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The paper claims that a physics-informed neural network can recover the proliferation rate in a porous-medium tumor growth model from sparse, noisy binary images, and that the inferred rate predicts future tumor radius with errors around 2–

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PINNs recover the proliferation rate in a nonlinear tumor-growth PDE with ~1% error on synthetic data and predict real tumor radii within ~5%, though validation choices limit the strength of that claim.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection The real-data demonstration is genuinely new and the paper is worth engaging, but the theoretical overclaims and the untested model-misspecification leave the central biological claim unproven. the 5 major comments →

arxiv 2511.15940 v2 pith:3W3WFNV2 submitted 2025-11-20 math.AP

Data-Driven Parameter Identification for Tumor Growth Models

Liu Liu , Yifei Wang , Qinyu Xu , Xiaoqian Xu This is my paper
classification math.AP MSC 35Q9235K6568T0792C50
keywords tumor growth modelingporous medium equationphysics-informed neural networksparameter identificationinverse problemsbinary observationsradial symmetry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper tries to establish that a physics-informed neural network can recover the proliferation rate in a porous-medium tumor growth model from sparse, noisy, binary observations of tumor presence, without directly measuring cell density. The claim is that the PDE's structure regularizes the inversion enough that the inferred rate v is reliable, even though the data are only 0/1 labels and cover only early times. If true, this would give a practical route to estimating tumor growth parameters from standard lab images and predicting future tumor radius. The paper supports this with synthetic recovery tests, noise robustness tests, and a real-data demonstration in which the inferred v predicts held-out tumor radii with errors of 4.7% and 2.3%.

Core claim

For the model ρ_t − Δρ^3 = vρ with radial patch initial condition and zero Dirichlet boundaries, the paper reports that training a PINN with a combined PDE-residual, initial/boundary, and binary cross-entropy data loss yields a convergent estimate of v. On synthetic data with known v in [1.7, 2.2], recovered values are within about 1%. After adding Gaussian noise, relative errors stay below 1% after 30,000 epochs. Applied to real lab images, the inferred proliferation rate v ≈ 3.1264, combined with forward simulation, predicts tumor radii 2.2308 and 2.4426 at held-out times where observed radii are 2.13 and 2.5, i.e., relative errors 4.732% and 2.296%. The paper argues this shows the inferre

What carries the argument

The central object is the porous medium equation ρ_t − Δρ^m = g(x,t)ρ, with m=3, in which the proliferation rate v enters linearly as a source term and is treated as a trainable scalar. The machinery is the PINN loss, a weighted sum of PDE residual, initial condition, boundary condition, and data misfit, where the data term is binary cross-entropy because observations are 0/1 tumor-presence labels. The PINN simultaneously fits the density field uθ(t,x,y) and the parameter v, and the paper supplies convergence results—loss tending to zero and L1 contraction to the PDE solution—to justify that the regularization is not ad hoc.

Load-bearing premise

The real tumor is assumed to follow the specific equation ρ_t − Δρ^3 = vρ with radial symmetry, a fixed patch initial condition, zero Dirichlet boundaries, and no necrotic core; if the true growth law differs, the inferred v is a fitting artifact rather than a biological parameter.

What would settle it

Generate synthetic data from a non-radial or nutrient-limited tumor model and run the same pipeline; if the inferred v is inaccurate or unstable, or if a nutrient-limited model fits the held-out radii with substantially lower error, then the claimed reliability of v is specific to the model class and not a biological identification.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the method is correct, proliferation rate can be estimated from routine lab images without density calibration, since binary labels suffice.
  • Inferred v from early-time data can be used to predict future tumor boundary, enabling treatment-timing simulations.
  • The same framework extends to spatially varying proliferation rates and unknown initial densities, though with somewhat larger errors.
  • The convergence theorems imply that PINN-based inversion inherits stability from porous-medium L1 contraction, so noisy sparse data does not destroy recovery.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial: The recovered v is a parameter of the assumed model class; the paper does not compare against nutrient-limited or different-m models, so v should be read as an effective rate, not a directly biological constant.
  • Editorial: The binary cross-entropy loss discards density magnitude; a natural test is to see whether adding density information from imaging tightens the estimate or reveals identifiability limits.
  • Editorial: The same pipeline could be used backwards to select among competing PDE models by comparing held-out prediction errors across exponents m and source forms; the paper does not do this.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The manuscript proposes a PINN-based method for estimating the proliferation rate v in the tumor-growth porous-medium equation ρ_t − Δρ^3 = vρ. It first tests the method on synthetic data for v = 1.7, 1.8, 1.9, 2.0, 2.1, 2.2 and reports relative errors around 1%, then adds Gaussian noise and shows convergence. The method is then applied to binary lab images of tumor radius: using early-time data to infer v and predicting two held-out times gives relative radius errors 4.732% (t = 0.875) and 2.296% (t = 1.0) (Table 3). The manuscript also extends the framework to spatially varying proliferation parameters and to an unknown initial density. Section 4 contains convergence theorems based on universal approximation and L1 contraction estimates.

Significance. If the real-data claims were properly supported, the paper would provide a useful benchmark for inverse parameter identification in nonlinear tumor-growth PDEs from scarce, binary observations. Strengths include the reproducible code/data repository, the use of a separate numerical solver for synthetic ground truth, the explicit treatment of noisy data in Section 5.2, and the held-out time-point predictions in Section 6.3. However, the central biological claim — that the inferred v ≈ 3.13 is the proliferation rate of the real tumor — is currently under-supported because the real-data validation does not exclude model misspecification, the radius threshold is chosen post hoc, and the training/testing split is inconsistent with the data table. The theoretical section also overclaims relative to what is proved. These issues are load-bearing but can be addressed with additional experiments and revised claims, so major revision is appropriate.

major comments (5)
  1. [§6.3, Figs. 8–9, Table 3] The predicted tumor radius is read from the contour u = 0.1, while the observed radii in Table 2 come from image boundaries. No sensitivity analysis is reported for this threshold. Since the level-set radius is threshold-dependent, the claimed 4.732% and 2.296% errors are not robust evidence. Please report predicted radii for several thresholds (e.g., 0.05, 0.1, 0.2, 0.5) and show whether the relative errors remain below 5%.
  2. [§6.2, Table 2] The text states that the first five time points t = 0, 0.125, 0.25, 0.375, 0.5 are used for training and t = 0.875, 1.0 for testing, but Table 2 contains no observed data at t = 0 or 0.125 and includes t = 0.625 and 0.75. This inconsistency makes the actual training window and the meaning of the held-out prediction unclear. Please specify exactly which rows of Table 2 enter L_data, and whether t = 0.625 and 0.75 are used for training, validation, or excluded.
  3. [§4, Theorems 2 and 3] Theorem 2 asserts that a two-layer neural network can approximate the solution ρ in the C^2(Ω) ∩ C^1([0,T]) norm, citing the classical universal approximation theorem (Cybenko), which only gives C^0 approximation on compact sets. No proof or reference for the stronger norm approximation is provided. Since Theorem 3's convergence of the loss relies on ∥ρ − ρ_NN∥_{C^2∩C^1} → 0, this is a genuine gap. Either prove a refined approximation theorem under stated assumptions on σ and ρ, or explicitly weaken the convergence claim to hold under the additional hypothesis that such a C^2∩C^1 approximation exists.
  4. [§6.1, §6.3, §7] The real-data experiment is conducted under a fixed model class: m = 3, radial symmetry, patch initial condition, zero Dirichlet boundary, and no necrotic core. The paper never runs an alternative model through the same pipeline. If a deliberately wrong model (e.g., m = 2, nutrient-limited growth, or a nonsymmetric initial condition) also fits the binary data and predicts held-out radii within a few percent, then the experiment has low discriminative power and v ≈ 3.13 may be a fitting artifact rather than a biological parameter. Please add such a wrong-model comparison and discuss what would distinguish the assumed model.
  5. [§6.2, Eq. (27)] The binary cross-entropy loss in Eq. (27) requires ŷ_i ∈ [0,1], but the network output is described as having 'absolute value added' to ensure nonnegativity; no sigmoid or other probabilistic mapping is mentioned. If the output can exceed 1, the logarithms in the BCE are ill-posed. Please clarify the exact output transformation used in the real-data experiments; if a sigmoid is applied, state it explicitly and give its role in extracting the u = 0.1 contour radius.
minor comments (4)
  1. [§5.1, Figs. 2–3] The synthetic recovery results are reported only as 'around 1%' relative error without exact numeric values or standard deviations across random seeds. Please add a table with the final v_pred for each v_true and, ideally, repeated runs with different initializations.
  2. [§4.2, Remark 1] The proof uses a boundary loss in terms of ρ^m while the discrete loss (11) uses u directly. The remark says the difference is negligible in practice; a brief explanation or an experiment supporting this would improve the paper.
  3. [§7.2, Fig. 13] The inferred initial density a = 0.3754 is substantially different from the value a = 1 used in the earlier sections. Please discuss whether this is biologically plausible and whether it changes the interpretation of the inferred v.
  4. [General] Minor text issues: 'parametermas' should be 'parameter m'; 'Xu'an Dou' should likely be 'Xuan Dou'; 'iterations' appears as 'interations'; references [4] and [5] are duplicates, as are [23] and [24]. The train/test description in Section 7.1 ('training data up to t=0.75') should be made consistent with Section 6.2.

Circularity Check

0 steps flagged

No circular derivation: held-out radius predictions are genuine forward extrapolations; only minor non-load-bearing self-citations.

full rationale

The derivation chain is not circular. The unknown proliferation rate v is a trainable parameter in the PINN loss (Eqs. 8–12; binary-data version Eq. 27) and is fit using early-time binary observations. The reported test errors (Table 3) come from fixing the inferred v, solving the same PDE forward with the numerical solver, and comparing the density-0.1 contour radius to observed radii at t=0.875 and t=1, which are not used in the fit. This is a genuine extrapolation in time, not a refit or a definitional identity. Synthetic validation (Section 5) uses a solver from Ref. [16] with no author overlap, and serves as a consistency check rather than as evidence for the real-data claim. The self-citations ([11], [18], [28]) are background or methodological and are not load-bearing: none is used to impose uniqueness or to smuggle in the central ansatz. The m=3 model, radial symmetry, patch initial condition, and 0.1 radius threshold are explicit assumptions; if they are wrong, the inferred v may be a fitting artifact, but that is a model-misspecification or validity risk, not a circularity. Minor inconsistencies (the stated training split vs. Table 2, and the lack of threshold sensitivity) reduce transparency but do not make the prediction equivalent to its inputs. Score 2 reflects only the presence of minor self-citations that are not load-bearing.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The empirical results depend on inferred proliferation parameters (v, v1, v2), an inferred initial-density amplitude a, a post hoc boundary threshold, per-case tuned loss weights, and strong modeling assumptions (PME with m=3, radial symmetry, patch IC, zero Dirichlet, no necrosis). No new physical entities are introduced.

free parameters (5)
  • v (proliferation rate) = 3.1264 (real-data Fig 6); approximately 2.9-3.0 (multi-init Fig 7); 1.7-2.2 in synthetic cases
    Central unknown coefficient in ρ_t−Δρ^3=vρ, inferred jointly with network weights.
  • v1, v2 (spatially varying proliferation) = 7.0968, -5.9086
    Coefficients in g(x,y)=v1+v2 sin(r), inferred in Section 7.1.
  • a (initial tumor density) = 0.3754
    Inferred initial density within r<0.5 in Section 7.2.
  • Tumor-boundary density threshold = 0.1
    Hand-chosen after training to define predicted tumor radius from density maps (Section 6.3); no sensitivity analysis is reported.
  • Loss weights w1-w4 = w4=50-100 per vtrue in Table 1; w4=5 for real data
    Selected by hyperparameter tuning trials; w4 changes with vtrue, so synthetic recovery accuracy is partly tuned.
axioms (6)
  • domain assumption The real tumor is governed by ρ_t − Δρ^3 = vρ with a single constant v in the observed window.
    Central to the inversion; no model selection against nutrient-limited or other growth laws (Sections 2 and 6).
  • domain assumption Radial symmetry, patch initial condition ρ0=1 for r<0.5, zero Dirichlet boundaries.
    Stated in Sections 2 and 6.1; used to convert 2D images to radius and to define the computational domain.
  • domain assumption Darcy's law and pressure law p=m/(m−1)ρ^{m−1} with m=3.
    Brings the model to Eq (5); m=3 is chosen to complement Bayesian work in [10] (Section 2).
  • ad hoc to paper Two-layer neural networks can approximate the PME solution in C^2(Ω)∩C^1([0,T]) norm.
    Theorem 2 asserts this; cited Cybenko [9] only proves uniform approximation, so the C^2 statement is an unproved assumption used for the convergence proof.
  • standard math The entropy L1 contraction estimate (23) from Kobayasi [15] applies to neural-network residuals of the form (22).
    Used in Theorem 4 without verifying that the neural-network residual e_r satisfies the hypotheses of the cited theorem.
  • domain assumption Binary image labels (1=tumor, 0=no tumor) with necrosis neglected faithfully represent tumor presence for radius measurement.
    Section 6.1: 'Tumor necrosis at the center is neglected for simplicity'; the BCE loss equates predicted density to binary labels.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Data-Driven Parameter Identification for Tumor Growth Models." pith.science (2026). https://pith.science/paper/3W3WFNV2

@misc{pith2026251115940,
  author       = {Pith},
  title        = {Pith review of: Data-Driven Parameter Identification for Tumor Growth Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3W3WFNV2}},
  note         = {Machine review of arXiv:2511.15940}
}
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read the original abstract

Modeling tumor growth accurately is essential for understanding cancer progression and informing treatment strategies. To estimate the parameters in the tumor growth model described by a nonlinear PDE, we adopt Physics-Informed Neural Networks (PINNs) and DeepONet, which show advantages especially when the observation data is scarce and contains noise. With the help of real-life lab data, we have demonstrated the potential of applying deep learning tools to address data-driven modeling for tumor growth in biology.

Figures

Figures reproduced from arXiv: 2511.15940 by Liu Liu, Qinyu Xu, Xiaoqian Xu, Yifei Wang.

Figure 1
Figure 1. Figure 1: PINNs setup and framework 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Predicted values of parameter v by PINNs model (blue) compared with ground truth (red) [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative errors for each value of v. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relative errors between learning parameter and its target value. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Observed images of tumor growth in lab, in which the tumor growth profile is marked in red circle. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The evolution of v across successive iterations, showing a convergent trend to 3.1264 over 0 to 80000 iterations [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The convergence of v with different initial values across iterations 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The left panel displays the spatial distribution of tumor density at time t = 0.875, where color [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The left panel displays the spatial distribution of tumor density at time t = 1, where color intensity [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The evolution of the variables v1 and v2 across successive iterations, showing a convergent trend to 7.0968 and -5.9086 over 0 to 80000 iterations. After the training, we simulate tumor growth forward using the learned values of v1 and v2 and compare the predicted tumor boundaries at t = 0.875 and t = 1 against the ground truth values, which are recorded in [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The left panel displays the spatial distribution of tumor density at time t = 0.875, where color [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The left panel displays the spatial distribution of tumor density at time t = 1, where color intensity [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The evolution of the variables v and a across successive iterations, showing a convergent trend to 3.1441 and 0.3754 over 0 to 80000 iterations. This extension highlights the flexibility of the PINNs framework in handling additional un￾known parameters, even when they are embedded in the initial conditions rather than the governing PDE. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.