Biomedical Engineering Reference
In-Depth Information
magnetic resonance imaging (fMRI) that rely on changes in blood
flow or content (e.g. radioactive labeling or oxygenation) and
therefore produce indirect correlates of neuronal activity with
delays that are, at best, a good fraction of a second in the case
of fMRI and minutes in the case of PET.
Finally, the forward problem is linear as a direct consequence
of the linearity of the laws of electromagnetism. In other words,
the electric and magnetic field generated by any combination of
instantaneous current elements is simply the sum of individual
contributions from each element. In the case of continuous pri-
mary current density, the instantaneous electric and magnetic field
can be computed by integrating the contributions from each small
volume element in the source space. In the case of a spherical
model, the source space for MEG includes only regions where
neurons and possibly white matter exists, any intervening regions
and boundaries are not part of the source space as long as they do
not generate primary currents.
3.2. Inverse Problem
In contrast to the forward problem, the inverse problem has no
unique solution, a mathematical fact that was already demon-
strated over 150 years ago
(3)
. Simply stated, it is impossible
to reconstruct uniquely the electrical current density inside the
head from MEG and/or EEG measurements. Even if we knew
exactly the electrical potential on the surface of the head and
the magnetic field everywhere outside the head, we would still
be unable to determine the currents inside the head. In practice,
non-uniqueness is much less of a problem than would appear from
the dry mathematical statements. By definition, silent sources can-
not be recovered and noise and sparse sensor coverage further
limit what can be reliably extracted about the non-silent part of
the current density vector. Nevertheless, what is often required of
the data is to provide reliable estimates about which areas of the
brain were preferentially activated by some stimuli or tasks and
when. This limited objective is often satisfied with estimates of
the timecourse of the non-silent part of the source configuration.
The key question in practice is how accurately and reliably one
can recover the non-silent part of the primary current density.
A unique solution of the biomagnetic inverse problem can
be obtained by introducing constraints for the form of the gen-
erators. Two types of constraints are particularly popular
(6)
.
The first assumes that the generators are one or more point-like
sources, or current dipoles. Dipole source localization solutions
are often interpreted as representatives for their neighborhood
and are referred to as equivalent current dipoles (ECD). The sec-
ond family of popular source localization methods assumes that
the continuous current density can be written as a linear sum of
(weighted) functions, each defining the sensitivity profile or lead
fields for one of the sensors. These methods, known as minimum
