Introduction

MR-guided focused ultrasound (tcMRgFUS) procedures for breast cancer need faster methods of accurately measuring temperature simultaneously in adipose tissue and glandular tissue. Since the proton resonance frequency shift (PRFS) method is not effective at measuring temperature change in fat^1,2, the T₁-to-temperature relationship has been investigated for this need due to its versatility across all tissue types. In 2019, a single reference (SR) method was published for the dual-angle variable flip angle (VFA) method of T₁-mapping, which method cut the necessary number of required dynamic acquisitions in half³. However, this SR-VFA method neglected changes in T₂* with temperature³. This study eliminates that assumption and provides an understanding, through simulation, of the calculation bias produced by T₂* changes with temperature. In addition, appropriate scanning parameters to minimize or eliminate this bias were determined to make the SR-VFA method a more viable choice towards fast T₁ thermometry during MRgFUS procedures.

Materials and Methods

Monte Carlo simulations were performed 1000 times on noisy signals of a single simulated voxel generated using the SPGR equation, $$$ (1-E_1) \sin (\alpha) / (1-E_1 \cos (\alpha)) , (E_1 = e^{(-TR/T_1)}) $$$ to compare the VFA method with the SR-VFA method and the T₂*-corrected SR-VFA method in terms of accuracy and precision relative to the true T₁ value. SNR was adjusted to be 100 relative to the signal at the Ernst angle for each TR/T₁ baseline pair for all signals. Results were parameterized based on % change in T₁ from the baseline T₁, TE/T₂*, TR/T₁ baseline, and a temperature sensitivity ratio Z, defined as (T₂* % change) / (T₁ % change), both relative to their baseline values. Parameters were varied within the following ranges: T₁ % change - [0 200%]; TE/T₂* - [0 0.5]; TR/T₁ - [0.01 0.3]; Z - [0 1]. Z was kept positive because in general T₂ and T₁ both increase with temperature¹. The bias and standard deviation of the 1000 T₁ estimates were calculated with each parameter set. The bias value was found by taking the difference between the true T₁ value and the mean of the noisy estimates.

For the flip angle sensitivity simulations, the reference and dynamic flip angles α and β were varied from 0 to 90°. The angles that produced the minimum variance for a simulated parameter set were compared with the theoretical minimum obtained through the theory of propagation of errors applied to the SR-VFA T₁ calculation. The “optimal” angles were defined as those that produced the minimum average variance over the 0-200% change in T₁ for each parameter set.

The T₂* correction was simulated via additional noisy signals at TE₂ > TE₁ for each TE tested and fitting a mono-exponential curve to the two points for a T₂* estimate.

Results

Figure 1 shows a representative example of the effects of changes in T₂* on the SR-VFA calculation.
When T₂* effects are neglected, a negative bias is produced that increases in magnitude with larger changes in exp(-TE/T₂*).

When noisy T₂* estimates were applied to the SR-VFA method’s calculations on the bolded lines’ data from Figure 1, the absolute bias dropped to <0.2% of the baseline T₁, shown in Figure 2. While this 2-point estimate eliminated the bias, Figure 3 shows that substantial noise was introduced due to the T₂* measurements. Additionally, a weighted combination of T₁ measurements at each echo to minimize noise is only partially effective due to the correlation between each echo’s corrected T₁ after applying the same T₂* correction to each echo prior to combination.

Further, it was found that the “optimal” reference flip angle produces ~77% of the signal at the Ernst angle of the baseline TR/T₁, while the “optimal” dynamic angle produces 88-92% of the baseline TR/T1 Ernst angle signal.

Discussion and Conclusions

It is shown that the SR-VFA method produces substantial calculation bias when T₂* effects are neglected, which could lead to substantial temperature errors in T₁ thermometry. Minimizing TE/T₂* for the SR-VFA method is crucial to its accuracy, with best results produced when TE/T₂* < 0.2. However, this bias can be removed by applying a reasonable estimate of the relative changes in T₂* with temperature. Since only accurate relative changes in T₂* are required for the T₂* correction to the SR-VFA method, a properly spaced two-point approximation of T₂* is sufficient for a reasonable correction to the bias. To mitigate the problem of substantially increased noise, further work will determine the best strategies for estimating T₂* while keeping noise uncorrelated, perhaps through changing which signals are used to produce each estimate of T₂*, or by optimizing the number and spacing of echo times.

While the SR-VFA method is most accurate when TE << T₂*, PRFS thermometry performs best when TE~ T₂* ². However, the SR-VFA method with T₂* correction can utilize both these needs by utilizing multi-echo acquisitions. If the first echo TE << T₂* and the second TE is closer to T₂*, simultaneous PRFS and the T₂* - corrected SR-VFA method could both be performed with reasonable speed. In future work, these relationships will be verified in experiment, and echo times for optimizing scan time in tandem with T₁ accuracy and PRFS viability will be determined.