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 Figure 1. The values of the resistors, capacitors, and inductors are given in Appendix A. The heart cycle period is determined physiologically by the vagal-sympathetic mechanism. With the support of this cardiopulmonary platform, heart valve closure timing can be defined by the differential blood pressure (BP) across the valves.

1) Ventricular model The ventricular model is based on the work of Chung et al. [13], wherein the left and right ventricles are modeled as a time-varying elastic function. The elastic function consists of three curves, i.e., the end-systolic pressure-volume relationship (ESPVR), the end-diastolic P-V relationship (EDPVR), and a time-varying activation function e_v(t). For example, the blood pressure in the left ventricle, P_lv, with respect to volume V_lv and time t, is given as

where P_{lv_ES}(V_lv) denotes the ESPVR and P_{lv_ED}(V_lv) denotes the EDPVR. V_d is the constant volume intercept. E_es 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. e_v(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. e_a(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 P_lv(V_lv, t), P_rv(V_rv, t), P_la(V_la, t) and P_ra(V_ra, 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 C_i of the activation function used in this paper were modified according to Braunwald’s research [14]. The left ventricle contraction onset was earlier than that of the right ventricle by 13 ms; meanwhile, the right atrium contraction onset was earlier than that of the left atrium by 20 ms, as shown in Table 2 for C_i. The curves of the modified activation functions for the cardiac chambers are shown in Figure 2.

Table 1. Parameters of the ventricular and atrial model.

Table 2. Parameters of activation function.

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 [10], in which the compliances are described by the P-V relationship and the resistances of the vena cava and proximal arteries are nonlinear functions of blood volume.

Systemic veins. Vessels stiffen as their volume increases. The P-V relationship of systemic veins can be represented as follows,

where P_sv and V_sv are the pressure and volume of systemic veins, K_v is a scaling factor, and V_max is the maximal volume of the systemic veins.

Vena cava. The P-V relationship of the vena cava is as follows,

where P_vc and V_vc are the pressure and volume of the vena cava. V₀ and V_min 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 K_R 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

Table 3. Parameters for nonlinear P-V curves and resistances in systemic veins, the vena cava and proximal arteries.

where and denote the fully activated and passive pressures of proximal arteries; V_sap is the volume; V_sap,0 is the minimal volume; K_c, K_p1 and K_p2 are scaling factors; D₀ is a volume parameter; and τ_p is constant. The resistance is determined by V_sap as follows,

where R_sap,0 is offset resistance, V_sap,max is the maximal volume, and K_r is a scaling factor. The all parameters used in this subsection are listed in Table 3.

The cardiopulmonary model shown in Figure 1 can be solved numerically using an iteration scheme. An iteration loop is initialized from the onset of LV pressure rise at t = 0. All initial values for the volumes and flows are given in Appendix B. The simulated BP curves of one heart beat cycle for the left ventricle, left atrium and aortic artery are presented in Figure 3(a). The simulated BP curves for their counterparts in the pulmonary system are presented in Figure 3(b).

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 Figure 3. The closure timings of the mitral valve, aortic valve, tricuspid valve and pulmonary valve are denoted as t_m, t_a, t_t and t_p, respectively. It is widely accepted that acoustic vibrations due to the closures of the mitral and tricuspid valves are major components of S1. Similarly, the closures of the aortic and pulmonary valves are the main contributors to S2. Therefore, heart sounds collected from the thorax surface can be used as timing markers of heart valve closures. Heart valve closure timings can thus be noninvasively monitored. Although the mechanism of heart sound generation is still in dispute, the timing correlations between heart sounds and heart valve closures have been widely accepted by researchers. Based on these facts, the timing intervals of any two valve closures are shown in Figure 4. For example, the timing interval between the mitral valve and tricuspid valve closures (TIMT) is calculated as the time interval between the mitral and tricuspid components of the first heart sound, i.e., Δt_mt = t_t – t_m. Similarly, the other three timing intervals TIAP, TIMA and TIAM can be defined as Δt_ap = t_p – t_a, Δt_ma = t_a – t_m, and Δt_am = t_m – t_a, respectively, as illustrated in Figure 4. The mitral valve closes at the onset of left ventricular contraction, while the aortic valve closes at the onset of left ventricular extension. Therefore, Δt_ma denotes the systolic time interval (STI) of a heart beat cycle, Δt_am denotes the diastolic time interval (DTI), Δt_ap may be represented by the time split of the second heart sound, and Δt_mt may be represented by the time split of the first heart sound. The objective of this paper is to investigate the relationships between LVSBP and heart valve closure timings, which can be non-invasively accessed by the timing intervals of heart sounds.

1) Heart rate control Heart rate, characterized by Sunagawa as a three-dimensional response [15], is controlled by vagal and sympathetic neural activity. The human heart rate response to vagal and sympathetic input is further characterized by Lu and Clark [10]

where h₁ – h₆ are constants that can be found in Table 4. F_Hrs and F_Hrv are normalized sympathetic and vagal discharge frequencies, respectively.

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 F_con. A rising F_con increases maximum elastance and shortens the systolic

Table 4. Parameter values for control of heart rate and contractility.

period. Eq.2 for the end-systolic P-V relationship then becomes

and the activation function e_v(t) is rewritten as

where

,

.

The constants a_min and b_min represent the minimum values of the functions a and b. K_a and K_b 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 F_vaso. 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, C_aop, decreases as blood pressure increases, indicating that diastolic pressure increases more slowly than systolic pressure. This phenomenon was also verified in the authors’ experiments. C_aop is defined as a function of the sympathetic frequency F_vaso,

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 (F_Hrs, F_con, F_vaso) and vagal (F_Hrv) frequencies are initialized at 0.5. During exercise, sympathetic (F_con, F_Hrv, F_vaso) frequencies become dominant; meanwhile, vagal (F_Hrv)

frequency decreases, as shown in Figure 5 for the first 100 seconds. Once the subject stops exercising after 100 seconds, sympathetic (F_con, F_Hrs, F_vaso) frequencies immediately begin to decrease, and vagal (F_Hrv) frequency begins to increase. Finally, all frequencies converge at 0.5. As we expected, the subject’s heart rate and LVSBP increase during the first phase and decrease during the second phase. Simulation results based on this platform are given in Figure 6 and show that the heart rate begins at 76 beats per min, increases to 107 beats per min, and finally returns to 76 beats per min. LVSBP starts at 120 mmHg and increases to 160 mmHg at the end of the exercise period. LVSBP returns to 120 mmHg as the subject recovers to the resting state.

Using the definitions of timing intervals given in Section 2.3, the responses of Δt_ma, Δt_am, Δt_ap and Δt_mt to LVSBP in the two-phase routine can be calculated, as illustrated in Figure 7. Solid lines represent the relationships in the exercising phase, and dashed lines represent the relationships in the recovery phase. The fact that the solid and dashed lines nearly overlap means that the two-phase physiological routine is reversible. The subject begins from a resting physiological condition, reaches a maximum load at the 100th second, and comes back to the original resting state at the end of the simulation following a reversible physiological routine. It can be seen in Figure 7 that Δt_ma, Δt_am and Δt_ap have strong negative relationships with LVSBP in both phases, while Δt_mt has a slightly negative relationship. These relationships imply that hemodynamic variations in the left ventricle can be reflected in heart valve closure timing intervals. These intervals tend to decrease with increasing LVSBP.

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 Δt_ma, Δt_am, Δt_ap and Δt_mt 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 [22], was used to extract the intervals between peaks. Each cycle of a heart sound signal contains a group of Δt_ma, Δt_am, Δt_ap and Δt_mt. Hundreds of groups of intervals were thus determined from a 300-second recording. The scatter plots of the intervals with respect to LVSBP for one subject are depicted in Figure 8, showing that all of the intervals have negative relationships with LVSBP. From visual observation, Δt_ma, Δt_am, Δt_ap have much stronger negative relationships to LVSBP, while the LVSBP relationship with Δt_mt is much less negative. The experimental results for Δt_ma, Δt_am and Δt_ap are very close to the simulation results. The result for Δt_mt agrees with that of the simulation, but the experimental Δt_mt shows a slightly stronger negative relationship than that of the simulation.

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∙P_LVSBP+ Offset.         (16)

The slopes and offsets are listed in Table 5. The slopes and offsets vary among the subjects. It can be seen that Δt_ma, Δt_am and Δt_ap have strong negative relationships with LVSBP, while Δt_mt has a slightly negative relationship.

Table 5. Slopes and offsets of the best-fit lines for the 6 subjects.