Evaluating the physiological significance of respiratory sinus arrhythmia: looking beyond ventilation-perfusion efficiency

Abstract

We conducted a theoretical study of the physiological significance of respiratory sinus arrhythmia (RSA), a phenomenon used as an index of cardiac vagal tone and wellbeing, whereby the heart rate (HR) increases during inspiration and decreases during expiration. We first tested the hypothesis that RSA improves gas exchange efficiency but found that although gas exchange efficiency improved with slow and deep breathing and with increased mean heart rate, this was unrelated to RSA. We then formulated and tested a new hypothesis: that RSA minimizes the work done by the heart while maintaining physiological levels of arterial carbon dioxide. We tested the new hypothesis using two methods. First, the HR for which the work is minimized was calculated using techniques from optimal control theory. This calculation was done on simplified models that we derived from a previously published model of gas exchange in mammals. We found that the calculated HR was remarkably similar to RSA and that this became more profound under slow and deep breathing. Second, the HR was prescribed and the work done by the heart was calculated by conducting a series of numerical experiments on the previously published gas exchange model. We found that cardiac work was minimized for RSA-like HR functions, most profoundly under slow and deep breathing. These findings provide novel insights into potential reasons for and benefits of RSA under physiological conditions.

Figure 1. The hydrodynamic model analogy of gas transport

The left container represents the lungs and the right container represents the pulmonary capillaries.

Figure 2. Typical calculations of optimal heart rate and partial pressures in Models 1a and 1b

The inspired ventilation, q_in, is the same for both models (top panel). The shaded areas show the inspiration period. The averaged arterial partial pressure of O₂ () was constrained to 104 mmHg, the averaged arterial partial pressure of CO₂ () was constrained to 41 mmHg, and V_T= 0.5 l. All other parameters are the same as in Appendix A. Heart rate (HR) calculated for Model 1a – continuous line, HR calculated for Model 1b – dashed line. T_I and T_E are the inspiration and expiration periods, respectively. Note that O₂ and CO₂ are uncoupled.

Figure 7. Increased T_I/T_E ratio leads to a weaker RSA and a growing phase-shift between the maximum heart rate and the end of inspiration

The calculation was performed using Model 1b. The shaded areas show the inspiration period for each case. The respiratory cycle is 5 s and V_T= 0.5 l in all cases. All other parameters are as in Appendix A. In blue T_I/T_E= 1:1, in red T_I/T_E= 1:2, and in black T_I/T_E= 2:1. The inspired ventilation, q_in, is shown in the top panel.

Figure 3. The RSA-like shape of the R-R interval is preserved by all three models

The calculation of the R-R interval for the different models was performed using the optimal control technique described in Appendix D and taking the inverse of the optimal heart rate. V_{T =} 1.0 l, T= 10 s, the averaged arterial partial pressure of CO₂ was constrained to 41.5 mmHg. T_I and T_E are the inspiration and expiration periods, respectively. The shaded area shows the inspiration period.

Figure 4. Stronger RSA as a result of increased average arterial partial pressure of CO₂

The optimal heart rate (HR) was calculated for Model 1a (top three curves) and Model 1b (bottom three curves) with V_T= 0.5 l. In Model 1b, the averaged arterial partial pressure of CO₂ was constrained to three values: 39 mmHg (dashed–dotted line), 40 mmHg (dashed line), 41 mmHg (continuous line). In Model 1a, the averaged arterial partial pressure of O₂ was constrained to: 100 mmHg (dashed–dotted line), 104 mmHg (dashed line), 108 mmHg (continuous line). T_I and T_E are the inspiration and expiration periods, respectively. The shaded area shows the inspiration period.

Figure 5. Stronger RSA with increased tidal volume when the arterial partial pressure of CO₂ is controlled (Model 1b)

The optimal heart rate (HR) was calculated for Model 1a (labelled ‘O₂’) and Model 1b (remaining three curves). The inspired ventilation, q_in, is the same for both models (top panel). In Model 1b, the averaged arterial partial pressure of CO₂ was constrained to 40 mmHg. In Model 1a, the averaged arterial partial pressure of O₂ was constrained to 104 mmHg. Tidal volume took on the following three values: V_T= 0.4 l (dashed–dotted line), V_T= 0.7 l (dashed line), V_T= 1.0 l (continuous line). Only the case of V_T= 1.0 l is shown for Model 1a. T_I and T_E are the inspiration and expiration periods, respectively. The shaded area shows the inspiration period.

Figure 6. Stronger RSA with slow and deep breathing

The optimal heart rate (HR) was calculated for Model 1a (labelled ‘O₂’) and Model 1b (remaining three curves). The shaded areas show the inspiration period for each breathing pattern. Only one cycle is shown. The inspired ventilation, q_in, is the same for both models (top panel). In Model 1b, the averaged arterial partial pressure of CO₂ was constrained to 41 mmHg. In Model 1a, the averaged arterial partial pressure of O₂ was constrained to 104 mmHg. Respiratory period and tidal volume were: T= 2.5 s, V_T= 0.25 l (dashed–dotted line); T= 5 s, V_T= 0.5 l (dashed line); T= 10 s, V_T= 1 l (continuous line). The minute ventilation was the same for all simulations: V_T/T= 0.1 l s⁻¹= 6 l min⁻¹. Only the case T= 10 s, V_T= 1 l is shown for Model 1a.

Figure 8. A saw-tooth HR function used to mimic RSA

The parameter Δ governs the degree of RSA while m is the mean heart rate (HR). Here we took m= 1.2 Hz = 72 beats min⁻¹, Δ= 0.2 Hz = 12 beats min⁻¹. The shaded area shows the inspiration period.

Figure 9. Mimicking pulsatile blood flow (discrete heart rate)

The arterial partial pressure of CO₂ (p_c, middle panel) and the arterial partial pressure of O₂ (p_o, top panel) are initialized for every heart beat. The heart rate shown here is taken at the minimum of the continuous curve in Fig. 10 with Δ= 0.84 Hz, mean HR = 1.9605 Hz. T_I and T_E are the inspiration and expiration periods, respectively. The shaded areas show the inspiration period.

Figure 10. A minimum in the work done by the heart is associated with RSA and increases under slow and deep breathing when CO₂ is constrained

Simulations are performed with the Ben-Tal (2006) model, using prescribed HR functions. Positive Δ represents RSA, negative Δ represents inverse RSA, constant heart rate is marked by a cross at Δ= 0. E/E₀ curves are plotted as functions of Δ (E is calculated with eqn (5), E₀ is the value of E at Δ= 0), constraining the partial pressure of CO₂ along each curve. Data points are obtained from the simulation, and the continuous lines are least-squares quadratic fits. Data points represented by circles (fitted with a continuous line) were taken with T= 10 s, V_T= 1.0 l, constraining the averaged arterial partial pressure of CO₂ to 38.7124 mmHg and normalizing E by 39.98 Hz. Squares with a dashed line were taken with T= 5 s, V_T= 0.5 l, constraining the averaged arterial partial pressure of CO₂ to 38.5611 mmHg and normalizing E by 4.968 Hz. Diamonds with dashed–dotted line were taken with T= 2.5 s, V_T= 0.25 l, constraining the averaged arterial partial pressure of CO₂ to 38.5004 mmHg and normalizing E by 0.2225 Hz. Triangles with a dotted line were taken with T= 10 s, V_T= 1.08 l, constraining the averaged arterial partial pressure of CO₂ to 38.5995 mmHg and normalizing E by 62.482 Hz.

Figure 11. Insignificant energy saving when O₂ is constrained

Simulations are performed with the Ben-Tal (2006) model, using prescribed HR functions. Positive Δ represents RSA, negative Δ represents inverse RSA, constant heart rate is marked by a cross at Δ= 0. E/E₀ curves are plotted as a function of Δ (E is calculated by eqn (5), E₀ is the value of E at Δ= 0), constraining the partial pressure of O₂ along each curve. Data points are obtained from the simulation, and the continuous lines are least-squares quadratic fits. Data points represented by circles (fitted with a continuous line) were taken with T= 10 s, V_T= 1.0 l, constraining the averaged arterial partial pressure of O₂ to 105.7143 mmHg and normalizing E by 13.2576 Hz. Squares with a dashed line were taken with T= 5 s, V_T= 0.5 l, constraining the averaged arterial partial pressure of O₂ to 105.8179 mmHg and normalizing E by 4.974 Hz. Diamonds with dashed–dotted line were taken with T= 2.5 s, V_T= 0.25 l, constraining the averaged arterial partial pressure of O₂ to 105.1352 mmHg and normalizing E by 1.069 Hz.

Figure 12. Gas exchange efficiency improves with slow and deep breathing and with increased mean heart rate but this is unrelated to RSA

Volumes of O₂ (left panels) and CO₂ (right panels) taken up or removed by the blood over a minute normalized by the minute ventilation and converted to percentage, calculated with prescribed heart rate functions (shown in Fig. 8). Upper panels are done with human parameters, and for these, data in dark grey (blue online) have T= 10 s, V_T= 1 l, data in light grey (red online) have T= 5 s, V_T= 0.5 l, and data in black have T= 2.5 s, V_T= 0.25 l. The minute ventilation was the same for all these simulations: V_T/T= 0.1 l s⁻¹= 6 l min⁻¹. Data represented by circles connected with a continuous line have m= 0.6 Hz, squares with dashed line have m= 1.2 Hz, and diamonds with dashed–dotted line have m= 2.4 Hz. Lower panels are performed with dog parameters, and for these, data in dark grey (blue online) have T= 8 s, V_T= 0.42 l, data in light grey (red online) have T= 4 s, V_T= 0.21 l, and data in black have T= 2 s, V_T= 0.105 l. The minute ventilation was the same for all these simulations: V_T/T= 0.1 l s⁻¹= 3.15 l min⁻¹. Data represented by circles connected with a continuous line have m= 0.5 Hz, squares with dashed line have m= 1.0 Hz, and diamonds with dashed–dotted line have m= 2.0 Hz. All other parameters are given in Appendix A.

Figure 13. The calculation of the RSA-like heart rate is not affected qualitatively by changes to the expression of the work done by the heart

The calculations were performed by minimizing where n= 2 (dashed line), n= 1.5 (continuous line) or n= 3 (dashed–dotted line) and by using Model 1b.