A hybrid model for tumor spheroid growth in vitro I: Theoretical development and early results

Kim Yangjin, Magdalena A. Stolarska, Hans G. Othmer

Research output: Contribution to journalArticlepeer-review

136 Scopus citations


Tumor spheroids grown in vitro have been widely used as models of in vivo tumor growth because they display many of the characteristics of in vivo growth, including the effects of nutrient limitations and perhaps the effect of stress on growth. In either case there are numerous biochemical and biophysical processes involved whose interactions can only be understood via a detailed mathematical model. Previous models have focused on either a continuum description or a cell-based description, but both have limitations. In this paper we propose a new mathematical model of tumor spheroid growth that incorporates both continuum and cell-level descriptions, and thereby retains the advantages of each while circumventing some of their disadvantages. In this model the cell-based description is used in the region where the majority of growth and cell division occurs, at the periphery of a tumor, while a continuum description is used for the quiescent and necrotic zones of the tumor and for the extracellular matrix. Reaction-diffusion equations describe the transport and consumption of two important nutrients, oxygen and glucose, throughout the entire domain. The cell-based component of this hybrid model allows us to examine the effects of cell-cell adhesion and variable growth rates at the cellular level rather than at the continuum level. We show that the model can predict a number of cellular behaviors that have been observed experimentally.

Original languageEnglish (US)
Pages (from-to)1773-1798
Number of pages26
JournalMathematical Models and Methods in Applied Sciences
Issue numberSUPPL. 1
StatePublished - 2007

Bibliographical note

Funding Information:
Research supported by NSF grants 0317372 and 0517884, and by the Minnesota Supercomputing Institute.


  • Hybrid model
  • Mechanical effects
  • Nutrient limitations
  • Tumor growth


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