Synopsis

Conventional Extended Phase Graphs (EPGs) allow for an efficient analysis of signal behavior in MRI sequences involving spoiling gradients. The concept has been adapted to spatially resolved EPGs, where spoiling and spatial information is assumed to be fully uncoupled. In this work, we formulate an approach for the combination of configuration state theory and the conventional k-space concept, by re-coupling spoiling and spatial information. The concept not only permits investigating the interference of spoiled signals leading to additional ringing artifacts, but also allows to study arbitrary readout trajectories in the realm of EPGs.

Introduction

The Extended Phase Graph (EPG) allows for the efficient calculation of the transient and steady-state response of MRI sequences involving spoiling gradients.^1,2 In the original EPG formulation, the Bloch equations are transformed from the spatial to the Fourier domain, where spoiling becomes a shift of “configuration state (CS)” populations. Fundamental to this derivation is the underlying uniformity assumption of the considered spin ensemble, i.e. spatial dependencies except for those arising from the application of gradient fields are omitted.² A spatially resolved EPG has been proposed, where spatial variations are simulated by independent EPG calculations.³

In this work, we propose to re-combine the "spoiling axis" of the EPG and the "spatial" dependencies into one common k-space. In this way, the concept of EPG and readout is unified, which allows to study image and contrast formation with spatial dependencies, the interference of spoiled signals and allows to discuss readout gradients in the framework of EPGs.

Theory

The dynamics of a magnetization distribution$$\vec{M}(\vec{r})=\left(\begin{array}{c}M_+(\vec{r})\\M_-(\vec{r})\\M_z(\vec{r})\end{array}\right),$$with$$$\;M_\pm=M_x\pm{}iM_y\;$$$are governed by the Bloch equations,⁴ which are fully uncoupled in the image domain, if exchange and motion is neglected. Excitation, relaxation, and precession as well as de-phasing by gradients can be described by operators$$$\;\mathbf{T}_{\phi(\vec{r})}(\alpha(\vec{r}))$$$,$$$\;\mathbf{E}(\vec{r})\;$$$and$$$\;\mathbf{S}_{\Delta k}(\vec{r})\;$$$(see [2] for expressions).

To keep track of spoiling, the magnetization is represented by an integral over CSs$$$\;F_\pm\left(\vec{k},\vec{r}\right)\;$$$,$$$\;Z\left(\vec{k},\vec{r}\right)\;$$$given by$$\vec{M}(\vec{r})=\int\text{d}^3k\;{e^{i\vec{k}\cdot\vec{r}}\left(\begin{array}{c}F_+(\vec{k},\vec{r})\\F_-(\vec{k},\vec{r})\\Z(\vec{k},\vec{r})\end{array}\right)}.$$Insertion into the spatial Bloch operators allows to recover the form of the propagators:$$\mathbf{T}:\quad\left(\begin{array}{c}F_+\left(\vec{k},\vec{r}\right)\\F_-\left(\vec{k},\vec{r}\right)\\Z\left(\vec{k},\vec{r}\right)\end{array}\right)^+=T_{\phi\left(\vec{r}\right)}\left(\alpha\left(\vec{r}\right)\right)\cdot\left(\begin{array}{c}F_+\left(\vec{k},\vec{r}\right)\\F_-\left(\vec{k},\vec{r}\right)\\Z\left(\vec{k},\vec{r}\right)\end{array}\right)^-$$ $$\mathbf{E}:\quad\left(\begin{array}{c}F_+\left(\vec{k},\vec{r}\right)\\F_-\left(\vec{k},\vec{r}\right)\\Z\left(\vec{k},\vec{r}\right)\end{array}\right)^+=\left[\begin{array}{*{20}{c}}E_2\left(\vec{r}\right)&0&0\\0&E_2^*\left(\vec{r}\right)&0\\0&0&E_1\left(\vec{r}\right)\end{array}\right]\cdot\left(\begin{array}{c}F_+\left(\vec{k},\vec{r}\right)\\F_-\left(\vec{k},\vec{r}\right)\\Z\left(\vec{k},\vec{r}\right)\end{array}\right)^-+\delta\left(\vec{k}\right)\left(\begin{array}{c}0\\0\\Z_0\left(\vec{r}\right)\left(1-E_1\left(\vec{r}\right)\right)\end{array}\right),$$where$$$\;\delta(\cdot)\;$$$is the Dirac delta distribution. Equivalently to the classical EPG, the Ansatz of "spoiling tracking" leads to the same update equations$$\mathbf{S}\left(\Delta\vec{k}\right):\quad{}F_\pm\left(\vec{k},\vec{r}\right)^+=F_\pm\left(\vec{k}\mp\Delta\vec{k},\vec{r}\right)^-,$$thus gradients lead to up-shifting of$$$\;F_+\;$$$states and down-shifting of$$$\;F_-\;$$$states, respectively. N.B., other than interpretation of the EPG, the above equations do not differ from the ones by Malik.³

Definition of the combined k-space: The signal$$$\;S\left(\vec{k}'\right)\;$$$in "k-space" (here k' is used to distinguish it from spoiling tracking k) is given by the Fourier transform (FT) of the transverse magnetization$$$\;M_+(\vec{r})\;$$$$$S\left(\vec{k}'\right)=\int\text{d}^3r\;e^{i\vec{k}'\cdot\vec{r}}M_+\left(\vec{r}\right)=\int\text{d}^3k\;\int\text{d}^3r\;e^{i\left(\vec{k}'+\vec{k}\right)\cdot\vec{r}}F_+\left(\vec{k},\vec{r}\right)=\int\text{d}^3k\;F_+\left(\vec{k},\vec{k}'+\vec{k}\right),$$where$$$\;F_+\left(\vec{k},\vec{k}'\right)\;$$$denotes the spatial FT of a CS. Thus, during readout, the echo amplitude is not only given by $$$\;F_0\;$$$, but by a superposition of all CSs' k-spaces.

Results and Discussion

In Figure 1, a k-space view for spoiling is shown. Here, three repetitions of a spoiled SSFP sequence are demonstrated, where imaging gradients (red) are used to move through k-space. Spoiling gradients (pink) "move" the observer out of k-space, where the next RF pulse creates a new FID. The previous FID populates the$$$\;F_{+1}\;$$$CS at negative spatial frequencies relative to the FID, whereas the 180°-component of the RF pulse leads to population of the$$$\;F_{-1}\;$$$state at positive spatial frequencies. In the third row, mixing of the$$$\;F_{-1}\;$$$CS and the FID can be observed, which adds up to the$$$\;F_{0}\;$$$state.

In Figure 2, the combination of EPG and k-space is graphically depicted. Here, the vertical axis corresponds to the discrete CSs, whereas the horizontal axis depicts their spatial frequency (k'). The k-space, which is read out in an MRI experiment, is given by the integral over the discrete CSs along the black, diagonal lines. The combined k-space is illustrated at the top right, which differs for low spoiling significantly, when compared to the pure$$$\;F_0\;$$$state (pink). The finite readout bandwidth is depicted as a band centered around the$$$\;F_0\;$$$CS. Depending on the spoiling$$$\;\Delta{}k$$$, the spacing of the CSs along the vertical axis changes, leading to less overlap and thus less interference between states.

In Figure 3, the concept is applied to a single voxel (a) to demonstrate the importance of finite-bandwidth readout and impact of spoiled signals. The FT of the voxel is given by a sinc (b), which is read with finite bandwidth$$$\;2\pi/\Delta{}x\;$$$and spoiling (colored squares). After a discrete reconstruction without k-space apodization (c), Gibb's ringing can be observed for the unspoiled signal (black). While spoiling leads to attenuation, it also causes ringing artifacts, which are of higher amplitude and long-ranged compared to Gibb's ringing.

Figure 4 shows the "true" k-space of a spoiled SSFP sequence in steady state formed from 10.25-fold oversampling. Two cases of spoiling are compared to direct simulation without oversampling and no echo mixing. Again, the ordering of CSs in k-space is reversed. In addition to Gibb's ringing, additional ringing in the less spoiled case is observed, reminiscent of the mixing of spoiled signals. The k-space depiction is equivalent to what is found in experiments, e.g. when the TESS echoes$$$\;F_{-1,0,+1}\;$$$are read out using a single acquisition.⁵ Figure 5 shows an animation of a moving window reconstruction of the same phantom going from$$$\;F_{+2}\;$$$to$$$\;F_{-2}$$$.

Conclusion

Acknowledgements

References

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