent diffusion coefficient’ or ADC, to be distinguished from the diffusion coefficient of free water molecules. In tissues like white brain matter, an additional complication arises from the fact that molecular mobility is not the same in all directions, i.e. the diffusion process is anisotropic and the scalar diffusion coefficient must be replaced by a tensor quantity [4,5]. Diffusion imaging thus provides a window on the microscopic structures and processes (presence and permeability of membranes, equilibrium intracellular extracellular water, …) inside the tissues as reflected by the motion of the water molecules.

b= k 2

2.2. Imaging diffusion

where i and j can be any of the three spatial directions x, y, z in an orthogonal frame of reference. The bij factors characterize the sensitizing gradients along the i and j directions [6]:

2.2.1. Scalar diffusion model In an isotropic environment molecular mobility can be described by a scalar diffusion coefficient, reflecting the fact that the brownian motion is similar in all spatial directions. Description of the effect of isotropic diffusion on the spin echo signal is relatively simple in this case. In the absence of magnetic field gradients, the signal is unaffected by the presence of incoherent motion. As soon as field gradients are switched on during any stage of the signal preparation, the motion leads to spin dephasing that, due to the random nature of the successive trajectories of each individual molecule, cannot be undone. The result is an exponential attenuation of the original signal S0(N(H), T1, T2) obtained in the absence of field gradients: S =S0(N(H), T1, T2) e − bD,

(1)

where D is the (apparent) diffusion coefficient of the medium and b is a scalar reflecting the properties of the gradient G(t) that was present during the experiment [2,5]:

0

G(t%) dt%

dt.

(2)

0

In this expression, G(t%) is replaced by − G(t%) for gradients switched on after the 180° pulse at t=TE/2. For a constant linear gradient of strength G [1], b= k 2G 2TE3/12,

(3)

while for sensitization using two identical rectangular pulses (duration l, spacing Z) placed on either side of the 180° pulse [2]: b= k 2G 2l 2(Z −l/3).

(4)

Both schemes lead to complete rephasing of static spins. The second arrangement (due to Stejskal and Tanner) is now routinely employed for quantitative diffusion work. It has the advantage that no strong gradient needs to be present during the echo sampling leading to improved signal-to-noise ratios. Using several (at least two) diffusion weighted images obtained for different b-values, the local ADC values can be calculated by fitting the signal values to Eq. (1).

2.2.2. Tensor diffusion model As a consequence of their morphology, many tissues exhibit anisotropic diffusion behavior: the ADC values measured using the Stejskal –Tanner sequence depend on the direction of the sensitizing gradient. For an anisotropic diffusion process, Eq. (1) must be replaced by a more complicated one: S=S0(N(H), T1, T2) e − % bij Dij,

bij = k 2

& & TE

0

t

0

&

t

Gi (t%) dt%

(5)



Gj (t%) dt% dt,

(6)

0

while the Dij are elements of the apparent diffusion tensor. This tensor is symmetrical and contains only six independent elements, the determination of which needs the acquisition of images with at least two different diffusion weightings for each of at least six independent directions of the sensitizing gradient. The information in the diffusion tensor may be conceptualized using the ‘diffusion ellipsoid’ picture: the portion of space within which we can expect a molecule to end up due to its aleatory motion expands around the point of origin as time goes by and, in general, has the shape of a flattened cigar, reflecting anisotropic mobility [7].

2.3. DWI in neuro-6ascular imaging Neuro-vascular applications are in general not interested in the anisotropy aspects of the diffusion pro-

R. Luypaert et al. / European Journal of Radiology 38 (2001) 19–27

cesses. However, as some important structures (e.g. the white matter) exhibit anisotropic diffusion, the results obtained with a scalar Stejskal – Tanner approach can be expected to depend on the orientation of the sensitizing gradient. A way to overcome this problem is to repeat the scalar diffusion measurement for three independent sensitizing directions x, y, z and calculate the mean of the resulting ADC’s: D=(Dxx + Dyy + Dzz )/3.

(7)

As this average is proportional to the trace of the diffusion tensor, it is independent of the choice of the sensitizing directions. The same information is of course available after any measurement following the full tensor approach. A further consideration for chosing a protocol for DW imaging is the role played by motion artifacts. As in diffusion weighted imaging the signals have been sensitized to microscopic motion of the water molecules, other sources of motion (e.g. macroscopic patient motion) may lead to large phase errors and corresponding artifacts in the images. One solution consists in measuring and correcting the phase errors in the raw data before image calculation (the navigator echo technique [8]). It allows the use of conventional spin echo sequences, yielding high signal-to-noise ratios. Unfortunately, the scheme does not work for all types of motion to be expected in the clinical setting. An alternative but technically more demanding solution is to use rapid imaging in order to minimize potential macroscopic motion during image acquisition. The sequence of choice has become single shot echo planar imaging, with a Stejskal – Tanner preparation part (Fig. 1). Typical results obtained on a standard clinical imager (here a Siemens Magnetom Vision) are shown in Fig. 2. The diffusion weighted images in Fig. 2a (images with sensitizing gradient along the x, y and z axes of the scanner, respectively, being shown in the consecutive rows) were part of the results obtained by a full tensor acquisition using EPI with following sequence settings: TR = 800 ms, TE= 123 ms, slice thickness 6 mm, FOV 240×240 mm, matrix 128× 128, five slices and b-values of: (a) 0 s/mm2; (b) 300 s/mm2; (c) 1200 s/mm2 for sensitizing gradients along six directions. The acquisition time was 3 minutes for five measurements of each image. As expected, on the diffusion weighted images the tissues with very mobile water (e.g. cerebro – spinal fluid) are dark while structures with reduced mobility (e.g. white matter tracts perpendicular to the sensitizing gradient) are bright (weak dephasing and less signal attenuation). Fig. 2b shows the ADC maps Dxx, Dyy and Dzz obtained on the basis of these diffusion weighted images using pixel-by-pixel fitting to Eq. (1). In the ADC maps tissues with mobile water are bright (high ADC),

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while structures with reduced mobility are dark (low ADC). One of the advantages of these calculated maps is that they contain pure diffusion information: the so-called ‘T2 shine-through’ effect, due to the presence of T2 contrast in the diffusion weighted signals (Eq. (1)), is completely absent. Note that structures with anisotropic diffusion show marked variation in brightness from one map to the other, reflecting the fact that water diffusion is stronger along the nerve fibers than perpendicular to them. Finally, Fig. 2b also presents the trace image calculated following Eq. (7) and illustrating that in this image all directional aspects of the diffusion process are averaged out.

3. Perfusion weighted imaging

3.1. Background One of the early approaches to perfusion weighted MRI was proposed by Le Bihan [9]. Comparing the aleatory nature of the motion of blood through the randomly oriented capillaries to that of brownian motion, this approach tried with some success to use the principles of diffusion weighted MR for estimating blood flow. This ‘intravoxel incoherent motion’ (IVIM) technique has more recently been replaced by methods relying on magnetic susceptibility and inflow effects. Susceptibility PWI is based on the passage of intravascular tracers like Gd-DTPA through the capillaries, producing a transient signal loss due to

Fig. 1. Schematic overview of diffusion weighted MRI methodology.

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Fig. 2. Typical images obtained for a healthy volunteer using DW-EPI: (a) Diffusion weighted images corresponding to three different diffusion weighting gradients (b-factors indicated expressed in s/mm2) along each of the three spatial axes x (1st row), y (2nd row), z (3rd row); (b) ADC maps for sensitization along x, y and z, respectively, and the corresponding mean diffusion (trace) map.

susceptiblity effects and allowing first-pass kinetics of the agent to be applied [10]. PWI with arterial spin tagging uses the blood itself, with suitably prepared magnetization, as an endogenous tracer [11]. With a number of hypotheses and limitations to be discussed later, perfusion imaging allows the estimation of several important hemodynamic parameters: cerebral blood volume (CBV), defined as the fraction of the total tissue volume within a voxel occupied by blood; cerebral blood flow (CBF) or perfusion, defined as the volume of arterial blood delivered to the tissue per minute per tissue volume and the mean transit time (MTT), which corresponds to the average time it takes a tracer molecule to pass through the tissue studied. The complementary role played by PWI next to DWI is well illustrated by the application of both techniques

to acute stroke. There are strong indications that an area of decreased CBV, decreased CBF and increased MTT corresponds to both the infarct core and the surrounding reversible ischemic tissue, while an area of decreased ADC is limited to the irreversibly ischemic core [12].

3.2. Imaging perfusion 3.2.1. Tracer kinetics In MR perfusion imaging, the models introduced in the past for nuclear medicine studies [13] remain useful, except that now the tracer is either an intravascular contrast agent or a volume of blood that has been tagged using RF excitation. Excellent reviews of the results of kinetic theory that are of interest for MR

R. Luypaert et al. / European Journal of Radiology 38 (2001) 19–27

have been presented in the past [14,15]. The reader is referred to those texts for more detailed discussions of the assumptions needed for the conclusions to be valid. Before summarizing these results, we define two additional quantities describing the interaction between blood, tracer and tissue. The blood – tissue partition coefficient p of a tracer expresses the equilibrium distribution of tracer between blood and tissue. For an intravascular tracer, it equals CBV while for a freely diffusable tracer it is about 1. The residue function R(t) describes the probability that a molecule of tracer, that entered a voxel at t= 0, is still inside that voxel at a later time t. It depends on the transport of the tracer between blood and tissue and the subsequent clearance from the tissue volume.

3.2.2. General assumptions Following assumptions are tacidly made in any analysis applying tracer kinetics: 1. The perfusion must be constant and unaffected by the tracer. 2. The tracer must be thoroughly mixed with the blood. 3. The concentration of the tracer can be monitored accurately. 4. Recirculation of the tracer can be corrected for when present.

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3.2.3. Useful relationships from kinetic theory Four important expressions provide the foundation for most of the perfusion applications met in clinical MR: CT(t)= CBF MTT =

&

&

t

Ca(t%) R(t− t%) dt%,

(8)

0

R(t) dt,

(9)

0

&

CBF = p/MTT,

CT(t) dt= p

0

&

(10)

Ca(t) dt.

(11)

0

The first expression states that the local tissue concentration at time t equals the sum of all amounts of tracer that entered the voxel at some previous time, weighted by the probability that these amounts are still there at time t, taking into account imperfect bolus administration through the arterial tracer concentration Ca(t). Note that the fact that R(0)= 1 and a perfect (i.e. instantaneous) bolus imply: CT(0)/CBF.

(12)

Eq. (9) simply puts the definition of MTT under mathematical form. [10] is the well-known central volume principle. It follows from Eq. (8) for Ca(t) held constant (equilibrium conditions). Finally, Eq. (11), which can be derived from Eq. (8) using the properties of convolution integrals, provides a link between the

Fig. 2. (Continued)

R. Luypaert et al. / European Journal of Radiology 38 (2001) 19–27

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area under the arterial and tissue concentration curves and the partition coefficient.

3.3. Application to microspheres For tracers consisting of particles that remain stuck in the capillaries after administration, R(t) = 1 and from Eq. (8) we see that after the bolus has reached the tissue, CT(t)=CT = CBF

&



Ca(t) dt.

(13)

0

The perfusion can be calculated on the basis of local tissue concentration and the area under the arterial concentration curve. The calculation is robust and provides the gold standard for CBF measurements.

3.4. Application to intra6ascular tracers Tracers that remain in the blood can yield model independent information on CBV on the basis of Eq. (11) and the knowledge that for an intravascular tracer and intact blood –brain barrier p = CBV:

CBV =

& &



CT(t) dt

0

.

(14)

Ca(t) dt

0

When the arterial concentration time curve Ca(t) cannot be measured, relative CBV values can still be calculated, assuming all the capillaries in the region of interest are fed by the same artery. CT(t) and Ca(t) can in principle also lead to estimates of CBF using Eq. (8), but in this case sufficient knowledge about R(t) is necessary. In addition, for MR imaging of an intravascular tracer, there is no simple relationship between the MTT, as defined in Eq. (9), and the first moment of the tissue concentration time course:

& &



MTT "

Fig. 3. Schematic overview of perfusion weighted MRI methodology for intravascular tracer studies.

t CT(t) dt

0

,



(15)

CT(t) dt

0

which means that only Eq. (9) and a detailed knowledge of R(t) can lead to direct estimates of MTT. Most frequently, detailed knowledge of the tissue vasculature and, consequently, R(t) is not available. However, based on mathematical models and computer simulations, Weisskoff et al. [16] have indicated that, even in this case, useful semi-quantitative relative values for the MTT and CBF may still be derived from the tissue concentration data, provided microvascular topology (hence R(t)) is reasonably constant in the region of interest (at most variations in the number of

perfused capillaries and their diameters, with only moderate variation in the capillary length distribution). If this is the case, the first moment of CT(t) is expected to behave approximatively like the MTT multiplied by a constant factor common for all pixels, and a relative value of CBF may be derived from the relative CBV and this relative MTT using Eq. (10). If this assumption does not apply, large systematic errors can arise when comparing regions of interest with different residue functions [17]. The use of non-parametric deconvolution techniques in combination with Eq. (8) has been advocated as an alternative solution for the failing knowledge about R(t). In how far it can be applied in the clinical situation remains unclear, although recently published data on animal models seems promising [18].

3.4.1. Dynamic Gd-DTPA perfusion imaging In order to apply these results to MR, we need first to be able to derive tissue concentrations from the signal intensities measured. Villringer et al. [19] have experimentally established that the susceptibility related changes in signal intensity introduced by a bolus of Gd-DTPA have the form: S=S0 exp(− TE DR2),

(16)

where S0 is the signal obtained in the absence of contrast agent, TE is the echo time of the sequence used and DR2 is the change in relaxation rate R2 = 1/T2 due to the agent. The basic mechanism underlying these

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changes is illustrated in Fig. 3. Due to the large difference in susceptibility between the capillaries containing the paramagnetic Gd-DTPA and the surrounding tissues, strong field gradients exist in the neighbourhood of the vessel walls, leading to direct signal dephasing in gradient echo images and diffusion mediated dephasing in spin echo images. Experimental data [20] show that for the concentration range expected in clinical applications, DR2 =k2CT.

(17) The proportionality constant k2 can be expected to depend on the particular tissue, field strength and pulse sequence. Spin echo based PWI shows reduced appearance of large vessels and may therefore be more representative of capillary perfusion, while gradient echo based techniques exhibit higher contrast-to-noise ratio [21] and are usually preferred in the clinic for that reason. Assuming both relationships Eq. (16) and Eq. (17) to be valid, the tissue concentration during bolus transit can be monitored using following expression: ln(S/S0) . (18) CT = k2 TE

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A bolus of Gd-DTPA administered as a short venous injection of a few seconds duration will have a width of up to 10 s by the time it reaches the brain, creating a signal dip of about 15 s or longer. Adequate coverage of the whole brain with T2* weighted images at a time resolution of B 2 s needs rapid imaging sequences like EPI. In Fig. 4a results for an acute stroke patient are shown obtained on a Philips Gyroscan Intera using a GE-EPI sequence with following settings: TR= 650 ms, TE = 30 ms, flip angle 30°, slice thickness 7 mm, FOV 230× 230 mm, matrix 128×128, 11 slices, one acquisition, leading to a total acquisition time of 1 min and 15 s and a time resolution of 1.9 s. The figure shows one time point out of every four for one slice, covering the whole time course of 32 images and clearly illustrating the transient signal drop caused by the susceptibility differences introduced by the tracer. Typical data processing makes use of Eq. (18) to determine CT(t) on a pixel-by-pixel basis. In practice, k2 is not usually known and assumed to be the same for all tissues of interest, which means that the concentration values will only be relative. A typical problem that needs to be solved when doing these calculations is that

Fig. 4. Typical results obtained using T2* weighted EPI on a patient with acute stroke: (a) First pass of a Gd-DTPA bolus: one time point out of each four is shown for a single slice, illustrating the transient signal drop due to the susceptibility effect introduced by the tracer; (b) Corresponding relative CBF, relative CBV and relative MTT maps (color scale from 0 (blue) to 100% (red)).

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commonly their magnetisation is inverted) at the level of the large feeding vessels. The resulting image reflects how, after a delay, these protons reach the capillaries in the slice of interest and diffuse in the tissue water space. The second image is obtained without inversion. In ideal conditions, the difference signal is proportional to the amount of blood delivered to the slice during the delay period and therefore should reflect perfusion. Several variations of this basic scheme have been investigated: in the oldest method due to Williams et al. [22] images with and without continuous adiabatic inversion in the neck are subtracted, Kwong et al. [23] introduced subtraction of two inversion recovery images obtained with a slice-selective and a non-selective 180° pulse, respectively, and Edelman et al. [24] put forward the EPISTAR technique in which images with and without an inversion pulse below the slice of interest are subtracted. A general kinetic model for analysing the quantitative application of these and related techniques [25] leads to the following expression for the difference in longitudinal magnetisation in the tissue due to labeled blood: Fig. 5. Schematic overview of perfusion weighted MRI methodology for endogenous tracer studies.

DM(t)= 2M0bCBF

&

t

Ca(t%) R(t− t%) m(t−t%) dt%,

0

of recirculation. In order to obtain the true first bolus passage concentration time curve, we need to eliminate any contribution from tracer re-entering the volume of interest. The standard approach [14] is to fit the data curve (with recirculation cut-off) using a gamma variate function of the form: C(t) = K(t− t0)r e − (t − t0)/b.

(19)

K, r and b are fit parameters and t0 is the time at which the tracer first appears in the data. In most clinical settings, no Ca(t) curve is obtained and only relative values of CBV are calculated using Eq. (14). For situations with relatively uniform microvascular topology, based on the conclusions of Weisskoff [16], relative MTT values are derived from the tissue concentration curve and combined with the relative CBV data using Eq. (10) to yield a relative CBF map. This approach was followed for the data in Fig. 4b, which show a strong reduction of the relative values for CBV and CBF and prolongation of the relative MTT in a region of the left brain, caused by the presence of acute stroke due to thrombosis of the arteria cerebri media.

3.4.2. Perfusion imaging using arterial tagging A second family of perfusion techniques makes use of RF labeled water in the blood as endogenous, freely diffusable tracer. In all these techniques, subtraction of two images isolates the signal of inflowing arterial blood in the slice of interest (Fig. 5). Typically, for one image the water protons in the blood are tagged (most

(20) indicating that the magnetization difference due to tagging is proportional to the equilibrium magnetization of the blood and to blood flow. The integral shows that in addition to the normalized arterial concentration Ca(t) of magnetization arriving in a voxel at time t, the residue function R(t) of tagged water molecules and their clearance m(t) due to (mainly) relaxation will in general affect the result obtained. The basic features of these methods can be understood using following simplified expressions obtained by just considering the amount of blood entering the voxel and the magnetization carried by that blood, neglecting the difference in T1 between blood and tissue [15]: DM = 2M0b CBF t e − t/T1 (pulsed tagging at t=0), (21) DM = 2M0b CBF T1 (continuous tagging).

(22)

In each case, the measured signal difference, which is proportional to DM, is proportional to CBF, the equilibrium magnetization of blood M0b and a factor with the dimension of time. Typical values for CBF, t and T1 lead to estimated magnetization differences of about 2% of the equilibrium magnetization, stressing the importance of sufficient SNR in these methods. Another complication that affects some of the spin tagging methods is magnetisation transfer. When the water spins are labeled in a slab preceding the slice of interest, the spins in that slice undergo off-resonance excitation that selectively saturates the broad resonance peak of

R. Luypaert et al. / European Journal of Radiology 38 (2001) 19–27

macromolecule-bound protons. As this saturation gets transferred to the free protons, it results in a severe loss of brain signal (up to 60%). This effect can be compensated by using RF excitation of a symmetrical slab following the slice of interest during the baseline acquisition [24]. Although a short discussion of inflow perfusion methods has been included here for completeness, their clinical application has been limited in comparison to that of intravascular tracer methods, mainly due to their sensitivity to patient motion and the insufficient signal-to-noise ratios often met in practice, especially in low flow conditions.

4. Conclusion Diffusion and perfusion weighted MR imaging are rapidly gaining acceptance as clinical tools. DWI makes use of the signal attenuation due to the random motion of water molecules in a strong gradient and allows insight in the microscopic behaviour of the tissues as reflected by the mobility of the water molecules (e.g. in the changes in the intracellular vs extracellular water balance in acute stroke). PWI enables assessment of regional cerebral hemodynamics using a variety of methods, among which the first pass endovascular bolus studies are presently the most common in the clinic. Typically, they lead to relative values for CBV and, for situations where the vascular topology can be assumed to be sufficiently constant in the whole region of interest, to semi-quantitative relative values for MTT and CBF. Both imaging techniques provide complementary information that is expected to be of prime importance for the diagnosis and treatment of cerebrovascular disease, tumors and other disorders.

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