Zhibo Zhu^{1}, R. Marc Lebel^{2}, Yannick Bliesener^{1}, and Krishna S. Nayak^{1}

Quantitative
DCE-MRI requires fast pre-contrast T_{1} mapping (scan time <3 min)
with matching resolution and coverage. Recent advances in imaging have
substantially improved resolution and coverage of DCE-MRI but without matched
improvements in the pre-contrast T_{1} data. Here, we demonstrate
a sparse T_{1} mapping method and characterize a tradeoff between data acquisition and _{ }T_{1} statistics, using a variable flip angle (VFA)
approach and sparse Cartesian spiral sampling pattern, with image domain
wavelet sparsity constraint. This method provides the necessary high-resolution
whole-brain T_{1}/M_{0} maps for DCE-MRI tracer kinetic
analysis.

Data were acquired on
a glioblastoma subject using a GE MR750 scanner with a 12-channel head coil. The
vendor provided 3D spoiled gradient echo sequence was modified to include
sparse undersampling. Six increasing flip angles each with 5000 TRs and a final
flip angle with 20000 TRs were performed. Note that the last flip angle has a
longer duration since it is used in the DCE scan and precedes
the contrast injection. The flip angle increased from 1.5° to 15° logarithmically^{2}. Other settings: 5 ms TR, 1.9 ms TE, 240×240×240
mm^{3} FOV, 2 mm slice thickness, and 256×240×120 matrix size.

Pre-contrast T_{1}/M_{0} mapping is
performed by solving the following constrained inverse problem^{2}:

$$p = \min_p \Vert F_u S m - d\Vert_{\ell_2}^2 + \lambda \Vert \Psi m\Vert_{\ell_1}$$

where the estimated anatomic image magnitude is

$$\Vert m\Vert = Dp = M_0\left(1-e^{-\frac{TR}{T_1}}\right)\frac{\sin{B_1\alpha}}{1-\cos{B_1\alpha}}$$

$$$p$$$ concatenates T_{1} and M_{0}
into a vector, $$$F_u$$$ is
the undersampled Fourier transform, $$$S$$$ is the coil sensitivity, $$$m$$$ is VFA images, $$$D$$$ converts T_{1} and M_{0}
values into VFA images, $$$\Psi$$$ is a sparsifying transform (e.g., wavelet), $$$d$$$ is measured $$$k$$$-space data, and $$$\lambda$$$ is a regularization parameter. This problem is
solved using POCS iterations, which alternate between thresholding wavelet
coefficients and forcing data consistency. A flowchart of this workflow is
shown in Figure 1.

We used wavelet
transform as the spatial sparsifier as it preserves subtle features as well
as denoises data. $$$\lambda$$$ was empirically chosen as 0.3 and remained
constant in all experiments. To avoid model failure impacting the estimation
process, we impose both non-negative and non-infinite constraints on each
pixel’s T_{1} and M_{0} values.

We explored T_{1} mapping accuracy as a function of
the number of TR periods included at each flip angle. Faster acquisitions were
simulated by discarding samples at each flip angle, as illustrated in Figure 2.
We measure and report the error in T_{1} and M_{0} maps as functions of the amount of
retained data.

Figure 3 shows the
ROI outline and difference maps for T_{1}, M_{0}, and anatomic
images within the ROI. These maps were obtained by computing differences
between results of (5000, 20000) setting and results of any (A, B)
setting. Note that M_{0} difference
decreases as B decreases when A is small. In addition, there was at most 85 ms
T_{1} difference per pixel, and the differences in quantity of
magnetization and image energy were 1.873% and 3.290% at most.

Figure 4 shows means
and standard deviations of estimated T_{1} in normal white matter. T_{1}
values were selected from the ROI in Figure
3. All results show similar mean T_{1}. Interestingly, increasing A or decreasing B both result in a subtle decrease in standard deviation of T_{1}.

Figure 5 illustrates a $$$k$$$-space
error curve, a ROI T_{1} absolute change curve and a ROI image increment
curve. Both T_{1}
change and image increment became trivial after 80 iterations,
while residuals in $$$k$$$-space were quite significant.

This approach enables
adequate pre-contrast T_{1} estimation matched to the resolution and
coverage of modern high-resolution whole-brain DCE-MRI in a reasonable scan
time (<3 minutes), in stark contrast to a fully sampled VFA acquisition that
would require more than 20 minutes.
Despite different subsample
settings, both T_{1} values mean and standard deviation were stable, and
no substantial bias was observed. This indicates that the proposed approach is
flexible and allows a trade-off between data acquisition time and T_{1} statistics.

Another important
observation is significant $$$k$$$-space residuals. One possible cause is that wavelet sparsity regularization can over-smooth images. One way to
compensate for these residuals is to impose TGV constraints on vascular parameters directly^{3}, or to enforce data
consistency after each iteration (used in this work).

1. Guo, Y., Lebel, R. M., Zhu, Y., Lingala, S. G., Shiroishi, M. S., Law, M. and Nayak, K. (2016), High‐resolution whole‐brain DCE‐MRI using constrained reconstruction: Prospective clinical evaluation in brain tumor patients. Med. Phys., 43: 2013-2023. doi:10.1118/1.4944736

2. Lebel, R. M., Guo, Y., Lingala, S. G., Frayne, R., Nayak, K. S. "Highly Accelerated DCE imaging with integrated T1 mapping." Proc. ISMRM 25th Scientific Sessions, Honolulu, April 2017, p138.

3. Maier O., Schoormans J., Schloegl M., *et al*.
Rapid T1 quantification from high resolution 3D data with model‐based reconstruction. Magn Reson Med.
2018;00:1–18. https://doi.org/10.1002/mrm.27502