Fractal theory calculates the dimension of a signal, called the fractal dimension (FD), by considering the signal as an object. Fractal dimension of normal IHR by different methods FD of normal and noisy IHR by rescaled range method. Fractal theory states that any signal can be viewed as an object and its calculated dimension, called the fractal dimension (FD), can be used to understand the internal mechanism of a system that generates and controls the signal.
Historical Background
Therefore, let's consider some metric peculiarities of some unusual mathematical objects, which we later use to describe some biomedical systems. Fractal modeling is an alternative way to describe some self-similar or self-affine signals and is a non-parametric modeling. The measured length of the fractal line will vary depending on the size of the ruler used.
Aim of the Project
Fractal modeling technique has been applied to a variety of fields ranging from one-dimensional signal study [8] to image processing [9], from sea-scattered radar signal [10] to medical image analysis [II]. Fractal dimension is also widely used to understand the characteristics of strange attractors, the case of a chaotic signal. The methods are also applied to calculate the fractal dimension of normal and pathological HR data with an ultimate goal of distinguishing abnormal HR from a normal one.
Organization of the Dissertation
ELECTROCARDIOGRAM
Cardiac Activity
SA node), which is the pacemaker or initiator of the electrical impulses that stimulate the heart. Since the ventricle has to perform a more powerful pumping action, its walls are thicker than those of the atrium and its surfaces are corrugated. Between the anterior wall of the ventricle and the septum is a ridge of muscle that is part of the heart's electrical conduction system, known as the Bundle of His.
At the junction between the right and left atrium and the right ventricle of the septum, there is another node, the atrioventricular node. The right atrium and left ventricle are connected by a fibrous tissue known as the atrioventricular ring, to which are attached the three cups of the tricuspid valve, which is the connecting valve between the two chambers. Its walls are about three times as thick as the walls of the right ventricle because of this feature.
Conduction to the left ventricle occurs through the left bundle branch, which is the ventricular muscle on the septum side. A discussion of the anatomy of the heart and the electrical excitation system necessary to produce and control the muscular contraction should help round out the background materials needed to understand cardiac dynamics. Blood enters the heart on the right side through two main veins: the superior vena cava, which leads from the upper extremities of the body, and the inferior vena cava, which leads from the extremities of the body's organs below the heart.
The right ventricle pumps blood through the pulmonary artery to the lungs where it is oxygenated.
The Electrocardiogram
Every action potential in the heart originates near the top of the right atrium, at a point called the pacemaker or SA node. The activation wave travels parallel to the surface of the atria to the junction of the atria and ventricles. The wave ends at a point near the center of the heart, the AV node.
At this point, some special fibers act as a "delay line" to ensure proper timing between the action of the atria and ventricles. The front of the wave in the ventricle does not follow along the surface, but is perpendicular to it and moves from the inside to the outside of the wall of the ventricle, ending at the top or apex of the heart. The QRS complex is the combined result of atrial repolarization and ventricular depolarization, which occur almost simultaneously.
The shape and polarity of each of these features varies with the location of the measuring electrodes in relation to the heart, and a cardiologist usually bases his diagnosis on readings taken from various electrode locations. The P-R interval is taken from the beginning of the P wave to the beginning of the QRS complex. The Q-T interval is taken from the beginning of the QRS complex to the end of the T wave.
The S-T segment is the period between the end of the QRS complex and the beginning of the T wave.
ECG Lead Systems
It is the time required for depolarization to travel from the SA node through the atria, the AV node, and the His-Purkinje system to the ventricles. The QRS represents the time it takes for depolarization to pass through the His-Purkinje system and the ventricular muscles. Einthoven postulated that at any given time of the cardiac cycle the frontal plane representation of the electrical axis of the heart is a two-dimensional vector.
Further, the ECG measured from any of the three basic limb lists is a one-dimensional time variant. The sides of the triangle represent the lines along which the three projections of the ECG vector are measured. Based on this, Einthoven showed that the instantaneous voltage measured from any of the three limb plug positions is approximately equal to the algebraic sum of the other two or that the vector sum of the projections on the three lines is equal to zero.
To actually make this true, the polarity of the lead II measurement must be reversed. The three standard, three augmented, and six V leads allow the spatial direction of excitation and recovery of the cardiac chambers to be identified. For example, the chest wires (V) provide a very local examination of the ventricular myocardium, and the activity recorded is the projection in the horizontal plane.
This lead system simulates the VI position with electrode placement as follows: positive electrode, fourth intercostal space, right sternal border, negative electrode just below the outer portion of the left clavicle, with ground just above anywhere but usually below the right clavicle.
FRACTAL MODELING TECHNIQUES
- Introduction
- Box Counting Method
- Rescaled Range Method
- Relative Dispersion Method
- Fourier Method
Theoretically, FD is independent of the resolution (sampling rate) of the series due to the scale-invariant properties of fractals, i.e., it has the problem of estimating the dam height that can hold the entire water flow in input under a constant output condition. . When the dam discharge is a stationary random variable, the lake water level is the integral of the differences of the constant inflow and outflow.
Range, R(u) is defined as the maximum value minus the minimum value of the integral of the differences of each data from the arithmetic mean of a data series averaged at a different starting point. When the range is divided by the standard deviation, S(u) of the corresponding data set, we get RlS. In this analysis we take advantage of the fact that the variance of a variable changes as the measurement resolution changes.
The above is an equation of a straight line whose slope is directly related to the fractal dimension of that time series. Power index is given by the slope of the spectral density measured in logarithmic scale. The power index is one of the measures of the irregularity of time series data.
The slope of the power spectrum is determined by fitting a straight line using the least-squares method.
FRACTAL DIMENSION OF SIMULATED DATA
- Introduction
- Description of Data
- Estimation of Fractal Dimension
- Discussion
We applied four methods, as previously described, to estimate the FD on generated data with s=1.5 and 1-.=1.5. Since the calculation of PSD using FFT requires a data length that must be a power of 2, we considered the same data length type for all methods. The dimensions were estimated by each method for different data lengths to observe the convergence, that is, to estimate that the data length has an FD of 1.5.
From the obtained results, we see that in the box counting method there is a tendency to increase the FD with the increase of the data length and the best result is found when the data length is 6000. It is interesting to note that the rescaled range method also produces the best results. result when the data length is 6000, but there is no monotonic variation in the estimated FD with this method. The relative dispersion method tends to estimate lower FD as the data length increases.
Given the trend of the result, we might expect that a data length of 7000 or slightly more will yield a useful result. But due to the limitation of the software we used, we rejected the idea of testing the Fourier method with a huge amount of data. In this figure, the abscissa indicates the data length and the ordinate indicates the estimated FD.
FRACTAL DIMENSION OF HEART RATE
- Introduction
- Data Description
- Fractal Dimension of Normal Heart Rate
- Fractal Dimension of Abnormal Heart Rate
- Discussion
- CONCLUSIONS
- Discussions
- Future Perspectives
It makes no sense to calculate the FD of unevenly distributed heart rate data as long as we want to manipulate this dimension by making heart state decisions. In the previous chapter, it was found that this method gives the best result for a data length of about 6000. On the other hand, the other 3 methods give variable FD for different data sets, even though all IHR are of normal subjects.
Since the purpose of applying fractal dimensional analysis IS to detect cardiac abnormality, it is hoped that the FD of an abnormal HR time series will be different from that of normal one. The results are shown in Table 5.3 together with the FD of normal IHR reproduced from Table 5.2. The calculation of FD from normal and noisy HR time series data is presented in this chapter.
The results show that the rescaling method gives consistent FD for all 5 normal IHR data sets. Different types of heart defects are expected to give different FDs if an appropriate data length is chosen. To use fractal theory to detect actual abnormalities, we need to calculate the FD IHR of different types of pathology.
Since the calculation of FD requires a long length of IHR data and it is not available at the moment, we are unable to calculate the FD of pathological lHR.
APPENDIX
