Nikolaos Kallistis^{1}, Ian Rowe^{1}, and Steven Sourbron^{2}

The purpose of the study is to compare a direct model-based reconstruction with an indirect compress sensing reconstruction for the estimation of T1-map, from simulated radial sampled datasets. Comparisons are performed for the binning strategy that is optimal in each case as measured by T1-errors.

The direct reconstruction solves the nonlinear-least-squares optimization problem with a gradient-based L-BFGS algorithm without regularization, while for the indirect method the images are reconstructed using the iGRASP technique.

The accuracy for both methods is similar, however the computational time of the model-based reconstruction is a limiting factor for clinical applications.

Model-based reconstruction of spin density M_{o} and relaxation time T_{1}-maps, after
a saturation recovery (SR) magnetization preparation, solves the
nonlinear-least-squares optimization
problem

$$(M_0 (\textbf{r}),T_1 (\textbf{r}))=\underset{(M_0 (\textbf{r}),T_1 (\textbf{r}))}{argmin}∑_t‖y(\textbf{k},t)-f(M_0 (\textbf{r}),T_1 (\textbf{r}),t)‖_2^2 \qquad [1] $$

where $$$y(\textbf{k},t)$$$ is the raw data and *f* is the operator which converts the M_{o} and T_{1} maps to ($$$(\textbf{k},t)$$$-space by performing a NU-FFT
on the signal after a saturation pulse at time point t:

$$f(M_0,T_1,t)=NUFFT [ S(M_0,T_1,t)] \qquad [2]$$

$$S= (M_0 (1-exp(-t/T1) \qquad [3] $$,

where the mono-exponential relation (eq.3)
represents the model for the signal behaviour after the SR magnetization
preparation.
Optimisation is performed with the gradient-based L-BFGS
algorithm^{4}, subject to bound constraints,
allowing only positive values. The maximum number of L-BFGS
iterations was set to 200. For the indirect method, the signal model (eq.3) is
fitted on the images reconstructed by the iGRASP^{5}, by a trust-region-reflective algorithm.
All calculations were performed in MATLAB (The MathWorks, MA) on a PC (Intel i5@3.30GHz,
8GB RAM). For the iGRASP reconstruction we used the available online Matlab
implementation.

Two digital reference objects were created by simulating the signal evolution after a single saturation pulse assuming free recovery, using a pseudo-continuous sampling interval TS.

a) A modified
Shepp-Logan phantom (Matrix:256x256;Overall Acquisition Time:1000ms;TS:1ms;Total no. Spokes:1000) with predefined values for M_{o} and T_{1} (Fig.1)

b) An abdomen phantom (Matrix:256x256;Overall
Acquisition Time:4000ms;TS:3 ms; Total no. Spokes:1333),
with predefined M_{o} and T_{1} maps (Fig.5) calculated using the MOLLI^{6}
sequence. The simulated signal images S(**r**,t)
were converted to radial S(**k**,t)-data
where each spoke was acquired for each TS using the NU-FFT and the golden-angle
scheme.
The data were retrospectively combined into
frames S(**k**, t_{f}) of
binning spokes (1-89) leading to sampling rates of TR =1xTS to TR = 89xTS,
where t_{f} is defined as the time point of the central spoke in the
frame.

ROIs were selected for the estimation of the relative errors(RE=|Rec – Ref|/Ref). The optimum value of spokes per frame is estimated by finding the minimum value of the RE across the sampling rates.

Figure 1 presents the reconstructed maps using the direct and indirect methods. The image quality of the Model-based reconstruction
is better for all sampling rates than the indirect method, preserving the
features of the T1-map (boundaries, shape) without contamination of excessive
noise. The M_{o}-map is less
affected by the number of the binning spokes for both reconstructions.

Figure 2 shows the mean values of the ROIs. The Model-based reconstruction fails to estimate the low T1 values for sampling rates over 21ms (21xspokes), while the iGRASP accuracy drops due to the noise induced as the result of binning less than 21 spokes/frame .

Figure 3, shows that the reduction of the number of binning spokes, increasing the sampling rate, improves the accuracy of the Model-based reconstruction. The optimum number of binning spokes for the model-based reconstruction seems to be one, while for the iGRASP is fifty-five.

Figure 4, presents the reconstructed maps using the optimum number of binning spokes per dynamic frame. Using one-fifth of the data (acquiring 200 spokes), we notice that the Model-based method is more robust to reduced sampling duration than iGRASP.

Figure 5 displays the reconstructed maps of the simulated abdominal
phantom. The iGRASP reconstruction achieves better results than the Model-based
reconstruction, however this is mainly due to the early termination of the optimization algorithm.

1. Wang, X. et al. Model-based T1 mapping with sparsity constraints using single-shot inversion-recovery radial FLASH. Magn Reson Med 79, 730-740, doi:10.1002/mrm.26726 (2018).

2. Tran-Gia, J., Bisdas, S., Kostler, H. & Klose, U. A model-based reconstruction technique for fast dynamic T1 mapping. Magn Reson Imaging 34, 298-307, doi:10.1016/j.mri.2015.10.016 (2016).

3. Roeloffs, V. et al. Model‐based reconstruction for T1 mapping using single‐shot inversion‐recovery radial FLASH. International Journal of Imaging Systems and Technology 26, 254-263 (2016).

4. Schmidt, M. minFunc: unconstrained differentiable multivariate optimization in Matlab, <http://www.cs.ubc.ca/~schmidtm/Software/minFunc.html> (2005).

5. Feng, L. et al. Golden-angle radial sparse parallel MRI: combination of compressed sensing, parallel imaging, and golden-angle radial sampling for fast and flexible dynamic volumetric MRI. Magn Reson Med 72, 707-717, doi:10.1002/mrm.24980 (2014).

6. Messroghli, D. R. et al. Modified Look-Locker inversion recovery (MOLLI) for high-resolution T1 mapping of the heart. Magn Reson Med 52, 141-146, doi:10.1002/mrm.20110 (2004).

Figure 1. Reconstructed T1-maps (a) and Mo-maps (b) using the model-based
reconstruction and iGRASP (indirect method).

Figure 2. The mean values of
the ROIs shown in the central image for the Model-based reconstruction and
iGRASP.

Figure 3. (a) The relative error of the reconstructed maps for both methods. (b) The
absolute differences of the reconstructed maps with the ground truth.

Figure 4. Reconstructed maps using the optimum number of spokes for the original
dataset and by using only the one-fifth of this dataset.

Figure 5. Reconstructed maps of the simulated abdominal phantom using the optimum
number of spoke.