Evolutionary dynamics of cancer: from epigenetic regulation to cell population dynamics -- mathematical model framework, applications, and open problems

Jinzhi Lei

REVIEW 3 major objections 3 minor 47 references

Evolutionary dynamics of cancer: from epigenetic regulation to cell population dynamics -- mathematical model framework, applications, and open problems

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proposes that heterogeneous, plastic stem cell regeneration—captured by a delay integro-differential equation with an epigenetic inheritance kernel—is a general mathematical framework for cancer evolution.

desk verdict A mostly synthetic framework paper that is honest about its open problems but leans heavily on the author's prior work, with the unmeasured Beta transition kernel as the main load-bearing weakness. read the letter →

arxiv 1908.07048 v1 pith:IOYD24OB submitted 2019-08-19 q-bio.CB

Jinzhi Lei This is my paper · ORCID

classification q-bio.CB MSC 92D2545K0592C50

keywords stemcellregenerationdifferential-integralequationepigeneticplasticityheterogeneitycancerdevelopmentcomputationalbiologyCAR-Ttherapyrelapseopenproblems

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 proposes a general mathematical framework in which cancer evolution is driven by stem cells whose epigenetic state changes randomly at each division. Starting from a classical G0-phase cell-cycle model, the author derives a delay differential equation for a homogeneous stem cell pool, then generalizes it to a delay integro-differential equation, Eq. (11), in which cells are labeled by an epigenetic state $x$ and a kernel $p(x,y)$ gives the probability that a daughter cell of state $x$ comes from a mother of state $y$. The framework connects single-cell epigenetic state to population-level rates of proliferation, apoptosis, and differentiation, and it is extended to include genetic mutations in Eq. (17). Hybrid computational implementations of the framework are applied to inflammation-induced tumorigenesis and to relapse after CD19 CAR-T therapy, reproducing observed behavior in both settings. The author also identifies open mathematical problems—steady-state existence, entropy decomposition, inverse estimation of $p(x,y)$, and local state transitions—that must be solved before the framework can be used predictively.

What carries the argument

The load-bearing object is Eq. (11), a delay integro-differential equation for the resting-phase stem cell density $Q(t,x)$ over an epigenetic state space $\Omega$. Its nonlocal term $$2\int_\$\Omega$ \$\beta$(\hat{Q}_{\tau(y)},y)Q(t-\tau(y),y)$e^{{-\mu(y)\tau(y)}}$p(x,y)\,dy$$ carries the effect of cell division: a mother at state $y$ divides after duration $\tau(y)$, suffers apoptosis at rate $\mu(y)$, and produces daughters whose epigenetic state is drawn from the inheritance kernel $p(x,y)$. The kernel, named the transition function, is the mechanism that converts single-cell epigenetic plasticity into population heterogeneity. In the concrete one-dimensional model (14) it is a conditional Beta distribution with mother-state-dependent shape parameters $a(y)=\eta(y)\varphi(y)$ and $b(y)=\eta(y)(1-\varphi(y))$, fixed by the mean and variance functions $\varphi(y)$ and $\eta(y)$ obtained from the author's simulations of histone-modification inheritance. The characteristic-line integration that converts the age-structured PDE (10) into the delay equation (11) is the same technique that produces the homogeneous delay equation (4) from the G0 model; later, a local-transition approximation replaces the global kernel by a drift-and-diffusion expansion and yields the second-order equation (40).

What would settle it

Track individual labeled stem cells through one division in a relevant tissue (e.g., hematopoietic stem cells) by single-cell sequencing or reporter imaging, and compare the empirical daughter-state distribution for many mother states $y$ with the conditional Beta kernel (14) fitted through $\varphi(y)$ and $\eta(y)$. If the observed distributions are not Beta-shaped or the fitted shapes vary with mother state outside the assumed functional forms, the central kernel of Eq. (11) fails. A complementary experimental check: the CAR-T model predicts that combined CD19/CD123 CAR-T administration at specific ratios prevents CD19-positive relapse, a prediction testable directly in the mouse model that motivated the paper.

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Extended reading notes

Core claim

The central claim is that heterogeneous, plastic stem cell populations—the engine of cancer development—are governed by the delay integro-differential equation (11), in which the cell population $Q(t,x)$ at epigenetic state $x$ changes through loss by proliferation and differentiation and through influx from mothers of all states $y$ whose daughters randomly transition to $x$. The kernel $p(x,y)$ encodes epigenetic plasticity: it is the inheritance probability of daughter state given mother state. The paper argues that this equation places intrinsically nonlocal, non-measurable epigenetic transitions at the center of cancer dynamics and unifies single-cell gene-expression states, cell-cycle duration, apoptosis, differentiation, and feedback-dependent proliferation into one population-level description. In the one-dimensional stemness case (13) the kernel takes the explicit Beta-distribution form (14), and with genetic heterogeneity the framework becomes (17), a system over $m$ genetic types coupled by mutation probabilities. The same framework, discretized in hybrid single-cell simulations, reproduces the multi-pathway progression from chronic inflammation to malignancy and predicts that CAR-T-induced plasticity can drive CD19-positive relapse.

Load-bearing premise

The whole framework stands on the unmeasured inheritance kernel $p(x,y)$: the paper admits that biology cannot measure this function directly, and its concrete Beta-distribution form comes from the author's own simulations, so if real daughter-state distributions differ from that family every heterogeneous-model prediction built on Eq. (11) would shift.

Editorial extensions

If this is right

In the homogeneous G0 model, abnormal growth requires at least one of three dysregulations—loss of differentiation/senescence, sustained proliferative signaling, or evasion of apoptosis—which the paper identifies with known cancer hallmarks.
With heterogeneity included, cell-to-cell variance in behavior is not noise layered on a homogeneous population but a consequence of random epigenetic inheritance, so therapy resistance and relapse become intrinsic outcomes of the same regenerative dynamics.
Gene mutations can be layered onto the framework through Eq. (17): each genetic type has its own kinetic rates and the mutation network $p_{i,j}$ couples the types, so mutation-driven intratumoral heterogeneity is a corollary of the same equation.
The applications imply that chronic inflammation can drive tumorigenesis through several distinct pathway combinations, and that CAR-T-induced plasticity toward hematopoietic stem-like and myeloid-like states is sufficient to produce CD19-positive relapse.
The four open mathematical problems—steady-state existence and stability, a nonnegative entropy production/dissipation decomposition, the inverse problem for $p(x,y)$, and local-transition limits—define what must be proved before the framework can be used for personalized prediction.

Reading between the lines

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

If the framework is right, the first practical task is not model refinement but measurement: single-cell lineage-tracing time series should be used to estimate $p(x,y)$ directly, and the inverse-problem formulation in Section 5.3 already sketches how that could be done.
The local-transition expansion (40) turns the global-kernel equation into a Fokker-Planck-type equation with drift $2a$ and diffusion $D$, suggesting a concrete bridge to classical mutation-selection balance and niche-construction models in evolutionary biology that the paper does not pursue.
The entropy problem has a testable corollary: if cancer development is an entropy-increasing process, then the entropy production rate under the framework should be large exactly during transitions from precancerous to malignant states, a quantity that could be estimated from sequentially sampled single-cell distributions.
The model's pathway-mutation treatment suggests a natural therapeutic extension not tested here: using the same hybrid equations to rank combinations of targeted agents by how far they push the population away from the malignant steady state, rather than by immediate cell kill.

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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 knowing: this is not a new-results paper. It is a framework-and-open-problems paper, and it mostly says so. The central equation (11) is the author's bioRxiv [22] and is closely related to the PNAS model [23]; the inflammation and CAR-T applications are condensed summaries of [13] and [46]. What is genuinely new here is the local-transition limit in Section 5.4 and the four mathematical problem statements. That is modest novelty, but the paper does not oversell it as a breakthrough.

What it does well: the derivation of (11) from the age-structured model is a standard but clean characteristic-line calculation, and the notation is serviceable. More importantly, the paper is candid about its own weakest point: Section 3.2 explicitly says the transition function p(x,y) cannot be measured directly and that the possible form is unknown. That honesty matters. The open problems are also sensible and could be a useful starting point for someone entering this area.

The soft spots are real but proportionate. The load-bearing assumption is the conditional Beta kernel, and it comes from the author's own histone-modification simulations [18,19], not from independent lineage-tracing or single-cell data. Since p multiplies the whole renewal flux in (11), every prediction about heterogeneity, tumorigenesis timing, and CAR-T relapse is conditional on that kernel. The paper contains no sensitivity analysis and no model-data comparison in this manuscript. That is a serious limitation, though not a fatal one if the applications are read as illustrative rather than validated.

There are also algebraic slips. Equation (7) puts eta outside the bracket where theta should be; the correct steady state should have theta times the bracket, as the condition eta = beta0 theta^n/(theta^n + Q*^n) + beta1 gives directly. Equation (14), and the CAR-T kernel in Section 4.3, drop the -1 exponents from the general Beta form used in Section 3.2. Minor, but they need fixing. The local-transition derivation in Section 5.4 also deserves a careful check in review; the expansion of the convolution appears to mix the F term and the diffusion coefficient conventions.

The self-citations are not padding; they are the actual sources of the main equation and applications. But the paper should clearly mark which parts are prior results and which are new.

Bottom line: this is a paper for readers who want a single entry point to Lei's modeling framework and its open mathematical questions. It deserves a serious referee, yes, but as a framework/exposition paper with requested revisions, not as a primary research contribution. I would send it to review, ask the author to fix the algebra, mark prior results, and either provide code and data or temper the predictive claims.

Referee Report

3 major / 3 minor

Summary. The manuscript proposes a general mathematical framework for the evolutionary dynamics of heterogeneous and plastic stem cell populations, with applications to cancer. The authors first present a homogeneous G0 cell-cycle model, derive a delay differential equation, and then extend it to a delay integro-differential equation (Eq. 11) for the density of resting-phase stem cells with epigenetic state, where inheritance is described by a transition kernel. The kernel is specified as a conditional Beta distribution motivated by the authors' own histone-modification simulations. The framework is extended to genetic heterogeneity (Eq. 17) and to a local-state-transition approximation (Eq. 40). Two hybrid computational applications are described: inflammation-induced tumorigenesis and relapse after CAR-T therapy, with details referred to prior and preprint papers. The paper closes with four open mathematical problems: existence and uniqueness of steady states, entropy production and dissipation, inverse problems, and local transition approximations.

Significance. If the mathematical framework is accepted, it provides a coherent bridge between single-cell epigenetic state and population-level cancer dynamics, and the open problems may stimulate useful analysis. The derivation of Eq. (11) is a standard characteristic-line calculation and is clearly presented. However, the biological relevance of the framework hinges on the transition kernel, whose Beta form is not independently validated; the application results are not demonstrated in this manuscript. The framework is therefore more a proposal of a modeling paradigm than an established predictive model.

major comments (3)

[Section 3.2, Eq. (14)] The general conditional Beta form is stated with exponents a(y)-1 and b(y)-1, but Eq. (14) omits the -1 terms. With a=eta*phi and b=eta*(1-phi), the density in Eq. (14) corresponds to Beta(eta*phi+1, eta*(1-phi)+1), whose mean is (eta*phi+1)/(eta+2) and whose variance is not phi*(1-phi)/(1+eta) as claimed. This inconsistency affects the parameterization used in the applications and must be corrected. Please specify whether the intended shape parameters are eta*phi and eta*(1-phi) (with exponents minus one) or eta*phi+1 and eta*(1-phi)+1.
[Sections 3.2 and 4.2-4.3] The transition kernel p(x,y) multiplies the entire renewal flux in Eq. (11), so all model predictions depend on its form. The paper states that 'biologically we cannot measure this function directly,' and the chosen Beta form is justified only by the authors' own simulations in references [18,19]. The application sections (inflammation-induced tumorigenesis and CAR-T relapse) do not present sensitivity analyses, quantitative model-data comparisons, or independent validation within this manuscript; they refer to prior and preprint papers for details. As a result, the claim that the framework reproduces these biological processes is not supported by the evidence presented here. The authors should either provide independent experimental support for the kernel, perform a systematic sensitivity analysis, or explicitly label the applications as conditional illustrations.
[Section 3.2, multi-dimensional extension] In the generalization to multiple epigenetic states, the text says 'we can extend the above Beta-distribution by the multiply rule p(x,y) = sum_i p_i(x_i,y).' A sum of marginal densities is not a joint density on the product space and does not satisfy the normalization condition unless the p_i are weighted categories. The intended independent-component joint distribution should be the product of the marginal densities. This point should be corrected to avoid ambiguity in applying the framework to high-dimensional states.

minor comments (3)

[Section 2, Eq. (7)] In the steady-state formula, the prefactor should be theta rather than eta; solving beta(Q*)=eta with beta from Eq. (5) yields Q* = theta [ (beta0+beta1-eta)/(eta-beta1) ]^{1/n}.
[Section 5.2, Eq. (25)] The definition of total cell number is missing the differential: it should read Q(t) = integral over Omega of Q(t,x) dx.
[Throughout] There are numerous typographical errors (e.g., 'challenge issue' and 'computations models' in the abstract, 'propose model' in Sections 4.1 and 6, 'serous' in Section 4.2, 'my promoted' in Section 4.3), and reference [46] is listed only as 'Preprint, 2019'; these should be corrected.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: Eq. (11) is obtained by a self-contained characteristic-line calculation, and the unmeasured Beta kernel is an acknowledged modeling ansatz; the main caveat is heavy reliance on self-citations, not constructional circularity.

full rationale

The central derivation is mathematically self-contained: integrating the age-structured PDE (10) along characteristics and substituting the boundary condition yields the delay integro-differential equation (11); no parameter is fitted to the quantity that Eq. (11) is then said to predict. The transition kernel p(x,y) is introduced as an unmeasured modeling ingredient, and Section 3.2 explicitly states 'biologically we cannot measure this function directly, and do not know the possible form neither,' then proposes a conditional Beta form motivated by the author's own simulation studies [18,19]. This is an acknowledged ansatz rather than a disguised output of Eq. (11); the framework is stated for arbitrary p(x,y), so the mathematical claim does not reduce to the Beta choice. The applications in Sections 4.2 and 4.3 are hybrid implementations whose details and validations are delegated to prior papers [13,46]; they are presented as applications, not as the derivation of the framework. Extensive self-citation is present, and the unvalidated Beta kernel is a genuine correctness and identifiability concern, but no equation-level constructional equivalence or fitted-parameter-as-prediction step was found. The score of 2 reflects the presence of self-citations for the kernel and applications, not a circular reduction of the main derivation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central contribution is a mathematical structure, so the ledger records the modeling choices, especially the unmeasured transition kernel and the ad hoc rate functions, that carry the biological content. No fundamentally new physical entity is introduced.

free parameters (4)

Hill parameters theta, n, beta0, beta1 in proliferation rate beta(Q) (Eq. (5)) = not fitted in this paper
These shape the negative-feedback proliferation term used in the G0 model; the steady-state condition (6)-(7) and the three-hallmarks interpretation depend on them.
Rate functions beta(hatQ,x), kappa(x), mu(x), tau(x) = generic; concrete forms in Eqs. (15)-(16) and Section 4.3
Cell behavior depends on epigenetic state x through ad hoc Hill-type expressions (a1,a2,a3,b1, 5.8, 2.2, 3.75, 4.0, etc.) chosen for qualitative behavior; no parameter estimation or error analysis is given.
Transition kernel shape functions phi(y), eta(y) = unspecified; pre-defined functions in Section 3.2
The Beta-distribution kernel p(x,y) is the core mechanism for plasticity, but phi and eta are not measured; they come from the author's histone-modification simulation [18,19].
CAR-T model parameters (gamma19, gamma22, X0, X1, n0, n1, m, alpha34, mu0, mu1) = calibrated in prior preprint [46]
The Section 4.3 application has numerous numerical parameters used to reproduce mouse relapse data and to predict combination therapy; only the formulas, not the fitted values or uncertainty, appear in this paper.

assumptions (5)

domain assumption The G0 compartment model: resting cells either enter proliferation at rate beta or leave at rate kappa, and proliferating cells divide after fixed duration tau with apoptosis probability mu (Eqs. (1)-(3)).
This is the standard Burns-Tannock model [2], treated as the base truth for all subsequent population equations.
domain assumption Epigenetic state x changes only at cell division, and inheritance is described by a transition probability p(x,y) that integrates to 1 over daughter states (Section 3.1, Eqs. (10)-(11)).
The entire heterogeneous model depends on inheritance being Markovian in x at division; no mechanism such as time-continuous state switching is modeled.
domain assumption Proliferation beta, apoptosis mu, differentiation kappa, cycle duration tau, and cytokine secretion zeta depend only on current state x and global effective population hatQ (Eqs. (8)-(11)).
This neglects spatial structure, cell-cell contact, immune cells, and microenvironment, which the author acknowledges in Section 6.
ad hoc to paper p(x,y) has the conditional Beta-distribution form with mean phi(y) and variance phi(1-phi)/(1+eta) (Eqs. (13)-(14)).
The paper states p cannot be measured and proposes this form based on simulations in the author's own histone-modification model [18,19]; the choice is not derived from independent experimental data.
domain assumption In the applications, mutations and CAR-T responses act through specified pathway changes and CD34/CD123 state transitions (Table 1, Section 4.3).
The inflammation and CAR-T models import mutation taxonomies and plasticity assumptions from [13] and [46]; if those pathway assignments are wrong, the simulated agreements would not transfer.

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Cite this review

Pith. "Pith review of Evolutionary dynamics of cancer: from epigenetic regulation to cell population dynamics -- mathematical model framework, applications, and open problems." pith.science (2026). https://pith.science/paper/IOYD24OB

@misc{pith2026190807048,
  author       = {Pith},
  title        = {Pith review of: Evolutionary dynamics of cancer: from epigenetic regulation to cell population dynamics -- mathematical model framework, applications, and open problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IOYD24OB}},
  note         = {Machine review of arXiv:1908.07048}
}

abstract

Predictive modeling of the evolutionary dynamics of cancer is a challenge issue in computational cancer biology. In this paper, we propose a general mathematical model framework for the evolutionary dynamics of cancer with plasticity and heterogeneity in cancer cells. Cancer is a group of diseases involving abnormal cell growth, during which abnormal regulations in stem cell regeneration are essential for the dynamics of cancer development. In general, the dynamics of stem cell regeneration can be simplified as a $\mathrm{G_0}$ phase cell cycle model, which lead to a delay differentiation equation. When cell heterogeneity and plasticity are considered, we establish a differential-integral equation based on the random transition of epigenetic states of stem cells during cell division. The proposed model highlights cell heterogeneity and plasticity, and connects the heterogeneity with cell-to-cell variance in cellular behaviors, e.g. proliferation, apoptosis, and differentiation/senescence, and can be extended to include gene mutation-induced tumor development. Hybrid computations models are developed based on the mathematical model framework, and are applied to the process of inflammation-induced tumorigenesis and tumor relapse after CAR-T therapy. Finally, we give rise to several mathematical problems related to the proposed differential-integral equation. Answers to these problems are crucial for the understanding of the evolutionary dynamics of cancer.

Figures

Figures reproduced from arXiv: 1908.07048 by the authors.

**Figure 1.** The G0 model of stem cell regeneration. During stem cell regeneration, cells in the resting phase either enter the proliferating phase with a rate β, or be removed from the resting pool with a rate κ due to differentiation, aging, or death. The proliferating cells undergo apoptosis with a probability µ. following partial differential equations[2] (1) ∇s(t, a) = −µs(t, a), (t > 0, 0 < a < τ ) dQ dt = 2s(t, τ ) − (β(… view at source ↗

**Figure 2.** Illustration of the model for heterogeneous stem cell regeneration. Similar to [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Framework of the the mathematical model of heterogeneous stem cell regeneration. 3.2. The transition function p(x, y). In the above equation, the transition function p(x, y) is important to connect cell heterogeneity with plasticity. However, biologically we cannot measure this function directly, and do not know the possible form neither. Here, we propose a possible form of the transition function through numerical … view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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