Static Electric Charge And Electric Fields

Physical objects may acquire an electric charge when they have a net excess or deficit of electrons relative to the number of protons. When the number of electrons exceeds the number of protons, the object is said to have a negative charge, whereas a net deficit of electrons results in the object acquiring a positive charge. An electrically charged object is surrounded by an electric field such that charged objects act at a distance on other objects having similar or opposite charge. The strength of the electric field is related to the magnitude of its charge. A gravitational field also acts at a distance, but an electric field has an important difference, polarity. Thus, electric fields have directionality with the convention that field lines are drawn away from positively charged and toward negatively charged objects. The electric fields surrounding charged objects interact with each other such that the presence of two electrically charged objects results in a force that either attracts the two objects (if they have opposite charges) or repels the two objects (if they have the same qualitative charge). This attractive or repulsive force acts at a distance, not requiring that the charged objects be in contact, and the attractive or repulsive force (F) is given by Coulomb’s law:

where Q₁ and Q₂ are the magnitude of the charges (measured in coulombs), r is the distance separating the charged objects, and k is a constant equal to with ε being the permittivity of free space. According to Coulomb’s law, the force attracting or repelling charged objects increases with the magnitude of charge and decreases with the square of the distance that separates them.

Electric Circuit

An electric circuit is an electric charge–conducting pathway that ends at its beginning. For electric circuits involved in myocardial stimulation by pacemakers or ICDs, the voltage difference in the circuit is provided by a battery. The difference in voltage generated by the battery results in a flow of charge (current) from the pulse generator through the conductors in the leads, electrodes, extracellular electrolytes, cell membranes with highly regulated transmission of charged ions in both directions through the membrane, and intracellular ions and charged molecules. As current flows through the complete electric circuit, the net voltage change must be zero (Kirchoff’s voltage law). Thus, the electromotive force (voltage) generated by the battery (an increase in potential energy) must be completely dissipated (a decrease in potential energy) as current flows through all the elements of the circuit to end at the battery.

Electric circuits in the clinical practice of pacing have multiple elements, including the pulse generator battery, the lead conductor(s), the electrodes, the myocardium, and blood within the great veins and cardiac chambers. All these elements introduce opposition to the flow of current.

Electrode Polarity

All defibrillator and pacemaker electric circuits have both a positively charged electrode (the anode) and a negatively charged electrode (the cathode). The negatively charged cathode is typically the tip electrode on a pacing lead. Electrons from the pulse generator flow through the cathode-tissue interface and return to the anode, which may be located on a pacing lead or the pulse generator casing.

The terminology used for electrode polarity may be confusing as applied to lead electrodes and the electrodes of a battery. The electrode in a battery at which oxidation occurs (e.g., oxidation of lithium to yield Li⁺ plus an electron, e⁻) is the battery anode. The battery anode, by continuing oxidation, furnishes electrons to the circuit external to it. Therefore, in contradistinction to the terminology used for pacing leads, the terminal of the battery where electrons are provided to the circuit is the battery anode. From the battery anode, electrons flow through the circuitry and eventually enter the pacemaker lead electrode that is in contact with the myocardium. This electrode, receiving electrons from the pulse generator and furnishing electrons to the tissue, is the lead cathode. The return electrode located in the heart or on the pulse generator casing is the lead anode. It collects electrons from the tissue and returns them through the pulse generator circuitry to the positive electrode of the battery, the battery cathode, where reduction occurs (e.g., I₂ + 2e⁻ yields 2I⁻). The consistency in the terminology is that, when oxidation occurs, it occurs at an anode, and in the circuitry, an anode connects to a cathode that subsequently connects to another anode, and so on.

Opposition to Flow of Electric Current

In cardiac pacing and defibrillation, several elements in the electric circuit oppose the flow of electric current. As current flows through some of these elements, such as the conductor wires in the leads and pulse generator, the opposition to current flow results in energy being lost as heat. These elements are known as ohmic, and the opposition to current flow is called resistance (R). The instantaneous voltage developed across a perfect resistor is linearly proportional to the instantaneous current flow through the resistor. If a steady voltage across the resistor is represented by V, the current by I, and an unchanging resistance by R, the relationship is expressed as follows in Ohm’s law:

In reality, when all the elements of a cardiac pacing circuit are considered, these factors are much more complex than can be represented by Ohm’s law alone. These other factors include capacitance and inductance. For calculations involving a pulsed (e.g., pacemaker stimulus) or an alternating current (AC) circuit that contain reactive elements (e.g., electrodes in contact with electrolytes), the term impedance (Z) must be used in place of resistance. Impedance is a vector sum of resistance (R) and reactance (X). The following sections discuss the major components of reactance.

Capacitance

As mentioned earlier, current can be carried in different ways. Electrolyte is used as a generic term for the electrically conductive extracellular and intracellular fluids near pacemaker or defibrillator electrodes and elsewhere in the heart and blood vessels. The electrolyte conducts ions but not electrons. The difference between the conduction of electric currents by electrolytic fluids and the flow of electrons over metal wires is crucial to pacing. Because a negatively charged pacing electrode in contact with the endocardium is surrounded by blood and interstitial fluid, positively charged ions move toward that electrode during a pacing stimulus. This results in the phenomenon of polarization, which develops rapidly and dissipates slowly after the end of the stimulus. The opposite effect occurs on the positively charged anode. This effect of ions moving to oppose the flow of electric current has the effect of a capacitor in the circuit.

A capacitor is an object that stores energy in an electric field by holding positive charges apart from closely approximated negative charges. A capacitor requires a material or space between the layers of negative and positive charges that is normally nonconducting (the dielectric). A cell membrane, although leaky, acts as a capacitor by separating the negatively charged inside of the cell from the more positively charged outside. Cell membranes have very high capacitance per unit area of cell membrane. The interface between a pacing electrode and the charged electrolytes that surround the electrode at its surface in the myocardial tissue acts, in part, as a capacitor. The terms Helmholtz capacitor and Helmholtz capacitance are used in this chapter for capacitor-like effects that occur at pacemaker and defibrillator electrode-electrolyte interfaces.

Capacitance (C) is the term that specifies, for a given voltage applied across a capacitor, how much electrical charge (Q) can be stored by the capacitor. If V represents a steady voltage applied across the capacitor, then Q = CV. (If E is used instead of V as the symbol for electric potential, the relationship may be expressed as Q = CE.) The unit for capacitance is the farad. One farad is the capacitance of a capacitor that, on being charged to 1 volt, will have stored 1 coulomb of charge. Again, a coulomb is the amount of charge delivered by 1 ampere flowing for 1 second. Coulombs delivered can be expressed as follows:

in which Q_t is the total charge delivered between time 0 and time t, and i_t is the instantaneous current at each time segment between time 0 and time t. The integral is the net area under the instantaneous current-versus-time plot.

Inductance

When an alternating current flows through a wire, a magnetic field surrounding the wire is induced. An inductor is an object that stores or releases energy in or from a changing magnetic field. The voltage difference across an inductor is proportional to the rate of change of current flowing through the inductor. Energy is stored during the formation of the magnetic field and is released when the magnetic field decreases or disappears. Inductance is the term that specifies the relationship between the voltage across an inductor and the rate of change of current traversing the inductor. The magnitude of the inductance can be represented by the symbol L. If V_t represents the instantaneous voltage across the inductor, and i_L represents the instantaneous current flowing through the inductor, the relationship is given by the following equation:

Note that the voltage across the inductor is directly proportional to the rate of change of current flowing through the inductor. Cell membrane currents have some of the current- and voltage-versus-time characteristics of an inductance in parallel with a capacitance.¹ These inductance-like effects are related to the timing and magnitude of potassium ions moving into and out of the cell.²

Reactance (Capacitance and Inductance)

For reactive elements connected in series, net reactance is the scalar sum of inductive reactance (positive in the mathematical complex plane) and capacitive reactance (negative in complex plane). Pure reactance values depend on the rates of change of current and voltage, whereas pure resistance values do not. Phenomena of this type are caused by differences in the timing of the peaks (phase angles) of the voltages or currents in the various reactive components. These reactive effects can be important in biventricular pacing threshold measurements (see later discussion). The component of reactance that is most relevant to both pacing electrodes and cell membranes is capacitance, with inductance being much less important. For example, the cardiac action potential spreading throughout the heart generates a changing magnetic field that transiently stores a very small amount of energy. However, the changing magnetic field generated by spread of the action potential is so small that it is not clinically significant except in the research setting.³

For circuits with resistance R, capacitance C, and inductance L in series, where q_t represents the charge accumulated across the capacitance at any time t, and where the current through these combined elements at time t is i_t, the voltage V_t at time t is described by the following equation:

(remembering that and that the voltage across a capacitor at time t is ).

These equations indicate that, for an instantaneous current i_t, the instantaneous voltage across the series circuit is the sum of the effects at that instant in time of the resistance, capacitance, and inductance of the circuit. Note especially that the instantaneous effects are highly related to the net amount of charge, , that has accumulated in the capacitor from time 0 to the instantaneous time t. The equations show that the capacitance effect on the voltage decreases as the capacitance increases. This has clinical relevance in that, for example, the polarization voltage that interferes with autosensing pulse generators decreases as the electrode capacitance increases.

The Phospholipid Bimembrane

Living cells maintain or regenerate a difference in electric potential across the cell membrane. Excitable tissues such as myocardium respond to electrical stimulation of one or more cells with a wave of electrical depolarization—a transient reversal of the voltage gradient across the membrane—that can propagate from cell to cell. Excitable cells respond to a relatively small applied change in electric potential difference across the cell membrane by triggering a series of biochemical and biophysical events (described later). These depolarization events result in myocyte contraction. Depolarization propagates along excitable cell membranes, as in the transmission of an action potential along a squid axon, and from myocyte to myocyte through gap junctions.

Cell Membrane Characteristics

Cell membrane characteristics are major determinants of tissue excitability. The membrane of the cardiac myocyte is composed principally of phospholipids, cholesterol, and proteins.⁴ The membrane phospholipids have a charged polar headgroup and two long hydrocarbon chains arranged as shown in Figure 1-1. The cell membrane comprises two layers of phospholipids with their hydrophobic aliphatic chains oriented toward the central portion of the bilayer membrane and their polar headgroup regions toward the outside boundaries of the membrane. Because the membrane is composed of two layers of phospholipids, the polar regions of the phospholipid molecules interface with the aqueous environments inside and outside the cell. The lipid-soluble hydrocarbon chains are forced away from the aqueous phase to form a nonpolar interior.

Figure 1-1 The phosphomembrane bilayer of a cardiomyocyte contains ion channels and structures that maintain distinct ion concentrations inside and outside of the cell. Charge pumps maintain an ion concentration gradient across the membrane while ion channels allow for charge in the form of ions to pass through the membrane. The ion current flow causes changes in transmembrane potential and cardiac activation.

Determinants of the Resting Transmembrane Potential

Relatively large gradients of individual ion concentrations exist across the cardiac cell membrane.⁵ The gradient of sodium ions (Na⁺) across the membrane is approximately 145 millimoles per liter (mmol/L) outside to 10 mmol/L inside. In contrast, the potassium ion (K⁺) concentration outside the cell is approximately 4.5 mmol/L, whereas the inside concentration is 140 mmol/L. In the absence of a cell membrane, both Na⁺ and K⁺ would rapidly move in a direction determined by the concentration gradient. The diffusion force tending to move K⁺ out of and Na⁺ into the cell is proportional to the concentration gradients of those ions. The potential energy attributable to the diffusion force (PE_d) tending to move K⁺ out of the cell is given by the following equation:

[1-1]

where R is the gas constant, T is the absolute temperature, ln is the natural logarithm operator, [K⁺]_i is the concentration of potassium ion inside the cell, and [K⁺]_o is the concentration of potassium ion outside the cell. If the ratio of [K⁺]_i to [K⁺]_o is large, the potential energy across the membrane is large. For each ion species, the difference in concentration between the inside and the outside of the cell results in that ion’s contribution to the difference in electric potential across the cell membrane.

In resting cardiac cells, the intracellular cytoplasm has a measured potential of about −90 millivolts (mV), relative to the extracellular fluid. This electric force tends to move positively charged ions such as K⁺ and Na⁺ to the inside of the cell and negatively charged ions such as chloride (Cl⁻) to the outside of the cell in proportion to the potential gradient. The potential energy attributable to the electric force (PE_e) tending to move K⁺ into the cell is expressed as follows:

[1-2]

where z here is the valence (the number of positive or negative electrical charges) of the ion, F is the Faraday constant (96,500 coulombs/equivalent), and V_m is the transmembrane potential difference (transmembrane voltage, measured in millivolts). During equilibrium, the total of the potential energies from diffusion and electric forces is zero, and no net ionic movement occurs. Therefore, the sum of Equation 1-1 and Equation 1-2 may be set to zero. This yields the Nernst equation, which describes (in measurable electrical units) the potential that must exist for a single ionic species, here K⁺, to be in equilibrium across the membrane of a resting cardiac cell:

[1-3]

or, in log base 10 terms,

Using known values for extracellular K⁺, V_m(K⁺) = −90 mV. When Equation 1-3 is solved using Na⁺ concentrations, a V_m(Na⁺) of +50 mV is obtained. Therefore, it is the equilibrium potential for potassium ion (not sodium ion) that is the major factor responsible for the resting transmembrane potential. This suggests that the resting membrane is more permeable to K⁺ than to Na⁺.

To calculate the transmembrane potential when multiple ionic species exist in different concentrations across the membrane, the Goldman constant field equation (modified by Hodgkin and Katz⁶) is used:

[1-4]

where P_K+, P_Na+, and P_Cl− are the cell membrane permeabilities for the respective ions. At physiologic concentrations, this equation yields a transmembrane potential of −90 mV (the equilibrium potential for K⁺). Equation 1-4 describes how resting potentials vary as sodium and potassium ion concentrations are changed. Because there is a passive leak of charged ions through the membrane, the resting potential would not exist at the level of −90 mV unless it were actively maintained. This is accomplished by two active transport mechanisms that exchange Na⁺ ions for K⁺ and calcium (Ca²⁺) ions.

One might ask: with all the potassium ions and other positively charged ions in the cell, and with a relatively small amount of negatively charged chloride ions in the cell, how can the interior of the cell be negative with respect to the outside? The answer is that an array of intracellular organic and inorganic anions inside the cell—molecules that do not cross the membrane—carry net negative charges sufficient to make the overall balance of charge negative.

Voltage-Gated Channels

Voltage-gated channels open in response to an applied electric potential. The source of this voltage can be an action potential propagated from an adjacent cell or the electric field of an artificial pacemaker electrode. If depolarization of the membrane exceeds a threshold voltage, an action potential is triggered, resulting in a complex cascade of ionic currents flowing across the membrane into and out of the cell. As a result of this flow of charge across the membrane, the potential gradient across the membrane changes in a characteristic pattern of events that produce the cardiac action potential (Fig. 1-3).

Figure 1-3 Microelectrode recording of transmembrane potential (V_m) from left ventricular endocardium of human heart.

In phase 0 (depolarization), sodium ions (Na⁺) rapidly enter the cell through fast channels. In phase 1, the initial repolarization is primarily the result of activation of a transient outward potassium ion (K⁺) current and inactivation of the fast Na⁺ current. In phase 2 (plateau), the net current is very small, although the individual Na⁺, Ca²⁺, and K⁺ currents are about an order of magnitude larger. Phase 3 (final repolarization) completes the cycle, with the Na⁺-K⁺ pump bringing the membrane potential to a stable point at which inward and outward currents are again in balance. During phase 4, the cell is polarized and gradually undergoes slow depolarization.

(From Stokes K, Bornzin G: The electrode-biointerface [stimulation]. In Barold SS [ed]: Modern cardiac pacing. Mt. Kisco, NY, Futra, 1985, pp 33-78.)

Selective membrane-bound proteins (ion channels) determine the passive transmembrane flux of an individual ion species. The transmembrane currents determine or influence cellular polarization at rest, action potential depolarization and repolarization, conduction, excitation-contraction coupling, and myofibril contraction. The channels that regulate transmembrane conductance of Na⁺ and Ca²⁺ are voltage gated. The sodium channel is a large protein molecule composed of approximately 1830 amino acids.¹⁰ It contains four internally homologous repeating domains, believed to be arranged around a central water-filled pore lined with hydrophilic (“water-loving”) amino acids. It is estimated that there are 5 to 10 Na⁺ channels per square micrometer (µm²) of cell membrane. When an alteration changes the membrane potential to about −70 to −60 mV (the threshold potential), four to six positively charged amino acids move across the membrane in response to the change in electric field. This causes a change in the conformations of the channel proteins, resulting in opening of the channel. After a single Na⁺ channel changes to the open conformation, about 10⁴ Na⁺ ions enter the cell. On depolarization of the membrane, the Na⁺ channels remain open for less than 1 millisecond (msec). After rapid depolarization of the membrane, the Na⁺ channel again changes to the closed conformation. In addition to the Na⁺ channel, specialized proteins are suspended in the cell membrane that have differential selectivity for K⁺, Ca²⁺, and Cl⁻ ions, with much different time constants for activation and inactivation.

Sinoatrial and Atrioventricular Nodal Cells

In contrast to Purkinje fibers and working myocardial cells, sinoatrial (SA) and atrioventricular (AV) nodal cells are characterized by no true resting potential. Following repolarization, the transmembrane potential of the nodal cells reach a minimum of approximately −60 mV. The SA and AV nodal cells have a slow, inward, depolarizing, combined Na⁺/K⁺ current known as the “funny” current (I_f). These I_f channels open and cause the transmembrane potential to rise slowly until Ca²⁺ channels are activated and an action potential is generated. The rate of this rise in transmembrane potential is affected by factors such as autonomic nervous stimulation and hormones.

In these nodal structures, depolarization is primarily mediated by inward Ca²⁺ conductance through specialized Ca²⁺ channels. There are two types of Ca²⁺ channels in the mammalian heart. The L-type channels are the major voltage-gated pathway for entry of Ca²⁺ into the myocyte, and they are heavily modulated by catecholamines.¹¹ The T-type channels contribute to spontaneous depolarization of the cell associated with automaticity (pacemaker currents). The pore of the Ca²⁺ channel has a functional diameter of about 0.6 nanometers (nm), larger than that of the Na⁺ channels (0.3-0.5 nm).¹² The selectivity for Ca²⁺ is high, up to 10,000-fold greater than that for Na⁺ or K⁺. The key elements are high-affinity binding sites for Ca²⁺, positioned along a single file pore. “Elution” of a Ca²⁺ ion occurs when another Ca²⁺ ion enters and is selectively bound.

In normal situations, the SA node has the most pronounced automaticity and highest spontaneous firing rate in the heart. If the SA node is not functioning properly, however, or if block develops between the SA node and the AV node, the AV node will display pacemaker activity at a slightly slower rate than the SA node. In the absence of a pacemaker signal from the AV node, Purkinje fibers will activate at an even slower rate than either the SA or the AV node.

Maintenance of Resting Membrane Potential

The resting membrane potential is maintained by the pumping of Na⁺ ions out of the cell and K⁺ ions into the cell. The Na⁺,K⁺—adenosine triphosphatase (ATPase) pump moves three Na⁺ ions out of the cell in exchange for two K⁺ ions moved into the cell.^13–15 The basic unit of the Na⁺,K⁺-ATPase protein (pump) consists of one alpha (α-) and one beta (β-) subunit. The α-subunit is large (1016 amino acids) and spans the entire membrane, whereas the β-subunit is a smaller glycoprotein. There appear to be about 1000 pump sites/µm² of cardiac cell membrane. The fully activated pump cycles about 50 to 70 times per second (interval of 15-20 msec/cycle). Similarly, the Na⁺-Ca²⁺ pump moves three Na⁺ ions out of the cell in exchange for one Ca²⁺ ion.^16,17 Therefore, both transport mechanisms result in the net movement of one positive charge out of the cell, polarizing the membrane and maintaining a negatively charged interior. The function of both exchange mechanisms depends on the expenditure of energy in the form of high-energy phosphates and is susceptible to interruptions in aerobic cellular metabolism (e.g., during ischemia).

The Cardiac Action Potential

When the voltage gradient across the membrane of a myocyte decreases so that the inside of the cell becomes less negatively charged with respect to the outside of the cell, a critical transmembrane voltage difference is reached, the threshold voltage. At threshold, the cell membrane suddenly undergoes a further depolarization that is out of proportion to the intensity of the applied stimulus. This abrupt change in the potential across the membrane is the start of a cascade of inward and outward currents that together are known as an action potential.¹⁸

The cardiac action potential is an enormously complex event and consists of five phases¹⁹ (see Fig. 1-3): phase 0, the upstroke phase of rapid depolarization; phase 1, the overshoot phase of initial rapid repolarization; phase 2, the plateau phase; phase 3, the rapid repolarization phase; and phase 4, characterized by a slow, spontaneous depolarization of the membrane in cells with spontaneous pacemaker activity, until the threshold potential is again reached and a new action potential is generated.

Phase 4: Automaticity and the Conduction System

Automaticity is the property of certain cells by which they are able to initiate an action potential spontaneously. It has been known for centuries that the heart can exhibit spontaneous contraction even when completely denervated. Leonardo da Vinci observed that the heart could “move by itself.”²⁴ William Harvey reported that pieces of the heart could “contract and relax” separately.²⁵ Many cells within the specialized conduction system have the potential for automaticity.

Not all parts of the heart, however, possess this property. In fact, cells in different areas of the heart have different transmembrane potentials, thresholds, and action potentials. Fast responses are characteristic of ordinary working ventricular muscle cells and His-Purkinje fibers, with resting membrane potentials of −70 to −90 mV and rapid conduction velocities. Normal SA and AV nodal cells have slow responses, with resting potentials of −40 to −70 mV and slow conduction velocities. Cells, or group of cells, with the fastest rate of spontaneous membrane depolarization during phase 4 are the first to reach threshold potential and initiate a propagated impulse. Therefore, cells with the steepest slope in phase 4 become the heart’s natural pacemaker. Ordinary working myocardial cells usually are not automatic.

Normally, depolarization is initiated at the SA node.^26,27 Figure 1-4 shows action potentials from different types of cardiac cells.²⁸ Rather than maintaining a stable resting membrane potential, the repolarization of the action potential is followed by a slow depolarization from about −71 to −54 mV, the threshold required to initiate another action potential. This slow, spontaneous depolarization drives cardiac automaticity and is related to a specialized current (I_f). In the case of AV nodal cells, the fast upstroke is carried predominantly by an inward Ca²⁺ current. Repolarization is caused by delayed activation of the K⁺ current. The balance of inward and outward currents determines the net “pacemaker” current and is finely regulated by both adrenergic and cholinergic neurotransmitters. In the presence of AV block or abnormal SA nodal function, AV junctional cells in the region of the proximal penetrating bundle usually assume the role of pacemaker at rates slower than that of the sinus node. In the absence of disease in the AV junction, the escape rhythm occurs with a frequency that is about 67% of the sinus rate.²⁹

Figure 1-4 Action potentials from different cell types in the heart.

Specialized pacemaker and conduction cell types have distinct action potential characteristics. Activation times for a normal heartbeat demonstrate the spread of activation throughout the whole heart. AV, Atrioventricular.

(From Malmivuo J, Plonsey R: Bioelectromagnetism: principles and applications of bioelectric and biomagnetic fields. New York, 1995, Oxford University Press.)

Myocardial Cell Electrical Properties

A single Purkinje fiber typically has an internal resistance that is two to three times greater than that of blood. The specific membrane resistance of a Purkinje fiber is on the order of 10⁴ ohms–square centimeter (Ω-cm²