A Hybrid Model Integrated with Correction of Susceptibility Induced Phase Error in Magnetic Resonance Thermometry

Kexin Deng^{1}, Yuxin Zhang^{1}, Yu Wang^{1}, Bingyao Chen^{2}, Xing Wei^{2}, Jiafei Yang^{2}, Shi Wang^{3}, and Kui Ying^{3}

** Theory** The susceptibility change induced phase error can be expressed as in (1),$${\Delta}{\phi}=-{\gamma}TE({\Delta}{B_{mac}}-\frac{2}{3}{{\Delta}}{\chi}B_{0}),$$where $$$\gamma$$$ is the
gyromagnetic ratio. TE is the echo time, $$${\Delta}{\chi}$$$ is the
susceptibility difference and $$$B_{0}$$$ is the external
magnetic field. Here, the phase difference $$$\Delta{\phi}$$$ is induced by both local ($$$\chi$$$) and nonlocal ($$$B_{mac}$$$) susceptibility changes. $$${\Delta}B_{mac}$$$, the change in the macroscopic magnetic field in the
object, can be given as:$${\Delta}{B_{mac}}=F^{-1}(k^{'}{\cdot}{F(\chi)}),$$where $$$k^{'}=\frac{B_{0}}{k^{2}}(k_{x}k_{x}, k_{y}k_{x}, -({k_{x}}^{2}+{k_{y}}^{2}))$$$. $$$F$$$ denotes the Fourier transform operator while $$$k^{2}={k_{x}}^{2}+{k_{y}}^{2}+{k_{z}}^{2}$$$ is the k-space vector.

To correct
the susceptibility induced phase error, the original hybrid model^{2} is modified as$${\widetilde{y}_j}=(\sum\limits^{N_{b}}_{b=1}x_{j,b}w_{b})e^{i(\left\{Ac\right\}_{j}+\theta_{j}+\Delta\phi_{j})}+\epsilon_{j},$$where $$$j$$$ is the
image pixel indexes. $$$\left\{x_{b}\right\}_{b=1}^{N_{b}}$$$ are complex images
from the baseline library and the $$$w_{b}$$$ are weighting
factors of the baseline images, $$$A$$$ is a
matrix of smooth basis functions, $$$c$$$ is a polynomial coefficient vector, $$$\theta$$$ is a
temperature-induced phase shift, $$$\Delta\phi$$$ is the susceptibility induced phase shift and $$$\epsilon$$$ is complex Gaussian noise. The
model is then fit by an iterative gradient-based algorithm. $$$\Delta\phi$$$ is initialized by the susceptibility induced
phase estimated from the previous image frame.

** Simulation**
A rectangular object was synthesized in MATLAB in the
similar way as in (2), as shown in Fig.2. The round region (black arrow in
Fig.2 (a)) is fat while the rest region is water. Then we simulated the heating
process of this object by assigning different susceptibility change rates on
water and fat

(1) SM. Sprinkhuizen, MK. Konings, et al. Temperature-Induced Tissue Susceptibility Changes Lead to Significant Temperature Errors in PRFS-Based MR Thermometry During Thermal Interventions. Magnetic Resonance in Medicine 2010, 64:1360–1372.

(2) WA. Grissom, V. Rieke, AB. Holbrook, Y. Medan, M. Lustig, J. Santos, et al. Hybrid referenceless and multibaseline subtraction MR thermometry for monitoring thermal therapies in moving organs. Medical Physics. 2010, 37(9): 5014-5026.

(3) Y. Wang, T. Liu. Quantitative Susceptibility Mapping (QSM): Decoding MRI Data for a Tissue Magnetic Biomarker. Magnetic Resonance in Medicine 2015, 73:82–101.

Fig.1
Simulated susceptibility change under the temperature distribution of the hot
spot (left) and the susceptibility induced phase error (right).

Fig.2 Simulation
experiment. Magnitude image of simulated data (a). The simulated reference with
hot spot (b). The estimated phase maps (c and d) and error maps (e and f) of
hybrid model and SIPEC hybrid model are shown, respectively.

Fig.3 Phase
curve along the red dash line in Fig. 2 for true PRF phase and estimated phase
of hybrid and SIPEC hybrid model, respectively.

Fig.4 Phantom
experiment. Magnitude of the phantom (a). The estimated phase of the region of
interest (ROI marked in (a) with dashed square) by SIPEC hybrid model (d) shows
better hot spot than the original hybrid (c) model. The difference
map of the phase maps estimated by the two models is shown in (f). Susceptibility induced phase difference
($$$\Delta\phi$$$) map (e) between the two time frames is estimated
simultaneously with the temperature map.

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

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