Experiments in Fluids

, Volume 43, Issue 6, pp 823–858

Magnetic resonance velocimetry: applications of magnetic resonance imaging in the measurement of fluid motion

Authors

    • Mechanical Engineering DepartmentStanford University
  • Marcus T. Alley
    • Department of Radiology, Lucas MRI/S CenterStanford University
Review Article

DOI: 10.1007/s00348-007-0383-2

Cite this article as:
Elkins, C.J. & Alley, M.T. Exp Fluids (2007) 43: 823. doi:10.1007/s00348-007-0383-2

Abstract

Magnetic resonance velocimetry (MRV) is a non-invasive technique capable of measuring the three-component mean velocity field in complex three-dimensional geometries with either steady or periodic boundary conditions. The technique is based on the phenomenon of nuclear magnetic resonance (NMR) and works in conventional magnetic resonance imaging (MRI) magnets used for clinical imaging. Velocities can be measured along single lines, in planes, or in full 3D volumes with sub-millimeter resolution. No optical access or flow markers are required so measurements can be obtained in clear or opaque MR compatible flow models and fluids. Because of its versatility and the widespread availability of MRI scanners, MRV is seeing increasing application in both biological and engineering flows. MRV measurements typically image the hydrogen protons in liquid flows due to the relatively high intrinsic signal-to-noise ratio (SNR). Nonetheless, lower SNR applications such as fluorine gas flows are beginning to appear in the literature. MRV can be used in laminar and turbulent flows, single and multiphase flows, and even non-isothermal flows. In addition to measuring mean velocity, MRI techniques can measure turbulent velocities, diffusion coefficients and tensors, and temperature. This review surveys recent developments in MRI measurement techniques primarily in turbulent liquid and gas flows. A general description of MRV provides background for a discussion of its accuracy and limitations. Techniques for decreasing scan time such as parallel imaging and partial k-space sampling are discussed. MRV applications are reviewed in the areas of physiology, biology, and engineering. Included are measurements of arterial blood flow and gas flow in human lungs. Featured engineering applications include the scanning of turbulent flows in complex geometries for CFD validation, the rapid iterative design of complex internal flow passages, velocity and phase composition measurements in multiphase flows, and the scanning of flows through porous media. Temperature measurements using MR thermometry are discussed. Finally, post-processing methods are covered to demonstrate the utility of MRV data for calculating relative pressure fields and wall shear stresses.

1 Introduction

Magnetic resonance velocimetry (MRV) is a versatile experimental method capable of measuring laminar and turbulent fluid motion in highly complex geometries in which no other measurement modalities can be applied. In this review, MRV will be used as a general term referring to all measurements of fluid motion as opposed to its traditional use in referring to measurements of mean velocities only. MRV uses magnetic resonance imaging (MRI) and other nuclear magnetic resonance (NMR) techniques. It can be implemented in specialized magnets used for research and conventional MRI magnets used for clinical imaging. MRV is non-invasive and requires no optical access or flow markers, and measurements can be obtained in clear or opaque MR compatible flow models and fluids. Numerous flow quantities can be measured including mean velocities, Reynolds stresses, and diffusion coefficients and tensors. Moreover, MRV can be combined with other techniques based on NMR principles during the same experiment to measure additional quantities such as chemical species or properties of multiple phases. MRV and MR thermometry can be used in convective heat transfer experiments to measure temperature and flow quantities.

These quantities can be measured along single lines, in planes, or in full 3D volumes with resolutions ranging from 1 cm to 10 μm. It is possible to measure velocities ranging from ∼10 m/s down to 1 cm/day and diffusion coefficients greater than 10−14 m2/s (Gladden 2003). 2D image acquisitions with velocity measurements can be completed in tens to hundreds of milliseconds using fast imaging sequences or in several seconds up to several minutes with more conventional techniques. 3D velocity fields require slightly more time (∼minutes), and in most cases, high quality MRV experiments measuring three-components of velocity throughout a 3D volume can be completed in less than a few hours. Conventional medical MRI systems can image objects up to 30 cm in diameter with sub-millimeter resolution while more specialized systems are capable of better resolution but typically fit only smaller objects.

A particularly powerful application of MRV is to combine data with results from computational fluid dynamics (CFD) simulations to produce highly detailed information about very complicated flows. Engineers now use CFD simulations to predict fluid motion in a variety of applications ranging from air flow in gas turbine blade cooling passages to blood flow in human arteries to the multiphase flow of gases and liquids in bioreactors. Most of these realistic flows are in complex geometries and contain both separated and strong secondary flow. Such conditions are challenging for modern CFD to predict for two reasons. First, adequate resolution of the flow in complicated geometries requires a very large computational grid, especially for moderate to high Reynolds numbers. Second, turbulence modeling of even simple separated flows is already challenging without the added complications of multiple interacting separation zones. Because MRV is capable of providing highly resolved, concrete information throughout an entire geometry, MRV results can be used in a point by point comparison to evaluate the CFD results. In many cases, the CFD may need to be tuned or the computational grid refined in order to produce accurate results. Once this is done, the CFD results can be used to complement the MRV results by providing information not measurable with MRV such as wall shear rates and pressure. In most realistic flows, the wall shear layers cannot be resolved well enough with MRV to calculate wall shear rates, and only relative pressure may be calculated using MRV velocity fields.

Combining MRV and CFD data, of course, relies on the ability of MR to measure flow with the same fluid in the same geometry as in the simulated conditions. Modern rapid prototyping (RP) technologies based on layered manufacturing processes such as stereolithography (SL) facilitate the combination of MRV and CFD by manufacturing a model with the exact same flow geometry used in the simulation. This has been utilized in a few studies by mechanical engineers interested in machinery and more extensively by biomedical engineers studying blood flow. In addition to the realistic flow models artificially manufactured, realistic materials can also be used with MRV if they are MR compatible. The use of real materials is especially important for single and multiphase flows in porous media. For instance, studies of oil flow through fine grained sandstone can be made using the real materials.

Magnetic resonance compatible materials include plastic, rubber, glass, ceramic, and wood as well as living and dead tissue. While non-magnetic metals such as copper and other noble metals, titanium, and aluminum can be placed in the magnet, the imaging artifacts created by their distortion of the magnetic field preclude their use close to regions of interest in flow experiments. Generally, electrical devices such as induction and stepper motors can be used to power flow pumps inside the magnet room as long as they are placed a safe distance from the magnet. These types of motors typically do not create noise in the MR measurements even if they are unshielded. However, some devices such as servo motors do create noise and require shielding.

Magnetic resonance velocimetry is a fast growing field primarily due to its applications in medical imaging. However, MRV has numerous applications in studying flows found in other disciplines. The review by Fukushima (1999) describes the use of MRV for a wide variety of flows. Gladden (2003), Mantle and Sederman (2003), and Koptyug and Sagdeev (2002) provide numerous examples of MR applications in chemical engineering flows. More recently, Tyszka et al. (2005) review MR microscopy and its applications in biological, geological, and engineering flows.

The past few years have seen significant expansion of the use of MRV. Measurements of mean velocities and turbulent velocity fluctuations have been made in turbulent flows in complex geometries such as those found in gas turbine blade internal cooling passages. Blood flow in nearly every section of the human cardiovascular system from the heart to the fingers has been studied. MRV measurements in flowing gases have been made in human airways, in pipes, and over wing and bluff body models in pipe test sections.

Magnetic resonance velocimetry is seeing increasing application, and its capabilities are being expanded constantly because of its versatility and the widespread availability of MRI scanners. The purpose of this review is to provide a base from which to start MRV experiments by describing many of the basic principles of MRI and by surveying recent MRV experiments of interest to the fluid mechanics community. So as not to duplicate information contained in previous review articles, the primary focus of this work will be publications from the past 6–8 years. In addition, many articles are listed at the beginning of each section as additional sources for information that may not be included in this review.

The first section provides a general description of the principles behind MRV and sources of uncertainty and errors. The next several sections review applications of MRV to a few areas of fluid mechanics research: engineering flows, physiological and biological flows, multiphase flows, and flows through porous media. A section describing engineering applications of MR thermometry follows. The final section addresses post-processing issues: methods for visualizing and analyzing the data, methods for comparing MRV and CFD data, and methods for processing the data to calculate quantities such as relative pressure and wall shear stress.

2 MRI basic principles

Magnetic resonance velocimetry is an experimental technique that uses MRI to measure the three-dimensional velocity distribution within an object. MRI was implemented by Lauterbur and Mansfield (Lauterbur 1973; Mansfield 1977), and is based on the phenomenon of NMR that was developed independently by both Bloch and Purcell (Bloch et al. 1946; Purcell et al. 1946). The presence of a strong magnetic field B0 (which for clinical MRI systems is on the order of 0.5–3 T) will induce nuclei with non-zero magnetic moments to align along the field direction. These nuclei then precess at the Larmor frequency ω0, which is defined as
$$ \omega_{0} = \gamma B_{0} $$
(1)
Here the gyromagnetic ratio γ is a constant determined by the type of nucleus under study. Due to the abundance of hydrogen within biological tissues, 1H protons are the most common source of the signal detected in MRI. 1H imaging is often referred to as proton imaging. Other nuclei of interest are 13C, 23Na, 31P, 19F, and 2H. It is important to note that the resonant frequencies depend on the molecular environment. This gives MRI the capability of not only distinguishing different molecules containing the same signal generating nuclei (e.g., water and lipids) but also different nuclei within the same molecule. The MR signal is created using an RF excitation pulse that also oscillates at the Larmor frequency. As a result of the excitation, the aligned spins are tipped into the plane perpendicular to the main field to create transverse magnetization. The time-varying fields that result from the precession of this transverse magnetization induce voltages in nearby receiver coils which are then detected by the MR system.

2.1 MRI signal

Because the resonant frequency of the spins is directly proportional to the external magnetic field, spatial localization is done using weak magnetic field gradients \({\vec{G}}\) that are played out along the three physical axes of the magnet. For an object with a spatial density profile \(\rho ({\vec{r}}),\) the acquired signal as a function of time can be shown to have the form (Haacke et al. 1999):
$$ s(t) = {\int {\rho ({{\vec{r}}})\exp^{{- {\rm i}\varphi ({{\vec{r}}},t)}}}}{\rm d}{{\vec{r}}}, $$
(2)
where the phase \(\varphi ({{\vec{r}}},t)\) is determined by the main magnetic field B0 and the imaging gradients \({\vec{G}}\) as
$$ \varphi ({{\vec{r}}},t) = \gamma {\int\limits_0^t {(B_{0} + {\vec{G}}(t') \cdot}}{{\vec{r}}}){\rm d}t'$$
(3)
The first term is the phase accumulation due to the Larmor frequency of the nuclei and is removed during demodulation by the receiver subsystem. Equation 2 can be cast into a Fourier transform pair with the introduction of the spatial frequency, or k-space, term \({\vec{k}}(t)\) (Likes 1981):
$$ {\vec{k}}(t) = (\gamma \mathord{\left/{\vphantom {\gamma {2\pi)}}} \right.\kern-\nulldelimiterspace} {2\pi)}{\int\limits_ 0^t {{\vec{G}}(t'){\rm d}t'}} $$
(4)
Using this, the demodulated signal can be written as
$$ s({\vec{k}}) = {\int {\rho ({{\vec{r}}})\exp^{{-{\rm i}2\pi {\vec{k}} \cdot {{\vec{r}}}}}}}{\rm d}{{\vec{r}}} $$
(5)
From this equation, under the assumption that the encoding gradients \({\vec{G}}\) are linear, the underlying spin density \(\rho ({{\vec{r}}})\) can be recovered using an inverse Fourier transform
$$ \rho ({{\vec{r}}}) = {\int {s({\vec{k}})\exp^{{2\pi {\vec{k}} \cdot {{\vec{r}}}}} {\rm d}{\vec{k}}}} $$
(6)

2.2 Sampling requirements

During data acquisition, the MR receiver samples the signal discretely at time steps Δt selected by the user. Therefore, in a one dimensional example, k-space is sampled at the intervals
$$ \Updelta k = (\gamma /2\pi){\int\limits_t^{t + \Updelta t} {G(t')}}{\rm d}t', $$
(7)
which for a constant gradient reduces to
$$ \Updelta k = (\gamma /2\pi)G\Updelta t. $$
(8)
As a result of the discrete nature of this process, an insufficient sampling rate will produce aliasing in the final images, so that an object will appear as if part of it has folded back upon itself. This can be avoided by assuring that the sampling occurs at or above the Nyquist rate. The Nyquist condition can be stated as a relationship between the sampled interval in k-space Δk and the image field-of-view (FOV) in the corresponding direction as
$$ \Updelta k = 1/{\rm FOV}$$
(9)
The necessary extent in k-space that must be covered during an acquisition is dictated by the desired resolution of the final image. If N samples are taken along a particular direction, the amount of k-space covered is
$$ k_{{\max}} = N\Updelta k= 1/\Updelta x, $$
(10)
where the underlying spatial resolution of the reconstructed image is Δx = FOV/N.

2.3 The MRI sequence

Two- or three-dimensional MR images are created by sequential repetitions of the excitation and signal reception process, i.e., by repeated execution of a combination of RF pulses and time-varying magnetic field gradients in what constitutes a pulse sequence. Each repetition has a length referred to as the sequence repetition time, or TR. TR intervals are typically on the order of milliseconds and are determined by a combination of logistics (the minimum amount of time needed to play out the required set of gradients) and the desire for particular types of contrast in the final images.

All sequences begin by using an RF excitation pulse to create transverse magnetization over some region of the object being imaged. In a 2D MR sequence, a plane or slice of spins is excited using a pulse with a narrow bandwidth of frequencies. The RF pulse is played concurrently with a gradient along the slice select direction in order to establish the relationship between spatial position and precession frequency. Only those spins within the bandwidth of the pulse are tipped into the plane transverse to the main magnetic field direction and will contribute to the formation of the image. Conventionally, the slice direction corresponds to the Z axis in k-space (kz).

After slice selection, the transverse spins are manipulated using a combination of gradients and (possibly) further RF pulses to form an echo. During the echo formation, the MR receiver is turned on and k-space data are sampled. The period between the RF excitation and the center of the refocused echo is called the echo time (TE). The TE of a sequence can be set to produce various types of contrast, but for the purposes of measuring flow, a shorter TE is preferred. This is because the amount of transverse magnetization will decay over time, leading to a loss of signal and a decrease in signal-to-noise ratio (SNR) as the TE increases.

There are a large number of ways in which a pulse sequence can cover k-space during an acquisition (Hennig 1999). The most common approach is that of Cartesian sampling, in which a constant gradient (referred to as the readout gradient) is used during the echo formation and the resulting data are acquired on rectangular grid points in k-space. This process is referred to as frequency encoding, because as in the case of slice selection, the readout gradient establishes a relationship between spatial position and precession frequency. Conventionally the frequency direction corresponds to the X axis in k-space, so that during the readout process, data along kx are acquired at time steps determined by the bandwidth of the receiver. The design of the readout gradient involves producing enough gradient area during each time step to fulfill the Nyquist criterion (Eq. 9), while the length of the readout gradient is determined by the number of resolution points Nx needed along this direction.

In Cartesian sampling, spatial information is resolved along the direction orthogonal to the frequency encoding axis using a process known as phase encoding. The phase encoding axis corresponds to the Y axis in k-space. Before the readout gradient and receiver are turned on, a phase encoding gradient is applied that determines the ky location for the subsequent set of kx data. From TR to TR, the area of the phase-encode gradient changes by \({\varvec{\Updelta}}k_ y,\) where \({\varvec{\Updelta}}k_y\) is determined by the appropriate Nyquist criterion (Eq. 9). In this way, every TR acts to acquire one line of k-space data. The number of TRs, and hence the length of the sequence, is determined by the number of resolution points Ny along the phase encoding direction.

After the Nx × Ny k-space matrix has been completely filled, the Fourier transform is used to reconstruct the 2D image. Typically the raw data matrix is expanded to a larger, square dimension Nimag, where Nimag is a power of two (common values are 256 or 512), in order to take advantage of the efficiency of the Fast Fourier Transform. This process is referred to as zero-filling, because the extra k-space matrix elements are filled with zeroes before the reconstruction. The Fourier transform process acts to sinc-interpolate the data, resulting in an apparent increase in resolution (Liang et al. 1992). While zero-filling does not change the underlying spatial resolution, it can improve the continuity and visibility of small structures (Du et al. 1994).

In 3D MR imaging, the RF pulse is used to excite a volume of spins rather than a single slice. Resolution in the slice direction is then accomplished using phase encoding in the same way as the in-plane dimension. A sequence diagram illustrating the timing of the 3D imaging gradients is shown in Fig. 1. The dimension of the 3D volume in the slice direction is split into the number of desired slices denoted as Nz, and the duration of the 3D sequence will be approximately Nz × Ny × TR. The 3D sequence has several advantages over a multi-slice, 2D approach. Thin 2D slices require a significant slice selection gradient which will limit the minimum achievable width. The 3D volume excitation places less burden on the scanner hardware, and phase encoding can be used to make arbitrarily thin slices. Multi-slice 2D acquisitions can have continuity problems from slice to slice, especially if the object being imaged moves during the study. Finally, as will be shown below, 3D studies generally will have higher SNR than 2D examinations.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-007-0383-2/MediaObjects/348_2007_383_Fig1_HTML.gif
Fig. 1

The basic components of a 3D imaging pulse sequence are shown above. The gradients are played along the readout (RO), in-plane phase encode (PE), and slice-select (SS) directions. The sequence begins with an RF excitation pulse (shown on the RF axis) which plays concurrently with the slice-select gradient (blue, SS axis). If the sequence is designed to do phase-contrast imaging the flow-encoding bipolar gradients, shown in red, will be present. If the sequence is designed simply for imaging, these gradients are absent and the overall sequence length (TR) will be shorter. Prior to readout, phase encoding gradients (green) along the PE and SS directions are applied (the dotted lines indicate that the gradient amplitudes will change from TR to TR). The readout gradient and prewinder are shown in blue on the RO axis. The data acquisition timing is shown along the ACQ axis, and the echo time (TE) is defined as the time between the peak of the RF pulse and the peak of the acquired echo signal. After the data acquisition ends, spoiler gradients (yellow, RO direction) and phase encode rewinders (yellow, PE and SS directions) are often applied to remove any unwanted residual signal

The above discussion has concentrated on the use of Cartesian sampling for imaging in MR. Other k-space trajectories such as spiral (Meyer et al. 1992) and projection-reconstruction (Glover and Pauly 1992) can offer more efficient sampling strategies and thus significantly reduce scan time. However, since the data are not acquired on a rectilinear grid, extra processing must be done during the reconstruction prior to the Fourier transform (Jackson et al. 1991). In addition, the longer readout times that are often employed can result in greater image artifacts due to off-resonance effects (Block and Frahm 2005).

Frequency encoding methods rely on a relatively long (i.e., several milliseconds) readout gradient for acquiring a line of k-space. While they are efficient, they are inappropriate for some imaging applications. Alternative single point imaging (SPI) methods sample a single point of data for every RF excitation and phase encode. SPI methods are time inefficient, but they have advantages for imaging gases with short relaxation times (see Sect. 2.5) and high molecular diffusivity or imaging high speed flows (Balcom et al. 1996; Prado et al. 1999). Often in gas imaging or other low signal experiments, SPI is combined with multi-point averaging in which the single-point k-space signal is sampled several times for each RF excitation and gradient pulse in order to improve signal-to-noise ratio (SNR).

2.4 Signal-to-noise ratio

An important consideration in any imaging experiment is the achievable signal-to-noise ratio. In MR imaging, the root mean square noise electromotive force (EMF) over a frequency bandwidth Δν is (Hoult and Richards 1976):
$$ N_{{\rm RMS}} = {\sqrt {4kTr\Updelta \nu}}, $$
(11)
where k is Boltzmann’s constant, T is the sample temperature, and r is the sample resistance. Generally, the temperature is fixed, as it is usually impractical to decrease noise through cooling the sample. The resistance term r has contributions from both the resistance of the sample and the resistance of the coil. For field strengths greater than 0.5 T, the resistance term is dominated by the conductive losses in the sample (McVeigh and Atalar 1993), and a well designed receiver coil should not contribute resistive losses to the noise.
It is possible under certain idealized cases to derive the exact expressions for the SNR from an imaging experiment. In the case of a voxel with volume ΔV within a conducting sphere of radius b, the SNR of the free induction decay can be shown to be (Chen and Hoult 1989):
$$ {\rm SNR} = M_{0} \Updelta V{\sqrt {15\rho /(8\pi kTb^{5} \Updelta \nu)}} ,$$
(12)
where M0 is the magnetization and ρ is the resistivity of the sample. However, exact expressions such as these are not generally useful, since experiments are rarely done under ideal circumstances. Instead, it is more instructive to focus on the parameters within this expression that are under the control of the operator. These include the choice of receiver coil, the frequency bandwidth Δν, and the size of the sample voxels ΔV.

The appropriate choice of receiver coil can have a significant effect on the SNR in the final images. Larger coils (such as a body coil) will see more of the sample, and therefore contribute more thermal noise from the sample resistivity ρ. If one is only interested in imaging a subset of the sample, using a coil that is tailored to the size of the experiment will reduce noise from extraneous regions that do not contribute to the signal and thus improve the SNR.

If the sampling time of the receiver is denoted as Δt, the acquisition bandwidth, BW, is simply:
$$ \hbox{BW} = 1 / \Updelta t $$
(13)
Typically, acquisition bandwidths range from ±15.625 kHz (Δt = 32 µs) to ±125 kHz or higher (Δt = 4 µs). The voxel bandwidth Δν is the range of frequencies present within the sample volume ΔV, and is defined as BW/Nx, where Nx is the number of samples acquired during the readout. One can then see from Eq. 12 that
$$ {\rm SNR} \propto M_{0} \Updelta V{\sqrt {1/\Updelta \nu}} = M_{0} \Updelta V{\sqrt{N_{x} \Updelta t}}.$$
(14)
This is an important concept in that the SNR is proportional to the square root of the length of time over which data are acquired. A higher receiver bandwidth will reduce acquisition time at the expense of achievable SNR. This expression applies to the data acquired during a single TR interval, and a complete experiment consists of multiple TRs. Specifically, if the number of in-plane phase encodes in the acquisition is denoted as Ny, the number of slice-direction phase encodes denoted as Nz (note that for 2D sequences Nz is simply 1), and the number of times the entire scan is repeated is denoted as NSA, the total acquisition time is NSA × N× Ny × TR. Over the course of the scan, the signal will increase linearly with the number of repetitions, while the noise will increase as the square root. This implies that
$$ {\rm SNR} \propto M_{0} \Updelta V{\sqrt {{\rm NSA} \times N_{x} \times N_{y} \times N_{z} \times \Updelta t}} $$
(15)
Therefore, the SNR of an exam is proportional the square root of the time during which the receiver is turned on.

The last parameter that can be used to influence SNR is the choice of voxel size ΔV. Because the noise in the MR image does not depend on the signal, SNR cannot be recovered by averaging pixels to form an image with lower resolution. This is a fundamental difference between magnetic resonance and other imaging modalities (such as computed tomography and nuclear medicine) that depend on counting statistics. As an example, two pixels of size ΔV containing signal S and noise σ can be averaged to form a larger pixel with size 2ΔV with an SNR of \({\sqrt 2}S/\sigma.\) However, it can be seen from Eq. 15 that if the scan had been initially performed with the larger voxel size the increase in SNR would have been a factor of 2. Therefore, if SNR is an issue, it is important to choose the image resolution with the largest acceptable voxel size.

2.5 Relaxation times

After the creation of transverse magnetization, the excited spins lose the ability to contribute signal through two mechanisms. The first is the re-growth of longitudinal magnetization (Mz) due to the interaction of the spins with the main magnetic field B0. This process is described as the spin-lattice relaxation time, and is quantified with the time constant T1. In the case of Mz = 0 (e.g., following a 90° excitation pulse), it can be shown that the magnetization will recover exponentially as:
$$ M_{z} (t) = M_{0} (1 - {\rm e}^{{- t/T_{1}}}) .$$
(16)
The second mechanism is the loss of coherence of the transverse magnetization itself due to local spin-spin interactions. Immediately following the RF excitation pulse, the transverse magnetization \(M_{\bot}\) will begin to dephase with a time constant T2 according to the relationship
$$ M_{\bot} (t) = M_{0} {\rm e}^{{- t/T_{2}}}. $$
(17)
Generally, the decay of the transverse magnetization is also affected by external gradient inhomogeneities in the local environment. These effects will always decrease the effective T2, and depending on the conditions this reduction can be significant. This effective T2 is referred to as T2*.

Typically, the T1s of tissue are in the order of hundreds of milliseconds, while the tissue T2 values are usually 100 ms or less. Pure water has a significantly longer T1 and T2 of approximately 4 and 2 s, respectively. The variation in T1 and T2 values between fluids or tissue types is what makes MRI such a powerful technique for generating images with high differential contrast. MRV experiments in unheated single phase flows, however, involve imaging homogenous substances so that contrast manipulation is a relatively unimportant consideration. Instead, a greater concern is a shorter scan time. This is usually accomplished by using a sequence with as short a TR as possible. Unfortunately, a short TR sequence will saturate any substance with a T1 value that is significantly larger than the TR because insufficient time is available between excitations to allow the longitudinal magnetization to fully recover (see Eq. 16). Therefore, contrast agents are often used with fluids to artificially reduce the T1 to values that are typically on the order of 100 ms. The result is that saturation effects are almost completely eliminated, so that even very slowly moving spins will produce signal when imaged with a short TR sequence.

2.6 Measuring flow in MR

Because the measured signal in MRI is complex (Eq. 2), there are several ways in which flow can be imaged. Tagging sequences use RF pulses and magnetic field gradients to spatially modulate the spin magnetization in order to tag and track material volumes. These sequences produce images with bright and dark gridlines that deform with the motion of the underlying object. Velocities are found by quantifying this grid distortion over time. Errors in determining displacements from the grid images make tagging less accurate than alternative MRV methods such as phase contrast (Moser et al. 2000). However, tagging is useful for visualizing flow patterns, and it has applications for anatomical depiction and tissue motion.

More accurate quantitative velocity information can be obtained using the phase of the MR signal. This can be seen by considering the effects of an applied gradient \({\vec{G}}\) on the phase of a group of moving spins. If the time-dependent position of this spin set is expanded as
$$ {{\vec{r}}}(t) = {{\vec{r}}}_{0} + {\vec{v}}_{0} t + 1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2{\vec{a}}_{0} t^{2} + \cdots, $$
(18)
then using Eqs. 1 and 3, the phase of the signal in the presence of a gradient \({\vec{G}}(t)\) is given as
$$ \phi (t) = \phi_{0} ({{\vec{r}}}) + {\int\limits_0^t {\omega (t'){\rm d}t'}} $$
(19)
$$ \phi (t) = \phi_{0} ({{\vec{r}}}) + \gamma {\int\limits_0^t {{\vec{G}}(t') \cdot {{\vec{r}}}(t'){\rm d}t'}} $$
(20)
$$ \phi (t) = \phi_{0} ({{\vec{r}}}) + {{\vec{r}}}_{0} \cdot \gamma {\int\limits_0^t {{\vec{G}}(t'){\rm d}t'}} + {\vec{v}}_{0} \cdot \gamma {\int\limits_0^t {t'{\vec{G}}(t'){\rm d}t'}} + \tfrac{1}{2}{\vec{a}}_{0} \cdot \gamma {\int\limits_0^t {t^{\prime 2} {\vec{G}}(t'){\rm d}t'}} + \cdots , $$
(21)
where \(\phi_{0} ({{\vec{r}}})\) is the initial phase at time t = 0. For a gradient \({\vec{G}}(t),\) the gradient moment \({\vec{M}}_{i}\) is defined as
$$ {\vec{M}}_{i} = {\int\limits_0^t {t^{\prime i} {\vec{G}}(t')}}{\rm d}t', $$
(22)
and the phase of the MR signal can be written as
$$ \phi (t) = \phi_{0} ({{\vec{r}}}) + {{\vec{r}}}_{0} \cdot \gamma {\vec{M}}_{0} + {\vec{v}}_{0} \cdot \gamma {\vec{M}}_{1} + \tfrac{1}{2}{\vec{a}}_{0} \cdot \gamma {\vec{M}}_{2} + \cdots. $$
(23)
Velocity encoding in a particular direction typically is done using a set of gradients that are designed to have a net area of zero. Because \({\vec{M}}_{0} = 0,\) these gradients are often referred to as a bipolar pair. If the acceleration and higher order terms are ignored, the phase due to the application of the bipolar gradient is dependent on the spin velocity \({\vec{v}}_{0}\) as:
$$ \phi (t) = \phi_{0} ({{\vec{r}}}) + {\vec{v}}_{0} \cdot \gamma {\vec{M}}_{1} .$$
(24)
Since the Fourier transform resolves signal amplitudes and phases as functions of spatial location, the phase in each pixel can be measured.

Note that it cannot be assumed that the measured phase in any single acquisition is due solely to the application of the bipolar gradient. The phase term ϕ0 is voxel dependent and arises from a variety of sources, such as B0 inhomogeneity, susceptibility differences, off-resonance effects, eddy currents, and RF inhomogeneity (Pelc 1995). As a result, resolution of the flow in one direction requires the use of two acquisitions with different gradient \({\vec{M}}_{1}\) values.

If we consider a voxel composed of both static spins and spins with a constant velocity \({\vec{v}}_{0},\) then the acquired signal from an acquisition with no bipolar gradient (M1 = 0) can be written as
$$ S_{1} = (S_{s} + S_{v}){\rm e}^{{i\phi_{0}}}, $$
(25a)
where Ss is the contribution from static spins and Sv is the contribution from moving spins. With the application of the bipolar gradient, the second acquisition produces the signal:
$$ S_{2} = (S_{s} + S_{v} {\rm e}^{{{\rm i}\phi_{v}}}){\rm e}^{{{\rm i}\phi_{0}}}, $$
(25b)
where \(\phi_{v} = \gamma {\vec{v}}_{0} \cdot {\vec{M}}_{1}.\) A phase map can then be created by taking the phase difference at each point in the image:
$$ I_{{\Updelta \phi}} = \arg (S_{2} /S_{1}), $$
(26)
where arg (Si) is the phase of the complex signal Si. Finally, an apparent velocity image can be created using the magnitude of the bipolar first moment:
$$ I_{v} = I_{{\Updelta \phi}} /\gamma M_{1}.$$
(27)
This type of imaging is referred to as phase contrast (PC) imaging (Dumoulin et al. 1989; Pelc et al. 1991a). An illustration of the timing of the bipolar gradients with respect to the imaging gradients is shown in the sequence diagram in Fig. 1. Note that when imaging flow, most voxels will not have contributions from static spins unless they contain portions of a wall or boundary. In cases with static spins, so-called partial volume effects can be problematic as will be discussed below in Sect. 2.11.
An important parameter in the design of a PC imaging experiment is the velocity encoding value, or VENC. The VENC value is defined as the velocity that produces a phase shift of π radians, or
$$ {\rm VENC} = \pi /\gamma M_{1}. $$
(28)
It can be seen that including sign information, unambiguous velocity measurements can only be made for phase shifts between ±π. Therefore some knowledge of the velocities being imaged must be used to avoid velocity aliasing in the reconstructed images. Typically the VENC is set to be slightly greater than the largest velocities under study. In periodic flows such as blood flow in the human arterial system, the velocities vary significantly throughout the cardiac cycle. In order to improve the measurement accuracy, Ringgaard et al. (2004) implemented a sequence in which variable VENCs are tailored to the different phases of the cardiac cycle. In turbulent flows, the VENC must be chosen with the turbulent velocity fluctuations in mind. Although the VENC may be above the mean velocity, high speed turbulent fluctuations can occasionally make the velocity greater than the VENC. In this case, because most 2D or 3D measurements require several seconds to minutes to fully sample k-space, the measured phase will be an inaccurate average of aliased and non-aliased samples.
PC techniques can easily be extended to measure flow in all three physical directions, in which case a minimum of four measurements are required to eliminate any underlying phase offsets (Pelc et al. 1991b). The most efficient configuration for measuring flow in all directions is the “balanced four-point” method (Pelc et al. 1991b), in which the flow-encoding M1 values are altered in pairs. If it is assumed that the complex signal Si has equal and uncorrelated noise in each channel with variance σ2, it can be shown that the variance in the phase image arg (Si) is (Conturo and Smith 1990):
$$ \sigma^{2}_{{\phi_{i}}} = \sigma^{2} /{\left| {S_{i}} \right|}^{2}, $$
(29)
where |Si| is the magnitude of the signal in any given voxel. Each velocity image from the balanced four-point acquisition will then have equal variance of the form:
$$ \sigma^{2}_{v} = \sigma^{2} /{\left| {\gamma M_{1} S} \right|}^{2}, $$
(30)
in which it is assumed that each measurement Si has comparable signal (|Si| ≈ |S|, for all i). This can also be written as
$$ \sigma^{2}_{v} = (\sigma^{2} {\rm VENC}^{2})/{\left| {\pi S} \right|}^{2}. $$
(31)
This equation shows that the noise in a phase contrast image is dependent on the local signal strength, but is independent of the measured phase. Furthermore, the noise will increase linearly as the VENC increases. Therefore, the optimal PC acquisition will incorporate a large enough VENC to prevent aliasing, but not one so large as to unnecessarily increase the image noise. It is possible to collect data with an artificially low VENC, and then un-wrap the aliased velocity data using a second scan performed with an appropriate VENC (Lee et al. 1995). This approach takes advantage of the higher velocity-to-noise ratio of the low VENC scan at the cost of extra imaging time. This technique must be applied carefully in turbulent flow for the reasons described above.

2.7 Time-resolved flow imaging

The PC-MRI sequence described above can be gated in order to measure periodic flows. These time-resolved PC sequences are used primarily to measure physiologic blood flow. Typically, a trigger or gating signal such as the ECG signal from a human or animal is sampled along with the velocity data which is sorted either retrospectively or prospectively into a prescribed number of time bins spaced throughout the signal period. The achievable temporal resolution in this type of imaging depends on the TR of the sequence and the way in which the flow-encoding measurements are interleaved. The result is a time-resolved movie of flow motion constructed from the average of several periods of the flow, and it is often called “cine” PC-MRI.

2.8 Measuring turbulent diffusion and Reynolds stress in MR

The time evolution of the nuclei magnetization vector is described by the traditional Bloch–Torrey equation and depends on the main magnetic field, magnetic field gradients, bulk nuclei motion, and molecular diffusion. Kuethe (1989) presents a modified Bloch–Torrey equation for the complex “transverse magnetization” density function \(m({\vec{x}},t)\) in a moving turbulent fluid,
$$ \frac{{\partial m}}{{\partial t}} = - {\rm i}\omega_{0} m - {\rm i}\gamma {\vec{x}} \cdot {\vec{G}}m - \frac{m}{{T^{*}_{2}}} - \nabla (\vec{V}_{0} m) + \nabla ((D_{\rm m} + D_{\rm t})\nabla m),$$
(32)
where \({\rm i} = {\sqrt {- 1}},\; {\vec{x}}\) is the position vector, \(\vec{V}_{0}\) is the mean velocity of the fluid, Dm is the molecular diffusion coefficient, and Dt is the turbulent diffusion coefficient. In a simplified analysis for turbulent flow, the turbulent diffusion coefficient is assumed to be constant and much larger than Dm.
By applying a sequence of RF pulses, a bipolar magnetic field gradient, and the appropriate imaging field gradients (see Fig. 1), an image of a flow can be created in which the signal magnitude S in each voxel is a measure of the average value of \(m({\vec{x}},t).\)S is related to the quantities above by the relation
$$ S = S_{o} {\rm e}^{{- g(\tau, G)\gamma^{2} D_{t}}},$$
(33)
where τ is the duration of the bipolar gradient, G is the amplitude of the bipolar gradient, g(τ, G) is a function that describes the bipolar gradient waveform, and S0 is the signal magnitude measured with G = 0. This relation indicates that the turbulent diffusion coefficient can be calculated by measuring the signal loss between images acquired with and without a bipolar gradient. The measurable signal loss can be increased by increasing the bipolar gradient amplitude or the duration of the bipolar gradient.
Moreover, the variance of the turbulent velocity can be measured if the timing of the sequence is sufficiently short relative to the turbulent diffusion time scale. Batchelor (1949) modeled the diffusion coefficient for isotropic turbulence as
$$ D_{\rm t} (t) = {\left\langle {u^{\prime 2}} \right\rangle}{\int\limits_{t_{0}}^t {R_{\rm L} (t'){\rm d}t'}}, $$
(34)
where 〈u′2〉 is the variance of the fluctuating velocity and \(R_{\rm L} = {{\left\langle {u'(t)u'(t + t')} \right\rangle}} \mathord{\left/{\vphantom {{{\left\langle {u'(t)u'(t + t')} \right\rangle}} {u^{\prime 2}}}} \right.\kern-\nulldelimiterspace} {u^{\prime 2}}\) is the Lagrangian velocity autocorrelation coefficient. If the time (tt0) = τ is sufficiently short so that RL ≈ 1, then Dt = σ2uτ, where \(\sigma_{u} \equiv {\sqrt {{\left\langle {u^{\prime 2}} \right\rangle}}},\) the square root of the Reynolds normal stress (for unity density) in the streamwise direction. Substituting for Dt gives the result
$$ S = S_{o} {\rm e}^{{- g(\tau, G)\gamma^{2} \sigma^{2}_{u}}}, $$
(35)
and the variance of the fluctuating velocity can be found from the equation σ2u = ln(S0/S)/γ2g(τ, G) once g(τ, G) is known. Several authors present different results for this function depending on their assumptions about the characteristics of the turbulence and the timing of the gradient sequence (De Gennes 1969; Gao and Gore 1991; Gatenby and Gore 1994; Kuethe and Gao 1995; Dyverfeldt et al. 2006; Elkins et al. 2007). For instance, Dyverfeldt et al. derive the formula σ2u = 2 ln(S0/S)/kv2, where kv = γM1.

The variance for the other components of velocity can be measured by applying bipolar gradients in different directions. Moreover, bipolar gradients can be applied simultaneously along multiple axes, and the result can be combined with the single axis results to calculate the absolute value of the turbulent shear stresses. For instance, \(\sigma_{{u + v}} \equiv {\sqrt {{\left\langle {u' + v'} \right\rangle}^{2}}}\) is measured by applying bipolar gradients in the streamwise and cross-stream directions, and \({\left| {\overline{{u'v'}}} \right|}\) is calculated from the formula \({\left| {\overline{{u'v'}}} \right|} = (\sigma^{2}_{{u + v}} - \sigma^{2}_{u} - \sigma^{2}_{v})/2.\) The reader is directed to the cited references for discussions of the inaccuracies in these methods for measuring Reynolds stresses that are related to turbulence time scales, gradient timing, imaging resolution, and other sources of errors.

2.9 Fourier velocity encoding

As described above, phase-contrast imaging is an efficient technique for measuring moving spin velocities. However, this technique can be inaccurate when applied to voxels containing spins with a range of velocities. In this case, the standard two-point (or four-point, for all three flow directions) measurement is a weighted average that is the result of a complex sum of the individual velocity components present. In voxels containing both moving and stationary spins, the results can significantly underestimate the velocity distribution.

To address this problem, the velocity spectrum can be resolved using a technique referred to as Fourier velocity encoding (FVE). First described by Moran (1982), FVE extends the idea of phase encoding to the velocity dimension. Specifically, the phase produced from a moving spin in the presence of a bipolar lobe with a first moment \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{M}_{1}\) is \(\phi = \gamma {\vec{v}} \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{M}_{1}\) (Eq. 24). If we consider a gradient bipolar lobe of constant amplitude \(|{\vec{G}}|,\) lobe duration δ and temporal separation Δ (the time between the leading edges of each lobe), the gradient first moment is \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{M}_{1} = {\vec{G}}\Updelta \delta .\) Note that for most velocity measurements Δ = δ. The velocity k-space component \({\vec{k}}_{v}\) is defined as \({\vec{k}}_{v} = {(\gamma} \mathord{\left/{\vphantom {{(\gamma} {2\pi}}} \right.\kern-\nulldelimiterspace} {2\pi}){\vec{M}}_{1}.\) The application of a bipolar gradient is equivalent to measuring the velocity-dependent spin density at \({\vec{k}}_{v}\) (Haacke et al. 1999):
$$ \rho ({{\vec{r}}},{\vec{k}}_{v}) = {\int {\rho ({{\vec{r}}},{\vec{v}}){\rm e}^{{- {\rm i}2\pi {\vec{k}}_{v} \cdot {\vec{v}}}} {\rm d}{\vec{v}}}}, $$
(36)
and the velocity spectrum is found from the inverse Fourier transform of \(\rho ({{\vec{r}}},{\vec{k}}_{v}).\) In order to resolve the range of velocities \(\pm v_{{\max}}, |{\vec{k}}_{v} |\) must be stepped according to the Nyquist criteria: Δkv = 1/(2vmax), and if 2N velocity encoding values are used, the velocity resolution will be Δv = vmax/N. Note that FVE can be combined with normal imaging in spatial k-space to create multi-dimensional datasets with velocity spectra at each voxel in 1D, 2D, or 3D images.
An alternative notation is often used in the literature based on the work of Callaghan (1991) who introduced q-space as a reciprocal space to spin displacement (this is in analogy to k-space, which represents the reciprocal space to spin position). In q-space, \({\vec{q}} = (\gamma \mathord{\left/{\vphantom {\gamma {2\pi}}} \right.\kern-\nulldelimiterspace} {2\pi}){\vec{G}}\delta.\) Therefore, \({\vec{k}}_{v} = {\vec{q}}\Updelta,\) and Eq. 36 becomes
$$ \rho ({{\vec{r}}},{\vec{q}}) = {\int {\rho ({{\vec{r}}},{\vec{v}}){\rm e}^{{- {\rm i}2\pi \Updelta {\vec{q}} \cdot {\vec{v}}}} {\rm d}{\vec{v}}}}. $$
(37)
The advantage of FVE is that the velocity spectrum can be resolved in areas of complicated flow without the need to worry about partial volume effects (see Sect. 2.11). However, this comes at the cost of additional scan time, which can be significant if many encoding steps are used. For this reason, FVE is often done using only one spatial dimension to produce velocity-resolved projection images. Several authors have investigated an alternative to FVE by using a sparse, non-uniformly sampled q-space (Xing et al. 1995; Wise et al. 1996). The resulting image sets are processed using a Bayesian probability analysis to extract the best estimate of the velocity and diffusion or dispersion coefficient within each voxel. The advantage to this approach is not only a reduced scan time when compared to FVE, but the probability analysis yields an estimate of the associated errors in the calculated velocity.

2.10 Measuring molecular displacement and dispersion (q-space MRI)

When implemented with the appropriate pulse sequence, q-space MRI can be used to measure molecular displacement and dispersion. Because of this, it has a variety of applications in the study of flows through porous media. Consider the bipolar pulse sequence from Sect. 2.9 with two short gradient pulses of amplitude \(|{\vec{G}}|\) each applied for time δ and separated in time by Δ (the time between the leading edges of each pulse). A spin in the measurement volume will move from \({{\vec{r}}}_{0}\) to \({{\vec{r}}}\) in the time Δ. Not all of the spins will have the same displacement vector \({\vec{R}} = {{\vec{r}}} - {{\vec{r}}}_{0}\) due to molecular diffusion in stationary fluids, advective dispersion due to flow variations, and restricted motion due to small pore size and limited interconnectivity in porous media. Seymour and Callaghan (1997) combined imaging with displacement gradients and showed the resulting signal is given by
$$ S({\vec{k}},{\vec{q}}) = \iint {\rho ({{\vec{r}}})P_{s} ({{\vec{r}}},0|{{\vec{r}}} + {\vec{R}},\Updelta){\rm e}^{{{\rm i}2\pi {\vec{k}} \cdot {{\vec{r}}}}} {\rm e}^{{{\rm i}2\pi {\vec{q}} \cdot {\vec{R}}}} \, {\rm d}{\vec{R}}}\, {\rm d}{{\vec{r}}}. $$
(38)
Here \(P_{s} ({{\vec{r}}},0|{{\vec{r}}} + {\vec{R}},\Updelta)\) is the conditional probability that a spin at position \({{\vec{r}}}\) moves to the position \({{\vec{r}}} + {\vec{R}}\) in the time Δ and can be found using the inverse FFT and dividing by the spin density ρ which can be measured using conventional MRI imaging. \(P_{s} ({{\vec{r}}},0|{{\vec{r}}} + {\vec{R}},\Updelta)\) yields the distribution of displacements for each voxel in the image. The voxel-averaged velocity is found by dividing the mean displacement by Δ, and the variance in the displacements can be used to estimate the voxel-averaged dispersion coefficient.

In many porous media, it is not possible to resolve the pore space with MRI, so average displacement distributions for larger volumes (several mm3) are often measured by eliminating the spatial encoding gradients used for imaging. An advantage of using q-space to measure the average displacement distributions is that information about the flow and pore structure can be obtained with resolution that is independent of voxel size. The interested reader is directed to Seymour and Callaghan (1997) for their explanation of q-space applications in flows through porous media.

2.11 Limitations in MR flow measurements

Measuring flow with MR has been shown to be reliable and accurate (Rebergen et al. 1993; Pelc et al. 1994; Elkins et al. 2003), and yet successful results depend on an understanding of the limitations of the technique. In this section, we review the major sources of error when using PC-MRI (Firmin et al. 1990).

2.11.1 Partial volume effects (Tang et al. 1993)

For a voxel with both stationary and moving spins, the phase difference method of image reconstruction (Eq. 26) can lead to partially offsetting errors in flow calculations. In this case, the flow area in the partially occupied voxel will be assumed to be that of the entire voxel and will thus be overestimated. On the other hand, the phase map creation IΔϕ in Eq. 26 gives an average value for the voxel and will underestimate the phase shift due to the moving spins. The degree of underestimation will depend on the amount of stationary signal present, and this is generally problematic only in regions with small areas of flow. While these two effects can compensate for each other, the effects are unpredictable, and it is best to minimize the combination of stationary and flowing spins as much as possible. One can increase the number of pixels across the flow region to the extent that SNR will allow. In addition, it is best to image in planes perpendicular to the flow direction to minimize the volume of wall or boundary within the voxel.

2.11.2 Flow-related artifacts (Wood et al. 1993; Pelc et al. 2002)

Artifacts due to flow can be classified into two categories. Time-of-flight (TOF) effects occur when the signal magnitude in a given pixel changes over the course of the scan. Typically these errors are the result of movement in the object being imaged, or in the case of MRV, changes in the velocity distribution over time. Phase effects (which will be explored in more detail below) can change both the phase of the signal as well as its magnitude. The impact of these effects on images depends on whether or not these signal changes are consistent throughout the scan. The best way to ensure consistency is to ensure that the velocity distributions are temporally steady, or that the system is synchronized to the variability.

If the signal changes are not consistent as the k-space data are acquired, the modulation of the k-space data will disperse signal from one pixel to other image pixels creating artifacts dependent on the signal errors across k-space (Henkelman and Bronskill 1987). With 2D or 3D Fourier transform imaging, the traversal across k-space in the readout direction is generally very fast compared to the signal modulation time. Therefore the inconsistencies appear primarily in the phase-encoded directions of the image. If the inconsistencies are relatively periodic as a function of k-space location, as can be the case with pulsatile flow, the artifacts will appear as discrete replicates or ghosts of the moving structures or interfaces. If the signal changes are more random, the artifacts will be less structured.

Phase errors can affect image quality in three distinct ways. In the first, consider a voxel containing spins all moving at the same velocity ν0. Most imaging (phase encode and readout) gradients are not designed to have zero first moment (M1) values, and, as a result, these spins will accumulate a phase proportional to ν0 over the TR. If the velocity changes from one readout to the next, the corresponding phase change will produce ghosting in the same manner as magnitude changes. The second type of error is referred to as intravoxel phase dispersion, and it occurs when a given voxel contains spins with a range of velocities. In this case, the presence of a gradient with a non-zero first moment will produce a corresponding range of phases which will add destructively, leading to a reduction of signal within the voxel.

Both of these errors can be reduced by minimizing the higher moments of the gradients within a sequence. A sequence can be designed so that the imaging gradients are designed to have zero first moments (Bernstein et al 1992). This is referred to as flow compensation, and is typically done for the slice-select and readout gradients. The advantage of this approach is a reduction in motion sensitivity, which will reduce both ghosting and intravoxel phase dispersion. While phase-contrast imaging relies on first moment differences between two sets of acquisitions, flow compensation can be applied to reduce the extraneous phase contributions from gradients other than those responsible for flow-encoding. Flow-compensation is done using additional gradient lobes, and as a result will tend to increase sensitivity to higher order motion such as acceleration. The extra gradients will increase the sequence TR and TE, which can lead to longer scan times and less signal due to T2* decay.

The third type of error occurs in the presence of moving spins traveling in a direction that is oblique to the orientation of the phase-encoded and frequency-encoded axes. In this case, a displacement of the signal relative to the true location can occur. This mismapping occurs because the phase encoding gradient encodes the spin position at the time the pulse is applied, while the frequency encoding gradient encodes position at the time of the center of the echo (Nishimura et al. 1991). Any motion during this interval (Δt) will make the encoded X and Y locations inconsistent with each other. This problem can be minimized by moving the phase encoding gradient as close to the readout as possible to reduce Δt.

Misregistration artifacts can also be a problem in the slice direction for both 2D and 3D imaging. In 2D imaging the slice location is defined during RF excitation, while the position in the readout direction is encoded TE ms later. In 3D imaging the problem is only marginally better, since the slice-direction phase encode is typically done immediately after the slice-selection gradient. A displacement of the flow signal will occur in any image with flow that is not perpendicular to the imaging plane. In 2D imaging, this artifact can only be reduced by reducing TE, while for 3D imaging the slice-direction phase encoding can be placed as close as possible to the readout gradient.

2.11.3 Acceleration and higher moment terms

In the derivation of PC imaging above (Eq. 24), the presence of higher order flow terms has been specifically ignored. This is clearly a valid assumption for constant flow, but in the presence of more complicated flows these higher order terms will produce both misregistration errors and errors in the velocity estimates (Frayne and Rutt 1995). It can be shown that the error in the velocity measurement is proportional to the amount of local acceleration and the time difference between the moment center of the velocity encoding gradients and the center of the echo. As with spatial displacement errors, the most effective way of reducing these errors is to place the velocity encoding gradients as close to the readout as possible (Stefansic and Paschal 1998). Furthermore, as can be seen from Eq. 23, the acceleration-induced phase depends on the second moment of the gradients. If possible, reducing the duration of the velocity encoding gradients will also reduce the potential for acceleration-induced artifacts (Oshinski et al. 1992).

2.12 Post-processing considerations

Fundamentally, creating an image in MR is as simple as performing a Fourier Transform on the collected k-space data. However, more processing is required to produce satisfactory images both from a purely visual standpoint and as a source of quantitative information. In this section, we discuss the post-FT corrections necessary to produce accurate image and flow information.

Magnetic resonance imaging is based on the use of weak linear gradients to perform spatial encoding. Practically, the design of the gradient coil can only produce truly linear gradients over a small volume at the center of the coil. As a result, the reconstructed images will contain non-linear distortions that increase with distance from the isocenter of the magnet. Depending on the gradient coil, gradient strengths can deviate by more than 60% over a 40-cm3 volume (Markl et al. 2003). This problem can be corrected with post-processing algorithms that are part of the standard reconstruction software packages (Glover and Pelc 1986). However, it should be noted that this correction typically is not done in the slice direction for either multi-slice 2D or volumetric 3D data acquisitions. Therefore, one should always attempt to place the center of the volume under study as close as possible to the isocenter of the magnet to minimize these effects.

The non-linearity of the imaging gradients also has consequences for flow measurements. Because of this distortion, there can be significant deviations between the intended and actual flow encodings (Markl et al. 2003). This will affect the magnitude and direction of the encoded velocities. Angular deviations can be as much as 13° at the edge of a 20 cm3 volume and 45° at the edge of a 40 cm3 volume. One can predict these three-dimensional deviations with knowledge of the gradient coil profile, and this information can then be used to fully correct 3D velocity data. However, because of time considerations, often only one-directional flow is measured through the slice plane. In this case, the distortion causes the local slice-select gradient to be parallel to the local flow encoding direction, and a magnitude correction of the velocity data can be shown to be sufficient.

Anytime a gradient is activated, additional non-linear gradients appear as a result of Maxwell’s conditions that the divergence of the field is zero and the curl of the field is negligible. To lowest order, these additional gradients produce the field (Bernstein et al. 1998):
$$ B_{m} (x,y,z,t) = (G^{2}_{x} z^{2} + G^{2}_{y} z^{2} + G^{2}_{z} (x^{2} + y^{2})/4 - G_{x} G_{z} xz - G_{y} G_{z} yz)/(2B_{0}). $$
(39)
The separate terms are all non-linear and inversely proportional to the main field strength B0. Because PC-MRI depends on the difference of at least two acquisitions with different gradient configurations, the result is that these so-called Maxwell terms will introduce shading into the reconstructed phase images. While this shading can be eliminated with a judicious gradient configuration, the penalty is a loss of efficiency and an increase in the achievable sequence TE. Instead, these phase effects can be corrected using a post-processing correction as part of the image reconstruction. It can be shown that four coefficients are necessary to calculate the induced Maxwell phase as a function of spatial location. These terms depend on the strength and timing of the imaging gradients; therefore, these terms can be easily calculated by the sequence before each scan for later use in the reconstruction.

Gradient activity during scanning also induces eddy currents in the magnet. Eddy currents appear as small amplitude residual gradients of opposite polarity that decay exponentially and are present in all MR scans (Ahn and Cho 1991). Unlike the Maxwell phase effects, which are a function of gradient amplitude, eddy currents depend on the slew rate of the applied gradients. In addition, the appearance of the Maxwell phase terms is due to the underlying physics of the magnetic fields, while the appearance of eddy currents is dependent on the design of the magnet and its internal components. In phase contrast imaging, residual eddy currents will produce a spatially varying background phase that is mostly linear and is characterized by one or more exponential decay time constants (Glover and Pelc 1987; Glover et al. 1990). In most commercial systems, eddy current compensation is done using pre-emphasis filters that shape the gradients to reduce the major effects. However, it is common to find residual phase offsets that will affect quantitative measurements. Because eddy current effects are system specific, this residual background error must be removed using post-processing algorithms. For in-vitro measurements in which the flow can be stopped, the ideal approach is to acquire pairs of data sets under both flow-on and flow-off conditions. The images with no flow will contain an accurate measure of the eddy current induced phase which can then be used to correct the flow-on data. For in-vivo research, the most common approach is to measure phase in the static background areas surrounding the flow, and use the measurements to estimate the phase correction throughout the image (Walker et al. 1993).

2.13 Fast imaging

Fast imaging techniques are clearly appealing in that they can greatly reduce the acquisition time necessary for a given experiment. However, as was demonstrated above in Eq. 15, the SNR of an experiment will be proportional to the voxel size and the time spent acquiring data. Therefore, fast imaging techniques are only useful if the conditions of an experiment tolerate the subsequent loss of SNR. Some of this loss can be made up with the use of better coil arrays or with contrast agents, but ultimately SNR considerations will limit scan time reductions and the applicability of fast scanning techniques. These sequences also tend to have a greater sensitivity to artifacts which can make them unsuitable for certain applications. Many authors have their own specialized fast sequences, but most of these are related to the EPI or spiral sequences described below.

2.13.1 Echo planar imaging (EPI)

Echo planar imaging was first described in 1977 by Sir Peter Mansfield at the University of Nottingham (Mansfield 1977), and with the evolution of better and more sophisticated MR imaging systems the original technique has evolved considerably. Of the several variations in EPI, the simplest is that of a single-shot, gradient-echo (GRE) acquisition (Edelman et al. 1994).

In GRE-EPI, each TR begins with a 90° RF excitation pulse to create transverse magnetization. Multiple k-space lines are then collected by oscillating the readout gradient continuously to refocus the transverse magnetization into a successive series of echoes. As each echo is acquired, the phase encode gradient is increased slightly, or “blipped”, in order to move to the next line of k-space (Stehling et al. 1989). In a single-shot GRE-EPI acquisition, all of the lines of k-space for a complete image are acquired in a single TR. The minimum TR is dependent on the image resolution and the number of slices desired, but a single slice low resolution (64 by 64 pixels) image can be completely acquired in as little as 40 ms.

This reduction in imaging time often comes at the cost of a reduction in image quality and an increase in the number of artifacts. One of the greatest disadvantages to single shot GRE-EPI is the limitation on the achievable image resolution. Because each echo is formed through gradient refocusing, the transverse signal evolution is dominated by T2* decay. T2* time constants tend to be quite short making it difficult for single shot techniques to achieve image matrix sizes greater than 128 × 128 because of the rapid decrease in transverse magnetization (Poutschi-Amin et al. 2001). In addition, the falloff in signal from T2* decay will modulate the acquired k-space data in such a way that the resulting image will suffer from increased blurring (Farzaneh et al. 1990).

A characteristic of many fast imaging sequences is their long readouts. While this added acquisition time helps increase the efficiency of the sequence, it also tends to make the sequence more susceptible to off-resonance artifacts. A common problem in in-vivo imaging is the signal from lipids, which at 1.5 T resonate at a frequency difference of about 220 Hz from water. During the long EPI readout, these off-resonant spins will produce phase errors in the phase-encode direction of the sequence. As a result, any lipids that are present in the imaging volume will be shifted in the phase-encode direction, often by a significant fraction of the field-of-view. This problem is resolved with the use of special RF pulses that are designed to excite only water (Meyer et al. 1990). In addition to these chemical shift artifacts, susceptibility differences between adjacent regions of different structures will also produce artifacts in EPI images. These artifacts are most common at air-tissue interfaces, and will appear to geometrically distort the local anatomy (Ojemann et al. 1997).

All of these artifacts can be reduced by decreasing the amount of time spent during readout. One of the most successful ways of achieving this has been through the development of MR imaging technology. MRI systems have gone from gradient limits of 10 mT/m and slew rates of 17 mT/m/ms to clinical systems today that have typical configurations of 40 mT/m gradient limits and slew rates of 230 mT/m/ms. The result has been a decrease in readout duration with a corresponding decrease in artifacts.

2.13.2 Multi-shot EPI

While single shot EPI sequences provide great reductions in scan time, their long readouts limit the achievable resolution and lead to an increase in artifacts. If slightly longer scan times are permissible, multi-shot techniques can be used to address these limitations. The concept behind a multi-shot implementation is very simple: instead of covering all of k-space in one excitation, the complete k-space data set is acquired over several TRs. As an example, consider a single shot technique that acquires 128 phase encodes with a readout of 100 ms and a TR of 1 s. A 4-shot implementation would acquire 32 phase encodes in a single TR using a readout time of just 25 ms, and the complete set of data would be acquired over a period of 4 TRs. Provided that the TR length is kept constant, the scan time would increase by a factor of four. But the advantage of this approach is that there is less time in the shorter readout over which phase errors can accrue. As a result, multi-shot techniques are less affected by chemical shift and susceptibility artifacts. And because there is less time during data acquisition for T2* decay, multi-shot images will contain less blurring than single-shot techniques. Higher resolution images can then be acquired by adding more TRs as necessary at the expense of an increase in scan time.

2.13.3 Spiral imaging

A second type of fast imaging sequence is the spiral sequence. The first spiral images were published by Ahn et al. (1986) and then further developed by Meyer et al. (1992) for coronary imaging. Spiral sequences use oscillating waveforms on both in-plane axes during readout. As a result, the beginning of the data acquisition starts at the origin of k-space and spirals outward (hence the name of the sequence). In using both in-plane gradients equally, the spiral sequence is more efficient from a hardware point of view than an EPI sequence. Because of its geometry the spiral trajectory is capable of sampling a given region of k-space more efficiently than a Cartesian acquisition, and as a result shorter readout times are necessary. Finally, the oscillating gradients are inherently moment compensated, which means that flow-induced phase errors will be periodically refocused.

Like any rapid imaging technique with long readouts, spiral imaging is more susceptible to image artifacts than standard imaging techniques. The presence of off-resonant spins such as lipids will result in blurring throughout the image (compared with the shift of the fat signal in EPI). Therefore spiral acquisitions generally also use spatial-spectral pulses to eliminate the signal from these unwanted spins.

Spiral imaging tends to be less widely adopted than acquisitions that use traditional Cartesian sampling. Part of this reason is the spiral reconstruction is more complicated because the k-space data are not acquired on a rectangular grid. However this has become less of an issue with the development of more sophisticated reconstruction algorithms. In addition, partial Fourier and rectangular FOV acquisitions are difficult to implement because of the symmetry of the gradient activity on both in-plane axes. Imaging artifacts due to susceptibility can be considered in some cases to be unacceptable (Block and Frahm 2005). However, spiral imaging has been shown to have some advantages over EPI in functional magnetic resonance (fMRI) studies due to its lower sensitivity to brain motion and its more efficient acquisition (Noll et al. 1995).

2.13.4 Parallel imaging

Parallel imaging (PI) is a relatively new development in MRI that speeds up image acquisition by using the spatial information from each element of multiple coil arrays to replace data that would normally be acquired through phase encoding (Bammer and Schoenberg 2004; Larkman and Nunes 2007). The implementation of a PI acquisition involves two steps. In the first step, the desired imaging sequence simply acquires some fraction of the number of phase encodes that would normally be acquired. Because less data are acquired, the imaging time is reduced by a corresponding amount. It is important to note here that PI is not restricted to a particular type of sequence, as any type of acquisition can be reconfigured to acquire less data. However, since only a fraction of the needed phase-encodes have been acquired, the resulting images will be severely aliased unless these missing data are somehow replaced.

This is done using multiple coil arrays, the use of which forms the second part of the PI acquisition strategy. The idea is that each coil element has a different spatially dependent RF sensitivity profile, and this information can be used to supplement the gradient localization done through conventional phase encoding. Currently, there are several different reconstruction methods which synthesize the missing data during the reconstruction. However, all of these methods fall into two general categories. The first encompasses reconstructions that attempt to fill in the missing k-space data before the Fourier transformation. A well-known example of this approach is the GeneRalized Auto-calibrating Partially Parallel Acquisition (GRAPPA; Griswold et al. 2002). The second category involves reconstructions that first perform the Fourier transformation and then attempt to remove the aliased signal in the image domain. The most well-known of these techniques is SENSitivity-Encoded MRI (SENSE) (Pruesmann et al. 1999).

The clear advantage with PI techniques is the reduction in scan time. One is theoretically able to reduce the scan time by a factor of anywhere up to the number of coils in the array (e.g., for a 4-element coil array reduction factors from 1 to 4 are possible). Image quality typically suffers at the extreme end of this range, and so more typical reduction factors would be 2–3 (for four- and eight-channel coils, respectively).

The most obvious disadvantage is the reduction in the image SNR. As the scan time is lowered the subsequent image SNR is reduced by the square root of the reduction factor. In actuality the SNR is reduced slightly more due to a so-called geometry factor which reflects imperfections in the coil coverage. This has the added implication that the SNR in the resulting image will be spatially dependent. Using coil arrays specifically designed for PI help mitigate these effects. All PI reconstruction techniques require a low resolution k-space data set that is used to calculate the RF sensitivity profile for each coil. These data can either be acquired with a separate calibration scan prior to the PI sequence or as extra data during the PI sequence itself. In the latter case some of the scan time reduction advantage is lost, but this is not usually a significant problem.

2.14 High field imaging

MRI systems are available at a wide range of main magnetic field strengths, ranging from 0.015 to 7 T (Hu and Norris 2004). While 1.5 T units traditionally have been the state-of-the art for clinical whole-body scanners, higher field units of 3 and 4 T are becoming more common, and several manufacturers have recently introduced 7 T research machines. Similarly, small-bore animal research systems are now operating at field strengths of up to 17.6 T.

The primary benefit of a higher field is the accompanying increase in SNR. The signal increase can be expected to be linear with field strength up to 3 T, at which point complications due to B1 inhomogeneity make further gain at higher fields position dependent. Nonetheless, an average linear increase in SNR up to 7 T has been reported in brain imaging studies (Vaughan et al. 2001). The drawbacks at increasing field strengths include greater RF power requirements and power deposition, greater susceptibility artifacts (which will increase distortions at flow channel boundaries), and greater B1 inhomogeneity (which will limit the size of the phantoms that can be imaged). For in-vivo work, tissue T1 values tend to get longer (leading to longer sequence TRs and longer scan times) while tissue T2 values tend to get shorter (requiring stronger gradients to achieve shorter TEs).

3 Engineering flows

Engineering flows are found in engineering devices and include flows through internal cooling passages in gas turbine blades, heat exchangers, and manifolds. Typically, engineers are interested in areas of pressure loss found in separation regions in diffusers or 180° bends, areas of high heat transfer characterized by high turbulence or impinging flow on a wall, and areas of low heat transfer in stagnation regions with no flow. Accurate measurements of these conditions can help assess the efficiency of these devices.

The use of MRV measurements for rapidly acquiring details in complex flows is an area of great potential that has yet to be exploited. The design of these flow passages are often based on inherited design practices, textbook analysis derived from simple flows, or CFD, and yet none of these may be accurate enough for real engineering flows. Optimization of complex internal flow geometries is difficult because these tools fail to give the designer concrete information about the flow behavior. MRV is capable of providing highly resolved, accurate information about the flow throughout the entire geometry. These results can be used directly or in conjunction with analysis or CFD to improve the performance of flow devices.

The following survey of MRV experimental applications in engineering flows is split into two sections: one describing mean velocity measurements and one describing measurements of instantaneous velocity, Reynolds stresses and turbulent diffusion coefficients. Phase contrast (PC) techniques are the most commonly used for measuring mean velocity. PC and other pulsed gradient sequences are used for measuring diffusion and Reynolds stresses. Instantaneous velocities are imaged using fast imaging techniques which can capture snapshots of turbulence given the appropriate flow speeds and imaging resolution.

3.1 Mean flow

Early MRV experiments looked at simple flows with low Reynolds numbers. Ku et al. (1990) measured mean velocities in 2D transverse slices in laminar and turbulent steady flow through a pipe, a curved pipe, a converging–diverging nozzle, and an orifice and compared the results to laser Doppler anemometry (LDA) measurements. Li et al. (1994) used a time-averaging technique to make measurements in a 2D axial slice in pipe flow for Reynolds numbers between 1,200 and 9,400. Their technique produced one-component mean streamwise velocity profiles across the pipe diameter with images representing a 2D space–velocity plane. Fukushima (1999) presented a validation experiment in which MRV and LDA data for turbulent flow around a bend are compared for Dean numbers of 100 and 200.

Early papers also looked at stenotic (partially obstructed) flows which are relevant to diagnosing and treating cardiovascular disease. They can be highly turbulent and present problems when imaging blood vessels. Siegel et al. (1996) studied stenotic (partially obstructed) pipes. Their measurements in several 2D upstream and downstream axial slices compared well to LDA data. Gach and Lowe (1998) used an ultrafast sequence, RUFIS described in Madio et al. (1998), to characterize flow through gradual and abrupt 75 % area reduction stenoses and to measure reattachment lengths. Moser et al. (2000) measured velocities in pipe flow with a step stenosis for Re = 100 and 258 based on pipe diameter using tagging and phase contrast (PC) sequences. They showed the PC measurements were more accurate than the tagging results. This clinically relevant flow has received much attention from the MRI community, and more papers investigating turbulence and its effect on imaging stenotic flows will be discussed in Sect. 4.1.

More recent experiments are exploiting the capabilities of MRV for capturing full three-component 3D velocity fields in higher Reynolds number flows in highly complicated geometries. Elkins et al. (2003) presented measurements for the full 3D velocity field in an idealized serpentine passage created to model a gas turbine blade internal cooling passage. Elkins et al. (2004) presented results for a similar geometry with complementary PIV data taken in a 180° bend. Both papers utilized stereolithography (SL) manufacturing to create the flow models. The resolution of the MRV and PIV measurements were both approximately 1 mm yielding a detailed point by point comparison in the centerplane of the flow channel. The MRV and PIV in-plane velocities agreed to within ±10% in magnitude and ±10° in the direction throughout most of the flow excluding the thin shear layer bounding the separation region downstream of the bend.

Iaccarino and Elkins (2006) compared CFD with the MRV results for the serpentine internal cooling passage studied in Elkins et al. (2004). Direct point by point comparisons highlight the strengths and weaknesses of each method for studying these types of flows and point to the advantages of combining the two methods. Figure 2 shows the 3D serpentine passage model and compares velocity magnitude contours in the center plane of the model.
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Fig. 2

Study of the flow in the serpentine passage shown in the 3D view. Comparison between the velocity magnitude in the horizontal plane at the mid-height of the channel computed using an immersed boundary (IB) technique with a Reynolds averaged Navier–Stokes code (top) and measured with MRV (bottom). There is good agreement in the first bend while the MRV shows higher velocities just upstream of the second bend, and the separation regions after each bend compare well (work conducted with Iaccarino and Kalitzin, Iaccarino et al. 2003)

Despite the ability of MRV to measure flow in real devices, there are few examples in the literature. Amin et al. (2003) fabricated an MR compatible device which can serve as a single-screw extruder or a concentric-cylinder Couette device. They measured three components of velocity and mixing of two fluids in 2D transverse slices in the flow of 1% aqueous sodium carboxymethylcellulose (CMC) with two different concentrations of MnCl2. They used velocity measurements in the Couette device to study the rheological properties of pure glycerol, 1% aqueous sodium CMC, and other fluids for a range of rotation speeds and temperatures. Moser et al. (2003) measured the three components of velocity in the swirling flow in a rotating partitioned pipe mixer phantom. They used an EPI sequence with phase contrast velocity encoding to measure each component of velocity in several 2D axial slices from which they constructed the three-dimensional flow field.

Examples of the application of MRV to non fully turbulent flows include Seymour et al. (1999) who characterized the flow instabilities for Taylor vortex flow in a Couette cell using water and pentane at supercritical Taylor numbers. They measured spatially resolved three component velocities in a 2D slice to study the vortex structures. In addition, they presented non-spatially resolved q-space data that captures the coherence of the vortex structures in flows with and without instabilities. Moser et al. (2001) also investigated flow instabilities in the classic laminar Taylor–Couette–Poiseuille flow through a wide gap annulus with a spinning inner cylinder. They used a spin tagging sequence to measure velocity in several 2D planes with different orientations from which they created a cinematographic visualization of the helical and toroidal structures in the flow. Song et al. (2001) imaged the convective flow cells inside falling water droplets in air. Their 3D, three component velocity measurements were constructed using frequency encoding methods and accumulating data for 10 h from approximately 33,800 droplets which were assumed to be identical. Amar et al. (2005) measured the internal velocity fields in single liquid drops that were stably levitated in the magnet bore with a custom flow system. Tsushima et al. (2003) created a high pressure (30 MPa) flow rig to study the flow of non-Newtonian clay through a sudden contraction. Raguin et al. (2005) measured the velocity for flow with a Reynolds number on the order of 1 in a network of staggered microchannels 2 mm by 0.9 mm wide. Measurements were made in a 2D slice with an approximate resolution of 100 µm.

One of the most significant advances in MRI in recent years has been the development of gas flow imaging techniques. MR images can now be made using several types of gases: hyperpolarized 3He, 129Xe, and 13C; 1H gases such as butane, propane, and acetylene; fluorinated gases such as SF6 and C2F6; and gases containing 17O. While most gas studies have been made in porous media, the subject of another section, or involve the physiological study of lungs, a few are related to engineering flows. Certainly, the number of gas flow MRV experiments will increase significantly in the near future due to the numerous possibilities created by these new methods. The study of compressible flows is one important potential application.

Hyperpolarized gases contain a large fraction of nonequilibrium polarized nuclei, typically 104 to 105 times greater than the equilibrium polarization in water. This offsets the factor of 103 loss in signal due to the low density of the gas (Middleton et al. 1995). The MR signal from a given volume of these gases, therefore, can be an order of magnitude larger than an equal volume of water. On the downside, hyperpolarized gas is expensive to produce and store, and interaction of the gas with surfaces can quickly destroy the polarization. In addition, imaging sequences tend to use small flip angles to avoid destroying the polarization, and this limits the SNR. Hyperpolarized gases have been used mostly for imaging in the lungs, but applications for these gases are growing in both medical and nonmedical areas. Mair and Walsworth (2002) review hyperpolarized gas imaging for nonmedical fields.

Measurements can be made in the non-hyperpolarized gases listed above at standard temperature and pressure (STP), but the signals are 102 to 103 times weaker than water. The signal strength can be enhanced by pressurizing the gas, but this complicates the experiment. Instead, signal is increased by using longer scan times in which multiple measurements are acquired and averaged. Signal also is increased by sacrificing spatial resolution for larger voxels. Employing both of these tricks can improve the signal by 2–3 orders of magnitude which results in a signal magnitude comparable to the signal from liquids.

Mair et al. (2000) measured convective velocities in the evaporation of laser-polarized Xenon from a liquid drop. 1D and 2D spin density, diffusion, and velocity measurements were made in this two-phase gas-liquid mixture. Koptyug et al. (2000) imaged the flow of gaseous butane, propane, and acetylene through millimeter size channels in monolithic alumina catalysts. Axial velocities up to 93 cm/s were measured in 15 mm thick 2D slices of the flow with 400 µm true in-plane spatial resolution.

Newling et al. (2004) measured streamwise velocities in 2D slices in the flow of SF6 over objects placed in a pipe test section. They studied flow over a Clark-Y wing section airfoil (v = 12 m/s Re = 150,000 based on chord length) and a bluff obstruction (v = 17.0 m/s, Re = 210,000 in the upstream pipe flow). They used a velocity-sensitized pure phase encoding sequence (SPRITE, as described in Balcom et al. 1996). Figure 3 compares their measurements and computations with contour plots of the streamwise velocity in the flow over a Clark Y-wing airfoil. Although the resolution of the measurements is coarse, the agreement is good.
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Fig. 3

The streamwise velocity component in the flow of SF6 gas over a Clark-Y wing section placed in a pipe test section measured with MRV (a) and computed with CFD (b). Mean flow velocity in the pipe was 12 m/s giving Re = 150,000 based on chord length (36 mm; reprinted courtesy: Newling et al. 2004 copyright (2004) by the American Physical Society)

De Rochefort et al. (2006) measured velocities in the laminar flow of hyperpolarized 3He gas through a straight and curved pipe. All three velocity components were measured in 2D cross sections of the curved pipe, and results compared well with computations.

3.2 Turbulence and diffusion

There have been several studies of methods for capturing “instantaneous” measurements of velocity in turbulent flows. The duration of the imaging sequences discussed in the Basics of MRI prevents real instantaneous MR velocity measurements. However, if the flow is slow enough, nearly instantaneous measurements are possible for fast imaging times, typically 10–50 ms for a 2D image. Kose (1991) conducted a study to capture a snapshot of the transitional flow in a pipe at Reynolds number of 2,250. He used an ultrafast imaging sequence to measure two velocity components in a 2D axial slice of the flow in order to visualize and quantify velocities associated with intermittent turbulent vortices. Similarly, Gatenby and Gore (1996) used EPI to study pipe flow for Reynolds numbers up to 6,300, and Gach and Lowe (2000) used their rotating ultrafast imaging sequence (RUFIS) to image turbulent flow through a 75% reduction in area stenosis.

A more recent study by Sederman et al. (2004) used a modified EPI sequence called GERVAIS to acquire “instantaneous” measurements of the three-components of velocity in a 2D axial slice over a timescale of 60 ms in pipe flow for Reynolds numbers ranging from 1,250 to 5,000. Figure 4 shows measurements of the in-plane velocities (vectors) and through plane velocities (contours) for six Reynolds numbers. Vortices associated with low-Reynolds number turbulence are clearly seen for Reynolds number > 2,000. They acquired 128 sequential images for each Reynolds number and calculated circumferentially averaged mean velocity profiles. They also acquired 16 sequential images of one component of velocity (imaging time ∼ 20 ms) and calculated the acceleration, between pairs of images. Finally, they demonstrated the possibility of measuring the spatially resolved temporal velocity autocorrelation using the series of images.
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Fig. 4

Three orthogonal component velocity images acquired at increasing Re of (a) 1,250, (b) 1,700, (c) 2,500, (d) 3,300, (e) 4,200, and (f) 5,000. The color scale identifies the magnitude of the z-velocity, and ranges from zero to twice the average velocity for ac and from zero to 1.5 times the average velocity for df. The flow velocity in the plane of the image (i.e., XY) is shown by the vectors on each image. The vector scale bar on each image corresponds to 1 cm/s. (reprinted courtesy: Sederman et al. 2004 copyright (2004) Elsevier)

In their experiment to measure velocity and turbulence intensity in turbulent pipe flow for Reynolds numbers between 1,200 and 9,400, Li et al. (1994) used MRI in a more traditional way by averaging hundreds of signal acquisitions. They combined spatial and velocity-space imaging to acquire images of the velocity profile across the diameter of the pipe. Their technique produced one-component mean streamwise velocity profiles and turbulence intensity, which they showed to be related to the time-averaged pixel intensities.

Methods such as the ones described above which measure statistical quantities from many samples of velocity at a single point, line, 2D slice, or 3D volume are very time consuming, especially if one wants measurements in complex 3D geometries, and they fail to take advantage of the ability of MRI to efficiently measure quantities such as the velocity variance during image acquisition. Sequences for efficiently measuring diffusion coefficient and velocity variance are discussed in subsections of the Basics of MRI, and specific details can be found in the following papers.

Kuethe (1989) measured the turbulent diffusion coefficient for water in a turbulent jet for Reynolds numbers ranging from 8,000–18,000 and in turbulent pipe flow for Reynolds numbers as high as 29,500. Newling et al. (2004) measured the turbulent diffusivity in the high speed flow of gaseous SF6 over a bluff object placed in a pipe.

Gao and Gore (1991) and Kuethe and Gao (1995) measured the velocity variance, σ2 ≡ 〈u′ 2〉 in pipe flow. Gatenby and Gore (1994) used a method based on fractional echo acquisitions to vary TE in order to map values for the turbulent integral time scale and turbulent intensity in the flow downstream of a 50% and a 70% reduction in area obstruction in pipe flow. Dyverfeldt et al. (2006) measured turbulence intensity in flow through a tube with a 75% area reduction cosine shaped stenosis and compared their results to laser Doppler anemometry (LDA) measurements. Elkins et al. (2007) implemented a 3D method based on the methods in Gao and Gore (1991) and acquired the full field streamwise and cross-stream velocity variances in the flow downstream of a backward facing step in a square channel with a Reynolds number of 48,000 based on the step height and the freestream velocity at the step edge. Figure 5 shows the results from this study for the standard deviations of the streamwise and cross-stream velocities compared to particle image velocimetry (PIV) measurements made in centerplane of the square channel. As an example of the efficiency of these methods versus the more traditional averaging technique of Li et al. (1994), the time to complete the velocity variance measurements in the nearly 800,000 points in the 3D volume enclosing the backward facing step flow was less than 1 h versus 10 h it took Li et al. to measure one profile in pipe flow.
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Fig. 5

Velocity fluctuations measured with MRV and PIV in the centerplane of backward facing step flow in a square channel (Re = 48,000 based on step height, H, and the bulk mean velocity at the step). a Magnitude image from sequence with bipolar gradient amplitude G = 5 G/cm. Contour plots show the standard deviation of the streamwise velocity normalized by the bulk mean velocity at the step, measured with MRV (b) and PIV (c) and normalized cross-stream velocity standard deviation measured with MRV (d) and PIV (e). The agreement is excellent except in the thin developing shear layer close to step where spatial resolution and large velocity gradients create problems for both measurement techniques (reprinted courtesy: Elkins et al. 2007)

4 Physiologic, biomedical, and biologic flows

This section surveys recent papers in the vast subject of physiologic flows. Most of these papers are focused on the cardiovascular system. Others investigate gas flows related to the respiratory system, and one considers flow in the digestive system. Papers studying biomedical engineering flows relating to biomedical devices and bioreactors are included. Finally, several papers are reviewed that examine fluid motion related to living plants and animals.

4.1 Physiologic flow

The primary use for MRI is medical imaging, and there are many MRI flow papers concerned with physiologic flows. While the primary research focus has been related to blood flow, there are other applications for measuring motion in the body. In particular, measurements of fluid–structure interactions such as those found in the heart and its valves are gaining attention. MRI data are used to study these flows in numerous ways. Anatomical geometries can be extracted for further study either through CFD or in vitro experimentation. MRV measurements of blood flow are used to study flow through vasculature and to validate CFD simulations of pulsatile flow. In particular, MRV is used to look at flow through diseased arteries, a condition which is difficult to simulate since the pathologic processes of the tissue must be simulated in addition to the flow. Finally, important information such as relative pressure and wall shear stress can be determined noninvasively from in vivo MRV data.

Taylor and Draney (2004) review MRI blood flow measurements in the cardiovascular system. The reader is directed to the blood flow imaging section of that review for explanations of the implementations of PC-MRI methods for imaging pulsatile flows. The goal of the present review is to provide a few references for measuring fluid motion in each region of the body to which the reader can refer and use as a starting point for further study.

4.1.1 In vitro

Since MRV is capable of making non-invasive measurements, it is ideal to study physiologic flows in vivo. However, in vivo measurements are often difficult due to long scan times which are uncomfortable for the subjects. Results from long scans are often compromised by respiratory motion and subject movement. The measurements may be complicated due to the location of the site and the presence of other tissues, and it may be difficult to obtain the desired resolution and data quality. In these cases, in vitro experiments are set up to test sequences or study physiologic flows with well controlled parameters.

Kortright et al. (2001) evaluated their turbo BRISK sequence for measuring pulsatile flow around a curved pipe created to mimic the geometry of the aortic arch. They accurately measured the velocities in a 2D plane compared to LDA measurements. The BRISK sequence afforded a time savings of as much as 94% over a conventional velocity encoded cine sequence. Zhang et al. (2004) used a rapid, segmented k-space phase velocity mapping sequence and a control volume method for quantifying mitral valve regurgitant volumes in an acrylic idealized model of the left ventricle and mitral valve.

Elkins et al. (2005) performed a validation experiment by comparing time-resolved 3D PC-MRI and PIV measurements in an anatomically accurate human thoracic aorta phantom with a prosthetic bileaflet valve. In several previous papers, the time-resolved PC-MRI methods have been validated using volumetric flow rates, but few have compared the MRV phase averaged velocity vector fields to instantaneous vector fields measured with more established techniques like PIV. Figure 6 shows a photograph of the optically clear aortic arch with a laser sheet illuminating an oblique cross section and a point-by-point comparison of the velocity vectors in that plane for peak systolic flow.
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Fig. 6

Results from a validation experiment comparing time-resolved PC MRV and PIV measurements in pulsatile flow through an anatomically accurate aortic arch. The photograph shows the laser sheet illuminating the measurement plane in the model. The vector plot compares in-plane velocities measured with MRV (red arrows) and PIV (blue arrows) during the peak of systole. Only half the data in the Y direction is shown. The agreement between MRV and PIV results lend confidence in time-resolved PC MRV measurements in pulsatile flows. Note the PIV results in the branches and lower right hand corner are inaccurate due to glare in the PIV images (work conducted with Iyengar, Draney, Medina, and Wicker, Elkins et al. 2005)

Martin et al. (2005) studied a disease of the spinal canal called syringomyelia in which the spinal chord has a fluid filled cyst which obstructs flow of the cerebrospinal fluid (CSF). They used in vivo 3D MR imaging to create a model of a human spinal chord and PC-MRI to measure the flow of CSF. The in vitro model allowed them to investigate the fluid pressure and wall motion in addition to the flow characteristics. Isoda et al. (2006) investigated flow patterns in a silicone model of an intracranial aneurysm derived from a 3D CT dataset. Time resolved 3D PC-MRI measurements of the flow in the silicone model were analyzed and 3D streamlines were calculated to help visualize flow patterns.

4.1.2 In vivo

In vivo studies using MRI to make flow measurements are too numerous to completely cover. The papers described below survey flow measurements made in several regions of the body and are grouped accordingly.

Ebbers et al. (2002) made time resolved 3D PC-MRI measurements of the velocity field in the left atrium and ventricle of a human subject and used the measurements to calculate the relative pressure field in the heart. They used the data to examine the function of the heart through the cardiac cycle. Figure 7 shows the combined pressure field and velocity vectors in a plane in the left ventricle for several phases of the heart cycle.
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Fig. 7

A long axis slice of the relative pressure field (color contours), and the velocity field (black streamlines) in the left side of the normal human heart at the onset, peak, and end of the early phase of diastolic inflow (ac), the late phase of diastolic inflow (df), and ventricular systole (gi); (reprinted courtesy: Ebbers et al. 2002 copyright (2002) ASME)

Varaprasathan et al. (2002) discussed using velocity encoded cine (VEC) MRI, or time resolved PC-MRI, measurements in patients with congenital heart disease for quantifying flow rates and estimating pressure gradients for preoperative diagnosis and planning and postoperative monitoring. The pitfalls of VEC imaging were discussed as well. Nayak et al. (2003) used a sequence based on spiral phase contrast to measure high speed jets found in patients with valvular stenosis. Movie loops of 2D flow images were obtained with 22 ms temporal resolution in a 10 heartbeat breathhold. Westenberg et al. (2004) measured blood flow through the mitral valves in 10 healthy subjects. They made time-resolved PC-MRI measurements of three-component velocities in 6 radially stacked 2D images in the left ventricle and constructed a three-dimensional velocity field. The flow rates through the valves showed good agreement with flow rates in the aorta unlike conventional methods based on simpler one-component velocity PC-MRI measurements.

Markl et al. (2005) used time-resolved 3D PC-MRI to measure three-component velocities in the thoracic aortas of 10 healthy volunteers and 12 patients that had undergone root sparing aortic valve replacement. Visualization tools such as 3D time-resolved particle traces and 3D streamlines were created from the velocity fields to help identify the presence of flow structures such as vortices in the cusps at the root of the aorta as well as retrograde and helical flow in the ascending aorta.

Didier (2003) reviewed the application of MR techniques to diagnosing and characterizing valvular heart disease and discussed a number of quantitative metrics that can be calculated with PC-MRI flow measurements. Waters et al. (2005) evaluated the use PC-MRI for examining aortic valve disease. They acquired through-plane velocities in 2D cross planes approximately parallel to the plane of the valve root.

Sakuma and Higgins (2004) reviewed MR flow measurements in coronary arteries which are complicated by respiratory and cardiac motion and the small diameters (typ. 2–4 mm dia.) of the vessels. They discussed measuring velocities and volumetric flow rates in the arteries as well as total coronary flow in the coronary sinus. Schiemann et al. (2006) used time-resolved PC-MRI to measure velocities in the right and left coronary arteries of 83 patients who had aortic valve replacement. Maximum velocities during systole and diastole were quantified and the time varying velocity profiles were characterized. Langerak et al. (2002) made in vivo flow measurements in stenotic and nonstenotic coronary artery bypass grafts. They validated their MR results using Doppler flow measurements of peak velocity and velocity reserve.

Cheng et al. (2003) used time-resolved 2D PC-MRI to quantify blood flow through supraceliac and infrarenal slices in the abdominal aorta of a male volunteer during rest and moderate lower limb exercise on an MR compatible stationary cycle. Yokosawa et al. (2005) presented 2D time-resolved PC-MRI velocity measurements in a plane 2–5 cm downstream of the aortic valve for each of three healthy volunteers. They observed non-uniform flow patterns and the presence of secondary flows.

Bammer et al. (2007) used time-resolved 3D PC-MRI to measure flow in the major cerebral vessels in 14 healthy volunteers. They identified a number of interesting flow characteristics using streamline flow visualization. Figure 8 shows streamline visualization of the flow through the right internal carotid artery (ICA) bifurcation into the anterior and middle cerebral arteries (ACA and MCA) for systole and late-diastole. They also evaluated the quality of data for different parallel imaging parameters in both 1.5 and 3 T magnets.
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Fig. 8

Streamline visualization of blood flow. Blood flow at four different cardiac phases is shown through the carotid siphon (curved arrows) and cavernous segment (asterisk) of the right ICA and through the bifurcation of the ICA into the MCA (arrows) and the ACA (open arrow). Top row systolic phase; bottom row end-diastolic phase. Red and yellow streamline colors reflect high velocities, and green and blue colors reflect slower velocities. Units are m/s. During the systolic phase, the tailing edge of the pulse wave can be very well appreciated as it travels distally toward the MCA and ACA. Both proximal and distal to the ACA/MCA bifurcation, the velocity remains increased in the end-diastolic phase due to the vessel caliber reduction. No retrograde flow is apparent (reprinted courtesy: Bammer et al. 2007 copyright (2007) Wiley-Liss, Inc., a subsidiary of John Wiley & Sons, Inc.)

Wu et al. (2004) measured time-resolved velocities in 2D planes with high spatial resolution (in plane resolution 0.25 mm × 0.25 mm, slice thickness 3 mm) in the brachial, common carotid, and superficial femoral arteries in 20 healthy volunteers. Their sequence varies the velocity encoding for each heart phase to improve accuracy during the low velocity diastolic phases. They used these high resolution measurements to calculate wall shear rates and the oscillatory index for all three arteries.

Oelhafen et al. (2006) investigated reactive hyperemia, the instantaneous increase in blood flow in response to the sudden removal of a flow blockage, in the superficial femoral artery using a real-time 2D sequence to measure velocity with an in-plane resolution of 1 mm by 1 mm and temporal resolution of 50 ms. In addition, they used an image processing algorithm to segment out the vessel area in order to track vessel shape and blood flow during vessel responses to occlusive stimuli.

Klarhofer et al. (2001) obtained high resolution (in plane: 0.1 mm × 0.2 mm) velocity measurements in the index fingers of five volunteers using PC MRI and a strong small bore gradient system positioned in a whole body 3 T MR scanner. Boulby et al. (1999) used a flow sensitive EPI sequence to measure flow in the gastric antrum of eight healthy volunteers after eating.

4.1.3 MRV and CFD

Magnetic resonance velocimetry and CFD results can be combined to provide a more accurate and complete picture of physiologic flows than is possible from each method alone. Aside from providing validation data, imaging data are used to create geometric models of realistic anatomies for simulations. MRV measurements are useful for defining realistic boundary conditions, and it is the best method for measuring in vivo flow conditions for prescribing inlet conditions. Once validated, CFD provides better resolved information regarding wall shear stress and relative pressure. CFD also can be used for modeling physiologic flows for surgery planning or simulating medical devices. A few papers combining MRV and CFD for in vitro and in vivo physiologic flows are reviewed below.

Glor et al. (2002) reviewed several combined CFD and MRI studies. They discussed the use of MRI measurements for boundary conditions and presented results for the flow through a U bend that approximates flow through the aortic arch. Botnar et al. (2000) and Zhao et al. (2003) investigated flow through carotid artery models using CFD and 2D PC-MRI. Ku et al. (2005) investigated the flow in a model of a large arterial bypass graft and compared results in 2D planes throughout the model. Marshall et al. (2004) studied flow in models of normal and stenosed carotid arteries. Using time-resolved 3D PC-MRI velocity fields, they estimated the wall shear stresses which compared qualitatively well with CFD results.

Leuprecht et al. (2003) made a combined CFD and in vivo MRI study of flow in the ascending aorta in a healthy volunteer and a patient with a prosthetic valve. Velocities measured in vivo in a plane just downstream of the valves were used as inlet conditions for the computations. Results were compared for both subjects at multiple planes in the ascending aorta.

4.1.4 Fluid–structure applications

Since MR methods are sensitive to motion of fluid and tissue, they are also being applied to the study of fluid–structure interaction in the cardiovascular system to help understand the mechanical forces in the walls of the heart and vessels created by the pulsatile blood flow and their role in the development of diseases such as atherosclerosis. Masood et al. (2000) reviewed MR methods for measuring myocardial motion, namely, MR tagging, phase velocity mapping, and diffusion imaging. These methods can be combined with flow measurement techniques to study heart mechanics as in the study by Wise et al. (2005) which measured the wall and fluid motion in rat hearts with PC-MRI using sparse q-space sampling. Ozturk et al. (2003) reviewed MR methods and presented measurements for motion of the heart and blood flow inside of it. The method of Boese et al. (2000) for estimating aortic compliance from MR pulse wave velocity measurements may find potential application in the study of fluid–arterial wall interaction as well.

4.1.5 Gas imaging

The recent advances in MR gas imaging of hyperpolarized (HP) and other gases have extended MR applications into the respiratory system. Traditionally, this has been a difficult region to image because of the lack of water in the airway and lungs and the large susceptibility difference at the air-tissue interface. As these relatively new methods advance, a dramatic increase is expected in MRV measurements of the flow through the respiratory system.

Möller et al. (2002) reviewed HP gas MRI for use in lung imaging and pulmonary function evaluation. They also described the use of HP gas in injectable solutions for intravascular applications such as angiograms and measuring blood flow rates in veins of mice. Kadlecek et al. (2005) provided an extensive review of methods for using HP noble gases and MRI to investigate lung function. Oros and Shah (2004) reviewed the use of hyperpolarized 129Xe and MRI in biological systems. Golman and Petersson (2006) reviewed the more recent use of HP 13C and detailed its use in an injectable solution to visualize vasculature, map perfusion, and study metabolic pathways. De Rochefort et al. (2006) used HP 3HE gas to make velocity measurements in a 2D plane in a human trachea during inhalation. De Rochefort et al. is one of a few studies that have applied gas imaging methods to making velocity measurements in vivo.

Kuethe et al. (2000) used SF6 to image obstructions in rat lungs, and Schreiber et al. (2001) used SF6 to investigate pulmonary ventilation in pigs. Zhu et al. (2005) described the use of 17O, a stable oxygen isotope detectable by NMR, gas as a tracer for measuring cerebral blood flow and cerebral metabolic rate of oxygen utilization.

4.2 Biomedical applications

Fluid–structure interactions are also important for the design of many biomedical devices such as blood filters, intravascular prosthetic devices such as stents and stent-grafts, and many others. Methods for imaging the flow inside these devices are under development. Kuehne et al. (2001) presented a method for quantifying pulmonary blood flow through nitinol stents placed across the pulmonary valve in growing swine. Walsh et al. (2005) made phase velocity measurements of flow through peripheral sized (9 mm dia.) nitinol stents placed inside plastic tubing. Heese et al. (2005) measured the complex three-dimensional flow through a commercially available blood filter.

Magnetic resonance imaging also provides the potential for measuring fluid velocities and wall transport phenomena from biofilms and tissues. Paterson-Beedle et al. (2001) reviewed the use of MRI in bioreactors and described its many potential uses in these multiphase environments. Manz et al. (2003) applied multiple MR techniques to measure flow and image the biofilm in a biofilm tube reactor. Nott et al. (2005a, b) used MR to measure flow and monitor reactions in their bioreactor.

4.3 Biologic flow

Since MRI is versatile, non-invasive and relatively harmless to living things, it has numerous applications in biological flows in and around plants and animals. In the study by Windt et al. (2006), a custom MR system was built to fit around potted plants in order to quantify and compare the phloem and xylem flows in poplar, castor bean, tomato, and tobacco plants. Maps of velocity, flow conducting area and volume flow were made on a per pixel basis with resolution ∼0.1 mm over the course of tens of hours to track the diurnal changes.

In cases in which it is not possible to adapt the MRI system to the plant or animal, realistic models and flows can be created with methods similar to those used for in vitro modeling of physiologic flow. Chang (2007) combined MRV and CFD to create simulation techniques for studying flows around corals. Skeletons of Stylophora pistillata coral and others were scanned by x-ray computed tomography (CT). These CT images were processed with segmentation software, and a 3D-solid CAD model was created. A solid plastic replica of the coral was made using fused deposition manufacturing (FDM), a rapid prototyping technique and placed inside an MR compatible flow channel. The same CAD model was used in the CFD computations, and MRV velocity measurements were used for inlet flow conditions. MRV measurements were compared with results from several different simulations. A comparison between the results from MRV and the time averaged results from an immersed boundary large eddy simulation is shown in Fig. 9. The non-invasive capabilities of MRV make it ideal for measuring flow in the interior spaces of the coral between its branches.
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Fig. 9

MR compatible flow channel and plastic replica of a Stylophora pistillata coral are shown. Color contour plots compare MRV and time averaged CFD results for the streamwise velocity (m/s) in cross planes with vertical (z) and streamwise (x) axes through two sections of the coral. Both methods capture regions of separated flow behind and between the branches as well as regions of acceleration where open flow paths exist between the branches (Chang et al. 2007)

5 Multiphase flow

Multiphase flows consist of combinations of solids, liquids, and gases and typically present difficult environments for measurement systems due to their destructiveness to sensors and/or opacity. MR is well suited to making measurements in these flows because it is non-invasive and sensitive to chemical constituents and state properties. MR measurements are capable of separating the signals from each phase by exploiting relaxation characteristics, chemical shifts, or nuclear resonance frequencies. The literature contains numerous experiments including flow studies of the sedimentation of solid particles in liquids, concentrated suspensions of solid particles in liquids, stable and unstable stratification with two immiscible fluids, emulsions such as oil/water mixtures, liquid and gas combinations such as air and water, and granular particles in gas. Granular flows can be imaged if the particles contain oils or liquids with MR signal. MR has applications in three phase liquid and gas combinations such as flows containing oil, water, and gas commonly experienced in petroleum pipelines and three phase solid, liquid, and gas combinations found in chemical and bio reactors. Often the motion and distribution of the gas or solid phases are found by measuring liquid phase quantities and employing subtraction methods. Recent developments in the imaging of gases and solids create new possibilities for probing the behavior of these constituents.

Magnetic resonance imaging applications in multiphase flows are too broad of a topic to be covered completely in this review. Fukushima (1999) is a good source for MRI multiphase experiments performed in the 1990s. The review by Mantle and Sederman (2003) includes a discussion of gas/liquid and liquid/liquid flows in porous media and gas/liquid flows through small millimeter sized channels. The following is a brief survey of recent MRI applications in multiphase flow studies.

Beyea et al. (2003) studied sedimentation of buoyant low density polyethylene (LDPE) particles (dia. 150–180 µm) and negatively buoyant borosilicate glass particles (dia. 250–300 µm) in a perfluorinated oil. MRI is useful for studying highly concentrated suspensions due to the opacity of the flows. They used multiple excitation frequencies tuned to the 1H resonance of the LDPE particles and 19F resonance of the oil to image these two phases and used subtraction to obtain information about the glass particle concentration distribution.

Moraczewski et al. (2005) measured the particle concentration and velocity field for the flow of a suspension of particles through an abrupt 1:4 axisymmetric expansion. They investigated the effects of particle volume fraction, particle diameter, and Reynolds number using three different particle sizes and Newtonian fluids based on water and glycerin mixtures. Figure 10 shows concentration images for various particle bulk volume fractions and two particle sizes. The image brightness indicates fluid volume fraction, so the darker central regions show where the particles are flowing. As particle volume fraction is increased, the particles move closer to the center of the expansion.
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Fig. 10

Spin-echo concentration images (longitudinal cross sections) of initially homogeneous monodisperse suspensions in an abrupt 1:4 area ratio expansion. Images were obtained after the flow reached steady state. ad 85 μm PMMA particles, a ϕbulk = 0.2 (bulk volume fraction), Re = 0.6, Rep = 0.005; b ϕbulk= 0.4, Re = 0.7, Rep = 0.023; c ϕbulk = 0.48, Re = 0.01, Rep = 0.006; d ϕbulk = 0.5, Re = 0.01, Rep = 0.016. eh 255 μm polystyrene particles; e ϕbulk = 0.2, Re = 0. 1, Rep = 0.004; f ϕbulk = 0.4, Re = 0.1, Rep = 0.009; g ϕbulk = 0.45, Re = 0.04, Rep = 0.008; h ϕbulk = 0.5, Re = 0.004, Rep = 0.004. ϕbulk is the bulk volume fraction of the particles. Flow Reynolds number is Re = ρUR/η(ϕ): ρ is density of suspension, R is radius of large tube, U is average axial velocity, and η(ϕ) is suspension viscosity. Particle Reynolds number is Rep ρUa/ηfluid: a is particle radius and ηfluid is the viscosity of the suspending fluid (reprinted courtesy: Moraczewski et al. 2005 copyright (2005) by the Society of Rheology)

Chen et al. (2002) reviewed MR methods for use in single phase and multiphase flow measurements in porous media. In porous media, the relaxation times of fluids are highly dependent on fluid–surface interactions and pore geometry thereby eliminating one method for differentiating phases. However, one liquid phase can be marked by adding small concentrations of paramagnetic ions such as Cu2+ (CuSO4) or Mn2+ (MnCl2). For a range of tracer concentrations, there is typically a linear relationship between MR signal and tracer concentration which enables mass balance calculations from MR measurements. Chen et al. studied stable and unstable stratified flow using fresh water and salt water marked with Cu2+. They also presented results for an oil recovery experiment in sandstone. They flooded oil saturated sandstone with aqueous MnCl2 and measured the flow of the water solution over time.

Gotz and Zick (2003) studied the steady flow of a water–oil emulsion through a tube. They used a fast imaging sequence to visualize the water and oil droplets and chemical shift selective imaging (CHESS) to obtain all three velocity components and concentration measurements for both liquids. Amin et al. (2004) measured the index of mixing in the flow of two fluids each composed of 1% aqueous sodium carboxymethylcellulose (CMC) with different concentrations of MnCl2 in their MR compatible single-screw extruder.

Sederman et al. (2003) used their ultra-fast SEMI-RARE sequence to study air-water bubble train flow through a narrow tube. They acquired eight successive images each separated by 72 ms in time from which they quantified bubble size and velocity. They also reported the first observations made with MRI of liquid recirculation zones in the bubble train flow. Using similar methods, Gladden et al. (2003) studied air–water flow through channels in a ceramic monolith. They visualized bubble-train flow in real time and measured bubble size and velocity. They also quantified liquid hold-up and wetting efficiency in trickle flow through a bed of packed cylindrical porous alumina extrudate.

More recently, Gladden et al. (2005) studied trickle to pulse transition in the mixed flow of air and water through a trickle bed column packed with cylindrical porous alumina pellets of length and diameter 3 mm. They captured the liquid motion and quantified the frequency and spatial extent of local pulsing events using time series of ultrafast 2D and 3D images,

Nguyen et al. (2006) investigated the trickle flow of a gas-liquid (nitrogen–diesel oil) mixture at elevated pressures and temperatures (up to 10 bar and 125°C) through a cylindrical packed bed with 2 mm dia. nonporous spherical glass beads. They also studied a bed with 1.3 mm dia. porous Al2O3 trilobe catalysts which resembled a bench-scale reactor used for the hydrodesulfurization of diesel oil. They measured the porosity of the beds, the liquid saturation, and wetting efficiency using 3D images of the liquid phase.

Koptyug et al. (2004) combined multiple MRI and NMR spectroscopic imaging techniques to study an operating gas–liquid–solid model catalytic reactor. They made spatially resolved measurements of the reactant to product conversion within the catalyst bed. Manz et al. (2003) measured flow and imaged the biofilm in a biofilm tube reactor, and Nott et al. (2005a, b) measured flow and monitored reactions in their bioreactor.

6 Flow through porous media

Non-invasive MR methods are particularly well-suited for studying flow through porous media. These flows are encountered in the fields of engineering, physics, geology, biology, medicine, and materials science. Porous media include porous catalysts in chemical reactors, porous scaffolds in bio-reactors, granular media, rocks, foams, living tissues, and many others. In these flows, it is important to characterize a material’s porosity, its pore sizes and shapes and to measure fluid velocities and dispersion coefficients. MRI can be used to visualize interstitial space. In some cases, phase contrast MRI can be used to measure velocity vectors. In other cases, MR methods associated with q-space (molecular displacement) measurements are particularly well suited for studying dispersion phenomena (Seymour and Callaghan 1997). Q-space measurements can be made without requiring voxel sizes on the order of the pore sizes which in some cases can be just a few microns or less and nearly impossible to resolve.

Measurements in porous materials present a number of difficulties for MR methods. Due to the typically small pore volumes, there is not much fluid to create a measurable signal. Velocities can be extremely slow and can require long scan times. Even so, Candela et al. (2000) have presented results for velocities on the order of cm/day. The solid phase of many porous materials distorts the magnetic fields, an effect that gets worse with field strength. Therefore, one must balance this effect with the need for a higher field strength to increase the low amount of signal coming from small volumes of fluid. There are significant partial volume effects when the pore sizes are too small to be resolved, and partial volume corrections may be necessary. The solid internal surfaces of the pores can drastically change the relaxation characteristics of the fluids being studied, and often the relaxation times need to be calibrated with the fluid inside the material. Nevertheless, MR methods are some of the few methods for internally measuring flow through porous media and providing data to evaluate and develop analytic models and computer simulations.

Koptyug and Sagdeev (2002) provided an extensive review of MR applications in these flows complete with a descriptive list of previous experiments. Another source for past and present developments is the Proceedings of the International Meeting on Recent Advances in MR Applications to Porous Media which is published in Magnetic Resonance Imaging every several years. The reviews by Fukushima (1999) and Mantle and Sederman (2003) also described MR measurements made in flows in porous media. A few experiments are reviewed here to supplement the information provided in these reviews.

Many studies use packed beds of spherical beads and other materials to simulate porous materials. Often these idealized flow models are used to develop and evaluate measurement techniques. Candela et al. (2000) reviewed MRI and q-space methods. In particular, they discussed measuring distributions of displacements averaged over a sample to yield local and non-local dispersion coefficients. Ogawa et al. (2001) studied water flow through a cylindrical tube packed with crushed glass particles and one packed with spherical glass beads. Several series of consecutive 0.5 mm thick 2D slices were acquired and analyzed to quantify the porosity and pore size distribution for both packed beds. Three-dimensional velocity fields were measured using MR tagging. Chen et al. (2002) reviewed MR methods for single phase and multiphase flows in porous media and presented porosity and velocity measurements in single phase flow through a column of packed glass beads. They discussed the pitfalls of using phase-shift methods for measuring velocity in porous materials. Khrapitchev et al. (2002) measured the 2D time correlations for the velocity field in two flows using water and a 70% wt. glycerol solution through randomly packed spherical beads using a technique based on velocity exchange spectroscopy. Results are presented for a wide range of values for the Peclet number, bead diameter, and correlation times. Moser and Georgiadis (2004) used q-space MRI methods based on a pulse-gradient stimulated echo sequence to measure the displacement spectra for water flow through a pipe packed with spherical acrylic beads. From these measurements, they obtained interstitial velocity autocorrelation lengths which they confirmed using results from phase contrast velocity measurements in the same flow. Ren et al. (2005) investigated the effects that varying particle diameter has on the flow through a cylindrical packed bed. Two-dimensional MRI was used to look at the structure of the beds and measure velocities, and a tagging method was used to visualize flow patterns. Mertens et al. (2006) investigated low Reynolds number flows of Newtonian and non-Newtonian flow through tubes packed with 1.7 and 4 mm dia. glass spheres. They measured both spatially resolved velocities and integral displacement distribution functions.

In an application for micro MRI in the presence of electrical fields, Locke et al. (2001) looked at electro-osmotic flow in a 1 cm o.d. glass tube filled with glass spheres (dia. 2–4 mm). They made high resolution (78 μm × 78 μm in-plane, 100 μm slice thickness) velocity measurements in several 2D planes. They eliminated interfering electrical signals by turning off the electrical circuit during the readout gradients. Figure 11 shows velocity contours in three 100 μm thick slices with 1 mm spacing. There is a complicated pattern of forward and reverse flow throughout the interstitial spaces between beads and between the beads and the cylinder wall.
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Fig. 11

Velocity profiles in three slices 100 μm thick located 1 mm apart in the electro-osmotic flow through a column of particles: a bottom slice 1, b middle slice 2, and c top slice 3. The circular voids indicate the spherical beads. Forward and reverse flow is seen throughout the interstitial spaces between beads (reprinted courtesy: Locke et al. 2001 copyright (2001) American Chemical Society)

Many workers are interested in measuring the distribution of flow through packed beds of catalysts or channels in monolithic catalysts in order to optimize their utilization in reactors. Koptyug et al. (2000) imaged the flow of gaseous butane, propane, and acetylene through millimeter size channels in monolithic alumina catalysts. Axial velocities up to 93 cm/s were measured in 15 mm thick 2D slices of the flow with 400 μm true in-plane spatial resolution.

Other investigators have applied MR methods to flows through real materials. Benscik and Ramanathan (2001) measured 1D profiles of local permeability in samples of Indiana limestone and North Sea oil reservoir sandstone. They measured the pressure profile for heptafluoropropane gas flowing through the rocks using a calibration of signal strength versus pressure, and they calculated the permeability from a relation between pressure gradient and permeability. Gore et al. (2001) pointed out that biological tissues are porous media and described methods for measuring the apparent diffusion coefficient of water and surface to volume ratio in both biological and non-biological porous flows. Zhang et al. (2005) used gas imaging of H and SF6 to measure diffusion across a model polymeric membrane of potential interest as a gas separator in metal hydride batteries. Recently, Harel et al. (2006) studied the flow of 129Xe gas through a silica aerogel.

Finally, in an older paper included to motivate new applications, Tseng et al. (1998) presented early results for using low magnetic fields (21 G) and hyperpolarized 3He to image voids in porous and grannular media and fissures and cavities inside or near paramagnetic and metallic objects. The use of low magnetic field strength provides the advantage that it significantly reduces susceptibility artifacts making it more practical for imaging in porous materials and around paramagnetic and metallic objects.

7 MR thermometry

Magnetic resonance imaging is capable of making non-invasive temperature measurements and is currently being used clinically for monitoring hypo- and hyperthermia during tumor treatment. MR thermometry can measure 3D temperature maps in fluids and can be combined with flow imaging to study heat transfer. Three-dimensional temperature measurements are also being used for applications in food science (Nott and Hall 2005c; Shaarani et al. 2006).

There are three methods for measuring temperature, all based on the temperature dependencies of different MR observables. These temperature sensitive quantities are the diffusion coefficient, the T1 spin lattice relaxation time, and the proton resonant frequency (PRF) shift [sometimes referred to as chemical shift (CS)]. Several papers compared these techniques in phantoms and tissues (Bertsch et al. 1998; Nott et al. 2000; Wlodarczyk et al. 1999), and concluded that the method based on the PRF shift is the most accurate and easiest to apply. Wlodarczyk et al. showed that MR temperature measurements in a gel phantom with each of these techniques can be made with uncertainties of 1°C or less (0.4°C for PRF shift) provided that accurate calibrations are performed.

Methods based on the diffusion coefficient suffer from decreasing sensitivity with increasing temperature differences and high sensitivity to motion, qualities that make this technique difficult to use in flow experiments. Relaxation methods [also referred to as inversion recovery (IR)] based on T1 also suffer from signal losses due to flow and decreasing SNR with increasing temperature. However, Ogawa et al. (2000) described an IR tagging method for simultaneous flow and temperature measurements. The method utilizes MR tagging methods to measure the flow while using T1 relaxation for the temperature measurements. The results in slow laminar flow are good; however, the use of MR tagging in highly turbulent flows is difficult because of the rapid tag dispersal.

The PRF shift is due to a molecular shielding effect on the local magnetic field around a nucleus that is linearly proportional to temperature. This shift has a coefficient of α ≈ −0.01 ppm/°C in water over a fairly large temperature range, 0–80°C (Hindman 1966). As described by Ishihara et al. (1995), MRI can measure temperature from the phase of the MR signal using the equation,
$$ T - T_{{\rm ref}} = {(\phi - \phi_{{\rm ref}})} \mathord{\left/{\vphantom {{(\phi - \phi_{{\rm ref}})} {{\left({\gamma \alpha B_{0} TE} \right)}}}} \right.\kern-\nulldelimiterspace} {{\left({\gamma \alpha B_{0} TE} \right)}},$$
(40)
where ϕref is the phase in a reference scan from a fluid at a reference temperature Tref. Because the shift coefficient is small, accurate temperature measurements depend on eliminating phase signals unrelated to temperature. Subtracting the phase in the reference image should reduce these errors. However, extraneous phase angles can arise from drift in the magnetic field and field gradients or from distortions in the field caused by changes in temperature dependent magnetic properties in the imaged fluid or object. Typically, stationary reference objects are included in the images from which the changes in baseline phase maps can be measured and corrected. Rieke et al. (2004) presented a correction method for in vivo imaging that can be applied to in vitro experiments.

The temperature and velocity in non-isothermal flow can be measured using the phase of the MR signal. A velocity phase map is measured in a flow at an isothermal reference temperature and subtracted from a second velocity phase map of the heated or cooled flow to obtain the temperature field. It is assumed that the velocity profile is the same for both temperature conditions. Of course, the temperature dependence of the viscosity of water or other fluids complicates this method since the isothermal reference and heated flows will have different Reynolds numbers and possibly different velocity fields. Sun and Hall (2001) used phase measurements to measure radial profiles of velocity and temperature in 2D planes in the laminar heated starting length flow in a circular cylinder. Elkins et al. (2004) presented velocity and temperature profiles from 3D measurements in fully developed turbulent pipe flow.

The phase mapping technique requires accurate knowledge of the reference phase. Alternative techniques used in MR spectroscopy and reviewed by Kuroda (2005) offer the possibility of making CS measurements using internal references in a gel or fluid to determine temperature change. One big disadvantage of these methods is they typically require long scan times compared to phase mapping. Samson et al. (2006) used the CS method to investigate the minimum detectable temperature difference in a phantom containing DSS sodium salt as a chemical shift reference. They found that detecting temperature differences less than 0.3°C is realistic, and the minimum detectable difference is 0.1°C.

8 Post-processing MRV data

Magnetic resonance velocimetry efficiently acquires large amounts of data in short amounts of time. Such large amounts of data can be difficult to analyze and process. In addition, the quality and quantity of the data provide the potential for calculating wall shear rates and relative pressure fields if accurate processing methods are available. This section reviews several papers concerned with these and other post-processing issues.

Relative pressure information is useful for analyzing pressure losses in internal fluid passages as well as trying to understand the pressure forces within the heart or an aneurysm. Tyszka et al. (2000) used time-resolved 3D PC-MRI data to calculate the relative pressure field for pulsatile flow in a compliant tube and in the descending aorta of a healthy volunteer. Tasu et al. (2000) used a time-resolved 2D Fourier acceleration encoding sequence to measure the total flow acceleration and calculate pressure gradients in the pulsatile flow through a compliant tube and the aorta of a healthy volunteer. They presented pressure gradient maps in an axial cross section of a compliant tube, an axial cross section of the ascending aorta, and the plane of the aortic arch. Ebbers et al. (2002) measured the velocity field in the left atrium and ventricle of a healthy human heart using time-resolved PC-MRI and calculated the time varying pressure field. Thompson and McVeigh (2003) used a time-resolved 2D sequence to acquire in-plane velocities from which they calculated the in-plane pressure differences. With the appropriate choice of the 2D plane, physiologically relevant pressure differences can be calculated using short scans that can be completed with two breath-holds. They validated their calculations with invasive pressure catheter measurements in vitro and in a canine heart. In vivo results are presented for a human heart using slices in the standard three chamber image orientation. Nasiraei-Moghaddam et al. (2004) investigated the effects of noise, imaging resolution, and flow rate on the accuracy of intravascular pressure calculations from PC-MRI velocity measurements made in flows through phantoms with 50, 75, and 90% area stenoses. Buyens et al. (2005) calculated 2D maps of pressure distributions from 2D acceleration measurements in the left ventricles of five healthy volunteers.

Wall shear rates (WSR) or wall shear stresses (WSS) are of particular interest to those studying cardiovascular flows since there is a direct link between WSS and atherosclerosis. Robertson et al. (2001) measured time-resolved velocities in 2D axial slices of pulsatile cylindrical tube flow and calculated WSS. They evaluated the accuracy using analytic models based on the Womersley solution. Cheng et al. (2002) made time-resolved 2D measurements in vitro using pulsatile flow through a tube and in vivo in the supraceliac and infrarenal regions of a healthy aorta. They calculated WSS from Lagrangian interpolation functions fit to elements close to the tube and vessel walls. Wu et al. (2004) calculated wall shear rates from multi-sectored 3D paraboloid fits to time-resolved 2D measurements in normal carotid, brachial, and femoral arteries. Marshall et al. (2004) acquired time-resolved 3D PC-MRI measurements in models of healthy and stenosed carotid bifurcations and calculated WSS. They compared their results with CFD values for the same flows and geometries.

Visualization of MRV data, especially the time-dependent 3D three-component velocity fields, is computationally intensive and good visualization software is a necessity. Several papers used a commercial 3D visualization software called EnSight (CEI, Morrisville, NC, USA) to view vector fields and contours in user specified cross planes and to create 3D streamlines and particle traces. Wigstrom et al. (1999) acquired time-resolved 3D velocity data for a volume surrounding an entire human heart and reported visualization techniques for the pulsatile intracardiac flow using EnSight. Markl et al. (2004) presented visualization results for time-resolved 3D PC-MRI measurements in healthy and diseased thoracic aortas. Similarly, Markl et al. (2005) showed helical flow, vortices, and other complex flow structures in patients with valve repairs. Bammer et al. (2007) used EnSight to visualize time-resolved 3D flow in the intracranial blood vessels (see Fig. 8.)

While EnSight provides many useful visualization tools, its methods for calculating streamlines and particle traces are hidden to the user. Other investigators have developed their own methods for generating streamlines. Yang et al. (2000) presented a method for noise removal and the formation of transient streamlines. They applied this method to process time-resolved velocities measured in a normal right atrium, dilated left ventricle, and ascending aortic aneurysm. Fatouraee and Amini (2000) presented an algorithm that enforces continuity to improve the accuracy of PC-MRI data. They visualized data from the pulsatile flow through a model of an axisymmetric abdominal aneurysm, and they compared results to simulations made using Fluent, a commercial CFD code.

Visualizing flow or motion in particular regions of the images can be important and requires methods for accurate segmentation. Oelhafen et al. (2006) developed auto segmentation methods for tracking real time vessel distortion and motion during reactive hyperemia, the instantaneous increase in blood flow in response to the sudden removal of a flow blockage.

As described in Sect. 2, PC-MRI measurements have some inherent inaccuracies that can be reduced with appropriate correction schemes. In most PC measurements, it is assumed that the acceleration of the flow is negligible during the time it takes to make the measurement. As previously discussed, this can lead to displacement artifacts. Thunberg et al. (2000) provided a correction scheme for this effect and demonstrated its utility in an in vivo flow through a repaired aortic coarctation. Flow profiles and the images of vessel walls acquired with PC-MRI suffer from distortion and blurring due to limitations in the sampling of k-space. The methods of Lagerstrand et al. (2002) improve the accuracy of measurements of vessel size and flow rate by removing these effects. They demonstrated their method with in vitro flow data and velocity fields acquired in a healthy human descending aorta.

In analyzing MR velocity data, better resolution than that in the original images is often desirable. For instance, one may want use contiguous 2D slices of data to create a 3D velocity field with improved interslice resolution. Frakes et al. (2004) described a method for creating the 3D velocity field and presented results for flow in the total cavopulmonary connection.

Methods for statistical analysis are necessary to quantitatively compare MR with other types of data. Box et al. (2002) provided error analysis based on the Kolmogorov-Smirnov method and quantitatively compared 2D PC-MRI velocity profiles and finite-element method (FEM) results in the flows through a curved tube and a carotid bifurcation model. Iaccarino and Elkins (2006) evaluated MRV and CFD results with traditional methods such as comparing velocity vectors point by point in a 2D plane and comparing velocity profiles along lines in the 3D geometry. Certainly, more sophisticated methods are needed to help facilitate the evaluation and comparison of full 3D data sets.

9 Summary

The purpose of this review is to introduce the diverse capabilities of MRV, to provide a general description of the principles behind MRV, to discuss its accuracy and limitations, and to review the applications of MRV in these areas of fluid mechanics research: turbulent flows in complex geometries, physiological flows, multiphase flows, and flows through porous media. The topic of MR thermometry is included due to its potential use in convective heat transfer applications.

Magnetic resonance velocimetry techniques possess many possibilities and advantages. Flow quantities that can be measured include mean velocities, Reynolds stresses, and diffusion coefficients and tensors. MRI methods can differentiate between multiple liquids, gases, and solids. The measurements are non-invasive and can be made in opaque fluids in and around opaque objects. Data acquisition is fast and can cover extremely complicated 3D geometries. With MRV, it is possible to capture a complete picture of a realistic flow within a realistic geometry. Finally, these methods can be implemented in clinical MR systems which are becoming increasingly abundant at universities, hospitals, and imaging centers around the world.

The field of MRI has seen exciting new developments just in the last few years which will broaden its impact on the study of fluid mechanics significantly. Recent experiments demonstrate the ability to study gas flows and turbulent flows in realistic geometries. For decades, engineers and scientists have studied simplified flows meant to represent more complicated real flows. MRI has great potential for providing advanced measurement techniques to study real flows.

Acknowledgments

The authors wish to thank Professor John K. Eaton and Professor Norbert Pelc for their helpful discussions. Support for Christopher J. Elkins was provided by a grant from General Electric Aircraft Engines as part of the GE-University Strategic Alliance. Support for Marcus Alley came from a National Institutes of Health grant (P41 RR09784). Both authors were also supported by the National Science Foundation under grants CTS-0432478 and OCE-0425312.

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© Springer-Verlag 2007