Properties of Tumor Spheroid Growth Exhibited by Simple Mathematical Models

Abstract

Solid tumors, whether in vitro or in vivo, are not an undifferentiated mass of cells. They include necrotic regions, regions of cells that are in a quiescent state (either slowly growing or not growing at all), and regions where cells proliferate rapidly. The decision of a cell to become quiescent or proliferating is thought to depend on both nutrient and oxygen availability and on the presence of tumor necrosis factor, a substance produced by necrotic cells that somehow inhibits the further growth of the tumor. Several different models have been suggested for the basic growth rate of in vitro tumor spheroids, and several different mechanisms are possible by which tumor necrosis factor might halt growth. The models predict the trajectory of growth for a virtual tumor, including proportions of the various components during its time evolution. In this paper we look at a range of hypotheses about basic rates tumor growth and the role of tumor necrotic factor, and determine what possible tumor growth patterns follow from each of twenty-five reasonable models. Proliferating, quiescent and necrotic cells are included, along with tumor necrosis factor as a potential inhibitor of growth in the proliferating pool and two way exchange between the quiescent and proliferating pools. We show that a range of observed qualitative properties of in vitro tumor spheroids at equilibrium are exhibited by one particular simple mathematical model, and discuss implications of this model for tumor growth.

Experimental method and statistical analysis to fit tumor growth model using SPECT/CT imaging: a preclinical study

“…[1][2][3], as described previously in the section for assessment of inversion algorithm. We used parameters recommended by Wallace-Guo (18). However, we changed the m parameter in Eqs.…”

“…The model captures the interdependence of the different cell pools that make up a tumor with different combinations of functions for growth with other terms considered. In (18), of the 25 different models explored, the one designated 3E was concluded to perform the best, given in the following equations.…”

“…Thus necrosis formation is an important factor to be accounted for in serial images to obtain accurate parameters such as growthrate. Necrosis effects have been modeled theoretically by moving boundary methods modeling nutrient concentration (mainly by oxygen diffusion) (13)(14)(15)(16)(17), but on the other hand, in these models, typically, tumor cell density or volume effects (6)(7)18) are not explicitly considered. Another model that incorporates necrosis is the theoretical ode-compartment volume model proposed by Wallace and Guo (18).…”

“…The Wallace-Guo model considers an overall effect of necrosis on the proliferating layer and may provide a more accurate picture of tumor proliferation than models which do not consider necrosis. Several different combinations of growth and necrosis models are proposed in theory in (18) and while some are forward simulated, none are fitted to any experimental data.…”

“…Therefore, to enable our in vivo analysis, we made some small modifications to the main structure of the theoretical model in (18) to obtain a twocompartment version, to better fit the experimental data. Also, in addition to the surface-area based and logistic growth considered in (18), we added for consideration a Gompertzian growth because of its strong results for in vivo use (7).…”