Introduction

For the purpose of T₂ relaxometry, the slice selective CPMG multi-echo signal can be modelled accurately using a slice-profile corrected EPG model. Assuming a single T₂ component this model would typically be parametrized in terms of T₁, T₂, overall amplitude and B₁f, which is a percentage scale factor accounting for B₁ transmit inhomogeneity, having a value of 100% for a slice profile with 180° refocusing angle at its centre. B₁f determines the precise excitation and refocusing flip angles. Early work fitting these models constrained B₁f as 0 < B₁f<100%^1,2; i.e. it was assumed that for example a nominal 170° refocussing pulse produces the same echo train as a 190° pulse. Later work showed that this assumption fails for slice-selective pulses³. This may cause redundancy and hence instability in the parameter estimates, since B₁f equally above or below the correct value produces similar but not identical signal models (Figure 1). Since transmit B₁ typically varies smoothly and slowly across in vivo images, we expect B₁f to do the same. Therefore we hypothesise that locally constraining pixels to have the same polarity of B₁f (i.e. either all above, or all below, the prescribed value) will yield more accurate parameter estimates.

Aims

Our purpose was to characterise this source of error and test a spatial regularization scheme to reduce parameter estimation bias. To clearly elucidate these effects we first investigated their importance, and the advantage of a spatial regularization approach in single component CPMG signal simulations, and then illustrated an application in real world muscle T₂ CPMG imaging data where the presence of fat in diseased tissue necessitates the use of a multi-T₂-component model.

Methods

Simulations were performed in MATLAB using a single component sEPG model, with Bloch equation simulated excitation and refocusing profiles using RF pulse shapes from our 3T scanner (Siemens Prisma). Simulated 17 echo sEPG signals with 10ms echo spacing and varying Rician noise levels were used to create synthetic images with varying B₁f for a ground truth T₂ of 30ms, typical of healthy muscle at 3T; fitting was performed using maximum likelihood estimation (MLE) (Figure 2) explicitly accounting for Rician noise. The MLE method seeks the parameter values yielding a global minimum in the log-likelihood. Due to noise contamination, in practice each pixel may return a log-likelihood surface with, with respect to B₁f, a single minimum close to the true value, or due to the near symmetry of the CPMG signal behaviour for B₁f values with equal positive or negative deviations from the nominal value, 2 local minima corresponding to these respective conditions. In the later case one of these local minima is identified as the global minimum, but random errors may in fact cause incorrect assignment, producing an error in the parameter estimates (Figure 3).
To mitigate this problem B₁f spatial regularisation was introduced for those pixels where two local minima with respect to B₁f were found. The B₁f polarity for a given pixel was fixed to a value corresponding to a majority vote (median filter) of the dominant B₁f polarity, calculated over a surrounding 20 pixel diameter disc for the simulations, and a 40 pixel diameter disc for the in vivo datasets. The diameter was determined empirically to avoid either over- or under-smoothing (Figure 4).
To investigate in vivo applicability the spatially regularized sEPG-MLE method was implemented with an extended multi-component model accounting for muscle-water and intra-muscular fat compartments⁴, and applied to in vivo human thigh-level CPMG data (22 echoes, other parameters as above) providing muscle water T₂ (T_2m), apparent fat fraction (ff_a), and B₁f estimates. For corroboration, maps of estimated B₁f where compared with dual angle method (DAM) B₁ field maps⁵.

Results

Applying the spatial regularization when fitting the simulated data yielded more accurate T₂ estimates (Figure 3) than the preliminary non regularized fit, in fact returning T₂ values identical to those obtained for a fit with B₁f polarities fixed to the a priori correct values. B₁f polarity maps were consistent with B₁ maps produced with the established DAM method (Figure 4).
On fitting the multi-component fat-water sEPG model to the in vivo data, similar B₁f polarity related fit instability was seen, producing biased parameter estimates (Figure 5). Consistent with simulations, the direction of this bias was dependent on the polarity of the B₁f error. Applying the B₁f polarity spatial regularization scheme to the in vivo data improved the parameter estimation, exemplified by the more physically consistent B₁f behaviour seen in the figure.

Conclusion

Addition of a B₁f polarity spatial regularisation step to the sEPG-MLE fitting approach resolves the B₁f polarity ambiguity issue and increases accuracy of the estimated parameters.