arXiv subjects

Sampsa Pursiainen

Publications and source records attributed to Sampsa Pursiainen.

At least 19 recordsLinked to original sources

arXiv2026

In Silico Study for Optimizing Intensity and Focality Electrode Configurations for Directional DBS Under Uncertainty Using Metaheuristic L1L1 Method

Background and Objective: As Deep Brain Stimulation (DBS) advances toward directional leads and optimization-based current steering, selecting electrode contact configurations becomes complex. This study formulates configuration selection as an inverse mapping between target activation and electrode currents using metaheuristic L1-norm regularized L1-norm fitting (L1L1). L1L1 incorporates lead-field uncertainty arising from electrode placement, tissue properties, and forward modeling assumptions. Methods: The framework introduces lead-field perturbations and restricts the controllable domain through a sensitivity-based feasibility criterion within a finite element formulation derived using the Complete Electrode Model. Current distributions were optimized for 8- and 40-contact leads. Performance was evaluated using focused current density, nuisance current density, and their ratio under safety and sparsity constraints. Results: L1L1 was evaluated using noiseless and noisy lead fields, with noise selected to reflect attenuation within the volume of tissue activated. The method produced sparse, spatially selective stimulation patterns across perturbation levels. Hyperparameter optimization yielded bipolar or multipolar configurations. Compared with the Reciprocity Principle (RP), which produced strictly bipolar configurations, and Tikhonov-regularized least squares (TLS), which produced more distributed solutions, L1L1 enabled controlled transitions between sparse and multipolar patterns. It concentrated stimulation within the target while limiting unintended current spread, particularly under noisy conditions. Conclusions: L1L1 can assist specialists in optimizing DBS configurations. By incorporating uncertainty directly into optimization, it provides robust and interpretable current steering across lead configurations while accounting for forward-model variability.

math.OC↗

arXiv2026

Advanced EEG Source Models from the Perspective of FEM and Inverse Solutions

In this study, we compare forward solutions computed with finite element methods in Zeffiro Interface and DUNEuro, using the different source models they provided. We compared two of the source models from DUNEuro, called Whitney basis and Local subtraction, and the divergence-conforming model of Zeffiro Interface. For source estimation, we applied sparsity-promoting standardized hierarchical adaptive L1 regression (SHAL1R), standardized Kalman filtering (SKF), classical sLORETA, and dipole scanning. Analyses include Earth Mover's Distance, depth bias scatter plots, and qualitative assessments of amplitude distribution and focality. Preliminary experiments with source interpolation for each method revealed that Local subtraction closely matches expectations for the local behavior of the lead field at various depths. The main results reveal that the success of an inverse method depends strongly on the compatibility between its assumptions about the focality of the source and the chosen source model, with point-source models performing best when paired with methods designed for such sources, i.e., sparsity-promoting methods and methods that scan for a single source. Moreover, source models that admit patch sources with a wide distribution are more sensitive to additional noise.

math.NA↗

arXiv2026

Forward--Inverse Interplay in FEM-Based EEG Source Imaging: Distributional Signatures of Advanced Source Models and Inverse Solvers

Electroencephalography (EEG) source imaging aims to infer brain activity from electrical potentials measured on the scalp. This is a difficult problem because many different source patterns can explain the same measurements. The result depends strongly on two things: the forward model and the inverse method. In this work, we study how these two parts work together. We focus not only on where the activity is located, but also on how the reconstructed activity is distributed in space. We suggest that different source models create different signatures in the reconstructed activity. We use realistic head models and compute forward solutions with the finite element method using Zeffiro Interface and DUNEuro. We test different source models, including 2 implementations of a divergence-conforming model, and one implementation of Local subtraction approach. For inverse methods, we use advanced methods such as standardized hierarchical adaptive L1 regression (SHAL1R), standardized Kalman filtering (SKF), and classical dipole scanning. To understand the complex interplay between the forward and inverse approaches, we analyze the inverse source localization results using distributional quantitative measures, including Earth Mover's Distance and depth bias scatter plot, and qualitatively assess the amplitude distribution and focality. The results show that there is a strong dependence between the choice of source model and the success rate of a given inverse method: a source model that corresponds well with a single point-like source is a good match with an inverse method that presupposes such a source.

math.NA↗

arXiv2026

A Complete-Electrode-Model-Based Forward Approach for Transcranial Temporal Interference Stimulation with Linearization: A Numerical Simulation Study

Background and Objective: Transcranial temporal interference stimulation (tTIS) is a promising non-invasive brain stimulation technique in which interference between electrical current fields extends the possibilities of electrical brain stimulation. The objective of this study is to develop an efficient mathematical tTIS forward modelling scheme that allows for realistic and adaptable simulation and can be updated accurately when the contact resistance is modified in one or more electrodes. Such a model is vital, for example, in optimization processes that seek the best possible stimulation currents to exhibit or inhibit a given brain region. This study aims to establish and evaluate the complete electrode model (CEM), i.e., a set of boundary conditions incorporating electrode impedance and contact patch, as a forward finite-element-method-based simulation technique for tTIS and investigate linearized CEM as a surrogate. Results: The CEM-based forward simulation successfully reproduced the volumetric stimulating fields induced by tTIS. Sensitivity analysis showed that variations in electrode resistance affects the field distribution, especially in regions where the interfering currents have nearly equal amplitudes. The linearized CEM model closely matched the full nonlinear model within a predefined peak signal-to-noise ratio (PSNR) threshold for relative error. Both models exhibited the highest sensitivity near the focal region.

math.NA↗

arXiv2026

Multi-Compartment Volume Conductor with Complete Electrode Model: Simulated Stereo-EEG Source Localization using Brainstorm-Zeffiro Plugin

This study introduces a novel integration of the Brainstorm (BST) software and the Zeffiro Interface (ZI) to enable whole-head, multi-compartment volume conductor modeling for electroencephalography (EEG) source imaging, with a particular focus on stereotactic EEG applications. We present the BST-2-ZI plugin, a MATLAB-based tool that facilitates seamless transfer of tissue segmentations and anatomical atlases from BST into ZI for finite element (FE) mesh generation as well as forward and inverse modeling. The generated FE meshes support variable spatial resolution and implement the complete electrode model (CEM), allowing for precise modeling of both invasive depth electrodes and non-invasive scalp electrodes. Using the ICBM152 template and synthetic source simulation, we demonstrate the end-to-end pipeline from MRI data to lead field (LF) computation and source localization in a stereotactic EEG (stereo-EEG) setting. Our numerical experiments highlight the capability of the pipeline to accurately model multi-compartment head geometry and conductivity with a stereotactic CEM-based electrode configuration. Our preliminary source localization results show how a synthetic stereo-EEG probe corresponding to a bidirectional deep brain stimulation (DBS) probe with four omnidirectional contacts can, in principle, be coupled with scalp electrodes to improve source localization in its vicinity.

physics.med-ph↗

arXiv2025

Tracking EEG Thalamic and Cortical Focal Brain Activity using Standardized Kalman Filtering with Kinematics Modeling

Kalman filtering has proven to be effective for estimating brain activity using EEG recordings. In particular, the introduced post hoc standardization step of the algorithm, inspired by the sLORETA time-invariant method, reduces the depth bias and thus allows the estimation to appear at the correct depth from the electrode surface. In the current work, we propose first and second-order kinematic evolution models, where the state-space vector includes not only the dipolar source activity but also its velocity and acceleration. Compared to our previous study, this motion model yields smoother and more physically plausible estimates of brain activity even when the measurement noise is high, for both superficial and deep sources. In addition, we introduce a tunable power parameter that enhances the computational efficiency of the algorithm. Our simulation study, which involves thalamic and cortical activity in the somatosensory region, demonstrates that accurate estimation and tracking of both superficial and deep brain activity are feasible.

math.NA↗

arXiv2025

The Effect of Prior Parameters on Standardized Kalman Filter-Based EEG Source Localization

EEG Source localization is a critical tool in neuroscience, with applications ranging from epilepsy diagnosis to cognitive research. It involves solving an ill-posed inverse problem that lacks a unique solution unless constrained by prior knowledge. The Bayesian framework enables the incorporation of such knowledge, typically encoded through prior models. Various algorithms have been proposed for source localization, and they differ significantly in how prior knowledge is incorporated. Some approaches rely on anatomical or functional constraints, while others use statistical distributions or sampling-based techniques. In this landscape, the Standardized Kalman Filter (SKF) represents a dynamic Bayesian approach that integrates temporal modeling with a Gaussian prior structure. It addresses the depth bias, a common limitation in source localization, through a post-hoc standardization step that equalizes sensitivity across cortical depths and makes deep activity detection feasible. This study focuses on the development and optimization of Gaussian prior models within the SKF framework for simultaneous cortical and sub-cortical activity detection. Synthetic data similar to the P20 / N20 component of the somatosensory evoked potentials (SEP) was used to identify effective prior parameter configurations for reconstructing both deep and superficial sources under different noise levels. We also investigated the role of RTS smoothing in enhancing source separability. Our results indicate that raising the standardization exponent to 1.25, along with smoothing, significantly improves depth localization accuracy at low noise levels.

math.NA↗

arXiv2025

The Local Subtraction Approach For EEG and MEG Forward Modeling

EDIT: A revised version of this article has been published in the SIAM Journal on Scientific Computing, see https://epubs.siam.org/doi/full/10.1137/23M1582874. In the revised version, the name of the approach was changed from "localized subtraction" to "local subtraction". In FEM-based EEG and MEG source analysis, the subtraction approach has been proposed to simulate sensor measurements generated by neural activity. While this approach possesses a rigorous foundation and produces accurate results, its major downside is that it is computationally prohibitively expensive in practical applications. To overcome this, we developed a new approach, called the local subtraction approach. This approach is designed to preserve the mathematical foundation of the subtraction approach, while also leading to sparse right-hand sides in the FEM formulation, making it efficiently computable. We achieve this by introducing a cut-off into the subtraction, restricting its influence to the immediate neighborhood of the source. In this work, this approach will be presented, analyzed, and compared to other state-of-the-art FEM right-hand side approaches. We perform validation in multi-layer sphere models where analytical solutions exist. There, we demonstrate that the local subtraction approach is vastly more efficient than the subtraction approach. Moreover, we find that for the EEG forward problem, the local subtraction approach is less dependent on the global structure of the FEM mesh when compared to the subtraction approach. Additionally, we show the local subtraction approach to rival, and in many cases even surpass, the other investigated approaches in terms of accuracy. For the MEG forward problem, we show the local subtraction approach and the subtraction approach to produce highly accurate approximations of the volume currents close to the source.

cs.CE↗

arXiv2024

A Coupled Diffusion Approximation for Spatiotemporal Hemodynamic Response and Deoxygenated Blood Volume Fraction in Microcirculation

Background and Objective: This proof of concept study investigates mathematical modelling of blood flow and oxygen transport in cerebral microcirculation, focusing on understanding hemodynamic responses. By coupling oxygen transport models and blood flow dynamics, the research aims to predict spatiotemporal hemodynamic responses and their impact on blood oxygenation levels, particularly in the context of deoxygenated and total blood volume (DBV and TBV) fractions. Methods: A coupled spatiotemporal model is developed using Fick's law for diffusion, combined with the hemodynamic response function derived from a damped wave equation. The diffusion coefficient in Fick's law is based on Hagen-Poiseuille flow, and arterial blood flow is approximated numerically through pressure-Poisson equation (PPE). The equations are then numerically solved with the finite element method (FEM). Numerical experiments are performed on a high-resolution 7-Tesla Magnetic Resonance Imaging (MRI) dataset for head segmentation, which facilitates the differentiation of arterial blood vessels and various brain tissue compartments. Results: The applicability of the model is further demonstrated through numerical experiments utilizing a 7 Tesla magnetic resonance imaging (MRI) dataset for head segmentation, which facilitates the differentiation of arterial blood vessels and various brain tissue compartments. By simulating hemodynamical responses and analyzing their impact on volumetric DBV and TBV, this study offers valuable insights into spatiotemporal modelling of brain tissue and blood flow. Conclusions: This study utilizes spatiotemporal modelling with high-resolution 7 Tesla-MRI head data to explore cerebral blood flow, oxygen transport, and brain dynamics. It enhances understanding of cardiovascular conditions, improves simulation accuracy, and offers potential clinical applications for targeted interventions.

math.NA↗

arXiv2024

Anomaly distinguishability in an asteroid analogue using quasi-monostatic experimental radar measurements

This study conducts a quantitative distinguishability analysis using quasi-monostatic experimental radar data to find a topographic and backpropagated tomographic reconstruction for an analogue of asteroid Itokawa (25143). In particular, we consider a combination of travel-time and wavefield backpropagation tomography using the time-frequency representation (TFR) and principal component analysis (PCA) approaches as filtering techniques. Furthermore, we hypothesise that the travel time of the main peaks in the signal can be projected as a topographic imaging of the analogue asteroid while also presenting a tomographic reconstruction based on the main peaks in the signal. We compare the performance of several different filtering approaches covering several noise levels and two hypothetical interior structures: homogeneous and detailed. Our results suggest that wavefield information is vital for obtaining an appropriate reconstruction quality regardless of the noise level and that different filters affect the distinguishability under different assumptions of the noise. The results also suggest that the main peaks of the measured signal can be used to topographically distinguish the signatures in the measurements, hence the interior structure of the different analogue asteroids. Similarly, a tomographic reconstruction with the main peaks of the measured signal can be used to distinguish the interior structure of the different analogue asteroids.

astro-ph.EP↗

arXiv2023

Pressure-Poisson Equation in Numerical Simulation of Cerebral Arterial Circulation and Its Effect on the Electrical Conductivity of the Brain

This study considers dynamic modelling of the cerebral arterial circulation and reconstructing an atlas for the electrical conductivity of the brain. While high-resolution 7-Tesla (T) Magnetic Resonance Imaging (MRI) data allow for reconstructing the cerebral arteries with a cross-sectional diameter larger than the voxel size, electrical conductivity cannot be directly inferred from MRI data. Brain models of electrophysiology typically associate each brain tissue compartment with a constant electrical conductivity, omitting any dynamic effects of cerebral blood circulation. Incorporating those effects poses the challenge of solving a system of incompressible Navier-Stokes equations in a realistic multi-compartment head model. We postulate that circulation in the distinguishable arteries can be estimated via the pressure-Poisson equation, which is coupled with Fick's law of diffusion for microcirculation. To establish a fluid exchange model between arteries and microarteries, a boundary condition derived from the Hagen-Poisseuille model is applied. The relationship between the estimated volumetric blood concentration and the electrical conductivity of the brain tissue is approximated through Archie's law for fluid flow in porous media. Through the formulation of the PPE and a set of boundary conditions based on the Hagen-Poisseuille model, we obtained an equivalent formulation of the incompressible Stokes equation. Thus, allowing effective blood pressure estimation in cerebral arteries segmented from open 7T MRI data. As a result of this research, we developed and built a useful modelling framework that accounts for the effects of dynamic blood flow on a novel MRI-based electrical conductivity atlas. The electrical conductivity perturbation obtained in numerical experiments has an appropriate overall match with previous studies on this subject.

math.AP↗

arXiv2023

The Effects of Peeling on Finite Element Method -based EEG Source Reconstruction

The problem of reconstructing brain activity from electric potential measurements performed on the surface of a human head is not an easy task: not just because the solution of the related inverse problem is fundamentally ill-posed (not unique), but because the methods utilized in constructing a synthetic forward solution themselves contain many inaccuracies. One of these is the fact that the usual method of modelling primary currents in the human head via dipoles brings about at least 2 modelling errors: one from the singularity introduced by the dipole, and one from placing such dipoles near conductivity discontinuities in the active brain layer boundaries. In this article we observe how the removal of possible source locations from the surfaces of active brain layers affects the localisation accuracy of two inverse methods, sLORETA and Dipole Scan, at different signal-to-noise ratios (SNR), when the H(div) source model is used. We also describe the finite element forward solver used to construct the synthetic EEG data, that was fed to the inverse methods as input, in addition to the meshes that were used as the domains of the forward and inverse solvers. Our results suggest that there is a slight general improvement in the localisation results, especially at lower noise levels. The applied inverse algorithm and brain compartment under observation also affect the accuracy.

math.NA↗

arXiv2023

Multi-compartment human head modeling: generating adaptive tetrahedral mesh with GPU acceleration

This paper introduces a highly adaptive and automated approach for generating Finite Element (FE) discretization for a given realistic multi-compartment human head model obtained through magnetic resonance imaging (MRI) dataset. We aim at obtaining accurate tetrahedral FE meshes for electroencephalographic source localization. We present recursive solid angle labeling for the surface segmentation of the model and then adapt it with a set of smoothing, inflation, and optimization routines to further enhance the quality of the FE mesh. The results show that our methodology can produce FE mesh with an accuracy greater than 1 millimeter, significant with respect to both their 3D structure discretization outcome and electroencephalographic source localization estimates. FE meshes can be achieved for the human head including complex deep brain structures. Our algorithm has been implemented using the open Matlab-based Zeffiro Interface toolbox with it effective time-effective parallel computing system.

math.AP↗

arXiv2023

Navier-Stokes Modelling of Non-Newtonian Blood Flow in Cerebral Arterial Circulation and its Dynamic Impact on Electrical Conductivity in a Realistic Multi-Compartment Head Model

Background and Objective: This study aims to evaluate the dynamic effect of non-Newtonian cerebral arterial circulation on electrical conductivity distribution (ECD) in a realistic multi-compartment head model. It addresses the importance and challenges associated with electrophysiological modalities, such as transcranial electrical stimulation, electro-magnetoencephalography, and electrical impedance tomography. Factors such as electrical conductivity's impact on forward modeling accuracy, complex vessel networks, data acquisition limitations (especially in MRI), and blood flow phenomena are considered. Methods: The Navier-Stokes equations (NSEs) govern the non-Newtonian flow model used in this study. The solver comprises two stages: the first solves the pressure field using a dynamical pressure-Poisson equation derived from NSEs, and the second updates the velocity field using Leray regularization and the pressure distribution from the first stage. The Carreau-Yasuda model establishes the connection between blood velocity and viscosity. Blood concentration in microvessels is approximated using Fick's law of diffusion, and conductivity mapping is obtained via Archie's law. The head model used corresponds to an open 7 Tesla MRI dataset, differentiating arterial vessels from other structures. Results: The results suggest the establishment of a dynamic model of cerebral blood flow for arterial and microcirculation. Blood pressure and conductivity distributions are obtained through numerically simulated pulse sequences, enabling approximation of blood concentration and conductivity within the brain. Conclusions: This model provides an approximation of dynamic blood flow and corresponding ECD in different brain regions. The advantage lies in its applicability with limited a priori information about blood flow and compatibility with arbitrary head models that distinguish arteries.

math.AP↗

arXiv2022

CutFEM forward modeling for EEG source analysis

Source analysis of Electroencephalography (EEG) data requires the computation of the scalp potential induced by current sources in the brain. This so-called EEG forward problem is based on an accurate estimation of the volume conduction effects in the human head, represented by a partial differential equation which can be solved using the finite element method (FEM). FEM offers flexibility when modeling anisotropic tissue conductivities but requires a volumetric discretization, a mesh, of the head domain. Structured hexahedral meshes are easy to create in an automatic fashion, while tetrahedral meshes are better suited to model curved geometries. Tetrahedral meshes thus offer better accuracy, but are more difficult to create. Methods: We introduce CutFEM for EEG forward simulations to integrate the strengths of hexahedra and tetrahedra. It belongs to the family of unfitted finite element methods, decoupling mesh and geometry representation. Following a description of the method, we will employ CutFEM in both controlled spherical scenarios and the reconstruction of somatosensory evoked potentials. Results: CutFEM outperforms competing FEM approaches with regard to numerical accuracy, memory consumption and computational speed while being able to mesh arbitrarily touching compartments. Conclusion: CutFEM balances numerical accuracy, computational efficiency and a smooth approximation of complex geometries that has previously not been available in FEM-based EEG forward modeling.

cs.CE↗

arXiv2022

Investigation of Wavelength-induced Uncertainties in Full-Wave Radar Tomography of High Contrast Domain: An Application to Small Solar System Bodies

This paper aims to reconstruct the internal structure of a two-dimensional test object via numerically simulated full-wave time domain radar tomography with the presence of wavelength-induced (WI) uncertainties, following from a complex domain structure, and domain diameters 21 or 64 times the wavelength of the signal propagating inside the target. In particular, we consider an application in planetary scientific studies of reconstructing the interior structure of an arbitrary high contrast small Solar System Body (SSSB), i.e., an asteroid, with a probing signal wavelength limited by the instrument and mission payload requirements. Our uncertainty reduction model finds the reconstruction via averaging multiple inverse solutions assuming that the WI deviations in the solutions correspond to random deviations, which we assume to be independent and identically distributed (IID). It incorporates error marginalisation via a randomised signal configuration, spatial-averaging of candidate solutions, frequency-based error marginalisation, and the truncated singular value decomposition (TSVD) filtering technique, based on our assumptions of the phase discrepancy of the signal, domain parameters, and the full-wave forward model. The numerical experiments are performed for 20 and 60 MHz centre frequencies proposed for CubeSat-based radars, the latter being the centre frequency of the Juventas Radar which will be aboard Hera mission to investigate the interior structure of asteroid Dimorphos. A benchmark reconstruction of the target was obtained with the spatial averaging, sparse point density and frequency randomised configuration for both 20 and 60 MHz frequency systems.

astro-ph.EP↗

arXiv2022

L1-norm vs. L2-norm fitting in optimizing focal multi-channel tES stimulation: linear and semidefinite programming vs. weighted least squares

This study focuses on Multi-Channel Transcranial Electrical Stimulation, a non-invasive brain method for stimulating neuronal activity under the influence of low-intensity currents. We introduce mathematical formulation for finding a current pattern which optimizes a L1-norm fit between a given focal target distribution and volume current density inside the brain. L1-norm is well-known to favor well-localized or sparse distributions compared to L2-norm (least-squares) fitted estimates. We present a linear programming approach which performs L1-norm fitting and penalization of the current pattern (L1L1) to control the number of non-zero currents. The optimizer filters a large set of candidate solutions using a two-stage metaheuristic search in from a pre-filtered set of candidates. The numerical simulation results, obtained with both a 8- and 20-channel electrode montages, suggest that our hypothesis on the benefits of L1-norm data fitting is valid. As compared to L1-norm regularized L2-norm fitting (L1L2) via semidefinite programming and weighted Tikhonov least-squares method, the L1L1 results were overall preferable with respect to maximizing the focused current density at the target position and the ratio between focused and nuisance current magnitudes. We propose the metaheuristic L1L1 optimization approach as a potential technique to obtain a well-localized stimulus with a controllable magnitude at a given target position. L1L1 finds a current pattern with a steep contrast between the anodal and cathodal electrodes meanwhile suppressing the nuisance currents in the brain, hence, providing a potential alternative to modulate the effects of the stimulation, e.g., the sensation experienced by the subject.

math.OC↗

arXiv2022

Conditionally Exponential Prior in Focal Near- and Far-Field EEG Source Localization via Randomized Multiresolution Scanning (RAMUS)

In this paper, we focus on the inverse problem of reconstructing distributional brain activity with cortical and weakly detectable deep components in non-invasive Electroencephalography. In particular, we aim to generalize the previously extensively used conditionally Gaussian prior (CGP) formalism to achieve distributional reconstructions with higher focality. For this purpose, we introduce as a hierarchical prior, a general exponential distribution, refered to as conditionally exponential prior (CEP). The first-degree CEP corresponds to focality enforcing Laplace prior that suffers from strong depth bias making the deep activity unrecoverable. We sample over multiple resolution levels via RAMUS to reduce this bias as it is known to depend on the resolution of the source space. Moreover, we introduce a procedure based on the physiological a priori knowledge of the brain activity to obtain the shape and scale parameters of the gamma hyperprior that steer the CEP. The posterior estimates are calculated using iterative statistical methods, expectation maximization and iterative alternating sequential algorithm, which we show to be algorithmically similar and to have a close resemblance to the iterative reweighting methods. The performance of CEP is compared with the recent sampling-based dipole localization method Sequential semi-analytic Monte Carlo estimation (SESAME) in numerical experiments of simulated somatosensory evoked potentials related to the human median nerve stimulation. Our results suggest that a hybrid of the first-degree CEP and RAMUS can achieve an accuracy comparable to the second-degree case (CGP) while being more focal. Further, the proposed hybrid is shown to be robust to noise effects and compare well to the dipole reconstructions obtained with SESAME.

math.NA↗