{"id":"c822f732-32b4-4771-bcca-52d32e539ab4","arxiv_id":"2502.05039","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A spatially-resolved, lipid-structured model of early atherosclerosis reproduces the observed internal lipid and macrophage peaks only when macrophage mobility drops steeply with lipid load while lifespan changes little.","lead":"The authors build a mathematical model of early atherosclerosis tracking how macrophages move and die as they fill with lipid, and compare it to human coronary artery images. The model shows that the spatial pattern of lipid and macrophage buildup only matches the images when lipid-laden macrophages slow down sharply but do not die much faster.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The model's stated lipid-dependent egress boundary condition is not what Eqs. (3)/(41) implement, so the central ψD>0.98 conclusion may depend on which version was simulated.","rationale":"The paper is transparent about its limitations, and the mathematical derivations (e.g., the monotonicity proof in §3.1, the concavity argument in §3.2) are sound. The central inference is nevertheless a numerical inference: the claimed necessity of ψD near 1 and ψβ near 0 comes from the scanned parameter regions in Figs. 12–13, not from a theorem. That makes the exact PDE specification load-bearing. I found an internal inconsistency between the stated egress assumption and the printed equations. The reader's weakest_assumption focused on the linear-in-ℓ mobility law, which the authors themselves flag as a possible artefact in Discussion Q1; that is a reasonable robustness concern, but it is a modelling choice the authors openly acknowledge. The egress mismatch is stronger because it is an unresolved contradiction inside the model statement. Without code, one cannot tell which version generated the figures. I am not proposing rejection: the qualitative mechanism (lipid-laden MDMs lose motility and are retained) is biologically plausible and robustly supported by the contrast between Fig. 4 and Fig. 11. The issue is that the specific quantitative threshold—ψD>0.98, ψβ<0.2—may be an artefact of an unstated implementation choice. This warrants the same CONDITIONAL outcome the reader gave, with the added request to correct or justify the egress boundary condition and provide code or an explicit re-run under both interpretations.","tokens_in":33192,"tokens_out":7396,"duration_ms":74900,"concrete_test":"Re-run the steady-state (ψD, ψβ) sweep of Fig. 12 under both boundary specifications: (i) the printed Eq. (3)/(41) with egress γmℓ, and (ii) the text-described egress γ(1−ψDℓ/ℓmax)mℓ at x=0 and x=1, with the corresponding boundary conditions for M and LM derived consistently. Compare the set of (ψD, ψβ) pairs where M(x) and Ltot(x) both have global internal maxima. If the region is essentially unchanged, the ambiguity does not affect the central conclusion; if it shifts or disappears, the manuscript must state which boundary condition was used and re-evaluate the claimed parameter threshold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that internal global maxima require high ψD and low ψβ—rests on the numerical phase diagram in Fig. 12. That diagram is not uniquely specified because §2.2 describes MDM egress as proportional to mobility, γ(1−ψD ℓ/ℓmax), but Eq. (3) (and dimensionless Eq. (41)) impose egress flux γmℓ, independent of ℓ. The text is explicit: 'we assume that MDMs at x = 0, X exit the lesion at a rate proportional to their mobility: γ(1 − ψD ℓ/ℓmax).' The equations just above do not contain that factor. This is not cosmetic: the trapping mechanism that generates peaks is the balance between lipid-dependent diffusion and lipid-dependent egress. If the numerics used Eq. (3), then the analysis omits the stated egress-mobility coupling; if it used the text version, the moment equations (26)–(27) and the M/LM boundary conditions are inconsistent. Either way, the link between the stated biology and the simulated model is ambiguous, and no code is provided to resolve it. A secondary concern, already flagged by the authors in Discussion Q1, is that the linear D(ℓ) assumption may drive the transient deep infiltration; but the boundary-condition ambiguity is more immediately load-bearing because it affects the very simulations used to infer the ψD>0.98, ψβ<0.2 region.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a one-dimensional, spatially resolved PDE model in which monocyte-derived macrophages are structured by a discrete lipid-content index m_l(x,t), coupled to extracellular free LDL, retained LDL, apoptotic and necrotic lipid, HDL, and inflammatory mediators. The model is calibrated to the Nakashima et al. (2007) human coronary artery images through a sigmoidal LDL-retention capacity K_r(x) with steepness theta=10. The principal results are: (i) when MDM kinetics are lipid-independent (psi_D=psi_beta=0), the steady-state total MDM density M(x) is strictly decreasing, proved by a closed-form solution; (ii) when only lifespan depends on lipid content (psi_D=0, psi_beta>0), M satisfies M''>=0 and therefore cannot have an internal maximum; (iii) when only mobility depends on lipid content (psi_D>0, psi_beta=0), internal maxima are possible if the ingested-lipid density L_M-M is sufficiently concave (Eq. 71); and (iv) numerical sweeps over (psi_D,psi_beta) and (L*,H*) identify a small region near (psi_D,psi_beta)=(1,0) where both M and total lipid L_tot have internal global maxima at steady state, matching the images. The authors conclude that lipid-induced loss of MDM mobility, rather than lipid-induced apoptosis, is the mechanism responsible for the internal MDM and lipid peaks.","tokens_in":33522,"tokens_out":8175,"duration_ms":82402,"significance":"The paper's main strengths are the structured-population framework that combines spatial position and lipid content, and the two rigorous negative results: the closed-form monotonicity proof for psi_D=psi_beta=0 (Eqs. 63--66) and the convexity obstruction for psi_beta>0, psi_D=0 (Eq. 69). These are clean and checkable. The positive result, captured in the Fig. 12 phase diagram, is biologically plausible and useful, but it is established only by numerical sweeps with a qualitative shape classification. The boundary-condition inconsistency described below, together with the absence of code, means the numerical evidence for the central claim is not yet fully reproducible. The paper is transparent about several limitations, including the linear mobility assumption and the calibration of K_r(x), which is a distinct strength.","major_comments":[{"comment":"The text following Eq. (3) states that MDMs at x=0 and x=X exit the lesion at a rate proportional to their mobility, gamma(1-psi_D l/l_max), but Eq. (3) and its dimensionless counterpart Eq. (41) impose the lipid-independent flux gamma m_l at both boundaries (with the R(t) delta_{l,0} term at x=0). The moment equations (26)--(27) are derived by summing Eq. (3) and therefore also contain gamma M and gamma L_M, not the mobility-weighted sums that the prose would imply. This is a load-bearing issue: if the simulations used Eq. (3), the stated egress--mobility coupling was not part of the model that produced Fig. 12; if the simulations used the text version, the governing equations in the manuscript do not match the numerics. Please reconcile the text and equations, state explicitly which boundary condition was implemented, and ideally provide the numerical code or the discretized form used.","section":"§2.2, Eqs. (3) and (41)"},{"comment":"The phase diagram in Fig. 12 is the primary evidence for the central conclusion that internal global maxima require psi_D>0.98 and psi_beta<0.2. However, the paper does not specify how the categories 'monotone', 'local maximum not equal to global maximum', and 'local maximum equal to global maximum' are computed from the numerical steady states. No threshold for identifying a local maximum, no statement of how many extrema are found, and no convergence check for the classification under the reported Delta x=0.02 discretization are given. Because the claimed region is a small neighbourhood of (psi_D,psi_beta)=(1,0), the classification procedure must be reproducible and robust; please provide the algorithm or the code used to generate Fig. 12.","section":"§3.4, Fig. 12"}],"minor_comments":[{"comment":"The abstract says the model 'predicts' deep initial lipid accumulation and internal peak formation, but feature 1 is imposed through the calibrated retention profile K_r(x) (Eq. 7) and feature 2 is used to select the parameter region in Fig. 12; consider using language such as 'reproduces' or 'is consistent with' and explicitly distinguishing calibrated features from emergent ones.","section":"Abstract and §4 Q1"},{"comment":"In Table 2, the row for the dimensionless lifespan sensitivity parameter lists '~psi_beta psi_D Sensitivity of MDM lifespan to lipid load'; the parameter name should be psi_beta, not psi_D.","section":"Table 2"},{"comment":"The Discussion already flags that the transient deep infiltration could be an artefact of the linear mobility decrease with lipid content; it would strengthen the paper to include a sensitivity test with a nonlinear or saturating mobility function, such as D(l)=D_M(1-psi_D(l/l_max)^p), to show that the Fig. 12 region is not an artefact of linearity.","section":"§4 Q1"},{"comment":"The text states that peak formation in both M and L_tot requires psi_D>0.98 and psi_beta<0.2, but Fig. 12 uses a grid resolution of only 0.1 for psi_D<0.9; it would be useful to state explicitly how the reported thresholds were derived from the grid and whether they are sensitive to the grid resolution.","section":"§3.4"}],"recommendation":"major_revision","confidential_remarks":"The egress boundary-condition ambiguity is the key obstruction to accepting the paper in its current form. If the authors can show that the same phase diagram is obtained under both interpretations of the boundary condition, or can unambiguously specify which version was solved and provide code, a shorter revision path is possible. The manuscript fits the journal's scope and the analytical parts are solid; the main positive result needs to be made reproducible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper generalizes the authors' earlier lipid-structured ODE models to a spatially resolved, lipid-structured PDE model and asks when the model reproduces two features of Nakashima's human coronary images: early deep lipid accumulation and later internal peaks in lipid and macrophage densities. The key new result is qualitative: internal global maxima appear only when macrophage mobility is strongly lipid-dependent and apoptosis is weakly lipid-dependent. That is a sensible organizing insight for atherosclerosis modelers, and the negative results are the strongest part. The paper proves analytically that with lipid-independent kinetics the steady-state MDM density is monotone decreasing, and that lipid-dependent apoptosis alone cannot produce an internal maximum. Those proofs are clean and correct. The numerical phase diagram in Fig. 12 is a reasonable extension, and the discussion is honest about the linear-mobility assumption and the transient deep infiltration being a possible artefact.\n\nBut there is a load-bearing inconsistency that the stress-test note caught and that I confirmed on reading. The text states that MDMs exit at x=0 and x=X at a rate proportional to their mobility, γ(1−ψDℓ/ℓmax). The actual boundary conditions in Eqs. (3) and (41) impose egress flux γmℓ, with no lipid dependence. The numerical scheme is described as using boundary conditions (42)–(46), which do not include the mℓ boundary condition (41) at all. So it is genuinely unclear which egress rule was simulated. This matters because the trapping mechanism that creates peaks is precisely the interplay between lipid-dependent diffusion and lipid-dependent egress. If the numerics used the text's mobility-proportional egress, the moment equations and boundary conditions are inconsistent; if they used Eq. (3), the stated biology is not what was simulated. No code is provided to resolve this. The central conclusion—that ψD>0.98 and ψβ<0.2 are required—may survive, but as written it is not reproducible.\n\nThere is also a calibration issue: Kr(x) is chosen to match feature 1, and ψD, ψβ are swept to match feature 2, so the abstract's language that the model \"predicts\" these features overstates independent verification. That is a framing problem, not a fatal one.\n\nThe intended audience is mathematical biologists working on atherosclerosis and structured population models. The paper deserves serious peer review because the question is important and the analytical core is sound, but the boundary-condition ambiguity and missing code must be fixed before the numerical claims can be trusted. I would send it to review with a request for clarification and code, not desk-reject it.","headline":"The qualitative peak-formation result is plausible and the negative proofs are solid, but a boundary-condition mismatch in the text means the central phase diagram needs verification before I'd trust the ψD>0.98 conclusion.","tokens_in":33987,"tokens_out":2914,"would_cite":false,"duration_ms":33800,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["92C37","92C50","35Q92"],"pacs":[],"model":"deepseek-v4-flash","headline":"Lipid-laden macrophages stall rather than die faster, and that stalling is what creates the internal lipid and macrophage peaks of early coronary lesions, according to a new spatial model.","keywords":["atherosclerosis","lipid-structured population model","macrophage","foam cells","LDL retention","HDL","spatial model","lipid-dependent mobility"],"falsifier":"Measure single-cell migration speed as a function of intracellular lipid content in monocyte-derived macrophages under controlled LDL/HDL conditions; if the speed-versus-load relationship is not approximately linear (for example, mobility drops sharply only beyond a threshold load, or saturates before zero), the model's necessity claim for $\\psi_D>0.98$, $\\psi_\\beta<0.2$ is a linearity artefact. A second check: scan many early human lesions for the transient deep MDM infiltration that the model produces before the peak forms; never observing it would support the authors' own suspicion that this wave is an artefact of the linear assumption.","tokens_in":33022,"feed_emoji":"🫀","tokens_out":8410,"duration_ms":84441,"temperature":0.7,"pith_summary":"The paper builds a one-dimensional, lipid-structured model of the artery-wall intima in which monocyte-derived macrophages (MDMs) are labelled by their lipid content as well as their position, and asks which biological mechanisms can produce the two hallmark features of early human coronary lesions seen in Nakashima's images: lipid that first collects deep in the intima, and later a lipid-plus-macrophage peak inside the wall rather than at either boundary. It finds the deep lipid accumulation is explained by a spatially non-uniform LDL retention capacity that rises toward the media. The central result is that the interior peaks appear only when MDM mobility falls steeply with lipid load (diffusivity sensitivity $\\psi_D$ close to 1) while MDM lifespan depends only weakly on load ($\\psi_\\beta$ small); lipid-dependent death alone provably cannot produce an internal maximum. The authors conclude that what creates the lesion's characteristic peaks is that lipid-laden macrophages lose motility and become trapped, not that they die faster. They also find that mean MDM lipid content rises with depth for every blood LDL/HDL combination tested.","feed_headline":"Stalled foam cells, not faster death, shape early plaque peaks","feed_subtitle":"A spatial model reproduces the lesion's internal peaks only when mobility drops steeply as macrophages load lipid.","key_machinery":"The central object is the lipid-structured MDM density $m_\\ell(x,t)$, where $\\ell=0,\\dots,\\ell_{\\max}$ indexes the discrete lipid content $a_0+\\ell\\Delta a$ and $x\\in[0,1]$ is depth into the intima. The carrying mechanism is the lipid-content-dependent mobility coefficient $D_M(1-\\psi_D \\ell/\\ell_{\\max})$ in the diffusion term of the equation for $m_\\ell$, which makes fully loaded cells immobile when $\\psi_D=1$. The argument works because this term couples the total MDM density $M(x,t)$ to the second derivative of the ingested-lipid density $L_M-M$: at steady state $d^2M/dx^2 = (\\psi_D/\\kappa)\\,d^2(L_M-M)/dx^2 + M/D_M$, so an internal peak requires $\\kappa(L_M-M)$ to be sufficiently concave. A sigmoidal LDL retention capacity $K_r(x)$ supplies the initial deep-lipid gradient, and a fast-mediator quasi-steady reduction closes the recruitment loop by relating endothelial mediator density to a weighted integral of retained LDL.","core_discovery":"On its own terms, the paper establishes a necessary condition on the biology: to match the spatial structure of early human atherosclerotic lesions, the effective diffusion coefficient of an MDM must decrease linearly from $D_M$ to $D_M(1-\\psi_D)$ as its lipid index $\\ell$ goes from 0 to $\\ell_{\\max}$, with $\\psi_D$ so large that only a narrow neighbourhood of $(\\psi_D,\\psi_\\beta)=(1,0)$ reproduces global internal maxima in both the MDM density $M(x)$ and total lipid density $L_{\\rm tot}(x)$. The proof has two parts. With $\\psi_D=\\psi_\\beta=0$, the steady-state MDM profile is exactly $M(x)=A\\cosh((1-x)/\\sqrt{D_M})+B\\sinh((1-x)/\\sqrt{D_M})$ with positive $A,B$, so $M'(x)<0$ everywhere. With $\\psi_\\beta>0=\\psi_D$, the steady-state equation gives $d^2M/dx^2\\ge 0$, so $M$ is concave-up and cannot have an interior maximum. Only for $\\psi_D>0$ can the ingested-lipid density $\\kappa(L_M-M)$ be sufficiently concave to make $d^2M/dx^2<0$ and produce the observed internal peak; the paper verifies numerically that this requires $\\psi_D$ above roughly 0.92 and, for the peak to be the global maximum, $\\psi_D>0.98$ and $\\psi_\\beta<0.2$ with blood LDL sufficiently high relative to HDL.","pith_inferences":["Because the model's necessity result rests on the linear decrease of mobility with lipid load, a nonlinear mobility-loss curve (saturating or threshold-like) could broaden or shift the $\\psi_D$ window; the authors themselves flag the linearity assumption as a possible source of the transient deep-infiltration artefact, so the 'stall, not die' conclusion should be tested against measured speed-vers","If maximal MDM infiltration depth really can retreat as the lesion matures, then depth-based staging of early lesions may be unreliable; a prospective counting of MDM positions across many Nakashima-stage lesions would tell whether the transient deep wave is real.","The model's prediction that high LDL relative to HDL is required for peak formation suggests that interventions raising HDL capacity might suppress the internal macrophage peak; this consequence is untested because the paper varies HDL only as a boundary condition, not as a therapy."],"forward_implications":["The early deep-intima lipid accumulation is set by the LDL retention profile $K_r(x)$ before macrophages arrive, matching Nakashima's observation that the deep intima is the first site of lipid deposition.","When $\\psi_D>0.98$ and $\\psi_\\beta<0.2$, both $M(x)$ and $L_{\\rm tot}(x)$ develop internal global maxima that approximately coincide, reproducing the Nakashima images; outside this region the profiles are monotone or have only a local maximum that is not global.","Increasing $\\psi_\\beta$ alone lowers MDM density, MDM lipid content, and infiltration depth while raising total lesion lipid, because dying cells deposit necrotic lipid; increasing $\\psi_D$ alone does the opposite for MDM lipid content because trapped lipid-laden cells are retained.","Mean MDM lipid content increases with depth for all tested blood LDL and HDL levels, so deep cells are not just more numerous but also more lipid-laden.","HDL lipid capacity stays nearly uniform across the lesion at all times, so its spatial profile cannot explain spatial heterogeneity; only its time course changes, rising then falling as MDM efflux takes over."],"supporting_citations":[{"why":"Supplies the human coronary lesion images (lipid and MDM densities across DIT, fatty streak, and PIT stages) that define the two qualitative features the model must reproduce.","marker":"[1]"},{"why":"Provides the lipid-structured uptake/efflux formulation (incremental lipid index, capacity-limited uptake, HDL-mediated efflux) that the present model extends to space.","marker":"[44]"},{"why":"The spatially homogeneous lipid-structured model with lipid-dependent kinetics whose predictions for apoptosis- and emigration-dependent lipid content are compared in the Discussion.","marker":"[41]"},{"why":"Source for the observation that foam cells lose migration ability, motivating the lipid-dependent mobility term $\\psi_D$.","marker":"[14]"},{"why":"Source for free-cholesterol cytotoxicity and apoptosis of lipid-laden macrophages, motivating $\\psi_\\beta$.","marker":"[13]"},{"why":"Provides the macrophage migration-path data used to estimate the unladen diffusivity $D_M$.","marker":"[85]"},{"why":"Human aortic interstitial fluid data used to justify negligible LDL lipid density in the tunica media boundary condition.","marker":"[54]"},{"why":"Supports the assumption that MDMs do not traverse the internal elastic lamina into the media in early human lesions.","marker":"[12]"}],"fun_headline_variants":["Deep lipid first, then foam cells stall to make plaque peaks","Mobility drop, not lifespan, creates inner plaque lipid peaks","Steep mobility decline reproduces lesion's internal lipid maxima","Foam cells must stall sharply to match early plaque architecture"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole 'stall, not die' conclusion rests on the model's assumption that MDM mobility falls linearly with lipid content, so that a fully loaded cell is immobile when $\\psi_D=1$; if the real mobility-versus-load curve is nonlinear or saturating, the narrow parameter window that reproduces the lesion peaks could shift or vanish.","fun_headline_variants_meta":{"raw":{"variants":["Deep lipid first, then foam cells stall to make plaque peaks","Mobility drop, not lifespan, creates inner plaque lipid peaks","Steep mobility decline reproduces lesion's internal lipid maxima","Foam cells must stall sharply to match early plaque architecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000494,"raw_usage":{"total_tokens":2529,"prompt_tokens":1155,"completion_tokens":1374,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":771,"completion_tokens_details":{"reasoning_tokens":1305}},"tokens_in":771,"tokens_out":1374,"duration_ms":11213,"temperature":1.0,"reasoning_tokens":1305,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:28:24.874172+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure single-cell migration speed as a function of intracellular lipid content in monocyte-derived macrophages under controlled LDL/HDL conditions; if the speed-versus-load relationship is not approximately linear (for example, mobility drops sharply only beyond a threshold load, or saturates before zero), the model's necessity claim for $\\psi_D>0.98$, $\\psi_\\beta<0.2$ is a linearity artefact. A second check: scan many early human lesions for the transient deep MDM infiltration that the model produces before the peak forms; never observing it would support the authors' own suspicion that this wave is an artefact of the linear assumption.","supporting_citations":[{"cited_title":"Mathematical Biosciences, 108971 (2023)","cited_arxiv_id":null,"evidence_quote":"Provides the lipid-structured uptake/efflux formulation (incremental lipid index, capacity-limited uptake, HDL-mediated efflux) that the present model extends to space."},{"cited_title":"Bulletin of Mathematical Biology 85(9), 85 (2023)","cited_arxiv_id":null,"evidence_quote":"The spatially homogeneous lipid-structured model with lipid-dependent kinetics whose predictions for apoptosis- and emigration-dependent lipid content are compared in the Discussion."},{"cited_title":"a mechanism for atherosclerosis regression? Arteriosclerosis and Thrombosis: A Journal of Vascular Biology 12(8), 936–944 (1992)","cited_arxiv_id":null,"evidence_quote":"Source for the observation that foam cells lose migration ability, motivating the lipid-dependent mobility term $\\psi_D$."},{"cited_title":"Journal of Biological Chemistry 276(45), 42468–42476 (2001)","cited_arxiv_id":null,"evidence_quote":"Source for free-cholesterol cytotoxicity and apoptosis of lipid-laden macrophages, motivating $\\psi_\\beta$."},{"cited_title":"Euro- pean Biophysics Journal 45, 301–309 (2016)","cited_arxiv_id":null,"evidence_quote":"Provides the macrophage migration-path data used to estimate the unladen diffusivity $D_M$."},{"cited_title":"Proceedings of the Royal Society of London","cited_arxiv_id":null,"evidence_quote":"Human aortic interstitial fluid data used to justify negligible LDL lipid density in the tunica media boundary condition."}],"review_version":1}