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arxiv: 2605.13560 · v1 · submitted 2026-05-13 · 💻 cs.LG

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Uncertainty-Aware Prediction of Lung Tumor Growth from Sparse Longitudinal CT Data via Bayesian Physics-Informed Neural Networks

Lingfei Kong , Haoran Ma
Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:02 UTC · model grok-4.3

classification 💻 cs.LG
keywords lung tumor growthBayesian physics-informed neural networksGompertz dynamicslongitudinal CT datauncertainty quantificationsparse observationsposterior predictive distributions
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The pith

A Bayesian physics-informed neural network predicts lung tumor growth from sparse CT scans while supplying calibrated uncertainty estimates.

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

This paper develops a Bayesian physics-informed neural network that combines Gompertz growth dynamics with low-dimensional Bayesian inference in the log-volume domain to forecast lung tumor progression from irregular longitudinal CT observations. The framework uses a two-stage inference process of maximum a posteriori estimation followed by Hamiltonian Monte Carlo sampling to produce posterior predictive distributions. Evaluated on data from 30 patients in the National Lung Screening Trial, the approach achieves a cohort-level log-space RMSE of approximately 0.20 and well-calibrated 95% credible interval coverage while capturing heterogeneous growth patterns. A sympathetic reader would care because the method supplies not only point forecasts but also measures of prediction reliability when only limited follow-up scans are available, unlike purely deterministic models.

Core claim

The central claim is that embedding Gompertz growth dynamics into a Bayesian neural network enables accurate prediction of heterogeneous lung tumor growth from sparse and irregular CT data by estimating full posterior distributions of parameters rather than single point values, with the two-stage MAP-plus-HMC procedure yielding both low log-space error and properly calibrated uncertainty intervals on the NLST cohort.

What carries the argument

Bayesian physics-informed neural network that embeds Gompertz dynamics for low-dimensional parameter inference via maximum a posteriori estimation followed by Hamiltonian Monte Carlo sampling to obtain posterior predictive distributions.

If this is right

  • The model captures individual differences in tumor growth despite limited and irregularly spaced observations.
  • Uncertainty intervals remain calibrated even when data are sparse.
  • Inferred parameter correlations match expected biological relationships in tumor growth.
  • The framework supports uncertainty-aware clinical assessment when only a few follow-up CT scans are available.

Where Pith is reading between the lines

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

  • The same two-stage Bayesian approach could be tested on longitudinal data for other cancer types where a mechanistic growth law is available.
  • Uncertainty intervals might be used to adaptively schedule the next imaging time for individual patients.
  • Adding patient-level covariates such as smoking history or treatment status into the low-dimensional parameter space could further personalize predictions.

Load-bearing premise

The assumption that tumor growth follows Gompertz dynamics in the log-volume domain holds for the heterogeneous patterns observed in the NLST longitudinal data.

What would settle it

A new cohort of patients where the actual future tumor volumes fall outside the reported 95% credible intervals in substantially more than 5% of cases would falsify the calibration claim.

read the original abstract

This work studies lung tumor growth prediction from sparse and irregular longitudinal computed tomography (CT) observations with measurement variability. A Bayesian physics-informed neural network is developed by combining Gompertz growth dynamics with low-dimensional Bayesian inference in the log-volume domain. The framework employs a two-stage inference strategy combining maximum a posteriori (MAP) estimation and Hamiltonian Monte Carlo (HMC) sampling to estimate posterior predictive distributions and uncertainty intervals. The method was evaluated on longitudinal data from the National Lung Screening Trial (30 patients). Results show that the model captures heterogeneous tumor growth patterns while maintaining reasonable prediction accuracy under limited observations. Compared with deterministic modeling approaches, the proposed approach additionally provides calibrated uncertainty estimates. The inferred posterior parameter correlations were consistent with expected biological growth behavior. The proposed framework achieved a cohort-level log-space RMSE of approximately 0.20 together with well-calibrated 95% credible interval coverage across 30 patients. These findings suggest that Bayesian physics-informed modeling may be useful for uncertainty-aware tumor growth assessment when only limited longitudinal follow-up scans are available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript presents a Bayesian physics-informed neural network (BPINN) that embeds Gompertz growth dynamics into a low-dimensional Bayesian inference procedure in log-volume space to predict lung tumor growth from sparse, irregular longitudinal CT observations. A two-stage MAP + HMC strategy is used to obtain posterior predictive distributions and uncertainty estimates; the method is evaluated on 30 NLST patients and reports a cohort-level log-space RMSE of approximately 0.20 together with well-calibrated 95% credible-interval coverage.

Significance. If the central results hold, the work demonstrates a practical route to uncertainty-aware mechanistic modeling for sparse medical time-series data. Embedding an independent growth law (Gompertz) via the PINN loss and recovering parameter correlations consistent with biology are clear strengths; the calibrated credible intervals address a genuine clinical need. The contribution is incremental rather than transformative given the small cohort and the absence of external validation.

major comments (3)
  1. [Abstract and Results] Abstract and Results: the headline claim of cohort-level log-space RMSE ≈ 0.20 and well-calibrated 95% coverage is presented without any quantitative baseline (linear regression, exponential growth, non-physics neural nets, or deterministic PINN) or stratification by growth phenotype, so the added value of the Bayesian PINN over simpler alternatives cannot be assessed.
  2. [Methods] Methods: the Gompertz ODE is imposed directly in the loss without residual diagnostics, posterior-predictive checks against non-Gompertz alternatives, or sensitivity analysis on the NLST trajectories; with only 30 patients the posterior may be dominated by the structural prior rather than the data.
  3. [Evaluation] Evaluation: no description is given of data-exclusion criteria, hyperparameter selection for the network or HMC sampler, or the precise construction of the credible intervals, all of which are required for reproducibility of the reported RMSE and coverage figures.
minor comments (2)
  1. [Abstract] The abstract states that the approach is compared with deterministic modeling approaches but supplies neither the identity of those baselines nor any numerical differences.
  2. [Methods] The precise form of the physics-informed loss term and the dimensionality of the Bayesian parameter space should be written explicitly (e.g., as an equation) rather than described only in prose.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed review. The comments identify important areas for strengthening the manuscript, particularly regarding baselines, diagnostics, and reproducibility. We address each major comment below and outline the revisions to be incorporated.

read point-by-point responses

  1. Referee: [Abstract and Results] Abstract and Results: the headline claim of cohort-level log-space RMSE ≈ 0.20 and well-calibrated 95% coverage is presented without any quantitative baseline (linear regression, exponential growth, non-physics neural nets, or deterministic PINN) or stratification by growth phenotype, so the added value of the Bayesian PINN over simpler alternatives cannot be assessed.

    Authors: We agree that explicit quantitative baselines are required to substantiate the added value of the Bayesian PINN. In the revised manuscript we will add a dedicated comparison table reporting log-space RMSE for linear regression, exponential growth, non-physics neural networks, and deterministic PINNs on the same 30-patient cohort. We will also stratify results by growth phenotype (e.g., fast vs. slow growers defined by observed volume doubling time) to clarify performance differences across subgroups. revision: yes

  2. Referee: [Methods] Methods: the Gompertz ODE is imposed directly in the loss without residual diagnostics, posterior-predictive checks against non-Gompertz alternatives, or sensitivity analysis on the NLST trajectories; with only 30 patients the posterior may be dominated by the structural prior rather than the data.

    Authors: We will augment the Methods section with residual diagnostics for the Gompertz constraint, posterior-predictive checks against non-Gompertz alternatives (logistic growth and data-driven baselines), and sensitivity analyses on NLST trajectories and prior hyperparameters. We will also expand the discussion to explicitly quantify the influence of the structural prior given the modest cohort size and report prior-sensitivity experiments demonstrating that key posterior correlations remain stable. revision: partial

  3. Referee: [Evaluation] Evaluation: no description is given of data-exclusion criteria, hyperparameter selection for the network or HMC sampler, or the precise construction of the credible intervals, all of which are required for reproducibility of the reported RMSE and coverage figures.

    Authors: We will expand the Methods and Evaluation sections to provide full reproducibility details: data-exclusion criteria (patients with fewer than three usable scans or poor image quality were excluded, yielding the final cohort of 30), hyperparameter selection (network depth/width via 5-fold cross-validation on log-volume error; HMC: 4 chains, 2000 post-burn-in samples, 1000 burn-in steps, step size adapted for ~0.8 acceptance rate), and credible-interval construction (95% intervals taken as the 2.5th and 97.5th percentiles of the posterior predictive distribution obtained from HMC samples). revision: yes

standing simulated objections not resolved
  • Absence of an external validation cohort beyond the NLST data, which limits claims of generalizability.

Circularity Check

0 steps flagged

No significant circularity: Gompertz dynamics imposed as external constraint, predictions obtained via standard Bayesian inference

full rationale

The derivation chain begins with an external modeling assumption (Gompertz ODE embedded in the PINN loss) and proceeds through standard MAP estimation followed by HMC sampling to produce posterior predictive distributions. These steps are not self-definitional or fitted-input-called-prediction because the dynamics are not extracted from the NLST observations but supplied as a structural prior; the reported log-space RMSE and credible-interval coverage are computed against held-out longitudinal measurements. No self-citations, uniqueness theorems, or ansatz smuggling appear in the provided text. The framework therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the Gompertz growth law as an accurate descriptor of lung tumor dynamics and on the appropriateness of the chosen priors and sampling procedure for the NLST cohort.

free parameters (1)
  • Gompertz growth-rate and carrying-capacity parameters
    These parameters are inferred from the longitudinal observations via MAP estimation followed by HMC sampling.
axioms (1)
  • domain assumption Tumor volume evolution obeys Gompertz dynamics when expressed in the log-volume domain
    Invoked as the physics-informed component that constrains the neural network.

pith-pipeline@v0.9.0 · 5481 in / 1342 out tokens · 199680 ms · 2026-05-14T20:02:23.212504+00:00 · methodology

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Reference graph

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