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(Hypertension. 1996;27:1079-1089.)
© 1996 American Heart Association, Inc.

Articles

Wave Propagation in Coupled Left Ventricle–Arterial System

Implications for Aortic Pressure

David S. Berger; Kimberly A. Robinson; Sanjeev G. Shroff

From the Cardiology Section, Department of Medicine, University of Chicago (Ill).

Correspondence to Sanjeev G. Shroff, PhD, University of Chicago Medical Center, Room M-507, MC-5084, 5841 S Maryland Ave, Chicago, IL 60637. E-mail sshroff@medicine.bsd.uchicago.edu.

	Abstract

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Abstract The objective of this study was to examine the effects of wave propagation properties (global reflection coefficient, _G; pulse wave velocity, c_ph; and characteristic impedance, Z_o) on the mechanical performance of the coupled left ventricle–arterial system. Specifically, we sought to quantify effects on aortic pressure (P_ao) and flow (Q_ao) while keeping constant other determinants of P_ao and Q_ao (left ventricular end-diastolic volume, V_ed, and contractility, heart rate, and peripheral resistance, R_s). Isolated rabbit hearts were subjected to real-time, computer-controlled physiological loading. The arterial circulation was modeled with a lossless tube terminating in a complex load. The loading system allowed for precise and independent control of all arterial properties as evidenced by accurate reproduction of desired input impedances and computed left ventricular volume changes. While propagation phenomena affected P_ao and Q_ao morphologies as expected, their effects on absolute P_ao values were often contrary to the current understanding. Diastolic (P_d) and mean (P_m) P_ao and stroke volume decreased monotonically with increases in _G, c_ph, or Z_o over wide ranges. In contrast, these increases had variable effects on peak systolic P_ao (P_s): decreasing with _G, biphasic with c_ph, and increasing with Z_o. There was an interaction between _G and c_ph such that _G effects on P_m and P_d were augmented at higher c_ph and vice versa. Despite large changes in system parameters, effects on P_m and P_s were modest (<10% and <5%, respectively); effects on P_d were always two to four times greater. Similar results were obtained when the single-tube model of the arterial system was replaced by an asymmetrical T-tube configuration. Our data do not support the prevailing hypothesis that P_s (and therefore ventricular load) can be selectively and significantly altered by manipulating _G, c_ph, and/or Z_o.

Key Words: pulse wave, propagation • pulse wave, velocity • blood pressure • rabbit • heart • ventricular function • compliance

	Introduction

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The arterial system Z_in, the hydraulic load opposing ejection, consists of two components: a steady component composed of R_s and a pulsatile component consisting of distributed compliant and inertial properties (as well as, to a lesser extent, R_s). Together with the cyclic nature of ventricular ejection, these pulsatile load properties give rise to wave propagation phenomena; pressure and flow waves originate at the left ventricle and propagate along the arterial tree at finite wave propagation velocities, and the waves are partially reflected at sites of impedance mismatch (eg, bifurcations, arterioles).

That wave reflections and propagation affect P_ao and Q_ao pulses has long been established, both theoretically and experimentally; of this there is little dispute.^{1 2 3 4 5} For example, the amplification of P_ao as it propagates along the aorta is one manifestation of finite propagation velocities and peripheral reflections.^{4 6} Also, the morphological differences between P_ao and Q_ao are due to reflections, as evidenced by their morphologies becoming more similar as reflections are reduced.^{3 6 7 8} The shape of P_ao itself also contains some information regarding the magnitude of wave reflections and their timing.⁹ It has been suggested that in addition to the morphological changes in P_ao, reflected waves reaching the ascending aorta in systole act to significantly augment measured systolic pressure, ie, ventricular load, and that this pressure elevation is greater with increased magnitude and/or earlier return of reflections.^{10 11 12 13 14 15} On the basis of these interpretations, it has been postulated that certain types of hypertension, particularly those with elevated systolic pressure, can be treated through reduction of reflections and wave velocity.^{15 16 17} Although these hypotheses seem reasonable, there exists no direct physical evidence that observed pressure reductions are due to reduced reflections and/or reduced wave velocity This is because of attendant experimental difficulties; studies designed to address these issues, in both humans and intact animals, leave uncontrolled one or more other factors that might contribute to changes in pressure and flow. Moreover, recent theoretical findings lead to the hypothesis that wave reflections themselves do not necessarily increase the load faced by the contracting left ventricle nor do they substantially affect mean ventricular outflow, especially when compared with the effects of R_s.⁸

In this study, we used an isolated heart preparation with real-time loading of the left ventricle^{18 19} to examine experimentally the effects of various wave propagation properties on the mechanical performance of the coupled left ventricle–arterial system (ie, P_ao and Q_ao). This system allowed independent control of arterial system load, preload, and HR, thus permitting targeted changes in arterial properties. The results presented in this report will be compared with the previous model-based findings⁸ and discussed in light of experimental work in humans and intact animals. Finally, we will question the extent to which reduction of wave reflections and/or c_ph is beneficial to the left ventricle in terms of P_ao and SV.

	Methods

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All protocols were reviewed and approved by the University of Chicago Institutional Animal Care and Use Committee and conform with the Guide for the Care and Use of Laboratory Animals (National Institutes of Health publication 85-23, revised 1985).

Experimental Preparation
Experiments were performed on hearts isolated from normal, adult male rabbits (New Zealand White) weighing 2 to 3 kg. Rabbits were preanesthetized with 5.0 mg/kg xylazine (Vedco) and 0.2 mg/kg atropine (Elkins-Sinn) and after 10 minutes were anesthetized with 30 to 50 mg/kg ketamine (Kedalar, Parke-Davis) and 1.0 mg/kg acepromazine (Vedco). Tracheotomy was performed after anesthesia, and rabbits were artificially ventilated (model 683, Harvard Apparatus) with room air at a respiratory rate of 43 breaths/min and tidal volume of 25 to 30 mL. After median sternotomy and ligation of great vessels, a metal cannula connected to the perfusion system was inserted into the brachiocephalic artery and immediately flushed with heparinized saline (3.0 mL, 1000 U/mL). Retrograde perfusion of the coronary arteries was then begun at a constant perfusion pressure of 80 mm Hg and temperature of 37°C. The heart was perfused with oxygenated modified Krebs-Henseleit solution,¹⁹ which was not recirculated. Connective tissue was cut away and the heart removed from the chest while being constantly perfused. Therefore, coronary circulation was not interrupted at any time.

A thin latex balloon, secured at the end of a piston-cylinder device, was positioned in the left ventricle via the mitral orifice. A purse string tied around the mitral orifice secured the heart to the piston-cylinder device attached to a linear motor. A catheter-tip pressure transducer (model MPC-500, Millar Instruments, Inc) was advanced into the left ventricle via a side port in the piston-cylinder device. The piston position was sensed by a linear voltage displacement transformer (model 294, Transtek). More extensive details of the isolated heart setup can be found elsewhere.^{18 19} All hearts were paced with unipolar electrodes attached to the apex of the left ventricle.

Arterial System Model and Loading System
Fig 1A depicts schematically the features of the system that allow physiological loading of the isolated left ventricle. Instantaneous P_v is continuously monitored by the computer and acts as an input to the model of arterial circulation, in this case a single, uniform, elastic tube terminating in a complex load (Fig 1B). The arterial system tube parameters are Z_o, tube length (L), and c_ph. The ratio L/c_ph is , the one-way transmission time. Thus, propagation characteristics can be set by prescribing either or some combination of L and c_ph. The relative contribution of reflected waves at the left ventricle–arterial system interface is quantified by _G. It is important to note that the model formulation is such that Z_o, , and _G can be varied independently. This allowed precisely controlled experiments on wave propagation that are, for all practical purposes, not possible in the real system because a change in a physical property (eg, vessel wall stiffness) produces simultaneous changes in Z_o, c_ph, and _G. The terminal load, a variation on the three-element Windkessel,²⁰ is given by C, high-frequency resistance (R_o), and R_s. The tube and load are matched at high frequencies, therefore,

View larger version (22K): [in this window] [in a new window]   Figure 1. A, Schematic of real-time, computer-controlled loading system. Command signal was updated and data were sampled every 1.0 millisecond. B, Single-tube model with complex terminal load. Abbreviations are as defined in the text.

(1)

Following Campbell et al,²¹ this coupled system can be described by the following delay-differential equation,

(2)

(3)

(4)

where P_c is the pressure in the terminal compliance chamber and constants and ß are given by

(5)

With measured P_v as input, Q_ao was calculated from numerical integration of the system (fourth-order explicit Adams predictor-corrector algorithm; step size=1.0 millisecond). The transitions from isovolumic contraction to ejection and from ejection to isovolumic relaxation occurred when P_vP_ao and when P_v<P_ao, respectively. Calculated Q_ao was in turn integrated to yield the desired reduction in instantaneous V_v. The piston was moved exactly by an amount equal to this desired volume change, and the whole process was continued in real time with P_v samples taken every 1.0 millisecond. After ejection, preset filling pressure (P_f) and filling resistance (R_f) were used for calculation of filling flow (Q_f):

(6)

The onset of filling occurred when P_fP_v, and filling continued until the desired V_ed was reached, at which time filling was stopped. Finally, HR was independently controlled and systemic nervous and humoral influences were not present. Thus, the experimental apparatus was able to reproduce the conditions of the computer experiments described in Berger et al⁸ with the same degree of precision and independent control of arterial load parameters.

Experimental Protocols
Several protocols were performed in which a single property of the arterial system was altered, through adjustment of specific model parameters, while everything else was kept constant. These protocols were (1) protocol : adjusting _G() while keeping R_s, Z_o, and c_ph constant; (2) protocol c_ph: adjusting c_ph while keeping |_G|, R_s, and Z_o constant; and (3) protocol Z_o: adjusting Z_o while keeping c_ph and R_s constant. A total of 12 experiments were performed; six rabbits were used for protocols and c_ph combined and six for protocol Z_o. Experimentally, _G() was altered by adjusting only the value of C, thereby keeping the nonpulsatile hydraulic load constant.⁸ Changes in c_ph were accomplished by altering the wave transmission time, (Equations 2 through 4), in equispaced increments. These transmission times were converted to c_ph changes assuming a fixed tube length of 0.1 m. Altering c_ph in this manner results in a denser collection of points in the lower third of the c_ph range.

In all protocols, HR was fixed at 120 beats/min. V_ed was first adjusted under isovolumic conditions to yield end-diastolic P_v of 5 to 10 mm Hg and thereafter was held constant throughout the experiment. A single experimental "run" consisted of eight different arterial loads in which only one wave propagation property or parameter was changed, eg, eight different _G() values for protocol . The order of the eight predefined loads within a given run was randomly chosen and administered by the computer. Data were collected under numerical and physiological steady-state conditions only. For the first of eight loading conditions, the preparation was given 60 seconds to reach steady state; the seven subsequent load changes were given at least 30 seconds. Once steady state was reached, one ejection beat followed immediately by one isovolumic beat from the same V_ed were sampled. This data collection scheme is illustrated in Fig 2 with P_ao and P_v (top), Q_ao (middle), and V_v (bottom) for the first two loading conditions in a run of protocol c_ph. Data from several runs were collected within each protocol, specifically, different c_ph for protocol , different _G() for protocol c_ph, and different c_ph for protocol Z_o. The Table contains ₁, c_ph, and Z_o values of all runs for protocols , c_ph, and Z_o, respectively.

View larger version (31K): [in this window] [in a new window]   Figure 2. Representative data record from protocol cph showing Pao (solid) and Pv (dashed, top), Qao (middle), and Vv (bottom) for the first two conditions from a single run. After the start of the run, the preparation was allowed to reach steady state with at least 60 seconds of undisturbed ejection. At this time, one ejection beat followed immediately by one Piso from the same end-diastolic volume were sampled. After sampling, the load was changed by the computer, and sampling took place after at least 30 seconds of undisturbed ejection. This scenario was repeated for eight different loads per run.

View this table: [in this window] [in a new window]   Table 1. 1, cph, and Zo Values Used in Protocols , cph, and Zo

Occasionally, a premature ventricular contraction or other arrhythmia would occur during ejection, thereby disturbing the transition to steady state. Regardless of when this occurred, a full 30 seconds was given after the arrhythmia for steady state to be reached. The total time for a single run was 5 to 6 minutes, short enough so that natural degradation of the preparation was not a factor within a given run. The total time required for all runs reported here was approximately 1 hour, potentially long enough for degradation to affect ventricular function. To minimize any effects of degradation, we changed the order of protocols and runs within a protocol from heart to heart.

Data Analysis
We subjected P_ao and Q_ao data to traditional wave reflection analysis to obtain Z_in() and _G(), where Z_in()=P_ao()/Q_ao(), and _G()=[Z_in()-Z_o]/[Z_in()+Z_o]. The magnitude of the first and most significant _G() harmonic is denoted ₁. Z_in()⁼⁰ was compared with the desired R_s. If the difference between the two values was greater than 2%, an indication that either the preparation was not in steady-state ejection or the closed-loop servocontrol was inadequate, that specific condition was excluded from further analysis. This was a rare occurrence. On the basis of P_iso, P_dev was determined as the difference between peak and end-diastolic P_iso. P_dev was used to evaluate changes in left ventricular contractility following each loading condition and over time.

The effects of changes in wave propagation properties on P_ao and SV were analyzed. Specifically, we were interested in P_s, P_m, and P_d. Data from different hearts were analyzed together on the basis of percent changes from a common reference condition: smallest ₁ (or equivalently largest C) for protocol , largest c_ph for protocol c_ph, and smallest Z_o for protocol Z_o.

	Results

Top Abstract Introduction Methods Results Discussion References

 
First we evaluated the performance of the closed-loop servocontolled system. Fig 3A shows model-derived (analytic) magnitudes ||Z_in|| and phases (_Zin) of Z_in() superimposed on values calculated from experimental measurements of protocol . At all frequencies, an excellent agreement between model-derived and experimental Z_in() can be seen. Fig 3B shows a close correspondence between command V_v (integral of Q_ao) imposed on the left ventricle and actual V_v measured by the linear voltage displacement transformer during ejection.

View larger version (17K): [in this window] [in a new window]   Figure 3. A, Model-derived (solid line) and experimentally determined (symbols) Zin for the range of C in protocol . Both magnitude and phase are reproduced for the entire frequency range. B, Command (solid line) and actual (symbols, shown at 10-millisecond intervals for clarity) Vv during ejection. These data demonstrate the accuracy of the loading system.

Next we examined whether a variation in a specific wave propagation property of the model yielded the desired variation in the arterial system loading property. Z_in spectra and _G() magnitude (|_G|) from one experiment are shown in Figs 4 and 5, respectively. In protocol , diminished oscillations of both |Z_in| and _Zin (Fig 4, left) as C increases indicate reduced wave reflections, an observation confirmed by the data in Fig 5 (left). Furthermore, unchanged high-frequency |Z_in| and little change in the zero-crossing of _Zin (Fig 4, left) confirm constant Z_o and c_ph. In protocol c_ph, |Z_in| exhibited diminished oscillations as c_ph increased and a more gradual convergence toward a common Z_o, whereas the zero-crossing of _Zin was shifted toward higher frequencies; ie, the system becomes more Windkessel-like (Fig 4, middle). Despite these changes in Z_in(), |_G| was completely unaffected by a change in c_ph (Fig 5, middle). In protocol Z_o, increasing Z_o yielded the expected increase in |Z_in| for all harmonics, with no change in the zero-crossing of _Zin (Fig 4, right). Increasing Z_o also yielded a reduction in |_G| (Fig 5, right), which is consistent with the definition of _G().²² Also note that R_s was unchanged for all conditions; thus, R_s cannot be a factor in resultant P_ao and Q_ao changes.

View larger version (22K): [in this window] [in a new window]   Figure 4. Magnitude [|Zin()|] and phase (Zin) of Zin for the range of C in protocol (left), of cph in protocol cph (middle), and of Zo in protocol Zo (right) from a single experiment. Results are representative of all experiments.

View larger version (13K): [in this window] [in a new window]   Figure 5. Magnitude of G for the range of C in protocol (left), of cph in protocol cph (middle), and of Zo in protocol Zo (right) from a single experiment. Results are representative of all experiments.

Within any given run, P_dev was fairly constant, never varying more than a few millimeters of mercury, and the time courses of P_iso were practically superimposable. Over the course of the experiment (approximately 1 hour), a small but noticeable (approximately 5%) decrease in P_dev occurred. No other indications of diminished ventricular performance such as altered coronary flow or elevated end-diastolic P_v were observed in the hearts from which these data are reported.

Protocol
Fig 6 shows typical P_v, P_ao, Q_ao, and V_v from a single heart for control (middle), increased reflections (left), and reduced reflections (right). Morphological changes in P_ao were as expected: increased _G() yielded increased pulse pressure and end-ejection P_ao (P_ee), with a systolic shoulder developing at high _G(). Furthermore, P_ao morphology begins to approach that of Q_ao at low _G(). Both P_s and P_d increased with decreasing _G(), with relatively larger changes in P_d. Q_ao was also affected by _G(). Increased _G() yielded a shorter ejection period, higher peak flow, and narrower flow profile. Specifically, ejection started and ended earlier with greater change in end-ejection time. Fig 7A shows P_s, P_m, and P_d with SV plotted against ₁ over the full range tested. The trends for this heart are clear: decreased P_s and P_d with increasing ₁ and changes in P_d becoming larger at high ₁; relatively constant P_m, which eventually decreases at high ₁; and changes in SV identical to those of P_m, which is mandatory because R_s and HR are unchanged.

View larger version (28K): [in this window] [in a new window]   Figure 6. Results from one experiment in protocol : Pao and Pv (top), Qao (middle), and Vv (bottom) for 1=0.81 (left), 1=0.49 (middle), and 1=0.23 (right).

View larger version (15K): [in this window] [in a new window]   Figure 7. A, Ps, Pm, and Pd (top) and SV (bottom) for the range of 1 in protocol for a single experiment. B, Percent changes (mean±SE) in Ps, Pm, and Pd for all six experiments in protocol . Note that Pd is by far the most affected variable.

The results from all six hearts in protocol are pooled in Fig 7B, which shows percent changes (mean±SE) from the point of the minimum reflection coefficient. Despite the different absolute pressure levels between hearts, which were as great at 20 mm Hg, changes are quite consistent along the entire range of increasing ₁: a steady and small decrease in P_s not exceeding 4%; a larger decrease in P_d approaching 14%; and an almost constant P_m until the highest values of ₁, at which P_m decreased less than 2%. Since the P_d decrement was much greater than that of P_s, pulse pressure increased by 44% over the range of increasing ₁. Note that although the choice of reference ₁ is arbitrary, the directional changes observed are not dependent on the reference value of ₁. This comment applies to all subsequent percent change results.

Protocol c_ph
Fig 8 depicts typical P_v, P_ao, Q_ao, and V_v from a single experiment for c_ph=7.58 m/s (left), c_ph=3.79 m/s (middle), and c_ph=2.16 m/s (right). Altering c_ph, which affects the timing of the reflected wave, has clear, consistent effects on P_ao morphology: distinct diastolic oscillations with reduced c_ph and a gradual elimination of these oscillations as the reflected wave arrives earlier with increasing c_ph. Both peak Q_ao and ejection period were affected little by c_ph, whereas the flow profile was more symmetrical at low c_ph. Fig 9A shows P_s, P_m, P_d, and SV over the full range of c_ph. Both P_m and P_d showed initial rapid decreases with increasing c_ph (SV followed P_m), leveling off at high velocities. P_s showed a biphasic response, with a minimum value occurring at approximately 5 m/s.

View larger version (28K): [in this window] [in a new window]   Figure 8. Results from one experiment in protocol cph: Pao and Pv (top), Qao (middle), and Vv (bottom) for cph=7.58 m/s (left), cph=3.79 (middle), and cph=2.16 (right).

View larger version (15K): [in this window] [in a new window]   Figure 9. A, Ps, Pm, and Pd (top) and SV (bottom) for the range of cph in protocol cph for a single experiment. B, Percent changes (mean±SE) in Ps, Pm, and Pd for all six experiments in protocol cph. Note that Ps response is biphasic and Pd is by far the most affected variable.

Percent changes (mean±SE) in P_s, P_m, and P_d for all six experiments in protocol c_ph are shown in Fig 9B, with the highest velocity as the reference condition. These changes are remarkably consistent, with P_d being the most sensitive, showing a 12% change across the entire c_ph range. Changes in P_s and P_m were smaller: less than 4% for the former and 5% for the latter. Consequently, pulse pressure consistently increased with increasing c_ph. The biphasic P_s response was evident in all hearts.

Combined and c_ph
Percent changes in P_s, P_m, and P_d due to changes in ₁ at three different c_ph and due to changes in c_ph at three different ₁ are shown in Fig 10. For each experiment, a single reference condition (lowest ₁ and highest c_ph) was used for computation of these percent changes. P_d and P_m decrements after an increase in ₁ were augmented as c_ph increased (Fig 10A). Similarly P_m and P_d decrements in response to increasing c_ph were augmented by ₁ increases (Fig 10B). Regarding P_s, its biphasic response to changing c_ph also was augmented, and the point of minimum P_s was shifted to lower c_ph by increased ₁ (Fig 10B). In contrast, different c_ph had no effect on the slope of the P_s- ₁ relationship until large values of ₁ were reached. Also note that for any combination of ₁ and c_ph changes, the percent change in P_d was by far the largest of the three pressures.

View larger version (28K): [in this window] [in a new window]   Figure 10. Percent changes (mean±SE) in Ps (top), Pm (middle), and Pd (bottom) from a single reference condition for three different values of cph in protocol (A) and three different values of 1 in protocol cph (B). Effects of increased 1 are augmented by increased cph and vice versa.

Protocol Z_o
Fig 11 shows typical P_v, P_ao, Q_ao, and V_v for a single experiment for Z_o=5.0 mm Hg·s/mL (left), Z_o=7.0 mm Hg·s/mL (middle), and Z_o=10.0 mm Hg·s/mL (right). Peak Q_ao decreased while ejection period increased (combined earlier start- and later end-ejection times) with increasing Z_o. Increasing Z_o over a wide range yielded increased pulse pressure through combined decreases in P_d and increases in P_s (Fig 12A). The pooled data from six hearts in Fig 12B again show very consistent percent changes for all pressures. As in protocols and c_ph, P_d was the most sensitive variable, changing almost 16% over the Z_o range. The percent change in P_s was also similar in magnitude to those from protocols and c_ph, increasing 4% as Z_o increased over the range. The percent change in P_m (or equivalent in SV) was large compared with those from protocols and c_ph, decreasing 10% over the range.

View larger version (28K): [in this window] [in a new window]   Figure 11. Results from one experiment in protocol Zo: Pao and Pv (top), Qao (middle), and Vv (bottom) for Zo=5.0 mm Hg·s/mL (left), Zo=7.0 mm Hg · s/mL (middle), and Zo=10.0 mm Hg · s/mL (right).

View larger version (16K): [in this window] [in a new window]   Figure 12. A, Ps, Pm, and Pd (top) and SV (bottom) for the range of Zo in protocol Zo for a single experiment. B, Percent changes (mean±SE) in Ps, Pm, and Pd for all six experiments in protocol Zo. Again, Pd is the most affected variable.

Unlike in protocols and c_ph, in which rates of pressure changes varied over their respective ranges (ie, biphasic or eventually plateauing), the changes in pressures due to changes in Z_o were linear over the entire range. Changes in c_ph shifted the P_d-Z_o and P_m-Z_o relationships in a parallel manner (Fig 13); the effect on the P_s-Z_o relationship was more subtle because of the biphasic response of P_s to changes in c_ph.

View larger version (20K): [in this window] [in a new window]   Figure 13. Percent changes (mean±SE) in Ps (top), Pm (middle), and Pd (bottom) from a single reference condition for three different values of cph in protocol Zo.

	Discussion

Top Abstract Introduction Methods Results Discussion References

 
Our overall goal is to understand better the ways in which wave reflections and c_ph affect cardiovascular performance, from both mechanical and energetic perspectives. Here we report the effects of _G(), c_ph, and Z_o on P_ao and Q_ao generated in the coupled left ventricle–arterial system. This is the first experimental study that examines systematically the independent effects of the three aforementioned wave propagation properties under rigorously controlled conditions.

Adequacy of Experimental Control
For study of the direct effects of wave propagation phenomena on P_ao and Q_ao, several experimental constraints must be met⁸ : (1) complete control of arterial system properties such that _G() and c_ph can be altered independently of each other and Z_o; (2) ability to alter propagation properties without affecting the steady component of the arterial load, ie, constant R_s; (3) control of preload, V_ed, and HR; (4) elimination of neurohumoral feedback; and (5) a heart whose contractile properties do not change over time.

The isolated heart preparation readily fulfilled requirements 3 and 4. Data presented in Figs 3 through 5 indicate that requirements 1 and 2 are also met. Regarding requirement 5, a genuine potential source of error was unavoidable degradation of the preparation, some of which was evident in the decreased P_dev as experiments progressed. Although the magnitude of this decrease was small (5%), it may have affected late in an experiment the sensitivities of P_ao and Q_ao to load changes. To minimize the systematic effects of such degradation, we rearranged the order of protocols and runs within a protocol between hearts.

Are Results Specific to the Chosen Arterial Model?
The single-tube model treats all peripheral reflections as arising from one equivalent reflection site. To test the possibility that multiple reflection sights will yield different results, we loaded two hearts with the asymmetrical T-tube model^{21 23} consisting of two parallel circulations: head-end and body-end. Protocols , c_ph, and Z_o were implemented by changing parameters of the two circulations simultaneously and in proportion. Results with T-tube loading (Fig 14) were the same as those with single-tube loading. Thus, with uniform changes in the arterial system properties, as would occur in most cases, the number of reflection sites does not seem to matter. The issue of whether the same results will emerge with nonuniform changes (eg, localized alterations resulting in new, discrete reflection sites) needs further investigation.

View larger version (13K): [in this window] [in a new window]   Figure 14. Ps, Pm, and Pd (top) and SV (bottom) for protocol (left), protocol cph (middle), and protocol Zo (right) from a single experiment with the heart beating into an asymmetrical T-tube model. These results are entirely consistent with those of the single-tube model.

Extending this argument further, we believe that results would be minimally affected if more complex models of the arterial system were used, eg, more branches, elastic and/or geometric taper, lossy tubes, etc. It is sufficient to use a minimal model that reproduces Z_in accurately under a variety of vasoactive conditions and allows for independent changes in reflection coefficient and wave velocity. Single- and T-tube models with uniform, lossless tubes and complex terminal load satisfy these criteria.^{22 24 25 26}

Reflection Coefficient and Pulse Wave Velocity: Comparison With Model-Based Study
Except for P_d, which was by far the most sensitive variable, changes in all pressures and SV were modest, especially considering the wide ranges of ₁ and c_ph examined. These results, along with the morphological changes in P_ao and Q_ao due to changes in _G() and c_ph, are entirely consistent with those found in the model-based study.⁸ The physical reasons for these changes were discussed in detail elsewhere.⁸ Briefly, alterations in _G() and c_ph result in a redistribution of pressure between diastole and systole such that P_m/SV is maintained (a consequence of constant HR and R_s). So, with a reduction in ₁, for example, the simultaneous reduction of diastolic pressure decay and slight increase in SV yield a proportional increase in P_m and a redistribution of pressure from diastole to systole such that P_s increases slightly.

Characteristic Impedance
Unlike protocols and c_ph, changes in Z_o simultaneously alter a second arterial system property, namely _G(). Thus, the effects of Z_o on pressure and flow can be considered the result of changes in both Z_o and _G(), and it is instructive to compare the results in Fig 12B with those in Fig 7B. The increase in Z_o from 3.0 to 10.0 mm Hg·s/mL yielded a decrease in ₁ from 0.62 to 0.20 (Fig 5). Over this range of decreasing ₁, P_s increased by approximately 3%, P_m was unchanged, and P_d increased by approximately 6% (Fig 7B). With protocol Z_o, corresponding changes after an increase in Z_o were (Fig 12B) approximately 4% increase in P_s, approximately 9% decrease in P_m, and approximately 15% decrease in P_d. Thus, changes in ₁ certainly contribute to P_s changes in protocol Z_o. Since P_m was unaffected by ₁ over this range, the P_m changes in protocol Z_o are likely due to changes in Z_o alone. Finally, note that the directional changes in P_d with decreasing ₁ are opposite between protocols and Z_o. Therefore, the effects of Z_o alone (ie, with invariant _G) on P_d are expected to be greater than these observed in protocol Z_o ( Z_o and ₁).

Interaction Among ₁, c_ph, and Z_o Effects
Results presented in Fig 10 indicate that the sensitivities, and in some cases the directionality, of pressure responses due to changes in one parameter are greatly modified by changes in another. In general, effects of reflections are augmented at higher wave velocities and vice versa. On the other hand, percent changes in P_s, P_m, and P_d due to changes in Z_o are unaffected by c_ph (ie, no slope change in Fig 13). In contrast to the experiments described here, (patho)physiological events in the intact circulation (eg, exercise, disease processes, pharmacological treatment, aging) can simultaneously affect multiple arterial system properties. That is, certain arterial system properties tend to correlate, because of either shared underlying physical properties or dependence on common physiological responses. For example, aortic c_ph and Z_o tend to follow one another because both are inversely related to vessel wall stiffness.¹ By traversing along and between the families of curves (Figs 10 and 13), changes in pressures due to simultaneous changes in wave propagation properties can be predicted. In doing so, one would still observe that changes in P_s are much smaller than those in P_d.

Comparison With Previous Studies
Morphological changes in P_ao and Q_ao observed in this study are consistent with those of previous reports. Specifically, development of a systolic shoulder in P_ao is a classic morphological indicator of increased contribution of reflected waves,⁴ and the disappearance of diastolic oscillations and gradual similarity between P_ao and Q_ao morphologies with reduced reflections has been observed in human and intact animal experiments.^{6 7}

With respect to pressure and flow magnitudes, the general consensus is that wave reflections act to increase the pressure load on the ventricle. Consequently, reducing the magnitude of the reflections, delaying the return of reflections, or both have been associated with significantly reduced P_s with relatively little effect on P_d.^{11 12 13} Our results do not agree with these observations. We show small changes in P_s and P_m (or SV), and in some cases increases, with large reductions in _G() and c_ph. Furthermore, changes in P_d are much larger than changes in both P_s and P_m.

The role of wave propagation has been investigated with one of two approaches: intact animal experiments or modeling experiments in which both the arterial system and left ventricle are represented by mathematical constructs. The present study represents a hybrid of the two: a real heart coupled to a physical/mathematical model of the arterial system. We will reconcile our observations with those in the literature on the basis of two issues: (1) the left ventricle is neither an ideal flow nor an ideal pressure source²⁷ and (2) properties other than wave reflection and propagation are often left uncontrolled.

Regarding the first issue, making any change in the arterial system, however small, will change Z_in() in some way, thereby changing the hydraulic load into which the heart ejects. Therefore, the effects on P_ao and Q_ao arising from changes in _G(), c_ph, and/or Z_o are twofold: ejection and subsequent wave modification. In the dynamically coupled system, model or animal, the heart interacts with the load until steady-state solutions of P_ao and Q_ao emerge. Therefore, all aspects of P_ao and Q_ao are subject to change when the load changes, as seen in Figs 6, 8, and 11. Changes in _G() and c_ph provide information regarding only the nature of the backward wave relative to the forward wave; effects on the forward wave itself are not addressed. Thus, one cannot simply make predictions a priori about the effects of wave propagation properties on measured P_ao and Q_ao based solely on the known effects of these properties on the relationship between forward and backward waves; the effects on ejection should not be overlooked.

Theoretical Model Studies
Although many studies have used wave transmission–type arterial models, only those in which a dynamic ventricular source is coupled to the arterial model are suitable for comparison with the results presented here. McIlroy and Targett²⁸ showed both elevated P_s and P_d with increased |_G|. Since |_G| was increased in their study by increasing R_s, both wave reflections and the steady component of load contributed to the observed changes in pressure.⁸ Using a multibranching arterial model, Fitchett²⁹ decreased arterial compliance by increasing the elastic modulus for all segments (which corresponds to a simultaneous increase in c_ph and Z_o and possibly a decrease in ₁) and observed pressure changes that are consistent with our data (Figs 10 and 13). However, his observation that P_s changed more than P_d after increased wave reflections is contrary to our results. Although the difference cannot yet be fully explained, it may be due to the way in which wave reflections were manipulated: 80% occlusion and a fourfold increase in elastic modulus of a single segment. As mentioned before, such nonuniform, discrete changes in wave reflections will be the subject of future investigations. Finally, Latson et al³⁰ and Burkhoff et al³¹ analyzed P_ao generated by a left ventricle model connected to measured Z_in and P_ao generated by the same model left ventricle connected to an equivalent three-element Windkessel. Their Windkessel equivalent corresponded to not only an increase in c_ph to infinity but also a reduction in |_G|, a consequence of maintaining the total arterial compliance (ie, redistribution from tube to load). Results from these studies are consistent with our data, keeping in mind that these changes in c_ph and |_G| have competing effects on P_d and P_m and variable effects on P_s (Fig 10).

Isolated Heart Studies
Most isolated heart studies have used the three-element Windkessel model, in which c_ph is infinite.^{27 32 33 34} Nevertheless, changes in Windkessel compliance, C_w, are equivalent to changes in _G(); decreased C_w results in increased |_G|. In all these studies, a reduction in C_w caused the following: a small decrease in P_m and SV, a decrease in P_d, and either an increase^{27 34} or a biphasic response^{32 33} in P_s. Changes in P_s relative to P_d were always smaller. These observations are entirely consistent with our results from protocol at the highest c_ph (15.15 m/s, Fig 10A).

In contrast to the Windkessel loading, Kirkpatrick et al¹⁸ successfully loaded the isolated heart with wave transmission models; our loading system is identical to theirs. Although Kirkpatrick et al primarily focused on presenting a new methodology, one can observe effects of load compliance (Fig 7 in Kirkpatrick et al) and wave velocity (Fig 6 in Kirkpatrick et al) on P_ao and SV that are similar to what we report here.

Intact Heart Studies
Several methods have been used to alter arterial wave propagation properties in the intact circulation: providing an alternative ejection path via either an artificial Windkessel³⁵ or a stiff tube placed in the ascending aorta³⁶ ; applying plaster or Lucite ferrules to various portions of the aorta and large arteries³⁷ ; and replacing a section of native aorta with glass³⁸ or Tygon³⁹ tubing. These manipulations cause an increase in "aortic" wall stiffness, resulting in an increase in c_ph and in some cases an increase in Z_o. All of these studies show that increasing aortic stiffness in these manners yields decreased P_d, increased pulse pressure, and a relatively smaller reduction in SV, observations consistent with the results of the present study. However, increments in P_s were significantly greater than those from the present study, their magnitudes being more comparable to the decrements in P_d. These magnitude discrepancies can be explained by changes in factors other than those associated with wave propagation. For example, Randall et al³⁶ showed trends of increasing R_s and V_ed (inferred from their left ventricular end-diastolic diameter and pressure) with increased aortic stiffness. Similarly, it can be inferred from the data of Kelly et al³⁹ that R_s and V_ed increased by approximately 20% and 13%, respectively. Since P_ao is highly sensitive to R_s and V_ed, even small increases in these quantities can significantly augment P_s, thus reconciling the discrepancies.

Another class of studies uses drug effects, aging, and disease processes to make inferences regarding the effects of wave propagation phenomena. In general, elevated systolic pressure (eg, isolated systolic hypertension in the elderly) is accompanied by increases in wave reflections and/or increased c_ph.^{40 41 42} Conversely, vasodilator drugs, such as nitroprusside and nitroglycerin, that lower systolic pressure also tend to reduce wave reflections.^{10 11 40 43} On the basis of these observations, it has been hypothesized that reducing wave reflections, especially with greatly elevated P_s, is beneficial in that P_s can be reduced significantly with little change in P_d. Our results are clearly inconsistent with these hypotheses. Once again, we propose that changes in factors other than wave reflections are responsible for this discrepancy. For example, Yaginuma et al¹¹ report that in normotensive patients without cardiac disease, nitroglycerin reduces P_s, P_m, and SV significantly without any change in P_d. They attribute these changes to a reduction in peripheral reflections only; since R_s, Z_o, c_ph, C_w, and HR did not change. We offer an alternative explanation. All pressure and SV changes observed by Yaginuma et al can be reproduced simply by a 20-mL reduction of V_ed, consistent with vasodilator-induced venous pooling, without invoking any other cardiovascular changes. For a normal human left ventricle, this change in V_ed would cause left ventricular end-diastolic pressure to decrease by less than 4 mm Hg,⁴⁴ a pressure change that was actually observed by Yaginuma et al. Similar concerns are applicable to subsequent studies regarding effects of vasodilator-mediated changes in wave reflections and consequent alterations in P_ao.^{12 13 14 16} That wave reflections are reduced with nitroglycerin and other vasodilators is not disputed; however, this reduction of reflections in itself cannot preferentially reduce P_s.

Physiological Significance
One must keep three things in mind when extrapolating to the intact circulation our results regarding the effects of wave propagation properties on P_ao and Q_ao. First, we purposely examined very wide ranges of ₁, c_ph, and Z_o. Typical changes in these parameters under various (patho)physiological conditions are smaller, especially for c_ph (as much as threefold change) and Z_o (as much as twofold change). Therefore, in the intact circulation, expected changes in P_ao and Q_ao would be smaller than the extremes shown in this study. Second, effects are small for P_m (or SV) and P_s. Furthermore, since ₁, c_ph, and Z_o usually change in the same direction and their individual effects on P_s compete, the net response of P_s may be attenuated when two or more properties change simultaneously. Third, P_d is by far the most affected, especially with combined changes in which individual effects are in the same direction. Thus, our data do not support the hypothesis that P_s (hence, ventricular load) can be selectively and significantly reduced by reducing reflections.

In this study, we evaluated the relevance of wave propagation and reflection in normal hearts and solely in terms of the effects on P_s, P_m, P_d, and SV. It is worthwhile to also examine the effects of wave reflections on other variables, eg, left ventricular relaxation and filling, coronary flow, and myocardial oxygen consumption. Energetics and coronary flow are particularly interesting because of the greater sensitivity of P_d. Finally, the effects of wave reflections in diseased and hypertrophied hearts as well as the role of wave propagation phenomena, if any, in the remodeling process also remain to be elucidated.

	Selected Abbreviations and Acronyms

 

C = terminal load compliance of arterial system cph = pulse wave velocity G = global reflection coefficient 1 = magnitude of the first and most significant G() harmonic HR = heart rate Pao = aortic pressure Pd = minimum diastolic aortic pressure Pdev = peak developed left ventricular pressure in isovolumic beat Piso = left ventricular pressure in isovolumic beat Pm = mean aortic pressure Ps = peak systolic aortic pressure Pv = left ventricular pressure in ejecting beat Qao = aortic flow (left ventricular outflow) Rs = peripheral resistance SV = stroke volume Ved = left ventricular end-diastolic volume Vv = left ventricular volume = angular frequency Zin = arterial system input impedance