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1Cardiology Department, Leiden University Medical Center, Leiden; and 2Department of Physics and Medical Technology, Vrije Universiteit Medical Center, Amsterdam, the Netherlands
Submitted 7 February 2006 ; accepted in final form 11 December 2006
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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autonomic nervous system; cardiovascular variability; Mayer waves; spectral analysis; transfer function
0.1 Hz/period of 10 s). Effective baroreflex blood pressure buffering occurs below the Mayer frequency (10, 16). Besides a sympathetic limb that modulates peripheral resistance, the baroreflex has also sympathetic and parasympathetic (vagal) limbs that influence cardiac contractility, venous return, and cardiac rhythm. Usually, baroreflex functioning is characterized by baroreflex sensitivity (BRS). This index of baroreflex vigor is defined as the reflex-induced change in interbeat interval in milliseconds per millimeter of Hg blood pressure change (14, 34, 36, 44). The prognostic value of BRS and the favorable consequences of successful interventions with BRS have amply been demonstrated (27, 28).
Little is known, however, about the representativeness of this index for the efficacy of blood pressure buffering. There are two reasons to be skeptical in this respect: 1) by definition—interbeat interval change per unit blood pressure change—BRS is bound to characterize baroreflex-mediated effects on the heart, while the baroreflex buffers blood pressure mainly by controlling peripheral resistance (2, 30); and 2) oftentimes being assessed in the Mayer frequency range of spontaneous heart rate and blood pressure fluctuations (15, 39), BRS might represent resonance rather than buffering baroreflex characteristics.
We addressed these skepticisms by simulations with a hybrid mathematical model of baroreflex blood pressure and heart rate control composed of hemodynamic elements that are evaluated on a beat-to-beat basis, linked to a time-continuous modeled neural control part. By changing some parameter settings, the model mimics physiological as well as pathological hemodynamic and autonomic conditions.
By simulating with various gain combination values, we quantified the role of the sympathetic and parasympathetic gains in the three baroreflex limbs for blood pressure variability (BPV) and heart rate variability (HRV) under physiological and pathological conditions. From the obtained systolic blood pressure and interbeat interval values, relations between BRS and blood pressure buffering and between blood pressure buffering and resonance were examined.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Model Description
A gross overview of the autonomically controlled model is given in Fig. 1 (see Tables 1 and 2 for abbreviations and model parameters). The model represents the systemic circulation and consists of three sections: a hemodynamic section, a baroreceptor section, and an autonomic control section. The model generates output in the form of time-dependent systolic blood pressure (SBP) values (mmHg) and interbeat interval (IBI) values (ms) by using a sinusoidal pressure probe (frequency adjustable, amplitude fixed at 1 mmHg) as an input signal. This apparently small perturbation at the input of the baroreflex produces reduced SBP fluctuations (amplitude <1 mmHg, buffering) but also Mayer waves (amplitude >1 mmHg, resonance), depending on the frequency of the pressure probe.
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n is a left ventricle filling factor:
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n = 1.
Stroke volume Qn, assuming a constant arterial compliance (CA), determines pulse pressure Pn by:
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Baroreceptor section. The baroreceptors are modeled linearly within the range between a threshold of 90 mmHg and a saturation level of 150 mmHg. At the baroreceptors, the systolic blood pressure Sn is compared with a low-pass filtered systolic blood pressure reference value. This value functions as a dynamic blood pressure set point, mimicking the physiological process of baroreceptor resetting (47, 50, 51). The pressure variability source is added at the input of the baroreflex, rendering a sample-and-hold systolic blood pressure variability signal SBP, the first model output signal.
Autonomic control section.
In the time-continuous autonomic control section, SBP is converted into an afferent neural signal NE by factorizing this signal by the baroreceptor sensitivity coefficient 
N. This signal serves as input for three effectors: vagal heart rate control (output: vagal signal n); sympathetic heart rate control (output: sympathetic signal m); and sympathetic peripheral resistance control (output: Windkessel time constant T). The vagal signal n represents vagal heart rate deceleration (0 < n < 1), while the sympathetic signal m represents sympathetic heart rate acceleration (m > 1).
The three effectors are modeled in frequency-dependent functional blocks, with specific sensitivity coefficients, time constants, time delays and by autonomic tones (N, MH, and MPR; see Table 1 for actual values). In addition to these model parameters, extra baroreflex gain multipliers (SH, V, and SPR) were added to strengthen or weaken the role of each baroreflex effector.
The neural time-continuous part and the hemodynamic beat-to-beat part are interconnected by an integral pulse frequency modulator (IPFM), which simulates cardiac pacemaker function (18). Rosenblueth and Simeone (40) have demonstrated that combined sympathetic and vagal influences on the sinus node contribute to the actual heart rate R according to the following relationship: R = R0·m·n, where R0 is the intrinsic heart rate. Integration of incoming neural activity results in the generation of the heart interval length In (18). This interbeat interval IBI is the second model output signal.
Adjustable Model Parameters
Thus the model is controlled by seven parameters: one (Vref) for stroke volume, three (MH, N, and MPR) for autonomic outflow, and three (SH, V, and SPR) multipliers for the gains in the three baroreflex limbs.
The first four parameters for stroke volume and autonomic outflow were set as two fixed combinations (Table 1) to represent either normal physiological or abnormal pathological resting conditions. With an increased sympathetic tone to the heart and to the peripheral resistance and decreased parasympathetic tone and reference stroke volume, the pathological parameter settings represent a serious pathological condition resembling congestive heart failure. Compared with the physiological conditions, the resting heart rate is higher (90 beats/min instead of 60 beats/min), and the average SBP is slightly lower (114 mmHg instead of 120 mmHg).
The last three parameters serve as potentiometers (multipliers) on the vagal and sympathetic baroreflex gains to the heart and to the peripheral resistance; V = SH = SPR = 1 is the reference value that is to represent a normally working baroreflex. When one of these parameter values equals 0, the corresponding limb of the baroreflex does not react to changes of SBP with respect to the reference value, and the corresponding effector output becomes the (fixed) tone. A value of 0.5 corresponds to weak involvement. The maximum value of these parameters is 3; this value corresponds to a strong involvement of a given baroreflex limb, e.g., as found in highly trained subjects.
Simulations and Frequency Characteristics
For a given combination of the seven model parameter values, 100 simulation runs were done. A single simulation run served to determine one SBP variability (BPV) frequency component, one IBI variability (HRV) frequency component, and the modulus of the SBP-to-IBI transfer function (TF, necessary to compute BRS) frequency component, at a given frequency of the sinusoid pressure probe. A single simulation run was executed as follows. First, the model was run until steady-state conditions were met. Cubic splines were then fitted through the resulting output signals to obtain the amplitudes of the SBP and IBI fluctuations caused by the pressure probe. Finally, the corresponding TF frequency component was computed by dividing HRV (the amplitude of the IBI fluctuations) by BPV (the amplitude of the SBP fluctuations). The 100 simulation runs were done to construct the complete frequency characteristics of BPV, HRV, and TF by computing all frequency components between 0.003 and 0.300 Hz (step 0.003 Hz).
A total of 162 frequency characteristics of HRV, BPV, and TF were made for both the physiological as well as for the pathophysiological conditions. These 162 frequency characteristics were made to represent 162 different combinations of baroreflex gain multiplier settings. One-hundred fifty gain multiplier combinations were randomly chosen to simulate uncoupled baroreflex gains (values between 0 and 3 from uniform distributions for V, SH, and SPR). In addition, 12 other V/SH/SPR combinations were made to simulate coupled baroreflex gains (0.5/0.5/0.5, 1/1/1, 1.5/1.5/1.5, 2/2/2, 2.5/2.5/2.5, 3/3/3). Besides these multiplier combinations, an extra set of simulation results (obtained with V/SH/SPR combinations 0/1/1, 3/1/1, 1/0/1, 1/3/1, 1/1/0, 1/1/3) was made for the generation of Fig. 2.
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After having computed a full BPV, HRV, and TF characteristic, we determined the following variables. BRS was computed as the averaged magnitude of TF in the low-frequency (LF, 0.05–0.15 Hz) band (15, 37, 39). This band incorporates the Mayer frequencies. SBP-buffering capacity was expressed as the amplitude of the original perturbation (the 1-mmHg sinusoidal pressure probe) divided by the BPV amplitude at the lowest simulated frequency component [0.003 Hz, which is still well above the baroreceptor resetting frequency (47)]; e.g., when the BPV at the lowest frequency component had an amplitude of 0.25 mmHg, the buffering capacity was 4. Maximal SBP resonance (in the LF band) was expressed as the maximal BPV divided by the amplitude of the original perturbation. To determine the relative importance of V, SH, and SPR for baroreflex sensitivity and blood pressure buffering and resonance, multiple linear regressions were done. For these regressions, only the simulations made with random generated baroreflex gain multipliers were used.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Figure 2A3 shows an unexpected influence of sympathetic heart rate control on IBI: the transfer function in the LF band (i.e., BRS) even decreases when control is strengthened (strong SH). Obviously, blood pressure buffering and resonance are completely insensitive for changes in the sympathetic gain to the heart (Fig. 2A2).
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Figure 2, C1–C2, shows the striking effect of a strengthened sympathetic peripheral resistance control (multiplier SPR assumes the value of 3; multipliers V and SH are kept at a value of 1) on the HRV and BPV frequency characteristics. Figure 2C2 shows that the original sinusoidal disturbance of SBP by the 1-mmHg pressure probe (see Fig. 1) is strongly weakened (buffered) for the lowest frequencies, is amplified (resonance) over nearly the whole LF band, and returns to 1 mmHg for higher frequencies. The larger part of this effect, especially the resonance phenomenon, disappears under normal control (multiplier SPR assumes the value of 1). The frequency characteristic is almost flat when control is absent (multiplier SPR assumes the value of 0). The shapes of the HRV frequency characteristics in Fig. 2C1 grossly follow the BPV characteristics. As expected, the TF frequency characteristics (Fig. 2C3) are very much similar for all three SPR values, 0, 1, and 3. In summary, from the frequency characteristics shown in Fig. 2, C1–C3, the impression arises that sympathetic peripheral resistance control strongly influences resonance and buffering while it does not affect the TF or BRS.
Figure 3 depicts the strongest relations between vagal and sympathetic baroreflex gains, SBP-buffering capacity, SBP resonance, and BRS on the basis of the results of multiple linear regression analysis. It is pointed out that in a physiological setting, 83% of the variance in SBP buffering was attributable to sympathetic peripheral resistance control (multiplier SPR); under pathological conditions this was 78%. Also, 99% of the variance in BRS was attributable to vagal heart rate control (multiplier V); under pathological conditions this remained 99%.
The scatterplot of the SBP-buffering capacity as a function of SPR, together with linear fits for the physiological and the pathological data (Fig. 3A), shows close to perfect linear relationships. Also, there is little difference between the linear fits for the physiological and the pathological simulation results. Obviously, heart rate control, but also the settings of Vref and MPR, were of minor importance for blood pressure buffering.
The scatterplot of BRS as a function of V, together with linear fits for the physiological and the pathological data (Fig. 3B), shows nearly perfect linear relationships. Here, the physiological fit (slope 6.9 ms/mmHg) and the pathological fit (slope 4.0 ms/mmHg) differ considerably: with equal vagal gain multipliers, BRS is much larger in physiological conditions.
Figure 3C shows that SBP-buffering capacity and SBP resonance have a convex relationship and that the resonance phenomenon is much more prominent in physiological circumstances compared with pathological conditions. The strong link between buffering and resonance follows directly from regression analysis: also here, multiplier SPR attributes the most to variance in SBP resonance (95% under normal conditions, 91% under pathological conditions).
Finally, Fig. 3D shows that BRS was almost unrelated to SBP-buffering capacity, unless coupled baroreflex gains (simulation results represented by the open and solid squares) are assumed. The squared correlation coefficients of the linear regressions of SBP-buffering capacity on BRS were as low as 0.037 (physiological conditions) and 0.083 (pathological conditions).
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Simulations with various combinations of baroreflex gains, under physiological as well as under pathological conditions (increased sympathetic tone, decreased vagal tone, reduced cardiac stroke volume), yielded frequency characteristics of the transfer function, of HRV, and of BPV, and values of BRS, blood pressure-buffering capacity and baroreflex resonance in a wide range of conditions that may be met in real life. All frequency characteristics had a smooth character, and even with striking resonance the transfer function did not show discontinuous or deviant behavior compared with its value below and above the frequency band of resonance (see Fig. 2). In the following, the simulation results will be discussed in the order they have been presented in Fig. 3.
Baroreflex Gains and Blood Pressure-Buffering Capacity
Our results suggest a predominant role for the sympathetic limb to the peripheral vasculature for the blood pressure-buffering capacity of the arterial baroreflex (Fig. 3A). There is almost no difference in buffering capacity between the physiological and the pathological conditions. This result clearly illustrates the fact that efficacy of baroreflex-mediated blood pressure control rests on dynamic control of the peripheral resistance. Modulation of heart rate by baroreflex-mediated modulation of the vagal and sympathetic tone to the heart is not very relevant for blood pressure control in the frequency range of interest for this study (0.05–0.3 Hz).
Obviously, the simulation results may not be interpreted in such a way that baroreflex-mediated blood pressure buffering in patients is not different from that in healthy subjects. The sympathetic feedback gain to the peripheral vasculature is the decisive factor here. We speculate that this gain will be lower in patients. Hence, it may have been somewhat unrealistic to extend the simulations in pathological conditions to a similar value of SPR than the simulations in physiological conditions. The consequence of our speculation would be that the blood pressure-buffering capacity in patients is smaller than that in healthy subjects.
Baroreflex Gains and BRS
Baroreflex sensitivity is linear with and depends almost exclusively on the vagal feedback gain to the heart (Fig. 3B). The slopes of the linear regressions (6.9 and 4.0 ms/mmHg with physiological and pathological conditions, respectively) are merely to be explained on the basis of heart rate differences between these two situations and the way the IPFM (18) reacts to fluctuations in vagal tone. The fact that BRS depends on heart rate has been recognized earlier (1), and proposals have been done to normalize BRS on heart rate, or, alternatively, to express BRS in beats per minute per millimeter Hg instead of in milliseconds per millimeter Hg. Such arithmetic operations would change the linear relationships in Fig. 3B to curved ones but leave the conclusions unaffected that BRS increases with increasing vagal feedback gain and that the vagal feedback gain almost uniquely determines BRS.
The predominant role of the vagal feedback gain on the baroreflex sensitivity (38) can also be formulated in a slightly different way: due to the differences in the time constants of the vagal and the sympathetic branches (in our model 0.1 and 4.0 s, respectively), the greater part of HRV is simply vagal transmission of blood pressure variability to the sinus node. This is easily perceived in Fig. 2, B1–B3 and C1–C3, and in accordance with the findings of Cevese et al. (9). When vagal feedback gain is zero (dotted lines in Fig. 2, B1–B3), there is almost no HRV (Fig. 2B1) while there still is appreciable BPV (Fig. 2B2). When there is appreciable vagal baroreflex feedback gain (solid and dashed lines in Fig. 2, B1–B3, and all lines in Fig. 2, C1–C3), the HRV frequency characteristics in Fig. 2, B1 and C1, have the same shape as the BPV frequency characteristics in Fig. 2, B2 and C2. In the case of overt (sympathetically mediated) blood pressure resonance, where the BPV frequency characteristic has a narrow peak (Fig. 2C2), a similar "monochrome" HRV frequency characteristic is seen in Fig. 2C1. Alternatively, when there is no outspoken resonance (Fig. 2B2), there is "broad band" HRV (Fig. 2B1).
Blood Pressure-Buffering Capacity and Mayer Waves
Figure 3C illustrates the principle that a negative-feedback control system with feedback delay buffers the controlled variable at certain frequencies at the cost of resonance at other frequencies. The baroreflex blood pressure control system as simulated in this study behaves in a way that is very similar to what was experimentally observed (10). Blood pressure buffering, a major function of the baroreflex, occurs at frequencies below the Mayer waves (resonance in the LF band, Fig. 2C2). Typically, the arterial baroreflex could dampen blood pressure and heart rate responses to stressors that last several minutes. On the one hand, neural control of blood pressure by sympathetically induced vasoconstriction is relatively fast (seconds). On the other hand, baroreceptor resetting (47, 50, 51) limits the maximal duration during which baroreflex mediated buffering of a stressor may continue. In our simulations, the BPV frequency characteristics in Fig. 2, A2, B2, and C2, shows that dampening (reduction of the sinusoidal pressure probe disturbance) is strongest for the lowest frequencies.
Although there still exists some controversy about the origin of the observed spontaneous blood pressure and heart rate variations around 0.1 Hz (32), we assume that this phenomenon is due to the dynamics of the closed-loop vasomotor control (arterial peripheral resistance), in which the time delay of a few seconds plays the major role. Building a baroreflex model with negative-feedback control, and with parameters estimated from physiologically known data, results in a model that simply shows such resonance behavior, without the need to postulate centrally driven oscillators or (strong) nonlinearities.
Resonance, the price to be paid for buffering, is likely to be useless in terms of homeostasis. At the same time, it may be an innocent phenomenon without any negative impact for the organism (22). The fact that Mayer waves, useless or not, exist, facilitates spectral BRS assessment in the LF band, by creating an input signal (blood pressure variability) for the baroreflex of which the output signal (heart rate variability) can easily be measured. There is no inherent signal analysis problem in measuring BRS by the transfer function around the resonance frequency. However, the 180° phase shift caused by the time lag in the sympathetic efferent baroreflex limb to the heart with respect to the phase shift in the efferent vagal limb, that has a much shorter time lag, may cause the sympathetic and vagal limbs to the heart to counteract in the LF band. This effect will become prominent with increased sympathetic gain to the heart (see, e.g., Fig. 2A1, dashed line). In this respect, lower TF frequencies would constitute a more realistic BRS estimate, because here vagal and sympathetic feedback to the heart is concordant (Fig. 2A3, dashed line). In general, TF values in the LF band are not too different at higher frequencies; TF values increase for lower frequencies (Fig. 2, A3, B3, and C3).
Baroreflex Sensitivity and Blood Pressure-Buffering Capacity
One of the major reasons to perform this study was the question of whether there is a relation between the primary function of the baroreflex, i.e., blood pressure buffering, and the generally accepted clinical index for baroreflex vigor, BRS. Figure 3D shows that this relation does almost not exist. The correlation coefficients of the regression lines of SBP-buffering capacity on BRS are very low, and the data are diffusely distributed.
Indeed, vagal control of heart rate (major cause of the BPV-to-HRV transfer and, hence, major determinant of BRS) and sympathetic modulation of the peripheral vasculature (major cause of peripheral resistance adaptations and, hence, a major determinant of blood pressure buffering) become effective via separate efferent pathways of the baroreflex. There should not necessarily be a fixed relationship between the feedback gains in both reflex limbs (43).
To our knowledge, there are no data regarding the relative strengths of the gains in the three baroreflex effector limbs. It might well be that subjects with a low gain in the vagal limb have also low gains in the sympathetic limbs, amongst others, because part of the origin of these gains is to be found in the common afferent pathway of the reflex starting at the baroreceptors in the arterial wall up to and including the nucleus tractus solitarii in the brain stem. Inspection of the simulation results obtained under coupled gains (closed and open squares in Fig. 3D) reveals that in such cases there is a seemingly linear relationship between BRS and blood pressure-buffering capacity in healthy subjects as well as in patients.
Limitations of the Model
Basic to our study is the representativeness of the mathematical model. The original model has extensively been validated (46), among others by comparing the results of modeled vagal blockade and of standing with real-world observations. The modified model, however, has a simplified hemodynamic structure. Since the simulations addressed blood pressure and heart rate control in the supine posture only, the dynamic control capabilities of cardiac contractility and venous return on cardiac output and hence, blood pressure, have completely been removed (obviously, such a simplification cannot be made in cases where the average IBI changes because of an altering circulatory load). Elimination of these feedforward mechanisms enabled us to concentrate on the role of the various baroreflex gains, especially in the LF band, rather than steady-state phenomena in the lower frequencies. As our simulation results are still comparable with the various spectra produced by the original model, we do believe that our model still produces relevant spectra.
The modified model that was used for our present study generates and explains some situations that are known from the clinic. It is obvious that the resonance phenomenon in the LF band, generally known as Mayer waves (33), is strongly under influence of the baroreflex. The only situations in which Mayer waves hardly appear is when the sympathetic baroreflex gain to the peripheral resistance is small (see Fig. 2C2, dotted line). This simulation result parallels studies in rats (19) and in humans (42, 45). The relevance of the model is underscored by the observation that SBP variability in the LF band decreases for a fixed heart rate (results not shown here). This phenomenon was described by Taylor and Eckberg (45) in a study in humans. The authors demonstrated that baroreflex-mediated heart rate control was not effective in reducing blood pressure variability that had a larger amplitude in sinus rhythm compared with fixed-interval atrial pacing.
Within the operating space constituted by the ranges of the parameters as given in Table 1, in combination with baroreflex gain multiplier values between 0 and 3, our model can be used without any difficulty. For example, as the baroreflex gain to the peripheral resistance (SPR) should not have any influence on the transfer function, Fig. 2C3 shows indeed that only varying SPR produces almost the same transfer functions. The minimal differences between those functions can be explained by nonlinearities in the model. Further expansion of the operating space may therefore be not allowed. Furthermore, higher baroreflex gains would no longer be realistic and lead to, e.g., unacceptably high blood pressure variability values.
BRS can be enhanced by training (7), and the beneficial effects of a thus increased BRS have convincingly been demonstrated (27). How this effect is accomplished remains uncertain. Inhibition of stressor-induced heart rate increases may be one reason; the vagal feedback gain in the cardiac efferent limb may predominantly cause this effect. Inhibition of stressor-induced blood pressure increases may be another reason; the sympathetic feedback gain in the baroreflex efferent limb to the peripheral vasculature may predominantly cause this effect. Both effects could help to inhibit a stressor-induced raise of myocardial oxygen consumption, which is proportional to the product of heart rate and systolic blood pressure (3, 29).
A final remark regards the phenomenon as seen in Fig. 2A3. It appears that BRS (the TF between 0.05 and 0.15 Hz) may lower with high sympathetic gain to the heart. This is caused by the differences in the latencies/time constants in the sympathetic (17, 41) and vagal (6, 48) limbs, bringing the vagal and the sympathetic effects in counterphase in the BRS frequency band. Hence, there are situations thinkable in which cancellation of vagal effects on heart rate by concurring sympathetic effects on heart rate in counterphase incorrectly suggest a deficient baroreflex. For higher frequencies, the influence of the sympathetic feedback gain weakens and disappears because of a low pass filtering effect caused by slow neurotransmitter diffusion at the synaptic clefts (17).
In conclusion, our simulation study suggests that, within the limitations of the model, BRS and baroreflex-mediated blood pressure buffering are unrelated baroreflex features unless there is a good physiological reason to assume a fixed relation between the baroreflex feedback gains in the efferent baroreflex limbs to the heart and peripheral vasculature.
Also, we conclude that baroreflex-mediated blood pressure-buffering capacity is almost uniquely determined by the sympathetic baroreflex feedback gain to the peripheral vasculature, while BRS is almost uniquely determined by the vagal feedback gain to the sinus node.
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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