_{1}

^{*}

This research is performed based on the modeling of biological signals. We can produce Heart Rate (HR) and Heart Rate Variability (HRV) signals synthetically using the mathematical relationships which are used as input for the Integral Pulse Frequency Modulation (IPFM) model. Previous researches were proposed same methods such as one model of ECG signal synthetically based on RBF neural network, a model based on IPFM with random threshold, method was based on the estimation of produced signals which are dependent on autonomic nervous system using IPFM model with fixed threshold, a new method based on the theory of vector space that based on time-varying uses of IPMF model (TVTIPMF) and special functions, and two different methods for producing HRV signals with controlled characteristics and structure of time-frequency (TF) for using non-stationary HRV analysis. In this paper, several chaotic maps such as Logistic Map, Henon Map, Lorenz and Tent Map have been used. Also, effects of sympathetic and parasympathetic nervous system and an internal input to the SA node and their effects in HRV signals were evaluated. In the proposed method, output amount of integrator in IPFM model was compared with chaotic threshold level. Then, final output of IPFM model was characterized as the HR and HRV signal. So, from HR and HRV signals obtaining from this model, linear features such as Mean, Median, Variance, Standard Deviation, Maximum Range, Minimum Range, Mode, Amplitude Range and frequency spectrum, and non-linear features such as Lyapunov Exponent, Shanon Entropy, log Entropy, Threshold Entropy, sure Entropy and mode Entropy were extracted from artificial HRV and compared them with characteristics as extracted from natural HRV signal. Also, in this paper two patients that called high sympathetic Balance and Cardiovascular Autonomy Neuropathy (CAN) which is detected and evaluated by HRV signals were simulated. These signals by changing the values of the some coefficients of the normal simulated signal and with extracted frequency feature from these signals were simulated. For final generation of these abnormal signals, frequency features such as energy of low frequency band (EL), energy of high frequency band (HL), ratio of energy in low frequency band to the energy in high frequency band (EL/EH), ratio of energy in low frequency band to the energy in all frequency band (EL/ET) and ratio of energy in high frequency band to the energy in all frequency band (EH/ET) from abnormal signals were extracted and compared with these extracted values from normal signals. The results were closely correlated with the real data which confirm the effectiveness of the proposed model. Various signals derived from the output of this model can be used for final analysis of the HRV signals, such as arrhythmia detection and classification of ECG and HRV signals. One of the applications of the proposed model is the easy evaluation of diagnostic ECG signal processing devices. Such a model can also be used in signal compression and telemedicine application.

According to the electrical activity of the heart and considering that the heart produces a series of electrical potentials, these electrical signals of the heart can be recorded by installing electrodes on the chest, left hand, and on the right foot as well. This type of signal is called electrocardiogram or ECG.

Components of this signal are: P, Q, R, S and T. The R component is considered as the component with am- plitude greater than the other components. If we could extract the peak of R by adopting a suitable method, this peak would represent a heartbeat. If the time between successive extracted R peaks is plotted as a function of time, we can have a diagram like the one in

This research is carried out based on modeling of biological signals. Thus, HRV signals can be synthetically produced by mathematical equations that are used as input for the IPFM model. Hence, this project is performed via modeling and analyzing mathematical relations. This method was chosen because by changing the inputs of IPFM model we can produce synthetic HRV signal which is similar to natural signals (the one taken from human body), and we can model several types of diseases that show their effect on the signal. So it can be concluded that this model is a comprehensive model for generating synthetic HRV signal because it is capable of producing several types of HRV signals. Hence, this type of study and its implementation is considered applied. This research requires several sequences of chaotic signals which can be obtained via relations and initial conditions of the existing resources. In this study, Guyton’s physiology book can be used as the reference to know the function and physiology of the heart and all the nerves of the sympathetic and parasympathetic effects on cardiac signals

Jafarnia Dabanloo et al. in 2004 [

By applying the Zeeman nonlinear model, Jafarnia Dabanloo et al. (2006) produced HRV signal with a cycle of ECG signals using neural network. In the results of their survey they reported the effects of breathing signals and production of Mayer waves in the power spectrum of the obtained HRV signal [

In another study Jafarnia Dabanloo et al. (2013) presented a model based on IPFM but with random threshold. They used random sequences with normal distribution [

In a research by Bailon et al. (2011), to produce HRV signal a method based on IPFM was introduced. Time threshold used in this study included non-static values. The proposed method is based on the estimation of produced signals which are dependent on autonomic nervous system using methods from IPFM model with fixed

threshold and this case demonstrates the need for a fixed time-dependent amount to produce signals of the heart rate. Finally, it was shown that the results of this method matched with the values obtained from physiological signals of human body [

In a research, by using physiological IPFM model, Seydnejad et al. [

In 2011, Michele Orini et al. was presented two different methods for producing HRV signals with controlled characteristics and structure of time-frequency (TF) for use in non-stationary HRV analysis [

In 2012, Ali Almasi et al. were presented a dynamic model to generate synthetic Phonocardiogram signals [

In 2013, Diego Martin et al. were presented a stochastic model for Photoplethysmogram (PPG) signal [

In this paper, without taking electrocardiogram signal from human body (and then obtaining HRV signal from it), this signal was obtained through a proposed mathematical model. Various signals derived from the output of this model can be used for final analysis of the HRV signals such as arrhythmia detection and classification of ECG. One application of a dynamic model which is able to produce synthetic ECG signals is the easy evaluation of diagnostic ECG signal processing devices. Such a model can also be used in signal compression and telemedicine. The overall objective of this article is to achieve an applied mathematical model for modeling and synthesizing HR and HRV signals using IPFM model in which chaotic maps are considered as its threshold level.

In this paper, a mathematical model for synthesizing heart rate variation signals (HRV) has been proposed. In the proposed model, the effect of sympathetic and parasympathetic nerves, and also, an internal input to the SA node that all of which are effective in HRV signals are evaluated. First the input of IPFM model is considered as

This section represents the method of generating chaotic maps used in the present study.

Equation (1) is called the function of logistic map which is widely used today in modeling especially for natural systems.

Parameter A specifies the chaotic state of this map. This parameter is checked for the state of 0 ≤ x ≤ 1 and 0 ≤ A ≤ 4. At first, the function

The function

It is a nonlinear system that is a simplified model of convection in the fluids. This model was proposed by meteorologist Edward Lorenz in 1963. Lorenz Model is based on simplification of Navier-Stokes equations for fluids. The fluid motion and temperature disturbances can be mentioned with the three variables X (t), Y (t), and Z (t). These variables are not spatial variables. The variable X is related to the time dependence of the fluid flow function. Variables Y and Z are related to time dependence of temperature deviations in areas far from linear areas of temperature that are obtained for the non-convective mode. By using these variables, the equations of Lo- renz Model can be expressed as three interdependent differential equations as follows (Equation (2)).

In the first equation,

Henon map is a two variables map which is defined by Equation (3).

The diagram in

Tent map was defined based on Equation (4).

The bifurcation diagram of tent map has been show in

In this section we introduce a normal ECG signal and how we read it in MATLAB software. The initial calling signal was gained as

In the following figure, the first 10 seconds of ECG signal have been drawn for seeing more clearly (see Fig- ure 10).

Then, to have a clean signal, noise deviation from the baseline, AC noise (power) and frequency interference were eliminated by applying the linear filters.

Also,

After performing the above operations, the R peak of the ECG signal can be extracted.

Then, according to the intervals of these extracted peaks, HR and the HRV (from the time difference between the peaks of the extracted R waves) are obtained from the HR.

Then, the density of power spectrum of the obtained HRV signal is calculated and it is represented in the output.

In the proposed model we first produce the desired chaotic maps with respect to relations and descriptions set forth in Section 3.2. In this scheme, the chaotic maps including Logistic, Henon, Lorenz, and Tent were used. After coding each formula of these chaotic maps, the output of each chaotic map was saved in a separate matrix.

1) Logistic Map

Logistic map is defined with the initial value of x_{0} = 0.1, and a matrix sequence and the values obtained from the model can be finally saved in a matrix. To generate chaotic sequence by logistic mapping, the following equation was used. In this case, the initial value x (0) was set to 0.1. A is the control parameter that should be a positive, real and as 0 ≤ A ≤ 4. If the Logistic mapping equation is generated by above relation in which A = 1.2, the system will be stable. If this quantity goes up to 3.57, the system will be chaotic.

2) Henon Map

Henon map was produced based on the Equation (4). Initial values x (0) and y (0) were set to 0.0239 and 0.0239, respectively. The chaotic parameters “a” and “b” were set 1.4 and 0.3, respectively. Then, by using these values and these relations, the chaotic sequence was obtained. The values resulted from the chaotic sequences were saved until they could be used as a threshold in IPFM model.

3) Lorenz Map

Lorenz maps were produced by Equation (2). We’ll check and calculate the fixed points of this model. In

proposed model, the 0.3 as the initial value for x (0) and y (0) and z (0) and values of p = 10, r = 11 and

have been allocated to the parameters of the Lorenz model. In the results chapter, the outputs of Lorenz model are shown.

4) Tent Map

Tent map was produced by Equation (5). In the proposed model, the initial value for x_{n} of this mapping was considered 0.05 and this mapping can be defined as a sequel to 1000 entries. The parameter “a” was also defined in the range of 0 to 1. As a result, after the values obtained from the chaotic mapping are normalized, they are used as threshold values in the IPFM model.

Heart Rate (HR) is a signal that controlled by the autonomic nervous system (ANS), and it contains of information about changes and signs of heart activity. ANS has two subsystems, sympathetic nerves and parasympathetic nerves. The HR may be increased by sympathetic activity or decreased by parasympathetic activity [_{0} will be deemed as an assumed index in this model. In addition, the input S_{1} produces VLF component in HRV signal. As we know, the VLF is related to peripheral vascular and mechanisms of temperature regulation that have been produced by the sympathetic nervous system [_{1} and S_{2} respectively can determine the effectiveness of parasympathetic and sympathetic mechanisms that are related to the pressure sensor (baroreceptory) that appear in the LF (or Mayer) [_{2} input is another part of the parasympathetic nervous impact on the heart, which its influence is visible in the power spectrum of HRV signal. This input is due to frequency changes of breathing during inhalation and exhalation. These respiratory changes are effective in the HF (or RSA) related to power spectrum of HRV signal [

All the above mentioned effects of increasing and decreasing heart rate by sympathetic and parasympathetic nerves have been presented in this model by Equation (5).

The input of the IPFM model was considered as

Then the integrator output with a threshold level that was previously generated by chaotic maps is compared. In fact, every chaotic mapping to an IPFM model is defined as a threshold to be considered in that model. There- fore, IPFM model for different chaotic threshold level, the output will be different. These different outputs are obtained from the output of model as the assimilated different HRV signals of the output. Where the value of the integrator output exceeds the threshold level, a pulse is generated from the IPFM model output that will be considered as a heartbeat. In fact, each pulse represents the occurrence or production of an R wave, which is calculated by Equation (7). Then, the HRV signal is achieved by calculating the difference in time of occurrence of the R wave.

In this section, from the all signals linear features such as Median, Mean, Variance, Standard Deviation, Maximum Amplitude, Minimum Amplitude, Amplitude Range and Mode, and non-linear features such as Lyapunov Exponent, Shanon Entropy, log Entropy, Threshold Entropy, sure Entropy and mode Entropy were extracted. This features extraction process was done in order to compare the values of the extracted features with artificial signals generated in this method by which the similarities and accuracy percentage of the proposed model output with the normal ECG signal will be calculated.

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

66.5143 | 66.5108 | 0.1225 | 0.3501 | 67.0760 | 65.9024 | 1.1682 | 65.9024 |

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

0.9772 | 0.9773 | 2.6459 × 10^{−5} | 0.0051 | 0.9863 | 0.9691 | 0.0172 | 0.9691 |

In this paper, tow patient that called high sympathetic Balance and Cardiovascular Autonomy Neuropathy (CAN) which is detected and evaluating by HRV signals were simulated. These signals by changing the values of the some coefficients of the normal simulated signal and with extracted frequency feature from these signals were simulated. To modeling HRV signal for these patients, some coefficients basic equation of IPFM model such as

For final generation of these abnormal signals, frequency features such as energy of low frequency band (EL), energy of high frequency band (HL), ratio of energy in low frequency band to the energy in high frequency band (EL/EH), ratio of energy in low frequency band to the energy in all frequency band (EL/ET) and ratio of energy in high frequency band to the energy in all frequency band (EH/ET) from abnormal signals were extracted and compared with these extracted values from normal signals.

In case of diseases high sympathetic balance, feature as ratio of energy in low frequency band to the energy in high frequency band (EL/EH) in compared with the same ratio in normal signal should increase at about twice, and also feature as ratio of energy in low frequency band to the energy in all frequency band (EL/ET) in compared with the same ratio in normal signal should increase and finally feature as ratio of energy in high frequency band to the energy in all frequency band (EH/ET) in compared with the same ratio in normal signal should re- main unchanged. Also, for simulation of Cardiovascular Autonomy Neuropathy (CAN) signal, feature as ratio of energy in low frequency band to the energy in high frequency band (EL/EH) in compared with the same ratio in normal signal should has value is close to zero, and feature as ratio of energy in low frequency band to the energy in all frequency band (EL/ET) in compared with the same ratio in normal signal should increase and has value is close to zero, and finally feature as ratio of energy in high frequency band to the energy in all fre- quency band (EH/ET) in compared with the same ratio in normal signal should be show a slight decrease. These proportions and values of characteristics of the diseases in compared with the normal HRV signals based on the physiological changes that cause these diseases are on the cardiovascular system (and subsequent created changes on ECG, HR and HRV signals) were included are calculated (from understanding of References [

First, the output of chaotic maps used in this study is presented.

This part represents how the results of X (IPFM model input) in the proposed model were calculated. In the IPEM model, the threshold level obtained from the different chaotic maps is compared with the outputs of IPFM model integrator. If the integrator output is more than threshold level, one pulse will be produced in the output. The pulse time (t_{i}) is in fact the time that R wave occurred in HR. The difference between the times obtained re- sults in the production of HRV signal. The results of the proposed model based on the different chaotic maps have been indicated in Figures 23-27.

After achieving the necessary results of the proposed model, these results will be compared statistically with the results of normal signal. Features from the results of the proposed model such as mean, median, variance, standard deviation, maximum amplitude, minimum amplitude, mode and amplitude range were compared with the normal sample. As a result, the model error and the accuracy of the model were obtained.

Then the above mentioned features were obtained from the simulated HRV signals achieved in the previous stage. This extraction of features is done to compare the features resulted from simulated HRV signals with HRV signal which has been obtained from normal and natural ECG signal to find the similarities and accuracy of output in proposed model with normal and natural ECG signal. Tables 3-10 show the linear features extracted from the simulated HR and HRV signal.

Also, some non-linear features of time series sequence were extracted. These feature are Lyapunov Exponent, Shanon Entropy, log Entropy, Threshold Entropy, sure Entropy and mode Entropy. Figures 28-31 were shown results of these features extracted.

The results of the proposed model and comparing to the results achieved from normal and natural HRV signal showed that the proposed model has a good performance in modeling the HRV signal.

In this study it has presented an applied mathematical model to generate artificial HRV signal based on IPFM model with using chaotic threshold levels. As a result this project will be performed as model using mathematical analysis. This method has been chosen because by means of changing input of IPFM model, we can produce synthetic HRV signal which is similar to the natural signal (the one taken from human body), and we can also model several types of diseases that show their effects on the signal. Considering that this model is capable of producing several types of HRV signals it can be concluded that it is a comprehensive model for synthesizing this type of signal (with comparison to results of references [

The results of this study indicated that the proposed model has a qualified functioning and there are a lot of similarities between the signals of this model with HRV ones. In this study, the coupling between the sympathetic and parasympathetic nervous system was not intended, however, we achieved acceptable results. The results were closely correlated with the real data which confirm the effectiveness of the proposed model. Various

signals derived from the output of this model can be used for final analysis of the HRV signals, such as arrhythmia detection and classification of ECG and HRV signals. One of the applications of the proposed model is the easy evaluation of diagnostic ECG signal processing devices. Such a model can also be used in signal compres-

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

65.8266 | 66.1161 | 25.6498 | 5.0646 | 73.9739 | 59.4060 | 14.5679 | 59.4060 |

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

0.9114 | 0.9127 | 0.0048 | 0.0696 | 1.0100 | 0.8111 | 0.1989 | 0.8111 |

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

64.8976 | 66.3771 | 29.6145 | 5.4419 | 79.0626 | 59.4079 | 19.6546 | 59.4079 |

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

0.9244 | 0.9097 | 0.0051 | 0.0711 | 1.0100 | 0.7589 | 0.2511 | 0.7589 |

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

65.4290 | 65.0236 | 9.3284 | 3.0542 | 73.4711 | 34.1603 | 39.3108 | 34.1603 |

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

0.9170 | 0.9252 | 0.0035 | 0.0590 | 1.7564 | 0.7971 | 0.9593 | 0.7971 |

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

65.9189 | 66.1631 | 18.1799 | 4.2638 | 74.1641 | 59.4059 | 14.7582 | 59.4059 |

Median | Mean | Variance | SD | Max Amplitude | Min Amplitude | Amplitude Range | Mode |
---|---|---|---|---|---|---|---|

0.9102 | 0.9105 | 0.0034 | 0.0585 | 1.0100 | 0.8048 | 0.2052 | 0.8048 |

sion and telemedicine. The major advantage of this study is simplicity and good performance of the proposed model. Figures 28-31 were shown the comparison of all results in this study. Following charts shown the comparison of all the features extracted from natural signals and the simulated signals with proposed method (See Figures 28-31).

As in Figures 28-31 were clear, linear feature of normal HR signal such as median, mean, max amplitude, min amplitude and mode were nearly to these feature that extracted from all simulated HR signal. Variance, standard deviation and amplitude range are features in all simulated HR signals that not close to natural HR sig- nal. Also, non-linear feature of normal HR signal such as lyapunov exponent, Shanon entropy, log entropy, threshold entropy, sure entropy and norm entropy were very nearly to these feature that extracted from all simu- lated HR signal.

About HRV signal, linear feature of normal HRV signal such as median, mean, variance, standard deviation, max amplitude, min amplitude and mode were nearly to these feature that extracted from all simulated HR signal (except results of lorenz map in max amplitude). Also, non-linear feature of normal HRV signal such as lyapunov exponent and average of all entropy were very nearly to these feature that extracted from all simulated HRV signal. But each of entropy singly (especial Shanon and log entropy), is not very closely to the natural HRV signal. It should be noted that these extracted features was not exist in previous researches and in this paper we extracted these features for showing acceptable results of proposed method with comparison to similar studies.

In general we can generate artificial HR and HRV signals that results of this method are very closely with normal HR and HRV signals that obtained from human heart. Various signals derived from the output of this model can be used for final analysis of the HRV signals, such as arrhythmia detection and classification of ECG and HRV signals. One of the applications of the proposed model is the easy evaluation of diagnostic ECG signal processing devices. Such a model can also be used in signal compression and telemedicine. Because signals generated by this model due of the high number of cardiac cycles and low volume of these, they can easily upload them to the internet for educational purposes and produce different models of cardiac signals. Also by changing the coefficients of the parameters in Equation (5), production different normal and patient signals from a prototype model, that this is due to the education mode of these signals.

This work was supported by Research Fund of Islamic Azad University, Dezful Branch, under research project: “Application Mathematical Model for Artificial Generation of Heart Rate Variability Using IPFM Model with Chaotic Maps”.