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REVIEW 4 major objections 5 minor 87 references

Coupled Eikonal problems to model cardiac reentries in Purkinje network and myocardium

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A partitioned Eikonal scheme with a new pseudo-time solver can reproduce electrical reentries between the Purkinje network and the cardiac muscle in pathological states.

desk verdict The capability this paper targets is real and worth pursuing, but Equation (6) as written appears to undercut its own active-set logic, so the central numerical claim needs a fix before I would trust the reentry results. read the letter →

arxiv 2412.13837 v1 pith:XT6PADHL submitted 2024-12-18 math.NA cs.NA

Samuele Brunati , Michele Bucelli , Roberto Piersanti , Luca Dede' , Christian Vergara This is my paper · ORCID
classification math.NAcs.NA MSC 92C3065M6065N30
keywords cardiacelectrophysiologyPurkinjenetworkelectricalreentriesEikonalequationEikonal-diffusionpseudo-timemethodPurkinje-musclejunctionsbundlebranchblock
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that Eikonal-based models of cardiac electrical activation can reproduce reentrant circuits in which a wavefront leaves the Purkinje network, enters the heart muscle, and later re-enters the network at a different junction, something previous Eikonal coupling algorithms could not do. The authors build a partitioned scheme that solves a one-dimensional Eikonal problem on the Purkinje fibers and a three-dimensional Eikonal-diffusion problem in the myocardium, exchanging activation times at Purkinje-muscle junctions. The enabling device is a new pseudo-time method for the muscle problem that enforces only the stimuli still active when the wavefront arrives, so redundant junctions do not distort the activation map. The central claim, tested in a real biventricular geometry, is that this scheme captures all possible reentries under pathological and therapeutic conditions such as left bundle branch block and cardiac resynchronization therapy. If the claim holds, modelers get a computationally cheap tool for studying reentrant arrhythmias that involve the cardiac conduction system.

What carries the argument

The load-bearing mechanism is the active-stimulus pseudo-time iteration in Eqs. (5)-(6) together with Algorithm 1. A Purkinje-muscle junction (PMJ) is the terminal point where the one-dimensional Purkinje network connects electrically to the three-dimensional myocardium; Algorithm 2 labels each PMJ as orthodromic, antidromic, or collision by comparing activation times and the two propagation delays. Algorithm 1 repeatedly solves the 1D network Eikonal problem with Fast Marching and the 3D muscle Eikonal-diffusion problem, each time feeding the other the activation times at the PMJs. The pseudo-time method makes this iteration safe: by imposing Dirichlet conditions only on active stimuli, it lets the muscle ignore a junction that would be activated after the wavefront has already passed, so the alternating solves see the true earliest-arrival structure.

What would settle it

Run the same LBBB and CRT simulations with $N_{\max}$ increased beyond 3, say $N_{\max}=4,5,6$, and compare the counts of antidromic, orthodromic-from-antidromic, and collision PMJs together with total activation time; if new reentries appear or the classification changes, the 'all reentries' claim is false. A sharper test would compare the predicted reentry sites and timings against a Monodomain or Bidomain reference solution on the same geometry.

Watch

Extended reading notes

Core claim

The core discovery is that bidirectional Purkinje-muscle propagation, and therefore reentry, can be obtained from two Eikonal solves connected by an iterative exchange of junction information, provided the muscle problem uses a pseudo-time iteration that selectively enforces stimuli. In the proposed method, at each pseudo-time step the Dirichlet set $S^{n+1}$ contains only those junctions whose prescribed activation time satisfies $u^0_m(x_i) < t_{n+1}$ and $u^0_m(x_i) < u^n(x_i)$; all other junctions are left free, which removes the non-physical delays that a classical pseudo-time solver would produce with redundant stimuli. The coupling classifies each junction as orthodromic, antidromic, or collision, and alternates the network and muscle solves for a fixed number of iterations. In the reported simulations this produces antidromic activation of the left Purkinje network under left bundle branch block and lets the faster Purkinje conduction re-emerge orthodromically from other junctions, a reentry pattern that the earlier algorithm in [6] could not represent.

Load-bearing premise

The load-bearing premise is that the active-stimulus criterion in Eq. (6) correctly identifies the stimuli the muscle should obey and that the pseudo-time iteration converges to the physically right activation pattern, with three coupling iterations enough to reveal every reentry; the paper leaves the continuous limit of this iteration and a convergence test for future work.

Editorial extensions

If this is right

  • With the proposed scheme, the left bundle branch block simulation shows antidromic PMJs near the septum and many orthodromic PMJs activated by antidromic propagation, reproducing a reentry that the single-pass algorithm of [6] could not show.
  • The new pseudo-time method discards inactive CRT stimuli: in the resynchronization scenario, a right-ventricular lead firing after the orthodromic wavefront passes produces no effect, while the left-ventricular lead changes the activation of the free wall.
  • The 531 PMJs can be summarized by four counts (orthodromic from AV node, orthodromic from antidromic propagation, antidromic, collision), giving a compact diagnostic signature for each scenario.
  • Because the two Eikonal problems exchange only activation times, the Purkinje network model can be swapped for a patient-specific one without altering the muscle solver.
  • With $N_{\max}=3$ iterations the coupled algorithm completes a biventricular simulation in under two hours, so reentry-capable Eikonal modeling remains computationally light.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same active-stimulus idea could be exported to any earliest-arrival problem with competing point sources, such as multi-site pacing or ectopic foci, where a later prescribed source should be suppressed by an earlier wavefront.
  • The paper fixes $N_{\max}=3$ without a convergence test, so the 'all reentries' claim is only as strong as that choice; checking whether junction counts or activation times change for $N_{\max}=4,5,\dots$ would turn the claim into a quantitative one.
  • A natural next step is to compare these Eikonal predictions with a Monodomain or Bidomain solution of the same scenarios, since those models handle bidirectional propagation intrinsically and would reveal any timing or location error introduced by the Eikonal approximation.
  • The collision-PMJ category may be more than bookkeeping: junctions where muscular and network fronts meet are exactly the sites where a reentrant circuit could anchor, so the counts might serve as a predictor of arrhythmia susceptibility.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a partitioned scheme coupling a 1D Eikonal model of the Purkinje network with a 3D Eikonal-diffusion model of the myocardium, introduces a pseudo-time method whose Dirichlet set is restricted to stimuli classified as active (Eq. 6), and iterates network and muscle solves to account for orthodromic and antidromic propagation and reentries (Algorithm 1). The method is demonstrated on a real biventricular geometry in four scenarios: healthy, WPW syndrome, complete LBBB, and CRT pacing. The authors claim that this is the first Eikonal-based model able to capture all possible signal reentries between the Purkinje network and the myocardium under pathological conditions.

Significance. If the approach is made rigorous, it would be a useful contribution: the partitioned architecture is modular, the pseudo-time active-set idea is novel, and the simulations are carried out in a realistic, publicly derived geometry with full anisotropic fiber fields and a rule-based Purkinje network. The reported wall-clock times under two hours for the full coupled problem are a genuine practical strength. However, the central methodological novelty is currently supported only by heuristic reasoning: the active-set criterion in Eq. (6) appears self-defeating for stimuli that should remain enforced, and no convergence or validation evidence is provided for the claim of capturing all possible reentries. The paper's value hinges on fixing these load-bearing points.

major comments (4)
  1. [Section 2.2.2, Eq. (6)] The active-stimulus criterion xi ∈ S^{n+1} ⇐⇒ u0_m(xi) < t^{n+1} and u0_m(xi) < u^n(xi) cannot keep a genuinely needed stimulus active: if xi ∈ S^n, the Dirichlet condition in (5) forces u^n(xi)=u0_m(xi), so the strict inequality u0_m(xi) < u^n(xi) is false and xi is removed at the next pseudo-time step. For an earliest source at that point, the unconstrained solution will then drift above u0_m(xi), reactivating xi, and nothing in Section 2.2.2 or Figure 3 explains why this two-cycle does not occur. Since Algorithm 1 relies on this pseudo-time solver to discard redundant stimuli and the text explicitly disclaims a rigorous continuous limit, this is a load-bearing gap. Please correct the criterion (e.g., non-strict comparison, a latch for already-enforced stimuli, or evaluation of u^n at xi without the Dirichlet constraint) and demonstrate that the active set becomes eventually constant on the reported examples.
  2. [Algorithm 1 and Section 3.3] The iteration count Nmax=3 is fixed with no stopping criterion and no sensitivity study; Section 3.3 only says the choice balances reliability with cost. The paper's central claim that the algorithm captures 'all possible signal reentries' requires either a convergence criterion, a demonstration that the PMJ classification and activation maps do not change when Nmax is increased, or a precise characterization of the class of reentries for which three iterations are sufficient. Without one of these, the reported reentries in T-WPW, T-LBBB, and T-CRT are outputs of an unverified finite iteration.
  3. [Section 4] The Abstract and Section 5 claim 'accuracy' and 'robustness', but all four scenarios are qualitative demonstrations: no comparison is made against a Monodomain or Bidomain reference, against the earlier Eikonal-Eikonal coupling of [6], or against clinical measurements. Since the active-set pseudo-time method deliberately produces a solution inconsistent with the original problem (2), a quantitative benchmark (e.g., activation times, PMJ classification, or isochrones against a reference model) is needed to support the accuracy claim. This is a load-bearing omission for the stated contributions.
  4. [Abstract and Section 5] The phrase 'all possible signal reentries' overstates what is demonstrated: only four scenarios on a single geometry are shown, and the algorithm's ability to find reentries depends on the untested active-set dynamics and the fixed Nmax. Please either soften the claim to 'the considered reentry mechanisms' or provide an exhaustive or formal argument for the 'all possible' wording.
minor comments (5)
  1. [Section 4.1] The sentence 'The latest activation time, recorded in Table 4, is slightly elevated' refers to total activation time, which is reported in Table 3, not Table 4.
  2. [Equation (5)] The BDF term is typeset ambiguously: 'αBDFun+1 − un BDFσ / ∆t' should be written as (α_BDF u^{n+1} − u^n_{BDFσ})/Δt to make the denominator clear.
  3. [Algorithm 1] The symbol M is overloaded: it denotes the collection of muscular sources in the input line and the number of antidromic PMJs in line 9; please use distinct names.
  4. [Algorithm 1] The comment on line 6 says 'collsion PMJs' instead of 'collision PMJs'.
  5. [Section 2.1.2] The statement that Fast Marching reduces to Dijkstra's algorithm in 1D is correct but would benefit from a brief explanation of how the FMM update handles multiple sources, since this behavior is later used for disregarding redundant stimuli.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the coupling scheme is a forward simulation with literature-derived parameters, and the reentry results are not fitted outputs; the active-set criterion in Eq. (6) is a formal consistency concern but not an input-to-output circularity.

full rationale

The derivation chain is not circular. The physical parameters in Table 2 (conductivities, conduction velocities, PMJ delays) are taken from prior literature ([14], [7], [71]) rather than fitted to the presented simulations, and the reentry scenarios in Section 4 are forward outputs of Algorithm 1, not quantities used to calibrate the model. Self-citations to [6], [7], [14], and [48] provide physiological constants and the baseline coupling idea, but the new partitioned scheme and pseudo-time solver are formulated explicitly in Section 2 with their own equations, so no load-bearing claim reduces to a self-citation. The paper itself flags limitations that are relevant to correctness but not to circularity: Section 2.2.2 states that 'A rigorous derivation of the pseudo-time-continuous counterpart to (5) is not addressed in this paper' and that the solution of (5) is 'deliberately not consistent' with (2), and Section 3.3 fixes Nmax=3 iterations without a convergence safeguard. One formal concern is that Eq. (6) defines activity by the strict inequality u0_m(x_i) < u^n(x_i), while Eq. (5) sets u^{n+1}(x_i)=u0_m(x_i) on the active set; once enforced, the strict inequality fails at the next pseudo-time step, so the active set may oscillate rather than converge. This is a well-posedness and robustness issue in the proposed solver, not a case of a predicted quantity being equivalent to a fitted input or a self-citation chain, and it does not change the circularity verdict.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard Eikonal modeling assumptions and on two ad-hoc-to-paper assumptions: the correctness of the active-set pseudo-time solver and the sufficiency of the fixed iteration count. No new physical entities are introduced. The model parameters are not fitted to the presented results.

free parameters (2)
  • Nmax = 3
    Maximum number of coupling iterations in Algorithm 1, chosen by hand 'to balance reliable results with reduced computational time'; no convergence criterion or sensitivity study.
  • Pseudo-time discretization parameters = not specified
    The pseudo-time method (Eq. 5) requires a step size dt and BDF order sigma, but these are not reported, making results dependent on unreported solver choices.
assumptions (5)
  • domain assumption Eikonal equation is a valid approximation of cardiac activation times in the Purkinje network (Eq. 1)
    Standard in cardiac modeling; accepted from prior literature ([6,7]).
  • domain assumption Eikonal-diffusion equation (Eq. 2) is a valid model for myocardial activation
    Standard model; parameters taken from [14,52].
  • domain assumption PMJ delays are constant and known (do=10ms, da=2ms)
    From literature [7]; the network and muscle activation times at PMJs are related through these fixed delays.
  • ad hoc to paper The active-set criterion (Eq. 6) yields a convergent, physically meaningful solution of the pseudo-time problem
    No proof is provided; the paper states that a rigorous derivation of the pseudo-time-continuous counterpart is not addressed.
  • ad hoc to paper Nmax=3 iterations of Algorithm 1 are sufficient to capture all reentries
    Set by hand; no convergence study or stopping criterion is given.

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Cite this review

Pith. "Pith review of Coupled Eikonal problems to model cardiac reentries in Purkinje network and myocardium." pith.science (2026). https://pith.science/paper/XT6PADHL

@misc{pith2026241213837,
  author       = {Pith},
  title        = {Pith review of: Coupled Eikonal problems to model cardiac reentries in Purkinje network and myocardium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XT6PADHL}},
  note         = {Machine review of arXiv:2412.13837}
}
read the original abstract

We propose a novel partitioned scheme based on Eikonal equations to model the coupled propagation of the electrical signal in the His-Purkinje system and in the myocardium for cardiac electrophysiology. This scheme allows, for the first time in Eikonal-based modeling, to capture all possible signal reentries between the Purkinje network and the cardiac muscle that may occur under pathological conditions. As part of the proposed scheme, we introduce a new pseudo-time method for the Eikonal-diffusion problem in the myocardium, to correctly enforce electrical stimuli coming from the Purkinje network. We test our approach by performing numerical simulations of cardiac electrophysiology in a real biventricular geometry, under both pathological and therapeutic conditions, to demonstrate its flexibility, robustness, and accuracy.

Figures

Figures reproduced from arXiv: 2412.13837 by the authors.

Figure 1
Figure 1. Left: representation of the components of the cardiac conduction system (picture adapted from [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Ventricular myocardial domain Ωmus (left) with three sources at S0 and corresponding Purkinje domain Ωp (right) with a source at Γ0. 2.2. Electrical activation of the myocardium 2.2.1. Continuous problem The propagation of the electrical signal in the myocardium Ωmus ⊂ R 3 , see [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Numerical solution returned by the novel pseudo-time method in a one-dimensional example, compared to the classic [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Purkinje-muscle junctions (PMJs) classification in presence of opposite wavefronts propagating between the Purkinje [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: (a) Real biventricular geometry reconstructed from four-chamber heart of a patient [ [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Comparative view of the four scenarios: T-H, T-WPW, T-LBBB, T-CRT. First column: simulation setup; second [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Graphical visualization of the Purkinje-muscle junctions (PMJs) classification in the four test cases. PMJs are [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Comparison between T-H (a) and T-WPW (b). The activation time scale was saturated in the range [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: A zoom on the reentries of the electrical signal in the muscle in T-LBBB (a) and T-CRT (b): the muscular wavefront [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.