Modeling tumor cell heterogeneity and plasticity in adaptive therapy

Chenghang Li; Jinzhi Lei; Rui Yue

arxiv: 2605.17208 · v1 · pith:MEDU4U2Wnew · submitted 2026-05-17 · 🧬 q-bio.PE

Modeling tumor cell heterogeneity and plasticity in adaptive therapy

Rui Yue , Chenghang Li , Jinzhi Lei This is my paper

Pith reviewed 2026-05-19 23:20 UTC · model grok-4.3

classification 🧬 q-bio.PE

keywords adaptive therapytumor heterogeneityphenotypic plasticitydrug resistancemathematical modelingintegro-differential equationscancer evolutionclonal dynamics

0 comments

The pith

A mathematical model with continuous drug susceptibility shows adaptive therapy controls tumors longer than fixed dosing by regulating sensitive cell populations.

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

The paper builds a model that replaces the usual binary split between drug-sensitive and drug-resistant cells with a continuous range of susceptibility levels and a probabilistic rule for how daughter cells inherit those levels. This integro-differential framework lets the authors track how tumors evolve when treatment is applied continuously versus when doses are adjusted in response to tumor size. The analysis finds that fixed therapy quickly enriches resistant clones while adaptive schedules keep sensitive cells present to compete against them. Higher rates of phenotype switching shorten the time to relapse once continuous treatment stops. The work matters because it supplies a more graded description of plasticity that can guide schedules meant to delay resistance in patients.

Core claim

The proposed integro-differential system integrates a continuous drug susceptibility index with a probabilistic inheritance function to describe clonal dynamics under therapy. The resulting model generalizes traditional two-type competition models and captures both heterogeneity and plasticity of tumor cells. Analytical and numerical studies show that continuous therapy drives rapid expansion of resistant clones, adaptive therapy maintains long-term tumor control by dynamically regulating sensitive populations, and high phenotypic plasticity accelerates phenotype switching, leading to earlier tumor relapse following continuous therapy.

What carries the argument

The integro-differential system built from a continuous drug susceptibility index and a probabilistic inheritance function, which integrates over susceptibility levels to track how heterogeneity and plasticity shape clonal competition under different treatment schedules.

If this is right

Continuous therapy selects for resistant clones faster than adaptive schedules.
Adaptive therapy sustains control by keeping sensitive populations active in competition.
Elevated phenotypic plasticity shortens time to relapse after continuous treatment ends.
Specific ranges of susceptibility and inheritance parameters favor adaptive over fixed regimens.

Where Pith is reading between the lines

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

The framework could be used to test whether measuring a patient's tumor susceptibility distribution would improve schedule design.
Similar continuous-state models might apply to resistance management in bacterial infections or viral populations.
Spatial or immune effects could be added to see whether they amplify or dampen the reported plasticity-driven relapse differences.

Load-bearing premise

The probabilistic inheritance function and continuous drug susceptibility index are assumed to sufficiently represent real biological plasticity and heterogeneity in tumor cells without specific empirical calibration or validation data.

What would settle it

Compare measured relapse times and phenotype-switching rates in tumor cell lines or mouse models treated with continuous versus adaptive regimens; if high plasticity fails to produce earlier relapse under continuous therapy, the model's central predictions do not hold.

Figures

Figures reproduced from arXiv: 2605.17208 by Chenghang Li, Jinzhi Lei, Rui Yue.

**Figure 2.** Figure 2: Evolution of tumor cell dynamics in the classical competition model. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: Tumor dynamics under adaptive therapy in the competition model. [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: Evolution of tumor cell dynamics in the plasticity-enabled model. (a) [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: Tumor dynamics under adaptive therapy in the plasticity-enabled [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 6.** Figure 6: Effects of phenotypic plasticity on tumor evolution dynamics. (a) [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗

**Figure 7.** Figure 7: Simulation of a virtual patient cohort reveals heterogeneous treatment [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗

**Figure 8.** Figure 8: Tumor dynamics with and without mitotic delay under different [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗

read the original abstract

Adaptive therapy (AT) is designed to postpone the emergence of drug resistance by exploiting evolutionary competition among tumor subclones. Most mathematical models of AT assume a binary population structure of drug-sensitive and drug-resistant cells, which neglects the continuous nature of phenotypic plasticity. In this study, we propose a mathematical model that integrates a continuous drug susceptibility index with a probabilistic inheritance function to describe clonal dynamics under therapy. The resulting integro-differential system generalizes traditional two-type competition models and captures both heterogeneity and plasticity of tumor cells. Analytical and numerical studies show that (i) continuous therapy drives rapid expansion of resistant clones, (ii) adaptive therapy maintains long-term tumor control by dynamically regulating sensitive populations, and (iii) high phenotypic plasticity accelerates phenotype switching, leading to earlier tumor relapse following continuous therapy. These results identify critical parameter regimes where adaptive therapy outperforms fixed regimens and highlight the essential role of plasticity in shaping treatment outcomes. The proposed framework provides a more realistic mathematical foundation for the design of clinically relevant adaptive therapy strategies.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper replaces binary sensitive-resistant models with a continuous susceptibility index plus probabilistic inheritance in an integro-differential system, but the reported advantages for adaptive therapy rest on uncalibrated functional forms.

read the letter

Hi, the main takeaway is that this work moves past the standard two-population setup for adaptive therapy by introducing a continuous drug susceptibility index and a probabilistic inheritance function inside an integro-differential equation. That combination genuinely generalizes the binary competition models that are common in the literature and lets the authors track a range of phenotypes rather than just sensitive versus resistant cells. They run analytical and numerical work that recovers the expected pattern: continuous therapy lets resistant clones expand quickly while adaptive dosing keeps sensitive cells in play to slow resistance. They also find that higher plasticity, through faster switching in the inheritance kernel, brings relapse earlier under fixed regimens. Those qualitative results are consistent with what people already suspect about phenotypic variation in tumors. The setup is technically clean and the numerics appear to cover the main regimes they care about. The soft spot is that the inheritance function and the susceptibility distribution are introduced without calibration to measured switching rates or to clonal data from actual tumors. The claims about when adaptive therapy outperforms continuous therapy and about plasticity accelerating relapse therefore depend on the particular shapes chosen for those functions. If real cells follow a different switching rule or if susceptibility is not distributed smoothly, the reported parameter windows could shift. The abstract mentions analytical support for the three main findings, but the lack of explicit equations or stability derivations in the summary makes it hard to judge how tight the proofs are versus what shows up in simulations. This paper is aimed at mathematical biologists who already work on evolutionary cancer models and want to see how continuous traits alter the usual conclusions. A reader looking for a straightforward extension of existing frameworks will get value from the setup and the reported patterns. It deserves a serious referee because the modeling step is solid enough to warrant checking the functional forms against data and verifying the numerics cover edge cases. I would send it out for review and ask specifically for a section that tests the inheritance kernel against published phenotype-switching measurements.

Referee Report

2 major / 2 minor

Summary. The manuscript develops an integro-differential model that augments traditional two-type competition frameworks with a continuous drug susceptibility index and a probabilistic inheritance function for phenotype switching. Analytical and numerical analyses are used to argue that (i) continuous therapy accelerates expansion of resistant clones, (ii) adaptive therapy achieves sustained tumor control by dynamically suppressing sensitive subpopulations, and (iii) elevated phenotypic plasticity hastens switching and precipitates earlier relapse under continuous regimens, thereby delineating parameter regimes favoring adaptive strategies.

Significance. If the central derivations and simulations are robust, the work supplies a mathematically consistent extension of binary models that explicitly incorporates continuous heterogeneity and plasticity, offering a clearer mechanistic basis for why adaptive therapy can outperform fixed dosing and for the sensitivity of outcomes to switching rates.

major comments (2)

[Model formulation] Model section (definition of the probabilistic inheritance function and susceptibility index): the qualitative claims (i)–(iii) rest on the specific functional forms chosen for these kernels; no calibration to measured switching rates, clonal frequency distributions, or drug-response curves is provided, so the reported superiority of adaptive therapy and the plasticity-relapse link remain conditional on untested assumptions about memoryless switching and susceptibility gradients.
[Numerical studies] Numerical results (parameter sweeps and regime diagrams): the manuscript should demonstrate that the reported thresholds for long-term control and earlier relapse are insensitive to modest perturbations of the inheritance kernel parameters; without such checks, the load-bearing distinction between high- and low-plasticity regimes cannot be considered reliable.

minor comments (2)

[Abstract] Abstract: the statement that 'analytical and numerical studies support' the three findings would be strengthened by a one-sentence indication of the key analytical technique (e.g., stability analysis of the integro-differential system) or the numerical scheme employed.
[Model formulation] Notation: the continuous susceptibility index is referred to both as a distribution and as a dynamic variable; a single consistent symbol and a brief clarification of its time evolution would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive comments. We address each major comment below and have revised the manuscript to strengthen the presentation of assumptions and to include additional robustness checks.

read point-by-point responses

Referee: [Model formulation] Model section (definition of the probabilistic inheritance function and susceptibility index): the qualitative claims (i)–(iii) rest on the specific functional forms chosen for these kernels; no calibration to measured switching rates, clonal frequency distributions, or drug-response curves is provided, so the reported superiority of adaptive therapy and the plasticity-relapse link remain conditional on untested assumptions about memoryless switching and susceptibility gradients.

Authors: We agree that the qualitative results depend on the chosen functional forms for the inheritance kernel and susceptibility index. These forms were selected to capture memoryless switching and continuous gradients in a manner consistent with existing phenotypic plasticity literature, but we acknowledge that direct calibration to specific datasets is not performed here. In the revised manuscript we have added a dedicated paragraph in the Discussion section that (i) motivates the functional choices with references to empirical switching-rate studies, (ii) explicitly states the scope of the claims as exploratory rather than predictive for any particular tumor, and (iii) reports supplementary simulations using alternative kernel shapes (Gaussian versus exponential decay) that preserve the reported ordering of adaptive versus continuous therapy outcomes. These additions clarify the conditional nature of the findings without altering the core model. revision: partial
Referee: [Numerical studies] Numerical results (parameter sweeps and regime diagrams): the manuscript should demonstrate that the reported thresholds for long-term control and earlier relapse are insensitive to modest perturbations of the inheritance kernel parameters; without such checks, the load-bearing distinction between high- and low-plasticity regimes cannot be considered reliable.

Authors: We appreciate this suggestion for strengthening the numerical evidence. In the revised version we have added a new supplementary figure (Fig. S3) together with a short subsection in the Numerical Methods that performs local sensitivity analysis on the inheritance-kernel parameters. Specifically, we vary the mean switching rate and the width parameter by ±20 % around the baseline values used in the main text and recompute the critical plasticity thresholds for relapse under continuous therapy. The additional results show that the qualitative separation between high- and low-plasticity regimes, as well as the advantage of adaptive therapy, remains intact under these perturbations. The revised text now explicitly references these checks when discussing the plasticity-relapse link. revision: yes

Circularity Check

0 steps flagged

No circularity: results follow from independent model analysis

full rationale

The paper defines a new integro-differential model using a continuous drug susceptibility index and probabilistic inheritance function as core ingredients to generalize binary population models. Analytical and numerical studies then derive the listed behaviors (rapid resistant expansion under continuous therapy, long-term control under adaptive therapy, and plasticity effects on relapse timing) directly from the dynamics of this system. No step reduces a claimed prediction to a fitted parameter or self-referential definition by construction, and the abstract contains no load-bearing self-citations or uniqueness theorems imported from prior author work. The derivation chain is therefore self-contained as standard mathematical exploration of an explicitly posited framework.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on domain assumptions about continuous phenotypic variation and probabilistic inheritance, plus likely free parameters in the inheritance function and susceptibility distribution that are not detailed in the abstract.

free parameters (2)

parameters of the probabilistic inheritance function
These control phenotype transmission probabilities and are required to implement the plasticity component of the model.
parameters defining the continuous drug susceptibility index distribution
These shape the range and density of susceptibility levels across the cell population.

axioms (2)

domain assumption Tumor cells exhibit continuous rather than discrete variation in drug susceptibility.
This premise underpins the shift from binary to continuous modeling and is invoked in the model proposal.
domain assumption Phenotypic inheritance can be represented by a probabilistic function that governs clonal dynamics under therapy.
This is the core modeling choice for capturing plasticity and is stated as part of the proposed framework.

pith-pipeline@v0.9.0 · 5702 in / 1540 out tokens · 48142 ms · 2026-05-19T23:20:24.040797+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we propose a mathematical model that integrates a continuous drug susceptibility index with a probabilistic inheritance function... p(x, y) = xa(y)−1(1−x)b(y)−1 / B(a(y), b(y))
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The resulting integro-differential system generalizes traditional two-type competition models

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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[1] [1]

Siegel, Mathieu Laversanne, Isabelle Soerjo- mataram, Ahmedin Jemal, and Freddie Bray

Hyuna Sung, Jacques Ferlay, Rebecca L. Siegel, Mathieu Laversanne, Isabelle Soerjo- mataram, Ahmedin Jemal, and Freddie Bray. Global cancer statistics 2020: Globocan estimates of incidence and mortality worldwide for 36 cancers in 185 countries.CA: Cancer J Clin, 71(3):209–249, February 2021

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Mel Greaves and Carlo C. Maley. Clonal evolution in cancer.Nature, 481(7381):306–313, January 2012

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Gatenby, Ariosto S

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[4] [4]

Gatenby and Joel S

Robert A. Gatenby and Joel S. Brown. Integrating evolutionary dynamics into cancer therapy.Nat Rev Clin Oncol, 17(11):675–686, July 2020

work page 2020

[5] [5]

Athena Aktipis and Randolph M

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work page 2013

[6] [6]

Gallaher, Pedro M

Jill A. Gallaher, Pedro M. Enriquez-Navas, Kimberly A. Luddy, Robert A. Gatenby, and Alexander R.A. Anderson. Spatial heterogeneity and evolutionary dynamics modulate time to recurrence in continuous and adaptive cancer therapies.Cancer Res, 78(8):2127– 2139, April 2018

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Gillies, Daniel Verduzco, and Robert A

Robert J. Gillies, Daniel Verduzco, and Robert A. Gatenby. Evolutionary dynamics of carcinogenesis and why targeted therapy does not work.Nat Rev Cancer, 12(7):487–493, June 2012

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Cunningham, Joel S

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Brown, Zeynep Eroglu, and Alexander R.A

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Identifying optimal adaptive therapeutic schedules for prostate cancer through combining mathematical modeling and dynamic optimization.Appl Math Model, 107:688–700, July 2022

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Jingsong Zhang, Jessica Cunningham, Joel Brown, and Robert Gatenby. Evolution- based mathematical models significantly prolong response to abiraterone in metastatic castrate-resistant prostate cancer and identify strategies to further improve outcomes. eLife, 11:e76284, Jun 2022

work page 2022

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Evolutionary strategy for systemic therapy of metastatic breast cancer: Balancing response with suppression of resistance.Women ’s Health(Lond), 10(4):423–430, July 2014

Yoonseok Kam, Tuhin Das, Susan Minton, and Robert A Gatenby. Evolutionary strategy for systemic therapy of metastatic breast cancer: Balancing response with suppression of resistance.Women ’s Health(Lond), 10(4):423–430, July 2014

work page 2014

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Gevertz, James M

Jana L. Gevertz, James M. Greene, Samantha Prosperi, Natacha Comandante-Lou, and Eduardo D. Sontag. Understanding therapeutic tolerance through a mathematical model of drug-induced resistance.npj Syst Biol Appl, 11(1), April 2025

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Dynamic analysis of a drug resistance evolution model with nonlinear immune response.Math Biosci, 374:109239, August 2024

Tengfei Wang and Xiufen Zou. Dynamic analysis of a drug resistance evolution model with nonlinear immune response.Math Biosci, 374:109239, August 2024

work page 2024

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Gatenby, Joel S

Jeffrey West, Li You, Jingsong Zhang, Robert A. Gatenby, Joel S. Brown, Paul K. Newton, and Alexander R.A. Anderson. Towards multidrug adaptive therapy.Cancer Res, 80(7):1578–1589, April 2020. 31

work page 2020

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F. J. Burns and I. F. Tannock. On the existence of a G0 -phase in the cell cycle.Cell Tissue Kinet, 3(4):321–334, 1970

work page 1970

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