The previous studies on respiratory physiology have indicated that inspiration and expiration have opposite effects on heart hemodynamics. The basic reason why these opposite hemodynamic changes cause regular timing variations in heart sounds is the heart sound generation mechanism that the acoustic vibration is triggered by heart hemodynamics. It is observed that the timing of the first heart sound has nonlinear relation with respiratory phase; that is, the timing delay with respect to the Rwave increases with inspiration and oppositely decreases with expiration. This paper models the nonlinear relation by a HammersteinWiener model where the respiratory phase is the input and the timing is the output. The parameter estimation for the model is presented. The model is tested by the data collected from 12 healthy subjects in terms of mean square error and model fitness. The results show that the model can approximate the nonlinear relation very well. The average square error and the average fitness for all the subjects are about 0.01 and 0.94, respectively. The timing of the first heart sound related to respiratory phase can be accurately predicted by the model. The model has potential applications in fast and easy monitoring of respiration and heart hemodynamics induced by respiration.
Heart sounds are commonly considered as a series of mechanical vibrations produced by heart vascular system [
It can be found from the previous works that the relation between respiratory phase and the timing of S1 was highly nonlinear. This paper tries to approximate the relation in time domain by quantitative nonlinear HammersteinWiener model. The timing of the first heart sound may therefore be predicted by the model. Cardiovascular status in respiratory process may be possible to be monitored by the relation. These results suggest that a quantitative analysis of the relation could be used as a noninvasive continuous monitoring of hemodynamic state during respiratory cycles.
The experimental protocol was approved by the Ethics Committee of the Department of Biomedical Engineering, Dalian University of Technology. Twelve young male subjects aged 24 ± 1.8 years participated in the experiments. All subjects provided their consent to participate in the experiments. They were asked to remain at rest for 10 min before data collection. Each subject was asked to lie on his back in a bed during data sampling. Heart sounds, ECG lead II, and respiratory signals were simultaneously recorded at a sampling frequency of 2 kHz (PL3516B111, ADinstruments, Australia). A heart sound microphone sensor (MLT201, ADinstruments, Australia) was placed at the left third intercostal space. The breathing transducer (MLT1132, ADinstruments, Australia) was a belt sensor positioned at the middle of the thorax to record respiratory movement. One data collection period lasted 150–180 s. Data were collected thrice for each subject with 3minute intervals. A portion of collected signals is given in Figure
A portion of the collected signals. ECG: ECG signal of the lead II; HS: heart sound signal; RES: respiratory signal.
Definitions of the timing and the phase.
Nonlinear relation between the timing and the phase. (a) Two series of the timings and the phases are shown in time domain. (b) Scatter plot in the joint plane of timing and phase. EXP: expiration, INS: inspiration. (c) The nonlinear relation is approximated by a HammersteinWiener model.
Sampling frequency transform from nonuniform discrete to uniform discrete by interpolation.
A HammersteinWiener model generally consists of three parts [
Structure of the HammersteinWiener model.
The iterative approach is implemented based on the preceding estimation for internal variables. It is assumed that the internal variables of the
The steps to implement the iterative procedure may be summarized as follows.
Initialization: the parameter
A least mean square (LMS) algorithm is used to minimize the error in the
Starting from
An iterative procedure is used to obtain the optimal parameter for the
Once the iteration converges or reaches the maximum number of iterations, the iteration stops. The convergence condition is generally defined as the degree of variation of the parameters in the process, such as the dynamic variance. Once the degree of variation reaches a low level, the iteration is believed to converge successfully. The optimal parameter vector,
Let
Repeat the steps (2)–(4).
In this model, respiratory phase is the model input and the timing is the model output. Two indicators are used to evaluate the consistence between the predicted timing and the observed timing. One indicator is mean square error (MSE) which is written as
The digital sequence of respiratory phase and timing are both slow varying. They are downsampled to 50 Hz to reduce the number of samples. The time domain nonlinear relation between respiratory phase and timing of S1 is virtually seen in Figure
An example of model evaluation. (a) Approximation between
There are 12 subjects involved in the experiments to collect data. To evaluate the model adaption to other subjects, the MSE and fitness were listed in Table
MSE and fitness between
Subject number  1  2  3  4  5  6  Average 

MSE  0.008  0.009  0.011  0.006  0.014  0.006  0.009 

0.963  0.957  0.946  0.970  0.926  0.959  0.954 


Subject number  7  8  9  10  11  12  Average 


MSE  0.019  0.016  0.010  0.011  0.014  0.010  0.013 

0.921  0.925  0.952  0.945  0.934  0.949  0.938 
The degree of nonlinearity that the HammersteinWiener model can approximate is directly related to the order of the polynomials in the first and third modules. The basic understanding to the model shows that a higher degree of nonlinearity needs higher order polynomials. However, the degree of nonlinearity between the respiratory phase and the timing of S1 varies from person to person. Then the question of how the order of the polynomials can be selected arises. The authors analyzed the nonlinearity that the HammersteinWiener model approximates for all subjects. The results showed that the approximation performance of the model increased a little once the order of the polynomials is greater than 3. For example, for subject #2 in Table
Approximation performance of the HammersteinWiener model for subject #2 with varied polynomial order. (a) MSE decreased with the polynomial order; (b) fitness value increased with the polynomial order.
The studies in respiratory physiology indicated that respiration has regular effect on heart hemodynamics. These hemodynamic variations lead to the fact that the closure of mitral valve occurs late in inspiration and early in expiration. The opposite behaviors of mitral valve close are further reflected by the timing of first heart sound due to the mechanism of heart sound generation. A nonlinear relation is observed between the timing and the respiratory phase based on data collected from healthy subjects. A HammersteinWiener model is used to approximate the nonlinear relation in time domain. The parameter estimation for the model is presented in this paper. The performance tests show that the timing of first heart sound can be accurately predicted by the respiratory phase based on the model. This has potential applications in fast and easy monitoring of respiration and heart hemodynamics induced by respiration.
The authors claim that there is no conflict of interests.
This work was supported by the 2010 Yeungnam University Research Grant.