## Abstract

Because systole and diastole are coupled and systolic ventricular-vascular coupling has been characterized, we hypothesize that diastolic ventricular-vascular coupling (DVVC) exists and can be characterized in terms of relaxation and stiffness. To characterize and elucidate DVVC mechanisms, we introduce time derivative of pressure (dP/d*t*) vs. time-varying pressure [P(*t*)] (pressure phase plane, PPP)-derived analogs of ventricular and vascular “stiffness” and relaxation parameters. Although volume change (dV) = 0 during isovolumic periods, and time-varying left ventricular (LV) stiffness, typically expressed as change in pressure per unit change in volume (dP/dV), is undefined, our formulation allows determination of a PPP-derived stiffness analog during isovolumic contraction and relaxation. Similarly, an aortic stiffness analog is also derivable from the PPP. LV relaxation was characterized via τ, the time constant of isovolumic relaxation, and vascular (aortic pressure decay) relaxation was characterized in terms of its equivalent (windkessel) exponential decay time constant κ. The results show that PPP-derived systolic and diastolic ventricular and vascular stiffness are strongly coupled . In support of the DVVC hypothesis, a strong linear correlation between relaxation (rate of pressure decay) indexes κ and τ (κ = 9.89τ − 90.3, *r* = 0.81) was also observed. The correlations observed underscore the role of long-term, steady-state DVVC as a diastolic function determinant. Awareness of the PPP-derived DVVC parameters provides insight into mechanisms and facilitates quantification of arterial stiffening and associated increase in diastolic chamber stiffness. The PPP method provides a tool for quantitative assessment and determination of the functional coupling of the vasculature to diastolic function.

- ventricular-vascular coupling

assessment of systolic ventricular-vascular coupling (SVVC) has focused on ventricular output and arterial properties. There is a wide range of SVVC studies evaluating left ventricular (LV) function via the end-systolic pressure-volume relationship (ESPVR), stroke volume (SV), and cardiac output (CO) (8, 37, 41, 43). Arterial properties have been characterized in terms of pressures via the windkessel model and the windkessel metric [often described as effective arterial elastance (*E*_{a})] (8, 24, 25, 27, 41, 53). A preponderance of data led Nichols et al. (42) to conclude that aging alters myocardial function because the “ventricle is coupled to an abnormally high arterial load or input impedance.” Furthermore, O'Rourke et al. (46) conclude that this ventricular alteration is “aggravated by hypertension and disorders of large arteries.” Taken as a whole, this is consistent with long-term adaptation consistent with homeostatic requirements for maintenance of CO.

Although some have considered the effect of afterload on diastolic relaxation (16, 21, 23), much of this research concerns coupling of ejection to vascular properties. Although there is direct evidence that systolic and diastolic function are mechanically coupled via titin (17, 23, 26) and the extracellular matrix (40, 50), through the end-diastolic pressure volume relationship (EDPVR) (8, 37) and the triple control of relaxation (5), the relationship between chamber and vascular properties during diastole, i.e., diastolic ventricular-vascular coupling (DVVC), has not been fully characterized. Ventricular-vascular coupling (VVC) has recently reemerged as a potentially important diastolic function determinant. For example, diabetes is often associated with hypertension and is known to affect diastolic function (10, 49). However, it remains unclear how diabetes modulates VVC properties (25, 49). Arterial stiffness has also recently been suggested as a major determinant of chamber properties in subjects having heart failure with a normal ejection fraction (EF) (27). One reason that diastolic stiffness properties, or isovolumic stiffness properties in particular, remain unexplored relates to how time-varying stiffness is defined. Stiffness is traditionally expressed as the change in pressure per unit change in volume (dP/dV), but during isovolumic relaxation, when dV = 0, stiffness expressed in this form is undefined.

The clinical utility of cardiac catheterization is well established, but full utilization and exploitation of the hemodynamic data available have been generally underappreciated. Stroke work (SW), maximum elastance (*E*_{max}) or the ESPVR, the time constant of isovolumic relaxation (τ), and the EDPVR are the generally accepted, invasively obtained indexes of cardiovascular function (1, 14, 24, 37, 41, 43, 54). However, analysis of hemodynamic data in the pressure phase plane (PPP) has been limited to selected research themes. The PPP is a plot of the time derivative of pressure (dP/d*t*) vs. time-varying pressure [P(*t*)] (see Fig. 1*B*). The PPP has been utilized to aid in improved geometric determination of τ by determining the closeness of a monoexponential relation: (1) to the PPP contour, where P_{LVo} is a pressure constant and P_{LV∞} is the pressure asymptote (12, 24). More recent applications using the phase plane have elucidated relations between peak positive and negative dP/d*t* and the pressures at which they occur, the linearity of *Eq. 1*, and a relation between the phase plane area (limit cycle area) and ventricular stiffness (11, 12). The conceptual generalization of “physiological hyperspace” as the analytical arena for LV function characterization, spanned by P, V, dP, and dV axes, views the PPP as a two-dimensional slice of this four-dimensional hyperspace (11).

We previously used a harmonic oscillator as a kinematic analog for both diastolic (18, 19, 28, 34) and systolic (44) physiology. The kinematic approach was used to derive and validate the causal, linear relation between the peak transmitral flow-to-peak mitral annular velocity ratio (E/E′) and LVEDP (35), the noninvasive determination of stiffness (dP/dV via the kinematic parameter *k*) (32, 34), and the load independence of *E*_{max} (44).

In this work we utilize the known relation between model stiffness and the geometric attributes (i.e., semimajor and minor axes) of the loops generated by the motion of the oscillator in the kinematic phase plane (Fig. 1*A*) and apply them to the similarly shaped loops observed in the PPP generated by physiological pressure data (Fig. 1, *B* and *C*). The kinematic phase plane for a simple harmonic oscillator plots velocity (d*x*/d*t*) vs. position [*x*(*t*)] of the oscillator. The harmonic oscillator's kinematics is governed by the (mass normalized) equation of motion: (2) where *c* is damping and *k* is stiffness. Both *c* and *k* are mass-normalized (per gram) constants allowing determination of two independent parameters from the data instead of requiring the a priori specification of a third parameter (mass) and rescaling these accordingly (29, 30, 34). It is straightforward to proceed by analogy and apply the kinematic phase plane-derived algebraic expression for stiffness (*k*) in the PPP (by replacing displacement *x* with pressure P), deriving novel, analogous stiffness parameters that utilize data from isovolumic and nonisovolumic periods, thereby quantitating ventricular-vascular properties.

### Glossary

- SVVC
- Systolic ventricular-vascular coupling
- DVVC
- Diastolic ventricular-vascular coupling
- P
_{LV}(*t*) or P_{Ao}(*t*) - Pressure of left ventricle (LV) or aorta (Ao) (mmHg)
- dP
_{LV}/d*t*or dP_{Ao}/d*t* - Time derivative of LV or Ao pressure (mmHg/s)
- dP
_{LV}/d*t*^{+}or dP_{Ao}/d*t*^{+} - Peak positive value of time derivative of LV or Ao pressure (mmHg/s)
- dP
_{LV}/d*t*^{−}or dP_{Ao}/d*t*^{−} - Peak negative value of time derivative of LV or Ao pressure (mmHg/s)
- PPP
- Pressure phase plane
- dP/dV
- Chamber stiffness (mmHg/ml)
*K*_{LV}^{+}or*K*_{Ao}^{+}- Stiffness analog derived from dP/d
*t*^{+}(upper half of PPP) for LV or Ao (s^{−2}) *K*_{LV}^{−}or*K*_{Ao}^{−}- Stiffness derived from dP/d
*t*^{−}(lower half of PPP) for LV or Ao (s^{−2}) - τ
- Time constant of ventricular isovolumic relaxation (ms)
- κ
- Time constant of vascular (aortic root) relaxation; related to windkessel resistor-capacitor (RC) constant (ms)
- EF
- Ejection fraction (%)
- SW or
- Stroke work—the area of the pressure-volume loop (mmHg × ml)
- P
_{LV max}and P_{Ao max} - Maximum LV or Ao pressure (mmHg)
- P
_{LV min}and P_{Ao min} - Minimum LV or Ao pressure (mmHg)
- P
_{es} - End-systolic pressure (mmHg)
- LVEDP
- Left ventricular end-diastolic pressure (mmHg)
- SV
- Stroke volume (ml)
- ESPVR
- End-systolic pressure-volume relationship
- EDPVR
- End-diastolic pressure-volume relationship
*E*_{max}- Maximum elastance (mmHg/ml)
*E*_{a}- Windkessel metric of arterial function, commonly referred to as effective arterial elastance; defined as the ratio of P
_{es}and SV (mmHg/ml) *k*- Doppler E wave-derived LV stiffness; computed via parameterized diastolic filling formalism (s
^{−2})

## METHODS

#### Data acquisition.

Data from 13 patients were selected from an existing Cardiovascular Biophysics Laboratory database of simultaneous high-fidelity micromanometric pressure and Doppler echocardiography recordings (3, 9, 34, 38, 39, 49). Subjects were chosen specifically because of good-quality (low noise) data; demographics are listed in Table 1. All data were collected according to a protocol approved by the Washington University Medical Center Human Studies Institutional Review Board with informed consent obtained from all subjects. Data were acquired during routine cardiac catheterization for evaluation of coronary artery disease.

Data acquisition methodology has been previously described (3, 9, 34, 38, 39, 49). Briefly, high-fidelity simultaneous LV pressure and aortic root pressure and volume measurements are recorded via 6-Fr dual micromanometer-tipped pigtail pressure-volume conductance catheters (Millar Instruments, Houston, TX). Two transducer control units (TC-510 and TCB-500, Millar Instruments) are used to hydrostatically calibrate the transducers via submersion just below the surface of a 37°C normal saline bath. Pressure data from the transducers are fed into a clinical amplifier and imaging system (Quinton Diagnostics, Bothell, WA). Conductance catheterization signals are fed into a Custom PC via a standard research interface (Sigma-5, CD Leycom, Zoetermeer, The Netherlands). After appropriate preparation, femoral arterial access is obtained, a suitable valved sheath is placed in the artery, and the catheter is advanced under fluoroscopic control in a retrograde fashion across the aortic valve and into the mid-LV. The distal of the catheter's two pressure transducers is located in the LV for acquisition of ventricular pressure [P_{LV}(*t*)], while the second (proximal) transducer is located in the aortic root for acquisition of simultaneous vascular pressures [P_{Ao}(*t*)]. All hemodynamic data were acquired at a sampling rate of 200 Hz before the administration of any contrast agents. After data acquisition, routine clinical catheterization ensued, including ventriculography and selective coronary angiography. Conductance-derived volume signals were calibrated via angiographically derived, appropriately calibrated volumes. EF and SV were similarly calculated via angiographically derived, calibrated volumes.

#### Phase plane-derived stiffness analog.

Time-varying elastance [*E*(*t*)] is defined as *E*(*t*) = P(*t*)/[V(*t*) − V_{o}] where P(*t*) and V(*t*) are the time-varying pressure and volume and V_{o} is the empirically derived unstressed volume. (1, 7, 44, 51). Although this defines a time-varying elastance (or stiffness) that varies during isovolumic periods, physiological stiffness is conventionally defined as the slope dP/dV of the P-V relation. Equivalently, it can be computed from (dP/d*t*)/(dV/d*t*) for simultaneously measured values of pressure and volume. However, during isovolumic contraction or relaxation dV = 0; thus the conventional definition of stiffness fails.

From the physiological and kinematic modeling perspective the heart functions as an oscillator whose pressure and volume outputs have well-defined phase relations (18, 19, 28, 34, 44). By considering the geometric features of the loop inscribed in the kinematic phase plane for an ideal oscillator having known stiffness (frequency), we propose the same method for deriving an analog stiffness from the same geometric features of the PPP loop generated by the LV. The method avoids the “isovolumic catastrophe” generated by dP/dV as the definition of stiffness. For undamped (*c* = 0) oscillation and suitable initial conditions, the solution to *Eq. 2* is: (3a) and (3b) where *A* is the initial displacement and ω is the frequency of oscillation given by the mass-normalized stiffness (ω = ). Note that the ellipse-shaped loop in the kinematic phase plane traces a clockwise trajectory (Fig. 1*A*); the loop in PPP is also inscribed in a clockwise fashion, reflecting events of the cardiac cycle (Fig. 1, *B* and *C*). From the kinematic phase plane, we note that the intercept of the loop on the velocity d*x*/d*t* and displacement *x*-axis is related to oscillator stiffness via the relation: (4) Thus the square of the ratio of the maximum *y*-axis position of the loop (*A*ω) to one-half the width on the *x*-axis (*A*) allows computation of stiffness.

By analogy (replacing *x* and d*x*/d*t* by P and dP/d*t*), we define the PPP-derived stiffness analog as the square of the peak derivative of pressure divided by one-half the difference between the minimum and maximum pressure [ΔP = (P_{max} − P_{min})/2] or: (5) PPP-derived ventricular stiffness can be obtained from data in the upper half (+) or lower half (−) of the phase plane. For the LV, this includes isovolumic contraction (*K*_{LV}^{+}) via dP_{LV}/d*t*^{+} in the upper half of the phase plane, or from the lower half of the phase plane it includes isovolumic relaxation (*K*_{LV}^{−}) via dP_{LV}/d*t*^{−} (see Fig. 1). This allows us to utilize geometric features, i.e., characteristic dimensions of phase plane loops. The form of *Eq. 5* suggests that greater dP/d*t* indicates greater stiffness. Because our derivation (*Eqs. 3*–*5*) of analog stiffness uses an undamped oscillator, it does not require a forcing function to maintain motion. A less idealized and more physiological model would use a forced, damped nonlinear oscillator. However, in the steady-state situation (stable heart rate and stable blood pressure), the analog of a forced, damped nonlinear oscillator inscribes a steady-state loop having stable characteristic dimensions, similar to those of the ideal unforced oscillator, permitting the use of *Eq. 5* (57). Utilizing analogous features of the loop inscribed by the aortic root pressure contour in the PPP, we can obtain PPP-derived values for vascular stiffness during early ejection (*K*_{Ao}^{+}) via dP_{Ao}/d*t*^{+} (upper half of aortic phase plane loop) and during late ejection and diastole (*K*_{Ao}^{−}) via dP_{Ao}/d*t*^{−} (lower half of aortic phase plane loop). In contrast, the PPP-derived analog stiffness parameter, obtained from the dimensions of the loop defined by *Eq. 5* and encompassing isovolumic intervals, can also serve as a functional stiffness analog based on its kinematic derivation (Fig. 1*A*) and its physiological interpretation in the PPP.

We note that these phase plane-derived parameters depend only on the typical dimensions (height, width) of the loop and not on where, relative to the coordinate origin, the loop is located. Accordingly, they constitute relative rather than absolute indexes. Similarly, we note that this analog parameter is reported as a constant, whereas physiological stiffness is actually time varying. Because an oscillator having constant stiffness (i.e., a fixed spring constant) inscribes a closed loop (ellipse) in the kinematic phase plane (Fig. 1*A*) and because the heart, with time-varying stiffness, similarly inscribes a closed loop in the PPP (44), our ability to determine stiffness analogs based on loop dimensions reconciles the apparent inconsistency of characterizing time-varying stiffness as a (lumped) constant. In mathematical terms, approximating the actual PPP loop as an idealized ellipse (44) can be viewed as the leading term in a series expansion.

#### Phase plane-derived relaxation analog.

The terminal portion of the isovolumic pressure contour is characterized in terms of the time constant of isovolumic relaxation via *Eq. 1*. This is the current standard measure of LV relaxation. Differentiating, we note that dP_{LV}/d*t* is a linear function of P(*t*): (6) By convention, pressure data commencing ≈5 ms after peak negative dP/d*t* (dP/d*t*^{−}) to ≈5 ms before mitral valve opening (12, 24) was used to determine the slope of the linear regression in the PPP. Whereas the rate of pressure decay in the LV (relaxation) is well known via τ, the rate of pressure decay of the periphery (vascular relaxation) is characterized as a windkessel (45). Whereas the pressure decay in the LV is associated with Ca^{2+} sequestration and cross bridge uncoupling (9, 26) modulating actual relaxation of myocytes, pressure decay in the aorta is not specifically and strictly related to “relaxation” of smooth muscle. However, diastolic pressure decay in the aortic root corresponds to “recoil” or “relaxation” of elastic (windkessel) elements of the vasculature (4) whose kinematic behavior—manifesting as a pressure decay, in analogy to LV relaxation also manifesting as a pressure decay—can be assessed in the PPP. In the ascending aorta (root), the measured pressure contour is conventionally modeled as a two-element windkessel and can be solved for a resistor-capacitor (RC) time constant (53), which we define as κ. Thus the pressure decay (relaxation) in the aorta (after aortic valve closure) is fit by the mathematically identical relationship: (7) where κ is a time constant of aortic “relaxation,” P_{Aoo} is a pressure constant and P_{Ao∞} is the pressure asymptote. The method is identical to determination of τ for the LV and is applied to the appropriate segment of the aortic root PPP loop (≈5 ms after aortic valve closure to ≈5 ms before aortic valve opening) (Fig. 1*C*).

#### Data analysis.

Data were analyzed offline via a custom analysis program (Matlab 6 Mathworks, Natick, MA). For each subject, at least 10 cardiac cycles of simultaneous LV and aortic root pressure data were analyzed and the results were averaged. Pressure was recorded, and dP(*t*)/d*t* was calculated numerically for both the LV and the aorta. The parameters determined directly from the PPP were (3) maximum (P_{LVmax} and P_{Aomax}) and minimum (P_{LVmin} and P_{Aomin}) pressure, peak positive dP/d*t* and peak negative dP/d*t* for the LV (dP_{LV}/d*t*^{+} and dP_{LV}/d*t*^{−}) and aorta (dP_{Ao}/d*t*^{+} and dP_{Ao}/d*t*^{+}), left ventricular end-diastolic pressure (LVEDP), and end-systolic pressure (P_{es}), defined as the minimum pressure of the dicrotic notch (Fig. 1*C*).

PPP-derived stiffness was derived via *Eq. 5*, including both isovolumic contraction (*K*_{LV}^{+}; upper half of PPP) and relaxation (*K*_{LV}^{−}; lower half of PPP) periods and for initial ejection *(K*_{Ao}^{+}; upper half) and late ejection (*K*_{Ao}^{−}; lower half) for the aortic PPP loop. Ventricular relaxation τ was measured via the phase plane (12, 24); vascular relaxation κ was measured from ≈5 ms after the dicrotic notch to ≈5 ms before aortic valve opening.

Traditional catheterization-based measures of stiffness were also considered. Arterial stiffness was calculated via effective arterial elastance *E*_{a}: (8) *E*_{a} is actually a windkessel metric, not a direct measure of arterial elastance, being more dependent on arterial resistance than compliance (51). However, it has generally been characterized in the literature as an acceptable clinical and experimental surrogate of elastance; we used *E*_{a} in order to facilitate comparison to previous studies of arterial stiffness (8, 24, 25, 27, 41, 54). Maximum elastance (*E*_{max}) and EDPVR were not measured because of the lack of load variation (volume) during data acquisition; single-beat and noninvasive methods of calculating *E*_{max} (7) resulted in unstable or inconsistent (*E*_{max} < 0.1 or >100) results.

To obtain validated LV diastolic stiffness values, we applied the parameterized diastolic filling (PDF) formalism (10, 19, 25, 31, 32, 34, 38, 49) to transmitral Doppler E waves. Briefly, in the PDF formalism, *Eq. 2* is solved (*c* ≠ 0) for the E-wave transmitral Doppler velocity contour, *v*(*t*), as: (9) where ω is the mass normalized frequency, *ω* = /2. The PDF formalism predicts transmitral flow velocity during early rapid filling (E wave). The viscoelastic damping/relaxation parameter *c* computed from the Doppler E wave has been shown to play a role in chamber stiffness (32, 34), causing a phase shift between pressure and flow (58). It also manifests in characterizing the cardiovascular effects of diabetes (10, 49), hypertension (31), caloric restriction (39), exercise, and heart failure (38, 48). Also, recent results show that isovolumic relaxation, expressed as 1/τ, is linearly related (*r* = 0.71) to the viscoelastic damping/relaxation of the ventricle (*c*) determined from the E wave (9). We analyzed E waves from five cardiac cycles from each subject, using model-based image processing via a custom LabVIEW (National Instruments, Austin, TX) interface to determine the E wave-determined chamber stiffness parameter *k* (3, 9, 19, 34, 49).

To validate our PPP-derived analog indexes of stiffness, we determined the following linear correlations: stiffness analog for the LV (*K*_{LV}^{+}, *K*_{LV}^{−}) vs. early filling-derived stiffness (*k*) and traditional parameters of LVEDP and EF and vascular stiffness analog (*K*_{Ao}^{+}, *K*_{Ac}^{−}) vs. E_{a}. Because single-beat data-based *E*_{max} determination (7) yielded inconsistent results, we used (kinematic, E wave-derived chamber stiffness) *k* (32, 34) and EF as standard LV indexes and E_{a}, which is commonly referred to as arterial elastance. Contraction-relaxation coupling was assessed by comparing *K*_{LV}^{+} vs. *K*_{LV}^{−}. τ is a well-established index of relaxation (12, 15), and κ is the two-component windkessel. However, the windkessel is considered as an interplay between stiffness and relaxation properties. Therefore, we also examined the relations between relaxation parameters (τ and κ) and PPP stiffness analog parameters (*K*_{LV}^{±}, *K*_{Ao}^{±}) and early filling-derived stiffness (*k*).

#### Diastolic ventricular-vascular coupling hypothesis.

Previous work in SVVC shows that the heart adapts to load in a long-term, steady-state manner. For example, for altered, chronic vascular loads (hypertension), the heart will adapt and alter its ESPVR. Evidence of contraction-relaxation coupling also exists at the organ level (22) and the cellular level (5, 6). Thus we hypothesize that the diastolic properties of stiffness and relaxation must also be coupled.

Our hypothesis of DVVC was assessed by comparing LV and aortic PPP-derived stiffness (*K*_{LV}^{−}, *K*_{Ao}^{−}) indexes; vascular stiffness (*K*_{Ao}^{±}) was compared to LVEDP and early diastolic stiffness *k* (32, 34). SVVC was also evaluated via comparison of *K*_{LV}^{+} and *K*_{Ao}^{+}. The hypothesis that LV relaxation and peripheral relaxation rates (i.e., pressure decay) are coupled was tested by determining the correlation between τ (LV relaxation) and κ (vascular “relaxation”).

## RESULTS

Table 1 lists clinical variables along with ranges of measured variables for all subjects. Although the PPP-based loops have many features that can be designated as “parameters,” we restricted our comparisons to features of the loops having established physiological analogs such as stiffness and relaxation.

#### Validation of PPP-derived analogs.

Analog stiffness during isovolumic relaxation (*K*_{LV}^{−}) was correlated with early filling-derived *k* [*k* = 0.283(*K*_{LV}^{−}) − 2.67, *r* = 0.79, *P* < 0.004]. This relation to early filling-derived stiffness, which is known to correlate with dP/dV (32, 34), further validates our PPP-based analog of stiffness. The observed ventricular stiffness relations and their correlation with standard indexes suggest that the PPP-based approach can determine stiffness during isovolumic periods and is not hindered by the fact that dV = 0. *K*_{LV}^{+} had significant correlation with *K*_{LV}^{−} [*K*_{LV}^{−} = 0.919 We expect that our PPP-based analogs of stiffness during contraction and relaxation should correlate with each other, based on previous work showing correlation between dP_{LV}/d*t*^{+} and dP_{LV}/d*t*^{−} (12) and as a feature of contraction-relaxation coupling (22). It is expected that and correlate with EF; this is in keeping with the Frank-Starling mechanism in that a higher EF associated with an increased SW requires that the ventricle generate sufficient contractility during the isovolumic periods to generate the required dP/dV to overcome the load and eject at valve opening.

Vascular stiffness during early ejection achieved reasonable correlation with *K*_{Ao}^{−} [*K*_{Ao}^{−} = 0.272 (*K*_{Ao}^{+}) + 271, *r* = 0.59, *P* < 0.04] and with *E*_{a} [*K*_{Ao}^{+} = −571(*E*_{a}) +1,090, r = 0.67, P < 0.02]. The relation between early and late aortic stiffness is expected as they measure nearly the same parameter. The windkessel metric (*E*_{a}) is typically used as a pressure-volume loop-derived surrogate of aortic stiffness (8, 24, 25, 27, 41, 54). Our observation of significant correlation between *E*_{a} and provides reassuring, independent corroboration that our phase plane-derived index has validity as a measure of aortic (vascular) stiffness.

τ correlated with both K_{LV}^{+} [τ = −0.129 (K_{LV}^{+}) + 111, r =0.76, P < 0.003] and K_{LV}^{−} [τ = −0.137 (K_{LV}^{−}) + 144, and with early filling-derived *k* [τ = −0.275(*k*) + 104, *r* = 0.69, *P* < 0.02], providing evidence of coupling between stiffness and relaxation. Although independent in the mathematical modeling sense (30, 52), in the in vivo setting relaxation and stiffness are expected to be causally related via calcium-mediated mechanisms. For example, relaxation is controlled, in part, by calcium cycling (26); titin, which modulates stiffness, also has calcium-dependent properties (6). Vascular relaxation κ was more dependent than τ on other measured parameters; κ did not correlate (*P* > 0.05) with , , or early filling-derived *k*. However, *κ*'s modest correlation to *E*_{a} [κ = −228(*E*_{a}) + 582, *r* = 0.56, *P* < 0.05] suggests that stiffness, which affects wave reflections and windkessel parameters (45), also affects relaxation. This observation is reassuring and has been previously described (6, 26, 44).

#### Diastolic ventricular-vascular coupling of stiffness.

Figure 2*A* illustrates the expected SVVC relationship via our analog measures of stiffness , whereas Fig. 2*B* shows a clear DVVC relationship . A very strong relationship is observed between the vasculature and ventricular early diastolic stiffness *k* and LVEDP[LVEDP = 0.015 (K_{Ao}^{+}) + 25.7, *r* = 0.63, *P* < 0.03], providing further evidence for aortic and ventricular diastolic coupling. These observations support our hypothesis that DVVC can be characterized in terms of the stiffness analog indexes derived from the PPP.

#### Diastolic ventricular-vascular coupling of relaxation.

Figure 3 illustrates DVVC in terms of relaxation indexes. Ventriculography indicated significant wall motion abnormalities in three subjects, two of whom had abnormally high values of τ (τ > 80 ms); these subjects were separated from the group because of these abnormalities. These pathological examples imply that there is a “normal” DVVC relation, whereas impaired wall motion modifies the normal DVVC relationship. In the setting of normal LV function, there is a clear and significant correlation (*r* = 0.81) between the rate of isovolumic ventricular relaxation (τ) and the rate of relaxation (κ) of the vasculature during diastole.

## DISCUSSION

#### PPP-derived stiffness parameters.

Stiffness is conventionally defined via a differential change in volume, e.g., *K*_{LV} = dP/dV (28, 32, 34, 36). However, this definition makes it impossible to assess global chamber stiffness during isovolumic (dV = 0) periods. In contrast, linear oscillators have constant stiffness coefficients measured in units of mass per second and can be easily derived from Hooke's law. By applying the analog of the kinematic phase plane-derived geometric features of an ideal oscillator's loop in the PPP, we determined new, analogous PPP-derived parameters of stiffness that utilize PPP data and encompass both isovolumic relaxation and filling periods. Furthermore, because P and V can have different phases as a result of viscoelastic effects, stiffness, defined either as dP/dV or *E*(*t*) = P(*t*)/[V(*t*) − V_{o}], relies on the assumption of minimal viscosity (or nearly steady state) change in P and V. However, previous work on viscous and elastic properties of the myocardium (55, 56) shows that viscoelastic effects, are in general, not negligible. In diastole, for example, viscoelastic effects during relaxation cause a phase shift between pressure and flow (volume) (58). Therefore, using an alternative expression for lumped stiffness (such as *Eq. 5*) may improve physiological characterization by allowing for the presence of phase differences between elastic and viscous effects. Vascular stiffness correlated well with the gold-standard measure of effective arterial elastance (*E*_{a}). Furthermore, ventricular stiffness during isovolumic relaxation correlated well with early diastolic stiffness (*k*).

The kinematic modeling approach for characterization of physiology has previously shown that filling velocity can be accurately characterized via the motion of a harmonic oscillator (18, 19, 28). Furthermore, by modeling the whole heart kinematically, new insight into load independence of maximum elastance (*E*_{max}) can be elucidated by demonstrating that the (kinematically derived) ESPVR (*E*_{max}) analog depends only on intrinsic properties of the oscillator (such as stiffness and damping parameters), rather than on initial conditions (load) (44). Kinematic features and springs are also part of the three-component modified Hill or Maxwell model (2, 13, 20), and linear spring features are evident at the molecular level, manifested by ankyrin repeats (33). Additional legitimacy for the kinematic (bidirectional, linear spring) approach to characterizing diastolic suction-initiated filling in the intact heart resides in its ability to anticipate (28) the relationships ultimately observed in the kinematic behavior (recoil) of myocytes due to titin, which has been shown to behave as a bidirectional linear spring that obeys Hooke's law and generates a “pushing” force during early filling (17).

#### Vascular stiffness and relaxation.

Although aortic stiffness may be continuously measured via dP/dV, it is difficult to assess dV because while volume enters the aorta in systole, volume simultaneously leaves the arterial circulation and enters the venous system. Thus we opted for assessing vascular stiffness via the PPP. It is mechanically reasonable that ventricular dP/d*t*^{+} and dP/d*t*^{−} should be related to vascular stiffness. Mechanically, as the LV ejects blood a more compliant aorta would accommodate the volume and thereby reduce ventricular dP/d*t*^{+}, whereas a stiffer vessel would require increased rate of pressure development and resist distension, thus requiring a greater ultimate pressure and dP/d*t*^{+}. Vascular properties are often measured via arterial impedance and windkessel attributes. Our approach to stiffness characterization is not meant to replace these methods but to complement them and to facilitate comparison of ventricular and vascular properties. The relaxation phase of the aortic pressure phase plane loop is measured in accordance with the two-element (RC) windkessel model. Thus our PPP-based analogs of stiffness and relaxation can characterize vascular properties.

#### Ventricular vascular coupling of stiffness.

To further characterize the systolic coupling between ventricular and vascular properties, investigators have used *E*_{max} and the windkessel metric or effective vascular elastance *E*_{a} (8, 24, 25, 27, 41, 54), which is supported by our SVVC finding (Fig. 2*A*). In general, characterization of simultaneous diastolic ventricular and diastolic aortic attributes has not been considered. To more fully characterize the ventricular and aortic stiffness and relaxation relationships we assessed their correlation via analogous measures of stiffness and relaxation extracted from the PPP. Our findings unambiguously indicate that diastolic ventricular-vascular stiffness indexes are related (Fig. 2*B*). We caution that these results are based on averaged data, from multiple consecutive beats obtained in a clinical setting, in a near steady-state condition. Thus the observed coupling should not be interpreted as providing information about short-term, beat-to-beat coupling or instantaneous response to acute changes but rather apparent long-term adaptation between the ventricle and vasculature. However, it is reasonable from a systems physiology perspective that an increase in vascular stiffness would require an increase in ventricular stiffness to maintain cardiac output. Similarly, an increase in ventricular stiffness would necessitate an increase in vascular stiffness to accommodate the pressure and volume load.

These observations reveal and quantify the coupling between the ventricle and vasculature (Fig. 2). Our (analog) measures of stiffness and relaxation incorporating data from isovolumic periods are consistent with prior observations. For example, indirect evidence of coupling between systolic and diastolic measures of stiffness was observed by Chen et al. (8). We recently showed (9) that 1/τ, determined during isovolumic relaxation, is related to the decay rate (parameter *c*) of the Doppler E wave (i.e., relaxation during filling). Our current results show that the diastolic aortic (*K*_{Ao}^{−}) and ventricular relaxation analogs of stiffness correlate with early diastolic stiffness *k*. These observations indicate that VVC occurs in the context of systolic-diastolic coupling.

The methods proposed here are general. They may be useful in phenotypic characterization of ventricular-arterial diseases. For example, it has been shown that diastolic function is altered before the onset of systolic dysfunction, manifesting as diminution of the LV EF, in diabetic cardiomyopathy (47). However, it is unclear whether diastolic stiffness and relaxation properties similarly influence vascular properties. Similarly, hypertension with LV hypertrophy is known to alter diastolic function. Our proposed methods will assist in elucidating the causal relations that explain the difference between “diastolic heart failure,” attributed to intrinsic diastolic myocardial properties (15), and “heart failure with normal EF,” whose proponents provide evidence that it is the properties of the vasculature that affect ventricular stiffness and relaxation properties (27).

#### Ventricular-vascular coupling of relaxation.

A novel finding is that the relaxation analogs for both the ventricle and the vasculature are correlated. Ventricular relaxation (τ) is measured after dP/d*t*^{−} has been attained and after the aortic valve has closed; vascular relaxation (κ) is measured in the aorta, from aortic valve closure until aortic valve opening. The coupling, manifesting as the observed correlation in Fig. 3, may seem counterintuitive because of the mechanical barrier (aortic valve) between how fast pressure drops in the closed aortic root vs. how fast the pressure drops in the isovolumically relaxing ventricle. Because our analysis determines averaged, rather than beat-to-beat, values of the parameters (16, 21), the observed correlation indicates that coupling exists, which likely conveys time-averaged, long-term response to loading. In the systems physiology sense, the ability to homeostatically maintain a steady mean arterial pressure places constraints on the rates at which LV pressure and aortic pressure must decay. If LV pressure drop took too long relative to the time available for filling, the LV could not aspirate blood from a low-pressure pulmonary system without undue elevation of filling pressures. Similarly, if aortic pressure decay were too fast or too slow, a steady-state mean arterial pressure could not be maintained and would diminish or escalate. When considered in the context of these homeostatic control mechanisms, the observed correlation between τ (LV) and κ (vasculature) becomes more meaningful.

Interestingly, the exact physiological explanation and mechanism of the observed coupling are as yet unclear. Also, three of our subjects displayed segmental wall motion abnormalities (hypokinesis) visible on ventriculography (2 severe, 1 slight). Their κ-τ relation appears to deviate from the regression relation observed for the normal group, suggesting a modified method of coupling between the ventricle and vasculature. Because of this deviation, we speculate that the observed τ-κ correlation is an intrinsic property of normal tissue. However, it would be possible to test whether the τ-κ correlations is an inotropy-dependent extrinsic coupling or a remodeling-dependent intrinsic coupling by examining it in experimental conditions involving inotropic stimulation or exercise, in pre- and postinfarction cases, or in the setting of peripheral vascular disease. Furthermore, aortic pressure decline is not solely based on relaxation of vascular smooth muscle but also involves elastic recoil of the large vessels. This, along with the apparent modification of the κ-τ relation in the setting of wall motion abnormalities, suggests that this relaxation-based measure of DVVC is distinct from the stiffness-based DVVC observed.

#### Limitations.

The kinematic modeling strategy uses idealized, global, lumped-parameter methods. Therefore, our indexes convey global (rather than regional or segmental) chamber and vascular (no branches) attributes. For example, our assumptions for deriving the PPP analog for stiffness (*Eq. 5*) from the kinematic phase plane assume an idealized steady state. Another approach for model-based stiffness determination during isovolumic contraction may employ a “forcing” function as the analog for systole (44) but should not affect the steady-state PPP-derived loop dimensions. Furthermore, the shape of the aortic PPP loop is only approximately similar to the kinematic phase plane shape in its relaxation (dP/d*t* < 0) portion. For technical reasons we did not determine *E*_{max}. In an effort to limit arterial time associated with data acquisition, we relied solely on angiographically derived and suitably calibrated volumes. We did not “overdetermine” LV volume by using aortic root flow velocity or transit-time flow probes in conjunction with ventriculography. Although such data may assist in computing more accurate volumes and flows, the requirement to minimize arterial time was deemed a higher priority.

Additional analysis via altered preload by Valsalva or Muller maneuvers and direct measures of beat-to-beat variation could further elucidate the transient effects of coupling, as opposed to the long-term, steady-state results seen in this study. Although heart rate per se was not varied in this study and heart rate variation is likely to affect load transiently, it is unlikely that slight heart rate variations over the long term would affect the observed, time-averaged relations. Studies altering heart rate or load, via infusion of positive inotropes or vascular constrictors/dilators, may further elucidate the short-term features of coupling mechanisms but were not within the scope of the present study.

This physiological study used in vivo human data, therefore limiting the interventions available. Future animal studies may allow increased characterization of VVC during isovolumic periods that cannot be performed in human studies. For example, a quick alteration of pressure in the ventricle or the vasculature just after aortic valve closure would help elucidate the short-term (beat to beat) vs. long-term (steady state) behavior of these “independent” systems and most clearly elucidate the source of the DVVC relation of relaxation. An alteration in vascular pressure or stiffness, independent of ventricular pressure, may also assist in determining whether vascular properties lead to development of heart failure with a normal EF.

#### Conclusions.

We investigated the coupling between the vasculature and the ventricle, including explicit contributions by isovolumic and diastolic ventricular properties. Using kinematic analogs, we derived and validated relative, rather than absolute, stiffness and relaxation analogs for the LV and the vasculature. Our derivation allows determination of a ventricular stiffness constant encompassing isovolumic (dV = 0) periods, thereby avoiding the “isovolumic catastrophe” imposed by defining stiffness as dP/dV. We found that systolic and diastolic LV stiffness and LV stiffness and aortic stiffness were significantly correlated. We also observed significant correlations between diastolic ventricular (τ) and diastolic vascular (κ) relaxation indexes. These results, in aggregate, underscore the utility of kinematic modeling and analysis of physiological data in the PPP and validate our hypothesis that DVVC can be characterized via PPP-derived indexes of stiffness and relaxation. Our observations elucidate mechanisms of ventricular-vascular and contraction-relaxation coupling by shedding light on the underlying physiological relationships in quantitative terms.

## GRANTS

This work was supported in part by the Whitaker Foundation, the National Heart, Lung, and Blood Institute (Grants HL-54179 and HL-04023), the Affiliate of the American Heart Association, the Barnes-Jewish Hospital Heartland Foundation, and the Alan A. and Edith L. Wolff Charitable Trust.

## Acknowledgments

The assistance of the Barnes-Jewish Hospital cardiac catheterization laboratory staff is gratefully acknowledged; we thank Peggy Brown for echocardiographic data acquisition and Yue Wu for assistance in the use of Matlab.

## Footnotes

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- Copyright © 2006 by the American Physiological Society