SAFT: Split-Algorithm for Fast T2 Mapping

Tom Hilbert^{1,2,3}, Jean-Philippe Thiran^{2,3}, Reto Meuli^{2}, Gunnar Krueger^{2,3,4}, and Tobias Kober^{1,2,3}

It is common practice
for model-based methods to define a cost-function that incorporates the model
behavior directly within a data fidelity term^{1-3}. This is also done
in the “Model-Based Accelerated Relaxometry by Iterative Nonlinear Inversion”
(MARTINI)^{2}, a T2 mapping algorithm using a multi-echo spin-echo
(MESE) sequence. MARTINI’s cost-function is defined as follows:

$$\phi(M_{0},T2)=\frac{1}{2}\sum_{c=1}^N\sum_{t\in{TE}}^{}{\parallel}PF\left\{S_{c}M_{0}\exp\left(-\frac{t}{T2}\right)\right\}-Y_{t,c}{\parallel}_2^2$$

with TE being
the echo times, N the number of coil elements, P a binary mask representing the
sampling pattern, F the Fourier transform operator, S the coil sensitivities, M_{0}
the equilibrium magnetization, T2 the transverse relaxation and Y the
acquired k-space data. Minimizing this cost-function will result in an
estimation of T2 and M_{0}. However, the minimization of this nonlinear
problem is numerically challenging and may lead to image artifacts and long
reconstruction times. We suggest splitting up the problem similarly to what was
proposed for compressed sensing^{4}, resulting in a 2-step algorithm
which we term “Split-Algorithm for Fast T2 mapping” (SAFT).

**Step 1:**
The
MESE magnetization is calculated based on an initial guess of T2 and M_{0}
using the forward signal model:$$\hat{M}_{t}=M_{0}\exp\left(-\frac{t}{T2}\right)$$Using this first
guess of the magnetization, the following problem is solved with a linear least-squares
algorithm:$$\phi_{1}(M)=\frac{1}{2}\sum_{c=1}^N\sum_{t\in{TE}}^{}{\parallel}PF\left\{S_{c}M_{t}\right\}-Y_{t,c}{\parallel}_2^2+\sum_{t\in{TE}}^{}\alpha{\parallel}M_{t}-\hat{M}_{t}{\parallel}_2^2$$estimating the
magnetization M that best fits the acquired data. The second l_{2}-norm of
the cost function forces the magnetization to be similar to the previously
calculated $$$\hat{M}$$$ with the
similarity weighted by α.

**Step 2:** The MESE signal model
is fitted onto the previously estimated M by solving the following problem with
a nonlinear least-squares algorithm,$$\phi_{2}(T2,M_{0})=\frac{1}{2}\sum_{t\in{TE}}^{}{\parallel}M_{t}-M_{0}\exp\left(-\frac{t}{T2}\right){\parallel}_2^2$$yielding a new
estimate of T2 and M_{0}. Subsequently, steps 1 and 2 are iteratively
repeated until the algorithm converges to a minimum, providing an approximation
of T2 and M0. Optionally, a spatial regularization can be added
to Φ_{2}. Here, we performed an additional reconstruction using a
wavelet sparsity constraint for both T2 and M_{0}.

^{1}Block,
Kai Tobias, Martin Uecker, and Jens Frahm. "Model-based iterative
reconstruction for radial fast spin-echo MRI." Medical Imaging, IEEE
Transactions on 28.11 (2009): 1759-1769.

^{2}Sumpf, Tilman J., et al.
"Model-based nonlinear inverse reconstruction for T2 mapping using highly
undersampled spin-echo MRI." Journal of Magnetic
Resonance Imaging 34.2 (2011): 420-428.

^{3}Sumpf, Tilman J., et al. "Fast T2
Mapping with Improved Accuracy Using Undersampled Spin-echo MRI and Model-based
Reconstructions with a Generating Function." Medical Imaging, IEEE
Transactions on 33.12 (2014): 2213-2222.

^{4}Huang, Feng, et al. "A rapid and
robust numerical algorithm for sensitivity encoding with sparsity constraints:
Self-feeding sparse SENSE." Magnetic Resonance in Medicine 64.4 (2010):
1078-1088.

^{5}Hilbert,
Tom, et al. "MARTINI and GRAPPA-When Speed is Taste." Proc. Intl.
Soc. Mag. Reson. Med.. 22.4077 (2014).

^{6}Hardy,
Peter A., and Anders H. Andersen. "Calculating T2 in images from a phased
array receiver." Magnetic Resonance in Medicine 61.4 (2009): 962-969.

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

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