MAGNETIC RESONANCE IN MEDICINE 5, 502-507 (1987)

Reciprocal Space Description of MR Experiments*

S. MULLER

Biocenter of the University, Klingelbergstrasse 70, CH-4056 Basel, Switzerland

Received June 15, 1987; revised August 31, 1987

The phase space description of Fourier-based imaging methods is extended to describe imaging of heterogeneous distributions of chemical shift, J coupling, and T, and T, relax- ation. Simple graphical illustrations for a general class of spectroscopic imaging experiments are provided. 0 1987 Academic Press, Inc.

The phase space concept has proven to be a powerful and elegant tool for describing and classifying all Fourier-based MRI experiments (1, 2). Its popularity is based mainly on the simple graphical representation of MR signal phase for Bo gradient governed imaging techniques. The phase of spins, however, is influenced not only by Bo gradients but also by intrinsic spin properties like chemical shift and Jcoupling. These evolutions are usually described by vector models or quantum mechanical calculations, the one exception being the paper of Brown et al. (3) where the chemical-shift evolution was discussed in reciprocal space.

In this communication we demonstrate, however, that under certain circumstances Fourier space trajectories can be used as descriptors for chemical-shift evolution, J coupling, and relaxation as well. As a result graphical representations similar to the k space trajectories will be obtained allowing a straightforward overview of many imaging, spectroscopic imaging, and high-resolution techniques.

Consider a heterogeneous sample containing spins which are characterized by their location r = (x, y, z) , chemical shift u, a J multiplet pattern, and K~ = 1/T, and K~

= l/T2 for the relaxation rates. We introduce a multidimensional space {x, y, z, u, J, K ~ , K ~ } spanned by the coordinates x, y, z, (r, J, K ~ , K ~ . In this space the sample can be described by a seven-dimensional density function +(x, y , z , u, J, K ~ , K ~ ) which is defined such that its value at a given place (Xg, yo, zo, uo, Jo, K ~ ~ , K ~ ~ ) is proportional to the number of spins in the sample which are located at ro = ( X g , yo, zo), have a chemical shift UO, a multiplet component at J = Jo, and relaxation rates K ~ ~ , and K ~ ~ .

(Note that the Jaxis does not represent the Jcoupling but stands for the offset frequency caused by J splitting; i.e. the density function along the J axis forms the multiplet patterns.) One could state that the final and most general goal of MR experiments is the complete determination of +, since all MR information about the sample can be extracted from at will.

We now describe the connection between the measured MR signal and +. In a first example we apply repetitive rf excitation pulses to the sample (repetition time: TR),

* A preliminary account ofthis work has been made at the 6th Annual Meeting of the Society of Magnetic Resonance in Medicine, New York, 1987, book of abstracts, p. 768.

0740-3194/87 $3.00 502 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.

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in addition a BO gradient G(t) = (GJt), G,(t), G,(t)) will be applied during the evolution. In complex notation and with reference to the Larmor frequency of the nucleus under study the MR response S(t, TR) of the sample can be written as

where we have used the vector

This signal S(t, TR) can be regarded as specific values of a general transform @* of @ which is defined as

@*(r*, u*, J*, KT , K?)

The signal S(t, TR) then is proportional to

S(t, TR) 0~ @*(k(t), t, t, TR, 0; [41

i.e., the signal S(t, TR) is proportional to the amplitude values of @* measured at (r*, u*, J*, KT, K ? ) = (k(t) , t , t, TR, t). In other words, the time evolution in the experiment can be identified with a defined sampling path (k(t), t, t, TR, t ) in the space {r*, u*,

In this example there is no reason to extend the conventional three-dimensional imaging k space to seven dimensions, since the evolution along the additional u*, J*, KT, K? axes is trivial. There are, however, a number of ways to influence also the spin evolutions by the remaining parameters:

J*, KT, K ? } .

-Insertion of a broadband 180" pulse in the evolution period causes a phase inversion with respect to r* and u*. This phase inversion is described as a reflection of the sampling path at r* = 0 and u* = 0, respectively.

-For weak coupling, a reflection at J* = 0 can be provided if only one of the coupling partners of the spin pair under investigation is affected by the 180" pulse (e.g., in heteronuclear experiments).

-Evolution along J* can be stopped with decoupling experiments. -Modification of G(t) allows an almost free design of k(t). -Measurements with varying TR scan @* at different KT positions.

Thus a great number of scanning paths can be designed in {r*, u*, J*, K T , K ? ) . The idea in a general MR experiment then is to scan @* as appropriately as possible for the problem under investigation and to reconstruct @ from @*, e.g., by Fourier trans- forms with respect to r, u, J, and subsequent fitting procedures for K~ and K ~ , by MEM reconstruction ( 4 ) or postfiltering techniques (5).

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In the following, these ideas will be fixed by briefly discussing some well-known experiments using the @-a* concept. For clarity the examples will be illustrated in two-dimensional subspaces of a*.

1. PULSE ACQUIRE EXPERIMENT WITH 7'R = co

Under high-resolution conditions (k(t) = 0) and with TR = co Eq. [I] reduces to

s(t) cx s - * s @(r, U, J , K I , K2)e'"'e'2"J'e-'"drdUd~dK2.

Comparison with Eq. [4] yields

S(t) oc @*(O, t, t, 0, t).

In Fig. la the projection of the tracing track through @* onto the a*-J* subspace is shown. @* is scanned from its center along a straight line a* = J* = t, which results in the free induction decay signal S(t). For this example the projection theorem which says that a cross section through the center of @* yields after Fourier transform a projection of @ onto the same straight line can be applied. In fact, Fourier transforming the FID yields a spectrum which can be interpreted as a projection of a 2D J-resolved spectrum onto the line a = J. In Fig. Ib the u * - K ~ subspace is shown. The ~f evolution causes the dampening of the signal due to transverse relaxation.

2. SPIN ECHO SEQUENCE

The 90z-7- 180,0-7-SE experiment is extensively used in MR imaging where, for the r* axes of @*, graphical illustrations have been given in (2). Here, the influence of the sequence on u* and J* will be discussed for weak spin-spin coupling. Figure 2a shows the scanning track in the a*-J* subspace. The broadband 180" pulse inverts the phase angle of the spins caused by chemical-shift evolution; J evolution, however, is not affected if the 180" pulse is applied to the whole spin system. This phase behavior is shown in Fig. 2a, where the 180" pulse only causes a reflection at u* = 0. Variation of 7 yields data sets which correspond to a sampling grid in a*-J*. From such a data set two-dimensional J-resolved high-resolution spectra can be obtained (6).

Especially in heteronuclear cases (i.e., AX systems) the 180" pulse can be applied either to the spin species under investigation (A) or to the coupling partners alone (X) .

J'

/ b

/ FIG. 1 . Scanning path for pulse-acquire experiment. (a) In the u*-J* subspace of a* the free induction

decay corresponds to the amplitude values of a* along u* = J* = t . (b) The projection of the scanning path onto the u * - K ~ plane corresponds to the straight line 6 = K: = t.

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J’ t a b

FIG. 2. Sampling track in the reciprocal space for spin-echo experiment. An AXspin system is regarded; the A spins are observed. (a) Evolution for a 902-~-180&-7-SE experiment. (b) Evolution for a 902-7- 1802-7-SE experiment. (c) Evolution for a 902-~-180j-~-SE experiment. (d) Evolution for a 9O,”-s-BBx experiment.

In Fig. 2b the a*-J* scanning track for the A spins is shown if the 180” pulse is applied only to the A spins, while Fig. 2c shows the corresponding track for a 180; pulse. Figure 2d reveals the effect of X broadband decoupling applied after a delay 7: J evolution is stopped.

3. THREE DIFFERENT SPECTROSCOPIC IMAGING SEQUENCES ARE NOW SHOWN IN THE x*-U* SUBSPACE OF a*

a. 1 + I-Dimensional Spectroscopic Imaging (3, 7)

In a SE experiment, 90,O-7- 180,”-7-acq, a phase-encoding gradient is applied during the first r period. This results in a scanning track of @* as shown in Fig. 3a. @* is sampled along an equidistant mesh and a two-dimensional Fourier transform directly yields the +(x, a) distribution. Note also that J evolution takes place at the same time and influences the signal shape; the scanning track in the a*-J* subspace is the same as that in Fig. 2a. The T2 influence corresponds to that in Fig. 1 b.

b. Spectroscopic Projection Reconstruction Imaging (8)

In this experiment the transverse magnetization evolves in a Bo gradient which is incremented stepwise for each acquisition. The concentric sampling traces for this pulse-acquire experiment are shown in Fig. 3b. In order to obtain equal angles for each projection, the gradient strength must be increased tangensoidally. Since gradient strength cannot be increased to infinity, there will be an area around the x* axis which

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a b

X'

O* t C

X.

FIG. 3. Sampling tracks for spectroscopic imaging experiments. (a) 1 + 1-dimensional imaging. In the first half of the spin-echo experiment an x phase-encoding gradient is applied, which is stepwise incremented. (b) Spectroscopic projection reconstruction imaging. After excitation the spins evolve in a Bo gradient which is incremented in successive experiments such that an isotropic sampling of the x*-u* plane is provided. (c) Sampling path with EPSM. The rapidly switched gradients provide the sampling of the x* axis, while chemical-shift evolution is illustrated by the movement along u*.

is not sampled and the measured data must be extrapolated into this region (8). Back- projection of the data with the PR algorithm results in the @(x, a) density function, which was termed pseudo-object in (8). In this experiment each FID is J modulated as in Fig. la; T2 influence is as described in Fig. lb.

c. Echo Planar Shift Mapping, EPSM (9)

The EPSM track is shown in its simplest version in Fig. 3c. The rapidly switched gradient allows the scanning of one-half of the a*-x* plane in a single acquisition. J* and K? evolution are again represented as in Figs. la and lb, respectively.

These examples demonstrate how the elegant phase space description of Fourier- based imaging methods can be applied directly to a much broader class of experiments. Our experience has shown that in many cases the sketch of the sampling path through @* has a higher conceptual transparency than the pulse timing diagram. In addition, the discussion of artifacts, as well as sensitivity and efficiency considerations, appear in a different light using the @-@* picture and simple explanations for many obser- vations are provided.

Due to the fact that with respect to r, CT, and J, @ and @* are linked via a Fourier transform, Fourier theorems like the projection, the shift, and the convolution theorems can be applied to these subspaces (cf. example 1). Unfortunately, the relaxation pa- rameters do not share these properties, for these coordinates @ and @* are linked via a real transform (Eq. [3]) which requires that the K~ and K~ values of @ usually must

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be determined by fitting the experimental data. This asymmetry is the reason that the relaxation parameters are often ignored in the phase space description of imaging techniques, or they are roughly included into the scanning of @* by multiplying the scanned signal with (1 - e-TRKl)e-fKz. Note, however, that this is only correct if one single TI and T2 value exists for the whole sample, a rather special case.

Of course, the proposed description cannot illustrate all observations made in spec- troscopy or imaging experiments. Steady-state free precession, an important feature in rapid imaging, effects of strong spin-spin coupling or multiquantum spectroscopy are examples which are not included directly in the @-@* picture. It seems, nevertheless, that a very broad class of spectroscopic imaging experiments is well described by this concept and that an understanding of the experimental mechanisms is greatly facili- tated.

In summary, the k space description of Fourier-based imaging methods has been extended to include spectroscopic imaging and high-resolution experiments. This ex- tension is based on a unified description of spin phase evolution in the transverse plane, combining effects of chemical shift, applied field gradients, weak spin-spin, coupling and relaxation. The introduction of a generalized density function @ and its transform @* has allowed a convenient interpretation of the nuclear signals as am- plitude values of @* along paths defined by the pulse and gradient sequence. Since many in vivo MR experiments in the future will consist of a combination of sophis- ticated imaging and spectroscopy techniques, the @, @* concept may be a valuable tool for a discussion of the underlying concepts and will illustrate many methods by sketching the corresponding sampling path through @*.

ACKNOWLEDGMENTS

The author thanks N. Beckmann for useful discussions. This work was supported by the Swiss National Science Foundation Grant 4.889.085.18 and the Kommission zur Forderung der wissenschaftlichen For- schung, Projekt 1462.

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