## Abstract

Wave intensity analysis provides detailed insights into factors influencing hemodynamics. However, wave intensity is not a conserved quantity, so it is sensitive to diameter variations and is not distributed among branches of a junction. Moreover, the fundamental relation between waves and hydraulic power is unclear. We, therefore, propose an alternative to wave intensity called “wave power,” calculated via incremental changes in pressure and flow (dPdQ) and a novel time-domain separation of hydraulic pressure power and kinetic power into forward and backward wave-related components (Π_{P±} and Π_{Q±}). Wave power has several useful properties: *1*) it is obtained directly from flow measurements, without requiring further calculation of velocity; *2*) it is a quasi-conserved quantity that may be used to study the relative distribution of waves at junctions; and *3*) it has the units of power (Watts). We also uncover a simple relationship between wave power and changes in Π_{P±} and show that wave reflection reduces transmitted power. Absolute values of Π_{P±} represent wave potential, a recently introduced concept that unifies steady and pulsatile aspects of hemodynamics. We show that wave potential represents the hydraulic energy potential stored in a compliant pressurized vessel, with spatial gradients producing waves that transfer this energy. These techniques and principles are verified numerically and also experimentally with pressure/flow measurements in all branches of a central bifurcation in sheep, under a wide range of hemodynamic conditions. The proposed “wave power analysis,” encompassing wave power, wave potential, and wave separation of hydraulic power provides a potent time-domain approach for analyzing hemodynamics.

- wave intensity
- hydraulic power
- reservoir
- wave separation
- hemodynamics

## NEW & NOTEWORTHY

We propose an alternative to wave intensity termed “wave power” (dPdQ), which is conserved at junctions, and present techniques for decomposing hydraulic power into forward and backward components. We establish conceptual and quantitative links between waves, wave potential, and hydraulic power for the first time, verified numerically and experimentally.

wave intensity analysis is a powerful and increasingly popular method for assessing the timing, directionality, and magnitude of hemodynamic waves in the cardiovascular system (31, 32). Defined as the product of incremental changes in pressure and velocity (i.e., dPd*U*), wave intensity can be separated into components related to forward- and backward-running waves, thus clearly distinguishing between waves arising from upstream (e.g., ventricular) vs. downstream (e.g., arterial) events. Although now widely used to analyze experimental, numerical, and, increasingly, clinical research data (2, 6, 12, 13, 15, 22, 34, 39, 43, 47), there are two drawbacks to the method. First, in some clinical and many experimental settings, volumetric blood flow, rather than blood velocity, is the primary measured quantity. Although velocity can be calculated by dividing flow by vessel cross-sectional area (CSA), the common assumption of a constant CSA, applied due to the technical difficulty in obtaining accurate, continuous measurements of internal CSA, may introduce errors. Second, since blood velocity is not a bulk flow property, wave intensity (with units of Watts per square meter) is not a conserved quantity and is therefore sensitive to vessel diameter variations and cannot be used to study the relative distribution of waves to the respective branches of a vascular junction.

To overcome these limitations, we propose a new quantity called wave power, defined as incremental changes in pressure and flow (dPdQ), which is calculated directly from measurements of flow (e.g., via perivascular flow probes). We show that wave power is a conserved quantity under most circumstances, which is important when studying wave transmission and reflection dynamics at vascular junctions and leads to a more robust measure of wave magnitude in the presence of diameter variations (Fig. 1). Moreover, the units of wave power are simply those of power (i.e., Watts), which implies a relationship to hydraulic power.

Hydraulic power is an extremely important physiological quantity related to the rate of energy production and expenditure in the cardiovascular system. Total power is composed of pressure power and kinetic power, which in turn are conventionally analyzed in terms of steady and pulsatile components (16, 28). A key limitation of this conventional analysis is that it does not provide direct insight into the nature and time course of ventricular and vascular events that together determine cardiovascular efficiency. Although a major advantage of wave separation analysis is that such events may be studied in a relatively straightforward manner (i.e., by quantifying the timing and magnitude of certain waves, and their effects on pressure and flow), this technique has not been extended to the analysis of hydraulic power, aside from one nascent study (40).

The lack of wave separation methodology for hydraulic power is most likely due to the common view that fluctuating pressure/flow wave components have no connection to the absolute values of pressure and flow (which are fundamental to the calculation of power). However, we recently proposed a new hemodynamic paradigm, involving a concept termed “wave potential,” that allows wave separation to be meaningfully performed on absolute pressure and flow signals (24). Two important implications arising from this paradigm, which are explored herein for the first time, are that *1*) hydraulic power may also be subjected to wave separation analysis, thereby allowing calculation of hydraulic power components related to forward- and backward-running waves and wave potential; and *2*) wave power provides a quantitative link between pressure-flow waves and hydraulic power.

This paper is structured as follows. First, we present methods for calculating wave power, its forward/backward components, and its relationship to wave intensity. Second, we demonstrate that, unlike wave intensity, wave power is a quasi-conserved quantity, and we explore the implications of this for hemodynamic analyses at vascular junctions. Third, we describe methods for performing wave separation on hydraulic power, including separation of pressure power and kinetic power. Fourth, we elucidate the relationship between hydraulic power and wave power. Fifth, we explore the physiological interpretation of wave potential with respect to wave power and hydraulic power. Finally, we verify the developed techniques and principles using simple numerical models and also experimentally in a central arterial bifurcation under a wide range of hemodynamic conditions induced by mechanical and drug interventions.

## MATERIALS AND METHODS

### Wave Intensity

Wave intensity (d*I*) is calculated from pressure (P) and cross-sectional mean velocity (*U*) signals via (32)
(1)

Using the method of characteristics, Parker and Jones (32) showed that wave intensity directly relates to changes in so-called characteristic variables (*w*_{±}) that separately govern the propagation of pressure/velocity waves in the forward (*w*_{+}) and backward (*w*_{−}) directions,
(2)

where ρ is blood density, *c* is wave speed, and d*w*_{±} are related to pressure and velocity via
(3)

Here, ρ*c* is the velocity-based characteristic impedance (32), which defines the relationship between pressure and velocity associated with forward or backward waves (dP_{±} = ±ρ*c*d*U*_{±}). This quantity is determined by ρ and vascular size and stiffness, noting that wave speed is related to vessel distensibility (*D*) via *c*^{2} = 1/(ρ*D*).

From *Eq. 2*, one of the key properties of wave intensity is evident, namely that backward-running waves have negative wave intensity, and forward waves have positive wave intensity. Where forward and backward waves overlap, it is useful to separate net wave intensity into forward- and backward-running components, which can be achieved without any linearizing assumptions via
(4)

For further details regarding this nonlinear separation, see Ref. 19. In practice, calculation of d*w*_{±} is usually not possible, as this requires knowledge of the pressure-area relation. However, wave separation is still possible by making several linearizing assumptions (i.e., d*I* = d*I*_{+} + d*I*_{−} and constant wave speed), which lead to (32)
(5)

### Wave Power

Starting with the one-dimensional equations governing pressure and flow (Q), in the appendix we show that the method of characteristics can also be used to define a quantity we have termed “wave power” (dπ), (6)

where *Z*_{c} = ρ*c*/*A* is the flow-based cross-sectional area (*A* is cross-sectional area), which governs the relationship between pressure and flow wave perturbations (45, 46). *Equation 6* is a similar expression to that for wave intensity (*Eq. 2*), but involves *Z*_{c} rather than wave speed and now
(7)

As with wave intensity, a nonlinear separation of wave power is possible via *Eq. A19* (see appendix), or a linear separation can be performed by assuming constant *Z*_{c} and dπ = dπ_{+} + dπ_{−}, leading to
(8)

Under these linear flow conditions, *Z*_{c}dQ = ρ*c*d*U*, and the only difference between *Eqs. 5* and *8* is the use of flow-based rather than velocity-based characteristic impedance in the denominator; hence wave power and wave intensity are simply related via dπ = *A*d*I* for linear flow.

One of the potential benefits of wave power is that its units are that of a conserved quantity (W), unlike wave intensity (whose units are W/m^{2}). For example, a wave propagating in a single vessel that encounters a local decrease in CSA will experience an increase in d*U* and wave intensity, but no change in dQ or wave power. Thus, unlike wave intensity, wave power is relatively insensitive to diameter variations (Fig. 1*B*). Note, however, that, according to Bernoulli's principle, convective acceleration leads to a fall in dP (conversion of pressure power to kinetic power), which may reduce wave power; however, except in pathological settings such as a severe stenosis, this effect is likely to be small since typically dP ≫ 1/2 ρd*U*^{2}.

### Wave Power at Junctions

Another setting where wave power (but not wave intensity) is expected to be conserved is branch junctions. Consider a single bifurcation containing one parent vessel and two daughter vessels (Fig. 1*A*). Any change in flow in the parent vessel (dQ_{1}) must equal the sum of corresponding flow changes in the daughter vessels (dQ_{2} and dQ_{3}),
(9)

Wave power in the parent vessel is therefore (10)

If we assume continuity of dP across the junction (i.e., dP_{1} = dP_{2} = dP_{3}), thereby ignoring the generally small pressure losses that occur at junctions (27), then
(11)

This may be generalized to any multibranch junction as follows, (12)

where *i* and *j* refer to parent and daughter indexes, respectively. It follows that the forward and backward components also add up at a junction,
(13)

In the special case where the junction is well-matched and there are no backward waves in the daughter vessels, the sum of incident and transmitted forward waves balance, (14)

Therefore, violation of *Eq. 14* indicates an influence of backward waves. For example, for a nonmatched bifurcation (dπ_{1,−} ≠ 0) with well-matched outlets (i.e., dπ_{j,−} = 0),
(15)

where subscript *i* = 1 refers to the parent vessel, and subscripts *j* = 2 and *j* = 3 refer to the daughter branches. Here, the degree of pressure and flow wave reflection depends on the characteristic admittance (*Y* = 1/*Z*_{c} = *A*/ρ*c*) of parent and daughter branches, according to the well-known expression for a junction reflection coefficient,
(16)

A wave power reflection coefficient (Γ_{π}) may also be defined by multiplying pressure (Γ_{P}) and flow (Γ_{Q} = −Γ_{P}) reflection coefficients to yield
(17)

Substituting *Eq. 17* into *Eq. 15*,
(18)

where *T*_{π} is a power transmission coefficient. Hence, for a well-matched junction where Γ_{π} = 0, dπ_{1,+} = dπ_{2,+} + dπ_{3,+}. Conversely, the limiting cases where combined daughter vessel admittance approaches zero (no transmission of pressure perturbations) or infinity (no transmission of flow perturbations), both lead to Γ_{π} = −1 and no transmission of wave power (i.e., *T*_{π} = dπ_{2,+} + dπ_{3,+} = 0).

### Forward and Backward Components of Pressure, Flow, and Hydraulic Power

The forward and backward components of pressure (P_{±}) and flow (Q_{±}) signify the pressure and flow that would exist if only forward or only backward waves existed (24, 46). Extending preliminary work in Ref. 40, in this section we show that hydraulic power can also be separated into such components. A nonlinear separation of pressure and flow can be achieved by integrating *Eqs. A11* and *A12* or *A17* (see appendix). However, in most settings, a linear separation is used such that
(19)
(20)

where the constant P_{ud} is undisturbed pressure and represents the equilibrium pressure that is attained when no wave potential or waves exist in the vascular system, typically at a value of ∼10 mmHg in human adults (24). The linear separation makes use of the water-hammer equation that relates incremental changes in P_{±} and Q_{±} via *Z*_{c} as follows,
(21)

Combining *Eqs. 19* and *20* with *21*, and integrating, it can be shown that (24)
(22)
(23)

Ignoring gravitational effects, hydraulic power (Π) at a given location is equal to the sum of pressure (Π_{P}) and kinetic (Π_{Q}) components,
(24)

Pressure power represents the potential for hydraulic work per second, while kinetic power represents the rate of energy transfer due to the flowing fluid. Note that the sign of Π, which may be negative when Q < 0, simply indicates the direction of power transfer. From *Eqs. 19* and *20*, pressure power can be expressed in terms of P_{±} and Q_{±} as
(25)

From *Eqs. 22* and *23*, an absolute form of the water hammer equation can be derived,
(26)

By making use of *Eq. 26*, *Eq. 25* can then be expanded and simplified to yield
(27)

Forward and backward components of Π_{P} power can now be defined as
(28)

where the right-hand side is obtained by substituting *Eqs. 22* and *23*. Importantly, the pressure power components are additive,
(29)

Kinetic power may also be expressed in terms of forward and backward flow components, (30)

Expanding, (31)

In the case where Π_{Q∓}, the forward and backward components of kinetic power are
(32)

However, the two middle terms in *Eq. 31* show that the sum of these components do not equal the net kinetic power. Rather,
(33)

where Π_{X} = (3Q_{−}Q_{+}^{2} + 3Q_{−}^{2}Q_{+})ρ/2*A*^{2} describes a nonlinear interaction whereby the influence of Π_{Q±} on Π_{Q} depends on Π_{Q∓}. If we define the ratio *K*_{±} = −Q_{∓}/Q_{±}, substitution into *Eq. 31* leads to
(34)

From *Eq. 23*, we see that *K*_{±} → 1 when Q ≪ (P − P_{ud})/*Z*_{c}, which is usually the case in the arterial system. Thus, except in settings where *Z*_{c} is large and/or pressure is small, the impact of Π_{Q±} is, therefore, likely to be largely negated by Π_{X}. Total hydraulic power may be written as
(35)

However, based on the arguments presented above, the impact of the kinetic power components is likely to be small compared with the pressure power components. Coupled with this are prior findings that kinetic power is generally very small compared with pressure power (5, 16, 28).

Aside from energy dissipation due to viscous effects, hydraulic power must be conserved. At junctions, expressions for the reflection and transmission of power perturbations are, therefore, the same as for wave power (see *Eqs. 17* and *18*). However, it should be remembered that the constituents of hydraulic power may change; for example, in a tapering or stenosed vessel, convective acceleration causes a transfer of power from Π_{P} to Π_{Q}.

### Relationship Between Hydraulic Pressure Power and Wave Power

Changes of Π_{P±} are related to corresponding changes in the pressure and flow components as follows,
(36)

From this it can be shown that (37)

By making use of *Eqs. 21* and *26*, this can be reduced to
(38)

Finally, given that 2P_{±}/dP_{±} ≫ 1, therefore
(39)

Alternatively, it can also be shown that (40)

These equations show that, although dπ_{±} is small compared with dΠ_{P±}, the two quantities are quantitatively linked via the fractional change in pressure or flow wave components. For example, if over a small time interval, the forward component of pressure or flow increases by 0.1%, then the associated change in the forward component of pressure power will be equal to two times wave power divided by 0.1%.

Note that, in general P_{±} > 0, Q_{+} > 0, and Q_{−} < 0, so forward compression waves (FCWs) (dπ_{+} > 0, dP_{+} > 0, dQ_{+} > 0) and backward expansion waves (dπ_{−} < 0, dP_{−} < 0, dQ_{−} > 0) increase Π_{P}, while forward expansion waves (dπ_{+} > 0, dP_{+} < 0, dQ_{+} < 0) and backward compression waves (dπ_{−} < 0, dP_{−} > 0, dQ_{−} < 0) decrease Π_{P}.

### Wave Potential, Wave Power, and Pressure Power Components

We recently proposed the concept of wave potential, which represents the potential for waves to be produced and is revealed in the absolute values of the forward and backward pressure/flow components (i.e., if these are zero, there is no wave potential) (24). Wave potential exists when pressure and/or flow differ from P_{ud} and zero, respectively. We here extend this novel paradigm by showing that the absolute values of the forward and backward components of hydraulic pressure power represent the potential for hydraulic work to be performed by a distended vessel.

Consider a long uniform vessel (reference area 5 cm, reference wave speed 4 m/s) with a closed inlet (*x* = 0) and a closed valve at (*x* = 15 cm), as shown in Fig. 2 (similar to the model used in Fig. 4 in Ref. 24). At *t* = 0, the proximal part of the vessel (*x* < 15) is pressurized to P = 10 mmHg, while distal to the valve, P = P_{ud} = 0. According to *Eq. 28*, under such conditions, although Q = 0, Π = 0, and no waves exist (i.e., dπ_{±} = 0), it remains that Π_{P±} = ±52 mW. Although these nonzero equilibrium values may initially seem meaningless, their physical meaning is revealed when the valve is opened at *t* = 0.02 s (dashed vertical lines in Fig. 2). This event gives rise to two waves that propagate in opposite directions: a FCW that increases flow and pressure (from 0 to 5 mmHg) distal to the valve, and a backward expansion wave that increases flow and decreases pressure (from 10 to 5 mmHg) proximal to the valve (pressure/flow waveforms not shown). Simultaneously, proximal Π_{−} falls from −52 mW to 0, and distal Π_{+} rises from 0 to 52 mW. The waves and associated changes in Π_{±} then propagate away from the valve. When the backward expansion wave reaches the closed inlet, it is entirely reflected, resulting in a forward expansion wave that continues to reduce pressure (from 5 to 0 mmHg), but now reduces outflow and Π_{+} to zero as it propagates back toward the open valve. When this wave reaches the valve, the fluid volume stored in the compliant walls has been emptied, and Π_{±} = 0, signifying an absence of wave potential. We previously showed (24) that the volume (V) emptied is equal to
(41)

where *L* is the length of the reservoir (in this case, 15 cm) and Q_{±}^{0} = ±(P^{0} − P_{ud})/(2*Z*_{c}) are the initial values of Q_{±}. In addition, integration of the total pressure power arising from the emptying reservoir is a measure of the released hydraulic energy (see shaded area in Fig. 2). Multiplying *Eq. 41* by the initial pressure components (P_{±}^{0}) predicts the amount of stored energy (*E*_{R}) as
(42)

In a more general setting where *c* = *c*(*x*) and Π_{±} = Π_{±}(*x*), the energy stored in a compliant pressurized reservoir is
(43)

This is also the energy required to fill the reservoir (i.e., to pressurize it from P_{ud} to P). Hence, the absolute values of Π_{±} have a specific and potentially important physiological significance.

### In Vivo Experiments

#### Experimental preparation.

Experiments were approved by the Murdoch Childrens Research Institute Animal Ethics Committee and conformed with guidelines of the National Health and Medical Research Council of Australia. Five young adult wethers (i.e., male castrated sheep, weight 24.8 ± 1.7 kg) were studied using similar methods to those described previously (22, 24). Animals were anesthetized with intramuscular ketamine (5 mg/kg) and xylazine (0.1 mg/kg), followed by 4% isoflurane delivered by mask. Anesthesia was maintained with isoflurane (2–3%), nitrous oxide (∼30%), and O_{2}-enriched air delivered via a mechanical ventilator and supplemented by an intravenous infusion of ketamine (1–1.5 mg·kg^{−1}·h^{−1}) and midazolam (0.1–0.15 mg·kg^{−1}·h^{−1}). Ventilation was adjusted to maintain arterial O_{2} tension at 100–120 mmHg and arterial CO_{2} tension at 35–40 mmHg.

After incision of the left side of the neck, a polyvinyl catheter was inserted into the external jugular vein for fluid and drug infusion. A 6-Fr sheath was passed into the left common carotid artery for passage of a 5-Fr pigtail micromanometer catheter with a central lumen (model SPC-454D, Millar Instruments, Houston, TX) into the aortic trunk (AoT) to obtain fluid-filled and high-fidelity pressures. A left thoracotomy was performed in the fourth intercostal space, and 3.5-Fr micromanometer catheters (model SPR-524, Millar Instruments) were inserted via purse-string sutures into the origin of the brachiocephalic trunk (BCT) and the descending thoracic aorta (DAo) just beyond the aortic arch, and transmitrally into the left ventricular (LV) cavity through the roof of the left atrium. Transit-time flow probes (Transonic Systems, Ithaca, NY) were placed around the AoT (20 mm), the BCT close to its origin (12 or 14 mm), and the DAo (16 mm) just above where the hemizygous vein crossed this vessel. Finally, the DAo was encircled with an adjustable snare 2–3 cm caudal to the site of flow probe and micromanometer placement.

#### Experimental protocol.

After completion of instrumentation, baseline variables were recorded onto computer at a sample rate of 1,000 Hz. To produce a wide range of pressure and flow (and thus power), hemodynamics were then altered with the following steady-state interventions: *1*) constriction of the DAo via tightening of the snare to raise mean AoT pressure by ∼10 mmHg and then ∼25 mmHg, thereby inducing substantial wave reflection; *2*) after release of the DAo snare and return of hemodynamics to baseline levels, stepwise infusion of dobutamine at rates of 0.5, 1.0, 2.5, 5, and 10 μg·kg^{−1}·min^{−1} to increase LV contractility; and *3*) following cessation of dobutamine, infusion of esmolol (100–200 μg·kg^{−1}·min^{−1}) to reduce LV contractility and AoT pressure. Data recording was repeated once hemodynamics had stabilized during each intervention. Each animal thus provided nine data points for inclusion in the analysis. At the end of the study, animals were euthanized with an overdose of pentobarbitone sodium (100 mg/kg).

#### Data analysis.

Offline analysis was performed in Spike2 (Cambridge Electronics Design, UK). High-frequency noise was removed with a spectral filter, being careful not to remove any signal harmonics. Mean AoT, BCT, and DAo micromanometer pressures were equalized to the mean aortic fluid-filled catheter pressure. Noting that the BCT is the only supra-aortic branch in sheep, and to ensure that AoT, BCT, and DAo flows were internally consistent, LV output (minus coronary blood flow) was derived in each animal as the sum of mean BCT and DAo flows. The measured mean AoT flow was then matched to this calculated LV output using a derived scaling factor (1.35 ± 0.27). All subsequent hemodynamic analyses were performed with this adjusted AoT flow, using ensemble-averaged signals generated from a bloc of recorded data containing more than 10 beats. Vessel CSAs were calculated from external diameters, measured via calipers. For each of the three measurement locations, phase lags between pressure and flow were corrected as in Ref. 34, and *Z*_{c} was calculated as the slope of the PQ relation during early systole, when reflected waves are assumed to be negligible (7, 11). The peak positive rate of change of LV pressure (dP/d*t*_{max}) was calculated as an index of LV contractility. In addition, systemic vascular resistance (SVR) was derived as [AoT P_{mean} − 5 mmHg]/AoT Q_{mean}, and total arterial compliance was calculated via the pulse pressure method (41). As per convention (4), cycle average pressure power (i.e., active power = steady + oscillatory power) was calculated as mean(Π_{P}), steady pressure power as mean(P) × mean(Q), and power efficiency as steady power divided by average active power.

Since dπ is dependent on sample rate, in some figures we also present a time-corrected form of wave power (*wp*), calculated via the time-derivatives dP/d*t* and dQ/d*t*, rather than the incremental changes dP and dQ. This form of wave power has units of W/s^{2} and, as with time-corrected wave intensity (36), may be compared with data acquired at different sample rates. Numerical simulations were performed using a nonlinear one-dimensional blood flow modeling code, details of which have been described previously (20, 21).

#### Statistical analysis.

Experimental data were analyzed using least squares linear regression for each animal, with the regression equation calculated as the line of symmetry through the data since both *X* and *Y* data points are subject to unknown errors (3). Regression coefficients from each individual animal were then averaged to obtain an overall regression equation for the study group. Results are expressed as means ± SD, and significance was taken at *P* < 0.05.

## RESULTS

### Experimental Studies

Baseline mean and pulse AoT pressures were 90 ± 8 and 18 ± 3 mmHg, respectively, with substantial variations in these pressures during interventions (mean, 51–125 mmHg; pulse, 13–51 mmHg). Baseline values (mean ± SD) and range over all interventions for other hemodynamics quantities were as follows: heart rate, 114 ± 7 and 82–243 beats/min; LV output, 3.2 ± 0.5 and 2.2–4.8 l/min; LV stroke volume, 28.0 ± 4.7 and 16.2–36.0 ml; LV dP/d*t*_{max}, 1,879 ± 622 and 698–9,354 mmHg/s; SVR, 1.63 ± 0.20 and 0.75–2.55 mmHg·s·ml^{−1}; total arterial compliance, 1.25 ± 0.17 and 0.32–1.58 ml/mmHg; average active power, 0.67 ± 0.15 and 0.29–1.02 W; steady power, 0.65 ± 0.15 and 0.25–0.95 W; and power efficiency, 95 ± 1 and 67–97%.

Figure 3 shows an illustrative example of wave power analysis under baseline conditions (*left*) and during constriction of the DAo (*right*). At baseline, wave power was characterized by two main peaks: a FCW in early systole that increased Π_{P+} and Π_{Q+}, and a forward expansion wave in late systole that decreased Π_{P+} and Π_{Q+} (see green dashed lines in Fig. 3). While backward waves were small at baseline, a backward compression wave that decreased Π_{P−} and Π_{Q−} was clearly visible during DAo constriction (between red dashed lines in Fig. 3). Peak Π_{Q} was small compared with peak Π_{P}, and the FCW-related increase in Π_{Q+} (green line, Fig. 3), although nonnegligible compared with the associated change in Π_{P+}, was almost entirely negated by a decrease in Π_{X} (blue dashed line), as predicted by *Eq. 34*. This was also true of the entire data set, in which the increase in Π_{Q+} amounted to 35 ± 17% of the total rise in Π_{+}, but 93 ± 9% of this contribution was negated by a decrease in Π_{X}, such that peak Π_{Q} accounted for only 3.2 ± 1.8% of peak total hydraulic power. Figure 4*A* shows that a linear relationship exists between FCW peak dπ_{+} and the associated change in Π_{P+} (*R*^{2} = 0.89 ± 0.05, *P* < 0.001), albeit with a notable variability of individual slopes and dπ_{+} being several orders of magnitude smaller than ΔΠ_{P+} (average slope 3.19 × 10^{−3} ± 1.87 × 10^{−3}). When dπ_{+} is corrected for the overall fractional change in P_{+} (i.e., mean P_{+}/total change in P_{+} due to the FCW), this relationship is also linear (*R*^{2} = 0.90 ± 0.08, *P* < 0.002, Fig. 4*B*), but with a highly consistent slope between animals (3.40 × 10^{−3} ± 0.6 × 10^{−3}), noting that this is a “gross” evaluation of the analytic relation dΠ_{P+} = 2(P_{+}/dP_{+})dπ_{+} (*Eq. 39*).

A linear relationship was present between FCW dπ_{+} area and LV dP/d*t*_{max} (*R*^{2} = 0.84 ± 0.12, *P* < 0.01, Fig. 5*A*), indicating a dependence of FCW power on LV contractility. However, the slope of this relation was variable (2.20 × 10^{−3} ± 1.84 × 10^{−3}). A strong relation with a considerably less variable slope was found between end-diastolic wave potential (i.e., |Π_{±}|) and the ratio of SVR and *Z*_{c} (*R*^{2} = 0.88 ± 0.12, *P* < 0.005, Fig. 5*B*). This is consistent with the principle that wave potential relates to windkessel function (24), given that reservoir filling increases with an increase in SVR (i.e., greater outflow resistance) and a decrease in *Z*_{c} (i.e., less stiff and/or larger compliant reservoir).

To test for conservation of wave power and pressure power at the AoT-BCT-DAo junction in sheep, Fig. 6 shows the forward components of (time-corrected) wave intensity, wave power and pressure power in these vessels at baseline and during infusion of dobutamine (5 μg·kg^{−1}·min^{−1}). Dashed gray lines correspond to the sum of BCT and DAo forward component waveforms, while dashed black lines also account for backward components (not shown, see *Eq. 15*). It can be seen that the sum of BCT and DAo wave power and pressure power are almost equal to the corresponding AoT signals, suggesting conservation of these quantities, whereas wave intensity is not conserved. Figure 7 shows a close 1:1 relationship between AoT FCW peak dπ_{+} and the sum of BCT and DAo FCW peak dπ_{+} (slope 0.98 ± 0.14, *R*^{2} = 0.98 ± 0.01, *P* < 0.001) for all animals. Similar relationships were also found for FCW area (i.e., integrated dπ_{+}), and the change in Π_{P+} related to the FCW, with a slope somewhat less than 1.0 suggesting a minor influence of wave reflection, emerging after the peak in FCW dπ_{+}.

### Numerical Simulations

To elucidate the influence of local wave reflection on power transmission at junctions, we designed a simple one-dimensional model of a single bifurcation. When the bifurcation is well-matched (Γ_{P} = 0), perturbations of pressure, flow, power and wave power are all transmitted through the junction without reflection (Fig. 8, *left*). However, if the daughter branches have a combined admittance (*Y* = 1/*Z*_{c}) that is one-third that of the parent admittance, resulting in Γ_{P} = 0.5 via *Eq. 16* (Fig. 8, *middle*), 50% of the incident pressure wave is positively reflected (i.e., further increasing pressure), and 50% of the flow wave is negatively reflected (reducing net flow); hence 25% of the pressure power and wave power is reflected, while transmitted power is also reduced by this amount. Conversely, if the daughter branches have a combined admittance that is three times that of the parent admittance, resulting in Γ_{P} = −0.5 (Fig. 8, *right*), 50% of the incident pressure/flow wave is negatively/positively reflected; although this produces reflected waves that decrease/increase pressure/flow (which is opposite to the Γ_{P} = 0.5 case), the effect on wave power is the same (a 25% decrease). Hence, optimal power transmission occurs when Γ_{P} = 0.

## DISCUSSION

This paper has proposed an alternative to wave intensity that is calculated with blood flow in place of velocity. This wave power has units of Watts, shares all of the useful properties of wave intensity but, in contrast to wave intensity, has the additional desirable property of being a quasi-conserved quantity. Following on from our preliminary work (40), we also showed that hydraulic power can be separated into forward and backward components and that perturbations of pressure power components are directly linked to wave power. Moreover, the absolute values of hydraulic power components signify wave potential and the hydraulic energy stored in a compliant pressurized vessel. These advances for the first time enable detailed time domain analysis of cardiovascular processes contributing to hydraulic power and establishes the fundamental links between wave phenomena and hydraulic power.

Originally proposed by Parker and colleagues (32, 33) in 1988, wave intensity (dPd*U*) has become a valuable and widely used tool for analyzing hemodynamics in the time domain (2, 6, 12, 13, 15, 22, 34, 39, 43, 47). More recently, three alternative definitions of wave intensity have been proposed. For settings where noninvasive measurements of blood velocity and vessel diameter (*D*) or vessel area (*A*) are available (e.g., echocardiography and phase contrast MRI), Feng and Khir (9) proposed the use of d*D*d*U*, Tanaka et al. (42) advocated d[ln*D*]d*U*, while Biglino et al. (2) used d[ln*A*]d*U*. Although similar to wave intensity, these quantities have units of m^{2}/s, m/s, and m/s, respectively, whose physical meaning is unclear in this context (the term “wave intensity” was retained, even though the units were no longer those of intensity).

The quantity proposed in this study, wave power (dPdQ), has the readily understandable units of power (Watts) and is most suited to settings where pressure and volumetric flow are available, such as invasive experimental studies employing micromanometer-tipped catheters and perivascular flow probes (38–40). Use of flow may also be preferred in the setting of phase-contrast MRI, where calculation of velocity from flow requires an extra step (measurement of CSA, which unlike flow is sensitive to the segmentation); dPdQ could then be calculated with a pressure waveform measured directly or estimated via a calibrated CSA waveform (35). In settings where velocity is measured (e.g., Doppler ultrasound), but the beneficial properties of wave power are desired, wave power could be derived after estimating flow from velocity via a variety of available methods (8, 14, 26). In accord with prior wave intensity studies (29), our experimental data indicated that the initial forward compression wave power in the aorta was related to LV contractility, as assessed by LV dP/d*t*_{max}, (Fig. 5), although the extent to which the scatter of data points (average *R*^{2} = 0.84) and the wide variability of individual slopes observed in this relationship is due to a load dependence deserves further study.

We have shown that wave power is a (quasi-)conserved quantity, which is an advantage over wave intensity. This is of particular relevance at junctions where, although total wave power is conserved, its transmission from parent to daughter vessels is degraded by any impedance mismatching (whether positive or negative; see Fig. 8), with the remaining power being reflected. However, experimental analysis of the FCW in the AoT, BCT, and DAo of sheep suggested that this bifurcation is exceptionally well-matched in the forward direction. Although such matching is an oft-assumed property of normal arterial bifurcations (21, 30, 44), the data in Fig. 7 are, to our knowledge, the clearest evidence of this principle presented to date. Finally, while not specifically addressed in this paper, conservation of wave power means that the relative distribution of wave power in both diverging and converging vascular junctions can be assessed. This is highly relevant in settings such as the fetal circulation, where the right ventricular output splits into ductal and pulmonary arterial streams, with the ductal component then merging with aortic isthmus flow to form total flow within the DAo (39).

While wave power is conserved under most circumstances, it must be acknowledged that several phenomena may lead to a loss or gain of wave power. First, in a tapering or stenosed vessel, convective acceleration leads to an increase in kinetic power and a corresponding decrease in pressure power and wave power. To assess the magnitude of this effect, we performed some additional simple numerical simulations and found that an ∼40% stenosis decreases wave power by ∼10% for a model with parameters representing a normal aorta, or ∼3% for a stiff aorta. Hence, except in significantly stenotic vessels, convective acceleration is likely to have a negligible effect on wave power conservation. Second, nonlinear effects (specifically, a pressure dependence of *Z*_{c}) will lead to amplification of compression waves and attenuation of expansion waves as they propagate; this phenomenon was previously investigated in the context of wave intensity (17), but also applies to wave power (as is evident in Fig. 2) and may be appreciable where vessels are highly compliant, pulse pressure is high, and/or the propagation distance is significant. Finally, viscous effects (arising from blood and vessel wall viscosity) cause dissipation of energy and thus wave power (10, 17).

Another useful property of wave power is its relation to hydraulic pressure power. Although wave power is several orders of magnitude smaller than pressure power (see Figs. 3 and 4*A*, depending on the sample rate), we showed that wave power bears a direct relation to changes in pressure power (via *Eqs. 39* and *40*), i.e., in inverse proportion to the fractional changes in pressure or flow components. As might be expected, any wave that has a flow-increasing effect also has a power-increasing effect (either a FCW or a backward expansion wave), while any wave that has a flow-decreasing effect also has a power-decreasing effect (forward expansion wave or backward compression wave). Despite the approximately linear relationship found in Fig. 4*A*, it is important to note that a large instantaneous value of wave power indicates a rapid change in pressure power; the overall change in pressure power is related to the size and duration of the wave. As an example, the dashed red lines in Fig. 3, *right*, indicate a prominent backward compression wave that causes a rapid but relatively small decrease in Π_{P−}. However, following this is a series of small backward compression waves that appear relatively insignificant in the wave power signal, but in fact have a greater (albeit slower) effect on Π_{P−} than the initial large wave. Similarly, during diastole, dπ_{±} is very small since Π_{P±} changes slowly, but the accumulated change over the whole time period is appreciable. We, therefore, recommend combining assessment of wave power with analysis of the forward and backward components of hydraulic power to obtain a comprehensive “wave power analysis.”

In the past, hydraulic power has been separated into different types of components, which each provide specific insights into properties of the arterial system and the efficiency of blood flow. These include potential (including pressure and hydrostatic) and kinetic components (4, 16), steady and pulsatile (or oscillatory) components (16, 28), and in-phase and reactive (or quadrature) power (1, 4, 16). However, previous analyses of hydraulic power have been almost exclusively performed in the frequency domain (i.e., via Fourier analysis), which, although providing important global information, does not allow investigation of how physiological events at specific times in the cardiac cycle contribute to the heart's power output and its interaction with the arterial load. On the other hand, wave intensity and wave separation analyses have enabled insightful investigations of propagating pressure/flow perturbations (waves) in the time domain, but the effects of these waves on hydraulic power have not been addressed.

In this paper, we, therefore, proposed a simple time-domain method for separating pressure power into forward and backward components. A key benefit of this analysis is that, as with P_{±} and Q_{±}, changes in Π_{±} arise from propagating waves. Hence the timing, directionality, and magnitude of hemodynamic events contributing to instantaneous hydraulic power at a given location can be quantified and visualized in an intuitive and mathematically robust fashion (Fig. 3). Moreover, the degree and distribution of power transmission at junctions can be assessed (Figs. 6 and 8). Wave separation can also be applied to kinetic power, although this is complicated by interaction terms. However, since forward and backward kinetic components are mostly negated by these interaction terms, and since net kinetic power is small compared with pressure power, kinetic power may be neglected in most settings.

We recently introduced the concept of wave potential to provide a comprehensive wave-based paradigm of arterial hemodynamics, unifying explanations of wave phenomena, windkessel function, and mean pressure in a single analytic technique without requiring the problematic separation of pressure into so-called “reservoir” and “wave” components (18, 22, 23, 25). A key element underpinning this concept was the recognition that the absolute values of the forward and backward components, which were previously considered meaningless, do have an important physiological significance. Thus, whereas changes in these components indicate the effects of propagating disturbances (waves), their absolute values represent the potential for producing such disturbances (wave potential), such that any spatial gradient in wave potential produces waves.

In the present work, we have shown that the absolute values of Π_{P+} and Π_{P−} also represent wave potential (see Fig. 2), which, when combined with wave speed, indicate how much hydraulic energy is stored in the compliant pressurized reservoir (*Eq. 43*). For example, at a given pressure, the stored energy in a stiff vascular network is less than that stored in a more compliant network. Conversely, it takes more energy to pressurize a stiff network. Indeed, our experimental data showed that wave potential increases with increased reservoir function, i.e., with greater pooling of blood in the conduit arteries, occurring when SVR increases or vascular stiffness decreases (reduced *Z*_{c}, see Fig. 5*B*).

Separation of pressure, flow, wave power, and hydraulic power into their forward and backward components requires knowledge of *Z*_{c}, which differs from velocity-based analyses that require wave speed. However, *Z*_{c} can be calculated using equivalent methods to those used to obtain wave speed, with the added benefit that measurement of CSA is not required if flow is the measured quantity. For example, *Z*_{c} can be estimated as the slope of the pressure-flow relation during early systole, if backward waves are minimal during this time, as originally described by Dujardin and Stone (7), which is equivalent to the pressure-velocity loop method for calculating wave speed (11).

In conclusion, the proposed wave power analysis links a flow-based alternative to wave intensity with a novel time domain analysis of hydraulic power and the concept of wave potential. Wave power provides added value compared with previous methods in that it can be used to study the distribution of wave power at junctions, is relatively insensitive to diameter variations, and is quantitatively linked to transient changes in hydraulic pressure power. The novel wave separation applied to hydraulic power is relatively simple to perform and provides insights into upstream and downstream events contributing to the ventricle's power output and its dependence on arterial load, along with energy storage and discharge (i.e., windkessel function), as expressed by wave potential. Future work is needed to fully explore the benefits of these techniques for studying ventriculo-vascular coupling in health and disease.

## GRANTS

J. P. Mynard was funded by a CJ Martin Early Career Fellowship from the National Health and Medical Research Council of Australia. This work was supported by the Victorian Government's Operational Infrastructure Support Program.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

Author contributions: J.P.M. conception and design of research; J.P.M. analyzed data; J.P.M. interpreted results of experiments; J.P.M. prepared figures; J.P.M. drafted manuscript; J.P.M. and J.J.S. edited and revised manuscript; J.P.M. and J.J.S. approved final version of manuscript; J.J.S. performed experiments.

## ACKNOWLEDGMENTS

We thank Magdy Sourial, Sarah White, Amy Tilley, and Aaron Mocciaro for assistance with experimental studies.

## APPENDIX

This section derives a new variant of wave intensity calculated from blood pressure and volumetric blood flow rather than blood velocity. Sherwin et al. (37) derived the following pressure-flow form of the one-dimensional Navier-Stokes equations, obtained under the assumption that spatial gradients of material properties and reference CSA are negligibly small, (A1) (A2)

where *U* = Q/*A* is cross-sectional mean velocity, *D* = (1/*A*)dP/d*A* is distensibility, *f* represents viscous friction, and α is the momentum flux correction factor, which depends on the velocity profile. Neglecting viscous friction and assuming a flat velocity profile (α = 1), these equations can be rewritten in nonconservative form as follows,
(A3)
(A4)

where wave speed (*c*) is given by
(A5)

*Equations A3* and *A4* can then be written in quasi-linear matrix form as
(A6)

where

The eigenvalues of this system are (A7)

These can be used to derive the left eigenvectors (**l**_{i}) by solving **l**_{i}**H** = λ**l**_{i}, enabling diagonalization of the equation system,
(A8)

Using the identity d**W** = **l**_{i}d**U**, the equation system can be cast in terms of Riemann invariants, or characteristic variables, **W** = [*w*_{+}; *w*_{−}], such that
(A9)

The characteristic variables *w*_{+} and *w*_{−} separately govern pressure/flow information that propagates forwards and backwards, respectively. After calculation of **l**_{i}, it can be shown that
(A10)

where *Z*_{c} is characteristic impedance (which in a general analysis is a pressure-dependent variable), and *M* is the Mach number. With some manipulation,
(A11)
(A12)

Wave power (dπ) can now be defined as (A13)

In a setting where only forward waves exist, (A14)

Similarly, if only backward waves exist, then (A15)

Hence, forward wave power is always positive (for subcritical flow, *M* < 1, which is relevant to most physiological flows), while backward wave power is always negative. Although strictly, dπ = dπ_{+} + dπ_{−} + *MZ*_{c}d*w*_{+}d*w*_{−}/2, the last term is very small; hence dπ_{±} are essentially additive without any linearizing assumptions. Moreover, in most physiological settings, *U* ≪ *c*, and hence *M* ≪ 1, such that Eq. A10 becomes
(A16)

so *Eq. A12* reduces to
(A17)

and the expression for wave power reduces to (A18)

with the forward and backward components, (A19)

- Copyright © 2016 the American Physiological Society