Martin Soellradl^{1}, Lukas Pirpamer^{1}, Jan Sedlacik^{2}, Franz Fazekas^{1}, Stefan Ropele^{1}, and Christian Langkammer^{1}

Macroscopic field inhomogeneities increase the effective transverse relaxation rate R_{2}^{*}. In contrast to conventional models assuming ideal rectangular pulses, we developed an R_{2}^{*} correction model for Gaussian excitation pulses. After demonstrating the validity of the model in phantom measurements we measured 10 volunteers with 2mm and 4mm slice thickness, respectively. Uncorrected and corrected R_{2}^{*} values were assessed regionally and a significant effect of the correction was observed. An advantage of the proposed method is that it only requires two echoes, rendering it useful in clinical MRI.

The
effective transverse relaxation rate R_{2}^{*}
is a sensitive measure for brain iron concentration^{1} and can be
easily obtained by gradient echo (GRE) sequences. Besides its sensitivity for
microscopic field variations R_{2}^{*} is also affected by
macroscopic field variations causing an enhanced signal decay^{2},
which increases with slice thickness Δz. This hampers the comparison of R_{2}^{*}
rates measured with different Δz, which might be an issue in longitudinal
studies. Correction methods that are assuming an ideal rectangular slice
profile result in sinc-shaped signal decay^{2}. However, when using a
Gaussian excitation pulse for the GRE the signal decay deviates from the ideal
one^{3} (Figure 1).

Therefore,
this work presents a model describing the signal attenuation for a Gaussian
excitation pulse, which in turn allows to correct for R_{2}^{*} estimation
errors in the presence of macroscopic field variations.

The
transverse magnetization over time t from a gradient echo, assuming a constant field
gradient G_{z} within a voxel due to macroscopic field variations^{4},
can be described along z as^{3}:

$$S(t,z_0)=\int_{-\infty}^{\infty}\rho(z)e^{-R_2^*~t}e^{-i\gamma(\Delta B_0(z_0)+ G_z\cdot[z-z_o])t}SRF(z-zo)dz$$

with spin
density ρ(z), the field offset ΔB_{0}(z_{0}) and the spatial
response SRF(z-z_{0})
for the center of the slice z_{0}. For a Gaussian excitation pulse the SRF is modelled as a Gaussian function
with standard deviation σ_{Δz}:

$$SRF(z-z_0)=e^{-\frac{(z-z_0)^2}{2~\sigma_{{\Delta}z}}}$$

Assuming a constant spin density along z the solution becomes:

$$S(t,z_0)=\hat{\rho}e^{-i\gamma\Delta B_0(z_0)t}e^{-R_2^*~t}e^{-\frac{(\gamma~G_z~\sigma_{\small\Delta_z})^2}{2}}$$

with scaled spin density $$$\hat{\rho}$$$.

Phantom and in-vivo measurements were performed on a clinical 3T scanner (Magnetom PRISMA, Siemens).

**Phantom MRI: ** To determine σ_{Δz} of equation (3) a spherical agar-phantom (diameter=10cm, concentration=10g/l) was measured twice, first with standard shim and then with manually changed linear z-shim, with a 2D multi-echo gradient echo sequence (mGRE) with varying slice thickness Δz=2/3/4/5/6mm. Sequence settings: FA=20° (Gaussian excitation pulse); 32 echoes; TE1=3.1ms; ΔTE=9ms; TR=300ms; resolution=1x1xΔz mm^{3}. From standard shim images (Δz=2mm) a reference R_{2}^{*}_{ref } was determined from voxels with negligible small gradient magnitude (γ|G|< 2rad/(ms mm)). With R_{2}^{*}_{ref}, and the images with changed shim, it was possible to determine σ_{Δz} with G_{z} derived from the field-map ΔB_{0}. With σ_{Δz} and G_{z} corrected R_{2}^{*}_{corr }maps were estimated by fitting equation (3) and then compared with R_{2}^{*}_{ref}.

Gradients G_{x,y,z} were calculated by differentiation of the ΔB_{0}-map obtained from a linear fit of the temporal unwrapped phase.

**In-vivo MRI:** Ten healthy
volunteers (26-31 years) were scanned with an anatomical MPRAGE (1mm³) and twice
with a mGRE sequence with Δz=2/4mm. (Gaussian FA=20°; 22 echoes; TE1=4ms; ΔTE=4.2ms;
TR=2710ms; resolution=1x1x Δz mm^{3}). Corrected R_{2}^{*}_{corr}
maps were obtained by fitting equation (3) using the previous estimated σ_{Δz}
and G_{z} obtained from the
ΔB_{0}-map, which was calculated from the unwrapped phase of first and third
echo.

**Regional evaluation:** Numerical
differences were assessed by calculating uncorrected R_{2}^{*}
and corrected R_{2}^{*}_{corr} mean values in the basal
ganglia, obtained from the segmentation of the MPRAGE with FSL FIRST^{5} and
registration to mGRE-space with FSL FLIRT^{5}. Mean values were
regional compared with a Paired Student's t-test.

**Phantom measurements:** Resulting σ_{Δz }–maps estimated from G_{z} -maps and R_{2}^{*}_{ref} =7.6 s^{-1} are shown in Figure 2a-c and indicate that σ_{Δz} is differentiable and increasing with Δz (σ_{Δz}=0.88/1.34/1.76/2.25mm).
Uncorrected
R_{2}^{*} values are increasing with Δz (Figure 2d), whereas
the corrected R_{2}^{*}_{corr} values (Figure 2e)
minimize differences to R_{2}^{*}_{ref} (Figure 2a)
with growing noise levels for larger Δz (Figure 2e).

**In-vivo measurements:** Figure 3 shows the uncorrected R_{2}^{*}-map and corrected R_{2}^{*}_{corr }-map with a strong correction effect adjacent to the sinuses and cavities for Δz=4mm. Estimated mean values between R_{2}^{*} and R_{2}^{*}_{corr} were significant different (p<0.05) in all regions for each Δz. After
correction, R_{2}^{*}_{corr} in the caudate nucleus and the pallidum
remained significantly different (p<0.05) between Δz=2mm and Δz=4mm.

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2. Fernández-Seara, M.A. and Wehrli, F.W. (2000), Postprocessing technique to correct for background gradients in image-based R*2 measurements. Magn. Reson. Med., 44: 358–366.

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6. Yang, X., Sammet, S., Schmalbrock, P. and Knopp, M. V. (2010), Postprocessing correction for distortions in T2* decay caused by quadratic cross-slice B0 inhomogeneity. Magn. Reson. Med., 63: 1258–1268.

Figure 1: (Left) Ideal rectangular and Gaussian spatial response function (SRF). (Right) Signal decay of the transverse magnetization due to a constant field gradient (Gz=200µT/m) over echo times TE for both SRFs: The signal decay depends clearly on the SRF.

Figure 2: Parameter maps obtained from multi-echo
gradient echo data acquired with different slice thickness Δz and changed linear z-shim: (a) reference R_{2}^{*}_{ref }-maps, (b) calculated gradient maps G_{z}, (c) σ-maps estimated from the R_{2}^{*}_{ref} value and gradient G_{z}, (d) uncorrected R_{2}^{*}_{ref }-maps, (e) corrected R_{2}^{*}_{corr }-maps, (f) relative error estimated with the R_{2}^{*}_{ref} value (a).

Figure 3: Uncorrected R_{2}^{*}-maps, corrected R_{2}^{*}_{corr }-maps and difference-maps for slice thickness Δz=2mm (row a) and Δz=4mm (row b).