Detecting malignant dynamics on very few blood sample using signature coefficients

R\'emi Vaucher; St\'ephane Chr\'etien

REVIEW 4 major objections 5 minor 13 references

Detecting malignant dynamics on very few blood sample using signature coefficients

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Signature coefficients from seven blood samples can detect malignant ctDNA dynamics with high accuracy.

desk verdict Novel application of signature theory to ctDNA monitoring, but the null distributions are algebraically wrong and the p-values are uncalibrated, so the headline detection results don't hold as stated. read the letter →

arxiv 2506.09097 v1 pith:ODEVDKU7 submitted 2025-06-10 q-bio.QM cs.LGstat.ML

Rémi Vaucher , Stéphane Chrétien This is my paper

classification q-bio.QMcs.LGstat.ML

keywords SignaturetheoryCirculatingtumorDNABirthanddeathprocessesAnomalydetectionCancerearlyMultipletestingLévyareaSkellamdistribution

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

The paper proposes that signature coefficients—numbers that encode the shape of a time-ordered signal through iterated integrals—can act as test statistics for detecting aggressive tumor dynamics from as few as seven blood samples per patient. Under a stochastic model in which the benign ctDNA signal at each sampling time is an independent centered Poisson variable—a Poisson count shifted so that its mean is zero—the authors derive the null distributions of the first- and second-order signature coefficients and of the Lévy area; these distributions are all Skellam laws with parameters proportional to the expected benign ctDNA count. These laws feed a multiple-testing procedure that compares each patient's observed coefficients with the benign hypothesis and applies a Benjamini–Hochberg correction. On simulated benign and malignant ctDNA trajectories, the corrected procedure reports accuracy above 0.94 and an F1-score of 0.999, suggesting that a non-invasive, low-burden test for malignancy could be built from very sparse blood draws. The central claim is that signature features, rather than fitted birth–death parameters, carry enough information to separate benign from malignant dynamics.

What carries the argument

The Signature transform of the two-dimensional path $X_t=(t,C^B_t)$—a feature map that records the shape of a path as a hierarchy of iterated integrals—restricted to first- and second-order coefficients and to the Lévy area $LA=\tfrac12(S^{12}-S^{21})$. The shuffle product identity expresses products of lower-order coefficients as sums of higher-order ones, letting the authors compute $S^{12}$ as a linear combination of the sampled $C^B_{t_i}$ values; Chen's identity gives the signature of the piecewise-linear interpolation of the irregularly sampled observations. Under the null, those linear combinations are differences of independent Poisson sums, hence Skellam distributed. This machinery converts a sparse, irregularly sampled biomarker trajectory into a small set of coefficients with known null laws, avoiding the need to estimate birth–death parameters from only seven observations.

What would settle it

Simulate benign-only trajectories in which consecutive samples are spaced close enough that the ctDNA level has not returned to equilibrium between draws, then run Algorithm 1; if the reported p-values are not uniform on [0,1], the claimed false-positive control fails.

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Extended reading notes

Core claim

Under the null hypothesis that the patient has only benign tumors, the benign ctDNA signal is modeled as a centered Poisson variable with parameter $\Lambda$, in equilibrium and independent across sampling times. The paper derives the null distributions of the signature coefficients of the path $X_t=(t,C^B_t)$: the first coefficient is $S^1=C^B_T-C^B_0\sim\mathrm{Skellam}(\Lambda,\Lambda)$, while the second-order coefficients $S^{12}$, $S^{21}$, and the Lévy area $LA=\tfrac12(S^{12}-S^{21})$ each follow Skellam distributions whose parameters are explicit functions of $\Lambda$, the total time $T$, and the sampling step $\Delta t$. These laws turn the signature coefficients into test statistics: Algorithm 1 computes the observed coefficients after centering, evaluates p-values against the corresponding Skellam null with $\Lambda$ estimated by the patient's own sample mean, and rejects the benign hypothesis when the Benjamini–Hochberg-corrected p-values are extreme. On 4000 benign and 36000 benign-then-malignant simulated trajectories sampled seven times over 2000 days, the authors report accuracy above 0.94 for each individual test and 0.958 for the corrected multiple test, with F1-scores reaching 0.999. They also state that no alternative feature-extraction approach they found approaches these goals.

Load-bearing premise

The test's error rates depend on the assumption that the benign signal seen at each blood draw is an independent random fluctuation around the same constant average, and that substituting the patient's own seven-sample average for that constant leaves the p-values valid.

Editorial extensions

If this is right

A malignancy test based on seven blood draws spread over roughly seven years could replace or complement imaging and biopsy as a low-burden detection route.
No per-patient fitting of birth–death model parameters is needed: the signature coefficients themselves are the test statistics, sidestepping an otherwise impossible small-sample estimation problem.
Pooling the four signature tests with a Benjamini–Hochberg correction gives 0.958 accuracy and a 0.999 F1-score in the paper's simulations.
The feature-extraction step is unsupervised, so the method does not depend on large labeled training cohorts.
The explicit null distributions provide the starting point for the authors' proposed extensions to higher-order signature coefficients, random sampling intervals, and protocols with fewer than seven samples.

Reading between the lines

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

An implicit extension is that the same Skellam-null machinery could apply to any biomarker whose benign concentration is a Poisson-type equilibrium with a known mean, not only ctDNA.
A testable prediction beyond the paper is that accuracy should degrade as sampling intervals approach the ctDNA elimination half-life, because consecutive samples would no longer be independent as the null model requires.
The near-perfect F1-score in simulation may reflect the large separation between the benign and malignant trajectories chosen in the experiments; a harder benchmark would include smaller net growth rates and measurement noise.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a look if you care about either signature methods or ctDNA monitoring, but the central statistics are off. The genuinely new part is deriving null laws for low-order signature coefficients of a benign birth-death ctDNA process, and the signature pipeline for very few samples is a reasonable idea. The exposition of signature theory is clear, and the GitHub code is a plus. Those are real strengths.

The problem is the math behind Algorithm 1. In Eqs. (19)-(21), they treat constant multiples of Poisson counts as Poisson: if C ~ Poisson(Λ), then cC is not Poisson(cΛ) unless c=1. The probability generating function is exp(Λ(z^c − 1)), not the Poisson PGF. So S12, S21, and the Lévy area are linear combinations of Poissons with unequal coefficients, not Skellam variables. The empirical histograms in Figs. 7-10 cannot actually match those laws, so the claimed "adequation" is suspect. On top of that, Algorithm 1 plugs the patient's seven-point sample mean into Λ and then computes p-values from a distribution derived for a known mean. The estimation error in Λ is material at n=7 and changes the null distribution of the centered signature. The p-values are therefore not uniform under the null, and the accuracy/F1 scores in Table 1—which look internally consistent to me, contrary to the reader's note—are computed from uncalibrated decisions.

The numerical validation is also in-sample: the simulated benign and malignant trajectories come from the same BDP family that motivates the null. That's a circularity, not fatal on its own, but combined with the missing real data and the absence of any comparison with alternatives (the paper itself says those are future work), the headline "first of its kind... very high detection" claim is well beyond what the evidence supports.

Who gets value from this? Someone thinking about signature-based hypothesis testing on biomarker traces would find the setup interesting and the flaws instructive. But as a paper claiming a working detection method, it's not there. The errors are specific and fixable in principle—compute the correct null via characteristic functions, calibrate with Λ estimated—but the current manuscript would need substantial revision.

My recommendation: send it to peer review anyway. The idea is novel and deserves a serious referee who can force the corrected derivations and a real-data or at least an independent-model validation. Desk rejection would be defensible, but a thorough review could turn this into a useful contribution.

Referee Report

4 major / 5 minor

Summary. The paper proposes a statistical test for detecting malignant ctDNA dynamics from a small number of blood samples, using signature coefficients of the time-rescaled, centered ctDNA path. The benign ctDNA level is modelled as a centered Poisson process at equilibrium, and the authors derive purported Skellam null distributions for the second-order signature coefficients S12, S21, and the Lévy area. These laws are then used in a Benjamini-Hochberg multiple-testing procedure (Algorithm 1) applied to simulated birth–death trajectories with seven samples per patient. The paper reports accuracy above 0.94 and F1-scores near 0.999, and claims that this is the first method of its kind for early detection from time-ordered blood samples.

Significance. If the null distributions were correct, the paper would offer a genuinely novel application of signature theory to a clinically important problem, with the practical appeal of using very few samples and avoiding supervised feature learning. The authors provide reproducible code and build on an established birth–death model of ctDNA shedding, which are clear strengths. However, the core distributional derivation contains algebraic errors, and Algorithm 1 introduces an unaccounted plug-in estimate of the nuisance parameter. The reported detection performance therefore rests on statistically invalid p-values. The underlying idea is promising, but the current implementation does not establish a reliable test.

major comments (4)

[Section 4.1.1, Eqs. (19)–(21)] The derivation treats a constant multiple of a Poisson random variable as Poisson with scaled mean. For example, (T−Δt/2)C_T^B is not Poisson((T−Δt/2)Λ) unless the multiplier is 1. Consequently S12, S21, and LA are not Skellam, and the p-values computed from these laws are invalid. The restriction T∈N and Δt∈2N is also incompatible with the later choice T=1 and Δt=1/n, because the coefficients are then non-integer and the stated Poisson-summation argument does not apply.
[Algorithm 1 and Section 4.1] Algorithm 1 centers the observed path by the sample mean C and then sets Λ=C when computing the p-value. The theoretical null distribution in Section 4.1.1 assumes a known Λ and independent centered Poisson observations. Replacing Λ by a seven-point estimate changes the null distribution of the test statistic in a way that is not accounted for; with n=7 the estimation error is material. The Benjamini-Hochberg decisions in Table 1 are therefore not calibrated, and the reported accuracy and F1 scores do not support the paper's central claim.
[Section 4.1.1, Eq. (16) and footnote 1] The paper defines Poisson_c as the centered Poisson distribution, but then labels scaled versions of the variables as "P(cΛ)", which is only correct for uncentered Poisson variables. If the variables are centered, their means are zero and the stated means cΛ are wrong; if they are uncentered, the scaling still does not preserve the Poisson law. This inconsistent use makes the distributional claims ambiguous and invalidates the equations independently of the scaling issue.
[Section 4.2 and Figures 7–10] The empirical validation compares simulated trajectories to the same incorrectly derived theoretical laws, so it cannot rescue the distributional claims. The simulation description does not state whether Λ is known or estimated, which is essential for evaluating Algorithm 1. Without a valid null distribution and a correct treatment of the plug-in step, neither the distributional fits nor the downstream classification scores in Table 1 are meaningful.

minor comments (5)

[Abstract and Introduction] There are several typographical errors, including "scarsity" for "scarcity" and "kniowledge" for "knowledge" in Section 4.4; these should be corrected.
[Algorithm 1] The definition C = (1/n) Σ_{i=0}^n C_{t_i} uses n+1 terms, while the text defines n as the number of observations; the indexing should be made consistent.
[Section 4.4] The claims that the method is "the first of its kind" and that "no alternative feature extraction approach" can approach its goals are not supported by any comparison with existing methods; these statements should be removed or substantiated with experiments.
[Section 4.1.1] The relationship between the normalized time T=1, Δt=1/n and the actual sampling scheme (seven samples every 300 days over 2000 days) should be stated explicitly, since the current presentation is confusing.
[Figures 7–10] The figure captions should state the number of simulations used and whether Λ is known or estimated, so that the empirical comparisons can be interpreted correctly.

Circularity Check

1 steps flagged · score 6.0 of 10

Detection claim rests on p-values whose null parameter is estimated from the same seven samples, making the evaluation in-sample and the reported performance partly an artifact of the plug-in.

self definitional [Algorithm 1 and Section 4.1.1 (Eqs. 16-21)]
"Compute C= 1/n Σ_{i=0}^n C_{t_i} with n the number of observations of X. Compute S_w(\tilde X) with \tilde X = (t, C_t - C). Compute the p-value corresponding to the observed value of S_w with Λ = C. ... For the sake of simplicity, we will set Λ = E[(C_t^B)_{t∈[0;T]}]."

Section 4.1.1 derives the null laws (16)-(21) for centered signature coefficients under the assumption that C_t^B are independent centered Poisson variables with fixed mean Λ = E[C_t^B], and the signal is centered by that expectation. Algorithm 1 instead centers by the sample mean C and then evaluates the p-value by setting Λ = C, the very same seven observations being tested. The Skellam parameter in Eqs. (16)-(21) is therefore not the fixed expectation used in the derivation but a random function of the data; the null distribution of the centered statistic is no longer the derived law, and the p-values are not uniform under H0.

full rationale

The paper's theoretical null distributions (Eqs. 16-21) are derived from an externally sourced birth-death ctDNA model [7], so the mathematical derivation is not equivalent to its inputs and there are no load-bearing self-citations. However, the testing procedure in Algorithm 1 replaces the fixed expectation Λ used in the derivation by the sample mean C of the same seven observations, and computes p-values from distributions whose parameter is therefore a function of the data being tested. This is a concrete self-referential step: Eq. (19) states S12 ~ Skellam(TΛ,TΛ) for Λ = E[C_t^B], while Algorithm 1 evaluates the p-value 'with Λ = C'. The plug-in estimator is correlated with the centered observations, so the null distribution of the test statistic is not the derived Skellam law and the p-values are not calibrated. Table 1's accuracy and F1 are computed from Benjamini-Hochberg decisions on these p-values, so the headline detection capability is partly an artifact of this in-sample parameter estimation. The simulation-based 'confirmation' in Figures 7-10 uses trajectories drawn from the same model family that motivates the null laws, which does not provide independent support for the incorrectly applied plug-in. No ad hominem or self-citation issue is present; the circularity is in the estimation-testing loop, not in the mathematical derivation itself.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a biological model borrowed from prior ctDNA literature, on a Markov-equilibrium assumption about benign samples, and on several mathematical shortcuts that are either approximate or incorrect. The only fitted number is the benign mean Lambda, estimated from the same seven samples used for testing. No new physical or biological entities are postulated.

free parameters (1)

Lambda (benign ctDNA mean) = sample mean of the patient's n observations, denoted C_bar in Algorithm 1
The null distributions in Equations 16, 19, 20, and 21 are parameterized by Lambda, and Algorithm 1 sets Lambda equal to the sample mean computed from the same blood samples used for testing, without correcting for estimation uncertainty.

assumptions (4)

domain assumption Benign ctDNA levels at the n sampling times are independent centered Poisson variables with common mean Lambda.
Used at the start of Section 4.1.1 and in Algorithm 1; if sampling times are too close relative to ctDNA elimination, autocorrelation breaks the Skellam derivations.
domain assumption The ctDNA shedding dynamics follow the two-type branching process with rates b, d, q_d, and epsilon described in Section 2.1.
The null and alternative distributions are inherited from this model; if the biological mechanism differs, the test may not transfer to real patients.
ad hoc to paper Replacing the unknown mean Lambda by the sample mean C_bar leaves the null distribution of the test statistics valid.
Algorithm 1 uses Lambda equal to C_bar with no derivation; for n=7 and small Lambda this is an approximation that can mis-calibrate p-values.
ad hoc to paper Scaling a Skellam random variable by 1/2 yields a Skellam distribution.
Invoked after Equation 21 to pass from 2LA to LA; this is false in general and is a derivation error, not a standard result.

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Cite this review

Pith. "Pith review of Detecting malignant dynamics on very few blood sample using signature coefficients." pith.science (2026). https://pith.science/paper/ODEVDKU7

@misc{pith2026250609097,
  author       = {Pith},
  title        = {Pith review of: Detecting malignant dynamics on very few blood sample using signature coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ODEVDKU7}},
  note         = {Machine review of arXiv:2506.09097}
}

read the original abstract

Recent discoveries have suggested that the promising avenue of using circulating tumor DNA (ctDNA) levels in blood samples provides reasonable accuracy for cancer monitoring, with extremely low burden on the patient's side. It is known that the presence of ctDNA can result from various mechanisms leading to DNA release from cells, such as apoptosis, necrosis or active secretion. One key idea in recent cancer monitoring studies is that monitoring the dynamics of ctDNA levels might be sufficient for early multi-cancer detection. This interesting idea has been turned into commercial products, e.g. in the company named GRAIL. In the present work, we propose to explore the use of Signature theory for detecting aggressive cancer tumors based on the analysis of blood samples. Our approach combines tools from continuous time Markov modelling for the dynamics of ctDNA levels in the blood, with Signature theory for building efficient testing procedures. Signature theory is a topic of growing interest in the Machine Learning community (see Chevyrev2016 and Fermanian2021), which is now recognised as a powerful feature extraction tool for irregularly sampled signals. The method proposed in the present paper is shown to correctly address the challenging problem of overcoming the inherent data scarsity due to the extremely small number of blood samples per patient. The relevance of our approach is illustrated with extensive numerical experiments that confirm the efficiency of the proposed pipeline.

Figures

Figures reproduced from arXiv: 2506.09097 by the authors.

**Figure 2.** Illustration of reparametrization invariance. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Geometric illustration of shuffle product [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Benign trajectories C B t as a function of time [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: Benign then malignant trajectories C BM t as a function of time. dataset. The empirical and theoretical distributions are displayed in [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Pure malignant trajectories as a function of time. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: Empirical distribution of S 1 (X) against its theoretical distribution from Equation 16 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: Empirical distribution of S 12(X) against its theoretical distribution from Equation 19 4.3. Our testing scheme Based on the explicit form of the distribution of our pivotal quantities under the Null Hypothesis, i.e. Benign state of ctDNA dynamics we obtain the testin…

**Figure 9.** Figure 9: Empirical distribution of S 21(X) against its theoretical distribution from Equation 20 Algorithm 1 Testing for S w with w ∈ {2, 12, 21, 12 − 21} Require: X and w ∈ {2, 12, 21, 12 − 21} (with the notation 12 − 21 refering to the Levy Area). Ensure: (H0) : the tumor is…

**Figure 10.** Figure 10: Empirical distribution of LA(X) against its theoretical distribution from Equation 21 the statistically impossible task of estimating the coefficients of a Birth and Death Model before testing possible malignancy of a patient’s ctDNA level dynamics. The performance o…

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Chevyrev, A

I. Chevyrev, A. Kormilitzin, A primer on the signature method in ma- chine learning, arXiv preprint arXiv:1603.03788 (2016). 21

arXiv 2016

[2] [2]

Fermanian, Learning time-dependent data with the signature trans- form, Ph.D

A. Fermanian, Learning time-dependent data with the signature trans- form, Ph.D. thesis, Sorbonne Université, 2021

work page 2021

[3] [3]

Chen, Iterated integrals of differential forms and loop space ho- mology, Annals of Mathematics 97 (1973) 217–246

K.-t. Chen, Iterated integrals of differential forms and loop space ho- mology, Annals of Mathematics 97 (1973) 217–246

work page 1973

[4] [4]

P. K. Friz, N. B. Victoir, Multidimensional stochastic processes as rough paths: theory and applications, volume 120, Cambridge Univer- sity Press, 2010

work page 2010

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Baudoin, Diffusion processes and stochastic calculus, 2014

F. Baudoin, Diffusion processes and stochastic calculus, 2014

work page 2014

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Durrett, R

R. Durrett, R. Durrett, Branching process models of cancer, Springer, 2015

work page 2015

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Avanzini, D

S. Avanzini, D. M. Kurtz, J. J. Chabon, E. J. Moding, S. S. Hori, S. S. Gambhir, A. A. Alizadeh, M. Diehn, J. G. Reiter, A mathematical model of ctdna shedding predicts tumor detection size, Science advances 6 (2020) eabc4308

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Lyons, A

T. Lyons, A. D. McLeod, Signature methods in machine learning, arXiv preprint arXiv:2206.14674 (2022)

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