INTRODUCTION

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RIGHT VENTRICULAR (RV) diastolic abnormalities such as reduced myocardial compliance (MC) may impair systolic function by limiting filling and play an integral role in cardiac adaptations to disease. Nevertheless, the development of indexes for RV function has evolved slowly, and numerous gaps remain in our understanding of RV diastolic function (10, 13, 28). Limitation of our understanding of the right ventricle can be attributed to the paucity of information from animal analogs of human RV disease and the lack of mathematical models for the analysis of RV performance. In contrast, comparable aspects relating to the left ventricle (3, 9, 17, 23) are much more highly developed. However, the simple application of indexes developed for the left ventricle to RV function has led to confusing and conflicting results.

The primary aim of the current study is to derive and validate new mathematical models and indexes specifically for the analysis of RV diastolic dynamics and the quantitative assessment of diastolic RV dysfunction and failure. Recognizing that a consensus has not been reached on what constitutes a correct mathematical descriptor of diastolic pressure-volume relations even for the extensively studied left ventricle, we propose that these new conceptual approaches will also contribute to the study of left ventricular function. Our present focus, however, is on RV diastolic pressure-volume relations and corresponding myocardial properties utilizing newly developed, chronically instrumented, awake dog models of RV volume overload (VO) and RV free-wall ischemia (IS) and a conventional model of RV pressure overload (PO).

	METHODS

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Sensor Implantation

View larger version (19K): [in this window] [in a new window]   Fig. 1.   Diagram of the instrumented dog heart demonstrating the sonomicrometric implanted sensors (left) and measured dimensions of the shell subtraction model (right). VRVFW denotes the right ventricular (RV) free wall volume, determined by water displacement during autopsy. Inset, the derivation of the passive filling pressure (Ppf) from the measured pressure (PM) by subtracting the relaxation pressure (PR). LV, left ventricular; Vtotal, total volume.

CL Data Acquisition

Attenuation of autonomic reflexes was accomplished by intravenous propranolol and atropine. The catheter was advanced, under fluoroscopy, ensuring that the two micromanometers were located inside the right atrium and ventricle. Multiple data sets were recorded under steady-state conditions and digitized at 400 Hz. Each set was 20-30 s long to provide sufficient beats for analysis. After CL data acquisition, RV PO or VO was induced.

Induction of PO

Induction of VO

View larger version (60K): [in this window] [in a new window]   Fig. 2.   A: pressure waveforms at the time of induction of RV volume overload through surgical tricuspid chordal rupture. Note the elevation of the right atrial (RA) pressure level and the striking shape similarity between RA and RV pressures. B: hematoxylin and eosin histopathological staining. The arrows point to infarcted myocardium.

Induction of IS

Mathematical Modeling of RV Filling Pressure-Volume Relations

(1)

(2)

View larger version (43K): [in this window] [in a new window]   Fig. 3.   Representative ensemble averages of measured diastolic RV pressure obtained under control (CL), pressure overload (PO), volume overload (VO), and ischemia (IS) conditions. Superimposed on each ensemble average are the individual diastolic pressures in the ensemble. Insets, typical RV pressure pulse trains.

View larger version (25K): [in this window] [in a new window]   Fig. 4.   Representative shapes of Ppf-V relationships. A: sigmoidal Ppf-V in PO. B: Ppf-V in VO fitted as an exponential; this can also be viewed as the upper portion of a sigmoidal curve. Experimental data points are displayed along with the fitted curves. C: demonstration of how the sigmoidal pressure-volume [P(V)] curve is derived from the logistic curve after substituting pressure (P) for time and volume [V(P)] for population [p(t)] by a 90° counterclockwise rotation and a horizontal 180° flip. D and E: representative myocardial compliance data and curves calculated using curve-fit parameters from a sigmoidal and an exponential Ppf-V relationship, respectively. Approximate maximum and end-diastolic myocardial compliances are labeled.

Calculation of MC. Global MC (17) was defined as
obtained by taking the derivative of the inverse function of Eqs. 1 and 2. For the simple exponential model, MC can be modeled using Eq. 3

(3)

Similarly, for the sigmoidal model, MC is

(4)

Throughout the passive filling process, both V and P_pf are positive and increasing. Equation 4, therefore, is always positive. On the right-hand side of Eq. 4, the exponential terms in the numerator and the denominator are both positive. To keep its right side positive, the signs of K₂, C, and (P_pf + B) had to be selected properly and consistently, so as to fall into one of the four categories outlined in Table 4. This allowed direct comparability of the estimated model parameters.

Regression and Statistics Methods

	RESULTS

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RV Hemodynamics

P_pf-V Relations and MC

Figure 4 also shows representative curves of RV MC, calculated using the results obtained from the P_pf-V relationships for CL, PO, VO, and IS. Figure 4D shows the curve resulting from the sigmoidal model (Eq. 4) and Fig. 4E shows the curve using a simple exponential model (Eq. 3). The most striking difference between these curves is the chamber volume at which MC_max occurs. For an exponential P_pf-V relationship, the maximum is found right at the beginning of the process and declines throughout the entire filling period. With the sigmoidal relationship, the MC reaches a maximum well into the filling process. The maximum in compliance corresponds to the inflection point of the sigmoidal curve. The curves derived from both models show a continuous decline in MC after their respective maxima.

We compared the sensitivity of the conventional simple exponential model and the new sigmoidal model in detecting the changes in the P_pf-V relationship resulting from RV disease. As shown in Fig. 5, substantially better fits to the data points were obtained with the sigmoidal model than with the conventional exponential. The sum of the squares of the residuals with the former was smaller by one order of magnitude than with the latter. Moreover, as shown in Fig. 5, top, the residuals of the exponential (but not of the sigmoidal) fit are characterized by a strong correlation of sequential observations, indicating that a systematic effect is neglected by the exponential model. Statistical analysis (see Table 1) showed that in contradistinction to the sigmoidal model (see Table 1), the simple exponential model was not sensitive enough to detect the important P_pf-V alterations ensuing in RV disease. These included changes in sigmoidality and in relative elevation or depression of the P_pf-V data in the midrange of operating volume, leftward rotation, and shift to higher operating volumes. Accordingly, the data for MC derived from the exponential model were not included in further analyses. Figures 6 and 7 summarize significant changes from CL that were caused by the three RV disease modes in the sigmoidal model parameters and compliance values.

View larger version (16K): [in this window] [in a new window]   Fig. 5.   Bottom: the much closer agreement between the Ppf-V data points and the least-squares fitted curve when the sigmoidal model is used compared with the exponential. The residual sum of squares (SSRes) for the former was smaller by one order of magnitude than for the latter. Top: the residuals of the exponential fit are characterized by a strong correlation of sequential observations.

View larger version (20K): [in this window] [in a new window]   Fig. 6.   Top: all three of the RV disease modalities studied significantly changed parameters B, C, and K1 of the sigmoidal model. Bottom: the impact of corresponding parametric changes on the overall shape of the sigmoidal curve. Bottom left, an increase in B resulting in a submersion of the lower portion of the curve. As a result, the portion of the sigmoidal curve remaining above the zero pressure line may resemble an exponential. Bottom middle, an increase in C resulting in elevation with leftward rotation of the middle portion of the curve without altering the position of the asymptotes; as a result, the slope dP/dV increases, implying reduced myocardial compliance. Bottom right, an increase in K1 resulting in a parallel, rightward displacement of the entire curve to higher operating volumes.

View larger version (29K): [in this window] [in a new window]   Fig. 7.   Top: representative shapes of RV Ppf-V relationships and corresponding global curves of myocardial compliance at CL, RV pressure overload, volume overload and RV free wall ischemia. Middle: changes in maximum myocardial compliance (MCmax) and the associated filling pressure (PMCmax at the inflection point). Bottom: changes in myocardial compliance (MCed) and the associated pressure (PMCed) at end diastole.

CL Condition

Pressure Overload

Volume Overload

Ischemia

In summary, statistical analyses were performed on the impact of each RV disease modality on model parameters and indexes of RV diastolic function. ANOVA and Bonferroni-Holm statistics (Table 1 and Fig. 6, top) show that significant differences exist for three of the model coefficients: B, C, and K₁. PO induced significant increases from CL in C and K₁ and a decrease in B. VO provoked increases in B, C, and K₁. IS caused a decrease in B and C from CL and an increase in K₁. Coefficients A and K₂ did not differ significantly from CL for any disease modality. Figure 6, bottom, shows the effects of parametric changes in each of these model coefficients on the shape and location of the sigmoidal curve on the P_pf-V plane.

Important changes in P_pf-V relations and diastolic properties are induced by the three RV failure modes (Table 3 and Fig. 7). PO alters RV MC properties more significantly than the other two conditions. It caused a decrease in RV MC_max as well as an increase in P_pf levels at both MC_max and MC_ed. VO exhibited a strong increase in MC_ed and a somewhat decreased maximal compliance compared with CL values. IS decreased the maximum compliance that was attained during the early filling process.

	DISCUSSION

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The material properties of the working myocardium during diastolic filling are important in the assessment of both diastolic and systolic function (24), but their determination presents challenging difficulties (8, 12, 15, 17, 21-26). A consensus has not been reached on what constitutes a correct mathematical descriptor of diastolic pressure-volume relations even for the extensively studied left ventricle. It is common practice to curve fit diastolic pressure-volume data in exponential form. Ostensibly, the reason behind the practice is because papillary muscle exhibits an exponential stress-strain relationship, so should the pressure-volume relations for the intact ventricle. This approach was adopted in 1969 by Noble et al. (19), whose work was often cited by subsequent investigators. However, even Noble cautioned that "pressure often differed from that predicted by the exponential equation," suggesting that it is not a reliable descriptor of ventricular diastolic dynamics. In a comprehensive survey on the clinical assessment of diastolic ventricular function, Mirsky and Pasipoularides (17) noted several drawbacks of using an exponential model for pressure-volume relations during the filling period. In the present study, RV P_pf-V relations were examined on alert dogs at CL and in three models of RV disease.

Sigmoidal Model for RV P_pf-V Relations

The most significant difference between the sigmoidal and the exponential model is that the sigmoidal model is concave toward the abscissa in the lower range of operating volume. The exponential model corresponds to the upper portion of the universal sigmoidal curve depicted in Fig. 8. Adjusting the values of the sigmoidal parameters shifts the portion of the sigmoidal curve that is concave toward the x-axis higher above or lower below the 0 P_pf level. As illustrated in Fig. 8, submerging the sigmoidal curve below the x-axis is equivalent to removing the lower operating volume segment of the sigmoidal curve. Therefore, the general sigmoidal model is robust and can describe specific P_pf-V curves manifesting an exponential form.

View larger version (15K): [in this window] [in a new window]   Fig. 8.   The universal sigmoidal curve: if a sigmoidal curve is displaced as shown from A to B, the portion of the overall sigmoidal Ppf-V relation actually manifested above the x-axis may appear to be quasiexponential. Inset, MCmax corresponds to the inflection point of the universal sigmoidal curve where  = dP/dV is at its minimum.

Quantitative Analysis of RV Diastolic Physiology

An essential advantage of using the sigmoidal equation to model the P_pf-V relation is the inclusion of the pressure shift parameter B, which effectively determines the vertical position of the curve. An upward shift (decreased B), regardless of compliance, indicates that a higher pressure is required to fill the ventricle. In the present study, most interesting changes were quantifiable by parameter B. The decrease in B in RV PO reflects a P_pf-V relationship with strongly manifest sigmoidality accruing from the rise of the universal sigmoidal curve out of the x-axis (Fig. 6). On the other hand, the increased B in VO reflects a less pronounced sigmoidality resulting from a submersion of the concave portion of the universal sigmoidal curve below the x-axis (Fig. 6). In IS, no significant change occurred in B.

In PO, higher filling pressure was required to force blood into the hypertrophied RV chamber in early filling, resulting in the increased sigmoidality of the P_pf-V relation. In VO, the change in the P_pf-V relation accrued from the increase in operating volume range, which caused a reduction in relative wall thickness. This reduction makes the right ventricle more compliant during early diastole, as manifested in the submersion of the P_pf-V curve below the x-axis (Fig. 8).

In IS, the factors considered in the previous two states work in directions opposite to each other. On one hand, there is an increase in the operating volume (i.e., reduction in relative wall thickness), resulting in a submersion of the P_pf-V relation. On the other hand, previous studies (8, 9, 17) have shown that regional hypertrophy eventually ensues under IS because nonischemic muscle compensates for pumping capability lost due to ischemic regions. On balance there was a statistically significant elevation of the P_pf-V relation, i.e., a decrease in B.

The impact of any individual sigmoidal parameter on the overall shape of the curve depends not only on its own value, but also on the values taken by the other parameters. To obtain comparability and a consistent interpretation of changes in the sigmoidal parameters, investigators must commit to one of the four parameter-combination cases shown in Table 4, because the impact of individual parameters on the overall curve is case dependent. This is another manifestation in cardiac mechanics of the critical need to pay attention to the range of applicability of model parameters (22). In the present study, the sign combination of parameters B, C, and K₂ corresponded to case 2.

Myocardial Compliance

Myocardial viscoelastic creep and compliance. The aspect differentiating the sigmoidal from the exponential model is that the sigmoidal curve is concave toward the abscissa in the lower range of operating volume. The behavior evident in the lower portion of the sigmoidal curve may be conceived as a manifestation of viscoelastic creep. In general, viscoelastic behavior may be imagined as a spectrum having the elastic deformation as one limiting case and viscous flow the other extreme, with changeable combinations of the two spread over the intervening range. Thus viscoelastic creep embodies response patterns that characterize behavior blends of elastic deformation and viscous flow. According to the Boltzmann superposition principle (7), the behavior of a viscoelastic material loaded in a series of steps can be analyzed by summation of the time-dependent effects of each step. The creep deformation (t) is thus determined for incremental loads ₁, ₂, ₃,... applied at times t₁, t₂, and t₃, ... as

(5)

where J(t  t₁) is the creep-response function that is analogous to compliance. Therefore, the deformation response of the viscoelastic myocardium is not manifested instantaneously in response to applied load; moreover, at any time t, it depends on all of the foregoing series of loading steps up to that time. Figure 9 provides a simplified depiction of the phenomenon using two pressure loading steps in time, the incremental viscoelastic response to each of them (inset) and the resulting chamber volume response in the pressure-volume diagram. In view of these considerations, it is to be expected that evaluation of myocardial diastolic properties is more likely to be clinically useful when it involves compliance curves spanning the entire diastolic period, as is shown in Figs. 4 and 6. With this in mind, the examination of specific indexes derived from such curves (MC_max, MC_ed, etc.) is operationally helpful in evaluating diastolic function in a clinical setting, as is illustrated in a subsequent segment.

View larger version (20K): [in this window] [in a new window]   Fig. 9.   Schematic representation of how a highly nonlinear pressure-volume curve embodies superposition of consecutive viscoelastic (time dependent) responses to successively imposed pressure steps. V(t), time-dependent change in volume.

Morphomechanical correlations. It is premature at present to attempt to identify structural components of the RV muscle with the exhibited viscoelastic response because much more information is required for such an identification. A very broad interpretation of the viscoelastic relaxed myocardial response may, however, be offered: the viscoelastic creep embodies response patterns that characterize the interactions of elastic myocyte, elastin, and collagenous connective tissue formed elements with the viscous cardiac myocyte intracellular and tissue fluids. Furthermore, it is instructive to recall the demonstration by H. B. Bull in Remington's (27) classic book on tissue elasticity that although a single nylon fiber is elastic, a stocking woven from nylon fibers exhibits viscoelastic behavior. Viscoelasticity may result from the histoarchitectonic rearrangement of elastic and viscous structural elements of RV muscle in response to the imposed pressure load.

Clinical implications of MC changes. The present study showed that the passive compliance of RV muscle changed significantly as a result of PO, VO, and IS. PO and IS resulted in decreased MC_max. Decreased MC_max in early diastole will have the direct clinical consequence of increasing central venous pressure and, in turn, decreasing cardiac output. Both of these effects will be exacerbated at higher heart rates. Because the P_pf-V relations were shifted upward in PO, the pressure levels applying at MC_max and at end diastole were both significantly increased, again tending to elevate central venous pressure clinically. In VO, a significant increase in operating end-diastolic compliance was observed, whereas the level of EDP remained unchanged. This is consistent with the increase in C observed in the P_pf-V relations, which causes an elevation of the middle segment of the sigmoidal curve. This is complemented by a reduction in the slope of the final segment of the curve and an increased end-diastolic compliance. Because of this diastolic compliance rise in VO there may be less of a tendency to elevate central venous pressure than in PO or IS. In VO due to tricuspid regurgitation it may be the regurgitant tricuspid flow (v wave) itself that will directly elevate central venous pressure and impair forward cardiac output.