Franz Patzig^{1}, Bertram Wilm^{1}, and Klaas Pruessmann^{1}

Due to their short read-out time single-shot techniques are frequently used for several imaging modalities but they are prone to static B0 off-resonance artifacts. To avoid separately acquired field maps joint estimation of the object and the B0 map has been proposed as a potential solution alternating between updating an object and a field map guess. A measure to compare cost functions is introduced and two different joint estimation cost functions are investigated whereby a new cost function in image space is suggested. It shows its potential if only a less reliable B0 map guess is given.

*Cost function in k-space*

A previous implementation [1] proposed to minimize the following least-squares problem on the signal data $$$s(t)$$$

$$L(\omega_0(x),o(x))=\left\lVert s(t)-E(k(t),\omega_0(x))\,\cdot\,o(x)\right\rVert^2$$

$$\hat{\omega}_0(x),\hat{o}(x)=\text{argmin}\;L(\omega_0(x),o(x))$$

where $$$k(t)$$$ describes the k-space coordinate, $$$o(x)$$$ the imaged object, and $$$E$$$ the encoding matrix with entries

$$E_{mn}=\exp(-i\omega_0(x_n)t_m)\exp(-i2\pi k(t_m)\cdot x_n)$$.

One reason for the non-convexity of the problem are phase wraps of the complex signal.

*Magnitude cost function in image space*

* *

Another cost function is proposed based on the fact that the magnitude of the point spread function (PSF) of EPI and spiral trajectories is a one to one mapping for different field offsets $$$\omega_0(x)$$$:

$$L(\omega_0(x), o(x)) = \left\lVert \mid E^+(k(t),0)\cdot s(t)\mid - \mid E^+(k(t),0)\cdot E(k(t), \omega_0(x)) \cdot o(x)\mid\right\rVert^2$$

$$\hat{\omega}_0(x), \hat{o}(x) = \text{argmin}\;L(\omega_0(x), o(x))$$

Here, $$$E^+$$$ describes the pseudo-inverse of the operator $$$E$$$. The considered images are in the distorted space as the $$$E^+$$$ operator is computed for $$$\omega_0(x) = 0$$$.

* *

*Gradient Descent and Attractor Size*

* *

* *A quality measure
of a cost function can be defined by the attractor size defined as the maximum
disturbance of the point of interest from which a gradient descent still
converges to (or close to) the point of interest. To assess the performance of
the cost function attractor sizes were evaluated by differing the quality of
the initial guess and a gradient descent algorithm was used to perform the B0 optimization
of the JE problem (Fig. 1).

*Measurement parameters and B0 fitting*

MR scanning was performed on a 3T MR system (Philips Healthcare, Best, The Netherlands) using an 8-channel head coil array equipped with 16 magnetic field sensors (Skope MR Technologies, Zurich, Switzerland) [2] to record the actual k-space trajectory. A spiral in/out sequence with TE: 43 ms, FOV: 22 cm, resolution: 1x1 mm was played out. An initial field map guess was obtained by splitting the acquired data into two images and fitting the image phase. For reference, a field map was fitted from a 2-echo gradient echo (GRE) sequence.

[1] Sutton et al. – Dynamic Field Map Estimation Using a Spiral-In / Spiral-Out Acquisition

[2] Kennedy et al. – An industrial design solution for integrating NMR magnetic field sensors into an MRI scanner

[3] Barmet et al. - Sensitivity encoding and B0 inhomogeneity - A simultaneous reconstruction approach, Proceedings of the ISMRM 2005

Investigation
of attractor size for the k-space and the here proposed magnitude cost
function: for larger mean perturbations the magnitude cost function converges
to a result with a smaller mean error which hints for a larger attractor size;
for smaller perturbations both cost function yield similar deviation in the
found field maps.

(A)
GRE measured field map; (B) reconstructed object using the GRE field map (A);
(C) distorted field map from spiral in/out images; (D) reconstructed object
using the distorted field map (C).

Results
of B0 optimization employing different cost functions starting from an initial
map of constant 0 Hz (mean deviation to GRE field map = 13 Hz). The top row
shows the found field map (A) and the according reconstructed image (B)
minimizing the cost function in k-space. The bottom row shows the found field
map (C) and the according reconstructed image (D) minimizing the cost function
of the magnitudes in image space.

Results
of B0 optimization employing different cost functions starting with the
distorted field map (Fig. 2C) as initial field map guess (mean deviation to GRE
field map = 10 Hz). The top row shows the found field map (A) and the according
reconstructed image (B) minimizing the cost function in k-space. The bottom row
shows the found field map (C) and the according reconstructed image (D)
minimizing the cost function of the magnitudes in image space.

Results
of B0 optimization employing different cost functions starting with a perturbed
version of the GRE field map (Fig. 2A) as initial field map guess (mean
deviation to GRE field map = 4 Hz). The top row shows the found field map (A)
and the according reconstructed image (B) minimizing the cost function in
k-space. The bottom row shows the found field map (C) and the according
reconstructed image (D) minimizing the cost function of the magnitudes in image
space.