This work introduces two computational models of heart failure with preserved ejection fraction based on a lumped-parameter approach and finite element analysis. These models are used to evaluate the changes in the hemodynamics of the left ventricle and related vasculature induced by pressure overload and diminished ventricular compliance.
Scientific efforts in the field of computational modeling of cardiovascular diseases have largely focused on heart failure with reduced ejection fraction (HFrEF), broadly overlooking heart failure with preserved ejection fraction (HFpEF), which has more recently become a dominant form of heart failure worldwide. Motivated by the paucity of HFpEF in silico representations, two distinct computational models are presented in this paper to simulate the hemodynamics of HFpEF resulting from left ventricular pressure overload. First, an object-oriented lumped-parameter model was developed using a numerical solver. This model is based on a zero-dimensional (0D) Windkessel-like network, which depends on the geometrical and mechanical properties of the constitutive elements and offers the advantage of low computational costs. Second, a finite element analysis (FEA) software package was utilized for the implementation of a multidimensional simulation. The FEA model combines three-dimensional (3D) multiphysics models of the electro-mechanical cardiac response, structural deformations, and fluid cavity-based hemodynamics and utilizes a simplified lumped-parameter model to define the flow exchange profiles among different fluid cavities. Through each approach, both the acute and chronic hemodynamic changes in the left ventricle and proximal vasculature resulting from pressure overload were successfully simulated. Specifically, pressure overload was modeled by reducing the orifice area of the aortic valve, while chronic remodeling was simulated by reducing the compliance of the left ventricular wall. Consistent with the scientific and clinical literature of HFpEF, results from both models show (i) an acute elevation of transaortic pressure gradient between the left ventricle and the aorta and a reduction in the stroke volume and (ii) a chronic decrease in the end-diastolic left ventricular volume, indicative of diastolic dysfunction. Finally, the FEA model demonstrates that stress in the HFpEF myocardium is remarkably higher than in the healthy heart tissue throughout the cardiac cycle.
Heart failure is a leading cause of death worldwide, which occurs when the heart is unable to pump or fill adequately to keep up with the metabolic demands of the body. The ejection fraction, i.e., the relative amount of blood stored in the left ventricle that is ejected with each contraction is used clinically to classify heart failure into (i) heart failure with reduced ejection fraction (HFrEF) and (ii) heart failure with preserved ejection fraction (HFpEF), for ejection fractions less than or greater than 45%, respectively1,2,3. Symptoms of HFpEF often develop in response to left ventricular pressure overload, which can be caused by several conditions including aortic stenosis, hypertension, and left ventricular outflow tract obstruction3,4,5,6,7. Pressure overload drives a cascade of molecular and cellular aberrations, leading to thickening of the left ventricular wall (concentric remodeling) and ultimately, to wall stiffening or loss of compliance8,9,10. These biomechanical changes profoundly affect cardiovascular hemodynamics as they result in an elevated end-diastolic pressure-volume relationship and in a reduction of the end-diastolic volume11.
Computational modeling of the cardiovascular system has advanced the understanding of blood pressures and flows in both physiology and disease and has fostered the development of diagnostic and therapeutic strategies12. In silico models are classified into low- or high-dimensional models, with the former utilizing analytical methods to evaluate global hemodynamic properties with low computational demand and the latter providing a more extensive multiscale and multiphysics description of cardiovascular mechanics and hemodynamics in the 2D or 3D domain13. The lumped-parameter Windkessel representation is the most common among the low-dimensional descriptions. Based on the electrical circuit analogy (Ohm's law), this mimics the overall hemodynamic behavior of the cardiovascular system through a combination of resistive, capacitive, and inductive elements14. A recent study by this group has proposed an alternative Windkessel model in the hydraulic domain that allows the modeling of changes in the geometry and mechanics of large vessels-heart chambers and valves-in a more intuitive way than traditional electrical analog models. This simulation is developed on an object-oriented numerical solver (see the Table of Materials) and can capture the normal hemodynamics, physiologic effects of cardiorespiratory coupling, respiratory-driven blood flow in single-heart physiology, and hemodynamic changes due to aortic constriction. This description expands upon the capabilities of lumped-parameter models by offering a physically intuitive approach to model a spectrum of pathologic conditions including heart failure15.
High-dimensional models are based on FEA to compute spatiotemporal hemodynamics and fluid-structure interactions. These representations can provide detailed and accurate descriptions of the local blood flow field; however, due to their low computational efficiency, they are not suitable for studies of the entire cardiovascular tree16,17. A software package (see the Table of Materials) was employed as an anatomically accurate FEA platform of the 4-chamber adult human heart, which integrates the electro-mechanical response, structural deformations, and fluid cavity-based hemodynamics. The adapted human heart model also comprises a simple lumped-parameter model that defines the flow exchange among the different fluid cavities, as well as a complete mechanical characterization of the cardiac tissue18,19.
Several lumped-parameter and FEA models of heart failure have been formulated to capture hemodynamic abnormalities and evaluate therapeutic strategies, particularly in the context of mechanical circulatory assist devices for HFrEF20,21,22,23,24. A broad array of 0D lumped-parameter models of various complexities has therefore successfully captured the hemodynamics of the human heart in physiological and HFrEF conditions via optimization of two or three-element electrical analog Windkessel systems20,21,23,24. The majority of these representations are uni- or biventricular models based on the time varying-elastance formulation to reproduce the contractile action of the heart and use a non-linear end-diastolic pressure-volume relationship to describe ventricular filling25,26,27. Comprehensive models, which capture the complex cardiovascular network and mimic both the atrial and ventricular pumping action, have been used as platforms for device testing. Nevertheless, although a significant body of literature exists around the field of HFrEF, very few in silico models of HFpEF have been proposed20,22,28,29,30,31.
A low-dimensional model of HFpEF hemodynamics, recently developed by Burkhoff et al.32 and Granegger et al.28, can capture the pressure-volume (PV) loops of the 4-chamber heart, fully recapitulating the hemodynamics of various phenotypes of HFpEF. Furthermore, they utilize their in silico platform to evaluate the feasibility of a mechanical circulatory device for HFpEF, pioneering computational research of HFpEF for physiology studies as well as device development. However, these models remain unable to capture the dynamic changes in blood flows and pressures observed during disease progression. A recent study by Kadry et al.30 captures the various phenotypes of diastolic dysfunction by adjusting the active relaxation of the myocardium and the passive stiffness of the left ventricle on a low-dimensional model. Their work provides a comprehensive hemodynamic analysis of diastolic dysfunction based on both the active and passive properties of the myocardium. Similarly, the literature of high-dimensional models has primarily focused on HFrEF19,33,34,35,36,37. Bakir et al.33 proposed a fully-coupled cardiac fluid-electromechanics FEA model to predict the HFrEF hemodynamic profile and the efficacy of a left-ventricular assist device (LVAD). This biventricular (or two-chamber) model utilized a coupled Windkessel circuit to simulate the hemodynamics of the healthy heart, HFrEF and HFrEF with LVAD support33,37.
Similarly, Sack et al.35 developed a biventricular model to investigate right ventricular dysfunction. Their biventricular geometry was obtained from a patient's magnetic resonance imaging (MRI) data, and the model's finite-element mesh was constructed using image segmentation to analyze the hemodynamics of a VAD-supported failing right ventricle35. Four-chamber FEA cardiac approaches have been developed to enhance the accuracy of models of the electromechanical behavior of the heart19,34. In contrast to biventricular descriptions, MRI-derived four-chamber models of the human heart provide a better representation of the cardiovascular anatomy18. The heart model employed in this work is an established example of a four-chamber FEA model. Unlike lumped-parameter and biventricular FEA models, this representation captures hemodynamic changes as they occur during disease progression34,37. Genet et al.34, for example, used the same platform to implement a numerical growth model of the remodeling observed in HFrEF and HFpEF. However, these models evaluate the effects of cardiac hypertrophy on structural mechanics only and do not provide a comprehensive description of the associated hemodynamics.
To address the lack of HFpEF in silico models in this work, the lumped-parameter model previously developed by this group15 and the FEA model were readapted to simulate the hemodynamic profile of HFpEF. To this end, the ability of each model to simulate cardiovascular hemodynamics at baseline will be first demonstrated. The effects of stenosis-induced left ventricular pressure overload and of diminished left ventricular compliance due to cardiac remodeling-a typical hallmark of HFpEF-will then be evaluated.
1. 0D lumped-parameter model
2. The FEA model
3. Aortic valve stenosis
NOTE: Aortic stenosis is often a driver of HFpEF as it leads to pressure overload and ultimately, to concentric remodeling and compliance loss of the left ventricular wall. The hemodynamics observed in aortic stenosis often progress to those seen in HFpEF.
4. HFpEF hemodynamics
NOTE: To simulate the effects of chronic remodeling, the mechanical properties of the left heart were modified.
Results from the baseline simulations are illustrated in Figure 3. This depicts the pressure and volume waveforms of the left ventricle and the aorta (Figure 3A) as well as the left ventricular PV loop (Figure 3B). The two in silico models show similar aortic and left ventricular hemodynamics, which are within the physiologic range. Minor differences in the response predicted by the two platforms can be noticed during the ventricular emptying and filling phases, where non-linearities are better captured by the FEA model compared to the lumped-parameter platform. In physiology, such non-linear effects arise mainly as a result of the hyperelastic response of the heart tissue and are therefore more accurately reproduced by multidomain and high-order computational models18.
Ventricular and aortic hemodynamics were obtained for aortic stenosis, as this often leads to left ventricular pressure overload and ultimately, to the development of HFpEF. Pressure and volume waveforms at a 70% reduction of the aortic valve orifice area are shown for both models in Figure 4. Stenosis resulted in an elevated pressure gradient across the aortic valve. For the 70% stenosis considered in this work, peak transaortic pressure gradients of 41 mmHg and 54 mmHg were obtained with the lumped-parameter (Figure 4A) and FEA (Figure 4B) models, respectively. This moderate variation likely arises as another consequence of the lack of a constitutive equation defining the material properties of the cardiac tissue in the lumped-parameter model, in which compliance is simply defined by an array of numerical values. This model therefore does not capture fluid-structure interactions, which are instead accurately represented by the FEA model. Nevertheless, the results from both models are consistent with the American Society of Echocardiography (ASE) and the European Association of Echocardiography (EAE) classifications of moderate aortic valve stenosis, which denote peak transaortic gradients of 40-65 mmHg for aortic constrictions of approximately 60-75%38,39,40.
Left ventricular PV loops at baseline, 70% aortic stenosis, and of HFpEF following stiffening of the ventricular wall are summarized in Figure 5. Similar patterns can be observed in Figure 5A, depicting the results from the lumped-parameter model, and in Figure 5B, which shows the hemodynamics obtained via FEA. These PV loops are consistent with those in the scientific and clinical literature of HFpEF1,11,28,32. In particular, both models are able to capture the increase in the systolic left ventricular pressure due to the rise in afterload induced by aortic stenosis. Furthermore, the end-systolic volume is increased in the stenosis PV loop, leading to a drop in stroke volume. Upon remodeling and loss of left ventricular compliance, the end-diastolic pressure-volume relationship (EDPVR) becomes elevated, resulting in higher end-diastolic pressures and lower end-diastolic volumes. These phenomena, which are due to the inability of the left ventricle to relax and fill adequately, are successfully captured by the HFpEF PV loops in both the low- and high-dimensional models.
As another indication for diminished diastolic function, the flow through the mitral valve is shown in Figure S2, which highlights both the early relaxation (E) and atrial contraction (A) phases. Compared to the normal and stenosis profiles, HFpEF flow is characterized by a slightly higher peak E-phase mitral flow and significantly diminished peak A-phase flow, highlighting that passive stiffening of the left ventricle results in an elevated E/A ratio, which is consistent with the scientific literature30. Finally, Figure 6 shows changes in the myocardium stress map in the normal and HFpEF hearts during both systole and diastole. The long-axis view of the left ventricle illustrates the volumetric averaged stress distributions and shows elevated stresses in the HFpEF heart due to the characteristic loss of ventricular compliance. From baseline values of (61.1 ± 49.8) kPa and (0.51 ± 7.35) kPa for the healthy heart during peak-systole (t = 0.2 s) and end-diastole (t = 1.0 s), respectively, the mean stress correspondingly increased to (97.2 ± 205.7) kPa and (2.69 ± 16.34) kPa in HFpEF, suggesting that the hemodynamic changes observed in HFpEF are rooted in profound structural changes affecting the failing heart.
Figure 1: Domain of anatomically derived lumped-parameter model in the object-oriented numerical solver (see the Table of Materials), showing the four-chamber heart, the aorta, and the upper body, abdominal, lower body, and pulmonary circulations. Abbreviations: LV = left ventricle; RV = right ventricle; LA = left atrium; RA = right atrium; R1 = arterial resistance; R2 = venous resistance; C = compliance; IVC: inferior vena cava; SVC: superior vena cava. Please click here to view a larger version of this figure.
Figure 2: Finite element analysis model of the human heart. (A) 3D representation of the finite element analysis model of the human heart. (B) Simplified lumped-parameter representation of the blood flow model in the model coupled with the structural fluid exchange models18. Abbreviations: LV = left ventricle; RV = right ventricle; LA = left atrium; RA = right atrium; Raortic = aortic valve resistance; Rmitral = mitral valve resistance; Rpulmonary = pulmonary valve resistance; Rtricuspid = tricuspid valve resistance; Carterial = systemic arterial compliance; Rsystem = systemic arterial resistance; Cvenous = systemic venous compliance, Rvenous = systemic venous resistance; Cpulmonary = pulmonary compliance; Rpulmonary-system = pulmonary resistance. Please click here to view a larger version of this figure.
Figure 3: Baseline simulations and pressure-volume waveforms for the lumped-parameter and finite element analysis models of the human heart. (A) Left ventricular pressure and volume waveforms and aortic pressure calculated by the lumped-parameter and FEA models at baseline. (B) Left ventricular PV loop obtained through both platforms at baseline. Abbreviations: FEA = finite element analysis; LV = left ventricular; PV = pressure-volume. Please click here to view a larger version of this figure.
Figure 4: Left ventricular pressure and volume waveforms and aortic pressure calculated at 70% reduction of the aortic valve orifice area. (A) Lumped-parameter model, (B) FEA model. Abbreviations: FEA = finite element analysis; LV = left ventricular. Please click here to view a larger version of this figure.
Figure 5: Left ventricular PV loops of the healthy heart, under acute stenosis-induced pressure overload, and of the HFpEF heart following chronic remodeling and stiffening. (A) Lumped-parameter, (B) FEA models. Abbreviations: EDPVRH = end-diastolic pressure-volume relationship in the simulated healthy heart; EDPVRHFpEF: end-diastolic pressure-volume relationship in the simulated HFpEF physiology; PV – pressure-volume; FEA = finite element analysis. Please click here to view a larger version of this figure.
Figure 6: von Mises stress (avg: 75%) under physiologic conditions and of the HFpEF heart during peak-systole and diastole, as predicted by the FEA model. The color maps indicate stress levels in MPa. Higher stresses can be seen in HFpEF (92.7-2.7 kPa) compared to the healthy heart (61.1-0.5 kPa) during peak-systole (t = 0.2 s) and end-diastole (t = 1.0 s). Please click here to view a larger version of this figure.
Figure S1: Input signals for (A) centrifugal pump, (B) left ventricle, (C) right ventricle, (D) left and right atria for the lumped-parameter simulation. Please click here to download this file.
Figure S2: (A) Aortic and (B) mitral flow signals for the baseline, stenosis, and HFpEF profiles, obtained by FEA. Abbreviations: E = early relaxation phase; A = atrial contraction; FEA = finite element analysis; HFpEF = heart failure with preserved ejection fraction. Please click here to download this file.
Table S1. Geometric and mechanical parameters of baseline lumped-parameter simulation. Please click here to download this Table.
Table S2. Extensive set of parameters of baseline lumped-parameter simulation. Please click here to download this Table.
Table S3. Fluid cavities values in the mechanical finite element analysis (FEA) model18. Please click here to download this Table.
Table S4. Boundary conditions of fluid exchange links for the finite element analysis (FEA) model18. Please click here to download this Table.
Table S5. The required simulation files for the finite element analysis (FEA) model18. Please click here to download this Table.
Table S6. Parameters for the aortic-stenosis lumped-parameter simulation. Please click here to download this Table.
Table S7. Fluid exchange link definitions in the finite element analysis (FEA) model18. Please click here to download this Table.
Table S8. Parameters for the HFpEF lumped-parameter simulation. Please click here to download this Table.
The lumped-parameter and FEA platforms proposed in this work recapitulated the cardiovascular hemodynamics under physiologic conditions, both in the acute phase of stenosis-induced pressure overload and in chronic HFpEF. By capturing the role that pressure overload plays in the acute and chronic phases of HFpEF development, the results from these models are in agreement with the clinical literature of HFpEF, including the onset of a transaortic pressure gradient due to aortic stenosis, an increase in the left ventricular pressure, and the reduction in the end-diastolic volume due to wall stiffening41. Furthermore, this FEA model was able to capture elevations in myocardial stress in HFpEF throughout the cardiac cycle. To ensure a correct setup of these simulations, the steps outlined in the protocol section above must be followed rigorously. For the lumped-parameter model, it is essential that the network of hydraulic elements is recreated correctly as shown in Figure 1 and that the prescribed values are provided as input parameters (Table S1 and Table S2). In addition, the solver block must be defined and connected to the network at any node.
Functioning of the FEA model requires all the simulation files that are packaged with the solver18 that are listed in Table S5. Omission of any of the prerequisite components might cause early termination of the simulation. For both platforms, it is critical to obtain the baseline simulation with the default input parameters prior to recreating the stenosis and HFpEF hemodynamic profiles. The original research article outlining the baseline simulation15 and the documentation linked to the simulation in the Supplemental Files can be consulted for troubleshooting the lumped-parameter model. Similarly, this FEA framework contains the software documentation and toolbox folder for troubleshooting18. In the event of a simulation error, the user can invoke the simulation diagnostics by executing the relative plug-ins in the toolbox folder18. Hemodynamic results from the lumped-parameter model were analogous to those calculated via FEA in each of the simulated conditions and consistent with the clinical literature of HFpEF. The high-dimensional FEA platform allows the capture of the complex biomechanical behavior of the heart and provides an accurate description of cardiovascular hemodynamics, albeit at the expense of the elevated computational demand. However, in the lumped-parameter model, the runtime is reduced from several hours to few minutes, constituting a significant advantage over higher-order in silico models.
In addition, by modeling a larger number of cardiovascular compartments, this lumped-parameter simulation allows the examination of blood flows and pressures at various sites of the cardiovascular tree and is therefore suitable for studies that extend beyond the heart chambers and the proximal vasculature. However, while being able to recapitulate global hemodynamics, this description fails to capture some minor effects of structural interactions and therefore lacks the accuracy typical of FEA representations. Analysis of the cardiac mechanics obtained in this study through the finite element approach corroborated those from previous investigations. Specifically, these mean stress values are in the same range as those predicted by growth models of the partially supported heart during chronic failure34,37. Compared to those models, the stress values found in these studies described herein were moderately higher due to the elevated level of aortic stenosis simulated to induce pressure overload. In addition, it was found that loss of left ventricular compliance in HFpEF has a major impact on endocardial stress.
However, diastolic stiffness and its sensitivity were not parametrically investigated in this study. In fact, this parameter was tuned to capture the physiologically relevant hemodynamic profile of chronic left ventricular pressure overload. Extensive sensitivity analysis should be performed to fully characterize the effects of diminished diastolic compliance. This computational model further suggests that biomechanical changes of the cardiac structure in the HFpEF may be a major driver of remodeling and may thus have considerable implications in the HFpEF hemodynamics and disease progression. Integration of a dynamic growth model with the fluid-structure interaction of the FEA simulation may be considered in future work to more comprehensively capture the dynamics of cardiac remodeling and hemodynamic aberrations induced by pressure overload. Moreover, further studies of the effects of active relaxation similar to Kadry et al.30 and electrical conduction and contractility might be needed to simulate different phenotypes of diastolic dysfunction.
The development of simulation platforms that are suitable for studies of HFpEF is largely underreported in the literature. In this context, this work provides a unique environment for studies of the HFpEF pathophysiology. The anatomically derived lumped-parameter model will allow rapid simulation of the effect that varying patient-specific hemodynamic parameters (e.g., vascular luminal area and compliance) play in the global hemodynamics for healthy and HFpEF conditions. In addition, FEA modeling permits detailed investigation of the effects of temporal changes in mechanical properties and excitability of the heart tissue as they change progressively during HFpEF. Furthermore, the proposed models have potential utility for the simulation of novel therapies for HFpEF, partly addressing the lack of reliable in vivo, in vitro, and in silico models of HFpEF, which may be responsible for the suspension of clinical trials due to inadequate device optimization42. Finally, future work may involve the integration of these models into a single simulation by replacing the simplified lumped-parameter description underlying the FEA approach with the numerical solver model. This may further enhance the accuracy of these models and further support computational studies of HFpEF and other cardiovascular conditions.
In summary, two distinct computational models of HFpEF were described in this study. The ability of the developed platforms to describe baseline hemodynamics under physiologic conditions was first demonstrated. Then, the changes arising from aortic stenosis and ultimately from HFpEF due to left ventricular remodeling were investigated, demonstrating that the results were consistent with those reported in the literature. Finally, the simulated hemodynamic conditions showed elevations in the cardiac wall stress in the HFpEF heart compared to physiologic conditions. In the context of the incredibly pressing healthcare challenge that HFpEF represents, these proposed platforms are among the first in silico descriptions that can provide insights into the hemodynamics and biomechanics of HFpEF. These computational models may be further used as a tool for the development of treatments for HFpEF, ultimately supporting translational research in the field.
The authors have nothing to disclose.
We acknowledge funding from the Harvard-Massachusetts Institute of Technology Health Sciences and Technology program, and the SITA Foundation Award from the Institute for Medical Engineering and Science.
Abaqus Software | Dassault Systèmes Simulia Corp. | Version used: 2018; FEA simulation software | |
HETVAL | Dassault Systèmes Simulia Corp. | Version used: 2018 | |
Hydraulic (Isothermal) library | MathWorks | Version used: 2020a | |
Living Heart Human Model | Dassault Systèmes Simulia Corp. | Version used: V2_1, anatomically accurate FEA platform of 4-chamber adult human heart | |
MATLAB | MathWorks | Version used: 2020a, object-oriented numerical solver | |
SIMSCAPE FLUIDS | MathWorks | ||
UAMP | Dassault Systèmes Simulia Corp. | Version used: 2018 | |
VUANISOHYPER | Dassault Systèmes Simulia Corp. | Version used: 2018 |