Noninvasive Electric Current Induction for Low-Frequency Tissue Conductivity Reconstruction: Is It Feasible With a TMS-MRI Setup?

Noninvasive quantification of subject-specific low-frequency brain tissue conductivity ( σLF ) will be valuable in different fields, for example, neuroscience. Magnetic resonance (MR)-electrical impedance tomography allows measurements of σLF . However, the required high level of direct current injection leads to an undesirable pain sensation. Following the same principles, but avoiding pain sensation, we evaluate the feasibility of inductively inducing currents using a transcranial magnetic stimulation (TMS) device and recording the magnetic field variations arising from the induced tissue eddy currents using a standard 3 T MR scanner. Using simulations, we characterize the strength of the incident TMS magnetic field arising from the current running in the TMS coil, the strength of the induced magnetic field arising from the induced currents in tissues by TMS pulses, and the MR phase accuracy required to measure this latter magnetic field containing information about σLF . Then, using TMS-MRI measurements, we evaluate the achievable phase accuracy for a typical TMS-MRI setup. From measurements and simulations, it is crucial to discriminate the incident from the induced magnetic field. The incident TMS magnetic field range is ±10-4 T, measurable with standard MR scanners. In contrast, the induced TMS magnetic field is much weaker (±10-8 T), leading to an MR phase contribution of ∼10-4 rad. This phase range is too small to be measured, as the phase accuracy for TMS-MRI experiments is ∼10-2 rads. Thus, although highly attractive, noninvasive measurements of the induced TMS magnetic field, and therefore estimations of σLF , are experimentally not feasible.

Introduction

Noninvasive mapping of tissue electrical properties in the megahertz range has recently become feasible with the development of magnetic resonance imaging (MRI)-based electrical property tomography (1–5). However, precise knowledge on tissue electrical conductivity at low frequency (LF: Hz–100 kHz) and the relation between electrical conduction and tissue composition in this frequency range is still limited. In the kilohertz range, the human body is electrically very heterogeneous (6–8) as cellular fraction, water-ionic content, and cell membranes modulate electrical conductivity. Unfortunately, pathologies change these factors, causing differences in tissue conductivity values () between healthy and nonhealthy subjects (9, 10). The ability to measure these subject-specific values of brain tissues is particularly a desired competence in neuroscience, as various diagnostic techniques and neurostimulation modalities like transcranial magnetic stimulation (TMS) operate in this frequency range (11–14).

TMS is an emerging technique that allows noninvasive modulation of cortical neurophysiology to diagnose and treat neurological disorders (15–20). Based on the Faraday induction principle, TMS uses a strong, time-varying magnetic field to inductively induce an electric field in the brain that can cause neuroactivation (Figure 1A) (21–24). Practically, TMS dosimetry is performed in a highly empirical fashion by using the “motor threshold” method (20, 23), where the motor cortex serves as reference area. However, because the electric field induced in the brain is modulated by the varying dielectric properties and the gyrification of the cortex (25, 26), the TMS dose varies for brain regions that are different from the motor cortex (27–30). Therefore, the motor threshold method is unreliable for most TMS purposes (31–33).

Figure 1.

A time-varying current (Icoil) running in a figure-8-shaped TMS coil creates a time-varying magnetic field (Bcoil) which, in turn, induces an electric field (Etissue) in brain tissues (A). Because of the conductive nature (σLF) of brain tissues, induced currents (Jtissue) arise from Etissue (B). These induced currents create an induced magnetic field (Btissue). Information on low-frequency (LF) tissue conductivity σLF is, therefore, imprinted only in Btissue and not in Bcoil.

To precisely guide TMS administration and to better understand the behavioral consequences of the deployed TMS electric field, different research groups are focusing their investigations on how stimulation parameters (number of TMS pulses, pulses' strength, coil models, and orientation) affect the induced TMS electric field by means of electromagnetic (EM) simulations (34–39). Although these valuable studies correctly adopt heterogeneous conductive brain models in the computation of the induced TMS electric field, the adopted conductivity values are simply derived from healthy group averages (40–42). Unfortunately, as argued in other studies (24, 30, 43–46), healthy group averages of cannot ensure optimal subject-specific dosimetry, as various factors such as ageing (47) and pathologies (10) induce variations in values. Moreover, because the induced electric field is also modulated by the tissue geometry, having subject-specific brain models would be valuable (32, 44, 48). Although this latter requirement can be satisfied by segmenting magnetic resonance (MR) images acquired before TMS administration, being able to noninvasively and nonpainfully determine subject-specific tissue values is still an unresolved issue.

LF tissue conductivity can be mapped using MR-electrical impedance tomography (MR-EIT) (49–51). In this technique, strong direct current (10 mA) is injected into the brain via skin surface electrodes (Figure 2A) while the subject is positioned in an MR scanner. The spatial pattern of these currents is modulated by the underlying tissue distribution. In turn, these injected currents lead to an induced magnetic field, in which information on is thus imprinted. By measuring this induced magnetic field using MR phase measurements, maps can be reconstructed (52, 53). However, strong currents and long injection times (10 milliseconds) are needed to achieve adequate MR phase accuracy in MR-EIT experiments. These requirements result in a sensation of pain that limits the in vivo applicability of MR-EIT.

Figure 2.

Current injection: the additional magnetic resonance (MR) phase contribution (ΦB_tissue) contains information on LF tissue conductivity (A). Current induction—quasi-static approximation: 2 additional MR phase contributions (B). ΦB_coil does not carry any information about σLF. Instead, ΦB_tissue contains information about σLF, as for the case of current injection.

To map tissue by avoiding direct current injection, using time-varying magnetic fields created by external coils to inductively induce currents has been suggested (Figure 1B and Figure 2B) (54). Subsequently, by following this inductive fashion, directly using the MR gradient coils to induce currents (55–57) has been suggested. Thus, high current density at injection points and, thus, pain sensation are avoided, making this approach very attractive and applicable to standard clinical MR scanners. However, tissue reconstructions were not feasible. In fact, the phase contribution arising from the induced magnetic field is too small to be accurately measured with standard MR systems (58, 59). In addition, it has also been shown that subtle, unavoidable imaging distortions hamper measurements of this phase contribution by creating a pseudo-LF conductivity contrast (60).

By following the appealing idea of inductively inducing currents in tissues, in this study, we use a TMS setup to induce much stronger currents (Figure 1A) in combination with an MR scanner used to measure the arising induced magnetic field (Figure 1B). Moreover, while standard MR gradient coils allow slew rates of 20 T/s at 10 cm from the gradients' isocenter, a TMS device can generate slew rates of up to 20 000 T/s. Thus, the reported 3 orders of magnitude increase that are needed to measure the induced magnetic field could be theoretically achieved (59). We have divided this study in 2 parts. First, using simulations, we characterize the strength of the induced magnetic field carrying information on the induced currents in the tissue and, thus, the LF conductivity and compare with that of the incident TMS magnetic field. This was evaluated for different conductive cylindrical models and realistic human brain models. Thus, we characterize the required phase accuracy to detect these induced magnetic fields with MRI. Moreover, we study the impact of different TMS waveforms. Second, MR measurements on phantoms are presented to investigate the achievable phase accuracy for a typical TMS-MRI setup. With this study, we investigate whether inductively inducing currents in tissues by using a TMS-MRI setup is a feasible methodology for performing noninvasive LF tissue conductivity reconstructions.

Theory

In TMS, the presence of conductive tissues such as the brain underneath the TMS coil leads to correction terms in the computation of the TMS EM field, which are a function of the tissue conductivity distribution (Figure 2B). In this kilohertz range, where displacement currents are negligible, these corrections can be modeled by the so-called quasi-static approximation (35). For this purpose, Maxwell equations are expanded in power series in the frequency domain ( and ), giving the following relations for a kth order (61): where a quasi-static condition is assumed in equation (2) (35, 62). For readability purposes, we do not explicitly write the spatial dependency (r) of the vector fields. In addition, the conductivity is also a tensor because of tissue anisotropy, but we can consider it as a scalar value for simplicity of derivation. From equation (3), the Biot–Savart law can be derived as follows:

In TMS, for k = 0, the zero-order vectors in the brain/object satisfy the static field equations , , , . is the incident TMS magnetic field arising from the current running in the TMS coil, thus not carrying any information about tissue conductivity. Throughout the paper, we will refer to this magnetic field as as follows: which gives an MR phase contributions defined as (63).

Higher-order field corrections of order k can be computed using the vectors of order k - 1 as sources (Figure 2B). For k = 1, the following equations are computed: where and are, respectively, the first-order electric field and current density induced in a conductive domain such as the brain, and is the first-order induced magnetic field arising from . Therefore, information on tissue conductivity is imprinted in .

The total induced electric field in brain tissues, called as throughout the paper, is:

In principle, is a solenoidal electric field induced by the time-varying incident TMS magnetic field [equation (6)]. However, because of the nonhomogeneous conductivity distribution of brain tissues, charge is accumulated at the boundaries between different conductive structures, leading to a conservative electric field that affects the incident, solenoidal electric field (25, 35, 48) as follows:

is proportional to the time-varying incident vector potential Acoil, which depends solely on the TMS coil configuration and level of current running into it . Thus, is always present, independently from the conductor underneath the TMS coil. Instead, , which arises from the charge accumulation at tissue boundaries between different conductive tissues (, with φ electrical potential), is directly modulated by the underlying tissue geometry and conductivity distribution (25, 35, 48). From equation (2), the total induced current density in tissue, called as , is, therefore, .

Analogous to equation (10), the total induced TMS magnetic field, called as , is: which gives an MR phase contributions defined as .

By combining equations (5) and (12), the total TMS magnetic field is, therefore, defined as follows:

Materials and Methods

Simulations

EM simulations aimed to characterize the strength of the incident and the induced TMS magnetic fields ( and , respectively) by using the quasi-static approximation described in the theory section. We then assessed the phase accuracy needed to detect in concurrent TMS-MRI experiments. In addition, we characterized the impact of different conductivity distributions of on .

Three simulations were performed in SCIRun (64), namely, 2 on conductive cylinders and 1 on a realistic human brain model. For the performed simulations, the TMS coil was modeled using 2 single-plane spiral wings (65), reflecting the geometry of the TMS coil used in the measurements. These wings were placed at 5 cm from the cylinders to mimic the actual position used in the measurements and in contact with the scalp to mimic the position in realistic TMS treatments.

In the first simulation, we characterized the strength of the z-component of the net (time average over the TMS pulse) incident TMS magnetic field , the only 1 component (parallel to the MR static magnetic field B0) measurable in an MR experiment. We also characterized the range of the net phase contribution that would arise from in an MR experiment. In an MR experiment, the phase contribution is proportional to the area underneath the TMS current waveform (63). The same phase contribution can be obtained by using the time average value of the TMS current waveform (see online supplemental Appendix A, ) computed from independent oscilloscope measurements. For this simulation, a typical bipolar TMS pulse that lasts for a full period was used (Figure 3A). The TMS output was set to 1%, leading to a . By applying the Biot–Savart law, the net was computed. This simulation was performed using a homogeneous conductive cylinder (Figure 4A) with the same geometry and electric conductivity as that of the phantom used in the measurements (Figure 4D). Thus, consistent comparison with measurements could be performed. However, for a bipolar TMS pulse that lasts for a full period, the net induced current in tissue is zero (see online supplemental Appendix A and Figure 3B) (66). Thus, obviously, the induced magnetic field and its related phase contribution are zero.

Figure 3.

Net transcranial magnetic stimulation (TMS)-coil current, net incident TMS magnetic field, and its related phase contribution for a full TMS pulse shape (A) and for a truncated TMS pulse shape (C). The pulse shape of the current running in the TMS coil was derived from oscilloscope measurements. Net time derivative of the TMS-coil current (proportional to the induced electric field), net induced TMS magnetic field, and its related phase contribution for a full TMS pulse shape (B) and for a truncated TMS pulse shape (D). Four concurrent TMS-magnetic resonance imaging (MRI) measurements (E).

Figure 4.

Cylindrical models used in simulations: one homogeneous (A) and one with 2 compartments (B). The 3-dimensional (3D) mesh was performed using Gmsh. Realistic human brain model and 3 sets of tissue conductivity used in simulations (C). Geometry and composition of the experimental phantoms (D). Conductivity values were confirmed by dielectric probe measurements (85070E Agilent Technology, Santa Clara, California).

Because information on is imprinted solely in , to induce a non-zero net , a truncated TMS waveform should be used (see online supplemental Appendix A and Figure 3, case 2). Consequently, in the second simulation, we used the same waveform adopted in the first simulation but truncated at the first quarter (63), TMS output 1%, and t1 = 0.1 ms, leading to and a rate of change of the coil current of 0.55 × 106 A, in line with other studies (36, 42). We characterized the strength of the net and for such a truncated TMS pulse. Then, the 3-dimensional mesh model, the conductivity distribution, and the vector potential (computed using the rate of change of the coil current) (35) were given as input to the finite element method (FEM) solver to compute Etissue and Jtissue. From Jtissue, the strength of and the range of were characterized. By performing this simulation on 2 different conductive cylinders (one homogeneous and one consisting of 2 different conducting compartments, (Figure 4, A and B) (67), the impact of different conductive compartments was evaluated.

Then, in the third simulation, we defined the strength of and for realistic in vivo situations by using a realistic human brain model (68) and the truncated TMS waveform adopted in the second simulation. We, therefore, explored the feasibility of measuring in vivo by characterizing the required phase accuracy for concurrent TMS-MRI experiments. In addition, we evaluated the impact of different distributions on Etissue (relevant quantity for TMS dosimetry) and . Finally, we characterized the phase accuracy needed to detect subtle variations in arising from these variations in . The phase accuracy determines the feasibility of this technique. The adopted values reflect the conductivity variations reported in other studies (Figure 4C) (35, 38, 42).

Measurements

Concurrent TMS-MRI measurements were conducted in a clinical 3 T MR scanner (Achieva, Philips Healthcare, Best, The Netherlands) with elliptical surface MR receive coils (flex-M) and using a standard TMS stimulator (Magstim Rapid2, Whitland, UK) with an MR-compatible figure-8-shaped TMS coil (28, 69). Using a typical TMS-MRI setup, the phase accuracy characterized by these measurements is representative.

The MR sequence adopted was a spin-echo sequence with the following parameters: relaxation time/echo time = 1000/50 milliseconds, field of view = 160 × 160 × 2.5 mm3, resolution = 2.5 × 2.5 × 2.5 mm3, voxel/bandwidth = 0.15/3 kHz, and number of signal averages = 2. The TMS device was synchronized with the MR sequence by using the MR-transistor–transistor logic signal delivered at every radiofrequency (RF) excitation as a reference time point. TMS pulses were delivered before each readout gradient (69). The surface of the TMS coil was placed at 4.5 cm from the phantoms. For each experiment, 4 measurements were performed to correctly isolate the phase contributions and (see online supplemental Appendix B and Figure 4D, measurement numbers M1–4). For the measurements with TMS-on, the TMS outputs were 1% and 4%, for the first and the second experiments, respectively. Because a standard TMS stimulator was used, only bipolar pulses that lasted for a full period could be used (Figure 3, case 1). For these measurements, we prepared 2 agar phantoms sturdy enough to prevent motion artifacts (60) as follows: 1 conductive (1.6 S/m) and 1 nonconductive, as a reference to compensate for (see online supplemental Appendix B, Figure 4D). The 2 phantoms were carefully placed at the same position in the scanner by using a dedicated phantom holder designed for this purpose.

With these experiments, we characterized the phase range of , which, in principle, includes both the contributions and , by using the conductive phantom (see online supplemental Appendix B). We also characterized the phase range of only by using the nonconductive phantom. This allowed direct comparison with the first simulation. We finally characterized the achievable MR phase accuracy (inverse of the signal-to-noise ratio [SNR]) (59) to enable measurements in concurrent TMS-MRI experiments for a realistic TMS-MRI setup.

Results

The impact of a realistic TMS pulse shape that lasts for a full period (Figure 3, case 1) on the TMS-related phase contribution, , is characterized by using the homogeneous cylinder (Figure 5A). The results of this first simulation are shown on the same plane where measurements were performed, thus mimicking the experimental setup and allowing direct comparison with the measurements. From simulations, is on the order of 10−5 T (Figure 5B), leading to a in the order of radians (Figure 5C). This result suggests that , and thus , can be measured in an MR experiment.

Figure 5.

Phantom simulation (full TMS waveform): the displayed maps are extracted from 3D simulations on the same plane where measurements were performed. Homogeneous cylindrical model (A). z-Component of Bcoil (B) and ΦB_coil maps (C), both independent from the sample conductivity.

Figure 6 shows the results from the second simulations performed on 2 conductive cylinders (one homogeneous and one with 2 different conductive compartments, Figure 6A and Figure 6F, respectively) and using the truncated TMS pulse waveform to induce a nonzero (Figure 3, case 2). From these simulations, we observe that the use of a truncated TMS waveform leads to an increase in (10−4 T) and, consequently, in (102 rads), with respect to the use of a full TMS waveform (Figure 5). In addition, by comparing the results obtained from the 2 different conductive cylinders, we observe that the incident magnetic field (Figure 6, B and G) and its related phase contribution (Figure 6, C and H) are not affected by the presence of different conductive compartments. This is because the incident magnetic field does not depend on the conductivity of the structure underneath the TMS coil. Instead, as shown by these simulations, the conductivity distribution modulates the induced magnetic field (Figure 6, D and I) and thus its related phase contribution (Figure 6, E and J). The impact of variations is clearly visible from the discrepancy between the histograms of the 2 maps (Figure 6K). However, it is important to note that the induced magnetic field (∼10−8 T) is about 4 orders of magnitude lower than the incident magnetic field (10−4 T). As shown in Figure 6, E and J, is in the range of ∼10−4 rads. This result characterizes the phase accuracy needed to measure in concurrent TMS-MRI experiments. Moreover, these results highlight the challenge of correctly disentangling the phase contributions arising from and , as the latter field constitutes a very small fraction of the former.

Figure 6.

Phantom simulations (truncated TMS waveform). Homogeneous (A) and 2-compartment cylindrical models (F). z-Component of Bcoil (B, G) and ΦB_coil maps (C, H), independent from the sample conductivity. z-Component of Btissue (D, I) and ΦB_tissue maps for the 2 cylindrical models (E, J). Histograms of Btissue for the 2 models (K). Clearly visible variation in Btissue patterns between the 2 models.

In the third simulations, the impact of inter-subject variations of on and and the range of and are characterized for a realistic human brain model (Figure 7 and Figure 8, respectively). For the adopted 3 different conductive brain models, the norm of Etissue is shown on the gray matter and white matter surfaces (Figure 7). Our results correspond with the results presented in a previous valuable study (35). By comparing the mean and standard deviation of the top 30% values of ∥Etissue∥ for the 3 models (calculated independently for each brain model in the gray matter and the white matter), we observe that different values induce significant variations in the deployed Etissue (Figure 7, bar-plots). This highlights the importance of accurately predicting subject-specific tissue conductivity values to correctly guide TMS dosimetry.

Figure 7.

Norms of Etissue in the gray matter (GM—top row) and white matter (WM—bottom row) for the 3 brain models. Mean and standard deviation of the top 30% values of the norm of Etissue for each brain model in GM and WM. It is visible how different σLF distributions lead to significant variations in the induced electric field.

Figure 8.

Realistic head model and cut plane used for visualization purposes (A). z-Component of Bcoil and ΦB_coil maps, both independent from σLF (B). For each model, the z-component of Btissue and ΦB_tissue is computed (C). Variations in the z-component of Btissue because of different conductivity distributions (D). These maps were computed by subtracting Btissue of model 3 and Btissue of model 1 (upper part), and by subtracting Btissue of model 3 and Btissue of model 2 (bottom part).

For each brain model (Figure 8A), and the related phase contribution are computed (Figure 8C). The induced magnetic field (10−8 T) is about 4 orders of magnitude lower than the incident magnetic field (10−4 T) (Figure 8B), in line with the value observed for the cylindrical structure (Figure 6). In addition, maps show slightly different patterns between the 3 different brain models because of the different conductivity distributions. This is a direct consequence of the previously observed variations in the Etissue maps. As shown in Figure 8D, variations in lead to magnetic field variations in the range of nanotesla. From these results (Figure 8C), we conclude that the necessary MR phase accuracy needed to measure for in vivo TMS-MRI experiments is about 10−4 rads. However, an even higher accuracy will be needed to actually detect variations in tissue conductivity distributions.

In Figures 9 and 10, the results of the 2 experiments performed by using a realistic TMS-MRI setup and a full TMS waveform are proposed. With these experiments, we assess the attainable MR phase accuracy for concurrent TMS-MRI experiments. In both experiments, 2 phase maps were acquired for each phantom (one conductive and one nonconductive): one with TMS-on (Figure 9 and Figure 10, A and C) and one with TMS-off (Figure 9 and Figure 10, B and D). The significant impact of different (RF phase contribution, ∼3 rads, see online supplemental Appendix B), which scales with the RF conductivity (megahertz), is visible in the measurements with TMS-off. This highlights the importance of correctly compensating for . By subtracting the phase maps measured with TMS-on and TMS-off [subtraction inside parentheses in equation (B.2); see online supplemental Appendix B], ΦTMS (Figure 9E and Figure 10E) and (Figure 9F and Figure 10F) are computed, respectively, for the conductive and the nonconductive phantoms. As shown in the plots (Figure 9G and Figure 10G), ΦTMS coincides with . From maps, the z-component of Bcoil is reconstructed (Figure 9H and Figure 10H). The range of the measured (radians) and (10−5 T) quantitatively reflects the values previously observed in simulations. Finally, maps can be, in principle, obtained by subtracting ΦTMS and maps [see online supplemental Appendix B, subtraction between parentheses in equation (B.2)] (Figure 9I and Figure 10I).

Figure 9.

Experiment 1: TMS = 1%. Phase maps with TMS-on for the conductive (A) and nonconductive (C) phantoms. Phase maps with TMS-off for the conductive (B) and nonconductive (D) phantoms. Reconstructed ΦTMS map for the conductive phantom (E). Reconstructed ΦB_coil map for the reference phantom (F). Comparison between ΦTMS and ΦB_coil profiles (G). Reconstructed Bcoil_z map (H). ΦB_tissue map (subtraction between ΦTMS and ΦB_coil) (I).

Figure 10.

Experiment 2: TMS = 4%. Phase maps with TMS-on for the conductive (A) and nonconductive (C) phantoms. Phase maps with TMS-off for the conductive (B) and nonconductive (D) phantoms. Reconstructed ΦTMS map for the conductive phantom (E). Reconstructed ΦB_coil map for the reference phantom (F). Comparison between ΦTMS and ΦB_coil profiles (G). Reconstructed Bcoil_z map (H). ΦB_tissue map (subtraction between ΦTMS and ΦB_coil) (I).

For the performed measurements, the actual MR phase accuracy for detection is estimated to be in the order of 10−2 rad, which is 2 orders higher than what is required from simulations. In addition, for the adopted full TMS waveform, we should observe that is zero, as the net induced Jtissue is zero. However, in maps, we can observe a certain pattern in the range of 0.1 rads. This pattern is caused by an imperfect compensation of while performing the subtraction between and (relative error ∼1%). Therefore, this result highlights that very high precision is required to correctly compensate for .

Finally, to evaluate whether a stronger net incident TMS magnetic field could be of benefit, we performed a second experiment with a TMS output of 4%. From our measurements (Figure 10), significant image corruption can be observed in the region underneath the TMS coil. This corruption arises from the intra-voxel dephasing created by the stronger incident, highly nonuniform TMS magnetic field Bcoil.

Discussion

Being able to measure subject-specific would be valuable for different fields of research such as oncology and neuroscience (11–14, 70). In MR-EIT, in vivo conductivity measurements require direct injection of eddy currents in tissue and measurements of their impact on the MR phase (49–51). In this study, we explored whether inductive generation of currents using an MR-compatible TMS setup could be a less painful alternative to MR-EIT. Such a setup is able to generate much stronger time-varying magnetic fields than switching MR gradient coils previously proposed in other studies (58–60). However, as shown by our analysis, 3 main challenges hamper measurements of the induced magnetic field arising from inductively induced currents in tissues.

First, for such an inductive technique to work, it is crucial to correctly disentangle the incident magnetic field from the induced magnetic field. This is because, only this latter field contains information on tissue . For this purpose, subtractions between different phase images are needed (see online supplemental Appendix B). The fundamental problem is that the induced magnetic field Btissue is about 10−4 lower that the incident magnetic field Bcoil. Therefore, very high precision and reproducibility is required to correctly disentangle the phase contribution arising from Btissue from the phase contribution arising from Bcoil.

Second, as demonstrated, information on is only imprinted in Btissue. Therefore, to reconstruct , the net Btissue has to be nonzero. As discussed in this work, this requirement is satisfied if a truncated TMS pulse is used. However, for standard TMS setups, only TMS pulses that last for a full period can be used; thus, the net Btissue is zero (see online supplemental Appendix A, Figure 3, case 1) (66). Therefore, an additional setup (pulse modulator) should be used to comply with this requirement.

Third, supposing that it would be possible to correctly isolate from all the other phase contributions, from our simulations, the phase range of is in the order of 10−4 rads. This phase range is about 2 orders of magnitude lower than the detectable phase in concurrent TMS-MRI experiments. In addition, to distinguish small variations in tissue conductivity, an even higher phase accuracy would be needed.

To bring into a measurable phase range, one should increase Btissue of at least 2 orders of magnitude. To strengthen Btissue, one can increase the strength of the induced current Jtissue by increasing the TMS pulse frequency and pulse strength.

By increasing the pulse frequency, that is, reducing the induction time (ti), stronger Jtissue and, therefore, Btissue can be achieved. In contrast, attention has to be paid to not enter into a different dispersion band. Brain tissues exhibit the beta dispersion centered between 105–107 Hz. Above this dispersion band (100 MHz), the cell membranes exhibit a negligible impedance, so currents are capable of passing through both the extracellular and intracellular media (71). To avoid conductivity reconstructions in a different frequency dispersion band, and thus not directly translatable to LF tissue conductivity, the TMS-pulse frequency (1–10 kHz) can be increased by an additional factor of 10 (ie, ti = 0.01 milliseconds). However, in an MRI experiment, the measured net is proportional to both and the induction time ti. Hence, in the computation of , the increase in is cancelled by the reduction in the induction time ti, leading to an unchanged range. For this reason, the increase of the TMS-pulse frequency would not be a beneficial solution.

As aforementioned, another strategy to increase the induced Jtissue can be to strengthen the TMS output. Btissue increases with a stronger TMS output, but Bcoil also increases. Unfortunately, as already observed in Figure 10, this leads to considerable signal dephasing. From our results, a truncated TMS pulse and much stronger TMS outputs should be used to bring above the noise level. To comply with such a requirement, one should measure a much stronger, highly nonuniform . Consequently, to avoid signal dephasing, one should therefore considerably reduce the voxel size (in the range of micrometers).

This latter observation brings us to a final consideration. A smaller voxel size comes quickly at the cost of SNR loss. Instead, to detect very small magnetic field fluctuations such as the one produced by Btissue, the SNR should be considerably increased. Thus, only the number of scan repetitions can be increased. However, in practice, unfeasible scan time would be required to achieve enough SNR for measurements.

As discussed, the unsuccessful ability to measure Btissue by inductively inducing currents in the brain using a combined TMS-MRI setup arises from the physical limitations behind the physics of the induction principle. On the contrary, by injecting currents in tissues (MR-EIT), direct measurements of Btissue and, consequently, reconstructions are feasible. The first macroscopic difference between the 2 techniques is that in MR-EIT, images subtractions between different conductive phantoms are not needed. This is because currents are directly injected into the brain, thus there is no incident magnetic field Bcoil. Second, for both techniques, is proportional to and to ti, time of injection/induction. Despite the comparable range (±10−8 T), the relevant difference in the time of injection/induction (10 milliseconds in MR-EIT and 0.1 milliseconds in TMS-MRI) leads to a measurable/nonmeasurable in MR-EIT and TMS-MRI, respectively (72).

These observations define the physical limitations hampering the feasibility of noninvasively measuring subject-specific . Hence, future studies should focus on alternative methodologies to noninvasively and nonpainfully measure Btissue for subject-specific reconstructions.

Conclusions

LF tissue conductivity reconstructions can only be performed by measuring the phase contribution arising from the induced magnetic field, in which information on is imprinted. However, despite stronger currents being inductively induced using a TMS stimulator compared with MR gradient coils, these measurements are not feasible with a standard TMS-MRI setup. This is because, the induced magnetic field is very weak; thus, very high SNR is required to correctly measure it. If a higher level of current running through the TMS coil is used to strengthen the induced currents in tissues and to increase the induced magnetic field, considerable image dephasing would be observed because of the strong, highly nonuniform incident TMS magnetic field. In light of our observations, we believe that direct reconstructions performed by inductively inducing currents in the brain are not feasible even if a TMS-MRI setup is used.