Spin-Echo MRI Using /2 and Hyperbolic Secant Pulses

Transcription

1 Magnetic Resonance in Medicine 6:75 87 (2009) Spin-Echo MRI Using /2 and Hyperbolic Secant Pulses Jang-Yeon Park* and Michael Garwood Frequency-modulated (FM) pulses have practical advantages for spin-echo experiments, such as the ability to produce a broadband rotation, with an inhomogeneous radiofrequency (RF) field. However, such use leads to a nonlinear phase of the transverse magnetization, which is why FM pulses like the hyperbolic secant (HS) pulse are not commonly used for multislice spin-echo magnetic resonance imaging (MRI). Here, a general theory and methods are described for conventional spin-echo imaging using a HS pulse for refocusing. Phase profiles produced by the HS pulse are analytically described. The analysis is extended to yield the specific relationships between pulse parameters and gradients, which must be satisfied to compensate the nonlinear phase variation produced with a spin-echo sequence composed of /2 and HS pulses (the /2 HS HS sequence). The latter offers advantages for multislice spin-echo MRI, including excellent slice-selection and partial compensation for RF inhomogeneity. Furthermore, it can be implemented with a shorter echo time and lower power deposition than a previously described method using a pair of HS pulses. Magn Reson Med 6:75 87, Wiley-Liss, Inc. Key words: frequency-modulated pulse; nonlinear phase compensation; hyperbolic secant pulse, spin-echo MRI The Center for Magnetic Resonance Research and Department of Radiology, University of Minnesota Medical School, Minneapolis, Minnesota, USA Presented in part at the ISMRM at Seattle in the Unites States, 2006, and at Berlin in Germany, Grant sponsors: NIH; Grant numbers: P4 RR008079, P30 NS05709; Grant sponsor: Keck Foundation; Grant sponsor: the MIND Institute. *Correspondence to: Jang-Yeon Park, Center for Magnetic Resonance Research, 202 6th Street SE, Minneapolis, Minnesota 55455, USA. jypark@cmrr.umn.edu Received August 5, 2007; revised August 20, 2008; accepted August 2, DOI 0.002/mrm.2822 Published online in Wiley InterScience ( Wiley-Liss, Inc. 75 Frequency-modulated (FM) pulses, including adiabatic full-passage (AFP) pulses, are commonly used in pulse sequences to invert spins. In the classical rotating frame analysis of FM pulses, where the effective magnetic field vector B eff sweeps an angle relative to the z -axis, it can be shown that a magnetization vector M will remain approximately aligned with B eff when satisfying the adiabatic condition, B eff (t) d /dt, where is the gyromagnetic ratio. AFP pulses are usually preferred for accomplishing spin inversion because the rotation remains invariant in the presence of spatial inhomogeneity in the RF field (B ), once B exceeds a threshold level needed to satisfy the adiabatic condition. Another advantage of FM pulses is their ability to produce extremely wideband excitation or inversion because the bandwidth is not constrained by the available B. Unfortunately, when B is reduced to accomplish excitation (flip angle /2), standard AFP pulses cannot function adiabatically (because B eff (t) d /dt), and hence, the flip angle becomes approximately a linear function of B, similar to conventional pulses. Exceptions to this include the class of composite adiabatic pulses that were designed to produce a uniform rotation of any desired angle (,2). When compared to a single AFP pulse, most composite adiabatic pulses have the disadvantages of requiring a longer pulse duration and greater specific absorption rate (SAR). The manner in which an RF pulse influences the phase of transverse magnetization (M xy ) is an important consideration in many applications. For example, despite its ability to produce a B -insensitive rotation, a single AFP pulse is not often used to refocus magnetization in spinecho sequences because the phase of the magnetization following the pulse is a nonlinear function of the frequency offset, and this leads to signal loss in certain types of acquisitions including multislice MRI. To overcome this problem, some composite AFP pulses have been proposed to eliminate the nonlinear phase profile produced by a single AFP pulse (3 5). A double spin-echo sequence has also been developed using two identical AFP pulses, in which the nonlinear phase variation generated by the first AFP pulse is cancelled out by the second AFP pulse (6). An alternative approach uses FM pulses for both excitation and refocusing, because under certain conditions the nonlinear phase produced by the AFP refocusing pulse can be compensated by creating transverse magnetization with an FM excitation pulse, which also produces a nonlinear phase (7 ). In the first demonstration of this approach, Kunz (7,8) employed a chirp pulse (a linear frequency sweep with constant B ) for both excitation and refocusing in 2D spin-echo imaging, realizing that the quadratic phase profile produced by the chirp pulse can be compensated by appropriately setting the amplitude of a slice-selective gradient or the frequency-sweep velocity used in the individual chirp pulses. In other words, in a spin-echo sequence consisting of two chirp pulses with frequency-sweep velocities v and v 2 in the presence of slice-selective gradients G and G 2, complete rephasing occurs for G G 2 and v 2 v 2,orforG G 2 and v v 2.Böhlen et al. (9) also used the chirp pulse for both excitation and refocusing in spectroscopy. They independently reached the conclusion that the quadratic phase compensation can be attained when the pulse length of chirp excitation (T p, ) is twice as large as that of chirp refocusing (T p,2 ), that is, T p, 2T p,2, which is the same as the first of two conditions Kunz already derived. Cano et al. (0) obtained the same condition, that is, T p, 2T p,2,

2 76 Park et al. for the nonlinear phase compensation when a hyperbolic secant (HS) pulse is used for both excitation and refocusing in spectroscopy. Although Cano et al. (2 4) analyzed a spin-echo sequence using the HS pulse, their analysis was based on the assumption that the frequency sweep of the HS pulse is approximately linear over the most intense portion of the pulse and, thus, is not substantially different from the analysis of the chirp pulse previously done by Kunz and Böhlen et al. On the other hand, simply borrowing Kunz s idea, Shmueli et al. () proposed another condition to achieve rephasing when the HS pulse is used for both excitation and refocusing, which controls the frequency-sweep rate of each HS pulse by changing a pulse truncation factor ( ), that is, Although the works mentioned above all focused on methods of removing the nonlinear phase, it has been recently reported that adiabatic 3D spin-echo imaging can benefit from using a single AFP pulse for refocusing without nonlinear phase compensation when spatial-encoding is performed in the slab-selective direction (5 8). To avoid sacrificing signal-to-noise ratio (SNR) in conventional 3D MRI, the dynamic range of the analog-to-digital converter (ADC) must be sufficient to digitize the large signal arising from the slab of in-phase magnetization, while leaving an adequate number of bits to define the thermal noise level. The nonlinear phase variation produced by a FM pulse helps to reduce this problem because the isochromats are never in phase at the same time, and thus, the peak echo amplitude is reduced (9 2). In addition, the potential for SNR loss due to B inhomogeneity during the pulse can be minimized by operating the FM pulse in the adiabatic condition. In this paper, a general theory and methods are analytically described when a single AFP pulse is used for refocusing in spin-echo imaging, specifically focusing on the use of the HS pulse, which delivers an excellent slice or slab profile as well as B insensitivity. One purpose of this work is to present a theoretical analysis of the phase profiles produced when a single HS pulse is used for slabselective refocusing, which is relevant to previous works in adiabatic 3D spin-echo imaging. Second, the phase profiles are analytically described for the case of a spin-echo sequence that uses HS pulses for both excitation and refocusing, which will be referred to as the /2 HS HS sequence. Finally, based on this analysis, general conditions for complete rephasing with nonlinear phase compensation across the slice are proposed for multislice 2D /2 HS HS imaging. The analysis and methods presented here are demonstrated using Bloch simulations and by performing imaging experiments on a phantom and human brain at 4T. THEORETICAL ANALYSIS AND SIMULATIONS Nonlinear Phase Profiles Produced by an HS Pulse Used for Refocusing After some period of time following an excitation pulse, suppose an AFP pulse is used to create a spin echo. The initial orientation of M xy is assumed to be perpendicular to the B direction of the excitation pulse (it should be noted that, in the case of an adiabatic half-passage [AHP] pulse, M xy and B are collinear) (22). After excitation, the effect of the AFP pulse on M xy can be easily visualized in a frame of reference (x, y, z ) that rotates in synchrony with the time-dependent pulse frequency RF (t). In this frame of reference called the RF frame, the orientation of B remains stationary during the pulse. When B and the main external magnetic field (B 0 ) are applied along the x and z axis, respectively, the effective magnetic field vector eff is given by the vector sum of the field components and, eff t t xˆ t ẑ, [] where (t) is the time-dependent pulse amplitude ( B [t]) and (t) is a time-dependent resonance offset, that is, (t) 0 RF (t), where 0 is the Larmor frequency of a given isochromat. Provided that the adiabatic condition is well satisfied, the spin isochromats initially perpendicular to eff will rotate about eff and will remain almost perpendicular to eff throughout the pulse (see Appendix for details). Thus, after the frequency sweep, the isochromats in the transverse plane experience a rotation with an additional nonlinear phase variation ( nl ) due to their nutation about eff, where nl is given by nl 0 T p eff t dt 0 T p t 2 t 2 dt. [2] Here, the sign of nl depends on the direction of frequency sweep, that is, positive if the sweep is from high to low frequency and negative if the sweep is from negative to positive. Now take the case of the HS pulse. The HS pulse was originally introduced in a complex form to analytically solve the Bloch-Riccati (4), that is, t max sech t t 0 i, [3] where max is the maximum pulse amplitude in rad/s, is a real constant, and in rad/s determines the pulse amplitude at which the pulse is truncated. From a practical viewpoint, it is convenient to express a frequency-swept pulse like the HS pulse in terms of amplitude-modulation (AM) and FM functions. By introducing a normalized time 2t/T p for t in the range [0, T p ], where T p is the pulse length, the AM and FM functions of the HS pulse can be expressed as and max sech [4] RF c A tanh, [5] where c is the center frequency in the frequency sweep range (or pulse carrier frequency), A is the amplitude of the frequency sweep in rad/s (A ), and is treated as a dimensionless truncation factor ( 0.5T p ). Typically, is set so that sech( ) 0.0 (i.e., 5.3), which means the pulse is truncated so that the pulse amplitude is % of max

3 Spin-Echo MRI Using Hyperbolic Secant Pulses 77 at its extremes. From Eqs. [], [4], and [5], eff of the HS pulse can be described as eff max sech xˆ A tanh ẑ, [6] where is a time-independent resonance offset defined as 0 c, [7] and thus, from Eq. [2], nl of the HS pulse is given by nl (, max ) T p 2 max 2 sech 2 A tanh 2 d. [8] In the limit that tanh( ) 3 (in fact, tanh( ) for the value of mentioned above), Eq. [8] can be analytically evaluated and, fortunately, max -dependent terms can be separated from -dependent terms, that is, where nl, nl, max nl, nl,2 max, [9] AT p ln 2 T p A 2 nl,2 max AT p log max T p max 2 A ln A A 2 tan 2A max 2 A 2 2A 2 max Here, nl, ( ) determines the shape of nl along the resonance offset ( ) axis while nl,2 ( max ) adds a constant phase offset, for a pulse of given max, that shifts the whole nl curve vertically on a graph. At first glance the nonlinear terms in nl, ( ) appear to have a complicated form due to their logarithm functions, but these are simply quadratic functions of to a first-order approximation, as can be shown by Taylor expansion (2). It is worthwhile to note in Eq. [9] that max does not affect the shape and the position of nl along the frequency direction due to the separation of and max, which can lead us to an alternative approach that analyzes a spinecho sequence using a HS pulse in a more intuitive and descriptive way, affording an analytic expression for nl with no dependency on max. This alternative approach, which will be presented below, is similar in form to the previous analysis of a gradient-echo sequence using an HS pulse for excitation (2). Specifically, it considers only the FM function of the HS pulse for the calculation of the nonlinear phase profile produced by the /2 or HS pulse in the reference frame, which rotates at the center frequency in the sweep range (the c frame). This alternative approach may be effective and useful to analyze pulse sequences using the FM pulses including the HS pulse as long as max -variation is not relevant to the analysis, or is small relative to the effects of -variation. Using a Single HS Pulse for Refocusing In this section, a spin-echo sequence is analyzed that uses a single HS pulse for refocusing, following any conventional /2 excitation pulse (e.g., a sinc or AHP pulse). The nonlinear phase profile produced by a HS pulse is calculated using the alternative analysis mentioned above, and it is demonstrated that a pseudo spin-echo is generated during the readout, by chasing the position of the vertex of the nonlinear phase profile along a frequencyencoded (or phase-encoded) direction, which is the same as the slice-selective direction. As previously done for excitation (2), the present analysis assumes that during the frequency sweep an isochromat with frequency achieves resonance at a unique time t that is determined by the FM function. As shown in Figure, results from Bloch simulation (M z [t]) provide a good rationale for this assumption in the case of both /2 and rotations performed by the HS pulse. The present analysis also assumes that the isochromat experiences a rotation instantaneously along an axis in the transverse plane at the exact time of resonance, t. Based on these assumptions, t can be written in terms of and T p during HS refocusing, t T p 4 ln A A T p 2, [0] by taking the inverse of the FM function of the HS pulse (i.e., Atanh( ), where 2t/T p ). With the additional assumption that the isochromat freely evolves in the transverse plane before the rotation (t t ), the phase immediately after the rotation is given by t 2 HS (, t ), where HS (, t ) is the phase of the HS pulse at t t given by HS,t 0 t RF t c dt AT p 2 ln A sech A 2 2, [] when using Eq. [0] to obtain t. Here, 2 HS (rather than simply HS ) has to be added to the phase negation (i.e., t ) because a rotation produces a reflection of M xy ( ) about the B direction given by HS (, t ) during the pulse (Fig. 2). In contrast, when the HS pulse is used for excitation, only HS is considered for calculating the phase immediately after excitation (2). One interesting point of note is that, according to the identity of logarithmic function, Eq. [] can be separated into two parts without and with -dependence: HS,t AT p 2 A sech AT p 2 ln A 2 2. [2] Because the first term in Eq. [2] does not affect the shape and the position of HS along the frequency direction with no dependency on, it can be ignored in the context of the present analysis for convenience. Thus, Eq. [2] can be rewritten in terms of the -dependent term only, that is,

4 78 Park et al. Then, using Eqs. [0], [3], and [4], the total phase accrued at time t immediately following HS refocusing (t 0) becomes,t AT p ln A 2 2 T p 2 ln A t. [5] A In Eq. [5], the first and second nonlinear terms are in exact agreement with nl, ( ) in Eq. [9], which supports the validity of the present approach in an analytical manner. The third linear term shifts the vertex of as t progresses, as illustrated in Figure 3a. In the presence of a field gradient G, the vertex moves in the direction of the gradient (x) as time progresses, because Gx. In this case, the -vertex position can be determined by taking the partial derivative of (, t ) with respect to to be zero, that is, T p 2 ln A t 0. [6] A Hence, the time at which the vertex occurs for a given isochromat is t T p 2 ln A. [7] A FIG.. Amplitude modulation (AM), frequency modulation (FM), and time-dependent longitudinal magnetization of the HS pulse. The HS pulse shown has T p 2msandBW/2 0 khz. a: AM functions ( [t]/2 ) are shown for /2 excitation and inversion ( max /2.35 and 4 khz, respectively). b: The FM function, ( RF [t] c )/2 ) shows that the frequency sweeps from 5 to 5 khz during the pulse. c: The time evolution of M z, as obtained from Bloch simulation, is plotted for three different isochromats with offset frequencies /2 3, 0, and 3 khz, in the event of /2 excitation. The horizontal dotted line is drawn to indicate the M z value of cos( /4) 0.7, which corresponds to the middle of /2 excitation. d: The time evolution of M z is plotted for three different isochromats ( /2 3, 0, and 3 khz) in case of inversion. c and d offer a good justification for the assumption that, during the frequency sweep, an isochromat with frequency achieves resonance at a unique time t, which is determined by the FM function. HS,t AT p 2 ln A 2 2, [3] and, hereafter, Eq. [3] will be used for the description of HS. During the remaining time of the pulse ( T p t ), it is also assumed that the isochromat freely evolves in the transverse plane. Therefore, given a time delay between excitation and refocusing, the total phase accrued at the end of HS refocusing is given by, T p t 2 HS,t T p t. [4] Equation [7] shows that each isochromat has a unique local rephasing time (Fig. 3b), and thus, in the presence of the gradient, a pseudo-echo would be produced because the isochromat corresponding to the vertex is locally rephased with nearby isochromats corresponding to the neighborhood of the vertex (2). It can be also expected from Eq. [7] that the isochromats excited early or late in the frequency sweep will be rephased early or late in the pseudo-echo, respectively. Rephasing takes place only locally (near the shifting vertex), and thus the signal energy is spread in time. The latter advantage has been exploited in 3D spin-echo imaging to maintain maximum SNR when the dynamic range of the ADC is limiting (5,6). Insensitivity to B -inhomogeneity is also possible because the HS pulse can be operated in the adiabatic condition. If apodization is needed (e.g., to improve SNR), Eq. [7] can be used to determine the local rephasing times of isochromats for applying a sliding window function (2). The /2 HS HS Spin-Echo Sequence The analysis performed so far considers the phase profile only when a single adiabatic HS pulse is used to produce the rotation in a spin-echo sequence. The next step is to consider the case in which both excitation and refocusing are performed with HS pulses. Because both /2 and HS pulses yield a nearly identical and uniform frequencyresponse profile (Fig. 4), this /2 HS HS sequence not only enables an excellent slice selection in multislice 2D spin-echo imaging, but it also provides a desirable degree

5 Spin-Echo MRI Using Hyperbolic Secant Pulses 79 FIG. 2. An illustration of the phase relationship between the transverse magnetization vector before (M ) and after (M ) applying rotations about different axes, as viewed in the reference frame of the on-resonant isochromat (i.e., the c rotating frame). a: When B is applied along the x axis ( RF 0), the magnetization vector is reflected about the x axis and experiences a rotation (i.e., its phase changes from a to a ). b: When the direction of B is not along the x axis ( RF 0), the magnetization vector is reflected about B, to yield a final magnetization phase equal to b RF. c: Following rotations about different axes, the angle between magnetization vectors is c ( b RF ) ( a ) RF ( a b ) 2 RF, where a b RF from a and b. In the case of conventional pulses like a sinc pulse, when B is applied along the x axis (at the c rotating frame) with a zero initial phase, the B direction is fixed along the x axis during the pulse, and thus, M ( ) simply experiences a phase negation, as illustrated in a. In the case of FM pulses like a HS pulse, because the B direction is not fixed in the x axis but is determined by RF (, t ) during the pulse, 2 RF (, t ) has to be added to the phase negation for the proper calculation of the M ( ) phase, as shown in b and c. of tolerance to B inhomogeneity due to the adiabatic property of the HS pulse. Suppose that /2 excitation is performed with a HS pulse with some arbitrary choice of T p, and parameters (below, subscripts of and 2 denote excitation and refocusing pulses, respectively). With the support of the result from Bloch simulation (Fig. c), like before (2) the analysis assumes that the magnetization rotates to the transverse plane instantaneously at the time of resonance, t, T p, A 4 A T p, 2, [8] which has the same form as Eq. [0]. It is also assumed that the isochromat freely evolves in the transverse plane during the remaining time of the pulse ( T p, t, ). Based on these assumptions, the phase accumulated by the end of HS excitation (t T p, ) becomes,t p, HS,t, 2 T p, t,, [9] where the additional /2 phase term arises because the /2 rotation of the magnetization takes place in a plane perpendicular to the direction of B. Following HS excitation, suppose that a HS pulse with parameters T p,2 and 2 is applied to refocus the magnetization after a time delay. During this pulse, the isochromat is assumed to freely evolve in the transverse plane before and after the rotation taking place at t T p, t,2. Thus, the phase immediately after the magnetization experiences the rotation is (, T p, ) ( t,2 ) 2 HS (, t,2 ), and the total phase accrued at the end of the HS pulse can be written as,t p, T p,2,t p, t,2 2 HS,t,2 T p,2 t,2. [20] Here, as mentioned in the previous section, 2 HS arises because the isochromat is reflected about the axis of B (Fig. 2). Ignoring the constant term /2 in Eq. [9], the total phase at any given time t after the refocusing pulse can be written as,t A 2T p,2 2 A A T p, 2 A T p,2 2 A 2 A 2 T p, 2 A A t T p, 2, [2] where t t T p, T p,2. According to Eq. [2], provided that 2 and A A 2, the first through fourth nonlinear terms are cancelled out when the specific condition T p, 2T p,2 is satisfied. Under this condition, complete rephasing occurs at t T p, /2 for A, which is in agreement with the previous works in spectroscopy using the chirp pulse by Böhlen et al. (9) and the HS pulse by Cano et al. (0). On the other hand, if T p, T p,2 and A A 2, all the nonlinear terms also disappear when 0.5 2, which explains why the condition Shmueli et al. () previously obtained by simulation functioned. If T p, 2T p,2 and 2, the nonlinear terms in Eq. [2] do not cancel, and due to the fifth (linear) term, the nonlinear phase profile moves in frequency space as t progresses. In this case, the vertex times (or local rephasing times) of isochromats can be obtained by taking the partial derivative of (, t ) with respect to to be zero, which yields t 2 T p,2 A 2 A 2 T p, 2 ln A A T p, 2. [22]

6 80 Park et al. frequency sweep of the /2 pulse will have its vertex occur early in the pseudo echo produced by the pulse). The General Condition for Nonlinear Phase Compensation across a Slice Now suppose that slice-selective gradients G and G 2 are applied during HS excitation and refocusing, respectively. By ignoring a constant term (e.g., /2) and substituting G x, where x is the spatial coordinate in the direction of the field gradient G, from Eqs. [3], [8], and [9] the phase profile at the end of slice-selective HS excitation can be obtained, x,t p, A T p, 2 G T p, x 4 2 A 2 G x A G x A G x G T p,x. [24] 2 If a rephasing gradient ( G ) is applied for one-half of the excitation time ( T p, /2) immediately after HS excitation, which compensates for the third (linear) term, the phase profile at the end of the rephasing gradient becomes x, 3 2 T p, FIG. 3. Phase profiles and local rephasing times (or vertex times) of pseudo-echoes in a spin-echo sequence using a single HS pulse for refocusing. a: Phase profiles from the analytical derivation (Eq. [5]) and Bloch simulation are compared during acquisition, that is, at t 0.05T p,, and 0.05T p. b: Local rephasing times of isochromats calculated from the analytical derivation (Eq. [7]) and simulation are compared. HS pulse parameters were T p msand A/2 5 khz. For excitation, a sinc pulse with T p 0.6 ms was used. Bloch simulations were performed using EXOR cycle. The plots show that predictions based on the analytical approximation and Bloch simulations are in reasonable agreement, which supports the validity of the analytical descriptions and approximations presented in this work. The residual difference between the analytical approximation and Bloch simulations can be ascribed to the assumption of an instantaneous resonance of a spin isochromat at a unique time determined by the FM function. In addition, the asymmetry of the nonlinear phase profiles in the simulation results may be explained by the fact that the trajectories of isochromats excited symmetrically in time about the pulse midpoint (e.g., 3 and 3 khz) are not exactly the same during the pulse. A T p, 2 2 G T p,x A 2 G x 4 A G x A G x. [25] Equation [22] shows that, when T p, 2T p,2, each isochromat has a unique local rephasing time and a pseudoecho is generated. For example, with the choice of parameters, T p, T p,2 T p and A A 2 A, Eq. [22] becomes t T p 4 ln A A T p 2 [23] Interestingly, Eq. [23] has exactly the same form as Eq. [8] except for the time delay, which means that the isochromats undergo local rephasing in the same order as they are excited (i.e., an isochromat that is excited early in the FIG. 4. Inversion frequency-response profiles (M z /M 0 ) obtained from Bloch simulation as a function of a resonance offset ( ) and a maximum pulse amplitude ( max ). The HS pulse was used in the simulation, with T p msandbw/2 0 khz. It is shown that the HS pulse is able to deliver an excellent slice or slab profile for both excitation and refocusing because the HS pulse yields nearly identical and highly uniform profiles for both excitation and refocusing. (In the case of excitation, M xy profile must be as desirable as M z because M 2 xy M 2 z.) It is also shown that, due to the adiabatic property of the HS pulse, spin isochromats within pulse bandwidth experience effectively the same rotation above the threshold max at which inversion begins to occur.

7 Spin-Echo MRI Using Hyperbolic Secant Pulses 8 On the other hand, from Eqs. [0], [3], and [4] the phase function due to HS refocusing in the presence of G 2 ( G 2 x) can be obtained, x,t p,2 A 2T p,2 2 2 G 2T p,2x A 2 2 G 2 x 2 2 A 2 G 2 x A 2 G 2 x. [26] From a comparison of Eqs. [25] and [26], the general condition can be derived for compensating the non-linear phase profiles across the slice produced by HS excitation and HS refocusing. Provided that 2, the nonlinear phase terms in Eqs. [25] and [26] cancel each other when the following conditions are satisfied, A T p, 2A 2 T p,2, [27] G T p, 2G 2 T p,2. [28] From a practical point of view, Eq. [27] describes the relationship between a fundamental parameter of RF pulses, which is the product of the pulse bandwidth in Hz (BW/2 ) and T p in seconds. Here, this product is called R, which allows Eq. [27] to be rewritten as R 2R 2. [29] The pulse parameters A ( BW/2) and T p are not independent of each other (i.e., R is fixed once the pulse has been generated). Therefore, the relationship between G and G 2 given by Eq. [28] is constrained by the choice of R and R 2. However, an infinite number of solutions to Eqs. [27] and [28] are possible because there are more undetermined variables than equations. According to Eqs. [27] and [29], any pair of T p, and T p,2 can be chosen as long as the R value of the HS excitation pulse is twofold larger than that of the HS refocusing pulse. Once T p, and T p,2 have been chosen, Eq. [28] can be used to obtain the values of G and G 2. Among all possible solutions to Eqs. [27] and [28], one simple solution is [ 2, T p, 2T p,2, A A 2, G G 2 ], which is the same as that previously shown to work for 2D spin-echo imaging with chirp pulses (8). Unfortunately, this solution requires the excitation pulse to be twofold longer than the refocusing pulse, which in certain applications can be a disadvantage (e.g., the requirement for T p, 2T p,2 increases the minimum echo time [TE]). The potential disadvantage of the first solution described above does not occur with an alternative new solution, which is [ 2, T p, T p,2, A 2A 2, G 2G 2 ]. For the sake of convenience, these two simple solutions, [ 2, T p, 2T p,2, A A 2, G G 2 ] and [ 2, T p, T p,2, A 2A 2, G 2G 2 ], will be referred to as condition I and II, respectively. On the other hand, in the case that 2, specifically 0.5 2, the nonlinear phase terms in Eqs. [25] and [26] also disappear when the following conditions are satisfied, A T p, A 2 T p,2 or R R 2, [30] FIG. 5. The /2 HS HS sequence diagrams for multislice 2D spin-echo imaging, satisfying the conditions for compensating nonlinear phases across the slice. For convenience, only the sliceselective direction is illustrated with AM and FM functions of the /2 and HS pulses, and the peak amplitude of the AM functions in the figure reflects the actual max for /2 excitation and refocusing. a: The sequence diagram is shown satisfying condition I, that is, [ 2, T p, 2T p,2, A A 2, G G 2 ]. b: The sequence diagram is shown satisfying condition II, that is, [ 2, T p, T p,2, A 2A 2, G 2G 2 ]. c: The sequence diagram is shown satisfying condition III, that is, [ 0.5 2, T p, T p,2, A A 2, G G 2 ]. Although HS excitation is not adiabatic, the /2 HS HS sequence offers a high degree of tolerance to B inhomogeneity due to full adiabaticity of HS refocusing. All the gradients are shown in dark gray, except for crusher gradients which are shown in light gray. G T p, G 2 T p,2. [3] Although there are also an infinite number of solutions that satisfy these conditions, the simplest one is [ 0.5 2, T p, T p,2, A A 2, G G 2 ], which is identical to the condition proposed by Shmueli et al. (). Like condition II, this solution has an advantage over the condition I in terms of TE because it also requires the same pulse length for both excitation and refocusing. This solution will be referred to as condition III hereafter. Spin-echo pulse sequence diagrams satisfying these three simple conditions are illustrated in Figure 5. Chemical-Shift Effects The analysis above predicts phase compensation, and thus spin-echo formation, when using HS pulse parameters and

8 82 Park et al. FIG. 6. Remaining nonlinear phase profiles and signal loss when condition II is employed. It was assumed that there exists only one chemical species with a chemical shift offset in the slice and its spin density is constant across the slice. a: The remaining nonlinear phase profiles were attained from both Bloch simulation and the analytical approximation (Eq. [32]) when the ratio of the chemical shift and the pulse bandwidth ( /BW) is equal to 0, 0.2, and 0.5. The remaining nonlinear phase profiles were plotted in the reduced slice thickness given by Eq. [37], which shifts along the slice-selective direction when 0. The range of the nonlinear phase variation increases as /BW increases. The nonlinear phase profiles from the analytical approximation are in good agreement with the ones from Bloch simulations. b: Signal loss obtained from Bloch simulations is plotted for different /BW, which results from nonlinear phase variation and mismatched slice profiles of excitation and refocusing. In all cases, HS pulses with BW/2 20 khz and BW2/2 0 khz were used for excitation and refocusing, respectively, having the same T p ( ms) and ( 5.3). Bloch simulations were performed using EXOR cycle. gradient amplitudes that conform to the constraints of Eqs. [27] and [28], or Eqs. [30] and [3]. However, to this point no consideration has been given to the potential consequences of a frequency offset that might arise from some sources other than the field gradient ( Gx). Examples of other sources of frequency shifts include B 0 inhomogeneity, chemical shift ( ), and variations in magnetic susceptibility within the sample. In the present analysis, the specific source of the frequency shift is not important, because all of these types of frequency offsets will be indistinguishable in the data. Therefore, here only chemical shift will be considered to produce a frequency offset which is secondary to that produced by G and G 2, but it should be realized that this analysis applies to other sources of frequency shifts as well. The change in frequency (chemical shift) is expressed in rad/s by the symbol. The phase evolution due to in the /2 HS HS sequence can be evaluated with the help of Eqs. [2], [25], and [26]. Accordingly, at time t immediately after the pulse (t 0),,x,t A 2T p,2 2 A 2 2 G 2 x 2 A T p, 2 A 2 G x 2 T p,2 2 2 G 2 x A 2 G 2 x A 2 G 2 x T p, 4 G x A G x A G x t T p 2, [32] where is a time delay between the excitation and refocusing pulses. Equation [32] shows that the nonlinear phase terms are nonzero when condition II is employed, whereas they cancel out in the case of conditions I and III. In other words, if 2, T p, T p,2, A 2A 2, and G 2G 2 (condition II), the nonlinear terms do not cancel because the frequency encoding in the presence of G ( 2G) and G 2 ( G) becomes 2 Gx and Gx, respectively, and thus, 2 Gx 2( Gx ) in the arguments of the logarithmic functions in Eq. [32]. Because the remaining nonlinear phase terms do not depend on time, but rather rely only on x and, Eq. [32] can be separated into two parts, without and with time dependence:,x,t,x 2,t, [33] where, for a given, (, x) represents the remaining nonlinear phase variation which is fixed in time across the slice after slice selection, and 2 (, t ) describes the free evolution of the -spin species after the pulse. In this case, the resultant spin-echo signal can be expressed as S t exp t /T 2 THK/2 exp i 2,t d THK/2,x exp i,x dx, [34] where (, x) is the spin density profile in the slice and T 2 is the transverse relaxation time constant. In Figure 6a, (, x) is shown for different values of /BW and is compared with data obtained from Bloch simulations. Figure 6a shows that the remaining nonlinear phase profiles from the analytical approximation (Eq. [32]) and from Bloch

9 Spin-Echo MRI Using Hyperbolic Secant Pulses 83 simulations are in good agreement. It also shows that the range of the remaining nonlinear phase variation increases as the value of /BW increases. In addition to the remaining nonlinear phase variation, a reduction in slice thickness also takes place with 0 when the condition II is employed. With condition I or III, where A A 2 A and G G 2 G, slice selection occurs for x values satisfying A A x G G, [35] and therefore, the slice shift is simply /( G) and the spin-echo slice profile is unaltered. On the other hand, the -dependence of the spin-echo slice position and profile is slightly more complicated to predict in the case of condition II. Specifically, when using A 2A 2 2A and G 2G 2 2G, slice-selective excitation occurs for x values satisfying 2A 2A x 2 G 2 G, [36] whereas in refocusing, slice selection occurs for x values given by Eq. [35]. Hence, the slice produced with condition II is narrower than that produced with condition I, because 2A A x 2 G G. [37] To calculate the amount of the reduced slice thickness when condition II is employed, let the slice thickness of conditions I and II be THK I and THK II, respectively. In this case, from Eqs. [35] and [37], THK I and THK II are THK 2A G, THK 4A 2 G. [38] Thus, the reduction of the slice thickness due to the mismatched slice selection of excitation and refocusing is given by THK /THK 4A 4A 2BW. [39] For example, the reduction of the slice thickness will be 0% when /2 500 Hz and BW/ Hz. In Figure 6b, the signal loss, which arises from both the remaining nonlinear phase variation and the mismatched slice selection of excitation and refocusing, is plotted versus /BW.It was evaluated from Bloch simulations, assuming that T 2 is infinitely long and there is only a single spin having uniform spin density across the slice (i.e., (, x) constant). When compared to the effect of the slice-thickness reduction given by Eq. [39], Figure 6b shows that the main signal loss of condition II when 0 can be attributed to the dephasing of spins across the slice due to the remaining nonlinear phase variation, especially for /BW 0.2. Bloch simulations were performed to investigate the shift and shape of slice profiles produced with the /2 HS HS sequence under conditions I, II, and III using different values of. EXOR cycle was employed to simulate the effect of crusher gradients alongside the refocusing pulse (i.e., like crusher gradients, EXOR cycle eliminates nonrefocused components of transverse magnetization) (23). For the HS pulses used in simulation for conditions I and II, the truncation factor was set to be 5.3. In the case of simulation for condition III, for the /2 HS pulse and 2 for the HS pulse were 5.3 and 0.6, respectively. Figure 7a and b shows that conditions I and II deliver excellent (square) slice profiles. However, as shown in Figure 7c, the sharpness of the M xy transition regions is diminished in condition III because the transition regions of the slice profile are broadened when is increased, that is, the cutoff amplitude at the edges of the pulse is decreased ( 2 2 ). In Figure 7b (condition II), a slight asymmetry in the M xy profiles and a reduction in the slice thickness are noticeable when /2 0.5 khz. These distortions are due to the different slice-selective gradients amplitudes (i.e., G 2G 2 ), as well as a slight difference in the width of the M xy transition regions produced with /2 HS versus HS pulses. In all three cases, the slice position shifts when 0 (i.e., the chemical-shift displacement). As shown in Figure 7d, e, and f, complete rephasing occurs inside the slice ( A) when 0. On the other hand, when 0, a nonlinear phase variation remains in the slice with condition II (Fig. 7e), whereas complete rephasing still occurs with conditions I and III (Figs. 7d,f), in agreement with theoretical predictions. EXPERIMENTS Experimental measurements were performed to test the theoretical prediction that conditions I, II, and III can compensate the nonlinear phase induced by the individual HS pulses in the /2 HS HS sequence. The threshold value where the adiabatic condition began to be satisfied by the HS pulse was determined experimentally by stepping the power level until the signal intensity in projections appeared to plateau. In both phantom and human experiments, the power of the HS pulse was set 3 db higher than this adiabatic threshold value, to gain partial insensitivity to B inhomogeneity. All experiments were performed with 90 cm 4T magnet (Oxford Magnet Technology, Oxfordshire, UK) interfaced to an imaging spectrometer (model Unity Inova, Varian, Palo Alto, CA). RF transmission and reception was performed with a TEM head resonator (24). Human studies were performed according to the procedure approved by the Institutional Review Board of the University of Minnesota Medical School. First, 2D spin-echo imaging was performed on a water phantom (Fig. 8). For condition I, HS pulses with T p, 20 ms and T p,2 0 ms were used for excitation and refocusing, respectively, having the same A/2 ( 0.5 khz) and ( 5.3) (Fig. 8b). For condition II, HS pulses with A /2 khz and A 2 /2 0.5 khz were used for excitation and refocusing, respectively, having the same T p ( 0 ms) and ( 5.3) (Fig. 8c). For condition III, HS pulses with 5.3 and were used for excitation and refocusing, respectively, having the same A/2 ( 0.5 khz) and T p ( 0 ms) (Fig. 8d). For comparison, the

10 84 Park et al. FIG. 7. M xy and phase profiles from Bloch simulations of the /2 HS HS sequence with nonlinear phase compensation across the slice. A D spin-echo sequence was simulated satisfying condition I (a and d), condition II (b and e), and condition III (c and f). A uniform D object was assumed for the simulation. For a and d, HS pulses with T p, 8msandT p,2 4 ms were used for excitation and refocusing, respectively, having the same A/2 (.25 khz) and ( 5.3). For b and e, HS pulses with A /2 2.5 khz and A 2 /2.25 khz were used for excitation and refocusing, respectively, having the same T p ( 4 ms) and ( 5.3). For c and f, HS pulses with 5.3 and were used for excitation and refocusing, respectively, having the same A/2 (.25 khz) and T p ( 4 ms). Bloch simulations were performed using EXOR cycle. As shown in d, e, and f, complete rephasing occurs for A with no chemical shift offset ( 0) in the slice. On the other hand, with 0, a nonlinear phase variation remains in the slice with condition II, whereas complete rephasing still occurs with conditions I and III. a and b show that conditions I and II deliver an excellent slice profile despite a shift in slice position for /2 0.5 khz. However, as shown in c, condition III produces a degraded slice profile in terms of the sharpness of the M xy transition regions because the transition regions of the slice profile are broadened when is increased. In b, the slight asymmetry of the M xy profiles and the slight reduction of the slice thickness for /2 0.5 khz appears not only because slice selection by HS excitation does not agree with that by HS refocusing when they have different slice-selective gradients, that is, G 2G 2, but also because the M xy profiles produced by HS excitation and HS refocusing have different sharpness in their transition regions. experiment was repeated using a sinc pulse with five lobes and the same bandwidth ( khz) (Fig. 8a). In all the cases, pulse repetition time (TR) 2 s, echo time (TE) 50 ms, matrix size , field of view (FOV) 5 5 cm 2, and slice thickness 5 mm. Figure 8 shows that the full adiabatic property of the HS pulse significantly reduced the signal loss due to B inhomogeneity, especially near the periphery of the phantom where B was weakest. In order to illustrate this more clearly, D image profiles were plotted along the line i and ii (Figs. 8e,f), respectively. Although the SNR at the center of the phantom was nearly the same in all three cases, the maximum SNR increases in Figure 8b d were about 30% near the periphery of the phantom. Multislice 2D spin-echo imaging of human brain was performed using the /2 HS HS sequence under condition II (A /2 2A 2 /2 2 khz; T p, T p,2 5 ms; 2 5.3) (Fig. 9b). For comparison, the experiment was repeated using sinc pulses of the same bandwidth ( 2 khz) (Fig. 9a). In all cases, TR 0.5 s, TE 22 ms, matrix size , FOV cm 2, and slice thickness 5 mm. Five slices were acquired with mm gap ( 20% of slice thickness) between slices. The first slice was chosen to compare SNR of acquisitions performed with sinc and HS pulses. To easily detect differences in sequence performance, the images were processed without applying intensity correction or any other filtering or smoothing. Upon close inspection of the first slice, it can be seen that the /2 HS HS sequence affords better SNR, particularly in the periphery of the brain, because the adiabatic HS pulse used for refocusing provided insensitivity to B inhomogeneity. The SNR improvement was 7% and 22% in white matter and cortical gray matter, respectively. DISCUSSION AND CONCLUSIONS In this work, a general theory and methods were described for spin-echo imaging using frequency-swept pulses, specifically HS pulses. Although only the HS pulse was treated here, it should be straightforward to apply the same analysis and methods to other frequency-swept pulses. The equations derived in this work are expected to be useful for analyzing spin-echo sequences that exploit the multiple desirable features of HS pulses, including an ability to excite and refocus spins within a sharply delineated (square) bandwidth using only low peak power. The SAR of the /2 HS HS sequence can be roughly estimated by integrating the pulse power over time in order to compare it with the SAR produced by a conventional spin-echo sequence (e.g., one performed with sinc pulses). Provided that the same pulse bandwidths are used in the comparison and assuming that the HS pulses have the minimum power needed to achieve /2 and rotations, the SAR of the sequences is approximately the same. Another advantage of the /2 HS HS sequence comes

11 Spin-Echo MRI Using Hyperbolic Secant Pulses 85 FIG. 8. 2D spin-echo imaging of a water phantom using a volume coil at 4T. a: The image was obtained using sinc pulses (five lobes) with T p 4.45 ms. T p of 4.45 ms was chosen to have the same BW ( 2A) in refocusing as the HS pulse, that is, A/2 0.5 khz. b: The image was obtained using HS pulses satisfying condition I. HS pulses with T p, 20 ms and T p,2 0 ms were used for excitation and refocusing, respectively, having the same A/2 ( 0.5 khz) and ( 5.3). c: The image was obtained by using HS pulses satisfying condition II. HS pulses with A /2 khz and A 2 /2 0.5 khz were used for excitation and refocusing, respectively, having the same T p ( 0 ms) and ( 5.3). d: The image was obtained by using HS pulses satisfying condition III. HS pulses with 5.3 and were used for excitation and refocusing, respectively, having the same T p ( 0 ms) and A/2 ( 0.5 khz). In all the cases, TR 2s,TE 50 ms, matrix size , FOV 5 5 cm 2, and slice thickness 5 mm. The dimensions of the cylindrical water phantom were: length 8 cm, diameter 9 cm. The phantom contained 0.45% NaCl solution and trace amounts of Gd-DTPA. The power of the HS pulse was set 3 db higher than the adiabatic threshold value in order to gain partial insensitivity to B inhomogeneity. The image obtained with HS pulses displays better quality than that obtained with sinc pulses, especially in the periphery of the phantom, due to the adiabatic property of the HS pulse. In order to illustrate this more clearly, D profiles are shown along the line i and ii, in e and f, respectively. Although the SNR at the center of the phantom is almost the same in all cases, the maximum SNR increases with HS pulses are about 30% near the periphery of the phantom where B is weakest. with operating the HS pulse in the adiabatic condition, because then it partially compensates for spatial B variation, which can be a noticeable problem in spin-echo MRI of humans at high magnetic fields (B 0 3T), even when irradiating with a volume coil. To compensate for B inhomogeneity throughout the sample, it is necessary to set the HS pulse power to a value that exceeds the adiabatic threshold throughout the volume of interest. Of course, the latter increases the SAR over the conventional (B -sensitive) approach, and thus, the advantages of using the HS pulse in the adiabatic condition can only be realized when operating sufficiently under SAR guidelines. When a single HS pulse is used in a spin-echo sequence (when only the HS pulse is used), a pseudo-echo can be generated using a gradient to shift the position of the vertex of the nonlinear phase profile ( nl ) as shown in Figure 3. In Figure 3a, the asymmetry of nl in the simulation results may be explained by the fact that the trajectories of isochromats excited symmetrically in time about the pulse midpoint (e.g., 3 and 3 khz) are not exactly the same during the pulse. In the pseudo-echo, the transverse magnetization (M xy [ ]) is never in phase. Instead, only local rephasing around the plane of the vertex occurs at any given time, and thus, the signal energy is spread in time. In 3D spin-echo imaging, the reduced amplitude of the pseudo-echo can be exploited to maintain SNR when the dynamic range of the ADC is limiting due to the large signal contained in the selected slab (5 8). Although certain constraints apply (8), images can be reconstructed from pseudo-echoes by standard Fourier transformation, as in conventional image reconstruction. Besides improving SNR, this type of 3D imaging can provide partial B - insensitivity because the pulse can be operated in the adiabatic condition. By also using an AHP pulse for excitation, the sequence can provide full compensation for B inhomogeneity (5,6,8), although it should be noted that an AHP pulse is not slab-selective. When HS pulses are used for both excitation and refocusing (i.e., the /2 HS HS sequence), according to the analysis of the phase profiles, complete rephasing at TE occurs for [ 2, T p, 2T p,2 A A 2 ]or[ 0.5 2, T p, T p,2 A A 2 ] because the nonlinear phase terms are completely cancelled out for these specific conditions. These conditions can be regarded as condition I and III, respectively, without gradients (i.e., G G 2 0). Although the former condition for complete rephasing was already proposed and used in spectroscopy experiments by Cano et al. (0), they derived it under the assumption

12 86 Park et al. FIG. 9. Multislice 2D spin-echo imaging of human brain using a volume coil at 4T. a: The image was obtained using sinc pulses (five lobes) with T p 2.3 ms. T p of 2.3 ms was chosen to have the same BW ( 2A) in refocusing as the HS pulse, that is, A/2 khz. b: The image was obtained using HS pulses for both excitation and refocusing, with nonlinear phase compensation across the slice (condition II). HS pulses with A /2 2 khz and A 2 /2 khz were used for excitation and refocusing, respectively, having the same T p ( 5 ms) and ( 5.3). In all the cases, TR 0.5 s, TE 22 ms, matrix size , FOV cm 2, and slice thickness 5 mm. Five slices were acquired with mm gap ( 20% of slice thickness) between slices. The first slice was chosen to compare SNR of acquisitions performed with sinc and HS pulses. Flow artifacts were suppressed using presaturation in a 5 cm-wide slab inferior to the slices. Because the adiabatic property of the HS pulse significantly reduces signal loss related to B inhomogeneity, images obtained using the /2 HS HS sequence display better quality in terms of SNR, especially in the periphery of the brain. The numbers given in the figure represent the SNR in white matter, gray matter, and ventricle (indicated by small white boxes). When HS pulses were used, SNR improvement was 7%, 22%, and 22% in white matter, gray matter, and ventricle, respectively. that the frequency sweep of the HS pulse is linear. This assumption essentially regards the HS pulse as a chirp pulse, and therefore, the analysis of Cano et al. is not substantially different from that by Kunz s (8) and Böhlen et al. (9). The analysis presented here derived the former condition without assuming the frequency sweep of the HS pulse is a linear function. Although other assumptions were used in the present derivations, comparisons with Bloch simulations showed that the derived equations describe M xy (, t) with a high degree of accuracy. Our analysis also produced the specific relationships between the parameters used with the /2 and pulses (specifically,, T p,, A, G, and 2, T p,2, A 2, G 2, respectively) which must be satisfied to compensate the nonlinear phase produced with slice selection. Among all the sets of parameters which satisfy the conditions of Eqs. [27] and [28], or Eqs. [30] and [3], detailed analysis was performed on three simple choices, [ 2, T p, 2T p,2, A A 2, G G 2 ], [ 2, T p, T p,2, A 2A 2, G 2G 2 ], and [ 0.5 2, T p, T p,2, A A 2, G G 2 ], referred to as condition I, II, and III, respectively. Although condition II is new, condition I was also never exploited for spin-echo imaging with HS pulses, and, concerning condition III, only slice profiles acquired in phantom experiments were reported by Shmueli et al. (), at least to the best of the authors knowledge. Compared with the alternative way to compensate the nonlinear phase using two identical HS pulses, the /2 HS HS sequence allows shorter TE and lower power deposition. Condition I has limited benefit in reducing TE because T p, 2T p,2, whereas with conditions II and III, the minimum TE is restricted by the same factors that limit TE in conventional spin-echo sequences. A disadvantage of condition II is the potential for signal loss in the presence of a frequency offset (e.g., chemical shift or change in magnetic susceptibility). On the other hand, the signal reduction that occurs with such a frequency shift appears to offer an interesting new possibility for spinecho imaging with susceptibility-weighted contrast (25), which will be investigated in the future. Condition III has a disadvantage that the HS pulse with 2 requires 2 3 db higher power to reach an adiabatic threshold and the sharpness of the spin-echo slice profile is reduced, compared to conditions I and II. Unfortunately, the /2 HS HS sequence does not offer full compensation for B inhomogeneity because the /2 HS pulse cannot be operated adiabatically. Despite this limitation, the adiabatic HS pulse imparts partial B -insensitivity to this sequence, and therefore, it is expected to be advantageous in a number of applications where B distortions are unavoidable. One final advantage of the sequence is the excellent (square-shaped) slice profiles of the /2 and HS pulses, especially in conditions I and II. Examples of MRI applications which are likely to benefit from the use of the /2 HS HS sequence include multislice versions of spin-echo EPI and diffusionweighted imaging. ACKNOWLEDGMENTS The authors thank one of the referees for providing the analytical solution of the integral of Eq. [8] (i.e., Eq. [9]). With the help of it, we were able to confirm the validity of Eq. [5] (thus, our analysis itself) in an analytical way. We also thank Drs. Djaudat Idiyatullin and Curt Corum for insightful discussions. APPENDIX When an AFP pulse is applied along the x direction in the presence of B 0 ẑ, inthe RF frame eff changes its orientation during the frequency sweep in the x z plane at an angular velocity d /dt (Fig. 0a), where is the angle between eff and the z -axis and is given by t tan t t. [A.] In this case, M follows eff if the magnitude of eff is much larger than its angular velocity, that is, eff t d dt, [A.2] which is known as the adiabatic condition. In order to explain why M follows eff when the adiabatic condition is satisfied, it is convenient to introduce another reference frame that rotates in concert with eff (i.e., the eff frame). In the eff frame, with coordinates denoted as x, y, and z, eff appears to be stationary (along the z -axis), and the