Weighted-peak assessment of exposure to MRI gradient and static fields – supplemental material
Weighted-peak assessment of occupational exposure due
to MRI gradient fields and movements in a nonhomogeneous static magnetic field
D. Andreuccetti1, G. M. Contessa2, R. Falsaperla2, R. Lodato3, R. Pinto3, N.
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Zoppetti1 and P. Rossi2
1
IFAC-CNR ("Nello Carrara" Institute for Applied Physics of the Italian
National Research Council), via Madonna del Piano 10, 50019 Sesto Fiorentino
(Florence), Italy.
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2
INAIL (Italian Workers' Compensation Authority), Via di Fontana Candida 1,
00040 Monte Porzio Catone (Rome), Italy.
3
ENEA (Italian Agency for New Technologies, Energy and Sustainable
Economic Development), Unit of Radiation Biology and Human Health,
Casaccia Research Centre, via Anguillarese 301, 00123 Rome, Italy.
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Corresponding author e-mail: D.Andreuccetti@ifac.cnr.it
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Weighted-peak assessment of exposure to MRI gradient and static fields – supplemental material
Supplemental material (to be published online only)
The basic theory and the software development procedure adopted to implement and validate
the numerical filters used for the calculation of the weighted-peak indexes in time domain are
20
presented here. Attention is focused on filters relative to the ICNIRP reference levels for
occupational exposures and magnetic flux density, as defined in both 1998 and 2010
guidelines.1,2
Filter development
The ICNIRP-1998 and ICNIRP-2010 guidelines behave differently below 1 Hz; in particular,
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ICNIRP-1998 reference levels are constant between 0 Hz and 1 Hz, while ICNIRP-2010 ones
are not defined. The amplitude of the numerical transfer function adopted in this study was
assumed to be proportional to 1/f2 below 1 Hz. The discrepancy with the reference levels as
defined in the guidelines is not critical in the case of exposures to gradient fields, since their
spectral contents are much higher than 1 Hz. In the case of movements in SMF, where spectral
30
components below 1 Hz are very important, the 1/f2 trend was assumed for 2010 reference
levels only, while the actual constant value was assumed for 1998 ones.
The transfer function W1998 in the Laplace domain of a filter is reported in Eq. (1), where s =
j2πf and j is the imaginary unit. This filter implements the inverse of the ICNIRP-1998
reference levels (scaled to peak values) for occupational exposures to magnetic flux density. In
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this expression, a and b are the terms that represent the two ICNIRP-1998 corner frequencies at
8 Hz and 820 Hz.
This filter does not take into account the corner frequency at 65 kHz, because it is intended to
be used for impressed fields having frequency spectrum contained well below that limit.
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Weighted-peak assessment of exposure to MRI gradient and static fields – supplemental material
s2
W1998 ( s)  Af
s  a s  b
Af 
 
1
T 1
30.7 10  6  2
(1)
with
a  2 8Hz
b  2 820 Hz
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A digital version of this filter can be implemented with the so called “zero-pole matching”
technique. The transfer function U1998 of this numerical filter is reported in Eq. (2), where z-1 is
the delay unit. 3 This is a IIR (Infinite Impulsive Response) filter, that lets the current output
sample depend on the current input sample, the previous last two input samples and the
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previous two output samples. In this expression, Tc is the sampling interval and fn is a
normalization frequency (at which the transfer functions of the analog and the numerical filters
have the same amplitude) that, in the present study, was chosen equal to 1/200 of the sampling
rate 1/Tc.
This particular technique has been preferred due to its simplicity and especially because the
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coefficients of the filter depend explicitly from the sample frequency 1/Tc. This is important
because the numerical filter is implemented in a software procedure that can be applied to a
generic input waveform, once the sample frequency is known.
U1998( z )  K  V1998( z )
V1998( z ) 
1  2  z 1  z 2
1  a1  b1 z 1  a1  b1  z  2
with
a1  e  j 2 aTc b1  e  j 2 bTc K 
(2)
W1998s  j 2f n 
V1998 z  e j 2f nTc


Where a, b and W1998 have been defined in Eq. (1).
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Filter validation
Two tests were performed on the numerical filters, in order to validate their implementation.
The first one, carried out in the frequency domain, consisted in the comparison of the amplitude
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Weighted-peak assessment of exposure to MRI gradient and static fields – supplemental material
(Fig. 1 and Fig. 2) and the phase (Fig. 3 and Fig. 4) of the analog and the numerical transfer
functions. This comparison showed that the agreement between the transfer functions is good
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on the whole digital filter bandwith only when the higher filter corner frequency is sufficiently
lower than the upper limit of this bandwidth, that is the so called Nyquist limit fNyq (i.e. half the
sampling rate) (see Fig. 2). More generally, a better agreement is achieved in the lower part of
the numerical filter bandwidth; this is particularly evident for the phase response of the
numerical filter that is always zero at fNyq (Fig. 3). In order to let the the numerical filter work in
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the lower part of its bandwidth, a sampling rate sufficently higher (at least double) than the
higher filter cut-off frequency should be selected.
The second test, executed in the time domain, consisted in feeding the analog and the numerical
filters with a waveform composed by four sinusoids with different frequencies and phases. The
expression of the input waveform is reported in Eq. (3), while the values chosen for the
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amplitudes and the phases of its spectral components are listed in Table I (these values were
chosen in the frequency range of interest for what concerns the MR gradient fields).
4
f (t )   Ai  sin 2 f i t   i 
i 1
(3)
In this case, the output of the analog filter (once the transient is finished) is analytically known
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and can be compared with the output of the numerical filter when its input is feeded with a
sampled version of the waveform of Eq. (3). The results are shown in Fig. 5 for the ICNIRP2010 filter.
As it can be noted, the absolute difference of the two sequences decreases almost three order of
magnitudes in less than 1 ms. The initial discrepancy is generated by the fact that we are
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comparing the steady-state response of the analog filter with the transient response of the
numerical filter.
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Weighted-peak assessment of exposure to MRI gradient and static fields – supplemental material
References
ICNIRP (International Commission on Non-Ionizing Radiation Protection), “Guidelines for
1
limiting exposure to time-varying electric, magnetic, and electromagnetic fields (up to 300
85
GHz)”, Health Phys. 74, 494-522 (1998).
ICNIRP (International Commission on Non-Ionizing Radiation Protection), “Guidelines for
2
limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz)”, Health Phys.
99, 818-836 (2010).
A. V. Oppenheim and R. W. Schafer, “Discrete-Time Signal Processing” 3rd edition, Upper
3
90
Saddle River Pearson (2010).
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Weighted-peak assessment of exposure to MRI gradient and static fields – supplemental material
TABLE I. Amplitudes and phases of the spectral components of the test signal of Eq. (3).
i
fi
Ai
φi
1
204
60 µT
0°
2
587 Hz
20 µT
31°
3
1015 Hz
9 µT
156°
4
2312 Hz
6 µT
77°
FIG. 1. Amplitude responses of the ICNIRP-2010 filters in continuous (c)
and discrete domain, for various sampling rate fs.
95
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Weighted-peak assessment of exposure to MRI gradient and static fields – supplemental material
FIG. 2. Percentage relative differences of analog and digital amplitude
responses of ICNIRP-2010 filters, for various sampling rate fs.
FIG. 3. Phase responses of the ICNIRP-2010 filters in the continuous (c)
and discrete domain, for various sampling rate fs.
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Weighted-peak assessment of exposure to MRI gradient and static fields – supplemental material
FIG. 4. Percentage relative differences of analog and digital phase
responses of ICNIRP-2010 filters, for various sampling rate fs.
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FIG. 5. Results of the filter validation test in time domain: comparison of the steady-state response of the
analog filter (teo) and of the numerical filter (num) to the test waveform defined in Eq.(3) and Table I.
The absolute difference between teo and num is also reported (secondary axis).
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