Analysis of a free-boundary tumor model with angiogenesis

We consider a free boundary problem for a spherically symmetric tumor with free boundary r < R(t). In order to receive nutrients u the tumor attracts blood vessel at a rate proportional to α(t), so that ∂u ∂r +α(t)(u− ū) = 0 holds on the boundary, where ū is the nutrient concentration outside the tumor. A parameter μ in the model is proportional to the ‘aggressiveness’ of the tumor. When α is a constant, the existence and uniqueness of stationary solution is proved. For the more general situation when α depends on time, we show, under various conditions (that are always satisfied if μ is small), that (i) R(t) remains bounded if α(t) remains bounded; (ii) limt→∞R(t) = 0 if limt→∞ α(t) = 0; and (iii) lim inft→∞R(t) > 0 if lim inft→∞ α(t) > 0. Surprisingly, we exhibit solutions (when μ is not small) where α(t) → 0 exponentially in t while R(t) → ∞ exponentially in t. Finally, we prove the global asymptotic stability of steady state when μ is sufficiently small.