# The Impact of Left Ventricular Performance and Afterload on the Evaluation of Aortic Valve Stenosis: A 1D Mathematical Modeling Approach

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{es}) and end-diastolic (E

_{ed}) elastance) and principal afterload indices (total vascular resistance (TVR) and total arterial compliance (TAC)) on the TPG for different levels of aortic stenosis. In patients with critical aortic stenosis (aortic valve area (AVA) ≤ 0.6 cm

^{2}), a 10% increase of E

_{ed}from the baseline value was associated with the most important effect on the TPG (−5.6 ± 0.5 mmHg, p < 0.001), followed by a similar increase of E

_{es}(3.4 ± 0.1 mmHg, p < 0.001), in TAC (1.3 ±0.2 mmHg, p < 0.001) and TVR (−0.7 ± 0.04 mmHg, p < 0.001). The interdependence of the TPG left ventricular performance and afterload indices become stronger with increased aortic stenosis severity. Disregarding their effects may lead to an underestimation of stenosis severity and a potential delay in therapeutic intervention. Therefore, a comprehensive evaluation of left ventricular function and afterload should be performed, especially in cases of diagnostic challenge, since it may offer the pathophysiological mechanism that explains the mismatch between aortic severity and the TPG.

## 1. Introduction

_{es}) and end-diastolic elastance (E

_{ed}), accordingly), total vascular resistance (TVR) and total arterial compliance (TAC) on the TPG for different levels of aortic valve stenosis is aimed to be quantified.

## 2. Materials and Methods

#### 2.1. 1D Mathematical Model of the Cardiovascular System

#### 2.2. Boundary Conditions

_{t}), proximal resistance (R

_{1}) and distal resistance (R

_{2}), as depicted in Figure 1, while the proximal aorta is coupled with the left ventricle, which is modeled according to the varying elastance model, as described by Sagawa et al. [52]. According to this approach, the instantaneous elastance of left ventricle E(t) is defined with the following relation:

_{es}, E

_{ed}, heart period, and maximum elastance time of a given case/patient, as described in Figure 1B [54]. The entire set-up of the equations is implicitly solved for the entire cardiac cycle, providing the pressure and flow waveforms throughout the entire arterial tree.

#### 2.3. Stenotic Aortic Valve Model

#### 2.4. Validation of the Coupled Model

_{es}, E

_{ed}, left ventricular end-diastolic pressure (LVEDP), dead volume (V

_{0}) and AVA) were acquired and given as an input to our model (Figure 2 panels A and B) in order to create a patient-specific simulation. In our cardiovascular model, the TAC of a patient is the total sum of the compliances of the main systemic arteries (C

_{1-D}) and the terminal compliance (C

_{t}), while the TVR is the sum of the proximal resistance (R

_{1}) and the distal resistance (R

_{2}), as described in Figure 1B. The patient’s TVR and TAC were first estimated by using an initial TVR value, estimated by the ratio of the mean aortic pressure and divided by the cardiac output and TAC value as the ratio of SV to pulse pressure. In the following step, values are iteratively increased or decreased within the physiological limits until the predicted mean aortic flow is comparable to the one measured. The simulation results of the model are presented by comparing them with the measured values in parentheses in Figure 2 (panel C for the aortic flow waveform and panel D for the left ventricle and aortic pressure waveforms). The estimations of the model were validated by comparing the results with the measured values of pulse pressure and the maximum TPG and are presented in Figure 2D.

#### 2.5. Simulation Strategy of the Aortic Stenosis Cases

^{2}) were created, presenting a progressive increase of E

_{es}for a range of physiological values (min 0.5 mmHg/mL to max 6 mmHg/mL) [48,58]. Since we aimed to estimate the independent effect of E

_{es}on the TPG, the diastolic performance of the left ventricle (E

_{ed}) and afterload (TVR and TAC) remained constant during these simulations. In order to explore the potential interaction with the severity of the aortic stenosis, the same hemodynamic settings were applied for 30 additional cases but with different values of AVA (1.0 cm

^{2}, 1.5 cm

^{2}and 2.0 cm

^{2}, n = 10 cases per AVA stenosis level). The same strategy was applied for an incremental change in E

_{ed}(min 0.03 mmHg/mL to max 0.31 mmHg/mL) [48,59], a change in TVR (min 0.6 mmHg×s/mL to max 1.8 mmHg×s/mL) [60] and change in TAC (min 0.5 mL/mmHg to max 2 mL/mmHg) [60]. Simulations were run by letting each key parameter vary from the lowest to the highest possible, with 10 evenly spaced values in the predefined range, while at the same time, maintaining all other parameters as constant. In addition, since the TAC and TVR do not change independently from each other in vivo [61], we simulated 40 additional cases with a progressive increase in afterload but with the TAC and TVR being coupled according to a hyperbolic relation linking the two variables.

_{es}enhances the pressure in the arterial system, which in turn decreases the TAC), during our simulations, we iteratively increased or decreased the parameters of the arterial tree until they converged to the initial set values within a given error threshold. Finally, it should be noted that the capacity of the model to simulate situations with an extreme discrepancy between the TAC and TVR while keeping the left ventricular performance parameters unaffected is limited since the ventricular–arterial interaction would counterbalance these effects by modifying E

_{es}in the physiological cases.

#### 2.6. Echocardiography

#### 2.7. Statistical Analysis

^{2}, 1.0 cm

^{2}, 1.5 cm

^{2}and 2.0 cm

^{2}) by the use of linear regression analysis. The values are expressed as either the regression beta coefficient ± standard error (Figure 3 and Figure 4) or the pressure gradient change for every 10% increase of the independent variable (Table 1). Statistical significance was assumed at a two-sided P-value level of 0.05. Statistical analysis was performed in IBM SPSS statistics (IBM Corp. Released 2020. IBM SPSS Statistics for Windows, Version 27.0. Armonk, NY, USA: IBM Corp.).

## 3. Results

#### 3.1. E_{es} and E_{ed} Impact on Mean TPG

_{es}and E

_{ed}changes on the mean TPG for different levels of aortic stenosis is presented in Figure 3 and Table 1. A decrease in left ventricular myocardial contractility, assessed by the E

_{es}, was associated with a lower mean TPG (AVA 0.6 cm

^{2}(beta 7.46 ± 0.29, p < 0.001), AVA 1.0 cm

^{2}(beta 3.04 ± 0.20, p < 0.001), AVA 1.5 cm

^{2}(beta 1.6 ± 0.12, p < 0.001), and AVA 2.0 cm

^{2}(beta 1.12 ± 0.36, p < 0.012)). A significant interaction with the AVA was seen, with the relation between E

_{es}and the mean TPG being stronger in the most severe aortic stenosis cases. Accordingly, an increase in left ventricular stiffness/relaxation, assessed by the E

_{ed}, was associated with a lower mean TPG for a given AVA (AVA 0.6 cm

^{2}(beta −427 ± 41, p < 0.001), AVA 1.0 cm

^{2}(beta −217 ± 17, p < 0.001), AVA 1.5 cm

^{2}(beta −100 ± 9, p < 0.001), and AVA 2.0 cm

^{2}(beta −48 ± 4, p < 0.012)). A significant interaction with the AVA was seen, with the relation between E

_{ed}and the mean TPG becoming stronger in parallel with the progression of aortic valve stenosis severity.

#### 3.2. TVR and TAC Impact on Mean TPG

^{2}(beta −5.5 ± 0.35, p < 0.001), AVA 1.0 cm

^{2}(beta −2.6 ± 0.33, p < 0.001), and AVA 1.5 cm

^{2}(beta −0.5 ± 0.23, p = 0.032)). This was not seen in cases with an AVA of 2.0 cm

^{2}(beta −0.4 ± 0.35, p = 0.321). A significant interaction with the aortic valve area was seen, with the relation between TVR and the mean TPG becoming stronger as the aortic stenosis grew more severe. Accordingly, an increase in TAC was associated with an increase in the TPG for a given AVA (AVA 0.6 cm

^{2}(beta 11.8 ± 1.9, p < 0.001), AVA 1.0 cm

^{2}(beta 5.9 ± 0.9, p < 0.001), AVA 1.5 cm

^{2}(beta 5.7 ± 0.6, p < 0.001), and AVA 1.0 cm

^{2}(beta 2.9 ± 0.3, p = 0.005)). A significant interaction with the AVA was seen, with the relation between TAC and the pressure gradients becoming stronger with the progression of aortic severity. Similar results were obtained when coupled TAC and TVR values were used as input variables (Figure 4 and Table 1).

#### 3.3. Relative Contribution of E_{es}, E_{ed}, TVR, and TAC on Mean TPG

^{2}, the E

_{ed}change was associated with the most important effect on the TPG (−5.6 ± 0.5 mmHg, p < 0.001), followed by E

_{es}(3.4 ± 0.1 mmHg, p < 0.001), TAC (1.3 ± 0.2 mmHg, p < 0.001), and TVR (−0.7 ± 0.04 mmHg, p < 0.001). Similar classifications were noted with a higher AVA; however, the magnitudes of the effect seem to become weaker as aortic stenosis severity decreases.

#### 3.4. SV and Mean TPG Relationship for a Given AVA

_{es}, E

_{ed}, TVR, and TAC independently affect the TPG through a flow-dependent manner, we explored the association directly between SV and the mean TPG (Figure 5). A significant increase in the mean TPG was seen with an increase in SV for any given AVA. A significant interaction with an AVA severity is also observed, with the dependence being stronger as the aortic valve severity increases.

## 4. Discussion

_{ed}) is, at least, as important as left ventricular myocardial contractility (E

_{es}) in determining the TPG in the presence of aortic stenosis; (2) TAC, as well as TVR, affect the TPG independently for a given aortic stenosis level; and (3) the interaction between the left ventricular performance, afterload indices, and the TPG becomes stronger as aortic valve severity increases.

_{ed}was even higher than the one of E

_{es}, especially in patients with critical AVA. This is in accordance with clinical studies showing strong associations between the diastolic dysfunction indices and the presence of severe aortic stenosis without an increased TPG and with a normal ejection fraction (paradoxical low-flow, low-gradient aortic stenosis) [63]. This is of major clinical relevance since aortic stenosis and diastolic dysfunction often coexist and evolve parallel with aging [64,65]. It follows that a comprehensive evaluation of diastolic dysfunction should be part of the routine examination when evaluating aortic valve disease since it may significantly blunt the TPG, especially when aortic valve stenosis becomes critical.

_{es}and E

_{ed}but different TAC and TVR combinations are plotted against peripheral blood pressures. It follows that the afterload evaluation (and thus the potential effect on the TPG) cannot be accurately predicted by measuring only peripheral blood pressures. For this reason, a comprehensive evaluation of left ventricular afterload could be particularly relevant when measuring the TPG. This can easily be achieved non-invasively through wave separation analysis by combining the pressure data obtained from a high-fidelity tonometer (the carotid or radial artery) and the aortic flow obtained by transthoracic echocardiography.

_{es}, E

_{ed}, TAC, and TVR on TPG presents a significant interaction with the actual level of aortic stenosis since the effects seem to be more important as the aortic stenosis becomes more severe. This is of particular clinical importance, suggesting a greater risk for TPG underestimation in patients presenting with lower AVAs. At the same time, it is exactly these patients who require the most precise evaluation since the indication for valve replacement (vs. clinical follow-up) depends heavily on pressure gradients. Our proposed methodology offers a promising tool for distinguishing pseudo-stenosis from true stenotic cases. Future clinical studies are needed to validate our proposed methodology under different physiological conditions, such as decreased myocardial contractility or increased afterload. Additionally, the use of the model can be extended to other valve diseases as well.

## 5. Limitations

_{es}, E

_{ed}, TAC, and TVR, although allowing for the precise estimation of the independent effect of each variable on pressure gradients, do not represent pure physiological states since significant interactions take place normally. Finally, the mathematical model does not incorporate physiological reflexes in response to blood and flow changes that may impact the TPG.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Yu, J.; Wang, Z.; Bao, Q.; Lei, S.; You, Y.; Yin, Z.; Xie, X. Global Burden of Calcific Aortic Valve Disease and Attributable Risk Factors from 1990 to 2019. Front. Cardiovasc. Med.
**2022**, 9, 1003233. [Google Scholar] [CrossRef] [PubMed] - Yadgir, S.; Johnson, C.O.; Aboyans, V.; Adebayo, O.M.; Adedoyin, R.A.; Afarideh, M.; Alahdab, F.; Alashi, A.; Alipour, V.; Arabloo, J.; et al. Global, Regional, and National Burden of Calcific Aortic Valve and Degenerative Mitral Valve Diseases, 1990–2017. Circulation
**2020**, 141, 1670–1680. [Google Scholar] [CrossRef] [PubMed][Green Version] - Go, A.S.; Mozaffarian, D.; Roger, V.L.; Benjamin, E.J.; Berry, J.D.; Borden, W.B.; Bravata, D.M.; Dai, S.; Ford, E.S.; Fox, C.S.; et al. Executive Summary: Heart Disease and Stroke Statistics—2013 Update: A Report From the American Heart Association. Circulation
**2013**, 127, 143–152. [Google Scholar] [CrossRef] - Lindman, B.R.; Clavel, M.-A.; Mathieu, P.; Iung, B.; Lancellotti, P.; Otto, C.M.; Pibarot, P. Calcific Aortic Stenosis. Nat. Rev. Dis. Primers
**2016**, 2, 16006. [Google Scholar] [CrossRef] [PubMed][Green Version] - Roth, G.A.; Mensah, G.A.; Johnson, C.O.; Addolorato, G.; Ammirati, E.; Baddour, L.M.; Barengo, N.C.; Beaton, A.Z.; Benjamin, E.J.; Benziger, C.P.; et al. Global Burden of Cardiovascular Diseases and Risk Factors 1990–2019. J. Am. Coll. Cardiol.
**2020**, 76, 2982–3021. [Google Scholar] [CrossRef] - Pibarot, P.; Clavel, M.-A. Live Longer and Better without Aortic Valve Stenosis. Lancet Healthy Longev.
**2022**, 3, e573–e574. [Google Scholar] [CrossRef] - Vahanian, A.; Beyersdorf, F.; Praz, F.; Milojevic, M.; Baldus, S.; Bauersachs, J.; Capodanno, D.; Conradi, L.; De Bonis, M.; De Paulis, R.; et al. 2021 ESC/EACTS Guidelines for the Management of Valvular Heart Disease. Eur. Heart J.
**2022**, 43, 561–632. [Google Scholar] [CrossRef] - Baumgartner, H.; Hung, J.; Bermejo, J.; Chambers, J.B.; Edvardsen, T.; Goldstein, S.; Lancellotti, P.; LeFevre, M.; Miller, F.; Otto, C.M. Recommendations on the Echocardiographic Assessment of Aortic Valve Stenosis: A Focused Update from the European Association of Cardiovascular Imaging and the American Society of Echocardiography. J. Am. Soc. Echocardiogr.
**2017**, 30, 372–392. [Google Scholar] [CrossRef] - Otto, C.M.; Nishimura, R.A.; Bonow, R.O.; Carabello, B.A.; Erwin, J.P.; Gentile, F.; Jneid, H.; Krieger, E.V.; Mack, M.; McLeod, C.; et al. 2020 ACC/AHA Guideline for the Management of Patients With Valvular Heart Disease: A Report of the American College of Cardiology/American Heart Association Joint Committee on Clinical Practice Guidelines. Circulation
**2021**, 143, e72–e227. [Google Scholar] [CrossRef] - Reymond, P.; Bohraus, Y.; Perren, F.; Lazeyras, F.; Stergiopulos, N. Validation of a Patient-Specific One-Dimensional Model of the Systemic Arterial Tree. Am. J. Physiol.-Heart Circ. Physiol.
**2011**, 301, H1173–H1182. [Google Scholar] [CrossRef][Green Version] - Kondiboyina, A.; Harrington, H.A.; Smolich, J.J.; Cheung, M.M.H.; Mynard, J.P. Optimized Design of an Arterial Network Model Reproduces Characteristic Central and Peripheral Haemodynamic Waveform Features of Young Adults. J. Physiol.
**2022**, 600, 3725–3747. [Google Scholar] [CrossRef] - Charlton, P.H.; Mariscal Harana, J.; Vennin, S.; Li, Y.; Chowienczyk, P.; Alastruey, J. Modeling Arterial Pulse Waves in Healthy Aging: A Database for in Silico Evaluation of Hemodynamics and Pulse Wave Indexes. Am. J. Physiol-Heart Circ. Physiol.
**2019**, 317, H1062–H1085. [Google Scholar] [CrossRef][Green Version] - Blanco, P.J.; Watanabe, S.M.; Passos, M.A.R.F.; Lemos, P.A.; Feijoo, R.A. An Anatomically Detailed Arterial Network Model for One-Dimensional Computational Hemodynamics. IEEE Trans. Biomed. Eng.
**2015**, 62, 736–753. [Google Scholar] [CrossRef] [PubMed] - Bikia, V.; Adamopoulos, D.; Pagoulatou, S.; Rovas, G.; Stergiopulos, N. AI-Based Estimation of End-Systolic Elastance from Arm-Pressure and Systolic Time Intervals. Front. Artif. Intell.
**2021**, 4, 579541. [Google Scholar] [CrossRef] - Pagoulatou, S.; Rommel, K.-P.; Kresoja, K.-P.; von Roeder, M.; Lurz, P.; Thiele, H.; Bikia, V.; Rovas, G.; Adamopoulos, D.; Stergiopulos, N. In Vivo Application and Validation of a Novel Noninvasive Method to Estimate the End-Systolic Elastance. Am. J. Physiol. Heart Circ. Physiol.
**2021**, 320, H1554–H1564. [Google Scholar] [CrossRef] - Bikia, V.; Pagoulatou, S.; Trachet, B.; Soulis, D.; Protogerou, A.D.; Papaioannou, T.G.; Stergiopulos, N. Noninvasive Cardiac Output and Central Systolic Pressure From Cuff-Pressure and Pulse Wave Velocity. IEEE J. Biomed. Health Inform.
**2020**, 24, 1968–1981. [Google Scholar] [CrossRef][Green Version] - Clark, C. Relation between Pressure Difference across the Aortic Valve and Left Ventricular Outflow. Cardiovasc. Res.
**1978**, 12, 276–287. [Google Scholar] [CrossRef] [PubMed] - Garcia, D.; Pibarot, P.; Durand, L.-G. Analytical Modeling of the Instantaneous Pressure Gradient across the Aortic Valve. J. Biomech.
**2005**, 38, 1303–1311. [Google Scholar] [CrossRef] [PubMed] - Garcia, D.; Kadem, L.; Savéry, D.; Pibarot, P.; Durand, L.-G. Analytical Modeling of the Instantaneous Maximal Transvalvular Pressure Gradient in Aortic Stenosis. J. Biomech.
**2006**, 39, 3036–3044. [Google Scholar] [CrossRef] - Hatoum, H.; Mo, X.-M.; Crestanello, J.A.; Dasi, L.P. Modeling of the Instantaneous Transvalvular Pressure Gradient in Aortic Stenosis. Ann. Biomed. Eng.
**2019**, 47, 1748–1763. [Google Scholar] [CrossRef] - Ringle Griguer, A.; Tribouilloy, C.; Truffier, A.; Castel, A.-L.; Delelis, F.; Levy, F.; Vincentelli, A.; Bohbot, Y.; Maréchaux, S. Clinical Significance of Ejection Dynamics Parameters in Patients with Aortic Stenosis: An Outcome Study. J. Am. Soc. Echocardiogr.
**2018**, 31, 551–560.e2. [Google Scholar] [CrossRef] [PubMed] - Kim, S.H.; Kim, J.S.; Kim, B.S.; Choi, J.; Lee, S.-C.; Oh, J.K.; Park, S.W. Time to Peak Velocity of Aortic Flow Is Useful in Predicting Severe Aortic Stenosis. Int. J. Cardiol.
**2014**, 172, e443–e446. [Google Scholar] [CrossRef] [PubMed] - Altes, A.; Thellier, N.; Bohbot, Y.; Ringle Griguer, A.; Verdun, S.; Levy, F.; Castel, A.L.; Delelis, F.; Mailliet, A.; Tribouilloy, C.; et al. Relationship Between the Ratio of Acceleration Time/Ejection Time and Mortality in Patients With High-Gradient Severe Aortic Stenosis. JAHA
**2021**, 10, e021873. [Google Scholar] [CrossRef] [PubMed] - Virag, Z.; Lulić, F. Modeling of Aortic Valve Dynamics in a Lumped Parameter Model of Left Ventricular-Arterial Coupling. Ann. Univ. Ferrara
**2008**, 54, 335–347. [Google Scholar] [CrossRef] - Aboelkassem, Y.; Savic, D.; Campbell, S.G. Mathematical Modeling of Aortic Valve Dynamics during Systole. J. Theor. Biol.
**2015**, 365, 280–288. [Google Scholar] [CrossRef] [PubMed] - Laubscher, R.; van der Merwe, J.; Liebenberg, J.; Herbst, P. Dynamic Simulation of Aortic Valve Stenosis Using a Lumped Parameter Cardiovascular System Model with Flow Regime Dependent Valve Pressure Loss Characteristics. Med. Eng. Phys.
**2022**, 106, 103838. [Google Scholar] [CrossRef] - Korakianitis, T.; Shi, Y. A Concentrated Parameter Model for the Human Cardiovascular System Including Heart Valve Dynamics and Atrioventricular Interaction. Med. Eng. Phys.
**2006**, 28, 613–628. [Google Scholar] [CrossRef] - Korakianitis, T.; Shi, Y. Numerical Simulation of Cardiovascular Dynamics with Healthy and Diseased Heart Valves. J. Biomech.
**2006**, 39, 1964–1982. [Google Scholar] [CrossRef] - Mynard, J.P.; Davidson, M.R.; Penny, D.J.; Smolich, J.J. A Simple, Versatile Valve Model for Use in Lumped Parameter and One-Dimensional Cardiovascular Models. Int. J. Numer. Methods Biomed. Eng.
**2012**, 28, 626–641. [Google Scholar] [CrossRef] - Mynard, J.P.; Smolich, J.J. One-Dimensional Haemodynamic Modeling and Wave Dynamics in the Entire Adult Circulation. Ann. Biomed. Eng.
**2015**, 43, 1443–1460. [Google Scholar] [CrossRef] - Briand, M.; Dumesnil, J.G.; Kadem, L.; Tongue, A.G.; Rieu, R.; Garcia, D.; Pibarot, P. Reduced Systemic Arterial Compliance Impacts Significantly on Left Ventricular Afterload and Function in Aortic Stenosis. J. Am. Coll. Cardiol.
**2005**, 46, 291–298. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hachicha, Z.; Dumesnil, J.G.; Pibarot, P. Usefulness of the Valvuloarterial Impedance to Predict Adverse Outcome in Asymptomatic Aortic Stenosis. J. Am. Coll. Cardiol.
**2009**, 54, 1003–1011. [Google Scholar] [CrossRef][Green Version] - Côté, N.; Simard, L.; Zenses, A.; Tastet, L.; Shen, M.; Clisson, M.; Clavel, M. Impact of Vascular Hemodynamics on Aortic Stenosis Evaluation: New Insights Into the Pathophysiology of Normal Flow—Small Aortic Valve Area—Low Gradient Pattern. JAHA
**2017**, 6, e006276. [Google Scholar] [CrossRef] [PubMed] - Gardikioti, V.; Terentes-Printzios, D.; Iliopoulos, D.; Aznaouridis, K.; Sigala, E.; Tsioufis, K.; Vlachopoulos, C. Arterial Biomarkers in the Evaluation, Management and Prognosis of Aortic Stenosis. Atherosclerosis
**2021**, 332, 1–15. [Google Scholar] [CrossRef] - Gholampour, S.; Yamini, B.; Droessler, J.; Frim, D. A New Definition for Intracranial Compliance to Evaluate Adult Hydrocephalus After Shunting. Front. Bioeng. Biotechnol.
**2022**, 10, 900644. [Google Scholar] [CrossRef] - Gholampour, S.; Frim, D.; Yamini, B. Long-Term Recovery Behavior of Brain Tissue in Hydrocephalus Patients after Shunting. Commun. Biol.
**2022**, 5, 1198. [Google Scholar] [CrossRef] - Lancellotti, P.; Donal, E.; Magne, J.; Moonen, M.; O’Connor, K.; Daubert, J.-C.; Pierard, L.A. Risk Stratification in Asymptomatic Moderate to Severe Aortic Stenosis: The Importance of the Valvular, Arterial and Ventricular Interplay. Heart
**2010**, 96, 1364–1371. [Google Scholar] [CrossRef] - Mancusi, C.; de Simone, G.; Brguljan Hitij, J.; Sudano, I.; Mahfoud, F.; Parati, G.; Kahan, T.; Barbato, E.; Pierard, L.A.; Garbi, M.; et al. Management of Patients with Combined Arterial Hypertension and Aortic Valve Stenosis: A Consensus Document from the Council on Hypertension and Council on Valvular Heart Disease of the European Society of Cardiology, the European Association of Cardiovascular Imaging (EACVI), and the European Association of Percutaneous Cardiovascular Interventions (EAPCI). Eur. Heart J.—Cardiovasc. Pharmacother.
**2021**, 7, 242–250. [Google Scholar] [CrossRef] - Gotzmann, M.; Hauptmann, S.; Hogeweg, M.; Choudhury, D.S.; Schiedat, F.; Dietrich, J.W.; Westhoff, T.H.; Bergbauer, M.; Mügge, A. Hemodynamics of Paradoxical Severe Aortic Stenosis: Insight from a Pressure–Volume Loop Analysis. Clin. Res. Cardiol.
**2019**, 108, 931–939. [Google Scholar] [CrossRef] - Awtry, E.H.; Davidoff, R. Low-Flow Low-Gradient Aortic Stenosis. Circ. Cardiovasc. Imaging
**2012**, 5, 6–8. [Google Scholar] [CrossRef] [PubMed][Green Version] - Burwash, I.G.; Pearlman, A.S.; Kraft, C.D.; Miyake-Hull, C.; Healy, N.L.; Otto, C.M. Flow Dependence of Measures of Aortic Stenosis Severity during Exercise. J. Am. Coll. Cardiol.
**1994**, 24, 1342–1350. [Google Scholar] [CrossRef] [PubMed][Green Version] - Burwash, I.G.; Thomas, D.D.; Sadahiro, M.; Pearlman, A.S.; Verrier, E.D.; Thomas, R.; Kraft, C.D.; Otto, C.M. Dependence of Gorlin Formula and Continuity Equation Valve Areas on Transvalvular Volume Flow Rate in Valvular Aortic Stenosis. Circulation
**1994**, 89, 827–835. [Google Scholar] [CrossRef][Green Version] - Hayek, A.; Derimay, F.; Green, L.; Rosset, M.; Thibault, H.; Rioufol, G.; Finet, G. Impact of Arterial Blood Pressure on Ultrasound Hemodynamic Assessment of Aortic Valve Stenosis Severity. J. Am. Soc. Echocardiogr.
**2020**, 33, 1324–1333. [Google Scholar] [CrossRef] [PubMed] - Kadem, L.; Dumesnil, J.G.; Rieu, R.; Durand, L.-G.; Garcia, D.; Pibarot, P. Impact of Systemic Hypertension on the Assessment of Aortic Stenosis. Heart
**2005**, 91, 354–361. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pagoulatou, S.; Adamopoulos, D.; Rovas, G.; Bikia, V.; Müller, H.; Giannakopoulos, G.; Mauler-Wittwer, S.; Licker, M.-J.; Stergiopulos, N.; Noble, S. Arterial Wave Reflection and Aortic Valve Stenosis: Diagnostic Challenges and Prognostic Significance. Front. Cardiovasc. Med.
**2022**, 9, 863968. [Google Scholar] [CrossRef] - deFilippi, C.R.; Willett, D.L.; Brickner, M.E.; Appleton, C.P.; Yancy, C.W.; Eichhorn, E.J.; Grayburn, P.A. Usefulness of Dobutamine Echocardiography in Distinguishing Severe from Nonsevere Valvular Aortic Stenosis in Patients with Depressed Left Ventricular Function and Low Transvalvular Gradients. Am. J. Cardiol.
**1995**, 75, 191–194. [Google Scholar] [CrossRef] - Eleid, M.F.; Nishimura, R.A.; Sorajja, P.; Borlaug, B.A. Systemic Hypertension in Low-Gradient Severe Aortic Stenosis With Preserved Ejection Fraction. Circulation
**2013**, 128, 1349–1353. [Google Scholar] [CrossRef][Green Version] - Reymond, P.; Merenda, F.; Perren, F.; Rüfenacht, D.; Stergiopulos, N. Validation of a One-Dimensional Model of the Systemic Arterial Tree. Am. J. Physiol. Heart Circ. Physiol.
**2009**, 297, H208–H222. [Google Scholar] [CrossRef][Green Version] - Langewouters, G.J.; Wesseling, K.H.; Goedhard, W.J. The Static Elastic Properties of 45 Human Thoracic and 20 Abdominal Aortas in Vitro and the Parameters of a New Model. J. Biomech.
**1984**, 17, 425–435. [Google Scholar] [CrossRef] - Holenstein, R.; Niederer, P.; Anliker, M. A Viscoelastic Model for Use in Predicting Arterial Pulse Waves. J. Biomech. Eng.
**1980**, 102, 318–325. [Google Scholar] [CrossRef] - Womersley, J.R. An Elastic Tube Theory of Pulse Transmission and Oscillatory Flow in Mammalian Arteries; WADC Technical Report; Wright Air Development Center, Air Research and Development Command, Wright-Patterson Air Force Base: Dayton, OH, USA, 1957. [Google Scholar]
- Sagawa, K.; Maughan, L.; Suga, H.; Sunagawa, K. Cardiac Contraction and the Pressure-Volume Relationship, 1st ed.; Oxford University Press: New York, NY, USA, 1988; ISBN 978-0-19-504320-4. [Google Scholar]
- Senzaki, H.; Chen, C.-H.; Kass, D.A. Single-Beat Estimation of End-Systolic Pressure-Volume Relation in Humans. Circulation
**1996**, 94, 2497–2506. [Google Scholar] [CrossRef] [PubMed] - Segers, P.; Stergiopulos, N.; Schreuder, J.J.; Westerhof, B.E.; Westerhof, N. Left Ventricular Wall Stress Normalization in Chronic Pressure-Overloaded Heart: A Mathematical Model Study. Am. J. Physiol-Heart Circ. Physiol.
**2000**, 279, H1120–H1127. [Google Scholar] [CrossRef] [PubMed] - Young, D.F.; Tsai, F.Y. Flow Characteristics in Models of Arterial Stenoses—I. Steady Flow. J. Biomech.
**1973**, 6, 395–410. [Google Scholar] [CrossRef] [PubMed] - Young, D.F.; Tsai, F.Y. Flow Characteristics in Models of Arterial Stenoses—II. Unsteady Flow. J. Biomech.
**1973**, 6, 547–559. [Google Scholar] [CrossRef] - Dekker, A.L.A.J. Pressure-Volume Loops in Cardiac Surgery; Maastricht University: Maastricht, The Netherlands, 2003. [Google Scholar]
- Feldman, M.D.; Pak, P.H.; Wu, C.C.; Haber, H.L.; Heesch, C.M.; Bergin, J.D.; Powers, E.R.; Cowart, T.D.; Johnson, W.; Feldman, A.M.; et al. Acute Cardiovascular Effects of OPC-18790 in Patients with Congestive Heart Failure. Time- and Dose-Dependence Analysis Based on Pressure-Volume Relations. Circulation
**1996**, 93, 474–483. [Google Scholar] [CrossRef] - Chen, C.H.; Nakayama, M.; Nevo, E.; Fetics, B.J.; Maughan, W.L.; Kass, D.A. Coupled Systolic-Ventricular and Vascular Stiffening with Age: Implications for Pressure Regulation and Cardiac Reserve in the Elderly. J. Am. Coll. Cardiol.
**1998**, 32, 1221–1227. [Google Scholar] [CrossRef] [PubMed][Green Version] - Segers, P.; Stergiopulos, N.; Westerhof, N. Relation of Effective Arterial Elastance to Arterial System Properties. Am. J. Physiol. Heart Circ. Physiol.
**2002**, 282, H1041–H1046. [Google Scholar] [CrossRef] - Wohlfahrt, P.; Redfield, M.M.; Melenovsky, V.; Lopez-Jimenez, F.; Rodeheffer, R.J.; Borlaug, B.A. Impact of Chronic Changes in Arterial Compliance and Resistance on Left Ventricular Ageing in Humans. Eur. J. Heart Fail.
**2015**, 17, 27–34. [Google Scholar] [CrossRef][Green Version] - Baumgartner, H.; Hung, J.; Bermejo, J.; Chambers, J.B.; Evangelista, A.; Griffin, B.P.; Iung, B.; Otto, C.M.; Pellikka, P.A.; Quiñones, M.; et al. Echocardiographic Assessment of Valve Stenosis: EAE/ASE Recommendations for Clinical Practice. J. Am. Soc. Echocardiogr.
**2009**, 22, 1–23. [Google Scholar] [CrossRef] - Pibarot, P.; Dumesnil, J.G. Paradoxical Low-Flow, Low-Gradient Aortic Stenosis: New Evidence, More Questions. Circulation
**2013**, 128, 1729–1732. [Google Scholar] [CrossRef][Green Version] - Osnabrugge, R.L.J.; Mylotte, D.; Head, S.J.; Van Mieghem, N.M.; Nkomo, V.T.; LeReun, C.M.; Bogers, A.J.J.C.; Piazza, N.; Kappetein, A.P. Aortic Stenosis in the Elderly: Disease Prevalence and Number of Candidates for Transcatheter Aortic Valve Replacement: A Meta-Analysis and Modeling Study. J. Am. Coll. Cardiol.
**2013**, 62, 1002–1012. [Google Scholar] [CrossRef][Green Version] - Salmasi, A.-M.; Alimo, A.; Jepson, E.; Dancy, M. Age-Associated Changes in Left Ventricular Diastolic Function Are Related to Increasing Left Ventricular Mass. Am. J. Hypertens.
**2003**, 16, 473–477. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**(

**A**) 1D model of the cardiovascular system with main systemic arteries; (

**B**) at the proximal site, the left ventricular is modeled according to the varying elastance model. The instantaneous elastance of the left ventricle is calculated for a given E

_{es}, E

_{ed}, heart period (t

_{HP}) and maximum elastance time (t

_{max}) by exploiting the normalized varying elastance (E

_{N}). Left ventricular end-diastolic pressure (LVEDP) and dead volume (V

_{0}) are the additional parameters needed to construct the pressure–volume loop of a patient. At the distal boundaries, the 1D model is coupled to a three-element Windkessel model, which includes terminal compliance (C

_{t}), proximal resistance (R

_{1}) and distal resistance (R

_{2}). Compliances of the main systemic arteries are represented as C

_{1-D}.

**Figure 2.**1D mathematical model coupled with aortic valve stenosis model for the prediction of TPG. (

**A**) Predicted left ventricular and aortic pressure–volume curve from the model (input variables E

_{es}= 3.3 mmHg/mL, E

_{ed}= 0.1 mmHg/mL, V

_{0}= 39.3 mL and LVEDP = 11 mmHg, and maximal AVA = 0.6 cm

^{2}). (

**B**) Aortic valve opening as a function of time derived by the aortic valve stenosis model. (

**C**) Aortic flow, generated by the coupled 1D mathematical and the aortic valve stenosis model. (

**D**) Performance of the model in predicting TPG as compared to the actual measured values from the literature [57].

**Figure 4.**Impact of TVR and TAC on mean TPG for different levels of aortic stenosis. Adjusted TAC*: coupled TAC according to its hyperbolic relation with TVR [61].

**Figure 5.**Association between SV and mean TPG for different levels of AVA. Echocardiographic evaluation of a patient with severe aortic stenosis at rest and after dobutamine infusion (12.5 ug/min/kg). The myocardial recruitment observed after dobutamine leads to an increase in stroke volume (from 67 mL to 87 mL) with a concomitant increase in mean TPG (from 28 mmHg to 40 mmHg). The AVA remained stable, suggesting a true severe aortic valve stenosis (AVA 0.88 cm

^{2}for an LVOT diameter measured at 23 mm). Model predictions were accurate for both rest and dobutamine hemodynamic conditions.

**Figure 6.**Association between peripheral blood pressure and mean TPG for cases presenting different combinations of TAC and TVR. SBP: Systolic blood pressure, MBP: Mean blood pressure.

**Table 1.**Relative contribution of each determinant on TPG for different levels of aortic valve stenosis. Beta coefficients (Beta) are expressed in pressure (mmHg) for every 10% increase of each determinant from the baseline (lowest) value.

Aortic Valve Area | 0.6 cm^{2} | 1.0 cm^{2} | 1.5 cm^{2} | 2.0 cm^{2} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Beta | S.E. | p Value | Beta | S.E. | p Value | Beta | S.E. | p Value | Beta | S.E. | p Value | |

E_{es} (mmHg/mL) | 3.4 | 0.1 | <0.001 | 1.4 | 0.1 | <0.001 | 0.7 | 0.1 | <0.001 | 0.3 | 0.1 | 0.005 |

E_{ed} (mmHg/mL) | −5.6 | 0.5 | <0.001 | −2.8 | 0.2 | <0.001 | −1.3 | 0.1 | <0.001 | −0.6 | 0.1 | <0.001 |

TAC (mL/mmHg) | 1.3 | 0.2 | <0.001 | 0.7 | 0.1 | <0.001 | 0.6 | 0.1 | <0.001 | 0.2 | 0.1 | 0.05 |

TVR (mmHg×s/mL) | −0.7 | 0.04 | <0.001 | −0.3 | 0.04 | <0.001 | −0.1 | 0.03 | 0.032 | −0.04 | 0.04 | 0.321 |

Adjusted TVR * (mmHg×s/mL) | −1.1 | 0.04 | <0.001 | −0.8 | 0.1 | <0.001 | −0.4 | 0.03 | <0.001 | −0.2 | 0.02 | <0.001 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Çelikbudak Orhon, C.; Stergiopulos, N.; Noble, S.; Giannakopoulos, G.; Müller, H.; Adamopoulos, D. The Impact of Left Ventricular Performance and Afterload on the Evaluation of Aortic Valve Stenosis: A 1D Mathematical Modeling Approach. *Bioengineering* **2023**, *10*, 425.
https://doi.org/10.3390/bioengineering10040425

**AMA Style**

Çelikbudak Orhon C, Stergiopulos N, Noble S, Giannakopoulos G, Müller H, Adamopoulos D. The Impact of Left Ventricular Performance and Afterload on the Evaluation of Aortic Valve Stenosis: A 1D Mathematical Modeling Approach. *Bioengineering*. 2023; 10(4):425.
https://doi.org/10.3390/bioengineering10040425

**Chicago/Turabian Style**

Çelikbudak Orhon, Cemre, Nikolaos Stergiopulos, Stéphane Noble, Georgios Giannakopoulos, Hajo Müller, and Dionysios Adamopoulos. 2023. "The Impact of Left Ventricular Performance and Afterload on the Evaluation of Aortic Valve Stenosis: A 1D Mathematical Modeling Approach" *Bioengineering* 10, no. 4: 425.
https://doi.org/10.3390/bioengineering10040425