Synopsis

Keywords: MR Fingerprinting/Synthetic MR, Relaxometry

In MRF, it has been established – but often ignored – that the presence of diffusion causes bias in the estimated T2. We measure this effect in MR-STAT. Thereby we confirm the theoretical model that the bias is proportional to the square of T2. We also find that the bias level strongly depends on the RF sequence used. This opens the prospect of optimizing sequences on minimal diffusion-induced bias in T2.

Introduction

In MR-Fingerprinting¹ (MRF), it has been established that the presence of diffusion influences the estimated T2 value^2,3, an effect that is mostly – but doubtfully – ignored. In MR-STAT⁴, we can expect similar effects. Unfortunately, the findings by Kobayashi² are not immediately applicable to MR-STAT because of a substantial difference in TR; Freed³ is not directly applicable since it estimates the effect on the acquired signal. In this work, we estimate the diffusion-induced bias on the reconstructed T2 map. We show that the diffusion-induced T2 bias strongly depends on the applied RF flip angle sequence and therefore hypothesize that sequence design might mitigate this bias.

Theory

In fast multi-parametric relaxometry (e.g. MRF or MR-STAT), in good approximation, the effect of diffusion is expected to depend on the net integral of the gradient between successive excitations. Quoting Freed³ eq.(14):
$$$T_{\text{2eff}}^{-1}(p)\approx T_2^{-1}+\frac{p^2}{3T_D}$$$ (1)
where $$$T_{\text{2eff}}$$$ is the effective relaxation time, $$$p$$$ is the EPG mode⁵ and $$$T_D$$$ is a characteristic value given by $$$T_D\equiv(D\gamma^2 g^2 T_R^2)^{-1}$$$, with $$$D$$$ the diffusion constant (which we measured to be (2.1±0.2)µm²/ms for gel vials); for our purpose, $$$(gT_R)$$$ is interpreted as the net gradient area between two successive RF pulses. In default MR-STAT scans, $$$(gT_R)$$$ is on the order of $$$30\text{ms}\frac{\text{mT}}{\text{m}}$$$, so $$$T_D$$$ is on the order of 10s, i.e. much larger than $$$T_2$$$. This allows re-writing eq.(1) as $$$T_{\text{2eff}}(p)-T_2 \approx -\frac{p^2}{3T_D}T_2^2$$$. The left-hand side hereof could be seen as the bias to $$$T_2$$$, if $$$p$$$ were the only EPG state occupied during all of the measurement. Yet, in practice, non-balanced sequences continuously exhibit a varying mix of many EPG states. Without mathematical rigor, we model the total T2-bias of a sequence as $$$T_{\text{2eff}}-T_2 \approx -T_D^{-1}T_2^2 S$$$, with the “Sequence-dependent factor” $$$S=\sum{m(p)\frac{p^2}{3}}$$$ and $$$m(p)$$$ is the average occupancy of state $$$p$$$ during the sequence.
Re-expanding $$$T_D$$$, we model the bias as being proportional to $$$D\gamma^2 (gT_R)^2 T_2^2 S$$$, i.e., proportional to

1. The diffusion coefficient
2. The square of the spoiling gradient area
3. $$$T_2^2$$$
4. A sequence-dependent unitless factor $$$S$$$.
   
   

Methods

For known values of $$$d$$$, $$$(gT_R)$$$ and $$$T_2$$$, we would like to establish the sequence-dependent bias factor $$$S$$$. Since it is difficult to reliably measure $$$(T_{\text{2eff}}-T_2)$$$, we define a pair of scans that differ only in spoiling gradient area. The reconstructed T2 maps are called $$$T_{\text{2,FS}}$$$ (Full Spoiling) and $$$T_{\text{2,HS}}$$$ (Half Spoiling) respectively. Consequently, we have $$$(gT_R)_{\text{HS}}^2 \approx \frac{1}{4} (gT_R)_{\text{FS}}^2$$$ and so we can model $$$T_{\text{2,HS}}-T_{\text{2,FS}}\approx D\gamma^2 \left({(gT_R)_{\text{FS}}^2-\frac{1}{4} (gT_R)_{\text{FS}}^2}\right)T_2^2 S$$$.
By measuring $$$T_{\text{2,HS}}-T_{\text{2,FS}}$$$, we can calculate $$$S$$$ and we can estimate the bias $$$T_{\text{2eff}}-T_2$$$ as $$$-\frac{4}{3}(T_{\text{2,HS}}-T_{\text{2,FS}})$$$.
In our experiments, we made use of the Eurospin phantom⁶, which consists of 12 gel-vials with different relaxation properties. The base sequence consisted of a single-slice Cartesian pseudo-SSFP sequence on a 3T scanner (Philips Elition), with TR=10ms, TE=5ms, voxel size 1mm x 1mm, slice thickness of 5mm, with a FOV of 224mm, requiring 224 phase-encoding steps. The set of phase-encoding steps was repeated 6 times, allowing MR-STAT reconstruction of proton density, T1 and T2 maps. In total, 1344 readout-lines were acquired in 13.4 seconds.
For performance analysis, this base sequence was acquired 10 times (with 5s pause in-between), which led to 10 separate reconstructions; this enabled the calculation of the noise level at each pixel and it allowed to boost the precision of the outcome by $$$\sqrt{10}$$$.
To investigate the dependence of the diffusion-induced bias on the flip-angle train, the acquisitions were repeated for 6 different RF-pulse sequences. See figure 1A-F.

Results

As an example, figure 2 shows the reconstructed maps of RF sequence A for the two spoiling levels as well as their difference. Note the difference in T2. The difference in measured T1 values was not significant.
Figure 3 shows, for the 6 different RF sequences, the difference between $$$T_{\text{2,FS}}$$$ and $$$T_{\text{2,HS}}$$$ against $$$T_{\text{2,HS}}^2$$$. For most sequences, a linear relationship between $$$\Delta T_2$$$ and $$$T_2^2$$$ is apparent, highlighting the quadratic dependence between $$$\Delta T_2$$$ and $$$T_2$$$ as predicted in the Theory. The slopes were measured to have values between -0.36s^-1 and -0.053s^-1; corresponding to values of $$$S$$$ of 3.5 and of 0.51. These values clearly depend on the sequence (see also figure 4). For e.g. sequence B (slope of -0.36s^-1, $$$S=3.5$$$), for a T2-value of 200ms, the difference between the measured T2 values is about $$$(0.2\text{s})^2\cdot (-0.36\text{s}{-1})=14.4\text{ms}$$$; the estimate of the bias is $$$\frac{4}{3}$$$ times that value, i.e. close to 20ms.

Discussion

Diffusion causes bias in reconstructed T2. This bias can be substantial, up to 10% in our setup. Yet, in vivo, due to lower $$$D$$$ and T2, we may expect less than 5%. A very interesting finding is that this depends strongly on the sequence, e.g. the bias of RF sequence E is seven times lower than sequence B. Therefore, we hypothesize that sequences could be designed to mitigate that bias, by modelling the diffusion-sensitivity and optimizing thereupon.

Conclusion

As previously shown for MRF, also in MR-STAT diffusion influences the estimated T2. The diffusion-induced mis-estimate increases with $$$T_2^2$$$ and strongly depends on the RF-sequence applied. This opens the prospect of designing a sequence to mitigate the diffusion-induced bias.

Acknowledgements

This work has been financed by NWO grant number 17986