Dissertation Bălan S Marian Răzvan Mtp. Converted [607448]

UNIVERSITY OF CRAIOVA
FACULTY OF SCIENCE
DEPARTMENT OF PHYSICS
MASTER’S DEGREE STUDIES
THEORETICAL PHYSICS.

DISSERTATION THESIS

SCIENTIFIC COORDINATOR
Prof. univ. dr. (Ph.D) Radu Dan CONSTANTINESCU

GRADUATE
Marian -Răzvan BĂLAN

CRAIOVA
2020

UNIVERSITY OF CRAIOVA
FACULTY OF SCIENCE
DEPARTMENT OF PHYSICS
MASTER’S DEGREE STUDIES
THEORETICAL PHYSICS.

THEORETICAL METHODS IN
BIOELECTROMAGNETISM

SCIENTIFIC COORDINATOR
Prof. univ. dr. (Ph.D) Radu Dan CONSTANTINESCU

GRADUATE
Marian -Răzvan BĂLAN

CRAIOVA
2020

Theoretical Methods in Bioelectromagnetism
CONTENTS
Chapter 1
1.1 Introduction……………………… ……………………………………………… pag.
1.2 SUBDIVISIONS OF BIOELECTROMAGNETISM… ……………………….
1.2.1 Division on a Theoretical Basis ……………………………………………….
1.2.2 Division on an Anatomical Basis…………………………… ………………..
1.2.3 Importance of Bioelectromagnetism………………………… ……………….
Chapter 2 : Theoretical Methods for analyzing volume sources and volume conductors
2.1.a Introduction …………………………………………………………………….
2.1.b SOLID ANGLE THEOREM …………………………………………………
2.1.1 Inhomogeneous Double Layer ………………………………………………
2.1.2 Uniform double layer…………………………………………………..
2.2. MILLER -GESELOWITZ MODEL ……………………………………………
2.3. LEAD VECTOR …………………………………………………………………
2.3.1 Definition of the lead vector……………………………………………
2.3.2 Extending the concept of Lead Vector……………………………………..
Chapter 3 : Theory of Biomagnetic Measurements ………………………………..
3 BIOMAGNETIC FIELD …………………………………………………………..
Chapter 4 : Maxwell’s equations in bioelectromagnetism.
4.1. Introduction ……………………………………………………………… ………
4.2 MAXWELL'S EQUATIONS UNDER FREE SPACE CONDITIONS ……….

4.3 MAXWELL'S EQUATIONS FOR FINITE CONDUCTING MEDIA …………
4.4 SIMPLIFICATION OF MAXWELL'S EQUATIONS IN PHYSIOLOGICAL
PREPARATIONS ………………………………………………………………………….
4.4.1 Frequency Limit………………………………………………………………
4.4.2 Size Limitation………………………………………………………………….
4.4.3 Volume Conductor Impedance…………………………………………………
4.5 MAGNETIC VECTOR POTENTIAL AND ELECTRIC SCALAR POTENTIAL IN
THE REGION OUTSIDE THE SOURCES …………………………………………………..
4.6 STIMULATION WITH ELECTRIC AND MAGNETIC FIELDS
4.6.1 Stimulation with Electric Field……………………………………………
4.6.2 Stimulation with Magnetic Field……………………………………………
4.7 SIMPLIFIED MAXWELL'S EQUATIONS IN PHYSIOLOGICAL PREPARATIONS
IN THE REGION OUTSIDE THE SOURC ES……………………………………………
CONCLUSIONS………………………………………………………………… …………..
BIBLIOGRAPHY …………………………………………………………………………….

Theoretical Methods in Bioelectromagnetism
Chapter 1
1.1 Introduction
Bioelectromagnetism is a discipline that study of the interaction between the electric,
electromagnetic, and magnetic phenomena which arise in biological tissues.
These phenomena include:
– the behavior of excitable tissue (the sources) ;
– the electric currents and pot entials in the volume conductor;
– the magnetic field at and beyond the body ;
– the response of excitable cells to electric and magnetic field stimulation ;
– the intrinsic electric and mag netic properties of the tissue;
It is import ant to separate the concept of bioelectromagnetism from the conce pt of medical
electronics; the former involves bioelectric, bioelectromagnetic, and biomagnetic phenomena
and measurement and stimulation methodology, whereas the latter refers to th e actual devices
used for these purposes. By definition, bioelectromagnetism is an interdisciplinary since it
involves the as sociation of the life sciences with the physical and engine ering sciences.
Consequently, it have a special interest in those disciplines that combine engineering and
physics with biology and medicine. These disc iplines are briefly defined as it follows:
Biophysics: t he science that use the concepts of physics that find solution s of biological
problems.
Bioe ngineering: t he application of engineering to the development of healt h care
devices, analysis of biological systems, and manufacturing of products based on advances in this
technology. This term is also frequently used to include both biomedical engineering and
biochemical engineering (biotechnology).
Biotechnology: t he study of microbiological proc ess technology. t he mai n fields of
application of biotechnology are agriculture, and food and also drug production.
Medical electronics : A subdivision of biomedical engineering concerned with electronic
devices and methods in medicine.

Medical physics: A science based upon physical problems in clinical medicine is a branch
of physics that aims to preserve and restore human health with the help of medical equipment and
physical agents and to study for this purpose the physical processes of life, properties, structures
and functions of the body, causes and mechanisms of disease, as well as the fundamental laws of
physics applied in m edicine for diagnosing, treating and preventing diseases.
Biomedical engineering: An engineering discipline concerned with the application of
science and technology (devices and methods) to biology and medicine.

FIG 1.

Fig. 1. Currently recognized interdisciplinary fields that associate physics and engineering
with medicine and biology:
BEN = bioengineering,

BPH = biophysics,
BEM = bioelectromagnetism
,
MPH = medical physics,
MEN = medical engineering,
MEL = medical electronics.

Figure 1. illustrates the relationships between these disciplines. The coordin ate origin
represents the more theoretical sciences, such as biology and physics. As one moves away from
the origin, th e sciences become increasing ly applied. Combining a couple of sciences from
medical and technical fields yields interdisciplinary sciences s uch as medical engineering.
It must be understood that the discipline s are actually multidimensional, and thus their
two-dimensional d escription is only suggestive.

1.2 SUBDIVISIONS OF BIOELECTROMAGNETISM
1.2.1 Division on a Theoretical Basis

The discipline of bioelectromagnetism may be subdivided in many di fferent ways.
One such classification divides the field on theoretical grounds according to two universal
principles:
Maxwell's equations (the electromagnetic connection) and the pr inciple of reciprocity.
This philosophy is illustrated in Figure 1.2 and is discussed in greater detail below.

Maxwell's Equations
Maxwell's equations, i.e. the electromagnetic connection, connect time -varying electric
and magnetic fields so that when there are bioelectric fields there always are also biomagnetic
fields, and vice versa (Maxwell, 1865). Depending on whether we discuss electric,
electromagne tic, or magnetic phenomena, bioelectromagnetism may be divided along one
conceptual dimension (hori zontally in Figure 1.2) into three subdivisions, namely
(A) Bioelectricity,

(B) Bioelectr omagnetism (biomagnetism).
(C) Biomagnetism.

Subdivision B has historically been called "biomagnetism" which unfortun ately can be
confused with our Subdivision C. For Subdivision B it was also use d the conventional name
"biomagnetism" but, where appropriate, I think that the more precise term is
"bioelectromagnetism." H owever, because this field is not easily detected and does not have
any known value, I have om itted it from this dissertation ).

Reciprocity
Owing to the principle of reciprocity, the sensitivity distribution in the detection of
bioelectric signals, the energy distribution in electric stimulation, and the sensitivity
distrib ution of electric impedance measurements are the same. This is also true for the
corresponding bioelectromagnetic and biomagnetic methods, respectively. Depending on
whether we discuss the me asurement of the field, of stimulation/magnetization, or the
measurement of intrinsic properties of t issue, bioelectromagnetism may be divided within this
framework (vertic ally in Figure 1.2) as follows:
(I) Measurement of an electric or a magnetic field from a bioelectric source or (the
magnetic field from) magnetic material.
(II) Electric stimulation with an electric or a magnetic field or the magnetization of
materials (with magnetic f ield).
(III) Measurement of the intrinsic electric or magnetic properties of tissue.

Fig. 1.2. Organization of bioelectromagnetism into its subdivisions. It is first divided
horizontally to:
A) bioelectricity
B) bioelectromagnetism (biomagnetism), and
C) biomagnetism.
Then the division is made vertically to:
I) measurement of fields,
II) stimulation and magnetization, and
III) measurement of intrinsic electric and magnetic properties of tissue.
The hor izontal divisions are tied together by Maxwell's equations and the vertical divisions
by the principle of reciprocity.

Description of the Subdivisions
The aforementioned taxonomy is illustrated in Figure 1.2 and a detailed description of its
elements is given in this section.

(I) Measurement of an electric or a magnetic field refers, essentially, to the electric or
magnetic signals produced by the activity of living tissues. In this subdivision of
bioelectrom agnetism, the active tissues produce electromagnetic energy, which is measured either
electrically or magnetically within or outside the organism in which the source lies. This
subdivision includes also the magne tic field produced by magn etic material in the tissue. Examples
of these fields in the three horizontal subd ivisions are shown in Table 1.

Table 1.
Measurements of fields
(A) Bioelectricity
(B) Bioelectromagnetism
(C) Biomagnetism
Neural cells
electroencephalography (EEG)
magnetoencephalography (MEG)
electroneurography (ENG)
magneto neurography (MNG)
electroretinography (ERG)
magnetoretinography (MRG)
Muscle cells
electrocardiography (ECG)
magnetocardiography (MCG)
electromyography (EMG)
magnetomyography (MMG)

Other tissue
electro -oculography (EOG)
magneto -oculo graphy (MOG)
electronystagmography (ENG)
magnetonystagmography (MNG)
magnetopneumogram
magnetohepatogram .

Lead Field Theoretical Approach
As was mentioned in the beginning of section, Maxwell's equations co nnect time -varying
electric and magnetic fields, so that when there are bioelectric fields there are also biom agnetic
fields, and vice versa. This electromagnetic connection is the univers al principle link ing the se three
subdivisions – A, B, and C – of bioelectromagnetism in the horizontal direction in Figure 1.2. As
noted in the beginning of this section, the sensitivity distribution in the detection of bioelectric
signals, the energy dist ribution in electric stimulation, and the sensitivity distribution of the electric
impedance measurement are the same. All of this is true also for the corresponding
bioelectromagnetic and biomagnetic methods, respec tively. The universal principle that lin k
together the three subdivisions I, II, and III and unifies the discipline of bioelectromagnetism in
the vertical direction in Figure 1.2 is the principle of reciprocity.
I wanted to introduce this figure early, because it illustrates the fundamental prin ciples
governing the entire discipline of bioelectromagnetism.

Fig. 1.3. Lead field theoretical approach to describe the subdivisions of
bioelectromagnetism. The sensitivity distribution in the detection of bioelectric signals, the energy
distribution in electric stimulation, and the distribution of measurement sensitivity of electric
impedance are the same, owing to the principle of reciprocity. This is true also for the
corresponding bioelectromagnetic and biomagnetic methods. Maxwell's e quations tie time –
varying electric and magnetic fields together so that when there are bioelectric fields there are also
bioelectromagnetic fields, and vice versa.

1.2.2 Division on an Anatomical Basis
Bioelectromagnetism can be classified also along anatomical lines. This division is
appropriate especially when one is discussing clinical applications. In this case,
bioelectromagnetism is subdivided according to the applicable tissue. For example, one might
consider:
a) neurophysiological b ioelectromagnetism;
b) card iologic bioelectromagnetism;
c) bioelectromagnetism of other organs or tissues.

1.2.3 Importance of Bioelectromagnetism

Why should we consider the study of electric and magnetic phenomena in living tissues as
a separate discipline?
The main reason is that bioelectric phenomena of the cell membrane are vital functions of
the living organism. The cell uses the membrane poten tial in several ways. With fast opening of
the channels for sodium ions, the membrane potential is altered radically within a fraction of a
second. Cells in the nervous system communicate with one to another by means of such electric
signals that fastly travel along the nerve processes. In fact, life itself begins with a change in this
membrane potential. As the sperm merges with the egg cell at the instant of fertilization, ion
channels in the egg are activated. The resultant change in the membrane potential prevents access
of other sperm cells. Electric phenomena are easily measured, and therefore, this approach is direct
and feasible. In the investigation of other modalities, such as biochemical and biophysical events,
special transducers must be used to convert the phenomenon of interest into a measurable electric
signal. In contrast electric phenomena can easily be directly measured with simple electrodes;
alternatively, the magnetic field they prod uce can be detected with a magnetometer. In contrast to
all other biological variables, bioelectric and biomagnetic ph enomena can be detected in real time
by noninvasive methods because the information obtained from them is manifested immediately
throughou t and around the volume conductor formed by the body. Thei r source may be
investigated by applying the modern theory of volume sources and volume conductors, utilizing
the computi ng capability of modern computers. (The concepts of volume sources and volume

cond uctors denote three -dimensional sources and conductors, respectively, having large
dimensions relative to t he distance of the measurement.
Conversely, it is possible to int roduce temporally and spatially controlled electric
stimulations to activate paralyzed regions of the neural or muscular systems of the body.
The electric nature of biological tissues permits the transmission of signals for information
and for control and is therefore of vital importance for life. The first category includes suc h
examples as vision, audition, and tactile sensation; in these cases a peripheral transducer (the eye,
the ear, e tc.) initiates afferent signals to the brain. Efferent signals originating in the brain can
result in voluntary c ontraction of muscles to effect the movement of limbs, for example. And
finally, homeostasis involves clos ed-loop regulation mediated, at least in part, by electric signals
that affect vital physiologic functions such as heart rate, strength of cardiac contracti on, humoral
release, and s o on. As a result of the rapid development of electronic instrumentation a nd computer
science, diagnostic instruments, which are based on bioelectric phenomena, have deve loped very
quickly. Today it is impossible to imagine any hospital or doctor's office without
electrocardiography and electroencephalography. The development of microelectronics has m ade
such equipment portable and improv ed their diagnostic power. Implantable cardiac pacemakers
have allowed millions of people with heart problems to return to normal life. Biomagnetic
applica tions are likewise being fast developed and will, in the future, supplement bioelectric
methods in medical diagno sis and therapy. These examples reveal that bioelectromagnetism is a
vital part of our everyday life.
Bioelectromagnetism makes it possible to investigate the behavior of living tissue on both
cellular and organic levels. Furthermore, the latest scientific achievements now allow s cientists to
do research at the subcellular level by measuring the electric current flowing through a single ion
channel of the cell membrane with the patch -clamp method. With the latter approach,
bioele ctromagnetism can be also applied to molecular biology and to the development of new
pharmaceuticals.
Thus bioelectromagnetism offers new ways and important opportunities for the
development of diagnostic and therapeutic methods.

Chapter 2
Theoretical Methods for analyzing volume sources and
volume conductors
2.1.a Introduction
The two theoretical methods of this chapter (solid angle theorem an d Miller -Geselowitz
model) are used to evaluate the electric field in a volume conductor produced by the source – that
is, to solve the forward problem . After this discussion is a presentation of methods used to
evaluate the source of the electric field from measurements made outside the so urce, inside o r on
the surface of the volume conductor – that is, to solve the inverse problem . To obtain the desired
information at optimized capacity we can use these methods wich are important in designing
electrode configurations.
In fact, application of each of the following methods usually re sults in a particular ECG –
lead system. These lead systems are not discussed here in detail because the purp ose of this
chapter is to show that these methods of analysis form an independent theory of bioelectricity
that is not limited to particular ECG applications.
The biomagnetic fields resulting from the electric activity of volume source s are
discussed in detail as it follows.
2.1.b SOLID ANGLE THEOREM
2.1.1 Inhomogeneous double layer
PRECONDITIONS:
SOURCE: Inhomogeneous double layer
CONDUCTOR: Infinite, homogeneous, (finite, inhomogeneous)
The solid angle theorem was developed by the German physicist Herma nn von
Helmholtz in the middle of the nineteenth century . In this theory, a double layer is used as the
source , we now examine the structure of a double layer in somewhat greater detail.
Suppose that a point current source and a current sink (i.e., a negative source) of the same
magnitude are located close to each other. If their strength is i and the distance between them is

d, they form a dipole moment id. Consider now a smo oth surface of arbitrary shape lying within
a volume conductor. We can uniformly distribute many such dipoles over its surface, with eac h
dipole placed normal to the surface. In addition, we choose the dipole density to be a well –
behaved function of position – that is, we assume that the number of dipoles in a smal l area is
great enough so that the density of dipoles can be well approximat ed with a continuous funct ion.
Such a source is called a double layer (Figure 1 .4). If it is denoted by p(S) , then p(S) denotes a
dipole moment density (dipole moment per unit area) as a function of position, while its direction
is denoted by , the surface nor mal. With this notation, p(S)d is a dipole whose magnitude is
p(S)dS , and its direction is normal to the surface at dS.
An alternative point of view is to recognize that on one side of the double layer, the
sources form a current density J [A/m2] whereas on the other side the sinks form a current
density -J [A/m2], and that the conducting sheet between the surfaces of the double layer has a
resistivity ρ. The resistance across this sheet (of thickness d) for a unit cross -sectional area is
R = ρd (1)
where R = double layer resistance times unit area [Ωm2]
ρ = resistivity of the medium [Ωm]
d = thickness of the double layer [m]
Of course, the double layer arises only in the limit that d 0 while J such
that Jd p remains finite.

Fig. 1 .4 Structure of a double layer. The double layer is formed when the dipole density
increases to the point that it may be considered a continuum. In addition, we require that J
, d 0, and Jd p.
From Ohm's law we note that the double layer has a potential difference of
Vd = Φ 1 – Φ2 = Jρd (2)
where Vd = voltage difference over the double layer [V]
Φ1, Φ 2, = potentials on both sides of the double layer [V]
J = double layer current density [A/m2]
ρ = resistivity of the medium [Ωm]
d = double layer thickness [m]
By definition, the double layer forms a dipole moment per unit surface area of
p = Jd (3)
where p = dipole moment per unit area [A/m]
J = double layer current density [A/m2]
D = double layer thickness [m]
As noted, in the general case (nonuniform double layer), p and J are functions of position.
Strictly we require d 0 while J such that Jd = p remains finite. (In the case
where d is not uniform, then for e quation (2) to be a good approximation it is required that ΔΦ not
vary significantly over lateral distances several times d.)
Since p is the dipole moment per unit area (with the direction from negative to positive
source), dS is an elementary dipole. I ts field, given by e quation is:
(4)

since the direction of and d are the same. Now the solid angle d Ω, as defined by Stratton
(1941), is:
(5)
Thus
(6)

Fig. 1.5 . A sketch of some isopotential points on an isopotential line of the electric field
generated by a uniform double layer. That these points are equipotential is shown by the identity
of the solid angle magnitudes. Accordi ng to the convention chosen in e quati on (5), the sign of the
solid angle is negative.
The double layer generates a potential field given by equation (6), where dΩ is the element
of solid angle, as seen from the field point as the point of observation (Figure 1.5 ). This figure
provides an interpretation of the solid angle as a measure of the opening between rays from the
field point to the periphery of the double lay er, a form of three -dimensional angle.
Equation (6) has a part icularly simple form, which eas ily permits an estimation of the
field configuration arising from a given double layer source function.
This result was first obtained by Helmholtz, who showed that it holds for an infinite,

homogeneous, isotropic, and linear volume conductor. Later the solid angle theorem was also
applied to inhomogeneous volume conductors by utilizing the concept of second ary sources.
As well known , the inhomogeneous volume conductor may be represented as a
homogeneous volume conductor including secondary sources at the sites of the boundaries. Now
the potential field of a double layer source in an inhomogeneous volume con ductor may be
calculated with the solid angle theorem by applying it to the primary and secondary sources in a
homogeneous volume conductor.
The polarity of the Potential Field
We discuss shortly the polarity of the potential field generated by a double layer. This will
clarify the minus sign in e quations (5) and (6).
If the double layer is uniform, then the field point's potential is proportional to the total
solid angle subtended at the field point. It is therefore of interest to be able to determine this solid
angle. One useful l approach is the following: From the field point, draw lines (rays) to the
periphery of the double layer surface. Now construct a unit sphere centered at the field point. The
area of the sphere surface intercepted by the rays is the solid angle. If the nega tive sources
associated with the double layer face the field point, then the solid angle will be positive, according
to equation (5). This polarity arises from the purely arbi trary way in which the sign in e quation (5)
was chosen. Unfortunately, the litera ture contains both sign choices in the definition of the solid
angle (in this description I’ve adopted the one defined by Stratton, 1941).
For example, suppose a uniform double layer is a circular disk centered at the origin, whose
dipoles are oriented in the x direction. For a field point along the positive x -axis, because the field
point faces positive sources, the solid angle will be negative. However , because of the minus sign
in equation (5), the expression (6) also contains a minus sign. As a consequence, the potential,
evaluated from e quation (6), will be positive, which is the expected polarity.

2.1.2 Uniform double layer
PRECONDITIONS:
SOURCE: Uniform double layer
CONDUCTOR: Infinite, homogeneous
A uniform double layer exhibits some interesting properties that are discussed here in this
section.
To begin with, we note that e quation (6) describes the potential field in an infinite volume
conductor due to an inhomogeneous double layer; this reduces to the following simplified form
when the double layer is uniform:
(7)
Consider a closed uniform double layer. When such a double layer is seen from any point
of observation, it can always be divided into two parts. One is seen from the positive side and the
other is seen from the negative side, though each has exactly the same magnitude solid a ngle Ω,
as described in Figure 1.5 . (Double layer sources having more complex form can, of course, be
divided into more than two parts.) These both produce a potential of the same magnitude, but
because they have opposite signs, they cancel each other.
As a result, a closed uniform double l ayer produces a zero field, when considered in its
entirety.
Wilson (1931) applied this principle to electrocardiography, since he understood the
cardiac double layer source formulation. Suppose that the double layer formed by the
depolarization in the ventricles includes a single wavefront, which is represented by a uniform
doub le layer, and has the shape of a cup. If this cup is closed with a "cover" formed by a double
layer of similar strength, then a closed surface is formed, that does not generate any potential
field. From this we can conclude that the double layer having the shape of a cup can be replaced
with a double layer having the shape of the cup's cover, but with its double layer oriented in the
same direction as the cup, as described in Figure 1.5. From this example one can assert that two
uniform double layers with t he same periphery generate identical potential fields.

The field generated by a double layer disk at distances that are much greater than the disk
radius appears to originate from a single dipole. In fact, at large enough distances from any
dipole distri bution, the field will appear to originate from a single dipole whose strength and
orientation are the vector sum of the source components, as if they were all located at the same
point. This is the reason why the electric field of the heart during the act ivation has a dipolar
form and the concept of a single electric heart vector (EHV), as a description of the cardiac
source, has a wide application. This is particularly true when the activation involves only a single
ventricle. The true situation, where th e right and left ventricle are simultaneously active, is more
accurately repre sented by two separate dipoles. This same argument may be used in explaining
the effect of an infarct on the electric field of the heart. The infarct is a region of dead tissue; it
can be represented by the absence of a double layer (i.e., an opening in a double layer). As a
consequence, closing the double layer surface in this case introduces an additional cover, as
shown in Figure 1.7. The latter source is a direct reflection of the effect of the infarct. (The
paradox in this deduction is that the region of dead tissue is represented by an active
dipole directed inward.)
Finally, we summarize the two important properties of uniform double layers defi ned by
the solid angle theorem: A closed uniform double layer generates a zero external potential field
and t he potential field of an open uniform double layer is completely defined by the rim of the
opening (Wikswo et al., 1979).

Fig. 1.6 . A closed uniform double layer produces a zero potential field.

Fig. 1.7. The potential field of an open uniform double layer is completely defined by the
rim of its opening.
2.2. MILLER -GESELOWITZ MODEL
PRECONDITIONS:
SOURCE: Distributed dipole, cellular basis
CONDUCTOR: Finite, homogeneous
W. T. Miller and D. B. Geselowitz (1978) developed a source model that is based directly
on the generators associated with the activation of each cell. Their basic expression is patterned
after e quation:

which assigns a dipole source density to the spatial derivative of transmembrane voltage. For
three dimensions, instead of a derivative with respect to a single variable, a gradient (including
all three variables) is required. Consequently,
i = -σVm (8)
where
i = dipole source density [µA/cm2]
σ = conductivity [mS/cm]
Vm = spatial derivative of transmembrane voltage [mV/cm]
Miller and Geselowitz used published data to reveal and evaluate the action potential
waveforms at various sites throughout the heart as well as times of activation. They could thus
estimate Vm(x,y,z,t) and as a result, could evaluate the "actual" dipole moment per unit volume at
all points. For simplicity the heart was divided into a finite number of regions, and the ne t dipole
source strength in every region found by summing idV in that region.
In determining the surface potential fields the authors considered the number of dipole
elements to be a small set (of 21) and evaluated the contribution from each. This part of their
work constituted a relatively straightforward solution of the forward problem (dipole source in a
bounded volume co nductor). The reconstructed electrocardiograms showed very reasonable
qualities.
2.3. LEAD VECTOR
2.3.1 Definition of the lead vector
PRECONDITIONS:
SOURCE: Dipole in a fixed location
CONDUCTOR: Finite (infinite), inhomogeneous
We examine the potential field at a point P, within or at the surface of a volume
conductor, caused by a unit dipole (a unit vector in the x direction) in a fixed locati on Q, as
illustrated in Figure 1.5. (Though the theory, which we will develop, applies to both infinite and
finite volume conductors, we discuss here is only finite volume conductors, for the sake of
clarity.)

Suppose that at the point P the potential ΦP due to the unit dipole is cx. (The potential at
P must be evaluated relative to another local point or a remote reference point. Both choices ar e
followed in electrophysiology . For the present, we assume the existence of some unspecified
remote reference point.) Because of our linearity assumption, the potential ΦP corresponding to a
dipole px of arbitrary magnitude px is
ΦP = cx px (9)
A similar expression holds for dipoles in the y and z directions.
The linearity assumption ensures that the principle of superposition holds, and any
dipole can be resolved into three orthogonal components px, py, pz, and the potentials
from each superimposed. Thus we can express the potential Φ P at point P, due to any dipole at
the point Q
ΦP = cx px + cy py + cz pz (10)
where the coefficients cx, cy, and cz are found (as described above) by energizing the
corresponding unit dipoles at point Q along x-, y-, and z-axes, respectively, and measuring the
corresponding field potentials. Equations (9) and (10) are expressions of linearity, namely that if
the source strength is increased by a factor c, the resultant voltage is increased by the same
factor c. Since no other assumptions were required, equation (10) is valid for any linear volume
conductor, even fo r an inhomogeneous conductor of finite extent.

Fig. 1.8 . Development of the lead vector concept.
(A) Because of linearity, the potential at a point P in the volume conductor is linearly
proportional to dipoles in each coordinate direction.
(B) By superposition the potential at the point P is proportional to the sum of component
dipoles in each coordinate direction. This proportionality is three -dimens ional and can therefore
be considered as a vector , called lead vector .
(C) The potential at the point P is the scalar product of the source dipole and the lead
vector .
Equation ( 10) can be simplified if the coefficients cx, cy, and cz are interpreted as the
components of a vector . This vector is called the lead vector . Consequently, equation ( 10) can
be written as

ΦP = · (11)
The lead vector is a three -dimensional transfer coefficient which describes how a dipole source
at a fixed point Q inside a volume conductor influences the potential at a point within or on the
surface of the volume conductor relative to the potential at a reference location. The value of the
lead vector depends on:
• The location Q of the dipole
• The location of the field point P
• The shape of the volume conductor
• The (distribution of the) resistivity of the volume conductor
We tacitly assume that the potential at the reference is zero and hence does not have to be
considered. Note that the value of the lead vector is a property of the lead and volume conductor
and does not depend on the magnitude or direction of the dipole .
It can be shown that in an infinite, homogeneous volume conductor the lead vector is given
by the sum of components along lines connecting the source point with each of the two electrode
points (each scaled inversely to its physical length). The same also holds for a spherical,
homogeneous volume conductor, provided that the source is at the center.
2.3.2 Extending the c oncept of Lead Vector
In the previous section we considered the lead voltage to be measured relative to a remote
reference – as it is in p ractice in a so -called unipolar lead . In this section, we consider a bipolar
lead formed by a lead pair (where neither electrode is remote), and examine the corresponding
lead vector, as illustrated in Figure 1.9.
For each location P0 . . . P n of P, that lies within or at the surface of the volume
conductor, we can determine a lead vector 0 . . . n for the dipole at a fixed location, so that,
according to equation ( 11), we have
Φi = i · (12)

Then the potential difference between any two points Pi and Pj is
Vi j = Φ i – Φj (13)
This describes the voltage that would be measured by the lead whose electrodes are at
Pi and Pj. To what lead vector does this lead voltage correspond? Consider first the vector ij formed
by
i j = i – j (14)
Now the voltage between the points Pi and Pj given by Equation 13 can also be written, by
substitution from equation ( 12), as follows:
Vi j = Φ i – Φj = i · – j · = i j · (15)
hence identifying i j as the lead vector for leads Pi – Pj. From this result we can express any bipolar
lead voltage V as
(16)
where is a lead vector. We note that e quation (16) for bipolar leads is in the same form as
equation ( 11) for monopolar leads. But e quations (14-16) can be interpreted as that we may first
determine the lead vectors i and j corresponding to unipolar leads at Pi and Pj, respectively, and
then form their ve ctor difference, namely ij. Then the voltage between the points Pi and Pj, as
evaluated by a bipolar lead, is the scalar product of the vector ij and the dipole , as shown in Figure
1.9 and described by e quation (16).

Fig. 1.9 . Determination of the voltage between two points at or within the surface of a
volume conductor.
(A) The potentials Φi and Φj at Pi and Pj due to the dipole may be established with
scalar products with the lead vectors i and j , respectively.
(B) For determining the voltage Vi j between Pi and Pj, the lead vector i j = i – j is first
determined.
(C) The voltage Vi j is the scalar product of the lead vector i j and the dipole .

Chapter 3
Theory of Biomagnetic Measurements
3 BIOMAGNETIC FIELD
PRECONDITIONS:
SOURCE: Distribution of impressed current source elements i (volume source)
CONDUCTOR: Finite, inhomogeneous
The current density throughout a volume conductor gives rise to a magnetic field given
by the following relationship (Stratton, 1941; Jackson, 1975):
(1)
where r is the distance from an external field point at which is evaluated to an element of
volume dv inside the body, dv is a source element, and is an operator with respect to the so urce
coordinates. Substituting e quation which is repeated here,

into e quation (1) and dividing the inho mogeneous volume conductor intohomogeneous
regions vj with conductivity σj, we obtain

(2)
If the vector identity Φ = Φ + Φ is used, then the integrand of the
last term in equation ( 2) can be written σj [Φ (1/r)] – Φ (1/r). Since Φ = 0 for
any Φ, we may replace the last term including its sign by

(3)
We now make use of the following vector identity (Stratton, 1941, p. 604):
(4)
where the surface integral is taken over the surface S bounding the volume v of the volume
integr al. By applying ( 4) to equation ( 3), the last term in equation ( 2), including its sign, can now
be replaced by
(5)
Finally, applying this result to equation (2) and denoting again the primed and double –
primed regions of conductivity to be inside and outside a boundary, respectively, and orienting d
j from the primed to double -primed region, we obtain (note that each interface arises twice, once
as the surface of vj and secondly from surfaces of each neighboring region of vj )

(6)
This equation describes the magnetic field outside a finite volume conductor containing
internal (electric) volume sources i and inhomogeneities (σ" j – σ'j ). It was fi rst derived by David
Geselowitz(Geselowitz, 1970).
It is important to notice that the first t erm on the right -hand side of e quation (6),
involving i, represents the contribution of the volume source, and the sec ond term the effect of
the boundaries and inhomogeneities. The impressed source i arises from cellular activity and
hence has diagnostic value whereas the second term can be considered a distortion due to the

inhomogeneities of the volume conductor. These very same sources were identified when the
electric field generated by them was being evaluated as

(Just, as in the electric case, these terms are also referred to as primary source and
secondary source.)
Similarly, as discussed in connection with previous equation , it is easy to recognize that if
the volume conductor is homogeneous, the differences (σ" j – σ'j ) in the second expression are
zero, and it drops out. Then the equation reduces to the equation o f the magnetic field due to the
distribution of a volume source in a homogeneous volume conductor.
In the design of high -quality biomagnetic instrumentation, the goal is to cancel the effect
of the secondary sources to the extent possible. From an examina tion of equation (6) one can
conclude that the discontinuity in conductivity is equivalent to a secondary surface source j given
by j = (σ" j – σ'j )Φ where Φ is the surface potential on Sj. Note that j is the same secondary
current source for electric fields anterior equation as for magnetic fields.

Chapter 4
Maxwell’s Equations in B ioelectromagnetism.

4.1. INTRODUCTION
The behavior of time -varying and static electric and magnetic fields are governed by
Maxwell's equations formulated by James Clerk Maxwell (1865; 1873). These equations simply
summarize the mathematical consequences of the classical experiments of Faraday, Ampere,
Coulomb, Maxwell, and others.
Maxwell's equations can be found in general texts on electromagnetic theory. However,
they are essentially applicable to electromagnetic fields in free space (i.e., radiation fields).
Where conducting and/or magnetic media are involved, then, although the equations continue to
be valid, current sources can arise in other ways than specified under free space conditions.
These modifications must be introduced through a consideration of the particular nature of

current sources appropriate for the problem at hand.
The goal here, after introd ucing Maxwell's equations in the form valid for free space
conditions, is to specialize them so that they correctly describe conditions that arise in
bioelectromagnetism. Following this, our goal is to simplify the equations where possible, based
on practi cal electrophysiological considerations.
4.2 MAXWELL'S EQUATIONS UNDER FREE SPACE CONDITIONS
PRECONDITIONS:
SOURCES and FIELDS: Time -varying , ρ, ,
CONDUCTOR: Infinite, homogeneous free space σ = 0, µ = µ 0, ε = ε 0
Maxwell's equations are usually written in differential (and vector) form for free space
conditions as follows, where for simplicity a harmonic time variation is assumed:
(1)
(2)
(3)
(4)
(5)
Equation 1 is a statement of Faraday's law that a time-varying magnetic field induces an
electric field .
Equation 2 is a statement of Ampere's law that the line integral of magnetic field around a
closed loop equals the total current through the loop. The current is described as a displacement
current jωε0 plus source currents arising from the actual convection of ch arge in a vacuum.
Equation 3 arises from Coulomb's law and relates the electric displacement to the sources that
generate it, namely the charge density ρ.
Equation 4 is a statement of the conservation of charge , namely that its outflow from any closed
region (evaluated from ·) can arise only if the charge contained is dep leted.
Equation 5 recognizes that no magnetic charges exist , and hence the magnetic induction , must
be solenoidal.
4.3 MAXWELL'S EQUATIONS FOR FINITE CONDUCTING MEDIA
PRECONDITIONS:
SOURCES and FIELDS: Static or quasistatic emf , i, ,
CONDUCTOR: Finite, inhomogeneous sσ = σ(x,y,z), µ = µ 0, ε = ε 0

Our interest lies in describing electric and magnetic fields within and outside
electrophysiological preparations. Electrophysiological preparations are isolated regions (lying in
air) that involve excitable tissue surrounded by a conducting medium ( volume conductor ). The
conductivity σ of the volume conductor, in general, is a function of position [σ(x,y,z)]; that is, it
is assumed to be inh omogeneous. Its magnetic perme ability µ is normally assumed to be that of
free space (µ0), and, except for a membrane region the dielectric permittivity also has the free
space value (ε0).
If we consider for the moment a static conditi on, then we find that Equation 1 requires
that = 0. This means that must be conservative, a condition that is appropriate for
electric fields arising from static charges in free space (i.e., electrostatics). But in our conducting
medium, currents can flow only if there are nonconservative sources pr esent. So we must assume
the existence of electromotive forces. Thus for conducting media, Equation (1) must be modified
to the form of equation ( 6).
By the same reasoning, we must also recognize the presence
of impressed (applied) current fields, which w e designate i; these must be included on the right
side of equation ( 7), which correspond s to equation ( 2) as applied to conducting media. Such
sources may be essentially time -invariant as with an electrochemical battery that supplies an
essentially steady current flow to a volume conductor. They may also be quasistatic , as
exemplified by activated (excitable) tissue; in this case, time -varying nonconservative current
sources result which, in turn, drive currents throughout the surrounding volume conductor.
In a conducting medium there cannot be a convection current such as was envisaged by the
parameter i in Equation ( 2), and it is th erefore omitted from Equation (7) The convection
current is meant to describe the flow of charges in a vacuum such as occurs in high -power
amplifie r tubes. (For the same reason, equation ( 4) is not valid in conducting media.) In the
consideration of applied magnetic fields, one can treat the applied current flowing in a physical
coil by idealizing it as a free -space current, and hence accounting for it with the i on the right
side of equation ( 2). Since this current is essentiall y solenoidal, there is no associated charge
density. In this formalism the means whereby i is established need not be considered explicitly.
Because of the electric conductivity σ of the volume conductor we need to inc lude in the
right side of equation ( 7) the conduction current σ, in addition to the existing displacement
current jωε .

Another modification comes from the recognition that a volume charge density ρ cannot
exist within a conducting medium (though surface charges can accumulate at the interface
between regions of different conductivity – essentially equivalent to the charges that lie on the
plates of a capacitor). Therefore, equation ( 3) is not applic able in conducting media.
With these considerations, Maxwell's equations may now be rewritten for finite
conducting media as
(6)
(7)
(8)
(9)

In this set of equations, we obtain equation ( 8) by taking t he divergence of both sides of
equation ( 7) and noting that the divergence of the curl of any vector function is identically
zero · 0.
4.4 SIMPLIFICATION OF MAXWELL'S EQUATIONS IN
PHYSIOLOGICAL PREPARATIONS
PRECONDITIONS:
SOURCES and FIELDS: Quasistatic (ω < 1000 Hz) emf, i, ,
CONDUCTOR: Limited finite (r < 1 m) inhomogeneous resistive (ωε/σ < 0.15) µ = µ 0, ε = ε 0
Physiological preparations of electrophysiological interest have several characteristics on
which can be based certain simplifications of the general Maxwell's equati ons. We have already
mentioned that we expect the permittivity ε and permea bility µ in the volume conductor to be
those of free space (ε0, µ0). Three other conditions will be introduced here.

4.4.1 Frequency Limit
The power density spectra of signals of biological origin have been measured. These have
been found to vary depending on the nature of the source (e.g., nerve, muscle, etc.). The highest
frequencies are seen in electrocardiography. Here the bandwidth for c linical instruments
normally lies under 100 Hz, though the very highest quality requires an upper frequency of 200 –
500 Hz. In research it is usually assumed to be under 1000 Hz, and we shall consider this the
nominal upper frequency limit. Barr and Spach ( 1977) have shown that for intramural cardiac
potentials frequencies as high as 10 kHz may need to be included for faithful signal reproduction.
When one considers that the action pulse rise time is on the order of 1 ms, then signals due to
such sources oug ht to have little energy beyond 1 kHz. Relative to the entire frequency spectrum
to which Maxwell's equations have been applied, this is indeed a low -frequency range. The
resulting simplifications are described in the next section.
4.4.2 Size Limitation
Except for the very special case where one is studying, say, the ECG of a whale, the size of the
volume conductor can be expected to lie within a sphere of radius of 1 m. Such a sphere would
accommodate almost all intact human bodies, and certainly typical in vitro preparations under
study in the laboratory. A consequence, to be discussed in the next section, is that the "retarded"
potentials of general interest do not arise.
4.4.3 Volume Conductor Impedance
The volume conductor normally contains several di screte elements such as nerve, muscle,
connective tissue, vascular tissue, skin, and other organs. For many cases, the conducting
properties can be described by a conductivity σ(x,y,z) obtained by averaging over a small but
multicellular region. Since such a macroscopic region contains lipid cellular membranes the
permittivity may depart from its free -space value. The values of both σ and ε entering equations
(7) and ( 8) will depend on the particular tissue characteristics and on frequency. By making
macros copic measurements, Schwan and Kay (1957) determined that ωε/σ for the frequency
range 10 Hz < f < 1000 Hz is under 0.15. But in many cases it is possible to treat all membranes
specifically. In this case it is the remaining intracellular and interstitial space that constitutes the

volume conductor; and, since the lipids are absent, the medium will behave resistively over the
entire frequency spectrum of interest. In either case it is reasonable to ignore the displacement
current jωε0 within the volume conductor in equations ( 7) and ( 8). (One should always, of
course, include the capacitive membrane current when considering components of the total
membrane current.) Consequently, these equations can be simplified to equations ( 10) and (11),
respectively.
Thus Maxwell's equations for physiological applications have the form:
(6)
(10)
(11)
(9)
4.5 MAGNETIC VECTOR POTENTIAL AND ELECTRIC SCALAR
POTENTIAL IN THE REGION OUTSIDE THE SOURCES
PRECONDITIONS:
SOURCE: Quasistatic i (ω < 1000 Hz )
CONDUCTOR: Limited finite (r < 1 m) region outside the sources inhomogeneous
resistive (ωε/σ < 0.15), µ = µ 0, ε = ε 0
In this section we derive from Maxwell's equations the equations for magnetic vector
potential and electric scalar potential Φ in physiolo gical applications, equations ( 19) and
respectively (21) ,
Since the divergence of is identically zero (9), the magnetic field may be derived from
the curl of an arbitrary vector field , which is called the magnetic vector potential . This fulf ills

the requirement stated in equation ( 9) because the divergence of the curl of any vector field is
necessarily zero. Consequently,
(12)
Since = μ0, we can substitute equation ( 12) into equation ( 6). We consider only the
volume conductor region external to the membranes where the emf ’s are zero (note that the
emf’s are explicitly included within the membrane in the form of Nernst potential batteries), and
we consequently obtain
(13)
Now, when the curl of a vector field is zero, that vector field can be derived as the
(negative) gradient of an arbitrary scalar potential field (which we designate with the symbol Φ
and which denotes the electric scalar potential ). This assignment is vali d because the curl of the
gradient of any scalar field equals zero. Thus equation ( 13) further simplifies to
(14)

According to the Helmholtz theorem , a vector field is uniquely specified by both its
divergence and curl (Plonsey and Collin, 1961). Since only the curl of the vector field has
been specified so far (in equation 12), we may now choose
(15)
This particular choice eliminates Φ from the differential equation for (equation 17).
That is, it has the desirable effect of uncoupling the magnetic vector potential from the electric
scalar potential Φ. Such a consideration was originally suggested by Lorentz when dealing with

the free -space form of Maxwell's equations. Lo rentz introduced an equation similar to Equation
(15) known as the Lorentz condition , which is that
(16)
After w e have modified this expression since we have eliminated in equations ( 10) and
(11) the displacement current jωε in favor of a conduction current σ. This amounts to
replacing jωε by σ in the classical Lorentz condition (equation 16), resulting in equation ( 15).
The Lorentz condition can also be shown to have another important property, namely that
it ensures the sa tisfaction of the continuity condition.
Now, if we substitute Equations ( 12), (14), and ( 15) into equation (10), keeping in mind
that = /μ0, and if we use the vector identity that
(17)
we obtain
(18)

Just as emf ’s were eliminated by confining attention to the region external to the excitable
cell membranes, so too could one eliminate the nonconservative current i in equation ( 10). In
this case all equations describe conditions in the passive extracel lular and intracellular spaces;
The effect of sources within the membranes then enters solely through boundary
conditions at and across the membranes. On the other hand, it is useful to retain i as a
distributed source function in equation ( 10). While it is actually confined to cell membr anes
ensuring the aforementioned boundary conditions, it may be simplified (averaged) and regarded
as an equivalent source that is uniformly distributed throughout the "source volume." For field
points outside the source region which are at a distance that is large compared to cellular
dimensions (over which averaging of i occurs) the generated field approaches the correct value.

Equation ( 18) is known as the vector Helmholtz equation , for which solutions in integral
form are well known in classical electricity and magnetism (Plonsey and Collin, 1961). Adapting
such a solution to our specific equation gives
(19)
where

Note that r is the radial distance from a source element dV(x,y,z) (unprimed coordinates)
to the field point P(x',y',z' ) (primed coordinates), and is thus a function of both the unprimed and
primed coordinates.
To evaluate an upper bound to the magnitude of kr in the exponential terms in equation
(19) we choose:
rmax = 100 cm
ω = 2π·1000 1/s
µ0 = 4π·10-9 H/cm
σ = .004 S/cm
Then
krmax = .04
Since e-.04 = .96 , these exponential terms can be ignored a nd we get a simplification for e quatio n
(19), giving the magnetic vector potential under electrophysiological conditions:

(20)

The electric scalar potential Φ may be found from Ā by using e quation ( 15) with equation
(20). In doing so, we note that equation ( 20) involves an integration over the source
coordinates (x,y,z) while equation ( 15) involves operations at the field coordinates (x',y',z') .
Consequently, we get
(21)
where ' operates only on the field coordinates, which is why i is not affected. Since '(1/r) =
–(1/r), we finally get for the electric scalar potential:
(22)

Equation ( 22) is identical to static field expressions for the electric field, where i is
interpreted as a volume dipole density source function. This e quation corresponds exactly to
equation (7-5). Although a staticlike equation applies, i is actually time-varying, and
consequently, so must Φ be time -varying synchronously. We call this situation a quasistatic one.
When the source arises electrically (including that due to cellular excitation), a magnetic
field is necessarily set up by the resulting cur rent flow. The latter gives rise to a vector
potential , which in turn contributes to the resulting electric field through the term jω in
equation ( 14). However, under the conditions specified , |ω| is negligible compared to the term
|Φ| as discussed in Plonsey and Heppner (1967). Under these conditions we are left with the
scalar potential term alone, and equation ( 14) simplifies to

(23)
which also corresponds to a static formulation. It sh ould be kept in mind that equation ( 23) is not
exact, but only a good approximation. It corresponds to the quasistatic condition where the
electric field resembles that arising under static conditions. Under truly static conditions the
electric and magnetic fields are completely independent. Under quasistatic conditions, while the
fields satisfy static equations, a low frequency time variation may be superimposed (justified by
the low frequency conditions disc ussed earlier), in which case the magnetic field effects,
although extant, can normally be ignored.
Note that in this case, where the sources are exclusively bioelec tric and the simplification
of equation ( 23) is valid, equation ( 11) leads to Equation ( = i – σΦ).
4.6 STIMULATION WITH ELECTRIC AND MAGNETIC FIELDS
4.6.1 Stimulation with Electric Field
PRECONDITIONS:
SOURCE: Steady -state electric field
CONDUCTOR: Uniform fiber in volume conductor
The above comments notwithstanding, we are also interested in a situation where excitable tissue
is stimulated solely with an applied magnetic field. In this case the vector potential is large
and cannot be ignored. In fact, to ignore under these circu mstances is to drop the underlying
forcing function, which would leave an absurd result of no field, either electric or magnetic.
As we know for a single uniform fiber under steady -state conditions a homogeneous
partial differential equation arises:
(24)
where Vm = transmembrane potential

λ = space constant, characteristic of the physical and electric properties of the fiber
x = coordinate along the direction of the fiber
For a point source at the origin we have also know, that the solution to equation ( 24) is
equation
(25)
where V'(x) = deviation of the membrane voltage from the resting voltage.
In this equation
(26)
where Vm(0) = transmembrane potential at the origin
I0 = applied intracellular point current
ri = intracellular axial resistance per unit length
We remark, here, that for a more general applied scalar potential field, Φe, equation ( 24)
becomes
(27)
One can recognize in this equation that the second derivative of the applied potential field
along the fiber is the forcing function (in fact, it ha s been called the "activating function"),
whereas the dependent variable, Vm, is the membrane resp onse to the stimulation. Using equation
(23), one can write equation ( 24) as

(28)
where is the applied electric field.
4.6.2 Stimulation with Magnetic Field
PRECONDITIONS:
SOURCE: Time -varying magnetic field
CONDUCTOR: Uniform fiber in volume conductor
Electric stimulation may be produced by applying a time -varying magnetic field to the
tissue. As given in equation ( 12), this magnetic field is defined as the curl of a vector potential
. Now the stimulus is introduced solely through a magnetic field that induces an electric field .
Equation ( 27) is still completely valid except that the applied field is found from Equation
(14), namely whe re = –jω.
The determination of the vector field from a physical coil is found, basically, from
equation ( 20).
We also note that sin ce the differential equations ( 24), (27), and ( 28) are lin ear, and the
solution given in equation ( 25) is essentially the response to a (spatial) unit impulse at the origin
(set I0 = δ(x)), then linear systems theory describes the soluti on to equation ( 27) or (28), as
(29)
where denotes convolution. (The added factor of rm is required in orde r to convert the right
side of equation ( 29) into a current density.) The convolution operation can be performed by
taking the inverse Fourier transform of the product of the Fourier transform of V' and the Fourier
transform of the second derivative of δe. Such operations are readily carried out using the fast
Fourier transform (FFT).

4.7 SIMPLIFIED MAXWELL'S EQUATIONS IN PHYSIOLOGICAL
PREPARATIONS IN THE REGION OUTSIDE THE SOURCES
PRECONDITIONS:
SOURCES and FIELDS: Quasistatic (ω < 1000 Hz ) i, ,
CONDUCTOR: Limited finite ( r < 1 m) region outside the sources inhomogeneous
resistive (ωε/ε < 0.15) µ = µ 0, ε = ε 0
We finally collect the Maxwell's equations in their simplest form. These equations are
valid under quasistatic electrophysiological conditio ns outside the region of bioelectric sources:
(23)
(10)
(11)
(9)
CONCLUSIONS
In conclusion I tried to introduce and present the concept of bioelectromagnetism, and his
application as an important domain or field, composed by intersectation of other fields wich similar
proprieties that intefer, to help us understand the biological, electrical, magnetic properties and
functions of the living organisms (cell, tissues, organs, systems, etc) interconnected, to help us
improve studi es, techniques, methods and applications in different areas from the concept and
principles of physics.
In the second and third chapter I managed to present the few theoretical methods by which
we manage to measure through concepts and with the help of ph ysics – the different descriptions
of bioelectromagnetical fields with different specific equations.

And in the fourth chapter I presented the Maxwell’s equations applications in
bioelectromagnetism.
I think bioelectromagnetism was an important breakthrou gh in both ways, in evolution and
technology and it will also help us at the present moments, in the same time I think it will help us
in the nearly future, and I truly believe this. It will be easier to understand and live a better life
knowing more about ourselves and also with the help of its clinical applications especially in
medicine and other adiacent sciences.
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