Appendix A

Consistent System of Rectangular

and Spherical Coordinates for

Electrocardiology and Magnetocardiology

The basis for the AHA coordinate system arises from the basic research performed by Willem Einthoven. Einthoven defined the positive x axis as oriented from the right to the left side of the patient (which actually means from left to right when viewed from an observer, as traditionally defined in physics). Because the electric heart vector is typically directed to the left, back, and down, Einthoven chose the y axis to point down so that the main deflection of the QRS complex is positive. Einthoven investigated the ECG signal only in the frontal plane and therefore did not need the z coordinate.

When constructing his vectorcardiographic system, Frank decided to accept the x and y directions defined by Einthoven and defined the z coordinate to point to the back in order to have a right-handed coordinate system. This coordinate system is the one standardized by AHA.

The coordinate system of AHA includes the following shortcomings:

Relative to the natural observation planes of the patient (i.e., the frontal plane observed from the front, the sagittal plane observed from left, and the transverse plane observed from above), only the sagittal plane is observed from the positive side.

The spherical coordinate system, fixed to this coordinate system with the generally accepted axes, results in an unfamiliar orientation.

Additionally, Einthoven's attempt to obtain a positive deflection of the electric y signal results in a negative deflection in the magnetic signal.

For these reasons Malmivuo developed a consistent coordinate system for electrocardiology, which avoids the aforementioned drawbacks (Malmivuo, 1976; Malmivuo et al., 1977).

The rectangular coordinate system should be right-handed to be consistent with conventions in the physical sciences and to permit straightforward application of the standard equations used in vector analysis and electromagnetism.

The three coordinate planes are the

*xy*,*yz*, and*zx*planes.Each plane is viewed from positive side.

Angles in the

*xy*,*yz*, and*zx*planes are measured in the positive direction (counterclockwise) from the*x*,*y*, and*z*axes, respectively, with a range of either 0° to 360° or 0° to ±180° , with negative angles being measured in a clockwise direction from the axis.The four quadrants in each coordinate plane are specified in a positive, counterclockwise, sequence:

I: | 0° | to | 90° | |

II: | 90° | to | 180° | |

III: | 180° | to | 270° | |

IV: | 270° | to | 360° |

It is convenient to align the rectangular coordinate axes with the body axes (i.e., sagittal, transverse, and longitudinal axes). This means that the coordinate planes correspond to the frontal, left sagittal, and transverse planes. In order that the planes be viewed from their positive side and that these views be the same as those used in clinical vector electrocardiography, the positive directions of the

Note that selection of the positive directions of the axes in the above order provides the most practical orientation for the

(A.1) | |

The radius vector is described by the symbol *r*. The angles θ and φ are called *colatitude* and *longitude*, respectively. The angle θ is also called *polar* angle because it is an angle dimensioned from the pole (i.e. *z* axis). These requirements for rectangular and spherical polar coordinates are based on existing mathematical conventions. This mathematically consistent coordinate system is illustrated in Figure A.3A.

(A.2) | |

In the illustrative spherical coordinate system, the vector magnitude is represented by the symbol *M* (being the same as the radius vector *r* in the spherical polar coordinate system). The angles *E* and *A* are called the *elevation* and *azimuth*, respectively. This coordinate system is illustrated in Figure A.3B. (Note, that when the angles elevation and azimuth are those used in connection with the AHA coordinate system, they are represented by symbols *V* and *H* and they differ from those of the consistent system introduced in this chapter.)

The angles elevation and azimuth correspond exactly to the *latitude* and *longitude* angles, respectively, used in geography. Therefore, ordinary (and familiar) geographic map projection techniques can be immediately applied to maps of electric potential and magnetic field over the entire torso surface, as described in Figure A.4.

A) Mathematically consistent spherical polar coordinate system.

B) Illustrative spherical coordinate system.

the consistent coordinate system and the AHA coordinate system

Consistent coordinate system |
AHA coordinate system |

xy z E A N | -z+90°+x -y -V H M |

In the rectangular coordinates, the *x* and *z* coordinates in the consistent system have opposite polarity to those in the AHA system. However, the consistent system and the AHA system have *identical vector loop displays*.

In the spherical coordinates, the elevation angles (*E* and *V*) are the same in both systems except for different polarity. The azimuth angles (*A* and *H*) have the same polarity in both systems, but because of the different reference axis the values in the consistent system differ by 90° from the values in the AHA system. The vector magnitude *M* is, of course, the same in both systems. (Note, that in the existing literature one may find other definitions for the elevation and azimuth angle than those of the AHA.)

The components of a vector in the ABC coordinate system may be transformed to the XYZ coordinate system with the following linear transformation (Equation A.2):

(A.3) |

The components of a vector in the XYZ coordinate system may be transformed to the ABC coordinate system with the following linear transformation (Equation A.3):

(A.4) |

American Heart Association (1967): Recommendations for standardization of leads and of specifications for instruments in electrocardiography and vectorcardiography. *Circulation* 35: 583-7. (Report of Committee on Electrocardiography).

Frank E (1956): An accurate, clinically practical system for spatial vectorcardiography. *Circulation* 13:(5) 737-49.

Malmivuo JA (1976): On the detection of the magnetic heart vector - An application of the reciprocity theorem. *Helsinki Univ. Tech.*, Acta Polytechn. Scand., El. Eng. Series. Vol. 39., pp. 112. (Dr. Tech. thesis)

Malmivuo JA, Wikswo JP, Barry WH, Harrison DC, Fairbank WM (1977): Consistent system of rectangular and spherical coordinates for electrocardiography and magnetocardiography. *Med. & Biol. Eng. & Comput.* 15:(4) 413-5.

Morse PM, Feshbach H (1953): *Methods of Theoretical Physics.* Part I, 997 pp. McGraw-Hill, New York.

Smythe WR (1968): *Static and Dynamic Electricity*, 3rd ed., 623 pp. McGraw-Hill, New York.