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REVIEW 4 major objections 5 minor 30 references

Cancer model with moving extinction threshold reproduces real cancer data

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

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Pith's one-line read One simple model with a moving immune extinction threshold reproduces exponential and breast-cancer risk curves and predicts a cycle-length risk ratio close to observation.

desk verdict A genuinely novel modeling framework with a clean fix to Allee-type growth, but the breast-cancer story is oversold: the key immune-switch mechanism is fitted to the data and confounding explanations are left untested. read the letter →

arxiv 2506.05992 v1 pith:7RUXZNYK submitted 2025-06-06 q-bio.QM math.DS

Frank Bastian , Hassan Alkhayuon , Kieran Mulchrone , Micheal O'Riordain , Sebastian Wieczorek This is my paper · ORCID
classification q-bio.QMmath.DS MSC 92C5092D25
keywords cancermodellingextinctionthresholdimmunesystemAlleeeffectbreastmenstrualcyclehormonereplacementtherapycumulativerisk
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 claims that a single simple model—random mutations plus an immune response that acts as an extinction threshold—can reproduce several qualitatively different real cancer datasets. With a fixed immune response, the model yields the exponential age-specific cumulative risk seen in most cancers, illustrated by colorectal cancer in women. With an immune response that dips in the luteal phase and stops dipping after menopause, it reproduces the unusual polynomial-then-linear breast cancer risk curve, including the effect of hormone replacement therapy. The same mechanism predicts a 1.75-fold higher breast cancer risk for women with 22–25 day cycles than for women with 25–31 day cycles, close to the observed 1.86-fold difference. If correct, the paper establishes that a moving extinction threshold can explain these incidence patterns and produces testable predictions about cycle length and HRT.

What carries the argument

The load-bearing object is the moving extinction threshold, an Allee-type critical cluster size separating immune elimination from cancer establishment. In the model it is approximated by \(\bar{A}(t)\approx(s_m(t)-\overline{m}(t))/(\nu r-\mu(\nu+a\ln 10))\), where \(s_m(t)\) is the daily immune-elimination rate, \(\overline{m}(t)\) is the mean daily mutation count, and \(\nu,r,\mu\) are growth parameters. The threshold is what makes most mutated clusters die out and only threshold-crossing clusters become cancer; when \(s_m(t)\) varies with menstrual cycle phase and menopause, the threshold moves and changes the shape of the age-specific risk curve.

What would settle it

Measure age-specific cumulative breast cancer risk in a large cohort of women who never used HRT and were never screened: the model with distributed menopause but no HRT under-predicts risk after age 50, so if such a cohort still shows the same linear postmenopausal rise as the general population, the menopause/HRT switch cannot be the cause.

Watch

Extended reading notes

Core claim

The paper's central claim is that a single daily-update model, \(x(t+1)=x(t)+\nu r x(t)-\nu\mu x(t)^{(\nu+1)/\nu}-s_m(t)+m(t)\) clipped at zero, captures both carcinogenesis and tumour progression: \(m(t)\) is a binomially distributed number of new mutations whose per-cell probability rises linearly with age, and \(s_m(t)\) is the number of mutated cells the immune system removes each day. The immune term creates a critical cluster size, the extinction threshold \(A(t)\), below which the immune system eradicates the cluster and above which it grows into cancer. With a constant immune response the model produces the exponentially rising cumulative risk typical of most cancers and fits colorectal cancer data; with an immune response that weakens during the luteal phase of the menstrual cycle and returns to full strength after an effective menopause, it reproduces the polynomial-then-linear breast cancer risk curve once distributed menopause and HRT use are included. The same simulations fit breast cancer growth in two mouse datasets, reproduce a 1.75-fold relative risk for short versus longer cycles against the observed 1.86-fold value, and place most transitions to cancer in the luteal phase.

Load-bearing premise

The central explanation rests on the assumption that the postmenopausal flattening of breast cancer risk is caused by a switch in immune response at menopause; if the flattening instead comes from screening overdiagnosis, cohort changes, or another biological pathway, the breast-cancer explanation collapses even though the mathematics may remain consistent.

Editorial extensions

If this is right

  • A constant immune response is sufficient to explain why most cancers show exponentially rising age-specific cumulative risk.
  • The postmenopausal flattening of breast cancer risk can be explained dynamically by the end of cyclic progesterone-related immune weakening, without invoking separate biological mechanisms for young and old ages.
  • Shorter menstrual cycles raise risk because more days are spent with a lowered extinction threshold; the model quantifies this as a 1.75-fold effect for 22–25 versus 25–31 day cycles.
  • Most modelled transitions to breast cancer occur in the luteal phase (63–76%), consistent with progesterone as the main cycle-associated driver.
  • The same moving-threshold mechanism can be applied to other cancers and to time-varying factors such as infections, immune deficiencies, and immunotherapy.

Reading between the lines

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

  • Beyond the paper: if the extinction threshold is the true control point, strengthening immune clearance during the luteal phase should measurably shift the age of first threshold crossing in animal models.
  • Beyond the paper: the rescue-event mechanism implies that temporary immunosuppression need not be fatal—a cluster can cross the threshold and fall back below it—so timing-based immune interventions may deserve study.
  • Beyond the paper: the model offers a dynamical alternative to screening-artifact explanations of the postmenopausal breast cancer plateau; comparing never-screened, never-HRT cohorts with the model's predictions would separate the explanations.
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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

4 major / 5 minor

Summary. The paper proposes an ODE/stochastic model of carcinogenesis in which a cluster of mutated cells grows under a modified Richards' equation with an additional immune-elimination term (Eq. 5), random mutations are drawn from a binomial distribution whose mutation probability increases linearly with age (Eqs. 8-9), and a time-varying immune response s(t) produces a moving extinction threshold A(t) approximated by Eq. (11). The authors first show that the classical Volterra/Allee model (Model 1) cannot produce the growth-versus-saturation asymmetry typical of cancer, while their Model 2 can (Sec. 2.2, Supplement Prop. 1.1). They then fit Model 2 to two mouse breast-tumour data sets (Sec. 3.1), reproduce the exponential cumulative risk of colorectal cancer with a constant immune response (Fig. 6), and reproduce the polynomial-then-linear cumulative risk of breast cancer in Irish women by adding a menstrual-cycle immune switch, distributed menopause, and HRT-delayed menopause (Eq. 12, Fig. 7d). Finally, they report a model-predicted 1.75-fold increase in breast-cancer risk for 22-25-day cycles relative to 25-31-day cycles, compared with the 1.86-fold ratio observed by Yuan et al. 1988 (Sec. 3.2.3).

Significance. If established, the paper would make a useful contribution: a single process-based model with an extinction threshold generates several qualitatively different incidence patterns, and the moving-threshold concept is transferable to other time-varying immune contexts. The formal asymmetry result for Model 1 versus Model 2 is a clean and permanent mathematical contribution; the mice-data fits in Sec. 3.1 are genuine empirical tests with plausible estimated carrying capacities; and the colorectal exponential fit is a nontrivial qualitative success. The breast-cancer application, however, is at present an in-sample proof-of-concept rather than an established explanation: the key mechanism in Eq. (12) is not independently identified, the data are a single cross-section with known screening and cohort confounders, and the cycle-length and HRT 'predictions' are made with the same mechanism used to fit the curve. With a strengthened statistical and identifiability analysis, and with claims scaled to the evidence, this could become a solid paper.

major comments (4)
  1. [Abstract, §3.2, Tables 2-3, Fig. 7] The abstract and Conclusions claim that the model 'accurately reproduces' the breast-cancer cumulative risk and 'predicts' HRT and cycle-length effects, but every component of the breast-cancer fit is calibrated to the same 2016 Irish cumulative-risk curve: p0, pT, smin, and smax in Table 2, the menopause distribution, and the 26% HRT fraction and gamma-distribution parameters in Table 3 are all chosen so that Fig. 7(d) visually matches the black curve in Fig. 7. There is no holdout age band, no second population, no goodness-of-fit statistic, and no confidence interval on either R(t) or R-hat(t). With at least seven free parameters available for a 16-point curve, a visual match cannot support the strength of the reported claim. Please provide an out-of-sample check (for example, fit on ages 0-45 and predict the 50-85 age bands, or fit the NCRI 1994-2021 averaged data and predict the 2016 census-year curve) and quantify uncertainty with bootstrap or profile-likelihood methods.
  2. [§3.2.2, Eq. (12), Fig. 7] The post-menopausal flattening of the cumulative risk is produced by the discrete switch sm(t)=smax for t>M together with the HRT-delayed effective menopause M; without this switch (Cases 1 and 2 in Fig. 7b-c) the model under-predicts risk above age 50. This switch is not measured or independently estimated; it is supported only by qualitative progesterone-immunology citations and by the improved fit in Fig. 7(d). The observed polynomial-to-linear transition at ages 44-50 could instead arise from the organized biennial screening programme (overdiagnosis and lead-time), from birth-cohort differences in HRT use and reproductive history in the single-year 2016 cross-section, or from the proposed immune switch. Because the abstract claims 'new insights' into HRT, this identifiability problem is load-bearing. A concrete test would be to fit the model to screening-adjusted or age-period-cohort-decomposed incidence, or to a population with a different screening policy, and to compare the implied smin/smax switch with independent evidence on progesterone-related immune modulation in women.
  3. [§3.2.3 and Supplement §2.2.6] The 1.75-fold cycle-length ratio is not a free prediction: the menstrual-cycle mechanism in Eq. (12) is already the mechanism used to fit the overall breast-cancer curve in Fig. 7(d), so the cycle-length comparison in Fig. 9 is a consistency check of the same fitted mechanism rather than an independent test of the model. The quoted ratio also depends on modeling choices whose sensitivity is not reported: the lower bound of 22 days is set as approximately two standard deviations below the mean (Supplement §2.2.6), the luteal phase is fixed at 14 days, and the comparison target is a 1988 Shanghai study with a different birth cohort and screening context than the 2016 Irish data. Please report the ratio as a function of the chosen cycle-length bounds, give Monte Carlo uncertainty estimates for R-hat(51, tf), and state the population mismatch explicitly.
  4. [Supplement §3 and Table 3] The parameter values used in the complete model (Table 3) are obtained by fitting Model 2 to two measurements from a single untreated human breast cancer (Fornvik et al. 2016), with A and K fixed a priori. The uncertainties from this two-point boundary-value problem are not propagated into Fig. 7 or Fig. 9, and the text does not report how strongly the final breast-cancer curves depend on r, mu, s, and nu=10. Because these progression parameters are then held fixed while p0, pT, smin, and smax are tuned, the effective number of degrees of freedom in the breast-cancer fit is larger than a casual reading of Table 2 suggests. Please add a sensitivity analysis (for example, doubling or halving r, mu, and s and refitting the remaining parameters) or provide a joint uncertainty propagation.
minor comments (5)
  1. [Eq. (8)] The binomial distribution is printed as n!/(m!(n-m!)) in Eq. (8); the denominator should be m!(n-m)!.
  2. [Eqs. (7) and (11)] The variable a in A=10^a is used in Eqs. (7) and (11) but is not defined in the main text; please define it explicitly near the first occurrence.
  3. [Fig. 5 caption] The caption says 'black triangles in Fig. 5(a)' for Dataset 2, but Dataset 2 is plotted in Fig. 5(b).
  4. [Supplement §1.5.2] There is a duplicated article in the sentence 'falls again below the the (blue/red) moving extinction threshold'; please correct.
  5. [§3.2 and Supplement §4] The text describes the pre-menopausal breast-cancer curve as a '7th order polynomial', but the supporting analysis in Supplement Fig. 5 fits a power law R(t)=a t^b with b≈7; these are different functional forms, and the evidence for 'linear thereafter' is based on a cubic fit without tests of coefficient significance. Please clarify the terminology and report confidence intervals for the fitted exponents.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the breast-cancer curves are honest fits, and the 1.75-fold cycle-length prediction is an out-of-sample model output.

full rationale

The paper's derivation chain is not circular. The breast-cancer cumulative-risk curves in Fig. 7 are obtained by fitting the model parameters to the same NCRI incidence data; the Supplement (Table 2) explicitly states that the listed parameters were "obtained through fitting (red) Rhat(t) to (black) R(t)". The paper presents these as reproductions, not out-of-sample predictions, so there is no fitted-input-renamed-as-prediction at this step. The time-varying immune response in Eq. (12) is an input hypothesis supported by independent biological citations, not a quantity the model is claimed to derive from first principles; the polynomial-to-linear transition is an output of the assumed menopause switch, and the paper does not claim that the switch itself was predicted. The only explicit prediction, the 1.75-fold risk ratio for short cycles, is not a fitted parameter: all parameters were fixed by the Fig. 7(d) fit, and the ratio is obtained by varying follicular-phase length and comparing to an external observation [79] that was not used in the fit, so the prediction has independent content. Self-citations to [44] refer to the included supplementary material, which contains the technical proofs, parameter tables, and fitting details; these are checkable within the submission and do not constitute an unverified external authority. Limitations such as single-year 2016 Irish breast-cancer data, screening/cohort confounders, and the two-point human breast-growth fit (Supplement Sec. 3) are validity concerns, not circularity, and are outside the scope of this pass.

Assumptions & free parameters 8 free parameters · 6 assumptions · 1 invented entities

The model rests on a chain of parameter choices and domain assumptions. Four main parameters (p0, pT, smin, smax) are fitted to the same cumulative risk curves that the paper claims to reproduce, so the agreement is partly an in-sample fit. The breast-cancer-specific ingredients (menstrual cycle switch, distributed menopause, HRT) are inserted precisely where the previous model failed to match data, which increases circularity burden.

free parameters (8)
  • p0 = 2.7394e-5 to 2.7399e-5 (per dataset)
    Probability a stem cell mutates in one day at birth; fitted to cumulative risk curves (Table 2).
  • pT = 2.7408e-5
    Mutation probability at life expectancy; fitted to cumulative risk curves (Table 2).
  • smin = 274203 to 274295 cells/day
    Immune response during luteal phase; fitted to breast cancer cumulative risk in Cases 1-3 (Table 2).
  • smax = 274250 to 274300.9 cells/day
    Immune response outside luteal phase; fitted to cumulative risk (Table 2).
  • r, mu, s (growth and immune removal) = r=2.7878e-3/day, mu=1.5708e-4, s=239.32 cells/day (human full model); different fitted values for mice (Table 1)
    Growth and immune-removal parameters fitted to tumor growth data (mouse datasets and a two-measurement human case).
  • nu (shape parameter) = 10 (full model); 1 or 97196 for mice fits
    Shape parameter; fixed by choice in the full model, fitted in mice comparisons.
  • n (stem cell count) = 1e10 cells
    Assumed number of breast stem cells from literature; used in the binomial mutation process.
  • xd (detection threshold) = 3.05e9 cells (breast), 4.2e9 (colorectal)
    Converted from average screening detection diameter; not fitted but affects simulated incidence and thus the fitted p/s values.
assumptions (6)
  • domain assumption Stem cell mutations are independent Bernoulli trials, and the daily number of mutations follows a binomial distribution with n=10^10 cells and probability p(t) that increases linearly with age (Eqs. 8-9).
    This simplifies carcinogenesis to a one-step mutation process, ignoring multi-hit kinetics, repair mechanisms, and tissue architecture. Both n and p(t) are assumed.
  • domain assumption The immune system is represented solely by a daily removal rate s(t) of mutated cells, which sets an extinction threshold A(t) as in Model 2 (Eq. 5).
    The immune response is collapsed into one scalar rate; no explicit immune dynamics or spatial effects.
  • ad hoc to paper Progesterone during the luteal phase weakens the immune response, so sm(t) drops from smax to smin in luteal phases; this is the dominant mechanism for breast cancer risk (Eq. 12).
    The switch is introduced specifically to create the polynomial-to-linear transition at menopause and the cycle-length effect. Qualitative immunology references support the direction but not the magnitude or exclusivity.
  • domain assumption Population parameters for menopause (mean 51, sd 4.86) and HRT (26% use; gamma duration with mode 6 years, sd 4.8 years) apply to the 2016 Irish female cohort.
    These distributions are borrowed from UK/US studies and applied to Ireland; no uncertainty or cohort-adjustment is given (Sec 2.2.5).
  • ad hoc to paper The moving extinction threshold is well approximated by A-bar(t) = (sm(t) - m-bar(t)) / (nu*r - mu*(nu + a ln 10)) in Eq. (11), including for nu=10 used in simulations.
    Derived under an expansion in 1/nu and small s/r assumptions; for nu=10 and A=10^4, ln(A)/nu is about 0.92, so the nu much greater than 1 condition is marginal. Used for conceptual plots, not in the Monte Carlo itself.
  • standard math The incidence rate is constant within 5-year age bands when converting registry data to cumulative risk R(t) (Supplement 2.2.1).
    This is a standard piecewise-constant hazard approximation.
invented entities (1)
  • Moving extinction threshold A(t) independent evidence
    purpose: Represents the time-dependent critical cluster size below which the immune system eradicates mutated cells and above which cancer progresses; allows the model to couple carcinogenesis to changing immune states.
    The construct yields falsifiable predictions not used to set constants: a 1.75-fold risk ratio for short menstrual cycles (vs 1.86 observed) and 63-76% of transitions during the luteal phase. However, it is a model-derived quantity, not a directly measured biological entity.

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Cite this review

Pith. "Pith review of Cancer model with moving extinction threshold reproduces real cancer data." pith.science (2026). https://pith.science/paper/7RUXZNYK

@misc{pith2026250605992,
  author       = {Pith},
  title        = {Pith review of: Cancer model with moving extinction threshold reproduces real cancer data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7RUXZNYK}},
  note         = {Machine review of arXiv:2506.05992}
}
read the original abstract

We propose a simple dynamic model of cancer development that captures carcinogenesis and subsequent cancer progression. A central idea of the model is to include the immune system as an extinction threshold, similar to the strong Allee effect in population biology. We first identify the limitations of commonly used Allee effect models in reproducing typical cancer progression. We then address these limitations by deriving a new model that incorporates: (i) random mutations of stem cells at a rate that increases with age and (ii) immune response whose strength may also vary over time. Our model accurately reproduces a wide range of real-world cancer data: the typical age-specific cumulative risk of most human cancers, the progression of breast cancer in mice, and the unusual age-specific cumulative risk of breast cancer in women. In the last case, we use a moving extinction threshold to reflect the different immune response at different phases of the menstrual cycle and menopausal treatment. This provides new insights into the effects of hormone replacement therapy and menstrual cycle length. This moving threshold approach can be applied to a variety of other cancer scenarios where the immune response or other important factors may vary over time.

Figures

Figures reproduced from arXiv: 2506.05992 by the authors.

Figure 1
Figure 1. Two conceptual diagrams of cancer development incl [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The defining feature of cancer progression (the rati [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Population growth x(t) in the commonly used version of Model 1 (Eq. (3) with ν = 1) started from x(0) = 2 with a per capita decay rate r = 0.12, extinction threshold A = 1 and three different values of carrying capacity: (green) K = 10 , (blue) K = 104 and (red) K = 108 . For realistic values of K, the solutions show an unusual step-like behaviour that does not resemble the progression of any known cancer. For K = 1… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: (a) An example of breast cancer development [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: A comparison between real data on breast cancer prog [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The age-specific cumulative risk of a woman developi [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: The age-specific cumulative risk of a woman developi [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: The immune response s(t) in Eq. (12) plotted over the lifetime of a woman. The inset shows (red) the assumed changes in immune response along with (blue) the changes in progesterone levels during the menstrual cycle that occur between the age of 12 (the average age of …
Figure 8
Figure 8. Figure 8: Next, we compare the actual age-specific cumulative risk of breast cancer in women obtained from real data, R(t), with Rˆ(t) obtained from model (10) for three different cases of time-varying immune response (12). The model parameters are listed in [44, [PITH_FULL_IMA…
Figure 9
Figure 9. Figure 9: (Black) The model’s cumulative risk of a woman devel [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 1
Figure 1. Figure 1: (a) The binomial probability distribution [3, Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p032_1.png]
Figure 2
Figure 2. Figure 2: shows examples of rescue events in a solution to the complete cancer development model [3, Eq. (2.10)] 37.1 37.2 37.3 37.4 t[years] 5000 10000 15000 xt [cells] Rescue event Escape event xt A¯(t) [PITH_FULL_IMAGE:figures/full_fig_p033_2.png]
Figure 3
Figure 3. Figure 3: We also fix the lethal tumour size at xl = 1012 cancer cells, which corresponds to a tumour diameter of 124 mm or a tumour volume of 1 litre, indicated by the dotted horizontal line in [PITH_FULL_IMAGE:figures/full_fig_p038_3.png]
Figure 3
Figure 3. Figure 3: (Grey dots) The two measurements reported in [20], t [PITH_FULL_IMAGE:figures/full_fig_p039_3.png]
Figure 4
Figure 4. Figure 4: The age-specific cumulative risk of colorectal canc [PITH_FULL_IMAGE:figures/full_fig_p041_4.png]
Figure 5
Figure 5. Figure 5: The shape of the age-specific risk of breast cancer [PITH_FULL_IMAGE:figures/full_fig_p042_5.png]
Figure 6
Figure 6. Figure 6: The age-specific cumulative risk of (red) blood and ( [PITH_FULL_IMAGE:figures/full_fig_p043_6.png]

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Pith tools

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