Vol. 83, Issue 6, 1799-1813, December 1997

Coupled vs. uncoupled pericardial constraint: effects on cardiac chamber interactions

Masao Takata, Yasuhiko Harasawa, Sadek Beloucif, and James L. Robotham

Pathophysiology Research, National Children's Medical Research Center, Tokyo 154; Division of Cardiology, Kyushu University Hospital, Fukuoka 812-82, Japan; and Department of Anesthesiology and Critical Care Medicine, The Johns Hopkins Medical Institutions, Baltimore, Maryland 21205

ABSTRACT

INTRODUCTION

GLOSSARY

METHODS

RESULTS

DISCUSSION

APPENDIX

ACKNOWLEDGEMENTS

FOOTNOTES

REFERENCES

ABSTRACT

Takata, Masao, Yasuhiko Harasawa, Sadek Beloucif, and James L. Robotham. Coupled vs. uncoupled pericardial constraint: effects on cardiac chamber interactions. J. Appl. Physiol. 83(6): 1799-1813, 1997.The effects of pericardial constraint on cardiac chamber interactions were evaluated by mathematical model analyses based on a novel concept of coupled vs. uncoupled pericardial constraint. We hypothesized that the nature of pericardial constraint can be classified as a "coupled" constraint exerted by uniform liquid pressure or an "uncoupled" constraint exerted by regional surface pressure. The numerical solution of the model of atrioventricular interaction produced the characteristic waveforms in venous flows and right atrial/ventricular pressures in classical pericardial diseases. Coupled constraint accounted for the patterns in cardiac tamponade; uncoupled constraint accounted for those in constrictive pericarditis. Analytic solution of the model of ventricular interdependence demonstrated that coupled constraint (tamponade) produced greater gains in ventricular interdependence, increasing the occurrence of pulsus paradoxus, whereas uncoupled constraint (constriction) produced a greater effective right ventricular elastance, increasing the likelihood of Kussmaul's sign. Thus the concept of coupled vs. uncoupled constraint may offer a coherent framework to understand the characteristic steady-state and respiratory-induced hemodynamic events in multiple forms of pericardial diseases.

atrioventricular interaction; constrictive pericarditis; model; tamponade; venous return; ventricular interdependence

INTRODUCTION

CARDIAC CHAMBERS are physically linked to each other, hydrodynamically connected by blood flows, and situated in a space limited by the juxtacardiac structures. Filling and emptying of one cardiac chamber are thus influenced by changes in volume or pressure of other chambers. The pericardium has been shown to modulate such cardiac chamber interactions (13): the vertical interaction between an atrium and the corresponding ventricle within a cardiac cycle, i.e., atrioventricular interaction (4), and the horizontal interaction between the two ventricles, i.e., ventricular interdependence (6, 21). However, the mechanism whereby the pericardium in disease states influences such cardiac chamber interactions has not been well characterized.

The differences in hemodynamic profiles between cardiac tamponade and constrictive pericarditis have fascinated clinicians and physiologists for many years. There are classic observations of differences in vascular pressure waveforms in pericardial diseases, i.e., a prominent x-descent in atrial pressure with tamponade (18, 23) vs. a large y-descent in atrial pressure and a "dip-and-plateau" pattern in ventricular pressure with constriction (19, 23). Recent studies demonstrated that venous flow waveforms, normally biphasic with one peak during ventricular systole and the other during ventricular diastole (2, 25, 27), are also altered in pericardial diseases. With tamponade, we found in dogs that venous flows become mainly systolic with an almost absent diastolic flow (4), consistent with Doppler flow velocity studies in humans (3, 5, 8, 11). With constriction, an enhanced diastolic venous flow with a relatively small systolic component was observed by Doppler studies in humans (5, 8, 9, 11). These findings suggest that the manner in which the pericardium constrains the heart may be quite different between tamponade and constriction, despite similarly increased pericardial pressure. The differences in venous pressure and flow waveforms may be related to the unique nature of pericardial constraint in each pathology and its influence on atrioventricular interaction.

In addition to those steady-state pressure and flow changes, it has been well documented in the literature that respiratory-induced hemodynamic signs are manifested differently in cardiac tamponade and constrictive pericarditis. Pulsus paradoxus, i.e., an inspiratory decrease in arterial pressure associated with a decrease in left ventricular stroke volume, is observed much more often in tamponade than in constriction (22). Conversely, Kussmaul's sign, i.e., a paradoxical inspiratory increase in right atrial pressure, is occasionally associated with constriction but rarely with tamponade (18, 19). Respiration produces large changes in right heart volume due to an inspiratory increase in systemic venous return (25, 27), which then decrease left heart filling via ventricular interdependence, leading to pulsus paradoxus (7, 17). Thus it seems a reasonable assumption that the status of ventricular interdependence is different in nature or degree between tamponade and constriction. The specific manner of pericardial constraint and its effects on ventricular interdependence possibly could explain the differences in incidence of Kussmaul's sign in the two forms of pericardial diseases.

The goals of the present study are to better understand the characteristic hemodynamic signs in pericardial diseases and, more generally, to develop a conceptual framework characterizing the influence of the pericardial constraint on cardiac chamber interactions. We hypothesized that the nature of the pericardial constraint may be classified as one of two types: 1) a "coupled" constraint exerted by uniform liquid pressure over the heart (e.g., cardiac tamponade), which couples changes in all cardiac chamber volumes, and 2) an "uncoupled" constraint exerted by regional surface pressure over the heart (e.g., constrictive pericarditis), which independently restricts each chamber volume. On the basis of this hypothesis, the effects of coupled vs. uncoupled pericardial constraint on cardiac chamber interactions were studied by use of mathematical model analyses. Flow-mediated atrioventricular interaction was studied with a numerical model and computer simulation, whereas pressure-mediated ventricular interdependence was evaluated with an analytic model and symbolic mathematical analysis. The results suggest that the construct of coupled vs. uncoupled pericardial constraint may offer a physiological rationale to interpret the differences in the hemodynamic profiles of cardiac tamponade and constrictive pericarditis and provide insights necessary to develop better diagnostic and therapeutic strategies.

GLOSSARY

Model of atrioventricular interaction

Cp	Pulmonary arterial capacitance
Cv	Systemic venous capacitance
Epe	Pericardial elastance over RA and RV for coupled constraint
Epera	Pericardial elastance over RA for uncoupled constraint
Eperv	Pericardial elastance over RV for uncoupled constraint
Era	Time-varying elastance of RA
Eramax	Maximum elastance of RA
Eramin	Minimum elastance of RA
Erv	Time-varying elastance of RV
Ervmax	Maximum elastance of RV
Ervmin	Minimum elastance of RV
Lv	Systemic venous inertance
Pd	Downstream pressure source
Pp	Pressure at the pulmonary arterial capacitance
Ppe	Pericardial pressure over RA and RV for coupled constraint
Ppera	Pericardial pressure over RA for uncoupled constraint
Pperv	Pericardial pressure over RV for uncoupled constraint
Pra	RA pressure
Pratm	Transmural RA pressure
Prv	RV pressure
Prvtm	Transmural RV pressure
Pu	Upstream pressure source
Pv	Pressure at the systemic venous capacitance
PRI	P-R interval
p	Pulmonary arterial flow
t	Tricuspid flow
v	Combined vena caval flow
Rpd	Distal pulmonary arterial resistance
Rpp	Proximal pulmonary arterial resistance
Rt	Transtricuspid valve resistance
Rvd	Distal systemic venous resistance
Rvp	Proximal systemic venous resistance
RA	Right atrium
RV	Right ventricle
T	Cardiac cycle time
Tmaxra	Contraction time of RA
Tmaxrv	Contraction time of RV
Vra	Volume of RA above its unstressed volume
Vrv	Volume of RV above its unstressed volume
Vtotal	Total right heart volume above its unstressed volume

Model of ventricular interdependence

METHODS

Concept of Coupled vs. Uncoupled Pericardial Constraint

On the basis of this reasoning, we classified the nature of pericardial constraint as 1) a coupled constraint exerted by uniform pericardial liquid pressure over all four cardiac chambers or 2) an uncoupled constraint produced by independent and different local pericardial surface pressures over each cardiac chamber. The essence of this concept is that a coupled constraint restricts the volumes of four cardiac chambers together, whereas an uncoupled constraint independently restricts the volume of each cardiac chamber. The present study evaluates how such differences in the nature of pericardial constraint can be predicted to modify the status of cardiac chamber interactions.

Numerical Approach: Model of Atrioventricular Interaction

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Table  1.   Baseline parameter values used in simulation Abbreviations Values Systemic venous impedance Rvd (mmHg · s · ml1) 4 Rvp (mmHg · s · ml1) 0.14 Cv (ml/mmHg) 6.1 Lv (mmHg · s2 · ml1) 0.007 Pulmonary arterial impedance Rpp (mmHg · s · ml1) 0.2 Rpd (mmHg · s · ml1) 0.8 Cp (ml/mmHg) 1.2 Right heart parameters Rt (mmHg · s · ml1) 0.025 Eramax (mmHg/ml) 1.2 Eramin (mmHg/ml) 0.6 Tmaxra (s) 0.12 Ervmax (mmHg/ml) 4 Ervmin (mmHg/ml) 0.4 Tmaxrv (s) 0.18 T (s) 0.6 PRI (s) 0.12 See Glossary for definition of abbreviations.

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Table  2.   Changes in D/S v ratio and A/E t ratio with increases in pericardial constraint Pu, mmHg Pd, mmHg Prv,* mmHg Cardiac Output D/S v Ratio A/E t Ratio Control condition   (no elastances) 50 5 4.5 100 0.92 0.72 Coupled constraint   Epe = 0.5 mmHg/ml 40 5 7.0 73 0.62 0.59   Epe = 1.0 mmHg/ml 35 13 10.2 57 0.22 0.58   Epe = 2.5 mmHg/ml 30 14 12.5 39  0.02 0.52   Epe = 5.0 mmHg/ml 27 17 14.3 28  0.04 0.55 Uncoupled constraint   Epera = Eperv = 0.5     mmHg/ml 40 5 6.5 75 1.44 0.56   Epera = Eperv = 1.0     mmHg/ml 35 13 9.6 58 1.53 0.49   Epera = Eperv = 2.5     mmHg/ml 30 14 13.0 40 3.51 0.38   Epera = Eperv = 5.0     mmHg/ml 27 17 16.4 25 11.15 0.07 D/S v ratio, ratio of diastolic to systolic venous inflow volumes into RA; A/E t ratio, ratio of peak in A wave to peak in E wave in t. * Values at end-diastole. Expressed as relative values (control value = 100%). With increases in pericardial elastances, values for Pu and Pd were adjusted by approximately equal increments to produce desired changes in RV end-diastolic pressure.

Analytic Approach: Model of Ventricular Interdependence

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RESULTS

Results of Numerical Approach

With an increased coupled constraint (Figs. 4B and 5B), _v became a predominantly systolic flow with only a small diastolic component. P_ra showed a prominent x-descent with a small y-descent. Consistent with the change in _v, V_ra and V_rv changed almost reciprocally during a cardiac cycle. V_rv retained its biphasic pattern during diastole with an increasing phase due to the RA contraction. V_total was minimum at midsystole but almost constant from end systole to end diastole. In contrast, with an increased uncoupled constraint (Figs. 4C and 5C), _v became a predominantly diastolic flow. The A wave in _t was markedly attenuated. The y-descent in P_ra was prominent with a small x-descent, and P_rv exhibited a steep transient decrease at early diastole followed by a relatively unchanged portion in late diastole, i.e., a dip-and-plateau pattern. V_ra showed only a small change during a cardiac cycle. V_rv became monophasic during diastole with a minimal increase due to the RA contraction.

Figure 6 illustrates relationships of P_rv, transmural P_rv (relative to pericardial pressure), and pericardial pressure (P_pe or P_perv) to the increased coupled or uncoupled constraint. Under both conditions, most of P_rv during diastole is attributable to the pericardial pressure, with a small contribution from the transmural P_rv. With the increased coupled constraint, P_pe showed a transient decrease at midsystole but had returned to its end-diastolic level at end systole, i.e., a pattern similar to the changes seen in V_total. With the increased uncoupled constraint, P_perv showed its maximum at end diastole and minimum at end systole with a waveform similar to V_rv. Only the uncoupled constraint produced a "dip" in P_rv in early diastole.

Table 2 summarizes changes in the D/S _v and A/E _t ratios with increases in pericardial constraint. Under the control condition, the D/S _v ratio was 0.92 with an A/E _t ratio of 0.72. With increases in coupled constraint, D/S _v ratio decreased; i.e., the systolic flow became dominant. The A/E _t ratio showed a small decrease but remained >0.5. With an E_pe of >2.5 mmHg/ml, the D/S _v ratio was nearly zero, indicating almost absent diastolic flow in _v. With increases in uncoupled constraint, the D/S _v ratio increased and the A/E _t ratio decreased. With E_pera and E_perv of 5.0 mmHg/ml, the D/S _v ratio exceeded 10, indicating that the systolic component was less than one-tenth of the diastolic component in _v. The A/E _t ratio was only 0.07, consistent with the markedly attenuated A wave in _t.

Results of Analytic Approach

With no pericardial constraint

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With the coupled constraint

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With the uncoupled constraint

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Thus, G_V, G_P, and E_rveff would increase as the pericardial elastances increase with the coupled or uncoupled constraint. However, as evident in Eqs. 8 and 9, the manner in which the pericardial elastances affect these parameters was quite different between the two conditions. E_pe increases G_V, G_P, and E_rveff in a totally different fashion from any of ventricular elastances, whereas E_pelv and E_perv increase them in a manner essentially similar to the free wall elastances (E_lvf and E_rvf).

Figure 7 illustrates quantitative differences in the interdependence parameters between the coupled and uncoupled constraints. At a given level of E_petotal, G_V and G_P were higher with the coupled than with the uncoupled constraint (Fig. 7, A and B). On the other hand, E_rveff was higher with the uncoupled than with the coupled constraint (Fig. 7C). The differences in G_V, G_P, and E_rveff between the two constraint conditions became larger as E_s increased.

DISCUSSION

On the basis of the novel concept of "coupled vs. uncoupled" pericardial constraint, we evaluated the effects of pericardial constraint on cardiac chamber interactions by use of mathematical model analyses. The numerical model of atrioventricular interaction well approximated the steady-state venous flow and pressure waveforms observed in pericardial diseases. Increased coupled constraint accounted for the patterns in cardiac tamponade, and increased uncoupled constraint accounted for those in constrictive pericarditis. On the other hand, the analytic model enabled quantitative comparisons of the status of ventricular interdependence between the two constraint conditions. Increased coupled constraint (tamponade) produced greater interdependence gains, which should lead to manifestation of a pulsus paradoxus (17), whereas increased uncoupled constraint (constriction) was associated with a greater effective RV elastance and, hence, the increased likelihood of a Kussmaul's sign (25). These findings provide a basis for the pathogenesis of the characteristic steady-state and respiratory-induced hemodynamic signs seen in tamponade and constriction. Thus the construct of coupled vs. uncoupled pericardial constraint may offer a useful conceptual framework to understand the pathophysiology in various forms of pericardial diseases.

Pericardial Constraint and Atrioventricular Interaction

A question then arises as to why such different flow and pressure patterns take place with the increased coupled (tamponade) or uncoupled constraint (constrictive pericarditis). It is important to appreciate that, as the degree of tamponade or constriction increases, the pericardial pressure increases and approaches the intraluminal P_ra with a decrease in the transmural P_ra (18, 19, 23). In severe forms of both pericardial diseases, the intraluminal P_ra, i.e., the downstream pressure for systemic venous return, will be determined almost entirely by the pericardial pressure, which is the function of pericardial elastic recoil with minimal if any contribution of the intrinsic time-varying elastic properties of the underlying cardiac chambers.

With the increased coupled constraint, "elevated" and "uniform" P_pe all over the heart restricts the sum of the volumes of the RA and RV. P_pe is a function of the total right heart volume (V_total). During systole, ventricular ejection moves blood out of the intrapericardial space, reducing V_total and P_pe. The decrease in P_pe will lead to a decrease in the intraluminal P_ra (x-descent), enhancing venous return and producing a systolic antegrade _v. If sufficient volume can move into the RA during systole to replace the ejected stroke volume, the decreased P_pe and P_ra will again increase to the level of the upstream venous pressure, and _v should become zero at end systole. During diastole, the RA emptying and RV filling via the tricuspid valve do not directly influence V_total. P_pe is not affected by any intrapericardial volume transfer with the coupled constraint. Thus P_pe and P_ra will be almost unchanged during diastole, producing an almost absent diastolic _v and diminished y-descent in P_ra.

With the increased uncoupled constraint, elevated and "regional" P_pera and P_perv individually restrict the RA and RV. P_pera is a function of V_ra, whereas P_perv is determined by V_rv. During systole, ventricular ejection decreases V_rv and P_perv but does not affect P_pera. Thus the intraluminal P_ra shows little change during systole (diminished x-descent) and the systolic _v is small. During diastole, the RA emptying (the decrease in V_ra) into the RV produces a decrease in P_pera. The resultant decrease in the intraluminal P_ra (y-descent) will enhance venous return, producing a diastolic antegrade _v.

The simulation produced an early diastolic dip in P_rv only with the uncoupled constraint, but not with the coupled constraint (Fig. 5). This finding is consistent with the clinical observations that a dip-and-plateau pattern or a square root sign in P_rv is a characteristic phenomenon of constrictive pericarditis (18, 19, 23). An insight as to the pathogenesis of this sign may be derived from the fact that the pericardial pressure over the RV is a function of V_total with the coupled constraint, whereas it is determined only by V_rv with the uncoupled constraint. In tamponade, P_pe decreases initially during systole but returns to the previous end-diastolic level at end systole, along with the corresponding change in V_total. In constrictive pericarditis, however, P_perv continues to decrease during systole and reaches its minimum value at end systole, along with the change in V_rv. When the tricuspid valve is opened, the decreased P_perv will start to contribute to generation of a pressure gradient for venous return. In early diastole, P_perv increases as the RV fills, whereas the transmural P_rv decreases because the RV continues to relax. As a result of these changes, the intraluminal P_rv will exhibit a dip in early diastole. Because the combined elastance of the RV and pericardium is large, this dip in P_rv will soon be lost as the RV continues to fill rapidly and then abruptly cease filling, resulting in a plateau in P_rv in late diastole.

Pericardial Constraint and Atrial Function

With the increased coupled constraint, the reciprocal changes in V_ra and V_rv during a cardiac cycle imply that the RA fills with a volume nearly equal to stroke volume during the RV ejection and empties it into the RV during the succeeding diastole. Because _v is almost nil during diastole, direct filling of the RV from the venous system should be minimal during diastole. Thus, with cardiac tamponade, the atrium would not function well as a passive conduit for venous return during diastole, whereas it would serve as an efficient reservoir for ventricular filling by utilizing a decrease in V_total during ventricular ejection. Atrial function as a booster can still be preserved, as evidenced by the presence of the A wave in _t and the late diastolic increase in V_rv.

With the increased uncoupled constraint, V_ra exhibited only small changes during a cardiac cycle. Venous return occurred mainly during diastole, with the RA serving effectively as a passive conduit for blood flow from the venous system to the RV. The decreased A wave in _t and monophasic diastolic increase in V_rv suggest that the contribution of atrial contraction to ventricular filling was diminished. Because the difference between the maximum and minimum elastances of the RA is small compared with that of the RV, a large external elastance (E_pera) added over the RA would produce a combined RA-pericardium elastance that varies minimally during a cardiac cycle. Any influence of atrial contraction and relaxation on systolic venous flow would be masked by the increased uncoupled constraint. Filling of the right heart from the venous system would occur only during diastole when the tricuspid valve is opened and the elastances of the RV are connected parallel to the elastances of the RA. Thus, in constrictive pericarditis, the atrium appears to serve mainly as a conduit, and atrial function as a booster or a reservoir would be markedly attenuated, despite an unchanged atrial contractility.

Our results suggest that echocardiographic observation of atrioventricular volume changes within a cardiac cycle may provide an alternative diagnostic strategy for early detection of tamponade or constrictive physiology. Marked coupling between atrial and ventricular volume changes (i.e., reciprocal changes within a cardiac cycle) in addition to dominant systolic venous flow patterns should be useful signs of cardiac tamponade, whereas minimal changes in atrial volume and predominant diastolic venous flow should suggest a constrictive pericarditis.

Pericardial Constraint and Ventricular Interdependence

Our model highlighted similarities and differences among the interdependence mechanisms mediated by the myocardium per se, coupled pericardium, and uncoupled pericardium. Under conditions with no pericardial constraint, Eq. 7 clarified how the myocardial factors modulate the interdependence parameters. First, G_V and G_P were enhanced by E_s and E_lvf and E_rvf. Although the septal displacement is the sole element to produce myocardium-mediated interactions in the model, the free walls are still able to enhance them by augmenting the transseptal interaction (15). When the septum is shifted leftward, a stiffer LV free wall will augment the degree of constraint of a given LV volume, creating a greater increase in LV pressure. Second, E_rveff was influenced not only by the RV-related elastances (E_rvf and E_s) but also by the LV-related elastance (E_lvf). Thus factors determining E_rveff are not necessarily limited to the pure RV factors and will also be influenced by interdependence.

With the increased coupled constraint, P_pe restricts the LV and RV volumes together, increasing G_V and G_P. Because the RV and LV are exposed to the coupled pericardial elastance, E_rveff would also increase. It is important to note that the enhancement of interdependence produced by the coupled pericardium does not require involvement of the transseptal interaction. Equation 8 demonstrated that E_pe affected the interdependence parameters in a fashion totally different and independent from the septal or free wall elastances. This implies that the coupled pericardium provides a unique interdependence mechanism in addition to the myocardium-mediated interactions.

With the increased uncoupled constraint, P_pelv and P_perv impose local constraining forces on the LV and RV. As a result, the effective stiffness of each ventricle and, hence, E_rveff would increase. More interestingly, despite the uncoupled nature of the constraint, the intrapericardial volume coupling between the two ventricles was also enhanced, and G_V and G_P increased. The clue to understanding this apparently paradoxical finding is given in Eq. 9, in which E_pelv and E_perv affected the interdependence parameters essentially in a manner similar to E_lvf and E_rvf. Thus the uncoupled regional pericardium would behave as additional free walls, increasing the effective stiffness of the LV and RV free walls and augmenting the transseptal interaction. The interdependence mechanism mediated by the uncoupled pericardium can therefore be considered as an augmentation of the myocardium-mediated interactions.

The differences in the interdependence mechanism between the coupled and uncoupled pericardium may be further clarified by calculating changes in G_V, G_P, and E_rveff when E_s approaches infinity in Eqs. 7-9.

As E_s  , with no pericardial constraint

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with the coupled constraint

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and with the uncoupled constraint

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In this situation, the septum becomes extremely rigid, so that the myocardium-mediated interactions become negligible, as represented by zero values of G_V and G_P in Eq. 10a. With the coupled constraint, however, G_V and G_P did not become zero, still being under the influence of E_pe. In contrast, G_V and G_P approached zero with the uncoupled constraint. E_rveff became equal to E_rvf + E_pe with the coupled constraint, whereas it was equal to E_rvf + E_perv with the uncoupled constraint.

Understanding of these equations can provide an intuitive explanation as to why quantitative differences were observed in the degree of interdependence between the coupled and uncoupled constraint conditions (Fig. 7). As the degree of the pericardial constraint increases, the coupled pericardium produces a complete intrapericardial volume coupling between the two ventricles, whereas the uncoupled pericardium only provides a partial volume coupling by enhancing the already present transseptal interaction. G_V and G_P should therefore be larger with the coupled than with the uncoupled constraint. However, the RV is connected to the elastance of the total pericardium (E_pe) with the coupled constraint, whereas it is connected only to the elastance of the right-sided pericardium (E_perv) with the uncoupled constraint. The connection of the RV to the other half of the uncoupled pericardium (E_pelv) is indirect, mediated by the transseptal interaction. E_perv is substantially larger than E_pe, because it reflects only a regional portion of the pericardium. Thus it is likely that E_rveff increases more with the uncoupled than with the coupled constraint at a given level of pericardial stiffness. The differences between the two constraint conditions should increase as the influence of transseptal interaction is attenuated (i.e., E_s increases), as graphically shown in Fig. 7.

Pericardial Constraint and Respiratory-Induced Hemodynamic Signs

Although Kussmaul's sign is accepted as a useful clinical sign for pericardial pathology (18, 19), its pathogenesis had not been well explained. We recently demonstrated in canine experiments that a Kussmaul's sign should only occur during inspiration with an active diaphragmatic descent (25). When an inspiratory increase in systemic venous return is mainly attributed to a large increase in abdominal pressure, the rise in transmural RA pressure may exceed the fall in pleural pressure, leading to manifestation of a Kussmaul's sign. With a greater E_rveff under conditions of uncoupled pericardial constraint, the inspiratory increase in right heart volume would produce greater increases in transmural RV diastolic or RA pressure relative to pleural pressure. Thus the results are consistent with the classic observation that a Kussmaul's sign is observed relatively frequently in constrictive pericarditis (uncoupled) but rarely in cardiac tamponade (coupled). In an intuitive sense, with constriction only the elastances of the right heart and right-sided pericardium accept the enhanced venous return, whereas with tamponade the left-sided pericardium would also participate in buffering the effects of the increased right heart volume, resulting in a decreased likelihood of the manifestation of a Kussmaul's sign at a similar level of total pericardial constraint.

Critique

It is theoretically possible to model the overall cardiovascular system, including the four cardiac chambers as well as systemic/pulmonary arterial and venous beds, as a single closed circuit on a beat-to-beat basis. In this situation, however, the model must simultaneously fulfill two difficult and sometimes conflicting requirements. First, each vascular bed needs to be characterized as input impedance to simulate instantaneous venous and arterial pressure waveforms with reasonable precision. Second, the steady-state characteristics of each vascular bed must also be considered to simulate the effects of volume redistribution among the vascular beds in a closed-circuit circulation. Unfortunately, no model has been successful in perfectly reconciling the beat-to-beat (high-frequency) and closed-circuit (low-frequency or steady-state) characteristics into a single mathematical model mainly because of the lack of necessary experimental data of vascular properties from these two viewpoints. Thus our approach is appropriate, although not perfect, given the present level of experimental knowledge of circulatory parameters.

A potential problem may exist in our model because of the assumption of linear elements. The pressure-volume relationships of the pericardium and diastolic ventricular elastances are known to be highly nonlinear. Although it was possible to arbitrarily choose certain nonlinear mathematical functions for these elements, there are few experimental data on which we could reliably define such functions under the normal and pericardial disease conditions. Other limitations in our numerical model may include lack of inertances of the tricuspid and pulmonic valves and lack of the effects of descent of the base of the heart produced by ventricular contraction. All these factors may cause some problems in simulating accurate flow waveforms across the tricuspid valve or may underestimate venous flow during early ventricular systole. Nevertheless, the overall good agreement between the model and reality suggests that our simple model is valid as a first approximation, and the principles developed on the basis of the coupled vs. uncoupled constraint appeared useful in providing a rationale to interpret hemodynamic events in pericardial diseases.

ACKNOWLEDGEMENTS

The authors thank Drs. W. L. Maughan, W. Mitzner, K. Sunagawa, and K. Miyasaka for insightful and encouraging discussions during the development of this work.

FOOTNOTES

Address for reprint requests: M. Takata, Cardiovascular Research Center, Massachusetts General HospitalEast, 149 13th St., 4th Fl., Charlestown, MA 02129.

Received 4 June 1997; accepted in final form 31 July 1997.

APPENDIX

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Table  3.   Changes in D/S  v ratio with changes in each model parameter Changes in D/S v Ratio 0.5 × Baseline Baseline 2 × Baseline Systemic venous impedance   Rvd 0.97 0.92 0.82   Rvp 0.86 0.92 0.93   Cv 0.86 0.92 0.93   Lv 0.88 0.92 0.99 Pulmonary arterial impedance   Rpp 0.96 0.92 0.80   Rpd 1.03 0.92 0.74   Cp 0.88 0.92 0.93 Right heart parameters   Rt 0.95 0.92 0.87   Eramax 1.71 0.92 0.60   Eramin 0.56 0.92 2.93   Ervmax 0.64 0.92 1.06   Ervmin 1.01 0.92 0.83 Upstream and downstream pressure sources   Pu 0.81 0.92 0.98   Pd 0.98 0.92 0.81   Pu and Pd* 0.92 0.92 0.92 * Pu and Pd were simultaneously changed.

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REFERENCES