Magn Reson Mater Phy (2008) 21:5–14
DOI 10.1007/s10334-008-0105-7
RESEARCH ARTICLE
Parallel imaging in non-bijective, curvilinear magnetic
field gradients: a concept study
Juergen Hennig · Anna Masako Welz ·
Gerrit Schultz · Jan Korvink · Zhenyu Liu ·
Oliver Speck · Maxim Zaitsev
Received: 8 October 2007 / Revised: 1 February 2008 / Accepted: 4 February 2008 / Published online: 26 February 2008
© ESMRMB 2008
Abstract
Objectives The paper presents a novel and more generalized
concept for spatial encoding by non-unidirectional, nonbijective spatial encoding magnetic fields (SEMs). In
combination with parallel local receiver coils these fields
allow one to overcome the current limitations of neuronal
nerve stimulation. Additionally the geometry of such fields
can be adapted to anatomy.
Materials and methods As an example of such a parallel
imaging technique using localized gradients (PatLoc)system, we present a polar gradient system consisting of 2×8
rectangular current loops in octagonal arrangement, which
generates a radial magnetic field gradient. By inverting the
direction of current in alternating loops, a near sinusoidal
field variation in the circumferential direction is produced.
Ambiguities in spatial assignment are resolved by use of multiple receiver coils and parallel reconstruction. Simulations
demonstrate the potential advantages and limitations of this
approach.
Results and conclusions The exact behaviour of PatLoc fields
with respect to peripheral nerve stimulation needs to be tested
in practice. Based on geometrical considerations SEMs of
radial geometry allow for about three times faster gradient
J. Hennig (B) · A. M. Welz · G. Schultz · M. Zaitsev
Department of Diagnostic Radiology, Medical Physics,
University Hospital, Hugstetterstr.55, 79106 Freiburg, Germany
e-mail: juergen.hennig@uniklinik-freiburg.de
J. Korvink · Z. Liu
Department of Microsystem Engineering,
Laboratory for Simulation, Freiburg, Germany
O. Speck
Faculty for Natural Sciences,
Otto-von-Guericke University Magdeburg,
Magdeburg, Germany
switching compared to conventional head gradient inserts
and even more compared to whole body gradients. The strong
nonlinear geometry of the fields needs to be considered for
practical applications.
Keywords Gradients · Nonlinear high field ·
Rapid imaging · Image reconstruction
Introduction
Conventional magnetic resonance imaging employs temporally and spatially variable magnetic fields to encode position
by the local Larmor frequency of spins. Gradient systems
used for that purpose are designed to produce spatially linearly varying fields (=constant gradients) in three orthogonal directions x, y, z. Such constant gradients lead to a direct
mapping of the local resonance frequencies to spatial coordinates such that an image without distortions is produced after
Fourier transformation of the time domain signals. Constant
gradients have the benefit of constant voxel size and signal
intensities across the image can be directly compared without
the need for any volumetric correction. In addition, the use
of constant gradients to encode physical parameters, such
as flow or diffusion, allows for isotropic parameter encoding. Linearity is, therefore, a high priority in the design of
gradient systems. To achieve gradients with defined amplitude and slew rate in practice; however, a compromise has
to be made between linearity and efficiency of gradients,
since the demand for high linearity leads to designs requiring high voltage and currents. More important, at the current
stage of development, gradient system performance for MRI
is limited by safety concerns due to peripheral nerve stimulation (PNS) rather than by technical limitations of gradient
coils and/or power supplies [1,2]. Peripheral nerve stimulation increases with the local rate of change of the magnetic
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field. The maximum stimulation is induced at the location of
the largest field amplitude Bmax generated by the gradient
coil. Bmax and consequently the most severe rate of field
change Bmax /dt is located outside of the constant gradient
field and scales with the size of the gradient system for a
given gradient field geometry and slew rate. For a given limit
of Bmax /dt, a higher slew rate can only be achieved with
smaller gradient systems.
We have begun to explore the possibility of non-Cartesian
gradient encoding in the context of the recently proposed one
voxel one coil (OVOC) approach, which is based on the use
of spatial sensitivity profiles of multiple small coils as primary source for spatial localization [3]. A similar approach
has been called inverse MR imaging [4]. As already shown
in the application for high temporal resolution mapping of
brain physiology [3] additional spatial information at least in
one spatial direction can be introduced by one-dimensional
encoding with a gradient in a suitable direction. In the context
of this application it would be desirable to orient the direction of spatial encoding along the surface of the cortex (radial
or circumferential) rather than in Cartesian coordinates x, y
and z. In this paper we will present basic concepts of the use
of gradients with radial symmetry for imaging.
Methods
Since the term ‘gradients’ is firmly ingrained to represent
constant gradients in Cartesian coordinates, and also to avoid
confusion with the usage of the term ‘gradient’ as the spatial
derivative of the main magnetic field B0 , we use the term
spatial encoding magnetic field (SEM) for the general case
of non-Cartesian spatially variable fields. In general SEMs
with spherical or cylindrical geometry include multiple field
maxima and minima. Using such SEMs for spatial encoding,
thus, leads to a non-bijective correlation between frequency
domain and the location in object space. Therefore spatial
encoding is inherently ambiguous. Unambiguous encoding
is constrained to local subregions of the SEMs. The global
ambiguity can be resolved by parallel imaging approaches
using multiple local receiver coils. The number nc of coil elements has to be equal to or larger than the degree of ambiguity of the spatial encoding. We use the term parallel imaging
technique using local gradients (PatLoc) for this approach.
Arrangements of arrays of planar coils derived from a
three rung Cho design were used to generate the desired PatLoc fields [5]. A radial gradient can be achieved by placing ng such coils along the circumference of a cylinder. In
order to extend the radial field in the z-direction, a head-tohead arrangement of two coils was used as the basic building
block as suggested in [5]. The distance between the six central rungs was 8 mm, respectively, the length in z-direction
was 80 mm at 40 mm width. Figure 1 shows an octagonal
arrangement of such coils.
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Magn Reson Mater Phy (2008) 21:5–14
Fig. 1 Octagonal coil arrangement used in the simulations. 2×8 planar
coils are arranged in an octagonal arrangement around the direction of
the main field (z). r and φ denote the radial resp. azimuthal directions in
polar coordinates. Arrows next to the rungs indicate the current direction
Magnetic field simulations were performed using a MatLab software package (Mathworks Natick, MA, USA) for
solving the Biot-Savart law for one-dimensional current distributions on a cylinder. The program was adapted from MatLab code developed by Fa-Hsuan Lin at MGH available on
the internet (http://www.nmr.mgh.harvard.edu/~fhlin/tool_
b1.htm). For refinements the designs were transferred into
Maxwell3D (AnSoft Inc, Pittsburgh, PA, USA) for FDTDcalculations based on realistic materials and geometries
(copper wire with 2 mm thickness, rounded corners). The
maximum deviation between the idealized Biot-Savart calculations and the Ansoft simulation was ∼5% close to the
wire. For simulations of SEMs for imaging the analysis was
applied to the central plane, where concomitant fields can be
ignored due to the symmetry of the arrangement. The field
profile along the z-direction leads to a reduction in the maximum amplitude of Bz to 98.5, 92.2, 79.2 and 55% at 1, 2, 3
and 4 cm off-center in the z-direction, respectively.
Simulations of the one-dimensional signal behavior were
performed by using equidistant vectors of magnetization.
Typically between 64 and 256 isochromats per pixel and
1,024 pixels were used to avoid discretization artifacts.
T∗2 -decay as well as other physical parameters affecting the
signal were ignored.
For two-dimensional signal behavior 16 × 16 isochromats
per pixel were used in order to keep the matrix size and calculation times within a reasonable range for use with MatLab.
Simulation results show that beyond 4 × 4 isochromats per
pixel there is no significant change in results.
Magn Reson Mater Phy (2008) 21:5–14
Simulations
Figure 2 shows a radial field generated by a current I = 10 A
through each of the 2 × 3 rungs of the SEM coil elements.
The return path (larger arrow in Fig. 1) through the outer
rungs carried 30 A on each side. The radius of the cylindrical coil array was assumed to be 6 cm, which corresponded
to the target size aimed for in a first technical implementation. The field profile in the radial direction is shown in
Fig. 3a and b displays the gradient in the radial direction.
Figure 3b reveals that the local gradient along r is identical
to the mean gradient located at a position 57% outwards from
the center. For a given bandwidth and resolution in the frequency domain such a SEM will thus lead to higher spatial
resolution towards the periphery of the SEM, whereas spatial
resolution will deteriorate towards the center. To evaluate the
SEM performance regarding spatial resolution and localization, a simulation was performed using a one-dimensional
periodic spin density distribution. The period of the modulation pattern was 4 pixels (peak-to-peak) at a halfwidths of
approximately 1 pixel. The pattern was generated by starting with a periodic grid of deltafunctions, which was low
pass filtered to the desired halfwidth and convolved with a
bell-shaped filter to reduce ringing artifacts. Halfwidth was
chosen to be sufficiently narrow in order to demonstrate some
blurring at the mean gradient amplitude. A constant average
gradient Gconst of 9.11 mT/m as shown in Fig. 3a was applied
with data acquisition with a bandwidth of 40.96 kHz and a
resolution of 160 Hz/pixel. Sixty-four equidistant spins per
pixel were used in the simulation to yield a total of 256 × 64
isochromats. For each isochromat a signal was calculated
according to its Larmor frequency (Fig. 4a). The halfwidth
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a) Bz(mT)
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Fig. 3 a Field Bz along r. The dotted line represents a constant gradient
with identical maximum and minimum. b Gradient G corresponding to
the derivative of Bz along r. The dotted line represents the mean gradient
between the field maximum and minimum
Fig. 2 A color-coded field plot of magnetic field Bz produced by the
coil array in (Fig. 1) with a current of 10 A through each of the six rungs
at the center of the planar coils at a radius r = 6 cm. The maximum
field is 0.68 mT at a radius r = 5.56 cm from the center
of the observed modulation pattern is ∼ 2 pixels. For display
the signal was interpolated to 2,048 points by zerofilling to
avoid partial volume effects.
Figure 4b shows the spectrum of the same periodic onedimensional spin density distribution consisting of 256 pixels, when brought into a one-dimensional radial SEM R
aligned along r in Fig. 2 with a field and gradient profiles
as shown in Fig. 3. In order to avoid foldover artifacts, the
spin density outside of the field maximum (at r = 5.53 cm in
Fig 3a) was set to zero. Retransformation into spatial coordinates (Fig. 4c) reveals the higher resolution of the PatLoc
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a)
a)
Bz
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Fig. 5 a Magnetic field profile of SEMC generated from an octopole
arrangement of coils with a current of 10 A flowing in alternating directions through the coil configuration shown in Fig. 1. Arrows indicate
the primary direction of the field gradient. The wedge shaped outline
shows one of the eight subregions for unambiguous (but curved) spatial
encoding. b Gradient of the SEMC at equidistant positions in the dotted
ring shown in (Fig. 5a) as a function of the azimuth angle φ. Only two
of the four gradient lobes (0 < φ < 180◦ ) are shown for clarity
0.02
0.015
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r(cm)
Fig. 4 a Spectrum Fω of a periodically modulated spin distribution
with a peak-to-peak distance of 4 pixels and ∼1 pixel halfwidth measured with a constant mean gradient Gconst at a bandwidth ω =
40,960 Hz. The observed halfwidth of the spectrum is ∼2 pixels. b
Spectrum Fω of same spin distribution observed in the radial SEM R
shown in Fig. 3b at identical bandwidth. c Spectra from a and b mapped
to spatial coordinates
encoding at larger r with a cross-over at a position, where
SEM R equals Gconst . The integral over each peak is constant;
therefore, the peak amplitudes are increased at increased resolution.
If the proposed SEM R is used as readout gradient, an
additional SEMC with a circumferential gradient is required
to achieve two-dimensional image encoding. However, such
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circumferential gradients are fundamentally impossible to
realize in a monotonic fashion [6]. In addition to physical
constraints set by Maxwell’s equations the circumferential
field will thus be multipolar by necessity. A didactically
interesting and very straightforward, but in practice not
necessarily optimal approach to establish a multipolar
circumferential SEMC is achieved by alternating the current in the coil elements of the radial SEM R . The resultant
octopole field profile is shown in Fig. 5a. Figure 5b shows
the gradient dBz/dc = 1/r dBz/dφ in the circumferential
direction c normal to the radius r along the azimuth coordinate φ at radii 4, 4.25, . . ., 5.5 cm. Compared to the radial
SEM R (Fig. 3b) the maximum amplitude of the circumferential SEMC is increased, but decays more rapidly towards the
center due to interference between the opposing field lobes.
Magn Reson Mater Phy (2008) 21:5–14
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Fig. 6 Basic workflow for
PatLoc reconstruction of a spin
distribution Iρ(x, y) into its
PatLoc image Ix y (x, y). For
explanation see text
2
4
FPAT(kωR,kωC)nc
Fxy(kωx,kωy)nc
Iρ(x,y)
IPAT(ωR,ωC)nc
1
3
Using SEM R and SEMC for 2D-imaging requires modification of the usual reconstruction pipeline. The basic workflow of the reconstruction algorithm is shown in Fig. 6. The
black arrows indicate the pathway used in our simulations:
a given spin distribution Iρ (x, y) (Fig 6, 1) results in a
k-space dataset FPAT (kω R , kωC )nc (Fig 6, 2) for each of the
nc receiver coils. Each dataset represents the signals acquired
by using SEM R and SEMC for two-dimensional Fourier
encoding: 2DFT will transform FPAT (kω R , kωC )nc from frequency space into IPAT (ω R , ωC )nc (Fig 6, 3) where ω R , ωC
are defined by the Larmor frequencies at each point in the
fields generated by SEM R and SEMC . For known field profiles of the SEMs, IPAT (ω R , ωC )nc can be transformed into
Cartesian coordinates to yield Ix y (x, y)nc (Fig. 6, 5): ω R and
ωC are linked to x, y by
ω R = γ BSEMR (x, y)
(1)
and
ωc = γ BSEMC (x, y)
(2)
where BSEMR and BSEMC correspond to the fields generated
by the two SEMs.
Coordinate transformations are performed by calculating
the inverse functions of (1) and (2). Since (1) and (2) are
non-bijective, the inverse will lead to ambiguous mapping
of IPAT (ω R , ωC )nc to Ix y (x, y)nc . For the practical implementation the inverse function was calculated numerically:
for each Cartesian coordinate (x,y) the (ω R , ωC )-coordinates
were derived from (1) and (2) and the signal intensity was
calculated from IPAT (ω R , ωC )nc. using regridding with linear
interpolation.
6
Fxy(kωx,kωy)
Ixy(x,y)nc
5
Ixy(x,y)
7
The resulting image Ix y (x, y)nc demonstrates an eightfold
degeneracy due to the ambiguity of spatial encoding in the
multipolar fields. Based on the known sensitivity profiles of
each of the nc receiver coils the true image Ix y (x, y) (Fig. 6,
7) can be unwrapped by generalizing and adapting SENSE
reconstruction [7] or refinements thereof [8]. Alternatively,
warping of FPAT (kω R , kωC )nc into Cartesian k-space Fx y
(kωx , kω y )nc (Fig. 6, 4) can be performed using a transformation matrix based on the shift theorem. Data can then
be combined by a GRAPPA type of parallel reconstruction
[9,10] into Cartesian k-space (Fig. 6, 6). The final image
(Fig. 6, 7) is then reconstructed by 2DFT.
In order to avoid additional complexity from idiosyncrasies and imperfections of the parallel reconstruction algorithms, i.e., to remove the effects of the RF coil sensitivities,
we have used eight idealized wedge-shaped coil profiles (see
Fig. 5a) in our simulations. In this very artificial setup the
coil sensitivity is one within each wedge and zero outside. An
image can then be generated by performing a simple addition
of the individual coil images by simple PILS recombination
[11]. For display all images were expanded to 1,024 × 1,024
matrix size by sinc-interpolation.
In order to illustrate the imaging behavior for PatLoc
encoding (Fig. 7) shows a direct comparison of simulated
patterns for conventional gradients and encoding with SEM R
and SEMC , respectively. The amplitude of the conventional
gradients was assumed to be equal to the mean gradient
amplitude of BSEMR as shown in Fig. 3a. In order to avoid
ambiguities and, therefore, foldover in the radial direction,
the field-of-view was limited to 5.5 cm such that both PatLoc
gradients decay monotonously towards the center.
The circular pattern consisting of rings of finite widths
(Fig. 7a, b) demonstrates the higher resolution of PatLoc
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Fig. 7 Results of simulations
of circular (a, b), spokewheel (c,
d) and random (e, f) spin
distribution patterns imaged by
constant Cartesian gradients
(left column) versus PatLoc
gradients SEMC and SEM R
(right column). Linewidth of the
modulated geometrical patterns
was roughly 1 pixel at
256 × 256 matrix size
encoding towards the rim of the pattern as well as the widening of the rings towards the center. Simulated images of a
radial spoke wheel pattern (Fig. 7c, d) show the circumferential variation of the image resolution as a consequence of
the smoothness of the gradient near the field maxima and
minima (see Fig. 5b).
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Simulations of regular but non-linear discrete structures
can be misleading due to regridding artifacts and mathematical singularities near the field maxima and minima. Loss
of resolution at areas of low and/or non-orthogonal gradient
fields also appears to be rather patchy leading to enlarged
patchy areas more than to blurring. This is most likely a
Magn Reson Mater Phy (2008) 21:5–14
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consequence of the simulation using a large but still finite
number of isochromats where the intensity of isochromats
within each pixel is constant. For a realistic situation with a
homogeneous distribution of uncorrelated spins smooth blurring is expected in areas of low spatial resolution.
In addition to using regular structures we have thus used
a ‘noise image’ generated by a random distribution of spin
density in Cartesian space to visualize the overall transformation characteristics. An overall impression of the anisotropic
imaging behavior is shown in Fig. 7f. Substantial blurring
near the image center can clearly be identified. In addition,
degraded image quality is visible near eight spokes at
azimuthal positions corresponding to the gradient minima
and non-orthogonal (or even parallel) gradient directions of
BSEMR and BSEMC. .
In order to analyze the impact of such effects on a more
realistic case, we have used a standard MR-image as template
for the spin distribution. A 512 × 512 image of a human volunteer rescaled to the size of the gradient bore was used as
a model for Iρ(x, y) and ‘imaged’ with conventional gradients versus PatLoc gradients. As illustrated in Fig. 8, image
resolution of the PatLoc image is clearly degraded towards
the center of the head. In contrast good image quality can be
seen at the outer cortex close to the ‘spokes’ indicating the
minima and maxima of SEMs or vanishing local gradients
if data are acquired at 384 × 384 resolution (Fig. 8b). At
256 × 256 image resolution (Fig. 8c) the appearance of the
angular blurring is more pronounced.
The circumferential anisotropy could be compensated for
by repeating the experiment using the radial SEM R and an
interleaved circumferential SEMCI , where SEMCI is derived
from SEMC by a rotation of 22.5◦ . In practice, however, such
an approach would double the acquisition time. An alternative strategy to improve rotational anisotropy is to directly
use SEMC and SEMCI for spatial encoding. The orientation
of the gradient fields produced by both SEMC and SEMCI
changes continuously over space. Field direction is radial
along the radii connecting the extremity and circumferential
halfway between. In view of this continuous swirling around
it is remarkable that the gradient fields are homogeneously
orthogonal across the entire FoV (Fig. 9) with a maximum
and minimum crossing angle α of 109.5◦ and 70.2◦ , respectively. As shown in simulated images (Fig. 10) this leads to an
increase of the extent of the low resolution part at the center
due to the more rapid decline of the fields towards the center,
but to a beneficial rotational isotropy of the resulting images
along the circumference.
Discussion
Although the particular geometry of the SEMs discussed
in this paper is novel, there is abundant previous literature
Fig. 8 Spin distribution taken from a 512 × 512 volunteer image
imaged simulated to be imaged with conventional gradients (a) and
with PatLoc gradients SEM R and SEMC acquired at 384 × 384 (b) and
256 × 256 (c) matrix size
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a)
α(deg)
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120
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50
b)
ever not be treated as a simple perturbation any more, but
constitute an inherent part of the imaging process. Therefore a number of commonly used concepts which have been
developed to guide intuition for conventional imaging need
to be reconsidered. Frequency domain and Cartesian space
are disjoint in PatLoc imaging, such that basic properties of
signal sampling theory in the time/frequency domain cannot be simply translated into imaging properties in spatial
coordinates:
The use of a globally constant point spread function (PSF)
to characterize the imaging behavior fails. When using PatLoc encoding each pixel has its own PSF reflecting the fact
that spins at different spatial positions have different k-space
trajectories. Based on the curvature of the SEMs, the PSF
of each pixel will be anisotropic with variable preferential
directions. As a result spatial resolution will become a local
property and can be highly variable across the image.
The field of view (FoV) in conventional imaging is linked
to the distance between adjacent samples (a local property)
in k-space:
FoV = ω N /(γ G) = 1/(dwγ G),
Fig. 9 a Colour encoded display of the angle α between the gradients produced by the multipolar SEMC and the interleaved multipolar
SEMCI generated from SEMC by rotation by 22.5◦ . b Superposition of
isocontour lines of SEMCI and SEMC
regarding different aspects of PatLoc imaging. Nonlinear
spatial encoding fields have found numerous applications
outside clinical and biomedical MR. Flat gradient systems
[12,13] and inside-out encoding fields [14,15] are well
known. In the field of medical MR the use of sinusoidal varying gradients has been described by Patz [16–18]. Recently
non-bijective but still unidirectional fields have been suggested, which maintain the orthogonality of conventional
gradients [19].
The reconstruction workflow shown in Fig. 6 is formally
equivalent to the use of 2DFT with re-warping to account for
gradient non-linearities, which is commonly used in most
scanners today. In PatLoc imaging nonlinearities can how-
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(3)
where γ is the gyromagnetic ratio, G the gradient and ω N
the Nyquist frequency defined by the sampling rate 1/dw.
Application of this interrelationship to local variable gradients produced by SEMs is not correct and implies that the
FoV (in spatial coordinates) becomes spatially variable. This
somewhat defies intuition, where FoV clearly is perceived as
a property of the image as a whole.
So it needs to be noted that basic concepts derived from
signal theory like the Nyquist theorem are still applicable to
corresponding frequency domains in PatLoc coordinates ((2)
and (3) in Fig. 6) but can no longer be directly translated into
the spatial domain. Care needs to be taken when translating
imaging properties like over- and under-sampling, foldover
and others from the frequency domain into spatial coordinates.
The simulation presentd in this study have neglected the
influence of physical parameters such as field inhomogeneities, chemical shift effects, T∗2 and others to be accounted for
in a real implementation. One may also suspect that the use
of idealized wedge shaped sensitivity profiles makes images
look better than could be expected in a more realistic situation. For conventional 2DFT sampling strategy based on
the use of one SEM as readout gradient and the second as
phase encoding gradient, off-resonance effects due to field
inhomogeneities will produce a shift in the direction of the
local gradient of the SEM used as readout gradient. If SEM R
is used for that purpose, the chemical shift displacement in
the final image will increase radially towards the center. For
any of the multipole SEMs, the chemical shift displacement
will change its direction and extent all over the image. As
a result, it can be expected that fat/water-suppression and/or
Magn Reson Mater Phy (2008) 21:5–14
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Fig. 10 Simulation of a
phantom image produced by
combining the circular and
spoke wheel patterns in (Fig. 7a,
c) when imaged with
conventional gradients and with
SEMC and SEMCI . Linewidth of
the geometrical patterns was ∼ 1
pixel at 256 × 256 matrix size
more advanced techniques correcting for local field effects
may be mandatory for realistic applications of PatLoc imaging. The complexity will increase with the use of advanced
sampling strategies (EPIs, Spirals, VIPR, ...) in PatLoc experiments. Effects will be most dramatic near the maxima and
minima of the SEM fields, where local gradients are small
and changes in signal phase or frequency will result in artifacts across an appreciable distance.
Additional complications are expected, if the information
about the position of the probe in the SEM fields is incorrect or if the actual SEM field amplitudes deviate from their
nominal value by eddy current effects, or other instabilities.
Apart from the challenges to be met regarding measurement and image reconstruction, building PatLoc systems will
be far from trivial. Force and torque issues as well as eddy
current problems need to be considered in the practical realization.
The coil arrangement used for generating the SEMs is
geometrically different, but in principle similar to coil arrangements used in current shim systems. Time variable switching of high order shims is currently discussed for use in high
field imaging and spectroscopy and known to be a difficult,
but (hopefully) not insurmountable problem.
External interferences will be much smaller in multipolar coil arrangements. The cylindrical antipolar symmetry
of the induced effects is highly self compensating within the
cylindrical arrangement of conductive surfaces in current MR
systems. The self-compensating nature of mechanical forces
over short distances is also promising and may lead to benefits with respect to acoustic noise.
The efficiency of the coils as well as the geometry of the
fields can be further optimized using either analytical methods, such as the target field approach [20,21], or by numerical
optimization based on FEM- or FDTD-methods [22].
The octagonal arrangement described in this study reflects
a compromise between gradient amplitude/slew rate and
field-of-view: With increasing number of poles stronger and
faster gradients can be realized, but the useful imaging volume is more and more restricted to the rim of the circular FoV.
PatLoc SEMs are not meant to be used as stand-alone
devices, but in connection with the still existing conventional
gradients. A conventional z-gradient is a natural complement
to the multipolar SEMs to extend the measurement region
into a cylindrical volume. Slice selection will normally be
performed with a conventional gradient in order to select a
planar slice of homogeneous thickness. For any combination
of PatLoc, SEMs and conventional gradients the mutual nonorthogonality must be taken into account especially outside
of the magnet center, where concomitant gradients can be
significant.
Similarly, PatLoc SEMs may be useful to increase the
flexibility of conventional data acquisition. Using the radial
SEM R for slice selection can be useful to select a circular
slice at the center and/or a cylindrical tube as volume of
interest, which can then be imaged using conventional imaging techniques. The underlying radial geometry suggests that
back-projection techniques will be the natural approach for
imaging such volumes. Volumetric applications will be affected by the homogeneity of PatLoc fields along the z-direction.
Using PatLoc SEMs not for imaging but to encode parameters such as flow and diffusion is feasible, but will lead
to spatial variation of relevant parameters like venc and
b-factors.
Our simulations show that imaging with multipolar
SEMs is feasible, but it comes at a price of significant
anisotropy characterized especially by a hole at the center
of the imaging volume. Given this inherent limitation, one is
tempted to ask whether the complexity of the PatLoc encoding approach might outweigh its potential advantages. One
incentive already mentioned is the possibility to use the radial
SEM for ‘depth-encoding’ in a OVOC-experiment as in MREncephalography.
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Our simulations allow some estimates with respect to
physiological limitations set by peripheral nerve stimulation
(PNS). PNS depends on the local change of dB/dt and the
switching time ts [23,24]. Threshold values for PNS are individually variable and may also depend on the exact geometry
of the variable fields. For conventional gradients the physiologically relevant Bmax is located at the periphery or even
outside the usable field of view and scales with the size of
the gradient system. PatLoc SEMs will lead to considerably
reduced Bmax even at high gradient amplitude due to the
alternating nature of the fields between the multiple poles.
The relevant ‘pole distance’ rPNS in conventional gradients
defined as the distance between the maximum and minimum
field generated by the gradient is at least as large as the diameter 2r of the gradient system. In contrast, for an octopole
SEMC rPNS is smaller than π /4r due to the fact that the maxima and minima are located inside of the gradient volume.
From these geometrical considerations alone one can expect
that SEMs allow for considerable faster and/or stronger gradients by approximately a factor of three before the stimulation threshold is reached.
Currently ongoing developments at our institution focus
on the demonstration that such a system can be realized in
practice and that—most important—useful applications can
be developed to exploit the advantages of such an approach.
Since the advent of parallel imaging and more recently
receiver coil arrays becoming an integral part of the standard
imaging systems the fundamental concept of using homogeneous magnetic field gradients to achieve globally unambiguous spatial encoding was not critically reviewed. Indeed,
for unambiguous imaging using a coil array, it is sufficient
to achieve reasonably homogeneous encoding over the field
of view of each receiver coil. This degree of freedom, relaxing the requirement of global linearity of the encoding fields
may in our belief result in numerous advantages in switching
speed, nerve stimulation, acoustic properties, etc.
Although it may well be that the radicalness of the approach described in this paper may limit practical applications, we are confident that liberating gradient design from
the restrictive requirements regarding linearity may permit
realization of more flexible and efficient spatial encoding
schemes and may help to drive MR imaging beyond the current physiological limits.
Acknowledgments This project is supported by grant #13N9208 in
the project INUMAC supported by the BMBF (Federal Ministry for
Education and Research).
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