SCIENCE CHINA Technological Sciences, Volume 62 , Issue 8 : 1341-1348(2019) https://doi.org/10.1007/s11431-018-9483-7

A cell-based model for analyzing growth and invasion of tumor spheroids

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  • ReceivedDec 27, 2018
  • AcceptedMar 4, 2019
  • PublishedJul 11, 2019


Both chemical and mechanical determinants adapt and react throughout the process of tumor invasion. In this study, a cell-based model is used to uncover the growth and invasion of a three-dimensional solid tumor confined within normal cells. Each cell is treated as a spheroid that can deform, migrate, and proliferate. Some fundamental aspects of tumor development are considered, including normal tissue constraints, active cellular motility, homotypic and heterotypic intercellular interactions, and pressure-regulated cell division as well. It is found that differential motility between cancerous and normal cells tends to break the spheroidal symmetry, leading to a finger instability at the tumor rim, while stiff normal cells inhibit tumor branching and favor uniform tumor expansion. The heterotypic cell-cell adhesion is revealed to affect the branching geometry. Our results explain many experimental observations, such as fingering invasion during tumor growth, stiffness inhibition of tumor invasion, and facilitation of tumor invasion through cancerous-normal cell adhesion. This study helps understand how cellular events are coordinated in tumor morphogenesis at the tissue level.

Funded by

the National Natural Science Foundation of China(Grant,Nos.,11672161,11620101001)


This work was supported by the National Natural Science Foundation of China (Grant Nos. 11672161, 11620101001).


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  • Figure 1

    (Color online) (a) A tumor spheroid growing under environmental constraint. (b) Forces applied on interacting cells. (c) Illustration of pressure-regulated cell growth and division rule. During cell proliferation, cell will enter G1 phase and cell volume will increase at a constant rate to realize duplication. While cells undergoing strong compression will stop duplication and are arrested in G0 phase. After G1 phase and cell volume reaches a threshold, mother cell will enter M phase and divide into two daughter cells. (d) Experimental observation of fingering invasion of a MDA-MB-231 cell aggregate embedded in gels. Reprinted from ref. [33] with permission. Scale bar = 500 μm.

  • Figure 2

    (Color online) (a) Formation of a multicellular tumor spheroid. (b) Snapshot of pressure distribution inside a tumor spheroid at t=14τ. Light grey particles denote normal cells, otherwise cancerous cells. (c) Averaged pressure and packing fraction Φ as a function of radial distance r deviating from the centroid of the fixed spherical lumen. In all simulations, we take α=1, p0=148 Pa, and EN=1000 Pa.

  • Figure 3

    (Color online) Snapshots of fingering invasion during tumor growth. (a1)–(d1) Morphology of tumor spheroids. (a2)–(d2) Hemispherical cross-section of tumor invasion. (a3)–(d3) 3D view with a quarter of space removed. (a) α=1, t=14τ; (b) α=5, t=14τ; (c) α=20, t=14τ; (d) α=50, t=16τ. Red and green spheres denote normal cells and cancerous cells, respectively. In all simulations, we take p0=444 Pa and EN=1000 Pa.

  • Figure 4

    (Color online) Simulation results under different α. (a) Cancerous cell number NC versus time; (b) the radius of gyration of tumor spheroid Rg; (c) dependence of invasion degree ρ on the radial distance r at t=14τ; (d) invasion distance Δr as a function of time.

  • Figure 5

    (Color online) Impact of stiffness of normal cells on tumor growth. (a) EN=300 Pa; (b) EN=600 Pa; (c) EN=2000 Pa. Snapshot at t=14τ. Red and green spheres denote normal cells and cancerous cells respectively. (d) Cancerous cell number NC versus time. (e) The radius of gyration of tumor spheroid Rg. (f) Invasion degree ρ as a function of time. In all simulations, we take α=5 and p0=444 Pa.

  • Figure 6

    (Color online) Impact of heterotypic adhesion between normal cells and cancerous cells on tumor invasion under different β. (a) β=0.5; (b) β=1; (c) β=1.5. Snapshot at t=14τ. (d) Cancerous cell population size increases as a function of time. (e) Invasion degree versus time. In all simulations, we take α=5, p0=444 Pa and EN=1000 Pa.

  • Table 1   Model parameters






    Young’s modulus of cells



    450, 1000


    Poisson’s ratio of cells




    Adhesive energy between cells





    Reference diffusion constant


    cm2 s–1



    Diffusion constant of cells


    cm2 s–1



    Reference friction coefficient


    N s/m



    Friction coefficient between cells


    N s/m

    1, 0.02, 0.02


    Radius of cells





    Cell cycle





    Pressure threshold





    Radius of environment





    Growth rate of radius





    Time step




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