Predicting peripheral nerve stimulation for a gradient array

Supplementary note for the ERC Advanced Grant proposal — Ergin Atalar, Bilkent University.

Peripheral nerve stimulation (PNS) is set by the electric field in the whole body, not by the gradient field inside the volume being imaged. A conventional coil has one field pattern, so its PNS threshold is one number measured at installation. An array has as many field patterns as current combinations. The threshold becomes a function of the drive vector. To use the array safely, the scanner must predict PNS for every current combination it is about to play, in milliseconds. This note describes how we obtain that prediction.

The Huygens surface model

Davids, Guérin, and Wald compressed a full electromagnetic and neurodynamic body simulation into a single matrix defined on a Huygens surface, a mesh enclosing the body model at 5 cm from the skin carrying about 3000 small current loops as basis elements [1]. Two objects are precomputed once per body model and waveform family. The matrix \( \mathbf{B}_H \) holds the magnetic field of every basis loop at target points inside the body. The PNS oracle matrix \( \mathbf{P} \) holds the reciprocal PNS threshold of every basis loop at every nerve segment, in units of 1/A. Any coil is then characterized in seconds. Its field is projected onto the basis by a regularized least-squares fit, and the product with \( \mathbf{P} \) gives the oracle value of every nerve segment. The lowest current that fires any nerve is the PNS threshold of the coil. The model is validated against 21 body and head gradients and against the full simulation [1], which itself reproduces measured human thresholds [2]. The group at Massachusetts General Hospital shared the model and the precomputed matrices with us in 2026.

From one coil to an array

The model is linear in the source. This is the property that makes it fit the array. Let the array have \( N \) channels and let \( \mathbf{B}_S \) hold the field of each channel at unit amplifier current, one column per channel. One regularized solve maps all channels onto the Huygens basis at once,

\[ \mathbf{M} = \arg\min \; \lVert \mathbf{B}_H \mathbf{M} – \mathbf{B}_S \rVert^2 + \lambda^2 \lVert \mathbf{L} \mathbf{M} \rVert^2 , \]

and the product

\[ \mathbf{P}_A = \mathbf{P} \, \mathbf{M} \]

is the PNS oracle matrix of the array, one column per channel. For a channel-current vector \( \mathbf{i} \), the response of every nerve segment is the matrix-vector product \( \mathbf{P}_A \mathbf{i} \). No nerve reaches threshold if and only if

\[ \lVert \mathbf{P}_A \, \mathbf{i} \rVert_\infty \lt 1 . \]

This is a set of linear inequalities in the currents. Therefore the PNS limit enters the scan-time current optimization as an ordinary linear constraint, next to the amplifier current and voltage limits. The prediction for one drive vector is one matrix-vector product. That is fast enough to run inside the sequence.

The same inequality turns the third objective of the proposal into a linear program. For a target slice, we maximize the gradient strength over the slice subject to the PNS constraint, the field fidelity over the slice, and the amplifier limits. The gain over the conventional linear drive at equal gradient strength is

\[ Y = \frac{ \lVert \mathbf{P}_A \, \mathbf{u}_{\mathrm{lin}} \rVert_\infty }{ \lVert \mathbf{P}_A \, \mathbf{u}_{\mathrm{slice}} \rVert_\infty } , \]

where \( \mathbf{u}_{\mathrm{lin}} \) is the linear mode and \( \mathbf{u}_{\mathrm{slice}} \) is the slice-optimized mode. The oracle is calibrated per waveform family and rise time, so \( Y \) is reported per rise time and per slice position.

Status

References

1. Davids M, Guérin B, Wald LL. A Huygens surface approach to rapid characterization of peripheral nerve stimulation. Magn Reson Med 2022;87:377-393. doi:10.1002/mrm.28966
2. Davids M, Guérin B, vom Endt A, Schad LR, Wald LL. Prediction of peripheral nerve stimulation thresholds of MRI gradient coils using coupled electromagnetic and neurodynamic simulations. Magn Reson Med 2019;81:686-701. doi:10.1002/mrm.27382