Research ArticleCardiology

Ventricular stroke work and vascular impedance refine the characterization of patients with aortic stenosis

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Science Translational Medicine  11 Sep 2019:
Vol. 11, Issue 509, eaaw0181
DOI: 10.1126/scitranslmed.aaw0181

Hard work meets the path of least resistance

Transcatheter aortic valve replacement (TAVR) is a treatment for patients with aortic stenosis (narrow aortic valve) that reduces the transvalvular pressure gradient; however, only some patients experience improved quality of life after the procedure. To understand how valvular, ventricular, and systemic vascular conditions contribute to improvements after TAVR, Ben-Assa et al. studied 70 patients undergoing the procedure. Patients with lower preprocedural vascular impedance and higher left ventricular stroke work had greater improvements in quality of life after TAVR. This suggests that analyzing valve, ventricle, and arterial system hemodynamics could identify patients likely to benefit from TAVR and inform timing of intervention.


Aortic stenosis (AS) management is classically guided by symptoms and valvular metrics. However, the natural history of AS is dictated by coupling of the left ventricle, aortic valve, and vascular system. We investigated whether metrics of ventricular and vascular state add to the appreciation of AS state above valve gradient alone. Seventy patients with severe symptomatic AS were prospectively followed from baseline to 30 days after transcatheter aortic valve replacement (TAVR). Quality of life (QOL) was assessed using the Kansas City Cardiomyopathy Questionnaire. Left ventricular stroke work (SWLV) and vascular impedance spectrums were calculated noninvasively using in-house models based on central blood pressure waveforms, along with hemodynamic parameters from echocardiograms. Patients with higher preprocedural SWLV and lower vascular impedance were more likely to experience improved QOL after TAVR. Patients fell into two categories: those who did and those who did not exhibit increase in blood pressure after TAVR. In patients who developed hypertension (19%), vascular impedance increased and SWLV remained unchanged (impedance at zeroth harmonic: Z0, from 3964.4 to 4851.8 dyne·s/cm3, P = 0.039; characteristic impedance: Zc, from 376.2 to 603.2 dyne·s/cm3, P = 0.033). SWLV dropped only in patients who did not develop new hypertension after TAVR (from 1.58 to 1.26 J; P < 0.001). Reduction in valvular pressure gradient after TAVR did not predict change in SWLV (r = 0.213; P = 0.129). Reduction of SWLV after TAVR may be an important metric in management of AS, rather than relying solely on the elimination of transvalvular pressure gradients.


The symptoms of aortic stenosis (AS) were likely first presented some 350 years ago by the school of Rivierii (1). The dyspnea and heart failure they described remained an essential part of AS for years, making its way into the mortality triad that Ross and Braunwald published in 1968 (2). Over the last 50 years, the appearance of symptoms has become guide for intervention. The Placement of Aortic Transcatheter Valves (PARTNER) 1B trial (3), however, taught us that AS is more morbid than appreciated some half century ago: Current AS populations are profoundly different, older, and sicker than the patients described by Ross and Braunwald, and subjective reporting of symptoms could prevent optimal timing for intervention (4).

Quantifiable metrics could replace symptoms, adding precision and more timely triggers of intervention. For years, clinicians have sought to leverage the pressure gradient in AS, but the gradient alone cannot be used as determinant of extent of disease or need to intervene, and elimination of gradients does not always assure restoration of health. Aortic valve replacement virtually eliminates the transaortic pressure gradients in patients with AS, yet quality of life (QOL) does not improve in all patients. Some 35% of patients report no QOL benefit 1 year after transcatheter aortic valve replacement (TAVR) (5).

Accordingly, the field is searching for additional metrics to follow and use to discriminate subpopulations within the AS population. Aortic valve stenosis is a complex, systemic disease that is not solely limited to the aortic valve but is also dependent on the left ventricular (LV) state, vascular load, and ventricular-vascular coupling (610). The concept of low-gradient AS (LGAS) and the discriminatory potential of dobutamine (11) and nitroprusside (12) have further emphasized the importance of this paradigm in the pathophysiology of AS. Thus, many have called for the creation of a quantitative framework that integrates the interactive coupling of the ventricle, valve, and vasculature (68) to guide decision-making and treatment optimization.

Here, we performed a pilot study to test the hypothesis that interaction between the main components governing systemic perfusion—the LV, aortic valve, and arterial system—can more fully define AS state and the response to TAVR than valve gradient alone. We applied advanced computational models to calculate LV stroke work (SWLV) and vascular impedance in patients before and after TAVR. SWLV represents the work that the LV performs by displacing blood against the impedances of the aortic valve and the vascular system (1315). It is a measure of the energetic state of the LV directly influenced by the valvular and vascular compartments and might represent a comprehensive hemodynamic metric of LV energetic state in patients with AS before and after intervention. Vascular impedance is the load of the proximal aorta and distal arterioles: It adds to SWLV demands both before and after TAVR. We followed changes in SWLV and vascular impedance from baseline to 30 days after TAVR and evaluated the utility of these metrics in predicting improvement in QOL after TAVR. Together with valve gradients, SWLV and impedance might refine our understanding of AS.


Baseline characteristics

From April 2016 to April 2017, 70 patients with severe AS who underwent TAVR were prospectively enrolled from two large referral medical centers. The average age was 81 years, and 53% were female (Table 1). The majority had symptoms of heart failure at baseline [97% had New York Heart Association (NYHA) functional class ≥ II, and 88% had Kansas City Cardiomyopathy Questionnaire (KCCQ) summary score > 20]. None had clinically meaningful (≥moderate to severe) aortic regurgitation (AR), and only eight (11.4%) had LV ejection fraction (EF) ≤ 45%. Low-gradient severe AS (LGAS) (mean gradient, <40 mmHg and aortic valve area, <1 cm2) was found in 26 patients (37%), of which 13 had low flow, defined as stroke volume index (SVi) < 35 ml/m2. Sixty-two (88%) received a balloon-expandable valve, and eight (12%) received a self-expandable valve. One patient (1.4%) died within 30 days of procedure from cardiovascular causes. Clinical and echocardiographic data were available in all other patients at the 30-day post-TAVR time point. Noninvasive central blood pressure (BP) measurements and related analysis were available in 52 patients (74%).

Table 1 Baseline clinical and echocardiographic characteristics.

Values are presented as means ± SD or n (%). NYHA, New York Heart Association; KCCQ, Kansas City Cardiomyopathy Questionnaire; STS, Society of Thoracic Surgeons; CABG, coronary artery bypass grafting; PCI, percutaneous coronary intervention.

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Changes in hemodynamic metrics from baseline to 30 days after TAVR

In addition to classic valvular load indices, SWLV and the vascular impedance spectrum were determined before and 30 days after TAVR for each subject using noninvasive models (Fig. 1). Changes of ventricular, valvular, and vascular indices from baseline to 30 days after TAVR are presented in Table 2. As expected, valvular metrics improved significantly after TAVR (P < 0.001). There was no change in EF, SVi, or cardiac index from baseline to 30 days after TAVR. At 30 days after TAVR, only one patient had paravalvular leak > grade 2, and two patients had aortic pressure gradient > 20 mmHg.

Fig. 1 Analysis of ventricular, valvular, and vascular metrics.

(A) Schematic of a heart and great vessels undergoing TAVR and data acquisition workflow. Vascular: Photograph of noninvasive brachial BP measurements captured with the SphygmoCor XCEL device, used to derive peripheral (brachial) and central (aortic) pressure waveforms. Representative waveforms are shown. Valvular: Echocardiographic pulsed wave Doppler tracings captured at the left ventricular outflow tract (LVOT), used to derive central velocity waveforms. Representative waveform is shown. Ventricular: Computer-based lumped parameter model used patient-specific echocardiographic data to derive LV pressure and volume waveforms. Representative waveforms are shown (magnified image of the lumped parameter model can be seen in fig. S2). (B) Vascular impedance analysis. The aortic input impedance spectrum was derived using Fourier decomposition of the noninvasive central pressure and LVOT velocity waveforms. Z0 is the impedance modulus at the zero harmonic. Zc was calculated as the average of frequencies 2 to 10 Hz. (C) SWLV analysis. LV pressure-volume loop was constructed from the computer-based lumped parameter model. SWLV was calculated as the area of the simulated pressure-volume loop.

Table 2 Changes in hemodynamic metrics from baseline to 30 days after TAVR.

Values are presented as means ± SD. BP, blood pressure, Z0, vascular impedance at zeroth harmonic; Zc, characteristic impedance; LV, left ventricle; Zva, valvulo-arterial impedance; bpm, beats per minute.

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Systolic BP fell more than 15 mmHg from baseline in 12 patients (23%) and rose above 140 mmHg in 10 patients (19%). In the newly hypertensive patients, vascular impedance metrics—including impedance at zeroth harmonic (Z0), characteristic impedance (Zc), and total peripheral resistance (TPR)—significantly rose (P = 0.033, 0.039, and 0.036, respectively), but they did not change in the rest of the cohort (Fig. 2 and Table 3).

Fig. 2 LV stroke work and vascular impedance metrics in patients developing hypertension after TAVR.

Bar charts comparing (A) LV stroke work (SWLV), (B) characteristic impedance (Zc), and (C) vascular impedance (Z0) in patients who developed new hypertension after TAVR (n = 10; 19%, red) versus the rest of the cohort (n = 42; 81%, blue). Data are presented as the mean changes (Δ) from baseline to 30 days after TAVR. Error bars represent SD. P values represent the statistical significance of change from baseline to 30 days after TAVR and were tested with paired samples t test.

Table 3 Changes in SWLV and vascular impedance metrics in patients developing hypertension after TAVR.

Values are presented as means ± SD. Hypertensive response after TAVR was defined as systolic BP > 140 mmHg or diastolic BP > 90 mmHg 30 days after TAVR not present at baseline (28).

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In concert with the pathophysiological definition of SWLV in patients with AS, baseline SWLV correlated with central mean aortic pressure (r = 0.280 and P = 0.019), with SVi (r = 0.665 and P < 0.001), and with mean valvular pressure gradient (r = 0.372 and P = 0.002) (Fig. 3). Change in SWLV from baseline to 30 days after TAVR did not correlate with change in mean valvular pressure gradients (r = 0.213 and P = 0.129) (Fig. 4), whereas it did correlate with change in central mean aortic pressure (r = 0.432 and P = 0.001) and change in SVi (r = 0.735 and P < 0.001) (fig. S1). All the patients reduced their valve gradient after TAVR. SWLV was reduced in most patients (from 1.6 to 1.3 J; P < 0.001) but not in the 19% of our cohort who developed new hypertension after TAVR (1.5 to 1.4 J; P = 0.460) (Fig. 2 and Table 3).

Fig. 3 Properties of preprocedural SWLV.

Correlation of preprocedural SWLV with preprocedural mean transvalvular pressure gradient (A), stroke work index (B), and central mean aortic pressure (C). R represents the Pearson correlation coefficient; (n = 70).

Fig. 4 Changes in SWLV and valvular pressure gradients.

Minimum-maximum plots representing patient-specific changes from baseline to 30 days after TAVR in mean valvular gradient (top) and in SWLV (bottom) (each data point represents average of two measurements of a single patient). There is no correlation between the decrease in transvalvular pressure gradient after TAVR and the change in SWLV (R = 0.213 and P = 0.129; n = 52; fig S2). R represents the Pearson correlation coefficient

Predictors for QOL improvement 30 days after TAVR

Using a KCCQ-based QOL improvement end point, 58 patients (83%) improved and 12 patients (17%) did not improve 30 days after TAVR (Table 4). Patients whose QOL improved had higher baseline SWLV values (1.6 versus 1.2 J; P = 0.006) (Fig. 5). Other baseline metrics associated with QOL improvement were lower vascular impedance at the Z0 (4368.5 versus 6272.8 dyne·s/cm3; P < 0.001) and a higher body mass index (29.3 versus 24.5 kg/m2; P = 0.040). Low baseline Z0 was associated with QOL improvement at 30 days after TAVR, whereas baseline BP measurements did not (Table 4). Baseline EF, mean transaortic pressure gradient, NYHA functional class, and noninvasive central BP measurements did not differ significantly between the groups distinguished by QOL changes, although those with higher preprocedural KCCQ were more likely to have QOL improvement at 30 days after the procedure.

Table 4 Baseline differences between QOL improvement groups.

Values are presented as means ± SD or n (%). QOL improvement was defined as increase of ≥10 points in KCCQ score from baseline to 30 days after the procedure.

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Fig. 5 SWLV in QOL improvement groups.

Individual dot plot and overlaid box plot of baseline (pre-TAVR) SWLV segregated by QOL improvement groups. Box plots portray median values and 25th to 75th percentiles for the whole cohort (n = 70) and for the LGAS cohort (n = 26) (independent samples t test was used to compare the means).

Of the 26 patients with LGAS, 20 (77%) improved their QOL and 6 (23%) did not. As in the whole cohort, in this prespecified LGAS cohort, patients who showed improvements in their QOL had higher baseline SWLV values (1.60 versus 0.86 J; P = 0.005) (Fig. 5).


Our study was based on the premise that optimal definition of patients with AS cannot rely on the emergence of symptoms or valve gradient alone but rather should combine hemodynamic metrics of the valve, LV, and arterial subsystems. The increasing appreciation by the field that classifying the contribution of each subsystem to AS helps provide an integrated view of patient state may provide objective data as to when to intervene and how to optimize intervention with adjunctive therapies.

In this study, we used patient-specific data and computational analysis to include metrics of the ventricle proximal and vasculature distal to the stenosed aortic valve. These metrics allowed us to demonstrate that LV decompensation and recovery are simply driven not only by valve hemodynamics but also by the vasculature. There is, therefore, a subset of patients with new hypertension after TAVR due to increased vascular impedance that have persistently high SWLV even after valve replacement. Furthermore, we have demonstrated that these metrics affect QOL.

As a composite metric, SWLV represents the LV energetic state and incorporates ventricular geometry and its mechanical properties, as well as valvular and vascular impedances (1315). Accordingly, we observed that SWLV before TAVR correlated with preprocedural SVi, mean valvular gradient, and mean arterial pressure. Hence, SWLV might represent a hemodynamic metric that can be tracked in patients with AS before and after intervention. In our study, the higher the SWLV at baseline, the greater the QOL improvement after TAVR. In other words, patients with ventricles that can function at a higher energetic state before valve replacement might have a higher likelihood of improving their QOL after TAVR. The baseline values, and not the degree of change in SWLV, predicted improvement; changes in SWLV from baseline to 30 days after TAVR can only be appreciated once vascular impedance is incorporated because they cannot be explained by reduction in mean valvular pressure gradient alone. SWLV only fell in the cohort who did not show increased BP or vascular impedance after TAVR. In those patients with new hypertension, Z0 rose, hypertension emerged, and there was no fall in SWLV—the energetic load on the heart remained elevated even after the valve gradient was removed.

Herein lies an important distinction with the work of Perlman et al. (16) and Lindman et al. (17), who both reported on the benefit of hypertension after TAVR, and of the work of Yotti et al. (18), who demonstrated an increase in vascular load after TAVR associated with limited acute afterload relief. We add the full hemodynamic description, adding the coupling of vascular impedance with SWLV and implication on QOL at the 30-day time point, and in contrast to previous studies describing hypertension in general, our focus is on the emergence of hypertension and its associated failure of SWLV restoration.

The metrics of vascular impedance that we present here are derived from analysis of the frequency domain of the pressure and flow waveforms rather than the temporal domain alone. The use of impedance extracted from the frequency domain of the signal has the advantage of providing additional information regarding vascular load in the setting of pulsatile physiological flow and not just steady load, thus allowing better characterization of load opposing the LV.

The valvulo-arterial impedance (Zva) developed by the group of P. Pibarot (9, 19, 20) is a measure of total afterload, acknowledging the contribution of the vascular state to the physiology of AS. In our study, although Zva changed significantly after TAVR (P = 0.001), it did not discriminate between patients who improved their QOL and those who did not (P = 0.147). We speculate that the Zva index is very sensitive to aortic valve gradient changes that always reduce after TAVR.

Valvular and vascular impedances contribute to LV afterload in AS (9), and elevations in the latter might challenge SWLV even after successful valve replacement. In some patients, valvular stenosis might even mask the effects of vascular stiffness, which only emerges once flow is restored with removal of valve stenosis. Such thinking potentially explains how vascular impedance rose and SWLV did not fall as expected after TAVR in the cohort who developed hypertension after the procedure.

Patients with LGAS pose a diagnostic challenge in our daily practice (12, 21). This group can have depressed or preserved LV function and low or high transvalvular flows. Some patients are truly limited by the valve, whereas others are limited by the ventricular energetic state or by the compliance of the vascular tree. We believe that the integration of metrics that describes the state of the ventricular and vascular systems will improve our ability to manage these patients. In our study, the subpopulation of patients with LGAS who did not improve their QOL after TAVR had significantly lower baseline SWLV (P = 0.005). We also believe that SWLV could serve as an important metric in evaluating patient with AS and reduced EF. Because our cohort included mainly patients with normal EF, further studies are warranted.

These data support the emerging call to classify AS in more rigorous and precise terms that incorporate the individual contributions of the valve, ventricle, and distal vasculature. Temporal tracking of SWLV and vascular impedance metrics might direct the optimal timing of intervention (surgical AVR or TAVR) and help define the nature of adjunctive medical care thereafter. Moreover, these metrics might also identify patients who may not benefit from valve replacement, as the ventricle may not be likely to recover because of a patient’s increased vascular impedance.

Incorporating these data together allows us to appreciate that there are likely several subpopulations of AS patients. In some, the primary pathology is isolated to the valve, as in younger patients with bicuspid valves, where LV outlet flow is so restrictive that functional limitation can be attributed to the valve alone. Similarly, there are those patients for whom pathologies in the valve are linked and markedly exacerbated by LV failure or by excessive afterload. Accordingly, AS involves pathologic contributions from the ventricle and vasculature as well as the valve. Along this paradigm, the pathology AS in its most extreme and highly symptomatic form extends to all three elements, whereas in its earliest phases, one can define disease subsets where one aspect dominates. It is conceivable, then, that progressive validation of quantitative metrics will enable classification of AS with valve, ventricular, or vascular dominance well before the three combine to create end-stage disease. This perception might inform us as to what metrics to follow and what adjunctive therapies to apply with valve intervention (9, 18, 22).

There were several limitations to our study that warrant consideration. The relatively small sample size did not allow for investigation of clinical event rates after TAVR or for subgroup analyses. Clinical outcome measurement and improvement after TAVR are not straightforward to discern. We used QOL assessment that was shown to be an important end point in the current TAVR population. We acknowledge that 30-day QOL change might be early to fully appreciate the benefit or futility of valve replacement. However, a recent large study showed that QOL metrics that improved after TAVR at 30 days persisted for at least 1 year (5); in other words, the TAVR-related benefit in terms of symptom relief, functional capacity, and QOL observed at 12 months was totally attained at 30 days. Nonetheless, further studies with longer follow-up are needed. Such future studies might use more refined physiological assessments such as 6-min walk, exercise stress test, or VO2 max. There are multiple means of measuring vascular stiffness. We sought a noninvasive method that derives central BP from peripheral measurements, using the SphygmoCor XCEL device. Although this device has been used in other AS studies (23), it has not been fully validated in this population (24, 25). We did not have full data regarding out-of-hospital medication change during the first 30 days after TAVR. Nevertheless, the statistically significant increase in impedance in the newly hypertensive patients held whether patients were taking new BP medication.

In this pilot, hypothesis-generating study, our objective was to explore refined and comprehensive metrics to better characterize different subpopulations of patient with aortic valve stenosis. We further aimed to support the field in searching for advanced physiologic metrics that can become a part of the evaluation of patients with AS and may help guide clinicians about benefit and timing of intervention. SWLV and vascular impedance appear to be important metrics to classify patients with AS and meaningful discriminants of response to TAVR. Reduction of SWLV may be an important target in AS rather than reliance solely on the elimination of transvalvular pressure gradients. Because metrics of valve dynamics, ventricular function, and vascular impedance can readily and regularly be calculated, their use may become an essential part of the evaluation of AS patients. In addition, larger-scale studies are warranted to further validate these metrics and implement them in clinical practice.


Study design

We designed a study that examined the hypothesis that the physiological state of patients with AS can be more precisely defined by analyzing the interaction of the LV, aortic valve, and arterial system with specific metrics of each element. To evaluate this hypothesis, we prospectively enrolled patients with severe AS undergoing TAVR at the Massachusetts General Hospital or Brigham and Women’s Hospital. Patients were evaluated before the procedure and 30 days after TAVR. Data collection included QOL assessments and dedicated echocardiography and noninvasive central BP, allowing us to calculate SWLV and vascular impedance spectrums using in-house computational models. We investigated how dynamics in physiological metrics correlated with changes in QOL metrics. Informed consent was obtained as approved by the institutional review board. Eligibility for TAVR was determined by the local heart team.

Data acquisition

Demographic and procedural data were collected from local TAVR databases and patients’ records. KCCQ was collected at baseline and 30 days after TAVR. QOL improvement was defined as an increase in total KCCQ score of more than 10 points (5). Hypertensive response after TAVR was defined as systolic BP > 140 mmHg or diastolic BP > 90 mmHg 30 days after TAVR that was not present at baseline (16).

Echocardiograms were reviewed in a blinded fashion by two senior cardiologists. LGAS was defined as mean transvalvular gradient < 40 mmHg and aortic valve area < 1 cm2. Low flow state was defined as SVi < 35 ml/m2. The LV outflow track (LVOT) velocity waveform was extracted from the raw echocardiographic DICOM (Digital Imaging and Communications in Medicine) images by an in-house tool using the MATLAB software package (MathWorks Inc.) (Fig. 1A).

Noninvasive central pressure was captured with the SphygmoCor XCEL device (AtCor Medical). This device uses a generalized transfer function and standard BP measurements to derive the central (aortic) pressure waveform from the brachial volume displacement waveform. Pulse wave contours were formed by averaging several heartbeats. For those patients with a history of arrhythmias, care was taken during measurement to ensure that the waveforms that were chosen were best representing of sinus rhythm (Fig. 1A).

Protocol for noninvasive central pressure measurements with SphygmoCor XCEL device

A simple BP cuff connected to the XCEL device was placed on a subject’s arm, and an automated sequence measured pulse wave contours in the following manner. Three consecutive BP measurements were taken, and the average of the final two was used for pulse waveform calibration. The cuff was then reinflated to a subdiastolic pressure, and the pulse wave contours of several heartbeats were recorded and averaged. This averaged waveform was digitized and saved for offline processing and analysis.

Vascular impedance calculation

Aortic input impedance spectrums were calculated in the frequency domain using noninvasive central pressure waveforms recorded from the SphygmoCor and a velocity tracing from pulsed wave Doppler measured in the LVOT. Each waveform was decomposed into its Fourier harmonics. Moduli and phase at each harmonic were used to calculate input impedance via an in-house MATLAB program. Z0 was defined as the impedance moduli at the zeroth harmonic. Zc was calculated as the average of frequency 2 to 10 Hz, with frequencies greater than three times the median excluded (Fig. 1B). To allow for comparison of these vascular impedance metrics to other vascular load metrics used in the literature, we used standard equations to calculate TPR, systemic arterial compliance (SAC), and Zva.

Impedance is defined as resistance to pulsatile flow. This is the ratio between the frequency harmonics of the pressure waveform to those of the blood flow waveform (26).

Mathematically, this can be defined asZn=PnQnwhere Zn is the aortic input impedance at the nth harmonic. Pn and Qn are the pressure and flow harmonics at the nth harmonic.

The phase of the two waves with respect to each other can be calculated asPhase=phase flow(n)phase pressure(n)where phase(n) is the phase of the flow at the nth harmonics and similarly for pressure.

Lumped parameter model and SWLV calculation

A lumped parameter model of the left heart (Fig. 1A) incorporated the LV, aortic valve, extent of AR, and systemic circulation. Each submodel was validated with in vivo cardiac catheterization and magnetic resonance imaging data (2729). All input parameters were obtained as patient-specific echocardiographic measurements. Using this lumped parameter model, LV pressures and volumes waveforms were extracted. SWLV was calculated as the area of the simulated pressure-volume loop (Fig. 1C).

Heart-arterial model. The ventricle was filled by a normalized physiological mitral flow waveform adjusted for the required stroke volume. Coupling between LV pressure and volume was performed through a time-varying elastance E(t), a measure of cardiac muscle stiffnessE(t)=PLV(t)V(t)V0(1)where PLV(t), V(t), and V0 are LV time-varying pressure, time-varying volume, and unloaded volume, respectively. The amplitude of E(t) can be normalized with respect to maximal elastance, Emax, the slope of the end-systolic pressure-volume relationship, giving EN(tN) = E(t)/Emax. Time then can be normalized with respect to the time to reach peak elastance, TEmax (tN = t/TEmax)EmaxEN(t/TEmax)=PLV(t)V(t)V0(2)

A normalized curve of EN(tN) can be described using Fourier series. Therefore, the relationship between PLV(t) and V(t) can be determined for the LV.

Modeling aortic valve. AS was modeled using Eq. 3. This formulation expresses the instantaneous net pressure gradient across the stenotic valve (after pressure recovery) as a function of the instantaneous flow rate and the energy loss coefficient and links the LV pressure to the aorta pressureTPGnetAS=PLV(t)PA(t)=2πρELCoASQ(t)t+ρ2ELCoAS2Q2(t)(3)andELCoAS=(EOACOA)AAEOAAS(4)where ELCo∣AS, EOA∣AS, A, ρ, and Q are the valvular energy loss coefficient, the effective orifice area, the ascending aorta cross-sectional area, the fluid density, and the transvalvular flow rate, respectively. Variable aortic valve resistance (Rav) and constant aortic valve inductance (Lav) in the lumped parameter model are ρ2ELCoAS2Q(t) and 2πρELCoAS, respectively.

Modeling aortic valve regurgitation. AR was modeled using the same formulation as AS. AR pressure gradient is the difference between aortic pressure and LV pressure during diastoleTPGnetAR=2πρELCoARQ(t)t+ρ2ELCoAR2Q2(t)(5)andELCoAR=(REOA)ALVOTALVOTREOA(6)where ELCo∣AR, REOA and ALVOT are regurgitation energy loss coefficient, regurgitant effective orifice area, and LVOT area, respectively. Variable aortic valve regurgitation resistance (Rav) and constant aortic valve regurgitation inductance (Lav) in the model are ρ2ELCoAR2Q(t) and 2πρELCoAR, respectively.

Determining arterial compliance and peripheral resistance. The total systemic resistance was computed as the quotient of the average brachial pressure and the cardiac output [assuming a negligible peripheral venous pressure (mean ~ 5 mmHg) compared to aortic pressure (mean ~ 100 mmHg)]. This total systemic resistance represents the electrical equivalent resistance for all resistances in the current model. Because what the LV faces is the total systemic resistance and not the individual resistances, for the sake of simplicity, we considered the aortic resistance, Rao, and systemic venous resistance, RSV, as constants and adjusted the systemic arterial resistance, RSA, according to the obtained total systemic resistance. For each degree of hypertension, we fit the predicted pulse pressure to the actual pulse pressure (known by arm cuff sphygmomanometer) obtained from clinical study by adjusting compliances [aorta (Cao) and systemic (CSAC)].

Computational algorithm. A lumped parameter model (2729) was analyzed numerically by creating and solving a system of ordinary differential equations in MATLAB Simscape (MathWorks Inc.), enhanced by adding additional codes to meet demands of cardiac model in circuit. A Fourier series representation of an experimental normalized elastance curve for human adults was used to generate a signal to be fed into the main program. Simulations start at the onset of isovolumic contraction. LV volume, V(t), is calculated using LV pressure, PLV, and time-varying elastance values (Eq. 1). MATLAB’s ode23t trapezoidal rule variable-step solver was used to solve system of differential equations with initial time step of 0.1 ms. The convergence residual criterion was set to 10−5, and initial voltages and currents of capacitors and inductors were set to zero.

Equation for hemodynamics parameters

1. TPR was calculated asTPR=80*mean brachial pressurecardiac output

2. SAC was calculated asSAC=stroke volumebrachial pulse pressure

3. Zva (19) was calculated asZva=(Systolic brachial pressure+mean transvavular gradient)SVi

“Stroke work loss index” was not included in the analysis because it underestimates the hemodynamic significance of AS in patients with hypertension (30).

Statistical analysis

Continuous variables, presented as means (±SDs), were tested with the Student’s t test or Mann-Whitney rank sum test. Categorical variables are presented as number (percentage) and compared using chi-square or Fisher’s exact test. Normal distribution was assessed with the Shapiro-Wilk test. A bivariate Pearson correlation coefficient was computed to assess the relationship between continuous variables. All analyses were considered significant using two-tailed test with a P value of <0.05. The SPSS statistical package 20 was used.


Fig. S1. Properties of change in SWLV from baseline to 30 days after TAVR.

Fig. S2. Schematic diagram of the lumped parameter model.


Acknowledgments: We thank M. O’Rourke for guidance and insight; C. Ricciardi, C. Steinmetz, and T. Urman of the MIT/IMES Clinical Research Center for protocol management and data collection; and D. Furman, S. L. Capano, S. R. Devireddy, and B. J. Coronis for protocol management and patient data analysis. Funding: Supported, in part, by a research grant from Edwards Lifesciences to E.R.E. and a U.S. NIH (R01 49039) grant to E.R.E. Edwards Lifesciences had no influence on the design, data collection, or analysis of the study. Author contributions: E.B.-A.: conception and design, data collection and analysis, interpretation of data, manuscript writing, and critical revision of the manuscript; J.B.: conception and design, data collection and analysis, interpretation of data, manuscript writing, and critical revision of the manuscript; Z.K.-M.: data collection and analysis, manuscript writing, and critical revision of the manuscript; J.M.d.l.T.H.: conception and design, data collection and analysis, interpretation of data, and critical revision of the manuscript; B.L.: data collection and analysis; M.O.: data collection and analysis; F.K.: conception and design and critical revision of the manuscript; I.F.P.: interpretation of data and critical revision of the manuscript; I.I.: interpretation of data and critical revision of the manuscript; J.J.P.: interpretation of data and critical revision of the manuscript; P.B.S.: data collection and analysis, interpretation of data, and critical revision of the manuscript; S.E.: data collection and analysis, interpretation of data, and critical revision of the manuscript; M.B.L.: conception and design and critical revision of the manuscript; E.R.E.: conception and design, data collection and analysis, interpretation of data, manuscript writing, critical revision, and final approval of the manuscript. All authors read and approved the final manuscript. Competing interests: F.K. is an employee of Edwards Lifesciences; S.E. receives research funds from Edwards Lifesciences and received consulting fees from Medtronic and AstraZeneca; E.R.E. receives research support from Abiomed, Boston Scientific, Edwards Lifesciences, and Medtronic and is a founder of BioDevek and Panther Therapeutics. The other authors report no potential conflicts of interest. Data and materials availability: All data associated with this study are present in the paper or the Supplementary Materials. Code was deposited in Harvard Dataverse, available at:

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