Vascular Remodeling Processes.

<p>The dynamical processes of vascular network remodeling (A-E) are illustrated by exemplary state transitions with the current state shown in the left hand side box and the resulting state on the right. Preconditions are indicated above the center arrows, and transition probabilities are denoted below, respectively. (A) depicts the start of a new sprout (gray) which can happen on any site on the network provided that given preconditions are true. Here, one of several preexisting segments (blue) is split to create a new branching point. Space occupied by the tumor spheroid (orange), and space where angiogenic sprouting may take place (green), i.e. where <i>R</i>_<i>tum</i> < |<b><i>x</i></b>| <<i>R</i>_<i>tum</i> + <i>R</i>_<i>g</i>, is indicated by colored background. “In tissue” refers to the exterior of the spheroid, i.e. where |<b><i>x</i></b>| ><i>R</i>_<i>tum</i>. “In tumor” means the opposite, obviously. The path length on the network to the next branching point <i>d</i>_<i>br</i> must be larger than the lower limit <i>d</i>^{(<i>br</i>, <i>min</i>)}. The new segment is initialized with an associated life-time of <i>τ</i> = 1. (B) depicts the further extension of the sprout from (A). Additional segments inherit <i>τ</i> from the parent segment. Moreover <i>τ</i> is incremented, globally, for all sprouts once per time step Δ<i>t</i>. (C) depicts the degradation of vessel walls. The variable <i>w</i> represents the stability of the vessel wall, here depicted as varying wall thickness. It decreases continuously at the rate Δ<i>w</i>, resulting in a value of <i>w</i>′ at the next time step. In (D) an unstable vessel (red) is removed, representing occlusion of blood flow and complete disintegration. Such event is assumed to happen only to vessels with maximally degenerate walls <i>w</i> = 0 and low wall shear-stresses <i>f</i>, where <i>f</i> < <i>f</i>^{(<i>coll</i>)}. The emerging dead ends (gray) trivially have <i>f</i> < <i>f</i>^{(<i>coll</i>)}, and therefore collapse rapidly, resulting in a long ranged effect. Since collapses can happen anywhere, we left the background blank. (E) depicts the dilation of tumor vessels. Their radii increase at rate Δ<i>r</i> up to the upper limit <i>r</i>^(<i>max</i>).