Differential Effects of Isoflurane and Halothane on Aortic Input Impedance Quantified Using a Three-element Windkessel Model

All experimental procedures and protocols used in this investigation were reviewed and approved by the Animal Care Committee of the Medical College of Wisconsin. All procedures conformed to the American Physiologic Society Guiding Principles in the Care and Use of Animals* and were performed in accordance with the National Institutes of Health Guide for the Care and Use of Laboratory Animals.**

Surgical implantation of instruments has been previously described in detail. [13,18]In the presence of general anesthesia and using aseptic techniques, conditioned mongrel dogs underwent a left thoracotomy and heparin-filled catheters were placed in the proximal descending thoracic aorta and the right atrium for measurement of aortic pressure and fluid or drug administration, respectively. An ultrasonic transit-time flow probe (Transonic Systems, Ithaca, NY) was positioned around the ascending thoracic aorta for measurement of continuous aortic flow (Figure 2). A pair of miniature ultrasonic segment length transducers (5 MHz) were implanted in the left ventricular subendocardium for measurement of changes in regional contractile function (percentage segment shortening [%SS]). A high-fidelity micromanometer (P7, Konigsberg Instruments, Pasadena, CA) was positioned in the left ventricle for measurement of continuous left ventricular pressure and the maximum rate of increase in left ventricular pressure (dP/dt). A heparin-filled catheter was inserted directly into the left atrial appendage, and the left ventricular micromanometer was cross-calibrated in vivo against pressures measured with arterial and left atrial catheters (P⁵⁰pressure transducer, Gould Instruments, Oxnard, CA). All instrumentation was secured, tunneled between the scapulae, and exteriorized through several small incisions. The pericardium was left widely open, the chest wall closed in layers, and the pneumothorax evacuated by a chest tube. Each dog was fitted with a jacket (Alice King Chatham, Los Angeles, CA) to prevent damage to the instruments and catheters, which were housed in an aluminum box within the jacket pocket.

All dogs received systemic analgesics (fentanyl and droperidol [Innovar-Vet], Pittman-Moore, Mundelein, IL] as needed after surgery. Dogs were allowed to recover a minimum of 7 days before experimentation, during which time all were treated with intramuscular antibiotics [cephalothin (40 mg *symbol* kg sup -1) and gentamicin (4.5 mg *symbol* kg sup -1)] and were trained to stand quietly in an animal sling during recording of hemodynamics. Segment length signals were monitored with an ultrasonic amplifier (Crystal Biotech, Hopkinton, MA). End-systolic and end-diastolic segment lengths were measured at maximum negative left ventricular dP/dt and just before the onset of left ventricular isovolumic contraction, respectively. The lengths were normalized according to the method of Theroux et al. [19]%SS was calculated using the equation: %SS = (EDL - ESL) *symbol* 100 *symbol* EDL sup -1, where ESL = end-systolic segment length and EDL = end-diastolic segment length. Hemodynamic data were continuously recorded on a polygraph (7758A, Hewlett-Packard, San Francisco, CA) and digitized by a computer interfaced with an analog to digital converter.

Aortic input impedance spectra were determined from digitized, steady-state aortic blood pressure and aortic blood flow waveforms using the techniques of Taylor [20]and Burkhoff et al. [4]Data files consisting of 4,096 points were sampled at 200 Hz (20.48 s; frequency increment = 0.098 Hz) and were divided into five 2,048 point bins with 512 point overlap. A Hamming window was applied to each bin to reduce side lobe leakage. The autopower spectrum of the aortic blood pressure [P^pp(omega)], aortic blood flow [P^ff(omega)], and cross power spectrum between aortic pressure and blood flow waveforms [P^pf(omega)] were determined using a Welch periodogram technique. [21,22]The aortic input impedance [Zⁱⁿ(omega)] was calculated as a function of frequency using the formula Zⁱⁿ(omega) = P^pp(omega) *symbol* [P^pf(omega)] sup -1. Each impedance spectrum was calculated to a maximum frequency of 15 Hz because little spectral energy exists above this frequency in the cardiovascular system. [1]This range of frequency analysis encompassed 6-15 harmonics for the aortic pressure and blood flow waveforms evaluated in all dogs.

The calculated impedance spectra were corrected for the phase response of aortic flow probe and aortic pressure transducer (see appendix). The magnitude of the frequency response of the flow meter was flat from 0 to 15 Hz with the analog low-pass filter cutoff set to 30 Hz. The phase delay of this signal was 11.2 ms independent of frequency. The frequency response of the fluid-filled pressure transducer catheter system was determined by performing a quick-release, or "pop," test. [1]The undamped natural frequency of this system was found to be 15.1 Hz with a damping ratio of 0.16 (see appendix). Because of the nonlinear phase response of the pressure transducer system, the time delay of the aortic blood pressure waveform was frequency-dependent and was corrected in the frequency domain using the method of Milnor. [1]Additional error in the phase of the pressure signal because of the distance (approximately 4 cm) between the flow probe (ascending thoracic aorta) and the pressure transducer (descending thoracic aorta) was also corrected using standard techniques. [23].

Correlation of aortic pressure and blood flow waves at each frequency of the input impedance spectrum was determined using the magnitude squared coherence: MSC(omega) [right vertical bar] P^pf(omega) [left vertical bar]²*symbol* [P^pp(omega) *symbol* P sub ff (omega)] sup -1, where MSC = magnitude squared coherence. Magnitude squared coherence values vary between 0 (no correlation between pressure and flow) and 1 (matched correlation). Higher correlation between aortic pressure and blood flow waveforms increases the precision of the impedance value at a given frequency. Input impedance data with mean squared coherence values less than 0.8 were discarded. High coherence was observed at most frequencies in conscious dogs as a result of heart rate variability caused by the sinus arrhythmia (Figure 3), a finding that resulted in the elimination of less than 10% of Zⁱⁿ(omega) data points. Heart rate variability creates a large number of "fundamental" heart rates and corresponding harmonics, resulting in a nearly continuous input impedance spectrum. In contrast, little or no sinus arrhythmia occurred during isoflurane or halothane anesthesia and heart rate remained relatively constant at each anesthetic concentration. Thus, high coherence impedance measurements were grouped around harmonics of this fundamental frequency (i.e., heart rate) during each anesthetic intervention. Under these circumstances, a greater percentage of spectral points in Zⁱⁿ(omega) between the fundamental frequency and harmonics were eliminated resulting in a less continuous aortic input impedance spectrum (Figure 4). [22].

Z^cwas determined from the aortic input impedance spectra as the mean of the magnitude of Zⁱⁿ(omega) ([left vertical bar] Z sub in (omega) [right vertical bar]) between 2 and 15 Hz. [4,24,25]R was calculated as the difference between the value of [left vertical bar] Zⁱⁿ(omega) [right vertical bar] at zero frequency and Z^c. [left vertical bar] Zⁱⁿ(omega) [right vertical bar] at zero frequency is equal to systemic vascular resistance determined as the ratio of mean arterial pressure and mean aortic blood flow (i.e., cardiac output). [1]The C component of the Windkessel model was calculated using the method of Liu et al. [26]: Equation 1where A^d= the area under the diastolic portion of the arterial pressure curve above mean venous pressure (assumed to be 0 mmHg); MAP = mean arterial pressure; MAQ = mean aortic blood flow; P^es= end-systolic aortic pressure; and P^ed= end-diastolic aortic pressure. Because this method does not assume a monoexponential decay of the aortic pressure waveform, calculation of C can be performed in the presence of reflected waves from the distal arterial vasculature. The value of C was determined from the average of five consecutive beats for each intervention. A variable that predicts the distance to the net site of arterial wave reflections, the first minimum of [right vertical bar] Zⁱⁿ(omega) [left vertical bar] (F^min) was calculated at each intervention. Lastly, the arterial wave reflection factor (Delta Z/Z^c), defined as the ratio of the difference between [right vertical bar] Zⁱⁿ(omega) [left vertical bar] at F^minand the following maximum of [right vertical bar] Zⁱⁿ(omega) [left vertical bar] and Z^cwas also determined. Delta Z/Z^cis proportional to the magnitude of the reflected waves.

In two sets of experiments, dogs (n = 8; weight 26 plus/minus 1 kg, mean plus/minus SEM) were assigned to receive isoflurane or halothane in a random manner (coin toss) on separate days. Dogs were fasted overnight, and fluid deficits were replaced before experiments with crystalloid (500 ml 0.9% saline). Maintenance fluids were continued at 3 ml *symbol* kg sup -1 *symbol* h sup -1 for the duration of each experiment. After instruments were calibrated, baseline systemic hemodynamics were recorded in the conscious state. Continuous aortic blood pressure and aortic blood flow waveforms were recorded for later generation and analysis of Zⁱⁿ(omega). Dogs were then anesthetized by inhalation induction with either halothane or isoflurane in 100% oxygen. After tracheal intubation, anesthesia was maintained at 1.25, 1.5, or 1.75 minimum alveolar concentration (MAC) in a nitrogen (79%)-oxygen (21%) mixture. The order of MAC levels was assigned randomly (Latin square). End-tidal concentration of isoflurane and halothane were measured by mass spectrometry (Advantage 2000, Marquette Electronics, St. Louis, MO). The mass spectrometer was calibrated with known standards before and during experimentation. The canine MAC values for isoflurane and halothane used in this investigation were 1.28% and 0.86%, respectively. Systemic hemodynamics and aortic pressure and blood flow waveforms were recorded after 1 h equilibration at each end-tidal anesthetic concentration. Arterial blood gas tensions were maintained at conscious levels in all anesthetized dogs by adjustment of nitrogen and oxygen concentrations and respiratory rate throughout each experiment.

In a third set of experiments using an additional group of conscious dogs (n = 8; weight = 26 plus/minus 1 kg), continuous intravenous infusions of sodium nitroprusside were administered after baseline systemic hemodynamics and aortic pressure and blood flow waveforms had been recorded. The infusions of sodium nitroprusside (range 0.25-3 micro gram *symbol* kg sup -1 *symbol* min sup -1) were adjusted to decrease mean arterial pressure approximately 20, 30 and 50%, corresponding to decreases in mean arterial pressure observed during isoflurane and halothane anesthesia at 1.25, 1.5, and 1.75 MAC. After a 15-min equilibration period, cardiovascular parameters and aortic pressure and blood flow waveforms for subsequent Zⁱⁿ(omega) generation were recorded under steady state conditions at each infusion rate of sodium nitroprusside.

Statistical analysis of data within and between groups in the conscious state during anesthetic interventions and during sodium nitroprusside infusions was performed by multiple analysis of variance with repeated measures followed by applications of the students t test with Duncan's correction for multiplicity. Changes within and between groups were considered statistically significant when the P value was less than 0.05. All data were expressed as mean plus/minus SEM.

The effects of isoflurane and halothane on systemic hemodynamics and characteristics of arterial wave reflection are summarized in Table 1and Table 2, respectively. A small increase in arterial oxygen tension was observed in all anesthetized dogs during positive pressure ventilation. Isoflurane caused a significant (P < 0.05) increase in heart rate and dose-dependent decreases in systolic, diastolic, and mean arterial pressures, left ventricular systolic pressure, peak positive left ventricular dP/dt, %SS, and stroke volume (Table 1). Systemic vascular resistance also decreased during the administration of isoflurane, but this effect was not dose-related. A significant decrease in cardiac output occurred at 1.75 MAC. No change in left ventricular end-diastolic pressure was observed. Isoflurane caused a dose-related increase in C (0.55 plus/minus 0.02 during control to 0.73 plus/minus 0.06 ml *symbol* mmHg sup -1 during 1.75 MAC) and decrease in R (3,205 plus/minus 315 during control to 2,340 plus/minus 219 dyn *symbol* s *symbol* cm sup -5 during 1.75 MAC) (Figure 5). Isoflurane also increased Z^cat the highest dose. No changes in F^minor Delta Z/Z^coccurred during administration of isoflurane (Table 1), indicating that the distal site of net arterial wave reflection and the magnitude of the reflected waves were unaffected by this volatile anesthetic.

Halothane anesthesia produced hemodynamic alterations that were similar to those observed with isoflurane. Dose-related increases in heart rate and decreases in arterial pressures, left ventricular systolic pressure, dP/dt, %SS, and stroke volume occurred during administration of halothane (Table 2). In contrast to the findings with isoflurane, however, halothane caused dose-related decreases in cardiac output concomitant with maintenance of systemic vascular resistance at levels present in the conscious state. Halothane increased C (0.56 plus/minus 0.02 during control to 0.65 plus/minus 0.05 ml *symbol* mmHg sup -1 during 1.75 MAC), however, increases in C were not dose-related (Figure 5). No changes in R were observed during administration of halothane, in contrast to the observations during isoflurane anesthesia. Halothane also increased Z^cat 1.75 MAC. No changes in F^minand Delta Z/Z^coccurred with halothane, indicating that this anesthetic did not alter the pattern or magnitude of arterial wave reflections (Table 2).

Sodium nitroprusside caused dose-related increases in heart rate and decreases in systolic, diastolic, and mean arterial pressures, left ventricular systolic and end-diastolic pressures, and systemic vascular resistance (Table 3). No changes in dP/dt, %SS, or cardiac output occurred. Sodium nitroprusside caused dose-dependent increases in C (0.51 plus/minus 0.02 during control to 1.36 plus/minus 0.16 ml *symbol* mmHg sup -1 during the high dose) and decreases in R (3,275 plus/minus 417 during control to 1,812 plus/minus 252 dyn *symbol* s *symbol* cm sup -5 during the high dose) (Figure 5). In contrast to the findings during isoflurane and halothane anesthesia, no change in Z^cwas observed with the administration of sodium nitroprusside to conscious dogs. A significant decrease in F^minwas also observed with sodium nitroprusside (3.1 plus/minus 0.2 during control to 1.8 plus/minus 0.2 Hz at the high dose), indicating that this drug shifts the net arterial wave reflection site to a more distal location in the arterial vasculature (Table 3). However, no change in Delta Z/Z^cwere observed, indicating that the magnitude of reflected arterial waves was unaffected by sodium nitroprusside.

The ratio of mean arterial pressure to mean arterial blood flow (i.e., systemic vascular resistance) would completely describe the hydraulic resistance of the arterial vasculature opposing cardiac ejection if the heart was a constant flow pump or if the ventricle emptied into a rigid, nondistensible tube (assuming negligible inertia of the blood). Calculated systemic vascular resistance may provide qualitative information about left ventricular afterload in the intact cardiovascular system, but systemic vascular resistance ignores the frequency-dependent characteristics of the arterial system and the viscoelastic properties of the arterial walls. [27]A simple approximation of phasic hydraulic resistance obtained by dividing instantaneous aortic pressure by instantaneous blood flow is also strictly incorrect because a frequency-dependent phase difference (time lag) between pressure and flow waves occurs as well. Aortic input impedance (Zⁱⁿ) incorporates the viscoelastic and frequency-dependent properties of and the wave reflections that occur in the arterial vascular tree and has been shown to be a useful experimental tool for analysis of the mechanical function of the arterial system.

Zⁱⁿis determined by performing Fourier series or spectral analysis (using the fast Fourier transform or a periodogram technique) of aortic pressure and blood flow waveforms, transforming data from the time to frequency domain. [4,20]Zⁱⁿis the complex ratio of aortic pressure (the forces acting on the blood) to aortic blood flow (the resultant motion of the blood) [1,28]and is typically displayed by plotting the magnitude (modulus) and phase of Zⁱⁿas a function of frequency. These constitute the aortic input impedance spectrum. The Zⁱⁿ(omega) modulus describes the ratio of the magnitude of pressure to the magnitude of flow at each point in the frequency domain. The Zⁱⁿ(omega) phase is the difference between the phase angles of flow and pressure at each frequency. Phase angles are usually negative at low frequencies, indicating that blood flow precedes developed pressure under these conditions. Although it is clear that pathologic conditions or vasoactive drugs may alter Zⁱⁿ(omega) by affecting the mechanical properties of the arterial wall, [8]these changes in Zⁱⁿ(omega) are difficult to quantify. Thus, Zⁱⁿ(omega) is often interpreted using an analytical model known as the three-element Windkessel, an electrical analog of Zⁱⁿ(omega) that displays most of its general characteristics in the frequency domain. [5].

The Windkessel model describes three variables that are properties of the arterial system: Z^c, R, and C. Z^cis the input impedance of the aorta minus the effects of reflected waves from the distal arterial vasculature. [1]Although it is represented as a resistor in the model for simplicity, the value of Z^cis directly related to the Poiseullian resistance of the aorta and inversely related to compliance of this vessel. Net changes in Z^care determined by the viscoelastic properties of the aortic wall, aortic dimension, and intraaortic pressure. [1,29]R (systemic vascular resistance minus Z sub c) is generally an order of magnitude greater than Z^c, consistent with the concept of the aorta as a low-resistance, high-compliance conduit and the arterioles as primary resistance vessels. Because the radius of arterial vessels varies during the cardiac cycle, R and the resistive component of Z^ccan be considered mean hydraulic resistances.

The vast majority of C (> 90%) is determined by aortic compliance. [30,31]The viscoelastic characteristics of the proximal aorta allow this blood vessel to store part of the energy generated by the left ventricle during ejection and return it to the arterioles and capillaries during diastole. This rectifying quality of the aorta maintains diastolic pressure and decreases arterial pulse pressure for any given mean arterial pressure. The high compliance and large diameter of the proximal aorta are responsible for the low value of Z^cin the ascending aorta, properties that contribute to decreases in wasted left ventricular power. [8]In fact, diastolic arterial pressure would decrease to zero (and coronary perfusion would cease) if the aorta and peripheral arteries were rigid tubes. Age or disease states (e.g., essential hypertension or atherosclerosis) that decrease C result in lower mean diastolic pressure and increased pulse pressure. The value of C is nonlinearly related to intraarterial pressure and is determined primarily by the interrelation between collagen, elastin, and smooth muscle in the arterial wall. [8,24]Arterial distention resulting from increases in intraluminal pressure reduces C by shifting the load from the more compliant elastin fibers to the stiffer collagen fibers in the arterial wall. The Windkessel approximation of C represents a mean value for the entire arterial system because C has also been shown to vary during the cardiac cycle. [32].

In the current investigation, aortic input impedance spectra were quantified using the Windkessel parameters in the conscious state and during isoflurane and halothane anesthesia. Comparisons were made to results obtained during administration of sodium nitroprusside to decrease arterial pressure in the conscious state. The results demonstrated that isoflurane and sodium nitroprusside caused dose-related decreases in R, in contrast to the findings with halothane. Isoflurane-induced decreases in R were accompanied by concomitant decreases in calculated systemic vascular resistance as evaluated by the ratio of mean arterial pressure and mean arterial blood flow, confirming the findings of previous investigations in dogs [10-15]and humans. [16,17]Isoflurane and halothane caused equivalent increases in C, but the increase in C produced by the volatile anesthetics was considerably smaller than that caused by sodium nitroprusside. Isoflurane and halothane produced similar small increases in Z^cat 1.75 MAC, but equihypotensive doses of sodium nitroprusside did not change Z^c. These results indicate that the major difference in the actions of isoflurane and halothane on left ventricular afterload can be attributed primarily to effects on R, a property of the arterioles, and not to differential actions of these agents on the mechanical characteristics of the aorta. In contrast, sodium nitroprusside reduced R and caused pronounced increases in C, indicating that this vasodilator affects not only arteriolar tone but also aortic mechanical properties. The effects of sodium nitroprusside on aortic input impedance observed in the current investigation confirm and extend the findings of previous studies in animals and in humans. [24,28,29,33,34].

The small but significant increase in Z^cobserved with isoflurane and halothane at 1.75 MAC was probably related to anesthetic-induced decreases in intraaortic pressure. Decreases in aortic diameter associated with hypotension cause increases in the Pouiseullian resistance of the aorta, leading to increases in characteristic aortic impedance. In contrast, sodium nitroprusside did not alter Z^cat an equihypotensive dose. Under these conditions, the pronounced increase in C induced by sodium nitroprusside may have balanced increases in aortic resistance. The increase in C observed during the administration of isoflurane and halothane may have also occurred as a result of hypotension because of the inverse relation between compliance and pressure (Figure 6). However, other investigations have shown that C is independent of pressure over the range of pressures observed in this investigation. [35,36]Thus, a true anesthetic-induced increase in compliance remains a possibility. The volatile anesthetics examined in the current investigation produced very similar compliance-pressure relations; however, equihypotensive doses of sodium nitroprusside clearly increased the slope of this relation. This observation confirms the results of previous studies [26,32]and suggests that sodium nitroprusside may produce direct increases in C by affecting the mechanical compliance of the aorta, augmenting the relatively flat baseline compliance-pressure relation. [35].

The phenomenon of reflected waves in the arterial system was also evaluated using the aortic input impedance spectrum in the conscious and anesthetized states. When an arterial vessel branches, the characteristic impedance (input impedance minus the effects of wave reflection) of the proximal trunk may not be equivalent to the combined characteristic impedances of the distal branches. This hydraulic impedance mismatch causes some of the forward energy to be reflected back toward the heart. The reflected wave results in the nonisomorphic relation between arterial pressure and blood flow waveforms because the reflected wave adds to the pressure wave and subtracts from the blood flow wave. Wave reflections are manifested in aortic input impedance spectra as oscillations of Zⁱⁿ(omega) at higher frequencies. The magnitude of the oscillations of [left vertical bar] Zⁱⁿ(omega) [right vertical bar] is directly proportional to the magnitude of the wave reflections. [1]The frequency of F^mincorrelates with the distance to the major reflecting site. [1]The major reflecting site is not a true anatomic branching point but instead describes the average sum of all reflecting sites in relation to the aortic root. In the current investigation, no changes in the magnitude or net site of wave reflection were observed in the presence of isoflurane or halothane, suggesting that these volatile anesthetics have no effect on frequency-dependent, oscillatory arterial properties. In contrast, sodium nitroprusside-induced reductions in F^minindicate that this vasodilator shifts the net site of wave reflection to a more distal location in the arterial circulation.

The results of this investigation must be interpreted within the constraints of several potential limitations. Zⁱⁿ(omega) was calculated using arterial pressure waveforms measured with a fluid-filled catheter system implanted in the proximal descending aorta. A high fidelity micromanometer placed at the aortic root may have provided enhanced frequency response, however, the response of the fluid-filled system was determined using a de novo quick release ("pop") test and the magnitude and phase of Zⁱⁿ(omega) were appropriately corrected using established methods. [1]The distance between the pressure and blood flow transducers (approximately 4 cm) was also corrected by adjusting the magnitude and phase Zⁱⁿ(omega). [30]The primary error introduced by spacing between the transducers occurs in the phase of Zⁱⁿ(omega). In the current investigation, the Zⁱⁿ(omega) phase spectra recorded in the conscious state and during sodium nitroprusside infusions were comparable to previous reports [33,37]and also agreed qualitatively with the best-fit Windkessel frequency response (Figure 3). Furthermore, none of the afterload variables derived from the Windkessel model were determined from the Zⁱⁿ(omega) phase spectrum.

It may be observed from Figure 3that the frequency response of the Windkessel model (solid line) does not exactly match the measured aortic input impedance (open circles). The frequency response of the model (formula in Figure 1) has a magnitude equal to the sum of Z^cand R at zero frequency. The magnitude decays to Z^cat high frequencies. The rate of decay is determined by R and C. The phase of the frequency response of the model is zero at zero frequency. As frequency increases the phase reaches a minimum and then asymptotically approaches zero. The measured frequency response shows oscillations in the magnitude of the aortic input impedance spectra that are caused by reflected waves (Figure 3and Figure 4). The phase of the measured aortic input impedance may achieve positive phase angles because of the inertia of the blood. These aspects of the measured response cannot be accounted for by the Windkessel. The advantage of the three-element model is its parsimony, which allows the parameters of the model to be associated with real physical parameters (i.e., R, Zc, and C). [4].

Other models of the arterial system, including the transmission line (t-tube) [38-41]and other lumped parameter models, [42,43]have been proposed as alternatives to the three-element Windkessel for the quantitative interpretation of changes in left ventricular afterload assessed with Zⁱⁿ(omega). These models may describe the fine aspects of Zⁱⁿ(omega) in greater detail (particularly the wave reflection characteristics) but are considerably more complex to analyze and may be less physiologically intuitive than the Windkessel approach. In addition, evidence suggests that the details of Zⁱⁿ(omega) not depicted by the three-element Windkessel model probably do not have a significant impact on its overall interpretation in terms of afterload. [4]Thus, the three-element Windkessel model used in this investigation was an appropriate choice for analysis of the effects of volatile anesthetics and sodium nitroprusside on aortic input impedance.

The Zⁱⁿ(omega) modulus spectra obtained in anesthetized dogs were somewhat less continuous than those obtained in the conscious state (Figure 4) because some frequencies between the fundamental and corresponding harmonics were discarded on the basis of mean squared coherence criteria. This relative discontinuity in the Zⁱⁿ(omega) modulus spectra may have introduced an error in the calculation of Z^c, Delta Z/Z^c, and F^min. Generation of random heart rates by cardiac pacing during anesthesia would have provided a greater number of fundamental and harmonic frequencies, resulting in more continuous Zⁱⁿ(omega) modulus spectra in the presence of isoflurane and halothane. Nevertheless, Z^crepresents the arithmetic average of the Zⁱⁿ(omega) modulus between 2 and 15 Hz, and this average was probably not significantly affected by the exclusion of some points between the fundamental and harmonic frequencies because at least six harmonics were included in each spectrum. In addition, the calculation of Delta Z/Z^cand F^minwas probably accurate because the maximum amplitude of oscillations in Zⁱⁿ(omega) magnitude occurred around the single fundamental and corresponding harmonic frequencies [1]in the anesthetized state, locations in the spectrum where high coherence frequencies were grouped. Lastly, the mild spectral discontinuity observed between the fundamental and harmonic frequencies during anesthesia also resembled spectra generated with standard Fourier series analysis, an established method for evaluating aortic and pulmonary input impedance and wave reflection properties under a variety of physiologic conditions. [1,8].

In summary, the results of the current investigation demonstrated that the major difference between the effects of isoflurane and halothane on left ventricular afterload quantified using the Windkessel model of Zⁱⁿ(omega) was related to R, a property of arteriolar vessels, and not to characteristic aortic impedance or C, mechanical features of the aorta. No changes in arterial wave reflection patterns determined from Zⁱⁿ(omega) spectra occurred with isoflurane and halothane, also suggesting that these volatile anesthetics have no effect on frequency-dependent arterial properties. In contrast, sodium nitroprusside-induced reductions in R and marked enhancement of C indicate that this vasodilator affects not only arteriolar tone but also aortic mechanical characteristics.

The instruments used to measure aortic pressure and aortic blood flow in this investigation alter the pressure and flow waveforms used to calculate Zⁱⁿ(omega) and must be corrected in the frequency domain to account for their frequency response. The third-order Butterworth analog low-pass filter built in to the flow meter used to assess aortic blood flow has a linear-phase response [theta(omega)] that is a function of frequency: Equation 2where omega = angular frequency (omega = 2 pi f) and omega^c= the cutoff frequency. The slope of this function is equal to the time delay of the flow meter. The time delay equaled 11.2 ms with the cutoff frequency set a 30 Hz. Thus, the measurement of phase of Zⁱⁿ(theta^z(omega)) was corrected by: Equation 3where theta sub z (omega) *symbol* = the true phase of the flow signal. Because the magnitude of the frequency response of the flow meter is flat up to and beyond 15 Hz, no correction was required in the magnitude of Zⁱⁿ(omega).

The fluid-filled catheter-pressure transducer system used to measure aortic blood pressure behaves as an underdamped second-order system governed by the differential equation: Equation 4where M = the mass of the system; B = the viscous damping (resistance); and K = the stiffness. [8]The frequency response of the system is determined by the damping ratio (xi = B *symbol* [2 *symbol* (K *symbol* M)¹/2] sup -1) and the undamped natural frequency [omega^o= (K/M)¹/2]. Both of these variables were derived from an input step response function obtained by performing a quick release, or "pop," test. [1,8]The catheter-pressure transducer system was subjected to a constant hydrostatic pressure that was suddenly released, allowing the measured pressure to decrease rapidly to and oscillate around 0 mmHg (Figure 7). The magnitude of the first and second oscillations around 0 mmHg (p¹and p², respectively) were used to determine the logarithmic decrement (Lambda): Equation 5. The damping ratio (xi) was calculated from the "pop" test data using the equation: Equation 6. The undamped natural frequency (omega^o) was also determined empirically using the equation: Equation 7where T = the time between p¹and p². The pressure transducer system used in this investigation had a value of xi = 0.159 and omega^o= 94.9 rad *symbol* s sup -1 (15.1 Hz). The magnitude of the Zⁱⁿ(omega) spectrum was corrected (to [left vertical bar] Zⁱⁿ(omega) [right vertical bar]) using the expression: Equation 8where [left vertical bar] Zⁱⁿ(omega) [right vertical bar] = the measured magnitude of Zⁱⁿ(omega) and gamma = the relative frequency (omega/omega^o). [1]The phase of Zⁱⁿ(omega) (theta sub z (omega)) was also corrected (to theta^z(omega)*) using the formula: Equation 9.

Under ideal circumstances, Zⁱⁿ(omega) spectra should be calculated from aortic pressure and blood flow waveforms obtained from identical locations at the aortic root. In the current investigation, aortic blood flow was measured in the ascending aorta and aortic blood pressure was measured in the proximal descending thoracic aorta just distal to the aortic arch. The difference between these measurement sites was approximately 4 cm in all dogs. Over this distance, the magnitude of the pressure signal in the frequency domain may be altered slightly, but no significant changes in the magnitude of Zⁱⁿ(omega) would be expected. [23]An error in the phase of Zⁱⁿ(omega) between the flow meter and the pressure transducer must be corrected, however. This phase delay (theta^d(omega)) was corrected as a function of frequency: Equation 10where x = the distance between measuring devices and C^app(omega) = the apparent wave velocity as a function of frequency. The value of C^app(omega) is typically quite large at low frequencies (< 3 Hz) and remains fairly constant [C^app(omega) [nearly equal] 500 cm *symbol* s sup -1] at higher frequencies (> 3 Hz). [23]Within these constraints, the phase error is negligible at low frequencies and is directly proportional to frequency at higher frequencies: theta^d(omega) [nearly equal] 0.462 *symbol* omega. Thus, the phase of Zⁱⁿ(omega) (theta^z(omega)) was corrected (to theta^z(omega)*) such that: Equation 11theta^z(omega) phase calculations were included to insure completeness of the data, however, no Windkessel parameters were derived using theta^z(omega) measurements.

The corrections for the frequency response of the transducers as well as the distance between transducers were generalized for all experiments. The exact frequency response of the pressure transducer will vary from day to day and from dog to dog because of microbubbles, catheter bending, or slight differences in catheter length. The exact distance between the pressure and flow transducer also will vary slightly. However, these differences would be small and impossible to measure in the conscious animal. In addition, they most likely would have little effect on the applied spectral corrections.

The authors extend their gratitude to Dr. James Ackmann, Department of Biomedical Engineering, Marquette University, for his helpful comments and suggestions. The authors also thank John Tessmer and Dave Schwabe for technical assistance and Angela Barnes for preparation of the manuscript.

* Guiding Principles in the Care and Use of Animals. Bethesda, The American Physiological Society, revised 1991.

** Guide for the Care and Use of Laboratory Animals. Publication NIH 85-23. Bethesda, National Institutes of Health, Department of Health and Human Services, revised 1985.