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Am J Physiol Heart Circ Physiol 290: H1474-H1483, 2006. First published November 11, 2005; doi:10.1152/ajpheart.00663.2005
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Nonlinear isochrones in murine left ventricular pressure-volume loops: how well does the time-varying elastance concept hold?

T. E. Claessens,1 D. Georgakopoulos,2 M. Afanasyeva,3 S. J. Vermeersch,1 H. D. Millar,4 N. Stergiopulos,5 N. Westerhof,6 P. R. Verdonck,1 and P. Segers1

1Institute of Biomedical Technology, Ghent University, Gent, Belgium; Departments of 2Medicine and 3Pathology, Johns Hopkins University School of Medicine, Baltimore, Maryland; 4Millar Instruments, Houston, Texas; 5Laboratory of Hemodynamics and Cardiovascular Technology, Lausanne, Switzerland; and 6Laboratory for Physiology, Institute of Cardiovascular Research-Vrije University, Vrije University Medical Center, Amsterdam, The Netherlands

Submitted 20 June 2005 ; accepted in final form 9 November 2005


   ABSTRACT
TOP
 ABSTRACT
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
tThe linear time-varying elastance theory is frequently used to describe the change in ventricular stiffness during the cardiac cycle. The concept assumes that all isochrones (i.e., curves that connect pressure-volume data occurring at the same time) are linear and have a common volume intercept. Of specific interest is the steepest isochrone, the end-systolic pressure-volume relationship (ESPVR), of which the slope serves as an index for cardiac contractile function. Pressure-volume measurements, achieved with a combined pressure-conductance catheter in the left ventricle of 13 open-chest anesthetized mice, showed a marked curvilinearity of the isochrones. We therefore analyzed the shape of the isochrones by using six regression algorithms (two linear, two quadratic, and two logarithmic, each with a fixed or time-varying intercept) and discussed the consequences for the elastance concept. Our main observations were 1) the volume intercept varies considerably with time; 2) isochrones are equally well described by using quadratic or logarithmic regression; 3) linear regression with a fixed intercept shows poor correlation (R2 < 0.75) during isovolumic relaxation and early filling; and 4) logarithmic regression is superior in estimating the fixed volume intercept of the ESPVR. In conclusion, the linear time-varying elastance fails to provide a sufficiently robust model to account for changes in pressure and volume during the cardiac cycle in the mouse ventricle. A new framework accounting for the nonlinear shape of the isochrones needs to be developed.

ventricular function; curvilinear; end-systolic pressure-volume relationship

MANIPULATIONS OF THE MOUSE genome are increasingly being performed for the exploration and identification of the mechanisms underlying ventricular function in the healthy and diseased heart. However, the number of techniques for investigating the phenotypic expression of these alterations in vivo has been rather limited because of the small size of the murine heart and its rapid heart rate (17). Thanks to recent advances in biomedical engineering technology, there is the opportunity to accurately acquire intraventricular pressure and volume with a miniaturized combined pressure and conductance catheter (12).

Pressure-volume (P-V) loops have been in use for decades to describe the active and passive mechanical properties of the mammalian heart (16), its energy consumption (36), and its interaction with the arterial circulation (42). Suga and Sagawa (38) and Suga et al. (40) have contributed enormously to the understanding of ventricular function by introducing the concept of the time-varying elastance E(t). This elastance function is derived from the proportionality between intraventricular pressure and volume and describes the temporal course of the chamber stiffness throughout the cardiac cycle. They showed that E(t) was independent of end-diastolic volume (preload) and arterial pressure (afterload) within physiological ranges and that it was sensitively affected by inotropic interventions (40). The peak value of the linear time-varying elastance function, Emax, which approximates the slope Ees of the end-systolic pressure-volume relationship (ESPVR), is a commonly used measure of cardiac contractility in clinical practice and for research purposes (5, 30, 33). A prerequisite for using the linear E(t) concept is 1) a linear shape of all isochrones (lines connecting data acquired at the same time instant after the onset of systole) and the ESPVR in particular and 2) a common intercept of these isochrones with the volume axis (40).

The shape of the ESPVR, however, has always been subject to great interest but is also subject to controversy (26). Experiments in the dog (26, 35, 41, 43), mouse (12, 18), and rat (21) heart have shown a significantly curvilinear ESPVR when pressure and volume measurements were performed under a wide range of loading conditions. It has moreover been shown that the degree of nonlinearity (curvilinearity) of the ESPVR is dependent on the contractile state (7, 31). Whereas these studies challenged the initial concept of a linear ESPVR, the local slope of the ESPVR at low volumes is still considered a powerful index to assess the inotropic state (7).

Whereas previous studies mainly focused on the shape of the ESPVR and the assessment of its slope and the intercept, the purpose of this study was to analyze and describe the shape and shape change of all isochrones during the cardiac cycle. This study was undertaken because a pronounced curvilinear ESVPR has been observed in the mouse left ventricle, and we therefore expected the isochrones to deviate from linearity as well. A critical analysis and subsequent discussion about the aforementioned assumptions underlying the E(t) concept are provided. More specifically, we have 1) investigated the time-varying character of the volume intercept; 2) searched for the best regression algorithm for the ESPVR and the isochrones; and 3) analyzed time-dependent changes in the shape of isochrones throughout the cardiac cycle.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
Time-Varying Elastance Concept

Throughout the cardiac cycle, the P-V data points move counterclockwise in the P-V plane. The shape and size of the area within the trajectory (i.e., the P-V loop) change with loading conditions. This is illustrated in Fig. 1A, which shows three P-V loops under different preload conditions. Every P-V loop fits between two curves that define the intrinsic mechanical properties of the ventricle under a given contractile state: the end-diastolic P-V relationship (EDPVR), which describes the passive properties, and the ESPVR.


Figure 1
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  		Fig. 1. Overview of 6 regression algorithms (RA) for fitting isochronal pressure-volume (P-V) data. Lin, linear; Fix, fixed; Var, variable; Quad, quadratic; Log, logarithmic. A: RALin-Fix; conventional algorithm: linear isochrones converging to a fixed volume intercept V0,Lin. B: RALin-Var; linear isochrones resulting in a time-varying V0,Lin(t). C: RAQuad-Fix; quadratic isochrones converging to a fixed V0,Quad. D: RAQuad-Var; quadratic isochrones resulting in a time-varying V0,Quad(t). E: RALog-Fix; logarithmic isochrones converging to a fixed V0,Log. F: RALog-Var; logarithmic isochrones resulting in a time-varying V0,Log(t). ESPVR, end-systolic P-V relation; EDPVR, end-diastolic P-V relation; t1 and t2, two arbitrary time points during systole.

 
Figure 1A also shows two sets of isochronal P-V data points, i.e., P-V couples that occur at the same time instant after the onset of systole (e.g., t = t1 and t = t2). In the early 1970s, Suga and Sagawa (38) and Suga et al. (40) reported high correlation coefficients when using a linear function to fit the isochronal data in canine hearts. Moreover, at the end of systole, all regressed isochrones seemed to converge quite closely to a constant V0. Both observations led to the definition of a linear time-varying elastance E(t), which refers to the time-varying character of the global ventricular stiffness (40). In summary, the relation between intraventricular pressure and volume can be stated as follows: P(t) = E(t)·[V(t) – V0], where P(t) and V(t) are time-varying pressure and volume, respectively.

Experimental Protocol

Thirteen anesthetized, open-chest mice weighing 140 g (SD 18) (strains C57BL6 and C57BL6/129) were used in this study. The protocol was approved by the Animal Care and Use Committee of the John Hopkins University and conformed with the institutional guidelines. Anesthesia was initiated with methoxyflurane inhalation followed by intraperitoneal injection of urethane (750 mg/kg), etomidate (20–25 mg/kg), and morphine (1–2 mg/kg) dissolved in normal saline. A heating pad was placed underneath the animals, and the temperature was set to 37.5°C. All animals were ventilated by using a constant flow ventilator with 100% oxygen at 120 breaths/min (tidal volume, 200 µl).

An anterior thoracotomy was performed to enter the chest. An apical stab with a 26-gauge needle allowed for the placement of a custom-made, four-electrode conductance catheter with a dual-pressure sensor (Millar Instruments, Houston, TX). The catheter was advanced along the long axis of the left ventricle to place the distal tip in the aortic root and the proximal electrode just inside the endocardium. A correct position of the catheter was verified by online visualization of the shape and position of the P-V loops. The time-varying ventricular volume was determined by using the formula of Baan et al. (2). The gain factor {alpha}, used for calibration of the conductance catheter, was obtained by matching the conductance-derived stroke volume to that measured by a flow probe (1 RB; Transonic, Ithaca, NY), which was positioned around the thoracic aorta and filled with conducting gel on a beat-by-beat basis during transient vena cava occlusion (VCO). In addition to the determination of the gain factor {alpha}, the offset of the volume signal (parallel conductance Gp) is required to obtain a fully calibrated signal. Gp was assessed by an infusion of hypertonic saline (bolus injection of 5–10 µl, 35% saline), as described by others (2, 8). Pressure and volume signals were sampled at 2 kHz and transferred to an Intel Pentium IV PC for subsequent analysis.

Hemodynamic Analysis

Data acquisition and treatment. Data were obtained at baseline conditions and during gradual preload reduction, which was accomplished through manual VCO. The inotropic state was kept at basal level during the whole experiment. VCO generally yielded ~15–25 cycles, typically consisting of 180 samples each. To obtain an objective, automated determination of the onset of systole, this moment was taken as the time instant in the P-V plane where pressure was 4 mmHg higher than the pressure corresponding to a volume of 98% of the maximum volume (end-diastolic volume). Visual control of the obtained time points proved this algorithm to be sufficiently robust (see Fig. 2). These time points served as reference for the identification of the isochronal data points.


Figure 2
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  		Fig. 2. Representative example of a series of 24 P-V loops under transient preload reduction. Extrapolation of linear, quadratic, and logarithmic ESPVR yields V0,Lin, V0,Quad, and V0,Log, respectively. Isochronal ventricular P-V couples are shown as black dots with an interval of 2.5 ms. Four isochrones fitted with RAQuad-Var (compare Fig. 1D) during isovolumic relaxation (IVR) and isovolumic contraction (IVC) are shown. {circ}, onset and end of systole.

 
End-systolic data points were found by using an iterative way described previously (19, 32). Briefly, the P-V data points yielding the maximum P-to-V (P/V) ratio were linearly regressed. The obtained volume intercept V0 was subsequently used in the next iteration step to regress the data points corresponding to a maximum P/(V – V0) ratio. These steps were repeated until the slope of this regression line converges to a constant value Emax ({epsilon} < 0.1%), which typically occurred after three to four iterations. Both pressure and volume data were filtered by using a Savitsky-Golay smoothing filter (third-order, 15-samples window width), which preserved features of the original data, such as peak height and width (32). Further postprocessing was performed by using a custom-made application in Matlab Release 14 (Mathworks, Natick, MA).

Fitting ESPVR and isochrones. The end-systolic P-V data points were fitted to a linear (Pes = {alpha}1·Ves + {alpha}0), a quadratic (Pes = {alpha}2·Ves2 + {alpha}1·Ves + {alpha}0), and a logarithmic [Pes = ({alpha} + beta·Ves)–1·ln(Ves/V0)] function, where Pes is end-systolic pressure, Ves is end-systolic volume, and {alpha} and beta are parameter coefficients. The logarithmic model was chosen according to the elastance model of Mirsky et al. based on maximum systolic stiffness.

Isochronal data points were then fitted by using six different regression algorithms (RA): two linear (Lin), two quadratic (Quad), and two logarithmic (Log), each with either a fixed (Fix) or a variable (Var) time-varying intercept with the volume axis (all illustrated in Fig. 1).

Formula

For the RA with the fixed volume intercept (Fig. 1, A, C, and E), we extrapolated the linear, quadratic, and logarithmic ESPVR to the volume axis, yielding the constant volumes V0,Lin, V0,Quad, and V0,Log, respectively. These values were subsequently used to fit all other isochrones, such that they were mathematically restricted to go through V0,Lin,V0,Quad, or V0,Log.

For the remaining RALin-Var, RAQuad-Var, and RALog-Var (Fig. 1, B, D, and F), on the other hand, all isochronal P-V data were fitted by using linear, quadratic, and logarithmic functions, respectively. Next, every single isochrone was extrapolated to the volume axis to obtain the time-varying volumes V0,Lin(t), V0,Quad(t), and V0,Log(t).

In RALin-Fix and RALin-Var, coefficient {alpha}1 represents the slope of the linear isochrones. In algorithms RAQuad-Fix and RAQuad-Var, {alpha}2 represents the coefficient of curvilinearity. The coefficients {alpha} and beta that are used in the logarithmic description combine myocardial stiffness, chamber geometry, and other empiric constants (19).

Statistical Analysis

The appropriateness of applying a nonlinear model function (quadratic or logarithmic) to describe the ESPVR has been evaluated by using Akaike's information criterion (AIC), which is based on the principle of parsimony (20). AIC values are calculated as

Formula
where n is number of data couples (equals number of loops) and Pmeas,i and Pest,i are the measured and estimated pressures, respectively. The number of model parameters is represented by k (linear, k = 2; quadratic and logarithmic, k = 3). The k value that minimizes AIC corresponds to the best model.

For each isochronal regression algorithm, the difference between the estimated (fitted) and the measured pressures was assessed by root mean square error (RMSE) values, defined as

Formula

The goodness of fit was additionally assessed by the commonly used coefficient of determination R2. All time-dependent data were normalized for heart rate and subsequently averaged for all 13 animals. The results are expressed as means (SD). Statistics were performed with the use of SPSS 12 (SPSS, Chicago, IL). Differences between groups were analyzed by using paired t-tests. Statistical significance was assumed when P < 0.05.


   RESULTS
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 GRANTS
 REFERENCES
 
Table 1 summarizes the mouse hemodynamic data acquired at baseline and after preload reduction. End-diastolic pressure and volume, stroke volume, and end-systolic pressure were significantly different between baseline and VCO. A statistically significant difference was also seen for the heart rate (HR) [620 beats/min (SD 36) vs. 624 beats/min (SD 35)], because the majority of mice experienced a minute increase in HR. The estimated parameters derived from the linear, quadratic, and logarithmic ESPVR and the extrapolated volume intercepts are shown in Table 2. Note that {alpha}1 = Emax holds, by definition, for the linear model. The AIC values for the quadratic and logarithmic model are consistently smaller than those for the linear ESPVR, indicating that the ESPVR is indeed better modeled with a nonlinear function.


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  		Table 1. Summary of hemodynamic data during baseline and vena cava occlusion

 

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  		Table 2. Parameters derived from linear, quadratic, and logarithmic fitting of end-systolic pressure-volume data

 
Figure 2 shows a representative example of a series of 24 P-V loops obtained under gradual preload decline, demonstrating curvilinear isochrones (with an interval of 5 samples, i.e., every 2.5 ms), and a linear, quadratic, and logarithmic extrapolation of the ESPVR, yielding V0,Lin, V0,Quad, and V0,Log, respectively. The onset and the iteratively determined end of systole are also shown.

The time courses of curvilinearity ({alpha}2 = coefficient of V2) of the quadratic isochrones obtained with RAQuad-Fix and RAQuad-Var are given in Fig. 3. In RAQuad-Var, coefficient {alpha}2 decreases from virtually 0 (linear isochrones) at the onset of systole to approximately –0.1 mmHg/µl2 (concave to the volume axis) during ejection. The curvilinearity remains constant until the end of ejection, after which its coefficient {alpha}2 decreases rapidly to a minimum of –0.43 mmHg/µl2 (SD 0.40) in the first half of isovolumic relaxation (IVR). During IVR, the shape of the isochrones quickly shifts from concavity to the volume axis toward convexity [{alpha}2 = 0.86 mmHg/µl2 (SD 0.67)] and back to linearity during the filling phase. During isovolumic contraction (IVC) and ejection, the results for RAQuad-Fix are comparable with those for RAQuad-Var. The pronounced shift in curvilinearity observed during IVR, however, is not seen during IVC. In both algorithms, the isochrones simultaneously return to linearity (tN = 0.62).


Figure 3
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  		Fig. 3. Normalized time course of the degree of curvilinearity {alpha}2 (coefficient of V2) for quadratic regression algorithms RAQuad-Fix and RAQuad-Var. Data are averaged for all animals. Data are shown as means (SD).

 
The quality of the fit, quantified by RMSE values (Fig. 4, A and B) and the coefficient of determination R2 (Fig. 4, C and D) are shown in Fig. 4 as a function of normalized time in the cardiac cycle. As anticipated, nonlinear regression yields better results (i.e., lower RMSE values) than the commonly used linear regression. Moreover, when comparing the left (fixed V0) with the right (variable V0) panels, regression with a fixed V0 increases the RMSE because of fewer degrees of freedom. The worst fit is obtained with the conventionally used RALin-Fix, particularly during IVR. Quadratic and logarithmic regression algorithms perform similarly and thus could be used interchangeably for describing the shape of the isochrones. All algorithms perform comparably during filling, when the EDPVR is virtually linear. Similar conclusions can be drawn from the analysis of the time course of R2 during IVC and ejection, except that now RALin-Var, instead of RALin-Fix, yields the lowest R2 (poorest goodness of fit). During IVR and filling, RALin-Fix performs unacceptably (R2 < 0.75).


Figure 4
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  		Fig. 4. Normalized time course of the agreement between measured and fitted pressures, quantified by root mean square error (RMSE; A and B) and coefficient of determination (R2; C and D). A and C: regression algorithms with a fixed V0. Dashed line (C and D), R2 = 0.75. Data are averaged for all animals.

 
The time-varying character of V0,Lin(t), V0,Quad(t), and V0,Log(t), as well as the constant volume intercepts V0,Lin, V0,Quad, and V0,Log are presented in Fig. 5. From the onset of systole until mid-IVR, V0,Log(t) > V0,Quad(t) > V0,Lin(t). As can be predicted from the theoretical concept, the time-varying intercepts intersect their respective constant intercepts at the end of ejection (tN = 0.33). V0,Lin(t) ranges between –39.1 and 15.5 µl, whereas V0,Quad(t) acts in a smaller interval between –8.3 and 14.2 µl. V0,Log(t) changes considerably less during the cardiac cycle and ranges between 0.69 and 12.77 µl. The SD and the slope of V0,Log(t) at the end of ejection are considerably smaller than those for V0,Quad(t) and V0,Lin(t), suggesting that logarithmic regression is the most reliable fitting technique to estimate V0. During the second half of IVR and the filling phase, all algorithms result in comparable volume intercepts.


Figure 5
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  		Fig. 5. Normalized time course of V0,Lin(t), V0,Quad(t), and V0,Log(t) obtained with regression algorithms RALin-Var, RAQuad-Var, and RALog-Var, respectively. Dashed lines represent constant values V0,Lin (obtained with RALin-Fix), V0,Quad (obtained with RAQuad-Fix), and V0,Log (obtained with RALog-Fix).

 

   DISCUSSION
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
GRANTS
 REFERENCES
 
Our analysis revealed that 1) isochrones measured in the mouse left ventricle show a time-varying curvilinearity during IVC, ejection, and IVR; 2) the shape of the isochrones is best described by using a nonlinear function (quadratic or logarithmic) with a time-varying volume intercept, whereas a linear approximation with fixed volume intercept offers the poorest results, particularly during IVR and early filling; 3) the logarithmic fitting appears superior in estimating the fixed volume intercept of the ESPVR and moreover offers a physiological (i.e., positive) result; and 4) the intercepts V0,Lin(t), V0,Quad(t), and V0,Log(t) vary with time and differ from each other during IVC and ejection.

In the early 1970s, using a canine isolated heart preparation, Suga and Sagawa (38) and Suga et al. (40) reported very high coefficients of determination R2 when fitting the isochronal P-V data points with a linear function. Additionally, the extrapolated intercepts of the isochrones converged closely to a constant value V0, which is the minimal volume required for the left ventricle to generate supra-atmospheric pressure (38, 40). This V0 also equals the intercept of the isochrone with the highest slope, the ESPVR. On the basis of these experimental observations, they introduced the concept of a linear time-varying elastance E(t).

It should be realized that this definition of ventricular elastance is intrinsically based on the existence of linear isochrones and a common intercept V0. Whereas the concept allows for demonstrating many of the basic characteristics of the ventricle, the physiological interpretation behind this concept has remained unclear, because it is essentially that of a spring that alters its stiffness with time (maximum stiffness at end systole and minimum stiffness during diastole). Suga and Sagawa (39) were the first to link the performance of the ventricular chamber with myocardial cell properties by mathematically deriving the P/V ratio curve from the known myocardial force-velocity relation by using a series elastic and contractile element model. An inverse method was employed by Beneken et al. (3), who synthesized the P-V loops from physiological data on the force-velocity relation. In the late 1980s, Drzewiecki et al. (9) presented a direct relationship between the basic mechanisms of myofilament contraction and shape of the isochrones. They developed a thin-walled cylindrical model of the left heart to deduce the P-V relation and corresponding isochrones in a rabbit ventricle from the stress-strain relation in a contractile myofibril. In contrast to the serial model of Suga et al. (40), Drzewiecki et al. (9) used a structural model consisting of a contractile unit in parallel with a passive unit.

Various researchers (1, 25, 43) have reported some limitations of the conventional E(t) concept, particularly its sensitivity to afterload. Little et al. (23) investigated the adequacy of the E(t) to describe the difference between an ejecting and isovolumic beat and concluded that a flow-dependent term should be added to the time-varying elastance model, accounting for an "internal resistance." The variation in E(t) with heart size has also been a matter of discussion (37). Although these drawbacks are recognized by most researchers, the concept is still quite generally accepted for practical research purposes and is mainly applied to derive the maximum value Emax, which is used as an index of cardiac contractile function. This index is considered relatively insensitive to changes in loading conditions in isolated canine hearts (28, 29), conscious dogs (33), and humans (15). The size dependency has been corrected for by Beyar and Sideman (4), who introduced the ventricular mass (Vm) as a scaling factor for Emax (normalization is achieved via multiplication with Vm). Whereas during systole the E(t) concept has proven relatively accurate for predicting pressures from volumes in different loading conditions, the diastolic phase of the E(t) curve showed a much greater variation (38). In the literature, very little attention has been paid to this discrepancy and virtually no description of the shape change of isochrones during relaxation has been provided.

Although there was no a priori reason to expect that the ESPVR is linear—it was simply an experimental observation—it has been accepted for a long time, mainly because a straight line allows for uncomplicated definitions of the slope Emax and the intercept V0. In subsequent years, however, researchers who analyzed P-V loops in a much wider range of loading conditions than did Suga et al. (40) have shown a curvilinear ESPVR. Additionally, it has become evident that large alterations in contractile state can influence the curvilinearity of the ESPVR (7). Several authors reported the curvilinear shape of the ESPVR and thus proposed alternative mathematical descriptions for the ESPVR, such as quadratic (parabolic) (7, 19) or exponential (41) functions. Unfortunately, none of these fitting curves allows for a physiological interpretation. By combining stiffness, several geometric variables, and empiric constants, Mirsky et al. (26) introduced a logarithmic function to define the ESPVR. Because of the different existing ESPVR functions, however, a standardized definition of their (local) slope has been lacking, complicating comparisons between study groups or within individuals in experimental research or in clinical practice.

In this study, the P-V data acquired with a miniaturized combined pressure-conductance catheter demonstrated a markedly curvilinear ESPVR, according to Akaike's information criterion. Because the assumption of linear isochrones does not seem to be consistent with the presence of a nonlinear ESPVR, we systematically analyzed the isochrones in P-V diagrams.

The description of the shape change of the isochrones was based on the quadratic regression algorithms, because parameter {alpha}2 provides a direct quantification of their curvilinearity. Although the curvilinearity of both RAQuad-Fix and RAQuad-Var appears relatively small during IVC and ejection, it should not be underestimated because it is masked by the high degree of curvilinearity during IVR (Fig. 3). Drzewiecki et al. (9) attributed the curvilinear shape to the combination of a nonlinear active muscle function, the passive exponential stress-length relationship of myocardial tissue, and the geometry of the ventricle. Elzinga and Westerhof (10), on the other hand, assumed that the linear time-varying elastance can describe the mechanics of the whole canine left ventricle, but they found out that the concept was not applicable to isolated muscle. This discrepancy was attributed to "the complex organization of the cardiac muscle fibers in the wall of the heart" (10).

In our results, the time-varying V0,Lin(t), V0,Quad(t), and V0,Log(t) differed considerably from the constant intercepts, indicating that the assumption of a constant volume intercept is violated in murine ventricles, regardless of the regression algorithm used (Fig. 5). The most accurate and, more importantly, the only physiological volume intercept was obtained by using the logarithmic regression function, established by Mirsky et al. (26). The relatively large SD observed during filling for all regression algorithms was due to the shallow slope of the EDPVR. The time-varying character of V0 was previously explained by Drzewiecki et al. (9) as "apparently" time varying: In their theoretical study, the actual ventricular isochrones are obtained by mathematically adding a passive to an active component. The passive component represents the passive P-V relation of the elastic structure, which has a resting volume (equilibrium volume) Ve. The active component refers to the set of active function isochrones that have a common intercept, say Vd (i.e, the functionally dead volume, occurring at negative pressures). Because it is assumed that the active zero-pressure volume Vd is smaller than Ve, all isochrones (consisting of an active and a passive component) are concurrent at a negative pressure, which results in an "apparent" time-varying V0(t). Whether the time variation of this quantity has physiological meaning is not clear.

The curve fitting that was subsequently applied by using fixed intercepts as boundary conditions showed to what extent such a mathematical restriction reduces the quality of the fit (Fig. 4). The agreement between the measured and fitted data was assessed by calculating RMSE and R2 values. Although the time courses of RMSE and R2 were slightly different from each other [because of R2 being also dependent on the range in predictor values, i.e., the volumes (20)], both measures of agreement pointed out that during IVR and early filling, the conventional E(t) concept with fixed V0,Lin showed a poor agreement with the data, with R2 values far below 0.75. Figure 6 illustrates the deleterious effects of using the boundary condition P(V0,Lin) = 0 on the definition of elastance (instantaneous slope of the isochrones). During ejection, the conventional elastance (RALin-Fix) overestimates the slope by 26.41% on average. Surprisingly, during IVR, the time course of RALin-Var shows a striking dissimilarity to the time course of RALin-Fix, resulting in a significantly higher Emax [Emax = 8.41 mmHg/µl (SD 2.77)] before reaching its minimum value. On the other hand, a gradual decrease toward the end-diastolic elastance is observed in RALin-Fix. The latter time course is similar to what is seen in the literature, and the Emax [2.8 mmHg/µl (SD 0.7)] is in agreement with data from Reyes et al. (27) [3.3 mmHg/µl (SD 1.9)]. The unexpected difference between RALin-Fix and RALin-Var proves that the linear elastance concept is meaningless during IVR and early filling, although in the literature the elastance curve is frequently shown for the whole cardiac cycle. A simple exponential decay of the pressure waveform or Suga's logistic model would be sufficient to describe IVR (24).


Figure 6
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  		Fig. 6. Normalized time course of slope of isochrones fitted with RALin-Fix and RALin-Var. Data are averaged for all animals. Data are shown as means (SD).

 
In this study, we provided ample evidence that the linear elastance concept is not the ideal method to describe ventricular performance in mice during both systole and diastole. Whereas it is always feasible to construct an elastance curve once V0 is determined, it is not clear whether it represents the instantaneous stiffness of the ventricle. Consequently, various questions remain to be answered about the relation between ventricular elastance and isochronal P-V data. This debate can be summarized visually in Fig. 7, where the averaged parameters obtained from RALin-Fix (Fig. 7A) and RAQuad-Var (Fig. 7B) are used to reconstruct isochrones during IVR and IVC. In Fig. 7A, the lines simultaneously represent isochronal and "isostiffness" P-V data in the ventricle. The ventricle functionally operates at the same stiffness level during two moments in the cardiac cycle, once at a large volume during IVC and later at a smaller volume during IVR. In Fig. 7B, on the other hand, only isochronal P-V data can be seen and no information about stiffness can be obtained. Interestingly though, with the consideration of the isochrones during both IVC and IVR, at least the visual suggestion appears that the isochrones during IVC and IVR could/should be interconnected by means of a sigmoidal curve. It remains to be assessed whether this behavior is strictly a phenomenological observation (which we aimed to quantify in this study) or whether an alternative "nonlinear" time-varying elastance theory can be deployed, where ventricular behavior (stiffness) is described by a sigmoidal curve varying in time and changing its shape throughout the cardiac cycle. The sigmoidal shape (and the changes throughout the cardiac cycle) certainly resembles force-length relationships in isolated muscle experiments, where the shape of the sigmoidal curve is modulated by calcium concentrations. On the other hand, one can also follow a more pragmatic approach and describe diastole with a different time-varying quantity than the systole. If so, a new reference time point for the onset of diastole should be looked for, instead of the end ejection, which is used in our study. A reasonable but impractical reference point could be the time instant of transition from systole to diastole on myocardial level (i.e., when load ceases to sustain the cross bridges), as determined by Solomon et al. (34).


Figure 7
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  		Fig. 7. Reconstruction of isochrones during IVC and IVR by using average values of parameters. A: regression and reconstruction according to RALin-Fix. B: regression and reconstruction according to RAQuad-Var. ESV, end-systolic volume; EDV, end-diastolic volume; VCO, vena cava occlusion.

 
Finally, it is to be acknowledged that our study has the following potential limitations: 1) The data were obtained by using a single-frequency conductance system with the assumption of a constant volume signal offset (Vp) during the whole experiment. The validity and accuracy of this method has been questioned (11). It has been shown, however, that this method can certainly be used in mice (14) because nearly all of Vp can be attributed to near-field effects, i.e., the ventricle wall. The variation of Vp during the cardiac cycle is therefore very limited. In addition, if Vp was changing during the IVC occlusion, it would cause the P-V loops to shift leftward, because overall there would be a decrease in parallel conductance resulting from an decrease in total cardiac volume, including right ventricular volume. If the loops actually shifted to the left, the end-systolic point would also shift leftward, resulting in what would "appear" as a more linear or flatter ESPVR, which was clearly not observed in our experiments. 2) Although our automated detection of the onset of systole seems artificial, it was extremely robust in our measurements. However, false identifications may occur when the measured maximum volume is ~2% higher than the end-diastolic volume. Manual definition of the onset of systole is then required. An alternative technique for detecting the onset of systole has been proposed by Kass et al. (19). They defined it as the time point where dE/dt {approx} d(P/V)/dt exceeds 10% of dE/dtmax, where dE/dt is rate of change of elastance and dE/dtmax is the maximum rate of change of elastance (19). Even though both algorithms lack a physiological meaning, it is anticipated that they yield comparable results. We preferred our technique over others because of the difficulties in defining the elastance in case of a curvilinear ESPVR. 3) The experiments were not repeated under different inotropic conditions, so we were not able to assess the influence of contractility on the time course of all of the calculated variables. Nevertheless, we believe that this does not affect the general idea presented in this paper. 4) Significant decreases in the systemic pressure during the VCO could potentially result in myocardial ischemia and changes in contractility (and HR). Yet, Burkhoff et al. (6) stated that as long as the systemic pressure stays above 60 mmHg, the myocardial contractility is virtually unaffected. Because in our experiments the averaged end-systolic pressure (during the last loop of VCO) was 71 mmHg, we can reasonably expect that the effect of the lowered coronary artery pressure is negligible. 5) A statistically significant increase in HR between baseline and VCO was observed (620 vs. 624 beats/min). We believe, however, that these small changes in HR should not affect the contractility because the force-frequency response of the mouse has been shown to be flat at HR above 600 beats/min (13). 6) The data shown in Figs. 3–6 were averaged for all animals included in the study. Even though all data have been represented on a normalized timescale, the peaks of the curves could be slightly blunted if they do not occur at the same normalized time for each subject. 7) Finally, our experiments were done in open-chest mice. Even though it is the most frequently reported approach to date, it has some theoretical disadvantages compared with a closed-chest approach. In the closed chest, the lungs remain untouched and the cardiac position remains intact. Lips et al. (22) published significant differences in stroke volume, end-systolic volume, end-diastolic volume, and end-systolic pressure between the two approaches. Because our method produced physiological pressures and cardiac output, we assume that our findings can be extrapolated to the closed-chest approach.

In conclusion, we have demonstrated that the conventional linear time-varying elastance concept does not fully describe ventricular performance during the whole cardiac cycle in the mouse. In a recent review, Burkhoff et al. (6) have emphasized that accurate P-V analysis continues to be very important, because 1) P-V data often constitute the critical information in proving the consequences and the relevance of primary biochemical, molecular, or cellular discoveries, and 2) these P-V data may, in the end, be the basis for acceptance of new concepts. Our detailed description of the curvilinearity of the murine P-V isochrones provides important insights for the development of new standardized methods of P-V data analysis and also provides the basis for a coherent framework that needs to be developed to account for cardiac physiology and the variation in time of nonlinear isochrones throughout the complete cardiac cycle.


   GRANTS
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 GRANTS
REFERENCES
 
T. Claessens is funded by specialization grants of the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT 023228). S. Vermeersch is funded by the Fund for Scientific Research-Flanders (FWO G.0055.05).


   FOOTNOTES
 

Address for reprint requests and other correspondence: T. Claessens, Cardiovascular Mechanics and Biofluid Dynamics Research Unit, Ghent Univ., Sint-Pietersnieuwstr. 41, Gent B-9000 Belgium (e-mail: tom.claessens{at}ugent.be)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


   REFERENCES
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 

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