Magnetoencephalography of magnetic field is tiny, but still

Magnetoencephalography (MEG) is by far the most
advanced technique of measuring and evaluating the magnetic fields generated by
brain activities. When brain functions, the interaction among brain cells
generates tiny electrical voltages 1. The resulting flow of the intracellular
electrical current over the brain cortex produces magnetic fields. This amount
of magnetic field is tiny, but still can be detected and evaluated using
extremely sensitive magnetometers. Figure 1 shows the process that synchronized
neuronal currents induce weak magnetic fields over the brain. The magnetic
field generated from brain activity is in range of femtotesla (fT), and the
magnetic noise induced by ambient environment has the order of nanotesla (nT).
Therefore one big challenge with MEG is to detect the weak target signal from
the environmental noise.

The time resolution capabilities of MEG are in range
of 10 milliseconds or even faster. Comparing to a functional MRI, MEG can best
resolve events with a precision of several hundred milliseconds. Though MEG can
provide information of the activities of colonies of neurons of the cerebral
cortex (outmost thin layer that covers the cerebrum), it has a limitation in which
the neuron signals can only recorded from the scalp 2.

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One major challenge for MEG system is source
localization since a unique solution to the problem of reconstructing where
exactly the sources of these signals are localized within the brain does not
exist 3. The standard approach to overcome the non-uniqueness of the inverse
problem in MEG is to introduce constraints on the possible solutions to determine
the most suitable solution to interpret the data. Thus, the source localization
of the MEG signals depends to some degree on the models used and on the
corresponding assumptions, and therefore will have some degree of uncertainty.

1.2 Neurons – the origin of MEG
signals

As mentioned earlier, MEG signals are originated from
the interchange of ions among neurons. Neurons and glial cells are the major
building blocks of the brain. Neurons are the information processing unit there
their cell bodies and dendrites are concentrated in the grey matter. Glial
cells are the important unit for supporting the neuron structures and maintaining
the proper concentrations of ions. Glial cells can also transport nutrients and
other substances between blood vessels and brain tissue. Figure 2 is the
schematic drawing a neuron.

Neuron is a nerve cell which designated to transmit
information to other nerve cells and muscles. The process of transmitting
information among cells are referred as intra-cellular signaling and
extra-cellular signaling. Both types of neuronal signaling are essential to
maintain the functionality of the nervous system. When neuronal signaling takes
place, a voltage pulse is generated and travels along the axon. Once the
voltage at the axon reaches the firing threshold (about 40mV), an action
potential will be initiated and then causing a series of cascading events for
neurotransmission.   

There are over 10000 types of neurons which can be
categorized in three types: motor type, sensory type and interneurons. The
three types of neurons govern the brain functions such as movement, sensing
(visual, audio, olfactory, tactile) and recognitions. Figure 3 shows a more
illustrated brain functioning map.

1.3 Forward Problem

The forward problem in bio-magnetics studies the
magnetic field and electric potential arise from a known source. For practical
purposes, appropriate physics models must be selected for the source and the biological
object. In this section, the forward problem is discussed, however only fields within
a homogenous media due to electric source currents are considered. We describe
how the magnetic induction (B) and the electric potential (V) are computed
using the approximation of Maxwell’s equations and also by Biot-Savart law.

(1)

Suppose
that the conductor consists of the whole space with constant conductivity U.
Then equation (1) is Poisson’s equation with the solution 5:

(2)

where
the integration is over a region containing the source J’. Using the vector
identity V’. (J'(r’)Ir- r’1-I) = Ir-r’l-‘V’ – Ji(r’)+ Ji(r’) – V'(lr- r’1-I)
with V'(lrr’1-I) = Ir – r'(-3(r – r’) and the Gauss theorem we can transform
equation (1) to the form,

(3)

because
the surface integral jaGJ'(r’)lr – r’1-I * dS = 0 since J’ = 0 on the boundary
aG of G. Equation (2) is a convenient formula for V because it is also valid
for J’ which is not differentiable everywhere. Using the identity V’ x (J(
r’)lr – r’l”) = Ir – r’I-‘V’ x J( r’) +V'( (r – r’1-I) x J( r’) and Stokes’
theorem we obtain:

If the source current is evenly distributed on a
homogeneous line or surface, equations (2) and (3) remain valid if we replace
the volume density J’ by a line or a surface density and the volume integral by
a line or a surface integral, respectively 6.

(4)

To
measure the magnetic source within brain (treated as a homogenous media), we
derive a formula for B outside a spherically symmetric conductor and we assume
that J’ is a current dipole Q at located at distance ro from the
center of the sphere. The total current J outside from conduct is zero, and
thus V x B = 0. The magnetic field outside the conductor can then be expressed
in terms of a magnetic scalar potential U:

(5)

To
find an expression for U, we fix r outside G and consider a line integral of U
along the radius r+ter from 0 to infinity because U vanishes at
infinity, we obtain

(6)

The
last integral is easy to compute and we obtain and we got

(7)

Where
F = a(ra+r2-r0

r) and a = r-r0. Eventually the
magnetic general equation is derived as shown in equation (7)

Where

.
Equations
(6) and (7) show another important property of a spherical conductor: if the
source dipole is radial, then B outside G vanishes.

1.4 Inverse Problem

Inverse problem is much more difficult than forward
problem. For inverse problem, we need to search for the unknown sources by analysis
of the measured field data. The result may contain unlimited number of
solutions. Somethings the transfer of energy between magnetic and electric fields
within the brain media also complicates the inverse problem. To determine the
appropriate solution among all possible solutions, extra special conditions
must be added and we need to apply restrictions of possible source
configurations 7. 

 

1.5 Instrumentation

The most advanced and mature technology to implement
MEG signal measurement is using Superconducting quantum interference device (SQUID)
based imaging system. A SQUID system has gradiometers arranged to cover the
brain area and they have extremely high sensitivity and low noise which can
pick up even fetal brain signals 8. One of the major challenge for SQUID
based MEG signal measurement is environmental noise, which will be elaborated
in later sections. Approaches which have been adapted to solve the
environmental noise issue include noise cancellation, MEG source localization
which depends no noise cancellation and MEG inversion which contributes to
noise cancellation 9.

For noise cancellation, shield is the most straight
forward approach. A thin layer of highly conducted metal is applied to block
noise at low frequency. The noise from far distance and higher frequencies are
taken care by gradiometers. The higher order of the gradiometer more noise can
be attenuated. Usually a maximum of three order gradiometer is used because
higher order requires a large space 10.

MEG inversion contributes to noise cancellation
involves beam former which can be classified as linearly constrained minimum
variance (LCMV), synthetic aperture magnetometry (SAM) and signal space
projection (SSP). Also equivalent current dipole (ECD) which uses iterative
method for optimization of the result is also mostly used to model brain source
activity. But it is very sensitive to noise and correct result removal 11.