Sven Nouwens^{1}, Kemal Sumser^{2}, Maarten Paulides^{2,3}, Bram de Jager^{1}, and Maurice Heemels^{1}

^{1}Mechanical Engineering, Control Systems Technology, Technical University Eindhoven, Eindhoven, Netherlands, ^{2}Radiotherapy, Erasmus MC, Rotterdam, Netherlands, ^{3}Electrical Engineering, Electromagnetics for Care & Cure, Technical University Eindhoven, Eindhoven, Netherlands

Proton resonance frequency shift based MR thermometry is widely used to non-invasively monitor thermal therapies in vivo. Multi-echo gradient-echo sequences are being studied to increase the effective temperature to noise ratio, however, choosing the optimal echo-times is non-trivial. We developed an optimization framework based on measurement noise models to compute the optimal echo placement for schemes that either correct or do not correct for the conductivity bias. In a 4-echo phantom experiment, we demonstrated a reduction in the noise standard deviation of 22% when comparing optimal to equidistant echo placements.

$$T_\text{PRFS}:=\sum_{k=1}^{n}w_k\frac{\Delta \varphi_k}{\alpha\gamma\text{}B_0\mathrm{TE}_k}.\tag{1}\label{eq:1}$$Here, $$$w_k,~\Delta\varphi_k,~\mathrm{TE}_k$$$, and $$$n$$$ denote the $$$k$$$-th weight, the phase shift from the $$$k$$$-th echo, the $$$k$$$-th echo-time, and the number of echoes, respectively. Additionally, $$$\alpha,~\gamma,~B_0$$$ are defined as in

Finally, two remarks are in order. First, the presented framework assumes a fixed receiver bandwidth. Loosely speaking, the receiver bandwidth is correlated to the echo spacing and, hence, to TNR for each echo. Second, note that the PRFS noise model does not contain the Signal to Noise Ratio (SNR) of the MRI scanner as our framework is independent on this parameter. In other words, the results presented in this work are SNR independent, assuming the MR signal is sufficient large such that the phase noise is Gaussian distributed.

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Figure 1: TNR comparison for both optimal and equidistant echo placement with either not-corrected (left) or corrected (right) reconstructions (higher is better). Clearly, the optimal echo placement results in higher TNR in all cases. For the corrected reconstruction (right), the optimal echo-time placement is particularly successful compared to the equidistant placement strategy.

Figure 2: Optimal echo placement for both the not-corrected (left) and corrected (right) reconstruction of a multi-echo PRFS measurement. Each row on the y-axis displays the optimal echo placements relative to $$$T_2^\star$$$ for a specific number of echoes. For example, an optimal 6-echo sequence with conductivity bias correction has one echo at $$$0.1T_2^\star$$$ and five echoes after $$$T_2^\star$$$ at $$$[1.1,\ 1.2,\ 1.3,\ 1.4,\ 1.5]T_2^\star$$$.

Experimental verification of the optimal echo placement analysis using a 4-echo gradient-echo sequence. The echo-times are given by $$$[2.3,\ 16.3,\ 19.1,\ 21.9]\text{ms}$$$ and $$$[5.1,\ 10.7,\ 16.3,\ 21.9]\text{ms}$$$ for the optimal and non-optimal sequence, respectively. The standard deviation of the 4-echo PRFS measurements are $$$0.84^\circ\text{C}$$$ and $$$1.10^\circ\text{C}$$$ for the optimal and non-optimal MRT sequence, respectively. Note that the echo-times are constrained to $$$\Delta\mathrm{TE}$$$ and $$$\mathrm{TE}_\text{max}$$$

DOI: https://doi.org/10.58530/2022/2697