{"id":"1b5f2c2e-a420-4e2c-94d3-71c213abc83f","arxiv_id":"2412.00916","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"A growth-elasticity model with an energy-based feedback law predicts that unconfined tumor spheroids grow indefinitely while confined ones reach size equilibrium.","lead":"This paper derives a thermodynamically consistent model of how growing tissues respond to chemical and mechanical signals, and fits it to tumor spheroid experiments. Its main qualitative prediction is that tumors in free suspension keep growing instead of reaching a steady size, which differs from earlier models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unbounded free-growth prediction is an artifact of the specific zero-death growth-rate closure, not a unique consequence of thermodynamic consistency; adding a death term likely restores a finite equilibrium.","rationale":"The reader identified the same weakest assumption: the specific closure of the volumetric growth rate (Eq. 62) and the absence of an explicit apoptosis term. My reading of the paper confirms that the thermodynamic consistency argument only imposes a sign constraint on the coupling term Q, not the specific linear form of γ. The paper's central claim of unbounded free growth is therefore a modeling outcome, not a robust consequence of the framework. The analytical proof of no equilibrium at F_ext = 0 (Sec. 5.2.3) assumes γ ≡ 0 at equilibrium and β = 0; it does not exclude equilibrium branches with nonzero mass-source/sink balance. Given that the reader already issued a CONDITIONAL verdict based on this and other issues, my concern does not change the verdict; it strengthens the conditionality. The proposed concrete test—adding a death term and rechecking the equilibrium existence—would directly resolve whether the unbounded-growth prediction is robust. The paper is otherwise transparent about its choices and the modeling is careful, so I do not recommend rejection; the conditional verdict remains appropriate.","tokens_in":29545,"tokens_out":6556,"duration_ms":61215,"concrete_test":"Incorporate a constant or nutrient-dependent apoptosis term into the growth rate, e.g., γ = η c [1/2 kρc^2 + (K(Je − 1) − Je^{−1}W)] − γ_a c, and rerun the free-growth (F_ext = 0) case with the best-fit parameters from Table 1 (or the incompressible version). If any γ_a > 0 yields a finite steady-state radius—which the equilibrium condition would predict—then the unbounded-growth claim is not robust to an admittedly plausible modeling choice. Independently, re-derive Eq. (66) with the death term and solve for R; if a root appears for F_ext = 0, the original no-equilibrium proof fails for the augmented model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the derivation leading to Eq. (32), the authors show that the total energy rate contains the coupling term Q = -∫_Ω γ [ρ(E'_c c − E_c) + (σ_N − J_e^{−1}W)] dV, and that energy dissipation only requires γ to have the same sign as the bracket B. The 'simplest choice' γ = η B, with η = η c in Eq. (62), is one admissible closure, but the thermodynamic framework does not select it uniquely. In particular, adding an independent cell-death term γ̃ = γ − γ_a(c) (as in Greenspan-type models) preserves the dissipation inequality as long as the death term is non-negative, but it changes the equilibrium condition: Eq. (64) becomes η B − γ_a = 0, which can hold at a finite R even when F_ext = 0. Thus the central prediction that free-growing spheroids never plateau is a consequence of the zero-death closure, not a theorem of thermodynamic consistency. Moreover, the analytical no-equilibrium argument in Sec. 5.2.3 (Eq. 66) is restricted to β = 0 and assumes the equilibrium is characterized by γ ≡ 0; the paper does not rule out other equilibrium branches (e.g., a steady state with a proliferating rim and a necrotic core, which would keep the total mass constant while γ ≠ 0). Since this prediction is the paper's headline novelty, its reliance on a non-unique closure is the most load-bearing weakness.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a continuum model for chemomechanically regulated tissue growth in an Eulerian frame, using the multiplicative decomposition F = Fe Fg, a nutrient reaction-diffusion equation, and a dissipation-based closure for the volumetric growth rate. After deriving an energy balance, the authors choose the specific feedback law gamma = eta c [ (1/2) k rho c^2 + K(Je-1) - Je^{-1} W ] (Eq. 62), calibrate the model to tumor spheroid radius measurements in agarose gels [43] and under osmotic pressure [44], and study the dependence of growth on mechanical feedback strength, rearrangement rate, compressibility, and external loads. A central qualitative claim is that unconfined spheroids never reach a finite equilibrium radius and instead grow indefinitely, in contrast to Greenspan-type models. The paper also derives the linearized Maxwell-type viscoelastic limit and compares the framework with earlier biomechanical growth models.","tokens_in":29944,"tokens_out":8040,"duration_ms":77802,"significance":"If the central claims held as stated, the paper would be a valuable contribution: it gives a clean variational derivation of growth feedback, a frame-invariant evolution equation for the elastic Finger tensor, a transparent reduction to Maxwell-type linear viscoelasticity, and quantitative fits to two widely used spheroid experiments. The authors are also honest about parameter fitting, show robustness bands, and compare compressible and incompressible models with an information criterion. However, the headline prediction of unbounded free growth is conditional on a specific closure for gamma and on an unproven characterization of equilibrium, so the significance is currently lower than the abstract claims. The framework may still be useful as one admissible thermodynamically consistent model, but it is not uniquely selected by thermodynamics.","major_comments":[{"comment":"Thermodynamic consistency alone requires only that gamma and the bracket B in Eq. (31) have the same sign so that Q <= 0; it does not select the linear closure gamma = eta B. The later choices of quadratic chemical energy and eta(c,sigma) = eta c are modeling assumptions, not consequences of the variational principle. In particular, the absence of an independent death/apoptosis term means the bracket must vanish at equilibrium. Adding a non-negative death rate gamma_a(c) to gamma changes the equilibrium condition to eta B - gamma_a = 0 and permits a finite equilibrium even when F_ext = 0. Since Sec. 5.2.3 uses gamma = 0 as the equilibrium condition, the prediction of unbounded free growth is a property of the chosen closure, not a theorem of thermodynamic consistency. The manuscript should state this limitation explicitly and test whether the qualitative prediction survives under other admissible closures.","section":"Sec. 3.2, Eq. (32); Sec. 3.4, Eq. (62)"},{"comment":"The argument that no finite equilibrium exists for F_ext = 0 assumes gamma = 0 everywhere at equilibrium for beta = 0. However, stationarity of the radius only requires v(R) = 0, which with incompressibility gives the integral constraint Integral_0^R gamma(r) r^2 dr = 0; sign-changing gamma with a proliferating rim and a dying core is not excluded. The paper states that gamma becomes uniformly zero based on simulations, but does not prove this. The shooting analysis further assumes f_e_theta in (0,1] and uses heuristic root-selection rules when Eq. (82) has no positive root or multiple roots, so it cannot rule out other equilibrium branches. The phrase 'we can conclude' therefore overstates the result; the no-equilibrium conclusion is a numerical observation under a restricted ansatz.","section":"Sec. 5.2.3, Eq. (66) and Supplemental E"},{"comment":"The free-growth dataset is part of the calibration group (free, 0.7%, and 1% gel concentrations), so the predicted absence of a plateau in free growth is not an independent prediction. The other gel concentrations are tested only after fitting the tumor-associated parameters and then adjusting c_H, and the gel-release 'prediction' uses release times fitted to the same experimental data. With roughly eight effective free parameters and no genuine holdout set beyond the c_H scan, the agreement in Fig. 2 mainly demonstrates consistency rather than strong confirmation of the novel qualitative behavior. The manuscript should avoid calling the free-growth behavior a prediction and should report cross-validation or a parameter-identifiability analysis.","section":"Sec. 5.1.1, Tables 1-2, Fig. 2A"}],"minor_comments":[{"comment":"The author name is misspelled as 'Arajo' and later 'Arajuo'; it should be Araújo.","section":"Sec. 3.3, Refs [45]"},{"comment":"The AICc formula as printed is ambiguous and does not reproduce the reported positive AICc values with the given sample sizes and relative errors when compared with the standard form AICc = n log(RSS/n) + 2m + 2m(m+1)/(n-m-1). Please restate the formula unambiguously or justify the alternative definition used.","section":"Supplementary B, Eq. (67)"},{"comment":"The text describes the shaded bands as parameter sets within 10% of the best fit, while the caption says 'within 10% of the best fitting set of parameters'; these are different and should be clarified.","section":"Sec. 5.1.1 and Fig. 2 caption"},{"comment":"The rule that x(r_j) is set to x(r_{j+1}) when no positive real root exists is unphysical and may artificially force a solution; a documented branch-continuation or a more robust root-search procedure should be used instead.","section":"Supplemental E, paragraph on shooting method"},{"comment":"The artificial damping parameter beta-tilde is not specified, and no convergence study with respect to beta-tilde or the spatial grid is reported; a brief numerical validation would strengthen the reliability of the simulations.","section":"Sec. 4.2, Eq. (53)"},{"comment":"There are typographical errors in this section: 'deviotoric' should be 'deviatoric' and 'rearragement' should be 'rearrangement'.","section":"Sec. 2.4"}],"recommendation":"major_revision","confidential_remarks":"The derivation in the paper is internally sound and the reductions to earlier models are useful, but the central novelty is oversold. The authors should be asked to reposition the unbounded-growth prediction as a property of their specific closure rather than as a consequence of thermodynamic consistency, and to either prove or explicitly qualify the beta = 0 equilibrium analysis in Sec. 5.2.3. The AICc formula in Supplementary B also needs correction. With those changes, the paper could be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The energy-variational derivation of the chemomechanical feedback law is the real contribution, and it is new relative to their earlier Michaelis-Menten and adaptive-reference-map models. But the headline prediction—that free spheroids never plateau—is not a theorem of thermodynamic consistency. It follows from choosing the \"simplest\" closure, gamma = eta c B, with no independent apoptosis term.\n\nThe paper's strengths are substantial. The dissipation argument is internally consistent, frame invariance is checked in Supplemental C, and the linearized system recovers a Maxwell viscoelastic law. The authors show clean reductions of earlier growth models (Budday, Garcia, Araújo-McElwain) as special cases. The fits to Helmlinger's gel-confined spheroids and Montel's pressure-clamped spheroids are good: after fitting on free/0.7%/1% gels, they only refit c_H for the other gels, and the gel-release simulation tracks the data within 10% bands. For a continuum tumor model, that is credible evidence the framework captures something real.\n\nThe soft spots are real, and the stress-test note lands on the main one. Thermodynamics only fixes the sign of gamma relative to the bracket B. The \"simplest choice\" gamma = eta c B is an ansatz; a nonnegative death term gamma_a(c) preserves the dissipation inequality and changes Eq. (64) to eta B - gamma_a = 0, which can hold at finite R even with F_ext = 0. So the unbounded free-growth prediction is a property of the zero-death closure, not a unique consequence of the framework. The paper is candid about this: it says the new feedback \"does not allow an arbitrary apoptosis rate.\" But the abstract and conclusion still sell the no-equilibrium result as a prediction, and the equilibrium analysis in Sec. 5.2.3 restricts to beta=0 and the gamma-equivalent-to-zero branch. It does not rule out a steady state with a proliferating rim and a necrotic core, where net mass flux is zero but gamma is not zero. Also, the AICc formula in Supplemental B is misprinted; the model comparison should be redone. Minor: no code is shared, and the main validation is a grid-search fit with a dozen parameters, so \"prediction\" should be used carefully.\n\nWho is this for: groups building continuum models of tumor growth or tissue morphogenesis will want the derivation. It deserves a serious referee. The referee should ask for code, an out-of-sample test (e.g., predict gel-release without fitting release times), and a discussion that separates what thermodynamics forces from what the closure chooses.","headline":"Real derivation, real fits, but the no-plateau free-growth prediction is a chosen closure, not a thermodynamic necessity.","tokens_in":30431,"tokens_out":3357,"would_cite":true,"duration_ms":32960,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74L15","92C50","74A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A thermodynamic growth law predicts that tumor spheroids have no stable size unless the environment presses on them.","keywords":["growth","viscoelasticity","mechanical feedback","fluidity","nonlinear mechanics","tumor","thermodynamic consistency","growth-elasticity"],"falsifier":"Measure the radius of a tumor spheroid in long-term culture with no confining gel or applied pressure. If the radius approaches a finite plateau rather than continuing to increase with a positive but shrinking growth rate, the central claim of no equilibrium without external load is falsified. A second check: for $\\beta = 0$ the model predicts the volumetric growth rate is uniformly zero across the spheroid at the confined equilibrium, so measuring spatially resolved proliferation and finding a persistent central death zone would also count against the closure.","tokens_in":29348,"feed_emoji":"🧫","tokens_out":7326,"duration_ms":69501,"temperature":0.7,"pith_summary":"This paper builds a thermodynamically consistent continuum model of tissue growth in which the volumetric growth rate is not prescribed but emerges from the joint dissipation of elastic and chemical energy. The central message is that, in this model, a tumor spheroid growing without external confinement or pressure has no finite equilibrium radius: it keeps growing, with the proliferation zone narrowing to a rim near the surface. With a confining gel, the spheroid reaches a finite size that shrinks as the gel stiffens; under applied pressure, it can even vanish if the pressure exceeds half the chemical-energy coefficient. The model reproduces published spheroid-radius data under gel confinement, gel release, and osmotic pressure, and it traces the difference between uniform and rim-restricted growth patterns to the tissue rearrangement rate.","feed_headline":"Tumor spheroids never plateau without external force, model says","feed_subtitle":"A growth law derived from energy dissipation reproduces confined-spheroid data and makes a testable prediction about size control.","key_machinery":"The central machinery is an energy-variational argument on a growing elastic body with multiplicative decomposition $F = F_e F_g$, where $F_g$ is active growth and $F_e$ is elastic response, posed in the Eulerian frame. The total energy is the neo-Hookean elastic energy $W(F_e)$ plus a quadratic chemical energy $\\tfrac{k}{2}c^2$; requiring the total energy to decrease selects the deviatoric rearrangement rate $\\Gamma_D = \\beta\\big(F_e F_e^T - \\tfrac{1}{d}\\operatorname{tr}(F_e F_e^T)I\\big)$ and the volumetric growth rate $\\gamma = \\eta c\\big(\\tfrac{1}{2}k\\rho c^2 + K(J_e-1) - J_e^{-1}W\\big)$. At equilibrium, the zero-growth condition $\\gamma = 0$ combined with radial force balance reduces to a one-dimensional shooting problem whose boundary equations give the threshold $F_{\\rm ext} \\le k/2$ and the impossibility of a finite radius when $F_{\\rm ext} = 0$.","core_discovery":"The authors derive a coupled system for an Eulerian growth-elasticity model with nutrient diffusion. The load-bearing result is the chemomechanical feedback closure $\\gamma = \\eta c\\big(\\tfrac{1}{2} k \\rho c^2 + K(J_e-1) - J_e^{-1}W\\big)$, obtained by requiring that total elastic-plus-chemical energy is dissipated by volumetric growth, nutrient uptake, diffusion, and mass-conserving rearrangement. From the equilibrium analysis of this system, they show that when the external traction $F_{\\rm ext}$ is zero, force balance at the center cannot be satisfied for any finite radius, so the tumor grows indefinitely; when a gel provides confinement, an equilibrium always exists; and when a constant pressure is applied, an equilibrium exists only up to $P_{\\rm ext} = k/2$, above which the spheroid shrinks to zero. The authors also find that the inferred tissue rearrangement rate $\\beta$ is zero for gel-confined spheroids and positive for pressure-clamped spheroids, and that $\\beta$ controls whether the growth-rate profile is uniform or boundary-localized.","pith_inferences":["Editorial inference: if the unbounded-free-growth prediction holds, size homeostasis in avascular spheroids would itself be evidence of mechanical confinement from the microenvironment, not an intrinsic property of the tissue; plateau sizes would become environmental readouts rather than tissue-intrinsic constants.","Editorial inference: a decisive test would be long-time tracking of spheroids in non-adherent suspension with no gel; a clear plateau would falsify the no-death closure, while sustained growth with a positive but decreasing growth rate would support it.","Editorial inference: the threshold $P_{\\rm ext} = k/2$ suggests that measuring the critical osmotic pressure at which spheroids vanish could directly estimate the chemical-energy coefficient $k$, a parameter otherwise inferred only by fitting."],"forward_implications":["Free-growing spheroids have no finite equilibrium radius; growth continues with a narrowing proliferative rim, in contrast to classic nutrient-limited models that plateau via central cell death.","With a confining gel, a finite equilibrium radius exists for any gel stiffness, decreases as stiffness or mechanical feedback strength increases, and growth stops with $\\gamma$ uniformly zero when $\\beta = 0$.","With applied pressure, the equilibrium radius decreases as pressure increases, and the spheroid shrinks to zero once $P_{\\rm ext}$ exceeds $k/2$; gel confinement never produces this shrink-to-zero behavior.","The tissue rearrangement rate $\\beta$ controls the growth pattern: $\\beta = 0$ gives a uniform volumetric growth rate, while $\\beta > 0$ concentrates growth at the boundary and, under compression, produces negative growth in the core.","Both compressible and incompressible versions fit the spheroid-radius data; model selection favors the incompressible version despite a slightly larger fitting error, because it has one fewer parameter."],"supporting_citations":[{"why":"The classic nutrient-limited growth model with cell death; the baseline that reaches a finite size, which the paper's no-equilibrium free-growth prediction contrasts.","marker":"[17]"},{"why":"Supplies the gel-confinement spheroid radius measurements, including gel release, used to fit and validate the model.","marker":"[43]"},{"why":"Supplies the osmotic-pressure spheroid radius measurements used to fit the model under applied external stress.","marker":"[44]"},{"why":"Earlier tumor model with mechanical feedback and the analytic external gel traction formula that the current simulations adopt.","marker":"[33]"},{"why":"Earlier compressible tissue model whose linearized stress relaxation the current rearrangement closure recovers as a Maxwell-type fluid.","marker":"[34]"},{"why":"The continuum thermodynamics source for the energy-dissipation requirement that selects the growth and rearrangement closure forms.","marker":"[48]"},{"why":"The finite growth-elasticity decomposition $F = F_e F_g$ on which the Eulerian formulation is built.","marker":"[25]"}],"fun_headline_variants":["Tumor spheroids never stop growing without external force","Confinement, not internal cues, sets tumor spheroid size","Thermodynamic law predicts when tumor growth plateaus","External pressure caps tumor spheroid growth, model shows","No force, no finite size: tumor spheroids grow without limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The model assumes the volumetric growth rate has exactly the energy-dissipation form derived here and that dying cells are not explicitly included; add cell death or change that energy function and the unbounded-growth result could disappear.","fun_headline_variants_meta":{"raw":{"variants":["Tumor spheroids never stop growing without external force","Confinement, not internal cues, sets tumor spheroid size","Thermodynamic law predicts when tumor growth plateaus","External pressure caps tumor spheroid growth, model shows","No force, no finite size: tumor spheroids grow without limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00026,"raw_usage":{"total_tokens":1584,"prompt_tokens":935,"completion_tokens":649,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":563}},"tokens_in":551,"tokens_out":649,"duration_ms":6552,"temperature":1.0,"reasoning_tokens":563,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:51:50.213627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the radius of a tumor spheroid in long-term culture with no confining gel or applied pressure. If the radius approaches a finite plateau rather than continuing to increase with a positive but shrinking growth rate, the central claim of no equilibrium without external load is falsified. A second check: for $\\beta = 0$ the model predicts the volumetric growth rate is uniformly zero across the spheroid at the confined equilibrium, so measuring spatially resolved proliferation and finding a persistent central death zone would also count against the closure.","supporting_citations":[{"cited_title":"On the growth and stability of cell cultures and solid tumors","cited_arxiv_id":null,"evidence_quote":"The classic nutrient-limited growth model with cell death; the baseline that reaches a finite size, which the paper's no-equilibrium free-growth prediction contrasts."},{"cited_title":"Solid stress inhibits the growth of multicellular tumor spheroids","cited_arxiv_id":null,"evidence_quote":"Supplies the gel-confinement spheroid radius measurements, including gel release, used to fit and validate the model."},{"cited_title":"Stress clamp experiments on multicellular tumor spheroids","cited_arxiv_id":null,"evidence_quote":"Supplies the osmotic-pressure spheroid radius measurements used to fit the model under applied external stress."},{"cited_title":"Stress generation, relaxation and size control in confined tumor growth","cited_arxiv_id":null,"evidence_quote":"Earlier tumor model with mechanical feedback and the analytic external gel traction formula that the current simulations adopt."},{"cited_title":"An eulerian nonlinear elastic model for compressible and fluidic tissue with radially symmetric growth","cited_arxiv_id":null,"evidence_quote":"Earlier compressible tissue model whose linearized stress relaxation the current rearrangement closure recovers as a Maxwell-type fluid."},{"cited_title":"The mechanics and thermodynamics of continua","cited_arxiv_id":null,"evidence_quote":"The continuum thermodynamics source for the energy-dissipation requirement that selects the growth and rearrangement closure forms."},{"cited_title":"Stress-dependent finite growth in soft elastic tissues","cited_arxiv_id":null,"evidence_quote":"The finite growth-elasticity decomposition $F = F_e F_g$ on which the Eulerian formulation is built."}],"review_version":1}