In proton resonance frequency (PRF) based temperature mapping, the temperature of water-contained tissues is linearly dependent on the proton resonance frequency. However, the temperature can also be approximately linearly dependent on susceptibility, which could introduce errors in temperature estimation. The susceptibility change mainly happens when fat exists in the object and the error is hardly eliminated just by fat suppression¹. In this work, we propose a hybrid model integrated with susceptibility change induced phase, named susceptibility induced phase error corrected hybrid as SIPEC hybrid to correct the susceptibility change induced error in MR thermometry.

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 model² 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, 0.002 ppm/℃ and 0.01 ppm/℃, respectively. In this way, the Gaussian hot spot, the susceptibility change (Fig.1 (a)) induced phase error (Fig.1 (b)) under the temperature of the simulated hot spot were added to the phase of the synthesized image. Both hybrid model and the SIPEC hybrid model were applied to the synthesized image to validate the effect of the proposed model. RMSEs of the temperature maps were calculated to quantitatively compare the results.

Phantom experiment A phantom consisted of water and a large cylinder of pork fat was made. After heating in a microwave oven for five minutes, images of the phantom were acquired with GRE sequence (TE=15ms, TR=29ms, FOV=200mm×148mm) using an 8-channel head coil in a 3T MR scanner (Philips Healthcare, Achieva). Both two models were applied to the phantom images. Fig.2 shows the estimated phase maps (c and d) and error maps (e and f) of the SIPEC hybrid model and the original hybrid model, respectively. The phase shift error, especially in fat-water interface (white arrow in Fig.2 e and f) is decreased by the SIPEC hybrid method. The phase curve of the dashed red line shown in Fig.2 (b) is plotted in Fig.3. Compared with the original hybrid model, the phase curve acquired from SIPEC hybrid model better agrees with the true phase curve, as shown in Fig.3. The RMSE is about 0.19 radians in the SIPEC method, significantly decreased from the original hybrid model’s 1.13 radians. In the phantom heating experiment, zoomed-in estimated phase maps of hybrid and SIPEC hybrid model are shown in Fig.4 (c) and (d). The difference between the phase maps of the two models shows the improvement of SIPEC hybrid model in Fig.4 (f) and the estimated phase induced by susceptibility is shown in (e).A new hybrid model incorporating with susceptibility changes has been proposed. The susceptibility induced error in temperature estimation is reduced and validated by the simulation and phantom experiments.