Page 111 - Computational Modeling in Biomedical Engineering and Medical Physics

100 Computational Modeling in Biomedical Engineering and Medical Physics

The stem of the I-ECG problem here is the transfer equation
b 5 Cx; ð4:4Þ
which relates the potential on the epicardium (the source) to the potential on the
thorax surface (measurement, calculated potential). The outcome of the compan-
ion D-ECG is the availability (construction) of the transfer matrix, C,which maps
the electrical potential on the epicardium, x (m 3 1) vector of the potentials, onto
the electrical potential on the thorax, b (n 3 1) vector of the potentials on the
chest. The D-ECG is solved numerically, by successively assigning a unit potential
to each of the m nodes on the epicardium, while the rest of the nodes are of
zero potential, and the chest surface is electrically insulated (Mocanu, 2002). Each
column of C corresponds to the n potentials on the thorax calculated for a single-
activated node (on the epicardium). The selection of the n nodes on the epicar-
dium (electrodes for measuring the potential during open heart surgery) and the m
on the thorax (electrodes used to measure the potential) renders a collocation
approach to the solution.
Then the potential on the epicardium, x, is calculated by inverting C. Tikhonov
regularization (with l 2 constraint on the energy, gradient, or laplacian of the solu-
tion) was often used to stabilize the inverse solution (Iakovidis and Martin, 1991;
Velipasaoglu et al., 2000; Johnston et al., 1994; Mocanu et al., 2005), and the vali-
dation of reconstructions was performed experimentally, using electrolytic tanks
(Oster et al., 1998) or by measuring epicardial electrical potentials during open
heart surgery (Shahidi et al., 1994; Burnes et al., 2000). Other techniques encom-
pass more a priori information about the solution: local spatial regularization
(Johnson and MacLeod, 1996), constraints on the normal component of epicardial
current density (Velipasaoglu et al., 2000), constraints based on supraregularized and
subregularized solutions (Iakovidis and Martin, 1991). Another approach, the lapla-
cian electrocardiography (He et al., 1997; Mocanu, 2002)uses “Laplacian” (concen-
tric dipolar) electrodes (Carvalhaes and de Barros, 2014) to measure thoracic
potentials needed in the reconstruction of the epicardial potentials. Laplacian poten-
tial electrodes are surface reference free electrodes have been shown to improve the
spatial resolution of surface bioelectric signal recordings in EHG (electrohystero-
gram) and EEG measurements. Technically, their information is proportional to the
local surface charge (laplacian). These concentric, ring electrodes, show off better
immunity to noise but they are fainter and present a shorter distance coherence to
thesource(He et al., 1997; Li et al., 2005; Gao et al., 2017; Makeyev, 2017, 2018).
Fig. 4.4 depicts an array of dipolar laplacian electrodes used to measure the electric
field on the thorax surface in the I-ECG problem where the source of electric
activity is a current dipole.