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Inhomogeneity-informed Field-fitting for Quantitative Susceptibility Mapping (if-QSM)

Alexandra Grace Roberts¹, Alexey Dimov², Thanh Nguyen², Pascal Spincemaille², and Yi Wang^2,3
¹Electrical and Computer Engineering, Cornell University, New York, NY, United States, ²Radiology, Weill Cornell Medicine, New York, NY, United States, ³Biomedical Engineering, Cornell University, New York, NY, United States

Synopsis

Keywords: Bioeffects & Magnetic Fields, Artifacts

Motivation: Inhomogeneity in measured multi-gradient echo (mGRE) field data corrupts reconstructions of quantitative susceptibility maps by obscuring the tissue field of interest with strong background field.

Goal(s): To extend the voxel spread function (VSF) library implementation to a nonzero phase offset and demonstrate improvements on QSM.

Approach: The measured field was estimated and inhomogeneity contributions computed using the extended library implementation. The inhomogeneity field was then estimated and subtracted from the total field to reduce the influence of strong background fields in the QSM reconstruction

Results: Inhomogeneity-informed field-fitting for QSM is shown to improve total field reconstructions of the brain, carotid, and cervical spine.

Impact: The voxel spread function (VSF) library implementation is extended to include initial phase offset contributions and reduce the effect of field inhomogeneity in quantitative susceptibility map (QSM) reconstruction.

Introduction

The ill-posed problem of dipole deconvolution used in quantitative susceptibility mapping (QSMs)¹ relies on a robust estimate of the field map. Dipole-incompatible errors in the estimated field map generate nonlocal artifacts in QSM due to the wave propagator solution to the dipole deconvolution.² One source of such errors is field inhomogeneity, induced by the anatomical interfaces where susceptibility changes abruptly (e.g., air-tissue), and causing non-local image distortions. The effects of this distortion are typically ignored in in the background removal methods^3-12In the present work, we propose the extension of the voxel spread function (VSF) correction method to address this shortcoming. We demonstrated the improvement in QSM reconstruction using use QSM Challenge 2.0 data.¹³ Additionally, performance of this method in is assessed in cervical spine and carotid QSM reconstructions.

Theory

The measured mGRE signal $$$S_n(TE)$$$ can be decomposed as a product of the true signal $$$\sigma_n(TE)$$$ and field inhomogeneity contribution $$$F_n(TE)$$$, such that $$$S_n(TE)=\sigma_n(TE)F_n(TE)$$$.¹⁴ Here$$F_n(TE)=\frac{1}{|S_n(0)|}\sum_m \Psi_{nm}(TE)|S_m(0)|$$ Where $$$n$$$ and $$$m$$$ are the neighbor indices, is the complex mGRE signal at voxel $$$S_n(TE)$$$ and echo time $$$TE$$$, $$$|S_k(0)|$$$ is the signal magnitude at voxel $$$k$$$, $$$\Psi_{nm}$$$ is the voxel-wise kernel and phase of signal $$$S_m(TE)$$$ at neighboring voxel $$$m$$$. From this, inhomogeneity correction factor $$$F_n(TE)$$$ can be computed to correct field inhomogeneity as described in the original voxel spread function (VSF) implementation¹⁴ and the accelerated library implementation.¹⁵ The voxel-wise kernel is a function of the VSF $$$\eta(n,m;q_m(TE))$$$, which describes the effect of phase dispersion on the measured signal: $$\Psi_{nm}(TE)=\eta(n,m;q_m(TE))e^{i\phi_{0,m}+i\gamma b_m TE}$$ Here, $$$q_m(TE)$$$ is the phase dispersion, $$$\phi_{0,m}$$$ is the phase offset, $$$\gamma$$$ is the gyromagnetic ration and $$$b_m$$$ is the field map. Previously, this method has been shown to improve $$$R_2^*$$$ maps. However, the correction factor $$$F_n(TE)$$$ is complex and can therefore also be used to correct the signal phase, $$$\angle S_n(TE)$$$. The original library implementation¹⁵ consisting of matching each voxel of the field gradient with a precomputed kernel value to accelerate the calculation of $$$\eta(n,m;q_m(TE)$$$, is hereby extended by allowing $$$\phi_0 \neq 0$$$.

Methods

Complex multi-gradient echo (mGRE) data was simulated via dipole convolution with the QSM challenge 2.0 data¹³ in the presence of decay and complex Gaussian noise ($$$SNR=50$$$) at $$$7T$$$. To avoid introducing further phase wrapping from complex division of the measured signal by the inhomogeneity factor,^14,15 the phase correction was applied after fitting both the measured field and the field inhomogeneity maps. The simulated data was processed using the nonlinear field fitting algorithm¹⁶followed by ROMEO¹⁷ phase unwrapping and VSF correction. ROMEO phase unwrapping and field fitting was applied for both the field map estimation and the VSF estimation for the in vivo brain and spine and carotid. The corrected field map and VSF were computed iteratively for $$$K$$$ ($$$1$$$ for brain, $$$10$$$ for carotid, $$$15$$$ for brain and spine) iterations. The original and VSF-corrected field maps were used in nonlinear preconditioned total field inversion (npTFI)¹⁸ with preconditioner $$$P=20$$$ and $$$\lambda=100$$$. The cervical spine images were acquired using 3D mGRE at $$$3T$$$ with $$$6$$$ echo times and repetition time $$$TR=38ms$$$, flip angle $$$15°$$$, matrix size $$$256\times256\times72$$$ and voxel size $$$0.9375\times0.9375\times2mm^3$$$. The carotid images were acquired using 3D mGRE¹⁹ with $$$4$$$ echo times and $$$TR=20ms$$$, first echo time $$$TE_1=\Delta TE=4.7ms$$$, voxel size $$$0.63\times0.63\times2mm^3$$$, flip angle $$$30°$$$, scan time = $$$4$$$ minutes.

Results

The phase offset extension reduces the mean-squared phase error between the original implementation and library implementation by several orders of magnitude ($$$0.0024$$$ and $$$0.2866$$$, respectively) as shown in Figure 1. The inhomogeneity correction in the nonlinear field fitting reduces ground truth error ($$$0.43$$$ versus $$$0.42$$$) and the presence of nonlocal artifacts in the simulation (Figure 2). This method extends to improve inhomogeneity artifacts in the carotid (Figure 3) and cervical spine (Figure 4).

Conclusion

The proposed phase offset extension of the VSF library implementation reduces artifacts across multiple field strengths and improves accuracy of reconstructed QSM.

Acknowledgements

No acknowledgement found.

References

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7. Roberts A, Spincemaille P, Nguyen T, Wang Y. MEDI-FM: Field Map Error Guided Regularization for Shadow Reduction in Quantitative Susceptibility Mapping. presented at: International Society for Magnetic Resonance in Medicine; 2022; London, England. https://archive.ismrm.org/2022/2359.html 8. Roberts A, Spincemaille P, Nguyen T, Wang Y. Whole Brain Spherical Mean Value Filtering for Shadow Reduction in Quantitative Susceptibility Mapping. Paper No. 2172. presented at: International Society for Magnetic Resonance in Medicine; 2023; Toronto, Canada.

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Figures

Figure 1. Original VSF implementation (left), phase offset extension (middle) and original library implementation (right).

Figure 2. Ground truth (left) uncorrected (middle) and corrected (right) brain reconstruction.

Figure 3. Uncorrected (left) and corrected (right) carotid reconstruction.

Figure 4. Uncorrected (left) and corrected (right) joint brain and spine reconstruction.

Proc. Intl. Soc. Mag. Reson. Med. 32 (2024)

4897

DOI: https://doi.org/10.58530/2024/4897