Coupling models of resistive valves to muscle mechanics in cardiac fluid-structure interaction simulations

Luca Dede'; Michele Bucelli

REVIEW 2 major objections 6 minor 66 references

Coupling models of resistive valves to muscle mechanics in cardiac fluid-structure interaction simulations

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Conventional resistive valve models leave the wall reaction force out of the momentum balance, so heart FSI simulations drift and oscillate; this paper adds a distributed attachment force with matching resultant and shows the artifacts…

desk verdict A clean fix for a real linear-momentum imbalance in resistive valve FSI, but the 'Newton's laws' claim overreaches because only the resultant of the attachment force is balanced. read the letter →

arxiv 2506.03869 v1 pith:SZR7BQLG submitted 2025-06-04 math.NA cs.NA

Michele Bucelli , Luca Dede' This is my paper

classification math.NAcs.NA MSC 65M6074F1076Z0592C35

keywords fluid-structureinteractioncardiacmodelingvalveresistiveimmersedimplicitsurfaceNewton'slawsattachmentforcesleftheartsimulationALEformulation

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

Cardiac simulations that combine contracting walls with flowing blood need valves, but cheap 'resistive' valve models exert a force on the blood without passing the equal-and-opposite force to the wall that holds the valve in place. The paper argues that this omission violates Newton's laws: during isovolumetric contraction the unbalanced pressure force accelerates the whole simulated heart, producing drift in an idealized spherical benchmark and large longitudinal displacements, pulmonary-vein jets, and pressure-volume-loop oscillations in a realistic left heart. The fix is a distributed force added to the solid mechanics equation, spread over the region where each valve meets the wall and normalized so its resultant equals the force the fluid exerts on that valve. With the term included, the center of mass of the fluid-solid system stays put, the realistic traces become physiological, and the cost is small because the new term is evaluated explicitly from previous time-step data; the same correction extends to other resistive valve models.

What carries the argument

The load-bearing object is the distributed attachment force $\hat{\mathbf g}_k=(F_k/V_k)\,\delta_k(\varphi_k(\hat x+\hat d))$ added to the solid elastodynamics equation. The smoothed Dirac delta $\delta_k$ of the signed distance function $\varphi_k$ is the same RIIS kernel used in the fluid, so the force is spread over the valve-wall overlap region; $F_k$ is the integrated resistive force from the fluid momentum equation, and $V_k$ normalizes the distribution so its resultant equals $F_k$. Computing $\hat{\mathbf g}_k$ explicitly from the previous time step's velocity, pressure, and displacement keeps the added term almost free computationally while introducing a first-order error in time.

What would settle it

Run the spherical benchmark with the resistive plane tilted or displaced so its fluid force has nonzero torque about the system's center of mass; if the total angular momentum changes while no external torque acts, the model is not fully Newtonian. Equivalently, compute $\int_{\Omega_f} (x-x_0)\times R_k\,dx$ and the torque of $\hat{\mathbf g}_k$ about the same point at peak systole in the realistic heart: a nonzero difference directly contradicts the paper's Newton-law claim.

Watch

Extended reading notes

Core claim

The central claim is that the resistive immersed implicit surface (RIIS) valve model becomes consistent with Newton's laws in fluid-structure interaction only if the force the fluid exerts on each valve is transferred back onto the cardiac wall. The transfer is made explicit: with $R_k$ the resistive force density in the Navier-Stokes equation, the fluid force on valve $k$ is $F_k=\int_{\Omega_f} R_k\,dx$, and the wall receives $\hat{\mathbf g}_k=(F_k/V_k)\,\delta_k(\varphi_k)$, where $V_k$ is the integral of the same smoothed delta kernel over the solid domain. Matching the resultant of $\hat{\mathbf g}_k$ to $F_k$ restores linear momentum of the coupled fluid-solid system, and the numerical experiments confirm the consequence: no center-of-mass drift in the spherical benchmark and no unphysiological ventricular displacement, pulmonary-vein jets, or pressure-volume-loop oscillations in the realistic left heart.

Load-bearing premise

The repair assumes that transferring only the total push of the fluid on each valve to the wall is enough to restore Newton's laws; the fluid's twisting effect on the valve is never passed to the wall, so the system as a whole can still start to spin.

Editorial extensions

If this is right

Without the attachment force, the spherical benchmark shows a clear downward acceleration of the system's center of mass; with it, the center of mass remains stationary and only a small drift remains, shrinking like $\Delta t$ as the time step is refined.
In the realistic left heart, neglecting the force produces fast large longitudinal ventricular displacement, high atrial strains, strong pulmonary-vein jets, and sharp oscillations in pressure-volume loops, all absent when the force is included.
The pressure-volume loop of the left atrium becomes much more physiological when valve forces are included, matching the expected morphology.
The correction is not tied to RIIS specifically: the same force-balance principle extends to other resistive valve models used in FSI simulations.
The added computational cost is minor, with the linear solve dominating and the assembly of $\hat{\mathbf g}_k$ contributing only a small fraction of wall time.

Reading between the lines

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

The paper balances only the resultant of the fluid force on each valve, not its torque; a tilted or asymmetric valve would transmit a spurious twisting force to the wall, so angular momentum of the FSI system is not conserved in general.
A benchmark with an off-center resistive surface would expose this: if total angular momentum changes with no external torque, the model is not fully Newtonian even though it fixes the linear drift.
The near-linear relation between pressure jump and force density for closed valves could be used as a cheap surrogate for valve state in patient-specific models, since its slope encodes valve geometry.
Extending the explicit evaluation of $\hat{\mathbf g}_k$ to second-order extrapolation is a natural next step that should remove the residual first-order drift at essentially the same cost.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Punchline: this paper fixes a real, known problem—RIIS valve models in FSI violate linear momentum because the valve forces never reach the solid—with a simple, parameter-free attachment force. The spherical benchmark is convincing that the fix works for linear momentum. But the paper overclaims when it says the model 'satisfies Newton's laws': the construction only matches the resultant of the resistive force, not its torque, so angular momentum is not conserved in general.

The new piece is the explicit formula g_k = (F_k/V_k) delta_k, with F_k the integral of the resistive term and V_k a normalization. Earlier RIIS-in-FSI work papered over the imbalance with ad hoc boundary-condition calibration. Here the term is forced by the balance condition itself, so no parameters are tuned. The ideal sphere test is well chosen: no external forces, center-of-mass drift only vanishes when the term is added, and residual drift goes as dt, consistent with explicit treatment. The cost breakdown shows the overhead is negligible. The left-heart test shows a clear qualitative improvement in displacement and pressure-volume loops.

Soft spots, in proportion. The biggest is the torque issue, and it's real: the resistive force distribution on the valve generally exerts a torque, but g_k is determined only by its resultant. Assumption 2 (valve equilibrium) would require moment balance too, but only the force balance is used. The spherical benchmark has zero torque by symmetry, so it cannot expose the error. So the conclusion that the model 'satisfies Newton's laws' is too strong; what is established is linear-momentum balance. That is still a useful result, but the claim should be qualified or the method extended to match moments. Second, the spatial distribution of the attachment force is ad hoc—uniform scaling of the delta profile—with no argument that it's realistic. Third, the realistic test is qualitative; no quantitative comparison to a gold standard or sensitivity analysis. Fourth, no code or data are provided, which makes reproduction harder, though the method is simple enough to implement from the description.

Who should read it: anyone doing FSI with resistive valve models, or building whole-heart digital twins. It's a modest but solid contribution. It deserves peer review; a referee should ask for the Newton's laws claim to be scaled back, a discussion of torque, and ideally release of artifacts. I'd take it.

Referee Report

2 major / 6 minor

Summary. This paper proposes an extension of the resistive immersed implicit surface (RIIS) valve model for use in cardiac fluid-structure interaction (FSI) simulations. The authors add a distributed body force to the solid momentum equation, chosen so that its resultant equals the integral of the resistive force over the fluid domain for each valve, thereby restoring linear momentum balance to the coupled system. The modification is discretized explicitly, and numerical tests include a spherical benchmark (where center-of-mass drift is removed and the residual drift converges with the time step) and a realistic left-heart model (where valve attachment forces suppress unphysical ventricular displacements and pressure-volume loop oscillations).

Significance. The proposed modification addresses a real and practically important deficiency: standard RIIS, when embedded in FSI, violates linear momentum balance and can produce unphysical global motion of the heart. The spherical benchmark is well designed, and the observed first-order convergence of the residual drift with the explicit treatment is a strong consistency check. The implementation is based on the open-source lifex/deal.II stack, all parameters are reported, and no calibration is performed to obtain the physiological-looking PV loops. The main caveat is that the construction enforces only the resultant of the valve force, not the moment, so the paper's stronger claims about Newton's laws and physical consistency are not fully supported.

major comments (2)

[Section 2.2 (definition of C_k(t) and g_k; Eq. (3) for R_k)] The proposed construction enforces only the resultant of the fluid-valve force, not its moment. With g_k = (F_k/V_k) δ_k(φ_k(x̂+d̂)) and R_k = (R_k/ε_k)(u-u_ALE) δ_k(φ_k(x)), the torque on the solid is τ_g = (F_k/V_k) ∫_{Ω_s} x × δ_k dx, whereas the torque exerted by the fluid on the valve is τ_R = ∫_{Ω_f} x × (R_k/ε_k)(u-u_ALE) δ_k dx. These coincide only if u-u_ALE is constant on the support of δ_k, which is generally not the case. Hence the total angular momentum of the coupled system is not conserved in general. The spherical benchmark of Section 3.1 cannot reveal this because its azimuthal symmetry makes both torques zero, and the realistic test of Section 3.2 does not monitor angular momentum. Since the abstract and the conclusions claim that the model 'satisfies Newton's laws' and is 'physically consistent', the claim is stronger than what is established: only linear momentum balance is demonstrated. The authors should either modify the attachment force distribution so that it also matches the torque (e.g., by taking g_k = δ_k (C_k + D_k × (x-x̄)) and fixing D_k through the torque balance) or explicitly limit the claims to linear momentum conservation.
[Section 2.2, Assumption 3] Assumption 3 states that the spatial distribution of bg_k is the same as in the fluid, but the implemented formula bg_k = C_k(t) δ_k(φ_k(x̂+d̂)) is not the same vector distribution as R_k ∝ (u-u_ALE) δ_k(φ_k(x)); the direction of R_k varies with the local relative velocity, while bg_k has the constant direction of C_k. The assumption is therefore a new modeling ansatz for the shape of the attachment force, not a statement of equality of distributions. Because this choice determines the torque mismatch discussed in the previous comment, it is load-bearing and should be presented and justified as such, with a discussion of its consequences.

minor comments (6)

[Section 3.1, paragraph before Figure 3] The phrase 'both with and withouth the newly introduced term' contains a typo: 'withouth' should be 'without'.
[Table 1 caption] The word 'techinques' should be 'techniques'.
[Table 2] The row label 'computeg' should read 'compute g'.
[Section 3.2, first paragraph] The sentence 'we valves open and close according to...' is missing a verb; it should read 'we let the valves open and close according to...'.
[Appendix A] The activation time φ in the eikonal-diffusion model uses the same symbol as the valve signed-distance functions φ_k in Section 2.1; renaming one of them (e.g., τ_act) would avoid confusion.
[Table A.7] The Q0 entries are labeled with units 'mmHg' for initial flow rates; flow rates should have units of volume per time (e.g., mL/s) or the quantities should be defined differently.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: the attachment force is constructed, not fitted (C_k(t)=F_k(t)/V_k(t) is forced by the balance condition itself), so the stationary center of mass in the spherical benchmark is a consistency check of the implementation rather than an independent prediction; the without-g failure case and the realistic-heart comparison give the work independent content.

self definitional [Section 2.2 (derivation of C_k(t) and bg_k) and Section 3.1 (spherical benchmark); Conclusions, Section 4.]
"By Assumption 2, we choose Ck(t) such that, at every time t, the resultant of bgk equals the force that the fluid exerts on the k-th valve, denoted by Fk: ... Ck(t) = Fk(t)/Vk(t), bgk(bx, t) = Fk(t)/Vk(t) delta_k(phi_k(bx + bd)). ... These forces are computed in such a way to balance the forces exchanged by the fluid and the valves, thus making sure that the model satisfies Newton's laws."

The property showcased as the headline outcome -- that the FSI system is momentum-balanced (spherical benchmark: the center of mass remains stationary with bg and drifts without it) -- is exactly the property imposed when C_k(t) was defined so that the resultant of bg_k equals F_k, the fluid force on the valve. No parameter is fitted to data and no independent balance law is invoked: the stationarity check verifies that the construction and its explicit discretization were implemented correctly, not that the model predicts a new behavior.

full rationale

The derivation chain for the new term is self-contained: given the resistive force density R_k acting on the fluid (Eq. 3), the paper postulates bg_k = C_k(t) delta_k(phi_k) on the structure (Assumption 3), then fixes C_k(t) = F_k(t)/V_k(t) so that the resultant transmitted to the wall equals the fluid force on the valve (Section 2.2). Linear-momentum balance of the combined fluid-solid system therefore follows by construction, and the spherical-benchmark result (stationary center of mass with bg) is a consistency check of the implementation and discretization, not a fitted prediction. The comparison case without bg, which exhibits the unphysical drift, and the realistic left-heart test (PV loops, force-density traces) provide independent content, and no parameter is tuned to make the results look physiological. Self-citation is heavy (RIIS [24], the FSI model [13], eikonal-mechanics and active-stress models [44, 45, 52], and the discretization [12] are all from the authors' group), but the new construction is derived in-paper from stated assumptions, so the self-citations are substrate rather than load-bearing justification; nothing central reduces to an unverified self-citation. The appended skeptic note about torque is weighed here and classified as a correctness risk, not circularity: Assumption 2 (valve equilibrium) is used only for the force resultant, while bg_k is constrained only by its resultant, so a torque mismatch between R_k and bg_k is generally present, angular momentum is not conserved in general, and the spherical benchmark's symmetry hides this. That makes the abstract/conclusion statement that the model 'satisfies Newton's laws' overstated, but overclaiming is not a circular-derivation defect and does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The only new element is the balancing force g computed from the solution; it introduces no fitted constants. The central claim rests on four stated modeling assumptions, of which Assumptions 1-2 are domain assumptions inherited or natural for resistive valve surrogates, and Assumption 3 is an ad hoc distribution choice. No new physical entities are introduced.

assumptions (4)

domain assumption Assumption 1: In the deformed configuration, each surface Γ_k is in contact with the structure domain, so Γ_k ∩ Ω_s ≠ ∅.
Needed to ensure V_k(t) > 0 so the normalization C_k = F_k/V_k is well defined; if the valve's smoothed support does not reach the solid, the wall force cannot be transferred (Section 2.2).
domain assumption Assumption 2: Each valve remains in equilibrium at all times, i.e., the net force acting on it is zero.
This justifies setting the wall attachment force equal to the fluid force on the valve. It fails if valve inertia, chordae dynamics, or leaflet acceleration matter; it is standard for resistive surrogate valves.
ad hoc to paper Assumption 3: The spatial distribution of the attachment force on the wall matches the resistive force profile: g_k = C_k(t) δ_k(φ_k(x̂ + d̂)).
Chosen for convenience (same delta profile as the fluid force) with no anatomical basis; it determines local wall loading and affects the moment, but only the resultant is controlled (Section 2.2).
domain assumption Valve opening/closing velocity is neglected: ∂d_k/∂t ≈ 0, so the valve velocity equals u_ALE.
Inherited from prior RIIS work [13, 63]; removes explicit valve kinematics from the force balance (Section 2.1, Eqs. 2-3).

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Cite this review

Pith. "Pith review of Coupling models of resistive valves to muscle mechanics in cardiac fluid-structure interaction simulations." pith.science (2026). https://pith.science/paper/SZR7BQLG

@misc{pith2026250603869,
  author       = {Pith},
  title        = {Pith review of: Coupling models of resistive valves to muscle mechanics in cardiac fluid-structure interaction simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZR7BQLG}},
  note         = {Machine review of arXiv:2506.03869}
}

read the original abstract

To accurately simulate all phases of the cardiac cycle, computational models of hemodynamics in heart chambers need to include a sufficiently faithful model of cardiac valves. This can be achieved efficiently through resistive methods, and the resistive immersed implicit surface (RIIS) model in particular [Fedele et al., BMMB, 2017]. However, the conventional RIIS model is not suited to fluid-structure interaction (FSI) simulations, since it neglects the reaction forces by which valves are attached to the cardiac walls, leading to models that are not consistent with Newton's laws. In this paper, we propose an improvement to RIIS to overcome this limitation, by adding distributed forces acting on the structure to model the attachment of valves to the cardiac walls. The modification has a minimal computational overhead thanks to an explicit numerical discretization scheme. Numerical experiments in both idealized and realistic settings demonstrate the effectiveness of the proposed modification in ensuring the physical consistency of the model, thus allowing to apply RIIS and other resistive valve models in the context of FSI simulations.

Figures

Figures reproduced from arXiv: 2506.03869 by the authors.

**Figure 2.** Idealized spherical benchmark (Section 3.1). (a) Schematic representation of the domain. (b) Streamline representation of the fiber field, colored according to the transmural distance. We show clipped layers of the solid domain for clarity. 2.3. Valve-wall coupling in electromechanical models It is worth observing that issues in the balance of the pressure forces acting in chamber walls may also arise in models of c… view at source ↗

**Figure 3.** Idealized spherical benchmark (Section 3.1). Evolution over time of the system deformation and fluid pressure, without (top) and with (bottom) the force g. The red dot marks the position of the system’s center of mass. 3.1. Idealized spherical benchmark We consider a geometrically idealized setting in which the fluid domain is a sphere, the structure domain is a spherical shell surrounding it, and a resistive planar… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Idealized spherical benchmark (Section 3.1). Unless otherwise stated, the plot are obtained with ∆t = 5·10−4 s. (a) Vertical position z of the center of mass of the FSI system over time, with and without the attachment force g. (b) Vertical velocity of the center of ma…

**Figure 5.** Figure 5: Left heart test case. (a) Decomposition of the solid domain [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Left heart test case. Snapshots of the numerical solution without (left) and with (right) the valve forces [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Left heart test case. (a) Left ventricular pressure-volume loop. (b) Left atrial pressure-volume loop. (c) [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: Left heart test case, including valve forces [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

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