Fig 1.
Cardiovascular data and schematic illustration of vascular branching
(a) Mouse lung micro-CT images processed by Angicart. (b) Human head and torso MRI images processed by Angicart. (c) Schematic illustration of the asymmetric branching geometry and labeling. Parent vessel with radius r0 and length l0 branches into two daughter vessels with radius ri and length li with subscript i = 1 or 2. Branching angles, θi, are defined by the angle between the sides defined by the endpoints of the vessel pairs. Here, subscripts are determined by the non-adjacent vessel. (see Materials and Methods) (d) Optimization of local branching on a plane finds the optimal location of the branching junction J when the unshared endpoints (Vi) and the radii (ri) are fixed (see General framework for branching angle optimization and asymmetry).
Fig 2.
Comparison of real data for vascular networks versus random simulations of branching junctions.
The real and simulated networks (via local to global spatial constraints) are separated by different rows. A schematic small network is given to describe how different simulations are performed. The vessels and the fixed endpoints of the real branching network are represented in red. Vessels that result from random branching simulations are in black. The healthy mouse lung network and the simulated mouse lung networks are shown within a minimum spherical boundary that contains all branching data from the real network. Here, the red nodes for each figure correspond to the real data, whereas the black nodes correspond to the simulated data. Note that the terminal tips and the most upstream node (i.e., the source) are determined from real data and fixed throughout all simulations. The resulting asymmetry ratio distributions for length and branching angles are provided for the real network and for each of the simulations. The statistical comparisons of random branching simulations with empirical data are given in Table 1.
Fig 3.
Histograms or frequency distributions of the asymmetry ratios for radius (λr), length (λl), and branching angles (λθ) of vascular networks.
(a) mouse lung (1 individual) and (b) human head and torso (18 individuals). Note that radius and branching angle asymmetry ratios are both skewed towards perfect symmetry, whereas the length asymmetry ratio shows no skew and reveals much more asymmetry. (c) Histograms of branching angles for combined data of human and mouse networks appear to be unimodal both for θ0 and for θ1 & θ2 with peaks at 1.51 and 2.21 radians, respectively.
Table 1.
Statistical comparison of material-cost (MC) optimizations and random spatial constraints with empirical data.
Fig 4.
Histograms or frequency distributions of optimal asymmetry ratios for length (λl) and branching angle (λθ) derived from material-cost (MC) optimizations.
Surface-area (MC-SA) results are shown as solid lines and volume (MC-V) results are shown as dashed lines for (a) mouse lung and (b) human head and torso.
Fig 5.
Histogram of optimal branching angles for combined data of human and mouse networks for material-cost (MC) optimizations.
All histograms appear to have unimodal characteristics both for θ0 and for θ1 & θ2 with respective peaks at (a) 1.79 and 2.25 for the surface-area constraint and (b) 1.79 and 2.24 for the volume constraint.
Fig 6.
Junction-level comparison of optimal versus actual branching angles for the volume constraint of material-cost optimizations (MC-V).
(a) mouse lung and (b) human head and torso. The Pearson correlation coefficients and p-values are calculated for each plot.
Fig 7.
Comparison of approximate solutions with numerical solutions for the PC-1 (power-cost (PC) optimization beyond single branching).
Approximate solutions define linear boundaries on the c1c2-plane between different categories of the solution space: collapse to daughter endpoint, collapse to parent endpoint, and no-collapse. The different categories calculated from numerical simulation are marked by different colors as indicated in the figure. (a) An example of symmetric branching in vessel radius with parameter values: |V0V1| = |V0V2| = |V1V2| = 1, r0 = 1.20, r1 = 1, and r2 = 1, where c1 and c2 take values in the range [0,20]. (b) Zoomed version of (a) into the plane [0,2] × [0,2] with the same resolution as in (a). (c) An example of asymmetric branching in vessel radius with parameter values: |V0V1| = 0.8, |V0V2| = |V1V2| = 1, r0 = 1.1, r1 = 0.85, and r2 = 1, where c1 and c2 take values in the range [0,20]. (d) Zoomed version of (c) into the plane [0,2] × [0,2] with the same resolution as in (c).