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

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## 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

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**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 |

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## 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