Fast evaluation of the Biot-Savart integral using FFT for electrical conductivity imaging

doi:10.1016/j.jcp.2020.109408

Journal of Computational Physics

Volume 411, 15 June 2020, 109408

https://doi.org/10.1016/j.jcp.2020.109408 Get rights and content

Highlights

•
Detailed implementation of a computationally efficient numerical algorithm for the evaluation of the Biot-Savart integral based on FFT.
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Fast and efficient forward computation of magnetic resonance electrical impedance tomography (MREIT) and current density imaging (MRCDI).
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Opening the door to the development of new conductivity reconstruction methods.
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Providing a fast and open-source package for modeling of low frequency exogenic bioelectromagnetics by integrating the algorithm in SimNIBS.

Abstract

Magnetic resonance electrical impedance tomography (MREIT) and current density imaging (MRCDI) are emerging as new methods to non-invasively assess the electric conductivity of and current density distributions within biological tissue in vivo. The accurate and fast computation of magnetic fields caused by low frequency electrical currents is a central component of the development, evaluation and application of reconstruction methods that underlie the estimations of the conductivity and current density, respectively, from the measured MR data. However, methods for performing these computations have not been well established in the literature. In the current work, we describe a fast and efficient technique to evaluate the Biot-Savart integral based on the fast Fourier transform (FFT), and evaluate its convergence. We show that the method can calculate magnetic fields in realistic human head models in one minute on a standard computer, while keeping error below 2%.

Introduction

Novel magnetic resonance (MR) methods have been leveraging the high sensitivity of MR to inhomogeneities in the main magnetic field of the scanner to image the electrical impedance of biological tissue at low frequency (MR electric impedance tomography- MREIT) [1], [2] or to assess current density distributions inside the tissue (MR current density imaging - MRCDI) [3], [4]. In these applications, an external current source produces a static or quasi-static imaging current inside a body part, and the MR scan is optimized to be sensitive to the magnetic fields produced by these currents. Due to the nature of MR imaging, typically only a single component of the magnetic field can be measured. Therefore, inverse methods are needed to reconstruct the electrical impedance (MREIT) or the current flow (MRCDI) inside the volume conductor from the measurements [5].

The inverse methods typically require the calculation of magnetic fields caused by the externally applied imaging current (the forward problem) [3], [6], [7], [8], [9]. This can be done by direct evaluation of the Biot-Savart integral [6], [10], [11], solving a Poisson's equation for magnetic field [10], [11], or using the fast Fourier transform (FFT) to compute the Biot-Savart integral [11], [12], [13]. This last method has been shown to be the fastest among the three [11]. Other methods such as Fast multipole method (FMM) can also be applied to efficiently compute the Biot-Savart integral [14], but can be difficult to implement and is generally slower than FFT [15]. However, to the best of our knowledge, the numerical accuracy of the FFT method for the Biot-Savart integral calculations in MREIT and MRCDI forward calculation applications has not been evaluated in detail so far.

In the current work, we describe an FFT-based method for calculating the Biot-Savart integral and assess its speed and accuracy using both phantoms with analytical solutions and a realistic head model. We show that the method performs well in both conditions, being able to produce accurate solutions in reasonable time, and is therefore a good candidate to compute the magnetic field in forward simulation of MREIT and MRCDI. The method will be integrated in the free and open source software SimNIBS [16], which is currently capable of simulating electrical potential, electrical field, and current density field inside biological tissues, turning it into an integrated tool for MREIT and MRCDI forward calculations.

Section snippets

Low frequency current-injected bioelectromagnetic phenomena

Assuming the human body to be an ohmic conductor with electrical conductivity $σ (x)$ , the electric potential $u (x)$ caused by an externally injected low frequency current is governed by the Laplace equation with the Neumann boundary conditions: $\begin{matrix} \nabla \cdot (σ (x) \nabla u (x)) & = 0 in Ω, \\ - σ \nabla u \cdot n & = g on \partial Ω, \end{matrix}$ where the physical body corresponds to a bounded domain $Ω \subset R^{3}$ with a smooth boundary ∂Ω, x is a position vector in $R^{3}$ , g is the normal current flux through the boundary, and n is the outward unit normal vector on ∂Ω. A unique

Numerical experiments

In this section we present some numerical examples for the evaluation of the developed technique. In the first examples we have an analytic expression of the fields J and B, and in the final example we used a current density field calculated on a realistic head model using SimNIBS.

Benchmark tests

Fig. 2 shows slices of the $B_{3}$ field calculated analytically and numerically, as well as the errors in the smooth phantom, while Fig. 3 shows the same quantities for the piecewise smooth phantom. We have chosen to display $B_{3}$ because some MREIT and MRCDI methods are only sensitive to a single component of the magnetic field, usually denoted $B_{z}$ or $B_{3}$ [25], [26], [4]. Fig. 2a and Fig. 3a depict the analytical $B_{3}$ corresponding to smooth and piecewise smooth phantoms, respectively. As can be seen, by

Conclusion

In this paper, we developed a computationally efficient technique for evaluation of the Biot-Savart integral. This integral has the form of a convolution, and thus lends itself well to an efficient numerical evaluation by means of the FFT algorithm. We performed three numerical experiments to evaluate the technique, using two phantoms with known analytical solutions and in addition a realistic head model. The results show a good convergence in all three experiments, suggesting that the method

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was supported by the Lundbeck foundation (R244-2017-196 and R186-2015-2138), the Novonordisk foundation (NNF14OC0011413) and the Danish Council for Independent Research | Natural Sciences (grant number 4002-00123). The work was initiated while Hassan Yazdanian was a visiting PhD student at the Department of Applied Mathematics and Computer Sciences, Technical University of Denmark.

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View full text

Fast evaluation of the Biot-Savart integral using FFT for electrical conductivity imaging

Highlights

Abstract

Introduction

Section snippets

Low frequency current-injected bioelectromagnetic phenomena

Numerical experiments

Benchmark tests

Conclusion

Declaration of Competing Interest

Acknowledgements

NeuroImage

NeuroImage

J. Comput. Phys.

J. Comput. Phys.

NeuroImage

NeuroImage

Electrical tissue property imaging at low frequency using MREIT

IEEE Trans. Biomed. Eng.

Fast magnetic resonance electrical impedance tomography with highly undersampled data

SIAM J. Imaging Sci.

Current density imaging during transcranial direct current stimulation using DT-MRI and MREIT: algorithm development and numerical simulations

IEEE Trans. Biomed. Eng.

Analysis of recoverable current from one component of magnetic flux density in MREIT and MRCDI

Phys. Med. Biol.

Algebraic reconstruction for 3D magnetic resonance–electrical impedance tomography (MREIT) using one component of magnetic flux density

Physiol. Meas.

Low-frequency conductivity tensor imaging of the human head in vivo using DT-MREIT: first study

IEEE Trans. Med. Imaging

Numerical simulations of MREIT conductivity imaging for brain tumor detection

Comput. Math. Methods Med.

Software toolbox for low-frequency conductivity and current density imaging using MRI

IEEE Trans. Biomed. Eng.

Three-dimensional forward solver and its performance analysis for magnetic resonance electrical impedance tomography (MREIT) using recessed electrodes

Phys. Med. Biol.