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REVIEW 5 major objections 5 minor 23 references

Physiological neural representation for personalised tracer kinetic parameter estimation from dynamic PET

T0 review · 5 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that fitting an implicit neural representation to one patient's dynamic PET scan estimates the two-tissue compartment model parameters more accurately and with sharper anatomical boundaries than a self-supervised…

desk verdict A reasonable INR-for-kinetics idea whose evaluation only proves it can fit its own training loss; the parameter accuracy claim needs real validation. read the letter →

arxiv 2504.17122 v1 pith:P46XAXQM submitted 2025-04-23 eess.IV cs.AIcs.CV

Kartikay Tehlan , Thomas Wendler This is my paper · ORCID
classification eess.IVcs.AIcs.CV
keywords ImplicitNeuralRepresentationsTracerKineticModellingDynamicPETTwo-tissuecompartmentmodelParametricimagingAnatomicalpriors[18F]FDGTime-activitycurve
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

This paper tries to show that a small neural network trained on a single patient's dynamic PET scan can estimate the voxel-wise kinetic parameters of the two-tissue compartment model more accurately than a larger self-supervised deep network. The network maps spatial coordinates to four parameters — tracer influx K1, efflux k2, phosphorylation k3, and blood volume fraction Vb — by solving the kinetic model's differential equations inside the loss and comparing the reconstructed time-activity curve against measured frames. On a 24-patient [18F]FDG dataset, the authors report voxel-level mean-squared error of about 0.009 versus 0.066 for the baseline, with sharper edges in tumour and kidney regions. If this holds, personalised parametric imaging could be obtained per patient without large training corpora, which matters for data-scarce clinical settings.

What carries the argument

The load-bearing object is the implicit neural representation: a fully connected network with sinusoidal activations that takes normalised spatial coordinates, encoded by Gaussian Fourier Features, and returns K1, k2, k3, and Vb at that location. The two-tissue compartment model ODEs then synthesise a predicted time-activity curve from these parameters, and the MSE between predicted and measured curves is backpropagated through the ODE solve to update the network. This makes the physics the objective and turns the network into a continuous atlas of kinetic parameters; reconstructing a parametric image at any desired spatial resolution only requires querying the trained network at the relevant coordinates. Optional CT features enter as extra inputs, playing a supporting role rather than changing the final error.

What would settle it

Generate synthetic dynamic PET frames from known compartment parameters under the same 62-frame protocol, fit the INR to those frames, and compare the recovered K1, k2, k3, and Vb to the ground truth; if the time-activity-curve error stays low while the parameters drift from the truth, then the identifiability assumption is violated.

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Extended reading notes

Core claim

The central discovery is that a SIREN-based implicit neural representation, conditioned on Gaussian Fourier Features of spatial coordinates and optionally on CT-derived anatomical features, can encode the whole 3D+t dynamic PET signal of one patient as a continuous function whose output parameters obey the two-tissue compartment model. Because the forward model is differentiable, standard gradient descent on the time-activity-curve mean-squared error recovers the parameters. The authors report that all INR variants outperform the baseline in reconstruction error and produce parametric images with visibly higher spatial resolution, especially in the tumour and left kidney, where baseline errors concentrate; adding CT Hounsfield units or 4096-dimensional foundation-model features leaves the final error essentially unchanged while accelerating convergence.

Load-bearing premise

The four kinetic parameters must be identifiable from each voxel's measured time-activity curve alone under the 62-frame protocol; if different parameter sets give nearly identical curves, low reconstruction error will not guarantee correct parameters.

Editorial extensions

If this is right

  • Voxel-wise parametric images at the scanner's native 1.65 mm resolution become feasible from a single patient's dynamic scan without large training databases.
  • Tumour and highly vascularised regions, where the baseline concentrates most error, are precisely where INR fitting is claimed to improve: sharper boundaries in K1, k2, and k3 maps, aiding lesion delineation.
  • Per-patient optimisation makes the method personalisable: each patient's anatomy and physiology are encoded in the trained network, so downstream classification or segmentation could consume the INR directly.
  • Memory and training requirements (about 24 minutes and under 7 GB for the LoRes 3D variant) suggest the method can run on clinical workstation hardware, not only research GPU clusters.
  • CT priors (Hounsfield units or foundation-model features) do not change the final error but accelerate convergence, so anatomy can be used as a regulariser without sacrificing accuracy.

Reading between the lines

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

  • Because the paper reports MSE on reconstructed time-activity curves rather than agreement with ground-truth kinetic parameters, the strongest testable extension is to validate against an arterial input function or an independent reference method; low TAC error alone leaves an identifiability gap.
  • The near-identical MSE across HiRes and LoRes variants and with or without CT features suggests the INR's continuous spatial prior already captures most of the anatomical signal; CT features may mainly help convergence, implying a simpler spatial regulariser could obtain similar maps at lower cost.
  • If single-patient INRs prove reliable, dynamic PET analysis could shift from population-trained DNNs to per-patient fitting, which would make kinetic parameters portable across scanners and protocols because no cross-patient training distribution is assumed.
  • The authors note in Section 2.2 that all hyperparameters were taken from prior INR examples without ablation, so the reported error levels may not be the architecture's ceiling; testing whether tuning changes the comparison would be a direct next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper proposes per-patient implicit neural representations (SIREN with Gaussian Fourier features), optionally conditioned on CT Hounsfield values or on 4096-dimensional features from a 3D CT foundation model, to predict voxel-wise two-tissue compartment model (TCKM) parameters K1, k2, k3, and Vb from dynamic [18F]FDG PET. The parameters are optimized by minimizing the mean-squared error between TACs generated from the predicted parameters and the measured dynamic PET TACs. The authors evaluate multiple INR variants (2D/3D, HiRes/LoRes, with/without CT priors) against the self-supervised spatio-temporal neural network of De Benetti et al. on a 24-patient [18F]FDG dynamic PET/CT dataset, reporting lower voxel MSE, qualitative sharper parametric images, and lower training/inference resource footprints. The stated contributions are the introduction of a physiological INR for tracer kinetic modelling, its extension with CT foundation-model features, and its evaluation against a DNN baseline.

Significance. The idea of fitting a per-patient INR directly to dynamic PET data and reading out TCKM parameters is timely and potentially useful for data-efficient, personalized kinetic modelling. The paper makes its code available, which is a strength, and the qualitative figures suggest the INR can produce visually plausible parametric images. However, the present evaluation cannot support the abstract's claims of superior kinetic parameter estimation. The reported MSE in Table 1 is exactly the training loss minimized during optimization on the same data, there is no ground-truth kinetic parameter comparison, no nonlinear least squares baseline, no identifiability analysis, and the forward TCKM model is underspecified because the time-varying plasma input function is never given. As a result, the central claim that the INR estimates accurate physiological parameters is not established by the evidence presented.

major comments (5)
  1. [§2.2–2.3, Eq. (1)] The forward model is underspecified. The text states that the predicted TAC is generated by solving the TCKM ODEs, but it never writes the ODEs, the plasma input function Cp(t), or the delay/dispersion treatment; Section 2.2 only says the dynamic PET data were divided by the maximum of the image-derived input function. As written, Eq. (1) cannot be evaluated by a reader, and the outputs K1, k2, k3, Vb cannot be interpreted as physiological TCKM parameters. This is a core reproducibility and correctness issue.
  2. [Table 1, Eq. (1)] The reported voxel MSE is the same quantity minimized during training, evaluated on the same patient data used for optimization. Table 1 therefore reports training fit quality, which is expected to be low for an overparameterized per-patient network, and it does not validate the estimated kinetic parameters. The paper needs a comparison against voxel-wise nonlinear least squares fits (or otherwise reference parameter maps) with error and bias metrics, plus an out-of-sample or cross-validation scheme.
  3. [§2.3, Results] No identifiability analysis is provided for the four-parameter TCKM under the 62-frame protocol and the scalar-normalized input function described in Section 2.2. Multiple parameter sets can produce nearly identical TACs, so a low training MSE does not imply correct K1, k2, k3, and Vb. A simulation study with known ground-truth parameters should be included to demonstrate that the optimized parameters are recoverable and that low TAC error translates into parameter accuracy.
  4. [§3, Figs. 1–4, Table 1] The abstract claims superior spatial resolution and improved anatomical consistency, but no quantitative metrics for edge preservation, contrast, or anatomical consistency are reported; Figure 4 supports these claims only by visual inspection. In addition, Table 1 shows identical voxel MSE for inr-HiRes-2D, HU-INR-2D-HiRes, and FM-INR-2D-HiRes, so the claimed benefit of the CT and foundation-model priors is not evidenced. The Discussion's statement of "slightly faster convergence (not shown)" needs supporting convergence curves or a quantitative comparison.
  5. [§2.1, Table 1] The manuscript states that the dataset contains 24 patients, but Table 1 reports a single MSE value per variant and Figures 1–4 show one exemplary patient. It is unclear whether the quantitative results are averaged over all patients, over a single slice, or over a single patient. The aggregation and per-patient variability must be specified; otherwise the claim that the results apply to the [18F]FDG dynamic PET/CT dataset is not supported.
minor comments (5)
  1. [§2.2] The Gaussian Fourier feature construction is ambiguous: state whether the entries of B are drawn as N(0, σ²) and then multiplied by 10, and give the value of σ; currently "a standard deviation chosen to control frequency bandwidth" is not sufficiently precise for reproduction.
  2. [Table 1] The training and inference time columns are confusing because dashes appear for some entries and both training and inference times are listed for INR variants; clarify which times apply to which model and on which hardware they were measured.
  3. [Fig. 1 caption] The caption says "MMSE is higher" where "MSE" is intended; please correct the typo.
  4. [§1] The phrase "state-of-the-art DNNs" overstates the comparison, since only the De Benetti et al. network is evaluated as a baseline; consider saying "one previously proposed self-supervised DNN."
  5. [Throughout] There are minor typographical errors, including "de Benetti el al." (should be "de Benetti et al."), "signficantly" in Fig. 4, and inconsistent notation for Vb/V_B.

Circularity Check

1 steps flagged · score 6.0 of 10

Headline accuracy metric is the training loss itself, so the reported low TAC MSE is a fitted-input result rather than a validation of kinetic-parameter accuracy.

  1. fitted input called prediction [Section 2.2-2.3, Eq. (1); Section 3, Table 1]
    "Parameter estimation was performed by minimising the mean-squared error (MSE) between the time activity curve (TAC) generated from the predicted parameters (K1,k2,k3,Vb) and real TAC values measured at the input coordinate in the dynamic PET. ... This comparison employed an MSE loss, defined as: MSE = ... (1). As comparison metrics, we used MSE with its standard deviation (SD)."

    The network is trained to minimize exactly the per-voxel TAC MSE in Eq. (1), and the same Eq. (1) MSE is then reported in Table 1 as evidence of 'superior spatial resolution, lower mean-squared error'. The low TAC MSE is therefore the training objective evaluated on the training data: the 'prediction' of the TAC from estimated K1,k2,k3,Vb is, by construction, the fitted curve. Since no ground-truth kinetic parameters, identifiability analysis, or external generalization test is provided, the low MSE cannot independently support the central claim that the INR estimates accurate kinetic parameters.

full rationale

The comparison loop is closed at the loss function: the INR's only objective is to reproduce measured TACs via the TCKM forward model, and its only headline evaluation is the TAC MSE on the same measured TACs. This is a fitted-input-called-prediction pattern: the evaluation metric is identical to the optimized objective, so the low errors in Table 1 are expected from the fitting procedure and do not by themselves establish parameter accuracy. The baseline is De Benetti et al. [5], a prior work with overlapping authorship, but that alone is not circular; the circularity is the in-sample MSE metric. In addition, the paper does not specify the plasma input function Cp(t) or the TCKM ODEs, and it provides no identifiability analysis or reference kinetic parameters; these are serious reproducibility/validity concerns, but they are missing-support issues rather than additional circular steps. Under the proportionality rule, the central quantitative claim reduces to the fit, so score 6 is appropriate.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical or conceptual entities; it combines existing components: SIREN, Gaussian Fourier Features, the two-tissue compartment model, and a CT foundation model. The free parameters are fixed hyperparameters adapted from prior work rather than fitted to the test data, but they are not ablated or justified for this task. The main assumptions concern the validity and identifiability of the kinetic model and the unstated image-derived input function.

free parameters (5)
  • GFF frequency count = 256
    Chosen from Tancik et al. and not varied in experiments; controls the frequency bandwidth of the coordinate encoding.
  • GFF frequency multiplier = 10
    Adopted from literature recommendations in Section 2.2; scales the random frequency matrix B and affects the range of learnable frequencies.
  • SIREN architecture width and depth = 3 hidden layers, 512 units each
    Adapted from SIREN examples without ablation experiments, as stated in Section 2.2.
  • Learning rate and epoch count = 1e-5, 100 epochs
    Fixed hyperparameters for Adam optimization; no sensitivity analysis is reported.
  • GFF standard deviation = not specified
    Section 2.2 states a standard deviation was chosen to control frequency bandwidth, but the actual value is not given.
assumptions (5)
  • domain assumption The two-tissue compartment model with parameters K1, k2, k3, and Vb adequately describes voxel-level [18F]FDG kinetics.
    The entire loss function in Section 2.3 and all parametric images rely on this model; the paper does not test model misspecification or alternative compartment models.
  • domain assumption The image-derived input function (IDIF) is a valid surrogate for the arterial plasma input function.
    Section 2.2 divides TACs by the IDIF maximum but never states how the IDIF was derived or validated; errors in the input function propagate into all fitted parameters.
  • domain assumption The four TCKM parameters are identifiable from a single voxel TAC sampled at the 62-frame protocol.
    If identifiability fails, multiple parameter sets can yield nearly the same TAC, so low training error does not imply correct parameters; no identifiability analysis is provided.
  • domain assumption The measured TAC, after 2 mm Gaussian filtering and optional downsampling, is reliable enough that fitting it yields physiologically meaningful parameters.
    No noise model, no parameter regularization, and no handling of voxels with poor statistics is described in Sections 2.2-2.3.
  • domain assumption CT foundation model features (4096 per voxel) carry information relevant to the kinetic parameters.
    This is the premise of the CTFM variant in Section 2.2; the results only show similar MSE, not that the features improve parameter accuracy.

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Pith. "Pith review of Physiological neural representation for personalised tracer kinetic parameter estimation from dynamic PET." pith.science (2026). https://pith.science/paper/P46XAXQM

@misc{pith2026250417122,
  author       = {Pith},
  title        = {Pith review of: Physiological neural representation for personalised tracer kinetic parameter estimation from dynamic PET},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P46XAXQM}},
  note         = {Machine review of arXiv:2504.17122}
}
abstract

Dynamic positron emission tomography (PET) with [$^{18}$F]FDG enables non-invasive quantification of glucose metabolism through kinetic analysis, often modelled by the two-tissue compartment model (TCKM). However, voxel-wise kinetic parameter estimation using conventional methods is computationally intensive and limited by spatial resolution. Deep neural networks (DNNs) offer an alternative but require large training datasets and significant computational resources. To address these limitations, we propose a physiological neural representation based on implicit neural representations (INRs) for personalized kinetic parameter estimation. INRs, which learn continuous functions, allow for efficient, high-resolution parametric imaging with reduced data requirements. Our method also integrates anatomical priors from a 3D CT foundation model to enhance robustness and precision in kinetic modelling. We evaluate our approach on an [$^{18}$F]FDG dynamic PET/CT dataset and compare it to state-of-the-art DNNs. Results demonstrate superior spatial resolution, lower mean-squared error, and improved anatomical consistency, particularly in tumour and highly vascularized regions. Our findings highlight the potential of INRs for personalized, data-efficient tracer kinetic modelling, enabling applications in tumour characterization, segmentation, and prognostic assessment.

Figures

Figures reproduced from arXiv: 2504.17122 by the authors.

Figure 1
Figure 1. MSE (top row, red) and SD (lower row, blue) for a coronal slice of an exemplary patient. MMSE is higher on average over the complete slice (see FDB112 LoRes MSE Clipped vs. INR LoRes MSE), but particularly in the location of tumour (black arrow) and kidneys (green arrow), ignoring the blood pool (visible larger vessels). Also SD images show less variation on the MSE over time for our model. The comparison of LoRes a… view at source ↗
Figure 2
Figure 2. Comparison of different variants of HiRes models in a coronal slice of an exem￾plary patient. The first column shows the corresponding CT and static PET images. With increased input information, the degree of granularity slightly increases while MSE average and SD range remaining almost identical [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. MSE along segment crossing a mediastinal lymph node metastasis (depicted in corresponding PET slice, left). The plots show the MSE as a function of space (axis going from left to bottom of the graphs) and time (axis going from bottom to right). The center plot is the MSE for the 2D baseline, while the right plot shows the inr￾LoRes-2D variant. While the baseline shows the error in the center of the lesion at a late … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Coronal slice of parametric images generated with the 2D baseline (upper row second to last columns) and with the inr-LoRes-2D variant. The INR parameters show signficantly higher resolution in terms of more details and sharper edges. In concrete, the kidneys (green ar…

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