Limiting spheroid size as a function of growth factor source location

Solutions C(r) of the time-independent nonhomogeneous diffusion equation for three different piecewise-uniform source terms are used to examine the limiting size of multicell spheroids using a simple model which reproduces concentration-dependent mitotic behavior. A condition is derived under which nontrivial solutions do not exist (in all three cases), and a condition for the existence of a unique nontrivial solution is established for the case of growth-modifying factor (GMF) production throughout the spheroid. Qualitative behavior of the limiting size is established as a function of various physiological parameters. Of fundamental importance is the assumed GMF concentration threshold theta, near which (i.e., as C --> theta(-)) mitosis, and hence spheroid growth, is generally strongly inhibited.