# Valvular and Systemic Arterial Hemodynamics in Aortic Valve Stenosis

## A Model-Based Approach

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## Abstract

*Background *Assessment of the severity of a stenotic aortic valve has been confounded by reports of flow dependence of stenosis severity. We hypothesized that the pressure gradient at the stenotic aortic valve would be dependent on the characteristics of the arterial circulation. Therefore, clinically useful measures of the severity of aortic valve stenosis may have to take this dependence into account.

*Methods and Results *We developed an analog model of the systemic arterial circulation in the presence of a stenotic aortic valve. The model clearly describes the dependence of stenosis severity (described by coefficients A and B) on the resistive and capacitive properties of the arterial system. We used high-fidelity pressure recordings obtained at the time of diagnostic cardiac catheterizations and found that a highly significant relation between the measured mean transvalvular gradient and that predicted by the model was demonstrated both at rest (*r*^{2}=.90) and with exercise (*r*^{2}=.80). Furthermore, the relative constancy of stenosis coefficients A and B was validated.

*Conclusions *Transvalvular hemodynamics in patients with aortic valve stenosis are dependent on the properties of the arterial system. The current model describes such behavior, correctly predicts the transvalvular gradient from model parameters, and may be useful in the assessment of stenosis severity under various clinical and physiological conditions.

Assessment of the severity of aortic valve stenosis has traditionally relied on simultaneous measures of the transvalvular pressure gradient and transvalvular flow.^{1} These measures have been performed under resting conditions in routine clinical practice. However, physiological^{2} ^{3} ^{4} and pharmacological^{5} interventions in patients undergoing hemodynamic evaluation have indicated that these traditional indices of stenosis severity may be dependent on transvalvular flow. Because the aortic valve is inherently coupled to the systemic arterial vasculature, the aforementioned variability in derived stenosis severity suggests a dependence of the valvular pressure gradient on events occurring downstream (pressure-flow relation in the systemic arterial circulation). Therefore, a comprehensive analysis of stenosis severity under various physiological conditions must take into account this potential dependence of aortic valve pressure gradient on systemic arterial hemodynamics.

The purpose of the present study was to (1) develop a model of the systemic arterial circulation in the presence of a stenosed aortic valve and (2) analyze the effect of systemic arterial hemodynamics on measures of stenosis severity.

## Methods

### Model Development

Fig 1⇓ depicts an electrical analog model of a stenosed aortic valve interposed between the left ventricle and the systemic arterial circulation. The systemic arterial circulation is represented as a modified Windkessel model in which the lumped arterial compliance (C) is in parallel with the peripheral arterial resistance (R). This RC circuit is linked in series with the aortic characteristic impedance (Z_{c}), which, in turn, is coupled to the left ventricle via the stenotic aortic valve. In such a system, the relation between instantaneous flow Q(t) and distal arterial pressure (P_{down}) is given by Equation 1. In a circuit in which capacitance (C) and resistance (R) are in parallel, the relation between current (I) and voltage (V) is given by I(t)=CdV/dt+V/R. Root (ascending aortic) pressure, P_{Ao}, is given by Equation 2, in which the pressure (voltage) to the input to the arterial system is the sum of the presence (voltage) in the RC component of the circuit and the pressure (voltage) across the proximal limb of the circuit. Finally, the relation between the instantaneous pressure drop across the stenotic aortic valve is given in Equation 3, in which the coefficients A and B characterize the geometric and hydraulic properties of the stenotic aortic valve. Such a relation between pressure and flow across a stenotic orifice has been described and validated previously.^{6} ^{7} ^{8} Note that although the stenotic aortic valve in Fig 1⇓ is depicted as a resistance element, the relation described in Equation 3 is, by virtue of the quadratic term Q^{2}, highly nonlinear.

Manipulation of Equations 1 through 3 permits the development of a relation among coefficients A and B, the transvalvular pressure gradient, and the characteristics of the arterial bed, while eliminating flow. Thus, substituting

then

and

Substitution of Equations 7 and 8 in Equation 5 finally yields

Equation 9 indicates the dependence of stenosis severity on systemic arterial resistance (R) and compliance (C)—arterial properties that effectively characterize systemic arterial hemodynamics. Note also that the flow term Q(t) has been eliminated.

To derive coefficients A and B, several simplifications and assumptions were made: (1) The relation expressed in Equation 9 was examined at the instant t_{o}, wherein the transvalvular gradient (P_{v}−P_{Ao}) was maximum, that is, when the first derivative of the pressure gradient with respect to time was zero. (2) The aortic characteristic impedance is substantially less than the peripheral vascular resistance. With these simplifications, Equation 9 then reduces to the simpler expression shown in Equation 10.

Note that in Equation 10, the variables P_{Ao}, its derivative, and the gradient are instantaneous measures at time t_{o}. This latter expression, however, represents one equation with two unknowns: the coefficients A and B. Therefore, to solve for coefficients A and B, Equation 10 was evaluated at two different physiological states (rest and exercise), with the assumption that A and B remain constant from rest to exercise.

### Model Validation and Validation of Assumptions

As indicated above, several simplifying assumptions were made to allow for the derivation of stenosis coefficients A and B. With respect to the first simplifying assumption, that is, that conditions that exist at the time of peak gradient are met throughout the systolic ejection period, substitution of A and B into Equation 9 at all points in time should yield results not significantly different from zero. It will be recalled that Equations 4 through 10 express the instantaneous relation between variables. Ideally, then, the model should be validated using instantaneous measures of pressure and flow. Catheter-tip manometry of instantaneous pressure and flow across a stenotic aortic valve is technically demanding and highly position dependent within the region distal to the valve.^{9} Given the uncertainty of the instantaneous flow measurement and the goal of developing a clinically useful approach, we chose to examine as a first approximation the mean relation between variables. Thus, if A and B remain constant under rest and exercise conditions, then a close relation between the predicted mean transvalvular gradient (from Equation 3) and the measured mean transvalvular gradient should be demonstrable.

### Clinical Study Protocol

Fifteen patients (mean age, 59±7 years) were referred for diagnostic cardiac catheterization in the setting of clinically significant aortic valve stenosis. All patients were demonstrated to have normal left ventricular size and systolic function, angiographically normal coronary arteries, and no significant aortic regurgitation.

The study protocol consisted of micromanometric left ventricular and ascending aortic pressure determinations.^{9} Cardiac output was obtained using the thermodilution technique. Patients were studied under supine resting conditions as well as during 3 to 5 minutes of supine bicycle exercise. Data were reordered on FM tape and subjected to off-line analysis.

All subjects gave written consent for participation in the research protocol in accordance with guidelines established by the University of Pennsylvania Committee on Studies Involving Human Beings.

### Data Analysis

Total arterial compliance (C) was derived as described previously.^{10} Note that, as seen in Equations 9 and 10, with aortic valve closure Q=0 and CdP_{Ao}/dt+P_{Ao}/R=0. Thus, P_{Ao}(t)=Pe^{−t/RC}. Systemic vascular resistance (R) was derived as the relation between the mean aortic pressure and cardiac output. Aortic valve resistance was also derived after the method of Ford et al.^{11} The aortic valve area was derived according to the method of Gorlin and Gorlin.^{1}

Twenty to 30 consecutive analog beats were taken from FM tape and subjected to digital conversion at 4-ms intervals. Data were stored and signal-averaged as previously described.^{12}

### Statistical Analysis

Within-patient comparisons were accomplished with the paired *t* test. The relation between selected variables was estimated using a least-squares linear regression technique. Statistical significance was defined as *P*<.05.

## Results

Table 1⇓ delineates resting and exercise systemic arterial hemodynamics in the 15 patients with aortic valve stenosis. A significant decrease in systemic vascular resistance on exercise was noted (mean change, 20±9%), although no significant change in total arterial compliance was found.

Table 2⇓ summarizes aortic valve hemodynamics in the study population. A significant increase in the Gorlin-derived aortic valve area was noted on exercise (mean change, 10±3%), along with a significant decrease in the Ford-calculated valve resistance (mean change, 21±7%). By definition, the stenosis coefficients A and B remained constant from rest to exercise.

Fig 2⇓ demonstrates close agreement between the Gorlin-derived aortic valve area and stenosis coefficient A under resting conditions; however, less agreement is seen under exercise conditions. Fig 3⇓ also demonstrates close agreement between Ford aortic valve resistance and stenosis coefficient A under resting conditions; somewhat more scatter is noted with exercise. It is of note that the decrease in aortic valve resistance was strongly dependent on the resting level of valve resistance (*r*^{2}=.74, *P*<.001).

The validation of the model under mean conditions is seen in Fig 4⇓, which demonstrates a significant relation between the predicted mean transvalvular gradient and the measured mean transvalvular gradient. The former was obtained by integration of both sides of Equation 3 and substitution of the derived values for A and B. Note that the integral of the left side of Equation 3 over the ejection period divided by the ejection period yields the mean transvalvular gradient, while the integral of Qdt over the ejection period divided by the ejection period yields mean transvalvular flow. In accord with our original hypothesis, the constancy of coefficients A and B under resting and exercise conditions would necessitate the observed close relation between predicted and observed gradients.

The validation of the model under instantaneous conditions is seen in Fig 5⇓. The summation of the terms in Equation 9 for all 146 digitized time points throughout the ejection period (other than the moment of peak gradient) is shown for two randomly chosen beats in a single patient. It can be seen that the overall sum is not significantly different from zero (95% confidence interval, −11,+3) and supports the assumption that the values for coefficients A and B are relatively invariant over time.

## Discussion

In the present report, we developed a model that allows for the assessment of the interaction between a stenosed aortic valve and “downstream” hemodynamic events. By describing in one equation the relation between stenosis severity (represented by coefficients A and B) and systemic arterial compliance and resistance, the current model provides for a more comprehensive analysis of aortic valve function under various physiological and pathological conditions.

Traditional assessment of the severity of a stenotic aortic valve describes the relation between transvalvular pressure and flow. These measurements are usually obtained under resting conditions during diagnostic cardiac catheterization and have provided for reliable and prognostically useful indices in the management of such patients. However, the acquisition of hemodynamic data under varying physiological states in patients with aortic valve stenosis has, using these traditional methods, resulted in variable results for the calculated aortic valve area. Prior studies have reported consistent and significant increases in calculated aortic valve area with exercise.^{2} ^{3} ^{4} Explanations for this finding have varied from increased physical displacement of the stenotic aortic valve under conditions of increased transvalvular flow^{13} to limitations of the theoretical underpinning for valve area calculation.^{14} Several recent studies of the assessment of aortic valve stenosis under exercise conditions have emphasized the strong dependence of all derived indices of stenosis severity on transvalvular flow.^{15} ^{16}

Since both transvalvular flow and the pressure gradient (as demonstrated herein) are dependent on the characteristics of the systemic arterial circulation, a model providing a more flow-independent approach would be useful. As seen in Equation 9, the absence of the flow term Q(t) allows for such an approach. We have also demonstrated that the pressure gradient is a function of stenosis severity (A,B), aortic pressure, arterial compliance, and peripheral resistance. In the Gorlin model of valvular hemodynamics, the relation between the pressure gradient and flow is given by the quadratic relation ΔP=K_{1}Q^{2}. The approach used by Ford et al describes a linear relation between the pressure gradient and flow: ΔP=K_{2}Q. Our approach represents the summation of both the Gorlin and Ford expressions (Fig 6⇓). In this respect, all three models would predict similar pressure gradients at low values of Q. However, at higher values of Q, the pressure gradient is underestimated with both the Gorlin and Ford approaches. This underestimate of the pressure gradient would in the Gorlin model lead to the observed increase in calculated area and in the Ford model a decrease in valve resistance.

The current model describes the influence of the (analog) components of the peripheral arterial circulation on “upstream” events at the aortic valve. The model not only provides for evaluation of the severity of aortic valve stenosis under resting and exercise conditions but also may be used in the assessment of stenosis severity in other clinical settings. One such setting is the low cardiac output state. Under these circumstances, the Gorlin-derived aortic valve area may be problematic and potentially misleading.^{17} We have seen how, under high flow conditions, the Gorlin model would result in an underestimate of the pressure gradient. While all three models provide comparable estimates of the pressure gradient at lower flows, further consideration of our model under low flow conditions provides insight into this challenging clinical presentation. A common accompaniment of the low flow state is an elevated systemic vascular resistance. Rearranging the terms of Equation 10 reveals that the gradient is, at common levels of arterial pressure and compliance, inversely proportional to the systemic vascular resistance. Thus, the model predicts a lower gradient under these low flow–elevated resistance conditions for similar valvular stenoses (A,B). Patients 14 and 15 in our series were characterized by depressed cardiac outputs at rest and elevated systemic vascular resistance. Both were characterized by similar high values for stenosis coefficients A and B, although the patient with a higher systemic vascular resistance had a lower transvalvular gradient. Unfortunately, in this small group we could find no instances of lower values for coefficients A and B in the setting of low cardiac output, perhaps reflecting the biased nature of this sample in which only symptomatic patients with clinically severe valvular stenosis were chosen for study. Assessment of stenosis severity in the setting of severe arterial hypertension is another situation commonly encountered.^{18} Under such conditions, marked alterations in peripheral vascular resistance and total arterial compliance must be accounted for because the pressure gradient will vary directly with the compliance but inversely with the resistance.

The current model also indicates the importance of kinetic terms in the evaluation of stenosis severity. That is, the fraction of the overall pressure gradient due to the term BQ^{2} ranged from 15% to 35% of the total pressure drop (the sum of AQ+BQ^{2}). The kinetic terms clearly assume greater importance under high flow conditions such as encountered during exercise and therefore must not be discounted in the evaluation of stenosis severity. In fact, without the inclusion of the kinetic term, the relation between predicted and observed gradients was substantially inferior to that depicted in Fig 3⇑ (*r*=.20).

There are several limitations to the present study that must be acknowledged. In the current study, we develop a model based on the three-element Windkessel. Although this Windkessel model has been useful in the understanding of the behavior of the systemic arterial circulation, its limitations are acknowledged.^{19} However, given the ability to take into account such features of in vivo behavior as wave travel and wave reflection,^{20} the approximations made using the current approach allow for reliable model behavior.

A critical assumption was the constancy of stenosis coefficients A and B during rest and exercise. That A and B were essentially invariant is seen in the significant relation between predicted and observed transvalvular gradients. That is, if A and B were significantly different on exercise and these values were entered into Equation 3 (which describes the predicted behavior), a poorer correlation with the observed gradient would have been obtained.

One simplifying assumption was the relation between the transvalvular pressure gradient and flow (Equation 3). Although this relation is modeled under steady (nonpulsatile) conditions, application of the model to pulsatile hemodynamics has been validated.^{7} ^{8} As indicated previously, we chose, as a first approximation, to examine the relation in mean terms, given the uncertainty of catheter-tip measures of instantaneous flow velocity in the proximal ascending aorta and the need to develop a clinically useful method. Despite these limitations, both resting and exercise gradients could be reliably predicted by use of this model.

Another simplifying assumption relates to the solution of Equation 9 at the moment of the peak transvalvular gradient when the time derivative is zero. We chose this point to simplify our calculations. To establish the validity of Equation 9 at each instant in time over the ejection interval, we computed the exact solution for every digitized time point. Significant overall deviation from zero (less than 10% of all calculated values) was not seen in any instance in randomly chosen beats from all patients studied.

### Summary

The present study reports the development and preliminary validation of a model describing the relation between stenotic aortic valve hemodynamics and systemic arterial hemodynamics. The need for additional validation, particularly with respect to instantaneous measures of pressure and flow under carefully controlled experimental conditions, is recognized. The current approximate approach to understanding the relation between systemic arterial properties and valvular hemodynamics provides an alternative flow-independent method of assessing stenosis severity.

## Acknowledgments

The authors are indebted to the expert secretarial assistance of Nedra Ellis and Jeannette Forte. We also thank Dr Meir Shinnar for assistance with the mathematical solutions to the model.

- Received February 27, 1995.
- Accepted April 1, 1995.

- Copyright © 1995 by American Heart Association

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- Valvular and Systemic Arterial Hemodynamics in Aortic Valve StenosisWarren K. Laskey, William G. Kussmaul and Abraham NoordergraafCirculation. 1995;92:1473-1478, originally published September 15, 1995https://doi.org/10.1161/01.CIR.92.6.1473
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