This article investigates the relationships between heart valve closure timing intervals and left ventricular systolic blood pressure (LVSBP). For this investigation, the cardiopulmonary system is modeled as an analog circuit, including heart chambers, the distal and proximal aorta, distal and proximal systemic arteries/veins, systemic capillaries, the vena cava, the distal and proximal pulmonary artery, distal and proximal pulmonary arteries/veins, pulmonary capillaries and physiological control of heart rate and cardiac contractibility. In this model, the ventricles, atria and arteries were modeled as advanced pressur-volume relationships. A vagal-sympathetic mechanism was adopted to simulate transient systemic and pulmonary blood pressure. Four intervals, i.e., the timing interval between mitral and aortic valve closure (TIMA), the timing interval between aortic and mitral valve closure (TIAM), the timing interval be- tween aortic and pulmonary valve closure (TIAP) and the timing interval between mitral and tricuspid valve closure (TIMT), are further defined in a heart cycle to illustrate their relationships to LVSBP. Simula- tions showed that the TIMA, TIAM and TIAP have strong negative correlations with LVSBP; meanwhile, the TIMT has a slightly negative relationship with LVSBP. To further validate the relationships, 6 healthy male subjects were experimentally evaluated. The intervals were extracted from non-invasively sampled heart sound signals taken from the surface of the thorax. The experiments showed relationships consistent with those obtained by simulations. These relationships may have potential applications for noninvasively accessing LVSBP in real-time with a high time resolution of one heartbeat.
Left ventricular systolic blood pressure (LVSBP) is an important hemodynamic indicator of heart function. Developing a non-invasive method of accessing LVSBP is of interest to the medical community. However, the left ventricle is concealed in the thorax. Direct measurement by a catheter inserted to the left ventricle is commonly used for a final diagnosis. A non-invasive measurement is more widely accepted in regular medical checkups. LVSBP is often approximated by the aortic pressures measured in the left upper arm using a stethoscope and a sphygmomanometer. This approximation is inexpensive and easy to perform, but it is not a real-time measurement; it may take half a minute for one measurement. Additionally, it is uncomfortable. A non-invasive, comfortable, accurate and high time resolution LVSBP measuring method is desirable. To achieve this, the authors start from the hemodynamics of the systemic and pulmonary circulation. It is known that heart valve closure results from dynamic events associated with the interactions between the valves and differential pressures on each side [
This paper is organized as follows. Section 2 presents a modified cardiopulmonary model consisting of heart chambers, simplified arteries, simplified veins and a physiological mechanism to control heart rate and heart contractibility during dynamic exercise. Section 3 shows the simulation results. The experiments that were conducted are shown in Section 4 to validate the relationships. Sections 5 and 6 present the discussion and conclusions.
A modified model of the cardiopulmonary system was proposed in this paper and used as a platform to study the relationship between LVSBP and heart valve closure timing differences. The model was based on previous works from the last few decades [10-12]. The nonlinear pressure-volume (P-V) mathematical description of the heart ventricular ejection function can be simple and straightforward within a physiological framework. The nonlinear model of any part of an artery or vein generally consists of three elements: resistance, modeled by a resistor; compliance, modeled by a capacitor; and inertia, modeled by an inductor. The circulatory system is thus converted to an analog circuit, as shown in
1) Ventricular model The ventricular model is based on the work of Chung et al. [
where Plv_ES(Vlv) denotes the ESPVR and Plv_ED(Vlv) denotes the EDPVR. Vd is the constant volume intercept. Ees is the end-systolic elastance. V0 is the volume inter-
cept for end diastole. P0 is the pressure intercept. λ is an empirical constant quantifying the P-V relationship. ev(t) is the activation function consisting of 4 Gaussian curves,
2) Atrium model The left atrium and right atrium are modeled following the same principles as those used to model the ventricles. ea(t) is the activation function consisting of one Gaussian curve,
The parameters for both ventricle and atrium are given in Tables 1 and 2.
We can derive blood pressure Plv(Vlv, t), Prv(Vrv, t), Pla(Vla, t) and Pra(Vra, t) for the left ventricle, right ventricle, left atrium and right atrium, respectively, following Eqs.1 to 3. These equations have different coefficients, as illustrated in Tables 1 and 2. The parameters for Ci of the activation function used in this paper were modified according to Braunwald’s research [
3) Blood Vessels The P-V relationships of systemic veins, vena cava and proximal arteries are not linear, i.e., the compliances of these vessels are not constant. The nonlinear blood vessel models are based on Lu and Clark [
Systemic veins. Vessels stiffen as their volume increases. The P-V relationship of systemic veins can be represented as follows,
where Psv and Vsv are the pressure and volume of systemic veins, Kv is a scaling factor, and Vmax is the maximal volume of the systemic veins.
Vena cava. The P-V relationship of the vena cava is as follows,
where Pvc and Vvc are the pressure and volume of the vena cava. V0 and Vmin are the unstressed and minimum volumes, respectively. The parameters K1, K2, D1 and D2 can be tuned to adapt the P-V relationship for the human venous system. The resistance of the vena cava is given as follows,
where KR is a scaling factor and R0 is the resistance offset.
Proximal arteries. As the compliance and resistance of proximal arteries are related to vasomotor tone, P-V curves of arteries must be different under fully activated or passive conditions. We describe the P-V relationship of proximal arteries as
where and denote the fully activated and passive pressures of proximal arteries; Vsap is the volume; Vsap,0 is the minimal volume; Kc, Kp1 and Kp2 are scaling factors; D0 is a volume parameter; and τp is constant. The resistance is determined by Vsap as follows,
where Rsap,0 is offset resistance, Vsap,max is the maximal volume, and Kr is a scaling factor. The all parameters used in this subsection are listed in
The cardiopulmonary model shown in
According to hydrodynamics, the BP gradient across the heart valves causes the valves’ actions. Each valve opens or closes one time in a cardiac cycle depending on the differential pressure across the valve and thus contributes to the generation of heart sounds. The valve closure timings are thus defined by the relative pressures across the valves, as indicated by the arrows in
1) Heart rate control Heart rate, characterized by Sunagawa as a three-dimensional response [
where h1 – h6 are constants that can be found in
2) Ventricle contractility control From a physiological point of view, greater sympathetic tone increases myocardial elastance and shortens ventricular systole. Therefore, the ventricular activation function should be modified to describe the variation in the ventricular P-V relationship as a function of sympathetic efferent discharge frequency Fcon. A rising Fcon increases maximum elastance and shortens the systolic
period. Eq.2 for the end-systolic P-V relationship then becomes
and the activation function ev(t) is rewritten as
where
,
.
The constants amin and bmin represent the minimum values of the functions a and b. Ka and Kb are scaling parameters.
3) Vasomotor tone Proximal arteries. As mentioned in Section 2.1.3, arterial compliance and resistance are related to vasomotor tone, which is regulated by the normalized sympathetic frequency Fvaso. The P-V relationship of proximal arteries is characterized by
where and are given in (8a) and (8b). The resistance of proximal arteries becomes
As described in previous works [16,17], the compliance of the proximal aorta, Caop, decreases as blood pressure increases, indicating that diastolic pressure increases more slowly than systolic pressure. This phenomenon was also verified in the authors’ experiments. Caop is defined as a function of the sympathetic frequency Fvaso,
In the simulation, it is assumed that a virtual subject follows a two-phase physiological routine. The total simulation time for the two-phase routine is 300 seconds. The virtual subject begins in a resting state and is then asked to exercise in the first phase for the first 100 seconds and stop exercising to recover in the second phase for the next 200 seconds. In the resting state, the sympathetic and vagal nerves contribute to heart rate control in equal degrees. In the simulation, the normalized sympathetic (FHrs, Fcon, Fvaso) and vagal (FHrv) frequencies are initialized at 0.5. During exercise, sympathetic (Fcon, FHrv, Fvaso) frequencies become dominant; meanwhile, vagal (FHrv)
frequency decreases, as shown in
Using the definitions of timing intervals given in Section 2.3, the responses of Δtma, Δtam, Δtap and Δtmt to LVSBP in the two-phase routine can be calculated, as illustrated in
Six young male subjects participated in the experiments, and they had a mean age of 27 ± 3 years (means ± SE). The experiment protocol was approved by the local ethics committee. All of the subjects gave their consent to participate in the experiments. They were asked to stay at rest for 10 minutes before exercise. This resting state may be taken as a baseline for blood pressure. They were asked to perform a step-climbing exercise for 100 steps and lie on their backs on an examination table immediately afterwards. Heart sounds and ECG signals were simultaneously recorded at a sampling frequency of 2 KHz (Biopac MP150, USA), for which the heart sound microphone sensor was placed in an optimal position on the thorax to distinguish heart sound components. Blood pressure was measured immediately after each signal recording by a standard electronic oscillometric meter on the upper left arm (HEM-7200, OMRON, Japan). The signal recording lasted 300 seconds (5 minutes) so that the subjects had almost recovered to the resting state. Blood pressure values at any other time may be estimated using interpolation.
The signals were contaminated by breathing sounds at the beginning of the recordings. One may use our previously developed noise reduction technique to improve signal quality [18,19]. Although researchers have not come to a consensus on the mechanism of heart sound generation [20,21], the fact that the timings of the heart sounds reflect the timings of heart valve closures has been widely accepted. To the authors’ knowledge, no fully developed automatic scheme is available to extract the time intervals Δtma, Δtam, Δtap and Δtmt from heart sound signals. In this study, the authors adopted a time-frequency technique with the ECG signal as a reference to detect the timing intervals. Each heart sound signal was first segmented into cycles with the assistance of simultaneously sampled ECG R-waves. Then, continuous wavelet analysis, as illustrated in [
To further demonstrate the negative relationships between the intervals and LVSBP for the 6 subjects, the relationships were fitted to first-order equations based on the least squares criteriaΔt = Slope∙PLVSBP+ Offset. (16)
The slopes and offsets are listed in
From the physiological point of view, heart rate and cardiac muscle contractility will increase rapidly during exercise in the first phase, causing the systolic and diastolic time intervals to become shorter. These phenomena are reflected in heart sounds, i.e., Δtma and Δtam become shorter than in the resting state. Additionally, the increasing cardiac muscle contractility of the left ventricle leads to a shorter ejection period, causing aortic valve closure to shift to an earlier time. However, pulmonary valve closure shifts to an even earlier time. Therefore, the interval Δtap decreases in the final presentation. Both experimental and simulation results show that Δtma, Δtap and Δtam have strong negative relationships to LVSBP.
However, the results for Δtmt were less consistent. Simulations showed that Δtmt had a very small negative change with respect to LVSBP, while the experimental results showed a slightly stronger negative relationship. There are two possible factors that may have caused this result. 1) The parameter Δtmt was extracted with peak location using time-frequency analysis; location errors were unavailable; 2) The values of the resistors Rm and Rt in the analog circuit shown in
It may be concluded based on both simulations and experiments that Δtma, Δtam, Δtap are more sensitive to LVSBP and that Δtmt is less sensitive to LVSBP. These relationships may have potential applications in continuously monitoring LVSBP with a high time resolution of one heartbeat. It is possible to monitor relative variations in LVSBP without any calibration if heart sound signals can be collected during regular medical checkups. Hemodynamic disorders may be found in earlier stages of progression after performing long-term monitoring of the intervals.
In addition, another phenomenon, shown in
A modified mathematical model of the cardiopulmonary system was proposed as a platform to study the relationships between heart valve closure timing intervals and LVSBP, in which a vagal-sympathetic mechanism was introduced to control heart rate and ventricle contractibility. Four intervals, Δtma, Δtmt, Δtap and Δtam, based on the differential pressures across the valves, were defined to describe the valve closure timing intervals.
Using this platform, it was found that the hemodynamics vary as heart rate and ventricle contractibility increase. Simulation results show that Δtma, Δtap and Δtam bear strong negative correlations with LVSBP and that Δtmt has slight negative relation. To further validate these relationships, 6 subjects participated in a two-phase experiment. The four intervals were extracted from heart sound signals collected from the surface of the thorax. The experimental results agree with those obtained from simulations. This study helps in the understanding of the hemodynamic response to various physiological conditions. The method described may have potential applications in the continuous and non-invasive monitoring of heart chamber hemodynamics based on the heart sound features.
This work was supported, in part, by the Fundamental Research Funds for the Central Universities under grant DUT12JB07, DUT12JR03 and the National Natural Science Foundation of China under grant 81000643.
The model of the cardiovascular system presented in this article was simulated using the parameter values listed in Tables A1-A3. Because the hemodynamic parameters vary among human individuals, the values of the resistors, diodes, capacitors and inductors in this appendix reflect those of a typical human being and were obtained based on reasonable blood pressure values at reference knots in the circuit.
Initial conditions of the model include heart chamber and vascular volume, and systemic and pulmonary aorta flow, corresponding to the charges of the capacitors and the currents of the inductors. The initial values were obtained based on considerations of both reasonable blood distributions and total blood volume. A feasible set of parameters are listed in