## A �-space analysis of MR tagging (2000)

Venue: | J Magn. Reson |

Citations: | 2 - 1 self |

### BibTeX

@ARTICLE{Kerwin00a�-space,

author = {William S. Kerwin and Jerry L. Prince},

title = {A �-space analysis of MR tagging},

journal = {J Magn. Reson},

year = {2000},

volume = {142},

pages = {313--322}

}

### OpenURL

### Abstract

We present a k-space approximation that directly relates a pulse sequence to its residual pattern of z-directed magnetization M z, in a manner akin to the k-space approximation for small tip-angle excitation. Our approximation is particularly useful for the analysis and design of tagging sequences, in which M z is the important quantity—as opposed to the transverse magnetization components M x and M y considered in selective excitation. We demonstrate that our approximation provides new insights into tagging, can be used to design novel tag patterns, and, more generally, may be applied to selective presaturation sequences for purposes other than tagging. © 2000 Academic Press Key Words: tagging; pulse sequences; small tip-angle approximation; k space; presaturation.

### Citations

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(Show Context)
Citation Context ...l���k � l��. [16] m�1 Then, using R p(k) in the tagging k-space Eq. [14] yields the approximate SPAMM pattern where �ˆ m � �� l�1 N�1 M z� x� � � m�0 N�m N �ˆ 0 � 1 � 1 2 � l�1 � l� l�m �ˆ mcos�m�x�, =-=[17]-=- m � 1,...,N � 1 � l 2 . [18]sFIG. 3. Comparison of approximated tag profiles using Eq. [17] and actual tag profiles from Bloch equation simulation for four SPAMM sequences. The number and relative am... |

110 |
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Citation Context .... [10]) and calculate its autocorrelation function R p(k), as illustrated in Fig. 2 for the 1-1 SPAMM sequence. The general solution is given by R p�k� � 1 � 2 N�1 � l�1�N N��l� � �m� m��l���k � l��. =-=[16]-=- m�1 Then, using R p(k) in the tagging k-space Eq. [14] yields the approximate SPAMM pattern where �ˆ m � �� l�1 N�1 M z� x� � � m�0 N�m N �ˆ 0 � 1 � 1 2 � l�1 � l� l�m �ˆ mcos�m�x�, [17] m � 1,...,N ... |

94 |
MR imaging of motion with spatial modulation of magnetization. Radiology
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(Show Context)
Citation Context ...e and/or invert the Bloch equation for M z. Neglecting relaxation, the Bloch equation in the rotating frame is � M˙ � x M˙ y M˙ z �� � 0 G � x � B 1,y � G � x 0 B 1, x �� B1,y � B1, x 0 M � x My Mz , =-=[2]-=- where G is the applied gradient field, x is the spatial position, and B 1 � B 1,x � iB 1,y is the applied RF field. In this equation, M x, M y, and M z as well as G and B 1 are all implicit functions... |

49 |
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Citation Context ...x, t� � � i�x � G�t� M xy�x, t� � i�B 1�t�. [5] Solving this differential equation for any position x and time t yields where M xy�x, t� � i� �0 k�s, t� � � � �s t B 1�s�e ix�k�s,t� ds, [6] t G�u�du. =-=[7]-=- We explicitly include time as a variable in Eq. [6] because the expression is generally accurate at any time t in the pulse sequence. For a pulse sequence applied between 0 and T, the final transvers... |

41 |
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Citation Context ...iM y as M˙ xy�x, t� � � i�x � G�t� M xy�x, t� � i�B 1�t�. [5] Solving this differential equation for any position x and time t yields where M xy�x, t� � i� �0 k�s, t� � � � �s t B 1�s�e ix�k�s,t� ds, =-=[6]-=- t G�u�du. [7] We explicitly include time as a variable in Eq. [6] because the expression is generally accurate at any time t in the pulse sequence. For a pulse sequence applied between 0 and T, the f... |

32 |
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(Show Context)
Citation Context ...hape superimposed for reference. This new tag pattern has potential uses in motion tracking because it is functionally equivalent to tracking the intersections in a tagging grid, an approach taken in =-=(15)-=-, among others. The array of spots, however, has some distinct advantages over the grid. First, the well-defined circular features in the new pattern would likely permit better localization by image p... |

27 |
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(Show Context)
Citation Context ...0°, leaving no residual M z. Thus, the resulting pattern consists of regularly spaced dark bars located where x � 2m�/�. The complete transverse tag brightness profile is M z� x� � 1 2 � 1 2 cos��x�. =-=[1]-=- An image modulated by this pattern is shown in Fig. 1 along with a second image showing the effect of motion on the pattern. For SPAMM sequences with more than two RF pulses, the dark parallel bars a... |

27 |
A k-space Analysis of Small Tip-Angle Excitation
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(Show Context)
Citation Context ...ression for a SPAMM pattern generated with only three RF pulses is M z� x� � �cos� 1cos� 2cos� 3 � 1 2�1 � cos� 2�sin� 1sin� 3� ��sin� 2sin�� 1 � � 3��cos��x� �� 1 2�1 � cos� 2�sin� 1sin� 3�cos�2�x�, =-=[4]-=- where � 1, � 2, and � 3 are the tip angles associated with the three RF pulses (13). Such complicated relationships make it difficult to specify and utilize optimality criteria. For example, an impor... |

19 |
An introduction to NMR imaging: From the Bloch equation to the imaging equation
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Citation Context ...or t � T, which, in the k-space path notation, results in the final pattern Mz�x� � 1 � 1 2� � �K p�k�e ix�k 2 dk� . [13] A Fourier transform identity then yields Mz�x� � 1 � 1 2� 2 �K Rp�k�e ix�kdk, =-=[14]-=- where R p(k) is the autocorrelation function of p(k) given by R p�k� ��K p�u� p� �u � k�du. [15] We refer to Eq. [14] as the tagging k-space approximation. It shows that the critical information need... |

18 |
MRI of myocardial function: Motion tracking techniques
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(Show Context)
Citation Context ...be inverted to obtain the fields G and B 1 that produce a desired pattern of M z. In the special case of SPAMM sequences, an analytic solution exists and has the form N�1 M z� x� � � m�0 � mcos�m�x�, =-=[3]-=- where N is the number of RF pulses used, � is determined by the size of the gradient pulses, and � 0,...,� N�1 are coefficients determined by the sizes of the RF pulses (11, 13). The drawback of such... |

5 |
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(Show Context)
Citation Context ...s possible if M xy(x, t) is known for all time. The expression is determined by the bottom line of the Bloch equation, which may be written M˙ z�x, t� � 1 2i��B� 1�t� M xy�x, t� � B 1�t�M� xy�x, t��, =-=[11]-=- where the bar denotes a complex conjugate. To approximate M z(x, t), we replace M xy(x, t) by the small tip-angle approximation (Eq. [6]) and integrate Eq. [11] from 0 to t. The solution is Mz�x, t� ... |

4 |
A DANTE tagging sequence for the evaluation of translational sample motion
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- 1990
(Show Context)
Citation Context ...p-angle approximation has also been formulated using a kspace approach (4). In this approach, a “path” through k-space is defined by k-SPACE ANALYSIS OF MR TAGGING p�k� � �0 T B 1�s���k�s, T� � k�ds, =-=[8]-=- where the dimensionality of k is nominally 3, but may be less than 3 for some pulse sequences. With this definition, the final (t � T) transverse magnetization may be evaluated as M xy�x� � i� �K p�k... |

3 |
Simulations and demonstrations of localized tagging experiments
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(Show Context)
Citation Context ...imensionality of k is nominally 3, but may be less than 3 for some pulse sequences. With this definition, the final (t � T) transverse magnetization may be evaluated as M xy�x� � i� �K p�k�e ix�k dk, =-=[9]-=- that is, the resulting transverse magnetization and the k-space path are related by the Fourier transform, a concept commonly referred to as encoding excitation k-space. 3. METHOD: AN APPROXIMATION F... |

1 |
Bolster Jr., Improved sampling of myocardial motion with variable separation tagging
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(Show Context)
Citation Context ...g a constant M z, which we normalize to 1. Then, the Bloch equation may be written only in terms of the transverse magnetization M xy � M x � iM y as M˙ xy�x, t� � � i�x � G�t� M xy�x, t� � i�B 1�t�. =-=[5]-=- Solving this differential equation for any position x and time t yields where M xy�x, t� � i� �0 k�s, t� � � � �s t B 1�s�e ix�k�s,t� ds, [6] t G�u�du. [7] We explicitly include time as a variable in... |

1 |
Excitation of arbitrary shapes by gradient optimized random walk
- Sersa, Macura
- 1997
(Show Context)
Citation Context ...early demonstrated by a k-space analysis of the SPAMM sequence described in Section 2. For a SPAMM sequence with N RF pulses, the 1D path through k-space is given by p�k� � 1 N � �� m��k � �N � m���, =-=[10]-=- m�1 where � m is the tip angle (in radians) of the mth RF pulse and, again, ���G. See Fig. 2a for an example of a SPAMM path. Examining Eq. [10], we see that SPAMM paths are, in general, sequences of... |

1 |
Parameter relations for the Shinnar–Leroux selective excitation pulse design algorithm
- Pauly, Leroux, et al.
- 1991
(Show Context)
Citation Context ... To approximate M z(x, t), we replace M xy(x, t) by the small tip-angle approximation (Eq. [6]) and integrate Eq. [11] from 0 to t. The solution is Mz�x, t� � 1 � 1 2� � t �0 B1�s�e ix�k�s,t� 2 ds� , =-=[12]-=- which may be verified by differentiation. As above, we assume that the initial magnetization is entirely z directed and has been normalized to 1. We refer to Eq. [12] as the small tip-angle approxima... |

1 |
Frequency-domain simulation of MR tagging
- Crum, Berry, et al.
- 1998
(Show Context)
Citation Context ...proximation for M z. The final tag pattern M z(x) is found by evaluating Eq. [12] for t � T, which, in the k-space path notation, results in the final pattern Mz�x� � 1 � 1 2� � �K p�k�e ix�k 2 dk� . =-=[13]-=- A Fourier transform identity then yields Mz�x� � 1 � 1 2� 2 �K Rp�k�e ix�kdk, [14] where R p(k) is the autocorrelation function of p(k) given by R p�k� ��K p�u� p� �u � k�du. [15] We refer to Eq. [14... |