Obstructions in Vascular Networks: Relation Between Network Morphology and Blood Supply

<div><p>We relate vascular network structure to hemodynamics after vessel obstructions. We consider tree-like networks with a viscoelastic fluid with the rheological characteristics of blood. We analyze the network hemodynamic response, which is a function of the frequencies involved in the driving, and a measurement of the resistance to flow. This response function allows the study of the hemodynamics of the system, without the knowledge of a particular pressure gradient. We find analytical expressions for the network response, which explicitly show the roles played by the network structure, the degree of obstruction, and the geometrical place in which obstructions occur. Notably, we find that the sequence of resistances of the network without occlusions strongly determines the tendencies that the response function has with the anatomical place where obstructions are located. We identify anatomical sites in a network that are critical for its overall capacity to supply blood to a tissue after obstructions. We demonstrate that relatively small obstructions in such critical sites are able to cause a much larger decrease on flow than larger obstructions placed in non-critical sites. Our results indicate that, to a large extent, the response of the network is determined locally. That is, it depends on the structure that the vasculature has around the place where occlusions are found. This result is manifest in a network that follows Murray’s law, which is in reasonable agreement with several mammalian vasculatures. For this one, occlusions in early generation vessels have a radically different effect than occlusions in late generation vessels occluding the same percentage of area available to flow. This locality implies that whenever there is a tissue irrigated by a tree-like <i>in vivo</i> vasculature, our model is able to interpret how important obstructions are for the irrigation of such tissue.</p></div

Model for obstructions in a tree-like network.

<p>A: Illustration of a network with obstructions at level <i>n</i>, indicated by crosses. B: Electrical analogy for a <i>N</i>-level network with occlusions at level <i>n</i>.</p

Analytical approximation and numerical solution of the response for obstructed networks with a resistance’s jump.

<p>Analytical approximation and numerical solution of the quantity ln(Re[<i>χ</i>_<i>un</i>−<i>χ</i>]) for 20-level networks with a jump in resistance between levels 10−11 obstructed at level <i>n</i> as described in the text. A: The vessels have the typical dimensions of the dog arterioles for <i>n</i> ≤ <i>k</i> and of capillaries for <i>n</i> > <i>k</i>. B: The vessels have the typical dimensions of the dog terminal branches for <i>n</i> ≤ <i>k</i> and of arterioles for <i>n</i> > <i>k</i>.</p

Dynamic response for obstructed networks with equal vessels.

<p>Dynamic response for an 11-level network as a function of the level <i>n</i> in which obstructions occur. It is important to note that each point in this figure corresponds to a different network since we obstruct only one level at a time. The normalization is done with the network response without occlusions. The effect of the obstruction is more dramatic in the outer levels of the network. In this calculation, the vessels have the typical dimension of the dog arterioles (<i>r</i> = 1 × 10⁻⁵ m and <i>l</i> = 2 × 10⁻³ m).</p

Time-dependent flow for obstructed networks with a jump in resistance.

<p>Time-dependent flow for a network obstructed by 90% in area at level 3, for a network obstructed by 45% in area at level 11 and for a network without obstructions as reference. The network used was the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0128111#pone.0128111.g008" target="_blank">Fig 8</a>. The total pressure drop was set to 600 Pa.</p

Time-dependent flow for obstructed networks with equal vessels.

<p>Blood flow for an 11-level network with obstructions of 90% at levels 3, 8 and with no obstruction (reference). The sharp decrease in flow after obstructions at the outer level is clear. The network used was the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0128111#pone.0128111.g002" target="_blank">Fig 2</a>. The total pressure drop was set to 110 Pa.</p

Flow in single vessels of an obstructed network with equal vessels.

<p>Flow in single vessels in logarithmic scale as a function of the level they belong to for a network constituted by 11 levels and with obstructions of 60% in area at level 3. The curves shown are: the flow in the unobstructed path, the flow in the obstructed path and a reference curve for the flow in a path of an unobstructed network. For language clarification see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0128111#pone.0128111.g001" target="_blank">Fig 1</a>. Even though the total flow decreases with the obstructions, the flow in the non-obstructed vessels increases. The network used was the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0128111#pone.0128111.g002" target="_blank">Fig 2</a>. The pressure drop was set to 110 Pa.</p

Dynamic response for a network with vessel radii that follow Murray’s law.

<p>A: Real and imaginary parts of the response of the network (in <i>m</i>⁴) as a function of the level <i>n</i> at which obstructions occur. B: Real and imaginary parts of the ratio of two sequential resistances </p><p></p><p></p><p></p><p><mi>a</mi><mi>i</mi></p><mo>=</mo><p></p><p><mi>R</mi><mi>i</mi></p><p><mi>R</mi></p><p><mi>i</mi><mo>−</mo><mn>1</mn></p><p></p><p></p><p></p><p></p><p></p> as a function of the level <i>i</i> of the underlying network. Note that we use the subindex <i>i</i>, whenever we refer to a property of the underlying network, we use the subindex <i>n</i> whenever we refer to the response of the whole network when obstructions occur at level <i>n</i>.<p></p