JOURNAL OF MAGNETIC RESONANCE IMAGING 19:468 – 474 (2004)
Original Research
Time-of-Flight Quantitative Measurements of Blood Flow in Mouse Hindlimbs Shawn Wagner, PhD,1* Armin Helisch, MD,1 Georg Bachmann, MD,2 and Wolfgang Schaper, PhD, MD1 Purpose: To evaluate the feasibility of using time-of-flight (TOF) imaging to directly measure hindlimb blood flow in a mouse model of peripheral vascular disease. Materials and Methods: Four tubes were imaged simultaneously (diameters ⫽ 0.39 mm, 0.59 mm, and two at 1.46 mm) with a 1.0 mM copper sulfate solution for 19 flow velocities. In vivo measurements were performed in the hindlimbs of three mouse strains—C57BL/6 (N ⫽ 5), BALB/c (N ⫽ 5), and 129S2/Sv (N ⫽ 5)—three weeks after femoral artery ligation with a calibration standard. Results: The flow phantom showed that the intensity was linear (r2 ⫽ 0.92) over the pertinent blood flow velocities in the mouse hindlimbs. Measurements of the blood flow in the distal hindlimbs in different strains of mice (combination of both the venous and arterial flows) were obtained 21 days after right-sided femoral artery occlusion. The results showed that under similar conditions of anesthesia and temperature, SV129 mice on the nonligated side had the highest flows (0.50 ⫾ 0.07 mL/minute), followed by C57BL/6 (0.28 ⫾ 0.04 mL/minute) and BALB/c (0.23 ⫾ 0.05 mL/minute), P ⬍ 0.02. The ligated side measurements (SV129, 0.31 ⫾ 0.05 mL/ minute (P ⫽ 0.02); C57BL/6, 0.21 ⫾ 0.02 mL/minute (P ⫽ 0.13); and BALB/c, 0.12 ⫾ 0.02 mL/minute (P⫽ 0.06)) showed a trend in BALB/c and C57BL/6 and significant differences in SV129 for incomplete recovery three weeks after surgery, compared to the nonligated side. Conclusion: Two-dimensional TOF imaging permits quantitative in vivo measurements of hindlimb blood flow in a mouse model of peripheral vascular disease without the need of contrast injection, offering advantages of serial imaging not limited by tissue penetration. Key Words: time-of-flight; blood flow; vascular growth; mouse; strain J. Magn. Reson. Imaging 2004;19:468 – 474. © 2004 Wiley-Liss, Inc.
1 Department of Experimental Cardiology, Max-Planck-Institute for Physiological and Clinical Research, Bad Nauheim, Germany. 2 Department of Radiology, Kerckhoff Klinik, Bad Nauheim, Germany. *Address reprint requests to: S.W., Max-Planck-Institute for Physiological and Clinical Research, Department of Experimental Cardiology, Benekestrasse 2, 61231 Bad Nauheim, Germany. E-mail:
[email protected] Received August 26, 2003; Accepted December 15, 2003. DOI 10.1002/jmri.20025 Published online in Wiley InterScience (www.interscience.wiley.com).
© 2004 Wiley-Liss, Inc.
CURRENT BIOMEDICAL RESEARCH HAS concentrated on studying disease processes in mice because of the availability of genetically altered mice and because of the convenience and lower costs associated with keeping and handling smaller-size animals. New insights into the mechanisms of collateral vessel growth may be gained from a murine peripheral hindlimb ischemia model involving ligation of the femoral artery (1). A major difficulty, however, lies in accurately measuring physiological parameters in mice, like peripheral blood pressure and collateral-dependent blood flow. Quantitative measurements of distal hindlimb flow have not been reported, even though it would be a very important endpoint for assessing the development of the collateral circulation after arterial occlusion. Time-of-flight (TOF) experiments have been employed in the past using fast low angle shot imaging (FLASH) (2), multiple spin echo imaging (MSE) (3), and echo-planar imaging (EPI) (4) to assess blood flow. We will demonstrate that gradient echo TOF imaging can be utilized effectively to measure blood flow in small animals where noninvasive quantitative measurements are not currently available with any other technique. Some magnetic resonance imaging (MRI) techniques based on measuring the arrival time of injected contrast boluses have also been utilized in a pig coronary occlusion model (5) and a mouse hindlimb ischemia model (6). However, these methods require the administration of contrast agents. In mice, tail and jugular vein injections are difficult and result in local damage of the veins precluding serial studies. Furthermore, imprecision of the measurements may occur due to dispersion of the contrast bolus from the long tubing required to inject the animal while in the magnet. Experiments to measure the myocardial perfusion in rats have been accomplished using regional blood volume maps with and without contrast injections (7,8) and spin-labeling (9) techniques. We propose that two-dimensional TOF gradient echo sequences with large flip angles are advantageous since they incorporate spin labeling into a system with high fluid movement, resulting in adequate signal-to-noise ratios (SNRs). We will show that two-dimensional TOF is a valuable method of measuring blood flow in mouse hindlimb vessels. The data presented will further advance techniques and demonstrate that two-dimensional TOF measurements provide reliable assessment
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of blood flow recovery in a mouse hindlimb ischemia model without the use of contrast agents. Two-dimensional TOF imaging saturates a single image slice through the use of large flip angles and fast repetition times (10). The acquired signal is then highly contrast weighted from motions of fluid and blood entering the acquired slice in the interval between signal acquisition pulses. While this technique is not entirely new (11–14), the application advantage for small animals has not been explored. Quantitative imaging of small blood vessels becomes possible under conditions that limit the effects of artifacts. Through the selection of proper slice thickness and under conditions where the linear flow rate does not exceed or approach the slice thickness, flow velocities can be determined. A reevaluation of this method is necessary as a quantitative technique that has been previously disregarded for use in regions of high blood flow as a result of in-plane flow dropout and other reported artifacts from pulsatile flow, streaming, and signal misregistration (15–18). While arguments of nonlinear flow and artifacts in large animals and humans have merit, we show that in mice two-dimensional TOF is a useful method to measure blood flow in the peripheral vasculature with sufficient SNRs and limited artifacts. MATERIALS AND METHODS Imaging Coil An eight-leg 2.8-cm-diameter by 3.2-cm-length lowpass inductively coupled birdcage coil (19) was specially designed for mouse experiments and incorporated a water heating system in the probe housing to maintain a temperature of 37°C inside the magnet. The coil consisted of eight 5.1 pF capacitors (ATC multilayer ceramic) in the legs of the coil, resulting in a quality factor of 180. A highly sensitive probe was required to obtain sufficient SNRs for mouse experiments.
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Figure 1. Axial slice excitation in a tube or vessel filled with a moving solution. A: No flow results in a signal reduction or loss in the excitation/detection plane. B: Flowing solution results in an increase in signal from perturbed water moving out of the detection slice and unperturbed water entering the slice plane between excitation pulses.
by acquiring single lines in k-space every 22 msec with 90° pulses and echo times of 8 msec (Fig. 1) with a gradient echo sequence. The linear flow through the four tubes was calculated by measuring the volume of solution over a determined time period and dividing by the cross-sectional area. Flow intensities, Itotal, for the tubes were determined by a summation of the voxel intensities, In, in the area of the tube cross section minus an equivalent background value, bkg, taken from an adjacent area to approximate the solution background, which cannot be obtained from the images. These intensities were then divided by the cross-sectional area of the inner part of the tubing, Eq. [1].
Flow Phantom The flow phantom consisted of four tubes connected in series with inner diameters of 0.40 mm, 0.58 mm, and two of 1.46 mm. Connecting the tubes in series guarantees that there will be an equilibration of volumetric flow, while the varying diameters of the individual tubes will ensure different linear flow rates. Long tubing segments were left between the flow tubes of different diameters to allow for equilibration of proton nuclei of the solution before entering the joint detection and excitation plane. The tubes of equal diameters served as a control to verify that the same results can be obtained in different positions. TOF images were acquired in a 7.0-Tesla imager (Bruker PharmaScan 70/16) equipped with a 300 mT/m gradient for a square field of view (FOV) of 1.28 cm with a two-dimensional matrix of 128 ⫻ 128. Slice planes with widths of 0.62, 1, and 1.5 mm were acquired perpendicular to the axial flow of an aqueous 1.0 mM copper sulfate solution used to simulate the T1 and T2 values for blood (18). Flow velocities were varied by altering the height of a water supply reservoir and adding restrictions to the water outflow tube. The acquisition of the image plane was performed
冘 I total ⫽
共I n ⫺ bkg兲
n
Area
(1)
Femoral Artery Ligation and In Vivo Measurements Animals were handled in accordance with the American Physiological Society guidelines for animal welfare and the Bioethics Committee of the State of Hessen, Germany. Mice of three different strains were studied: C57BL/6 (N ⫽ 5), BALB/c (N ⫽ 5), and 129S2/Sv (N ⫽ 5). Narcosis for the surgery was achieved by intraperitoneal injection of xylazine (20 mg/kg) and ketamine (110 mg/ kg). The right femoral arteries were occluded by ligation with 6-0 silk sutures immediately distal to the origin of the deep femoral artery and proximal to origin of any other distal branch, like the a. genus descendens. Twenty-one days after occlusion measurements were performed with animals under narcosis with intraperitoneal injection of xylazine (15 mg/kg) and ketamine
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Figure 2. Theoretical plot for the acquired intensity for a tube or vessel using a two-dimensional TOF acquisition sequence with a linear bolus flow. A maximum intensity is reached when the flow velocities equal or exceed the replenishment time between rf pulses.
(85 mg/kg) at 37°C in a temperature-controlled MR probe after five minutes equilibration time. For the purpose of assessing the consequences of proximal collateral vessel growth in mouse hindlimbs, relative distal right-to-left blood flow ratios are desirable for quantitative evaluations. We acquired a 256 ⫻ 256 matrix with an in-plane resolution of 100 m, 0.62 mm thick for three axial two-dimensional TOF slices in the distal hindlimbs between knees and feet of both hindlimbs. A short repetition time of 22 msec and a 90° pulse angle was used to minimize the background signal with an echo time of 8 msec and 16 signal averages. Setup time and acquisition of three consecutive slices took less than 10 minutes per mouse. The acquisition of
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more than one slice plane serves to increase the accuracy of the measurements by allowing averaging. An infusion pump was used to control the flow of the calibration standard (1.0 mM CuSO4) through a 1.46-mmdiameter tube placed parallel to the distal hindlimbs. Mice from three different strains with a surgical femoral artery occlusion were imaged with one of four different standard flow velocities (0.2, 0.4, 0.6, or 0.8 mL/ minute) in the control tube. The cross sections through the legs of mice were evaluated by selecting a region of interest (ROI) encompassing the visible vessels. Voxels in the ROI were assumed to have flow if the intensities exceeded a noise threshold limit. Used regions inherently included partial-volume effects in the parameter of the vessels. The noise threshold limit was set by the mean of the noise in the surrounding tissue plus two times the standard deviation (SD) of the noise. Using the sum of these voxels (I) minus the mean of the noise of the tissue (bkg) for an equivalent volume of hindlimb tissue without visible vessels, a right-to-left flow ratio was obtained as described in Eq. [2]:
Ratio ⫽
I right calf ⫺ bkg I left calf ⫺ bkg
(2)
Equation [2] does not account for the signal intensity attributed to magnetization recovery by T1 processes within the slice plane, which is minimal since the signal repetition time is only 22 msec. With a T1 for tissue of about one second, the resulting image intensity is about 2% to 3% of the maximum, which would be acquired with long repetition times. While the linear flow
Figure 3. Example of a cross-sectional plane through the flow probe phantom with 1.0 mM CuSO4 solution. Slice thickness of acquisition planes: 0.62 mm (A), 1.00 mm (B), and 1.50 mm (C). The more defined flow over the entire area is achieved with the small slice thickness with less artifacts resulting from field homogeneity, as seen in the large-diameter tubing. The effects of laminar flow at the edges of the tubes are apparent in the smaller tubes.
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intensities are equivalent for all flow velocities, the volume flow dependence is not. The loss of the dependence is due to varying slice areas between the phantom and the vessels and differences in the background contributions, which will not be discussed here. However, if the slice area, Area, for an internal standard is similar to the area being measured, then volume flows can be approximated with good accuracy. A 1.46-mm-innerdiameter tube containing 1.0 mM CuSO4 at standardized flow velocities was imaged together with the mouse hindlimbs for the determination blood flow velocities. Different flow velocities were used with the mice to determine the slope, m, of the intensity per voxel to calibrate the system. The calibration tube served as an internal standard for variations in the setup of the equipment. Variations in the individual flow values arise from the effective homogeneity of the magnetic field, which varies between mice, as well as from possible differences related to matching and tuning of the MRI coil between experiments. The slope was then used to determine the flow in the individual mouse calf muscles of the hindlimbs with Eq.
Flow ⫽
共I calf ⫺ bkg兲 ⫻ Area calf vessel m ⫻ Area std tube
(3)
RESULTS In tissue with no flow, a loss of magnetization was observed due to the fast repetition time and large pulse angle. However, under conditions of fluid inflow, the available signal was determined by the amount of fluid that entered the slice between saturation/detection pulses. The theoretical maximum occurred when the linear flow equaled or exceeded the length of the detection slice per unit of time between radio frequency (rf) pulses (Fig. 2). For laminar flow this was not valid since the fluid in the center of the tube moved with a higher velocity than that near the walls. This effect was observed in images with low flow velocities. The intensity in the center volume is higher than that in the periphery (Fig. 3). In nonlinear tortuous vessels streaming has also been observed in past experiments in larger animals (15), i.e., higher intensities near the edge of the tubes. The smaller tubes of the flow phantom showed similar effects at wall edges. Increasing slice thickness resulted in phase artifacts from lower field homogeneity manifested as concentric rings in the cross-sectional view. These artifacts, however, did not significantly affect the results at the low flow velocities and were not observed in the mouse calibration standards. Flow intensities, Itotal, for the tubes were determined by a summation of the voxel intensities, In, in the area of the tube cross section minus an equivalent background value, bkg, taken from an adjacent area. These intensities were then divided by the cross-sectional area of the inner part of the tubing, Eq. [1]. The intensity values for the three tube sizes for all the linear flow velocities were plotted (Fig. 4). The evaluated intensities remained approximately linear for up to half the linear flow velocity that would be assumed
Figure 4. Flow intensities for three slices of differing thickness: 0.62 mm (A), 1.00 mm (B), and 1.50 mm (C). Data points plotted for three tube diameters: 1.46 mm (x, ⫹), 0.59 mm (䊐), and 0.39 mm (E).
for a plug-like flow, r2 ⫽ 0.94 in the worst case. The intensities for the calibration tubes were plotted and a least-squares fit (r2 ⫽ 0.92) was done to obtain the slope (Fig. 5). The distal hindlimb flow velocities (Table 1) were well below the slice plug flow replenishment time for a 0.62-mm slice thickness used in our experiments. The average flow volume velocities varied over a range of 0.1–1.0 mL/minute for the nonoccluded side, similar to the volume velocities used for the calibration standard. Thus, in a mouse femoral artery occlusion model the recovery of flow to the ischemic hindlimbs can be monitored noninvasively.
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Wagner et al. Table 2 Left Side Perfusion Estimate Mouse strain
Muscle weight (mg)
Maximum blood perfusiona [mL/(minute* 100 g)]
C57BL/6 BALB/c 129Sv
239.40 ⫾ 12.58b 224.00 ⫾ 5.47 170.40 ⫾ 3.90
117 ⫾ 18 103 ⫾ 22 293 ⫾ 42
a
Calculated maximum using the combined venous and arterial flow measured and the muscle weight assuming all blood flows though the tissue. b All values are the mean ⫾ SD.
Figure 5. Raw intensity values of the average cross-sectional area for the three consecutive slices of the flow standard. One datum point was acquired with each mouse; the calibration standard flow velocities were varied among the mouse strains to avoid possible bias among the strains.
DISCUSSION Direct flow measurements by TOF methods offer the advantage of being entirely noninvasive without the need for contrast agents. Sequences are fast, resulting in short examination times and permitting the measurement of many mice in a short time period. The method measures the flow in larger vessels, which should carry the majority of blood as conductance vessels through and to the calf muscle; it does not directly measure tissue perfusion related to the microcirculation. It should be noted that this method assumes blood flow to be perpendicular to the slice planes. This is true for the major vessels in the acquisition planes. Some smaller vessels may have different paths, however, which should not affect this larger-vessel flow-oriented method, and for the evaluation of mice from a homogeneous genetic background, the effects should be similar in all mice, still allowing direct comparison between different groups. A recovery time course after femoral artery occlusion is possible with this technique and will be subsequently published. Owing to the difficulties in obtaining reliable in vivo measurements of limb blood flow and perfusion in mice, relatively little data exist about the collateral dependent flow in mice after femoral artery ligation. The Table 1 Blood Flow for the Individual Strains
a
Mouse strain
Blood flow left hindlimb (mL/minute)
Blood flow right hindlimb (mL/minute)
Right-to-left ratioa
C57BL/6 BALB/c 129Sv
0.28 ⫾ 0.04b,c 0.23 ⫾ 0.05 0.50 ⫾ 0.07d
0.21 ⫾ 0.02c,d 0.12 ⫾ 0.02 0.31 ⫾ 0.05d
0.88 ⫾ 0.11 0.61 ⫾ 0.10 0.73 ⫾ 0.04
Calculated average ratio from the individual right-to-left blood flows for each mouse in the group. b All values are the mean ⫾ SD c Statistical difference when compared to 129Sv, P ⬍ 0.02. d Statistical difference when compared to BALB/c, P ⬍ 0.01.
addition of MRI to mouse studies may help to enhance the understanding of this small animal model of peripheral vascular disease. The interpretation of the data obtained by other methods is still unclear. Results obtained for blood flow (Table 2) in the mouse hindlimb in this study are consistent with data previously obtained. For rat muscle, tissue perfusion measurements in resting animals using microspheres determined values of 5– 6 mL/(minutes*100 g) in fast-twitch white (FTW) fiber sections and 10 –12 mL/(minute*100 g) in fasttwitch red (FTR) and slow-twitch red (STR) muscle (20). Studies in the limbs of humans have shown that perfusion can increase 10- to 20-fold after ischemic exercise (21,22). Therefore, maximum vasodilation values should be in the range of 100 –200 mL/(minute*100 g). The estimate provided in Table 2 shows a probable maximum tissue perfusion based on available blood flowing to the calf muscle, which is in line with what is expected physiologically. This estimate assumes that half of the flow is venous and is excluded. Gastrocnemius muscle weights are approximately half the total weight of muscle tissue; skin perfusion estimates are excluded. Additionally, the limb blood flow can be compared to data about hindlimb regional blood flow. A total of 6% of the cardiac output in a C57BL/6 mouse was determined to flow to the running muscles, gastrocnemius, and quadriceps (23). Cine MRI has shown a cardiac output of 15.7 ⫾ 0.5 mL/minute in C57BL/6 mice (24). With 3% of the total blood volume assumed to go to each side yields a flow of 0.47 mL/minute to the total limb; a slightly lower global flow should be expected in the more distal vessels supplying the gastrocnemius muscle as a result of losses in upper muscle perfusion. While a saturation pulse could have been used to remove the venous flow, this was not done to maintain higher SNRs. A saturation pulse would add more time to the sequence, resulting in increased background tissue noise. Moreover, the volume of blood flow in the venous system must be equal to that of the arterial blood flow since the technique measures blood flowing through the calf muscle and blood cannot return through any other structure not being measured. Inclusion of both vessels increases the number of detected voxels (left calf, 130 ⫾ 51; right calf, 106 ⫾ 34; mean ⫾ SD) and increases the detected signal. Table 3 shows that the measure flow rates did not approach the maximum flow level, which would diminish the signal as a result of fluid entering and leaving
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Table 3 Estimated Maximum Detectable Flow
a
Flow
Number of voxels
Maximum linear flow (cm/second)a
Maximum volumetric flow (mL/minute)a
Range of volumetric flow (mL/minute)
Right calf muscle Left calf muscle Standardization tube
106 130 177
2.76 2.76 2.76
1.79 2.20 2.99
0.07–0.45 0.09–1.00 0.2–0.8
Calculated based on the approximation of “plug-like” flow and a 22 msec repetition time.
the detection slice without being detected. Flow rates were well below the maximum plug-like flow and in the linear portion of the curve. Using the average blood flow and voxel count measured, the average flow rate per voxel was six times less than the maximum. Before, hindlimb blood flow was measured by MRI in mice by determining arrival time after a bolus contrast injection (6). The shortcomings of this technique are the required setup time, the difficulty in performing the injection, which also requires precise timing, and the limited repeatability due to local vascular damage after the injection. Significant differences between some mouse strains in normal hindlimb flows were obtained in our study. These may be physiologically important. Obviously, some of the differences could also be due to different cardiovascular susceptibilities of the mice to the narcotic agents. Thus, in studies involving femoral artery ligation, it is likely more useful to utilize the nonligated side as a control limb and compare group results as ratios of limb recovery. Pulsatile flow artifacts are minimal in the mouse hindlimbs, resulting in only occasional ghosting, which does not affect the results. Streaming and laminar flow problems associated with this method have not been observed at the low flow velocities and with the small vessel sizes occurring in the mouse hindlimb. In addition, the flow velocities in the distal murine hindlimb are well within the flow range where the measured intensity is linearly dependent on the flow, leading to accurate flow velocity assessments. This new fast MRI method for the measurement of hindlimb blood flow in mice has major advantages over the only other MR method that has previously been described for this purpose. It requires only about 10 minutes per mouse, including setup time, and thus allows for assessment of many mice in a short time period; furthermore, it does not require injection of a contrast agent, making it easy to perform repetitive studies on one animal without the need to take into account residual amounts of circulating contrast agent and without the necessity for obtaining vascular access. This MRI method also offers advantages over other current conventional techniques like x-ray angiography and fluorescent microsphere injections since quantitative information of the blood flow can be obtained in vivo and the animals can be repetitively studied. Laser Doppler imaging measures general tissue perfusion noninvasively but is more surface weighted and limited to measuring perfusion in arbitrary units. Microsphere and X-ray angiography methods in mice are limited to terminal studies. TOF inflow imaging should be a very helpful tool for mouse hindlimb ischemia studies, since
it permits serial assessments of large-vessel blood flow in the hindlimbs in vivo, unaffected by problems of tissue penetration, and without the need of contrast injection. ACKNOWLEDGMENT We thank Siegfried Langsdorf for the construction of the MR probe. We would like to thank Siegfried Langsdorf for the construction of the magnetic resonance probe. REFERENCES 1. Heil M, Ziegelhoeffer T, Pipp F, et al. Blood monocyte concentration is critical for enhancement of collateral artery growth. Am J Physiol Heart Circ Physiol 2002;283:H2411–H2419. 2. Matthaei D, Haase A, Merboldt KD, Hanicke W, Deimling M. ECGtriggered arterial FLASH-MR flow measurement using an external standard. Magn Reson Imaging 1987;5:325–330. 3. Stahlberg F, Henriksen O, Thomsen C, Stubgaard M, Persson B. Determination of flow velocities from magnetic resonance multiple spin-echo images. A phantom study. Acta Radiol 1987;28:643– 648. 4. Poncelet BP, Weisskoff RM, Wedeen VJ, Brady TJ, Kantor H. Time of flight quantification of coronary flow with echo-planar MRI. Magn Reson Med 1993;30:447– 457. 5. Pearlman JD, Hibberd MG, Chuang ML, et al. Magnetic resonance mapping demonstrates benefits of VEGF induced myocardial angiogenesis. Nat Med 1995;1:1085–1089. 6. Heeschen C, Jang JJ, Weis M, et al. Nicotine stimulates angiogenesis and promotes tumor growth and atherosclerosis. Nat Med 2001;7:835– 841. 7. Kahler E, Waller C, Rommel E, et al. Quantitative regional blood volume studies in rat myocardium in vivo. Magn Reson Med 1998; 40:517–525. 8. Kahler E, Waller C, Rommel E, et al. Perfusion-corrected mapping of cardiac regional blood volume in rats in vivo. Magn Reson Med 1999;42:500 –506. 9. Belle V, Kahler E, Waller C, et al. In vivo quantitative mapping of cardiac perfusion in rats using a noninvasive MR spin-labeling method. J Magn Reson Imaging 1998;8:1240 –1245. 10. Wehrli FW, Shimakawa A, Gullberg GT, MacFall JR. Time-of-flight MR flow imaging: selective saturation recovery with gradient refocusing. Radiology 1986;160:781–785. 11. Axel L. Blood flow effects in magnetic resonance imaging. Am J Roentgenol 1984;143:1157–1166. 12. Gullberg GT, Simons MA, Wehrli FW. A mathematical model for signal from spins flowing during the application of spin echo pulse sequences. Magn Reson Imaging 1988;6:437– 461. 13. Gao JH, Holland SK, Gore JC. Nuclear magnetic resonance signal from flowing nuclei in rapid imaging using gradient echoes. Med Phys 1988;15:809 – 814. 14. Wehrli FW. Time-of-flight effects in MR imaging of flow. Magn Reson Med 1990;14:187–193. 15. van Tyen R, Saloner D, Jou LD, Berger S. MR imaging of flow through tortuous vessels: a numerical simulation. Magn Reson Med 1994;31:184 –195. 16. Siegel Jr JM, Oshinski JN, Pettigrew RI, Ku DN. Computational simulation of turbulent signal loss in 2D time-of-flight magnetic resonance angiograms. Magn Reson Med 1997;37:609 – 614.
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