Synopsis

The convection-reaction partial differential equation (PDE) based algorithm for Magnetic Resonance Electrical Properties Tomograpy, cr-MREPT, does not suffer from internal boundary artifacts but has the Low Convective Field (LCF) artifact. The cr-MREPT PDE is rearranged such that H₁⁺ is the unknown variable and the coefficients depend on the EPs. The EPs are iteratively adjusted to minimize the difference between calculated and measured H₁⁺. The method can be applied to a Region-of-Interest without considering the whole object. Conductivity reconstructions for simulation objects and also for an agar phantom demonstrate that the proposed method does not suffer from boundary and LCF artifacts.

Introduction

Methods

The ROI is discretized by a regular Cartesian mesh. $$$U$$$ and $$$H$$$ are the vectors of nodal impedance and $$$H_1^+$$$ values respectively. PDE-4 is discretized at each interior node to obtain the set of linear algebraic equations $$$AH=0$$$, and one can solve for the interior values $$$H_{int}$$$, given the boundary values $$$H_{boun}$$$, as explained in Figure-1. Difference between this calculated $$$H_{int}^{calc}$$$ and measured $$$H_{int}^{m}$$$ is minimized iteratively to obtain $$$U$$$. Figure-1 defines the minimization problem where $$$\lambda$$$ is Tikhonov regularization parameter. At each iteration the sensitivity matrix $$$S$$$ (Jacobian) is calculated to approximate $$$H_{int}$$$ around the previously found EP distribution $$$U_0$$$ which is taken as uniform for the first iteration.

For std-MREPT $$$\gamma=\frac{\triangledown^2H_1^+}{i\omega\mu_0H_1^+}$$$ is used². For cr-MREPT, PDE-2 is regularized by adding an artificial diffusion term^3,4 ,discretized, and solved for $$$U$$$ in one step without iterations. 3D cylindrical simulation phantom placed in a birdcage RF coil is shown in Figure-2. For 2D simulations the object is placed in a uniform applied rotating RF magnetic field. Normalized RMS errors, $$$NRMSE_{\sigma}=\frac{\parallel \sigma_{calc}-\sigma_{real}\parallel_2}{\parallel \sigma_{real}\parallel_2}$$$ and $$$NRMSE_{H_{int}}=\frac{\parallel H_{int}^{calc}-H_{int}^{m}\parallel_2}{\parallel H_{int}^{m}\parallel_2}$$$ are used for performance assessment. Experiments are conducted with an agar phantom with the same dimensions and EP properties as the 3D simulation phantom. bSSFP and Bloch-Siegert sequences are used for $$$H_1^+$$$ phase and magnitude mapping respectively.

Results

Figure-2 shows for the 3D and 2D simulation phantoms that using the simulated $$$H_1^+$$$ data on the ROI boundary as a Dirichlet BC and the actual EPs in the ROI, the $$$H_1^+$$$ obtained in the ROI by solving PDE-4 is very similar to the simulated $$$H_1^+$$$ except for some minor (less than 0.5%) differences on the sharply varying anomaly boundaries (The ROI was selected to cover the central slice $$$\pm$$$5 slices for 3D application). These observations are in line with the uniqueness theorem proved by Ammari et al⁵.

Figure-3 shows the conductivity reconstructions for the 2D simulation phantom. std-MREPT has the notorious internal boundary artifacts. cr-MREPT has no boundary artifacts but the LCF-artifact¹ appears when Gaussian noise is added (SNR = 300) to the simulated $$$H_1^+$$$. With Inv-crMREPT after the 6th iteration $$$NRMSE_{H_{int}}$$$ does not decrease further whereas $$$NRMSE_{\sigma}$$$ still improves. With additive noise, a divergent behavior starts for $$$NRMSE_{\sigma}$$$ after the 6th iteration. In any case, Inv-crMREPT does not suffer from boundary and LCF artifacts. Each iteration of 2D Inv-crMREPT takes 45 seconds for a ROI with 4709 nodes (pixel size of 1.5mm).

Central slice reconstructions for the 3D simulation phantom and the experimental phantom are shown in Figures 4 and 5, respectively. In both cases, Inv-crMREPT does not have boundary and LCF artifacts. However some inaccuracies observed in conductivity values may be due to the fact that we have used the 2D version of PDE-4 for reconstruction.

Discussion

Conclusion

The “inverse problem approach” proposed in this study and the resulting iterative implementation, called Inv-crMREPT, provides conductivity reconstructions which are free from boundary and LCF artifacts. 3D implementation of the method is necessary for more accuracy, and is under way.

Acknowledgements

References

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