332 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



where i q is the current density penetrating the 

 active cross section of the myocardial fiber. The term 

 i/q may be called L ( = '"lead vector"). The recorded 

 potential W thus equals (if we neglect the constant 

 factor 4 tt) the scalar ("dot") product of the lead 

 vector L and the dipole moment Mj, and therefore 

 the product of the projection of M, on L and the 

 magnitude of L. E.xpressed in a simpler manner, this 

 product is identical with the product of the absolute 

 values of I Mi I and j L | and the cosine of the angle 

 between Mi and L. Obviously the physically deter- 

 mined lead vector replaces the geometrically deter- 

 mined lead line, i.e., the connection of the bipolar 

 electrodes or the connection between a unipolar 

 electrode and the zero point of the vector. Translated 

 into pictorial language, equation 3.6 means: the 

 recorded potential of the electromotive surface Q 

 depends upon a) the magnitude of the dipole moment 

 Mi (i.e., the membrane potential in the case of our 

 myocardial fiber, times the surface Q or the cross 

 section ot the fiber); h) the angle between Mi and 

 the direction of L (fig. 10); c) L itself, and this involves 

 the site and distance of the electrodes. The more 

 remote the electrodes, the more scattered the flow 

 lines of the current io introduced into the body (see 

 fig. 22). The portion recorded of Mi therefore is 

 greater the nearer Mi lies to one of the electrodes. 

 These relations are valid for every configuration oi 

 the field, every electrode position, and every peculi- 

 arity of the conductive medium, be it homogeneous 

 or not. 



The currents introduced into our consideration in 

 figures 8 and 10 of course do not exist in reality. 

 They only serve as a tool to demonstrate the mode of 

 recording potential differences on myocardial filjers 

 in the thorax. We therefore may replace the current 

 lines of our analysis by lines symbolizing the configura- 

 tion of what we call a "lead field." This means that 

 for each point in the interior of the ijody a vector L 

 may be defined, the direction and length of which 

 indicates one factor of the dot product determining 

 the recorded potential of equation 3.6. This lead 

 field, dependent in its configuration upon the position 

 of the electrodes and the form of the body, replaces 

 the simplified model of the projection laws in figure 6. 

 To every point of this field a lead vector can be 

 ascriijed, the scalar product of which, with the 

 moment Mi, immediately gives the amount of poten- 

 tial recorded for Mi at the given electrode positions. 

 The first outline of this theory was given by Burger 

 & van Milaan (141-143), and a detailed description 

 by McFee & Johnston (340-342). 



It may be mentioned that the l^ipolar ECG re- 

 corded from a strip of parallel muscle fibers may be 

 correctly interpreted as the difference between two 

 monophasic action potentials recorded at the sites of 

 the two bipolar electrodes ["Differenzkonstruktion," 

 (56, 69, 433, 435)]. This principle remains valid even 

 when the muscle strip is inserted into a volume con- 

 ductor. As soon as many strips going in various direc- 

 tions act together to build up an electric field, as in 

 the intact heart, the principle of forming the diff"er- 

 ence between two monophasic action potentials is 

 no longer applicable. It may be replaced by a 

 "multiple difference construction" (56), but this 

 leads to such complicated procedures that no simple 

 and correct construction can be obtained by such 

 methods. The superimposition of dipoles can be 

 handled in an appropriate manner onlv on the basis 

 of a vectorial concept. 



4. SUPERPOSITION OF DIPOLES .'>iND THEIR FIELDS 



The interaction of numerous muscular fibers can 

 be calculated in a sufticientlv e.xact manner, only to 

 the extent that the vectors Mi of each fiber are known 

 in respect to their site and time of appearance, and 

 only insofar as the simple law of \ectorial addition 

 remains applicable. It should be pointed out here 

 that, in general, this applicability does not exist. For 

 the majority of lead systems, multiple dipoles cannot 

 be represented, in a model. In' one single dipole 

 (363). Either complicated mathematics have to be 

 used (536) or a lead system adopted which corrects 

 the inultiple sites of electric sources by virtue of its 

 lead field (see sec. 5). A correct vectorial addition of 

 the various dipole moments would be possible only 

 if one and the same lead vector L were valid for all 

 different points of the myocardium. In such a case 

 only the projection of the sum of all individual dipole 

 moments on the lead \ector equals the sum of the 

 projections of the individual fibers. This condition 

 is appro.ximately met in e\ery derivation in which 

 the dimensions of the heart are small compared with 

 the electrode distance. This is valid in the special 

 "ideal" case of a centered dipole in a homogeneous 

 sphere with electrodes put on the surface of this 

 sphere. In such cases, the superposition of individual 

 dipole moments can be constructed through the 

 parallelogram of forces. Figures 11 and 12 indicate 

 the most common cases of such a superposition in two 

 examples. If such a superposition is extended to ail 

 myocardial fillers, the result is the "heart vector." 



