176 Mechanical and Electrical Character of the Heartbeat /9 : 7 



eccentric 2 as those occurring in normal humans. When one adds to this 

 the effects of the nonspherical shape of the human body, it seems very 

 reasonable that the use of the four standard leads loses a great deal of 

 the available information. 



An integral part of vector electrocardiography is the equivalent dipole 

 or heart vector. The discussion in the last section illustrated that a net 

 dipole is the first approximation to any distribution of current sources 

 whose net sum is zero. There are an infinite number of distributions 

 which have the same vector dipole as a first approximation. It is in no 

 way obvious that the heart should be well represented by the dipole 

 approximation. Two different types of experiments, which have shown 

 that this approximation is almost as good as the experimental data, are 

 discussed in the following paragraphs. 



Let the heart vector be denoted by p. It is conventional to represent 

 this as a sum of three vectors directed along the cartesian axes. One 

 may write 



P = Pxi + Pyj + pS 



where the subscripts refer to the scalar components of/), and i,j, and k 

 are unit vectors directed along the x, y, and z axes. Then at any point 

 on the periphery, the voltage V (relative to ground) may be written as a 

 linear sum of the three components of the heart vector; that is to say 



V = ap x + pp x + v p 2 



In general, the three constants a, jS, rj will depend on the location of the 

 dipole, the location of the observation point, and the shape of the torso. 

 The three quantities a, £, r\ will be constant for the entire QRS-complex 

 if the heart can be represented as a dipole. 



If V is measured at four points, one may write four equations 



V l = a lPx + PlPy + r\\Pz 



V 2 = 0. 2 p x + P 2 py + 7] 2 p 2 



V 3 = O-zPx + fizPy + V3pz 



V± = a 4 /> z + Pip y + r\±p z 



These may be regarded as four nonhomogeneous, linear equations in the 

 three unknowns p x , p y , and p z . For most sets of the (a, j8, v\)\ and the 

 V's, there are no consistent solutions for p x , p y , and p z . If, however, the 

 heart vector is a good approximation, the fourth equation should be a 

 linear combination of the first three. This, then, is a simple, un- 

 ambiguous test of the dipole approximation. 



Measurements on humans in which four pairs of wires are used, that 

 is, four independent leads, have shown that the QRS-complex can be 



2 Eccentric here means displaced from the vertical and horizontal center of 

 the torso. 



