E-fields in the human head due to time varying magnetic field gradients
Martin Bencsik, Richard W. Bowtell, Roger M. Bowley.
School of Physics and Astronomy., University of Nottingham, UK.
Abstract
A homogeneous spherical volume conductor is used as a model system for
the purpose of calculating electric fields induced in the human head by
externally applied timevarying magnetic fields. We present results for the
case where magnetic field gradient coils, used in MRI, form the magnetic
field, and use these data to put limits on the rates of gradient change with
time needed to produce nerve stimulation. Analytical results are shown for
ideal field gradients, and numerical results for a real whole body coil set.
Numerical analysis shows similar results when applied to a model human
head.
Introduction
A homogeneous spherical volume conductor is used as a model system for
,the purpose of calculating electric fields induced in the human head hy
externally applied time-varying magnetic field gradients. The electric field is
calculated analytically for the case of ideal longitudinal and transverse linear
field gradients, for any position of the center of the sphere. We also show
results from computer calculations yielding electric field maps in a sphere
when the field gradients are generated by a real MRI gradient coil set. In
addition, the effect of shifting the sphere within each gradient coil volume is
investigated. Numerical analysis yields similar results when applied to a
model human head, justifymg the utility of the presented analytical study.
The data generated is used to put limits on the rates of gradient change with
time needed to produce nerve stimulation in the human head,
Method
This problem has heen partly dealt with by Eaton"]. Here we exploit the
same formalism, applied to the special case where the B-field corresponds to
the linear field gradient used in MRI.
The frequency at which field gradients are alternated is typically of the order
of 1 H z , so that the quasi-static regime of Maxwell's equations is applicable
to systems spanning distances of the order of 1 m. Furthermore, typical
values for biological tissue conductivity (0.2 Sm-l) allow us to neglect the Bfield created hy induced currents. Finally, we may neglect displacement
currents relative to conduction currents because the typical values of tissue
electric permittivity (&-1.3xlO4) mean that ~,.&~w<<a for relevant
frequencies.
Calculations always involve: (i) evaluating the radial component of the
vector potential, A(r), at the surface of the volume conductor. (ii) deducing
the scalar potential from the boundary condition that the radial component of
the E-field is zero at the sphere surface. (iii) calculating the E-field,
accounting for the effect of the scalar potential via
E(r)=-VV(r)--, M r )
coil introduces further terms whose peak values scale with the product of the
sphere radius and shift distance. For the centred sphere, the peak E-field
when using a transverse gradient is greater hy a factor of 4/3 than the case of
a longitudinal one.
For the centred sphere, the gradient switching rate required to generate Efields above 6 Vm-l is 3770 Tm-ls-l for a longitudinal gradient, and 2827 Tm'
Is-' for a transverse gradient. For the sphere shifted by +13 cm (the greatest
shift before the sphere exits the homogeneous gradient volume of a typical
whole body gradient coil) the corresponding rates are 1037 Tm%' (2-shift)
and 1633 Tm%" (x-shift) for the G, gradient and 808 (2 or x-shift) for the G,
gradient.
Numerical solutions for real field gradient
Assuming a stimulation threshold of 6 V.m-', we find that for the G,
transverse gradient coil the greatest allowed rate of change (for the hot spots
at coordinates (x,y,z) = (?r24cm, Ocm, i 5 5 cm)) is 188 T.m-'.s-', whilst for
the longitudinal gradient coil the limit (for the hot spots at (x,y,z) =
( ~ 4 c m , O c m , ~ 6 c mis) 295 T.m-'.s-'. These hot spots are located close to the
regions of maximum current density in the gradient coils.
The analytical results in the homogeneous sphere (11.3 cm radius) are
compared with the results from the human head model, for identical field
strength and switching frequencies in the figure below. For this particular
case, we used a transverse field gradient along the x axis, (4= 16 mTm-'),
switched at 1 kHz. Despite considerable heterogeneity in the electric
conductivity and the deviation of the head shape from perfect sphericity, the
E-field distribution calculated in the human head matches that found
analytically in a sphere mimicking the top part of the head very well.
Slice wsition x .
=O m
IEI iV rn-')
at
Steps (ii) and (iii) require that the radial component of A(r) is decomposed
into a series of spherical harmonics. This is straightforward for the case of
pure field gradients.
Software was written (matlab, The Mathworks) to accomplish this
calculation for a homogeneous spherical conductor experiencing the field
gradient due to a whole-body, actively shielded, MRI gradient set (140 cm
length, 33 cm inner radius, region of less than 5 % deviation from linearity =
42 cm dsv). The radius of the sphere was kept constant (8 cm) and multiple
simulations were run where its centre was moved over the whole volume
inside the cylindrical gradient coil. It was thus possible to identify the
positions where induced E-fields are greatest (the 'hot spots' of the coil) and
to put a limit on the maximum rates of gradient change with time which
avoid nerve stimulation whatever the position of the sphere in the coil.
Finally, numerical calculations were also performed for the case of a digital
human head model (Medical VR Studio GmbH, Errach, Germany)
(isotropic spatial resolution = 3.6 mm), which accounts for 39 different
electrical conductivity values, using commercial software for solving
Maxwell's equations (MAFIA, C.S.T Darmstadt, Germany) via the finite
integration method.
Results
Analytical solutions for ideal field gradient
Here we list the main features of the calculated E-fields. The symmetry of
the spherical conductor means that the radial E-field is zero everywhere
within the sphere[']. The presence of charge density at the boundary of the
conductor strongly affects the E-field distribution.
The modulus of the induced E-field is always greatest at the periphery of the
sphere, and when the sphere is centred within the gradient coil, scales with
© Proc. Intl. Soc. Mag. Reson. Med. 10 (2002)
Conclusion
Analytic calculation of the E-field induced in a homogeneous spherical
conductor by time-vaying magnetic field gradients provides some new
insight into the magnitude and spatial variation of electric fields generated in
the human head during MRI. Computation times are of the order of a minute
for the homogeneous sphere, while application of the f ~ t eintegration
method to a human head model requires a time of a few days. The gradient
switching rates which we calculate to be necessary to cause E-fields in
excess of 6 Vm-' are considerably larger that those known to cause
stimulation in the human torso exposed to the fields of a whole-body gradient
coil. The calculations therefore bear out the fact that it is stimulation in the
torso that generally limits allowable gradient switching rates. With the
increased use of dedicated head gradient coils, which can generate much
larger rates of change of magnetic field, the potential for causing stimulation
in the head is increased. The observation that the induced electric field is
largest at the periphery of the sphere means that this stimulation is most
likely to occur in the scalp. At higher gradient switching rates nerves in the
brain will he stimulated. Given the increase of the induced electric field with
radial co-ordinate, it is likely that grey matter will be stimulated before white
matter, as occurs with transcranial magnetic stimulation (TMS). The effects
of such stimulation, which will clearly be of greater spatial extent than that
caused by the small coils in used in 'I'MS, merit further study.
[I] Eaton H , Med.Biol. Eng. Comp., 1992 433-440.
[Z] Branston N.M. and Tofb P.S., Phys. Med.Biol., 1991:161-168.