Damien Nguyen^{1,2}, Rahel Heule^{1,2}, Carl Ganter^{3}, and Oliver Bieri^{1,2}

In this work, we explore the decay of negative and positive steady state configurations as a mean to assess the tissue microstructure. Steady state configurations are retrieved up to a high order from an exhaustive sampling of the frequency response profile of low-angle balanced SSFP scans. Subsequently, the decay of configurations (termed DECO) is analyzed using a single-pole matrix pencil analysis yielding positive and negative DECO images. Any asymmetry in the configuration decay is directly linked to asymmetric frequency content within a voxel and is captured in the DECO difference image.

The aim of this work is to explore steady state configuration imaging to investigate tissue microstructure. To this end, we propose to sample the frequency response of balanced SSFP (bSSFP) for subsequent imaging and analysis of the "decay of configurations", termed DECO

Here
we make use of a seminal concept for motion-insensitive SSFP imaging introduced
by Zur *et al.* in 1990 [1]. To this
end, we acquired a series of N = 360
two-dimensional bSSFP images with different RF phase increments
$$$\varphi_j = j$$$ [°] (for
$$$j = 0, 1, ... N-1$$$).
SSFP
configurations or modes can then be retrieved from a $$$N$$$-point discrete
Fourier transform of the sampled bSSFP frequency response for subsequent DECO
image analysis.

Generally, the decay of positive and negative configurations follows, with increasing order $$$n$$$, a complex exponential behavior and the magnetization can be expressed as a sum over all configurations (cf. Eq. [11] by Ganter [2]) that reads for a centered echo:

$$M=E_2^{1/2}e^{i\left\langle\vartheta\right\rangle/2}\sum_{n=0}^{\infty}\left[\hat{p}\left(n+\frac{1}{2}\right)e^{in\left\langle\vartheta\right\rangle}+\frac{\Lambda}{E_2}\hat{p}\left(-\frac{1}{2}-n\right)e^{-i(n+1)\left\langle\vartheta\right\rangle}\right](-A)^nM^{(0)}$$

with the usual definition of $$$E_i \doteq \exp\left(-{\rm TR}/T_i\right)$$$ and

$$A=\frac{(a+b)/2}{a-\Lambda(a-b)/2}E_2$$$$M^{(0)}\doteq-i \frac{1-E_1}{a+b\Lambda}\sin\alpha$$$$\Lambda\doteq\frac{1}{a-b}\left[ a-E_2^2b-\sqrt{\left(a^2-E_2^2b^2\right)\left(1-E_2^2\right)} \right]$$$$a\doteq 1-E_1\cos\alpha\qquad b\doteq\cos\alpha-E_1$$

and where $$$\vartheta$$$ is the phase accumulated within one TR interval, distributed around some mean value $$$\left\langle\vartheta\right\rangle$$$ with probability density $$$p\left(\vartheta - \left\langle\vartheta\right\rangle\right)$$$ and associated characteristic function

$$\hat{p}(k) = \int_{-\infty}^{\infty} e^{ikx}p(x){\rm d}x$$

It is interesting here to note
that for a homogeneous probe with a symmetric
frequency distribution,
the decay of both positive and negative configurations has the same
characteristic time constant.

A sum-of-squares bSSFP image and the sampled complex frequency response for WM, GM and CSF are shown in Figure 1. From the image frequency-series, corresponding configuration images are derived and displayed in Figure 2 for configuration orders up to $$$\lvert n\rvert = 10$$$. Overall excellent image quality is achieved, and due to the low flip angle used, configurations can be traced to orders $$$\lvert n\rvert \gg 1$$$. Note that some susceptibility artifacts appear in the frontal part of the brain; especially for higher order positive modes, in contrast to negative modes where partial refocusing occurs.

For the selected ROIs (cf. Figure 1a), the decay of the configuration amplitudes is shown in Figure 3a, revealing that no aliasing occurs, even for CSF and modes $$$F_n$$$ with order $$$|n| > 20$$$. A logarithmic plot for the positive and negative configuration amplitudes (Figure 3b) indicates a close to mono-exponential decay thus corroborating the use of a single-pole matrix pencil analysis.

DECO images, illustrating the rate of the decay rate
of positive (R_{+}) and negative configurations (R_{-}) from a
matrix pencil analysis, are shown in Figure 4. Overall, positive configurations
decay slower than their negative counterparts for gray matter, whereas positive
modes decay faster than negative ones for deep white structures, such as the
corpus callosum (CC). In particular, for the selected ROIs, we obtain:
R±(WM)
0.27/0.28, R±(GM)
0.34/0.37 (R±(WM-CC) 0.31/0.29, R±(Putamen)
0.38/0.41).
Finally, a color-coded DECO difference image is shown in Figure 5,
accentuating the asymmetry in the configuration decay and revealing a
non-random tissue-related contrast.

For a homogeneous probe with a symmetric frequency distribution (around the on-resonance), the decay of positive and negative configurations with increasing mode order is essentially identical. Surprisingly, the decay of modes follows to a good approximation a mono-exponential decay and can thus be characterized by a single decay rate constant. Any asymmetry in the frequency distribution due to some tissue microstructure, will lead to an asymmetric bSSFP frequency profile, as already pointed out and discussed by Miller [4-5]. Here, we show that this asymmetry will generally lead to different decay rates for positive and negative configuration orders and is captured in the DECO difference image.

In conclusion, we
have introduced a new imaging method, termed DECO, that is sensitive to
intra-voxel frequency distributions caused by the underlying tissue
microstructure. Its sensitivity to subtle tissue alterations, such as the ones
commonly associated with neurodegenerative disorders, needs to be further investigated
and explored.

[1] Y. Zur *et al.* MRM 16
(1990) 444-459

[2] C. Ganter, MRM 56 (2006) 687-691

[3] Y.-Y. Lin *et al.* jMR 128 (1997) 30-41

[4] K. Miller, MRM 63 (2010) 385-395

[5] K. Miller, MRM 63 (2010) 396-406