ELECTROCARDIOGRAPHY 



327 



2 + J 



FIG. 2. The amount of potential recorded at P from the 

 charge at discs 2 and 3 of fig. i , the potential differences of 

 which are summed. The potential is proportional to the solid 

 angle 12, under which the discs appear from the electrode 

 point P. Mi is the individual total moment of the two discs. 



fiber during the "accession" process may be repre- 

 sented by only one single discoid charge with the 

 square unit moment 



m = mo -|- mj = — 



47r 



(2.3) 



where V = -V" + V' = | V" | + j V | is the 

 spike potential of the monophasic action potential, 

 counted in its absolute value. 



In figure i we closed the two ends of the fiber by 

 two discs, I and 4, a procedure which apparently 

 does not take into account the fact that between 

 the resting and the active portion of the fiber a certain 

 part of the fiber is in a transient state. Nevertheless, 

 a correct solution is possible if one extends the prin- 

 ciple of figure I to a subdivision of the fiber into 

 infinitely small slices, to each of which the same 

 procedure is applied (233). The general result of such 

 a consideration is that the dipole moment of the 

 single fiber can be represented by the simple expres- 

 sion of equation 2.3, where V equals the difference 

 between the membrane potentials at the beginning 

 and the end of the fiber. 



For many purposes equation 2.2, containing the 

 solid angle fi, is difficult to apply. Especially if 

 vectorial concepts are introduced, the dipole must be 

 represented by its total moment Mi^ which takes into 

 account tiie fact that the field potentials are pro- 



' The index i to M refers to the individual fiber, the moment 

 of which is represented by M,. 



portional to the surface q of the charge. 

 Mi = q ■ m 



(2.4) 



This moment M ■, can be represented by a vector which 

 stands perpendicular to discs i to 4 and lies along the 

 fiber directed from — to -|-, its length indicating the 

 scalar value of Mi. Since the membrane resting and 

 action potentials of the various cardiac fibers are 

 nearly identical, the length of the vector is dependent 

 only upon the cross section of the fiber, when identical 

 instants of the depolarization process are compared. 

 Using the total moment Mi, a second expression 

 of the local field potential generated by a single 

 fiber may be given: 



M, 



R2 



■cos 9 



(2.5) 



where R is the distance between the exploring elec- 

 trode and the center of the disc, and 6 the angle 

 between Mi and the "lead line," joining the exploring 

 electrode and the dipole center (fig. 2). 



The foregoing equations are valid only in infinite 

 homogeneous, linear resistive media. If the medium 

 has a boundary, the flow lines are forced to run 

 parallel to the boundary and are thereby compressed, 

 leading to a higher density of lines near the boundary 

 and therefore to higher local potentials. Simple equa- 

 tions for such limited fields can be derived only for 

 spherical forms. Even in cylinders they are rather 

 complicated (145, 362). The following is the equation 

 for the potential Vp at the point P inside a spherical 

 medium (151 ) : 



V„ = 



(^^£) 



cos 6 



(2.6) 



where r is the distance between the exploring elec- 

 trode and the dipole, R is the radius of the sphere, 

 mi = V/4 7r, if V is the spike potential, and q is the 

 cross section of the fiber. Figure 3 gives a picture of 

 the equipotential lines. The potentials at the surface 

 of the sphere (r = R) follow the much simpler 

 equation 



Vp = M,- -^cos e 

 R^ 



or, in the form of equation 2.2 



Vp = 3 • m, ■ fi 



(2.5a) 



(2.2a) 



This means that the potentials at the surfaces of the 

 sphere are three times as great as they could be if 



