INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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 end-diastolic P, peak-positive dP/dt (_+max), and peak-negative dP/dt (_min) and are illustrated in Fig. 1. Other indexes of LV function include the stroke volume (SV), the end-diastolic volume (V) (EDV), and the ejection fraction (EF) defined as EF = SV/EDV.

View larger version (17K): [in this window] [in a new window]   Fig. 1.   A: left ventricular pressure (P) vs. time (t) for 1 mouse. One cardiac cycle is shown. B: phase plane plot of the time derivative of P (dP/dt) vs. P. The closed trajectory form defines the limit cycle. Points of interest are maximum systolic P (1), peak-negative dP/dt (2), minimum diastolic P (3), peak-positive dP/dt (4), and left-ventricular end-diastolic P (5).

Additional physiological information can be obtained from the P-V loop, shown in Fig. 2. The area enclosed by the P-V loop for one cycle is the external work performed on the blood by the ventricle during one cardiac cycle (W)

(1)

Furthermore, dP/dV, defined at a point on the loop, is the dynamic stiffness of the LV at that point in the cardiac cycle. Time-varying elastance [E(t)] (18, 14) can be determined from a set of P-V loops. It is defined as E(t) = P(t)/V(t). E_max, the maximum value of E(t), which occurs at or near end systole (see Fig. 2), has been utilized as a load-independent measure of contractility (15).

View larger version (44K): [in this window] [in a new window]   Fig. 2.   Typical example of P-volume (V) data for 1 normal mouse. Maximum elastance (Emax) is determined in the conventional fashion and is given by the P = Emax (V  Vo) relation, where Vo is the intercept along the V axis. For these data, P = 3.7(V + 17.2) (r = 0.988). Note that linear best fit for Emax generates a negative value for Vo. See text for details.

Other parameters such as , the time constant of isovolumic relaxation, are computed by performing a least mean-square 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 against x 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 y-axis against the P plotted along the x-axis. 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 Fokker-Planck 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 respiratory-cardiac coupling and to characterize LV function (10). More recently, the phase plane method has been used to assess the discordance between dynamic and passive P-V relations in idiopathic cardiomyopathy (15) and for characterization of the relation between _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. 1A) can be deduced by analysis of limit cycle attributes in the phase plane (Fig. 1B). 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 of _+max and _min; 4) relationship between the P values at which _+max (P_C) and _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 systolic-diastolic 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 three-dimensional (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

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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 19-gauge needle, and respirated at a rate of 85 times per minute. The right carotid artery was then dissected free and cannulated. Next, a 1.4-F Millar pressure conductance catheter (model SPR-719, 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 rubber-insulated clamp. All P-V loops recorded during either steady-state 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, high-quality, short- and long-axis 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 long-axis 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 two-dimensional (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 long-axis echocardiographic view of the LV was obtained with the animal in the supine position. A series of 15-25 steady-state and IVC occlusion P-V loops was obtained by using the Millar ARIA-1 conductance system with the controller frequency set at 20 kHz and the low-pass cutoff frequency set at 50 Hz. Conductance and P signals were digitized by BioBench software (National Instruments, Austin, TX) and stored for later off-line analysis. Before analysis, the conductance catheter system's relative V data were recalibrated against the end-systolic V and EDV obtained for each mouse using the 2D echocardiographic technique described above.

Analysis of the data files was done off-line in the Cardiovascular Biophysics Laboratory using LabVIEW. The format of LabVIEW facilitates rapid, interactive plotting of the P-V contours and their analysis. Continuous or segmental sets of digitized, simultaneous P-V (conductance catheter) data were imported into DeltaGraph as column vectors. Programs were written in LabVIEW to compute dP/dt numerically from P(t) and to plot LVP limit cycles and P-V loops. The programs could also calculate the , E_max, and external work performed by the LV. An illustrative sample of 1 s of continuous P-V 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.

View larger version (37K): [in this window] [in a new window]   Fig. 3.   Illustrative example of continuous P-V data from 1 normal mouse. P-V data were obtained via conductance catheter. Time-varying elastance [E(t)] is computed from P(t)/V(t). dP/dt and the time derivative of V (dV/dt) are computed from P and V by differentiation, respectively. Note qualitative agreement of dV/dt tracing with transmitral Doppler E and A waves in diastole and Doppler aortic outflow contour in systole. See text for details.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Correspondence of LCA to external power. The external work (W = PdV) performed by the heart can be calculated from the area of the P-V loop for one cycle. The general (differential) expression for external mechanical power (_ex) is the time rate of change of work (7, 25)

(2)

which can be rewritten as

(3)

In this equation, power (_ex) is expressed as a function of the variables that define the phase plane (P, dP/dt) and the expression dV/dP = 1/(dP/dV), where dP/dV is the time-dependent (instantaneous) chamber stiffness.

min

(4)

(5)

(6)

(7)

(8)

(9)

(10)

View larger version (33K): [in this window] [in a new window]   Fig. 4.   Data relating P-V loop area to limit cycle area is shown for 5 mice (2 normal, 2 diabetic, 1 sick). Note very strong linear correlation (r = 0.94, 0.99, 0.98, 0.96, 0.99) for each individual. Linearity was predicted by energy-work criteria and underscores inotropy independence of the method. See text for details.

Physiological hyperspace. The relationship between the LVP limit cycle and the P-V loop can be further appreciated through the use of 3D embedding diagrams in physiological hyperspace spanned by dP/dt-P-V axes. A representative 3D embedding diagram for a normal mouse with 2D projections onto the phase plane, P-V plane, and V-dP/dt plane is shown in Fig. 5.

View larger version (42K): [in this window] [in a new window]   Fig. 5.   Three-dimensional (3D) embedding diagram in physiological hyperspace. Cardiac cycle contours are of V (x-axis) vs. dP/dt (y-axis) vs. P (z-axis). The 2-dimensional projections of the 3D contour onto the P-V plane (top left gray plot), the phase plane (dP/dt vs. P) (top right gray plot), and the dP/dt vs. V plane (bottom gray plot) are shown. Note the linear relationship (line) in 3D hyperspace, whose projection on the P-V plane is Emax, where linear regression yields P = 3.704271 * (V  16.67522) (r = 0.988). The projection of this line onto the phase plane, referred to as the "phase plane analog" of Emax, remains linear with linear regression dP/dt = 26.03880 * P + 1,898.363 (r = 0.834). See text for details.

(11)

(12)

(13)

View larger version (46K): [in this window] [in a new window]   Fig. 6.   P-V-dP/dt loops for the same mouse as Fig. 5 displayed stereographically to aid 3D visualization of physiological data embedded in hyperspace. Label along the vertical axis has been suppressed for clarity. The straight line is the hyperspace equivalent of the linear end-systolic P-V relation (Emax) encountered in the P-V plane. See text for details. (To generate 3D effect, view figure by moving it gradually from very close until images fuse.)

	DISCUSSION

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Embedding diagram in hyperspace: the arena for phase plane-to-P-V plane interaction. Assessment of ventricular function by using hemodynamic data is well established (26). In contrast, the technical difficulties associated with P-V data acquisition have limited its clinical acceptance. Development of the conductance catheter (1) and proposed methods to measure E_max by using single-beat 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.

min

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 (one-dimensional) analogs are force (F) and distance (x) and their time derivatives dF/dt and velocity (dx/dt). Because of the physiological relevance of the P-V plane and the phase plane, we investigated embedding data in 3D dP/dt-P-V space in this report. Three additional choices for 3D embedding using P and V and their time derivatives remain. These are P-V-dV/dt, P-dV/dt-dP/dt, and V-dV/dt-dP/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 P-V-dV/dt, providing work (P-V)-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 V-dV/dt-dP/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.

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. High-fidelity conductance (P-V) 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 LCA-to-P-V loop area (W) relation. As predicted by work-energy considerations, very strong linear correlation between W and LCA was observed (r = 0.97) for all five animals, indicating that power and work obey first-order kinetics. The equivalent of the end-systolic P-V 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.