Luis J. Gomez

Numerical Analysis and Design of Single-Source Multicoil TMS for Deep and Focused Brain Stimulation

Transcranial magnetic stimulation (TMS) is a tool for noninvasive stimulation of neuronal tissue used for research in cognitive neuroscience and to treat neurological disorders. Many TMS applications call for large electric fields to be sharply focused on regions that often lie deep inside the brain. Unfortunately, the fields generated by present-day TMS coils diffuse and decay rapidly as they penetrate into the head. As a result, they tend to stimulate relatively large regions of tissue near the brain surface. Earlier studies suggested that a focused TMS excitation can be attained using multiple nonuniformly fed coils in a multichannel array. We propose a systematic, genetic algorithm-based technique for synthesizing multichannel arrays that minimize the volume of the excited region required to achieve a prescribed penetration depth and maintain realistic values for the input driving currents. Because multichannel arrays are costly to build, we also propose a method to convert the multichannel arrays into single-channel ones while minimally materially deteriorating performance. Numerical results show that the new multi- and single-channel arrays stimulate tissue 2.4 cm into the head while exciting 3.0 and 2.6 times less volume than conventional Figure-8 coils, respectively.

Uncertainty Quantification in Transcranial Magnetic Stimulation via High-Dimensional Model Representation

A computational framework for uncertainty quantification in transcranial magnetic stimulation (TMS) is presented. The framework leverages high-dimensional model representations (HDMRs), which approximate observables (i.e., quantities of interest such as electric (E) fields induced inside targeted cortical regions) via series of iteratively constructed component functions involving only the most significant random variables (i.e., parameters that characterize the uncertainty in a TMS setup such as the position and orientation of TMS coils, as well as the size, shape, and conductivity of the head tissue). The component functions of HDMR expansions are approximated via a multielement probabilistic collocation (ME-PC) method. While approximating each component function, a quasi-static finite-difference simulator is used to compute observables at integration/collocation points dictated by the ME-PC method. The proposed framework requires far fewer simulations than traditional Monte Carlo methods for providing highly accurate statistical information (e.g., the mean and standard deviation) about the observables. The efficiency and accuracy of the proposed framework are demonstrated via its application to the statistical characterization of E-fields generated by TMS inside cortical regions of an MRI-derived realistic head model. Numerical results show that while uncertainties in tissue conductivities have negligible effects on TMS operation, variations in coil position/orientation and brain size significantly affect the induced E-fields. Our numerical results have several implications for the use of TMS during depression therapy: 1) uncertainty in the coil position and orientation may reduce the response rates of patients; 2) practitioners should favor targets on the crest of a gyrus to obtain maximal stimulation; and 3) an increasing scalp-to-cortex distance reduces the magnitude of E-fields on the surface and inside the cortex.

Low-Frequency Stable Internally Combined Volume-Surface Integral Equation for High-Contrast Scatterers

Volume integral equations (VIEs) are commonly used to analyze scattering from inhomogeneous dielectric objects. Unfortunately, when VIEs are applied to high-contrast scatterers, their discretization results in ill-conditioned systems of equations. Oftentimes volume-surface integral equations (VSIEs) are used to eliminate this effect. However, when the scatterer’s mesh has elements that are much smaller than the wavelength, VSIEs become ill-conditioned, too. This letter introduces a new set of internally combined VSIEs (ICVSIEs) that exhibit neither of these ill-conditioning phenomena. Just like in previous VSIE methods, surface currents are used to artificially increase the effective permittivity of the background medium in which volume polarization currents radiate. To remove ill-conditioning due to electrical size, coupling between the surface and volume is accounted for by judiciously adding contributions due to “exterior” and “interior” surface currents. Numerical data obtained by analyzing time-harmonic TE scattering from various 2-D layered cylinders suggests that discretization of the new ICVSIE yields matrices that are unaffected by the scatterer’s maximum permittivity and electrical size.

Volume-Surface Combined Field Integral Equation for Plasma Scatterers

The discretization of standard differential and volume integral equations (VIEs) pertinent to the electromagnetic analysis of negative permittivity media often results in ill-conditioned and slowly convergent systems of equations. We propose the use of a volume-surface combined field integral equation (VSCFIE) consisting of coupled surface combined field integral equations and VIEs to resolve this problem. The system leverages equivalence principles to artificially change the sign of the effective permittivity of the medium in which the polarization currents radiate. Numerical results obtained by analyzing time-harmonic TEz scattering from various 2-D layered cylinders suggests that discretization of the proposed VSCFIE yields well-conditioned and rapidly convergent systems of equations.

Internally Combined Volume-Surface Integral Equation for a 3-D Electromagnetic Scattering Analysis of High-Contrast Media

An internally combined volume surface integral equation (ICVSIE) for analyzing three-dimensional (3-D) electromagnetic scattering from heterogeneous high-contrast (HC) dielectric objects is proposed. Iteratively solved (generalized) volume integral equations [(g)VIEs] often converge slowly when applied to the analysis of electromagnetic interactions involving dielectric objects with permittivities much higher than those of the surrounding medium [HC breakdown] or discretized using mesh elements much smaller than the wavelength [low-frequency (LF) breakdown]. The proposed ICVSIE hybridizes a VIE and a surface integral equation (SIE) in a manner that eliminates both breakdown phenomena. Just as in previous hybrid VIE–SIE approaches, the ICVSIE invokes Huygens surface equivalence principle to transform the original problem into coupled equivalent exterior and interior scenarios. In the exterior scenario, surface currents radiate in the original surrounding medium; in the interior problem, surface and volume polarization currents radiate in a medium with permittivity closer to that of the object. By judiciously combining integral equations pertinent to both scenarios, the ICVSIE system is assembled. Numerical results show that the proposed ICVSIE is both HC and LF stable when applied to the analysis of 3-D scattering problems.

Sensitivity of TMS-induced electric fields to the uncertainty in coil placement and brain anatomy

A computational framework for statistically characterizing electric (E-) fields generated during transcranial magnetic stimulation (TMS) is presented. The framework combines a high dimensional model representation (HDMR) technique with a quasi-static finite-difference (QSFD) simulator to obtain statistics of E-fields due to uncertainty in the TMS setup and patients brain anatomy. Application of the proposed computational framework shows that E-fields induced by TMS are highly sensitive to the position and orientation of TMS coils, as well as the size of patients brain.

A simulation of focal brain stimulation using metamaterial lenses

Transcranial magnetic stimulation (TMS) is a relatively recent technique for studying brain function and treating neurological disorders [1]. In TMS, one or more coils carrying time varying currents located near the scalp generate magnetic fields inside the brain that in turn induce electric fields and eddy-currents in the conductive brain tissue (Fig.1). Whenever a nerve fiber is aligned with the induced electric field, a current is produced in the axon, which in turn depolarizes its membrane [2]. Unfortunately, standard TMS coils stimulate large lateral regions of tissue near the scalp. To broaden the scope of current TMS applications, new applicators capable of delivering focused magnetic fields deep into the brain are called for [3].

The ICVSIE: A General Purpose Integral Equation Method for Bio-Electromagnetic Analysis

Objective: An internally combined volume surface integral equation (ICVSIE) for analyzing electromagnetic (EM) interactions with biological tissue and wide ranging diagnostic, therapeutic, and research applications, is proposed. Method: The ICVSIE is a system of integral equations in terms of volume and surface equivalent currents in biological tissue subject to fields produced by externally or internally positioned devices. The system is created by using equivalence principles and solved numerically; the resulting current values are used to evaluate scattered and total electric fields, specific absorption rates, and related quantities. Results: The validity, applicability, and efficiency of the ICVSIE are demonstrated by EM analysis of transcranial magnetic stimulation, magnetic resonance imaging, and neuromuscular electrical stimulation. Conclusion: Unlike previous integral equations, the ICVSIE is stable regardless of the electric permittivities of the tissue or frequency of operation, providing an application-agnostic computational framework for EM-biomedical analysis. Significance: Use of the general purpose and robust ICVSIE permits streamlining the development, deployment, and safety analysis of EM-biomedical technologies.

Internally Combined Volume-Surface Integral Equation for EM Analysis of Inhomogeneous Negative Permittivity Plasma Scatterers

A well-conditioned internally combined volume-surface integral equation (ICVSIE) for analyzing electromagnetic scattering from perfect electrically conducting (PEC) surfaces coated with negative permittivity plasmas is presented. Existing integral equations pertinent to these problems oftentimes yield ill-conditioned systems of equations. The ill-conditioning stems from the presence of interfaces between a positive permittivity background medium and the negative permittivity plasma and/or is due to low-frequency breakdown. To tackle these issues, the proposed ICVSIE wraps the negative permittivity plasma into a Huygens surface and exploits surface equivalence principles to artificially change the sign of the permittivity of the medium in which equivalent volumetric polarization currents radiate. Next, it combines a Muller combined field integral equation on the Huygens surface with a volume-surface integral equation for the negative permittivity plasma that coats the PEC surface in a manner that circumvents low-frequency breakdown. The accuracy, efficiency, and stability of the proposed ICVSIE are demonstrated via its application to various canonical scatterers and a plasma-engulfed reentry vehicle.

Compression of Translation Operator Tensors in FMM-FFT-Accelerated SIE Solvers via Tucker Decomposition

Tucker decompositions (also known as higher order singular value decompositions) are used to lessen the memory requirements of translation operator tensors in fast multipole method-fast Fourier transform accelerated surface integral equation solvers. For many practical examples, the proposed drop-in code enhancement results in over 90% reduction in these tensors’ storage requirements while imposing negligible computational overhead, thus significantly enhancing the solvers’ application range on fixed computational resources.