Lumped-Parameter and Finite Element Modeling of Heart Failure with Preserved Ejection Fraction

Summary

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.

Abstract

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.

Introduction

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%, respectively^1,2,3. 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 obstruction^3,4,5,6,7. 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 compliance^8,9,10. 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 tree^16,17. 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 tissue^18,19.

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 HFrEF^{20,21,22,23,24}. 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 systems^20,21,23,24. 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 filling^25,26,27. 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 proposed^{20,22,28,29,30,31}.

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 HFrEF^{19,33,34,35,36,37}. 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 support^33,37.

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 heart^19,34. 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 progression^34,37. 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.

Protocol

1. 0D lumped-parameter model

1. Simulation setup
   NOTE: In the numerical solver environment (see the Table of Materials), construct the domain as shown in Figure 1. This is composed of the 4-chamber heart, the upper body, abdominal, lower body, and thoracic compartments, as well as the proximal vasculature, including the aorta, the pulmonary artery, and the superior and inferior venae cavae. The standard elements used in this simulation are part of the default hydraulic library. Details can be found in the Supplemental Files.
   1. Navigate the hydraulics library to find the required elements: hydraulic pipeline, constant volume hydraulic chamber, linear resistance, centrifugal pump, check valve, variable area orifice, and the custom hydraulic fluid.
      1. Drop the hydraulic pipeline elements into the workspace.
         ​NOTE: These account for frictional losses as well as wall compliance and fluid compressibility in blood vessels and heart chambers. Through this block, the pressure loss is calculated using the Darcy-Weisbach law, whereas the change in diameter due to wall compliance depends on the compliance proportionality constant, the luminal pressure, and the time constant. Finally, fluid compressibility is defined by the bulk modulus of the medium.
      2. Insert the constant volume hydraulic chamber elements to define wall compliance and fluid compressibility.
         ​NOTE: This block does not take into account pressure losses due to friction.
      3. Add the linear resistance elements to define resistance to flow.
         NOTE: This is independent of the geometrical properties of the vasculature, analogously to the resistive element used in electrical analog Windkessel models. Other blocks, such as the centrifugal pump, the check valve, the variable-area orifice, and the custom hydraulic fluid elements should be inserted to generate the desired pressure input to the system, model the effects of heart valves on blood flow, and define the mechanical properties of blood. Through these elements, the behavior of the cardiovascular system in both physiology and disease can be fully captured. The input signal for the centrifugal pump can be found in Figure S1A.
      4. Model the contractility of each heart chamber through the custom variable-compliance compliance chamber element.
         NOTE: This accepts compliance as a time-varying user-defined input signal and is based on the time-varying elastance model (Figure S1B-D).
   2. Provide the parameters relative to each element, as shown in Table S1, also found in Rosalia et al.¹⁵
   3. Insert a Physical Signal (PS) Repeating Sequence element for each of the blocks that require a time-varying user-defined input signal: the LV pump, the variable-compliance compliance elements, and the variable-area orifice blocks.
      NOTE: Input signals utilized for this simulation can be found in Figure S1.
   4. Select the default ODE 23t implicit solver and run the simulation for 100 s to reach a steady state.

2. The FEA model

1. Simulation setup
   NOTE: The FEA Model utilizes a coupled electrical-mechanical analysis in sequence. In this model, the electrical analysis is conducted first; then the resulting electric potentials are used as the excitation source in the following mechanical analysis. Therefore, the simulation setup contains two work domains: the electrical (ELEC) and the mechanical (MECH) domains, that are predefined in the FEA simulation software (Table of Materials)¹⁸. Hence, the following section only describes the analysis workflow. The FEA model uses the following user subroutines HETVAL, VUANISOHYPER, and UAMP for the electrical and mechanical material modeling¹⁸.
   1. Navigate the ELEC domain to perform electrical analysis using the predefined temperature procedure in the Standard module.
      1. Use a single-analysis step named BEAT. Set the duration of the cardiac cycle to 500 ms and apply an electrical potential pulse to a node set representing the sinoatrial (SA) node (node set: R_Atrium-1.SA_NODE).
      2. Review the default electrical waveform, which ranges from -80 mV to 20 mV over 200 ms with the smooth step amplitude definition, as described in the model guide¹⁸. Use the default values of material constants in the electrical analysis to adjust the AV delay.
      3. Launch the Job module, and create a job named heart-elec.
   2. Once the electrical analysis setup is completed, navigate the MECH domain to perform the fluid cavity-based mechanical analysis.
      NOTE: The mechanical simulation is performed after the electrical analysis, and the resulting electric potentials are used as the excitation source for the mechanical analysis. The mechanical analysis contains multiple steps.
      1. Use the three main steps named PRE-LOAD, BEAT1, AND RECOVERY1. In the PRE-LOAD step, review the boundary conditions of the pre-stressed state of the heart. Use 0.3 s as the step time to linearly ramp up the pressure in the fluid chambers.
         NOTE: The predefined fluid cavity pressure values are shown in Table S3. The pre-stressed state of the heart was already defined in the normal heart simulation setup, and the initial node conditions are provided in the external simulation files, as listed in Table S5. Recalculation of the zero-stress state using the inverse mechanical simulation is required whenever the boundary condition is modified, as explained in steps 3.2.2-3.2.4.
      2. In the BEAT1 step, use 0.5 s as the step time to simulate contraction.
      3. In the RECOVERY1 step, select 0.5 s for cardiac relaxation and ventricular filling for a heart rate of 60 bpm.
      4. Enable the subsequent steps, BEATX and RECOVERYX, to simulate more than one cardiac cycle to reach a steady state.
         NOTE: Three cardiac cycles will be sufficient to reach steady state. One cycle of the simulation is completed in ~8 h on a 24-core processor (3.2 GHz x 24).
      5. Launch the Job module, and create a job named heart-mech, enabling the double precision option.
2. Review the simplified lumped-parameter Windkessel model
   NOTE: The mechanical domain of the FEA model has a blood flow model, which is based on a simplified lumped-parameter circuit and is created as a combination of surface-based fluid cavities and fluid exchanges as seen in Figure 2¹⁸.
   1. Use the Windkessel representation mentioned in the above note to run the simulation. Review the blood flow model representation to adjust the values of the resistive and capacitive elements for flow resistances and structural compliances, respectively.
   2. Review the 3D finite element representation of four heart chambers, and ensure their geometrical positions are accurate.
   3. Check the heart assembly, and switch to the Interaction module to adjust the compliance and contractility values of each of the four heart chambers.
      NOTE: The default values in the Interaction module are configured to simulate an idealized healthy human heart beat cycle¹⁸.
   4. Review the following hydrostatic fluid cavities in the Interaction module, CAV-AORTA, CAV-LA, CAV-LV, CAV-PULMONARY_TRUNK, CAV-RA, CAV-RV, CAV-SVC, CAV-ARTERIAL-COMP, CAV-PULMONARY-COMP, and CAV-VENOUS-COMP (Table S3).
   5. Use the compliance chambers (CAV-ARTERIAL-COMP, CAV-PULMONARY-COMP, and CAV-VENOUS-COMP) as cubic volumes as they represent the compliance of the arterial, venous, and pulmonary circulations.
   6. Attach three compliance cubic volumes to a grounded spring, and review the stiffness value to model the pressure-volume response in the arterial, venous, and pulmonary circulations.
   7. Check the following fluid exchange definitions between the hydrostatic fluid cavities: arterial-venous, venous-right atrium, right atrium-right ventricle, right ventricle-pulmonary system, pulmonary system-left atrium, left atrium-left ventricle, and left ventricle-aorta (Table S4).
   8. Adjust the viscous resistance coefficient to modify the blood flow model in each fluid exchange link (see Supplemental files for more information about the viscous resistance effect).
3. Multiphysics simulation
   1. Locate the CAE database file in the working directory.
      NOTE: The FEA Model in this protocol is delivered in the database and is named as LH-Human-Model-Beta-V2_1.cae.
   2. Insert the input, object, and library files to the working directory to run the simulation. See Table S5 for the full list of input and library files.
   3. Launch the FEA model simulation software (see the Table of Materials).
      NOTE: Consult the software provider for compatibility with later versions¹⁸.
   4. Review the parts, assembly, and boundary conditions in both ELEC and MECH domains, as described in sections 2.2 and 2.3.
   5. First, run the electrical simulation job named heart-elec, as described in section 2.1.1.3. Visually inspect the electrical potential results to verify that the heart-elec simulation ran as expected. Then, ensure that the result file heart-elec.odb is in the working directory.
   6. Move to the second simulation phase by switching to the MECH domain. Review the values of the material constants used in the mechanical simulation to model the desired passive and active cardiac response.
   7. Ensure that the material library files for the mechanical analysis use the HYBRID- string name. To modify the material response of the heart chambers, adjust the appropriate hybrid material file, or replace the entire material response by defining a new material behavior in the Materials section in the CAE module.
      NOTE: Detailed information about the built-in constitutive laws can be found in the user guide¹⁸.
   8. In the PRE-LOAD step, set the pressures of the hydrostatic cavities to obtain the desired physiologic behavior. Use the built-in smooth amplitude option to ramp up from zero to the desired pressure level as described in step 2.1.2.1.
   9. Disable the pressure boundary conditions defined in 2.1.2.1 to run the blood flow model with a constant overall blood volume within the circulation system. Run the simulation job named heart-mech, as described in section 2.1.2.5.

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.

1. The lumped-parameter model
   1. Modify the input signal in the PS repeating sequence element relative to the aortic valve, located in the left ventricular compartment. Simulate a reduction of the orifice area equal to 70% compared to baseline (Table S6).
      NOTE: The input values will represent the orifice area of the stenotic valve during each heartbeat. The orifice area value can be easily adjusted by multiplying the start output values vector of the aortic valve PS element by a decimal value corresponding to the final orifice area with respect to its original value. In this work, a factor of 0.3 was used to achieve 70% constriction.
2. The FEA model
   1. Modify the fluid exchange definition of the LINK-LV-ARTERIAL parameter.
      NOTE: This parameter possesses a viscous resistance coefficient tuned to the blood flow between the left ventricle and the aorta. The effective exchange area can be modified to adjust the blood flow and create the appropriate aortic stenosis model (Table S7).
   2. Locate the toolbox folder and copy the files inside that folder to the main working directory.
   3. Perform an inverse mechanical simulation by executing the toolbox files¹⁸. To this end, change the suction pressures of the left ventricle and left atrium to 6 mmHg in the fluid cavity to adjust their initial volumetric state for the aortic stenosis model. Execute the inversePreliminary.py function.
      ​NOTE:Recalculation of the zero-stress state using the inverse mechanical simulation is required whenever the boundary condition is modified.
   4. Once the inverse mechanical simulation is completed, run the post-processing functions: calcNodeCoords.py and straight_mv_chordae.py. Use the default values for the other flow parameters, and run a new mechanical simulation as described in section 2.1.2.5.

4. HFpEF hemodynamics

NOTE: To simulate the effects of chronic remodeling, the mechanical properties of the left heart were modified.

1. The lumped-parameter model
   1. Modify the left ventricular diastolic compliance of the LV compliance element to mimic wall stiffening due to pressure-overload, using the value of end-diastolic compliance in Table S8.
      NOTE: Assume compliance to drop linearly from end-systole to end-diastole.
   2. Increase the leak resistance of the LV pump to 18 × 10⁶ Pa s m^-3 (Table S8) to capture the elevated left ventricular pressures observed in HFpEF.
2. The FEA model
   1. Edit the active material properties of the left ventricle geometry. Increase the stiffness component to tune the active tissue response affecting the stress components in the fiber and sheet directions in the constitutive model.
      1. Modify the material response of the left ventricle in the mech-mat-LV_ACTIVE file.
         ​NOTE: The magnitude of stiffness for the left ventricular chamber can be tuned to provide the appropriate diastolic compliance effects.
      2. Increase the stiffness parameters a and b in the anisotropic hyperelastic formulation to capture the increased stiffness response for the HFpEF physiology.
      3. In the PRE-LOAD step, set the fluid cavity pressures of the left ventricle and left atrium to 20 mmHg.
      4. Perform an inverse mechanical simulation to obtain the volumetric state of the left ventricle and atrium. Export the nodal coordinates from the heart-mech-inverse.odb file¹⁸.
      5. Execute the post-processing functions: calcNodeCoords.py and straight_mv_chordae.py, as described in step 3.2.4. Locate the new nodal input files in the working directory and perform a new mechanical simulation, as described in section 2.1.2.5.

Results

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 ventricula...

Discussion

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...

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
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