J. ELECTROCARDIOLOGY, 9 (2) 129-132

Angular Velocity of the QRS Loop of the Vectorcardiogram in the Normal Heart

BY EVAN FLETCHER, M.D., F.R.C.P.* AND SOAD BEKHEIT, M.D., PH.D., M.R.C.P.t

SUMMARY Angular ve loc i ty expressed in radians/

sec of the rotation movement of the QRS loop at intervals of 2.5 msecs was calcu- lated from a c o m p u t e r program wri t ten in Fortran IV. Frontal, horizontal and left sagittal planes were recorded in 125 nor- mal subjects for ana lys i s . The range o f a n g u l a r ve loc i ty for 375 QRS loops was from a few radians/sec to a maximum of 95 radians/sec . Average va lues o f m a x i m u m a n g u l a r v e l o c i t i e s were: f ronta l p lane , 46.2 rad ian s / s ec , h o r i z o n t a l p lane , 41 radians/sec , and left sagi t ta l plane, 34.1 r ad ian s / s ec . A n g u l a r v e l o c i t y e x p r e s s e d as a periodic funct ion o f the vector loop is characterized by polyphasic curves. In the frontal plane, curves t e n d to be more symmetrical with max imum values in the middle . A n g u l a r v e l o c i t y curves are an alternate expression of analysis of planar vector loops employed in clinical practice. Their n o r m a l r a n g e s are g iven in th i s paper.

Visual inspection of the QRS vector loop shows obvious differences in the length of arc distances inscribed by the time marker over known time intervals. On the assumption that

From the University of Texas Medical School, Houston, Texas. *Professor of Medicine, Director, Division of Car- diology. tAssistant Professor of Medicine, Division of Car- diology. Reprint requests to: Evan Fletcher, M.D., University of Texas Medical School at Houston, 226 Freeman Building, 6400 West Cullen Street, Houston, TX 77025.

the dipole is fixed, ~ these distances can be quantitated as the time marker inscribes a rota- tion motion relative to the dipole. The equations of motion of the time marker can be expressed as changes in the angle O projected on the di- pole by known arc distances of the loop (Fig. 1). The unit of measurement of the angle O is the radian, and the rotation motion of inscrip- tion of the loop is expressed in radians/sec. Angular velocity co can thus be expressed as a periodic function of the QRS vector loop. The definition of the radian and the equations for calculating angular velocity are shown in the appendix (A, B). The form of the angular velocity curve of the normal QRS loop has been described in a preliminary communica- tion. 2 Previously, angular velocity curves ex- pressed in degrees have been measured in infants, 3,4 and for part of the QRS loop in adults. 5 The purpose of this paper is to quan- titate angular velocity values in normal adults and describe the forms of curves for the planar QRS loops of the veetoreardiogram (VCG).

~--- Tangeni i al velocity

,5" " - " -- ~d S"--r

"o ~ ~ 7~,~ 7/- ,~ 47 ,' / ~ Angular v e!or (,.~)

~ ~-./~ J j - -

o z o % ~ I _ Oe

z ~ i ~ - ~ ~Tirne Lnterval-dt = 2"5 m s e c

NORMAL HORIZONTAL PLANE

Fig. 1: Representative QRS vector loop for com- putation of angular velocity. Measurements are made from the start of the loop (null point, illus- trated by the black dot) to the midpoint between the time markers as illustrated in vectors 1, 2, 3. Vector 2 is a composite of sequential vectors within distance ds in the time interval dt (2.5 msees). Angular velocity is the rate of change of the angle e expressed in radians/sec. (See Appendix for

derivation of the radian). Arrow indicates tangential velocity.

129

1 30 FLETCHER AND BEKHEIT

,~ 60+

.50'~

2O.

10,

0 11~25 21!25 31'25 &1"25 5~25 61'25 TI'2S

TIME rese ts

TO-

S0-

&O.

3 0 .

.+

o: 20-

.+++ ~, m!2 + ~'.2s ,+~'+2 + +;'-2+ m !2+ TIME m I I r

~'ZS

7O

I 0

4 0

1 i ~ +o

' 1 20

IO

0 | |

TIME mice&

Fig. 2: Angular velocity curves of the QRS vector loops in the normal heart (Frank VCG). A = frontal plane. B = horizontal plane. C = left sagittal plane. The curves are polyphasic. In the frontal plane (A) the curve is more symmetrical with max imum value of 62 radians/sec in contrast with max imum values of less than 30 radians/sec in hori- zontal plane (B) and left sagittal plane (C). Rad's/sec. = radians/sec.

J. ELECTROCARDIOLOGY, VOL. 9, NO. 2, 1976

ANGULAR VELOCITY OF QRS LOOPS 1 31

?0,

60~

so-

40-

'~ 30

20

10

11.25 21'.2S 31~25 TIME m.$ecs

M A T E R I A L S A N D M E T H O D S

The Frank VCG 6 was recorded in the frontal, left sagittal, and horizontal planes in 125 men, aged 20-37 years, with normal hearts, using the Sanborn Viso Scope Model 780-6A. A rectangular cartesian co-ordinate system centered at the start of the tracing (null point) was imposed on each vector loop. Thus the co-ordinates of each time marker were defined. When the null point was not clearly defined, an arbitrary location was chosen at the center of the "bright area." By transferring from cartesian to polar co-ordinates, the desired param- eters could be measured. A computer program written in Fortran IV was developed to calculate angular velocity in radians/sec at 2.5 msecs intervals.

R E S U L T S

Angular Velocity Curves When angular velocity was plotted as a peri-

odic function of time in the 375 QRS loops avail- able, the resul tant curves were invariably polyphasic (Fig. 2, A, B, C). In the 125 frontal QRS loops, however, angular velocity curves had a tendency to be more symmetrical with maximum values up to 60 radians/sec, distrib- uted in the middle third of the curve (Fig. 2, C).

Distribution of Angular Velocity Values The absolute range of angular velocity values

for the 375 QRS loops was from a few radians/ sec to 95 radians/sec with a tendency to a range of 10-50 radians/sec. Maximum values were abstracted from each loop, and their average values for 125 loops in each plane were: frontal plane, 46.2 radians/sec, left sagittal plane, 34.1 radians/sec, and horizontal plane, 41 radians/sec.

J. ELECTROCARDIOLOGY, VOL. 9, NO. 2, 1976

Fig. 3: Left bundle branch block. Angular velocity curve in frontal plane of the Frank VCG. Note the low angular velocity values less than 5 radians/sec in the time inter- val 21.25-71.25 msecs.

~5 D I S C U S S I O N

In the normal heart, ventrieular depolariza- tion begins at the termination of the branches of the bundle of His. 7,s Angular velocity curves of the QRS loops of the VCG therefore repre- sent the rate of rotation of sequential vectors during depolarization of the Purkinje-fiber- myocardial conducting system. However, angu- lar velocity curves abstracted from planar projections of the QRS loop are in part deter- mined by changes in direction, as are distances are components of the spatial vectors. Thus, the quantitative expression of angular velocity and periodic function curves in this paper repre- sents a supplementary method of analysis of the clinical VCG.

Angular velocity curves constructed from the QRS loops of the VCG do not give direct infor- mation concerning the spread of the activation front in the Purkinje-fiber-myocardial conduc- tion system, whether it is isotropie 9 or aniso- tropic, lo Slowdown of angular velocity has been reported in left bundle branch block. 5 Figure 3 is a representative example illustrating low angular velocity values less than 5 radians/see in the mid third of time interval 21.25-71.25 msees in the frontal plane.

In summary, the interpretation of angular velocity curves at present remains empirical. As constituted in this paper, they have the disadvantages of being planar and thereby eliminating vector direction. However, they do quantitate changes produced by ventrieular depolarization and may be a useful adjuvant to other vector notations. They may be of value

132 FLETCHER AND BEKHEIT

in q u a n t i t a t i n g abso lu t e c h a n g e s in v e c t o r loops in i n t r aven t r i cu l a r block or in change in env i ronment produced by drugs or metabolic imbalance.

Appendix

A The Radian. A radian is defined as the angle subtended at the center of a circle by an arc of the c i rcumference equal in length to the radius, i.e., the re are 2 w (6.28) radians in the circumference.

360 One radian there fore - 6.28 - 57.3 degrees .

Any angle O s u b t e n d e d by an arc S can c ~

be expressed in radians as O = ~ where R

is the radius. An angle measured in radians r ep resen t s a dimensionless number , being the rat io of a length to a length.

B A n g u l a r velocity ( ~ ) Radians /sec . I t is a m e a s u r e m e n t of anu la r d i s p l a c e m e n t ex- p ressed in radians over a finite t ime interval . Thus if angle O1 at t ime t l changes to angle 02 at t ime t2, then,

02 -- O1 ~o = = ~0_ = dO radians/sec

A t dt t2 - t l =

when O t approaches zero.

R E F E R E N C E S

1. GESELOWITZ, DB: Dipole theory of electro- cardiography. Am J Cardiol 14:301, 1964

2. BINNION, P F, FLETCHER, E, LAB, S AND MAGOWAN, J A: Velocity of ventricular depo- larization of the human heart determined from the QRS loop of the vectorcardiogram. J Physiol Soc 205, 1969

3. AINGER, LE: Spatial QRS curves of the new- born infant. Spatial magnitude, velocity and orientation (Frank system). Am Heart J 75:19, 1968

4. AINGER, L E AND SKINNER, W R: Normal maturation of the spatial QRS curve charac- teristic of early infancy. Am Heart J 77:5, 1969

5. SIMONSON, E, SCHMITT, OH, BLACKBURN, H W AND LEVINE, R B: The speed of ventricu- lar activation measured in the spatial vector- cardiograms. Circ Res 3:409, 1955

6. FRANK, E: An accurate, clinically practical system for spatial vectorcardiography. Circu- lation 13:737, 1956

7. TRUEX, R C AND SMYTHE, MQ: Recent obser- vations on the human cardiac conduction system with special considerations of the atrioventricu- lar node and bundle. In International Sym- posium on Electrophysiology of the Heart , B TACCARDI AND G MARCHETTI, eds. Pergamon Press, Oxford, 1965, p 177

8. ARNTZENIUS, A C: A model of excitation of the human heart. Cardiovasc Res 3:198, 1969

9. WOODBURY, J W: Cellular electrophysiology of the heart. Handbook of Physiology, Section 2, Circulation 1:237, 1961

10. SPERELAKIS, N: Additional evidence for high resistance intercalated discs in the myocardium. Circ Res 12:676, 1963

J. ELECTROCARDIOLOGY, VOL. 9, NO. 2, 1976