Tumor-Host Interaction with Contact Inhibition

L2 Tumor-Host Interaction with Contact Inhibition 

Linear diffusion satisfactorily describes the transport mechanism for a single population. For interacting populations, linear diffusion terms imply that the populations are able to mix completely, with the movement of one cell type unaffected by the presence of cells of the other type. The reality is exactly the opposite. Cell movement is typically halted by contact with another cell. This phenomenon is known as contact inhibition and is very well documented for many types of cells. Sherratt introduced a phenomenological model to account for contact inhibition [402]. Consider the interaction between normal and tumor cells with concentrations pm(xj) and Pt(x, t), respectively. The overall cell flux of both populations is given by x(Pn + Pt)- a fraction Pn/(Pn + Pt) of this flux corresponds to normal cells, so that the flux of normal cells is - [pn/(Pn + Pr)] x(Pn + Pr) a similar expression for the flux of tumor cells. These expressions indicate that the movement of one population is inhibited by the presence of the other. The system of dimensionless reaction-diffusion equations reads [402] 

The term -pj -pY in the kinetics of (8.8) represents the decrease in the cell division rate due to crowding. The constant y ( 1) expresses the proliferative advantage of the tumor cell population. Note that the model is in one-dimensional space and that the kinetic terms are of Lotka-Volterra competition type. 

Consider front-like initial conditions, such that the density of normal cells travels to the left connecting p = 0 to Pf = and the density of tumor cells travels to the right connecting pj. = 0 to /Oj. = y [402]. If y 1, fronts of normal and tumor cells move toward one another. As they approach each other, the tumor front continues 

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