Chapter 01
Electromagnetic fields in the body
The body is a wet, salty, layered conductor. Any changing electric or magnetic field that reaches it drives charges to move. What happens next depends mostly on one number: the frequency.
Tissue contains ions in water and cell membranes that store charge. Two material properties describe how a tissue responds to a field. Conductivity (σ, in siemens per metre) says how easily current flows. Relative permittivity (εr) says how much charge the tissue can store and how much it slows the wave. Both change strongly with frequency. That change is called dispersion.
Fields act on the body in two main ways, and they dominate at different frequencies:
- Stimulation. At low frequency, an induced electric field can push a nerve or muscle cell past its firing threshold. This is useful in TMS and other neuromodulation, and it is a hazard to limit near MRI gradient coils and power equipment.
- Heating. At radio frequencies, currents and oscillating water molecules lose energy as heat. The safety question becomes how much power each gram of tissue absorbs, and how much the temperature rises.
In between, from about 100 kHz to 10 MHz, both effects matter and the guidelines check both. Above 6 GHz the energy stays close to the surface, so the guidelines move from power per mass to power per skin area.
Measuring fields inside a living person is almost never possible. That is why exposure assessment, device safety and therapy planning rely on simulation: a digital body, a model of the device, and a solver for Maxwell's equations.
Chapter 02
The frequency map: from kilohertz to 300 GHz
Our solvers cover eight orders of magnitude in frequency. Across that range the wavelength in tissue shrinks from hundreds of metres to a fraction of a millimetre, and the physics changes with it.
Three length scales decide everything
Wavelength in tissue versus body size. If the wavelength is much longer than the body, the field inside follows the source almost instantly, like a static field: this is the quasi-static regime. If the wavelength is close to the size of the head or trunk, the field forms standing waves and hot spots: the resonant regime, where MRI at 3 T and 7 T and mobile phones live. If the wavelength is much shorter than the body, the energy is absorbed near the surface: the optical-like regime of millimetre waves.
Penetration depth. Conducting tissue absorbs the wave as it travels. The penetration depth δ is the distance over which the field falls to 1/e, about 37 %, of its value at the surface. It is metres at kilohertz, centimetres at MRI frequencies, and below a millimetre at 30 GHz and above.
Anatomical detail. Skin, the cerebrospinal fluid around the brain and the insulation of an implant lead are a millimetre thick or less. The grid must resolve them wherever the field changes across them.
Chapter 03 · Interactive
Frequency explorer
Move the slider or pick a preset. The page computes the tissue properties, the wavelength and penetration depth in that tissue, a rule-of-thumb voxel size, and which of our solvers applies.
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Wavelength and penetration depth against body sizes
Dispersion of the selected tissue
Source of the tissue values. Four-term Cole-Cole model of S. Gabriel, R. W. Lau and C. Gabriel, “The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues”, Phys. Med. Biol. 41 (1996) 2271–2293, Table 1, as stored in our solver's tissue table. The model is fitted to data from 10 Hz to 100 GHz; above 100 GHz (shaded) the values are an extrapolation and carry more uncertainty. Wavelength and penetration depth are for a plane wave in a large block of that one tissue. The voxel size is a common rule of thumb (a tenth of the wavelength, and a third of the penetration depth), not a guarantee of accuracy: in our studies the grid is chosen by convergence tests.
Chapter 04 · Band 1
Stimulation and current flow: kHz to 10 MHz
Here the wavelength is far larger than the body. The field inside is set by the source, the shape of the body and the conductivity of each tissue. Inductive and capacitive effects of the whole body can be neglected, which is why the regime is called quasi-static.
Current flow by conductivity. Electrodes drive it directly (electro-quasi-static); coils induce it through a changing magnetic field (magneto-quasi-static).
Huge conductivity contrasts, from CSF to skull, and thin folded layers. Linear systems with tens of millions of unknowns must converge fast at fine grids.
Induced electric field in the cortex or nerves, current density, temperature rise near electrodes, and lesion size for ablation.
The applications
EEG. Electrodes on the scalp record the brain's own currents. To locate a source, one needs the lead field: how a current at each point in the brain shows up at each electrode. By reciprocity, one solve per electrode gives the lead field for all sources at once.
TMS, transcranial magnetic stimulation. A coil on the head carries a brief, strong current pulse. The changing magnetic field induces an electric field in the cortex that can fire neurons. The question is where the field is strong, and by how much.
tDCS and TI. Transcranial direct current stimulation passes a weak current between scalp electrodes. Temporal interference (TI) uses two electrode pairs at slightly different kHz frequencies; where the two fields overlap, their sum beats at the difference frequency, which may reach deeper targets. Choosing the electrode positions is a search over many combinations.
Electroporation and ablation. Strong pulsed fields open pores in cell membranes, and tissue conductivity rises with the field. This makes the problem nonlinear. Pulsed-field ablation of heart tissue is an example; lesion size and heating are the outputs.
MRI gradient coils switch at kHz rates and can stimulate peripheral nerves, so the same solver checks them.
Chapter 05 · Band 1, cell scale
Cells in electric fields: from cable theory to bidomain BEM
Chapter 04 computes the field in the head or body, on a grid of about a millimetre. A neuron is a few micrometres thick. This chapter follows the field down to the cell: how it charges the membrane, when the classic cable model predicts the response well, and how a boundary-element model of the whole cell computes it when it does not.
A thin insulating membrane between two conducting fluids. The applied field charges it like a capacitor and changes the transmembrane voltage; ion channels respond to that voltage.
Micrometre membranes in a millimetre volume, field and membrane coupled both ways, and tens of thousands of time steps for one action potential.
Induced transmembrane voltage, activation threshold, where the cell fires first, and the error of the cable model.
The membrane is a capacitor
The cell membrane is a lipid bilayer about 5 nm thick. It hardly conducts, but the fluids on both sides do. Two conductors separated by a thin insulator form a capacitor: about 1 µF per cm² of membrane. At rest the inside sits about 70 mV below the outside. An applied field moves charge onto the membrane and changes this voltage. If the depolarization is large enough, voltage-gated sodium channels open and the cell fires an action potential.
A round cell in a uniform field
For a spherical cell of radius R in a uniform field E, the induced change in transmembrane voltage was derived by Schwan. It is largest on the two poles facing along the field, with opposite signs, and zero on the equator. It builds up with a time constant τ that is below a microsecond for a cell of 10 µm.
τ = R cm (ρi + ρe/2)θ is the angle from the field direction, cm the membrane capacitance per area, ρi and ρe the resistivities inside and outside. Example: R = 10 µm and E = 100 V/m give 1.5 mV at the poles. A long cylinder across the field gives 2 E R cos θ.
Long fibres: cable theory and the activating function
Axons and dendrites are long thin tubes, so the standard model treats them as one-dimensional cables: a chain of short compartments, each with its membrane capacitance and ion channels, joined by the resistance of the axoplasm. The applied field enters only through the extracellular potential Ve along the fibre, computed beforehand without the cell. Its second derivative along the fibre is the activating function: where it is positive the membrane depolarizes.
Cable models are fast and the basis of tools such as NEURON. They assume that the cell does not disturb the field, that the voltage is the same all around each compartment, and that the field changes slowly over the fibre's width.
When cable theory fails
The assumptions hold for distant sources such as TMS coils and scalp electrodes. They break down when a source is close to the cell, as with deep-brain-stimulation contacts and microelectrodes: the field then changes strongly across the fibre and along a few micrometres of it, and the cell itself distorts the field near it. In our comparison with the same membrane on both sides, the cable model agrees with the full model within 0.2 % at bends, branches and the soma, but for electrodes closer than about 50 µm the full model gives a lower threshold than the cable model.
How bidomain BEM computes it
The bidomain boundary-element method (BEM) keeps the cell in three dimensions. The inside and the outside of the cell are two conductors; only their boundaries are meshed with triangles: the membrane, the walls of the bath and the electrode surfaces. The unknown is the charge on these surfaces. The membrane voltage appears as a jump of the potential across the membrane. At each time step one linear system gives the charges, and therefore the field everywhere, including the field the cell itself produces. The membrane current then advances the voltage and the ion channels. The membrane can be passive or Hodgkin-Huxley. The Blue Brain channel set of a cortical layer 2/3 pyramidal cell is in review: on a membrane patch its spike times are within 0.07 to 8.5 µs of NEURON 9.0.2.
The matrix A is dense: every surface triangle interacts with every other. Hierarchical matrices (H-matrices) store the interaction between distant groups of triangles in low-rank form. In our solver the ranks stay between 28 and 58 up to 320 000 unknowns, and a block of 25 parallel axons needs 8.4 GB of memory instead of 37 GB. Chapter 10b explains how.
How we check it
- Exact solutions for a sphere, a cylinder, a cell with a nucleus and a prolate spheroid: the error falls with the square of the triangle size (second order).
- An independent finite-volume computation of the same cases agrees within 0.001 to 0.005 %.
- Our Blue Brain ion channels match the same channels in NEURON 9.0.2; our cable models run on the same membranes for the comparison above.
- The synthetic axon cases of a recent bidomain BEM study (Hasan et al., IEEE J-ERM 2026) are reproduced, except the threshold rise with packing density in their Fig. 15.
A field meets a membrane, step by step
- Current flows around the cell. The applied field drives ions through the fluid. The membrane hardly conducts, so the current bends around the cell instead of through it.
- Charge builds up on the membrane. Where the current meets the membrane, ions collect on its two faces. The membrane is a capacitor, so this charge changes the transmembrane voltage: up on one side of the cell, down on the other.
- Ion channels respond. Sodium and potassium channels open and close with the voltage. If the depolarization passes threshold, the cell fires an action potential.
- The cell changes the field. The membrane currents and the cell's own shape change the potential outside the cell. This is the two-way coupling that a bidomain model keeps and a cable model leaves out.
A cable model computes the field once, without the cell, and then feeds it to a one-dimensional fibre. A bidomain BEM solves steps 1 to 4 together, in three dimensions, at every time step.
Where the cable model and the full model differ
We compared the two on the same membranes, with the cable computed in our own code and in NEURON. On a straight axon, most of the difference between them comes from the mesh: the BEM approximates the round fibre by a polygon, which is slightly too thin electrically. Refine the polygon and the gap closes. Bends, branch points and the cell body agree with the cable model within the numerical uncertainty. The real difference appears where the field changes across the fibre: near an electrode within 5 to 20 fibre radii, the cable model overestimates the threshold by up to 5.6 %, and a field across the fibre cannot be represented in a cable model at all.
Building on the work of Gomez and Hasan
The bidomain BEM for neurons was developed by Czerwonky, Aberra and Gomez (J. Neural Eng., 2024) and extended to whole neurons and groups of neurons by Hasan et al. (IEEE J-ERM, 2026). Our solver follows the same formulation. We first reproduced their synthetic axon cases; the one result we could not reproduce is the threshold rise with packing density in their Fig. 15. Then we added:
- H-matrix algebra for large cells, with bounded ranks and memory close to the data (Chapter 10b).
- A charge constraint: a closed membrane carries no net charge, enforced exactly. It removes the one poorly conditioned mode of the system and leaves thresholds unchanged.
- A reconstructed layer 2/3 pyramidal cell against a NEURON cable model of the same tree, both with the Hodgkin-Huxley membrane. The reported 2 to 3.5 % gap appears only on the coarse mesh; converged, the BEM threshold lies 0.7 to 0.8 % below the cable for a field along the apical axis and a point source, and 2.7 % below it for a transverse field.
- Blue Brain ion channels in the BEM membrane (in review): spike times on a membrane patch within 0.07 to 8.5 µs of NEURON 9.0.2.
- Coupling to head models: the cell takes its field from our own head solution, for example under a TMS coil.
Chapter 06 · Band 2
MRI at 64, 128 and 297 MHz
An MRI scanner works with three fields: a strong static field B0, a radio-frequency field B1, and switched gradient fields. The radio-frequency field is the one that deposits heat, and its frequency is fixed by the magnet.
What the scanner does
Hydrogen nuclei in water behave like small magnets. In B0 they precess at the Larmor frequency, 42.58 MHz per tesla. That gives about 64 MHz at 1.5 T, 128 MHz at 3 T and 298 MHz at 7 T. A transmit coil, often a birdcage around the body or the head, produces a rotating field B1+ at that frequency, which tips the nuclei. Their signal is then received and turned into an image.
The coil also creates an electric field in the patient. Tissue absorbs power from it, measured as SAR. Scanners limit SAR in every sequence, using models of the coil and the body.
Why higher fields are harder
At 64 MHz the wavelength in brain tissue is about 40 cm, about twice the width of the head. At 297 MHz it is about 12 cm, smaller than the head. B1 becomes uneven, with bright and dark regions, and local SAR hot spots move with anatomy. Parallel transmit (pTx) at 7 T drives 8 or more coil elements with separate amplitudes and phases to even out B1. Local SAR inside the head cannot be measured, so it must be predicted by simulation, with a safety margin that covers the simulation's own errors.
Implants
A long conductive lead, for example of a pacemaker or a deep-brain stimulator, acts like an antenna. The electric field along the lead drives a current that leaves the lead at the exposed electrode and heats the tissue around the tip. ISO/TS 10974 describes how to assess this, and its most detailed tier uses a transfer function of the lead together with simulated fields along many lead paths in many body models.
Larmor calculator
Frequency from the proton gyromagnetic ratio, 42.577 MHz/T. Wavelength from the Gabriel 1996 Cole-Cole model of grey matter, as in the explorer. The head is roughly 15 to 20 cm across.
Chapter 07 · Band 3
Wireless and mobile: 0.4 to 6 GHz
Phones, tablets, wearables, Wi-Fi and Bluetooth devices sit close to the body. The antenna and the body form one system: the body detunes the antenna, and the antenna's near field sets where the body absorbs power.
Full-wave radiation and absorption. In muscle the wavelength shrinks from about 10 cm at 400 MHz to under 1 cm at 6 GHz, and absorption concentrates within a few centimetres of the surface nearest the antenna.
Resolving thin layers and the antenna gap while keeping the full head in the model. Grid error grows with frequency, roughly with the square of cell size over wavelength.
Peak spatial-average SAR over 1 g or 10 g, whole-body average SAR, and the antenna's accepted power and impedance.
Compliance testing for phones uses measured SAR in liquid-filled phantoms, and simulation supports design, worst-case search and anatomical studies. The computational method for SAR is set out in the IEC/IEEE 62704 series, which defines how to average SAR over a cube of tissue.
At these frequencies a whole head at millimetre resolution already means tens of millions of unknowns. Two things make it tractable. The first is a fast solver whose cost grows almost linearly with the number of cells. The second is a correction for the error the grid itself makes: a voxel grid makes waves travel at slightly the wrong speed, and the error grows with frequency.
Our dispersion-corrected volume integral equation adjusts the tissue permittivity for the cell size, so that a coarser grid gives the answer a much finer grid would.
Chapter 08 · Band 4
Millimetre waves for 5G and 6G: 30 to 300 GHz
At these frequencies the wave is absorbed within a millimetre or so of skin. What matters is the power entering each square centimetre of skin, and the temperature rise it causes there.
Surface absorption in skin layers. The stratum corneum, epidermis and dermis have different water content, and their thickness affects reflection and absorption.
A wavelength in skin of 2 mm or less needs cells of a few hundredths to a few tenths of a millimetre. A whole head at that size is out of reach, so models are local and layered.
Absorbed power density (APD) averaged over 4 cm², and also over 1 cm² above 30 GHz, and the skin temperature rise.
Above 6 GHz, international guidelines (ICNIRP 2020, IEEE C95.1-2019) stop using SAR and use the absorbed power density, the power crossing the skin surface per unit area, averaged over a square. IEC/IEEE 63195-2 sets out how to compute it.
The physical concern is still heating, so the link between APD and temperature rise is part of the science. For small, focused sources at the top of the band, a very small spot can heat more than a uniform exposure with the same averaged APD would suggest.
Published intercomparisons, where many groups compute the same case, show sizeable spread between groups. Reference values with error bars help laboratories check their own tools.
Tissue data above 100 GHz are extrapolated from models fitted at lower frequencies. Results there carry that extra uncertainty, and we state it.
Chapter 09
Building the model
Every simulation has three ingredients: a digital body, the properties of each tissue in it, and the device that creates the field.
Anatomical voxel models
A body model is a 3-D image in which each small cube, a voxel, carries a tissue label: skin, fat, muscle, bone, grey matter, CSF and so on. Models come from segmented MRI or CT scans of real volunteers, or from anatomical atlases. Typical sizes run from 2 mm down to 0.5 mm, and finer still for skin at millimetre waves.
Voxels are simple and robust, but curved surfaces become staircases. Our solvers reduce this with sub-voxel materials, which give a cell on a boundary the effective property of the tissues it contains, and with octree grids, which refine cells only where needed. On a head model the octree used about 0.4 times the unknowns of a uniform 1 mm grid at the same peak SAR within 0.3 %.
Tissue properties
Each label maps to a permittivity and conductivity at the study frequency, from the Gabriel 1996 parametric models that underlie the IT'IS Foundation database. Thermal studies add density, heat capacity, thermal conductivity and blood perfusion. Tissue values vary between people and measurements by tens of percent, which is why uncertainty analysis matters.
The sources
- MRI coils: birdcage coils (quadrature or linear), RF shields, and ported transmit arrays with matching networks, as at 7 T.
- Antennas: thin-wire dipoles and arrays with feed gaps and ports, solved together with the body so the body's effect on the antenna is included.
- Implant leads: insulated wires with exposed electrodes, along routes through the body.
- Electrodes: scalp pads with gel and contact resistance for tDCS, TI and EEG; catheter electrodes for ablation.
- Stimulation coils: TMS coils and MRI gradient coils, given as wire paths.
- Imported fields: a field computed by another tool on a surface around the body.
Chapter 10
The solvers, in plain words
No single method is best across eight decades of frequency. We built several solvers that share one body model, one tissue database and one set of safety readouts, and we use two of them on the same case wherever a result matters.
Volume integral equation (VIE)
Treats the body as a region of material that carries induced currents, each acting as a small radiator. Only the body is meshed; empty space and radiation to infinity are exact, with no absorbing boundary. A fast Fourier transform applies all cell-to-cell interactions at once, so the cost grows close to N log N. It runs on CPU threads or an NVIDIA GPU, works on uniform or octree grids, and couples to wire antennas, coils and implant leads. A dispersion correction removes most of the grid's speed error at gigahertz frequencies.
Finite-difference time domain (FDTD)
Fills a box around the body with a grid and steps the fields forward in time, cell by cell. It is the most widely used method in the field, which makes it a natural independent check on the VIE. Ours handles dispersive tissues, thin wires, ports and absorbing boundaries, and runs on GPU.
Multigrid quasi-static solver
Below about 10 MHz the full Maxwell equations reduce to a current-flow problem. An aggregation multigrid method solves it in a number of iterations that hardly grows as the grid gets finer, from 2 mm to 0.5 mm. It handles electrodes, floating metal, coil-induced fields, anisotropic white matter, and the field-dependent conductivity of electroporation.
Bioheat
The Pennes bioheat equation turns absorbed power into temperature. It includes heat conduction, cooling by blood perfusion and convection at the skin. For ablation it adds pulse trains and a tissue-damage model.
Uncertainty quantification
Tissue properties, anatomy, device position and component tolerances are never known exactly. Monte Carlo sampling and polynomial chaos expansions propagate these uncertainties to the result, and Sobol indices tell which input matters most. The outcome is a result with an error bar, or a safety margin.
Shared safety outputs
All solvers report through the same routines: peak spatial-average SAR over 1 g and 10 g, absorbed power density, induced-field metrics, per-tissue power and temperature rise. Using one routine for every solver keeps comparisons fair.
Chapter 10b · Fast algorithms
Hierarchical matrices: why a dense problem need not cost N²
Boundary-element and integral-equation methods couple every unknown with every other. Stored as is, the matrix grows with the square of the number of unknowns and its factorization with the cube. A hierarchical matrix (H-matrix) exploits the fact that distant interactions are smooth, and brings both close to N log N.
N surface nodes give N² matrix entries. 320 000 unknowns would need 819 GB for the dense matrix alone.
A block that couples two groups of nodes far apart, compared with their size, is numerically of low rank: a few columns times a few rows describe it.
Memory close to N log N, a factorization close to N log² N, and an error that is set in advance.
Low-rank blocks
Take a group of nodes on one side of a cell and another group on the far side. Seen from one group, the other looks almost like a single point, so the potential it produces changes smoothly across the first group. The block of the matrix that links them, m rows by n columns, can then be written as a thin m × k matrix times a thin k × n matrix, with a rank k of a few tens. Storing k(m + n) numbers instead of m n is the whole saving.
Which blocks: the cluster tree
The nodes are split in two by position, then each half again, until the groups hold about 64 nodes. Two groups are far enough apart, or admissible, when their distance is large compared with their size. Admissible blocks are stored in low-rank form; the rest are split further, down to small dense blocks along the diagonal. The result is the block pattern on the right: fine near the diagonal, coarse away from it.
Building the blocks without the full matrix: ACA
Computing a block in full and then compressing it would cost as much as the dense matrix. Adaptive cross approximation (ACA) avoids this. It computes one row of the block, picks its largest entry, computes the column through that entry, and subtracts their product. It repeats with the next row until the remainder is below the tolerance. After k steps it has the rank-k factors U and V and has computed only k rows and k columns. A final QR and SVD step removes any redundant rank, and a check on random entries confirms the error.
Solving with it: H-LU
Time stepping a cell model solves the same matrix thousands of times, so the matrix is factored once into lower and upper triangular parts, as in Gaussian elimination. In an H-LU, every block of the factors keeps the H-matrix format, and each update is written into an existing block and recompressed. Our factorization peaks at 1.04 to 1.17 times the memory of its data, and every later solve reads the compressed factor once.
Two kinds of partition
A simpler format, HODLR, compresses every off-diagonal block, including blocks of neighbouring groups that touch each other. On long, thin shapes this works well. On surfaces in three dimensions the touching halves share a long boundary, and the ranks grow: in our tests to 1 848 at 20 000 unknowns. The H-matrix only compresses blocks of groups that are well apart, and its ranks stay between 30 and 58 after the factorization up to 320 000 unknowns.
Chapter 11
Safety metrics and the standards
A field map is not a safety answer. Standards turn it into a few numbers, each averaged in a defined way, so that results from different laboratories can be compared with limits.
The metrics
SAR, specific absorption rate. Power absorbed per kilogram of tissue, in W/kg. Local SAR at one point is noisy, so the standards average it over a cube of 1 g or 10 g. The highest such average anywhere in the body is the peak spatial-average SAR (psSAR). The whole-body average is reported too.
APD, absorbed power density. Power entering the body per square metre of skin, in W/m², averaged over a square of 4 cm², and also 1 cm² above 30 GHz. It replaces SAR above 6 GHz.
Temperature rise. Heating is what the RF limits are designed to prevent, so the temperature rise itself is the most direct measure. For implants it is the main output: the rise next to the electrode tip.
Induced electric field. For stimulation, the limits use the electric field inside tissue, averaged over a small cube or along a short line and taken at a high percentile.
SAR cube and local SAR calculator
Cube side = (mass / density)1/3. Local SAR = σ|E|²/ρ with σ from the Gabriel 1996 model and density from the IT'IS Foundation database (as in our tissue table). Local SAR at one point is not the averaged psSAR the limits refer to.
| Standard or guideline | What it covers | What it asks for, in short |
|---|---|---|
| ICNIRP 2020 | Guidelines for RF exposure, 100 kHz to 300 GHz | Limits on whole-body SAR, local SAR over 10 g up to 6 GHz, and absorbed power density over 4 cm² (and 1 cm² above 30 GHz) above 6 GHz. For the general public: 0.08 W/kg whole-body, 2 W/kg local in head and trunk, 20 W/m² APD over 4 cm². |
| ICNIRP 2010 | Guidelines for low-frequency fields, 1 Hz to 100 kHz | Limits on the induced electric field in tissue, to prevent nerve stimulation. The field is averaged over small cubes of one tissue and taken at the 99th percentile. |
| IEEE C95.1-2019 | IEEE safety standard, 0 Hz to 300 GHz | Comparable limits; for low-frequency stimulation it uses the electric field averaged along 5 mm lines. |
| IEC/IEEE 62704-1 | How to compute SAR with FDTD | Rules for the model, the source and above all the averaging of SAR over 1 g and 10 g cubes, including cubes that touch the surface. |
| IEC/IEEE 63195-2 | How to compute power density, 6 to 300 GHz | Definitions of incident and absorbed power density, the averaging squares, and the checks a computation must pass. |
| ISO/TS 10974 | MRI safety of active implants | Test methods for RF heating, gradient-induced effects and more. Its tiered RF approach goes from simple bounds to full simulation of fields along realistic lead paths in body models (Tier 3), combined with the lead's transfer function. |
| IEC 60601-2-33 | Safety of MRI equipment | SAR limits for patients during scans, for example 2 W/kg whole-body and 3.2 W/kg head in normal operating mode. |
Limit values are quoted from the published guidelines for orientation only. Each standard has conditions, averaging times and exposure categories not shown here; a compliance study always works from the current text of the standard.
Chapter 12
What verification means, and how we do it
Verification asks whether the code solves its equations correctly. Validation asks whether those equations and the model describe reality. A trustworthy result needs both, and verification comes first.
The rungs in practice
Exact solutions. A sphere in a plane wave has an exact series solution (Mie). Layered slabs, spheres in coils and multi-shell EEG heads also have closed forms. On a 10 cm sphere of tissue at 128 MHz, our VIE matched the Mie field inside within 0.7 %, and a sphere in a birdcage matched the exact solution within 0.3 %.
A second solver. Two methods with different weaknesses rarely make the same mistake. We compute key cases with both the VIE and our FDTD, and compare with other established codes such as openEMS and the finite-element tool SimNIBS.
Benchmarks and intercomparisons. We rerun published standard cases, such as the dipoles over a flat phantom of IEC/IEEE 62704-1 and the 11-group absorbed power density intercomparison at 30 and 90 GHz.
Then on real anatomy
Convergence. Each anatomical result is repeated on finer grids until it stops changing, or extrapolated to zero cell size.
Uncertainty. Tissue values, segmentation, positioning and hardware tolerances are varied, and their effect is added up in an error budget. For a 7 T head array, this budget gave the 1.32 safety margin above.
Reproducibility. Results are identical bit for bit whatever the number of processor threads, every change runs through an automated test suite, and each study records the solver version and every input.
Validation against measurement is a separate step. Where a result depends on it, we say so and do not claim more than the verification supports.
Chapter 13
What we can do today, and what we offer
Corefield Biophysics is run by the two people who wrote the EMVIE solvers. We carry out each study ourselves, from model to report.
| Application | Band | Main outputs | Solvers |
|---|---|---|---|
| TMS and gradient-coil stimulation | kHz | Induced E field in cortex and nerves, ICNIRP and IEEE metrics, conductivity uncertainty | Multigrid MQS |
| tDCS, TI and EEG | DC to kHz | Field at targets, montage search, EEG lead fields | Multigrid EQS, UQ |
| Cell-scale stimulation (in final testing) | DC to kHz | Membrane voltage and thresholds of neurons and cells near electrodes and coils, and where cable models fail | Bidomain BEM with H-matrices |
| Electrosurgery and ablation | kHz to MHz | Current paths, heating, lesion size (pulsed-field ablation in development) | Nonlinear EQS, bioheat |
| MRI coils and implant heating | 64, 128 MHz | B1⁺, SAR, incident field along lead paths, tip heating | VIE, FDTD, bioheat |
| 7 T parallel transmit | 297 MHz | Q-matrices, virtual observation points, local-SAR safety margin | VIE, FDTD, UQ |
| Wireless devices near the body | 0.4 to 6 GHz | psSAR 1 g and 10 g, accepted power, temperature rise | VIE, FDTD |
| mmWave 5G/6G | 30 to 300 GHz | APD over 1 and 4 cm², skin temperature rise | FDTD, VIE |
Exposure and safety studies
SAR, absorbed power density, induced fields and temperature rise for a device and a body model, reported with the uncertainty that applies and the standard it follows.
MRI implant RF heating
Incident fields along lead paths in body models and coils at 1.5 T and 3 T, combined with a lead's transfer function, following the Tier 3 approach of ISO/TS 10974.
Transmit arrays and pTx safety
Coil and array modelling with ports, matching and shield, Q-matrices, virtual observation points and a safety margin built from an error budget.
Neurostimulation dose
Fields from TMS coils and scalp electrodes, montage optimisation for TI, and the effect of tissue uncertainty on the dose at a target.
Independent review
A second, independent computation of your own simulation results, with a statement of where they agree and why they differ.
Methods and training
Custom solver features, verification studies, and courses built on material like this guide.
Reza Gholami
Co-founder. Researcher at EPFL; formerly Senior R&D CEM specialist at Siemens EDA.
Nastaran Hendijani
Co-founder. Researcher at the University of Colorado Denver; formerly Senior R&D CEM specialist at Synopsys/Ansys.
To discuss a study, visit the Corefield Biophysics site or write to [email protected].
Chapter 14 · Reference
Glossary
Short definitions of the terms used in this guide.
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