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Accessibility of Cortical Regions to Focal TES: Dependence on Spatial Position, Safety, and Practical Constraints

Guilherme Bicalho Saturnino, Hartwig Roman Siebner, Axel Thielscher, Kristoffer Hougaard Madsen

arXiv:1909.04574v2physics.med-ph

TL;DR

TES targeting is limited by anatomy-dependent electric fields, which complicate focal stimulation. The paper introduces an efficient constrained optimization method and maps its performance across cortical positions, finding that focality and targeting depend on anatomy and safety constraints.

  • Problem

    Head conductivity and anatomy produce complex, variable TES fields, making it difficult to target regions reliably and interpret stimulation effects.

    Source: INTRODUCTION · p. 3

  • Method

    The paper develops an efficient multi-electrode optimization method that accounts for safety and channel limits, then maps targeting and focality across the cortex under varying constraints.

    Source: INTRODUCTION · p. 4, Optimization of Electrode Currents · p. 9

  • Results

    Focality, targeting accuracy, and achievable intensity are better for superficial than sulcal or deep targets; safety constraints further reduce focality and accuracy, especially for deeper targets.

    Source: Summary of findings · pp. 38–39

  • Takeaways & Limitations

    Under commonly implemented safety constraints, the paper finds little benefit from using more than eight electrodes for TES montages, while a fine planning grid remains useful.

    Source: Conclusion · p. 40

    The evaluation held the head model and safety constraints constant, and the framework does not address variability from brain state, age, genetics, or neurochemistry.

    Source: Limitations · p. 40

Abstract

from arXiv · show

Transcranial electric stimulation (TES) can modulate intrinsic neural activity in the brain by injecting weak currents through electrodes attached to the scalp. TES has been widely used as a neuroscience tool to investigate how behavioural and physiological variables of brain function are modulated by electric stimulation of specific brain regions. For an unambiguous interpretation of TES experiments, it is important that the electric fields can be steered towards one or several brain regions-of-interest. However, the conductive proprieties of the human head impose inherent physical limitations on how focal the electric fields in the brain produced by multi-electrode TES can be. As a rule of thumb, it is not feasible to target deep brain areas with TES, although focusing the field in some specific deeper locations might be possible due to favourable conductive properties in the surrounding tissue. In the present study, we first propose a novel method for the automatic calculation of electrode placements and stimulation intensities to optimally affect a given target position. We provide a computationally efficient and robust implementation of the optimization procedure that is able to adhere to safety constraints, while explicitly controlling both the number of active electrodes and the angular deviation of the field in the target area relative to the desired field direction. Leveraging the high computational efficiency of our method, we systematically assess the achievable focality of multi-electrode TES for all cortex positions, thereby investigating the dependence on the chosen constraints. Our results provide comprehensive insight into the limitations regarding the achievable TES dose and focality that are imposed by the biophysical constraints and the safety considerations of TES.

INTRODUCTION

TES fields depend on head conductivity and anatomy, making intuitive montages uncertain and manual optimization time-consuming. The paper presents an optimized multi-electrode approach and a framework for representing fields and target-related quantities.

INTRODUCTION

  • Head conductivity makes stimulation patterns anatomy-dependent and potentially unexpected, while broad montages complicate attribution to a particular brain region.Manual model-based montage optimization can require lengthy trial and error, especially with multi-channel systems.Source: INTRODUCTION · p. 3
  • Earlier fixed-grid methods varied in how they handled safety and channel limits; some fast approaches did not enforce a channel cap, while alternatives incurred much longer computation.The cited branch-and-bound and genetic-algorithm approaches were described as substantially slower.Source: INTRODUCTION · pp. 3–4
  • The proposed optimization incorporates TES safety limits and channel-count limits exactly while remaining computationally efficient.Its active-electrode constraint algorithm is reported to reduce computation from hours to under one minute while maintaining optimality.Source: INTRODUCTION · p. 4

Theoretical Framework

  • The extended lead-field representation maps currents across all electrodes, including the reference, to electric fields, with current conservation selecting a physically plausible solution.The current vector must sum to zero under Kirchhoff’s current law.Source: Theoretical Framework · p. 6
  • The framework evaluates target-directed mean field, total energy, and the angle between the mean target field and its desired direction.These quantities are introduced as measures relevant to the optimization procedure.Source: Theoretical Framework · p. 6

Forward Simulations of the TES Electric Fields

Forward fields were calculated with a finite-element head model and a fixed electrode grid. The model used tissue-specific conductivities and several grid densities for assessing grid-density effects.

Forward Simulations of the TES Electric Fields

  • Electric potentials were calculated with SimNIBS 2.1 using the finite element method and first-order tetrahedral elements, then fields were rescaled to unit current.Simulations applied Dirichlet boundary conditions at the stimulated and return electrodes.Source: Forward Simulations of the TES… · p. 8
  • The simulations used a healthy-volunteer MRI-derived head model with about 4 × 10^6 tetrahedra and isotropic tissue conductivities.The model included grey and white matter, CSF, bone, scalp, electrodes, and vitreous bodies.Source: Forward Simulations of the TES… · pp. 8–9
  • The main electrode grid contained 288 positions covering the head, neck, and face; additional grids contained 102, 190, and 381 positions.The alternative grid densities were used to assess how cortical fields depend on grid density.Source: Forward Simulations of the TES… · p. 9

Optimization of Electrode Currents

The section formulates electrode-current optimization around target field strength, focality, safety limits, electrode-count limits, and field direction. It also extends the formulations to multiple targets and uses branch-and-bound for problems with a limit on active electrodes.

Optimization without safety constraints

  • Problem 1 minimizes current energy while requiring a target-region field component and obeying Kirchhoff’s current law.Its reduced formulation cuts problem size by 3 to 4 orders of magnitude relative to formulations using the lead-field matrix directly.Source: 1) Without Safety Constraints · p. 11

Safety and focality constraints

  • Safety constraints can limit total injected current and the current through each electrode, while focal-field formulations constrain the target field.The focal-field formulation is a quadratic program; if its requested target field is infeasible, the implementation falls back to maximizing the target field under safety constraints.Source: 2) With Safety Constraints · pp. 12–13

Constraining the number of active electrodes

  • Limiting active electrodes makes the focal-field problem non-convex, so branch-and-bound partitions active and inactive electrode choices and bounds the objective to find a near-global solution.The method uses relaxed L1 constraints to obtain lower bounds.Source: 3) Constraining the number of active… · pp. 14–15
  • Within 10% of the global optimum, solutions were certified; with 288 electrode positions and six active electrodes, more than 20 branch-and-bound steps were needed at about 1% of positions.The optimal value was typically found within the first few iterations; the remaining steps certified proximity to the global solution.Source: 3) Constraining the number of active… · p. 15

Constraining the angular deviation of the electric field in the target

  • An angular constraint can control the maximum deviation of the target field from its desired direction, and can be combined with focality and electrode-count constraints.The angle constraint is non-convex in isolation but can be reformulated as a quadratic constraint with the target-field equality constraint.Source: 4) Constraining the angular deviation… · pp. 15–16

Optimization for multiple targets

  • For multiple targets, separate field constraints and target values avoid bias toward an easier-to-reach target; the paper does not address imposing angle constraints separately at each target.When target values are infeasible, a separate formulation maximizes fields across targets while preventing any target from exceeding its specified value.Source: 5) Optimization for multiple targets · pp. 17–19

Performance and Error Metrics

The section defines metrics for where the strongest field occurs and how broadly the field spreads. It explains how these measures complement one another when comparing optimization conditions.

Targeting error

Effective area

Stimulated area

  • Stimulated area measures the total area where field strength is at least 50% of the target field.Together, stimulated area and effective area describe focality, while targeting error tracks the location of the strongest field.Source: 3) Area with |𝐸⃗ | exceeding 50% of… · p. 20

Evaluation of Optimization Performance across the Brain

The evaluation maps optimized fields and performance metrics across cortical target positions. It tests successive optimization constraints and evaluates fields at the middle of the grey-matter layer, using the local surface normal as the target direction in the paper’s figures.

Evaluation setup

  • Optimization was tested first without safety constraints, then with safety, electrode-count, and target-field-angle constraints added successively.Fields were evaluated at the middle of the grey-matter layer, defined from the average of the pial and white-matter surfaces and down-sampled to 20,000 nodes.Source: Evaluation of Optimization Performance… · p. 20

Mapping procedure

  • The evaluation optimized a target at each cortical mesh node, one at a time, and calculated performance and error metrics for each solution.The mapping procedure computes the energy matrix, solves the selected optimization problem for each node and direction, then calculates and stores field metrics.Source: Evaluation of Optimization Performance… · p. 21

Target directions

Comparison to Ad-hoc Montages

Optimized montages improve focality over ad-hoc layouts for motor-cortex and cingulate-sulcus targets, while fields at the deeper cingulate target remain spatially extended. Across the comparisons, optimized montages have lower targeting errors and better focality.

Comparison to Ad-hoc Montages

  • For the left motor-cortex target, optimization makes the field more focal than the classical two-pad montage while matching its target field.Optimization also improves focality over a centre-surround ring montage for a target on the crown of the precentral gyrus.Source: Comparison to Ad-hoc Montages · pp. 21, 23
  • For the deeper left cingulate-sulcus target, optimization improves focality, but the strongest fields still extend across broad brain-surface areas.The authors attribute this limit to the head acting as an ohmic volume conductor.Source: Comparison to Ad-hoc Montages · p. 22
  • Optimized montages improve all reported focality metrics over ad-hoc montages for three Figure 2 targets, with lower targeting errors and better focality.The largest difference is for the standard motor-cortex montage; the metrics include total area above 50% of target field and Effective Area.Source: Comparison to Ad-hoc Montages · pp. 22, 24, Table 2 · p. 24

Maximally Achievable Focality without Safety Constraints

Without safety constraints, optimized focality varies substantially across cortical positions, with inferior, sulcal, and medial targets performing worse than superficial gyri. Increasing the electrode grid from 288 to 381 positions yields only marginal further improvement.

Maximally Achievable Focality without Safety Constraints

  • Even without safety constraints, inferior, sulcal, and medial cortical targets show greater targeting errors and worse focality than superficial gyral targets.The authors explain that currents must pass through other brain areas to reach deeper targets; Effective Area is used because it avoids an arbitrary cutoff.Source: Maximally Achievable Focality without… · pp. 24–25
  • Both targeting accuracy and focality improve as the electrode grid grows to 288 positions, with only marginal gains at 381 positions.The study therefore uses 288 electrode positions for its main analyses; similar patterns are reported for subcortical and tangentially oriented targets.Source: Maximally Achievable Focality without… · p. 25

Maximally Achievable Intensity in the Presence of Safety Constraints

With individual-electrode current limited to 1 mA and total current to 2 mA, the maximum achievable field is strongest at gyral targets and weaker at sulcal and deep targets. This intensity-maximizing strategy does not enforce focality.

Maximally Achievable Intensity in the Presence of Safety Constraints

  • Under 1 mA per-electrode and 2 mA total-current limits, maximum field strength reaches 0.6 V/m at some gyral positions, while sulcal and deep targets are weaker.Fields still exceed 0.3 V/m at most positions; because focality was not enforced, targeting error and Effective Area are much worse than in the unconstrained best-case scenario.Source: Maximally Achievable Intensity in the… · p. 26
  • Achievable target intensity rises as the electrode grid increases to 288 positions, but is almost unchanged at 381 positions.The same pattern is reported for subcortical targets and tangential target orientations.Source: Maximally Achievable Intensity in the… · p. 26

Effect of Safety Constraints on the Optimization of Focality

When focality is optimized under current limits, performance depends strongly on target position, with deep targets showing high targeting error and low focality. The constraints increase Effective Area across positions and limit field-cancellation strategies, especially for inferior, medial, and longitudinal-fissure regions.

Effect of Safety Constraints on the Optimization of Focality

  • With 1 mA per-electrode and 2 mA total-current limits at a 0.2 V/m target, deep targets are dominated by high targeting error and low focality.Relative to unconstrained optimization, error and Effective Area increase most in inferior and medial regions.Source: Effect of Safety Constraints on the… · p. 27
  • Higher target-field strengths leave less allowed current available for cancellation fields, creating a trade-off between target intensity and focality.The authors report the effect for target fields varied from 0.1 to 0.3 V/m.Source: Effect of Safety Constraints on the… · p. 28
  • Safety limits reduce the use of many electrodes to cancel undesired field directions, particularly near the longitudinal fissure, where CSF conductivity steers currents toward or away from the fissure.Inferior and medial regions are also among those most affected by the constraints.Source: Effect of Safety Constraints on the… · p. 28
  • Safety constraints consistently increase Effective Area across cortical positions because the optimization minimizes delivered energy, which is closely related to that metric.Small apparent targeting-error improvements at some positions are described as spurious because targeting error is not directly assessed during optimization.Source: Effect of Safety Constraints on the… · p. 28

Effect of Constraining the Number of Active Electrodes

Focality improves as the active-electrode limit rises, with most gains achieved by eight electrodes. Explicit field-direction constraints and electrode limits can be combined at relatively low additional cost, though direction constraints reduce performance in some regions.

Effect of Constraining the Number of Active Electrodes

  • The three electrode-selection approaches similarly reduce targeting error, but Projection preserves focality less well than either Branch-and-Bound method.Full B-B and Selected B-B perform very similarly; Selected B-B is faster and was used for later optimizations.Source: Effect of Constraining the Number of… · pp. 29–30
  • Four active electrodes generally reduce focality, six improve it substantially, and eight add further gains, mainly at gyral crowns; beyond eight, improvements are marginal.These comparisons used a target field of 0.2 V/m, individual-current limit of 1 mA, and total-current limit of 2 mA.Source: Effect of Constraining the Number of… · pp. 30–31
  • In inferior positions, limiting the montage to four electrodes does not affect the field because 0.2 V/m is near the local maximum achievable with four electrodes.The stated maximum number follows from the safety limits: ceil(2I_tot/I_ind) = 4.Source: Effect of Constraining the Number of… · p. 31
  • A 15°, 22.5°, or 30° direction constraint causes only slight accuracy and focality reductions in most areas, but larger reductions near the skull base and CSF.In those regions, fields tend to point laterally toward CSF, conflicting with the constrained direction normal to the cortical sheet.Source: Effect of Constraining the Angular… · p. 31
  • Combining a 22.5° direction constraint with limits of four, six, or eight electrodes adds little cost, and about eight electrodes remain sufficient.The reported results are similar to those with electrode limits alone.Source: Effect of Constraining the Angular… · p. 32

Multi-Target Optimization

The study evaluates simultaneous motor-cortex targeting, optimization runtimes, and the algorithm’s constraints and implementation. Two-target fields become more confined up to eight active electrodes, while runtimes remain practical across the tested problems and the method accommodates safety, channel, and field-angle limits.

Multi-Target Optimization

  • For two handknob targets, hotspots become more confined and decoupled as active electrodes increase from four to eight; ten adds little, though only two target combinations were tested.The tested configurations used 0.2 V/m at both targets, 1 mA individual and 2 mA total current limits, and outward/outward or inward/outward field directions.Source: Multi-Target Optimization · pp. 33–34

Time to run Optimizations

  • The most demanding tested optimization, combining field-angle and electrode-count constraints, takes a few seconds on average and a few minutes in the worst case.Focality with an electrode-count constraint alone had median runtimes only twice those without that constraint; angle-constrained problems were usually solved in under one second.Source: Time to run Optimizations · pp. 36–37

Summary of the algorithm and implementation

  • The common framework expresses practical montage objectives and constraints, including target field strength and direction, per-electrode and total-current limits, electrode count, and angular deviation.Most considered problems minimize brain field energy to obtain focal fields; Problems 2, 5, and 9 instead maximize the field in the target.Source: Summary of the algorithm and… · p. 37
  • Branch-and-Bound handles electrode-count limits efficiently and certifies that its solution is close to the global optimum; preselecting candidate electrodes accelerates the search.The authors attribute improved efficiency over prior Branch-and-Bound work probably to more suitable bound functions and state-space definition.Source: Summary of the algorithm and… · p. 38
  • The study explicitly controls target-field angle, unlike the cited LCMV approach, which requires the specified angle to be reached exactly.The QCQP implementation solves in under one second in most cases, though the authors note a cone solver would likely be more efficient.Source: Summary of the algorithm and… · p. 38

Summary of findings

Across the cortex, anatomy limits achievable TES focality, targeting accuracy, and intensity; safety constraints further reduce focality and accuracy, especially for deeper targets. The findings also identify practical trade-offs in electrode planning, current limits, and target spacing.

Summary of findings

  • Superficial targets allow better focality, targeting accuracy, and intensity than sulcal or deep targets; safety constraints further reduce focality and accuracy, especially for deeper targets.The anatomical differences persist even in a theoretical best-case scenario without safety constraints.Source: Summary of findings · pp. 38–39
  • Constraining the field direction adds little focality penalty for most target positions, and findings generalize to tangential directions and deep targets.The results also show a focality–intensity trade-off.Source: Summary of findings · p. 39
  • A fine electrode grid can improve montage planning, but only a few electrodes are used for stimulation, whose positions require accurate control in practice.The fine grid is needed for virtual-model planning, not for the final stimulation setup.Source: Summary of findings · p. 39
  • For single targets, total-current constraints severely limit benefits beyond about eight active electrodes by restricting secondary electrodes used to cancel fields outside the target.When the target intensity can be met, the total-current limit can counterintuitively increase energy delivered to the brain; the authors suggest considering current-density limits in relevant tissues.Source: Summary of findings · p. 39
  • The algorithm produced balanced focal fields for two distant targets, but closely spaced targets may not be fully decoupled, leaving stimulation between them.Stimulation of one target can produce a significant field at the other.Source: Summary of findings · pp. 39–40

Limitations

  • The study held the head model and safety constraints constant; anatomical variation, particularly in disease, may change exact optimal focality values.The authors expect the broader conclusions about optimization parameters and constraints to remain similar across head models.Source: Limitations · p. 40
  • The optimization framework does not address other sources of TES outcome variability, including brain state, age, genetics, and neurochemistry.Source: Limitations · p. 40

Conclusion

  • The study’s conclusions concern traditional TES; temporally interfering fields might generate more focal fields in deeper brain areas.The authors identify optimizing temporally interfering fields as future research.Source: Conclusion · pp. 40–41

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