COMMUNICATIONS
387
lo :
24,
[14] C. A. Harlow et al., "On pattern recognition and medicalimages," 4th Annual Princeton Conf. on Info. Sci. and Systems, 1970.
,o -
20 16
12
&
l0l
~0
Comparison of the Ballistocardiogram with the Electrocardiogram
'0 4 0 0
T. D. SCHICK AND E. K. FRANKE 40
60
120
160 200 240
Fig. 7. Automatically Traced Contour of Child's Femur.
Abstract-A ballistocardiogram (BCG) and a high frequency electrocardiogram (HF-ECG) were recorded from twenty-six volunteer human subjects. Seven of these subjects exhibited abnormal patterns in one (or both) of these signals. Significant matching was found when these signals were used to examine the correspondence of electrical and mechanical events in the heart.
able even when radioluscency in unossified cartilage is not a problem. The growth in long leg bones of children is on the order of 1 cm or less over periods of interest, whereas inaccuracy due to subjective measurement from film can be up to 5 mm. The problem of radioluscency of unossified cartilage was INTRODUCTION overcome with a two-step solution. A mammographic technique yielded a film with some slight contrast between cartilage The electrocardiogram (ECG) and ballistocardiogram (BCG) and surrounding muscle and fat. Computer processing then are simple, noninvasive techniques for deriving quantitative inredistributed the image brightness samples over a more rec- formation on cardiac function. Analysis of the BCG recording tangular histogram and emphasized higher spatial frequencies. of body displacement (or one of its time derivations) has led The overall result was to produce an image of sufficient clarity to what appears to be a sound theoretical explanation of the to allow accurate length measurement at the condylar edge genesis of the normal recording since calculations of stroke instead of the epiphyseal plate. volume made from the ballistocardiogram have been found to The magnitude of the estimated observer error prompted a be useful estimates of the actual stroke volume (1, 2). But the study into the feasibility of automatically tracing the contour agreement becomes progressively worse in BCG's of patients of the bone end and objectively relating it to the ruler grades. with cardiovascular disorders, which are commonly classified A tracing algorithm was developed and tested using a contrast (1) as Groups II, III, and IV. Group II exhibits beat variability enhanced radiogram of an excised infant's femur. The irregu- in a minority of ballistocardiographic complexes, Group III lar trace of Figure 7 follows the general line of the con- demonstrates beat variability in a majority of complexes but dyles but is not smooth enough to warrant placing confidence the characteristic waves are still identifiable. In Group IV, in the algorithm. Difficulties in accurately tracing the contour however, the patterns are so chaotic that the onset of systole are attributable to noise and poor edge definition. cannot be identified with certainty by using the ballistocardiogram alone. The question arises whether there might be correlates of the REFERENCES in abnormal BCG patterns discernible in the ECG irregularities [1] M. Anderson et al., "Distribution of lengths of the normal femur tracing. For both economy and convenience, the standard and tibia in children from 1 to 18 years of age," J. Bone andJoint electrocardiographic recorder does not extend to the high freSurgery, vol. 46A, pp. 1197-1202, 1964. [2] M. M. Maresh, "Linear growth of the long bones of the extremi- quency range where the irregularities of relevance might be ties from infancy through the adolescence," Am. J. Dis. Child., anticipated. It was shown as early as 1952 (3) that the QRS vol. 89, pp. 725-742, 1955. complex of the ECG contained high frequency notching and [3] D. Henderson, "Leg length measurement by scanography," Ap- slurring when observed by means of an oscillograph while the plied Radiology, vol. 3, pp. 34-37, 1974. conventional pen recording appeared normal. Later investiga[4] A. E. Burgess et al., "Analysis of errors in leg length measure- tions revealed significant correlations between these high frements," Wo be submitted to Investigative Radiology. events and heart disease (4, 5). Therefore, a link has [5] B. Blesser and D. Ozonoff, "A model for the radiologic process," quency been established between abnormal depolarization of the Radiology, vol. 103, pp. 515-521, 1972. (6] D. B. Darling, Radiography of Infants and Children, Springfield: ventricles demonstrated in the high frequency electrocardiogram (HF-ECG) and heart disease. It is reasonable then, to Charles C. Thomas, 1962. [7] M. M. Maresh, "Growth of the major long bones in healthy chil- investigate the relationship between the electrical and mechanical activity of the heart using the HF-ECG and the BCG. Put dren," Am. J. Dis. Child., vol. 66, 227-257, 1943. [8] 0. R. Hassanein, Towards Improvements in the Radiological Mea- another way: If the HF-ECG reveals notching and slurring in surement of Growth in Long Leg Bones of Children, Master's the QRS complex indicating abnormal depolarization pathThesis, The Univ. of British Columbia, Vancouver, 1974. ways through the ventricles (6), are these events reflected in [9] E. L. Hall et al., "A survey of preprocessing and feature extrac- the muscular contraction of the ventricles as monitored by tion techniques for radiographic images,"IEEE Trans. Computers, the ballistocardiogram? In order to answer this question, vol. C-20, pp. 1032-1044, 1971. methods for analysing both the HF-ECG patterns and the BCG J. D. PicStructure and the Representation of Campbell, Edge [101 patterns must be established. tures, Ph.D. Thesis, Univ. of Missouri, Columbia, 1969. [11] C. K. Chow and T. Kaneko, "Boundary detection of radiographic Manuscript received July 26, 1976; revised February 10, 1977, May images by a threshold method," IBM Report RC 3203, 1970. [12] C. A. Harlow etal., "Image analysis and line identification," Dept. 26, 1977, and August 22, 1977. Elect. Eng. and Dept. Radiol., Univ. Missouri, Columbia, Tech. T. D. Schick is with the Department of Physiology, Albany Medical Report, 1970.
[131 F. Roellinger et al., 'Computer analysis of chest radiographs," Computer Graphics and Image Proc., vol. 2, pp. 232-251, 1973.
Center, Albany, NY 12208. E. K. Franke is with the Department of Physics, University of Cincinnati, Cincinnati, OH 45221.
0018-9294/78/0700-0387$00.75 ( 1978 IEEE
IEEE TRANSACTIONS ON BIOMEDICAL
388
ENGINEERING, VOL. BME-25, NO. 4, JULY 1978
The present study was designed to investigate whether abnormal electrical events in the heart as reflected by the electrocardiogram produce abnormal muscular responses as monitored by the BCG. Thus, aberrant HF-ECG traces with no apparent .changes in the mechanical pumping of the heart are not as diagnostically significant as such traces with associated abnormal ballistocardiographic patterns. METHODS
Twenty-six volunteer subjects whose ages ranged from the mid-thirties to eighty years, participated in this project. Electrodes were placed on the left arm and leg for electrocardiographic recording (Lead III) and the electrode cables were connected to a differential preamplifier with a bandwidth of dc-I 000 Hz. Limb lead III was chosen since it showed a predominance of notching and slurring in this and other studies (3). The subject reclined on a longitudinal undamped, ultralow frequency ballistocardiograph (ULF-BCG); the acceleration of the subject's center of mass (1) was obtained using a Kistler Servo Accelerometer/Amplifier system. The ballistocardiogram and electrocardiogram were passed through Kiethly low noise amplifiers and simultaneously recorded on a 7channel electrocardiograph and transcribed onto a Sony FM Instrument tape recorder. The band pass of this recorder was over 1 kHz at 33 inches per second (ips) drive speed. This was more than sufficient for the frequencies of interest in this research. The entire recording system (excluding the electrocardiograph) was found to have flat response up to 900 Hz. A Faraday cage enclosed the ULF-BCG, and HE-ECG preamp and the accelerometer in order to minimize ac interference as much as possible. With the use of the FM tape recorder, two to three minutes of recording provided over 100 ECG and BCG complexes from each subject. The tapes were played back into a Tektronix storage oscilloscope to dissociate random transients from consistent notching or slurring in the QRS complex of the HF-ECG. In order to distinguish notching and slurring from artifacts, the tape speed w'as adjusted to 15 ips and the scope was triggered so that the QRS complexes were displayed beat by beat. Artifacts appeared randomly along the QRS complexes while notching and slurring remained stationary. Abnormal BCG's were more readily identified simply by scanning them on a strip chart recorder according to the criteria established by Starr (1). Of the twenty-six subjects studied, seven were found to have notching or slurring in the QRS complex or their BCG's fell into Groups II or III. These seven subjects were used to evaluate the correlation of electrical and mechanical activity of the ventricles. In order to proceed with the analysis, a total of sixty QRS complexes and their respective BCG complexes from one subject at a time were filtered, digitized, and stored on disk using the Hewlett-Packard 5451B Fourier Analyzer System. The HF-ECG was band passed to 210 Hz while the BCG was bandpassed to 20 Hz. The sampling rate was adjusted so that the maximum frequency of the display (or bandwidth) was over 250 Hiz. The fundamental frequency was 1.95 Hz. Each of the sixty QRS complexes were transformed into the frequency domain using the fast Fourier transform (FFT). This yielded 105 Fourier coefficients, an and bn. Similarly, each of the sixty BCG complexes were transformed using the FFT but the significant frequency content was much less (12 Hz) and yielded 6 Fourier coefficients. These sets of components for both the HF-ECG and BCG identified the corresponding time domain data. Since the frequency range of the HF-ECG was greater than the BCG, direct comparison was not possible. Therefore, to permit a comparative analysis between the two sets of recordings from each subject, the Fourier coefficients for each of the 60 QRS complexes were formed into a column array (or column vector) of increasing n, yielding a 105 component
D(X,PK)
x
'K
Fig. 1. The minimum distance between vectorsX andPk, D(X,Pk). vector representing the QRS complex. This will be referred to as the QRS pattern vector. These 60 pattern vectors were compared among themselves. Similarly, the Fourier coefficients for each corresponding BCG complex were formed into a vector column of 6 components, and the BCG pattern vectors compared among themselves. The comparison involved a search for the most similar pattern vector pairs for QRS complexes in the case of the HF-ECG and the most similar pattern vector pairs for the H-I-J-K complexes in the case of the BCG. This comparison was implemented using the minimum distance classification approach (7). The minimum distance classification scheme required a set of d real numbers which identified the pattern of interest. In the present case the pattern of interest was either the QRS complex of the HF-ECG or the H-I-J-K complex of the BCG (see ref. 1). As an example, consider the set of 60 QRS complexes. One of these QRS patterns was represented as a point x2, in a d dimensional pattern space with coordinates of x's were real (i.e., they were the Fourier X3* Xd) theandsetformed the so-called pattern vector X. Simcoefficients) ilarly each of the remaining 59 patterns were represented by 59). The pattern vector X was pattern vectors Pi(i = 1, then matched to one of the Pi pattern vectors. A straightforward method was to find the Euclidean distance between this vector X and each of the Pi's, and then find theminimum of this set of distances. The distance between X and Pi was
(xI,
2,*
given by:
D(X Pi)=
n =I
(xn Pni) -
d= 105 for the HF-ECG QRS complex = 6 for the BCG complex. x. was the nth component of X,Pni was the nth component of Pi. After these distances were calculated for all values of i, the minimum value of this set was found. If D(X, Pk) was the minimum value (here i = k), then the X pattern vector was "closest" to Pk and was said to be most similar to the pattern of Pk. (See Figure 1). The minimum distance classification program began with the first QRS complex pattern vector as X which was compared with each of the remaining 59 QRS complex pattern vectors, Pi. When the pattern which came closest to this first complex was found, these two were iden-
where
tified as a "match" of the patterns. Each successive pattern was independently run through the same comparative analysis. The same minimum distance classification program was also applied to the BCG pattern vector data to identify the BCG "matches." The computer printed out the numbered pair of patterns which were most alike for the HF-ECG's and for the For example, if the first QRS complex was found BCG's. most similar to the thirteenth QRS complex then the output was (1, 13). The output of these matches for the HF-ECG and BCG were then compared to see if there were cross matches between the electrical and mechanical signals. That match of (1, 13) of the HF-ECG occurs, was there the is,sameif amatch for the BCG? If there was such a cross match
389
COMMUNICATIONS
currence of cross matches for the abormal group were found significant at a level of a = .05 when compared to the normal group using the Welch's t-test (8). This indicates that the presence of aberrant electrical pathways represented as the notching and slurring of the QRS complex corresponds to a change in the muscular activity of the heart reflected by the BCG. Further studies on the irregularity problem of BCG's are desirable, since it is a major obstacle to a more refined classification of ballistocardiograms and has impeded a more widespread acceptance of ballistocardiography as a noninvasive diagnostic tool for detection of abnormal myocardial contraction.
TABLE I HF-ECG
BCG
(1,5)*
(1,5)*
(4,5) (5,1)*
(4,17) (5,1)*
(2,6) (3, 17)
(2,3) (3, 8)
(6,2) (7,10)* (8,17)
(6,11) (7,10)* (8,3)
(9,18)
(9,14)
(10,7)
(10,5)
(11, 10)
(11, 6) (12,10)
(12,8) (13,17)
(13,2)
(14,19) (15,5)
(14,15) (15,14)
(16,11)* (17,12) (18,19) (19,-12) (20,11)*
(16,11)* (17,8)
(18,3) (19,1) (20,11)*
Pattern classification table for twenty HF-ECG and twenty BCG complexes. Asterisks indicate the cross matches.
TABLE II No. of cross matches
Subject 1
6
2
14
3
14
4
5
5
7
6
13
7
3
The number of cross matches between HF-ECG and BCG in 7 subjects for 60 consecutive beats/subject. TABLE III Subject
Number of Matches
Subject
Number of
Matches
1
1
10
1
2
3
11
2
3
2
12
1
4
1
13
1
5
0
14
2
6
1
15
0
7
3
16
1
8
2
17
0
9
1
18
0
19
0
Matches for 19 normal subjects using data from 20 beats.
this would indicate a correlation between the two signals for those two beats.
ACKNOWLEDGMENT The authors wish to thank Professor H. H. Stratton for his helpful suggestions in the preparation of this manuscript.
REFERENCES 1. I. Starr and A. Noordergraaf: Ballistocardiography in Cardiovascular
Research. Philadelphia: J. B. Lippincott Co. (1967). 2. W. Scarborough, E. F. Folk III, P. Smith, J. Condon: "The nature of records from ultralow frequency ballistocardiographic studies and their relation to circulatory events." Am. J. Cardiol. 2:613-641, (1958). 3. P. Langner: "The value of high fidelity electrocardiography using the cathode ray oscillograph and an expanded time scale." Circ. 5 :249-256, (1952). 4. L. Horan, N. Flowers: "Diagnostic import of QRS notching in high frequency electrocardiograms of living subjects with heart disease." Circ. 44:605-611, (1971). 5. P. Langner, A. Lauer: "The relative significance of high-frequency and low-frequency notching in the electrocardiogram." Am. Heart J. 71 (1): 34-42, (1966). 6. D. Durrer, P. Formijne, R. van Dam, J. Bruller, A. van Lier, F. Meyler: "The electrocardiogram in normal and some abnormal conditions."Am. HeartJ. 61(3):303-314 (1961). 7. N. Nilsson: Learning Machines New York: McGraw-Hill Book Co. (1965). 8. Walpole, R. and R. Myers: Probability and Statistics for Engineers and Scientists. New York: Macmillan Co. (1972).
A Practical Algorithm for Solving Dynamic Membrane Equations STANLEY RUSH AND HUGH LARSEN Abstract-Many investigators work with the Hodgkin-Huxley model of membrane behavior or extensions thereof. In these models action potentials are found as solutions of simultaneous non-linear differential equations which must be solved using numerical techniques on a digital computer. Recent membrane models showing pacemaker activity, such as that of McAllister, Noble, and Tsien, involve solutions covering long periods of time, up to fisve seconds, and many ionic currents. Those added requirements make it desirable to have an efficient algorithm to minimize computer costs, and a systematic and simple solution method to keep the program writing and debugging to manageable levels.
RESULTS AND DISCUSSION Table I shows the computer matching of both HF-ECG Manuscript received February 8, 1977; revised July 27, 1977. This and BCG for the first twenty beats for one subject. The asterisk indicates a cross match. Note that the match (1, 5) work was supported by PHS under Grants HL-01486, HL-14614, and does not necessarily imply the match of (5, 1) (since the HL-09831. S. Rush is with the Department of Electrical Engineering, University first number in the bracket, the vector X, has been removed of Vermont, Burlington, VT 05401. from the pattern space). The cross matches for all seven subH. Larsen was with the Department of Electrical Engineering, Unijects are given in Table II. Table III shows the results of versity of Vermont, Burlington, VT 05401. He is now with Hewlettcross matches found for the group of 19 normals. The oc- Packard, Waltham, MA.
0018-9294/78/0700-0389$00.75 © 1978 IEEE
387
lo :
24,
[14] C. A. Harlow et al., "On pattern recognition and medicalimages," 4th Annual Princeton Conf. on Info. Sci. and Systems, 1970.
,o -
20 16
12
&
l0l
~0
Comparison of the Ballistocardiogram with the Electrocardiogram
'0 4 0 0
T. D. SCHICK AND E. K. FRANKE 40
60
120
160 200 240
Fig. 7. Automatically Traced Contour of Child's Femur.
Abstract-A ballistocardiogram (BCG) and a high frequency electrocardiogram (HF-ECG) were recorded from twenty-six volunteer human subjects. Seven of these subjects exhibited abnormal patterns in one (or both) of these signals. Significant matching was found when these signals were used to examine the correspondence of electrical and mechanical events in the heart.
able even when radioluscency in unossified cartilage is not a problem. The growth in long leg bones of children is on the order of 1 cm or less over periods of interest, whereas inaccuracy due to subjective measurement from film can be up to 5 mm. The problem of radioluscency of unossified cartilage was INTRODUCTION overcome with a two-step solution. A mammographic technique yielded a film with some slight contrast between cartilage The electrocardiogram (ECG) and ballistocardiogram (BCG) and surrounding muscle and fat. Computer processing then are simple, noninvasive techniques for deriving quantitative inredistributed the image brightness samples over a more rec- formation on cardiac function. Analysis of the BCG recording tangular histogram and emphasized higher spatial frequencies. of body displacement (or one of its time derivations) has led The overall result was to produce an image of sufficient clarity to what appears to be a sound theoretical explanation of the to allow accurate length measurement at the condylar edge genesis of the normal recording since calculations of stroke instead of the epiphyseal plate. volume made from the ballistocardiogram have been found to The magnitude of the estimated observer error prompted a be useful estimates of the actual stroke volume (1, 2). But the study into the feasibility of automatically tracing the contour agreement becomes progressively worse in BCG's of patients of the bone end and objectively relating it to the ruler grades. with cardiovascular disorders, which are commonly classified A tracing algorithm was developed and tested using a contrast (1) as Groups II, III, and IV. Group II exhibits beat variability enhanced radiogram of an excised infant's femur. The irregu- in a minority of ballistocardiographic complexes, Group III lar trace of Figure 7 follows the general line of the con- demonstrates beat variability in a majority of complexes but dyles but is not smooth enough to warrant placing confidence the characteristic waves are still identifiable. In Group IV, in the algorithm. Difficulties in accurately tracing the contour however, the patterns are so chaotic that the onset of systole are attributable to noise and poor edge definition. cannot be identified with certainty by using the ballistocardiogram alone. The question arises whether there might be correlates of the REFERENCES in abnormal BCG patterns discernible in the ECG irregularities [1] M. Anderson et al., "Distribution of lengths of the normal femur tracing. For both economy and convenience, the standard and tibia in children from 1 to 18 years of age," J. Bone andJoint electrocardiographic recorder does not extend to the high freSurgery, vol. 46A, pp. 1197-1202, 1964. [2] M. M. Maresh, "Linear growth of the long bones of the extremi- quency range where the irregularities of relevance might be ties from infancy through the adolescence," Am. J. Dis. Child., anticipated. It was shown as early as 1952 (3) that the QRS vol. 89, pp. 725-742, 1955. complex of the ECG contained high frequency notching and [3] D. Henderson, "Leg length measurement by scanography," Ap- slurring when observed by means of an oscillograph while the plied Radiology, vol. 3, pp. 34-37, 1974. conventional pen recording appeared normal. Later investiga[4] A. E. Burgess et al., "Analysis of errors in leg length measure- tions revealed significant correlations between these high frements," Wo be submitted to Investigative Radiology. events and heart disease (4, 5). Therefore, a link has [5] B. Blesser and D. Ozonoff, "A model for the radiologic process," quency been established between abnormal depolarization of the Radiology, vol. 103, pp. 515-521, 1972. (6] D. B. Darling, Radiography of Infants and Children, Springfield: ventricles demonstrated in the high frequency electrocardiogram (HF-ECG) and heart disease. It is reasonable then, to Charles C. Thomas, 1962. [7] M. M. Maresh, "Growth of the major long bones in healthy chil- investigate the relationship between the electrical and mechanical activity of the heart using the HF-ECG and the BCG. Put dren," Am. J. Dis. Child., vol. 66, 227-257, 1943. [8] 0. R. Hassanein, Towards Improvements in the Radiological Mea- another way: If the HF-ECG reveals notching and slurring in surement of Growth in Long Leg Bones of Children, Master's the QRS complex indicating abnormal depolarization pathThesis, The Univ. of British Columbia, Vancouver, 1974. ways through the ventricles (6), are these events reflected in [9] E. L. Hall et al., "A survey of preprocessing and feature extrac- the muscular contraction of the ventricles as monitored by tion techniques for radiographic images,"IEEE Trans. Computers, the ballistocardiogram? In order to answer this question, vol. C-20, pp. 1032-1044, 1971. methods for analysing both the HF-ECG patterns and the BCG J. D. PicStructure and the Representation of Campbell, Edge [101 patterns must be established. tures, Ph.D. Thesis, Univ. of Missouri, Columbia, 1969. [11] C. K. Chow and T. Kaneko, "Boundary detection of radiographic Manuscript received July 26, 1976; revised February 10, 1977, May images by a threshold method," IBM Report RC 3203, 1970. [12] C. A. Harlow etal., "Image analysis and line identification," Dept. 26, 1977, and August 22, 1977. Elect. Eng. and Dept. Radiol., Univ. Missouri, Columbia, Tech. T. D. Schick is with the Department of Physiology, Albany Medical Report, 1970.
[131 F. Roellinger et al., 'Computer analysis of chest radiographs," Computer Graphics and Image Proc., vol. 2, pp. 232-251, 1973.
Center, Albany, NY 12208. E. K. Franke is with the Department of Physics, University of Cincinnati, Cincinnati, OH 45221.
0018-9294/78/0700-0387$00.75 ( 1978 IEEE
IEEE TRANSACTIONS ON BIOMEDICAL
388
ENGINEERING, VOL. BME-25, NO. 4, JULY 1978
The present study was designed to investigate whether abnormal electrical events in the heart as reflected by the electrocardiogram produce abnormal muscular responses as monitored by the BCG. Thus, aberrant HF-ECG traces with no apparent .changes in the mechanical pumping of the heart are not as diagnostically significant as such traces with associated abnormal ballistocardiographic patterns. METHODS
Twenty-six volunteer subjects whose ages ranged from the mid-thirties to eighty years, participated in this project. Electrodes were placed on the left arm and leg for electrocardiographic recording (Lead III) and the electrode cables were connected to a differential preamplifier with a bandwidth of dc-I 000 Hz. Limb lead III was chosen since it showed a predominance of notching and slurring in this and other studies (3). The subject reclined on a longitudinal undamped, ultralow frequency ballistocardiograph (ULF-BCG); the acceleration of the subject's center of mass (1) was obtained using a Kistler Servo Accelerometer/Amplifier system. The ballistocardiogram and electrocardiogram were passed through Kiethly low noise amplifiers and simultaneously recorded on a 7channel electrocardiograph and transcribed onto a Sony FM Instrument tape recorder. The band pass of this recorder was over 1 kHz at 33 inches per second (ips) drive speed. This was more than sufficient for the frequencies of interest in this research. The entire recording system (excluding the electrocardiograph) was found to have flat response up to 900 Hz. A Faraday cage enclosed the ULF-BCG, and HE-ECG preamp and the accelerometer in order to minimize ac interference as much as possible. With the use of the FM tape recorder, two to three minutes of recording provided over 100 ECG and BCG complexes from each subject. The tapes were played back into a Tektronix storage oscilloscope to dissociate random transients from consistent notching or slurring in the QRS complex of the HF-ECG. In order to distinguish notching and slurring from artifacts, the tape speed w'as adjusted to 15 ips and the scope was triggered so that the QRS complexes were displayed beat by beat. Artifacts appeared randomly along the QRS complexes while notching and slurring remained stationary. Abnormal BCG's were more readily identified simply by scanning them on a strip chart recorder according to the criteria established by Starr (1). Of the twenty-six subjects studied, seven were found to have notching or slurring in the QRS complex or their BCG's fell into Groups II or III. These seven subjects were used to evaluate the correlation of electrical and mechanical activity of the ventricles. In order to proceed with the analysis, a total of sixty QRS complexes and their respective BCG complexes from one subject at a time were filtered, digitized, and stored on disk using the Hewlett-Packard 5451B Fourier Analyzer System. The HF-ECG was band passed to 210 Hz while the BCG was bandpassed to 20 Hz. The sampling rate was adjusted so that the maximum frequency of the display (or bandwidth) was over 250 Hiz. The fundamental frequency was 1.95 Hz. Each of the sixty QRS complexes were transformed into the frequency domain using the fast Fourier transform (FFT). This yielded 105 Fourier coefficients, an and bn. Similarly, each of the sixty BCG complexes were transformed using the FFT but the significant frequency content was much less (12 Hz) and yielded 6 Fourier coefficients. These sets of components for both the HF-ECG and BCG identified the corresponding time domain data. Since the frequency range of the HF-ECG was greater than the BCG, direct comparison was not possible. Therefore, to permit a comparative analysis between the two sets of recordings from each subject, the Fourier coefficients for each of the 60 QRS complexes were formed into a column array (or column vector) of increasing n, yielding a 105 component
D(X,PK)
x
'K
Fig. 1. The minimum distance between vectorsX andPk, D(X,Pk). vector representing the QRS complex. This will be referred to as the QRS pattern vector. These 60 pattern vectors were compared among themselves. Similarly, the Fourier coefficients for each corresponding BCG complex were formed into a vector column of 6 components, and the BCG pattern vectors compared among themselves. The comparison involved a search for the most similar pattern vector pairs for QRS complexes in the case of the HF-ECG and the most similar pattern vector pairs for the H-I-J-K complexes in the case of the BCG. This comparison was implemented using the minimum distance classification approach (7). The minimum distance classification scheme required a set of d real numbers which identified the pattern of interest. In the present case the pattern of interest was either the QRS complex of the HF-ECG or the H-I-J-K complex of the BCG (see ref. 1). As an example, consider the set of 60 QRS complexes. One of these QRS patterns was represented as a point x2, in a d dimensional pattern space with coordinates of x's were real (i.e., they were the Fourier X3* Xd) theandsetformed the so-called pattern vector X. Simcoefficients) ilarly each of the remaining 59 patterns were represented by 59). The pattern vector X was pattern vectors Pi(i = 1, then matched to one of the Pi pattern vectors. A straightforward method was to find the Euclidean distance between this vector X and each of the Pi's, and then find theminimum of this set of distances. The distance between X and Pi was
(xI,
2,*
given by:
D(X Pi)=
n =I
(xn Pni) -
d= 105 for the HF-ECG QRS complex = 6 for the BCG complex. x. was the nth component of X,Pni was the nth component of Pi. After these distances were calculated for all values of i, the minimum value of this set was found. If D(X, Pk) was the minimum value (here i = k), then the X pattern vector was "closest" to Pk and was said to be most similar to the pattern of Pk. (See Figure 1). The minimum distance classification program began with the first QRS complex pattern vector as X which was compared with each of the remaining 59 QRS complex pattern vectors, Pi. When the pattern which came closest to this first complex was found, these two were iden-
where
tified as a "match" of the patterns. Each successive pattern was independently run through the same comparative analysis. The same minimum distance classification program was also applied to the BCG pattern vector data to identify the BCG "matches." The computer printed out the numbered pair of patterns which were most alike for the HF-ECG's and for the For example, if the first QRS complex was found BCG's. most similar to the thirteenth QRS complex then the output was (1, 13). The output of these matches for the HF-ECG and BCG were then compared to see if there were cross matches between the electrical and mechanical signals. That match of (1, 13) of the HF-ECG occurs, was there the is,sameif amatch for the BCG? If there was such a cross match
389
COMMUNICATIONS
currence of cross matches for the abormal group were found significant at a level of a = .05 when compared to the normal group using the Welch's t-test (8). This indicates that the presence of aberrant electrical pathways represented as the notching and slurring of the QRS complex corresponds to a change in the muscular activity of the heart reflected by the BCG. Further studies on the irregularity problem of BCG's are desirable, since it is a major obstacle to a more refined classification of ballistocardiograms and has impeded a more widespread acceptance of ballistocardiography as a noninvasive diagnostic tool for detection of abnormal myocardial contraction.
TABLE I HF-ECG
BCG
(1,5)*
(1,5)*
(4,5) (5,1)*
(4,17) (5,1)*
(2,6) (3, 17)
(2,3) (3, 8)
(6,2) (7,10)* (8,17)
(6,11) (7,10)* (8,3)
(9,18)
(9,14)
(10,7)
(10,5)
(11, 10)
(11, 6) (12,10)
(12,8) (13,17)
(13,2)
(14,19) (15,5)
(14,15) (15,14)
(16,11)* (17,12) (18,19) (19,-12) (20,11)*
(16,11)* (17,8)
(18,3) (19,1) (20,11)*
Pattern classification table for twenty HF-ECG and twenty BCG complexes. Asterisks indicate the cross matches.
TABLE II No. of cross matches
Subject 1
6
2
14
3
14
4
5
5
7
6
13
7
3
The number of cross matches between HF-ECG and BCG in 7 subjects for 60 consecutive beats/subject. TABLE III Subject
Number of Matches
Subject
Number of
Matches
1
1
10
1
2
3
11
2
3
2
12
1
4
1
13
1
5
0
14
2
6
1
15
0
7
3
16
1
8
2
17
0
9
1
18
0
19
0
Matches for 19 normal subjects using data from 20 beats.
this would indicate a correlation between the two signals for those two beats.
ACKNOWLEDGMENT The authors wish to thank Professor H. H. Stratton for his helpful suggestions in the preparation of this manuscript.
REFERENCES 1. I. Starr and A. Noordergraaf: Ballistocardiography in Cardiovascular
Research. Philadelphia: J. B. Lippincott Co. (1967). 2. W. Scarborough, E. F. Folk III, P. Smith, J. Condon: "The nature of records from ultralow frequency ballistocardiographic studies and their relation to circulatory events." Am. J. Cardiol. 2:613-641, (1958). 3. P. Langner: "The value of high fidelity electrocardiography using the cathode ray oscillograph and an expanded time scale." Circ. 5 :249-256, (1952). 4. L. Horan, N. Flowers: "Diagnostic import of QRS notching in high frequency electrocardiograms of living subjects with heart disease." Circ. 44:605-611, (1971). 5. P. Langner, A. Lauer: "The relative significance of high-frequency and low-frequency notching in the electrocardiogram." Am. Heart J. 71 (1): 34-42, (1966). 6. D. Durrer, P. Formijne, R. van Dam, J. Bruller, A. van Lier, F. Meyler: "The electrocardiogram in normal and some abnormal conditions."Am. HeartJ. 61(3):303-314 (1961). 7. N. Nilsson: Learning Machines New York: McGraw-Hill Book Co. (1965). 8. Walpole, R. and R. Myers: Probability and Statistics for Engineers and Scientists. New York: Macmillan Co. (1972).
A Practical Algorithm for Solving Dynamic Membrane Equations STANLEY RUSH AND HUGH LARSEN Abstract-Many investigators work with the Hodgkin-Huxley model of membrane behavior or extensions thereof. In these models action potentials are found as solutions of simultaneous non-linear differential equations which must be solved using numerical techniques on a digital computer. Recent membrane models showing pacemaker activity, such as that of McAllister, Noble, and Tsien, involve solutions covering long periods of time, up to fisve seconds, and many ionic currents. Those added requirements make it desirable to have an efficient algorithm to minimize computer costs, and a systematic and simple solution method to keep the program writing and debugging to manageable levels.
RESULTS AND DISCUSSION Table I shows the computer matching of both HF-ECG Manuscript received February 8, 1977; revised July 27, 1977. This and BCG for the first twenty beats for one subject. The asterisk indicates a cross match. Note that the match (1, 5) work was supported by PHS under Grants HL-01486, HL-14614, and does not necessarily imply the match of (5, 1) (since the HL-09831. S. Rush is with the Department of Electrical Engineering, University first number in the bracket, the vector X, has been removed of Vermont, Burlington, VT 05401. from the pattern space). The cross matches for all seven subH. Larsen was with the Department of Electrical Engineering, Unijects are given in Table II. Table III shows the results of versity of Vermont, Burlington, VT 05401. He is now with Hewlettcross matches found for the group of 19 normals. The oc- Packard, Waltham, MA.
0018-9294/78/0700-0389$00.75 © 1978 IEEE
