## Abstract

Although the hydraulic work generated by the left ventricle (LV) is not disputed, how the work was dissipated through the systemic circulation is still subject to interpretation. Recently, we proposed that the systemic circulation should be considered as waves and a reservoir system (Wk). By combining the arterial and venous reservoirs, the systemic vascular resistance can be viewed as a series of resistors, which in sequence are the large-artery resistance, arterial reservoir resistance, the microcirculatory resistance, venous reservoir resistance, and large-vein resistance, and propelling blood through these resistance elements represents resistive losses. We then studied the changes in the fraction of the work consumed by each element when infusing methoxamine (MTX), a vasoconstrictor, and sodium nitroprusside (NP), a vasodilator. Results show that, under control condition, ∼50% of the LV stroke work was dissipated through arterial reservoir resistance (NP, ∼36%; MTX, ∼27%), another ∼25% was dissipated by the microcirculation (NP, ∼20%; MTX, ∼66%), and ∼20% of work by the large-artery resistances (NP, ∼37%; MTX, ∼6%). The energy dissipated by the venous resistances was small and had limited variation with NP and MTX, where the large-vein and venous reservoir resistances shared ∼1 and ∼3% of LV stroke work, respectively. Approximately 60% of LV stroke work is stored as the potential energy during systole under control, and the ratio decreases to ∼45% with NP and ∼80% with MTX.

- left ventricle circulatory coupling
- systemic vascular resistance
- arterial and venous reservoirs

the hydraulic work generated by the left ventricle (LV) pushes blood into the systemic circulation to provide the metabolic requirements of the organs. Although the amount of LV hydraulic work (i.e., the area of the pressure-volume loop) is not disputed, how this work is dissipated in the systemic circulation is still subject to interpretation. Thus far, the analysis of arterial hemodynamics has been dominated by the Fourier-based impedance method, in which aortic pressure and flow have been treated as the sum of sinusoidal wave trains oscillating about mean values; this construct led to the separation of LV hydraulic work into oscillatory work and work related to mean pressure and flow. However, it is difficult to directly relate either to the energy dissipated by the individual components of the serial resistive network.

We recently proposed a new approach to the systemic circulation, using the principle of Otto Frank's windkessel (8, 9), in which arterial reservoir pressure (P_{A-Res}) is proportional to the instantaneous blood volume. Viewed in the time domain, the arterial reservoir is a hydraulic integrator; the reservoir is charged when inflow exceeds outflow (during systole) and vice versa during diastole (22). The measured central aortic pressure (P_{Ao}) is equal to the sum of the P_{A-Res} and the pressure due to arterial wave motion (wave pressure; P_{A-Wave}). Our best analogy for this mechanism is that of a canal lock. The water level, which corresponds to P_{A-Res}, can increase or decrease depending on the balance of inflow and outflow, and, at any level, any perturbation such as throwing a stone into the lock will produce a propagating wave. Recently, we also showed that the venous system behaves simply as an inverse of the arterial system (21): in diastole, the arterial reservoir discharges, and the venous reservoir charges. To view the systemic circulation as a whole, we observed that, during diastole, P_{Ao} and P_{A-Res} decrease exponentially and asymptotically approach a nonzero level (P_{A-∞}), and inferior vena caval pressure (P_{IVC}) and venous reservoir pressure (P_{V-Res}) rise exponentially to approach P_{V-∞}. Generally, P_{V-∞} is different from P_{A-∞}.

By considering the arterial and venous wave-reservoir models together, systemic vascular resistance (SVR) can be viewed as a network of resistances arranged in series (21). In the absence of reflections, P_{A-Wave} is precisely proportional to the measured aortic flow (Q_{Ao}), and the ratio, P_{A-Wave} to Q_{Ao}, is numerically equal to characteristic impedance. We interpret P_{A-Wave}-to-Q_{Ao} ratio to be a measure of large-artery resistance (*R*_{Lrg-A}), the resistance separating the LV from the arterial reservoir. The arterial reservoir resistance (*R*_{A-Res}) is the resistance regulating the flow out of the reservoir and is equal to τ_{A}/C_{A}, where τ_{A} is the exponential time constant of the decline of diastolic P_{Ao}, and C_{A} is the arterial compliance. The large-vein resistance (*R*_{Lrg-V}) is the resistance separating the right atrium from the venous reservoir and is equal to the ratio of P_{V-Wave} to Q_{IVC,} where P_{V-Wave} is the venous wave pressure and Q_{IVC} is the inferior venal caval flow. The venous reservoir resistance (*R*_{V-Res}) is the resistance regulating the flow into the venous reservoir and is equal to τ_{V}/C_{V}, where τ_{V} is the exponential time constant of the rise of diastolic P_{IVC} and C_{V} is the venous compliance. Finally, we consider that the resistance defined by the difference between P_{A-∞} and P_{V-∞} divided by the mean flow [i.e., cardiac output (CO)] is the microcirculation resistance (*R*_{μcirc}). *R*_{μcirc} is also equal to SVR − ∑R_{i}, where ∑R_{i} is the summation of the above identified resistances. These incremental resistances correspond very well (21) to measurements of pressure drop across the vascular elements in the hamster cheek pouch (5). From that comparison, we suggested that *R*_{A-Res} involves arterial vessels as small as 60 μm and that *R*_{V-Res} also involves venous vessels as small as 60 μm.

Propelling blood through these resistance elements represents resistive losses, which can be quantified through the integration of the flow across a resistive element over time. In the steady state, LV stroke work equals the sum of the energy dissipated by all the resistive elements. Not all the LV stroke work, which is generated during systole, is dissipated during that interval; part of it is dissipated to produce systolic flow, but the remainder is stored as potential energy by the compliant arterial reservoir to be dissipated during diastole in pushing blood through the resistive components.

Differently from the impedance point of view in which LV stroke work is separated into mean and oscillatory components, we here demonstrate how LV stroke work can be separated into energy-dissipation components corresponding to each serial resistive element. Furthermore, we demonstrate how the distribution of this energy dissipation varies with the administration of a vasodilator, sodium nitroprusside (NP), and a vasoconstrictor, methoxamine (MTX).

## MATERIALS AND METHODS

### Theory

#### Determination of arterial and venous properties.

The experimental protocol was approved by the Institutional Animal Care Committee of the University of Calgary Faculty of Medicine and met the standards of the American Physiological Society. The values of resistive components were by-products of the calculations of the P_{A-Res} and P_{V-Res}. The methods of calculation were detailed elsewhere (21, 22). In brief, the arterial asymptotic pressure, P_{A-∞}, is initially determined by fitting P_{Ao} during the later part of diastole by using a three-parameter exponential-decay equation. P_{A-∞}, *R*_{A-Res}, and arterial reservoir compliance (C_{A-Res}) were determined by using a nonlinear search algorithm with appropriate initial guesses to minimize the mean-squared error between the calculated P_{A-Res} and the measured P_{Ao} during late diastole, during which wave action was assumed to be negligible. The logic of the venous reservoir is the inverse of the arterial reservoir. In late diastole, both P_{IVC} and P_{V-Res} rise exponentially, approaching P_{V-∞}, which had been determined by fitting a three-parameter exponential-rise equation to P_{IVC} in late diastole. P_{V-∞}, *R*_{V-Res}, and venous reservoir compliance (C_{V-Res}) were similarly determined by minimizing the mean-squared error between the calculated P_{V-Res} and the measured P_{IVC}.

#### Separation of SVR into serial resistive components.

The separation of SVR into its serial resistive components is shown in Fig. 1 (*t* = 0 corresponded to the onset of the atrial contraction for all figures). SVR was calculated as the pressure gradient between P̅_{A̅o̅} and P̅_{I̅V̅C̅} divided by Q̅_{A̅o̅}, where the overbar signifies the mean value. On the basis of P_{A-∞} and P_{V-∞}, SVR was separated into the arterial resistance (between P̅_{A̅o̅} and P_{A-∞}, red hatching), the venous resistance (between P_{V-∞} and P̅_{I̅V̅C̅},̅ blue hatching), and the *R*_{μcirc} (between P_{A-∞} and P_{V-∞}, purple crosshatching). The arterial resistance was further separated into the *R*_{Lrg-A} (i.e., the gradient between P̅_{A̅o̅} and P̅_{A̅-̅R̅e̅s̅}, P̅_{A̅-̅W̅a̅v̅e̅}, divided by Q̅_{A̅o̅}) and the *R*_{A-Res} (i.e., the gradient between P̅_{A̅-̅R̅e̅s̅} and P_{A-∞} divided by Q̅_{A̅o̅}). The venous resistance can also be further separated into the *R*_{Lrg-V} (i.e., the gradient between P̅_{V̅-̅R̅e̅s̅} and P̅_{V̅I̅C̅}, P̅_{V̅-̅W̅a̅v̅e̅}, divided by Q̅_{A̅o̅}) and the *R*_{V-Res} (i.e., the gradient between P_{V-∞} and P̅_{V̅-̅R̅e̅s̅} divided by Q̅_{A̅o̅}).

#### Energy dissipation by serial resistive components.

LV stroke work was calculated as the integral of (P_{Ao} − P_{IVC}) × Q_{Ao} during a cycle. The energy dissipated by individual serial resistive components was calculated as the integral of the pressure gradient across the given resistance multiplied by the flow (i.e., ΔP × Q) or, equivalently, the integral of Q^{2} × *R*.

The energy dissipated by the proximal *R*_{Lrg-A} is (1) and by the *R*_{A-Res} is (2) where Q_{A-Res} is the outflow from the arterial reservoir, calculated by dividing the gradient between the P_{A-Res} and P_{A-∞} by the arterial reservoir resistance, *R*_{A-Res}. *t*_{1} and *t*_{2} are the instants at which the aortic valve opens and closes, respectively.

Since the flow in microcirculation is generally steady, the energy dissipated during systole can be calculated as (3)

The energy dissipation by the *R*_{V-Res} and the *R*_{Lrg-V} is calculated as (4) and (5), respectively. Q_{V-Res} is the inflow of the venous reservoir, which is driven by the pressure gradient between P_{V-∞} and P_{V-Res} across *R*_{V-Res}. *t*_{3} and *t*_{4} are instants at which the right atrium begins to contract during sequential beats. Because we did not measure flow in the superior vena cava, Q_{IVC} was scaled so that its integrated volume over a cycle equaled that of the stroke volume.

All the LV stroke work is injected into the systemic circulation during systole, but only part of it is dissipated by resistive components during that interval; the remaining energy is stored as potential energy via the compliance of the arterial reservoir. The energy dissipated during systole can be calculated as the sum of *Eq. 1* to *Eq. 5* during systole (each equation integrated between *t*_{1} and the time of the minimum of the subsequent incisura). The energy stored as the potential energy at end systole is the difference between LV stroke work and the systolic energy dissipation. In the steady state, the potential energy remaining at end systole equals the energy dissipated during diastole.

#### Experimental preparation and protocol.

Studies were performed on eight healthy mongrel dogs weighing between 18 and 29 kg. They were anesthetized with thiopental sodium (20 mg/kg), followed by fentanyl citrate (30 μg·kg^{−1}·h^{−1}), and ventilated with a 1:1 nitrous oxide-oxygen mixture. The rate of a constant-volume respirator (model no. 607; Harvard Apparatus, Natick, MA) (tidal volume = 15 ml/kg) was adjusted to maintain normal blood gas tensions and pH. Body temperature was maintained at 37°C with a circulating-water warming blanket and a heating lamp. A lactated Ringer solution was infused through the jugular vein to manipulate LV end-diastolic pressure (P_{LVED}).

We measured pressure in the LV and right ventricle, and simultaneous pressure and flow were measured at the aortic root and IVC (within 2 cm of the right atrium). Pressures were measured with high-fidelity catheter-tip manometers (Millar Instruments, Houston, TX) and flows with ultrasonic flow probes (Transonic Systems, Ithaca, NY). The LV pressure catheter was introduced through the apex. Aortic root pressure was measured by introducing the catheter through the femoral artery and advancing it, retrogradely, into the root of the aorta, ∼1.5 cm from the valve. The IVC pressure catheter was inserted through the right jugular vein and was positioned in the IVC, ∼1 cm beyond the right atrium. The flow probes were located within 1 cm of the pressure transducers. After control recordings, NP (0.3 mg/min iv) was administered. After a 20-min period of recovery, MTX (5 mg iv bolus) was administered.

## RESULTS

Table 1 shows mean values of heart rate, CO, mean arterial and IVC pressures, and parameters (*R*, *C*, and P_{-∞}) for the arterial and venous reservoirs for the three conditions: NP administration, the control state, and MTX administration.

The constant determined to equalize IVC and aortic flow was 1.5 ± 0.1 (SE) among the different experiments.

In Fig. 2, *top*, the magnitude of SVR and all resistive components are presented for the control state and the administration of NP and MTX; the resistive components (*bottom*) are presented as fractions of SVR for each condition. These data are summarized in Table 2. Under control conditions, SVR was ∼60 mmHg·min·l^{−1}, which decreased to half (∼30 mmHg·min·l^{−1}) with NP and increased fivefold (∼300 mmHg·min·l^{−1}) with MTX. With respect to the response of each resistive component to the administration of NP and MTX, the arterial reservoir and the *R*_{μcirc} accounted for most of the changes in SVR; the *R*_{V-Res} and *R*_{Lrg-V} were relatively insensitive to these manipulations. Although small in absolute magnitude, *R*_{Lrg-A} increased with NP and decreased with MTX, expressed as a percentage of SVR.

In Fig. 3, power (*top*) and work (*bottom*) generated by the LV and dissipated by each serial resistive component are presented as functions of time during the cardiac cycle (data from Fig. 1, the control state). The LV-generated stroke power and stroke work (black curves), i.e., the inputs to the systemic circulation, are limited by the duration of systolic ejection. LV stroke work reaches its maximum when aortic flow stops; the vertical dashed line indicates the instant of maximum backflow. At the end of a cycle, ∼50% of the LV stroke work was dissipated through *R*_{A-Res}; another ∼25 and ∼20% of work were dissipated by the *R*_{μcirc} and *R*_{Lrg-A}, respectively. The energy dissipated by the venous resistances was small, where the *R*_{Lrg-V} and *R*_{V-Res} shared ∼1 and ∼3% of LV stroke work, respectively (see Table 3, control).

Figure 4 shows an example of power (*middle*) and work (*bottom*), generated by the LV and dissipated by each resistive component, as functions of time, and varied with NP and MTX (data from a different dog). (See Table 3 for the statistical work data from all 7 dogs.) With NP, the LV stroke work decreased and the fraction of energy dissipated by the *R*_{Lrg-A} and *R*_{A-Res} was comparable (∼35+%) and approximately twice the value of the *R*_{μcirc}; with MTX, the LV stroke work also diminished, the largest portion (∼65%) of which was dissipated by the *R*_{μcirc}. Figure 5 illustrates the absolute and relative values of LV stroke work and segmental energy dissipation under control conditions and with NP and MTX.

Figure 6 shows the potential energy stored in the reservoir, expressed as a percentage of LV stroke work, under control conditions and with the administration of NP and MTX (also see Fig. 4). Under control conditions, ∼60% of stroke work remained stored at the end of systole, to be discharged during diastole. This ratio decreased to ∼45% with the infusion of NP but increased to ∼80% with a bolus of MTX.

## DISCUSSION

For decades, arterial vasoconstriction and vasodilatation have been discussed in terms of increases and decreases in SVR, the sum of all the serial resistive components, from large arteries to large veins including the arterioles, the microcirculation, and the venules (2). Furthermore, to a great extent, it has been taught that changes in SVR are practically equivalent to changes in arteriolar resistance. Little was known with respect to the responses of other serial resistive components to the vasoconstrictors and vasodilators, and, as far as we are aware, no one has attempted to evaluate the energy dissipation of these individual components. The magnitude of individual resistive components has been evaluated by using micropipette techniques (5), but these techniques are difficult and often impractical.

In this study, we have proposed an operational method to break up SVR into its serial resistive components, on the basis of our understanding that the systemic circulation functions as waves propagating on time-varying arterial and venous reservoirs (21, 22). The measured arterial and venous pressures are respectively separated into a wave-related pressure and a reservoir pressure, both of which vary with respect to time during the cardiac cycle; each resistance is proportional to the pressure gradient across it. As shown elsewhere (21), by determining the two asymptotic pressures, P_{A-∞} and P_{V-∞}, SVR is divided into arterial, microcirculatory, and venous resistances (see Fig. 1). Furthermore, the arterial component can be divided into *R*_{Lrg-A} and *R*_{A-Res}, and, similarly, the venous component can be divided into *R*_{Lrg-V} and *R*_{V-Res}. The *R*_{μcirc} is equal to the difference between the two asymptotic pressures, divided by the mean flow. Therefore, SVR is considered as the serial summation of the *R*_{Lrg-A}, the *R*_{A-Res}, the *R*_{μcirc}, the *R*_{V-Res}, and the *R*_{Lrg-V}.

Although calculation of LV stroke work is straightforward, how this energy was dissipated by the systemic circulation was poorly understood. For decades, the interpretation of the systemic circulation has been dominated by Fourier-based methods, where the measured aortic pressures and flows were separated into a mean value and oscillatory components about it. The aortic pressure-flow relationship was presented via harmonics, where the modulus at 0 Hz was designated as the SVR and the average of the moduli of the high-frequency harmonics was defined as the characteristic impedance. Thus the energy dissipated by the systemic circulation was separated dichotomously into mean (which was a function of SVR) and oscillatory (which was a function of the summation of each harmonic term) components (12). So far, no attempt has been made to separate the energy dissipation with respect to the corresponding serial vascular structures.

Among the five different serial components of SVR identified in this study, the combination of the *R*_{A-Res} and the *R*_{μcirc} dominated the SVR under all conditions (control state, ∼90%; NP, ∼85%; and MTX, ∼95%; see Fig. 2, *bottom*, and Table 2). With the infusion of NP, CO increased ∼10% (from 1.3 to 1.4 l/min), and SVR decreased by ∼45% (from 62 to 34 mmHg·min·l^{−1}), relative to the control values. Most of the decrease in SVR was due to decreases in *R*_{A-Res} (by ∼50%, from 35 to 17 mmHg·min·l^{−1}) and the *R*_{μcirc} (by ∼45%, from 22 to 12 mmHg·min·l^{−1}). With a bolus of MTX, CO decreased by ∼80% (from 1.3 to 0.3 l/min) and SVR increased fivefold (from 62 to 307 mmHg·min·l^{−1}). Most of the increase in SVR was due to the increase in the *R*_{A-Res} (by 3.5-fold, from 35 to 123 mmHg·min·l^{−1}) and especially *R*_{μcirc} (by approximately eightfold, from 22 to 173 mmHg·min·l^{−1}). As mentioned in the introduction, by comparing our normal data to those of Davis et al. (5), who measured the distribution of the pressure drop in the vasculature of the hamster cheek pouch, the arterial reservoir would seem to involve vessels as small as 60 μm. We speculate that MTX constricts arterioles, increasing *R*_{μcirc} by making that component longer [i.e., as terminal arterioles become less than 60 μm in diameter, they functionally become part of the microcirculation, according to our comparison (21) with the data of Davis et al. (5)] and increasing *R*_{A-Res} by reducing arteriolar caliber (i.e., larger arterioles also constrict but remain >60 μm in diameter).

As shown in Table 1, the relative compliance ratios (i.e., C_{V-Res} to C_{A-Res}) were ∼22 for control conditions and for MTX administration. These values are consistent with previous estimates (11, 16, 17) and support the validity of our reservoir analysis (20). The value of the ratio for NP administration (i.e., ∼12) was less, perhaps because of an increase in C_{A-Res} due to the reduction in mean P_{Ao}.

As illustrated in Fig. 4 and Fig. 5, *top*, both the administration of NP and MTX decreased LV stroke work, relative to the control value. With NP infusion, this implies that LV systolic pressure decreased even more than stroke volume increased. With a bolus of MTX, this implies that stroke volume decreased even more than LV systolic pressure increased. The energy dissipated as blood flows through each serial resistance can be calculated by integrating the dissipation of power (see Fig. 3), which is the pressure difference across the resistance multiplied by flow or, equivalently, the square of flow multiplied by the resistance (Q^{2} × *R*). As shown in Table 3, under control conditions, the energy dissipated by the *R*_{Lrg-A} (60 mJ) is comparable with that of the *R*_{μcirc} (76 mJ) and is half the energy dissipated by the *R*_{A-Res} (144 mJ), although the magnitude of the *R*_{Lrg-A} is only ∼10% of the *R*_{μcirc} and ∼5% of the *R*_{A-Res} (see Table 2). With NP, the LV stroke work decreased ∼25% relative to the control value (from 298 to 221 mJ). Dissipation by the *R*_{Lrg-A} increased by ∼50% (from 60 to 88 mJ) and became comparable with the dissipation by the *R*_{A-Res} (82 mJ, down from 144 mJ, a 43% decrease) and approximately twice the dissipation by the *R*_{μcirc} (38 mJ, down from 76 mJ, a 50% decrease). Thus, with vasodilation by NP, the absolute and relative energy losses due to dissipation by the *R*_{Lrg-A} became substantial and perhaps physiologically important.

Although the concept of an arterial reservoir is not new, no one had previously demonstrated quantitatively how the energy produced by LV systolic ejection is stored as potential energy and subsequently dissipated during diastole. Our rationale to determine the potential energy is simple. In the steady state, the energy dissipated by all the resistive components (pink solid line, Figs. 3 and 4) must equal the LV stroke work (black line) at the end of the cycle. We define potential energy as the undissipated energy remaining at the end of systolic ejection, the vertical difference between the black line and the pink line, and define the arterial reservoir function as the percentage fraction of total stroke work that remains as potential energy at end systole. As shown in Fig. 6, arterial reservoir function decreases with the infusion of NP (40%, down from 60%) and increases with MTX (∼80%).

Based on Frank's concept of the windkessel (reservoir), we calculated the pressure (P_{A-Res}) that would prevail in the reservoir, assuming that the change in pressure was equal to the change in volume, divided by the compliance (21). Having thus calculated P_{A-Res}, we subtracted it from measured P_{Ao} and found that the difference (previously called excess pressure; now called wave-related pressure, P_{A-Wave}) was proportional to aortic inflow (Q_{Ao}). In the anesthetized dog, P_{A-Wave} was shown to be entirely due to forward-traveling waves generated by the LV (21) and due to the combination of forward- and backward-traveling waves, when such were generated by a counterpulsation balloon (21). Therefore, from these and other observations (unpublished data), we have concluded that P_{Ao} = P_{A-Res} + P_{A-Wave}, where P_{A-Wave} equals the pressure drop across *R*_{Lrg-A}, which is numerically equal to characteristic impedance (22).

Of course, this interpretation of characteristic impedance differs from the conventional one. Westerhof et al. (24) supplied the classical electrical circuit diagram of the three-element windkessel (a.k.a. the “westkessel”). Although they and others represent characteristic impedance as a resistance and acknowledge that impedance has units of resistance, they do not consider it to be truly a hydrodynamic resistance, i.e., that it dissipates energy. [In contrast, also see Fogliardi et al. (7), where they do equate characteristic impedance with a true resistance.] Earlier our laboratory (22) showed that characteristic impedance was numerically identical to the linear slope of the relation between wave-related (excess) pressure and flow, thus demonstrating fundamentally resistive behavior. In the present article, we demonstrate that *R*_{Lrg-A} (i.e., characteristic impedance) and its resultant energy dissipation are measurable and can increase substantially with NP administration.

Some people have argued that the “line loss” of the aorta is negligible. To be sure, careful measurements of the decrease in mean aortic pressure along its length show a decrease of only a few millimeters of mercury. However, we suggest that this number is more a function of mean flow than of peak ejection rate. Preliminary efforts to measure the pressure drop associated with water flowing at peak ejection rates through aorta-sized plastic tubes have indicated pressure differences of the same order of magnitude as predicted [i.e., ∼25 mmHg at 15 l/min; see Fig. 2 in Wang et al. (22)].

#### Ohmic or “waterfall” behavior?

We have analyzed the systemic circulation, assuming that it can be considered to be a series network of resistors, and we have defined that network by determining arterial and venous asymptotic pressures, which imply waterfall phenomena. Ohmic behavior is not obviously consistent with waterfall behavior. Although the resolution of this question is not clear, the systemic circulation does behave as if it is ohmic, and it also behaves as if there is a waterfall phenomenon. Our best evidence for ohmic behavior is how well our model predicts the pressure drops across serial elements in the systemic circulation. As shown in Fig. 7 in Wang et al. (21), the pressure drops across the serial elements in our model agree very well with the direct observations of Davis et al. (5), who made micropuncture measurements of pressure in the hamster cheek pouch. With respect to waterfall behavior, we are not the first to make this claim. Magder and his associates (18, 19) have described Starling-resistor behavior in skeletal muscle, and investigators (6, 16, 17) who tried to estimate mean circulatory filling pressure (MCFP) arrested the heart briefly (∼15 s) and observed an asymptotic decrease of arterial pressure and an asymptotic increase in venous pressure.

When we arrest the heart for a long time (∼45 s), we observe that pressure declines toward our estimated value of P_{A-∞} (normally 35–45 mmHg in our anesthetized dogs) for the first ∼4 s. The pressure then begins to decline faster and appears to approach a lower value (20–25 mmHg), as observed by the MCFP investigators mentioned above. These observations appear to be consistent with critical-closing pressure (4) and myogenic-reflex (1, 10, 15, 23) behavior. We speculate that a normal myogenic tone defines a critical closing pressure in the range of 35–45 mmHg, but, that (21) after several seconds of tissue ischemia, myogenic tone decreases such that critical closing pressure is reduced to 20–25 mmHg.

#### Effects of the arterial reservoir on the impedance spectra.

The relative importance of the arterial reservoir (i.e., windkessel) is greatest with MTX and least with NP (see Fig. 7; also note τ_{A} in Table 1). The top two rows show the pressure and flow rate in the ascending aorta. In the first row, P_{Ao} is shown in black, calculated P_{A-Res} in green, and calculated P_{A-∞} as a dashed horizontal line. Note that P_{A-Res} is minimal with NP but is almost identical to P_{Ao} with MTX. In the second row, Q_{Ao} is shown in black and calculated P_{A-Wave} in red. Note that P_{A-Wave} is scaled so that its peak value corresponds to the peak value of Q_{Ao}, thus emphasizing the similarity in their waveforms.

The next two rows show the impedance spectra, modulus and phase, calculated for the different pressures and flows. Impedance is calculated from measured values, P_{Ao}/Q_{Ao} (black), reservoir pressure, P_{A-Res}/Q_{Ao} (green), and wave-related pressure, P_{A-Wave}/Q_{Ao} (red). With NP, P_{A-Res} is very small, and so the impedance based on P_{A-Wave} is very similar to the impedance based on P_{Ao}. Also note that the modulus of the impedance is effectively constant at all frequencies and equal to the characteristic impedance. This follows from the relatively very small value of P_{A-Res} and from the close correspondence between P_{Ao} and P_{A-Wave} with NP. With MTX, the modulus of impedance on the basis of P_{Ao} shows the more common behavior of increasing substantially as the frequency decreases toward the fundamental frequency. Note, however, that the modulus based on P_{A-Res} is very similar to the measured modulus, and the modulus based on P_{A-Wave}, which is the difference since P_{Ao} = P_{A-Res} + P_{A-Wave}, is relatively constant for all frequencies and again equal to the characteristic impedance.

The fifth (Fig. 7, *bottom*) row shows measured Q_{Ao} plotted against measured P_{Ao}. With NP there is a very high correlation between P_{Ao} and Q_{Ao} (*r*^{2} = 0.97), indicating that almost all the variance in Q_{Ao} is explained by P_{Ao}. Since P_{Ao} = P_{A-Res} + P_{A-Wave} and we have seen that P_{A-Wave} ∝ Q_{Ao}, we conclude for NP that P_{Ao} is approximately equal to P_{A-Wave} and the effect of P_{A-Res} is negligible. In contrast, with MTX, the correlation is very poor (*r*^{2} = 0.05), indicating that virtually none of the variance in Q_{Ao} is explained by P_{Ao}. Thus we conclude that P_{Ao} is approximately equal to P_{A-Res} and that the effect of P_{A-Wave} is very small.

These observations suggest that the commonly seen increase in the modulus of measured impedance as the frequency decreases to the fundamental frequency (the input impedance) is, in fact, a result of treating the reservoir pressure as part of the aortic wave instead of a separate time-varying component of pressure induced by the windkessel mechanism. This observation is consistent in all of the analyses that we have performed but needs to be tested and explored more fully because of its implications in the interpretation of impedance spectra.

With the administration of NP, the nearly linear relationship between P_{Ao} and Q_{Ao} has been interpreted as evidence of improvement in ventriculo-arterial coupling (i.e., that reflected waves had become negligible) (3). However, our analysis suggests, instead, that the nearly linear relationship between P_{Ao} and Q_{Ao} is due to the reservoir effect having become negligible.

#### Possible clinical implications.

With respect to the treatment of hypertension, an important clinical implication of our approach pertains to characteristic impedance (i.e., *R*_{Lrg-A}). There is broad agreement that characteristic impedance is elevated in the presence of systolic hypertension (13, 14). If characteristic impedance is elevated, more wave-related pressure must be generated to accelerate the stroke volume, thus increasing the load on the LV and leading to systolic hypertension. This understanding might possibly lead to the development of novel and more useful therapeutic approaches.

## GRANTS

The study was supported in part by the Canadian Institutes for Health Research (Ottawa) sequential-operating Grants MOP-57801 and MOP-67223 (to J. V. Tyberg, N. G. Shrive, and I. Belenkie).

## Acknowledgments

We acknowledge with gratitude the incisive hemodynamic insights provided by Dr. I. Belenkie and the outstanding technical skill of Cheryl Meek.

## Footnotes

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.

- Copyright © 2008 by the American Physiological Society