Biomedical Engineering Reference
In-Depth Information
norm (MN) or weighted minimum norm (wMN) solutions, are
popular because they lead to a linear system of equations which
allows standard pseudoinverse techniques to define the inverse
operator that can then be applied directly to the data. Theoretical
scrutiny of the mathematical foundation of the inverse problem
shows that neither current dipoles nor linear solutions are ade-
quate. Minimum norm is not appropriate for tomographic local-
ization for a rather subtle reason; although the forward problem
is linear, the optimal algorithm for tackling the inverse problem
cannot be linear
(10)
. The laws of electromagnetism provide no
justification for expressing the full primary current density vec-
tor as a (weighted) sum of lead fields. Physics allows only the
direction of the primary current density to be so represented, but
this leads inevitably to a non-linear relationship between the mea-
surements and the distribution of generators. This conclusion was
reached first on the basis of simulation studies leading to the stan-
dard form of magnetic field tomography (MFT)
(9)
. In summary,
the basic assumption of MFT is that a (weighted) linear expan-
sion in terms of lead fields can represent only the
direction
of
the current density. This is as much as can be deduced from the
underlying physics for fixed detectors and conductivity profile.
The strength of the current density must be determined more
explicitly from the MEG signal itself. Specifically, the full current
density must be obtained from a highly non-linear system of equa-
tions for each snapshot of data. It is precisely because linearity
is lost, that a direct appeal to the data must be made on every
timeslice of the data and a new non-linear system of equations
must be solved each time. In this sense, MFT draws on all avail-
able information in the MEG signal. The advantage of the form
of non-linearity introduced by MFT is the ability to recover activ-
ity that can be either spatially sharp or distributed, thus leading
to tomographic description of the generators with practically no
a priori assumptions. However, non-linearity comes with a heavy
computational cost, but a rather affordable penalty today, thanks
to modern computers.
To appreciate the subtle difference between various inverse
problem approaches, it is necessary to consider in detail how the
lead fields can be used to construct estimates of the unknown cur-
rent density vector. The similar nature of different linear methods,
e.g. of MN and wMN and LORETTA
(19)
, and how they differ
from other non-linear methods like MFT and FOCUSS
(20)
can
be best demonstrated by expressing the unknown current den-
sity as a series expansion of lead fields with different orders in
the series modulated by the modulus of the current density raised
to some power, as described in detail elsewhere
(10, 21)
and in
outline in Appendix 1.
In the early 1990s, numerous comparisons between MFT
and ECD models with computer generated data
(9)
,MEG
