Synopsis

Proton resonance frequency thermometry is useful to estimate temperature change that is proportional to the resonance frequency change. In this study, we propose a dual-echo bSSFP thermometry method that generates a high intensity signal and linear phase to the frequency shift. Off-centered acquisition in the balanced steady-state free precession creates an imperfect linear phase with respect to the frequency shift. The dual-echo acquisition method compensates for the phase nonlinearity and generates phase information that is linearly proportional to the frequency shift. This phase linearization makes it possible to accurately measure the proton resonance frequency shift caused by temperature change.

INTRODUCTION

METHODS

The signal acquired at echo time (TE), TE=TR/2, of bSSFP sequence does not have a linear relation between the phase of the transverse magnetization and the off-resonance frequency. However, if the echo time is shifted from TR/2, the phase information becomes proportional to the off-resonance frequency and the shifted time. However linearity of the phase with respect to the off-resonance frequency is not perfect and depends on $$$E_{1}=e^{-TR/T_{1}}$$$ and $$$E_{2}=e^{-TR/T_{2}}$$$. The nonlinearity is explained by relationship between free induction decay (FID) signal and echo signal in the bSSFP sequence. The FID signal and echo signal are the main components of the bSSFP signal^2,3. Phase of the FID signal at TE is $$${\delta}={\triangle}f{\times}TE$$$ and that of the echo signal is $$${\delta}-{\theta}={\triangle}f{\times}TE-{\triangle}f{\times}TR$$$ where $$${\triangle}f$$$ is the off-resonance frequency. The FID signal always has a higher signal intensity than the echo signal at every time. This intensity difference between the FID and echo signal results in the phase nonlinearity of the bSSFP signal in Figure 1(c). The phase of the bSSFP signal is biased toward the FID phase. The phase nonlinearity can cause a measurement error when it is used for thermometry. The bSSFP signal at TE is described as follows:

$$M_{xy}(TE)=|M_{FID}|e^{-TE/T_{2}^{*}}e^{i\delta}-|M_{echo}|e^{-TE/T_{2}}e^{i({\delta}-{\theta})}+(higher~order~terms)\hspace{3em}[1]$$

$$|M_{FID}|=M_{0}\cdot{\tan{\alpha/2}}\cdot\left\{1-\left(E_{1}-{\cos{\alpha}}\right)\left(1-E_{2}^{2}\right)/\sqrt{p^{2}-q^{2}} \right\}\hspace{3em}[2]$$

$$|M_{echo}|=M_{0}\cdot{\tan{\alpha/2}}\cdot\left(1/E_{2}\right)\cdot\left\{1-\left(1-E_{1}{\cos{\alpha}}\right)\left(1-E_{2}^{2}\right)/\sqrt{p^{2}-q^{2}} \right\}\hspace{3em}[3]$$

where $$$M_{0}$$$ is the magnetization at thermal equilibrium, $$$\alpha$$$ is the flip angle, $$$p=1-E_{1}\cos{\alpha}-E_{2}^{2}\left(E_{1}-\cos{\alpha}\right)$$$, and $$$q=E_{2}\left(1-E_{1}\right)\left(1+\cos{\alpha}\right)$$$.

In order to solve this nonlinearity problem, we employ a dual-echo method where two echoes are obtained at different TEs, respectively. From the two echo signals, the phase evolution between two echo times is measured by division of complex signals acquired from two different TEs as follows:

$$\angle\frac{M_{xy}(TE_{2})}{M_{xy}(TE_{1})}=\angle\frac{M_{0}\cdot\frac{\left(1-E_{1}\right)\cdot\sin{\alpha}}{p-q\cos{\theta}}{\cdot}e^{-TE_{2}/T_{2}}{\cdot}\left(1-E_{2}e^{-i\theta}\right){\cdot}e^{i\theta\left(TE_{2}/TR\right)}}{M_{0}\cdot\frac{\left(1-E_{1}\right)\cdot\sin{\alpha}}{p-q\cos{\theta}}{\cdot}e^{-TE_{1}/T_{2}}{\cdot}\left(1-E_{2}e^{-i\theta}\right){\cdot}e^{i\theta\left(TE_{1}/TR\right)}}=\left(TE_{2}-TE_{1}\right)\cdot\triangle{f}\hspace{3em}[4]$$

The phase difference between dual echo signals represents a perfectly linear proportion to $$$\triangle{f}$$$ in Eq. [4]. With two echoes symmetrically apart from the center of TR, we can efficiently measure the temperature change.

RESULT

A numerical phantom consisting of a homogeneous circular object at the center is generated to verify the proposed MR thermometry method. At the center of the object, temperature increases 10℃. The temperature change at the other position of the object follows bioheat equation⁴ explaining heat dissipation. In addition, field inhomogeneity and noise are introduced in the simulation. TR=10ms and TE=0 and 10ms for two echoes, respectively, are used, which is an ideal scan condition for maximum efficiency, and phase cycling of π radian is applied. Figure 2(l) represents the result of the proposed method. The proposed method using dual echo signals effectively compensated for the nonlinearity and accurately estimated the temperature change.

A cylindrical CAGN phantom⁵ was made for experiments, which mimicked prostate tissue. Because actual temperature change could not be generated in our experimental conditions, the frequency shift from temperature change was substituted by a change of local field strength. The estimated frequency shift from the phase information using the proposed method are coincident with the applied local field.

DISCUSSION

Acknowledgements

References

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4. Pennes, H. H.. Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of applied physiology. 1948; 1(2): 93-122.

5. Hattori, K., Ikemoto, Y., Takao, W., et al. Development of MRI phantom equivalent to human tissues for 3.0‐T MRI. Medical physics. 2013; 40(3).