time resolved flow quantification with mri using phase methods: a linear systems approach

Time Resolved Flow Quantification with MRI Using Phase Methods: A Linear Systems Approach Frank Peeters, Robert Luypaert, Henri Eisendrath, Michel Osteaux

Phase-related unsteady (pulsatile) flow effects in MRI have been studied by means of linear response theory. These flow effects can be described in the frequency domain: the influence of the gradients on the phase shift is described by a transfer function, the spectrum of the gradient being the determining factor. An analysis of this transfer function is shown to provide information about the process of flow encoding: instant of encoding, induced distortions and how they are related to the gradient waveform. The connection with the traditional description in terms of the gradient moment expansion has also been investigated and clarified. This approach was applied to study the response of two time- resolved flow quantification techniques (Fourier flow method and phase mapping) by analyzing their amplitude and phase transfer functions. By simulation it is shown that a better interpretation of the measured velocity waveform is obtained and that Fourier analysis in combination with a 'correction by the inverse transfer function results in an accurate reconstruction of the velocity waveform studied. Key words: pulsatile flow; flow quantification; linear response theory; MRI.

INTRODUCTION

The effects of physiologic motion and in particular blood flow and cerebrospinal fluid (CSF) flow have often been perceived in MRI as unwelcomed sources of artifacts, and efforts aimed at minimizing them have received con- siderable attention in the literature (1). On the other hand, the signal changes resulting from flow have also been extensively exploited to provide quantitative data and vascular morphology (2). When studying flow effects, the fact that the flow in the macrocirculation is unsteady (pulsatile) or even worse (turbulent and complex flow patterns) (3) has led to complications that have not been fully resolved until now.

Flow effects in MR imaging are usually described by expanding the time-dependent position into the different orders of motion. The expansion is often truncated after the velocity term (or at most after the acceleration term): this is not a complete model leading to incomplete re-

MRM 33S37-354 (1995) From the Department of Physics, Faculty of Sciences, Magnetic Reso- nance, Vrije Universiteit Brussel (F.P., H.E.), and Biomedical MR Unit, Ac- ademic Hospital AZ-VUB, Brussels, Belgium (F.P., R.L., M.O.). Address correspondence to: Frank Peeters, Ph.D., Magnetic Resonance, Department of Physics, Faculty of Sciences, Vrije Universiteit Brussel, Plein- laan 2, 1050 Brussels, Belgium. Received May 2, 1994; revised November 11, 1994; accepted November 14, 1994. This text presents research results of the Belgian programme on interuni- versity attraction poles initiated by the Belgian state-Prime Minister's Office-Science Policy programming. The scientific responsibility is assumed by its authors.

Copyright 0 1995 by Williams 8 Wilkins All rights of reproduction in any form reserved.

0740-3194/95 $3.00

sults when studying unsteady (pulsatile) flow. This is illustrated by the fact that the influence of the gradients on flow effects in MFU, described by the gradient moment expansion, is not completely understood in the case of unsteady (pulsatile) flow. Yet, their role in designing sequences for eliminating artifacts (e.g., in MR angiography) or quantifying velocities is very important. This is especially true for phase methods (4) like the Fourier flow method (FFM), phase mapping (PMAP), and phase contrast angiography (PCA) that provide the most accurate results and are based on motion-induced phase shifts. When time-resolved measurements are performed using these techniques, the observed waveform deviates from the one obtained with Doppler U.S. measurements. A deeper insight into the process of flow encoding is necessary to explain (and to correct for) the observed deviations.

In this paper, we try to achieve this by using linear response theory for studying the influence of the gradients on measuring unsteady (pulsatile] flow. This allows us to answer a number of questions arising when trying to quantify this type of flow: which gradient waveforms can encode a certain order of motion, how should this encoding process be interpreted, at which instant is the order of motion encoded and to what extent can distortions arise. To achieve this end in practice, we explicitly derive the transfer functions for two types of time-resolved flow quantification techniques: FFM and PMAP (2D and ID). On the basis of the results obtained, the measurement of a typical (anthropomorphic] velocity waveform is simulated. Deviations between measurement and the "real" velocity waveform are estimated and a strategy for correcting them using the transfer function in combination with Fourier analysis is proposed and demonstrated to work in a variety of circumstances.

LINEAR SYSTEMS APPROACH TO FLOW ENCODING Flow Encoding as a Linear System

For a linear, time-invariant system (5), the output v,(f] and input vi(t) are connected by a convolution relation in the time domain:

with h( f ) the impulse response of the system. For frequency-domain analysis, use is made of a transfer function, which is the Fourier transform (6) of the impulse response:

337

338

I -T I 1 0

Peeters et al.

3‘

As a consequence, in the frequency domain, input and output are related by:

VAw) = Ho)Vi(a), [31

where Vo(w),Vi(w) are the Fourier transforms of the time signals. In general the transfer function is a complex function that can be written in phasor notation:

H(w) = IH ( w ) I e’”] [41

with IH(w)l the amplitude transfer function (amplitude attenuation) and O(w) the phase transfer function (phase dispersion).

In MRI, flow quantification by phase methods exploits the motion-induced phase shift acquired by spins that move along a magnetic field gradient. If we consider motion along the z-direction, this motion-induced phase shift is given by:

cp = r[:’Tg(Mt)dt [51

with g(t) the gradient along that direction, which is switched on from time ti to ti + T, with T the duration of the gradient. The time-dependent position depends on the velocity by:

[61

where to is the reference time and z(t,) the reference (initial) position of the moving spins. The motion- induced phase shift cp can be considered as a time-dependent function q(t) that depends on the instant t the gradient is switched on. Equation [51 can be rewritten as a convolution relation when an effective gradient g*(t) is introduced:

[71

= yg*(t) €3 z(t) [81

The original gradient g(t’) is defined for times t’E[t,t + T]. The effective gradient g* (t’) is obtained by shifting the original one to the time origin and then reflecting it with respect to the origin. As a result, g*(t‘) = g(t - t’) is defined for times t’E[-T,O]. This is illustrated in Fig. 1.

Equation [8] shows that h,(t) = yg*(t) is the impulse response for a linear system with the position z(t) of the moving spins as input and the motion-induced phase shift q(t) as output. However, for flow quantification, we are interested in a system with the velocity v(t) as input. To derive the impulse response for such a system, we start with Eq. [6]. This can be rewritten as a convolution relation by making use of the unit step function u(t):

z(t) = z(-m) + u(t) €3 v(t) [91

Substituting this in Eq. [8] and making use of the fact that, in flow quantification, the zeroth order moment of

a

C

g*(t - t ‘ )

d FIG. 1. (a) The original gradient g(t‘). The motion-induced phase shift is then dt) = y-fpTg(t’)z(t’)dt’. (b) The effective gradient g’(t’). (c) The reflected effective gradient g*(-t’). (d) The shifted, reflected effective gradient g’(t - t ’ ) = g(t’). The motion-induced phase shift is then: df) = yJ-Lm g*(t - t’)z(t’) dt’ = w*(t) C3 z(t).

the velocity encoding gradient is nulled (m, = O ) , results in:

[lo1 This equation shows that the impulse response for the velocity v(t) is given by:

[I11

This can also be interpreted as the step response for the position z(t). In the frequency domain, velocity and phase shift are related by:

@(w) = H v ( 4 v ( w ) [I21

q(t) = yg*(t) €3 u ( t ) @ v(t)

hJt) = yg*(tl €3 u( t )

Frequency Response of Flow Quantification with MRI 339

with

[131

As can be seen from Eq. [13], the transfer function H,,(w) for velocity quantification depends on the spectrum (Fourier transform) of the effective gradient g*(t) and thus on the spectrum of the gradient g(t) . Because the effective gradient g*(t) = g( - t ) and the gradient itself is a real function, the Fourier transform of the effective gradient is the complex conjugate of the spectrum of the gradient (note that here g(t) is the shifted gradient defined on [ O , T l ) . This means that in Eq. [13], G*(o) can be read as "complex conjugate of G(w)."

Interpretation of the Velocity Transfer Function

The phase e,, of the velocity transfer function H,,(o) tells us when the velocity is encoded. If the phase varies with frequency according to eJW) = ot,, linear response theory implies that the velocity is encoded at time t + t,, i.e., after a delay t, with respect to the switch on of the gradient. We shall now investigate when such an interpretation is valid.

The velocity encoding gradient g(t) on the interval can always be expressed with respect to the mid-

point t = T/2 of the interval: g(t - T/2). As a consequence, the transfer function can be written as:

where G(w) is now the spectrum of the shifted gradient (g(t) with tE [ - T/2,T/2]). In the time domain, this gradient consists of two contributions:

an even and an odd component. In the frequency domain, this becomes:

= + IT,(^) 1171

where GJw) is real and even and GJw) is purely imaginary and odd. As a result, the phase transfer function becomes:

Equation [18] shows that, when the gradient is odd (g, = 0) , OJo) = wT/2 and the velocity is encoded at time t + T/2, the midpoint of the gradient. When dt) is even, OJw) = wT/2 + &'sgn(w) (sgn is the sign-function). Because of the second term, one cannot say that the velocity is encoded at t + T/2 (in fact, as shown in Appendix A, such a gradient cannot encode velocity). In general, g(t) is neither odd nor even. The phase transfer function O,,(w) is then a nonlinear function of o and the concept of instant velocity encoding becomes meaning-

less at first sight. In this case (cfr. Appendix A) the transfer function can be written as:

[191

where E,,(o) represents the nonlinear terms. The delay 6,, is given by:

the moments of the shifted gradient g(t) . When the nonlinearities are small, the velocity is approximately encoded at t + T/2 + s,, and the nonlinearities introduce small phase distortions (phase dispersion). Larger nonlinearities can deform (falsify) the encoded velocity in a dramatic way.

In general, the process of velocity encoding can be described in the time domain by:

[211 d t ) = h v ( t ) €4 v(t),

or in the frequency domain:

@(o) = H,(w)V(w) with HJw) = A , ( O ) ~ ' ~ ~ ~ ) [22]

As a consequence, using Eq. [19], the flow encoding process can be written as:

q(t) = a,(t) €4 3-'(eiEJw)) €4 v ( t + - + 6,, ) 1231

On the basis of this formula, the flow encoding process is seen to consist of three steps: first, the velocity is encoded at At = t + T/2 + s,, (with t the onset of the gradient); secondly, it is smoothed by the amplitude attenuation function a,,(t) and, finally, the phase distortions E,,(o) produce a general deformation. An ideal encoding process should lead to &,, = 0 and A,,(w) = A = constant. Unfortunately, gradients that can achieve this do not exist in practice. However, odd gradients do not produce phase distortions; they only cause an amplitude attenuation and encode the velocity at a certain instant.

Equation [20] reveals that also in this context gradient moments are important. The connection between the linear systems and gradient moment approach is discussed in more detail in Appendix A. Furthermore, the analysis given above can be extended to the encoding process of the nth order of motion. The general results are summarized in Appendix B.

RESPONSE OF TIME RESOLVED PHASE METHODS FOR FLOW QUANTIFICATION

In the Linear Systems Approach to Flow Encoding section, we have shown how the process of flow encoding has to be interpreted for phase methods (instant of encoding, distortions, . . .). This was achieved on the basis of the impulse response h(t) or the transfer function H(w). In this section, we are interested in the consequences of the above phenomena on flow (velocity) quantification. This means that we now have a system with as input the real velocity vi(t) and as output the velocity v,,(t) mea-

340 Peeters et al.

sured on the basis of MR images. Attention will be fo- cused on the transfer functions for two specific flow quantification techniques: the Fourier flow method and phase mapping. In the macrocirculation, blood flow or CSF flow is periodic (synchronous to the heart cycle), which means that gating can be used to overcome ghosting. The influence of prospective gating on the transfer function will be investigated (the frequency response of retrospectively gated phase mapping has been reported by R. Frayne and B. K. Rutt, see ref. 7). Our attention will be restricted to gradient echo sequences because they allow in vivo measurements within an acceptable acquisition time. Furthermore, only flow perpendicular to the imaged slice will be considered.

Fourier Flow Method

In the Fourier flow method (FFM) (8 ) , the velocity v = v, perpendicular to the imaged plane is encoded by a phase encoding gradient with m, = 0, m, # 0. A simplified version of the pulse sequence is shown in Fig. 2. In plane, the position is encoded along the x-direction by the read out gradient. As a result, the amplitude images obtained directly display the velocity profiles v(x), while a projection takes place along the y-direction.

For FFM, the MR-signal S and the projected spin den- sity p (or image intensity) are related by:

S(k,,4,)-//p(x.v)eidx~v)dxdv 1241

with

cp(x,v) = cpx + cpv = k s + k,v 1251

where k, is determined by the read out gradient and k, by the velocity (phase) encoding gradient. The flow dependent phase shift cp, is given by

cpv = k,v = yn,v = yM,v, [261

where in the final expression the shifted gradient (with respect to the center) is used. By applying the Nyquist relations, the FOV (field of view, corresponding to the

Gz >t

>t Gx u TE

>t GY

FIG. 2. Pulse sequence for the Fourier flow method (FFM). The first line shows the RF-(sinc)-pulse. The second line shows the Gz gradients: slice select gradient, bipolar velocity (phase) encoding gradient and a spoiler gradient. The third line shows the Gx read out gradient. Finally, there are no Gy gradients.

maximum detectable velocity v,, in positive and nega- tive direction) and resolution can be determined:

with AMl the change in Ml due to the stepping of the phase encoding gradient and MI,,, the value of MI corresponding to the maximum gradient amplitude used. Knowledge of the FOV allows determination of unknown velocities by measuring the corresponding distances in the image. According to Eq. 1261, the transfer function for FFM is given by:

so that V,(o) = HFFM(o)Vi(o). The velocity transfer function H,(o) is given by Eqs. [13] or [14]. Remember that Vi is the real (input) velocity and V, the observed (output) velocity based on FFM images.

When a bipolar trapezoidal phase encoding gradient (see Fig. 3a) is used to encode the velocity, the transfer

i4 h 6

2T+z C

FIG. 3. Gradient waveforms: (a) trapezoidal velocity encoding gradient with amplitude g and duration 2T + 47, (b) sinusoidal velocity encoding gradient with amplitude g and duration 6, (c) trapezoidal acceleration encoding gradient with amplitude g and duration 4T + 77.


function is given by:

with sinc(x) = (sin(x))/x the sinc-function. Equation [29] shows that the phase transfer function varies linearly with frequency (no phase dispersion) and that the velocity is measured after a delay T + 27 with respect to the onset of the gradient or at the crossover of the bipolar gradient. The amplitude transfer function reveals an attenuation of the frequency components which is more pronounced for higher frequencies, as shown in Fig. 4 (in the figures, f = o / 2 ~ expressed in Hz will be used con- sistently). Equation [29] also shows that HFFM(w) does not depend on the gradient amplitude, but only on the durations. Note that for longer gradient durations, attenuation will be more pronounced at lower frequencies. This will be important when measuring small velocities (e.g., CSF), because of the underlimit to the gradient duration imposed by the maximum gradient strength available on the imager.

For a sinusoidal flow encoding gradient (see Fig. 3b), the transfer function is:

Again, there is no phase dispersion and the velocity is measured at the crossover of the gradient. The same conclusions as for the trapezoidal gradient can be drawn regarding the amplitude attenuation characteristics. A comparison of both gradients (see Fig. 5) shows that, for the same gradient duration, the sinusoidal gradient has a better amplitude transfer function (less attenuation at small frequencies). However, for the same FOV, the situation is just the reverse (Fig. 5). This can be understood if one compares the FOVs:

0 10 20 30 40 so f

FIG. 5. The FFM transfer function for bipolar trapezoidal and sinusoidal velocity encoding gradients. IH, (f)I is for a trapezoidal gradient with T = 40 ms and 7 = 1 ms. lH2(f)1 is for a sinusoidal gradient with the same duration: 6 = 84 ms. IH,(f)l is for a sinusoidal gradient with the same FOV as for IH,(f)l: thus now 6 = 104 ms.

with N the number of phase encoding steps and g,, the maximum gradient strength during the stepping of the velocity encoding gradient. Equations [31] and [32] im- ply that for equal Nand g,,, the FOV for the trapezoidal gradient is smaller than for the sinusoidal gradient with the same duration. In order to obtain the same FOV, the duration of the sinusoidal gradient should be 6 = d2~(T+7) (T+27) , which is longer than the corresponding duration 2T + 47 of the trapezoidal gradient. As a result, more attenuation can be observed due to this longer duration.

Instead of velocity, acceleration can also be measured, by using a tripolar gradient (Fig. 3c). In this case the FOV is given by:

47r - d N - 1) - FOV = 2a, =- YAM, yg,,(T+ T)(T+ 27)(2T+ 37) 1331

and the transfer function is:

0 10 20 30 40 so f

FIG. 4. The FFM transfer function for a bipolar trapezoidal velocity encoding gradient. lH1(f)l is for a gradient with constant amplitude duration T = 40 ms and ramp time T = 1 ms. For IH2(f)l, T = 10 ms and 7 = 1 ms. Note that IH(f)l is unitless and f is in Hz.

w(T + 27) W(2T+ 37) .w(4T+77) sinc[ sin^[ [MI

There is no phase dispersion and the acceleration is measured after a delay 2T + 727 with respect to the onset of the gradient (i.e., midpoint). Amplitude attenuation characteristics similar to those for the velocity encoding gradients can be observed in this case (more attenuation at higher frequencies and more attenuation at lower frequencies for longer gradient durations). Because in practice acceleration encoding gradients are longer than velocity encoding gradients, these attenuation effects will be more pronounced here.

For time resolved pulsatile flow measurements, the sequence should be synchronized to the cardiac cycle in order to overcome ghosting artifacts. This can be achieved by performing prospective gating (ECGtrigger- ing). For FFM, each image in an image series covering the

342 Peeters et al.

heart cycle is recorded with all phase encoding steps at the same instant during the cardiac cycle. As a result, prospective gating does not deform the transfer function for FFM.

A major drawback for FFM in practice is that it provides spatial resolution in only one dimension in the image plane (i.e., projection along the other). Although two dimensional resolution is possible by implementing a phase (position) encoding gradient, this would result in a very long acquisition time. Another problem with FFM is that, for a given FOV, relatively long gradient durations are needed. As a consequence, the time resolution of the measurements is relatively poor. In the next section we will discuss a technique with 2D spatial resolution and a better time resolution: phase mapping.

Phase Mapping

20 Phase Mapping. For a constant velocity v, the motion induced phase shift during a velocity encoding gradient is proportional to velocity: cp = ym,v. As a consequence, phase images that display this motion-induced phase shift +[-T,T] can be used to measure the velocity. To get rid of spurious phase shifts, two phase images are recorded: one scan with a velocity encoding gradient (m, = O,m, # 0 ) and a second scan with a velocity compensating gradient (m, = m, = 0) . After subtraction of the phase images, the phase in the difference image is in principle only induced by the motion and proportional to the velocity (an alternative method consists of the subtraction of two velocity-encoded phase images each with a different sensitivity of velocity encoding) (9, 10). The pulse sequences are shown in Fig. 6 (simplified version). The read out and phase encoding gradient encode the position along the x- and y-direction in plane so that now 2D spatial resolution is obtained. Because the phase is a cyclic variable (determined modulo 2 4 , the

- 0 - TR - - n

+P I Gx

R

FIG. 6. Pulse sequence for 2D phase mapping (PMAP). The first line shows the RF-(sinc)-pulse. The second line shows the Gz gradients: slice select gradient, velocity encodingkompensating gradients for the first and second scan and a spoiler gradient. The third line shows the Gx read out gradient and the (position) phase encoding Gy gradient is shown on the fourth line.

VENC = Zv,, (velocity encoding or velocity sensitivity) in a difference image is given by:

with m1,, and ml,* the first order gradient moments for the first and second scan. Velocities can be determined from the measured phase difference Acp according to the relation:

From this relation, the transfer function for phase mapping (PMAP) is found to be:

with H,,,(w) and H,,,(o) the velocity transfer functions (given by Eq. [13]) of the first and second scan.

When time resolved measurements are performed, the sequence has to be (prospectively) gated. Because if phase mapping images are subtracted, two cases have to be considered (contrary to FFM). The first triggering mode is illustrated in Fig. 7a, in which the timings during one heart cycle (RR-interval) are shown assuming that for both scans the repetition times TR are equal. For a certain view (phase encoding step), both scans are alternated. One can easily see that in this case a difference image is obtained by subtracting two images that are recorded at different instants during the heart cycle. Fur- thermore, the time resolution for the difference images is equal to 2TR. We shall call this mode of triggering "interleaved gating." In the other version (see Fig. 7b), for a certain view, all the encoding scans are performed during one RR and then all the compensating scans are performed during a second RR. In this case, images are subtracted that are recorded at the same instants of the corresponding heart cycles. The time resolution of the difference images is now TR, but the acquisition time is

R R

a R R

R R I I

b

FIG. 7. Timing diagram (one view) of the two gating modes: (a) interleaved gating, (b) sequential gating. RR = time interval between two R-waves on the ECG, RR = period of the pulsations.


twice that of interleaved gating (each view lasts 2RR). We shall call this mode “sequential gating.” We shall now discuss the transfer functions for the two triggering modes and different combinations of flow encoding/ compensating gradients (first/second scan). The time delay between the two flow encoding gradients (firstlsec- ond scan) will be called A . This means that for sequential triggering A = 0 and for interleaved triggering A = TR (if both gradients are not switched at the same instant during the scans, an extra delay should be included in A).

A first combination consists of two velocity encoding scans with identical bipolar gradients, but each with a different polarity. For the two cases shown in Figs. 8a, and 8b (trapezoidal and sinusoidal gradients), the transfer function becomes:

(trapezoidal) [38]

For sequential gating ( A = O), the same transfer function is obtained as for FFM. As a consequence, the same conclusions can be drawn with respect to the instant of

u u H

A H

a

H A

H

b

H A

H

C

FIG. 8. Flow encoding gradients for PMAP. For interleaved gating A = TR and for sequential gating A = 0 (details of the gradient waveforms can be found in Fig. 3). (a) Bipolar trapezoidal velocity encoding gradients of the opposite polarity. (b) Bipolar sinusoidal velocity encoding gradients of the opposite polarity. (c) Bipolar trapezoidal velocity encoding and tripolar trapezoidal velocity compensating gradient.

velocity measurement and with respect to attenuation characteristics. For interleaved gating, the velocity is now measured at t + T + 27 + A12 (trapezoidal gradient) or at t + 612 + A12 (sinusoidal gradient), with t the onset of the flow encoding gradient in the first scan. Thus, the velocity is measured halfway between the two flow encoding gradients of the two scans. The amplitude transfer function is the same as for FFM, except that it is now modulated by a cosine function that depends on the repetition time A = TR. As a consequence, IHPMAP(w)I becomes zero at o = n d T R (n integer). This means that these frequency components are lost in the measured (e.g., arterial) waveform. Note that for sequential gating (no modulation), the attenuation is determined by the largest time scale of the gradient: the first zero of IHw)l occurs at o = 2 d ( T + 27) or w = 4d6. For interleaved gating, the first zero (due to modulation) occurs at o = d T R . Because 2TR 2 T + 27 or 812, attenuation will be more pronounced at low frequencies for interleaved (as compared with sequential) gating. Figure 9 shows the amplitude transfer functions for sequential and interleaved phase mapping with trapezoidal gradients of the opposite polarity. For sinusoidal gradients, a similar behavior is obtained.

Another interesting combination consists of a bipolar trapezoidal velocity encoding gradient (first scan) and a tripolar trapezoidal velocity compensating gradient (second scan) as shown in Fig. 8c. For this combination, the amplitude and phase transfer functions can be written as:

[ o@T2+ 3 ~ ) ] ~ ~ ~ [ w(2T + 37 + A)] sln . (;A) - 2 - 1 - 8sin

2sin [ o(2T2+ 3r)] cos [ w(2T + 37 + A ) ] 2

1 -zsin[ w(2T + 37)

]sin[

[411

0 20 40 60 80 100 f

FIG. 9. Amplitude transfer functions for PMAP with trapezoidal velocity encoding gradients. Gradient parameters: T = 5 ms and T = 1 ms. \/-/#)I is for interleaved gating with A = TR = 30 ms and IH2(f)I is for sequential gating (A = 0).

344 Peeters et al.

For sequential gating (A = O), the phase transfer function can be simplified to OPmp(w) = w(3T + 57). The phase is linear in frequency and, as a result, the velocity is measured at t + 3T + 57, which can be written as t + (4T + 77) - (2T + 4~) /2 . This can be interpreted as follows: the velocity is measured after a delay (with respect to the instant the velocity encoding gradient is switched on), which is equal to the difference between the duration of the velocity compensating gradient and half the duration of the velocity encoding gradient. The amplitude transfer function in this case (A = 0) is the same as for sequential PMAP with two trapezoidal velocity encoding gradients (or FFM with a trapezoidal velocity encoding gradient), with identical attenuation characteristics as encountered before. Note that, because in PMAP shorter gradients are used as compared with FFM, there will be less attenuation at low frequencies. For interleaved gating (A = TR), Eq. [41] shows that the phase transfer function is now a nonlinear function of frequency. Due to this phase dispersion, the measured velocity will be deformed, as discussed in the Interpretation of the Velocity Transfer Function section. The amplitude transfer function is now that of sequential gating, but modulated by a factor that depends on the repetition time TR (see Eq. [40]). As opposed to the cases discussed previously, the amplitude transfer function exhibits attenuation as well as amplification. At low frequencies, there will be attenuation, but for higher frequencies there will be regions where the frequency components are amplified (up to a factor two or even more), alternating with regions where attenuation occurs. Some examples of the amplitude and phase transfer functions are shown in Fig. 10. Inspection of

3 , I I I I I

2.4 -

;. ,-\-,,-. ' ,'

0 10 20 30 40 50 f

~

I I I I

0 10 20 30 40 50 f

FIG. 10. Some examples of the transfer functions (IH(f)l = amplitude transfer function, argH(f) = phase transfer function in radi- ans) for interleaved PMAP with trapezoidal velocity encoding and compensating gradients. For H,(f): T = 5 ms, TR = 40 ms. For H,(f): T = 2 ms, TR = 30 ms. For H3(f): T = 10 ms, TR = 100 ms.

these plots shows that long gradient durations and long repetition times are disadvantageous with respect to phase dispersion and amplitude characteristics (oscilla- tory behavior, amplification/attenuation). In practice, short gradient durations and TR are favored.

To improve the repetition time (and thus the time resolution), one can modify the slice select gradient to encode or compensate the velocity (for flow perpendicular to the plane). In this case, the motion-induced phase shift cp, when calculated from the center of the RF-pulse also varies linearly with the velocity v (this is valid for small flip angles and remains a good approximation at larger flip angles (<goo) for most RF-pulses). Thus, velocity encoding and compensating slice select gradients as shown in Fig. 11 can be considered. In this case the transfer function can also be calculated analytically, but the expressions become rather complicated. Typical results are displayed in Fig. 11: H,( f ) (sequential gating)

gradl(t).103 "i -2.5

-5

-

Badz(t).103 -

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

t.10)

0.6 I I I I I 1 0 10 20 30 40 50

f

b

'1 .- sre(H3(0) I--:';, I,,; , ~.,l

I , ..?.-/ 0.25

............ ..~.. r - ................ , -----_,

0 , 0 10 20 30 40 50

f

C

FIG. 11. Transfer functions for PMAP with flow encoding and compensating slice select gradients. (a) Time diagram of the bipolar velocity encoding and tripolar velocity compensating slice select gradient (abscis is time in ms, ordinate is amplitude in mT/m). (b) Amplitude transfer functions. (c) Phase transfer functions. H,(f) is for sequential PMAP (A = 0). H2( f ) and H3( f ) are for interleaved PMAP with A = TR = 20 ms and A = TR = 50 ms, respectively.

Frequency Response of Flow Quantification with MRl 345

and Hzlf),H31f) (interleaved gating). H,(f) shows that for sequential triggering, attenuation can be observed in the amplitude transfer function and that the phase transfer function is, at first sight, linear in the frequency. How- ever, a closer inspection reveals that there is a small phase dispersion. For interleaved triggering (H2(f),H3(f)), again attenuation and amplification characteristics are observed for A(o), while @(a) displays some dispersion. Again, small gradient durations and repetition times are indicated.

For the first two combinations discussed in this section (two velocity encoding gradients and velocity encoding/ compensating gradients), sequential gating was found to exhibit no phase dispersion. However, for these combinations, the gradient timings were dependent on each other: the constant amplitude duration T was equal for both gradients (see Fig. 8). The question arises concern- ing what happens when the timings are different (e.g., velocity encoding and compensating gradients of the same duration). A first indication to the answer can be found in the last combination (slice select gradients). These asymmetrical gradients were found to give rise to (a small) phase dispersion, even for sequential gating. We shall now investigate when phase dispersion occurs in PMAP for a general combination of gradients (symmetrical or asymmetrical) and each of both gating modes.

The transfer function for PMAP was given by Eq. [37]. Combining this with Eq. [13] leads to:

with

AG(o) = S{Ag(t))t Ag(t) = gi(d - gz(t) [43I

Equations [42] and [43] show that the transfer function for PMAP is determined by the spectrum (complex conjugate of the Fourier transform) of the difference gradient Ag(t). The difference gradient is the difference between the flow encoding gradient gl(t) of the first scan and the flow encoding (compensating) gradient gz(t) of the second scan. Note that in Eq. [43], g,(t) is considered on the interval [O,T,] and g,(t) on [A,A + T,] with T,,T, the durations of the gradients; A = 0 for sequential gating and A = TR for interleaved gating. Following Eq. 1431, the transfer function for PMAP can be analyzed for different gradient combinations by using the results given in the Interpretation of the Velocity Transfer Function section, but now applied on the difference gradient Ag(t). This means that HpMAp(o) will not be dispersive when Ag(t) is odd, when centered around the middle of the time interval covered by the gradient.

Some examples for Ag(t) are shown in Fig. 12 for interleaved PMAP and in Fig. 13 for sequential PMAP. Figure 1 2 shows that for interleaved triggering, Ag(t) is odd (with respect to the midpoint 0 ' ) for case (a). For cases (b) and (c), Ag(t) is neither odd nor even and phase dispersion will result. This was also confirmed by the phase transfer functions given before (see Figs. 10 and 11). Thus, for interleaved PMAP, it is only valid to say that the velocity is measured at a certain instant, when two identical gradients of the opposite polarity are used.

/ / \ > 0 t

a

n - b

C

FIG. 12. Flow encoding gradients g,(t) and g2(f) of the first and second scan in interleaved PMAP (A = TR) and the difference gradient Ag(t): (a) bipolar velocity encoding gradients of the opposite polarity, (b) bipolar velocity encoding and tripolar velocity compensating gradient, (c) velocity encoding and velocity compensating slice select gradients.

In this case, the velocity is measured halfway between the two gradients. For sequential gating, Fig. 13 shows that, for case (a) (two identical bipolar gradients of the opposite polarity), Ag(t) is odd and thus the velocity is measured at the crossover of the gradients. Case (b) shows that for different gradient parameters, Ag(t) is no longer odd and phase dispersion will occur. An odd difference gradient will also result for case (c) with a bipolar velocity encoding and a tripolar velocity compensating gradient with the symmetry shown in Fig. 8c. Now, the velocity is measured at the crossover of Ag(t). Also in this case, for different gradient parameters, this is no longer valid (see Fig. 13d). Case (e) shows that, for the slice select gradients considered before, an asymmetric difference gradient results that gives rise to phase dispersion. Note that, as discussed in the Interpretation of the Velocity Transfer Function section, when the nonlinearities in the phase transfer function are small, the velocity is approximately measured after a delay given by Eq. [ZOI.

346

I I I I I 1 I I I I

Peeters et al.

I I I I I 1 I 1 I 0 1 2 3 4 S 6 7 8 9 10

-1

a t

tion, gradient waveforms and combinations can be designed that overcome this phase dispersion. ID Phase Mapping. In 2 D phase mapping, gating has to be performed to overcome ghosting artifacts. However, when the motion is nonperiodic (e.g., due to an irregular heart rate), ghosts will still arise. Another disadvantage is that with ECG-triggering, the acquisition time will become relatively long. These drawbacks can be circum- vented by eliminating the phase encoding gradient. The price for this is sacrificing spatial resolution along one dimension.

A first technique that avoids gating is RACE (real time acquisition and evaluation) (11). The pulse sequence for this method is the same as for 2 D PMAP (see Fig. 6), with the phase encoding gradient omitted. As a result, spatial resolution is only obtained along the x-direction and a projection is performed along the y-direction. Another version (12) of this line scan technique uses a 2 D W- pulse in order to excite a column of spins. For the re- mainder, the sequence is similar to RACE: after the W- pulse flow encoding or compensating gradients are used and spatial resolution is obtained along the read out direction. We shall call this version simply "line scan." Note that in RACE, the slice select gradient (which is necessarily asymmetrical) is always used to encode or compensate the flow, while in line scan symmetrical gradients can be used. Remember also that both techniques are run in interleaved mode.

When deriving the transfer function for RACE or line scan, due to the projection (along the y-direction), the velocity profile becomes important. For pulsatile flow in a straight cylindrical tube, the velocity profile is rather complicated: it depends on Bessel functions of a complex argument. In order to simplify the analysis, we only consider small Womersley numbers (13). In this case, the velocity profile is parabolic and can be written as:

with r the radical distance, R the radius of the tube, and V( t ) the time-dependent velocity factor. The motion- induced phase shift then becomes:

0 0.1 I 1.S 2 25 3 3.S 4 4.5 I 5.5

e 1

FIG. 13. Flow encoding gradients g,(t) and g2(t) of the first and second scan in sequential PMAP and t h e difference gradient (gradient amplitude in mT/m and time t in ms) Ag(t): (a) bipolar velocity encoding gradients of the opposite polarity, (b) idem but with different parameters, (c) bipolar velocity encoding and tripolar velocity compensating gradient, (d) idem but with different parameters, (e) velocity encoding and velocity compensating slice select gradients.

with q(t) position independent and given by the expressions quoted in the Flow Encoding as a Linear System section. For simplicity, we shall consider an isolated tube oriented along the z-axis (in vivo, surrounding tissue or vessels that are present along the projection direction will complicate the analysis for RACE) and a rectangular excitation profile for line scan. In this case, due to projection, the observed phase shift in a phase image is obtained as the argument of the complex number:

Of course, on the basis of the results given here and in the Interpretation of the Velocity Transfer Function sec-


with r = dx2 + J? and R,(x) = ?/R' - 2. The integral can be evaluated in terms of Fresnel integrals. The resulting expression for the observed phase shift is then:

with x the position along the read out direction, q(t) = h(t) €3 v(t) as stated before, and the Fresnel integrals defined as:

Equation [47] shows that, due to the projection mechanisms, the system becomes nonlinear. In this case, linear response theory is no longer valid and a nonlinear analysis should be performed (e.g., Volterra and Wiener theories (14)). Such an analysis is beyond the scope of this paper. However, for small arguments, the Fresnel integral can be expanded, giving rise to a linear approximation for the observed phase shift:

Jl(t) = ;[ 1 - (d'l - h(t) 63 v(t)

It is clear from Eq. 1491 that for each scan the velocity transfer function with projection is two thirds of HJw) obtained without projection. Combining this with Eq. [37], the transfer function for the RACE and line scan techniques becomes:

2 HRACE.LS(4 = p P h . f A P ( 4 [501

in the linear approximation. As a consequence, the results given for interleaved PMAP remain valid for RACE or line scan: the only difference is a factor 213. Remember that for large velocities (compared with the VENC), the linear approximation breaks down and nonlinear response theory is necessary.

Finally, to give another example where nonlinear effects can be important, we shall consider very briefly phase contrast angiography (PCA) (15). In 2D PCA, two bipolar velocity encoding gradients of the opposite polarity are used. After subtraction of the complex MR- signals obtained from both scans, a magnitude image is calculated where the magnitude (image intensity) is related to the phase shift according to at) = k sin q(t). For small phase shifts, the relation becomes linear but for large velocities (compared with the VENC) nonlinear response theory will be necessary.

INTERPRETATION OF FLOW MEASUREMENTS

In the Linear Systems Approach to Flow Encoding section, linear response theory was applied in order to un- derstand the process of phase-derived flow encoding. The transfer functions for some time resolved flow quantification techniques were then derived in the Response of Time Resolved Phase Methods for Flow Quantification section. In practice, it is interesting to find out how a

measured waveform vm(t) deviates from the real (e.g., arterial) waveform vreal( t), how these deviations are in- fluenced by the quantification technique, and how they can be avoided or corrected. The results obtained above will now be used to provide an answer to these questions.

Deviations between Real and Measured Waveforms

According to linear response theory, the flow measurement situation can be described by:

m

vEa(t) = v,, + CV,sin(nflt+a,) [511 n=1

CC

vhl~l(f) = Vo,m + xVn,msin(nflt + a,,m) [521 n=1

with

In these expressions, fl = 27r/Tp represents the funda- mental angular frequency, Tp being the pulsation period of the periodic waveform, which is represented as a Fourier sum (with amplitudes V, and phases an). As a result, because Hm(0) = 1 (except for ID PMAP), the D.C. component (net velocity) is not altered, and the amplitudes and phases of the velocity waveform are modified by the amplitude and phase transfer function of the MRI quantification technique. As seen in the previous sections, the linear terms in Om(w) correspond to a time shift (i.e., determines at what instant the velocity is measured), while IH-(w)l and the nonlinear terms in Om(w) cause a deformation of the velocity waveform.

Improving Measurement Timing

In time resolved measurements, an image series (or line series) is obtained on the basis of which the velocities are determined. Each image in the series corresponds to a certain delay after the trigger pulse (for prospective gating) or a delay with respect to the start of the measurement (for I D PMAP). Usually, the measured velocities are plotted against these delays, assuming that the velocities are measured at t = 0, A, 28, . . ., NA with N the number of images and A the delay between two images. For FFM and sequential 2D PMAP this delay is given by A = TR, and for interleaved 2D PMAP and 1D PMAP by A = 2 TR. Another common assumption is that the velocities are measured at the center of the echo: t = TE, A + TE, 2A + TE, . . ., NA + TE. However, as shown in the previous sections, in reality the velocities are measured at t = d, A + d, 2A + d, . . ., NA + d, with the time shift d being given by:

In this expression, ti, represents the instant the velocity encoding gradient is switched on in the first scan (with

348 Peeters et al.

the time origin at the start of the read cycle) and 6, is given by Eq. [ Z O ] , but applied on the difference gradient Ag(t]. Some interesting results for d are summarized in Table 1. Our study shows that velocity curves reported in the literature should always be shifted in time in order to represent the velocities at the correct instance during the cardiac cycle. Furthermore, the measured velocities can be affected by the attenuatiodamplification and phase dispersion phenomena, depending on the quantification technique (gradient parameters and sequence timing] and on the frequency content of the velocity waveform studied.

Improving the Measured Waveforms

A first point that can be made in this context concerns the common practice of plotting the measured velocities against time by connecting the measured points by straight lines. It is in fact a naive approach to reconstruct- ing the velocity waveform. A more meaningful result can be obtained by performing a Fourier analysis of the measured waveform, leading to a Fourier sum as given by Eq. [52]. The number of harmonics that can be determined is given by the Nyquist theorem: if N points are measured during the pulsation cycle, N / 2 harmonics can be reconstructed or if the sampling rate is fs and the heart frequency fH, fN/2fs harmonics can be retrieved. The sampling rate is determined by the temporal resolution of the measurement (the delay A]. As a consequence, A/2Tp harmonics can be reconstructed. As an example, for a heart rate of 1 Hz and a TR = 25 ms, 10 harmonics are found with 10 Hz as the highest frequency in the reconstructed waveform for 1D PMAP measurement. Maxi- mum frequencies encountered on patients are in the range 20-30 Hz. Such a Fourier analysis will lead to (determination of V,,,, and a,,MRI for each harmonic n] a reconstructed waveform that provides a more accurate approximation of the real velocity waveform. However, due to the limited number of reconstructed harmonics and deformations caused by the transfer function of the

measurement technique, some deviations will still be observed.

Deviations caused by the transfer function can be elim- inated provided HMRI(w) is known. In practice the gradient parameters and sequence timings are known and HMRI(w] can be calculated and used to correct the Fourier sum:

with

[581

[591

Table 1 Instant of Velocity Measurement for Some Flow Quantification Techniques

In principle, the corrected waveform should be identical to the real velocity waveform given by Eq. [51], apart from truncation effects. This correction scheme works only for those frequencies for which IH-(w)l # 0. This means that a frequency component, which has been completely removed by attenuation, cannot be recovered.

To illustrate the results given in this section, we have applied them to a real velocity waveform reported by Katz et al. (16) and corresponding to the ascending aorta in man. Figure 14 summarizes the data for a simulated interleaved 2D PMAP measurement. Plot (a] shows the bipolar velocity encoding and tripolar velocity compensating gradients chosen. For a repetition time TR = 40 ms, the temporal resolution is A = 2 TR = 80 ms, while the VENC for this sequence is vmaX = 100 cm/s. Plots (b) and (c) show the amplitude and phase transfer functions characterizing this sequence. In Fig. 14d four velocity waveforms are compared: the real velocity curve v(t), the naive velocity curve obtained by connecting the mea-

~~

Instant of velocity Gradient Condition measurement combination Sequential T See Figs 3a,b

Method Gating

ti + 5 FFM

2D PMAP Sequential T See Figs. 8a,b ti + 2

T,

ti + T'-?

See Fig. 8c

Ag(t) not odd Nonlinearities in e(o) small

T A See Figs. 8a,b t , + - + - 2 2

Interleaved

1D PMAP (Interleaved)

Ag(t) not odd Nonlinearities in

Ag(t) not odd Nonlinearities in e ( ~ ) small

e(o) small

Frequency Response of Flow Quantification with MRI

I I

349

I I L‘

0 10 20 30 40 50 60

ta1o3

a

b I I I I I

-5 ‘ I I I I I 0 10 20 30 40 50

f

C

sured points by straight lines vMRI(fR,) with tMRl = 0 , A, ZA, , . ,, NA, the reconstructed waveform vFouR(f) obtained by Fourier analysis of vIvw(tMRl) and the corrected velocity waveform vCOR( t) obtained after correction with the inverse transfer function as described by Eqs. [57]- [60]. Because measurements are performed over a 2 RR- cycle and the heart rate HR = 80, N = 18. As a result, nine harmonics can be reconstructed by Fourier analysis. As can be seen in Fig. 14d, vm(fivw) yields a relatively poor approximation to v(t). vFOUR(f) approaches v(t) closer, but still underestimates the peak velocities and shows some deformations. The corrected waveform vcoR(t) on the other hand provides a very good approximation to v(t). Plot (e) shows the same simulation but with HR = 120. Here, six harmonics can be retrieved. Similar conclusions as for plot (d) can be drawn. Note that in this case larger deformations can be observed due to the presence of higher frequencies and truncation effects. Figure 15 shows a FFM simulation. In these cases, A = TR = 60 ms and v,, = 100 cm/s. Plot (d) corresponds to HR = 120 and eight harmonics and plot (e) to HR = 80 and 10 harmonics. On these plots, the time shift d can be clearly observed. There is less deformation, as expected, because FFM does not cause phase dispersion. Figure 16 displays analogous simulations for the RACE technique. Here, A = 2 TR = 40 ms and v,, = 120 cm/s. For both cases 10 harmonics are used, while HR = 80 for plot (d) and HR = 120 for plot (e). The dramatic under- estimation of the velocities noted in these cases is mainly caused by the projection mechanisms leading to a factor 2/3 in the transfer function.

Finally, note that the observed deviations are very dependent on the response characteristics of the technique used (transfer function), the frequency content of the velocity waveform studied and the temporal resolution (thus the number of harmonics) used.

-- 0 02 0.4 0 6 08 I 1 2 1 4

~ , t -> t , t

d

0 0.2 0.4 0 6 0.8 1

t , t m . t . t

e

FIG. 14. Simulated interleaved 2D PMAP measurement (TR = 40 ms, HR = 80). (a) Time diagram of the velocity encoding and compensating gradients. (b) Amplitude transfer function. (c) Phase transfer function. (d) Comparison of four velocity waveforms: v(t) = real velocity waveform, vM,&.,,Rf) = velocity curve obtained by connecting the measured points by straight lines, vFOUR(f) = velocity waveform obtained by Fourier analysis of vMRI(tMRI), and vcoR(t) = velocity waveform obtained by Fourier analysis and

CONCLUSIONS

Our study shows that linear response theory is very helpful for obtaining a deeper insight in unsteady (pulsatile) flow effects in MRI. These effects can best be studied in the frequency domain: the influence of the gradients on the motion-induced phase shift is described by the transfer function H(o), the spectrum of the gradient being the determining factor. It was also shown that the traditional description in terms of the gradient moment expansion is very closely related to the transfer function. An analysis of H(o) provides a better insight in the process of flow encoding and an answer to practical questions like: which gradient waveforms can encode a certain order of motion, at which instant is it encoded, which distortions of the original waveform can be expected, . . . . Such an analysis can be very helpful in designing flow sensitive sequences.

This was illustrated through the application of the general results to two flow quantification techniques (FFM and PMAP), showing that each technique responds

correction with the inverse transfer function. (e) Comparison of the four velocity waveforms for the same measurement but now on a waveform with HR = 120.

350

lH(f)l 0 8

07

-

Peeters et al.

/-; - -

/

I I

I I I I l

a

b

C

(1 GJ -

grSd2(IG) Io3

-10 0 10 20 30 40 M 60

tG103

0 10 20 30 50 40 f

5 , I I I I

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0 10 20 30 40 50 f C

-20 ' 1 I I I 1 0 0.2 0.4 06 0.8 1

t.tlu(RI.1.1

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t.tlu(RI.t,t

d

-201 I I I I I I I 0 0.2 0.4 06 0.8 I 1.2 1.4

t.tlu(RI.1.t

e

FIG. 15. Simulated FFM measurement (TR = 60 ms, HR = 120). (a) Time diagram of the velocity encoding gradient. (b) Amplitude transfer function. (c) Phase transfer function. (d) Comparison of four velocity waveforms: v(f), vM~l(tMRl), vFOUR(~) and vCoR(t). (e) Comparison of the four velocity waveforms for the same measurement but now on a waveform with HR = 80.

I"

0 02 0 4 0 6 0 8 1 1,llu(RI>f.'

e

FIG. 16. Simulated RACE measurement (TR = 20 ms, HR = 80). (a) Time diagram of the velocity encoding and compensating gradients. (b) Amplitude transfer function. (c) Phase transfer function. (d) Comparison of four velocity waveforms: v(f), VMR~(~MRI), VFOUR(~) and vCOR(t). Note that the solid line for v(t) is not shown: this is because v(t) is identical to vcoR(t). (e) Comparison of the four velocity waveforms for the same measurement but now on a waveform with HR = 20. Note that the solid line for v(t) is not shown: this is because v(t) is identical to vcoR(f).


in a different way to a typical velocity waveform. The transfer function was found to depend on the gradient parameters and sequence timing and was shown to result in distortions determined by the amplitude transfer function (attenuation/amplification) and the phase transfer function (dispersion). On the basis of linear response theory, each phase method was analyzed, showing that the consequences can be very important, especially for interleaved gated PMAP. Sometimes, as in 1D PMAP, linear response theory was found to be insufficient, ne- cessitating the use of nonlinear response theory.

The results were then used for obtaining a better interpretation of the measured velocity waveform. Applica- tion of Fourier analysis in combination with a correction by the inverse transfer function was shown to result in an acceptable reconstruction of the velocity waveform studied. Note that, while our demonstrations were restricted to periodic waveforms, the analysis can also be applied to nonperiodic unsteady flow (Fourier sum becomes Fou- rier transform).

Our study of the frequency characteristics of time- resolved flow phenomena in h4RI is not complete. On the one hand, it contains a number of simplifications: it ignores the presence of noise, partial volume effects (for in-plane flow quantification), nonorthogonal orientation of the vessel with respect to the slice, intravoxel dephas- ing, etc., while ideal gradients (no eddy currents) were assumed. A full description of a realistic quantification situation must take these factors into account. On the other hand, the approach introduced here may allow further insights. The spectrum of the gradient does not only influence H(o), it also determines the attenuation observed in amplitude images caused by diffusion or turbulence (17,18). Design of gradients waveforms on the basis of frequency domain analysis could lead to methods for studying turbulent unsteady flow, the averaged time-dependent flow being studied through phase images and the turbulent contribution through amplitude images. Finally, a number of experiments must be designed and performed to validate our theoretical results.

APPENDIX A CONNECTION WITH THE GRADIENT MOMENT APPROACH

In flow imaging, it is customary to expand the time dependent position in a Taylor series:

n=O

1 + zj(to)(t - + . . .

with z(t,), v(to), a(to), j ( f o ) , . . . the different orders of motion (position, velocity, acceleration, jerk, . . .) at the

reference time (expansion point) to (19). The motion- induced phase shift then becomes:

n=O

with

A41

the nth order gradient moment (here the original, un- shifted gradient is considered). The gradient moments describe the contribution of the different orders of motion to the motion-induced phase shift. Flow compensation or quantification techniques are designed by adjust- ing these moments to attain the desired goal. Note that in Eq. [A4], the gradient moments depend on the expansion point to.

To investigate the influence of the gradient on flow quantification, the gradient is expressed in terms of a reference point t,:g(t) = g(t - tg). For example, the reference point can be chosen at the center of the interval during which the gradient is switched on: tg = t,. The motion-induced phase shift then becomes:

n=O

with

Mn = \:?f)t"dt, [A61

where g(t) is now the shifted gradient. The advantage of this strategy is that it ensures that the gradient moments M, will not depend on the expansion point to and on the instant the gradient is switched on. The moments only depend on the shape or symmetry of the gradient (note that g(t) is now centered around the time origin). Equa- tion [A51 implies that the motion-induced phase shift depends on these gradient moments M, and on the different orders of motion at the center of the gradient t,. We can therefore say that a flow encoding gradient has two effects. First, the time at which the gradient is switched on determines the instant of flow encoding: all orders of motion are encoded at the center of the gradient t,. Sec- ondly, the shape of the gradient (or its symmetry) determines which orders of motion are encoded and to what extent they contribute to the motion-induced phase shift: these effects are described by the gradient moments M,.

We shall now discuss how the gradient (or gradient moments) influences the orders of flow encoding. This can easily be done by studying the dependence of the transfer function H(o) on the gradient moments. As seen in the Flow Encoding as a Linear System section, the response to position for phase methods is described by the transfer function H,(o) = yG*(w). If one expands the spectrum of the gradient in a Taylor series and uses the

352 Peeters et al.

connection between the derivatives and the moments, one obtains:

As discussed at the end of the Flow Encoding as a Linear System section, these gradient moments m, are related to the interval [O,u. If the zeroth order moment is nulled (m, = O), a common factor iw can be isolated in Eq. [A8], and the transfer function becomes H, = iwH, with:

] [A91 m, + im2- - m3- - im4- + . . . w w2 0 3

2! 3! 4!

Thus with m, = 0 , the system becomes a differentiator and instead of position velocity is now encoded. If also m, = 0 , one obtains H, = iwH, with:

w w2 0 3

3! 4! 5! + im3- - m4- - im,- + . . .] [A101

Now, acceleration is encoded (if mo = rn, = 0) and so on. From now on, we will work with a shifted gradient,

centered around the origin (interval [- T/2,T/2]). This means that in the above expressions [A7]-[A10], a common factor exp[iwT/2] appears (as shown in the Interpre- tation of the Velocity Transfer Function section) and the moments m, convert to M,. The influence of the gradient shape can now be studied. As seen before, the gradient consists generally of two contributions: an even and an odd part (see Eq. [15]). These two components have the following property:

g,,(f)Pdt = 0 [ A l l ]

(the odd moments of an even gradient and the even moments of an odd gradient are equal to zero).

Let's first consider an even gradient. For such a gradient all odd moments are nulled. When M, # 0, position is encoded and the transfer function HJw) is real (see Eq. [A8]). This leads to phase O,(w) = 0 and thus the position is encoded at the center of the gradient. If M, = 0 and M , # 0 , acceleration is encoded and the transfer function HJw) is also real. Therefore, acceleration is now encoded at the center of the gradient. This analysis can be extended to cover all cases for an even gradient. Note that an even gradient cannot encode velocity, jerk, . . ., and so on (they are compensated). Finally, note that for such a gradient, the transfer function for velocity, jerk, . . . becomes purely imaginary. Thus their phases are 8, = Oj = . . . = &sgn(w).

For an odd gradient all even moments are nulled. When MI # 0 , velocity is encoded and H,(w) is real: velocity is encoded at the center of the gradient. When M , = 0 and M3 # 0, jerk is encoded at the center of the gradient and so on. Note that an odd gradient cannot encode position, acceleration, . . . (they are compensated). The phase transfer function for position, acceleration, . . . then becomes 8 ( w ) = 7r/2sgn(w).

When the gradient is neither even nor odd, it consists, as mentioned earlier, of an even and an odd component. As a result, the transfer function is complex with a real and an imaginary part. If M, # 0, position is encoded and the position transfer function H,(w) can be written as:

with

A2 A4 2! 4!

A ( w ) = A, + -0' + -w4 + . .

8, 8 5 3! 5!

e(w) = 6,0 + -w3 + -w5 + . . .

[A131

[A141

Table A1 Coefficients for the Expansion of the Amplitude and Phase Transfer Function for Gradients that Encode a Certain Order of Motion

A0 A2 81 s, Condition Order of motion

Mo # 0

Mo = 0 M, # 0

Mk = O(k = 0, 1,. . ., n - 1) M, # 0

M" 1 Y + 1 2 M"+2 1 M"+, 6 M"+3 2 M"+l ~ y- I - - - __ n nth order n! (n:)![n+l( M, 1 n+2 Mn ] n+l% - (n+3)(n+Z)(n+l) Mn (n+l)'( M. )


The coefficients Ai and 6i are given in Table Al: they depend on the different gradient moments M,. From the first row in Table A1 (position), the phase transfer function OJw) is seen to be a nonlinear function of o. As discussed in the Interpretation of the Velocity Transfer Function section (for velocity), the position is approximately encoded after a delay At = TI2 + 6, with respect to the time the gradient is switched on when the nonlinearities are small. When M, = 0 and M, # 0 (velocity encoding), the expansion can also be written for H J o ) and the same conclusions can be drawn for the encoded velocity (see the second row of Table Al). The analysis can be extended to acceleration encoding, . . . or more generally to nth order of motion encoding. The results are also shown in Table Al. On the basis of these results, special cases can be studied such as a phase encoding gradient that encodes position but compensates velocity (Mo # 0, M , = 0) and so on. Note that the results for even and odd gradients are special cases of the expressions given in Table Al.

From Table Al, we see that in general, for nth order of motion encoding, the time delay 6, is given by:

6, = L(5) n + l M,

and depends on the gradient moments for the encoded order of motion and one higher order. Thus for position encoding it will depend on MIIMo (velocitylposition), for velocity encoding on MJM, (accelerationlvelocity), and so on. It is also interesting to remark that in order to keep the nonlinearities (phase distortions) small, the ratio

1 Sk+2 o2 (k + i)(k + 2) R = [A161

should be kept small. Because an upper limit for the gradient moments is given by:

Mns*(') n+l ,

n + l 2 [A171

with g,, the maximum gradient amplitude and T the gradient duration, the ratio goes as R a (wTI2)'. As a consequence, phase distortions will be small for low frequencies and/or short gradients. Under these circumstances, amplitude attenuation effects (see A(o)) will also be small.

When the encoding process is described in terms of the moment expansion as in Eq. [As], in the presence of phase distortions, all consecutive terms will be present in the expansion. However, for odd or even gradients (no phase distortion), alternating terms will vanish. For example, velocity encoding with an odd gradient (with M, = 0) will result in:

i.e., only velocity-, jerk-, . . . related terms remain, and position- (of course), acceleration-, . . . related terms have disappeared. Note that even for velocity encoding with phase dispersion, the acceleration-related term will vanish when the delay 6, is included in the instance of encoding.

APPENDIX B GENERAL RESULTS FOR THE ENCODING PROCESS OF THE NTH ORDER OF MOTION

* Condition:

mk = 0 (k = 0,1,. . .,n - 1) and m, # 0

Mk = 0 (k = O , l , . . .,n - 1) and Mn # 0 or

* Impulse response:

hn(tl = yg*(t) €3 U(t)@" [Bll

[B21

(SCm1(t) is the mth derivative of the Dirac delta function)

* Transfer function:

[B51 k=n

* Flow encoding process:

T q(t) = a,(t) 8 3-1{e'e,(01} €3 bn)( t + y + [B7]

with

[B81

REFERENCES 1. J. L. Duerk, R. E. Wendt III, Motion artifacts and motion

compensation, in "Magnetic Resonance Angiography: Con- cepts & Applications" (E. J. Potchen, E. M. Haacke, J. E. Siebert, A. Gottschalk, Eds.), pp. 80-133, Mosby-Year Book, St. Louis, 1993.

2. E. J. Potchen, E. M. Haacke, J. E. Siebert, A. Gottschalk, Eds.,

354 Peeters et al.

“Magnetic Resonance Angiography: Concepts & Applica- tions,” Mosby-Year Book, St. Louis, 1993.

3. K. H. Parker, C. G. Caro, Flow in the macrocirculation: basic concepts from fluid mechanics, in “Magnetic Resonance Angiography: Concepts & Applications” (E. J. Potchen, E. M. Haacke, J. E. Siebert, A. Gottschalk, Eds.), pp. 134-145, Mosby-Year Book, St. Louis, 1993.

4. D. N. Firmin, C. L. Dumoulin, R. H. Mohiaddin, Quantita- tive flow imaging, in “Magnetic Resonance Angiography: Concepts & Applications” (E. J. Potchen, E. M. Haacke, J. E. Siebert, A. Gottschalk, Eds.), pp. 187-219, Mosby-Year Book, St. Louis, 1993.

5. J. D. Gaskill, “Linear Systems, Fourier Transforms and Op- tics,” John Wiley & Sons, New York, 1978.

6. R. N. Bracewell, “The Fourier Transform and Its Applica- tions,” 2nd ed., McGraw-Hill, New York, 1986.

7. R. Frayne, B. K. Rutt, Frequency response of retrospectively gated phase-contrast MR imaging: effect of interpolation. J. M a p . Reson. Imaging 3, 907-917, (1993).

8. J. Hennig, M. Mueri, P. Brunner, H. Friedburg, Quantitative flow measurement with the fast Fourier flow technique. Radiology 166, 237-240, (1988).

9. I. R. Young, G. M. Bydder, J. A. Payne, Flow measurement by the development of phase differences during slice forma- tion in MR imaging. Magn. Reson. Med. 3, 175-179, (1986).

10. G. L. Nayler, D. N. Firmin, D. B. Longmore, Blood flow im-

aging by cine magnetic resonance. J. Cornput. Assist. To- mogr. 10, 715-722, (1986).

11. E. Mueller, G. Laub, R. Grauman, et al., RACE-Real time acquisition and evaluation of pulsatile blood flow on a whole body MRI unit, in “Proc., SMRM, 7th Annual Meet- ing, Berkeley, CA, 1988,” p. 729.

12. K. Butts, N. J. Hangiandreou, S. J. Riederer, Phase velocity mapping with a real time line scan technique. Magn. Reson. Med. 29, 134-138, (1993).

13. W. W. Nichols, M. F. O’Rourke, “McDonald’s Blood Flow in Arteries,” 3rd ed., Edward Arnold, London, 1990.

14. M. Schetzen, “The Volterra and Wiener Theories of Nonlin- ear Systems,” John Wiley & Sons, New York, 1980.

15. C. L. Dumoulin, H. R. Hart, Magnetic resonance angiography. Radiology 161, 717-720, (1986).

16. J. Katz, R. M. Peshock, P. McNamee, et al., Analysis of spin-echo rephasing with pulsatile flow in 2D FT magnetic resonance imaging. Magn. Reson. Med. 4, 307-322, (1987).

17. J. Stepisnik, Measuring and imaging of flow by NMR. Progr. NMR Spectrosc. 17, 187-209, (1985).

18. P. T. Callaghan, “Principles of Nuclear Magnetic Resonance Microscopy,” Clarendon Press, Oxford, 1991.

19. 0. P. Simonetti, R. E. Wendt, J. L. Duerk, Significance of the point of expansion in interpretation of gradient moments and motion sensitivity. J. Magn. Reson. Imaging 1, 569-577, (1991).

time resolved flow quantification with mri using phase methods: a linear systems approach

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