Transcript electric field

International Workshop
ELECTROMAGNETIC FIELDS
AT THE WORKPLACES
SESSION 5. PROBLEMS AND PERSPECTIVES
FOR COMPUTATIONAL DOSIMETRY
(Session co-organised by EMF-NET / MT2)
Warszawa, Poland – 7 September 2005
Numerical techniques for quasi-static
electromagnetic dosimetry
Daniele Andreuccetti, IFAC-CNR, Firenze
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Numerical electromagnetic dosimetry
• Numerical techniques play a key role in
electromagnetic dosimetry, thanks to the
increasing performances of modern digital
computers.
• Applications to complex occupational
exposure problems, requiring multiple sources,
realistic environments and accurate modeling
of exposed subjects, are becoming possible.
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The quasi-static approach
The analysis of the interactions between biological
systems and low-frequency electric or magnetic fields
is greatly simplified if three conditions are satisfied.
1. The dimensions of
the involved objects
and their mutual
distances should be
small when
compared to the free
space wavelength.
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Propagation effects are
negligible: the electric and
magnetic fields can be
calculated by using the
methods of electrostatics
and magnetostatics.
W.T.Kaune, J.L.Guttman and R.Kavet: “Comparison of coupling of humans to
electric and magnetic fields with frequencies between 100 Hz and 100 kHz”,
Bioelectromagnetics vol.18, pp.67–76, 1997.
The quasi-static approach
2. The size of the
exposed object is
comparable to, or
smaller than, the
magnetic skin depth.
3. In the exposed
object, conduction
currents prevail
over displacement
currents.
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The applied magnetic field
will be essentially
unperturbed by the
exposed body.
Charge movements will follow
“istantaneously” the oscillations
of the fields. Body is
equipotential. The calculation of
the electric fields outside and
inside the body is separated into
two problems.
The quasi-static approach
condition n.2
Penetration depth [m]
1000
100
10
1
0,1
1
10
100
1000
10000
Frequency [kHz]
Blood
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Muscle
Fat
Bone (cortical)
Nerve
Skin (dry)
The quasi-static approach
condition n.3
1000
Loss tangent
100
10
1
0,1
0,01
1
10
100
1000
10000
Frequency [kHz]
Blood
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Muscle
Fat
Bone (cortical)
Nerve
Skin (dry)
Thanks to the quasi-static
approach…
• The electric and the magnetic field problems get
decoupled and can be solved separately.
• For each field, the external and internal problems
(with respect to the exposed body) are separated too.
• Usually, a multi-step approach is adopted, which
will be briefly described in the following.
• Remember: the quasi-static conditions are strictly
verified up to just a few hundred chilohertz, but the
quasi-static approach is often applied up to a few tens
of megahertz.
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Impressed electric and
magnetic fields
• Impressed fields are the electric and magnetic fields
which are present in the site of the exposed subject,
when the exposed subject itself is absent.
• Impressed fields are generated by the sources and
possibly are perturbed by objects in the exposure
theatre.
• Action values (or reference levels) specified by the
international RF safety standards (such as the
European Directive 2004/40/CE) are referred to the
intensity of the impressed fields.
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Impressed electric and
magnetic fields
• Exposures of humans to EM fields which are relevant
from the point of view of safety and health usually
occur very close to the sources, especially in
occupational environments.
• At lower frequencies, exposures hence involve the
close-field region, where the main contribution to the
fields is due to direct induction from charges and
currents on/in the source, while radiation fields are
negligible. In this case, impressed electric and
magnetic fields are "decoupled" and can be calculated
separately, by means of static-like techniques.
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Calculation of the impressed
electric field
• If in the close-field region, numerical calculation of
the impressed electric field is usually based on the
numerical solution of the Laplace equation for the
electric potential:
 0
2
( E  0, E   )
where the values of the potential Φ on the sources and
on other objects present in the exposure theater must
be assigned (or previously computed).
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Calculation of the impressed
electric field
• Applying the finite-difference technique in a
Cartesian grid of l-sized cubic cell, we obtain:
6   ( x , y , z )=  ( x + l , y , z )+  ( x - l , y , z )+
+ ( x , y + l , z )+ ( x , y - l , z )+
+ ( x , y , z + l )+ ( x , y , z - l )
Which is an algebraic system of equations, which can be
solved using standard iterative methods.
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Calculation of the impressed
electric field
• As an alternative approach, the surface-charge
integral equation can be used:
1
(Q) 
4 0
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
S
  P
dS
QP
The (known)
potential at point Q
is expressed as the
superposition of the
contributions of a
number of surfacecharge distributions
(unknown).
Calculation of the impressed
electric field
According to a standard approach based on the method of
moments, the surfaces of the sources and other objects are
subdivided into a collection of N small patches ΔSi having
known potential Φi and unknown (but assumed uniform)
charge density ηi, obtaining the following system of N
algebraic equations:
N
j
i
j
0 j 1
S j
i
j
 
1
4
i  1,..., N
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 
dS
P P
Pi  Si
Calculation of the impressed
magnetic field
• In the close-field region, numerical calculation of the
impressed magnetic field is usually based on the
(numerical) integration of the Biot-Savart equation
in differential form (also called the first Laplace
formula):
0 idP   Q  P 
dB  Q  
3
4
QP
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Contribution to
the MFD at point
Q due to the
current element
dP situated at
point P and
conducting the
current “i”.
Calculation of the impressed
magnetic field
• As an alternative approach, the magnetic
vector potential approach can be used :
0 J  P 
A Q  
dV

4 V Q  P
Similar expressions hold in
case of surface or linear
current density distributions.
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Vector potential
at point Q due
to a current
density J
distributed in
the volume V.
B   A
Segmentation
• Segmentation consists of building a discrete
model of the exposed subject (or of a part of
it), subdividing it into "segments", i.e. small
homogeneous elements with regular geometry
(as pixels in 2D problems or voxels in 3D
problems).
• Older methods were manual and relied on
atlases of anatomical cross sections.
• More recent ones make mostly use of the
Magnetic Resonance Imaging (MRI)
technique as a source of anatomic data, that
have to be (semi)automatically processed in
order to recognize the different tissue types.
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Segmentation: Norman
• The most significative example of the second group is
“Norman" (the NORmalized MAN, developed by
Dimbylow), which consists of a 3D parallelepipedshaped regular array of nearly 36 million homogeneous
voxels, modeling a standardized human body (1.76 m
height, 73 kg mass) and some surrounding space (8.3
million cells are in the body, the remaining in the
surrounding space). Each cell has a roughly cubic shape
(~ 2 mm size) and is labelled by a tag denoting its tissue
type, chosen from a palette of 37 different tissues. This
process is automatically achieved interpreting the gray
scale data of the MRI medical images.
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Segmentation
• A very high resolution
(voxel sizes down to 1 mm)
digital model of a man has
been developed at the Radio
Frequency Radiation Branch
of the Brooks US Air Force
Base Research Laboratory.
• This model is based on the
Library of Medicine Visible
Human Project dataset.
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Segmentation
The lower the segment size,
the higher the accuracy of the
results. On the other hand,
higher resolutions demand
higher computational
resources, as most numerical
methods have storage and
time requirements
proportional to the number of
segments.
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Dielectric properties of body tissues
• Values for the dielectric properties (relative
permittivity ε and conductivity σ) have to be
assigned to each segment, taking the field frequency
into account.
• Usually, this is accomplished through preliminary
assessment of the type of tissue which composes each
segment.
• C.Gabriel and colleagues developed a parametric
model able to represent the dielectric properties of
biological tissues in the frequency range from 10 Hz
to 100 GHz and have determined the values of the
model parameters for 45 different tissues.
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Dielectric properties of body tissues:
The Gabriel’s model
The Gabriel's model is based on the superposition of four ColeCole dispersion relations; the Cole-Cole equation is a
generalization of the well-known Debye theory of relaxation,
aimed to take into account the broadening of the dispersion
region due to the complexity of the structure and of the
composition of biological materials.
The 14 model parameters for each tissue have been determined
by means of experimental results and best fitting techniques.
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Dielectric properties of body tissues
• Based on this model, Internet applications have been
developed and are currently available, that allow the
on-line calculation and download of values for the
permittivity, the conductivity and a few derived
parameters of about fifty different human tissues in
the frequency range from 10 Hz to 100 GHz.
• There is also an alternative approach, able to directly
determine the dielectric properties of the voxels
without passing through the tissue recognition. Being
based on the evaluation of the tissue water content (by
means of MRI techniques), this approach is limited to
frequencies above 100 MHz approx.
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Numerical modeling
• A set of differential or integral equations, with
proper boundary conditions, is individuated.
• These equations are put in a “discrete” form,
i.e. are adapted to the segmented model. This
way, a system of linear algebraic equations is
derived.
• Finally, this system is numerically solved using
some standard computational algorithm.
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 
The “external” problem
 E  
B
 E  
t
 
 E
B
   J 
t

The solution of the
static field equations
(where time derivatives
are removed from
Maxwell equations)
leads to a sufficiently
accurate evaluation of
electric and magnetic
fields in presence of the
exposed subject and of
the surface-charge
density on its boundary.
Magnetic field and surface-charge density are the driving
terms for the calculation of the induced current density.
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External problem: magnetic field
• Because biological materials do not possess
ferromagnetic properties, they do not directly
distort the impressed magnetic field. The
scattered field produced by induced currents is
also assumed negligible (quasi-static
approach).
• Thus, the external problem for the magnetic
field does not exist, as it reduces to the
calculation of the impressed magnetic field.
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External problem: electric field
• For the external problem, the subject is regarded as
homogeneous, equipotential and perfectly conductive,
hence the internal electric field is null. The impressed
electric field is perturbed by the subject, so that the
field lines are perpendicular to its surface.
• Thus, solving the external problem for the electric field
is similar to calculating the impressed electric field,
but with two fundamental differences.
1. The exposed subject is present in the exposure theatre.
2. We are not interested in calculating the electric field
itself, we are interested in calculating the surface-charge
density on the exposed subject.
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External problem: electric field
Integral equation – moment method approach
• Surface-charge density integral equation for an
electrically grounded organism:
Q is a point of the
body, P is a point
1
 ( P)
 S (Q) 
d
on its surface, Φ0
4 0  Q  P
is the potential of
the impressed
 S (Q)   0 (Q)  0
field.

• This equation is usually solved by means of the
method of moments in order to calculate the
surface-charge density η(P).
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External problem: electric field
Laplace equation – finite difference approach
• The Laplace equation is solved (as for the
calculation of the impressed field, but in presence
of the exposed body):
 0
2
( E  0, E   )
• The surface-charge density on the body boundary
is computed from the normal gradient of the
potential:
   0 n̂ 
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The “internal” problem
• In order to determine the basic physical
quantities induced inside the body (current
density and SAR), the time derivatives are
reintroduced into the equations and internal
electric field and current density are computed,
using the results of the solution of the external
problem as boundary conditions.
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Internal problem: electric field
• The assumption of perfect conductor is
removed; the previously-calculated surfacecharge density serves as the boundary
condition for the calculation of the internal
electric potential distribution; from potential
differences the internal electric field and,
hence, the volume current density are
computed.
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Internal problem: magnetic field
• A time-changing magnetic field produces an
electric field whose lines form closed loops on
a plane perpendicular to the magnetic field and
have shapes determined by the boundary of the
medium. If the medium has a nonzero
conductivity, an induced (eddy) current will
flow: its distribution can be calculated with
methods based on the application of Faraday
law.
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Finite difference approach
The SPFD (Scalar Potential Finite
Difference) method
• This method is particularly suited to solve 3D
problems, because it always reduces to a scalar
equation.
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Finite difference approach
The SPFD (Scalar Potential Finite
Difference) method
• The resulting scalar equation can be solved using a
finite-difference technique. There are some
“discretization” problems which will be discussed in a
separate communication (N.Zoppetti).
• In case an applied magnetic field IS NOT present, the
second term (coming from the magnetic vector
potential) vanishes, thus leaving:
   0
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Finite difference approach
The SPFD (Scalar Potential Finite
Difference) method
• In case an applied electric field IS present, we
have to take into account the surface-charge
density calculated solving the external
problem. It represents a boundary condition for
the internal problem:
J  , internal = 2  f j  0 E  , external = 2  f j 
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The impedance network method
• In the three-dimensional impedance network method,
each cubic element ("cell") generated with the
segmentation process is differentiated into a 3D
network of impedances. Every cell is represented by
three impedances located on three of its edges,
sharing a common vertex.
• The value of each edge impedance is determined
averaging the complex conductivity values of its four
neighboring cells. At last, the whole exposed body is
represented by a linear circuit and the circuit theory is
applied to compute the currents in the impedances.
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The impedance network method
The “electric” analogy directly leads to formulate an algebraic
problem (a system of coupled Kirchhoff Current Law
equations), to be solved using an iterative process.
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The impedance network method
• In magnetic field exposure problems, the time-varying
applied magnetic flux density impresses a voltage in
each closed loop formed by four connected
impedances in one plane, according to the Faraday
law; this voltage is equated to the sum of the edge
currents times impedance around each loop.
• If an impressed electric field is also present, than the
subject surface-charge density is used to calculate the
current injected in the impedances at boundary cells,
by means of the continuity equation for the electric
charge as previously shown:
J  , internal = 2  f j  0 E  , external = 2  f j 
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Finite difference approach
The CVP (Current Vector Potential) method
  HS  J
J   Ei
B0
  Ei  
t
The basic equation of the
CVP method is obtained
combining the Ampere curl
equation for the scattered
magnetic field with the
Faraday-Maxwell curl
equation relating the induced
electric field to the impressed
magnetic flux density.
The method is not well suited for electric field exposure
dosimetric evaluations, because the surface-charge density
boundary conditions can not be easily accommodated.
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Finite difference approach
The CVP (Current Vector Potential) method
  Ei  2 fj 0 H 0
J
  Ei    


1

      HS 



Applications are usually limited to two-dimensional,
magnetic-field exposure problems, because in this
case only this technique leads to a scalar differential
equation, which can be solved by means of standard
finite-difference techniques.
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General remarks
• Both the impedance network and the SPFD methods
are suited for three-dimensional problems. The former is
basically a vector method, whereas the latter is
intrinsically scalar (i.e., it leads to a scalar equation even
in the most complicated 3D geometry). As a
consequence, the impedance method requires threetimes the memory storage necessary for the SPFD.
• Furthermore, the computational molecule for the SPFD
method is more compact and requests fewer arithmetic
operations, so that the time per iteration needed by the
impedance method is nearly twice that required by the
potential method.
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General remarks
• In an N cell-problem, the method of moments requires
computer storage proportional to N2 and computation
time proportional to N3; this situation becomes
prohibitive when dealing with high resolution
heterogeneous models. Finite difference-like methods
(including impedance methods), on the contrary, have
storage and time requirements proportional to N.
• The number of cells N is substantially larger in the
finite difference and impedance methods than in the
moment method because of the overhead of free-space
cells surrounding the body, often necessary to
guarantee the proper boundary conditions.
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An example of application
of the CVP method
Exposure to a 50 Hz
magnetic field point
source, such as a small
appliance (hair dryer)
Conductivity map
Tissue map
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An example of application
of the CVP method
Impressed field map
Calculated by non-linear
interpolation of measurements
(method of multipole)
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Current density map