Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids

Abstract

In order to determine the role of micromilieu in tumour spheroid growth, a mathematical model was developed to predict EMT6/Ro spheroid growth and microenvironment based upon numerical solution of the diffusion/reaction equation for oxygen, glucose, lactate ion, carbon dioxide, bicarbonate ion, chlorine ion and hydrogen ion along with the equation of electroneutrality. This model takes into account the effects of oxygen concentration, glucose concentration and extracellular pH on cell growth and metabolism. Since independent measurements of EMT6/Ro single cell growth and metabolic rates, spheroid diffusion constants, and spinner flask mass transfer coefficients are available, model predictions using these parameters were compared with published data on EMT6/Ro spheroid growth and micro-environment. The model predictions of reduced spheroid growth due to reduced cell growth rates and cell shedding fit experimental spheroid growth data below 700 microns, but overestimated the spheroid growth rate at larger diameters. Predicted viable rim thicknesses based on predicted near zero glucose concentrations fit published viable rim thickness data for 1000 microns spheroids grown at medium glucose concentrations of 5.5 mM or less. However, the model did not accurately predict the onset of necrosis. Moreover, the model could not predict the observed decreases in oxygen and glucose metabolism seen in spheroids with time, nor could it predict the observed growth plateau. This suggests that other unknown factors, such as inhibitors or cell-cell contact effects, must also be important in affecting spheroid growth and cellular metabolism.

Necrotic core in EMT6/Ro tumour spheroids: Is it caused by an ATP deficit?

“…The respective intracellular concentrations will be denoted asσ G ,σ O , andσ L . We assume: (i) glucose uptake follows a Michaelis-Menten kinetics and all glucose taken up by cells is consumed (Casciari et al, 1992, Venkatasubramanian et al, 2006; (ii) the exchange of pyruvate with the extracellular medium is disregarded; (iii) transmembrane transport of lactate is non-saturable in the concentration range of interest, and is driven by the gradient of lactate concentration; (iv) because of the high permeability of cell membrane to oxygen (Alberts et al, 1994), oxygen concentration inside the cell is equal to the extracellular concentration; (v) the pyruvate flow toward the Krebs cycle is a saturable function of pyruvate and oxygen concentrations (see also Aubert and Costalat, 2005).…”

“…Experimental data (Freyer and Sutherland, 1985, Casciari et al, 1992, Schroeder et al, 2005 have shown that glucose consumption increases when oxygen concentration decreases (the so-called Pasteur effect) and that, at physiological pH, oxygen consumption increases when glucose concentration decreases (the so-called Crabtree effect). Expressions for the glucose and oxygen consumption including the above effects were proposed on an empirical basis (Casciari et al, 1992).…”

“…Expressions for the glucose and oxygen consumption including the above effects were proposed on an empirical basis (Casciari et al, 1992). These effects seem to depend on the pH of cell environment: for instance, the Crabtree effect disappears in an acidic environment.…”

“…These effects seem to depend on the pH of cell environment: for instance, the Crabtree effect disappears in an acidic environment. Although the production of H + ions is related to lactic acid production, the description of pH changes is complicated by the presence of an intracellular buffer (see the mathematical models in Casciari et al, 1992, andWebb et al, 1999). Since a mechanistic representation of these phenomena appears to be rather complex, they are neglected in our model.…”

“…Almost all these models assumed that cell proliferation and cell death depend on the concentration of a single critical chemical that diffuses from the external medium into the cell aggregate. The diffusion of glucose, oxygen, lactate and other substrates has been included in a spheroid model proposed by Casciari et al (1992). In that model, the cell proliferation rate was taken as an empirical function of glucose, oxygen and hydrogen ion concentrations.…”