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Am J Physiol Heart Circ Physiol 289: H301-H307, 2005. First published February 11, 2005; doi:10.1152/ajpheart.01237.2004
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Prediction of circulatory equilibrium in response to changes in stressed blood volume

Kazunori Uemura,1 Toru Kawada,1 Atsunori Kamiya,1 Takeshi Aiba,1,2 Ichiro Hidaka,1,3 Kenji Sunagawa,4 and Masaru Sugimachi1

1Department of Cardiovascular Dynamics, National Cardiovascular Center Research Institute, Suita; 2Pharmaceuticals and Medical Devices Agency, Tokyo; 3Japan Association for the Advancement of Medical Equipment, Tokyo; and 4Department of Cardiovascular Medicine, Kyushu University Graduate School of Medical Science, Fukuoka, Japan

Submitted 7 December 2004 ; accepted in final form 4 February 2005


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Accurate prediction of cardiac output (CO), left atrial pressure (PLA), and right atrial pressure (PRA) is a prerequisite for management of patients with compromised hemodynamics. In our previous study (Uemura et al. Am J Physiol Heart Circ Physiol 286: H2376–H2385, 2004), we demonstrated a circulatory equilibrium framework, which permits the prediction of CO, PLA, and PRA once the venous return surface and integrated CO curve are known. Inasmuch as we also showed that the surface can be estimated from single-point CO, PLA, and PRA measurements, we hypothesized that a similar single-point estimation of the CO curve would enable us to predict hemodynamics. In seven dogs, we measured the PLA-CO and PRA-CO relations and derived a standardized CO curve using the logarithmic function CO = SL[ln(PLA – 2.03) + 0.80] for the left heart and CO = SR[ln(PRA – 2.13) + 1.90] for the right heart, where SL and SR represent the preload sensitivity of CO, i.e., pumping ability, of the left and right heart, respectively. To estimate the integrated CO curve in each animal, we calculated SL and SR from single-point CO, PLA, and PRA measurements. Estimated and measured CO agreed reasonably well. In another eight dogs, we altered stressed blood volume (–8 to +8 ml/kg of reference volume) under normal and heart failure conditions and predicted the hemodynamics by intersecting the surface and the CO curve thus estimated. We could predict CO [y = 0.93x + 6.5, r2 = 0.96, standard error of estimate (SEE) = 7.5 ml·min–1·kg–1], PLA (y = 0.90x + 0.5, r2 = 0.93, SEE = 1.4 mmHg), and PRA (y = 0.87x + 0.4, r2 = 0.91, SEE = 0.4 mmHg) reasonably well. In conclusion, single-point estimation of the integrated CO curve enables accurate prediction of hemodynamics in response to extensive changes in stressed blood volume.

logarithmic function; venous return surface; heart failure

ACCURATE PREDICTION of cardiac output (CO) and cardiac filling pressures after therapeutic interventions is indispensable for optimal management and improved prognosis of patients with compromised hemodynamics (4, 5, 13, 23). In the 1980s, Sunagawa's group (20, 27) extended the framework of circulatory equilibrium of Guyton and co-workers (9, 10) to analyze complicated hemodynamic conditions such as left-sided heart failure. The extended framework is composed of a venous return surface representing the venous return of the systemic and pulmonary circulations and an integrated CO curve representing the pumping ability of the left and the right heart (Fig. 1) (27). The intersection point of the venous return surface and the integrated cardiac curve gives the equilibrium CO, left atrial pressure (PLA), and right atrial pressure (PRA). Changes in stressed blood volume shift the venous return surface upward or downward, altering the equilibrium point accordingly.



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  		Fig. 1. Diagram of circulatory equilibrium for cardiac output (CO), venous return (COV), left atrial pressure (PLA), and right atrial pressure (PRA). Equilibrium CO, PLA, and PRA are obtained as the intersection point of the venous return surface and the integrated CO curve. [Modified from Uemura et al. (29).]

 
Our previous study (29) experimentally validated that venous return is a linear function of PLA and PRA and that this relation is expressed by a flat surface, i.e., the venous return surface (Fig. 1). In addition, because of the small intra- and interanimal variability in the slope of the surface, only a single set of CO, PLA, and PRA values is sufficient to estimate the venous return surface. Furthermore, it is possible to predict how the venous return surface shifts in response to a known amount of change in the stressed blood volume. These findings suggest that if the integrated CO curve can be estimated from a single set of CO, PLA, and PRA values, it is possible to predict hemodynamics in response to various therapeutic interventions, which induce changes in loading condition or in the pumping ability of the heart (29). The present study was therefore undertaken to develop a method to estimate the integrated CO curve from a single set of CO, PLA, and PRA values and to examine whether intersection of the integrated CO curve and the venous return surface thus estimated predicts hemodynamics in response to extensive changes in the stressed blood volume. Using our model, we were able to estimate the CO curve and predict the hemodynamics in anesthetized, open-chest dogs under conditions of left heart failure as well as normal cardiac function.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Integrated CO Curve

In our previous study, we showed that CO is closely related to PLA or PRA by a three-parameter logarithmic function (29)

(1)

(2)
where SL, FL, and HL and SR, FR, and HR are parameters.

To estimate the integrated CO curve from a single set of CO, PLA, and PRA values, we fixed the F and H parameters according to the following rationale. It is well known that the CO curve varies widely with changes in ventricular contractility, heart rate, vascular resistance, and diastolic stiffness (8, 10, 20, 21, 24, 30). As shown in the APPENDIX, these factors are mainly included in the S parameter rather than in the F or H parameters. The S parameter thus comprehensively represents the pumping ability of the left or right heart. Therefore, we hypothesized that variations in the CO curve can be explained exclusively by the S parameter. Once standard values of the F and H parameters are determined, we can estimate the integrated CO curve by calculating the S parameter from a single set of CO, PLA, and PRA values.

Animal Preparation

We used 15 adult mongrel dogs of either gender (20–30 kg body wt). Care of the animals was in strict accordance with the "Guiding Principles for the Care and Use of Animals in the Field of Physiological Sciences" approved by the Physiological Society of Japan. Anesthesia was induced with pentobarbital sodium (25 mg/kg), and endotracheal intubation was performed. Isoflurane (1.5%) was continuously inhaled to maintain an appropriate level of anesthesia during the experiment. Catheters (6-Fr) were placed in the right femoral artery and vein for withdrawal of blood and for administration of drugs and fluids. To stabilize autonomic tone, we isolated the carotid sinuses bilaterally and maintained the intrasinus pressure constant at 120 mmHg (22). The cervical vagosympathetic trunks were cut. Systemic arterial pressure was measured by a catheter-tipped micromanometer (model PC-751, Millar Instruments, Houston, TX) placed in the ascending aorta via the right carotid artery. After a median sternotomy, a small pericardial incision was made at the level of the aortic root. An ultrasonic flowmeter (model 20A594, Transonics, Ithaca, NY) was placed around the ascending aorta via the incision to measure CO. Fluid-filled catheters were placed in the left and right atria via the incision to measure PLA and PRA, respectively. They were connected to pressure transducers (model DX-200, Nihon Kohden, Tokyo, Japan). The junction between the inferior vena cava and the right atrium was taken as the reference point for zero pressure (22).

Experimental Protocol

Under normal control conditions, we first infused ~250 ml of 10% dextran solution via the right femoral vein. We withdrew blood from the femoral artery in steps of 2 ml/kg to a total volume of 16–22 ml/kg (8–11 steps per animal). In each step, after waiting for 1 min, we recorded CO, PLA, and PRA for ~10 s (Fig. 2). We assumed that this volume reduction alters only the stressed blood volume of the systemic and pulmonary circulation. Because we isolated the baroreceptors, baroreflex-related changes in unstressed blood volume were negligible. We defined the reference values of CO, PLA, and PRA when half of the volume reduction was attained.



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  		Fig. 2. Changes in arterial pressure, CO, PLA, and PRA throughout the examination. As PRA and PLA decrease after stepwise reduction of the stressed blood volume, CO also decreases (Frank-Starling mechanism).

 
To create left ventricular failure, we embolized the left circumflex coronary artery with 90-µm-diameter glass microspheres (28). We adjusted the number of microspheres injected so as to increase PLA by 20 mmHg. We then volume loaded the animals and repeated the protocol described above.

The data were recorded while respiration was temporarily suspended at end expiration. All analog signals were digitized at 200 Hz with a 12-bit analog-to-digital converter (model AD12-16UE, Contec, Osaka, Japan) using a dedicated laboratory computer system (model MA 20V, NEC, Tokyo, Japan) and were stored on a hard disk for subsequent analysis. All the recorded data were averaged over 5 s. All data, except pressure data, were normalized to individual body weight.

Data Analysis

Determination of standard values of F and H parameters. We determined the standard values of the F and H parameters in seven randomly selected dogs (group 1). Using the least-squares method, we fitted the PLA-CO and PRA-CO relations obtained under normal conditions to the three-parameter logarithmic functions (Eqs. 1 and 2). We averaged the F and H values of the left and right heart for the seven animals. The averaged values were used as the standard F and H parameters in subsequent analyses.

Estimation of the integrated CO curve. Using the standard F and H parameters, we examined whether we could estimate the integrated CO curve from a single set of CO, PLA, and PRA values. For each animal in group 1, we calculated the S parameter by substituting the reference values of CO, PLA, and PRA into Eqs. 1 and 2. This calculation was done under normal and heart failure conditions. After calculation of the S parameter, the PLA and PRA measured in each step were substituted into Eq. 1 to estimate CO of the left heart and into Eq. 2 to estimate CO of the right heart. The estimated and measured CO were compared by linear regression analyses.

Prediction of circulatory equilibrium. In the other eight dogs (group 2), we estimated the integrated CO curve and venous return surface. The CO curve was estimated as described above using the standard F and H parameters. Venous return surface was estimated according to our previous work (29) as follows

(3)
where V is the stressed blood volume, COV is the integrated venous return, and 0.129 (min), 19.61 (ml·min–1·kg–1·mmHg–1), and 3.49 (ml·min–1·kg–1·mmHg–1) are standard parameters characterizing the venous return surface (29). The reference CO, PLA, and PRA values were used to calculate V, which served as the reference stressed volume.

With altered V (from +8 to –8 ml/kg of the reference value), we numerically determined the intersection of the venous return surface (Eq. 3) and the integrated CO curve (Eqs. 1 and 2) to predict CO, PLA, and PRA. The predicted CO, PLA, and PRA were compared with the measured values. We considered the change in V (±8 ml/kg) to be substantial relative to the physiological stressed blood volume (~25 ml/kg) (17).

Statistics

Group data are expressed as means (SD). The level of statistical significance was defined as P < 0.05. To test the goodness of fit, the coefficient of determination (r2) and the standard error of estimate (SEE) were calculated.


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Determination of the Standard Parameters

Figure 3 shows the measured PLA-CO and PRA-CO relations in a representative dog. CO increases in response to increases in PLA or PRA by the Frank-Starling mechanisms. These relations could be fitted to the three-parameter logarithmic function as follows: CO = 66.7[ln(PLA – 2.08) + 0.1], r2 = 0.98, SEE = 5.9 ml·min–1·kg–1 (Fig. 3A) and CO = 112.7[ln(PRA – 1.39) + 0.19], r2 = 0.98, SEE = 5.5 ml·min–1·kg–1 (Fig. 3B).



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  		Fig. 3. Relation between CO and PLA (A) and between CO and PRA (B) in 1 dog. Solid curves, best fit of the 3-parameter logarithmic functions obtained by least-squares method.

 
Table 1 summarizes the results of the fit in seven dogs. Coefficients of determination were high for the left heart (r2 = 0.95–0.99) and the right heart (r2 = 0.90–0.99). These results indicated that the logarithmic functions represented the CO curves of the left and right heart with good accuracy. The averaged F and H values (FL = 2.03 mmHg, HL = 0.80, FR = 2.13 mmHg, and HR = 1.90) for seven animals were used as standard values in subsequent analyses.


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  		Table 1. Fit of CO-PLA and CO-PRA relations to three-parameter logarithmic functions

 
Estimation of the Integrated CO Curve

Figure 4 shows the estimated CO curves under normal and heart failure conditions for a single animal. From the reference values, we calculated individual values of the S parameter. Under normal conditions, the estimated CO curve accurately coincided with the measured points in the left and the right heart. A good agreement was also observed under left ventricular failure.



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  		Fig. 4. CO curves for 1 animal under normal conditions and left ventricular failure for the left heart (A) and the right heart (B). {bullet}, Reference hemodynamic values under normal conditions; {blacksquare}, reference hemodynamic values under left ventricular failure; {circ}, measured points under normal conditions; {square}, measured points under left ventricular failure. Estimated CO curves under normal conditions (solid lines) and under left ventricular failure (dashed lines) accurately correspond with measured points.

 
Figure 5 demonstrates the relation between estimated and measured CO of pooled data from seven animals (group 1). The estimated CO agreed with the measured CO in the left and right heart.



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  		Fig. 5. Relation between estimated and measured values of CO for left heart (A) and right heart (B) for 104 steps pooled over 13 CO curves. {bullet}, Normal cardiac function; {circ}, left heart failure; dashed line, line of identity. Regression analysis (solid line) reveals that estimated CO agrees well with measured CO in the left heart [y = 0.88x + 13.3, n = 104, r2 = 0.93, standard error of estimate (SEE) = 8.7 ml·min–1·kg–1] and right heart (y = 0.96x + 5.0, n = 104, r2 = 0.88, SEE = 12.1 ml·min–1·kg–1).

 
Prediction of Circulatory Equilibrium

Figure 6 illustrates the accuracy of prediction of hemodynamics in response to changes in stressed blood volume (8 dogs, group 2). Figure 6A shows the relation between predicted and measured CO. CO was predicted accurately over a wide range of CO values from 30 to 200 ml·min–1·kg–1. A small intercept value with a slope near unity also indicates the accuracy of prediction. Figure 6B shows the accuracy of the PLA prediction. Although variability increased in the high pressure range (>20 mmHg), the prediction was reasonably accurate. Similarly, PRA was also predicted with reasonable accuracy (Fig. 6C).



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  		Fig. 6. Relation between predicted and measured values for CO (A), PLA (B), and PRA (C) for 128 steps under normal cardiac function ({bullet}) and left heart failure ({circ}). Dashed line, line of identity. Prediction was done by intersecting the venous return surface and the integrated CO curve, both of which were estimated from a set of reference hemodynamic values. Regression analysis (solid line) reveals that predicted CO (y = 0.93x + 6.5, n = 128, r2 = 0.96, SEE = 7.5 ml·min–1·kg–1), PLA (y = 0.90x + 0.5, n = 128, r2 = 0.93, SEE = 1.4 mmHg), and PRA (y = 0.87x + 0.4, n = 128, r2 = 0.91, SEE = 0.4 mmHg) agree reasonably well with measured values.

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
APPENDIX
 GRANTS
 REFERENCES
 
The results of this study indicate that once a single set of steady-state CO, PLA, and PRA values is available, it is possible to predict the changes in hemodynamic variables resulting from a known amount of change in stressed blood volume. This prediction can be very helpful in management of patients under unstable hemodynamic conditions (13, 23).

Estimation of the Integrated CO Curve

We have shown that the integrated CO curve can be estimated with reasonable accuracy under normal and heart failure conditions (Figs. 4 and 5). By fixing the F and H parameters and by ascribing the changes in the CO curve exclusively to the S parameter, we were able to estimate the integrated CO curve from a single set of hemodynamic measurements. As shown in the APPENDIX, the F and H parameters are mainly related to the end-diastolic pressure-volume relation (Eq. A4). In advanced cardiac disorders seen clinically, the end-diastolic pressure-volume relation may vary drastically (6, 7). Hypertensive or idiopathic cardiomyopathy sometimes induces severe ventricular hypertrophy, thereby significantly altering the diastolic ventricular pressure-volume relation (14). In such cases, it may be desirable not to use fixed values but, rather, to estimate F and H parameters in individual patients. The cardiovascular properties shown in Eq. A4 can be estimated noninvasively under a steady-state hemodynamic condition (3, 12). Integration of these properties into our method may allow independent estimation of the three parameters in individual patients.

The following validations indicate that our mathematical model of the CO curve and its estimation are consistent with previous investigations. First, on the basis of Eq. A4 (see APPENDIX), using previously reported data (6, 11, 18, 25), we calculated the three parameters in the logarithmic function for the left heart. The values of the cardiovascular properties were chosen to be appropriate for a 20-kg dog (Table 2). The calculated SL (34 ml·min–1·kg–1), FL (3.2 mmHg), and HL (1.14) were compatible with those obtained in our experiment (Table 1). Second, Pouleur et al. (19) examined the CO curve of the left heart in dogs under various cardiac conditions (control, coronary occlusion, and nitroprusside infusion under control and coronary occlusion). Their CO curves could be approximated to our three-parameter logarithmic functions with reasonable accuracy (r2 = 0.94–0.99). When we applied the standard values of FL (2.03 mmHg) and HL (0.80) obtained in this study to their data and estimated their CO curve, the estimated CO closely correlated with the values measured (y = 0.67x + 29.0, r2 = 0.90, SEE = 5.0 ml·min–1·kg–1, from 40 to 150 ml·min–1·kg–1).


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  		Table 2. Values of the cardiovascular properties from previously reported data

 
Clinical Application of the Framework of Circulatory Equilibrium

Cardiac patients frequently receive empirical fluid challenges to treat low CO, unexplained hypotension, and oliguria (1, 32). Such empirical challenges sometimes exert deleterious effects by excessive volume expansion (1, 32). Our framework is free of such problems, because we can accurately estimate the stressed blood volume of the patient and predict hemodynamics resulting from the volume challenge, once we measure a single set of steady-state CO, PLA, and PRA values with, for example, Swan-Ganz catheters (2).

The outcome of acute or chronic heart failure has been related to the severity of reduced CO and elevated left ventricular filling pressure (4, 5, 13, 23). Several studies, however, indicate that patients with Forrester class IV hemodynamics are not necessarily condemned to a class IV prognosis. Even if the initial hemodynamics are classified as class IV, patients showing reduction in filling pressure after intensive medical therapy have a better prognosis than those without reduction in filling pressure (13, 23). With use of our framework for guidance, proper management of low CO and elevated filling pressures would improve the prognosis of such patients.

In clinical settings, the reference point for zero pressure is determined by an empirical external inspection (16). Changes in the patient's position relative to the pressure transducer may induce apparent changes in atrial pressures (16). These factors can result in a measurement error for atrial pressure and, consequently, an error in the prediction of the circulatory equilibrium. Accurate determination of the external reference point relative to the level of the right atrium and strict attention to patient position are required for clinical application of our framework.

Limitations of the Study

All the experiments of this study were conducted in anesthetized, open-chest dogs. Anesthesia and surgical trauma significantly affect the cardiovascular system (31). Whether this equilibrium framework can be applied to conscious, closed-chest animals (including humans) remains to be tested.

We isolated baroreceptors and fixed the autonomic tone in this study. This was necessary, because the baroreflex alters the CO curve and venous return surface through its effects on stressed blood volume, vascular resistance, heart rate, and cardiac contractility (8, 22). How changes in autonomic tone under the closed-loop condition affect the accuracy of hemodynamic prediction remains to be investigated.

Conclusion

The integrated CO curve can be estimated on the basis of a single set of hemodynamic measurements (CO, PLA, and PRA). The integrated CO curve thus estimated enables accurate prediction of hemodynamics (CO, PLA, and PRA) after extensive changes in stressed blood volume during heart failure and normal cardiac function.


   APPENDIX
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
GRANTS
 REFERENCES
 
Mathematical modeling of the CO curve. We derived the relation between CO and atrial pressure on the basis of the ventricular pressure-volume relation framework (15, 25) and the ventricular-arterial coupling framework (26) as follows.

The relation between the stroke volume (SV) and the ventricular end-diastolic volume (Ved) has been approximated with reasonable accuracy as

(A1)
where Ees is the slope (elastance), V0 is the volume axis intercept of the ventricular end-systolic pressure-volume relation, T is the heart period, and R is the arterial resistance (20, 25, 26). Dividing SV by T, CO can be expressed as

(A2)
Ved can be interrelated with end-diastolic pressure (Ped) by

(A3)
where k, {alpha}, and {beta} are constants (6, 7). If we approximate Ped by a scaled mean atrial pressure (PAt), {gamma}PAt ({gamma} is a proportionality constant), Eq. A2 can be rewritten as

(A4)
which can be simplified by lumping parameters for cardiovascular system properties into three constants, S, F, and H

(A5)


   GRANTS
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
REFERENCES
 
This study was supported by Health and Labor Sciences Research Grants for research on medical devices for analyzing, supporting, and substituting the function of the human body and research on advanced medical technology from the Ministry of Health, Labour, and Welfare of Japan, Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research A 15200040, C 14570707, and C 15590786, and Ministry of Education, Culture, Sports, Science, and Technology Grant-in-Aid for Young Scientists (B) 16700379, as well as the Program for Promotion of Fundamental Studies in Health Science of Pharmaceuticals and Medical Devices Agency of Japan.


   FOOTNOTES
 

Address for reprint requests and other correspondence: K. Uemura, Dept. of Cardiovascular Dynamics, National Cardiovascular Center Research Institute, 5-7-1 Fujishirodai, Suita 565-8565, Japan (E-mail: kuemura{at}ri.ncvc.go.jp)

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.


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TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 

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