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A ventricular-vascular coupling model in presence of aortic stenosis

¹Clinical Research Institute of Montreal, Montreal, Quebec, Canada; ²Department of Cardiothoracic Surgery, Academic Hospital, Maastricht, The Netherlands; and ³Quebec Heart Institute, Laval Hospital, Laval University, Quebec, Canada

Submitted 27 July 2004 ; accepted in final form 8 December 2004

	ABSTRACT

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
In patients with aortic stenosis, the left ventricular afterload is determined by the degree of valvular obstruction and the systemic arterial system. We developed an explicit mathematical model formulated with a limited number of independent parameters that describes the interaction among the left ventricle, an aortic stenosis, and the arterial system. This ventricular-valvular-vascular (V³) model consists of the combination of the time-varying elastance model for the left ventricle, the instantaneous transvalvular pressure-flow relationship for the aortic valve, and the three-element windkessel representation of the vascular system. The objective of this study was to validate the V³ model by using pressure-volume loop data obtained in six patients with severe aortic stenosis before and after aortic valve replacement. There was very good agreement between the estimated and the measured left ventricular and aortic pressure waveforms. The total relative error between estimated and measured pressures was on average (standard deviation) 7.5% (SD 2.3) and the equation of the corresponding regression line was y = 0.99x – 2.36 with a coefficient of determination r² = 0.98. There was also very good agreement between estimated and measured stroke volumes (y = 1.03x + 2.2, r² = 0.96, SEE = 2.8 ml). Hence, this mathematical V³ model can be used to describe the hemodynamic interaction among the left ventricle, the aortic valve, and the systemic arterial system.

mathematical modeling; cardiovascular system; cardiac catheterization; left ventricle

	MATERIALS AND METHODS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
Analytic model. Transvalvular flow rate was simulated using the V³ model (Fig. 1). The latter consists on the interaction among 1) the time-varying elastance model for the left ventricle (35), 2) the instantaneous net pressure gradient across the aortic valve (13), and 3) the three-element windkessel model for the systemic arterial load (38). The LV time-varying elastance (E) model relates the ventricular pressure (P_V) to the ventricular volume (V) and the unloaded volume, defined as the intercept of the end-systolic pressure-volume relationship with the volume axis (V₀) (35) as follows:

(1)

View larger version (21K): [in this window] [in a new window]   Fig. 1. Schema of the ventricular-valvular-vascular interaction mathematical model. PV, left ventricular (LV) pressure; V, LV volume; EOA, effective orifice area of the aortic valve; AA, cross-sectional area of the aorta at the sinotubular junction; PA, pressure in the ascending aorta; Z0, aortic characteristic impedance; C, total arterial compliance; R, systemic vascular resistance; PVE, central venous pressure.

(2)

where is the fluid density, Q is the transvalvular flow rate, and E_LCo is the energy loss coefficient of the aortic valve (12, 13). E_LCo is a function of the valve effective orifice area (EOA) and the aortic cross-sectional area (A_A) and is defined as E_LCo = EOA A_A/(A_A – EOA). The so-called aortic valve EOA is the minimal cross-sectional area of the flow jet, i.e., the cross-sectional area of the vena contracta, downstream of a native or bioprosthetic aortic heart valve (Fig. 1). Finally, the three-element windkessel model relates the P_A to the transvalvular flow rate (Q) as follows:

(3)

where Z₀ is the aortic characteristic impedance, R is the total vascular resistance, C is the total arterial compliance, and P_VE is the central venous pressure. Figure 1 depicts a schema of the complete coupling model. The sum (Eq. 2)/t + (Eq. 2)/(RC) yields:

(4)

The second term of Eq. 3 is substituted for the expression P_A/t + P_A/RC in the first term of Eq. 4, and P_V is replaced using Eq. 1. Also, during ejection, transvalvular flow rate can be written as: Q(t) = –V(t)/t. Therefore, Eq. 4 becomes:

(5)

where

At the onset of the ejection (t = t₀), V is equal to the LV end-diastolic volume (LVEDV) and Q = 0. Ejection begins when LV pressure reaches aortic pressure, i.e., when TPG_net is zero. According to Eq. 2, Q/t is thus also equal to zero at the ejection onset. Because Q(t) = –V(t)/t, the initial conditions are therefore:

(6)

The third-order nonlinear differential Eq. 5 and the corresponding initial conditions (Eq. 6) completely describe the LV volume during ejection in the presence of aortic stenosis under the conditions that the ventricular contractility, the arterial properties, and the aortic stenosis severity are known. The transvalvular flow rate is simply calculated using the negative derivative of V(t). It has been shown that the normalized elastance (E_N) is quasi-independent of the preload, the afterload, and the cardiac inputs (30, 35). Elastance E(t) can thus be defined from its peak value (E_max) and time to E_max (T_Emax) as follows: E(t) = E_max E_N(t/T_Emax) (30). Table 1 summarizes the independent parameters necessary for simulating LV volume during ejection using the V³ model.

Clinical protocol. Clinical materials and methods are described in detail by Dekker et al. (9). Briefly, LV and aortic pressures, LV volume, and ECG were measured in six symptomatic patients with severe aortic valve stenosis just before and just after the aortic valve replacement. All patients provided their consent, and the study was approved by the institutional committee of the Academic Hospital of Maastricht. LV volume and pressure were obtained by inserting a conductance catheter with a pressure sensor (Sentron ANP-223N, Roden, The Netherlands) into the left ventricle. The conductance catheter was connected to a Leycom Sigma 5DF system (CD Leycom, Zoetermeer, The Netherlands) used in the dual-frequency mode. P_A was measured using a pressure tip catheter (Sentron ANP-530N, Roden, The Netherlands) positioned in the aortic root.

Hemodynamic data (LV and aortic pressures and LV volume) were acquired over a few cardiac cycles before cardiopulmonary bypass when the hemodynamics conditions were stable (dataset 1), and then a series of pressure-volume loops were acquired during inferior caval vein occlusion (dataset 2). The same protocol was repeated 15 min after successful weaning from the cardiopulmonary bypass. Pressure-volume loops during caval vein occlusion could not be obtained preoperatively in two patients due to the presence of ectopic beats. The onset of the isovolumic contraction was detected from the peak of the QRS wave of the ECG signal.

Estimation of cardiovascular parameters in patients. For each patient, the normalized elastance was determined before and after valve replacement from the series of pressure-volume loops contained in dataset 2. E_max and V₀ were first calculated according to the method described by Kono et al. (16). With V₀ known, elastance was then determined from Eq. 1 for each loop, and the average value was computed and used thereafter. E_N was obtained by dividing elastance by E_max. T_Emax was calculated as the elapsed time between the onset of the isovolumic contraction and time when elastance is maximal; i.e., when E(t) = E_max. LVEDV was directly obtained from the LV volume waveforms measured during the steady-state condition (dataset 1). The other parameters E_max, V₀, R, C, Z₀, P_VE, and E_LCo were estimated using the Nelder-Mead minimization method (17). The function to be minimized was the sum of the relative errors between measured and simulated LV and aortic pressures, respectively. Typical values encountered in the literature were used as initial estimates. We chose to estimate E_max and V₀ with the use of the minimization method rather than to use those calculated from the LV pressure-volume loops of dataset 2, because LV contractility may vary significantly during operation (4, 9). Indeed, according to our preliminary calculations, we observed that E_max and V₀ measured during caval vein occlusion were different from those calculated during the steady-state condition. Once the minimization was completed, P_V and P_A and stroke volume (SV) estimated by the model were compared with the averaged experimental data to validate the V³ model.

Comparison of estimated and measured pressures. Estimated P_V and P_A were compared with measured pressures by calculating the total relative error defined as:

where |...| represents the vectorial 1-norm. The goodness of fit was evaluated by performing a linear regression using all the data points from the pressure waveforms, and the coefficient of determination (r²), as well as the root mean square error, were determined. We also calculated the mean of the residuals, defined as the differences between the measured pressures and the fitted pressures, and we studied their correlation with the measured pressure data to test whether the residuals displayed a systematic pattern. A similar analysis was performed with LV volumes measured and simulated during LV ejection.

Statistical analysis. Standard cardiovascular parameters (heart rate; stroke volume; cardiac output; LV end-diastolic and end-systolic volumes; LV ejection fraction; LV end-diastolic, peak, and end-systolic pressures; aortic systolic, diastolic and mean pressures; mean transvalvular pressure gradient; and Gorlin EOA) were measured for each patient. The data measured before valve replacement were compared with those measured after valve replacement with the use of a two-tailed paired t-test. A similar statistical analysis was also performed with the seven cardiovascular parameters estimated by the minimization method (E_max, V₀, R, C, Z₀, P_VE, and E_LCo).

	RESULTS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
Normalized elastance before and after aortic valve replacement. Aortic valve replacement did not significantly alter E_N for a given patient, as shown in Fig. 2. Because it was impossible to measure E_N in patients 3 and 5 before valve replacement and given that the E_N remained unchanged during surgery in all other patients, we chose to use the E_N determined postoperatively to simulate preoperative LV and aortic pressures in these two patients.

View larger version (18K): [in this window] [in a new window]   Fig. 2. Normalized elastance before (solid lines) and after (dotted lines) aortic valve replacement (AVR). Comparison is not possible in patients 3 and 5 because P-V loop data could not be obtained before surgery.

View larger version (28K): [in this window] [in a new window]   Fig. 7. Comparison between measured (solid lines) and estimated (dotted lines) LV and aortic pressure waveforms in the six patients after AVR.

View larger version (16K): [in this window] [in a new window]   Fig. 10. Comparison between EOA (in cm2) measured using the Gorlin formula and energy loss coefficient (ELCo) estimated from the minimization method. Dotted line, identity line. Solid and shaded circles correspond to data before and after surgery, respectively. Note the severe patient-prosthesis mismatch in patient 6 (isolated gray circle).

View larger version (21K): [in this window] [in a new window]   Fig. 3. Comparison between measured (solid lines) and estimated (dotted lines) LV volume waveforms during ejection in the six patients before AVR.

View larger version (20K): [in this window] [in a new window]   Fig. 4. Comparison between measured (solid lines) and estimated (dotted lines) LV volume waveforms during ejection in the six patients after AVR.

View larger version (17K): [in this window] [in a new window]   Fig. 5. Relationship between measured stroke volume (in ml) and that estimated from the minimization method. Dotted line, identity line. Black and grey dots correspond to data before and after surgery, respectively.

View larger version (27K): [in this window] [in a new window]   Fig. 6. Comparison between measured (solid lines) and estimated (dotted lines) LV and aortic pressure waveforms in the six patients before AVR.

(7)

and

(8)

where MAP, PP, and T represent the mean aortic pressure, the pulse pressure, and the cardiac period, respectively. We previously demonstrated that E_LCo is equivalent to the valve EOA calculated using the standard Gorlin formula (in cm²): EOA_Gorlin = Q_mean/(44.3 ), where Q_mean (in ml/s) is the mean transvalvular flow rate and TPG_net (in mmHg) is the mean net transvalvular pressure gradient, which includes the pressure recovery occurring downstream of the vena contracta (39). For the purpose of this analysis, E_LCo was thus compared with EOA_Gorlin.

As shown in Figs. 8–10, there was a strong correlation between the parameters estimated by the optimization method and the parameters obtained from experimental measurements (estimated vs. measured: resistance, y = 0.91x – 0.04, r² = 0.98; compliance, y = 1.12x – 0.15, r² = 0.94; energy loss coefficient, y = 1.00x + 0.04, r² = 0.95). This suggests that the minimization algorithm likely converges toward a consistent minimum.

View larger version (17K): [in this window] [in a new window]   Fig. 8. Relationship between resistance (in mmHg·s·ml–1) measured using the standard clinical formula (Eq. 7) and that estimated from the minimization method. Dotted line, identity line. Solid and shaded circles correspond to data before and after surgery, respectively.

	DISCUSSION

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

LV normalized elastance. The LV elastance normalized with respect to its amplitude and time to peak value is an essential element of the V³ model. Although normalized elastance was similar before and after aortic valve replacement in a given patient, it was different from one patient to the other (Fig. 2). When every patient is considered, the mean absolute difference between the individual normalized elastances and the averaged normalized elastance was 0.053 (maximum = 0.372). The interindividual variability observed in the present study was higher than that reported in the study of Senzaki et al. (30). In their study that included 52 patients, the variability of normalized elastance was minimal despite the presence of interindividual differences with regard to etiology of heart disease, LV myocardial contractility, and loading conditions. However, their study did not include patients with aortic stenosis. Moreover, their results are also in disagreement with those recently published by Kjørstad et al. (15), where a considerable variability of the normalized elastance was observed among the patients. Further studies in a larger number of patients with aortic stenosis are thus needed to assess the interindividual variability of normalized elastance in these patients.

Cardiovascular parameters. As expected, the systemic vascular resistance estimated by the minimization method was systematically lower than that calculated from experimental data (Fig. 8). Indeed, it can be shown from Eqs. 3 and 7 that R_clin is related to R as follows: R_clin = R + Z₀. The aortic characteristic impedance was not measured in the present study. Nonetheless, the value estimated by the minimization method (0.071 ± 0.014 mmHg·s·ml^–1) was very consistent with that observed in patients by other investigators (19).

Although Segers et al. (29) reported in an animal study that the stroke volume-to-aortic pulse pressure ratio (Eq. 8) systematically underestimates the compliance determined by the area method, Chemla et al. (7) showed that it provides an accurate estimation of the total arterial compliance in patients with a normal aortic valve. Our results (Fig. 9) are consistent with the results of Chemla et al., and they suggest that the stroke volume-to-aortic pulse pressure ratio may also be valid to estimate compliance in patients with aortic stenosis as well as in patients with aortic valve prosthesis. This observation, however, remains to be confirmed in a larger number of patients with aortic stenosis.

View larger version (16K): [in this window] [in a new window]   Fig. 9. Relationship between compliance (in ml/mmHg) measured using the standard clinical formula (Eq. 8) and that estimated from the minimization method. Dotted line, identity line. Solid and shaded circles correspond to data before and after surgery, respectively.

Limitations of the mathematical model. In the V³ model, we chose a three-element windkessel model even though it has been recently shown by Stergiopulos et al. (33) that a four-element model, including the so-called arterial inertance, is superior in describing the vascular properties. Inserting this fourth element in our coupling model would have resulted in a fourth-order differential equation for the LV volume, which would have therefore necessitated an additional initial condition (see Eq. 6), which is a priori unavailable. This would have therefore resulted in an implicit coupling model that is difficult to apply in the clinical situation. On the other hand, the utilization of the three-element windkessel model provided an explicit model of the LV volume during ejection, which is easily applicable in the context of clinical studies. According to our results, this simplification of our model was not a major limitation, because the estimated compliance and resistance were very consistent with the ones directly obtained from experimental data.

The estimated unloaded volume (V₀) was 0 in seven among the twelve datasets (Table 3). This parameter was estimated by the linear approximation of the pressure-volume relation given by Eq. 1. However, the pressure-volume relationship is not linear when LV volume becomes small (14, 21). Hence, V₀ has no physical or physiological meaning but is rather a virtual parameter computed by extrapolation. Despite this limitation, the time-varying elastance model remains one of the best simple models of the left ventricle dynamics, and its linear characteristic is appropriate over a large physiological range (14).

Potential clinical implications. It is often difficult to perform a comprehensive analysis of the interaction between different physiological systems in the context of a clinical study. Indeed, this approach usually requires the realization of a large number of physiological measurements in a large cohort of patients, and these measurements are often difficult or even impossible to perform in patients. The utilization of mathematical models may overcome this problem. Hence, the V³ model validated in the present study may be useful to investigate the interaction between the systemic arterial system, the aortic valve, and the left ventricle in different types of cardiovascular diseases. Moreover, as opposed to other descriptive models previously reported in the literature, the V³ model should be potentially applicable in the clinical setting.

Several recent studies suggested that the so-called degenerative aortic stenosis is probably due to an atherosclerotic process (22). Hence, in the vast majority of the patients, aortic stenosis is probably not a disease solely limited to the valve but rather one manifestation of an atherosclerotic process involving various components of the vascular system including the aorta. Accordingly, Briand et al. (3) have recently reported that 41% of the patients with aortic stenosis concomitantly have an abnormally low systemic arterial compliance. In addition, previous studies (1, 23) have reported that systemic hypertension is a frequent occurrence (30–40%) in patients with aortic stenosis. These findings underline that patients with aortic stenosis often have abnormal vascular properties. In such patients, the V³ model may be useful to better assess the respective contributions of the aortic valve and the systemic arterial system to the LV workload in patients having concomitantly aortic stenosis and vascular diseases (i.e., hypertension, atherosclerosis). This could contribute to determine whether the treatment should be targeted on the reduction of the valvular load by aortic valve replacement, on the reduction of the arterial load by medical treatment, or on both aspects. And in this context, the V³ model may be superior to existing simple methods for characterizing the two components of the LV load. For example, the transvalvular pressure gradient and systolic arterial pressure are often used to assess the magnitude of the valvular and vascular loads, respectively. However, these parameters are highly flow dependent and are therefore subject to pseudonormalization in patients with reduced cardiac output (3). This implies that, in such patients, these parameters may underestimate the impact of the aortic valve and/or of the systemic arterial system on the LV workload. The utilization of parameters that are less flow dependent such as the Gorlin EOA or the E_LCo for the characterization of valvular properties and the arterial resistance and compliance for the characterization of vascular properties may improve the ability to assess the two components of the LV load. However, when these parameters are analyzed separately, it is difficult to estimate what is the independent contribution of each parameter to the overall LV workload. The most frequently used approach to overcome this limitation in clinical studies is to perform multivariate regression analysis. However, this approach assumes a linear relationship between the different parameters, which is obviously not the case in the ventricular-valvular-vascular system. The utilization of a mathematical model such as the V³ model that relates the aforementioned vascular and valvular parameters with an appropriate formulation may be superior to conventional approaches for the assessment of the respective contributions of the aortic valve and systemic arterial system to the LV workload. This potential clinical usefulness of the V³ model, however, remains to be confirmed in a larger cohort of patients.

	APPENDIX

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
Determination of aortic pressure. We note t₀ as the time at which ejection begins and T as the cardiac period. The solution of Eq. 3 gives the aortic pressure:

where Q* is related to Q as follows:

and the diastolic aortic blood pressure is found using the periodic property of aortic pressure:

Ventricular pressure during the ventricular filling period. Ventricular pressure during the ventricular filling period is extrapolated using the following expression:

where time t* corresponds to the end of the isovolumic relaxation. In our study, according to the results of Senzaki et al. (30), we postulated that t* = 1.8T_Emax. Parameters a and n were calculated such that P_V is continuously differentiable. Reminding that time reference (t = 0) is fixed at the onset of the isovolumic contraction, it can be demonstrated that parameters a and n are thus given by:

	FOOTNOTES

 

Address for reprint requests and other correspondence: D. Garcia, Biomedical Engineering Laboratory, IRCM, 110 Pine West Ave., Montreal, Quebec, Canada H2W 1R7 (E-mail: Damien.Garcia{at}ircm.qc.ca)

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 MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 

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