Martin Soellradl^{1}, Stefan Ropele^{1}, and Christian Langkammer^{1}

The signal decay of a 2D gradient echo sequence is substantially influenced
by macroscopic field variations along the slice profile. Here we propose a
numerical model describing the signal decay due to a macroscopic field gradient
for arbitrary excitation pulses with large flip angles. Phantom and in-vivo
experiments show that accurate modelling requires inclusion of the phase along
the slice profile and the polarity of the slice selection gradient.
Additionally, we show that applying the model yields better results for R_{2}^{*}-mapping
and myelin water fraction estimation.

In
the presence of a varying macroscopic field Δω(z),
intravoxel dephasing along the slice profile yields
an overestimated effective transverse relaxation rate R_{2}^{*}^{1–6}. Based on the Fourier approximation of the slice
profile an accurate modeling for gradient echoes with flip angles α<60° was presented in^{3}. Here, we investigated the signal behavior for α>60° for applications that require long TR and thus
larger α and propose a numerical model for arbitrary
excitation pulses with linear approximated Δω(z).

We show that for a proper modeling the polarity of the slice selection
gradient G_{slice} and the phase along the slice profile have to be considered
and demonstrate that not only R_{2}^{*}-mapping but also clinical
interesting applications such as whole brain myelin water fraction (MWF) imaging
can be substantially improved with the proposed model.

_{}In
a 2D gradient echo sequence the measured signal at time t for slice position z_{i}
in slice selection direction z is described by:

$$S(z_{i}, t)=\int_{-\infty}^\infty|M_{xy}(z-z_i)|~e^{i\phi_{xy}(z-z_i)}~e^{i\gamma\Delta \omega(z)t}~ e^{i\phi_0(z)}~dz~S_{model}(t)$$

where |M_{xy}(z- z_{i} )| is the magnitude and φ_{xy}(z-
z_{i}) the phase of transverse magnetization, φ_{0}(z) the phase
offset and S_{model}(t) the signal model. Rewriting Equation
(1) by introducing F(z_{i},t), which describes all components
causing additional dephasing to S_{model}(t), leads to^{5}:

$$S(z_{i}, t)=~S_{model}(t)~F(z_i,t)$$

**Numerical
model F(z _{i},t): **|M

$$\lambda = \frac{G_{slice}}{G_{slice} + G_z}$$

The integral in Equation (1) was solved by numerical integration with
the following assumptions for each slice: linear field Δω(z,z_{i})=Δω_{0}(z_{i})+G_{z}(z_{i})*(z-z_{i}),
φ_{0}(z_{i})=const. and TR>>T1. G_{z} was obtained from the ΔB_{0}-map.

**MRI
sequence:** An adapted multi-echo gradient
echo sequence (mGRE) was developed with two different sinc-Hanning-windowed
pulses and variable polarity of G_{slice} (R_{2}^{*}-mapping: Bandwidth-Time-Product
BWT=2.7 and pulse duration T_{pulse}=2ms, MWF-mapping: BWT=2; T_{pulse}=1ms).

**Phantom R _{2}^{*}-mapping:**
A plastic
cylinder (Ø=12cm; length=25cm) was filled with 110µmol/l Magnevist® doped
agarose gel (5g/l) and scanned with positive and negative G

**In-vivo R _{2}^{*}-mapping:**
Three subjects were scanned with the mGRE
sequence with positive and negative G

**In-vivo MWF-mapping:
**One
subject was scanned with the mGRE sequence (positive G_{slice}; α=90°;
TE1/ΔTE/TR=2.37ms/2.2ms/2s; 32 bipolar echoes BW=500Hz/Px; resolution=1.1x1.1x4mm^{3}; 25 slices; two averages; scan time 11min).

All measurements were performed on a clinical 3T scanner (Magnetom PRISMA, Siemens).

**Analyses:** For R_{2}^{*}-mapping
F(z_{i},t) was calculated with and without B_{1}^{+}
and λ-correction. Then Equation (3) was minimized to fit R_{2}^{*}.
For comparison, R_{2}^{*} was estimated from a conventional fit
(F(z_{i},t)=1). MWF-maps were obtained by fitting once the measured
data and once the data divided by F(z_{i},t) (regularized) to a
multi-exponential model^{10}. Cut-off for myelin water T_{2}^{*}_{my} < 25ms^{11}.

**Phantom:** Figure 1 shows R_{2}^{*}
as function of |G_{z}| from a conventional fit (F(z_{i},t)=1). Compared
to α=30°, R_{2}^{*} for α=90° is different for +/- G_{z} and curves
are flipped when G_{slice} polarity is changed. The normalized signal decay (Figure
1) for |G_{z}|=100µT/m
reveals different signal decay for +/- G_{z}.
Incorporating F(z_{i},t) resulted
in a substantial, up to 5-fold, decrease of R_{2}^{*} compared to
conventional fitting (Figure 2). Given this strong effect, the difference
between using F(z_{i},t) with or without including B_{1}^{+}
and λ was smaller, but still visually observable.

**In-vivo: **Figure 3 shows R_{2}^{*}-maps
for +/- G_{slice} as well as their absolute difference maps. While conventional
fitted R_{2}^{*}
show substantial differences, results with F(z_{i},t) are less
influenced.
The difference between the models in Figure 4
illustrates that B_{1}^{+}, λ-correction is most effective in
areas with high B_{1}^{+}-deviations and strong G_{z}
(e.g.
frontal lobe).

Uncorrected and corrected MWF-maps with F(z_{i},t)
are illustrated in Figure 5. By using F(z_{i},t) it is possible to
recover MWF values from the frontal lobe where the conventional fit fails.

1. Hirsch, N. M. & Preibisch, C. T2* mapping with background gradient correction using different excitation pulse shapes. AJNR. Am. J. Neuroradiol. 34, E65-8 (2013).

2. Hernando, D., Vigen, K. K., Shimakawa, A. & Reeder, S. B. R2*mapping in the presence of macroscopic B0field variations. Magn. Reson. Med. 68, 830–840 (2012).

3. Baudrexel, S. et al. Rapid single-scan T2*-mapping using exponential excitation pulses and image-based correction for linear background gradients. Magn. Reson. Med. 62, 263–268 (2009).

4. Fernndez-Seara, M. A. & Wehrli, F. W. Postprocessing technique to correct for background gradients in image-based R2* measurements. Magn. Reson. Med. 44, 358–366 (2000).

5. Yablonskiy, D. A. Quantitation of intrinsic magnetic susceptibility-related effects in a tissue matrix. Phantom study. Magn. Reson. Med. 39, 417–428 (1998).

6. Preibisch, C., Volz, S., Anti, S. & Deichmann, R. Exponential excitation pulses for improved water content mapping in the presence of background gradients. Magn. Reson. Med. 60, 908–916 (2008).

7. Aigner, C. S., Clason, C., Rund, A. & Stollberger, R. Efficient high-resolution RF pulse design applied to simultaneous multi-slice excitation. J. Magn. Reson. 263, 33–44 (2016).

8. Reichenbach, J. R. et al. Theory and application of static field inhomogeneity effects in gradient-echo imaging. J. Magn. Reson. Imaging 7, 266–279 (1997).

9. Sacolick, L. I., Wiesinger, F., Hancu, I. & Vogel, M. W. B1 mapping by Bloch-Siegert shift. Magn. Reson. Med. 63, 1315–1322 (2010).

10. Whittall, K. P. & MacKay, A. L. Quantitative interpretation of NMR relaxation data. J. Magn. Reson. 84, 134–152 (1989).

11. Lenz, C., Klarhöfer, M. & Scheffler, K. Feasibility of in vivo myelin water imaging using 3D multigradient-echo pulse sequences. Magn. Reson. Med. 68, 523–528 (2012).

Figure 1: Comparison between R_{2}^{*} from a conventional
fit as function of the field gradient G_{z} for α=30° and α=90
with positive and negative slice selection gradient G_{slice}. Additionally, the normalized
signal decay is plotted for |G_{z}| = 100µT/m. R_{2}^{*} depends on the sign of G_{z} for α=90° (Note that
inverting the G_{slice} polarity flips R_{2}^{*} and magnitude curves), whereas for α=30° R_{2}^{*} solely depends
on |G_{z}|.

Figure 2: Phantom data showing the difference
between the R_{2}^{*}-maps estimated from a conventional fit and maps obtained from
the numerical approach for α=30°
and α=90° with
positive and negative slice selection gradient G_{slice}. For the numerical
approach F(z_{i}, t) was estimated with and without B_{1}^{+} and λ-correction.
Note that the R_{2}^{*} scale for the conventional fit is approx. 5-fold higher than
for the corrected models.

Figure 3: Comparison of R_{2}^{*}-maps obtained
from a conventional fit (upper row), the
proposed numerical approach without correction (center row) and with B_{1}^{+}
and λ-correction (bottom row) for positive and negative slice selection
gradient G_{slice}.
The absolute difference maps indicate that G_{slice }polarity has a substantial effect on R_{2}^{*} for the conventional fits and
less influence for the numerical models.

Figure 4: Influence
of B_{1}^{+}-correction and λ-correction on R_{2}^{*}-map
estimation. The first two left plots show the difference between the results
without and with B_{1}^{+} and λ-correction for positive and negative
slice selection gradient G_{slice}. The
third map shows the corresponding B_{1}-map and the right one the
estimated field gradient map G_{z}. Differences in the R_{2}^{*} maps
correlate with B_{1}^{+} variations and strength of G_{z}.

Figure 5: Comparison of the MWF
obtained by multi-exponential fitting of the GRE magnitude images (top row) and
by using the corrected data using F(z_{i}, t) (bottom row). Note that
the proposed model allows to recover MWF values in areas strongly affected by
the field gradient G_{z} (here especially pronounced in frontal areas and fine
subcortical fiber bundles).