The dynamics of growth-factor-modified immune response to cancer growth: One dimensional models

Abstract

By characterizing the effect of tumor growth factors as deviations from normal logistic-type growth rates, the spatio-temporal dynamics for a one-dimensional model of cancer growth incorporating immune response are studied. The growth rates considered are classified respectively as normal, activated, inhibited and delay activated. The homogeneous steady states are defined by relative extrema of a “free energy” function V( x) for each of the above four cases. This function is of particular importance in studying the coexistence of tumoral and cancer-free steady states, and in identifying the nature (progressive or regressive) of travelling wave solutions to the nonlinear partial differential equation governing the tumor cell dynamics in each case. Lower bounds on wave speeds for wavefronts linking stable tumoral states to unstable cancer-free states are established, as are estimates of wave speeds(~ 2 × 10 -7 − 2 × 10 -8 cm⧸ sec), corresponding to a tumor of size 1 cm being established in 50–500 days (depending on the value of the diffusion coefficient). Some exact solutions and wave speeds for analytic approximations to the system are obtained. An estimate is given for the tumor nucleation size in three dimensions, along with a lower bound on the system size necessary to support such tumor “outbreaks” (not unlike the budworm infestation problem). As more data becomes available on the nature of growth factors and the associated cellular response characteristics, models of the type developed here can be used as a basis for comparison of activation/inhibition interactions with the immune system.