Fig 1.
Chart summarizing the key components of the present study.
Bottom-left panels present representative vascular regions containing perfused (yellow) and hypoperfused (purple) vessels from Day 0 (just before irradiation) for one of the tumours for which irradiation-induced vessel pruning led to a decrease in perfusion (tumour 6; top) and one for which it led to an increase (tumour 1; bottom). Note that the latter vasculature contained many hypoperfused blunt-ended vessels, while the former contained more loops. Bottom-right panels show pruned synthetic (forking) networks exhibiting similar properties with perfused (red) and hypoperfused (blue) vessels.
Fig 2.
Representative vascular architectures from our experiments [15].
A single dose of 15 Gy X-rays was delivered to a tumour and changes to its vascular structure were monitored over the course of 4 days (reproduced from [50]). In the top row, endothelial cells are in cyan, while perfused vessels (qDot705) are in red. Every panel is a 2D representation of a 3D image in the form of a Z-stack approximately 300 μm tall. The depth of field for a single Z-slice was 2 μm. The middle and the bottom row show images of endothelial cells and perfused vessels, respectively. The scale bar corresponds to 100 μm.
Fig 3.
The first four days post radiotherapy (Day 0 is the day of irradiation).
Time evolution of (A) vessel count NV and (B) perfusion fraction for the 7 tumours studied in [15].
Table 1.
Key geometric and topological characteristics of tumour vasculatures on Day 0.
Note that all lengths are in μm, all geometric resistances (total, mean, and per loop) in μm−3, and all resistances with viscosity in cP⋅μm−3, where cP (centipoise) is a unit of dynamic viscosity equal to mPa⋅s).
Table 2.
Perfusion fractions on Day 0 and the final day of the pruning phase, the pruning-induced perfusion difference, and its relative counterpart for six studied tumours.
Tumours are ordered based on the relative change in PF as measured by .
Fig 4.
Flow chart summarising our model components.
Simulations were carried out using Microvessel Chaste [62, 63].
Table 3.
Heterogeneity in both hierarchical and non-hierarchical networks is modulated by a single parameter.
Fig 5.
Blood flows from left to right through a single inlet and a single outlet in the forking network.
Li and Vi denote the length and y-extent of a vessel in generation i, respectively. Note that our implementation of network heterogeneity results in vessels at the top of the network being the thickest in their generation and vessels at the bottom being the thinnest.
Fig 6.
Changes to the architecture of a forking network as (A) 0 vessels, (B) 50 vessels, (C) 100 vessels, and (D) 200 vessels are pruned.
Fig 7.
The two mechanisms of increase.
When a forking network (α = 1.1, μm) is pruned, perfusion (
) can increase, shown in (A), via Mechanism I when blood flow is rerouted from pruned vessels to other vessels, as seen in (B) where the number of perfused vessels increases and the number of hypoperfused vessels decreases. In contrast, perfusion (
) can also increase, shown in (A), via Mechanism II when hypoperfused vessels are pruned without any net increase in the number of perfused vessels, as evidenced in (B) by the constant number of perfused vessels.
Fig 8.
Flow is rerouted in this section of a forking network (α = 1.1, μm) when a single vessel is pruned (original position marked with an asterisk) between (A) and (B) and the flow rate increases in four hypoperfused vessels (blue) to the extent that they become perfused (red).
Fig 9.
Panel (A) shows the perfusion response of a single forking network (α = 1.2, μm). Considering the central portion of this network, panels (B)–(G) present the progression of flow distributions in the process of pruning, noting that the maximum of the colourmap is set equal to the perfusion threshold (i.e., vessels that are not dark red are hypoperfused). Having pruned (B) the 20 thinnest vessels, the network contains pairs of daughter vessels which experience flow rates slightly below the perfusion threshold (Qmin = 3 × 10−12m3s−1). Pruning one vessel of such a pair is likely to result not only in (C) fewer hypoperfused vessels but also in more perfused vessels due to flow rerouting. Between (D) and (E) and between (F) and (G), we primarily prune blunt ends which increases the perfusion fraction (
) via Mechanism 2. Between (E) and (F),
dramatically decreases because perfused paths are disrupted.
Fig 10.
The impact of the mean diameter and the diameter heterogeneity on the perfusion response.
Series of plots (A–I) showing, for the forking network, the influence of the mean diameter () and the diameter heterogeneity (α) of the unpruned network on how perfusion (
) changes during pruning. An increase in the number of perfused vessels is evidence of flow being rerouted into vessels that were previously hypoperfused. An increase in
without an increase in perfused vessels is evidence of Mechanism 2. Regions in which Mechanism 1 is primarily active are highlighted in grey.
Fig 11.
Perfusion response as measured by .
The initial perfusion fraction () depends on the vascular architecture (A–C). Note that, for a given diameter heterogeneity (α), the perfusion fraction (
) of all mean diameters (
) converges during the later stages of pruning, when the networks only contain thicker (low-resistance) vessels. Panel (D) shows the pruned architecture at a stage common to all values of
for α = 1.2 (147 pruned vessels). Here, we observe that all remaining vessels are either perfused or have blunt ends (no flow). No flow rerouting can occur in the hypoperfused vessels. Thus, subsequent pruning produces the same changes in
, regardless of the initial
. Note that the vessel colour indicates the local flow rate, with the range of the colourbar adjusted so that perfused vessels are dark red and vessels with little or no flow are dark blue.
Fig 12.
Since vessel conductivity depends on the fourth power of the mean network diameter (see Eq (3)), thinner forking networks offer much greater resistance () to blood flow regardless of α (A–C).
decreases monotonically during pruning because vessels are removed in order of increasing diameter.
Fig 13.
A drop in the number of loops per vessel () precedes a positive change in the perfusion fraction (
) in the forking networks (
μm) across all values of α (A–C). Minor differences in
arise between heterogeneities because α changes the order of pruning by virtue of changing the diameters of vessels.
Table 4.
The classification of forking networks into groups A and B varies based on the sampling point, i.e., how many vessels have been pruned when is measured.
Cells have been shaded green or red to represent positive or negative values of .
Fig 14.
The resistance per loop () follows trends inverse to those observed for the loops per vessel (
) for all values of α (A–C), but can also distinguish between networks of varying mean diameter.