Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations

1. Introduction

• Fiber orientation shapes electrical propagation and mechanical contraction, motivating accurate fiber representation in heart simulations.The passages link myofiber orientation to signal propagation and contraction.Source: §1 · p. 1
• In-vivo DTI produces noisy, low-resolution reconstructions and cannot resolve atrial fibers, while atlas-based mapping requires complex registration and depends on the original atlas.The atrial wall is thinner than the DTI voxel size; atlas-based methods also depend on previously reconstructed atlas data.Source: §1 · p. 2
• RBMs assign orthotropic fibers from myocardial geometry, while LDRBMs use Laplace-solution gradients for transmural and apico-basal directions, then rotate them to produce smooth changes.The rotations are intended to match histological observations.Source: §1 · p. 2
• Most ventricular RBMs focus on the left ventricle and use an artificial basal plane; right-ventricle orientations differ, and septal fiber discontinuity's effects on propagation remained unstudied.The cited LDRBM includes region-specific directions, while the septal discontinuity remained debated.Source: §1 · p. 2
• Atrial RBMs often require manual work or target specific morphologies, and no atrial LDRBM existed because overlapping bundles make ventricular methods nontrivial to extend.A general automatic pipeline for atrial fibers remained difficult.Source: §1 · p. 2
• The paper reviews and compares ventricular LDRBMs, including right-ventricle directions and the effects of orientation and septal discontinuity in EP simulations.The original LDRBM by Doste et al. already includes right-ventricle directions.Source: §1 · pp. 2–3
• The proposed atrial LDRBM qualitatively represents important fiber bundles, works on idealized and realistic geometries, and is used in a realistic four-chamber EP simulation.The authors also analyze the effect of complex atrial architecture on signal propagation.Source: §1 · p. 3

2. Rule-Based-Methods for ventricular fibers generation

• The shared pipeline labels ventricular boundaries, computes transmural distance and direction, builds a local coordinate frame, then rotates it to define fiber, cross-fiber, and sheet directions.The transmural distance gradient supplies the transmural direction; the final rotations are chosen to match histology and DTI observations.Source: §2 · p. 3
• R-RBM uses an artificial basal plane and a manually selected right-septum boundary, while its normal direction is the outward normal to that plane.This makes its septum selection partly arbitrary and less suitable for full biventricular geometry.Source: §2 · pp. 4, 5, 6, 9
• B-RBM adds an apico-basal direction and uses bislerp to interpolate coordinate frames smoothly, particularly near the septum and ventricular junctions.Its frames are formed from separate endocardial and epicardial coordinate systems.Source: §2 · pp. 4, 8
• D-RBM supports full biventricular geometry without an artificial basal plane and accounts for regional orientations in the septum and outflow tracts.Its normal direction combines apico-basal and apico-outflow-tract directions using an interpolation function.Source: §2 · pp. 4, 5, 7, 8
• A common Laplace-Dirichlet problem underpins the methods, with boundary conditions chosen to compute distances between ventricular boundary partitions.The unknown and boundary data vary by method and pipeline step.Source: §2 · p. 4
• The methods rotate local axes by position-dependent angles interpolated between epicardial and endocardial values; B-RBM and D-RBM also accommodate septal rotations, and D-RBM allows outflow-tract rotations.The angles are selected to match histological observations.Source: §2 · pp. 8–9
• The authors add a Laplace field that separates left from right ventricles for distinct rotations, and extend R-RBM’s output to include cross-fiber and sheet directions.Positive and negative values of ξ identify the left and right ventricles, respectively.Source: §2 · pp. 8–9
• B-RBM and D-RBM apply to full or based biventricular geometries, whereas R-RBM is less suitable for full geometry because its ring-normal direction is not strictly identified.R-RBM’s right-septum definition can also be arbitrary in a full biventricular geometry.Source: §2 · p. 9

• The atrial LDRBM combines gradients of several harmonic functions to represent differently oriented atrial bundles, since extending ventricular LDRBMs is not straightforward.It is proposed as a method that qualitatively reproduces important features, including fiber bundles.Source: §3 · p. 9
• The four-step workflow labels atrial boundaries, solves for transmural and intra-atrial distances, selects bundles, and builds a local coordinate system.Source: §3 · pp. 11, 13
• For each atrial point, bundle-selection rules assign a unique intra-atrial distance whose gradient defines the local normal direction.Bundle dimensions are parameterized separately for right- and left-atrial regions.Source: §3 · pp. 12–13
• The local coordinate system uses the transmural-distance gradient and the bundle-specific direction; without prescribed transmural variation, its axes are the final fiber, sheet, and cross-fiber directions.Source: §3 · p. 13
• Circular fibers encircle the pulmonary and caval veins, valve rings, and appendages; the isthmus fibers follow the tricuspid-valve direction.Source: §3 · p. 13
• The rules orient crista-terminalis fibers from the superior to inferior caval vein, while left roof and lateral-wall fibers descend perpendicularly to the mitral valve.Left septal-wall fibers transition smoothly toward the left septum and appendage.Source: §3 · p. 13

• Cardiac tissue is modeled as orthotropic, with conductivities specified along fiber, sheet, and cross-fiber directions.Source: §4 · p. 13
• The monodomain system evolves transmembrane potential and gating variables, using the CRN ionic model for atria and TTP for ventricles.The gating variables represent open-channel percentages and satisfy differential-algebraic equations.Source: §4 · p. 14
• Time stepping uses BDF3 with implicit diffusion and explicit ionic terms; spatial discretization uses continuous finite elements and Ionic Current Interpolation.Source: §4 · p. 14

5. Numerical results

• The planned studies compare ventricular LDRBMs, test atrial fiber reconstruction, assess atrial-fiber activation against isotropic conductivity, and simulate a realistic four-chamber heart.The realistic geometry is the Zygote model, a complete heart reconstructed from high-resolution CT scans and representing an average healthy heart.Source: §5 · p. 14

5.1. Setting of numerical simulations

• The preprocessing uses a semi-automatic meshing tool, but the authors state their tagging procedure is compatible with other tagging processes.Source: §5.1 · pp. 14–15
• Conductivities were fitted to target ventricular velocities of 60, 40, and 20 cm/s along fiber, sheet, and normal directions, and atrial velocities of 120 cm/s along fibers and 40 cm/s along sheet and cross-fiber directions.Source: §5.1 · p. 15
• Simulations used Q1 hexahedral finite elements with average mesh size h = 350 µm and BDF order 3 with time step Δt = 50 µs.Source: §5.1 · p. 15
• The ventricular and atrial LDRBMs and EP methods were implemented in the lifex finite-element library.Source: §5.1 · p. 15

5.2.1.1 Fiber generation in the idealized ventricles

• All three methods reproduce the left-ventricular helix, but their main idealized fiber-field differences occur near the septum, ventricular junctions, and lower right epicardium.The comparison quantifies fiber-field difference as 1 − |f_i · f_j|, with 0 for parallel and 1 for orthogonal fields.Source: Figure 7 · p. 16, §5.2.1.1 · pp. 16–17

5.2.1.2 Electrophysiology in the idealized ventricles

• 15 ms is the maximum B-RBM–D-RBM activation difference in the idealized model, versus 35 ms for R-RBM–B-RBM and 33 ms for R-RBM–D-RBM.The larger R-RBM comparisons differ most in the septum, especially toward the right endocardium.Source: Figure 8 · p. 17, §5.2.1.2 · p. 18

5.2.2. Realistic full biventricular model

• The realistic comparison uses the Zygote full biventricular geometry and compares B-RBM with D-RBM; papillary muscles and trabeculae carneae were removed from the model.The idealized biventricular geometry was built from intersecting truncated ellipsoids using prolate spheroid coordinates.Source: §5.2.1 · p. 15, §5.2.2 · p. 18

5.2.2.2 Electrophysiology in the realistic ventricles

• 14 ms (10% of the 140 ms total activation time) is the maximum B-RBM–D-RBM activation discrepancy in the realistic ventricles.It occurs in the lower endocardium above the right ventricular apex.Source: Figure 9 · p. 19, §5.2.2.2 · p. 19

5.2.2.1 Fibers generation in the realistic ventricles

• In the realistic model, both methods reproduce the left-ventricular helix to the mitral ring, but B-RBM is smooth across ventricular junctions while D-RBM is discontinuous.Their fiber fields also differ around the valve rings, septum, and right-ventricular apex.Source: §5.2.2.1 · p. 18, Figure 9 · p. 19

5.3. Atrial fibers generation and electrophysiology

• The atrial fiber-generation method was tested on idealized geometries and realistic Zygote and Riunet atria, with electrophysiology simulations performed on realistic geometries.The study also varies the size of the mitral-valve bundle to examine its effect on activation.Source: §5.3 · p. 20, §5.3.1 · pp. 20–21

5.3.1. Atrial fibers generation

• The atrial LDRBM qualitatively captures fiber arrangements in principal regions of both atria, including the caval veins, valve rings, appendages, septa, and atrial walls.Its rules produce circular arrangements around major veins and valve rings, with region-specific orientations in structures such as the crista terminalis and left atrial roof.Source: §5.3.1 · p. 21

5.3.2. Atrial electrophysiology

• Atrial LDRBM and isotropic simulations show significant differences in activation patterns and times, concentrated near the pulmonary veins and left appendage in LA and the right appendage and tricuspid valve in RA.The simulations use an SAN stimulus for RA and stimuli at the Bachmann’s Bundle, Fossa Ovalis, and Coronary Sinus Musculature for LA.Source: §5.3.2 · pp. 21–22
• Changing the mitral-valve bundle parameter from τmv=0.85 to τmv=0.65 thickens the bundle and produces a 28 ms maximum activation-time discrepancy in LAA.The discrepancy is 24% of the 116 ms total LA activation time.Source: §5.3.2 · p. 23

5.4.1. Whole heart fibers

• The whole-heart fiber field uses D-RBM for the ventricles and the proposed atrial LDRBM for both atria in the Zygote heart model.The ventricular and atrial transmembrane potentials are shown through a cardiac cycle in Figure 14.Source: §5.4.1 · p. 23, Figure 14 · p. 24

5.4.2. Whole heart electrophysiology

• The simulation models conduction-system connections as delayed spherical stimuli, with an AVN delay and multiple ventricular endocardial stimuli instead of a detailed Purkinje network.The prescribed sequence starts at SAN, proceeds through three inter-atrial connections, and then activates the ventricles.Source: §5.4.2 · pp. 24–25
• Atrial depolarization completes after about 120 ms and ventricular depolarization after about 270 ms; the last activated regions are LAA and the right-ventricular postero-basal area, respectively.The whole-heart activation maps are shown in Figure 13(c).Source: §5.4.2 · p. 25
• The simulated signal begins at SAN, spreads from right to left atrium, and reaches the ventricles after the AVN delay; atrial repolarization occurs during ventricular depolarization.The transmembrane-potential evolution is shown in Figure 14.Source: Figure 14 · p. 24, §5.4.2 · p. 25

6.1. Comparison among the three ventricular LDRBM

• R-RBM differs locally from B-RBM and D-RBM, and the methods’ septal definitions leave the anatomical ground truth for fiber continuity unresolved.B-RBM and D-RBM better describe right-endocardial fibers according to cited anatomical observations, while D-RBM has a strong discontinuity at the interventricular junction; further investigation is needed.Source: §6.1 · p. 25
• 28–29% total activation-time discrepancy separates R-RBM from B-RBM and D-RBM; B-RBM and D-RBM differ by only 10%.The differences are concentrated at the septum, especially the right endocardium; B-RBM's extensions yield an activation pattern close to D-RBM.Source: §6.1 · p. 25
• Septal activation differences may affect mechanical contraction, a consequence the authors say should be investigated in electromechanical simulations.They also identify right-ventricular orientations, junctional discontinuity, and outflow-tract orientations as features to investigate.Source: §6.1 · p. 25

6.2. Comparison of atrial fibers with pre-existing results and anatomical data

• The atrial LDRBM uses harmonic-problem distance functions and defined boundary sections; bundle dimensions remain adjustable through algorithm parameters.Unlike many existing atrial RBMs, the method is described as applicable to arbitrary geometries without manual or semi-automatic intervention.Source: §6.2 · p. 26
• Atrial LDRBM fibers agree closely with RBM and anatomical dissections, with localized differences at the Bachmann’s Bundle, Fossa Ovalis, and pectinate-muscle regions.The method approximates pectinate muscles by prescribing right atrial endocardial fibers perpendicular to the Crista Terminalis.Source: Figure 15 · p. 26, §6.2 · p. 27, Figure 16 · p. 27
• Atrial LDRBM fibers produce activation timings of 108 ms in RA and 116 ms in LA, compatible with cited prior predictions.The fiber-based activation pattern and timing differ consistently from the isotropic model.Source: §6.2 · p. 27

6.3. The whole heart case

• The realistic four-chamber simulation gives complete atrial depolarization at about 120 ms and ventricular activation at about 270 ms.The last activated regions are the LAA and the right-ventricular postero-basal area, respectively, in agreement with cited reports.Source: §6.3 · pp. 27–28

6.4. Limitations

• Atrial simplifications include surrogate rather than explicit inter-atrial connections and pectinate muscles; EP properties were also modeled homogeneously despite distinct atrial regions.The authors note that distinct atrial EP properties matter for realistic activation and repolarization, particularly in pathological conditions.Source: §6.4 · p. 28

7. Conclusions

• The unified framework generated cardiac fibers across a full-heart domain and supported physically meaningful EP simulations in a realistic four-chamber heart.The final simulation included LDRBM-generated fibers in both atria and ventricles.Source: §7 · p. 28
• Ventricular LDRBMs were given a communal mathematical description and improved for right-ventricular fiber orientations; EP comparisons highlighted the importance of proper right-ventricular orientation.The comparisons used numerical EP simulations.Source: §7 · p. 28
• The novel atrial LDRBM captured complex fiber directions across almost all anatomical atrial regions in comparisons with another RBM and anatomical dissections.It was applied to idealized and realistic geometries.Source: §7 · p. 28
• Realistic-geometry EP simulations showed a strong effect of atrial fiber-bundle architecture on electric-signal propagation.Source: §7 · p. 28

On the choice of physical parameters and numerical settings

• Conductivities were fitted to target ventricular velocities of 60, 40, and 20 cm/s along fiber, sheet, and normal directions, and atrial velocities of 120 cm/s along fiber and 40 cm/s along sheet and cross-fiber directions.The fitting procedure is described as iterative.Source: On the choice of physical parameters… · pp. 29–30
• The simulations used Q1 hexahedral finite elements with average mesh size h = 350 µm and BDF3 time stepping with ∆t = 50 µs.The passage calls the mesh size acceptable for linear finite-element approximation in physiological cases.Source: On the choice of physical parameters… · p. 30
• BDF3 with ∆t = 50 µs was compared against BDF1 with ∆t = 10 µs on a slab benchmark; the authors state BDF3 allows a larger step for the same accuracy.The passage identifies BDF1 as the more commonly used monodomain time discretization.Source: On the choice of physical parameters… · p. 30

Tagging procedure for the atrial and ventricular LDRBMs

• Atrial tagging extracts endocardium and epicardium, then connects border-zone points to tag pulmonary and caval veins, coronary sinus, and valve rings.In the right atrium, a manually defined top band connects the upper SCV and ICV elements.Source: Tagging procedure for the atrial and… · p. 31
• The right atrial tricuspid ring is clipped by a plane through the top band into septal-facing and free-wall portions.The two portions are Γtv−s and Γtv−f.Source: Tagging procedure for the atrial and… · p. 31
• Ventricular tagging extracts epicardium and both endocardia; R-RBM additionally divides the right endocardium using a distance threshold.Based-geometry tagging adds an upper basal plane, while complete biventricular tagging defines four valve rings by distance thresholds from corresponding atrial rings or outflow-tract roots.Source: Tagging procedure for the atrial and… · p. 31