Precision of Magnetic Resonance Velocity and Acceleration Measurements: Theoretical Issues and Phantom Experiments

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1 JOURNAL OF MAGNETIC RESONANCE IMAGING 3: (200) Original Research Precision of Magnetic Resonance Velocity and Acceleration Measurements: Theoretical Issues and Phantom Experiments Emmanuel P. Durand, MD, MSc,* Odile Jolivet, MSc, Emmanuel Itti, MD, MSc, Jean-Pierre Tasu, MD, PhD, and Jacques Bittoun, MD, PhD Magnetic resonance (MR) sequences have been developed for acquiring multiple components of velocity and/or acceleration in a reasonable time and with a single acquisition. They have many parameters that influence the precision of measurements: N S, the number of flow-encoding steps; NEX, the number of signal accumulations; and N D, the number of dimensions. Our aims were to establish a general relationship revealing the precision of these measurements as a function of N S,N D, and NEX and to validate it by experiments using phantoms. Previous work on precision has been restricted to two-step (N S 2) or D (N D ) MR velocity measurements. We describe a comprehensive approach that encompasses both multistep and multidimensional strategies. Our theoretical formula gives the precision of velocity and acceleration measurements. It was validated experimentally with measurements on a rotating disk phantom. This phantom was much easier to handle than fluid-based phantoms. It could be used to assess both velocity and acceleration sequences and provided accurate and precise assessments over a wide, adjustable range of values within a single experiment. Increasing each of the three parameters, N S,N D, and NEX, improves the precision but makes the acquisition time longer. However, if only one parameter is to be assessed, maximizing the number of steps (N S ) is the most efficient way of improving the precision of measurements; if several parameters are of interest, they should be measured simultaneously. By contrast, increasing the number of signals accumulated (NEX) is the least efficient strategy. J. Magn. Reson. Imaging 200;3: Wiley-Liss, Inc. Index terms: magnetic resonance imaging; phase contrast; velocity; acceleration; phantom; precision Abbreviations: rpm revolutions per minute, bpm beats per minute, SNR signal-to-noise ratio, SD standard deviation. MANY SEQUENCES FOR MEASURING VELOCITY AND ACCELERATION have been developed and can now be used in routine imaging. Data on several components of CIERM U2R2M ESA 808, Hôpital de Bicêtre, Le Kremlin-Bicêtre, France. Presented as a poster at the 7 th Annual Meeting of the International Society for Magnetic Resonance in Medicine, Philadelphia, 999. *Address reprint requests to: E.D., CIERM U2R2M ESA 808, Hôpital de Bicêtre, 78 rue du Général Leclerc, F Le Kremlin-Bicêtre, France. emmanuel.durand@cierm.u-psud.fr Received June 5, 2000; Accepted October 30, motion can be acquired simultaneously and within a reasonable time ( 3). In such sequences, many parameters have to be set and optimized to achieve maximum precision and accuracy in the minimum time. Theoretical considerations and experimental validation both have a part to play in achieving these goals. Throughout this work the term steps will be used for velocity (or acceleration)-encoding steps, and the term dimension will be used to denote the number of measured velocity (or acceleration) components. The numbers of steps and dimensions will be referred to as N S and N D, respectively. In 990, Conturo and Smith showed that the theoretical SD of a phase measurement is: I I where I is the signal magnitude (4). Their analysis of the phase noise in a stationary phantom matched their formula for a two-step D velocity measurement (Fig. a). In 99, Pelc et al described three strategies for making two-step 3D measurements (): a 6-point strategy (Fig. b) consisting of three independent two-step D measurements; a 4-point strategy (Fig. c), which differed from the former by a common point; and a balanced 4-point strategy (Fig. d), which used oblique gradients to maximize the gradient strength. Their theoretical analysis showed that a balanced 4-point strategy is the most efficient. They tested their results by computer simulation. In 997, Lamothe and Rutt (5) studied the N-step/D method (Fig. e). They showed that the SD of the velocity is inversely proportional to N S. They confirmed these results on flow experiments repeated 5 times in a straight-tube phantom. They concluded that increasing the number of steps is a very effective way of increasing the precision of the velocity measurement. However, other strategies, such as true multidimensional acquisitions (Fig. f) (3) or N-step multidimensional acquisitions (Fig. g), are also possible. The first aim of this work was therefore to establish a general formula for estimating the precision of multiple-compo- () 200 Wiley-Liss, Inc. 445

2 446 Durand et al. N S N S k N S N S (5) The contributions of the intermediate steps are used to avoid aliasing but have no influence on the variance. Hence, the velocity can be calculated as: Figure. Strategies for velocity (or acceleration) acquisition. The velocity (or acceleration) components are represented in the usual geometrical space: two-step, D (as studied by Conturo) (a); two-step, 3 D (6-point, according to Pelc) (b); twostep, 3 D with a shared point (4-point, according to Pelc) (c); 2-step, 3 D with a shared point (balanced 4-point, according to Pelc) (d); three-step, D (as studied by Lamothe) (e); two-step, true 3D (8-point) (f); and three-step, true 2D (9-point) (g). V g NS N S g The same analysis applies to acceleration with g A t 2 G(t)dt, being the total duration of the three-lobe 2 0 gradient pulse. Furthermore, in multidimensional acquisitions (N D ), when the whole acquisition is repeated NEX times for averaging, the difference NS for the velocity component in one direction is measured N times with (6) N NEX N S N D (7) nent sequences. The second aim was to test it against direct measurements on a phantom. We used a rotating phantom to obtain a direct assessment of the variance of measurements within a single experiment. MATERIALS AND METHODS Theoretical Precision of the Motion Measurements To encode one velocity (or acceleration) component, a bipolar (or tripolar) gradient is added to a standard imaging sequence. The acquisition is repeated N S times while this flow-encoding gradient is gradually increased, exactly as the space-encoding gradient used in standard imaging. Using a Fourier transform with zero padding allows a reduction in the number of steps (N S ) and hence a shortening of the acquisition time, assuming that the motion is uniform within each voxel (2). One can average several (NEX) acquisitions to increase the signal-to-noise ratio (SNR). Hence, the total acquisition time is T ACQ NEX N S N D t ACQ (2) where t ACQ is the acquisition time for a single image and N D is the number of velocity-encoded dimensions. If the step k (k ranging from to N S ) of the encoding gradient has a total duration and a magnitude G k (t), the velocity-related dephasing is k g k V (3) where g k 0 t G k (t)dt is the first moment of the gradient-to-time curve and V is the velocity. The phase difference between two consecutive steps is k V g k g k V g (4) where g is the first moment of the gradient increment. As shown by Lamothe and Rutt (5), computing the velocity is equivalent to considering the average phase difference : Indeed, each set of acquisitions for the other (N D ) directions encloses an independent measurement of the velocity component under consideration. Thus, these other (N D ) directions play the role of signal accumulations. Let us assume that, as usual, for a single acquisition, the real and imaginary images follow Gaussian laws centered on S R and S I, respectively, with the same uncorrelated SD (white noise). If is small compared to the amplitude M S 2 R S 2 I, then the variance of the phase angle tan S I S R (6) is: VAR 2 M 2 (8) From Eq. [6], [7], and [8], we can deduce that the variance for the velocity is: 2 2 VAR V N S g M 2 N NEX N D (9) S Finally, the SD for a velocity measurement is: 2 SD V N N S g M NEX N D /2 (0) S For the measurement of a single-component (N D ), Eq. [0] turns into the formula published by Lamothe and Rutt (5): V NEX N S () and for the two-step measurements (N S 2), Eq. [0] turns into: 2 SD V (2) g M NEX 2 ND /2

3 Precision of MR Velocity and Acceleration Measurements 447 Figure 2. Diagram showing the velocity V x on a rotating disk. The projection of the velocity along the x axis (V x ) is a constant on any horizontal line (y y 0 ): V x y 0. For single-dimension, two-step measurements, Eq. [0] gives the formula published by Pelc et al for the 2-point method (): 2 SD V g M NEX (3) One advantage of this phantom is that the velocity V x is a constant on every horizontal line (y y 0 ). Thus, to improve the precision of V x, the values can be averaged on such lines. Moreover, because V x is constant, its SD gives the precision of the measurement on this line in a single experiment (Fig. 2). According to Eq. [0], the SD does not depend on the velocity but on the encoding gradient g. Thus, the theoretical SD is the same over the whole phantom. The SD can therefore be averaged over the whole disk to obtain a very accurate assessment of measurement precision. The phantom was a 20-mm-thick, 270-mm-diameter cylinder made of Altuglass with Nylon screws and filled with an agar gel (T 200 ms, T 2 40 ms). It was moved by a steady speed motor. Except for this motor, the system was made entirely of nonmetallic, diamagnetic materials. Because our magnet was unshielded, the phantom was connected to the motor via a 3-m long axle made of PVC piping. This prevented both magnetic field damage to the motor and disturbances by the motor to the imager. This axle was supported by a glass ball bearing and held in place at its center by an antivibrating device made from three Teflon cylinders. This whole system was fixed to a long wooden board (Fig. 3). The motor was a brushless analogue servo drive motor DXM 208C, controlled by an LX 400 amplifier card from Emerson EMC (Chanhassen, MN). The maximum speed was 3000 rpm with a continuous torque of.3 Nm. A gearbox reduced its speed by a factor of 0. The motor was controlled by a continuous voltage from a stabilized power supply (from 0 to 5 V reduced to 0 V by a steady resistor wedge), the velocity of the motor being proportional to the input voltage. The amplifier card provided a transistor transistor logic (TTL) signal for each revolution of the motor. An LS90 TTL decade counter was used to record signal in every 0 to compensate for the gearbox. The signal amplitude was then reduced and connected to the electrocardiogram (ECG) leads, so providing the imaging system with a For Pelc s 4-point strategies (cf. Fig. c and d), the acquisition time would be T ACQ 4 NEX t ACQ and could be extended to T ACQ NEX ( N D (N S )) t ACQ for the general case of a common reference point. However, the measurement along one direction would carry no information for the other directions (N NEX). These techniques should therefore be considered as sharp schemes to repeat monodimensional acquisition, rather than true multidimensional ones. Phantom Measurements We used a phantom made of a rotating disk to obtain calibrated velocities (V) and accelerations (A). In such a rotating phantom, the in-plane velocity and acceleration components at any point of this object (x,y) are easily computed from the known angular velocity : V x y V y x and A x 2 x A y 2 y (4) Figure 3. Photographs of the phantom: the gel-filled cylinder (a) and the phantom inside the imager (b).

4 448 Durand et al. Figure 4. The four-step acquisition sequence measuring acceleration A x. gating signal for each revolution of the cylinder. The angular velocity of the phantom could be determined by the voltage delivered to the card (which was measured with a 0.5% precision digital voltmeter) and by the gating data on the imaging console, where 65 bpm corresponded to 65 rpm for the phantom (with a rpm precision). These methods were checked by manual counting over a timed period. The phantom provided a continuous range of adjustable velocities and accelerations from zero up to maximum values of 340 cm s and 8500 cm s 2. Magnetic Resonance Acquisitions All images were acquired on a.5 T magnetic resonance (MR) imager (GE Signa, Milwaukee, WI) with the standard body coil and conventional gradient coils. The axis of the phantom was aligned with the direction of the principal magnetic field. Transaxial monoslice images were acquired for in-plane (x,y) motion with a 40-cm field of view and a spatial matrix with a 60 flip angle gradient echo sequence (Fig. 4). Preliminary experiments were undertaken to measure either the in-plane velocity or acceleration in the phantom at 8, 40, 60, 20, 80, and 240 rpm with D flow-measurement sequences. The influence of gating was also tested at 20 rpm with a similar TR in both cases. We also checked that the motor induced no detectable noise in the imager. To assess the effects of the parameters N S,N D, and NEX on measurement precision, several experiments were carried out at 80 rpm (TR 333 ms, TE 20 ms) using various values of N S (from 2 to 8) and NEX (from to 4). The dependence of N D was also studied by acquiring successively images of V y alone (N D ), V x and V y (N D 2), V x, V y, and V z (N D 3), and V x, V y, A x, and A y (N D 4). Though using 3D velocity sequences was useful to simulate in vivo measurements, V z data were not exploited because the out-plane velocity is zero for such a phantom. Image and Data Processing The data processing comprised an autocorrelation of the phase evolution (3). This is equivalent to zero-padding Fourier transformation of the raw images in all directions to derive velocity (V x /V y /V z ) and acceleration (A x /A y /A z ) maps (2). After this processing, the image was subjected to bimodal segmentation to eliminate all pixels outside the phantom. As the readout encoding did not take place at the same time as the phase encoding, the image was distorted by a spatial misregistration (or displacement) artifact (7 9). Indeed, between the phase-encoding gradients and the signal readout, each pixel moves a V x T distance, where T is the delay between the encoding gradient center and the maximum of the echo center (8). This artifact was corrected by shifting the x position by ( - V x T) for each pixel of the image. To get an exact value of V x, we computed the theoretical V x from the known angular velocity (determined by the gating signal) and the radius from the mass center of the registered image (as the center of rotation). Such a correction is easy with this phantom because V x is known exactly for each point. Each parameter (e.g., V x ) was measured at every point (x,y) of the disk. Then, from each line of constant value (e.g., horizontal line y y 0 for the parameter V x ), the average and SD were calculated. The average (e.g., V X (y 0 )) was compared to the theoretical value (e.g., y 0 ) for accuracy. The SD (e.g., VX (y 0 )) represents the measurement precision on this line. Equation 8 states that V does not depend on the value of the velocity, but only on the signal value. As the phantom is homogeneous, the signal, and thus V, are assumed to be similar throughout the phantom. Therefore, the SD on every line could be averaged over the whole disk to yield a global estimate of the precision: N VX VX y 0 /N (5) y 0 Hence, this global SD, derived from a single experiment, is the average of about 00 SD, each one computed from about 00 velocity measurements. All the processing was done on an SPARC 20 workstation (Sun Microsystems, Mountain View, CA) with custom C software. RESULTS Preliminary Studies Figure 5 shows the acceleration A x images before (Fig. 5a) and after (Fig. 5b) correction and segmentation. The misregistration artifact, which is important at this high velocity (240 rpm), has been properly corrected. For each experiment, Table shows the effective angular velocity as determined by the input voltage and the gating, the maximal linear velocity, the field of speed, the slope of the regression line, the SD of the measurement, and the velocity resolution. Table 2 shows the same results for acceleration. Acceleration could not be determined reliably for angular velocities below 40 rpm because the acceleration-to-noise ratio was too low (acceleration under 50 cm s 2 ). For the other experiments, the results were similar to those for velocity. Tables and 2 show a good agreement between measurement and theory, irrespective of velocity. The

5 Precision of MR Velocity and Acceleration Measurements 449 Figure 5. Velocity images. Medium grey indicates zero velocity, lighter grey shades indicate positive velocities, and darker grey shades indicate negative ones. The readout axis appears horizontal in the image. a: A x at 240 rpm before correction and segmentation. The displacement artifact is clearly seen. Random noise values are present outside the phantom. b: A x at 240 rpm after correction and segmentation. The displacement artifact is now suppressed. c: V x at 20 rpm without gating before correction and segmentation. Motion artifacts impair the image. d: V x at 20 rpm without gating after correction and segmentation. Artifacts are still visible. squared correlation coefficients (r 2 ) were always over The slopes of the regression lines were very close to (under 2% error), and the intercepts were always close to 0. These error values remained close to the uncertainty about the true angular velocity; for example, 40 rpm was determined with a rpm precision, i.e., 2.5%. These data demonstrate that the measurements were accurate. The relative precision of measurements increased at higher velocities and accelerations. At 8 rpm, the mean SD for V x was 0.94 cm s (about 5% of maximum velocity), whereas it was 3. cm s (about % of maximal velocity) at 240 rpm. Without gating (the 20 rpm experiment), the quality of the images was greatly reduced by motion artifacts, as shown in Fig. 5c (before correction and segmentation) and 5d (after correction and segmentation). The motion artifacts were clearly visible both outside and inside the phantom, and V x was not constant on the horizontal lines. Moreover, the SD of velocity (V x ) and acceleration (A x ) without gating were biased by motion artifacts on the disk image: 3. cm s and 38 cm s 2, as compared to.0 cm s and 24 cm s 2, respectively, for the gated 20 rpm experiment. Unless the gel can be made perfectly homogeneous, the use of a rotating phantom therefore requires gating. Precision Studies The global SD of V x was determined for each strategy to test Eq. [0]. Figure 6a shows this SD plotted vs. the Table Numerical Results of the Velocity Measurements Angular velocity (rpm) Theoretical maximal linear velocity (cm/s) Field of speed (cm/s) Slope of the regression line for V x Slope of the regression line for V y Intercept for V x (cm/s) Intercept for V y (cm/s) Mean S.D. for V x (cm/s) Mean S.D. for V y (cm/s) Velocity resolution for V x (cm/s/pixel) Velocity resolution for V y (cm/s/pixel) The slopes and intercepts are those of the regression line of the measured values averaged on each line and plotted against the theoretical values. The standard deviations are calculated for each line, for which the velocity is constant, then averaged over the whole phantom. Table 2 Numerical Results for the Acceleration Measurements Angular velocity (rpm) Theoretical maximal linear acceleration (cm/s 2 ) Field of acceleration (cm/s 2 ) Slope of the regression line for A x Slope of the regression line for A y Intercept for A x (cm/s 2 ) Intercept for A y (cm/s 2 ) Mean S.D. for A x (cm/s 2 ) Mean S.D. for A y (cm/s 2 ) Acceleration resolution for A x (cm/s 2 /pixel) Acceleration resolution for A y (cm/s 2 /pixel) The slopes and intercepts are those of the regression line of the measured values averaged on each line and plotted against the theoretical values. The standard deviations are calculated for each line, for which the acceleration is constant, then averaged over the whole phantom.

6 450 Durand et al. DISCUSSION We have established a general relationship for assessing the precision of MR velocity and acceleration measurements using three parameters: N S, the number of encoding steps; N D, the dimension number; and NEX, the number of signals. MR experiments carried out on a rotating phantom showed that precision can be properly assessed by Eq. [0]. These experiments thus constitute an experimental validation of the formula. Each experiment included the measurement of about 00 SD, each of these being based on about 00 different measurements. This technique is more efficient for assessing precision than for simply repeating the experiment. Moreover, in trying to assess precision, simply repeating flow experiments introduces uncertainties over the velocity stability between experiments. Increasing each of these three parameters improves the precision, but it also increases the total acquisition time T ACQ. Thus, one can obtain an efficiency coefficient for the sequence by calculating the inverse of the product of the variance and the acquisition time (). This coefficient is: E 2 (6) T ACQ VX E NEX N S 2 N N D S N N D S NEX (7) E N S 2 N S N S 2 N S (8) Figure 6. Precision results. For each of these three parameters, the measured velocity SD is plotted against its theoretical dependence according to Eq. [0]. The regression line and the squared correlation coefficient are also indicated. a: N S, number of steps. b: N D, number of dimensions. c: NEX, number of averaging excitations. This coefficient shows that the number of encoding steps (N S ) has a big influence on efficiency. Increasing the number of steps is thus the most efficient way to increase precision without unacceptably increasing the acquisition time. This efficiency coefficient does not depend on the averaging number (NEX) or on the dimension number (N D ); so increasing NEX and N D does not change the efficiency (i.e., the square of velocity-tonoise ratio increases as the acquisition time increases). However, there is a powerful advantage in using multidimensional sequences, as opposed to increasing the NEX, because for the same acquisition time and precision one can get several motion parameters from a single acquisition. So, when focusing exclusively on precision, a multistep strategy is the best way of proceeding, but when several parameters are required, a multidimensional sequence should be considered. On the other hand, averaging is not an efficient strategy for the acquisition of motion components. As an example, the sequences indicated in Table 3 were all acquired in variable / N S (N S ), i.e., its theoretical dependence on N S, for a two-component strategy: N D 2, NEX. The graph shows a very strong linear correlation (r ). The theoretical dependence on N D (Fig. 6b) was tested by plotting the global SD vs. / 2 ND. Fig. 6c also shows a very high correlation (r 2 over 0.98) between and / NEX for three different single-component strategies (N D ). Table 3 Three Strategies With the Same Acquisition Time N S steps NEX N D Measured parameters S.D. of velocity A 2 2 V X 0 B 2 2 V X and V Y 0 C 4 V X 0 / 2

7 Precision of MR Velocity and Acceleration Measurements 45 the same time. The multistep sequence (C) offers the best precision. The averaging sequence (A) and the 2D sequence (B) have a similarly lower precision, but the latter yields two velocity components. Rotating phantoms, filled with either fluid (0 2) or gel (3 5) and moved by an electric motor (8,4) or a jet of compressed air (,2), have been used for motion tracking (3,5) or assessing the displacement artifact (8). Here, they were used to assess the precision of velocity MR measurements. Using a rotating disk as opposed to fluid-flowing phantoms to test velocimetry sequences has many advantages: the velocity distribution is stable, easily determined, and reproducible; rotating disks are easier and quicker to handle; a wide range of adjustable velocities and accelerations can be incorporated into a single experiment; and, unlike fluid phantoms, in which velocities may combine in the same voxel because of partial volume effects, in-plane velocity is not a problem. Lastly, such a rotating phantom provides assessments of measurement accuracy and precision (SD) within a single experiment. CONCLUSION This work describes a comprehensive multistep and multidimensional strategy for making MR measurements of velocity and acceleration. In assessing the precision of these measurements as a function of three parameters (N S, the number of encoding steps; N D, the dimension number; and NEX, the number of signal accumulations), a general relationship has been established. MR experiments were carried out on a rotating phantom and the results we have presented support our formula. This technique offers accurate and precise velocity and acceleration measurements from a single experiment. We conclude that the most efficient way of improving the precision of measurements is to increase the number of velocity (or acceleration) encoding steps (N S ). Increasing the number of dimensions (N D ) is not as precise. However, it has the advantage of providing several parameters from a single acquisition. Repeating acquisitions for signal averaging (NEX) is the least effective strategy because, while as efficient as taking more dimensions to improve precision, it provides only one parameter. ACKNOWLEDGMENTS We are indebted to the mechanical workshop of the Institut d Électronique Fondamentale and especially to Daniel Bouchon for building the mechanical part of the phantom. REFERENCES. Pelc NJ, Bernstein MA, Shimakawa A, Glover GH. Encoding strategies for three-direction phase-contrast MR imaging of flow. J Magn Reson Imaging 99;: Bittoun J, Jolivet O, Herment A, Itti E, Durand E, Tasu JP. MR mapping of multiple components of velocity and acceleration by interpolated Fourier encoding. In: Haase A, editor. ESMRMB. Brussels: Chapman and Hall; 997. p Bittoun J, Jolivet O, Herment A, Itti E, Durand E, Mousseaux E, Tasu JP. Multidimensional MR mapping of multiple components of velocity and acceleration by Fourier phase encoding with a small number of encoding steps. Magn Reson Imaging 2000;44: Conturo TE, Smith GD. Signal-to-noise in phase angle reconstruction: dynamic range extension using phase reference offsets. Magn Reson Med 990;5: Lamothe MJ, Rutt BK. Multistep phase difference phase contrast imaging. J Magn Reson Imaging 997;7: Gudbjartsson H, Patz S. The Rician distribution of noisy MRI data [published erratum appears in Magn Reson Med 996;36:332]. Magn Reson Med 995;34: Larson TCd, Kelly WM, Ehman RL, Wehrli FW. Spatial misregistration of vascular flow during MR imaging of the CNS: cause and clinical significance. Am J Roentgenol 990;55: Frayne R, Rutt BK. Understanding acceleration-induced displacement artifacts in phase-contrast MR velocity measurements. J Magn Reson Imaging 995;5: Nishimura DG, Jackson JI, Pauly JM. On the nature and reduction of the displacement artifact in flow images. Magn Reson Med 99; 22: Norris DG. Acceleration imaging by NMR. In: SMRM. London; Williams DM, Meyer CR, Schreiner RJ. Flow effects in multislice, spin-echo magnetic resonance imaging. Model, experimental verification, and clinical examples. Invest Radiol 987;22: Meyer CR, Williams DM. Bulk flow model for multislice magnetic resonance imaging sequences with phantom validation. Med Phys 988;5: Pelc NJ, Drangova M, Pelc LR, Zhu Y, Noll DC, Bowman BS, et al. Tracking of cyclic motion with phase-contrast cine MR velocity data. J Magn Reson Imaging 995;5: Nordell B, Stahlberg F, Ericsson A, Ranta C. A rotating phantom for the study of flow effects in MR imaging. Magn Reson Imaging 988;6: Lingamneni A, Hardy PA, Powell KA, Pelc NJ, White RD. Validation of cine phase-contrast MR imaging for motion analysis. J Magn Reson Imaging 995;5: