Computational modelling suggests complex interactions between interstitial flow and tumour angiogenesis

Abstract

Angiogenesis, the growth of capillaries from pre-existing ones, plays a key role in cancer progression. Tumours release tumour angiogenic factors (TAFs) into the extracellular matrix (ECM) that trigger angiogenesis once they reach the vasculature. The neovasculature provides nutrients and oxygen to the tumour. In the ECM, the interstitial fluid moves driven by pressure differences and may affect the distribution of tumour TAFs, and, in turn, tumour vascularization. In this work, we propose a hybrid mathematical model to investigate the influence of fluid flow in tumour angiogenesis. Our model shows the impact of interstitial flow in a time-evolving capillary network using a continuous approach. The flow model is coupled to a model of angiogenesis that includes tip endothelial cells, filopodia, capillaries and TAFs. The TAF transport equation considers not only diffusive mechanisms but also the convective transport produced by interstitial flow. Our simulations predict a significant alteration of the new vascular networks, which tend to grow more prominently against the flow. The model suggests that interstitial flow may produce increased tumour malignancies and hindered treatments.

Keywords: angiogenesis; interstitial flow; mathematical modelling; phase field.

Figure 1.

Angiogenesis starts with the activation of tip endothelial cells by tumour angiogenic factors, which are released in hypoxic regions and transported to nearby capillaries. Tumour angiogenic factor transport is usually assumed to be purely diffusive, but convective transport by interstitial flow may significantly influence tumour angiogenesis.

Figure 2.

Hydraulic conductivity function. We plot as a function of , that is, the one-dimensional solution to the phase-field equation on an infinite domain. The phase-field creates smooth interfaces that permit the differentiation of intravascular, transvascular and extravascular regions. Accordingly, the hydraulic conductivity displays three plateaus, one per region, connected by smooth transitions. In this plot, we take κ^e = 0.15, , and λ = 1. Although these values are not realistic, we use them in the plot to enable a simple visualization of the hydraulic conductivity as a function of c.

Figure 3.

Non-convective versus convective vascular patterns. Vascular patterns without VEGF convection (a–d) and with VEGF convection (e–p) create different vascular networks (q). The numerical method captures the fluid velocity and the phase-field, as shown by the velocity magnitude, u, and the phase-field profile, c, along a cut-line (r) and streamlines (s). We used a mesh composed of 512 × 512 elements in these simulations.

Figure 4.

Flow-mediated angiogenesis with a controlled, uniform fluid velocity. In the top-row simulation, there is no flow. In the bottom-row simulation, the fluid velocity is 6 μm s⁻¹. For both computations, time grows from left to right. The results show that our model predicts more prominent growth of the neovasculature against the flow, as observed in a recent experiment [18]. The VEGF was kept constant on the lateral boundaries with concentration f = f_hyc. Boundary conditions in the vertical direction were assumed to be periodic. The computation was performed on a tissue of size 1000 × 1000 μm. As this computation is performed on a larger piece of tissue, we added random perturbations to the TEC velocity field orientation [27] to produce a more realistic vascular pattern. We have verified (data not shown) that this does not have an impact on the symmetry breaking mechanism. Similar computations without random perturbations also produce more prominent capillary growth against the flow.

Figure 5.

Convection-increased angiogenesis. VEGF reaches the initial capillaries at different times when the tumour radius is varied (56.25, 37.50 and 25.00 μm). In the absence of convection, the smaller tumour is not able to trigger angiogenesis. The green colour scale represents VEGF concentration.

Figure 6.

Influence of intratumoral pressure. Intratumoral pressure alters the distribution of VEGF which, in turn, drives capillaries around the tumour. The fluid velocity inside the tumour is low, even within the capillaries that transverse the tumour.