Abstract
10.1152/japplphysiol.00560. 2001.—Our laboratory has previously shown that it is possible to elucidate novel physiological relationships by analyzing the left ventricular pressure (P) contour in the phase [time derivative of P (dP/dt) vs. P] plane (Eucker SA, Lisauskas JB, Singh J, and Kovács SJ, J Appl Physiol 90: 2238–2244, 2001). To further characterize cardiac physiology, we introduce a method that combines Pvolume (V) and phase planederived information in physiological hyperspace. From fourdimensional (P, V, dP/dt, time derivative of V) hyperspace, we consider threedimensional embedding diagrams having dP/dt, P, and V as coordinate axes. Our method facilitates analysis of physiological function independent of inotropic state and permits assessment of PVbased relationships in the phase plane and vice versa. To test feasibility, the method was applied to murine hemodynamic data. As predicted from first principles, the area of the PV loop (ventricular external work) correlated closely (r = 0.97) with phase plane limit cycle area (external power). The PV planederived linear (r = 0.99) endsystolic PV relationship (maximum elastance) appeared linear in the phase plane (r = 0.85). We conclude that analysis of data in physiological hyperspace is generalizable: it facilitates quantitative characterization of ventricular systolic and diastolic function and can guide discovery of novel physiological relationships.
 pressurevolume analysis
 phase plane analysis
 nonlinear dynamics
 systolicdiastolic coupling
invasive assessment of cardiac function utilizing the left ventricular (LV) pressure (P) (LVP) waveform is usually limited to the measurement of selected points of the LVP contour and its time derivative (dP/dt). These values, obtained during cardiac catheterization, include maximum LVP (P_{max}), minimum LVP, P at diastasis, LV enddiastolic P, peakpositive dP/dt (p˙_{+max}), and peaknegative dP/dt (p˙_{−min}) and are illustrated in Fig. 1. Other indexes of LV function include the stroke volume (SV), the enddiastolic volume (V) (EDV), and the ejection fraction (EF) defined as EF = SV/EDV.
Additional physiological information can be obtained from the PV loop, shown in Fig. 2. The area enclosed by the PV loop for one cycle is the external work performed on the blood by the ventricle during one cardiac cycle (W)
Other parameters such as τ, the time constant of isovolumic relaxation, are computed by performing a least meansquare fit of an assumed exponential decay to a selected portion of the P contour. The physiological and clinical significance of these parameters and the indexes derived from them have been established and form a firm basis for ventricular function analysis and clinical decision making (2, 7).
In a previous report (3), our laboratory advocated a physical scientist's or applied mathematician's perspective of the heart. Accordingly, the heart was viewed as a nonlinear oscillator that generates an output (P) as a function of time. We recorded and analyzed the output of the oscillator (P as a function of time) using the methods of nonlinear dynamics (8), which are familiar to physiologists and physical scientists. In analogy with the characterization of physical nonlinear oscillators whose equations of motion are known (8, 11), we selected the phase plane as the arena in which to perform our analysis of ventricular P data.
The phase plane is a graphical representation of a function having a variable, x, in which the time derivative, dx/dt, is the ordinate plotted againstx as the abscissa. For a precisely periodic or nearly periodic function, the phase plane plot forms a closed trajectory, called a limit cycle. In our application, the dP/dt is plotted along the yaxis against the P plotted along thexaxis. As shown in Fig. 1, the ventricular P completes one oscillation for each cardiac cycle; hence, the phase plane plot inscribes a limit cycle.
Phase plane methodology has been used in engineering and physics to characterize the trajectories of nonlinear systems described by differential equations when the relationship between velocity and distance, or between angular velocity and angle, is of interest. Applications include investigation of the nonlinear behaviors of a harmonically driven, linearly damped pendulum under various initial conditions and identification of the stable solutions (11). The phase plane has also been used to study a discrete nonlinear Schrödinger equation to identify bounded periodic orbits (24). In simulations of plasma collisions, numerical methods developed to model the effects of the collisions utilize the phase plane to solve a simplified FokkerPlanck operator (6).
Phase plane analysis has also been extended to applications in biology, medicine, and physiology. For instance, solutions to Volterra's population model representing the effects of toxin accumulation on a species for arbitrary intraspecies competition, birth rates, and levels of toxins can be analyzed in the phase plane (22). Biomechanical analysis of leg motion during drop landing has utilized the phase plane to characterize the damping performance of leg muscles when used for deceleration and power dissipation (13). Phase plane plots of physiological signals have been previously used to characterize selected aspects of the electrical activity of the heart. In this analysis, the myocardium is viewed as an excitable medium, thereby allowing the methods of nonlinear dynamics to be used to characterize the heart's propensity to certain arrhythmias (25).
Phase plane plots of canine LVP restricted to the isovolumic relaxation portion rather than the entire cardiac cycle have been used to compare a logistic model to the conventional exponential model of P decay (12). The phase plane has also been used to assess the consistency of an exponential fit in logistic models of P decay to the isovolumic relaxation portion of LVP in humans with heart failure (16). The method was also employed to discern respiratorycardiac coupling and to characterize LV function (10). More recently, the phase plane method has been used to assess the discordance between dynamic and passive PV relations in idiopathic cardiomyopathy (15) and for characterization of the relation between p˙_{−min} and ejected aortic blood momentum (20).
Attributes of ventricular function that are not easily discernible when the data are viewed in the usual P vs. time (t) format (Fig.1 A) can be deduced by analysis of limit cycle attributes in the phase plane (Fig. 1 B). Previous attributes analyzed (3) were 1) correspondence between limit cycle area (LCA) and EF; 2) validity of the exponential P decay assumption during isovolumic relaxation; 3) symmetry ofp˙_{+max} and p˙_{−min}; 4) relationship between the P values at which p˙_{+max}(P_{C}) and p˙_{−min} (P_{R}) occur; and 5) comparison of P_{C} and P_{R} to P_{max}. Information about diastolic and systolic function, global ventricular function, and systolicdiastolic coupling that could not be discerned from the usual P vs. t display format could be easily visualized and quantitated by using the phase plane.
In the analysis of dynamic (physical) systems (8, 18), the concept of canonical variables and n dimensional phase space is usually employed. It is in analogy to these methods that we introduce the concept of physiological hyperspace. Although the number of dimensions can be arbitrary, we restrict our analysis to four dimensions spanned by variables frequently encountered in physiology: P, V, and their time derivatives dP/dt and dV/dt, respectively. Specifically, we consider threedimensional (3D) embedding diagrams spanned by dP/dt, P, and V. Other possible choices for 3D embedding diagram coordinate axes are addressed in the discussion.
METHODS
Our focus in this report is methodological and concerns the application of nonlinear dynamic methods to the analysis of physiological data. The hemodynamic data that we employ were acquired during a course of experiments and were made available for our use courtesy of the Mouse Core Physiology facility at our institution.
The method utilized for acquisition of murine hemodynamic data is similar to that of Feldman et al. (4). Briefly, mice were anesthetized with an intraperitoneal injection of a mixture of ketamine (80 mg/kg) and xylazine (16 mg/kg), intubated with a blunt 19gauge needle, and respirated at a rate of 85 times per minute. The right carotid artery was then dissected free and cannulated. Next, a 1.4F Millar pressure conductance catheter (model SPR719, Millar Instruments, Houston, TX) was advanced into the ascending aorta proximal to the aortic constriction and then advanced further, retrograde across the aortic valve into the LV. An incision was made below the sternum, and the inferior vena cava (IVC) was identified and isolated. To produce acute reduction in cardiac preload, a transient occlusion of the IVC was produced by external compression of the vessel with a rubberinsulated clamp. All PV loops recorded during either steadystate or vena caval occlusion conditions were acquired during apnea at end expiration at a sampling rate of 500 Hz.
The mouse physiology core laboratory has a dedicated Acuson (Mountain View, CA) Sequoia 256 echocardiography system. Our experience with this system over several years has enabled us to obtain consistent, highquality, short and longaxis images of the mouse LV, such that accurate estimations of LV cardiac muscle and chamber V can be derived. By using the model of a cylinder hemiellipse (22) to determine LV muscle V, we have shown good agreement between LV V calculated from the longaxis echocardiographic image of the mouse LV at end diastole compared with LV V determined by postmortem weight (r = 0.933; P < 0.001). Given our ability to determine murine LV V accurately, we developed a scheme to calibrate the conductance catheter system based on absolute LV V values determined echocardiographically. Importantly, a recently published study has found close agreement between LV V values determined by conductance catheter calibrated using the traditional V calibration line method compared with V values obtained using twodimensional (2D) echocardiographic methods. (4)
In our approach, the Millar conductance catheter system was calibrated for relative V units, as specified by the manufacturer's directions, before insertion. After placement of the conductance catheter in the mouse LV such that the distal conductance electrode was located in the apex and the proximal electrode was located in the outflow tract, a longaxis echocardiographic view of the LV was obtained with the animal in the supine position. A series of 15–25 steadystate and IVC occlusion PV loops was obtained by using the Millar ARIA1 conductance system with the controller frequency set at 20 kHz and the lowpass cutoff frequency set at 50 Hz. Conductance and P signals were digitized by BioBench software (National Instruments, Austin, TX) and stored for later offline analysis. Before analysis, the conductance catheter system's relative V data were recalibrated against the endsystolic V and EDV obtained for each mouse using the 2D echocardiographic technique described above.
Analysis of the data files was done offline in the Cardiovascular Biophysics Laboratory using LabVIEW. The format of LabVIEW facilitates rapid, interactive plotting of the PV contours and their analysis. Continuous or segmental sets of digitized, simultaneous PV (conductance catheter) data were imported into DeltaGraph as column vectors. Programs were written in LabVIEW to compute dP/dtnumerically from P(t) and to plot LVP limit cycles and PV loops. The programs could also calculate the τ, E_{max}, and external work performed by the LV. An illustrative sample of 1 s of continuous PV data and its derivatives is shown in Fig.3. P and V constitute simultaneous digitized data from the conductance catheter system. E(t) computed as P(t)/V(t), dP/dt, and dV/dt is also shown. 3D plots and linear regression analyses were performed by using DeltaGraph.
RESULTS
Correspondence of LCA to external power.
The external work (W = ∮PdV) performed by the heart can be calculated from the area of the PV loop for one cycle. The general (differential) expression for external mechanical power (W˙_{ex}) is the time rate of change of work (7,25)
The LCA inscribed in the phase plane is bounded by the P differential (P_{max} − minimum LVP) and the dP/dtdifferential (p˙_{+max} −p˙_{−min}). The expression for this area is given by
Physiological hyperspace.
The relationship between the LVP limit cycle and the PV loop can be further appreciated through the use of 3D embedding diagrams in physiological hyperspace spanned by dP/dtPV axes. A representative 3D embedding diagram for a normal mouse with 2D projections onto the phase plane, PV plane, and VdP/dtplane is shown in Fig. 5.
The equation describing the endsystolic PV relation is usually written as
The equation of a straight line in 3D space, through an arbitrary pointU
_{1} (x
_{1},y
_{1}, z
_{1}), wherex
_{1} = V_{1},y
_{1} = dP/dt
_{1}, andz
_{1} = P_{1}, is
DISCUSSION
Embedding diagram in hyperspace: the arena for phase planetoPV plane interaction.
Assessment of ventricular function by using hemodynamic data is well established (26). In contrast, the technical difficulties associated with PV data acquisition have limited its clinical acceptance. Development of the conductance catheter (1) and proposed methods to measure E_{max} by using singlebeat methods (21) and bilinearly approximated E(t) (17) have provided major advances. However, broad clinical application has not yet been attained, in part because of the lack of clinical studies relying on E_{max} as an index for clinical decision making.
To characterize the information content of the LVP tracing more fully and thereby gain additional physiological insight, we have advocated analysis of LV hemodynamics in the phase plane (3). Novel insights pertaining to 1) LCA to EF relation; 2) validity of the exponential P decay assumption during isovolumic relaxation; 3) symmetry of p˙_{+max} andp˙_{−min}; 4) relationship between the P values at which P_{C} and P_{R} occur; and5) comparison of P_{C} and P_{R} to P_{max} have been characterized.
In this report, we introduce the concept of fourdimensional (4D) physiological hyperspace spanned by P, V, dP/dt, and dV/dt. The particular 3D subset used for embedding was motivated by the desire to elucidate the relation between the phase plane and PV planederived indexes of LV function. Although many possible PVderived parameters exist, we elected to focus on two items: the external W (PV loop area)toLCA relation and determining whether the analog of E_{max} is also linear in the phase plane. These choices were motivated by the significance of W as a physiological index and the importance of E_{max} as a loadindependent index of contractility.
The highly linear regression relationship observed between W and its time derivative power, as measured by LCA (Eqs. 59 ), has physiological significance. It implies that a differential relation of the form dW/dt = aW + bcan be determined. Solving this relation for W implies that W must have exponential time dependence. This is an independent prediction of the method, validated by the observed linear relationship between W and LCA.
Furthermore, application of the method provided a unique opportunity to determine whether the analog of E_{max} in the phase plane can also be approximated by a linear relationship. To facilitate 3D spatial visualization of the data embedded in hyperspace, we include a stereographic projection of the data of Fig. 5 in Fig. 6. The location of the regression line defining E_{max} is shown. It appears in the portion of the embedded surface as it curves from the PV plane into the dP/dtP plane. The proposed method is well suited for detailed characterization of topological aspects of this portion of the 2D surface in 3D space and determination of its physiological significance as projected onto other planes. Whether the endsystolic PV relation is curvilinear (9) or linear (12, 17) remains a topic of ongoing investigation. For data in the physiological range, a linear approximation is used, although reliance on linear regression for E_{max} may generate negative values of the “stressfree” V_{o}. In the present example, we note that the r value in the PV plane for the best linear fit for E_{max} wasr = 0.99, as can be observed in Figs. 2 and 5. The linear best fit for the same set of points in the phase plane yieldedr = 0.85. The (small) decrease in r value is consistent with curvilinearity of the endsystolic PV relation. Additional work remains to elucidate fully the extent to which a linear approximation to E_{max} is applicable in other coordinates of hyperspace.
Alternative choices for hyperspace coordinates.
From the perspective of nonlinear dynamics, considering P and V as canonical variables of ventricular function, it is natural to include their derivatives dP/dt and dV/dt. The 4D physiological hyperspace for ventricular function analysis is spanned by P, V, and their derivatives. The mechanical (onedimensional) analogs are force (F) and distance (x) and their time derivatives dF/dt and velocity (dx/dt). Because of the physiological relevance of the PV plane and the phase plane, we investigated embedding data in 3D dP/dtPV space in this report. Three additional choices for 3D embedding using P and V and their time derivatives remain. These are PVdV/dt, PdV/dtdP/dt, and VdV/dtdP/dt. The full richness of physiological relations that can be realized by analysis of 3D embedding diagrams using these axes has yet to be evaluated. In addition, the topological attributes of physiological data using these choices remain to be investigated. The particular 3D space spanned by PVdV/dt,providing work (PV)flow (dV/dt) information, is likely to be particularly interesting. This is in part due to our ability to obtain V and dV/dt data noninvasively in the form of echocardiographic transmitral Doppler (inflow) and aortic Doppler (outflow). Figure 3, bottom, shows features usually obtained by Doppler echocardiography. The two positive (upward) deflections in diastole correspond to the transmitral E and A waves. The negative (downward) deflection during systole corresponds to aortic outflow (the spike present on the tracing during outflow is an artifact). Similarly, embedding data in the VdV/dtdP/dt space is of interest because the instantaneous P gradient dP, which is a relative rather than absolute measure of P, can be estimated noninvasively via the Bernoulli equation and measurements of flow velocity via Doppler echocardiography. For these reasons, among others, analysis of hemodynamics in hyperspace and its 3D embedding represents a novel opportunity for discovery of new physiological relationships. An additional benefit of the proposed method is its generality. Specifically, the method is essentially topological. As such, it is independent of the presence or absence of inotropic stimulation. The method is ideally suited for investigation when response to negative or positive inotropic stimulation to determine how E_{max} varies in a particular preparation is the goal.
In conceptual terms, the method can facilitate the development of novel (mathematical) models of LV function whose viability can be tested based on agreement of model prediction with experimental data analyzed in hyperspace.
In experimental terms, application of the proposed method to human physiology can elucidate whether E_{max} remains linear in the phase plane and in the other 2D subsets of hyperspace, whether E_{max} occurs at the loci of points where the dP/dtV relation has maximum curvature (see Fig. 5), and whether the relation dP/dt (V) = P (dV/dt)remains valid, as required by the definition of E_{max}. Application in selected pathological states, such as diabetes, hypertension, or congestive heart failure, or states having primarily diastolic dysfunction or constrictive/restrictive physiology is particularly enticing.
Limitations.
The mouse data included preload alteration and permitted derivation of regression relations between W and LCA and determination of E_{max} and its analog in the phase plane. Afterload variation could have provided further information regarding the load dependence of these regression relations. Although data from additional animals are of definite interest, data from only five animals sufficed to illustrate how the method can be used. The strong linear correlation of W to LCA and linearity of E_{max} analog in the phase plane underscores the potential of the method.
Conclusion.
Highfidelity conductance (PV) catheter data from five mice (2 normal, 2 diabetic, 1 sick) were analyzed by using 3D embedding in 4D physiological hyperspace. The method is topological, is independent of inotropic state, and permitted determination of the LCAtoPV loop area (W) relation. As predicted by workenergy considerations, very strong linear correlation between W and LCA was observed (r = 0.97) for all five animals, indicating that power and work obey firstorder kinetics. The equivalent of the endsystolic PV relation (E_{max}) was determined in the phase plane and was found to be linear where its physiological and functional significance remains to be fully elucidated. The feasibility of our method of embedding physiological data in hyperspace has been demonstrated. It facilitates characterization of novel ventricular function relationships. Work regarding the range of possible hyperspace relationships and their physiological interpretation has begun.
Acknowledgments
We thank members of the Mouse Physiology Core Laboratory for provision of hemodynamic data; our colleagues Drew Bowman, Raj Shani, Mustafa Karamanoglu, Erik Lentz, and Tim Meyer; and anonymous reviewers for helpful comments and suggestions.
Footnotes

This study was supported in part by National Heart, Lung, and Blood Institute Grants HL54179 and HL04023, the Whitaker Foundation (Roslyn, VA), and the Alan A. and Edith L. Wolff Charitable Trust (St. Louis, MO).

Address for reprint requests and other correspondence: S. J. Kovács, Cardiovascular Biophysics Laboratory, Cardiovascular Division, 660 South Euclid Avenue, Box 8086, St. Louis, MO 63110 (Email: sjk{at}howdy.wustl.edu).

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 Copyright © 2002 the American Physiological Society