Angiogenesis: An Adaptive Dynamic Biological Patterning Problem

<div><p>Formation of functionally adequate vascular networks by angiogenesis presents a problem in biological patterning. Generated without predetermined spatial patterns, networks must develop hierarchical tree-like structures for efficient convective transport over large distances, combined with dense space-filling meshes for short diffusion distances to every point in the tissue. Moreover, networks must be capable of restructuring in response to changing functional demands without interruption of blood flow. Here, theoretical simulations based on experimental data are used to demonstrate that this patterning problem can be solved through over-abundant stochastic generation of vessels in response to a growth factor generated in hypoxic tissue regions, in parallel with refinement by structural adaptation and pruning. Essential biological mechanisms for generation of adequate and efficient vascular patterns are identified and impairments in vascular properties resulting from defects in these mechanisms are predicted. The results provide a framework for understanding vascular network formation in normal or pathological conditions and for predicting effects of therapies targeting angiogenesis.</p> </div

Concepts of microvascular pattern formation.

<p>(<b>A</b>) Microvascular networks are often conceptualized as mesh or hierarchical structures. A mesh minimizes diffusion distances between capillaries and tissue but has high flow resistance and results in non-uniform oxygen levels from the arterial to the venous side. In a hierarchical structure, larger supply vessels decrease flow resistance, but regions surrounding those vessels are inadequately supplied due to large diffusion distances. Colors (red - green - blue) indicate flow from arterial to venous vessels, with decline in oxygen levels. (<b>B</b>) Hypothesized steps in generation of functional vascular networks. A dense network of vessels is generated by over-abundant angiogenesis and refined by structural adaptation and pruning. Resulting networks combine features of mesh and hierarchical structures.</p

Simulated angiogenesis, showing oxygen and VEGF distributions.

<p>Oxygen levels in vessels and tissue are color coded according to scale at right. Dark blue indicates hypoxic tissue. Diagonal shading indicates VEGF above threshold concentration. Time (<i>t</i>) in days and total vessel length (<i>L</i>) in mm are indicated. Maximal rate of sprout formation is 2 mm⁻¹day⁻¹. Tissue oxygen demand is 2 cm³/100 cm³/min. (<b>A</b>) Initial configuration. (<b>B</b>) Sprouts (white arrows) and a short flow pathway between arterioles and venules (a-v shunt, purple arrow) are generated. (<b>C</b>) More complex flow pathways form, leading to improved oxygenation (lower right region). (<b>D</b>) Improved oxygenation leads to decreased VEGF levels. Structural adaptation causes pruning of the a-v shunt (purple arrow), but some redundant flow pathways remain (black arrows). Total vessel length reaches its maximum. (<b>E</b>) Structural adaptation leads to pruning of redundant flow pathways (black arrows). (<b>F</b>) Final refined network. Virtually no hypoxia remains and VEGF levels are generally below threshold. See online supplement, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002983#pcbi.1002983.s001" target="_blank">Video S1</a>, for movie clip.</p

Effect of tension-induced vessel migration on distribution of branching angles.

<p>(<b>A</b>) Example of network structure generated when tension-induced migration is suppressed, at <i>t</i> = 200 days. Other conditions are as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002983#pcbi-1002983-g004" target="_blank">Figure 4</a>. (<b>B</b>) Distribution of branching angles for network shown in A. (<b>C</b>) Distribution of branching angles for observed network structure. (<b>D</b>) Distribution of branching angles for network shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002983#pcbi-1002983-g004" target="_blank">Figure 4</a> at <i>t</i> = 200 days, including tension-induced migration.</p

Comparison of simulated networks with experimentally observed network.

<p>Comparison of simulated networks with experimentally observed network.</p

Steps in simulation approach.

<p>(<b>A</b>) Network of microvessels in rat mesentery, imaged using intravital microscopy. Shaded overlay highlights vessel positions, with arterioles (red), capillaries (green) and venules (blue). This region was selected for analysis because the outer loop of venules provides stable boundary conditions for the tissue domain. (<b>B</b>) Computer generated image of network structure, trimmed to reduce the number of network boundary nodes to five (arrows). (<b>C</b>) Network skeleton, used as initial condition for simulations. In (<b>D–I</b>), a small region within a typical simulation is shown at a sequence of times indicating aspects of the method. White triangles denote features mentioned in this caption. (<b>D</b>) The oxygen field surrounding the vessels is computed using the Green's function method. Blue shades denote low oxygen levels. VEGF is assumed to be generated in hypoxic regions and to diffuse according to local gradients, and the resulting VEGF field is computed. Diagonal hatching indicates VEGF concentration above a given threshold. (<b>E</b>) On vessels lying in regions with VEGF above threshold, sprouts are generated with probability dependent on local VEGF concentration. (<b>F</b>) A fixed rate of sprout elongation is assumed. Direction of growth is randomly varied at each time step. (<b>G</b>) If other vessels lie within a sector of radius 100 µm ahead of the sprout tip, the growth is biased towards them. (<b>H</b>) A sprout reaching another vessel forms a connection, allowing flow. (<b>I</b>) Diameters of flowing vessels adapt to metabolic and hemodynamic stimuli.</p

Comparison of simulated and observed network characteristics.

<p>Simulated oxygen and VEGF distributions in experimentally observed network (<b>A</b>), and in simulated angiogenesis at <i>t</i> = 200 days (<b>B</b>). The observed network structure is derived from the image shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002983#pcbi-1002983-g002" target="_blank">Figure 2A</a>. The network at <i>t</i> = 200 days was derived from the same initial network as shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002983#pcbi-1002983-g004" target="_blank">Figure 4A</a> but with a different seed for random number generation. Oxygen demand, rate of sprout formation, length scale, color coding of oxygen levels and diagonal shading indicating VEGF above threshold are as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002983#pcbi-1002983-g004" target="_blank">Figure 4</a>. (<b>C</b>) Frequency distribution of distance of tissue points to nearest vessel. (<b>D</b>) Frequency distribution of oxygen levels at tissue points. Results for simulated networks are mean ± standard deviation for n = 6 simulations with different seeds for random number generation.</p

Effects of inhibiting adaptation or conducted responses.

<p>(<b>A</b>) Angiogenesis without structural adaptation and pruning. Resulting structure is mesh-like, without hierarchical structure. (<b>B</b>) Simulation of angiogenesis with reduced conducted response strength. Short flow pathways from feeding to draining vessels carry most of the flow. Regions remote from feeding vessels receive inadequate perfusion. Length and color scales as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002983#pcbi-1002983-g004" target="_blank">Figure 4</a>.</p

Outline of methods.

<p>(A) Timeline of imaging events. Mice that were imaged under two imaging conditions were imaged on subsequent days. The order of imaging was scrambled to minimize order effects. (B) A 6 mM injection of 2-NBDG was given and imaged for at least 60 minutes, and the mean of the tumor region for each image was used to construct a kinetic curve. Images for the endpoints 2-NBDG₆₀ (2-NBDG intensity at 60 minutes) and the rate of delivery of 2-NBDG (R_D = 2-NBDG₆₀/T_max) are shown. (C) Trans-illumination images were collected in 10 nm increments from 500–600 nm and used to calculate hemoglobin saturation (SO2). (D) The table shows the number of mice used in each perturbation group. Each mouse was used for up to two imaging sessions, with 24 hours between sessions. The groups were randomized to minimize bias from imaging order, and an analysis of variance (ANOVA) was performed to test for order effects. No significant imaging order effect was observed for any experiment.</p

The ratio 2-NBDG/R_D facilitates assessment of glucose demand in heterogeneous regions of metastatic mammary tumors.

<p>(A) Representative images of vascular oxygenation (SO₂) and delivery-corrected 2-NBDG (2-NBDG₆₀/R_D) for a 4T1 tumor with low mean SO₂, a 4T1 tumor with intermediate mean SO₂, and a 4T07 with high mean SO₂. Adapted from Rajaram, et al. 2013. (B) Survival curves (1-cumulative distributions) show 2-NBDG₆₀, R_D, and 2-NBDG₆₀/R_D for regions of distinct SO₂ (%) in 4T07 and 4T1 tumors. For 4T1, 2-NBDG₆₀ is lower for 02,4T1<10 regions than for any other SO_2,4T1 (p = N.S.). Significantly lower rates of R_D are seen for the 02,4T1<10 group than for well-oxygenated 4T1 regions (p<0.05 or p<0.01 for 02,4T1<10 vs. 202,4T1<40 or 402,4T1<60, respectively). After correction for low R_D, 2-NBDG₆₀/R_D increased slightly but significantly in hypoxic regions (p<0.01 for 02,4T1<10 vs. 402,4T1<60). For 4T07, 2-NBDG uptake for the highest SO_2,4T07 regions decreased compared to the lowest SO_2,4T07 (p<0.01 for all 202,4T07<40 vs. 602,4T1<80). R_D is indistinguishable between SO_2,4T07 levels. After correction by R_D, 2-NBDG₆₀/R_D is lowest for 602,4T07<80 (p<0.01). Comparison between 4T1 and 4T07 shows that 2-NBDG₆₀ is higher for all SO_2,4T1 than all SO_2,4T07 (p<0.01). On the other hand, R_D for the best oxygenated 4T07 groups (402,4T07<60 and 602,4T07<80) is greater than for all 4T1 groups (p<0.01 for all groups except 402,4T1<60 vs. 602,4T07<80 where p<0.06). After correction by R_D, 2-NBDG₆₀/R_D is higher for all SO_2,4T1 than all SO_2,4T07 (p<0.01 for all SO_2,4T1 compared to all SO_2,4T07). Number of mice per group indicated by group name in legend.</p