Page 121 - Advances in Biomechanics and Tissue Regeneration

7.2 EQUATIONS THAT GOVERN THE ELECTRICAL ACTIVITY OF THE HEART 117

7.2 EQUATIONS THAT GOVERN THE ELECTRICAL ACTIVITY OF THE HEART

The heart shows two types of behavior: electrical and mechanical. All myocardial cells are similar with respect to the
mechanical function. However, from an electrical point of view, the cells may be classified into several types. The elec-
tric impulses transmitted through the heart are responsible for the rhythmic contraction of the heart muscle/cardiac
muscle. When the system works normally, the atria are contracted approximately a sixth of a second before the ven-
tricles, which enables the filling of the ventricles before pumping the blood into the lungs and the peripheral circulation
[32]. Another important point of the system is that the ventricles contract synchronously to generate proper blood
pumping. Therefore, all cells need to develop an AP in an ordered manner for which the cells must be excited conve-
niently along the cardiac cycle. To fully understand these phenomena, this chapter describes how the electrical activity
in the heart takes place, how it synchronizes, and the mathematical equations that rule them.

7.2.1 Governing Equations

This section describes the governing equations of the propagation of the heart electrical activity.

7.2.1.1 Bidomain Model
The electrical coupling of the cardiomyocytes and the conduction through the ventricles can be mathematically
described by a bidomain model [33]. In this model, the cardiac heart tissue is represented by two continuous domains
that share the space, that is, the intracellular and extracellular domains coexist spatially. This is opposite to reality
because each of them physically takes a fraction of the total volume. In this model, each domain acts as a volume con-
ductor with a different conductivity tensor and different potential, and the ionic currents flow from one domain to
another through the cell membrane that acts as a condenser.
The currents in the two domains are given by Ohm’s Law:

J i ¼ M i rV i , (7.1)
J e ¼ M e rV e , (7.2)

where J i is the intracellular current, J e is the extracellular current, M i and M e are the conductivity tensors, and V i and V e
are the intracellular and extracellular potentials, respectively.
The cell membrane acts as a condenser. Due to its small thickness, the charge stored on one side is compensated
immediately on the other side, by which the accumulation of charge at any point is zero, that is:

∂
q i + q e Þ ¼ 0, (7.3)
∂t
ð
where q i and q e are the charges in the intracellular and extracellular space, respectively.
In each domain, the flow of current in a point must equal the rate of accumulation plus the ionic current coming out
of the point, that is:

∂q i
+ χJ ion , (7.4)
∂t
r J i ¼
∂q e
χJ ion , (7.5)
∂t
r J e ¼
where J ion is the current through the membrane. The ionic current is measured by the unit of area of the cellular mem-
brane, whereas the density of charge and the flow of current are measured by unit of volume. The constant χ represents
the cell membrane area-to-volume ratio. On the other hand, the sign of the ionic current is defined as positive when the
current leaves the intracellular space and gets into the extracellular.
Introducing Eqs. (7.4), (7.5) in Eq. (7.3), we get the current conservation equation:
r J i + r J e ¼ 0: (7.6)

Replacing Eqs. (7.1), (7.2) in Eq. (7.6), we obtained
r M i rV i Þ + r M e rV e Þ ¼ 0: (7.7)
ð
ð

I. BIOMECHANICS