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Cite this article as: Pang Y, Zhang X. Precompensation for mutual coupling
between array elements in parallel excitation. Quant Imaging Med Surg
2011;1:4-10. DOI: 10.3978/j.issn.2223-4292.2011.11.02
Original Article
Precompensation for mutual coupling between array elements in
parallel excitation
1Department of Radiology and Biomedical Imaging, University of California San Francisco, San Francisco, CA, United States; 2UCSF/UC Berkeley Joint Graduate Group in Bioengineering, San Francisco & Berkeley, CA, United States
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Abstract
Parallel transmission or excitation has been suggested to perform multi-dimensional spatial selective excitation to shorten
the pulse width using a coil array and the sensitivity information. The mutual coupling between array elements has been a
critical technical issue in RF array designs, which can cause artifacts on the excitation profile, leading to degraded excitation
performance and image quality. In this work, a precompensation method is proposed to address the mutual coupling effect
in parallel transmission by introducing the mutual coupling coefficient matrix into the RF pulses design procedure of the
parallel transmission. 90° RF pulses have been designed using both the original transmit SENSE method and the proposed
precompensation method for RF arrays with non-negligible mutual coupling, and their excitation profiles are generated by
simulating the Bloch equation. The results show that the mutual coupling effect can be effectively compensated by using
the proposed method, yielding enhanced tolerance to insufficient mutual decoupling of RF arrays in parallel excitation,
ultimately, providing improved performance and accuracy of parallel excitation.
Key words
Parallel transmission; mutual coupling; transmit SENSE; transceiver array; Bloch equationQuant Imaging Med Surg 2011;1:4-10. DOI: 10.3978/j.issn.2223-4292.2011.11.02
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Introduction
Multidimensional spatially selective RF pulses have been used in MRI
to limit the electromagnetic signal emitted from the imaging object
within arbitrarily shaped and spatially restricted areas (1-6). This
approach requires homogeneous radiofrequency (RF) field to ensure
the accuracy of the excitation profile, especially for spin echo imaging.
This requirement has impeded the further development and application
of multidimensional spatial RF pulses to ultrahigh field MRI (7 Tesla
and beyond) (7-13) in which homogeneous RF field is difficult to
be achieved (8,14-18) due to the high frequency wave effect in the
conductive and high dielectric imaging object such as human body (19).
To achieve homogeneous RF field distribution and shorten the pulse
width, parallel transmission (20-28) is suggested to perform spatial
selective excitation using a coil array and the sensitivity information
of each element. The amplitude and phase of each element are
independently controlled to manipulate the RF field distribution within
imaging object to ultimately achieve B1 homogeneity (29,30). The
parallel transmission can be used not only in small-tip-angle excitation,
but also in large-tip-angle excitation and refocusing pulses at ultrahigh
fields (31-33). Although the average and local specific absorption ratio
(SAR) increase with the acceleration factor in parallel transmission (34),
the SAR can be optimized using different strategies such as variable
sampling rate or optimized k-space trajectories (21,35-38), providing
feasible ways to make tradeoffs between the acceleration factor and the
power deposition.
The mutual coupling has been a critical problem in RF coil array
design (39,40), especially at ultrahigh fields where the imaging sample
mediates the mutual coupling among the array elements (9,11,12,29,41-43). Comparing with the situation in parallel reception, this decoupling
problem becomes more critical in parallel transmission, deteriorating
the excitation accuracy and degrading the image quality. Although a
variety of decoupling schemes have been developed to address the issue
(12,14,29,40,42-46), insufficient decoupling among array elements is
still a major problem hindering the development and application of parallel transmission.
In this work, a precompensation method is proposed to address the
mutual coupling effect in transmit SENSE by introducing the mutual
coupling coefficient matrix into the RF pulses design procedure of the
transmit SENSE. Studies show that by using this precompensation
technique, the mutual coupling effect on the excitation pattern can be
compensated by using the corrected RF pulses as long as the mutual
coupling between array elements is not high enough to cause a clear
split of the element resonance peak, or in a weakly coupling case.
This precompensation technique provides a valuable approach to
performance optimization of transmit SENSE or parallel excitation
using multichannel RF transmit arrays with non-negligible mutual
coupling between array elements.
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Theory and methods
The transmit SENSE uses coil array to excite the spins to achieve the
desired profile. For an R-element array, the desired excitation pattern
can be described by the superposition of all the individual pulse profiles
weighted by the corresponding sensitivity profiles, which is described as
below:
 where Pdes(x) is the desired pattern, Sn(x) and Pn(x) are the complex sensitivity profile and the pulse profile of the nth coil, respectively. The profile Pn(x) of each coil can be expressed as:
 where M0 is the initial magnetization, B1,n(t) denotes the RF pulse of
the nth element of the array. G(t) and S(k(t)) are the accompanying
gradient and the spatial frequency sampling function respectively. k(t) is
the k-space trajectory defined by the gradient function G(t). By defining
 the Eq.[2] can be rewritten as
 Then, the mutual coupling between the array elements is introduced
into the calculating procedure. Assuming the mutual coupling
coefficient between the mth coil and the nth coil is cm,n, the excitation
pattern of the mth coil becomes:
 Thus, the desired excitation pattern Pdes(x) can be expressed as the
superposition of all the individual patterns weighted by the sensitivity
patterns:
 where Sm(x) is the sensitivity of the mth coil. Thus, from Eq.[4] and (6)
the Eq.[1] can be rewritten as:
 or
 where S*
n (x) is defined as
 Eq.[8] is similar to the central equation of transmit SENSE. The
difference is the sensitivity pattern S*n (x) equals the sum of all the
individual sensitivity patterns multiplied by the corresponding mutual
coupling coefficient. This corresponds to the compensation for the
mutual coupling effect in the excitation procedure. The following steps
for designing the pulses are the same as that in transmit SENSE: the
spatial coordinate x can be discretized to vectors first, and the Pdes(x),
Pn(x) and S*n (x) can be rewritten as vectors; then these three vectors
can be transformed to k-space style by convolution and by choosing
proper k-space trajectory and the full size variable Pfull(x) and S*full(x) can be obtained; finally, the individual excitation pattern Pn(x) under
the effect of mutual coupling can be calculated by solving the inverse
problem and each individual RF pulse can be obtained by the k-space
analysis method. The flowchart of the improved transmit SENSE with
precompensation method is shown in Figure 1.
To investigate the feasibility and the effectiveness of the proposed
method, a simple example of transmit SENSE is simulated by solving
the Bloch equation. Excitation pulses were designed using transmit
SENSE and the precompensation method respectively for comparison.
In both methods, the same spiral trajectory with 12 turns was used.
The max value of k-space extension was 0.5 cycle/cm. The desired
excitation pattern was a cylinder with 10 cm diameter, whose 2D profile
is shown in Figure 2A. The desired pattern was excited using a 4-element
array with the reduction factor of 2. The sensitivity patterns of each
element are shown in Figure 2B-E. The mutual coupling coefficient
matrix between the elements is shown in Figure 3. The proposed
precompensation method was applied to design the excitation pulses
for implementing the transmit SENSE using the RF arrays with nonnegligible
mutual coupling between elements.
Figure 1. Flowchart of the RF pulse design procedure using the precompensation method in parallel excitation with multichannel transmit
array.
Figure 2. (A) Desired excitation profile; (B-E) intrinsic sensitivity pattern of each array element.
Figure 3. Mutual coupling coefficients between array elements.
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Results
The conventional transmit SENSE method was applied to design the
excitation pulses for parallel excitation with the 4-element transmit array
in which non-negligible mutual coupling between array elements exists.
Based on these pulses, the excitation pattern of each array element was
generated and plotted, as illustrated in Figure 4.(A)-(D). The resulting
excitation patterns were also calculated. When the pulses were designed
based on conventional transmit SENSE, the artifacts in the resulting
excitation pattern is unable to be cancelled by each other. This leads to
the deteriorated excitation pattern with the residual aliasing artifacts,
as shown in Figure 4.(E). It is expected that the distortion of excitation
pattern will become worse when the electromagnetic coupling between
array elements increases.
With the use of the proposed precompensation method, the specific
RF pulses were designed by taking the mutual coupling coefficients into
calculation for each array element. By using these RF pulses, the mutual
coupling between the elements can be compensated. The artifacts in the
excitation pattern of each element (Figure 4.(F)-(I)) can be potentially
cancelled by each other through superposition, resulting in reduced
artifacts in the resulting excitation pattern, as shown in Figure 4.(J).
The improved parallel excitation performance using the proposed
method over the conventional transmit SENSE indicates that the
mutual coupling effect of the imperfectly designed transmit array can be
efficiently corrected by the precompensation technique.
Figure 4. The excitation profiles of a 4-element coil array with non-negligible mutual coupling between array elements: (A-D) individual excitation pattern of each element using conventional transmit SENSE; (E) the resulting excitation pattern from 4 elements, showing that the aliasing in excitation profile of each element is unable to be cancelled, leading to considerable artifacts; (F-I) individual excitation pattern of each element using the proposed precompensation method; (J) the resulting excitation pattern, showing that the artifacts caused
by the mutual coupling of array elements are significantly reduced due to the artifact cancellation of each element profile, yielding an
improved excitation profile.
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Discussion
In this work, the mutual coupling coefficients between array elements
have been taken into the pulses design procedure of transmit SENSE.
The sensitivity patterns of all the elements are multiplied by the
corresponding mutual coupling coefficient and then added together to
form new sensitivity patterns. The central equation of the superposition
of all the individual patterns multiplied by the sensitivity pattern is modified although the pulse design procedure is the same as that in
the conventional transmit SENSE. Thus, each individual pattern can
be obtained by solving the same inverse problem in transmit SENSE,
and each individual RF pulse can be calculated according to the k-space
analysis method.
The results of the transmit SENSE studies with a 4-element coil
array by simulating the Bloch equation have demonstrated the proposed
precompensation method is feasible to diminish the artifacts caused
by the mutual coupling between the elements, making the parallel
excitation method more tolerant to mutual coupling in RF transmit
arrays, ultimately improving the quality of excitation. This is very
helpful for parallel excitation because current decouple methods are
unable to thoroughly eliminate the mutual coupling in RF transmit
arrays. Combining this precompensation method and conventional
decouple schemes together for multi-channel excitation has potential to
decrease excitation artifacts and improve the excitation profile accuracy.
Although this method is validated by using transmit SENSE in this
work, it can also be applied to other parallel excitation methods such as the spatial domain method.
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Acknowledgment
This work was partially supported by NIH grants EB004453, EB008699,
EB007588-03S1 and P41EB013598, and a QB3 Research Award.
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References
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