Mitigating the impact of flip angle and orientation dependence in single compartment R2* estimates via 2-pool modeling

2.1 Simulating the sensitivity of $$$ {\mathrm{R}}_2^{\ast } $$$ to multiple compartments

Net SPGR signals were simulated with a TR of 19.50 ms and different flip angles (α = [6 9 12 15 19 26 31 36 42]°) and for a range of MWF between 0.02 and 0.20 with a step size of 0.02 representing different tissue conditions (encompassing WM at MWF_WM = 0.16 and gray matter [GM] at MWF_GM = 0.06).³⁰ Different exchange regimes were investigated by varying a directional (MW to IE) residency time between 100 and 500 ms in steps of 100 ms. A single compartment $$$ {\mathrm{R}}_2^{\ast } $$$ was computed, with log-linear fitting across TE = (2.56:2.38:14.46) ms of S_net(t), for every simulated tissue condition (i.e., each MWF and residency time combination). Simulation parameters are summarized in Supporting Information Table S1. Noisy simulated signals were generated by computing 10 000 instantiations of complex random noise added to the simulated signal with SNR (TE = 0) of 50 to investigate the variance of the $$$ {\mathrm{R}}_2^{\ast } $$$ estimate.

To investigate the effects of compartment-specific frequency offsets as described by the hollow cylinder fiber model,^{20, 21} we carried out additional simulations taking into account different fiber orientations (θ) with respect to B₀, ranging from 0° to 90° (with 10° intervals), for each of the 9 flip angles.

2.2 Linear model describing $$$ {\mathrm{R}}_2^{\ast } $$$ dependence on α

Finally, the sensitivity of $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ and $$$ \frac{d{R}_2^{\ast }}{d\alpha} $$$ to different tissue conditions was investigated and summarized as the maximum variation over the mean of $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ and $$$ \frac{d{R}_2^{\ast }}{d\alpha} $$$, respectively (Figure 3).

2.3 Acquisitions

2.4 Data analysis

All images were analyzed using SPM12 (http://www.fil.ion.ucl.ac.uk/spm/software/spm12/, Wellcome Centre for Human Neuroimaging, London, UK).

The T₁-weighted images, with α = 26°, for each participant and each session were co-registered to session 1 and segmented. Participant and session specific WM and GM masks were defined by those voxels with a probability of belonging to the respective tissue class >0.9. A single participant-specific WM or GM mask was obtained by combining the WM or GM masks across sessions via logical conjunction. To ensure equivalent processing, each $$$ {\mathrm{R}}_2^{\ast } $$$, $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$, and $$$ \frac{d{R}_2^{\ast }}{d\alpha} $$$ map was co-registered to the T₁-weighted image acquired with α = 26° in session 1 for each participant.

Finally, the per-α $$$ {\mathrm{R}}_2^{\ast } $$$, $$$ {\mathrm{R}}_2^{\ast } $$$ from ESTATICS, $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ and $$$ \frac{d{R}_2^{\ast }}{d\alpha} $$$ dependence on fiber orientation with respect to B₀ was analyzed using data from participant 1 from imaging sessions 2 and 3 (acquired at 7T) and are shown in Figure 7.

The $$$ {\mathrm{R}}_2^{\ast } $$$ maps for each scanning session were first resliced into the diffusion space (accounting for differences in FOV positioning and angulation). Subsequently, an affine transformation between the diffusion data and the maps was determined via co-registration to account for any head rotation between the different scanning sessions. The resulting transformation was applied to the primary fiber orientation directions ($$$ {v}_x $$$, $$$ {v}_y $$$, $$$ \mathrm{and}\ {v}_z $$$) in diffusion space and a new angle map, which represents the angle between the fiber orientation and B₀ at each head position (different scanning sessions), was computed with Equation (7).

$$$ {\mathrm{R}}_2^{\ast }(r) $$$, $$$ \hat{{\mathrm{R}}_2^{\ast }}(r) $$$, $$$ \frac{d{R}_2^{\ast }}{d\alpha}(r) $$$, and $$$ \uptheta (r) $$$ measurements were extracted in WM voxels with a WM probability >0.9 and a fractional anisotropy >0.6. The fiber angles $$$ \uptheta (r) $$$ were segregated into bins containing 200 voxels to have a sufficient number for calculation of reliable summary statistics. For each bin, mean $$$ {\mathrm{R}}_2^{\ast }(r) $$$, $$$ \hat{{\mathrm{R}}_2^{\ast }}(r) $$$, and $$$ \frac{d{R}_2^{\ast }}{d\alpha}(r) $$$ were calculated and plotted against $$$ \uptheta (r) $$$. The function $$$ {\mathrm{R}}_2^{\ast}\left(\uptheta \right)={R}_{2,\mathrm{Iso}}^{\ast }+{R}_{2,\mathrm{Aniso}}^{\ast }\ {\sin}^4\left(\uptheta \right) $$$, predicted by the hollow cylinder fiber model,^{20, 43} was fit to the data to extract the isotropic component of $$$ {\mathrm{R}}_2^{\ast } $$$ ($$$ {R}_{2,\mathrm{Iso}}^{\ast } $$$) and the proportional θ-dependence via $$$ {R}_{2,\mathrm{Aniso}}^{\ast }/{R}_{2,\mathrm{Iso}}^{\ast } $$$. Per single-α $$$ {R}_2^{\ast } $$$ was computed for α = 26° whereas ESTATICS, $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$, and $$$ \frac{d{R}_2^{\ast }}{d\alpha} $$$ were computed for the α pair = [6, 26]°.

The single α and α-pairs considered in the analysis were: 6°, 26°, 42° (for $$$ {\mathrm{R}}_2^{\ast } $$$ estimates), and [6, 26]°, [9, 42]°, and [9, 26]° (pairs suitable for R₁ mapping) for the estimates obtained with ESTATICS and $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$.

Reproducibility of $$$ {R}_{2,\mathrm{Iso}}^{\ast } $$$ and $$$ {R}_{2,\mathrm{Aniso}}^{\ast }/{R}_{2,\mathrm{Iso}}^{\ast } $$$ across 2 sessions (session 2 and 3) was measured as bias ± CI. Bias and 95% CI were respectively quantified as the mean and standard deviation (scaled by 1.96) across α or α-pairs of the difference of estimates between sessions.

3.1 Simulation results

Simulating $$$ {\mathrm{R}}_2^{\ast } $$$ for 9 different α and for tissues properties spanning MWF = 0.02:0.02:0.20 and residency time = 100:100:500 ms revealed an increase in $$$ {\mathrm{R}}_2^{\ast } $$$ as a function of α for every tissue condition, as shown in Figure 1A for 4 representative flip angles. The linear model shown in Equation (2) was used to fit the data. Good agreement was observed between the simulated data and the linear model fit, as evidenced by model errors <1 s⁻¹ (Figure 1B).

Simulations investigating the effects of the compartment-specific frequency offsets showed that the linear model (Equation [2]) approximated the data well for every simulated fiber orientation. Supporting Information Figure S2 shows the $$$ {\mathrm{R}}_2^{\ast } $$$ computed for each θ and α via log-linear fitting across TE (solid lines with diamonds), $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ computed via linear fitting pooling all flip angles for a given θ (turquoise solid line with circles) and $$$ {\mathrm{R}}_2^{\ast } $$$ estimated following application of the proposed linear model of the flip angle dependence (dashed lines). Good agreement was observed between the simulated data and the linear model fit for every flip angle (solid and dashed lines).

Simulations probing the dependence of the flip angle dependent $$$ \left(\frac{d{R}_2^{\ast }}{d\alpha}\right) $$$ and flip angle independent ($$$ \hat{{\mathrm{R}}_2^{\ast }} $$$) components of $$$ {\mathrm{R}}_2^{\ast } $$$ on MWF and residency times were repeated for θ = 90°. These results are shown in Supporting Information Figure S3, and are in keeping with those of Figure 1, which depicts the equivalent results for θ = 0°. $$$ {\mathrm{R}}_2^{\ast } $$$ increased with MWF and residency time. Furthermore, the proposed linear model continues to fit the data well, evidenced by residuals <2 Hz. $$$ {\mathrm{R}}_2^{\ast } $$$ also increased with θ, an effect that was accentuated at higher flip angles (Supporting Information Figure S2), whereas $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ had a greatly reduced dependence on θ.

RMSE and RMS percentage error of $$$ {\mathrm{R}}_2^{\ast } $$$ were computed for each α and collapsed across tissue conditions (Figure 2). This was done for 5 different α sets, containing variable numbers of flip angles, and used to estimate the linear model parameters. In the noise-free case, RMSE <0.6 s⁻¹ and RMS percentage error <1.5% were observed for each α set. The variability of the error, across tissue conditions, increased as the number of α used to fit the linear model decreased. The same trend was observed when noise was added to the simulations. In this case, RMSE increased and ranged from 2.1 to 3.2 s⁻¹ going from α set 1 (9 flip angles) to α set 5 (only 2 flip angles), respectively, whereas RMS percentage error increased to 6.2% and 9.1%, respectively. In agreement with simulation, the error in the experimental case for participant 1 (Figure 2B) increased as the number of flip angles included in the computation decreased. Consistent results were obtained for participant 2.

The dependence of the linear model coefficients $$$ \Big(\hat{{\mathrm{R}}_2^{\ast }} $$$ and $$$ \frac{d{R}_2^{\ast }}{d\alpha}\ \Big) $$$ on different tissue conditions is shown in Figure 3. The α-independent component, $$$ \hat{{\mathrm{R}}_2^{\ast }}, $$$ showed a high sensitivity of 12.6% to MWF as it ranged from 0.02 to 0.20. However, it was effectively independent of the residency time (0.74% maximal sensitivity). $$$ \frac{d{R}_2^{\ast }}{d\alpha} $$$ depended on both MWF and residency time. It had a maximal sensitivity of 13% to residency time, and a larger maximal sensitivity of 55% to MWF. The latter dependence was approximately quadratic.

3.2 In vivo results

$$$ {\mathrm{R}}_2^{\ast } $$$ maps were computed for 9 nominal α. Four representative $$$ {\mathrm{R}}_2^{\ast } $$$ maps, obtained with α = [6, 9, 26, 42]°, are shown in Figure 4A. An increase in $$$ {\mathrm{R}}_2^{\ast } $$$ is visually apparent with increasing flip angle, most notably in the corpus callosum (zoomed view). $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ and $$$ \frac{\mathrm{d}{\mathrm{R}}_2^{\ast }}{\mathrm{d}\upalpha} $$$ components obtained by fitting the linear model in Equation (4) are shown in Figure 4B. Consistent results were obtained for all the participants.

Figure 4C shows the mean measured $$$ {\mathrm{R}}_2^{\ast } $$$ in WM (red solid line), GM (blue solid line), and WM and GM combined (green solid line) obtained across the 9 nominal α. The dashed lines (with colors indicating tissue type as before) show the corresponding mean $$$ {\mathrm{R}}_2^{\ast } $$$ obtained by fitting the linear model (Equation [4]) in WM, GM, and WM + GM. The mean model residuals were <1 s⁻¹ across all α, showing good agreement with simulation results.

Figure 5 shows various $$$ {\mathrm{R}}_2^{\ast } $$$ maps computed using the multi-echo data obtained with α = 6°, 26° and 42°. These were either fit individually (blue), and subsequently used to derive the α-independent component ($$$ \hat{{\mathrm{R}}_2^{\ast }} $$$) of the linear model (green) using two α, or the data from two α were combined at the point of fitting $$$ {\mathrm{R}}_2^{\ast } $$$ using the ESTATICS approach (red). Histograms of values were consistent for $$$ \hat{\ {\mathrm{R}}_2^{\ast }} $$$, but variable for $$$ {\mathrm{R}}_2^{\ast } $$$, particularly in WM, when estimated on a per-α basis instead of using ESTATICS (Figure 5A). The reproducibility analysis, for each of the 3 participants, revealed $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ to be most robust to α variation in both WM and GM (Table 1, median COV of 0.98% and 1.42%, respectively), whereas the per-α $$$ {\mathrm{R}}_2^{\ast } $$$ estimates were least robust regardless of tissue type (Table 1, median COV of 9.61% and 11.98%, respectively). The robustness of the ESTATICS approach was intermediate (Table 1, median COV of 5.98% and 6.14%, respectively).

Bland–Altman analyses of inter-session repeatability (participant 1, sessions 2 and 3) are shown in Figure 6, with biases ± CI summarized in Supporting Information Table S3. $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ showed the smallest bias in WM, $$$ {\mathrm{R}}_2^{\ast } $$$ ESTATICS had the smallest bias in GM, whereas per-α $$$ {\mathrm{R}}_2^{\ast } $$$ showed the highest biases in both GM and WM. However, the CI was largest for $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ indicating poorest cross-session repeatability.

The $$$ {\mathrm{R}}_2^{\ast } $$$, $$$ {\mathrm{R}}_2^{\ast } $$$ ESTATICS, $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$, and $$$ \frac{d{R}_2^{\ast }}{d\alpha} $$$ dependence on WM fiber orientation with respect to B_0, (θ), is shown in Figure 7 for data acquired in sessions 2 and 3 (orientation data from session 4). The result of fitting the $$$ {\sin}^4\left(\uptheta \right) $$$ dependence predicted by the hollow cylinder fiber model is inset. $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ (green) had lower θ-dependence than $$$ {\mathrm{R}}_2^{\ast } $$$ estimated with a single α (α = 26°, blue) or the ESTATICS approach (red), with the ratio $$$ {R}_{2,\mathrm{Aniso}}^{\ast }/{\mathrm{R}}_{2,\mathrm{Iso}}^{\ast } $$$ for (session 2, session 3) being (0.133, 0.181), (0.223, 0.215), and (0.198, 0.205), respectively. $$$ \frac{d{R}_2^{\ast }}{d\alpha} $$$ showed the highest orientation dependence with $$$ {R}_{2,\mathrm{Aniso}}^{\ast }/{\mathrm{R}}_{2,\mathrm{Iso}}^{\ast } $$$ of 0.5815 and 0.264 in sessions 2 and 3, respectively. This observation agrees with simulations where the $$$ {\mathrm{R}}_2^{\ast } $$$ θ-dependence is greatly reduced in $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ and therefore must propagate into the $$$ \frac{d{R}_2^{\ast }}{d\alpha} $$$ component. $$$ {\mathrm{R}}_2^{\ast } $$$ depends not only on orientation, but also on the spatially varying microstructural composition of the tissue. Despite this, the dependence on fiber orientation predicted by the hollow cylinder fiber model is apparent in the data—as is the fact that this dependence increases with flip angle (Supporting Information Figure S4). Good agreement, in terms of pattern and effect size, was found between simulations (Supporting Information Figure S2) and in vivo results of $$$ {R}_2^{\ast } $$$ and $$$ \hat{R_2^{\ast }} $$$ dependence on θ (Supporting Information Figure S4).

Figure  8 summarizes the$$$ \uptheta $$$θ-dependence of the $$$ {\mathrm{R}}_2^{\ast } $$$ estimates across 2 sessions. $$$ {R}_{2,\mathrm{Iso}}^{\ast } $$$ was most robust, across both α-pairs and sessions, when derived from $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ (Figure  8A ) (Table  2 , COV _α  = 0.74%, Inter-session bias ± CI = −0.006 ± 0.980). $$$ \hat{{\mathrm{R}}_2^{\ast }} $$$ was also least θ-dependent (lowest $$$ {R}_{2,\mathrm{Aniso}}^{\ast }/{\mathrm{R}}_{2,\mathrm{Iso}}^{\ast } $$$) (Figure 8B ) and the most consistent as α varied (COV _α  = 3.88% versus 5.66% for ESTATICS and 14.70% for $$$ {R}_2^{\ast } $$$ per-α). However, the anisotropic component derived from $$$ \hat{R_2^{\ast }} $$$ had the lowest cross-session reproducibility (inter-session bias ± CI = 0.041 ± 0.010).

4.1 Limitations

A highly simplified tissue model with only two compartments, myelin and non-myelin water, was used to simulate the SPGR signal. The non-myelin water compartments merged the intra- and extra-cellular water such that compartment-specific frequency offsets caused by the myelin sheath were also neglected in the simulations and only characterized experimentally via the dependence on WM fiber orientation that results. Dephasing because of other macro- or microscopic field inhomogeneities was also ignored. Yet, even in this comparatively simple model, multiple fixed parameters had to be assumed, specifically the longitudinal and transverse relaxation rates of both the myelin and non-myelin water compartments.

MT effects are prominent in the human brain,⁴⁸ particularly in WM, but have not been included in these simulations. Although this is in keeping with previous works,^{16, 49, 50} further insights may be delivered by extending the EPG-X framework to account for this effect by modeling 3 exchanging pools. In addition, it has recently been shown that the excitation can be modified to minimize the effect empirically.⁵¹ Despite this confound, high levels of agreement were observed between our empirical measurements and our predictions based on two pool modeling.

It has been shown that combining micro-structural information, such as fiber orientation from diffusion tensor imaging, with multi-compartment relaxometry modeling can improve the description of the net signal and permit estimation of the MWF.¹⁶ The gains in model accuracy and robustness come at the cost of longer acquisition times, which are required to collect the data necessary to reduce the degrees of freedom of the model and avoid overfitting. Here, we instead aimed to characterize the impact that the true multi-compartment nature of tissue has on simpler single compartment $$$ {\mathrm{R}}_2^{\ast } $$$ estimates, which are typical of neuroscientific studies that sacrifice model complexity to maintain feasible acquisition times.