- Email: [email protected]
Optimization of the AC-gradient method for velocity profile measurement and application to slow flow
Accepted Manuscript Optimization of the AC-gradient method for velocity profile measurement and application to slow flow Ralf Kartäusch, Xavier Helluy, Peter Michael Jakob, Florian Fidler PII: DOI: Reference:
S1090-7807(14)00266-3 http://dx.doi.org/10.1016/j.jmr.2014.09.021 YJMRE 5522
To appear in:
Journal of Magnetic Resonance
Received Date: Revised Date:
24 June 2014 24 September 2014
Please cite this article as: R. Kartäusch, X. Helluy, P.M. Jakob, F. Fidler, Optimization of the AC-gradient method for velocity profile measurement and application to slow flow, Journal of Magnetic Resonance (2014), doi: http:// dx.doi.org/10.1016/j.jmr.2014.09.021
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Optimization of the AC-gradient method for velocity profile measurement and application to slow flow Ralf Kartäuscha, Xavier Helluyb Peter Michael Jakoba,b, Florian Fidlera
a
Research Center for Magnetic Resonance Bavaria e.V., Würzburg,
b
Germany Department of Experimental Physics 5, University of Würzburg, Würzburg, Germany
Word count: Abstract: 165 Body: 4281
Running Head: Optimization of the AC-gradient method for velocity profile measurement and application to slow flow
Corresponding author: Dipl. Phys. Ralf Kartäusch
Department of Experimental Physics 5, University of Würzburg Am Hubland D-97074 Würzburg Germany
E-Mail
[email protected]
Phone
+49 (0) 931 31- 86546
Fax
+49 (0) 931 31- 85851
Submitted to JMR Journal of Magnetic Resonance as an Article
1
Abstract This work presents a spectroscopic method to measure slow flow. Within a single shot the velocity distribution is acquired. This allows distinguishing rapidly between single velocities within the sampled volume with a high sensitivity. The technique is based on signal acquisition in the presence of a periodic gradient and a train of refocussing RF pulses. The theoretical model for trapezoidal bipolar pulse shaped gradients under consideration of diffusion and the outflow effect is introduced. A phase correction technique is presented that improves the spectral accuracy. Therefore, flow phantom measurements are used to validate the new sequence and the simulation based on the theoretical model. It was demonstrated that accurate parabolic flow profiles can be acquired and flow variations below 200 µm/s can be detected. Three post-processing methods that eliminate static background signal are also presented for applications in which static background signal dominates. Finally, this technique is applied to flow measurement of a small alder tree demonstrating a typical application of in vivo plant measurements.
Introduction The flow characteristic of xylem and phloem of plants is typically in the µm/s up to lower mm/s range. This slow flow velocities of phloem and xylem combined with a dominant amount of static background signal [1] make encoding of plant flow challenging. Additionally, plant measurements often require an accessible sensor and measurements spread over several weeks. MR has been proposed as a tool to enable the measurement of flow in plants over a prolonged period [2-9]. Additionally, mobile MR provides low-cost accessible sensors [10-16]. However, mobile sensors often suffer from low magnetic field homogeneity and, in general, do not provide high-gradient strength. Two types of flow-sensitizing techniques are mainly used to encode the coherent motion of spins and measure their average velocity. The time-of-flight technique labels the magnetization of a volume and enables the measurement of inflow or outflow displacement [17]. Alternatively, quantitative flow measurement techniques, which induce a flow-dependent phase shift of transverse magnetization subjected to tailored pulsed field gradient (PFG), are used for flow characterization [4, 18] in plants. Improvements of the previous techniques have led to the development of velocity-mapping MR measurements that enable the acquisition of a complete flow profile. This technique encodes analogue to the k-space for spatial encoding the q-space for displacement encoding [19]. Redpath et al. showed a technique that encodes the q-space by repeating the PFG measurement with different B0-gradient magnitude strength [20]. Therefore, the q-space is sampled, and by applying a Fourier transformation, a probability of displacement is reconstructed [7, 21-23]. A second way to acquire the q-space is the AC-gradient method, introduced in 1986 by Walton [24]. This technique uses an oscillating sinusoidal B0-gradient, which slightly increases the q-space encoding with each periodic analogue to a frequency encoding gradient in the k-space. During one period, the sinusoidal gradient acts as a flow encoding bipolar gradient. Walton has shown that the periodic repetition of bipolar gradients results in a linear phase encoding of flow with time, which, after Fourier transform, yields a velocity spectrum. Theoretically, there are no constraints on the shape of the bipolar gradient used and it is mainly because of gradient instabilities that the original method used a sinusoidal waveform. Walton et al. improved encoding efficiency per excitation by combining the AC-gradient method with a spin echo train in order to reduce signal decay caused by magnetic field inhomogeneity. The original publication focused on high-flow velocities. However, the technique has potential for slow flow encoding, as encountered in flow imaging of plants with portable MR. The original AC-gradient method used sinusoidal waveform mostly because of gradient timing stability concerns. Nevertheless, the continuous sweeping of the encoding gradient strength is associated with a series of undesirable properties. It is the objective of the first section of this work to identify those limitations and to propose techniques to remove those. This paper presents a series of improvements to the original AC-gradient method with refocusing pulse train technique in the Theory section. Velocity maps for phantom measurements with slow flow are acquired. 2
In the second part of this work, the optimized AC-sequence is applied to mobile MR in the case of slow flow encoding in plants. Three techniques are presented that remove the signal of the large background peak found in such biological systems. They are tested on phantom measurements, and as an application of these techniques, we present a three-day in vivo flow measurements of a small alder tree.
3
Theory The conventional AC-encoding for Fourier velocity encoding with periodic repetition of bipolar gradients was carried out with a sinusoidal shape of the motion-encoding gradient (Fig. 1a). The original AC-gradient method used sinusoidal waveform mostly because of gradient timing stability concerns. The sinusoidal gradient shape has low hardware requirements and is a compromise for old gradient amplifier.
Fig. 1 a) The original AC gradients sequence diagram is shown. A sinusoidal gradient carries out the flow encoding. b) The diagram of the modified sequence is plotted. The first echo has no phase encoding applied and is used to correct the phase of the remaining echoes. The magnetization is periodically rephased over a cycle of two echoes. Assuming perfect ࣊ pulses, the signal is dephased at the even echoes and the signal is acquired at the interval indicated by the red boxes.
Nevertheless, we note that the continuous sweeping of the encoding gradient strength is associated with a series of undesirable properties. However, applying trapeze gradients increases the slew rate and improves the encoding efficiency. Fig. 1b shows the sequence diagram. First, the smooth waveform of the sinusoidal function results in a loss of encoding efficiency per period compared to a rectangular bipolar gradient. Assuming the sinusoidal gradient shape is replaced with a rectangular shaped gradient without ramps using the same maximum gradient strength, the gradient area increases by 214%. Therefore, a factor of 214 % higher phase shift is introduced. Second, RF refocusing pulses are turned on while a linear sweep of the B0-gradient strength takes place. This results in a complex time and spatially dependent spectral broadening of the MR magnetization and is, therefore, associated with loss of efficiency of the refocusing pulses. 4
Third, signal acquisition occurs while the linear sweep of the B0-gradient takes place. This results in a complicated gradient strength dependent dephasing of transverse magnetization and a loss of signal. In fact, only one data point is sampled when gradient waveform crosses zero and can be acquired with the maximum intensity. Finally, in the original sequence, flow encoding starts immediately after the first 90° pulse, making it impossible to acquire a sample without flow encoding. In contrast to the sequences shown in [24, 25] and presented in Fig. 1a, no flow encoding is applied before the first echo in the new implementation. Therefore, samples without flow encoding are acquired. Those samples additionally provide a reference for automatic zero-phase correction. The flow encoding of the proposed gradient cycle differs from the original AC-method. In the following the encoding with trapeze gradients is calculated. The standard phase labelling of the spins by a gradient over a time period reads
Φ
(Eq. 1)
with …. Assuming a constant velocity of the spin species considered within the measurement time, one can approximate . Therefore, the phase encoding can be calculated by solving the Eq. 1 for the gradient cycle shown in Fig 1b using a maximum gradient strength of . The induced phase shift after (half gradient cycle) reads: Φ , 2 Δ
2 (Eq. 2)
Assuming perfect refocusing pulses, the signal is dephased in the even echo. Whereas the spatialdependent phase shift is removed after the full gradient cycle at 2 : Φ 2 , 2 2 Δ
(Eq. 3)
Where is the velocity phase encoding for a full gradient cycle. A full gradient cycle acts as a bipolar gradient and therefore the static magnetization is periodically rephased. The phase shift of constantly flowing spins leads to linear phase accumulation along the gradient cycles Φ , . And is the index of the gradient cycle. The signal can be characterized with the spin density , and the transversal -decay by: ∑ !∑ , "
# " ,
(Eq. 4)
This flow-dependent phase accumulation is the encoding of q-space [19]. By applying this formalism to Eq. 4, one can define: $
(Eq. 5)
The above equation 4 in combination with a % ∑ , "
can be rewritten as:
$ ∑ % " (Eq. 6) The definition of $ (Eq. 5) differs by a scaling factor of 2 from the usual q-space [19] method. In comparison, while the PFG method [20] corresponds to the phase encoding technique in k-space, the AC-gradient method is, in principle, analogous to the frequency encoding technique in k-space. The Fourier transform of $ from $ to $ yields the velocity spectrum &, 0 where 0 corresponds to the data points acquired exactly at the echo centre. However, the gradient cycle introduces only positive or negative phase encoding, though only half the q-space is encoded. Therefore, a single shot measurement acquires the half of the q-space and allows only half a Fourier transform, leading to the creation of a dispersive part in the spectrum. 5
& $ e
" $ e
& ) *)+, $ - ) *
(Eq. 7)
Here, A denotes the absorptive part; D, the dispersive. Therefore, the magnitude ./+& is a nonaccurate flow profile due to dispersive part. Though, the Fourier encoded flow-profile-function is element of reals and therefore equals the absorptive part A. However, a phase shift Φ besides the Fourier encoding causes a shift between the real and imaginary part of the spectrum and the absorptive part is not equal to 0".1& : 0"& - cos Φ * sin Φ (Eq. 8) The signal for the 1 data point away from the centre (1=0), using a dwell time Δ , reads: S$, 1 M , 1 e
$ M , 1 e
$
(Eq. 9)
Here, M , 1 is the magnitude of the signal decay due to inhomogeneity and Φ!"" 1Δ is the phase drift because of a spatially dependent magnet inhomogeneity and Φ # the receiver phase. Additionally, a phase shift is introduced, if the half Fourier transform interval does not start at time zero [26]. This can be solved by delaying the flow encoding by one echo (Fig 1b) and therefore providing the signal for the spectrum at time zero. A phase correction, then, becomes necessary to achieve an accurate and coherent spectrum. Additionally, this first echo provides a phase reference for all data points removing the phase shift shown in Eq. 9. Applying a full Fourier transformation to $, 1 results in a magnitude spectrum with only an absorptive part. Therefore, the spectrum becomes insensitive from phase errors by a symmetrical acquiring of the q-space. This can be achieved by repeating the measurement with a negative gradient magnitude that enables applying a full Fourier transformation cancelling out the dispersive part. Hence, the magnitude spectrum shows the correct flow spectrum. With the proposed gradient cycle an interval between the trapeze gradient with no gradient application is possible. Hence, in contrast to the methods presented earlier [24, 25], data points acquired during this interval show no dephasing from flow gradients, allowing a coherent summation by taking into account the phase correction mentioned above.
6
Materials and methods Simulation of a laminar flow model A slow water flow in a tube shows a parabolic velocity profile. Assuming a homogeneous density in the tube, one can show that the velocity is uniformly distributed between 0 and . Therefore, the velocity spectrum profile is equal distributed. Hence, the volume 9 of a velocity can be calculated ;< + , 0 = = ? by 9 : . 0, >"+
The signal of the parabolic profile can be simulated with Eq. 4. However, the displacement due to diffusion needs to be considered [4, 27]. For an unrestricted diffusion considering the outflow effect [25] the signal is simulated by: $ ∑ !∑ , "
# " , Cn, v" %$ (Eq. 10)
* is the diffusion coefficient and / is the diffusion weighting which depends on the gradient cycle and is defined by: /
B
CB
&& &&
D E
[25]. Here The very low outflow effect is considered by the multiplication with F , 1 ' denotes the echo time and L the coil length. The simulations were performed using Eq. 10 with the parameters described in the following subchapter ‘Data acquisition/processing’.
Method Hardware The AC-gradient method was implemented on an 18 MHz home-built accessible C-shaped mobile MRI scanner. The scanner has an accessible probe volume of 1x1x1 cm and a homogeneity of approximately 20 ppm in this volume. The scanner is equipped with a 3D gradient system. The measurements were acquired with a ∅10 mm solenoid coil of 10 mm length. A DriveL console (Pure Devices GmbH, Würzburg, Germany) was used for the measurement. Data acquisition/processing An oscillating 1 ms trapeze gradient was applied over 302 echoes using a maximum strength of 200 mT/m. In addition, the first echo without the flow encoding (Fig. 1) was used to correct the phase error (Eq. 6) for all following echoes. The whole echo was, therefore, used for averaging. 32 data points with a bandwidth of 16 kHz were acquired and averaged per echo. The total TR for a single-shot spectrum was set to 2 s and TE was 5.1 ms. In many cases a large background signal can cover small flowing parts. For example, the stem of a tree consists mainly of static spins and a minor part of flowing spins. Therefore, a large background peak occurs in the spectra and the change in flow is barely visible. The large stationary peak can be removed by subtracting a reference scan spectrum from the original flow spectrum. Three processing techniques to generate a reference spectrum and to produce net flow spectra were applied: 1. An additional measurement using a flow encoding direction orthogonal to the flow direction of interest provides a reference scan. The difference between those spectra, cancels out all orientation independent parts; removing the large stationary peak. Hence, only the flow parallel to the tube is shown in that spectrum difference. 2. Without this additional scan, the subtraction of a day spectrum from a night spectrum, shows the long term change between day and night flows and removes the stationary peak. 3. Alternatively, the symmetry of the static background in the spectrum’s centre can be used to remove the static background. The symmetrical part is removed by subtracting the positive velocities from the negative, leaving only the net flow spectrum [1]. 7
Phantom measurements Phantom experiments were performed to validate the simulations. The phantom flow was generated using a syringe pump (model A-99, Bioblock Scientific). The flow phantom was a tube with 6.2 mm inner diameter and was used to generate a flow spectrum without any additional stationary spins. A second phantom was built, consisting of water and oil capillaries within a tube of ∅5 mm. The capillaries provided stationary spins, and in combination with an additional tube of 1 mm inner diameter for flowing spins, the phantom mimicked a plant. Functional plant measurements The method’s feasibility was proven in vivo by acquiring the flow spectrum of the stem of young alder trees. This led to the creation of 3,000 single spectra over a period of three days. A 400 Watt radium lamp was used to directly control the light. The lamp was switched on in a 12-hour cycle with a short break in the first 'on' period. On the last day, no light except normal sun shine was applied. The processing two is used to reduce the static background without acquiring additional spectra. The result is the net velocity difference spectrum between xylem flow and phloem flow. By a summing up of the individual weighted velocity components, an average net flow can be calculated and the large number of spectra evaluated [21].
8
Results Simulation results The results of the simulation for a parabolic flux are shown in Fig 2. Fig. 2a shows the magnitude velocity spectrum obtained after applying a half Fourier transformation on the data points (grey line) like it is presented in [25]. Here, only the half q-space is acquired and the imaginary part has a dispersive part (Eq. 8). Therefore, only the real part of the spectrum can be used (Fig. 2a red line). An alternative way is acquiring the q-space symmetrically by repeating the measurement with a negative flow encoding gradient or to replace the missing part of the q-space with the complex conjugate of the measured part. Then the magnitude of the spectrum produces the result shown in Fig. 2a. The result of the parabolic flux simulation with the influence of diffusion (lines) is shown on top of the measured results (colons) in Fig 2b. The difference spectra are simulated by subtracting a spectrum for the velocity zero from the spectrum with flow. The results are shown in Fig 3b. As long as the outflow effect is negligible the amount of water remains constant in both spectra. Therefore, the spectrum with no flow has a higher static water amount and in the difference spectrum a negative peak occurs in the centre.
Fig. 2 a) Simulated spectra without diffusion applied are shown. The magnitude value of the curve is shown in grey (dashed). The red curve indicates the real part and the green the original spectrum. Filling up the q-space with the complex conjugate enables using the magnitude value of the curve and produces the same curve (red). b) Velocity spectra acquired from phantom with parabolic flow in a 6.2 mm tube. The colours indicates the velocities 0 µm/s (blue), 216 µm/s (green), 576 µm/s (red), 792 µm/s (turquois). The lines show the simulated spectra with the diffusion of water applied and the points correspond to the measured data.
9
Fig. 3 a) Velocity spectra of a phantom with a large static background signal (blue 2mm/s, red 4 mm/s and green static signal). The difference between a spectrum from a direction without flow and the flow spectrum is shown in blue und red (small peaks) as described in data processing 1. In the magnification is additionally the difference between the left and right side of the spectrum shown in black (processing 3, 4 mm/s). b) The lines show the simulations results of the difference spectra between 0 mm/s flow and mean velocities of 2 mm/s (blue) and 4 mm/s (red) on top of the measured difference spectra (colons) from Fig. 3a. The negative parts results from a higher stationary water amount in the spectra without flow.
Phantom measurement Four 1D spectra of a laminar flow measurement in the ∅6.2 mm tube for the mean velocity of 216 µm/s (green), 576 µm/s (red), 792 µm/s (turquois) and an additional measurement with no flow (blue) are shown in Fig. 2b (colons). The average flow velocity set by the pump is used for the simulation of the profile (lines). The measurement result is the real part after applying the phase correction (Eq. 9) using the first echo (no flow encoding). The measured results and the simulation results match with only minor deviations. The deviation is maximal for the highest velocity (792 µm/s). However, the accuracy of the pump is limited and can produce slightly deviation from adjusted velocity. Fig. 3a shows the results of the plant phantom measurements. The large peaks indicate the raw spectra. The red and blue straight lines indicate the difference between the spectrum of the gradient orthogonal to the flow direction (green line) and the one parallel to the flow line (blue 2 mm/s and red 4 mm/s) as described in processing 1. The magnification shows additionally the difference between the positive and negative velocities (processing 3; 4 mm/s flow) in the spectrum (black straight line). The dashed black line indicates the average flow velocity set by the pump. The simulations (lines) of the difference spectrum and the measurement results (points) are shown in Fig 3b. Plant measurement Fig. 4a shows the flow spectrum of an alder stem ∅10 mm. The large background signal of the static spins overlay the small velocity peak. The static signal is removed by subtracting a night spectrum from a day spectrum (green line), and the left side of the spectrum from the right (black line). The positions of the day and night spectra are indicated in Fig. 4b by black dashed lines.
10
Fig.4 a) Shows the velocity spectrum of a ∅10 mm alder stem at night (blue slashed), during the day (red), and the difference in green (processing 2). Additionally, the difference between the right and the left side of the day spectrum is drawn in black (processing 3). The points in time of the spectra are indicated by dashed black lines in b). b) The graph plotted shows the mean net flow velocity in an alder stem. The yellow rectangles highlight the interval with an illumination applied. The orange rectangle highlights the interval with only the sun shining. The black dashed lines indicate the position of the day and night spectra which were subtracted in Fig a).
The flow spectrum was used to calculate an mean velocity difference by summing up the weighted individual flow components [21]. Fig. 4b shows the mean velocity difference (Processing 2) over a measurement time of three days. The yellow surface indicates the periods with the light switched on. The orange rectangle indicates a region where only the sun lights the plant.
Discussion The original AC-method uses a sinusoidal gradient; here, the applied gradient magnitude is, for most of the time, low compared to the maximum applied gradient magnitude. In contrast to this, the trapeze gradient used in this work has a more effective encoding. A rectangular gradient at the maximum gradient magnitude increases the phase shift of constant velocity by 214 % in comparison to the sinusoidal gradient with the same maximum gradient magnitude. However, the slew rates are hardware-limited and eddy currents can influence the measurement. Therefore, the improvement depends on the gradient system and increases with higher echo times, as the slew duration, compared to the static gradient, becomes negligible. The setup used in this work, allows a ramp time of 100 µs until the maximum gradient strength of 200 mT/m was reached. Those parameters in combination with the continuous flow encoding during the echo train allowed the detection of flow variations lower than 200 µm/s (Fig. 2b). The presented results show the feasibility of acquiring spectra of slow flow. The proposed evaluation produced a spectrum without the artifacts present in previous papers [24, 25] as described in the simulation section. The delays, with gradients turned off between each bipolar gradient, enabled a coherent summing up of the data points, significantly increasing the SNR. Additionally, the refocusing efficiency of the RF pulse train is improved. Therefore, only minor signal of odd echoes occurred [24]. The combination of sampling starting from time zero and the phase correction removed artifacts such as baseline distortions that were present in [24]. The AC-gradients method has a long and strong encoding of displacement and therefore the diffusion is encoded [28] as well. The correct profile can be created by using the reconstruction method, as shown in the simulation. Experimental results differ from the pure laminar profile because of diffusion. The result is a folding of the laminar velocity profile in a given direction and the profile of water diffusion along the same direction [29]. Therefore, the single velocity components in the profiles correspond to the mean velocities within the measurement time. Owing to the very low flow velocities, the broadening of the diffusion strongly influences the rectangle shape of the laminar distribution. The influence of the unrestricted diffusion for the AC measurement is described by the equation 10. The transversal diffusion can influence the parabolic flow profile. The complex interplay of flow and orthogonal diffusion is described by the Taylor dispersion [30]. However, the Taylor dispersion starts 11
to play a role when #( H with the measurement time #( and the radius of the tube >)%# $ [31]. Therefore, due to relatively large tube sizes even with the high diffusion of free water the diffusion the measurement time needs to be in the time scale of minutes to influence the flow profiles. Hence, the assumption of laminar flow for the simulation is correct. The quality of the processing to remove static background signal depends on the circumstances in the plant. Subtracting two orthogonal flow encoded measurements (Processing 1) removes only static background as long as one of the measurements contains no flow. E.g. if the transversal flow in one direction of a plant was not negligible compared to the longitudinal flow, the difference spectrum would show a reduced net flow. The second processing would have the same problems, if in the reference measurement flow occurred. Additionally, significant changes in phloem or static water can indicate wrong flow changes. Processing three produces always in a plant the net flow spectrum between xylem and phloem as long as no additional spatial encoding is applied. Hence, a phloem flow with the same intensity as the xylem would produce zero net flow. This processing does not suffer from a change of static water or a change of T1 or T2. For a better understanding of the signal variation and quantification the implementation of 2D localized flow velocity measurements is required [10, 19]. An alternative means of acquiring the full q-space is the pulsed field gradient (PFG) method in which the q-space is sampled by repeating the measurement with the PFG of a different magnitude. A very high magnitude or very long gradient durations are needed to acquire a high encoding. For example an encoding of 500 µm/s is achieved by a PFG of 5 ms and 0.37 T/m strength, using a Δ of 1 ms. In contrast to this method, the technique presented enables a single-shot measurement. The same encoding strength is achieved by using 36 echoes of the AC trapezoid method with a single PFG duration of 1.5 ms and a gradient strength of 0.15 T/m. Hence, the AC-gradient method significantly lowers the effective inter-echo time and reduces the impact of diffusion incoherent dephasing caused by inhomogeneity. However, in addition to the flow encoding, T2 decay occurs during the echo train. In exchange for an increased sensitivity to noise, this can be reduced by applying deconvolution functions. The samples presented in this paper have high T2 values and therefore no deconvolution function was applied. This work focussed on acquiring slow flow profiles. However, fast changes in the high velocities can be traced by using very short echo time and acquiring velocity maps within few milliseconds. The proposed method is not limited to plant measurements. Owing to the permanent encoding of the flow, one can achieve an optimized encoding strength, as shown in the measurement of the parabolic-flow phantom (Fig. 2). This technique enables acquiring an accurate slow velocity spectrum within a singleshot measurement, obviating the need for additional phase correction.
12
References: [1] T.W. Scheenen, D. van Dusschoten, P.A. de Jager, H. Van As, Quantification of water transport in plants with NMR imaging, Journal of experimental botany, 51 (2000) 1751-1759. [2] E. Fukushima, Nuclear magnetic resonance as a tool to study flow, Annu Rev Fluid Mech, 31 (1999) 95-123. [3] G.B. Pike, C.H. Meyer, T.J. Brosnan, N.J. Pelc, Magnetic-Resonance Velocity Imaging Using a Fast Spiral Phase-Contrast Sequence, Magn Reson Med, 32 (1994) 476-483. [4] E.O. Stejskal, Use of Spin Echoes in a Pulsed Magnetic-Field Gradient to Study Anisotropic Restricted Diffusion and Flow, J Chem Phys, 43 (1965) 3597-&. [5] C.W. Windt, F.J. Vergeldt, P.A. de Jager, H. van As, MRI of long-distance water transport: a comparison of the phloem and xylem flow characteristics and dynamics in poplar, castor bean, tomato and tobacco, Plant, cell & environment, 29 (2006) 1715-1729. [6] E. Kuchenbrod, A. Haase, R. Benkert, H. Schneider, U. Zimmermann, Quantitative NMR microscopy on intact plants, Magn Reson Imaging, 13 (1995) 447-455. [7] P.T. Callaghan, C.D. Eccles, Y. Xia, Nmr Microscopy of Dynamic Displacements - K-Space and QSpace Imaging, J Phys E Sci Instrum, 21 (1988) 820-822. [8] Y. Xia, V. Sarafis, E.O. Campbell, P.T. Callaghan, Noninvasive Imaging of Water-Flow in Plants by Nmr Microscopy, Protoplasma, 173 (1993) 170-176. [9] H. Van As, J. van Duynhoven, MRI of plants and foods, J Magn Reson, 229 (2013) 25-34. [10] M. Rokitta, E. Rommel, U. Zimmermann, A. Haase, Portable nuclear magnetic resonance imaging system, Rev Sci Instrum, 71 (2000) 4257-4262. [11] T. Haishi, T. Uematsu, Y. Matsuda, K. Kose, Development of a 1.0 T MR microscope using a NdFe-B permanent magnet, Magn Reson Imaging, 19 (2001) 875-880. [12] S.M. Wright, D.G. Brown, J.R. Porter, D.C. Spence, E. Esparza, D.C. Cole, F.R. Huson, A desktop magnetic resonance imaging system, Magma, 13 (2002) 177-185. [13] B. Goodson, Mobilizing magnetic resonance, Phys World, 19 (2006) 28-33. [14] B. Blumich, J. Perlo, F. Casanova, Mobile single-sided NMR, Prog Nucl Magn Reson Spectrosc, 52 (2008) 197-269. [15] C.W. Windt, H. Soltner, D. van Dusschoten, P. Blumler, A portable Halbach magnet that can be opened and closed without force: The NMR-CUFF, J Magn Reson, 208 (2011) 27-33. [16] H. Vanas, J.E.A. Reinders, P.A. Dejager, P.A.C.M. Vandesanden, T.J. Schaafsma, In-Situ Plant Water-Balance Studies Using a Portable Nmr Spectrometer, Journal of experimental botany, 45 (1994) 61-67. [17] J.R. Singer, Blood Flow Rates by Nuclear Magnetic Resonance Measurements, Science, 130 (1959) 1652-1653. [18] F.W. Wehrli, A. Shimakawa, G.T. Gullberg, J.R. Macfall, Time-of-Flight Mr Flow Imaging Selective Saturation Recovery with Gradient Refocusing, Radiology, 160 (1986) 781-785. [19] T.W.J. Scheenen, D. van Dusschoten, P.A. de Jager, H. Van As, Microscopic displacement imaging with pulsed field gradient turbo spin-echo NMR, J Magn Reson, 142 (2000) 207-215. [20] T.W. Redpath, D.G. Norris, R.A. Jones, J.M.S. Hutchison, A New Method of Nmr Flow Imaging, Phys Med Biol, 29 (1984) 891-895. [21] A. Caprihan, J.G. Davis, S.A. Altobelli, E. Fukushima, A New Method for Flow VelocityMeasurement - Frequency Encoded Nmr, Magn Reson Med, 3 (1986) 352-362. [22] D.B. Twieg, J. Katz, R.M. Peshock, A General Treatment of Nmr Imaging with Chemical-Shifts and Motion, Magn Reson Med, 5 (1987) 32-46. [23] D.G. Nishimura, P. Irarrazabal, C.H. Meyer, A Velocity K-Space Analysis of Flow Effects in EchoPlanar and Spiral Imaging, Magn Reson Med, 33 (1995) 549-556. [24] J.H. Walton, M.S. Conradi, Flow velocity measurement with ac gradients, Magn Reson Med, 4 (1987) 274-281. [25] C. Schelhorn, P.M. Jakob, F. Fidler, Rapid spectroscopic velocity quantification using periodically oscillating gradients, J Magn Reson, 214 (2012) 175-183. 13
[26] A. Heuer, U. Haeberlen, A New Method for Suppressing Baseline Distortions in Ft Nmr, J Magn Reson, 85 (1989) 79-94. [27] D. Le Bihan, R. Turner, C.T. Moonen, J. Pekar, Imaging of diffusion and microcirculation with gradient sensitization: design, strategy, and significance, Journal of magnetic resonance imaging : JMRI, 1 (1991) 7-28. [28] E.O. Stejskal, J.E. Tanner, Spin Diffusion Measurements: Spin Echoes in the Presence of a TimeDependent Field Gradient, J Chem Phys, 42 (1965) 288-+. [29] J. Pope, S. Yao, Quantitative NMR imaging of flow, Concepts in Magnetic Resonance, 5 (1993) 281-302. [30] G. Taylor, Dispersion of Soluble Matter in Solvent Flowing Slowly through a Tube, Proc R Soc Lon Ser-A, 219 (1953) 186-203. [31] R. Aris, On the Dispersion of a Solute in a Fluid Flowing through a Tube, Proc R Soc Lon Ser-A, 235 (1956) 67-77.
14
Graphical abstract
Highlights This work describes a spectroscopic method for single shot flow quantification. It is based on continuous flow encoding by alternating gradients. Accurate flow velocity distributions of e.g. laminar flow profile are acquired. Spectral resolution of 200 µm/s enables in vivo plant flow measurements.
15
S1090-7807(14)00266-3 http://dx.doi.org/10.1016/j.jmr.2014.09.021 YJMRE 5522
To appear in:
Journal of Magnetic Resonance
Received Date: Revised Date:
24 June 2014 24 September 2014
Please cite this article as: R. Kartäusch, X. Helluy, P.M. Jakob, F. Fidler, Optimization of the AC-gradient method for velocity profile measurement and application to slow flow, Journal of Magnetic Resonance (2014), doi: http:// dx.doi.org/10.1016/j.jmr.2014.09.021
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Optimization of the AC-gradient method for velocity profile measurement and application to slow flow Ralf Kartäuscha, Xavier Helluyb Peter Michael Jakoba,b, Florian Fidlera
a
Research Center for Magnetic Resonance Bavaria e.V., Würzburg,
b
Germany Department of Experimental Physics 5, University of Würzburg, Würzburg, Germany
Word count: Abstract: 165 Body: 4281
Running Head: Optimization of the AC-gradient method for velocity profile measurement and application to slow flow
Corresponding author: Dipl. Phys. Ralf Kartäusch
Department of Experimental Physics 5, University of Würzburg Am Hubland D-97074 Würzburg Germany
[email protected]
Phone
+49 (0) 931 31- 86546
Fax
+49 (0) 931 31- 85851
Submitted to JMR Journal of Magnetic Resonance as an Article
1
Abstract This work presents a spectroscopic method to measure slow flow. Within a single shot the velocity distribution is acquired. This allows distinguishing rapidly between single velocities within the sampled volume with a high sensitivity. The technique is based on signal acquisition in the presence of a periodic gradient and a train of refocussing RF pulses. The theoretical model for trapezoidal bipolar pulse shaped gradients under consideration of diffusion and the outflow effect is introduced. A phase correction technique is presented that improves the spectral accuracy. Therefore, flow phantom measurements are used to validate the new sequence and the simulation based on the theoretical model. It was demonstrated that accurate parabolic flow profiles can be acquired and flow variations below 200 µm/s can be detected. Three post-processing methods that eliminate static background signal are also presented for applications in which static background signal dominates. Finally, this technique is applied to flow measurement of a small alder tree demonstrating a typical application of in vivo plant measurements.
Introduction The flow characteristic of xylem and phloem of plants is typically in the µm/s up to lower mm/s range. This slow flow velocities of phloem and xylem combined with a dominant amount of static background signal [1] make encoding of plant flow challenging. Additionally, plant measurements often require an accessible sensor and measurements spread over several weeks. MR has been proposed as a tool to enable the measurement of flow in plants over a prolonged period [2-9]. Additionally, mobile MR provides low-cost accessible sensors [10-16]. However, mobile sensors often suffer from low magnetic field homogeneity and, in general, do not provide high-gradient strength. Two types of flow-sensitizing techniques are mainly used to encode the coherent motion of spins and measure their average velocity. The time-of-flight technique labels the magnetization of a volume and enables the measurement of inflow or outflow displacement [17]. Alternatively, quantitative flow measurement techniques, which induce a flow-dependent phase shift of transverse magnetization subjected to tailored pulsed field gradient (PFG), are used for flow characterization [4, 18] in plants. Improvements of the previous techniques have led to the development of velocity-mapping MR measurements that enable the acquisition of a complete flow profile. This technique encodes analogue to the k-space for spatial encoding the q-space for displacement encoding [19]. Redpath et al. showed a technique that encodes the q-space by repeating the PFG measurement with different B0-gradient magnitude strength [20]. Therefore, the q-space is sampled, and by applying a Fourier transformation, a probability of displacement is reconstructed [7, 21-23]. A second way to acquire the q-space is the AC-gradient method, introduced in 1986 by Walton [24]. This technique uses an oscillating sinusoidal B0-gradient, which slightly increases the q-space encoding with each periodic analogue to a frequency encoding gradient in the k-space. During one period, the sinusoidal gradient acts as a flow encoding bipolar gradient. Walton has shown that the periodic repetition of bipolar gradients results in a linear phase encoding of flow with time, which, after Fourier transform, yields a velocity spectrum. Theoretically, there are no constraints on the shape of the bipolar gradient used and it is mainly because of gradient instabilities that the original method used a sinusoidal waveform. Walton et al. improved encoding efficiency per excitation by combining the AC-gradient method with a spin echo train in order to reduce signal decay caused by magnetic field inhomogeneity. The original publication focused on high-flow velocities. However, the technique has potential for slow flow encoding, as encountered in flow imaging of plants with portable MR. The original AC-gradient method used sinusoidal waveform mostly because of gradient timing stability concerns. Nevertheless, the continuous sweeping of the encoding gradient strength is associated with a series of undesirable properties. It is the objective of the first section of this work to identify those limitations and to propose techniques to remove those. This paper presents a series of improvements to the original AC-gradient method with refocusing pulse train technique in the Theory section. Velocity maps for phantom measurements with slow flow are acquired. 2
In the second part of this work, the optimized AC-sequence is applied to mobile MR in the case of slow flow encoding in plants. Three techniques are presented that remove the signal of the large background peak found in such biological systems. They are tested on phantom measurements, and as an application of these techniques, we present a three-day in vivo flow measurements of a small alder tree.
3
Theory The conventional AC-encoding for Fourier velocity encoding with periodic repetition of bipolar gradients was carried out with a sinusoidal shape of the motion-encoding gradient (Fig. 1a). The original AC-gradient method used sinusoidal waveform mostly because of gradient timing stability concerns. The sinusoidal gradient shape has low hardware requirements and is a compromise for old gradient amplifier.
Fig. 1 a) The original AC gradients sequence diagram is shown. A sinusoidal gradient carries out the flow encoding. b) The diagram of the modified sequence is plotted. The first echo has no phase encoding applied and is used to correct the phase of the remaining echoes. The magnetization is periodically rephased over a cycle of two echoes. Assuming perfect ࣊ pulses, the signal is dephased at the even echoes and the signal is acquired at the interval indicated by the red boxes.
Nevertheless, we note that the continuous sweeping of the encoding gradient strength is associated with a series of undesirable properties. However, applying trapeze gradients increases the slew rate and improves the encoding efficiency. Fig. 1b shows the sequence diagram. First, the smooth waveform of the sinusoidal function results in a loss of encoding efficiency per period compared to a rectangular bipolar gradient. Assuming the sinusoidal gradient shape is replaced with a rectangular shaped gradient without ramps using the same maximum gradient strength, the gradient area increases by 214%. Therefore, a factor of 214 % higher phase shift is introduced. Second, RF refocusing pulses are turned on while a linear sweep of the B0-gradient strength takes place. This results in a complex time and spatially dependent spectral broadening of the MR magnetization and is, therefore, associated with loss of efficiency of the refocusing pulses. 4
Third, signal acquisition occurs while the linear sweep of the B0-gradient takes place. This results in a complicated gradient strength dependent dephasing of transverse magnetization and a loss of signal. In fact, only one data point is sampled when gradient waveform crosses zero and can be acquired with the maximum intensity. Finally, in the original sequence, flow encoding starts immediately after the first 90° pulse, making it impossible to acquire a sample without flow encoding. In contrast to the sequences shown in [24, 25] and presented in Fig. 1a, no flow encoding is applied before the first echo in the new implementation. Therefore, samples without flow encoding are acquired. Those samples additionally provide a reference for automatic zero-phase correction. The flow encoding of the proposed gradient cycle differs from the original AC-method. In the following the encoding with trapeze gradients is calculated. The standard phase labelling of the spins by a gradient over a time period reads
Φ
(Eq. 1)
with …. Assuming a constant velocity of the spin species considered within the measurement time, one can approximate . Therefore, the phase encoding can be calculated by solving the Eq. 1 for the gradient cycle shown in Fig 1b using a maximum gradient strength of . The induced phase shift after (half gradient cycle) reads: Φ , 2 Δ
2 (Eq. 2)
Assuming perfect refocusing pulses, the signal is dephased in the even echo. Whereas the spatialdependent phase shift is removed after the full gradient cycle at 2 : Φ 2 , 2 2 Δ
(Eq. 3)
Where is the velocity phase encoding for a full gradient cycle. A full gradient cycle acts as a bipolar gradient and therefore the static magnetization is periodically rephased. The phase shift of constantly flowing spins leads to linear phase accumulation along the gradient cycles Φ , . And is the index of the gradient cycle. The signal can be characterized with the spin density , and the transversal -decay by: ∑ !∑ , "
# " ,
(Eq. 4)
This flow-dependent phase accumulation is the encoding of q-space [19]. By applying this formalism to Eq. 4, one can define: $
(Eq. 5)
The above equation 4 in combination with a % ∑ , "
can be rewritten as:
$ ∑ % " (Eq. 6) The definition of $ (Eq. 5) differs by a scaling factor of 2 from the usual q-space [19] method. In comparison, while the PFG method [20] corresponds to the phase encoding technique in k-space, the AC-gradient method is, in principle, analogous to the frequency encoding technique in k-space. The Fourier transform of $ from $ to $ yields the velocity spectrum &, 0 where 0 corresponds to the data points acquired exactly at the echo centre. However, the gradient cycle introduces only positive or negative phase encoding, though only half the q-space is encoded. Therefore, a single shot measurement acquires the half of the q-space and allows only half a Fourier transform, leading to the creation of a dispersive part in the spectrum. 5
& $ e
" $ e
& ) *)+, $ - ) *
(Eq. 7)
Here, A denotes the absorptive part; D, the dispersive. Therefore, the magnitude ./+& is a nonaccurate flow profile due to dispersive part. Though, the Fourier encoded flow-profile-function is element of reals and therefore equals the absorptive part A. However, a phase shift Φ besides the Fourier encoding causes a shift between the real and imaginary part of the spectrum and the absorptive part is not equal to 0".1& : 0"& - cos Φ * sin Φ (Eq. 8) The signal for the 1 data point away from the centre (1=0), using a dwell time Δ , reads: S$, 1 M , 1 e
$ M , 1 e
$
(Eq. 9)
Here, M , 1 is the magnitude of the signal decay due to inhomogeneity and Φ!"" 1Δ is the phase drift because of a spatially dependent magnet inhomogeneity and Φ # the receiver phase. Additionally, a phase shift is introduced, if the half Fourier transform interval does not start at time zero [26]. This can be solved by delaying the flow encoding by one echo (Fig 1b) and therefore providing the signal for the spectrum at time zero. A phase correction, then, becomes necessary to achieve an accurate and coherent spectrum. Additionally, this first echo provides a phase reference for all data points removing the phase shift shown in Eq. 9. Applying a full Fourier transformation to $, 1 results in a magnitude spectrum with only an absorptive part. Therefore, the spectrum becomes insensitive from phase errors by a symmetrical acquiring of the q-space. This can be achieved by repeating the measurement with a negative gradient magnitude that enables applying a full Fourier transformation cancelling out the dispersive part. Hence, the magnitude spectrum shows the correct flow spectrum. With the proposed gradient cycle an interval between the trapeze gradient with no gradient application is possible. Hence, in contrast to the methods presented earlier [24, 25], data points acquired during this interval show no dephasing from flow gradients, allowing a coherent summation by taking into account the phase correction mentioned above.
6
Materials and methods Simulation of a laminar flow model A slow water flow in a tube shows a parabolic velocity profile. Assuming a homogeneous density in the tube, one can show that the velocity is uniformly distributed between 0 and . Therefore, the velocity spectrum profile is equal distributed. Hence, the volume 9 of a velocity can be calculated ;< + , 0 = = ? by 9 : . 0, >"+
The signal of the parabolic profile can be simulated with Eq. 4. However, the displacement due to diffusion needs to be considered [4, 27]. For an unrestricted diffusion considering the outflow effect [25] the signal is simulated by: $ ∑ !∑ , "
# " , Cn, v" %$ (Eq. 10)
* is the diffusion coefficient and / is the diffusion weighting which depends on the gradient cycle and is defined by: /
B
CB
&& &&
D E
[25]. Here The very low outflow effect is considered by the multiplication with F , 1 ' denotes the echo time and L the coil length. The simulations were performed using Eq. 10 with the parameters described in the following subchapter ‘Data acquisition/processing’.
Method Hardware The AC-gradient method was implemented on an 18 MHz home-built accessible C-shaped mobile MRI scanner. The scanner has an accessible probe volume of 1x1x1 cm and a homogeneity of approximately 20 ppm in this volume. The scanner is equipped with a 3D gradient system. The measurements were acquired with a ∅10 mm solenoid coil of 10 mm length. A DriveL console (Pure Devices GmbH, Würzburg, Germany) was used for the measurement. Data acquisition/processing An oscillating 1 ms trapeze gradient was applied over 302 echoes using a maximum strength of 200 mT/m. In addition, the first echo without the flow encoding (Fig. 1) was used to correct the phase error (Eq. 6) for all following echoes. The whole echo was, therefore, used for averaging. 32 data points with a bandwidth of 16 kHz were acquired and averaged per echo. The total TR for a single-shot spectrum was set to 2 s and TE was 5.1 ms. In many cases a large background signal can cover small flowing parts. For example, the stem of a tree consists mainly of static spins and a minor part of flowing spins. Therefore, a large background peak occurs in the spectra and the change in flow is barely visible. The large stationary peak can be removed by subtracting a reference scan spectrum from the original flow spectrum. Three processing techniques to generate a reference spectrum and to produce net flow spectra were applied: 1. An additional measurement using a flow encoding direction orthogonal to the flow direction of interest provides a reference scan. The difference between those spectra, cancels out all orientation independent parts; removing the large stationary peak. Hence, only the flow parallel to the tube is shown in that spectrum difference. 2. Without this additional scan, the subtraction of a day spectrum from a night spectrum, shows the long term change between day and night flows and removes the stationary peak. 3. Alternatively, the symmetry of the static background in the spectrum’s centre can be used to remove the static background. The symmetrical part is removed by subtracting the positive velocities from the negative, leaving only the net flow spectrum [1]. 7
Phantom measurements Phantom experiments were performed to validate the simulations. The phantom flow was generated using a syringe pump (model A-99, Bioblock Scientific). The flow phantom was a tube with 6.2 mm inner diameter and was used to generate a flow spectrum without any additional stationary spins. A second phantom was built, consisting of water and oil capillaries within a tube of ∅5 mm. The capillaries provided stationary spins, and in combination with an additional tube of 1 mm inner diameter for flowing spins, the phantom mimicked a plant. Functional plant measurements The method’s feasibility was proven in vivo by acquiring the flow spectrum of the stem of young alder trees. This led to the creation of 3,000 single spectra over a period of three days. A 400 Watt radium lamp was used to directly control the light. The lamp was switched on in a 12-hour cycle with a short break in the first 'on' period. On the last day, no light except normal sun shine was applied. The processing two is used to reduce the static background without acquiring additional spectra. The result is the net velocity difference spectrum between xylem flow and phloem flow. By a summing up of the individual weighted velocity components, an average net flow can be calculated and the large number of spectra evaluated [21].
8
Results Simulation results The results of the simulation for a parabolic flux are shown in Fig 2. Fig. 2a shows the magnitude velocity spectrum obtained after applying a half Fourier transformation on the data points (grey line) like it is presented in [25]. Here, only the half q-space is acquired and the imaginary part has a dispersive part (Eq. 8). Therefore, only the real part of the spectrum can be used (Fig. 2a red line). An alternative way is acquiring the q-space symmetrically by repeating the measurement with a negative flow encoding gradient or to replace the missing part of the q-space with the complex conjugate of the measured part. Then the magnitude of the spectrum produces the result shown in Fig. 2a. The result of the parabolic flux simulation with the influence of diffusion (lines) is shown on top of the measured results (colons) in Fig 2b. The difference spectra are simulated by subtracting a spectrum for the velocity zero from the spectrum with flow. The results are shown in Fig 3b. As long as the outflow effect is negligible the amount of water remains constant in both spectra. Therefore, the spectrum with no flow has a higher static water amount and in the difference spectrum a negative peak occurs in the centre.
Fig. 2 a) Simulated spectra without diffusion applied are shown. The magnitude value of the curve is shown in grey (dashed). The red curve indicates the real part and the green the original spectrum. Filling up the q-space with the complex conjugate enables using the magnitude value of the curve and produces the same curve (red). b) Velocity spectra acquired from phantom with parabolic flow in a 6.2 mm tube. The colours indicates the velocities 0 µm/s (blue), 216 µm/s (green), 576 µm/s (red), 792 µm/s (turquois). The lines show the simulated spectra with the diffusion of water applied and the points correspond to the measured data.
9
Fig. 3 a) Velocity spectra of a phantom with a large static background signal (blue 2mm/s, red 4 mm/s and green static signal). The difference between a spectrum from a direction without flow and the flow spectrum is shown in blue und red (small peaks) as described in data processing 1. In the magnification is additionally the difference between the left and right side of the spectrum shown in black (processing 3, 4 mm/s). b) The lines show the simulations results of the difference spectra between 0 mm/s flow and mean velocities of 2 mm/s (blue) and 4 mm/s (red) on top of the measured difference spectra (colons) from Fig. 3a. The negative parts results from a higher stationary water amount in the spectra without flow.
Phantom measurement Four 1D spectra of a laminar flow measurement in the ∅6.2 mm tube for the mean velocity of 216 µm/s (green), 576 µm/s (red), 792 µm/s (turquois) and an additional measurement with no flow (blue) are shown in Fig. 2b (colons). The average flow velocity set by the pump is used for the simulation of the profile (lines). The measurement result is the real part after applying the phase correction (Eq. 9) using the first echo (no flow encoding). The measured results and the simulation results match with only minor deviations. The deviation is maximal for the highest velocity (792 µm/s). However, the accuracy of the pump is limited and can produce slightly deviation from adjusted velocity. Fig. 3a shows the results of the plant phantom measurements. The large peaks indicate the raw spectra. The red and blue straight lines indicate the difference between the spectrum of the gradient orthogonal to the flow direction (green line) and the one parallel to the flow line (blue 2 mm/s and red 4 mm/s) as described in processing 1. The magnification shows additionally the difference between the positive and negative velocities (processing 3; 4 mm/s flow) in the spectrum (black straight line). The dashed black line indicates the average flow velocity set by the pump. The simulations (lines) of the difference spectrum and the measurement results (points) are shown in Fig 3b. Plant measurement Fig. 4a shows the flow spectrum of an alder stem ∅10 mm. The large background signal of the static spins overlay the small velocity peak. The static signal is removed by subtracting a night spectrum from a day spectrum (green line), and the left side of the spectrum from the right (black line). The positions of the day and night spectra are indicated in Fig. 4b by black dashed lines.
10
Fig.4 a) Shows the velocity spectrum of a ∅10 mm alder stem at night (blue slashed), during the day (red), and the difference in green (processing 2). Additionally, the difference between the right and the left side of the day spectrum is drawn in black (processing 3). The points in time of the spectra are indicated by dashed black lines in b). b) The graph plotted shows the mean net flow velocity in an alder stem. The yellow rectangles highlight the interval with an illumination applied. The orange rectangle highlights the interval with only the sun shining. The black dashed lines indicate the position of the day and night spectra which were subtracted in Fig a).
The flow spectrum was used to calculate an mean velocity difference by summing up the weighted individual flow components [21]. Fig. 4b shows the mean velocity difference (Processing 2) over a measurement time of three days. The yellow surface indicates the periods with the light switched on. The orange rectangle indicates a region where only the sun lights the plant.
Discussion The original AC-method uses a sinusoidal gradient; here, the applied gradient magnitude is, for most of the time, low compared to the maximum applied gradient magnitude. In contrast to this, the trapeze gradient used in this work has a more effective encoding. A rectangular gradient at the maximum gradient magnitude increases the phase shift of constant velocity by 214 % in comparison to the sinusoidal gradient with the same maximum gradient magnitude. However, the slew rates are hardware-limited and eddy currents can influence the measurement. Therefore, the improvement depends on the gradient system and increases with higher echo times, as the slew duration, compared to the static gradient, becomes negligible. The setup used in this work, allows a ramp time of 100 µs until the maximum gradient strength of 200 mT/m was reached. Those parameters in combination with the continuous flow encoding during the echo train allowed the detection of flow variations lower than 200 µm/s (Fig. 2b). The presented results show the feasibility of acquiring spectra of slow flow. The proposed evaluation produced a spectrum without the artifacts present in previous papers [24, 25] as described in the simulation section. The delays, with gradients turned off between each bipolar gradient, enabled a coherent summing up of the data points, significantly increasing the SNR. Additionally, the refocusing efficiency of the RF pulse train is improved. Therefore, only minor signal of odd echoes occurred [24]. The combination of sampling starting from time zero and the phase correction removed artifacts such as baseline distortions that were present in [24]. The AC-gradients method has a long and strong encoding of displacement and therefore the diffusion is encoded [28] as well. The correct profile can be created by using the reconstruction method, as shown in the simulation. Experimental results differ from the pure laminar profile because of diffusion. The result is a folding of the laminar velocity profile in a given direction and the profile of water diffusion along the same direction [29]. Therefore, the single velocity components in the profiles correspond to the mean velocities within the measurement time. Owing to the very low flow velocities, the broadening of the diffusion strongly influences the rectangle shape of the laminar distribution. The influence of the unrestricted diffusion for the AC measurement is described by the equation 10. The transversal diffusion can influence the parabolic flow profile. The complex interplay of flow and orthogonal diffusion is described by the Taylor dispersion [30]. However, the Taylor dispersion starts 11
to play a role when #( H with the measurement time #( and the radius of the tube >)%# $ [31]. Therefore, due to relatively large tube sizes even with the high diffusion of free water the diffusion the measurement time needs to be in the time scale of minutes to influence the flow profiles. Hence, the assumption of laminar flow for the simulation is correct. The quality of the processing to remove static background signal depends on the circumstances in the plant. Subtracting two orthogonal flow encoded measurements (Processing 1) removes only static background as long as one of the measurements contains no flow. E.g. if the transversal flow in one direction of a plant was not negligible compared to the longitudinal flow, the difference spectrum would show a reduced net flow. The second processing would have the same problems, if in the reference measurement flow occurred. Additionally, significant changes in phloem or static water can indicate wrong flow changes. Processing three produces always in a plant the net flow spectrum between xylem and phloem as long as no additional spatial encoding is applied. Hence, a phloem flow with the same intensity as the xylem would produce zero net flow. This processing does not suffer from a change of static water or a change of T1 or T2. For a better understanding of the signal variation and quantification the implementation of 2D localized flow velocity measurements is required [10, 19]. An alternative means of acquiring the full q-space is the pulsed field gradient (PFG) method in which the q-space is sampled by repeating the measurement with the PFG of a different magnitude. A very high magnitude or very long gradient durations are needed to acquire a high encoding. For example an encoding of 500 µm/s is achieved by a PFG of 5 ms and 0.37 T/m strength, using a Δ of 1 ms. In contrast to this method, the technique presented enables a single-shot measurement. The same encoding strength is achieved by using 36 echoes of the AC trapezoid method with a single PFG duration of 1.5 ms and a gradient strength of 0.15 T/m. Hence, the AC-gradient method significantly lowers the effective inter-echo time and reduces the impact of diffusion incoherent dephasing caused by inhomogeneity. However, in addition to the flow encoding, T2 decay occurs during the echo train. In exchange for an increased sensitivity to noise, this can be reduced by applying deconvolution functions. The samples presented in this paper have high T2 values and therefore no deconvolution function was applied. This work focussed on acquiring slow flow profiles. However, fast changes in the high velocities can be traced by using very short echo time and acquiring velocity maps within few milliseconds. The proposed method is not limited to plant measurements. Owing to the permanent encoding of the flow, one can achieve an optimized encoding strength, as shown in the measurement of the parabolic-flow phantom (Fig. 2). This technique enables acquiring an accurate slow velocity spectrum within a singleshot measurement, obviating the need for additional phase correction.
12
References: [1] T.W. Scheenen, D. van Dusschoten, P.A. de Jager, H. Van As, Quantification of water transport in plants with NMR imaging, Journal of experimental botany, 51 (2000) 1751-1759. [2] E. Fukushima, Nuclear magnetic resonance as a tool to study flow, Annu Rev Fluid Mech, 31 (1999) 95-123. [3] G.B. Pike, C.H. Meyer, T.J. Brosnan, N.J. Pelc, Magnetic-Resonance Velocity Imaging Using a Fast Spiral Phase-Contrast Sequence, Magn Reson Med, 32 (1994) 476-483. [4] E.O. Stejskal, Use of Spin Echoes in a Pulsed Magnetic-Field Gradient to Study Anisotropic Restricted Diffusion and Flow, J Chem Phys, 43 (1965) 3597-&. [5] C.W. Windt, F.J. Vergeldt, P.A. de Jager, H. van As, MRI of long-distance water transport: a comparison of the phloem and xylem flow characteristics and dynamics in poplar, castor bean, tomato and tobacco, Plant, cell & environment, 29 (2006) 1715-1729. [6] E. Kuchenbrod, A. Haase, R. Benkert, H. Schneider, U. Zimmermann, Quantitative NMR microscopy on intact plants, Magn Reson Imaging, 13 (1995) 447-455. [7] P.T. Callaghan, C.D. Eccles, Y. Xia, Nmr Microscopy of Dynamic Displacements - K-Space and QSpace Imaging, J Phys E Sci Instrum, 21 (1988) 820-822. [8] Y. Xia, V. Sarafis, E.O. Campbell, P.T. Callaghan, Noninvasive Imaging of Water-Flow in Plants by Nmr Microscopy, Protoplasma, 173 (1993) 170-176. [9] H. Van As, J. van Duynhoven, MRI of plants and foods, J Magn Reson, 229 (2013) 25-34. [10] M. Rokitta, E. Rommel, U. Zimmermann, A. Haase, Portable nuclear magnetic resonance imaging system, Rev Sci Instrum, 71 (2000) 4257-4262. [11] T. Haishi, T. Uematsu, Y. Matsuda, K. Kose, Development of a 1.0 T MR microscope using a NdFe-B permanent magnet, Magn Reson Imaging, 19 (2001) 875-880. [12] S.M. Wright, D.G. Brown, J.R. Porter, D.C. Spence, E. Esparza, D.C. Cole, F.R. Huson, A desktop magnetic resonance imaging system, Magma, 13 (2002) 177-185. [13] B. Goodson, Mobilizing magnetic resonance, Phys World, 19 (2006) 28-33. [14] B. Blumich, J. Perlo, F. Casanova, Mobile single-sided NMR, Prog Nucl Magn Reson Spectrosc, 52 (2008) 197-269. [15] C.W. Windt, H. Soltner, D. van Dusschoten, P. Blumler, A portable Halbach magnet that can be opened and closed without force: The NMR-CUFF, J Magn Reson, 208 (2011) 27-33. [16] H. Vanas, J.E.A. Reinders, P.A. Dejager, P.A.C.M. Vandesanden, T.J. Schaafsma, In-Situ Plant Water-Balance Studies Using a Portable Nmr Spectrometer, Journal of experimental botany, 45 (1994) 61-67. [17] J.R. Singer, Blood Flow Rates by Nuclear Magnetic Resonance Measurements, Science, 130 (1959) 1652-1653. [18] F.W. Wehrli, A. Shimakawa, G.T. Gullberg, J.R. Macfall, Time-of-Flight Mr Flow Imaging Selective Saturation Recovery with Gradient Refocusing, Radiology, 160 (1986) 781-785. [19] T.W.J. Scheenen, D. van Dusschoten, P.A. de Jager, H. Van As, Microscopic displacement imaging with pulsed field gradient turbo spin-echo NMR, J Magn Reson, 142 (2000) 207-215. [20] T.W. Redpath, D.G. Norris, R.A. Jones, J.M.S. Hutchison, A New Method of Nmr Flow Imaging, Phys Med Biol, 29 (1984) 891-895. [21] A. Caprihan, J.G. Davis, S.A. Altobelli, E. Fukushima, A New Method for Flow VelocityMeasurement - Frequency Encoded Nmr, Magn Reson Med, 3 (1986) 352-362. [22] D.B. Twieg, J. Katz, R.M. Peshock, A General Treatment of Nmr Imaging with Chemical-Shifts and Motion, Magn Reson Med, 5 (1987) 32-46. [23] D.G. Nishimura, P. Irarrazabal, C.H. Meyer, A Velocity K-Space Analysis of Flow Effects in EchoPlanar and Spiral Imaging, Magn Reson Med, 33 (1995) 549-556. [24] J.H. Walton, M.S. Conradi, Flow velocity measurement with ac gradients, Magn Reson Med, 4 (1987) 274-281. [25] C. Schelhorn, P.M. Jakob, F. Fidler, Rapid spectroscopic velocity quantification using periodically oscillating gradients, J Magn Reson, 214 (2012) 175-183. 13
[26] A. Heuer, U. Haeberlen, A New Method for Suppressing Baseline Distortions in Ft Nmr, J Magn Reson, 85 (1989) 79-94. [27] D. Le Bihan, R. Turner, C.T. Moonen, J. Pekar, Imaging of diffusion and microcirculation with gradient sensitization: design, strategy, and significance, Journal of magnetic resonance imaging : JMRI, 1 (1991) 7-28. [28] E.O. Stejskal, J.E. Tanner, Spin Diffusion Measurements: Spin Echoes in the Presence of a TimeDependent Field Gradient, J Chem Phys, 42 (1965) 288-+. [29] J. Pope, S. Yao, Quantitative NMR imaging of flow, Concepts in Magnetic Resonance, 5 (1993) 281-302. [30] G. Taylor, Dispersion of Soluble Matter in Solvent Flowing Slowly through a Tube, Proc R Soc Lon Ser-A, 219 (1953) 186-203. [31] R. Aris, On the Dispersion of a Solute in a Fluid Flowing through a Tube, Proc R Soc Lon Ser-A, 235 (1956) 67-77.
14
Graphical abstract
Highlights This work describes a spectroscopic method for single shot flow quantification. It is based on continuous flow encoding by alternating gradients. Accurate flow velocity distributions of e.g. laminar flow profile are acquired. Spectral resolution of 200 µm/s enables in vivo plant flow measurements.
15