Yongquan Ye^{1} and Jingyuan Lyu^{1}

A novel relaxivity mapping method for MR transverse relaxivity mapping (e.g. T2*) is proposed and demonstrated. By extracting an overall complex signal ratio by means of multi-dimensional integration (MDI) , our method offers significantly improved SNR and homogeneous parametric mappings. With MDI, no explicit multi-channel combination operation is required, and calculation efficiency is extremely high for inline calculation.

Consider a data set (2D or 3D) with N_{e}
echoes and N_{c} channels. The signal of echo n_{e} and channel
n_{c} is:

$$$S(n_{e},n_{c})=C_{nc}S_{nc}e^{-TE_{ne} /T_{2}^{*}+i\Delta \omega TE_{ne}+i\varphi _{0}}$$$ [1]

where C_{nc} and S_{nc} are
coil sensitivity profile and baseline signal of channel n_{c}, Δω is off-resonance
frequency, and φ_{0} is baseline phase terms. Define the individual
complex signal ratio of echo n_{e} and channel n_{c} as:

$$$\Delta S(n_{e},n_{c}) = \frac{S(n_{e+1} ,n_{c})}{S(n_{e},n_{c})}=e^{-\Delta TE/T_{2}^{*}+i\Delta \omega \Delta TE}$$$ [2]

Where ΔTE is echo spacing. It is clear that the terms of C_{nc} and TEs have been eliminated, making the ratio independent of channel and echo. By solving the
following least square problem, an overall ΔS, with mathematically
identical form as Eq.2, can be obtained:

$$$argmin_{\Delta S}\sum_{n_{e}=1}^{N_{e}-1}\sum_{n_{c}=1}^{N_{c}}||S(n_{e}+1,n_{c})-S(n_{e},n_{c})\Delta S||_{2}^{2}$$$ [3]

Finally, T2* can be directly obtained as $$$T_{2}^{*}=-\Delta TE/ln|\Delta S|$$$.

For demonstration, 2D multi-echo GRE knee images were collected on a 1.5T scanner (uMR560, UIH, Shanghai) with a 12-channel knee coil, using following parameters: 8x echoes with monopolar readouts, TE=4.4~34.9ms with ΔTE=4.4ms, matrix size=205x256x10, voxel size=0.78x0.78x3mm. MDI results were obtained using uncombined images as well as ACC combined images. For comparison with curve fitting methods, two exponential models with (i.e. 3-parameter model) and without (i.e. 2-parameter model) noise offset terms and a 2-parameter linear model were tested on ACC images.

Fig.1 demonstrates the absence of coil
sensitivity effects (in terms of signal intensity and field homogeneity) in the
corresponding ΔS_{nc}, by solving Eq.3 with for each n_{c}.
However, coil sensitivity related SNR variation are still present in individual
ΔS_{nc}.
On the other hand, by solving Eq.3 for each
n_{e}, Fig.2 shows that the signal ratio over the channel dimension is spatially
uniform in SNR, albeit SNR decreases temporally, which is expected due to the
low signal of later echoes. Therefore by solving Eq.3 simultaneously along both
dimensions, spatial and temporal SNR variation will be eliminated.

Fig.3 compares T2* maps calculated using MDI vs. curve fitting. The calculation time of MDI on this data set was only a few seconds. Fig.4 shows the noise propagation of the compared methods.

1. Bidhult, S., C. G. Xanthis, L. L. Liljekvist, G. Greil, E. Nagel, A. H. Aletras, E. Heiberg and E. Hedstrom. Validation of a new T2* algorithm and its uncertainty value for cardiac and liver iron load determination from MRI magnitude images. Magn Reson Med, 2016; 75(4): 1717-1729.

2. Walsh, D. O., A. F. Gmitro and M. W. Marcellin. Adaptive reconstruction of phased array MR imagery. Magn Reson Med, 2000; 43(5): 682-690.

Fig.1 Examples of magnitude and phase
images of individual channels, as well as the corresponding ΔS_{nch},
obtained by solving Eq.3 with fixed n_{ch}. The combined images using
ACC and the final overall ΔS are also shown. For each individual channels, the corresponding ΔS_{nch}
shows spatial variation in SNR, which is eliminated in the final overall ΔS. Display
window settings are identical for each row.

Fig.2 Magnitude and phase of the images of
the signal ratio S_{nech+1}/S_{nech}, each obtained by solving
Eq.3 with fixed n_{ech} but all channels. No channel related spatial
SNR variation was present. However, there is the expected decrease of the
overall SNR as nech increases. Display window settings are identical
for each row.

Fig.3 Comparison of T2* mapping
results. MDI results: using channel uncombined data (a), SOS combined data (b) and
ACC combined data (c). Curving fitting results with SOS or ACC combined data: using
3-parameter exponential model (d, g), 2-parameter exponential model (e, h) and linear
model (f, i). Display window settings are identical for all results.

Fig.4 Comparison of computational noise
propagation for different T2* mapping methods. Complex Gaussian
noise was added to the under-sampled raw k-space data, then images of each
channel and echo were reconstructed using parallel imaging reconstruction. Then
with the same noise added images, corresponding T2* maps were
calculated using MDI (channel uncombined and ACC combined), as well as curve
fitting with aforementioned models. The above process was repeated 2000 times
with random but identical level of noise. Each noise propagation map was
calculated as the voxel-wise standard deviation of all T2* maps for the
corresponding mapping methods. Display window settings are identical for all
results.