Biomedical Engineering Reference
In-Depth Information
Equation (3.43) is popularly called the Poisson equation. The Poisson equation
describes the electrical field distribution through a volume conductor, and is the
equation utilized in analyzing electrical activity of various tissues. It is solved using
many computational tools by two approaches: surface-based and volume-based.
In both cases, the volume conductor is divided into small homogeneous finite ele-
ments. Finite elements are assumed to be isotropic for surface-based analysis (also
referred as the boundary-element method), and integral form of the equation is
used. For volume-based analysis, finite elements are anisotopic and a typically dif-
ferential form of equations is used.
Knowing the number, location, orientation, and strength of the electrical charge
sources inside the head, one could calculate the reading at an electrode on the sur-
face of the scalp with the Poisson equation. However, the reverse is not true (i.e.,
the estimation of the electrical charge sources using the scalp potential measure-
ments). This is called the inverse problem and it does not have an unique solution.
Hence, it is difficult to determine which part of the brain is active by measuring a
number of electrical potential recordings at the scalp.
3.4.4 Measuring the Electrical Activity of Tissues: Example of the
Electrocardiogram
The fundamental principles of measuring bioelectrical activity are the same for all
tissues or cells: the mapping in time and space of the surface electric potential cor-
responding to electric activity of the organ. However, for diagnostic monitoring,
electrical activity should be measured with less invasiveness, unlike intracellular
measurements (described in Section 3.3.5). For this purpose, the concept of volume
conduction is used (i.e., bioelectrical phenomena spread within the whole body
independent of electrical source position). As the action potential spreads across
the body, it is viewed at any instant as an electric dipole (depolarized part being
negative while the polarized part is positive). For example, electrical activity of the
heart is approximated by a dipole with time varying amplitude and orientation. An
electrode placed on the skin measures the change in potential produced by this ad-
vancing dipole, assuming that potentials generated by the heart appear throughout
the body. This forms the basis of the electrocardiogram (ECG), a simple noninva-
sive recording of the electrical activity generated by the heart.
In muscle cells, an active transport mechanism maintains an excess of Na
+
and Ca
2+
ions on the outside, and an excess of Cl
−
ions inside. In the heart, resting
potential is typically
90 mV for ventricular cells. The
positive and negative charge differences across each part of the membrane causes
a dipole moment pointing across the membrane from the inside to the outside of
the cell. However, each of these individual dipole moments are exactly cancelled
by a dipole moment across the membrane on the other side of the cell. Hence, the
total dipole moment of the cell is zero. In other words, the resting cell has no dipole
moment.
If a potential of 70 mV is applied to the outside of the cell at the left hand side,
the membranes' active transport breaks down at that position of the cell, and the
ions rush across the membrane to achieve equilibrium values. The sinoatrial (S-A)
node, natural pacemaker of the heart) sends an electrical impulse that starts the
−
70 mV for atrial cells and
−
