{"id":"0acbeabc-e085-4214-b45d-4e63c118bcb8","arxiv_id":"2412.13837","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A bidirectional Eikonal coupling scheme with a pseudo-time solver now models Purkinje-muscle reentries in cardiac electrophysiology, tested on a biventricular geometry in healthy and pathological cases.","lead":"This paper introduces an algorithm that lets two simplified models of heart electrical activation (one for the fast Purkinje fibers, one for the muscle) exchange signals in both directions, capturing abnormal reentry loops that earlier Eikonal-based models missed. It also adds a new pseudo-time solver that ignores redundant stimuli, and demonstrates the approach on a realistic heart geometry in four clinical scenarios.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (6) as written cannot keep a genuinely needed stimulus active: once enforced, u^n(x_i)=u0_m(x_i), so the strict inequality fails and the stimulus is dropped; nothing prevents active-set oscillation.","rationale":"The reader's weakest assumption already pointed at the active-stimulus criterion in Eq. (6) and the lack of convergence analysis. My concern sharpens that assumption into a concrete internal difficulty: strict inequality makes the criterion unable to keep an already-enforced source active, so the pseudo-time iteration may not converge at all. This is not a disagreement with the clinical consensus or a request for a different model; it is a correctness risk in the enabling algorithm. The paper deserves credit for the implemented lifex framework, the real biventricular geometry, and the four scenario simulations, which make the approach plausible and the numerical results qualitatively informative. However, the central claim of capturing all possible reentries rests on the very mechanism whose written form appears unstable, and the explicit disclaimer in Section 2.2.2 leaves that mechanism without a theoretical safety net. The fixed Nmax=3 in Algorithm 1 is a secondary concern: even if the active set stabilizes, three iterations are not shown to be sufficient. My recommendation is unchanged from the reader: the appropriate verdict is CONDITIONAL pending either a corrected and stable active-set formulation with empirical demonstration of eventual active-set constancy, or a rigorous analysis of the pseudo-time iteration. Should the proposed test reveal actual oscillation, the verdict would need to move to REJECT, but that is not established by the text alone.","tokens_in":17816,"tokens_out":9908,"duration_ms":102753,"concrete_test":"Instrument the active set S^n at every pseudo-time step in the 1D two-stimulus example of Figure 3 and in the T-H and T-LBBB scenarios, with Δt=0.10, 0.05, and 0.01 ms. Record whether any enforced source ever fails (6) immediately after being enforced, and whether S^n is eventually constant. If S^n oscillates or depends on Δt, the pseudo-time method has no well-defined steady state and the reentry maps of Section 4 are not credible; if S^n stabilizes under the strict inequality as printed, inspect the implementation—it likely uses ≤ or some latching rule, in which case Eq. (6) in the paper is misstated and should be corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central novelty is the pseudo-time solver (5) with the active-set criterion (6), which Algorithm 1 uses to decide which PMJ stimuli enter the muscle solve. As written, (6) is self-defeating for any stimulus that should remain active. If x_i was in S^n, the Dirichlet condition was imposed and u^n(x_i)=u0_m(x_i). At the next pseudo-time step, the condition u0_m(x_i)<u^n(x_i) is false, so x_i is removed from S^{n+1}. If x_i is the earliest source at that point, the free solution will drift above u0_m(x_i), making x_i active again. Nothing in Section 2.2.2 or Figure 3 explains why this two-cycle does not occur. The text explicitly disclaims a rigorous continuous limit ('A rigorous derivation of the pseudo-time-continuous counterpart to (5) is not addressed in this paper'), and Algorithm 1 uses a fixed Nmax=3 with no convergence safeguard. Since the reentries reported in Section 4 are outputs of this iteration, the central claim depends on the active set having a well-defined, stable steady state. The paper needs either a corrected inequality (e.g. non-strict comparison or a latch for already-enforced stimuli), an explanation of how u^n(x_i) is computed without the Dirichlet constraint, or a demonstration that membership of S^n becomes eventually constant.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1380,"tokens_out":1612,"duration_ms":48688,"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":[{"comment":"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.","section":"Section 2.2.2, Eq. (6)"},{"comment":"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.","section":"Algorithm 1 and Section 3.3"},{"comment":"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.","section":"Section 4"},{"comment":"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.","section":"Abstract and Section 5"}],"minor_comments":[{"comment":"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.","section":"Section 4.1"},{"comment":"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.","section":"Equation (5)"},{"comment":"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.","section":"Algorithm 1"},{"comment":"The comment on line 6 says 'collsion PMJs' instead of 'collision PMJs'.","section":"Algorithm 1"},{"comment":"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.","section":"Section 2.1.2"}],"recommendation":"major_revision","confidential_remarks":"The 'first time' claim in the abstract should be checked against the reaction-Eikonal and monolithic Purkinje-coupling literature cited in the introduction before publication; the main technical issue, however, is the active-set criterion in Eq. (6), which appears internally inconsistent as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: this paper promises a genuinely useful capability—bidirectional Purkinje-muscle reentries in an Eikonal framework—and the partitioned algorithm is a natural extension of Palamara et al. But the stress-test about Eq. (6) lands. As written, the active-set criterion drops any stimulus that was enforced at the previous pseudo-time step, because u^n(x_i) equals u0_m(x_i) on the Dirichlet set. The strict inequality u0_m(x_i) < u^n(x_i) is then false, so the stimulus is removed. Nothing in Section 2.2.2 or Figure 3 explains why the set doesn't oscillate. This is load-bearing: in the healthy T-H case the muscle has no other source, so if all PMJs are dropped, the activation time is never set. Either the inequality is a typo (should be non-strict, or the comparison should use an unconstrained solution), or the implementation does something the paper doesn't say. As published, the method does not work as specified.\n\nWhat's good: the paper is clear about what it does, extends [6] in a sensible direction, introduces a collision-PMJ classification that makes sense, and demonstrates the framework on four clinically relevant scenarios in a realistic geometry. The writing is honest—the authors explicitly flag the missing continuous derivation and the fixed Nmax=3.\n\nThe other soft spots are secondary: no comparison against a Monodomain or the original [6] algorithm, no convergence study, and parameters taken from literature. Those are typical for a methods paper and can be addressed with a benchmark and a sensitivity study.\n\nWho it's for: researchers in Eikonal-based cardiac electrophysiology who want to add Purkinje-network reentries to their models. The idea is worth engaging with. But the paper needs a corrected active-set rule and at least a sanity check on a simple 1D or 2D case showing the active set stabilizes. I'd send it to a serious referee, but only after the authors fix Eq. (6); if the fix is as simple as a non-strict inequality or a latch, the paper could become solid.\n\nBest,\n[You]","headline":"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.","tokens_in":18617,"tokens_out":3144,"would_cite":false,"duration_ms":30378,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["92C30","65M60","65N30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["cardiac electrophysiology","Purkinje network","electrical reentries","Eikonal equation","Eikonal-diffusion equation","pseudo-time method","Purkinje-muscle junctions","bundle branch block"],"falsifier":"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.","tokens_in":17577,"feed_emoji":"🫀","tokens_out":10212,"duration_ms":80067,"temperature":0.7,"pith_summary":"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.","feed_headline":"New Eikonal coupling captures all Purkinje-muscle reentries","feed_subtitle":"A pseudo-time solver plus iterative junction updates lets waves bounce back from muscle to network.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The earlier Eikonal-Eikonal Purkinje-muscle coupling algorithm that this work extends; it is the baseline that cannot capture network-to-muscle-to-network reentries.","marker":"[6]"},{"why":"Provides the coupled 3D-1D Monodomain Purkinje-muscle model and the PMJ delay values (10-15 ms orthodromic, 2-3 ms antidromic) used in the simulations.","marker":"[7]"},{"why":"One of the sources for the Eikonal-diffusion model of myocardial activation used in problem (2).","marker":"[13]"},{"why":"Supplies the conductivity values and rule-based myofiber architecture used to set the anisotropic muscle propagation.","marker":"[14]"},{"why":"Supplies the rule-based fractal-tree algorithm used to generate the Purkinje network on the endocardial surface.","marker":"[41]"},{"why":"Introduces the Eikonal-diffusion equation and the classical pseudo-time approach that the new method modifies.","marker":"[52]"},{"why":"Introduces the Fast Marching method used to solve the one-dimensional Purkinje Eikonal problem.","marker":"[53]"},{"why":"Provides the publicly available four-chamber heart geometry from which the biventricular mesh is extracted.","marker":"[62]"}],"fun_headline_variants":["Bidirectional Eikonal captures all cardiac reentries","Pseudo-time Eikonal makes reentry modeling complete","Coupled Eikonal scheme ends reentry blind spots","Eikonal solver now models muscle-to-network echoes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Bidirectional Eikonal captures all cardiac reentries","Pseudo-time Eikonal makes reentry modeling complete","Coupled Eikonal scheme ends reentry blind spots","Eikonal solver now models muscle-to-network echoes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000178,"raw_usage":{"total_tokens":1275,"prompt_tokens":900,"completion_tokens":375,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":310}},"tokens_in":516,"tokens_out":375,"duration_ms":3646,"temperature":1.0,"reasoning_tokens":310,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:44:44.256925+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Quarteroni, T","cited_arxiv_id":null,"evidence_quote":"One of the sources for the Eikonal-diffusion model of myocardial activation used in problem (2)."},{"cited_title":"Piersanti, P","cited_arxiv_id":null,"evidence_quote":"Supplies the conductivity values and rule-based myofiber architecture used to set the anisotropic muscle propagation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the rule-based fractal-tree algorithm used to generate the Purkinje network on the endocardial surface."},{"cited_title":"Colli Franzone, L","cited_arxiv_id":null,"evidence_quote":"Introduces the Eikonal-diffusion equation and the classical pseudo-time approach that the new method modifies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Fast Marching method used to solve the one-dimensional Purkinje Eikonal problem."},{"cited_title":"Strocchi, C","cited_arxiv_id":null,"evidence_quote":"Provides the publicly available four-chamber heart geometry from which the biventricular mesh is extracted."}],"review_version":1}