There is much more information contained in the ECG than is available from a single lead. Some of this information can be extracted from the vector electrocardiogram. The mathematical basis for the vector ECG comes from an understanding of the nature of a volume conductor. The human body is an inhomogeneous volume conductor, meaning that it is composed of electrically conductive material. If we assume that biological tissue is ohmic, there is a linear relationship between current and potential

P QRS T

Figure 14.1 Cellular transmembrane potential and electrocardiogram. The upper tracing represents the transmembrane potential of a single ventricular myocyte and the lower tracing shows the body surface potential during the same electrical event. The numbers on the upper tracing designate phases in the action potential cycle: 0: the upstroke, 1: the brief spike, 2: the plateau, 3: the rapid recovery, 4: resting potential. (Rushmer, 1976, Fig. 8-4, p. 286.)

Figure 14.1 Cellular transmembrane potential and electrocardiogram. The upper tracing represents the transmembrane potential of a single ventricular myocyte and the lower tracing shows the body surface potential during the same electrical event. The numbers on the upper tracing designate phases in the action potential cycle: 0: the upstroke, 1: the brief spike, 2: the plateau, 3: the rapid recovery, 4: resting potential. (Rushmer, 1976, Fig. 8-4, p. 286.)

The conductivity tensor a is inhomogeneous, because it is different for bone, lung, blood, etc., and it is anisotropic, because of muscle fiber striation, for example. Obviously, current is conserved, so that

where S represents all current sources.

The most significant current source in the human body is the spreading action potential wave front in the heart. The spreading cardiac action potential is well approximated as a surface of current dipoles. The rapid increase in membrane potential (of about 100 mV) translates into an extracellular decrease of about 40 mV that extends spatially over a distance (the wave front thickness) of about 0.5 mm. If the exact location and strength of this dipole surface and the conductivity tensor for the entire body were known, then we could (in principle) solve the Poisson equation (14.2) to find the body surface potential at all times during the cardiac cycle. This problem is unsolved, and is known as the forward problem of electrocardiography.

Figure 14.2 A collection of ECG recordings, including (a) Normal ECG recording (lead II) from a sedated 18-year-old male (JPK's son). (b) Atrial flutter showing rapid, periodic P waves, only some of which lead to QRS complexes. (Rushmer, 1976, Fig. 8-29, p. 316.) (c) Atrial fibrillation showing rapid, nonperiodic atrial activity and irregular QRS complexes. (Rushmer, 1976, Fig. 828, p. 315.) (d) (Monomorphic) ventricular tachycardia in which ventricular activity is rapid and regular (nearly periodic). (Davis, Holtz, and Davis, 1985, Fig. 17-24, p. 346.) (e) Ventricular fibrillation in which ventricular activity is rapid and irregular. (Rushmer, 1976, Fig. 8-30, p. 317.)

Figure 14.2 A collection of ECG recordings, including (a) Normal ECG recording (lead II) from a sedated 18-year-old male (JPK's son). (b) Atrial flutter showing rapid, periodic P waves, only some of which lead to QRS complexes. (Rushmer, 1976, Fig. 8-29, p. 316.) (c) Atrial fibrillation showing rapid, nonperiodic atrial activity and irregular QRS complexes. (Rushmer, 1976, Fig. 828, p. 315.) (d) (Monomorphic) ventricular tachycardia in which ventricular activity is rapid and regular (nearly periodic). (Davis, Holtz, and Davis, 1985, Fig. 17-24, p. 346.) (e) Ventricular fibrillation in which ventricular activity is rapid and irregular. (Rushmer, 1976, Fig. 8-30, p. 317.)

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Figure 14.3 ECG from the twelve standard leads, showing left bundle branch block (LBBB), diagnosed as such because of the lengthened QRS complex (0.12 ms), a splitting of the QRS complex in leads V1 through V4 into two signals, and a leftward deflection of the heart vector, indicated, for example, by the amplitude shift in lead V6. (Rushmer, 1976, Fig 8-46, p. 338.)

What we would really like to know is the operator, say T, called a transfer function, that solves (14.2) and yields the body surface potential denoted by

Even more useful, if the transfer function T were known, one could determine the sources by inverting the forward problem

This problem, known as the inverse problem of electrocardiography, is even harder to solve than the forward problem, because it is a numerically unstable mathematical problem.

Since these problems are yet unsolved, we do well to make some simplifications. Our first simplification is to view the action potential upstroke surface as a single current dipole, known as the heart dipole vector. We define the heart dipole vector as

Jv where J represents the dipole density at each point of the heart, and V is the heart volume. The heart dipole vector is assumed to be located at a fixed point in space, changing only in orientation and strength as a function of time.

Next we assume that the volume conductor is homogeneous and infinite with unit conductance. Then, from standard potential theory (see Exercise 1), at any point x in space, where the dipole is assumed to be located at the origin. Thus, at each point on the body surface,

where lx is a vector, called the lead vector, associated with the electrode lead at position x. Of course, for a real person, the lead vector is not exactly ^nxp, and some other method must be used to determine lx. However, (14.7) suggests that we can think of the body surface potential as the dot product of some vector lx with the heart vector H(t), and that lx has more or less the same orientation as a vector from the heart to the point on the body where the recording is made.

Since H is a three-dimensional vector, if we have three leads with linearly independent lead vectors, then three copies of (14.7) yields a matrix equation that can be inverted to find H(t) uniquely. In other words, if our goal is to determine H(t), then knowledge of the full transfer function is not necessary. In fact, additional measurements from other leads should give redundant information.

Of course, the information from additional leads is not truly redundant, but it is nearly so. Estimates are that a good three-lead system can account for 85% of the information concerning the nature of the dipole sources. Discrepancies occur because the sources are not exactly consolidated into a single dipole, or because the lead vectors are not known with great accuracy, and so on. However, for clinical purposes, the information gleaned from this simple approximation is remarkably useful and accurate.

The next simplification is to standardize the position of the body-surface recordings and to determine the associated lead vectors. Then, with experience, a clinician can recognize features of the heart vector by looking at recordings of the potential at the leads. Or sophisticated (and expensive) equipment can be built that inverts the lead vector matrix and displays the heart vector on a CRT display device.

Cardiologists have settled on 12 standard leads. The first three were established by Einthoven, the "father of electrocardiography" (1860-1927, inventor of the string galvanometer in 1905, 1924 Nobel Prize in physiology) and are still used today. These are the left arm (LA), the right arm (RA), and the left leg (LL). One cannot measure absolute potentials, but only potential differences. There are three ways to measure potential differences with these three leads, namely,

and of course, since the potential drop around any closed loop is zero,

With these three differences, there are three lead vectors associated with the orientation of the leads, and the potential difference is the amplitude of the projection of the heart vector H onto the corresponding lead vector. Thus, Lj = lj ■ H, and Vj = \Lj \ for j = I, II, III.

Einthoven hypothesized that the lead vectors associated with readings Vi, VII, Vm form an equilateral triangle in the vertical, frontal plane of the body, given by the unit vectors (ignoring an amplitude scale factor) lI = (1,0, 0), and lII = (1, 2V3,0). The Einthoven triangle is shown in Fig. 14.4. Here the unit coordinate vector (1, 0, 0) is horizontal from right arm to left arm, (0, 1, 0) is vertical pointing downward, and the vector (0, 0, 1) is the third coordinate in a right-handed system, pointing in the posterior direction, from the front to back of the chest. Associated with the frontal plane is a polar coordinate system, centered at the heart, with angle 0 = 0 along the x axis, and 0 = 90° vertically downward along the positive y axis.

Of course, the lead vectors of Einthoven are not very accurate. Experiments to measure the lead vectors in a model of the human torso filled with electrolytes produced measured lead vectors li = (0.923, —0.298, 0.241), and ln = (0.202,0.972, —0.121) (Burger and van Milaan, 1948), which are not in the frontal plane. These lead vectors are known as the Burger triangle.

It is fairly easy to glean information about the direction of the heart vector by recognizing the information that is contained in (14.7). The vector ECG is actually a time-varying vector loop (shown in front, top, and side views in Fig. 14.6), and deducing time-dependent information is best done with an oscilloscope. However, one can estimate the mean direction of the vector by estimating the mean amplitude of a wave and then using (14.7) to estimate the mean heart vector. The mean (or time average) of

Figure 14.4 The Einthoven triangle showing a typical heart vector H and associated lead vectors LI,LII and LIII. Because the body is approximately planar, the lead vectors are assumed to be in the frontal plane.

the QRS complex is approximately proportional to the sum of the (positive) maximum and the (negative) minimum.

Since the lead voltage is a dot product of two vectors, a change in mean amplitude of a particular wave suggests either a change in amplitude of the heart vector or a change in direction of the heart vector. For example, the normal QRS and T wave mean dipoles are oriented about 45° below horizontal to the left (see Exercise 5). This is close to orthogonal to the lead vector lm, and more or less aligned with lead vector ln. Thus, on a normal ECG, we expect the mean amplitude of a QRS to be small in lead III, large in lead II, and intermediate to these two in lead I. Shifts in these relative amplitudes suggest a shift in the orientation of the heart dipole. For example, an increase in the relative amplitude of the potential difference at lead III and a decrease in amplitude at lead II suggests a shift of the heart vector to the right, away from the left, suggesting a malfunction of the conduction in the left heart.

Although two orthogonal lead vectors suffice to determine the orientation of the heart vector in the vertical plane, for ease of interpretation it is helpful to have more leads. For this reason, there are three additional leads on the frontal plane that are used clinically. To create these leads one connects two of the three Einthoven leads to a central point with 5000 ^ resistors to create a single terminal that is relatively indifferent to changes in potential and then takes the difference between this central potential and the remaining electrode of the Einthoven triangle. These measurements are denoted by aVR, aVL, or aVF, when the third unipolar lead is the right arm, the left arm, or the left foot, respectively. The initial "a" is used to denote an augmented unipolar limb lead.

For standard cardiographic interpretation the lead vectors for leads I, aVR-, II, aVF, III, and aVL- are assumed to divide the frontal plane into equal 30° sectors. For example, li is horizontal, laVR- is declined at 30°, while laVF is vertical, etc. The superscript for aVR- denotes the negative direction of the lead vector laVR (Fig. 14.5).

With these six leads, vector interpretation of the frontal plane orientation of the heart dipole is fast. One looks for the leads with the largest and smallest deflections, and surmises that the lead vector with largest mean amplitude is most parallel to the heart dipole, and the lead vector with the smallest mean deflection is nearly orthogonal to the heart dipole. Thus in the normal heart situations, readings at leads II and aVR should be the largest in mean amplitude, with positive deflection at lead II, and negative deflection at lead aVR, while the mean deflections from leads III and aVL should be the smallest, being the closest to orthogonal to the normal heart dipole (Fig. 14.6). Deviations from this suggest conduction abnormalities.

Six additional leads have been established to obtain the orientation of the heart dipole vector in a horizontal plane. For these leads, the three leads of Einthoven are connected with three 5000 ^ resistors to form a "zero reference," called the central terminal of Wilson. This is compared to a unipolar electrode reading taken from six different locations on the chest. These are denoted by V1,V2,...,V6 and are located on

the right side of the sternum (Vi), the left side of the sternum (V2) between the third and fourth ribs, and proceeding around the left chest following just below the fourth rib, ending on the side of the chest directly under the armpit (V6) (Fig. 14.7).

While a detailed discussion of interpretation of a vector ECG is beyond the scope of this text, there are several features of cardiac conduction that are easy to recognize. Notice from Fig. 14.6 that the normal T wave and the normal QRS complex deflect in the same direction on leads I, II, and aVR (up on I and II, down on aVR). However, the QRS complex corresponds to the upstroke and the T wave to the downstroke of the action potential, so it must be that the activation (upstroke) and recovery (downstroke) wave fronts propagate in opposite directions. Said another way, the most recently activated tissue is the first to recover. The reason for the retrograde propagation of the wave of recovery is not fully understood. Second, an inverted wave (i.e., inverted from what is normal) implies that either the wave is propagating in the retrograde direction, or more typically with novice medical technicians, that the leads have been inadvertently reversed (see Exercise 4a).

The amplitude of the QRS complex reflects the amount of muscle mass involved in propagation. Thus, if the QRS amplitude is extraordinarily large, it suggests ventricular hypertrophy. If the ECG vector is leftward from normal, it suggests left ventricular hypertrophy (Fig. 14.8), while a rightward orientation suggests right ventricular hy-pertropy (Fig. 14.9). On the other hand if an amplitude decrease is accompanied by a rightward change in orientation, a diagnosis of myocardial infarction in the left ventricle is suggested, while a leftward orientation with decreased amplitude suggests a myocardial infarction of the right ventricle, as the heart vector is deflected away from the location of the infarction (see Exercises 6 and 7).

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