Journal of Nuclear Cardiology

, Volume 21, Issue 4, pp 723–729

Limitations of parametric modeling of the left ventricle using first harmonic analysis: Possible role for gaussian modeling


    • Section of Cardiology, Jefferson Heart InstituteThomas Jefferson University
Original Article

DOI: 10.1007/s12350-014-9905-5

Cite this article as:
Hansen, C.L. J. Nucl. Cardiol. (2014) 21: 723. doi:10.1007/s12350-014-9905-5



Fourier (cosine) analysis of time activity curves (TAC) of radionuclide ventriculography (RVG) may oversimplify the TAC and has limitations.


We identified 21 patients who had undergone 24 frame planar RVG with ejection fractions ranging from 8% to 76% (43% ± 19%). The TAC for each pixel was fitted to both a cosine and gaussian function then analyzed on a pixel-by-pixel basis then over the entire LV. Second, mathematical simulations were performed to analyze the stability of each fit in the presence of low amplitude and noise.


The fit was slightly but significantly better for the gaussian compared to the cosine function (RMS gaussian 13.0% ± 2.5% vs 13.5% ± 2.1% cosine; P = .016). There was near exact correlation with amplitude and between gaussian mu and cosine phase. The SD of phase from the cosine fit correlated strongly with the SD of the mu from the gaussian fit. The proposed new measure of dyssynchrony, the sigma parameter of the gaussian fit, correlated with the SD of the cosine phase (r = 0.520, P = .016). Simulations showed gradual but modest deviation of the sigma parameter from the gaussian fit with lower amplitudes whereas the deviation of the calculated SD of phase increased exponentially with decreasing amplitude.


First harmonic (cosine) fitting has significant limitations. Gaussian fitting is an alternative way to model the LV TAC. The sigma from the gaussian may provide additional information LV dyssynchrony and is less influenced by image noise. Gaussian fitting merits further evaluation for modeling LV function.


Image processinggated blood pool imagingcardiomyopathy


Planar radionuclide imaging is a well-accepted technique for the evaluation of left ventricular function. Parametric imaging has been frequently used to gain additional insights from these studies. Fourier analysis was introduced more than 30 years ago for ventricular analysis of both the time activity curve as well as individual pixels in the LV.1 This has been shown to be useful in multiple clinical situations including stroke volume ratio for detection of regurgitant lesions as well as phase analysis for identification of the source of ventricular arrhythmias or accessory pathways.2,3 With the development of other technologies, however, interest in the technique faded. Parametric modeling has received renewed interest with the recognition of the importance of ventricular dyssynchrony in heart failure and the introduction of biventricular pacing therapy.4-6 Although some patients appear to benefit substantially from this therapy, many patients do not, which has prompted strong interest in developing a noninvasive technique capable of identifying those patients most likely to benefit. Although many techniques have been proposed, such as tissue Doppler and speckle tracking with echocardiography, none has yet emerged as definitive.7,8 First harmonic Fourier analysis has more recently been proposed as a technique for identifying those patients most likely to benefit from resynchronization therapy.9,10

The Fourier transform can exactly model any periodic function as an infinite series of sines and cosines of monotonically increasing frequency. First harmonic Fourier analysis of gated blood pool images models the TAC as a cosine curve. Since the frequency is determined by the cardiac cycle, the time activity curve, as modeled by the cosine function has only two parameters: the amplitude or how far the curve deviates from 0 and the phase: at what point of the cycle the curve achieves it greatest amplitude. However, this model implicitly assumes systole occupies exactly one half of the cardiac cycle and is incapable of modeling either more or less. However, the relative ratio of time in diastole to systole varies with heart rate; under resting conditions, systole occupies significantly less than half of the cardiac cycle and should thus raises serious doubts that the cosine curve is the best way to model the cardiac cycle.11-13 Furthermore, systolic time intervals can change in heart failure.13 Although, mathematically, more accurate curve fitting can be achieved using higher harmonics, there is no intuitive clinical counterpart to the resulting parameters. A possible alternative is the gaussian function well recognized as the “normal” or “bell” curve. This curve has the parameters of mean (mu) and amplitude, which correlate directly with the phase, and amplitude of the cosine curve. It also has a third parameter, sigma, reflecting the width or standard deviation of the curve and thus introduces the ability to incorporate variations in the duration of systole. Because of this, it may be a more accurate way to model the cardiac cycle. Loss of synchrony may also be reflected in increased duration of systole and gaussian fitting could be particularly suited to characterizing patients with LV dysfunction.

The purpose of this study was to compare the goodness of fit between the first harmonic of the Fourier transform (i.e., cosine) and gaussian in a sample of patients with a wide variation of ventricular function.


We identified 21 patients who had undergone planar gated blood pool ventriculography who had a range of ejection fractions from 8% to 76% (43% ± 19%). Patients who demonstrated significant gating artifact or other technical limitations were not included. Best septal views were obtained in the conventional fashion. Images were obtained using 24 frames/cycle and a minimum of 200,000 counts per frame.

The images were filtered both temporally and spatially using a simple 1-2-1 and 9 point smooth filters, respectively; the images were then normalized to the maximal pixel in the image. The left ventricular region of interest (ROI) was manually identified. Time activity curves (TAC) were generated for each pixel in the ROI. Cosine and gaussian fitting was performed using a modification of the Levenberg-Marquardt algorithm using LAPACK.14 The Levenberg-Marquardt algorithm is a combination of the Gauss-Newton method and simple gradient descent. It is much more robust than Gauss-Newton and has become the most widely accepted algorithm for fitting non-linear equations.

For the gaussian fit, it should be noted that two TACs that are 180° out of phase will have the same mean but amplitudes of opposite sign. In order to facilitate comparison with the cosine parameters, the output of the gaussian fit was adjusted so that all amplitudes were positive; when the amplitude was negative, the mean was adjusted by half the cycle length. Goodness of fit was determined by calculating the percentage of the square root of the mean of the squares (%RMS) of the differences between the fitted model and the observed data for each model.

Second, mathematical simulations were performed to test the stability of the parameters of cosine and gaussian fits by creating 24 frame 64 × 64 pixel images set to a cosine or gaussian function. All pixels in each frame were set to the same value and each sequential frame followed a point on the cosine or gaussian curve, respectively. Peak counts were set to 100 and separate images were generated representing maximal amplitudes ranging from 2 to 50 for both functions. The standard deviation parameter (sigma) of the gaussian model was set to 5 for all amplitudes. Simulated Poisson noise was added to the images, and they were then filtered spatially and temporally, normalized then fit to a cosine or gaussian function, respectively, in the same way as the RVGs. The RMS error was calculated as the deviation of the fit from the noise-corrupted model. The sigma of the fitted gaussian would be expected to be 5 for all amplitudes, and the phase SD of should be 0 for the cosine model since, before the introduction of noise, all curves had the same standard deviation (for the gaussian) and phase (for the cosine).

Image analysis for both the RVGs and simulations was done on a Macintosh platform using software written in C++.

Statistical Analysis

Statistical analysis was performed using IBM SPSS Statistics version 20 (IBM, Armonk, NY). Differences were compared using a paired t test. Parameters were compared using linear regression. The variables in non-linear relationships were transformed to allow linear regression. A P value < .05 was considered significant. Categorization of the degree of correlation was as outlined by Dancey and Reidy.15


The results of a patient with an ejection fraction of 8% and apical dyskinesis are shown in Figure 1. There is near exact agreement between the amplitude images as well as the phase and mu images from the cosine and gaussian models. A more subtle change in sigma can be seen in the region of dyskinesis at the apex.
Figure 1

The parametric images from a patient with apical dyskinesis and an ejection fraction of 8% are shown. There is near exact correlation between the two amplitude images and the phase and mu images. The gaussian sigma image shows a change in the region of the apex. LAO, left anterior oblique; Amp, amplitude; Cos, cosine

The limitations of first harmonic and potential advantages of gaussian modeling can be seen in Figure 2. This shows the TAC of a single pixel; it can be seen that systole occupies less than half of the cardiac cycle. The difficulty of modeling this with a cosine curve is apparent and is reflected in the relatively high RMS error of 18%. The gaussian fit easily incorporates variations in the duration of systole and has a much lower RMS error of 6%. Of note, most of the error in the gaussian fit is in diastole and the period of atrial filling.
Figure 2

The time activity curve from a representative pixel along with its cosine and gaussian fits is shown. It can be seen that in this example systole occupies significantly less than half of the cardiac cycle and that the cosine fit cannot accurately model this. The approximate RMS error for the gaussian was 6%, for the cosine 18%

Comparing all patients, the average regional RMS error was slightly but significantly lower for the gaussian as compared to the cosine fit (13.0% ± 0.025% vs 13.5% ± 0.021%, P = .016). The 21 regions had a combined total of 5,356 pixels. Looking at all patients, there were near perfect correlations between the amplitude images regionally (r = 0.996, P < .001; Figure 3A) and on a pixel-by-pixel basis (r = 0.991, P < .001; Figure 3B). It should be noted that the slope of the regression line comparing amplitudes would be expected to be 0.5 because half of a cosine function is negatively deflected. That is, for a given total amplitude, half of a cosine curve is positive and the other half negative, whereas a gaussian curve will deflect the entire amplitude from the baseline.
Figure 3

The correlation of the amplitude from the gaussian and cosine fits are shown on a regional (A) and pixel-by-pixel (B) basis. There is near exact correlation noted

The phase of the cosine and mu of the gaussian also showed strong correlation on both a regional (r = 0.976, P < .001; Figure 4A) and pixel-by-pixel basis (r = 0.704, P < .001; Figure 4B). The generally accepted measure of dyssynchrony in cosine fitting, the SD of the phase, showed strong correlation with the SD of mu (r = 0.84, P < .001; Figure 5).
Figure 4

Same format as Figure 3 showing the regional (A) and pixel-by-pixel (B) relation between the mean (mu) of the gaussian and phase of the cosine
Figure 5

The standard deviation of the phase of the cosine is plotted against the standard deviation of the mean (mu) of the gaussian fit. There is a very strong correlation noted

Initial explorations of the potential implications of the sigma parameter of the gaussian fit showed a significant correlation with the SD of the phase (r = 0.520, P = .016; Figure 6). On a pixel-by-pixel basis, sigma correlated inversely with amplitude (r = 0.512, P < .001; Figure 7).
Figure 6

There was a significant correlation between sigma from the gaussian fit and the SD of the phase from the cosine fit
Figure 7

There was a nonlinear, inverse correlation between the amplitude of the gaussian fit and the standard deviation (sigma) demonstrating an increase in sigma with decreasing amplitude

Results of the mathematical simulations are summarized in Figure 8. The RMS error of both models increased as the amplitude decreased. RMS error (error was calculated as the difference between the fit and the noise-corrupted data) is shown on the X-axis. Starting on the left hand side (lowest RMS error) were the models with the greatest amplitude. Moving to the right, the amplitudes decreased, and the RMS error steadily increased. The gaussian sigma and the SD of the phase, in pixels, are shown on the Y-axis using a logarithmic scale. It can be seen that the deviation of the sigma fit increases modestly from the true value of 5 as amplitude decreases whereas the deviation of the SD of the phase deviates exponentially from the true value of 0 as amplitude decreases.
Figure 8

As the amplitude of the models decreased the RMS error of the fit (X axis) increased monotonically for both models. The deviation of sigma in the gaussian fit deviated slowly from the true value of 5 whereas the deviation of the phase SD of the cosine fit increased exponentially from the true value of 0. The Y axis is on a log scale


The results of this study show that the left ventricular time activity curve is at least as accurately modeled by a gaussian as by a first harmonic Fourier (cosine) function. The gaussian fit accurately replicates all of the parameters of a cosine fit (amplitude, phase and phase SD) and also provides the additional parameter, sigma. Although the current study was not designed to evaluate its utility, it is a practical and intuitive way to model changes in the duration of systole which have been shown to occur with different heart rates and disease states,11-13 an important characteristic that cannot be captured using first harmonic analysis.

The currently accepted parameter for identifying LV dyssynchrony is the standard deviation of the phase. The simulations performed here raise significant concerns as to exactly what this is measuring. We showed an exponential increase in error with the SD of the phase as amplitude decreased. This raises the obvious concern that some of the dyssynchrony that is being reported may be, in fact, a phenomenon of the increased noise at lower amplitude and not true dyssynchrony.

Sigma correlates with the SD of phase further supporting the possibility that it may be useful as a marker of LV dyssynchrony. Sigma, as a global parameter, would be expected to reflect dyssynchrony since differences in the onset of contraction will lead to an increase in the fraction of the cardiac cycle the heart spends in systole. As a local parameter, a better term for sigma would be dispersion because it reflects the duration of systole at that point. It is possible that sigma, or perhaps variations in sigma throughout the ventricle may prove useful in understanding pathology or predicting response to therapy.

Using higher harmonics of the Fourier transform can improve the accuracy of the fit which would then allow modeling of variations of the fraction of the cardiac cycle that the heart spends in systole and has also been described as a more accurate way to identify the actual onset of mechanical relaxation.16 The drawback to this is that although it provides a more accurate representation, the additional harmonics have no intuitive counterpart; they produce no additional clinical insight. They might not even be necessary if the only objective is to identify those points on the time activity curve where it crosses the mean. On the other hand, using a gaussian fit, the onset of mechanical systole or relaxation could be identified much more simply and intuitively by, for example, identifying the points at the full width half maximum or tenth maximum of the gaussian fit.

The primary purpose of parametric modeling is to produce clinically useful markers, parameters that “distill” the essence of physiologic or pathophysiologic process, parameters that provide insight into a disease state, parameters that assist in patient management. The strength of contraction, or amplitude, and timing of contraction, or phase, can be modeled with a first harmonic. Duration of contraction cannot. Although goodness of fit, in and of itself, is not the primary objective of parametric modeling, a more accurate fit certainly raises confidence that the parameters obtained have a more sound physiologic basis. It is not yet known whether the sigma parameter can help in this or not. The calculated sigma alone, variations of sigma between normal and diseased segments of a ventricle or even skewness or kurtosis may provide useful insight into a patient’s condition. This will need to be explored.


The current study is preliminary; there were no clinical endpoints or “gold standard” to assess the clinical utility, if any, of the sigma parameter from gaussian fitting. This needs further study. The current study also looked at only radionuclide ventriculography (RVG) whereas currently there is more interest in gated SPECT. We feel that the best place to begin this investigation is with RVG since it is a more accurate measure of LV function and, historically, phase analysis began with this. We are currently evaluating the utility of gaussian fitting with gated SPECT.

New Knowledge Gained

Parametric imaging using first harmonic analysis is limited by the inability of this method to model variations in systolic duration and by the very strong influence image noise has on its primary indicator of dyssynchrony, the standard deviation of phase. Parametric imaging using a gaussian model is as least as accurate as first harmonic analysis, accurately replicates the information provided by first harmonic analysis and has a novel parameter, sigma, which is much less influenced by image noise and potentially a useful marker of dyssynchrony. The clinical utility of this method has yet to be determined.


The author has nothing to disclose.

Copyright information

© American Society of Nuclear Cardiology 2014