Reconstructing all physical quantities from time-domain data of a very short sequence
Alessandro Sbrizzi1, Annette van der Toorn1, Hans Hoogduin1, Peter R Luijten1, and Cornelis AT van den Berg1

1UMC Utrecht, Utrecht, Netherlands

### Synopsis

Magnetic Resonance Spin TomogrAphy in Time-domain (MR-STAT) aims at reconstructing all physical quantities (T1,T2, PD, etc) directly from the data in the time-domain. By solving a large scale nonlinear inversion problem, the parameters can be inferred. The spatial encoding is entangled implicitly in the time response of the system. A very quick acquisition can be performed and processed to derive all system parameters, including electromagnetic field maps like B0 and B1. We illustrate MR-STAT at high resolution for a realistic numerical phantom and we show high precision reconstructions of quantitative maps.

### Introduction

Magnetic Resonance Spin TomogrAphy in Time-domain (MR-STAT [1]) aims at reconstructing all physical quantities ($T_1,T_2, PD$, etc) directly from the data in the time-domain by model-based reconstruction. By solving a large scale nonlinear inversion problem, the parameters can be inferred. Since the system equations are directly inverted from time-domain data, the signal can be acquired during transient states, resulting into a great flexibility in the design of the sequence. The link between temporal and spatial domain is still provided by the gradient fields, but the k-space data does no longer represent spatial frequencies. The spatial encoding, as in other tomographic techniques, is entangled implicitly in the time response of the system. As a result, a very quick acquisition can be performed and processed to derive all system parameters, including electromagnetic field maps like $B_0$ and $B_1$. MR-STAT can be seen as a generalization of conventional MRI and even MR Fingerprinting [2]. Some of its advantages are: 1) The parameters can be reconstructed with high precision, no need for look-up tables 2) the precision of the estimates is explicitly known for quality assessment, 3) reconstruction is not based on (highly) aliased magnetization images 4) various non-steady-states sequence types can be used (e.g. GRE with variable flip angles) with spiral or cartesian readout. Here, we illustrate MR-STAT at high resolution.

### Methods

We employ a numerical phantom [3] with realistic $T_1,T_2,PD,B_1,B_0$ and transceive phase values for a 3T headcoil driven in quadrature. See Table 1 and Figure 1(left column). The resolution is 1mm x 1mm resulting into a 216x216 voxels numerical phantom. Data is simulated for a fully balanced 2D GRE sequence with a gaussian slice-selective RF pulse. The flip angles are taken from a random normal distribution. The slice profile variation throughout the whole sequence is taken into account. The encoding gradients follow a cartesian coverage of the k-space for a 1mm x 1mm resolution. TR is the shortest possible: (TE,TR) = (2.3,4.8) ms. The k-space is filled 8 times. The total sequence duration is 8.3 seconds. Normally distributed noise is superimposed to obtain SNR = 40. The reconstruction is performed by solving:$$\begin{array}{crll} (\alpha^*,\vec{\beta}^*) & = \arg\min_{\alpha,\vec{\beta}}&\int_{t\in\tau} \left|d(t)-s(\alpha,\vec{\beta},t)\right|^2d t,& \mathrm{(Data}\,\,\mathrm{consistency)}\\& \mathrm{such}\,\,\mathrm{that} & s(\alpha,\vec{\beta},t) = \int_{V}\alpha\, m(\vec{\beta},t) d\vec{r},\quad t\in\tau& \mathrm{(Faraday's}\,\,\mathrm{law)}\\& & \frac{d}{d t}\vec{m}=\Pi\vec{m}+\vec{c}&\mathrm{(Bloch}\,\,\mathrm{equation)}\\& &\vec{m}(\vec{\beta},0) = \vec{e}_3 &\mathrm{(Initial}\,\,\mathrm{condition)}\\& &\vec{\beta} \in \mathbb{B} &\mathrm{(Physical}\,\,\mathrm{bounds)}\end{array}$$where the data in time, $d(t)$, is implictly related to the physical parameters $\alpha = PD\cdot B^-$ and $\vec{\beta}=(T_1,T_2,B_1,B_0)$. The number of unknowns is $216 \times 216 \times 6 \approx 2.8\cdot10^5$ while the number of time data points is $216 \times 216 \times 8 \approx 3.7\cdot10^5$ . This large scale nonlinear numerical inversion problem is separable into two smaller subproblems: a nonlinear one in $\vec{\beta}$ and a linear one in $\alpha$. The convergence behavior is enhanced by applying the variable projection method (VARPRO [4]). Computations are carried out in parallel by a cluster of 216 Linux computing cores with a Matlab/C implementation. The encoding capability of MR-STAT is reflected by the standard deviation (std) of the reconstructed maps: the smaller the values, the higher the expected precision. The std maps are directly available after reconstruction by calculating the covariance matrix ($\sigma^2(F^HF)^{-1}$ where $F$ is the sensitivity matrix and $\sigma$ the standard-deviation of noise) and taking the square roots of its diagonal elements.

### Results

All parameters $T_1,T_2,PD,B_1,B_0$ and transceive phase are correctly reconstructed from the same signal waveform (Figure 1, right column). In Table 1, the mean values (and corresponding std) per tissue types are reported. Note the high precision. MR-STAT is able to estimate the accuracy of the reconstructions. This is reflected in the standard deviation maps, Fig. 2. These maps display the same characteristics as the true error maps, given by $|T_1^{true}-T_1^{recon}|$ and $|T_2^{true}-T_2^{recon}|$, respectively. The reconstruction time is about 1.5 hours.

### Discussion and conclusions

We showed that: 1) MR-STAT can be applied to standard GRE sequences (with varying tip angles but constant TR) of few seconds; 2) reconstruction of high resolution multi-parameter maps is feasible 3) the accuracy of the reconstructions is monitored through the standard deviation maps, available with MR-STAT. The fundamental aspect for in-vivo application is that the forward model solver must accurately describe the MR scanner system, including the small imperfections in gradient trajectories, RF waveforms etc. However, experimental proof-of-principle results are encouraging [1]. MR-STAT could pave the way to shorter clinical protocols by synthetically reconstructing contrast images. Given the computation time, off-line reconstruction is the only option at this moment. This could take place overnight on a distributed computing facility. In the future, faster algorithms and more powerful computational hardware might lead to online reconstructions.

### Acknowledgements

[1] Sbrizzi A et al, ISMRM 2015 p3712.

[2] Ma D. et al, Nature 2013

[3] http://brainweb.bic.mni.mcgill.ca/brainweb/

[4] Golub GH and Pereyra V. Inverse Problems 2003.

### References

No reference found.

### Figures

Table 1 True and mean values of the reconstructed relaxation times per tissue type. The standard deviation is also reported

Figure 1 True and reconstructed parameter maps. From Top to Bottom: T1 [s], T2 [s], Proton Density [a.u.], |B1| map [a.u.], transceive phase [rad] and B0 [Hz].

Figure 2 Left: Estimated standard deviation maps as obtained from the reconstructions; Right: true error (i.e. difference) maps

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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