Analysis of the Quasistatic Impedance Approximation for Gradient Coil Peripheral Nerve Stimulation
M. E. Slinkard1, V. J. Srinivasan1, B. A. Chronik1
1
Stanford University, Stanford, CA, United States
Synopsis: The validity of the quasistatic impedance approximation for the
calculation of gradient field induced electric fields in biological tissues was
evaluated as a function of frequency. It was found that the approximation is
expected to fail for many tissues at frequencies below 1 kHz and above 1
MHz. This is counter to conventional wisdom, which states that the
approximation is acceptable for all frequencies below approximately 100
kHz, and has ramifications for the modeling and analysis of peripheral nerve
stimulation in gradient coils.
Introduction: The quasistatic impedance approximation [1] is commonly
applied in the analysis of electric fields induced in human tissues during the
operation of MRI gradient coils. It is typically assumed that this
approximation results in errors between 1% and 0.1% in normal tissues for
gradient switching frequencies below 100 kHz. There are three elements of
this approximation: (1) wavelength effects are negligible; (2) skin-depth
effects are negligible; (3) capacitive effects are negligible (i.e. displacement
current is much smaller than conduction current).
The first two
approximations are excellent for this application, wheras the third is
acknowledged to be the poorest. This third approximation can be expressed
as follows:
σ >> 2πυ ⋅ Kε o
[Eq. 1]
where σ is tissue conductivity, ν is frequency, K is relative permittivity, and
εo is permittivity of free space.
Methods: Permittivity and conductivity values of various biological tissues
were taken from the classic work of Gabriel, et al. [2]. The ratio of the RHS
to the LHS of Eq. 1, expressed as a percentage, was evaluated as a function
of frequency for the different tissues listed in Fig. 2. Values of this ratio
over 5% were considered to be a significant error.
Results: As an example, the dispersion curves of breast fat are shown in
Fig. 1 as a function of frequency from 0 to 1 MHz. There is a significant
dispersion in this frequency range evident in the figure. The error in the
quasi-static approximation due to neglecting the permitivity is shown as a
function of frequency in Fig. 1(b). There is a large increase in the error
(peak of approximately 50%) at approximately 10 Hz. Figure 2 summarizes
these results for all tissues evaluated It is very interesting to note the
presence of a significant window of frequencies, beween approximately 10
Hz and 1 kHz, where the effect of permitivity in many tissues is not
negligible in comparison to the effect of conductivity. From 1 kHz to 100
kHz, the permitivity effect is generally less than 5% of the conductivity
effect and the quasistatic approximation is acceptable.
Conclusion:
Although it is generally stated that the quasistatic
approximation is accurate to between 1% and 0.1%, our analysis indicates
the accuracy is more closely described as being between 10% and 1%. This
realization of the order of magnitude of the error associated with this
approximation has important ramifications on the way we approach the
analysis of PNS in gradient coils. We are not saying that the quasi-static
approximation is insufficiently accurate and should be abandoned. What we
are suggesting is that the actual accuracy of the approximation should be
kept in mind when deciding upon the precision of tissue models and
computational methods to be used in conjuction with the quasistatic
approximation.
References:
[1] R. Plonsey, D.B. Hepner. Bull Math Biophys, 29:657-664 (1967).
[2] S. Gabriel, R.W. Lau, C. Gabriel. Phys Med Biol, 41:2251 (1996).
Proc. Intl. Soc. Mag. Reson. Med. 11 (2003)
233
108
K
6
10
4
10
relative
permittivity
Example:
Breast Fat
102
σ
100
conductivity
10-2
100 101 102 103 104 105 106
freq. [Hz]
30
Percent Error
20
%
10
region of
< 5% error
0
100 101 102 103 104 105 106
freq. [Hz]
Figure 1: (Top) Dispersion curves of relative permitivity
and conductivity for breast fat tissue. (Bottom) The
percent error expected as a result of neglecting the
permitivity contribution to the electric field. The error is
unacceptably large for frequencies under 100 Hz, and over
1 MHz.
aorta
bone, cancellous
bone, cortical
bone, marrow
breast fat
cerebro spinal fluid
cervix
dura
fat (infiltrated)
fat (non infiltrated)
muscle
freq [Hz]
Figure 2: Bars indicate frequency regions over which
the permitivity is a negligible contribution (less than 5%)
of the total electric field. The quasistatic approximation
is acceptable over these regions. Most tissues are not
well approximated for frequencies between 10 and 1000
Hz, or over 1 MHz.