Nonlinear simulation of tumor growth

Vittorio Cristini, John Lowengrub, Qing Nie

Research output: Contribution to journalArticlepeer-review

251 Scopus citations


We study solid tumor (carcinoma) growth in the nonlinear regime using boundary-integral simulations. The tumor core is nonnecrotic and no inhibitor chemical species are present. A new formulation of the classical models [18, 24, 8, 3] is developed and it is demonstrated that tumor evolution is described by a reduced set of two dimensionless parameters and is qualitatively unaffected by the number of spatial dimensions. One parameter describes the relative rate of mitosis to the relaxation mechanisms (cell mobility and cell-to-cell adhesion). The other describes the balance between apoptosis (programmed cell-death) and mitosis. Both parameters also include the effect of vascularization. Our analysis and nonlinear simulations reveal that the two new dimensionless groups uniquely subdivide tumor growth into three regimes associated with increasing degrees of vascularization: low (diffusion dominated, e.g., in vitro), moderate and high vascularization, that correspond to the regimes observed in vivo. We demonstrate that critical conditions exist for which the tumor evolves to nontrivial dormant states or grows self-similarly (i.e., shape invariant) in the first two regimes. This leads to the possibility of shape control and of controlling the release of tumor angiogenic factors by restricting the tumor volume-to-surface-area ratio. Away from these critical conditions, evolution may be unstable leading to invasive fingering into the external tissues and to topological transitions such as tumor breakup and reconnection. Interestingly we find that for highly vascularized tumors, while they grow unbounded, their shape always stays compact and invasive fingering does not occur. This is in agreement with recent experimental observations [30] of in vivo tumor growth, and suggests that the invasive growth of highly-vascularized tumors is associated to vascular and elastic anisotropies, which are not included in the model studied here.

Original languageEnglish (US)
Pages (from-to)191-224
Number of pages34
JournalJournal of Mathematical Biology
Issue number3
StatePublished - Mar 2003


  • Boundary-integral simulations
  • Linear stability analysis
  • Self-similarity
  • Tumor growth

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics


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