Inference of weak-form partial differential equations describing migration and proliferation mechanisms in wound healing experiments on cancer cells

Srivastava, Siddhartha; Kinnunen, Patrick C.; Wang, Zhenlin; Ho, Kenneth K.Y.; Humphries, Brock A.; Chen, Siyi; Linderman, Jennifer J.; Luker, Gary D.; Luker, Kathryn E.; Garikipati, Krishna

doi:10.1371/journal.pcbi.1013607

Cited by 5Mathematical Biology Tumor GrowthCellular Mechanics and InteractionsCancer Cells and Metastasis

Inference of weak-form partial differential equations describing migration and proliferation mechanisms in wound healing experiments on cancer cells

Open access

Srivastava, Siddhartha, Kinnunen, Patrick C., Wang, Zhenlin, Ho, Kenneth K.Y., Humphries, Brock A., Chen, Siyi, Linderman, Jennifer J., Luker, Gary D. +2 more·2025·PLOS Computational Biology

DOI 10.1371/journal.pcbi.1013607PMC PMC11537331

Open this paper in CoChat — free account Open PDF

Abstract

Cancer metastasis, which requires migration of cancer cells away from the primary tumor, is responsible for approximately 65% percent of cancer-related deaths. Therefore, targeting signaling pathways that drive cancer cell migration or proliferation is a common therapeutic approach. Cell migration is commonly studied using experimental approaches which track cells or cell monolayers as they evolve over time. Computational modeling can then be used to fit partial differential equation (PDE) models to the data, providing mechanistic insights underlying the observed cell motion, including the contribution of various cellular behaviors such as random motion, directed motion, and cell division. A popular experimental technique, the scratch assay, measures the migration and proliferation-driven cell closure of a scratch in a confluent cell monolayer. However, these assays do not disambiguate between different drivers of scratch closure (for instance between cell proliferation and migration to open space). To improve analysis of this technique, we combine scratch assays, video microscopy, and PDE inference to gain quantitative insight to mechanisms of cell migration and proliferation. We capture the evolution of cell density fields over time using live-cell microscopy and automated image processing. Our PDE inference methods involve the use of weak form-based system identification techniques for cell density dynamics modeled with advection-diffusion-reaction systems. We then compare our method with recent modeling work, finding that our model discovery tool automatically identifies similar models including reaction and diffusion terms from a larger set of bases. We demonstrate the application of this framework on 2-dimensional scratch assays subject to the inhibiting effect of trametinib on wound closure and characterize the results in the context of the quantified uncertainty in our inference approach. Our integrated experimental and computational pipeline can be used to rapidly identify and refine models of cell migration in a variety of contexts, enabling the quantitative measurement of the effect of drugs and other perturbations on cell migration and proliferation with uncertainty accounted for.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper develops a weak-form variational system identification pipeline, followed by PDE-constrained optimization, to infer advection-diffusion-reaction PDEs from scratch-wound healing assays. Applied to 1D PC-3 data and 2D MDA-MB-231 breast cancer assays, it identifies parsimonious reaction-diffusion models and quantifies trametinib effects on random migration and proliferation.

Key findings

In 1D PC-3 scratch data, inferred 3-term models excluded advection and had constant diffusivity plus linear or quadratic reaction terms; for four of six initial seeding densities, RMSEs were significantly lower than those reported by Jin et al., and for two they were similar.
In 2D MDA-MB-231 assays, the Variational System Identification loss was essentially constant for two-term diffusion-plus-reaction models and increased only for models lacking reaction terms; the adopted three-term models for untreated and 100 nM trametinib had constant diffusivity and linear/quadratic reaction terms.
The inferred reaction rate was positive at low cell density and became negative above a critical density, interpreted as crowding-induced death; untreated diffusivity was around 50 μm²/hr and decreased with trametinib.
Low levels of trametinib inhibited wound closure primarily by decreasing random cell migration by approximately 20%; higher concentrations did not further inhibit migration, and RMSEs across conditions averaged about 13% of maximum cell concentrations.

Methods

MDA-MB-231 H2B-mCherry cells were seeded at 50,000 cells/well in 24-well glass-bottom plates, grown to confluence (~36 h), scratched with a p200 tip, and imaged every 20 min for 24–48 h; treatments included five trametinib concentrations from 10 μM to 100 nM and an untreated control. Automated image processing thresholded nuclear images, extracted centroids, binned positions (50–100 μm), and smoothed spatially (150 μm) and temporally (3 h 40 min) to obtain cell density fields C(x,t).
The 1D validation used previously published PC-3 scratch assay data from Jin et al., with initial seeding from 10,000 to 20,000 cells in 2,000-cell steps, density measured every 12 h for 48 h and averaged across three replicates.
Inference used the weak form of an advection-diffusion-reaction PDE with density-dependent D(C), vf(C), and r(C) up to quadratic order, discretized by finite elements in FEniCS; stepwise regression selected parsimonious terms, followed by PDE-constrained optimization with adjoint-based gradients/BFGS via dolfin-adjoint. The 2D mesh had 6,478 linear triangular elements, and 1D forward solutions used 42 linear elements.

Limitations

Paper-stated: inference proceeds by model selection from a candidate library, not by de novo discovery; future work aims to address experiments that may not resolve length scales with sufficiently high cell densities for continuum predictions and sparse data; at higher cell densities, Maxwell-Stefan diffusion may be more appropriate.
Paper-stated: in the 1D 18,000-cell case, the data were not sufficiently sensitive to the diffusive mechanism, requiring expansion to a four-term model.
Evident from the text: 2D experiments imaged one region of the scratch per well and did not track individual cells between frames; 2D results are from MDA-MB-231 cells only, and 1D validation from PC-3 cells, so generalizability across cell lines and conditions is not established here.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1371/journal.pcbi.1013607

Cites

61 of 63 references verified0 citation errors · 2 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

31 of 36 cited sources checked · 1 supported · 3 flagged · accuracy not yet measured

Premisecitation
The dataset, collected by Jin et al., consists of scratch assays performed on PC-3 prostate cancer cells with varying initial confluence prior to the scratch.
“The dataset, collected by Jin et al.,8 consists of scratch assays performed on PC-3 prostate cancer cells with varying initial confluence prior to the scratch.”
Quote in the paperSubstantial claim1 of 1 sources checked
- cites [8] NH Gopee, 2024, A prenatal skin atlas reveals immune regulation of human skin morphogenesis, Nature. source
  The Methods section, “Scratch wound assay of fibroblasts in co-culture with macrophages” (page unavailable), describes assays using skin-organoid-derived fibroblasts and macrophages, not PC-3 prostate cancer cells. It specifies a 5:1 seeding ratio and does not describe varying initial confluence. (LLM, conf=0.89)
  Checked by a reasoning model — may be wrong.
  Experimental — accuracy not yet measured
Findingcitation
Variational System Identification rapidly identified an accurate model describing the evolution of cell concentration over time for a variety of initial seeding densities, with accuracy that matched or improved upon results in the literature.
“With it, we were able to rapidly identify an accurate model describing the evolution of cell concentration over time for a variety of initial seeding densities with accuracy that matched or improved upon results in the literature8.”
Quote in the paperSubstantial claim1 of 1 sources checked
- cites [8] NH Gopee, 2024, A prenatal skin atlas reveals immune regulation of human skin morphogenesis, Nature. source
  Split into 2 atoms — 0 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 2 unverifiable. Expand the claim to see per-atom verdicts.
  Experimental — accuracy not yet measured
Findingown data
Low levels of trametinib inhibit wound closure primarily by decreasing random cell migration by approximately 20%.
“We demonstrate the application of this pipeline on high throughput 2-dimensional scratch assays and find that low levels of trametinib inhibit wound closure primarily by decreasing random cell migration by approximately 20%.”
Quote in the paperSubstantial claim
Findingown data
Variational System Identification successfully identified mechanisms of advection, diffusion and proliferation of cells in 2-D scratch assays by inferring the governing PDEs.
“We found that Variational System Identification could successfully identify mechanisms of advection (directed motion), diffusion (random motion) and proliferation of cells in 2-D scratch assays by inferring the governing PDEs.”
Quote in the paperSubstantial claim
Findingown data
The inferred reaction rate has a non-linear functional dependence on cell density: positive at low cell densities and becoming negative above a critical density, i.e., cell death sets in.
“This combination of techniques reveals a non-linear functional dependence of cell proliferation (reaction in the PDE) rate on cell density: positive at low cell densities and becoming negative above a critical density; i.e., cell death sets in(Fig. 6D).”
Quote in the paperSubstantial claim
- Rests on an assumption: The reaction term in the PDE corresponds to cell proliferation and death rather than other density-dependent processes.

Findingown data
All reaction terms have linear concentration dependence, except the four-term model for 18,000 cells.
“All the reaction terms have linear concentration dependence, except the four-term model for 18,000 cells (see Tab. 1).”
Quote in the paperNot a substantial claim
Findingown data
Untreated cells were able to almost close the wound after 24 hours.
“We observed that untreated cells were able to almost close the wound after 24 hours”
Quote in the paperNot a substantial claim
Findingown data
Expanding the model to include four terms, incorporating both linear and quadratic reaction terms, led to a significant reduction in the cost function during the PDE-constrained optimization.
“This expansion led to a significant reduction in the cost function during the PDE-constrained optimization.”
Quote in the paperSubstantial claim
Findingown data
For the case of 18,000 cells, minimal improvement in the cost function Eq. (18) from the PDE-constrained optimization was observed.
“For the case of 18,000 cells, we observed minimal improvement in the cost function Eq. (18) from the PDE-constrained optimization.”
Quote in the paperSubstantial claim
Findingown data
The effect of trametinib on cell diffusion and proliferation was quantitatively inferred.
“Extending the approach’s applications we quantitatively inferred the effect of trametinib, a MAPK pathway inhibitor, on cell diffusion and proliferation.”
Quote in the paperSubstantial claim
Findingown data
100 nM trametinib was enough to significantly inhibit cell migration.
“while 100 nM trametinib was enough to significantly inhibit cell migration.”
Quote in the paperSubstantial claim
Findingown data
For all initial seeding with 14000 and more cells, the center of the scratch (around x1=500μm) is occupied by cells after 36 hours.
“For all initial seeding with 14000 and more cells, the center of the scratch (around x1=500μm) is occupied by cells after 36 hours.”
Quote in the paperNot a substantial claim
Findingown data
The refined model for 18,000 cells with four terms demonstrates a strong agreement between the model solution and the experimental data.
“The refined model, presented in Tab. 1 under the label 18,000 (4 terms), demonstrates a strong agreement between the model solution and the experimental data.”
Quote in the paperSubstantial claim
Findingown data
For all six scratch assay datasets, the Variational System Identification loss (mean squared error between data and model predicted time derivatives) does not decrease for models having more than 3 terms.
“Fig. 3 shows that for all cases labelled by the initial number of cells seeded, the loss (mean squared error between data and model predicted time derivatives) does not decrease for models having more than 3 terms.”
Quote in the paperSubstantial claim
Findingown data
For both the untreated condition and 100 nM trametinib, the adopted three-term model terms corresponded to constant diffusivity and reaction terms that are linear and quadratic in cell density.
“For both the untreated condition and 100 nM trametinib, these terms corresponded to constant (concentration-independent) diffusivity, and reaction terms that are linear and quadratic in cell density.”
Quote in the paperSubstantial claim
- Rests on an assumption: The candidate library of constant, linear, and quadratic terms for diffusion, advection, and reaction contains the mechanisms needed to describe the observed cell migration dynamics.
Findingown data
Our models are comparable or slightly superior in accuracy to previously published, traditionally derived reaction-diffusion models applied to 1-D wound healing data.
“We benchmark this method by comparing inferred models with previous reaction-diffusion models applied to 1-D wound healing data, finding that our models are comparable or slightly superior in accuracy to the previously published, traditionally derived models.”
Quote in the paperSubstantial claim
Findingown data
There is an increase in cell density at the scratch edge (near x=0μm or 950μm) over time for all initial seeding densities.
“Furthermore, there is an increase in cell density at the scratch edge (near x=0μm or 950μm) over time for all initial seeding densities, indicating that the cells continue growing during the experiment.”
Quote in the paperNot a substantial claim
Findingown data
Higher concentrations of trametinib did not further inhibit cell migration.
“Higher concentrations of trametinib did not further inhibit cell migration (Fig. 5).”
Quote in the paperNot a substantial claim
Findingown data
The mean RMSE across conditions is approximately 13% of the maximum cell concentrations in the experiments shown in Fig. 6A.
“The mean value across conditions is ~ 13% of the maximum cell concentrations in the experiments shown in Fig. 6A.”
Quote in the paperNot a substantial claim
Findingown data
In every case, the three-term model excluded advection.
“In every case, the three-term model excluded advection.”
Quote in the paperSubstantial claim
Findingown data
The RMSEs between the forward predictions of the model and the data are essentially comparable across trametinib concentrations and for the untreated case.
“The RMSEs between the forward predictions of the model and the data are presented in Fig. 7, and are essentially comparable across trametinib concentrations and for the untreated case.”
Quote in the paperNot a substantial claim
Interpretationown data
The integrated experimental and computational pipeline can be adapted for quantitatively inferring the effect of biological perturbations on cell migration and proliferation in various cell lines.
“Our integrated experimental and computational pipeline can be adapted for quantitatively inferring the effect of biological perturbations on cell migration and proliferation in various cell lines.”
Quote in the paperSubstantial claim
Interpretationown data
Variational System Identification followed by PDE-constrained parameter optimization can be used to infer models of collective cell migration and proliferation/death in a wound healing assay.
“In summary, we demonstrate that Variational System Identification followed by PDE-constrained parameter optimization can be used to infer models of collective cell migration and proliferation/death in a wound healing assay.”
Quote in the paperSubstantial claim
- Rests on an assumption: The candidate library of constant, linear, and quadratic terms for diffusion, advection, and reaction contains the mechanisms needed to describe the observed cell migration dynamics.
- Rests on an assumption: The reaction term in the PDE corresponds to cell proliferation and death rather than other density-dependent processes.
Interpretationown data
The approach advanced here can be applied to 2-D wound healing to quantify effects of a targeted inhibitor on cell migration and density-dependent effects on proliferation.
“We further find that the approach advanced here can be applied to 2-D wound healing to quantify effects of a targeted inhibitor on cell migration and density-dependent effects on proliferation.”
Quote in the paperSubstantial claim
- Rests on an assumption: The candidate library of constant, linear, and quadratic terms for diffusion, advection, and reaction contains the mechanisms needed to describe the observed cell migration dynamics.
- Rests on an assumption: The reaction term in the PDE corresponds to cell proliferation and death rather than other density-dependent processes.
Premiseown data
The high cell densities that were achieved in the Jin et al. experiment justified the consideration of continuum advection-diffusion-reaction PDEs to model these data.
“The high cell densities that were achieved in this experiment justified the consideration of continuum advection-diffusion-reaction PDEs to model these data.”
Quote in the paperSubstantial claim
Interpretationown data
In a 1D setting, we infer models with accuracies that are competitive with other recent approaches.
“In a 1D setting, we infer models with accuracies that are competitive with other recent approaches.”
Quote in the paperSubstantial claim
Interpretationreasoning
Our estimates of diffusivity in the 1D and 2D settings are consistent with the prior literature.
“Thus, our estimates of diffusivity in the 1D and 2D settings are consistent with the prior literature.”
Quote in the paperNot a substantial claim
Interpretationreasoning
This pipeline can be used to rapidly identify parsimonious models for cell migration and proliferation from scratch assays.
“Our work demonstrates that this pipeline can be used to rapidly identify parsimonious models for cell migration and proliferation from scratch assays.”
Quote in the paperSubstantial claim
- Rests on an assumption: The candidate library of constant, linear, and quadratic terms for diffusion, advection, and reaction contains the mechanisms needed to describe the observed cell migration dynamics.
- Rests on an assumption: The reaction term in the PDE corresponds to cell proliferation and death rather than other density-dependent processes.
Interpretationreasoning
The inferred mechanism could be interpreted as crowding-induced death in the cell population.
“This inferred mechanism could be interpreted as a crowding-induced induced death in the cell population.”
Quote in the paperNot a substantial claim
- Rests on an assumption: The reaction term in the PDE corresponds to cell proliferation and death rather than other density-dependent processes.
Interpretationown data
In the 3-term model, the partial derivatives of the cost function with respect to the diffusivity coefficient remained low, indicating that the available data was not sufficiently sensitive to the diffusive mechanism of random cell motion.
“Furthermore, we noticed that in the 3-term model, the partial derivatives of the cost function with respect to the diffusivity coefficient remained low, indicating that the available data was not sufficiently sensitive to the diffusive mechanism of random cell motion.”
Quote in the paperSubstantial claim
Interpretationown data
Cells continue growing during the experiment.
“indicating that the cells continue growing during the experiment”
Quote in the paperSubstantial claim
Premisereasoning
The constant reaction term was omitted because it represents cell proliferation or death in a region with zero cell density, both of which are unphysical.
“The constant reaction term was omitted, since it represents cell proliferation (r>0) or death (r<0) in a region with zero cell density, both of which are unphysical.”
Quote in the paperSubstantial claim
Interpretationown data
Our PDE inference approach of Variational System Identification followed by PDE-constrained parameter optimization can infer parsimonious, quantitative and physics-based models for collective cell migration.
“Here, we demonstrate that our PDE inference approach of Variational System Identification followed by PDE-constrained parameter optimization can infer parsimonious, quantitative and physics-based models for collective cell migration.”
Quote in the paperSubstantial claim
Interpretationown data
The effect of trametinib treatment quickly saturates, as reflected in the coefficients for constant diffusivity and linear/quadratic reaction terms.
“Notably, the effect of trametinib treatment quickly saturates, as is reflected in the coefficients for constant diffusivity and linear/quadratic reaction terms in Tab. 2.”
Quote in the paperSubstantial claim
- Rests on an assumption: The candidate library of constant, linear, and quadratic terms for diffusion, advection, and reaction contains the mechanisms needed to describe the observed cell migration dynamics.
Interpretationown data
Our Variational System Identification and PDE-constrained parameter optimization modeling approach can be used to infer models for cell migration dynamics that are competitive with previous methods such as those by Jin et al.
“These results suggest that our Variational System Identification and PDE-constrained parameter optimization modeling approach can be used to infer models for cell migration dynamics that are competitive with previous methods such as those by Jin et al, which fit a model based on prior knowledge about the system.”
Quote in the paperSubstantial claim

Classified as probably not a substantial claim. 16
May combine two claims. 17
The quote only partly states this claim. 1

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation
Variational System Identification has been applied to identify governing equations for the evolution of materials, the spatiotemporal spread of COVID-19, and constitutive models of soft materials.
“Variational System Identification has been applied to identify governing equations for the evolution of materials17,20,19, the spatiotemporal spread of COVID-1921,18, as well as constitutive models of soft materials22,23.”
Quote in the paperNot a substantial claim7 of 7 sources checked
- cites [17] MJ Simpson, 2017, Quantifying the roles of random motility and directed motility using advection-diffusion theory for a 3T3 fibroblast cell migration assay stimulated with an electric field, BMC Syst Biol. source
  The Background and Methods > Modelling methods sections describe a 3T3 fibroblast migration assay using an advection–diffusion model. Our copy does not address Variational System Identification or its applications to material evolution, COVID-19 spread, or soft-material constitutive models (searched those terms); page numbers were not available. (LLM, conf=0.77)
  Experimental — accuracy not yet measured
- cites [20] NM Mangan, 2016, Inferring biological networks by sparse identification of nonlinear dynamics, IEEE Trans. Mol. Biol. Multi-Scale Commun. source
  The Conclusions describe implicit-SINDy applied to three biological models, while the SINDy section discusses identifying governing equations from data. Neither section addresses Variational System Identification applied to materials evolution, COVID-19 spread, or soft-material constitutive models; searches for those applications found no supporting passage in our copy. (LLM, conf=0.78)
  Experimental — accuracy not yet measured
- cites [19] SL Brunton, 2016, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc Natl Acad Sci. source
  Section 3 (p. 4) identifies the paper’s method as sparse identification of nonlinear dynamics (SINDy), not Variational System Identification. The Abstract (p. 1) lists demonstrations on oscillators, the Lorenz system, and fluid vortex shedding; it does not report the claimed applications to materials, COVID-19, or soft-material constitutive models. (LLM, conf=0.86)
  Checked by a reasoning model — may be wrong.
  Experimental — accuracy not yet measured
- cites [21] SH Rudy, 2017, Data-driven discovery of partial differential equations, Sci Adv. source
  The Introduction and Materials and Methods describe discovering governing PDEs through sparse regression, not applications of Variational System Identification to materials, COVID-19, or soft-material constitutive models. Searches for those applications found no passage addressing them in our copy (page numbers unavailable). (LLM, conf=0.76)
  Experimental — accuracy not yet measured
- cites [18] KKY Ho, 2023, Oscillatory ERK signaling and morphology determine heterogeneity of breast cancer cell chemotaxis via MEK-ERK and p38-MAPK signaling pathways, Bioengineering (Basel). source
  The cited paper’s Methods §2.7 applies Variational System Identification to breast-cancer cell migration, not to constitutive models of soft materials. Its References identify a separate Wang et al. (2021) paper on inferring the constitutive response of soft-material surrogates using that method; that is the work matching the clause to which this citation is attached. (LLM, conf=0.84) It matches “Inference of deformation mechanisms and constitutive response of soft material surrogates of biological tissue by full-field characterization and data-driven variational system identification” (2021) instead.
  Checked by a reasoning model — may be wrong.
  Experimental — accuracy not yet measured
- cites [22] M Quade, 2018, Sparse identification of nonlinear dynamics for rapid model recovery, Chaos. source
  The citation to this paper is attached to the COVID-19 application. The introduction (p. 1) mentions epidemiology only as a general example of a complex system, while the conclusions (p. 8) describe numerical demonstrations of abrupt-SINDy, not Variational System Identification. Our copy contains no account of identifying equations for the spatiotemporal spread of COVID-19 (searched for Variational System Identification, COVID-19, and epidemiology). (LLM, conf=0.76)
  Experimental — accuracy not yet measured
- cites [23] KP Champion, 2019, Discovery of nonlinear multiscale systems: Sampling strategies and embeddings, SIAM J Appl Dyn Syst. source
  The Abstract (p. 1) and Discussion (p. 17) describe SINDy and HAVOK methods for discovering dynamical systems, not applications of Variational System Identification to the listed subjects. Searches for variational identification, materials, and COVID-19 found no passage in our copy addressing those applications. (LLM, conf=0.77)
  Experimental — accuracy not yet measured
May combine two claims.
Premisecitation
Variational System Identification extends SINDy to inferring PDEs in weak form, thereby offering advantages in the presence of noise and high derivatives.
“It extends the popular SINDy approach12,13,14,15,16 to inferring PDEs in weak form, thereby offering advantages in the presence of noise and high derivatives17,18,19.”
Quote in the paperSubstantial claim3 of 3 sources checked
- cites [17] MJ Simpson, 2017, Quantifying the roles of random motility and directed motility using advection-diffusion theory for a 3T3 fibroblast cell migration assay stimulated with an electric field, BMC Syst Biol. source
  The Background and Modelling methods passages describe an advection-diffusion model for cell migration, not Variational System Identification. Searches for SINDy, weak-form PDE inference, noise, and high derivatives found no passage addressing the attributed claim. (LLM, conf=0.78)
  Experimental — accuracy not yet measured
- cites [18] KKY Ho, 2023, Oscillatory ERK signaling and morphology determine heterogeneity of breast cancer cell chemotaxis via MEK-ERK and p38-MAPK signaling pathways, Bioengineering (Basel). source
  Section 2.7 (page unavailable) says the authors used VSI to infer mechanisms and parameters of an advection–diffusion PDE. It does not say that VSI extends SINDy, uses a weak form, or offers advantages with noise or high derivatives; searches for those terms found no supporting passage. (LLM, conf=0.85)
  Experimental — accuracy not yet measured
- cites [19] SL Brunton, 2016, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc Natl Acad Sci. source
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation
First-principles modeling has established a variety of PDE models that accurately represent the behavior of scratch assays with a high density of cells and under a variety of conditions.
“First-principles modeling has established a variety of partial differential equation (PDE) models that accurately represent the behavior of scratch assays with a high density of cells and under a variety of conditions8,9.”
Quote in the paperNot a substantial claim2 of 2 sources checked
- cites [8] NH Gopee, 2024, A prenatal skin atlas reveals immune regulation of human skin morphogenesis, Nature. source
  The Methods section describes a fibroblast–macrophage scratch assay and its image-based analysis, but does not discuss PDE models of scratch assays. Searches for first-principles PDE modeling and cell-density-dependent scratch-assay models found no passage addressing the claimed modeling results. (LLM, conf=0.76)
  Experimental — accuracy not yet measured
- cites [9] PF Renz, 2024, In vivo single-cell CRISPR uncovers distinct TNF programmes in tumour evolution, Nature. source
  The Methods > Scratch assay passage describes an experimental cell-gap assay and image analysis, not a mathematical model. Searches for PDE models and first-principles modeling found no passage in our copy addressing whether such models accurately represent high-density scratch assays under varied conditions; page numbers were not provided. (LLM, conf=0.76)
  Experimental — accuracy not yet measured
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation
By estimating parameters for these models, past work has helped identify quantitative effects of biological perturbations in scratch assays.
“By estimating parameters for these models, past work has helped identify quantitative effects of biological perturbations in scratch assays8,10,11.”
Quote in the paperNot a substantial claim1 of 3 sources checked
- cites [8] NH Gopee, 2024, A prenatal skin atlas reveals immune regulation of human skin morphogenesis, Nature. source
  The ‘Scarless healing and potential macrophage contribution’ section reports that macrophage co-culture improved scratch wound-width closure over 72 hours, supporting a measured effect of a biological perturbation. The Methods section says the scratch images were analysed with an ImageJ macro and two-way ANOVA; it does not describe estimating parameters of a scratch-assay model to identify that effect. Page numbers are unavailable in our copy. (LLM, conf=0.85)
  Experimental — accuracy not yet measured
- cites [10] M Vang Mouritzen, 2018, Optimized scratch assay for in vitro testing of cell migration with an automated optical camera, JoVE. source
- cites [11] W Jin, 2016, Reproducibility of scratch assays is affected by the initial degree of confluence: Experiments, modelling and model selection, J Theor Biol. source
May combine two claims.
Premisecitation
Lagergren et al. inferred density-dependent reaction-diffusion equations for cell migration and identified non-linear functions of concentration for diffusivity and reaction terms that vary with initial seeding concentration, consistent with our findings.
“Lagergren et al. demonstrated the power of this approach on the Jin et al. dataset8 (also analyzed by Jin et al.), when they inferred density-dependent reaction-diffusion equations for cell migration10. They identified non-linear functions of concentration for diffusivity and reaction terms, and found that these functions vary with initial seeding concentration, consistent with our findings.”
Quote in the paperNot a substantial claim1 of 2 sources checked
- cites [8] NH Gopee, 2024, A prenatal skin atlas reveals immune regulation of human skin morphogenesis, Nature. source
  Split into 2 atoms — 0 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 2 unverifiable. Expand the claim to see per-atom verdicts.
  Experimental — accuracy not yet measured
- cites [10] M Vang Mouritzen, 2018, Optimized scratch assay for in vitro testing of cell migration with an automated optical camera, JoVE. source
May combine two claims.
Findingown data
The cell densities achieved in our MDA-MB-231 scratch assays are comparable to those observed by Jin et al. and admit treatment using continuum advection-diffusion-reaction PDEs.
“As seen in Fig. 6A, the cell densities achieved are comparable to those observed by Jin et al.,8 and admit treatment of the problem using continuum advection-diffusion-reaction PDEs.”
Quote in the paperSubstantial claim1 of 1 sources checked
- cites [8] NH Gopee, 2024, A prenatal skin atlas reveals immune regulation of human skin morphogenesis, Nature. source
  The Methods section describes scratch assays using skin-organoid-derived fibroblasts and macrophages, but gives no cell densities comparable to the citing paper's MDA-MB-231 assays. Searches for MDA-MB-231, Jin, and advection-diffusion-reaction PDEs found no passage addressing the attributed comparison or modeling rationale; no page number is available in our copy. (LLM, conf=0.77)
  Experimental — accuracy not yet measured
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation
Our past study showed that trametinib reduces both random and directed motion in chemotaxis.
“Our past study showed that trametinib reduces both random and directed motion in chemotaxis35.”
Quote in the paperNot a substantial claim1 of 1 sources checked
- cites [35] PC Kinnunen, 2022, Computational modeling implicates protein scaffolding in p38 regulation of Akt, J Theor Biol. source
  The Results section describes kinase activity and responses to CXCL12 or EGF, while Methods > Cell imaging lists p38 inhibitors and anisomycin as treatments. Searches for trametinib and for random and directed motion in chemotaxis found no passage addressing the claimed effect. (LLM, conf=0.75)
  Experimental — accuracy not yet measured
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation
Previous computational analyses of cell migration have identified diffusivity values from approximately 50 to 3000 μm2/hr.
“Previous computational analyses of cell migration have identified a wide range of diffusivity values, from approximately 50 to 3000μm2/hr9,40,41.”
Quote in the paperNot a substantial claim3 of 3 sources checked
- cites [9] PF Renz, 2024, In vivo single-cell CRISPR uncovers distinct TNF programmes in tumour evolution, Nature. source
  The Methods describe a scratch assay that images cells over 22 hours, but give no diffusivity estimates. Searches for the 50–3000 μm²/hr range and computational analyses of cell migration found no passage in our copy that addresses the claimed values. (LLM, conf=0.76)
  Experimental — accuracy not yet measured
- cites [40] S Mitusch, 2019, dolfin-adjoint 2018 .1: Automated adjoints for FEniCS and Firedrake, JOSS. source
  The Abstract describes software for deriving adjoint models for FEniCS and Firedrake; it does not discuss cell migration or report diffusivity values. Searches for cell-migration diffusivity and the 50–3000 μm²/hr range found no passage addressing the claim. No page number is provided in our copy. (LLM, conf=0.78)
  Experimental — accuracy not yet measured
- cites [41] Funke SW, Farrell PE. A framework for automated pde-constrained optimisation. ArXiv; 2013. source
  The Introduction (p. 2) discusses PDE-constrained optimisation, and the wetting-and-drying example (pp. 21–23) concerns tsunami modelling. Searches for cell migration, diffusivity, and the stated range found no passage reporting cell-migration diffusivity values. (LLM, conf=0.76)
  Experimental — accuracy not yet measured
Classified as probably not a substantial claim.
Premisecitation
Scratch assays have been miniaturized and mechanized, making them compatible with high-throughput screening.
“Scratch assays have been miniaturized and mechanized, making them compatible with high-throughput screening45,46.”
Quote in the paperSubstantial claim2 of 2 sources checked
- cites [45] H Narayanan, 2010, In silico estimates of the free energy rates in growing tumor spheroids, J Phys Condens Matter. source
  The Experimental methods section (pp. 8–9) describes growing and imaging tumor spheroids in hydrogels, not scratch assays. Searches for miniaturized or mechanized scratch assays and high-throughput wound-healing screening found no passage in our copy addressing the claim. (LLM, conf=0.76)
  Experimental — accuracy not yet measured
- cites [46] K Garikipati, 2017, Perspectives on the mathematics of biological patterning and morphogenesis, J Mech Phys Solids. source
  The passages from section 1.1 (pp. 3–6) discuss biological patterning and models of cell migration, not scratch assays. Searches for scratch assays, miniaturization, mechanization, and high-throughput screening found no passage in our copy addressing the claim. (LLM, conf=0.76)
  Experimental — accuracy not yet measured
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation
The treatment of cell migration via continuum PDEs is well established in biophysics, as is the role of cell arrangement in determining aspects of the progression of cancers.
“The treatment of cell migration via continuum PDEs is well established in biophysics36,37,38, as is the role of cell arrangement in determining aspects of the progression of cancers39.”
Quote in the paperNot a substantial claim2 of 4 sources checked
- cites [36] ST Vittadello, 2019, Mathematical models incorporating a multi-stage cell cycle replicate normally-hidden inherent synchronization in cell proliferation, J R Soc Interface. source
  The Introduction discusses mathematical models of cell proliferation, and §2.2 uses differential equations for cell-cycle-stage populations. Our copy does not address continuum PDE models of cell migration or the effect of cell arrangement on cancer progression; searches for those terms found no supporting passage. (LLM, conf=0.75)
  Experimental — accuracy not yet measured
- cites [37] TJ Hughes, 2003
  Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [38] A Logg, 2010, DOLFIN, ACM Trans Math Softw. source
  The citation is attached to the first clause, about continuum-PDE treatment of cell migration; citation 39 is attached separately to the cancer claim. The Introduction describes PDEs generally, and §5.3 gives a convection–diffusion example, but neither addresses cell migration or its established use in biophysics. Searches for cell migration and biophysics found no relevant passage in our copy. (LLM, conf=0.76)
  Experimental — accuracy not yet measured
- cites [39] A Logg, 2012 source
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation
Cell morphology and fluorescent reporters have been used to measure or infer cell states in migrating cells.
“Cell morphology42 and fluorescent reporters44 have been used to measure or infer cell states in migrating cells.”
Quote in the paperNot a substantial claim2 of 2 sources checked
- cites [42] S Srivastava, 2024, Pattern formation in dense populations studied by inference of nonlinear diffusion-reaction mechanisms, Numer Meth Eng. source
  The Introduction discusses cellular migration, and the Abstract describes inferring diffusion–reaction models from spatiotemporal data. Neither passage addresses using cell morphology or fluorescent reporters to measure or infer states in migrating cells. Searches for those measurement methods found no relevant passage in our copy. (LLM, conf=0.72)
  Experimental — accuracy not yet measured
- cites [44] A Akinleye, 2013, MEK and the inhibitors: From bench to bedside, J Hematol Oncol. source
  Citation 44 is attached to the fluorescent-reporters part of the sentence, not to cell morphology. The Introduction and MEK proteins sections discuss MAPK signaling and cell motility, but no passage found describes fluorescent reporters being used to measure or infer states in migrating cells (searched: fluorescent reporters, fluorescence, reporter assay, migration). (LLM, conf=0.76)
  Experimental — accuracy not yet measured
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation
MEK kinase is known to regulate cell migration and proliferation.
“MEK kinase is known to regulate cell migration and proliferation34.”
Quote in the paperNot a substantial claim1 of 1 sources checked
- cites [34] PC Spinosa, 2019, Short-term cellular memory tunes the signaling responses of the chemokine receptor CXCR4, Sci Signal. source
  The Introduction (page unavailable) says that ERK and Akt mediate cell proliferation and chemotaxis, and the Materials and Methods section places MEK upstream of ERK. This supports a pathway-level connection, but the paper does not specifically establish that MEK itself regulates cell migration and proliferation. (LLM, conf=0.78)
  Experimental — accuracy not yet measured
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation
At higher cell densities, more rigorous models such as Maxwell-Stefan diffusion would be more appropriate.
“In this work, we considered a polynomial cell concentration dependence in the diffusivity for modeling nonlinear diffusion, but at higher cell densities, more rigorous models, such as Maxwell-Stefan diffusion would be more appropriate33.”
Quote in the paperNot a substantial claim1 of 1 sources checked
- cites [33] PC Spinosa, 2020, Pre-existing cell states control heterogeneity of both EGFR and CXCR4 signaling, Cell Mol Bioeng. source
  The Materials and Methods describe an ordinary-differential-equation model of EGFR and CXCR4 signaling, not a cell-concentration-dependent diffusion model. Searches for Maxwell–Stefan diffusion and polynomial diffusivity found no passage addressing whether a different diffusion model is appropriate at higher cell densities. The passages provide section labels but no page numbers. (LLM, conf=0.78)
  Experimental — accuracy not yet measured
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation
There exist models that have considered density-dependent effects on migration and on cell proliferation.
“There exist models that have considered density-dependent effects on migration, for instance representing cells interacting with each other, and on cell proliferation, for instance, modeling its saturation at high cell density8,27.”
Quote in the paperNot a substantial claim2 of 2 sources checked
- cites [8] NH Gopee, 2024, A prenatal skin atlas reveals immune regulation of human skin morphogenesis, Nature. source
  The Methods sections on spatial data analysis and trajectory analysis describe models for estimating cell abundance and cell-state transitions, not models of density-dependent migration or proliferation. Searches for high-density proliferation saturation and density-dependent migration found no passage addressing either point in our copy. (LLM, conf=0.75)
  Experimental — accuracy not yet measured
- cites [27] Z Wang, 2021, System inference via field inversion for the spatio-temporal progression of infectious diseases: Studies of COVID-19 in Michigan and Mexico, Arch Comput Methods Eng. source
  The Introduction and Results discuss population-density gradients and mobility in a COVID-19 model, not cell migration or proliferation. Searches for density-dependent migration, cell interactions, and proliferation saturation found no passage in our copy addressing the attributed claim. (LLM, conf=0.74)
  Experimental — accuracy not yet measured
May combine two claims.
Findingown data
The Variational System Identification loss function is essentially constant for two-term models including diffusion and reaction terms, and only starts to increase for models that lack reaction terms.
“We observed that the Variational System Identification loss function is essentially constant for two-term models including diffusion and reaction terms, and only starts to increase for models that lack reaction terms (Fig. 6B).”
Quote in the paperSubstantial claim
May combine two claims.
Findingreasoning
Reaction and diffusion terms were identified underlying the observed behavior and consistent with previous results.
“Reaction and diffusion terms were identified underlying the observed behavior and consistent with previous results.”
Quote in the paperSubstantial claim
May combine two claims.
Findingown data
All resulting optimized models have a constant diffusivity term, and either a linear or quadratic dependence on concentration.
“All the resulting optimized models have a constant diffusivity term, and either a linear or quadratic dependence on concentration.”
Quote in the paperSubstantial claim
May combine two claims.
Findingown data
In the 2D wound healing experiments, we estimate diffusivity values for breast cancer cells around 50 μm2/hr in untreated conditions, with decreasing diffusivity in response to trametinib.
“In this setting, we estimate diffusivity values for breast cancer cells around 50μm2/hr in untreated conditions, with decreasing diffusivity in response to trametinib.”
Quote in the paperSubstantial claim
May combine two claims.
Findingown data
For four of the six initial seeding densities, our models have root mean squared errors significantly lower than the RMSE reported by Jin et al., and for two of the six cases they are similar.
“We found that for four of the six initial seeding densities, our models have root mean squared errors (RMSEs) that are significantly lower than the RMSE reported by Jin et al. and for two of the six cases they are similar (Fig. 4).”
Quote in the paperSubstantial claim
Classified as probably not a substantial claim.
Premisereasoning
Physics-informed neural networks can learn relationships to make predictions without any guarantee that the growth or diffusivities learned are physically realizable.
“However, in such approaches, neural networks can learn relationships to make predictions without any guarantee that the growth or diffusivities learned are physically realizable.”
Quote in the paperSubstantial claim
May combine two claims.
Premisereasoning
Variational System Identification and SINDy use weak and strong forms of differential equations in regression-based approaches, and therefore do not require repeated forward evaluations of the model, unlike traditional PDE-constrained inverse modeling, substantially reducing computational cost.
“Variational System Identification and SINDy respectively use the weak and strong forms of differential equations in regression-based approaches, and therefore do not require repeated forward evaluations of the model, unlike traditional PDE-constrained inverse modeling. This substantially reduces the computational cost.”
Quote in the paperSubstantial claim
Classified as probably not a substantial claim.
Premisenone given
Smoothing the data alleviates numerical noise in the calculation of derivatives.
“Thus smoothing the data alleviates numerical noise in the calculation of derivatives.”
Quote in the paperSubstantial claim
Classified as probably not a substantial claim. May combine two claims.
Interpretationreasoning
Neural networks can learn arbitrary relationships between cell density and diffusivity or reaction, but there is no guarantee that physically realizable diffusivity or reaction relationships are learnt unless physics-based constraints are built in.
“In the case of cell migration, while the neural networks can learn arbitrary relationships between cell density and diffusivity or reaction, there is no guarantee that physically realizable diffusivity or reaction relationships are learnt, unless a number of physics-based constraints are built in.”
Quote in the paperSubstantial claim
May combine two claims.
Premisenone given
In Variational System Identification, the equations learned are confined to the candidate library terms used, while neural networks can learn arbitrary relationships for equation parameters to best fit the data.
“In Variational System Identification, the equations learned are confined to the candidate library terms used, while neural networks can learn arbitrary relationships for equation parameters to best fit the data.”
Quote in the paperSubstantial claim
May combine two claims.
Premisenone given
PDE-constrained optimization with adjoint-based gradient computation does not change the selected operators but generates a new set of parameter values that fit the data better than those obtained by Variational System Identification with stepwise regression.
“PDE-constrained optimization with adjoint-based gradient computation does not change the selected operators but generates a new set of parameter values, which fit the data better than the those obtained by Variational System Identification with stepwise regression.”
Quote in the paperSubstantial claim
Probably not a substantial claim: may not pass S4 (based). May combine two claims.
Interpretationnone given
PINNs can discover relationships that maximize the fit to data, while Variational System Identification sacrifices fitting to data because it is constrained to selected terms which the modeler can ensure are physically meaningful.
“PINNs can discover relationships that maximize the fit to data, while Variational System Identification sacrifices fitting to data because it is constrained to selected terms which the modeler can ensure are physically meaningful.”
Quote in the paperNot a substantial claim
May combine two claims. The quote only partly states this claim.
Premisereasoning
The weak form of the PDE limits the maximum order of derivative evaluated on the data field to first order, which enhances the robustness of the inference against noise in the data field.
“This choice enhances the robustness of the inference against noise in the data field.”
Quote in the paperSubstantial claim
Classified as probably not a substantial claim.
Premisereasoning
Variational System Identification enables the modeler to restrict the candidate mechanisms and their mathematical forms to those that rigorously encode physical mechanisms.
“Variational System Identification, on the other hand, enables the modeler to restrict the candidate mechanisms and their mathematical forms to those that rigorously encode physical mechanisms.”
Quote in the paperSubstantial claim
May combine two claims.
Interpretationown data
The models inferred with and without trametinib suggest that even 100 nM trametinib causes a decrease in random cell migration and cell proliferation.
“The models inferred with and without trametinib suggest that even 100 nM trametinib causes a decrease in random cell migration and cell proliferation.”
Quote in the paperSubstantial claim
- Rests on an assumption: The candidate library of constant, linear, and quadratic terms for diffusion, advection, and reaction contains the mechanisms needed to describe the observed cell migration dynamics.

29 of 29

The paper relies on this without support.
Scratch assays do not measure dynamic effects.
“These assays do not measure dynamic effects.”
The paper relies on this without support.
Typically, the complexity of cell migration in scratch assays is reduced to a single number, which represents the amount of space filled in a given time or the distance traveled by the leading edge of cells.
“However, typically the complexity of cell migration is reduced to a single number, which represents the amount of space filled in a given time or the distance traveled by the leading edge of cells.”
The paper relies on this without support.
The candidate library of constant, linear, and quadratic terms for diffusion, advection, and reaction contains the mechanisms needed to describe the observed cell migration dynamics.
“We used the same library of candidate operators, including constant, linear, and quadratic terms for diffusion and advection, and linear and quadratic terms for reaction.”
The paper relies on this without support.
The reaction term in the PDE corresponds to cell proliferation and death rather than other density-dependent processes.
“This combination of techniques reveals a non-linear functional dependence of cell proliferation (reaction in the PDE) rate on cell density: positive at low cell densities and becoming negative above a critical density; i.e., cell death sets in(Fig. 6D).”

4 of 4

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

Matrix residual reverses the signs of the time-derivative and diffusion terms
Major
Equation (9) gives the residual positive time-derivative and diffusion terms. Equations (11), (12), and (14) instead put a negative time-derivative term in y and subtract a positive diffusion column from it, reversing both signs.
Advection library columns retain the velocity they are meant to parameterize
Minor
Equation (10b) defines vf through coefficients θ3-θ5, but equation (15) includes vf inside the columns corresponding to those coefficients. That makes those columns depend on the unknown coefficients, contrary to the stated matrix-vector regression R=y−Ξ⋅θ.

2 of 2

Discovering governing equations from data by sparse identification of nonlinear dynamical systems
Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1517384113 · Cited by 4,796 · see citing works
In vitro scratch assay: a convenient and inexpensive method for analysis of cell migration in vitro
Liang, C.-C., Park, A. Y., & Guan, J.-L. (2007) · Nature Protocols · doi:10.1038/nprot.2007.30 · Cited by 4,696 · see citing works
Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book
(2012) · Lecture Notes in Computational Science and Engineering · doi:10.1007/978-3-642-23099-8 · Cited by 2,204 · see citing works
A user’s guide to PDE models for chemotaxis
Hillen, T., & Painter, K. J. (2008) · Journal of Mathematical Biology · doi:10.1007/s00285-008-0201-3 · Cited by 1,618 · cited 1× · see citing works
“In this work, we have chosen a simple polynomial basis, however, more sophisticated functional forms [57] could lead to more biologically robust models.”
Data-driven discovery of partial differential equations
Rudy, S. H., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2017) · Science Advances · doi:10.1126/sciadv.1602614 · Cited by 1,557 · see citing works
The front and rear of collective cell migration
Mayor, R., & Etienne‐Manneville, S. (2016) · Nature Reviews Molecular Cell Biology · doi:10.1038/nrm.2015.14 · Cited by 879 · cited 1× · see citing works
“Cell migration is a complex, multiscale phenomenon that integrates many different inputs and cell behaviors, including directed and random motion, that are differentially regulated by signaling kinases, cell density, and other factors [1,2].”
DOLFIN: Automated finite element computing
Logg, A., & Wells, G. N. (2010) · ACM Transactions on Mathematical Software · doi:10.1145/1731022.1731030 · Cited by 750 · cited 1× · see citing works
“The numerical examples presented in this work have been posed and solved in two dimensions by the finite element method implemented on the FEniCS platform [38,39].”
The principles of directed cell migration
SenGupta, S., Parent, C. A., & Bear, J. E. (2021) · Nature Reviews Molecular Cell Biology · doi:10.1038/s41580-021-00366-6 · Cited by 654 · cited 1× · see citing works
“Cell migration is a complex, multiscale phenomenon that integrates many different inputs and cell behaviors, including directed and random motion, that are differentially regulated by signaling kinases, cell density, and other factors [1,2].”
Physics-informed learning of governing equations from scarce data
Chen, Z., Liu, Y., & Sun, H. (2021) · Nature Communications · doi:10.1038/s41467-021-26434-1 · Cited by 616 · cited 1× · see citing works
“[13,15,58,59] Delay terms in PDEs generally arise as phenomenological approximations, intended to capture latent biological processes that are difficult to model explicitly at the continuum scale.”
High-Sensitivity Measurements of Multiple Kinase Activities in Live Single Cells
Regot, S., Hughey, J., Bajar, B. T., Carrasco, S., & Covert, M. W. (2014) · Cell · doi:10.1016/j.cell.2014.04.039 · Cited by 542 · cited 1× · see citing works
“We engineered the MDA-MB-231 breast cancer cells used in this work to express a stable histone-2B (H2B) nuclear marker (mCherry), along with kinase translocation reporters [32] for the Akt and ERK kinases, as described previously [33,34].”
Collective cell migration in development
Scarpa, E., & Mayor, R. (2016) · The Journal of Cell Biology · doi:10.1083/jcb.201508047 · Cited by 459 · see citing works
Identification of genes that regulate epithelial cell migration using an siRNA screening approach
Simpson, K., Selfors, L., Bui, J., Reynolds, A., Leake, D., Khvorova, A., & Brugge, J. (2008) · Nature Cell Biology · doi:10.1038/ncb1762 · Cited by 448 · cited 1× · see citing works
“These measurements are commonplace in biological research, and have, for instance, been used to identify novel inhibitors of cell migration or genes responsible for regulating migration [6,7].”
Inferring biological networks by sparse identification of nonlinear dynamics
Mangan, N. M., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016) · IEEE Transactions on Molecular, Biological and Multi-Scale Communications · doi:10.1109/tmbmc.2016.2633265 · Cited by 436 · see citing works
Receptors
Lauffenburger, D. A., & Linderman, J. J. (1993) · doi:10.1093/oso/9780195064667.001.0001 · Cited by 429 · see citing works
Self-citation
Shifting the optimal stiffness for cell migration
Bangasser, B. L., Shamsan, G. A., Chan, C. E., Opoku, K. N., Tüzel, E., Schlichtmann, B. W., Kasim, J., Fuller, B., McCullough, B. R., Rosenfeld, S. S., et al. (2017) · Nature Communications · doi:10.1038/ncomms15313 · Cited by 330 · cited 1× · see citing works
“Past work focused on U251 glioma cancer cells measured a random motion coefficient (akin to diffusivity measured for a single cell) of between 60 and 300 𝜇 m 2 / hr across a range of substrate stiffnesses [51].”
Propagating Wave of ERK Activation Orients Collective Cell Migration
Aoki, K., Kondo, Y., Naoki, H., Hiratsuka, T., Itoh, R. E., & Matsuda, M. (2017) · Developmental Cell · doi:10.1016/j.devcel.2017.10.016 · Cited by 292 · cited 2× · see citing works
“For example, data about cell state (e.g. signaling activity, cell-cycle status, metabolic activity) gathered from fluorescent reporters or cell morphology could require updating a model to consider diffusivity as a function of both cell state and local cell density [3,52–54].”
MEK and the inhibitors: from bench to bedside
Akinleye, A., Furqan, M., Mukhi, N., Ravella, P., & Liu, D. (2013) · Journal of Hematology & Oncology · doi:10.1186/1756-8722-6-27 · Cited by 291 · cited 1× · see citing works
“MEK kinase is known to regulate cell migration and proliferation [44].”
Weak SINDy for partial differential equations
Messenger, D. A., & Bortz, D. M. (2021) · Journal of Computational Physics · doi:10.1016/j.jcp.2021.110525 · Cited by 245 · cited 1× · see citing works
“The weak form-based inference of PDEs presented as Variational System Identification and employed in this study [24,25] was also independently developed under a different name, Weak-SINDy, introduced by Messenger and Bortz [26].”
Traveling Wave Model to Interpret a Wound-Healing Cell Migration Assay for Human Peritoneal Mesothelial Cells
Maini, P. K., McElwain, D. L. S., & Leavesley, D. I. (2004) · Tissue Engineering · doi:10.1089/107632704323061834 · Cited by 240 · cited 2× · see citing works
“[ 11,12] These models commonly include reaction terms, which represent cell proliferation or death, and diffusion terms, which represent random cell motion, with various functional forms.”
Intercellular propagation of extracellular signal-regulated kinase activation revealed by in vivo imaging of mouse skin
Hiratsuka, T., Fujita, Y., Naoki, H., Aoki, K., Kamioka, Y., & Matsuda, M. (2015) · eLife · doi:10.7554/elife.05178 · Cited by 230 · cited 2× · see citing works
“For example, data about cell state (e.g. signaling activity, cell-cycle status, metabolic activity) gathered from fluorescent reporters or cell morphology could require updating a model to consider diffusivity as a function of both cell state and local cell density [3,52–54].”
dolfin-adjoint 2018.1: automated adjoints for FEniCS and Firedrake
Mitusch, S., Funke, S., & Dokken, J. (2019) · Journal of Open Source Software · doi:10.21105/joss.01292 · Cited by 208 · cited 1× · see citing works
“The PDE constrained optimization (16) was implemented in the dolfin-adjoint package [40,41].”
A Simplified Method for Quantifying Cell Migration/Wound Healing in 96-Well Plates
Yue, P. Y., Leung, E. P. Y., Mak, N. K., & Wong, R. N. (2010) · SLAS DISCOVERY · doi:10.1177/1087057110361772 · Cited by 196 · see citing works
Multi-scale modeling of a wound-healing cell migration assay
Cai, A. Q., Landman, K. A., & Hughes, B. D. (2006) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2006.10.024 · Cited by 182 · cited 1× · see citing works
“Previous computational analyses of cell migration, conducted on data gathered from a variety of experimental cell lines, have identified a wide range of diffusivity values, from approximately 50 to 3000 𝜇 m 2 / hr [12,49,50].”
Discovery of nonlinear multiscale systems: Sampling strategies and embeddings
Champion, K. P., Brunton, S. L., & Kutz, J. N. (2019) · SIAM Journal on Applied Dynamical Systems · doi:10.1137/18m1188227 · Cited by 171 · cited 1× · see citing works
“It extends the popular SINDy approach [19–23] to inferring PDEs in weak form.”
Biologically-informed neural networks guide mechanistic modeling from sparse experimental data
Lagergren, J. H., Nardini, J. T., Baker, R. E., Simpson, M. J., & Flores, K. (2020) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1008462 · Cited by 167 · cited 7× · see citing works
“Recently, delayed diffusion-reaction PDE models have been introduced to study scratch assay dynamics.”
Influential
Sparse identification of nonlinear dynamics for rapid model recovery
Quade, M., Abel, M., Nathan Kutz, J., & Brunton, S. L. (2018) · Chaos: An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.5027470 · Cited by 163 · see citing works
Screening for Cell Migration Inhibitors via Automated Microscopy Reveals a Rho-Kinase Inhibitor
Yarrow, J. C., Totsukawa, G., Charras, G., & Mitchison, T. J. (2005) · Chemistry & Biology · doi:10.1016/j.chembiol.2005.01.015 · Cited by 135 · cited 1× · see citing works
“These measurements are commonplace in biological research, and have, for instance, been used to identify novel inhibitors of cell migration or genes responsible for regulating migration [6,7].”
Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration
Sengers, B. G., Please, C. P., & Oreffo, R. O. C. (2007) · Journal of The Royal Society Interface · doi:10.1098/rsif.2007.0233 · Cited by 123 · cited 1× · see citing works
“Previous computational analyses of cell migration, conducted on data gathered from a variety of experimental cell lines, have identified a wide range of diffusivity values, from approximately 50 to 3000 𝜇 m 2 / hr [12,49,50].”
A prenatal skin atlas reveals immune regulation of human skin morphogenesis
Gopee, N. H., Winheim, E., Olabi, B., Admane, C., Foster, A. R., Huang, N., Botting, R. A., Torabi, F., Sumanaweera, D., Le, A. P., et al. (2024) · Nature · doi:10.1038/s41586-024-08002-x · Cited by 108 · see citing works
Reproducibility of scratch assays is affected by the initial degree of confluence: Experiments, modelling and model selection
Jin, W., Shah, E. T., Penington, C. J., McCue, S. W., Chopin, L. K., & Simpson, M. J. (2016) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2015.10.040 · Cited by 107 · cited 9× · see citing works
“There exist models that have considered density-dependent effects on migration, for instance representing cells interacting with each other, and on cell proliferation, for instance, modeling its saturation at high cell density [11,36].”
Influential
Time series modeling of live-cell shape dynamics for image-based phenotypic profiling
Gordonov, S., Hwang, M. K., Wells, A. H., Gertler, F. B., Lauffenburger, D. A., & Bathe, M. (2015) · Integrative Biology · doi:10.1039/c5ib00283d · Cited by 78 · cited 2× · see citing works
“For example, data about cell state (e.g. signaling activity, cell-cycle status, metabolic activity) gathered from fluorescent reporters or cell morphology could require updating a model to consider diffusivity as a function of both cell state and local cell density [3,52–54].”
Variational system identification of the partial differential equations governing the physics of pattern-formation: Inference under varying fidelity and noise
Wang, Z., Huan, X., & Garikipati, K. (2019) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2019.07.007 · Cited by 77 · cited 6× · see citing works
“Variational System Identification has been applied to identify governing equations for the evolution of materials [24,25,28], the spatiotemporal spread of COVID-19 [27,29], as well as constitutive models of soft materials [30,31].”
InfluentialSelf-citation
A framework for automated PDE-constrained optimisation
Funke, S. W., & Farrell, P. E. (2013) · arXiv · doi:10.48550/arxiv.1302.3894 · Cited by 58 · see citing works
Inference of deformation mechanisms and constitutive response of soft material surrogates of biological tissue by full-field characterization and data-driven variational system identification
Wang, Z., Estrada, J. B., Arruda, E. M., et al. (2020) · Journal of the Mechanics and Physics of Solids · doi:10.1016/j.jmps.2021.104474 · Cited by 56 · cited 1× · see citing works
“Variational System Identification has been applied to identify governing equations for the evolution of materials [24,25,28], the spatiotemporal spread of COVID-19 [27,29], as well as constitutive models of soft materials [30,31].”
Self-citation
In vivo single-cell CRISPR uncovers distinct TNF programmes in tumour evolution
Renz, P. F., Ghoshdastider, U., Baghai Sain, S., Valdivia-Francia, F., Khandekar, A., Ormiston, M., Bernasconi, M., Duré, C., Kretz, J. A., Lee, M., et al. (2024) · Nature · doi:10.1038/s41586-024-07663-y · Cited by 40 · see citing works
Examining Go-or-Grow Using Fluorescent Cell-Cycle Indicators and Cell-Cycle-Inhibiting Drugs
Vittadello, S. T., McCue, S. W., Gunasingh, G. P., Haass, N. K., & Simpson, M. J. (2020) · Biophysical Journal · doi:10.1016/j.bpj.2020.01.036 · Cited by 39 · see citing works
Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data
Wang, Z., Huan, X., & Garikipati, K. C. (2021) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2021.113706 · Cited by 39 · cited 5× · see citing works
“Variational System Identification has been applied to identify governing equations for the evolution of materials [24,25,28], the spatiotemporal spread of COVID-19 [27,29], as well as constitutive models of soft materials [30,31].”
InfluentialSelf-citation
Elastic Free Energy Drives the Shape of Prevascular Solid Tumors
Mills, K. L., Kemkemer, R., Rudraraju, S., & Garikipati, K. (2014) · PLoS ONE · doi:10.1371/journal.pone.0103245 · Cited by 38 · cited 1× · see citing works
“The treatment of cell migration via continuum PDEs is well established in biophysics [45– 47], as is the role of cell arrangement in determining aspects of the progression of cancers [48].”
Self-citation
System inference for the spatio-temporal evolution of infectious diseases: Michigan in the time of COVID-19
Wang, Z., Zhang, X., Teichert, G. H., Carrasco-Teja, M., & Garikipati, K. (2020) · Computational Mechanics · doi:10.1007/s00466-020-01894-2 · Cited by 33 · cited 1× · see citing works
“Variational System Identification has been applied to identify governing equations for the evolution of materials [24,25,28], the spatiotemporal spread of COVID-19 [27,29], as well as constitutive models of soft materials [30,31].”
Self-citation
Short-term cellular memory tunes the signaling responses of the chemokine receptor CXCR4
Spinosa, P. C., Humphries, B. A., Lewin Mejia, D., Buschhaus, J. M., Linderman, J. J., Luker, G. D., & Luker, K. E. (2019) · Science Signaling · doi:10.1126/scisignal.aaw4204 · Cited by 31 · cited 1× · see citing works
“We engineered the MDA-MB-231 breast cancer cells used in this work to express a stable histone-2B (H2B) nuclear marker (mCherry), along with kinase translocation reporters [32] for the Akt and ERK kinases, as described previously [33,34].”
Self-citation
<i>In silico</i>estimates of the free energy rates in growing tumor spheroids
Narayanan, H., Verner, S. N., Mills, K. L., Kemkemer, R., & Garikipati, K. (2010) · Journal of Physics: Condensed Matter · doi:10.1088/0953-8984/22/19/194122 · Cited by 31 · cited 1× · see citing works
“The treatment of cell migration via continuum PDEs is well established in biophysics [45– 47], as is the role of cell arrangement in determining aspects of the progression of cancers [48].”
Self-citation
SINDy for delay-differential equations: application to model bacterial zinc response
Sandoz, A., Ducret, V., Gottwald, G. A., Vilmart, G., & Perron, K. (2023) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2022.0556 · Cited by 31 · see citing works
Mathematical models incorporating a multi-stage cell cycle replicate normally-hidden inherent synchronization in cell proliferation
Vittadello, S. T., McCue, S. W., Gunasingh, G., Haass, N. K., & Simpson, M. J. (2019) · Journal of The Royal Society Interface · doi:10.1098/rsif.2019.0382 · Cited by 29 · see citing works
Perspectives on the mathematics of biological patterning and morphogenesis
Garikipati, K. (2017) · Journal of the Mechanics and Physics of Solids · doi:10.1016/j.jmps.2016.11.013 · Cited by 28 · see citing works
Self-citation
A perspective on regression and Bayesian approaches for system identification of pattern formation dynamics
Wang, Z., Wu, B., Garikipati, K. C., et al. (2020) · Theoretical and Applied Mechanics Letters · doi:10.1016/j.taml.2020.01.028 · Cited by 24 · cited 1× · see citing works
“Variational System Identification has been applied to identify governing equations for the evolution of materials [24,25,28], the spatiotemporal spread of COVID-19 [27,29], as well as constitutive models of soft materials [30,31].”
Self-citation
Quantifying the roles of random motility and directed motility using advection-diffusion theory for a 3T3 fibroblast cell migration assay stimulated with an electric field
Simpson, M. J., Lo, K.-Y., & Sun, Y.-S. (2017) · BMC Systems Biology · doi:10.1186/s12918-017-0413-5 · Cited by 23 · cited 2× · see citing works
“We included advection terms, even though they have not previously been required to model wound closure, to demonstrate the utility of our method as a model discovery tool, and because advection has been used in other cases to model cell migration tracked with live-cell microscopy [16,17].”
Modelling count data with partial differential equation models in biology
Simpson, M. J., Murphy, R. J., & Maclaren, O. J. (2024) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2024.111732 · Cited by 22 · see citing works
Scratch assay microscopy: A reaction–diffusion equation approach for common instruments and data
Gnerucci, A., Faraoni, P., Sereni, E., et al. (2020) · Mathematical Biosciences · doi:10.1016/j.mbs.2020.108482 · Cited by 18 · see citing works
Ogden material calibration via magnetic resonance cartography, parameter sensitivity and variational system identification
Nikolov, D. P., Srivastava, S., Abeid, B. A., et al. (2022) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2021.0324 · Cited by 12 · see citing works
Self-citation
Computationally efficient mechanism discovery for cell invasion with uncertainty quantification
VandenHeuvel, D. J., Drovandi, C., & Simpson, M. J. (2022) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1010599 · Cited by 11 · cited 2× · see citing works
“[ 11,13] By estimating parameters and quantifying parameter uncertainty for these models, past work has helped identify quantitative effects of biological perturbations in scratch assays [11,13–15].”
Pre-existing Cell States Control Heterogeneity of Both EGFR and CXCR4 Signaling
Spinosa, P. C., Kinnunen, P. C., Humphries, B. A., Luker, G. D., Luker, K. E., & Linderman, J. J. (2021) · Cellular and Molecular Bioengineering · doi:10.1007/s12195-020-00640-1 · Cited by 10 · cited 1× · see citing works
“We engineered the MDA-MB-231 breast cancer cells used in this work to express a stable histone-2B (H2B) nuclear marker (mCherry), along with kinase translocation reporters [32] for the Akt and ERK kinases, as described previously [33,34].”
Self-citation
System Inference Via Field Inversion for the Spatio-Temporal Progression of Infectious Diseases: Studies of COVID-19 in Michigan and Mexico
Wang, Z., Carrasco-Teja, M., Zhang, X., Teichert, G. H., & Garikipati, K. (2021) · Archives of Computational Methods in Engineering · doi:10.1007/s11831-021-09643-1 · Cited by 9 · see citing works
Self-citation
Oscillatory ERK Signaling and Morphology Determine Heterogeneity of Breast Cancer Cell Chemotaxis via MEK-ERK and p38-MAPK Signaling Pathways
Ho, K. K. Y., Srivastava, S., Kinnunen, P. C., Garikipati, K., Luker, G. D., & Luker, K. E. (2023) · Bioengineering · doi:10.3390/bioengineering10020269 · Cited by 8 · see citing works
Self-citation
A Device for Performing Cell Migration/Wound Healing in a 96-Well Plate
Poon, P. Y., Yue, P. Y. K., & Wong, R. N. (2017) · Journal of Visualized Experiments · doi:10.3791/55411 · Cited by 7 · see citing works
Structural identifiability of linear-in-parameter parabolic PDEs through auxiliary elliptic operators
Salmaniw, Y., & Browning, A. P. (2025) · Journal of Mathematical Biology · doi:10.1007/s00285-025-02225-w · Cited by 4 · cited 1× · see citing works
“From a more abstract perspective, however, it is also possible for non-uniqueness to be intrinsic to the model itself, regardless of data quality or resolution, as highlighted in recent works on structural identifiability of PDE models [43].”
Computational modeling implicates protein scaffolding in p38 regulation of Akt
Kinnunen, P. C., Luker, G. D., Luker, K. E., & Linderman, J. J. (2022) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2022.111294 · Cited by 4 · cited 1× · see citing works
“We processed fluorescent images of the MDA-MB-231 pHAEP cells as described previously [35].”
Self-citation
Pattern formation in dense populations studied by inference of nonlinear diffusion‐reaction mechanisms
Srivastava, S., & Garikipati, K. (2024) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.7475 · Cited by 2 · see citing works
Self-citation
Cellular Energy Cycle Mediates an Advection‐Like Forward Cell Flow to Support Collective Invasion
Zhang, J., Mosier, J. A., Wu, Y., Waddle, L., Taufalele, P., Wang, W., Sun, H., & Reinhart‐King, C. A. (2024) · Advanced Science · doi:10.1002/advs.202400719 · Cited by 2 · see citing works
Optimized Scratch Assay for In Vitro Testing of Cell Migration with an Automated Optical Camera
Mouritzen, M. V., & Jenssen, H. (2018) · Journal of Visualized Experiments · doi:10.3791/57691-v
A Device for Performing Cell Migration/Wound Healing in a 96-Well Plate
Poon, P. Y., Yue, P. Y. K., & Wong, R. N. S. (2017) · Journal of Visualized Experiments · doi:10.3791/55411-v
Inference of deformation mechanisms and constitutive response of soft material surrogates of biological tissue by full-field characterization and data-driven variational system identification
Wang, Z., Estrada, J. B., Arruda, E. M., et al. (2020) · doi:10.1101/2020.10.13.337964
Self-citation
The finite element method: Linear static and dynamic finite element analysis
Hughes, T. (2003)
Unresolved
Funke SW, Farrell PE. A framework for automated pde-constrained optimisation. ArXiv; 2013.
(n.d.)
Unresolved

Flagged first, then by citations

2 of the 60 references couldn’t be matched to a known paper.

Sorted by citations

Rheumatology in the digital health era: status quo and quo vadis?
Knitza, J., Gupta, L., & Hügle, T. (2024) · Nature Reviews Rheumatology · doi:10.1038/s41584-024-01177-7 · Cited by 66 · see citing works
A Novel Engineering-Based C¹ Piecewise Model for Wound-Healing Dynamics
Kontomaris, S. V., Torna, S., Petrou, M., et al. (2026) · Results in Engineering · doi:10.1016/j.rineng.2026.111770 · Cited by 1 · see citing works
Modeling Collective Cell Migration in a Data-Rich Age: Challenges and Opportunities for Data-Driven ModelingPreprint
Baker, R. E., Crossley, R. M., Falcó, C., & Martina-Perez, S. F. (2026) · Cold Spring Harbor Perspectives in Biology · doi:10.1101/cshperspect.a041757 · Cited by 1 · see citing works
The inference of Fokker-Planck equations via transport maps
Han, S., & Garikipati, K. C. (2026) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2026.118760
“Previous studies have demonstrated that cell migration can be modeled using variants of the Fokker-Planck equation [48].”
Self-citation
Nonlinear Modeling and Perturbation Analysis of Cell Population Dynamics Under External Factors—A Pilot Study of Low-Level Red Laser Irradiation on Lung Cancer Cells
Kontomaris, S. V., Pettermeridi, M., Malamou, A., & Stylianou, A. (2026) · Journal of Interdisciplinary Research Applied to Medicine · doi:10.3390/jdream6020010
“Integrated with live-cell imaging, these frameworks enable the quantitative distinction between migration and proliferation in wound closure [11].”

Authors: Srivastava, S., Kinnunen, P. C., Wang, Z., Ho, K. K. Y., Humphries, B. A., Chen, S., Linderman, J. J., Luker, G. D., Luker, K. E., & Garikipati, K.
Affiliations: University of Southern California (US)
University of Michigan (US)
Corresponding author: Garikipati, Krishna
ORCID: Kinnunen, Patrick C.
Ho, Kenneth K.Y.
Humphries, Brock A.
Chen, Siyi
Linderman, Jennifer J.
Luker, Gary D.
Luker, Kathryn E.
Garikipati, Krishna
Venue: PLOS Computational Biology
Publisher: Public Library of Science (PLoS)
Published: 2025-10-28
Volume: 21
Issue: 10
Pages: e1013607
ISSN: 1553-734X (ISSN-L)
1553-7358
DOI: 10.1371/journal.pcbi.1013607
PMC: PMC11537331
OpenAlex: W4404122244
Semantic Scholar: 29b8c159c9751da037d4585a7b8b035088398122
Cited by: 5
References: 60
Citations by year: 2026: 4 · 2024: 1
FWCI: 3.06
Type: journal article
Language: en
Update notices: new_version (2025-10-31)
Open access: gold
doi.org
Licence: cc-by
creativecommons.org/licenses/by/4.0/ (vor)
Locations: PLoS Computational Biology (publishedVersion, cc-by)
PubMed (publishedVersion)
PubMed Central (submittedVersion, cc-by-nc-sa)
Europe PMC (PubMed Central) (submittedVersion, cc-by)
DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
Funders: National Institutes of Health, United States (R01CA238042, R50CA221807, R01CA238023, R33CA225549, R37CA222563, R21AI173559)
W. M. Keck Foundation (N.A.)
Topics: Physical Sciences › Mathematics › Modeling and Simulation › Mathematical Biology Tumor Growth
Life Sciences › Biochemistry, Genetics and Molecular Biology › Cell Biology › Cellular Mechanics and Interactions
Health Sciences › Medicine › Oncology › Cancer Cells and Metastasis
Keywords: tumor cell migration, cell proliferation, wound healing, scratch assay, reaction-diffusion-advection model, trametinib, metastasis, model uncertainty, live-cell imaging

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-14.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-14.

Cited sources are checked by gpt-6-sol.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review

Inference of weak-form partial differential equations describing migration and proliferation mechanisms in wound healing experiments on cancer cells

Abstract

AI snapshot

TL;DR

Key findings

Methods

Limitations

Integrity signals

Claims the paper makes

Self-check

Cites

Cited by

Related

Record

About these checks