## Abstract

Introduction: In MR current density imaging (MRCDI) and MR electrical impedance tomography (MREIT) the current

density or conductivity is reconstructed from internal current-induced magnetic flux densities measured with MRI.

However, the current density and conductivity reconstruction is challenging due to low SNR, limited volume coverage,

and most importantly that only the component of the magnetic flux density parallel to the main field of the MR scanner

is measurable (Bz). The “projected current density” method [1] has been used in recent human in-vivo brain MRCDI

studies [2–5]. Comparing the results to simulated data we observed that the method only gives very coarse estimates of

the “true” current density.

Here we first analyze the accuracy of the projected current density algorithm when used to reconstruct currents in the

human head. Secondly, we propose to use an anatomically detailed head model and optimize the conductivities based on

the difference between simulated and measured magnetic fields. Parts of the work presented in this abstract have

previously been published in a journal article [6].

Methods: The projected current density algorithm attempts to reconstruct the current density from measured magnetic

flux density images Bz, and from a simulated current density J

0

and a magnetic flux density Bz

0 obtained from a model

with homogeneous conductivity. The equation derived from Ampère’s law is expressed as

𝑱

𝒓𝒆𝒄 = 𝑱𝟎 +

ଵ

ఓబ

ቂ

ఋ(ି

బ

)

ఋ௬

,

ିఋ(ି

బ

)

ఋ௫

, 0ቃ, [1]

where the directional derivatives of Bx and By are neglected since only Bz is measured in MRCDI. μ0 is the magnetic

permeability of free space. We analyzed the accuracy of the projected current density algorithm with simulated data using

the finite element method (FEM) implemented in SimNIBS

3.1.0 [7]. With a forward simulation using an anatomically

detailed head model, the current density was calculated and

used as the ground truth to evaluate the current density

reconstructed with the projected current density algorithm.

A simplified head model with no variation in the z-direction

(Fig. 1c) was also used to test the accuracy of the projected

current density algorithm for a simpler structure. For the

simplified head model, Jz as well as δBy/δz and δBx/δz

from the injected currents, are minimal, rendering all the

neglected terms in the projected current density method

insignificant.

Instead of reconstructing the current density from Bz, we

propose to compare the measured Bz with simulated Bz

obtained from a personalized head model with multiple

tissue types. The tissue conductivities are then estimated by

minimizing the difference between measured and simulated

Bz (fig 2). We scanned 5 subjects with two electrode

montages (right-left (RL) and anterior-posterior (AP))

using an acquisition-weighted multi-echo GRE acquisition

strategy [5]. The five variable tissue types used in the

optimization were white matter, gray matter, cortical CSF,

skull, and scalp. Ventricular CSF was kept constant to act

as an anchor point for the other conductivities since all

conductivities can be scaled with a common factor resulting

in the same current density and magnetic flux density. To

avoid overfitting of the conductivities to individual

subjects, we used leave-one-out cross-validation (LOOCV)

Figure 1: Evaluation of the accuracy of the projection current density

algorithm when used on simulated magnetic flux measurements from a

human head. a) Simulations based on the ernie head model (available as

example dataset in SimNIBS) showing the head model, tissue

conductivities, true current density, J

0 used in the algorithm and the

reconstructed current density. There is a remarkably clear difference

between true and reconstructed current density (R2=0.22). b) The true Bz

(simulated measurement), the neglected and used directional derivatives,

and the reconstructed current density, respectively. c) same figure as in a,

but with a simplified head model with no variation in the z-direction. The

similarity between true and reconstructed current density has greatly

improved (R2=0.63). d) The Bz and directional derivatives for the

simplified head model.

where four subjects were used for optimization and the error was evaluated

on the remaining subject with the obtained conductivities. Optimization was

performed both for RL and AP separately and combined. The error metric

was the relative root mean square difference between measurements and

simulations.

Results and Discussions: The results from the simulation of the projected

current density algorithm using a realistic head model and a simplified head

model with no variation in the z-direction are shown in fig. 1a,b and c,d,

respectively. The reconstructed current density for the simplified head model

(R2

= 0.63) outperforms the reconstruction for a realistic head model (R2

=

0.22). The reason for the poor performance with the realistic head model is

clear when visualizing the neglected terms as shown in fig. 1b and d. The

neglected terms are much stronger for the realistic head model than for the

simplified. Jz is additionally fully ignored by the projected current density

algorithm.

In fig. 2 a diagram of the proposed conductivity optimization is shown where

the difference between measured and simulated Bz is minimized. An example

optimization shows the improved similarity of measured and simulated Bz for

the RL montage. Less change is observed for the AP montage. This is also

apparent in fig. 3 where the RL errors are larger for all subjects while also

displaying a greater reduction of the error after optimization. The

improvements for the RL montage for all subjects when using LOOCV

indicates a systematic difference between head models and reality that needs

to be accounted for in simulations. We have here proposed to use conductivity

optimization to increase the robustness of the simulations. However, the

obtained conductivities are not necessarily accurate for the given tissue types

but do improve the current density simulations for a given electrode montage

that gives rise to the simulated Bz. This is further emphasized by the

difference in the tissue conductivities for the two electrode positions.

Conclusions: We have here shown that the projection current density

algorithm performs poorly for anatomically complex structures such as the

human head due to the neglected terms needed for an accurate current density

reconstruction. However, for simple structures with little variations in the zdirection, the algorithm provides reasonable results.

We have instead suggested using conductivity optimization of a personalized

head model to improve current density simulations for a given electrode

montage. Our results indicate a systematic difference between simulation and

reality for the RL electrode montage in all five subjects. This can be improved

using our proposed method

density or conductivity is reconstructed from internal current-induced magnetic flux densities measured with MRI.

However, the current density and conductivity reconstruction is challenging due to low SNR, limited volume coverage,

and most importantly that only the component of the magnetic flux density parallel to the main field of the MR scanner

is measurable (Bz). The “projected current density” method [1] has been used in recent human in-vivo brain MRCDI

studies [2–5]. Comparing the results to simulated data we observed that the method only gives very coarse estimates of

the “true” current density.

Here we first analyze the accuracy of the projected current density algorithm when used to reconstruct currents in the

human head. Secondly, we propose to use an anatomically detailed head model and optimize the conductivities based on

the difference between simulated and measured magnetic fields. Parts of the work presented in this abstract have

previously been published in a journal article [6].

Methods: The projected current density algorithm attempts to reconstruct the current density from measured magnetic

flux density images Bz, and from a simulated current density J

0

and a magnetic flux density Bz

0 obtained from a model

with homogeneous conductivity. The equation derived from Ampère’s law is expressed as

𝑱

𝒓𝒆𝒄 = 𝑱𝟎 +

ଵ

ఓబ

ቂ

ఋ(ି

బ

)

ఋ௬

,

ିఋ(ି

బ

)

ఋ௫

, 0ቃ, [1]

where the directional derivatives of Bx and By are neglected since only Bz is measured in MRCDI. μ0 is the magnetic

permeability of free space. We analyzed the accuracy of the projected current density algorithm with simulated data using

the finite element method (FEM) implemented in SimNIBS

3.1.0 [7]. With a forward simulation using an anatomically

detailed head model, the current density was calculated and

used as the ground truth to evaluate the current density

reconstructed with the projected current density algorithm.

A simplified head model with no variation in the z-direction

(Fig. 1c) was also used to test the accuracy of the projected

current density algorithm for a simpler structure. For the

simplified head model, Jz as well as δBy/δz and δBx/δz

from the injected currents, are minimal, rendering all the

neglected terms in the projected current density method

insignificant.

Instead of reconstructing the current density from Bz, we

propose to compare the measured Bz with simulated Bz

obtained from a personalized head model with multiple

tissue types. The tissue conductivities are then estimated by

minimizing the difference between measured and simulated

Bz (fig 2). We scanned 5 subjects with two electrode

montages (right-left (RL) and anterior-posterior (AP))

using an acquisition-weighted multi-echo GRE acquisition

strategy [5]. The five variable tissue types used in the

optimization were white matter, gray matter, cortical CSF,

skull, and scalp. Ventricular CSF was kept constant to act

as an anchor point for the other conductivities since all

conductivities can be scaled with a common factor resulting

in the same current density and magnetic flux density. To

avoid overfitting of the conductivities to individual

subjects, we used leave-one-out cross-validation (LOOCV)

Figure 1: Evaluation of the accuracy of the projection current density

algorithm when used on simulated magnetic flux measurements from a

human head. a) Simulations based on the ernie head model (available as

example dataset in SimNIBS) showing the head model, tissue

conductivities, true current density, J

0 used in the algorithm and the

reconstructed current density. There is a remarkably clear difference

between true and reconstructed current density (R2=0.22). b) The true Bz

(simulated measurement), the neglected and used directional derivatives,

and the reconstructed current density, respectively. c) same figure as in a,

but with a simplified head model with no variation in the z-direction. The

similarity between true and reconstructed current density has greatly

improved (R2=0.63). d) The Bz and directional derivatives for the

simplified head model.

where four subjects were used for optimization and the error was evaluated

on the remaining subject with the obtained conductivities. Optimization was

performed both for RL and AP separately and combined. The error metric

was the relative root mean square difference between measurements and

simulations.

Results and Discussions: The results from the simulation of the projected

current density algorithm using a realistic head model and a simplified head

model with no variation in the z-direction are shown in fig. 1a,b and c,d,

respectively. The reconstructed current density for the simplified head model

(R2

= 0.63) outperforms the reconstruction for a realistic head model (R2

=

0.22). The reason for the poor performance with the realistic head model is

clear when visualizing the neglected terms as shown in fig. 1b and d. The

neglected terms are much stronger for the realistic head model than for the

simplified. Jz is additionally fully ignored by the projected current density

algorithm.

In fig. 2 a diagram of the proposed conductivity optimization is shown where

the difference between measured and simulated Bz is minimized. An example

optimization shows the improved similarity of measured and simulated Bz for

the RL montage. Less change is observed for the AP montage. This is also

apparent in fig. 3 where the RL errors are larger for all subjects while also

displaying a greater reduction of the error after optimization. The

improvements for the RL montage for all subjects when using LOOCV

indicates a systematic difference between head models and reality that needs

to be accounted for in simulations. We have here proposed to use conductivity

optimization to increase the robustness of the simulations. However, the

obtained conductivities are not necessarily accurate for the given tissue types

but do improve the current density simulations for a given electrode montage

that gives rise to the simulated Bz. This is further emphasized by the

difference in the tissue conductivities for the two electrode positions.

Conclusions: We have here shown that the projection current density

algorithm performs poorly for anatomically complex structures such as the

human head due to the neglected terms needed for an accurate current density

reconstruction. However, for simple structures with little variations in the zdirection, the algorithm provides reasonable results.

We have instead suggested using conductivity optimization of a personalized

head model to improve current density simulations for a given electrode

montage. Our results indicate a systematic difference between simulation and

reality for the RL electrode montage in all five subjects. This can be improved

using our proposed method

Original language | English |
---|---|

Publication date | 2022 |

Number of pages | 2 |

Publication status | Published - 2022 |

Event | 2022 Joint Workshop on MR phase, magnetic susceptibility and electrical properties mapping - IMT School for Advanced Studies Lucca, Lucca, Italy Duration: 16 Oct 2022 → 19 Oct 2022 |

### Workshop

Workshop | 2022 Joint Workshop on MR phase, magnetic susceptibility and electrical properties mapping |
---|---|

Location | IMT School for Advanced Studies Lucca |

Country/Territory | Italy |

City | Lucca |

Period | 16/10/2022 → 19/10/2022 |