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A comprehensive and biophysically detailed computational model of the whole human heart electromechanics

Marco Fedele, Roberto Piersanti, Francesco Regazzoni, Matteo Salvador, Pasquale Claudio Africa, Michele Bucelli, Alberto Zingaro, Luca Dede', Alfio Quarteroni

arXiv:2207.12460v1math.NAcs.CEcs.DCcs.NAphysics.med-ph

TL;DR

Whole-heart electromechanical models have often omitted active atrial contraction, despite the complexity of atrial physiology and its interaction with ventricular function. This paper presents a biophysically detailed model of the whole human heart with active atrial and ventricular contraction. It reproduces multiple features of healthy atrial and ventricular function and shows the effects of atrial contraction and fiber-stretch-rate feedback.

  • Problem

    Whole-heart electromechanical models have often omitted atrial contraction, even though atrial physiology is complex and strongly influenced by ventricular function.

    Source: §1 · pp. 4–5

  • Method

    The authors extend a whole-heart electromechanical model with active atrial and ventricular contraction and couple its core models using feedbacks and stabilization techniques.

    Source: §4.2 · p. 25, §4.3 · p. 25, §6 · p. 40

  • Results

    The model captures atrial and ventricular physiology, including characteristic eight-shaped atrial pressure-volume loops, and omitting atrial contraction or fiber-stretch-rate feedback changes modeled function and semilunar fluxes.

    Source: §6 · p. 43

  • Takeaways & Limitations

    The model demonstrates a comprehensive computational representation of healthy whole-heart function, including atrioventricular interaction and the threefold atrial function.

    Source: §6 · p. 43

    The baseline simulation produces an atrial-contraction A-wave taller than the ventricular-passive-filling E-wave, which the authors relate to ventricular relaxation that is too slow during passive filling.

    Source: §5.2 · p. 31

Abstract

from arXiv · show

While ventricular electromechanics is extensively studied, four-chamber heart models have only been addressed recently; most of these works however neglect atrial contraction. Indeed, as atria are characterized by a complex physiology influenced by the ventricular function, developing computational models able to capture the physiological atrial function and atrioventricular interaction is very challenging. In this paper, we propose a biophysically detailed electromechanical model of the whole human heart that considers both atrial and ventricular contraction. Our model includes: i) an anatomically accurate whole-heart geometry; ii) a comprehensive myocardial fiber architecture; iii) a biophysically detailed microscale model for the active force generation; iv) a 0D closed-loop model of the circulatory system; v) the fundamental interactions among the different core models; vi) specific constitutive laws and model parameters for each cardiac region. Concerning the numerical discretization, we propose an efficient segregated-intergrid-staggered scheme and we employ recently developed stabilization techniques that are crucial to obtain a stable formulation in a four-chamber scenario. We are able to reproduce the healthy cardiac function for all the heart chambers, in terms of pressure-volume loops, time evolution of pressures, volumes and fluxes, and three-dimensional cardiac deformation, with unprecedented matching (to the best of our knowledge) with the expected physiology. We also show the importance of considering atrial contraction, fibers-stretch-rate feedback and suitable stabilization techniques, by comparing the results obtained with and without these features in the model. The proposed model represents the state-of-the-art electromechanical model of the iHEART ERC project and is a fundamental step toward the building of physics-based digital twins of the human heart.

1. Introduction

Whole-heart electromechanical models have often omitted atrial contraction, although atrial function is complex and strongly influenced by the ventricles. This paper presents a biophysically detailed four-chamber model that incorporates active atrial and ventricular contraction and aims to reproduce healthy cardiac function.

1. Introduction

  • Whole-heart electromechanics has lacked detailed models of both atrial and ventricular contraction, while atrial physiology is complex and influenced by ventricular function.Many earlier electromechanical whole-heart models treated atria as passive; the few including atrial contraction had limitations in the displayed atrial function or semilunar fluxes.Source: §1 · pp. 4–5
  • Atrial contraction remains underrepresented in prior electromechanical whole-heart models, and atrial models require calibrated biophysical descriptions because atrial function is shaped by ventricular contraction and relaxation.Earlier work reported atrial loops tending toward the characteristic eight-shape, but one prior model showed semilunar fluxes above physiological values, which the authors relate to omitted fiber-stretch-rate feedback.Source: §1 · p. 5
  • The proposed model combines detailed electrophysiology, passive mechanics, and active contraction models for the atria and ventricles.It extends prior ventricular models to the entire heart and accounts for tissue heterogeneity across cardiac components.Source: §1 · p. 5
  • Its components include anatomically detailed four-chamber geometry, a whole-heart fiber architecture with atrial bundles, chamber-specific ionic models, and microscale active-force generation.The model also couples mechanics to a 0D closed-loop circulation model and represents the pericardium and surrounding organs with spring-damper Robin boundary conditions.Source: §1 · pp. 5–6
  • The core models are calibrated by cardiac region and coupled through feedbacks, including mechano-electric and fiber-stretch-rate feedbacks.The numerical approach uses a segregated-intergrid-staggered scheme and stabilization terms for circulation and fiber-stretch-rate feedback to support a stable four-chamber formulation.Source: §1 · p. 6

2. Cardiac anatomy and physiology

The section reviews the heart's four-chamber anatomy, conduction system, and coordinated cycle of electrical, mechanical, and valvular events. It also explains how atrial and ventricular pressure-volume loops reflect distinct chamber functions and their interaction.

2. Cardiac anatomy and physiology

  • The anatomy and cardiac cycle involve coordinated events and interactions among chambers, tissue biophysics, and multiphysics processes.The review introduces the cardiac components represented in the model and distinguishes atrial from ventricular function.Source: §2 · pp. 6, 10
  • The heart comprises right and left atria and ventricles, with blood routed through the pulmonary and systemic circulations and valves regulating chamber and arterial flow.The atrioventricular valves also electrically insulate atria from ventricles.Source: §2 · pp. 6–7
  • The cardiac conduction signal begins at the sinoatrial node, reaches the left atrium through interatrial bundles, and is delayed at the atrioventricular node before activating the ventricles.The ventricular signal travels through the His bundle, bundle branches, and Purkinje fibers before propagating from endocardium toward epicardium.Source: §2 · p. 8
  • After passive ventricular filling, atrial contraction adds blood to the ventricles before ventricular contraction closes the atrioventricular valves and begins ejection.Ventricular relaxation and movement of the atrioventricular plane drive passive filling; ventricular ejection lowers ventricular volumes while the atria fill.Source: §2 · pp. 8–9
  • Atrial pressure-volume loops have a V-loop associated with passive ventricular influence and an A-loop associated with active atrial contraction.Together these phases reflect the atria's reservoir, conduit, and booster-pump functions.Source: §2 · pp. 9–10

3. Mathematical models

The mathematical model extends prior ventricular electromechanics models to the whole heart. It incorporates tissue heterogeneity across cardiac components and describes the model sections covering its domain, fibers, core models, and initial configuration.

3. Mathematical models

  • The model extends prior ventricular electromechanics models to the whole heart, representing heterogeneity in electrical conduction and active and passive mechanics across atria, ventricles, valves, and vessels.The earlier models considered only the left ventricle or both ventricles as three-dimensional domains, with homogeneous electrical and mechanical properties.Source: §3 · p. 10
  • The model builds on a cardiac electromechanics framework coupled to a lumped-parameter model of blood circulation.The paper describes the computational domain and fibers, presents its core models, and details recovery of the unloaded configuration and calculation of initial displacement.Source: §3 · p. 10

3.1. Computational domain

The computational domain represents the whole heart and separates chamber myocardium from the major arteries. Its atria include detailed anatomy and connections, while the ventricular endocardium is smoothed and lacks papillary muscles.

3.1. Computational domain

  • The computational domain is defined as the unloaded whole-heart geometry and subdivided into cardiac subdomains.The domain description identifies the four-chamber myocardium and the two arteries as components.Source: §3.1 · p. 10
  • The whole-heart domain includes detailed right and left atria with appendages, connected through the interatrial septum and Bachmann’s bundle.The two ventricles form a single subdomain with a smoothed endocardial layer without papillary muscles.Source: §3.1 · p. 10
  • The pulmonary trunk and aorta are modeled up to their main bifurcations so they can be fixed sufficiently far from the heart.At those distant boundaries, their movement is considered negligible.Source: §3.1 · p. 10

ARTIFICIAL BOUNDARIES

The domain includes simplified valves and atrial-pool caps, alongside boundary classes that distinguish cut vessels, epicardial surfaces, and internal cardiac surfaces.

ARTIFICIAL BOUNDARIES

  • Simplified valve geometries connect atria to ventricles and ventricles to arteries while representing stiff, insulating annuli and pressure differences across closed leaflets.The valve geometries fill the valvular orifices.Source: ARTIFICIAL BOUNDARIES · p. 11
  • Artificial caps close the entry veins to atrial blood pools, facilitating calculation of atrial volumes.Source: ARTIFICIAL BOUNDARIES · p. 11
  • Mechanical boundaries include rings at cut veins and arteries, epicardial regions contacting pericardial fluid or epicardial adipose tissue, and internal chamber and arterial surfaces.Source: ARTIFICIAL BOUNDARIES · p. 12

3.2. Modeling the cardiac fibers

The whole-heart Laplace-Dirichlet rule-based method constructs chamber-specific myocardial fiber orientations from boundary-value solutions and prescribed transmural angle variation. Its resulting architecture captures characteristic fiber patterns across all four chambers.

3.2. Modeling the cardiac fibers

  • The whole-heart Laplace-Dirichlet rule-based method prescribes myocardial fiber architecture using a method extended to atrial geometries.The authors use the whole-heart method proposed by Piersanti et al. in its improved version [55, Chapter 4].Source: §3.2 · p. 12
  • Laplace boundary-value problems define transmural and internal distances used to distinguish the four chambers and construct a local coordinate frame from their gradients.Source: §3.2 · p. 13
  • Fiber orientations are obtained by rotating the local frame according to helical and sheetlet angles that vary linearly between epicardial and endocardial values.Source: §3.2 · p. 13
  • The resulting architecture captures the LV helix, characteristic RV and outflow-tract fibers, and atrial bundles including inter-atrial connections, the Crista Terminalis, and Pectinate Muscles.Source: §3.2 · p. 13

3.3. The full electromechanical model

The full electromechanical model couples atrial and ventricular electrophysiology, active force generation, tissue mechanics, and closed-loop circulation across the four chambers. It links chamber pressures and volumes to myocardial mechanics and circulation while representing feedback among calcium, deformation, force, and electrical activity.

3.3. The full electromechanical model

  • The whole-heart model couples electrophysiology, active force generation, tissue mechanics, and circulation through calcium, active stress, fiber stretch and stretch rate, deformation-dependent electrical properties, and chamber-volume conservation.The electrical and ionic models exchange transmembrane potential and ionic currents; calcium drives force generation, mechanics feeds back to force and electrical activity, and chamber volumes couple 3D mechanics to the 0D circulation.Source: §3.3 · pp. 13–14, Figure 5 · p. 14

3.3.1. Electrophysiology (E )-(I )

  • Electrophysiology uses a monodomain model with separate atrial and ventricular ionic models, while deformation-dependent diffusion represents mechano-electric feedback.The Courtemanche et al. model is used for atria and the ten Tusscher and Panfilov model for ventricles; applied currents trigger activation at selected locations.Source: §3.3.1 · pp. 16–17

3.3.2. Active force generation

  • The RDQ20 force-generation model links calcium-sensitive crossbridge dynamics to active tension and captures sarcomere-length and force-velocity effects, producing stretch and stretch-rate feedback.The same model is parameter-calibrated for different chambers; sarcomere length depends on local fiber stretch, and its rate affects generated force.Source: §3.3.2 · p. 18

3.3.3. Active and passive mechanics

  • Tissue mechanics uses active stress with hyperelastic passive laws: an exponential Usyk model with quasi-incompressibility in myocardium and Neo-Hookean behavior in non-conductive regions.Active stress depends on force-generation tension and acts primarily along fibers, with cross-fiber components; epicardial Robin-like conditions represent pericardial support, and chamber blood pressures load the endocardium.Source: §3.3.3 · pp. 19–20

3.3.4. Blood circulation (C ) and 3D-0D coupling (V )

  • A closed-loop 0D circulation model uses RLC circuits for systemic and pulmonary vessels and non-ideal diodes for the four valves, with chamber pressures and volumes coupled to 3D mechanics by volume consistency.The 0D chamber pressure-volume relations are supplied by the 3D electromechanical model, and its chamber volumes must match volumes computed from the 3D endocardial surfaces.Source: §3.3.4 · pp. 20–21
  • Chamber volumes can be computed exactly because valves make the endocardial surfaces closed.Source: §3.3.4 · p. 22
  • The circulation-coupling model has nc + 4 equations and unknowns; its four chamber pressures act as Lagrange multipliers enforcing volume consistency and enter myocardial mechanics through endocardial normal-stress conditions.Pulmonary and systemic arterial pressures are applied at the pulmonary trunk and aorta, respectively.Source: §3.3.4 · p. 22

3.4. Reference configuration and initial displacement

The reference configuration is recovered from the imaged, pressure-loaded geometry by virtually deflating the heart, then inflating it under initial pressures to compute the mechanical model’s initial displacement. The procedure assumes quasi-static mechanics at a selected slow-motion point in diastole.

3.4. Reference configuration and initial displacement

  • The recovery assumes quasi-static mechanics at the end of ventricular passive filling, just before atrial contraction, when ventricular filling slows and chamber motion is negligible.This phase is often identifiable from the ECG and has better image quality; its small loading pressures also make the numerical problem less challenging.Source: §3.4 · p. 22
  • The imaged geometry is treated as pressure-loaded; virtual deflation under its internal pressures recovers the unloaded reference configuration.The pressures load the endocardium and endothelium in the right and left atria, ventricles, pulmonary trunk, and aorta.Source: §3.4 · p. 22
  • A fixed-point method with adaptive-step continuation solves the inverse problem of finding a reference geometry that deforms back to the imaged configuration.The forward deformation uses the imaging pressures and active tension; after recovery, the model applies initial-time pressures and tension to compute the initial displacement.Source: §3.4 · p. 23
  • The imaging phase and simulation’s initial phase can differ, so their active tension and chamber pressures may have different values.The recovered reference geometry is re-inflated using the pressures and active tension corresponding to the model’s initial time.Source: §3.4 · p. 23

4. Numerical approximation

The numerical approximation uses a segregated, multiresolution scheme to couple the heart’s core models, with different time steps for electrophysiology and the other blocks. Stabilization addresses oscillations in the staggered mechanics–circulation coupling while retaining the segregated approach.

4. Numerical approximation

  • The segregated-intergrid-staggered scheme solves core models sequentially at different spatial and temporal resolutions to represent their heterogeneous scales.The electrophysiology and ionic models take repeated smaller time steps; activation, mechanics, and circulation are then updated in sequence.Source: §4 · p. 23, Figure 6 · p. 24, §4.2 · p. 25

4.1. Numerical approximation of the core models

  • Electrophysiology and ionic models use BDF2 with IMEX treatment, while activation and mechanics use BDF1 and circulation uses explicit BDF1.Diffusion is implicit and reaction explicit in the electrophysiology IMEX scheme; mechanics is fully implicit, and ionic currents use interpolation.Source: §4.1 · p. 24
  • A shared tetrahedral mesh supports intergrid transfer and arbitrary finite-element choices, with P2 elements for electrophysiology and P1 elements for activation and mechanics.The mesh approach also represents atrioventricular valves as electrical insulators by omitting non-conductive regions from the electrophysiology system.Source: §4.1 · p. 25

4.2. Numerical coupling of the core models

  • The sequential update follows the main direction of model interactions, while opposite-direction feedbacks use the previous time step’s solution.An intergrid transfer operator evaluates feedbacks between ionic/electrophysiology and activation/mechanics models.Source: §4.2 · p. 25

4.3. Stabilizing the coupling of the core models

  • Segregated electromechanics can produce non-physical oscillations through mechanics–activation and circulation–mechanics feedbacks; monolithic methods avoid these instabilities but cost more and require one time step size.The stabilization terms target oscillations in the mechanics–activation and mechanics–circulation substep.Source: §4.3 · pp. 25–26
  • Fibers-stretch-rate feedback can destabilize the mechanics formulation because microscale activation and tissue mechanics describe strain in Eulerian and Lagrangian coordinates, respectively.The proposed correction adds a numerically consistent stabilization term to the formulation.Source: §4.3 · pp. 26–27
  • The staggered mechanics–circulation scheme is not unconditionally stable; the authors stabilize it by extrapolating circulation-derived chamber volumes to the next time step.The extrapolation accounts for how changing chamber pressures affect valve fluxes; the chamber pressures are determined alongside displacement under volume constraints.Source: §4.3 · pp. 26–27
  • The volume-constraint stabilization is reported to remove numerical oscillations and to be absolutely stable for any parameter choice and time step, without increasing computational cost.The scheme evaluates chamber pressures at the new time step while retaining circulation state variables from the previous step, preserving the staggered structure.Source: §4.3 · p. 27

5. Numerical Simulations and Discussion

The simulations assess the whole-heart model’s healthy-cycle behavior and test the roles of atrial contraction, fiber-stretch-rate feedback, and numerical stabilization. The baseline captures major chamber physiology, while the tests show how removing these features changes results or destabilizes the computation.

5.2. The baseline simulation

  • About 600 mL s−1 maximum semilunar-valve flux falls within the 500–600 mL s−1 range reported from PC-MRI.The authors attribute regulation of these fluxes partly to fiber-stretch-rate feedback in the RDQ20 model.Source: §5.2 · p. 31
  • The atrial pressure-volume loops are eight-shaped, with A- and V-loops similar in size, unlike the abnormally larger A-loops reported for some other models.The authors describe the loops as matching medical literature and note that this atrial behavior had not previously been shown so accurately computationally.Source: Figure 9 · p. 31, §5.2 · p. 32
  • The baseline reproduces physiological valve timing: right valves close after left valves, with near-synchronized atrioventricular closure and a longer semilunar-valve delay.The authors relate this behavior to the activation delay between the left and right ventricles.Source: Figure 10 · p. 32, §5.2 · p. 33
  • During ventricular passive filling, base motion increases ventricular volumes and decreases atrial volumes, with the atria acting as a passive conduit.During atrial contraction, the booster-pump function adds ventricular preload; during ventricular ejection, downward atrioventricular-plane motion drives ventricular emptying and atrial filling.Source: §5.2 · pp. 34–35

5.2.1. A quantitative analysis of volumetric indexes

  • All four chambers’ volumetric indexes fall within cardiac-MRI reference ranges, supporting the model’s ability to capture fundamental healthy-heart physiology.The paper describes this as an unusually comprehensive computational reproduction of cardiac function.Source: §5.2.1 · p. 36, Table 2 · p. 36

5.3. The impact of the atrial contraction and of the fiber-stretch-rate feedback

  • Without atrial contraction, the atrial A-loop disappears and ventricular preload decreases, producing a scenario the authors characterize as pathological rather than healthy.The comparison models the atria as passive tissue.Source: Figure 12 · p. 37, §5.3 · p. 38
  • Without fiber-stretch-rate feedback, semilunar fluxes rise to about 1200 mL s−1 in the AV and almost 1500 mL s−1 in the PV, despite little change in pressure-volume loops.The feedback locally reduces active force where fibers shorten rapidly, homogenizing shortening velocity and smoothing ejected blood flux.Source: §5.3 · p. 38, Figure 13 · p. 39

5.4. The need of the numerical stabilization

  • Removing active-stress stabilization causes non-physical oscillations and nonlinear-solver failure, while removing circulation-coupling stabilization leaves large, non-physical valve-flux oscillations.The authors report that reducing time step size does not resolve these instabilities; the stabilization terms remove them.Source: §5.4 · p. 40

6. Conclusions

The paper presents a stable, biophysically detailed electromechanical model of the whole human heart that includes active contraction in both atria and ventricles. Its simulations reproduce a broad set of healthy cardiac features, while comparisons show that atrial contraction, fiber-stretch-rate feedback, and numerical stabilization matter to the reported results.

6. Conclusions

  • The model combines an anatomically accurate four-chamber domain, myocardial fiber architecture, detailed electrophysiology and force generation, and a closed-loop circulation model.It uses chamber-specific laws and parameters and couples the core models through relevant feedbacks.Source: §6 · p. 40
  • Simulations capture atrial and ventricular physiology, atrial reservoir, conduit, and booster-pump functions, and atrioventricular interaction; all chamber volumetric indexes fall within physiological reference ranges.The reported features include eight-shaped atrial pressure-volume loops, atrial pressure waves, and three-dimensional cardiac deformation.Source: §6 · p. 43
  • Without atrial contraction, the model loses the atrial booster pump and ventricular preload; without fiber-stretch-rate feedback, semilunar fluxes exceed physiological levels.The authors link the latter feedback to reduced active force in rapidly shortening fibers and smoother ejected blood flux.Source: §6 · p. 43
  • Stabilization terms remove non-physical oscillations associated with the circulation model and fiber-stretch-rate feedback, supporting stable physiological simulations.The paper frames the interplay of accurate mathematical models and stable numerical methods as fundamental to reproducing heart physiology.Source: §6 · p. 43
  • The authors characterize the model as a fundamental step toward physics-based digital twins of the human heart.They report an unprecedented ability, to their knowledge, to reproduce healthy function in both atria and ventricles.Source: §6 · p. 44

Appendix A. Model and numerical parameters

Appendix A lists the parameters used for the baseline simulation, including electrophysiology, active force generation, mechanics, circulation, and numerical solvers.

Appendix A. Model and numerical parameters

  • The appendix reports parameters for the monodomain electrophysiology model and ionic models, using Courtemanche et al. for atria and ten Tusscher and Panfilov for ventricles.For the ventricular ionic model, the calcium peak is rescaled by 0.48.Source: Appendix A · p. 44, Table A.3 · p. 45
  • It lists chamber-specific RDQ20 active-force calibrations and crossbridge stiffness values, alongside passive-mechanics and reference-configuration recovery parameters.The atrial calibration is based on Mazhar et al.; modified active-force parameters are reported relative to Regazzoni et al.Source: Appendix A · p. 44, Table A.4 · p. 45, Table A.5 · p. 46
  • Circulation-model parameters and linear and nonlinear solver settings are provided in separate appendix tables.Source: Appendix A · p. 44, Table A.6 · p. 46, Table A.7 · p. 47
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