Seohee So^{1}, JaeJin Cho^{1}, Kinam Kwon^{1}, Byungjai Kim^{1}, and HyunWook Park^{1}

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.

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.

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
equation^{4} 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 phantom^{5} 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.

1. Hindman, J. C.. Proton resonance shift of water in the gas and liquid states. The Journal of Chemical Physics. 1966; 44(12): 4582-4592.

2. Gyngell, M. L.. The steady-state signals in short-repetition-time sequences. Journal of Magnetic Resonance 1969; 81(3): 474-483.

3. Freeman, R., & Hill, H. D. W.. Phase and intensity anomalies in Fourier transform NMR. Journal of Magnetic Resonance 1969; 4(3): 366-383.

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).

(a) A sequence diagram of the proposed dual-echo bSSFP imaging method. (b) and (c) are the magnitude and phase profiles of the conventional bSSFP for various TEs. When TE<TR/2, the phase decreases as θ increases. On the other hand, the phase and θ have a positive proportional relation when TE>TR/2. The phase difference between two echoes acquired with (a) shows the phase profile in (d), which is completely linear with respect to θ. The slope of the phase profile in (d) is proportional to the time difference between the two echoes.

Simulation data before and after heating and estimated temperature changes. (a, b) and (c, d) are the magnitude and phase of the reconstructed images from the first and second echoes, respectively, before heating. (e, f) and (g, h) are those after heating. The temperature changes are estimated from the first echo only (j), the second echo only (k) and the proposed dual echoes (l). (m) shows cutview graphs of temperature maps representing the center horizontal position marked as a dotted horizontal line in (i). (j) is underestimated and (k) is overestimated in comparison with the true temperature change (i).

Results of phantom experiments. The applied local field is horizontally modulated by a shim coil. For two methods of SPGR and the proposed method, five different frequency modulations from 1Hz/pixel to 5Hz/pixel are carried out. (a) and (c) are the phases for various frequency modulations. (b) and (d) represent cutviews of the temperature changes measured from the phase information along the horizontal dotted line in the first image of (a). Solid lines are the estimated frequency and dotted lines are the real frequency for every local field modulation (1~5Hz/pixel).