Age Structure Can Account for Delayed Logistic Proliferation of Scratch Assays

  • Ana Victoria Ponce BobadillaEmail author
  • Thomas Carraro
  • Helen M. Byrne
  • Philip K. Maini
  • Tomás Alarcón


Scratch assays are in vitro methods for studying cell migration. In these experiments, a scratch is made on a cell monolayer and recolonisation of the scratched region is imaged to quantify cell migration rates. Typically, scratch assays are modelled by reaction diffusion equations depicting cell migration by Fickian diffusion and proliferation by a logistic term. In a recent paper (Jin et al. in Bull Math Biol 79(5):1028–1050, 2017), the authors observed experimentally that during the early stage of the recolonisation process, there is a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. The authors did not identify the precise mechanism that causes the disturbance phase but showed that ignoring it can lead to incorrect parameter estimates. The aim of this work is to show that a nonlinear age-structured population model can account for the two phases of proliferation in scratch assays. The model consists of an age-structured cell cycle model of a cell population, coupled with an ordinary differential equation describing the resource concentration dynamics in the substrate. The model assumes a resource-dependent cell cycle threshold age, above which cells are able to proliferate. By studying the dynamics of the full system in terms of the subpopulations of cells that can proliferate and the ones that can not, we are able to find conditions under which the model captures the two-phase behaviour. Through numerical simulations, we are able to show that the interplay between the resource concentration in the substrate and the cell subpopulations dynamics can explain the biphasic dynamics.


Logistic growth model Cell proliferation Scratch assay Von Foerster–McKendrick type model Age-structured model 



AVPB would like to thank Gergely Rost, Maria Vittoria Barbarossa, Wang Jin and Matthew Simpson for helpful discussions. AVPB was supported by the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences, University of Heidelberg (DE) (DFG grant GSC 220 in the German Universities Excellence Initiative).

Supplementary material

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Supplementary material 1 (pdf 398 KB)


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Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Institute for Applied MathematicsHeidelberg UniversityHeidelbergGermany
  2. 2.Interdisciplinary Center for Scientific Computing (IWR)Heidelberg UniversityHeidelbergGermany
  3. 3.Wolfson Centre for Mathematical Biology, Mathematical InstituteUniversity of OxfordOxfordUK
  4. 4.ICREABarcelonaSpain
  5. 5.Centre de Recerca MatemàticaBellaterraSpain
  6. 6.Barcelona Graduate School of Mathematics (BGSMath)BarcelonaSpain
  7. 7.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain

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