Inżynieria biomedyczna/ Biomedical Engineering
MAGNETOACOUSTIC TOMOGRAPHY
WITH MAGNETIC INDUCTION
FOR LOW CONDUCTIVITY OBJECTS
TOMOGRAFIA MAGNETOAKUSTYCZNA
ZE WZBUDZENIEM INDUKCYJNYM DO BADANIA
OBIEKTÓW O NISKIEJ KONDUKTYWNOŚCI
Katarzyna Cichoń-Bańkowska1*, Marcin Ziółkowski2, Stanisław Gratkowski2,
Krzysztof Stawicki2, Andrzej Brykalski1, Barbara Szymanik2, Adam Żywica2
1
2
Zachodniopomorski Uniwersytet Technologiczny, Wydział Elektryczny,
Katedra Zastosowań Informatyki, 70-313 Szczecin, ul. Sikorskiego 37
Zachodniopomorski Uniwersytet Technologiczny, Wydział Elektryczny, Katedra
Elektrotechniki Teoretycznej i Informatyki, 70-313 Szczecin, ul. Sikorskiego 37
* e-mail: Katarzyna.Cichon-Bankowska@zut.edu.pl
ABSTRACT
This paper presents a new promising non-invasive electrical conductivity imaging method of
low-conductivity objects (e.g. biological tissues), called Magneto-Acoustic Tomography with Magnetic
Induction (MAT-MI). The described method combines ultrasound and magnetism into one unified hybrid
technique. The object is placed in a static and a pulsed (μs) magnetic field. The time-varying (pulsed) magnetic
field induces eddy currents in the object. Consequently, the object emit ultrasonic waves through the Lorentz
force produced by the combination of the eddy current and the static magnetic field. The ultrasonic waves are
then collected by the detectors located around the object for image reconstruction. MAT-MI combines the good
contrast of Electrical Impedance Tomography (EIT) with the good spatial resolution of sonography. The aim of
this paper is a review of theoretical studies of MAT-MI and discussion of possible future research directions.
Keywords: magnetoacoustic tomography with magnetic induction (MAT-MI), conductivity imaging of
biological structures
STRESZCZENIE
W pracy opisano nową obiecującą metodę nieinwazyjnego obrazowania przewodności elektrycznej obiektów
słabo przewodzących (np. tkanek biologicznych) zwaną tomografią magnetoakustyczną ze wzbudzeniem
indukcyjnym (ang. Magneto-Acoustic Tomography with Magnetic Induction – MAT-MI). Prezentowana metoda
jest techniką hybrydową i łączy badania ultradźwiękowe z magnetyzmem. Badany obiekt umieszczony jest
w statycznym oraz impulsowym (μs) polu magnetycznym. Zmienne pole magnetyczne indukuje w nim prądy
wirowe. W wyniku działania sił Lorentza powstałych dzięki interakcji prądów wirowych oraz statycznego pola
magnetycznego, obiekt będzie emitował fale ultradźwiękowe. Fale te są odbierane przez detektory umieszczone
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wokół obiektu. Metoda MAT-MI łączy dobry kontrast elektrycznej tomografii impedancyjnej (ang. Electrical
Impedance Tomography – EIT) z dobrą rozdzielczością przestrzenną metody ultradźwiękowej. Celem niniejszej
pracy jest przegląd badań teoretycznych metody MAT-MI oraz dyskusja przyszłych kierunków badawczych.
Słowa kluczowe: tomografia magnetoakustyczna ze wzbudzeniem indukcyjnym (MAT-MI), obrazowanie
konduktywności struktur biologicznych
1. Introduction
Medical imaging techniques play an increasingly significant role in biomedical research and clinical
diagnosis. Non-invasive electrical conductivity imaging of low-conductivity objects (e.g. biological
tissues) has been an active research area for the last several years [1, 2]. Up until now, many papers
have been published on Magneto-Acoustic Tomography with Magnetic Induction (MAT-MI),
however only a few referred to imaging of biological tissue’s conductivity, which is in the range
of 1 S/m. Figure 1 shows the plot of electrical conductivity versus electrical current frequency, for
different tissues [3, 4].
Fig. 1. Electrical conductivity versus electrical current frequency for different tissues [3, 4]
Electrical properties of biological tissues are known to be sensitive to their physiological and
pathological conditions. For example, human breast cancer or liver tumor has a meaningfully higher
electrical conductivity than the surrounding tissue [5, 6]. Table 1 presents a comparison of
conductivity values for various mice tissues (normal and cancerous) [7].
Table 1. Conductivity and density value for each tissue group measured from one random sample [7]
Samples
Normal mice breast
Cancerous mice breast
Conductivity (S/m)
0,239
0,547
Density (kg/m3)
1,121
1,319
In addition, the knowledge of biological tissue conductivity is of great interest to researchers
carrying out electromagnetic source imaging. In order to improve overall MAT-MI system sensitivity
and apply these methods in clinical diagnosis, further research needs to be carried out.
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2. Magnetoacoustic tomography with magnetic induction (MAT-MI) – theoretical
background
A number of efforts have been made to characterize tissue properties of biological systems through
the use of the acoustic theory. In such a method, traditional ultrasound imaging is used to image the
acoustic impedance distribution. Its high spatial resolution combined with a real-time imaging
capability, lack of side effects, and relatively low cost make it an attractive technique. However, with
an ultrasonic approach, it can be difficult to differentiate soft tissues because acoustic impedance
varies by less than 10% among muscle, fat, and blood (see Table 1) [7, 8]. Magnetoacoustic (MA)
technique was developed to improve spatial resolution and contrast of the existing tissue
characterization techniques. Using this technique, an electrical current is injected onto an object under
a static magnetic field, generating acoustic signals induced by the Lorentz force. To avoid current
injection and the shielding effect accompanied with current injection, a new MA approach, called
magnetoacoustic tomography with magnetic induction (MAT-MI), was recently proposed [2]. The
proposed method combines ultrasound and magnetism into one, unified hybrid technique. In this
method, the object is placed in a static and a pulsed (μs) magnetic field.
The pulsed magnetic field induces an eddy current in the object. Consequently, the object emits
ultrasonic waves through the Lorentz force produced by the combination of the eddy current and the
static magnetic field. The ultrasonic waves are then collected by the detectors located around the
object for image reconstruction. MAT-MI combines the good contrast of EIT with the good spatial
resolution of sonography [9]. Recently published articles report on high resolution 2 mm and
sensitivity in a range of 1 S/m, which seems to be highly promising in medical imaging diagnosis [10,
11]. At last, MAT-MI is compatible with the MRI setup. In both imaging modalities, the sample is
located in a static magnetic field and a time-varying magnetic field. However, MAT-MI is much less
demanding in terms of field homogeneity and stability than MRI.
2.1. Forward problem
The forward problem of MAT-MI consists of two physical processes, i.e. 1) magnetic induction in
a conductive object and 2) acoustic wave propagation with the Lorentz-force-induced acoustic sources.
The static magnetic field with flux density B0(r) (r is the position vector), the stimulating time-varying
magnetic field B1(r, t) (t is the time) and isotropic electrical conductivity distribution σ(r) of the
conductive object  are known, while the generated MAT-MI acoustic signal is unknown.
The magnetic stimulation B1(r, t) can represented by the curl of its vector potential A(r, t),
B1(r, t) =   A(r, t) [12]. In MAT-MI around μs level current pulses for driving the stimulating coil
are used. Assuming conductivity of the object of about 1 S/m and relative magnetic permeability of 1,
the magnetic induction problem can be considered quasi-static. This condition also indicates that the
total magnetic vector potential of the tested object can be very well approximated by the primary
vector potential Ap. The electric field inside the object and the eddy current density vector can be
expressed as [12]:
E r,t   
Ap  r , t 
t
 Φ  r , t  ,
J  r, t   σ  r  E  r, t 
(1)
Calculation of the primary magnetic vector potential for any coil configuration is a relatively easy
task (well known in the literature), so to solve the first physical process in the forward problem we
need to find the scalar potential (r, t) only. Following the procedure shown in the [12] forward
problem, one can find the partial differential equation governing the scalar potential function. The
final equation is:
   r  r , t   
A p r , t 
t
  r , t 
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with the condition at the outer boundary of the object:
Apn r , t 
 r , t 
,

n
t
(3)
where n is a direction of the unit vector normal to the boundary.
The quasi-static condition allows us to separate the spatial and temporal functions of the
time-varying fields, therefore: B1(r, t) = B1(r)f(t), Ap(r, t) = Ap(r)f(t), but: Φ(r, t) = Φ(r)f’(t),
E(r, t) = E(r)f’(t) and J(r, t) = J(r)f’(t), where f(t) is the temporal function of time, and the prime
denotes the first order time derivative. If the magnetic field stimulation pulse is short enough, it is
reasonable to assume that f(t) is a unit step function over time. In such a case f’(t) = δ(t), where δ(t) is
the delta function in time domain. After cancelling out the time functions, the equation (2) takes the
following form:
   r  r    Ap r    r 
Φ  r 
n
  Apn  r 
in 
(4)
at ∂
(5)
With the magnetically induced eddy currents J(r, t) = J(r)δ(t) and the static magnetic field B0(r),
the Lorentz force acting on the eddy currents over unit volume can be written as:
F  r , t    J  r   B0  r  δ  t 
(6)
The Lorentz force leads to mechanical vibrations and generates an acoustic wave governed by the
following wave equation [2]:
2 p  r, t  
2
1  p  r,t 
    J  r   B0  r   δ  t 
cs2
t 2
(7)
where p(r, t) is the acoustic pressure at spatial point r, cS = 1/√(ρ0 βS) is the acoustic speed, 0 is the
density of the material at rest and s is the adiabatic compressibility of the medium. By using Green’s
function, the acoustic field can be solved as [2]:

R
δ t  
cs 
1
pr , t      r '  J r '  B0 r ' 
dr '
4π V
R
(8)
where R = |r – r'| and the integration is over the sample volume. This equation gives the observed
acoustic pressure for an impulse source. In the physical experiments, the pressure can be measured
using ultrasonic transducers around the object. The measured or simulated pressure can be next used
for inverse reconstruction of the conductivity distribution.
2.2. Inverse problem
In the inverse problem the acoustic pressure signals around the object, either measured or simulated,
are used for reconstruction of the conductivity distribution σ(r) within the object. The inverse problem
can be essentially divided into two steps. In the first step, the source term ∙[J(r)  B0(r)], is
reconstructed from pressure. Different techniques can be used. The techniques will be studied in the
project. Using the time reversal method, the acoustic source can be described as follows [13]:
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1
   J  r   B0  r    
2πcs3

Sd
n   r  rd   2 p  rd , t 
r  rd
2
t 2
dS d
(9)
t  r  rd cs
where rd is the detection point on the surface Sd and n is a unit vector normal to the surface Sd at rd.
In the second step, the conductivity distribution σ(r) is reconstructed from ∙[J(r)  B0(r)]. The
problem is really challenging and different possibilities exist. In [13] the following approximate
formula is given:
σ r   
   J  r   B0  r  
(10)
B1  r   B0  r 
In last two years several improvements have been identified, however MAT-MI method is still in
its infancy. There are several universities and research institutions around the world studying
MAT-MI, including Minnesota University, Zhejiang University, Central South University for
Nationalities, Institute of Electrical Engineering, and Chinese Academy of Sciences. MAT-MI was
tested to image a wire phantom as well as real biological tissue (e.g. breast, liver) with different
conductivity in vitro [14]. The result shows that MAT-MI is sensitive to differentiating various types
of tissue with different conductivity in the image, even when tissues are acoustically inhomogeneous.
However, further research is needed to improve the inverse algorithm in order to obtain enhanced
imaging results.
3. Future research directions
The future research directions will focus mainly on building and testing MAT-MI laboratory station
(schematically shown in Figure 2).
preamplifier
Fig. 2. Proposed diagram of the measurement MAT-MI system setup
In parallel, analytical and numerical MAT-MI model formulations (so-called “forward problem”)
will be developed. Usually, solving a simplified test problem by using an analytical method is a useful
exercise to check the validity of the numerical model. Then, a full 3D numerical model will be
established using the finite element method (FEM). In order to perform such analysis, commercial
software Comsol Multiphysics will be used. Its open architecture allows the introduction of our own
unique procedures (e.g. infinite elements, absorbing boundary conditions [15], etc.). Based on
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retrieved data from measurements as well as from numerical calculations, the inverse problem (finding
a map of conductivity) will be formulated and finally solved. In magnetic induction tomography such
issue belongs to the class of ill-posed inverse problems which are numerically unstable. Obtained
information will give the rise to the application of artificial intelligence algorithms.
4. Conclusions
Medical imaging techniques play an increasingly significant role in biomedical research and clinical
diagnosis. Among all the related techniques, non-invasive electrical low-conductivity imaging
methods have been actively investigated for last several years because electrical properties of
biological tissues are known to be sensitive to their physiological and pathological conditions. For
example, it has been shown that tissues such as human breast cancer or liver tumor have
a meaningfully higher electrical conductivity than the surrounding tissue. It suggests that MAT-MI
merit further investigations, as they may become important non-invasive methods for imaging of
biological tissues’ electrical conductivity (e.g. cancerous tissue versus normal tissue). The data fusion
techniques may have potential applications to cancer and tumor imaging.
This topic has a rank of the original, theoretical and experimental basic researches which determine
new directions in science and will be helpful to get new knowledge about investigated phenomena.
According to the report FORESIGHT in 2008, screening-oriented technologies are one of the
priorities of research with a high potential for the desired economic development and social needs with
optimal use of limited public funds.
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otrzymano / submitted: 20.11.2014
wersja poprawiona / revised version: 02.12.2014
zaakceptowano / accepted: 20.12.2014
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