Hybrid modeling of tumor-induced angiogenesis

Abstract

When modeling of tumor-driven angiogenesis, a major source of analytical and computational complexity is the strong coupling between the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network, and the family of interacting underlying fields. To reduce this complexity, we take advantage of the system intrinsic multiscale structure: we describe the stochastic dynamics of the cells at the vessel tip at their natural mesoscale, whereas we describe the deterministic dynamics of the underlying fields at a larger macroscale. Here, we set up a conceptual stochastic model including branching, elongation, and anastomosis of vessels and derive a mean field approximation for their densities. This leads to a deterministic integro-partial differential system that describes the formation of the stochastic vessel network. We discuss the proper capillary injecting boundary conditions and include the results of relevant numerical simulations.

Ensemble Averages, Soliton Dynamics and Influence of Haptotaxis in a Model of Tumor-Induced Angiogenesis

“…Different from [9], we have included the step function θ(τ − t) in Equation (19). With τ = ∞ as in [9,10], the primary vessel keeps injecting tip density for all time. However, this may be artificial, as the primary vessel does not inject any more vessels after t = 0 + in many experiments on early stage angiogenesis.…”

“…Then, τ in Equation (19) may be a small time of the order of the time step used in a numerical code. The original boundary condition in [9] did not include the unit step function θ(τ − t) and, as a consequence, the deterministic description given by the tip density equation and its boundary conditions had an artificial injection of tip density at x = 0 for all t > 0. The anastomosis coefficient, Γ, has to be fitted using the stochastic description, [10].…”

“…However, the difference with the results of Equation (17) would be appreciable at the last stage when the vessel tips arrive at the tumor, which we do not study specifically in the present paper. The tip flux density at x = 0, j 0 (t, y), obtained by integrating Equation (16) over 0 < v < ∞ and −∞ < w < ∞, is [9] …”

“…Other possible choices change the birth term a little in Equation (13), but their effects on the marginal densityp(t, x, y) are small for the slab configuration. The boundary conditions for the tip density are [9] …”

“…Capasso and Morale also attempted to derive a continuum equation for the density of moving tip cells from the stochastic equations but could not account for branching and anastomosis [7]. Recently, we have derived a deterministic description for the average vessel tip density of a related hybrid stochastic tip cell model for tumor-induced angiogenesis that includes chemotaxis but not haptotaxis [9,10]. We have shown that the density of active vessel tips advances toward the tumor as a soliton-like wave [11,12].…”