Synopsis

We validate the use of non-linear signal phase evolution to estimate the susceptibility and orientation of sub-voxel structures. Capillary tubes filled with ferritin solutions of varying susceptibility were positioned in a water phantom. Multi-echo gradient-echo scans (100 echoes, echo-spacing=0.6ms) were acquired for multiple orientations of the capillaries with respect to the magnetic field. Fitting a 2-compartment model to the complex signal, processed using frequency difference mapping (FDM), resulted in susceptibility and signal fraction estimates in good agreement with theoretical predictions. Non-linear phase evolution, processed using FDM, may provide new insight into microvascular structure and physiology.

Introduction

Recently we reported that non-linear signal phase evolution can be used to characterise the physiology and structure of the cerebrovasculature¹. In this work we introduce a phantom model to better understand the complex multi-echo gradient-echo (MEGE) signal in the presence of multiple signal compartments at the sub-voxel level. We compare the results of fitting a two-compartment model to magnitude and phase data preprocessed to remove background field effects. Estimating susceptibility values and compartment volume fractions, we ascertain the feasibility and limitations of the technique for quantification of blood oxygenation and volume in-vivo.

Phantom design

Borosilicate glass capillary tubes ($$$\chi=-11\,\mathrm{ppm}$$$) with diameter d=1mm or 1.5mm, and wall thickness: 0.01-0.02mm, were filled with aqueous ferritin solution (iron concentrations 1mg/ml and 2mg/ml, respectively). The tubes were glued vertically onto a platform (6-8mm separation), and placed in a spherical shell filled with water (Fig.1A).

Signal acquisition and processing

Low-resolution ($$$4\times4\times4\,\mathrm{mm}^3$$$) single-slice bipolar MEGE data were acquired using a 7T Philips Achieva scanner (100 echoes; TE₁=1.1ms; ΔTE=0.6ms; TR=100ms; flip-angle $$$10^\circ$$$; 5 averages).

By rotating the phantom in the coronal plane we achieved 14 different orientations of the capillary tubes ranging from $$$0^\circ-90^\circ$$$ with respect to the magnetic field $$$\vec{B}_0$$$, (Fig.1C). The MEGE image slice was re-positioned to orthogonally transect the capillaries. Capillary orientations were estimated from a high-resolution ($$$0.75\times0.75\times0.75\,\mathrm{mm}^3$$$) gradient-echo scan (TE=30ms).

Complex signals acquired with opposite read-gradient polarities were multiplied together to remove phase inconsistencies between adjacent echoes². Frequency difference mapping (FDM) was applied to remove the non-local phase effects without corrupting the local phase evolution:

$$\mathrm{FDM}_n=\arg\left(\frac{S^+_n\cdot{S^-_n}\cdot\left[S^+_1\cdot{S^-_1}\right]^{n-2}}{\left[S^+_2\cdot{S^-_2}\right]^{n-1}}\right)\frac{1}{4\pi(n-2)\Delta\mathrm{TE}},\qquad[1]$$

where $$$S_n$$$ is the complex signal acquired at the nth echo-time $$$\mathrm{TE}_n=\mathrm{TE}_1+(n-1)\Delta\mathrm{TE}$$$, and $$$\pm$$$ denotes the read-gradient polarity. FDM was previously applied on unipolar MEGE data³; this is the first application to bipolar MEGE data, which enables closer echo spacing.

Finally, FDM and magnitude data were averaged over two ROIs containing the capillary tubes (Fig.1B), and magnitude data were normalised to the signal measured at TE₁.

Signal Modelling

We fitted a 2-compartment model of the local signal evolution, $$$F(\mathrm{TE}_n)$$$:

$$F(\mathrm{TE}_n)=(1-A)\cdot{e^{-R^*_{2,\text{water}}\mathrm{TE}_n}}+A\cdot{e^{i\omega\mathrm{TE}_n}}e^{-R^*_{2,\text{ferritin}}\mathrm{TE}_n},\qquad[2]$$

where $$$A$$$ is the ferritin solution signal fraction, $$$\omega$$$ is the orientation-dependent frequency offset between the water and ferritin compartments, and $$$R^*_{2,\text{water/ferritin}}$$$ represents the corresponding relaxation rates.

Results

Fig.2 shows values of $$$A$$$ and $$$\omega$$$ obtained by fitting Eq.[2] to FDM (Fig.2A.-B.) and magnitude data (Fig.2C.-D.). Dotted lines indicate the maximum volume fraction of a capillary within a voxel ($$$\pi(d/2)^2/4^2$$$). Solid lines show the predicted frequency offset $$$\omega(\theta)$$$ calculated using the susceptibility, $$$\Delta\chi$$$, (determined by the iron concentration⁴) of the ferritin solution:

$$\omega(\theta)=\gamma{B_0}\frac{\Delta\chi}{2}\cdot\left(\cos^2\theta-\frac{1}{3}\right),\qquad[3]$$

where $$$\theta$$$ denotes the orientation of the capillaries with respect to $$$\vec{B}_0$$$. Fig.3 maps the variation of $$$\omega$$$ as a function of orientation $$$\theta$$$ for the two groups of capillaries, obtained from voxelwise fitting of FDM data.

Discussions

The frequency offset of the capillary tubes, $$$\omega(\theta)$$$, obtained from FDM data (Fig.2B) shows extremely good agreement with the predicted evolution. This demonstrates that FDM data are very sensitive to the susceptibility difference between the two compartments and the orientation of the capillaries at the sub-voxel level.

In contrast, the magnitude data show insensitivity to the sign of $$$\omega(\theta)$$$ (red circles, $$$\theta>54.7^\circ$$$, Fig.2D), and the fitting procedure failed for the capillaries with higher iron concentration at $$$\theta>40^\circ$$$ (blue crosses, Fig.2D). We interpret these results to mean that the signal decay of the surrounding water can no longer be described as a single exponential for high iron concentrations and orientations closer to perpendicular. In these instances, our two-compartment model fails in the magnitude data. However, FDM data are less sensitive to relaxation times, and characterisation of $$$\omega$$$ is still achievable.

At low flip-angles, $$$A$$$ can be interpreted as the partial volume fraction of the capillaries. We observe larger signal fraction $$$A$$$ in larger capillaries. Small variations of $$$A$$$ with $$$\theta$$$ likely originate from variations in image slice positioning. We expect the maximum possible volume fractions (dotted lines, Fig2A-C) to overestimate the true signal fraction within our ROIs, which likely include edge voxels with small or no contribution from capillaries. However, the ratio of the signal fraction estimated in the two ROIs using the FDM data agrees well with prediction: $$$A_\text{small}/A_\text{large}=0.4545$$$, $$$(d_\text{small}/d_\text{large})^2=0.4444$$$.

Conclusion

Analysis of the evolution of phase and provides better estimation of sub-voxel frequency offsets than magnitude data. By applying FDM to bipolar MEGE data, we achieved shorter echo spacing, better sampling of any fast-decaying signal compartments. The phantom’s lower-concentration ferritin solution has comparable susceptibility to venous blood ($$$\Delta\chi\approx1\,\mathrm{ppm}$$$); changes in $$$\Delta\chi_\text{venous}$$$ reflect oxygenation status, an important neurophysiological biomarker. The biggest advantage of our method is that we can access the oxygenation and volume of vessels with known orientation, or investigate coherent venous orientation at the microscopic level, without requiring high-resolution scans that resolve individual microvessels.

Acknowledgements

References

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