Obstructions in Vascular Networks. Critical vs Non-critical Topological Sites for Blood Supply

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, that 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 in-vivo vasculature, our model is able to interpret how important obstructions are for the irrigation of such tissue.


Introduction
Occlusion of tubes has always represented a problem. From engines and filters to 2 arteries and bronchia, we can find countless systems where a reduction of the fluid 3 flow in a particular site due to the presence of an obstacle, results in the partial or 4 total failure of a process. Occlusion of bio-tubes in the human body represent an 5 The enormous amount of work involved in such computations is necessary when 23 one wants to describe specific zones of a vasculature, and to answer detailed questions 24 regarding flow profiles around obstacles, stenosis, bypasses, bifurcations, or flow in the 25 aortic arch. These complex computations are able to predict how the waveforms of 26 pressure and flow change in certain vessels due to obstructions, stenosis or vessel 27 suppression at particular sites [4,8,13]. Sophisticated models are also very interesting 28 from a theoretical and computational point of view. However, they involve too many 29 variables to allow for the derivation of analytical expressions when one is interested in 30 knowing the effect that obstructions have on the overall flow throughout an entire 31 network. Analytical expressions might be very powerful and are potentially useful 32 clinically, where a reduced number of parameters is often appreciated. 33 Knowledge about the structure of vascular networks, is key to predict the flow after 34 alterations in the vasculature, e.g. after the growth or introduction of new 35 vessels [14,15] or after the partial occlusion of vessels in the system [8,11,12,16]. The 36 correspondence between local structural network information and global flow through 37 a network after vascular alteration, was put forward in the work of Flores et al [14]; 38 the simplicity of the model allowed for analytical expressions that in turn lead to 39 conclusions not attainable otherwise. For instance, it was demonstrated that the  The purpose of the present study is to relate the basic, generic characteristics of an 47 arterial vasculature with the flow that goes through it after anatomical variations 48 caused by obstructions or vessel suppression occur. We deliberately keep a 49 reductionist approach in order to obtain analytical expressions for the system response 50 in which the roles played by the network structure, the degree of obstruction, and the 51 geometrical place where obstructions occur, can be clearly identified. 52 We study flow in three types of networks: one constituted by identical vessels, a 53 second one in which radii are given by Murray's law, and a third case in which large 54 changes in resistance exist within the network. We show how the underlying network 55 can lead to radically different behaviors of the hemodynamic response and identify 56 structural features present in tree-like vasculatures that are critical for the overall 57 capacity of the network to supply blood after obstructions. We demonstrate that our 58 results are local in the sense that they depend on the network structure around the 59 place where obstructions occur. This implies that whenever there is a tree-like network 60 in an in-vivo vasculature, our model is able to interpret the effect that an obstruction 61 has on flow.

63
Recently, a model has been introduced in order to study viscoelastic flow in a network 64 of tubes [17]. This model consists of a tree-like network in which rigid vessels bifurcate 65 always into identical vessels giving rise to identical branches of the network. At each 66 bifurcation step, the possibility of changes in the cross sectional area and the length of 67 the vessels is allowed. Each level (or generation) of the network is constituted by  The model considers a linear viscoelastic fluid with the rheological characteristics 73 of blood [18] in a range of shear rates where there is no shear thinning, and analyzes 74 the network hemodynamic response to a time-dependent periodic pressure gradient. A 75 Maxwell fluid [19] is used for this study, but the formalism can be easily generalized to 76 consider any linear viscoelastic fluid [20]. By considering mass conservation, and 77 assuming that the total pressure drop is the sum of individual pressure drops, the 78 dynamic response of the network, χ(ω), is written in terms of the dynamic 79 permeability of individual vessels K i (ω) as The sum is over the network levels, A i and l i are respectively the cross sectional 81 area and the length of the vessels at the i-th level, L and N are the total length of the 82 network and the total number of levels, respectively. The dynamic permeability for a 83 vessel of radius r i is where J 0 and J 1 are Bessel functions of 84 order zero and one, respectively, and , where ρ, t r , and η are the 85 density, relaxation time and the fluid viscosity respectively. In order to apply Eq. (1) 86 to a particular network of vessels, the network geometrical characteristics, namely, the 87 number of levels -that determine the number of vessels-, lengths and radii, are 88 required.

89
The network hemodynamic response relates viscoelastic flow and pressure drop in 90 frequency domain [14,17]. In order to have it explicitly in time domain one needs to 91 specify a time dependent pressure gradient. As the equations are linear, we can obtain 92 the fluid response to any time-dependent pressure gradient as a linear superposition of 93 sinusoidal modes. For a single-mode time-dependent pressure drop ∆p = ∆p 0 cos(ω 0 t), 94 the volumetric flow as a function of time is given by where the real and the imaginary parts of the response function χ (Eq. (1)) give the 96 flow in-phase and out-of-phase with the pressure gradient, respectively [21]. Eq. to study the hemodynamics of the system, without the requirement of considering a 103 particular pressure gradient. Our results are presented at 1.5 Hz, which is the resting 104 heart rate of the dog [22,23]. At such low frequencies the network response (and total 105 flow) is almost indistinguishable from the steady-state regime where the response is 106 real. However, we keep the formalism as general as possible to make it applicable 107 when external frequencies are imposed [21]. We use parameters for normal dog 108 blood [24], ρ = 1050 kg/m 3 , η = 1.5 × 10 −2 kg/(m s) and assume that the relaxation 109 time is similar to the one reported for human blood: t r = 1 × 10 −3 s [18,25].

126
The total resistance for an N -level network obstructed at level n is given by  Although we will focus on the overall behavior of the network, the analytical 131 approach can predict the local flow at each of the network vessels. We will 132 characterize the impact of vessel obstruction on tree-like networks by focusing on two 133 different types of paths. We will consider unobstructed paths, those which cross the 134 network without moving along any obstructed vessel, and obstructed paths, when an 135 obstructed vessel is crossed at some point in the network.

136
Obstructions in a network with equal vessels 137 We first treat the case of a network in which all vessels have approximately the same 138 radius, which is the case of several networks at the arteriole level, and approximate it 139 with a bifurcating network of equal vessels with resistance R 1 . We find that in this 140 case, the effect caused by occlusions is relatively small when it happens in the inner 141 vessels, and it is relatively large when it happens in the outer vessels. The network 142 response increases monotonically with the level number n in which occlusions occur 143 (see Fig. 2).

144
Physically, this implies that for a healthy tissue irrigated by a tree-like network, 145 occlusions are more dangerous when they occur in vessels of early generations since 146 blood supply is dramatically decreased, as illustrated in Fig. 3.

147
A mathematical analysis similar to the one presented in [14] for anastomosis, 148 allows us to have an analytical approximated expression for the dynamic response of a 149 network of equal vessels, χ, obstructed by a fraction of area f at level n, In this expression χ un is the response of the unobstructed network. The last term in  The theoretical prediction provides insight in the impact that the degree of 159 obstruction and its location inside the network has in its global response; in particular 160 the expression derived clearly shows that the change in the network's response due to 161 the presence of obstructions is highly determined by the structure of the unobstructed 162 network.  Assuming that the vascular system evolved to minimize the power required to 181 maintain and circulate blood [27], Murray derived, in 1926, the relationship known as 182 Murray's law. This one relates the parent radius, r p , and the two daughters vessels 183 radii, r d1 , r d2 , before and after a bifurcation, as According to an extensive study on the validity of Murray's law [28] and a review on 185 vascular flow of reference [29], physiological studies showed that, barring some anomalies, a large part of the mammalian vasculatures have reasonable agreement 187 with Eq. (5). According to [29], there is also a considerable mass of literature 188 comparing physiological studies in animals other than mammals, and even in 189 plants [30][31][32][33], that show good agreement with Murray's law. 190 We therefore consider Murray's law as an example of physiological relevance, in 191 which our analytical results illustrate how to explain the different tendencies in the 192 dynamic response in different sections of the network.

193
For our studies, we consider symmetrical branching, so in this case, radii of 194 subsequent levels are given by For lengths, we consider a power law decay with parameters that match the actual between this two quantities which will determine the value of a. It turns out that for a 206 bifurcating network, the response will be qualitatively different whenever a is smaller 207 or larger than 2, as we will see below. In order to gain insight into these results, we present analytical approximations for 215 networks in which the ratio of subsequent resistances is less than two or larger than 216 two. These ones agree well with numerical results whenever the real part of a is 217 considerably larger than its imaginary part. They are given by: for a < 2, which reduces to Eq. (4) when a = 1, and for a > 2. For values of a < 2, just as in the case of equal vessels (that has a = 1) presented in 229 the previous section, occlusions are more dangerous when they occur in vessels of early 230 generations since blood supply is dramatically decreased. On the other hand, when 231 a > 2, occlusions are more dangerous when they occur in vessels of late generations 232 since blood supply is smaller than for instance at vessels around the middle of the 233 network.

234
For the network presented here, that follows Murray's law for radii and a power law 235 for vessel lengths, two radically different behaviors are observed, one for external

Obstructions in a network with a jump in resistance 241
Finally, we consider a network for which vessels have a sudden jump in resistance.

242
This corresponds to physiological conditions in cases where vessels of small radii 243 branch from vessels of large radii. In particular, we consider a network for which we 244 have a resistance R 1 for i ≤ k and a resistance R 2 for i > k, where k is a level close to 245 the middle of the network. Therefore, the network has a jump in resistance a ≡ R2 R1 246 between levels k and k + 1. We obtain analytical approximations for real a that could 247 be useful when one analyzes jumps in resistance in the arterial tree of mammals. The 248 first case holds in the limit when (a − 1) ( 1   2 ) k−n << 1 for n ≤ k, and for a given a it is 249 better the farther away from the jump obstructions are. In this case, we find that the 250 network response is given by and 252 ln(χ un − χ) ≈ ln The right hand side of these expressions clearly highlights that the geometry of the 253 underlying network, the fraction of obstructed vessels, and the geometrical place, n, 254 where the obstruction is located contribute additively to the change in the network 255 response. As examples of physiological relevance for which these analytical expressions 256 could be useful, we find typical resistance jumps of the dog circulatory system [14] 257 between main arterial branches and terminal branches, and between arterioles and

263
We have generalized these results for the case when a vasculature has several 264 jumps, the approximated analytical expression is given by + n ln  We have worked out analytical expressions in the limit of larger resistance jumps, 279 to be precise in the limit where, (a − 1) can then be expressed as and 282 ln(χ un − χ) ≈ ln These expressions again show that the geometry of the unobstructed network, the for obstructions that block 90% of the area at level 3 and obstructions that block only 304 45% of the area at level 11, as displayed in Fig. 9 where we also plot the flow for an 305 unobstructed network as a reference. The figure clearly illustrates that relatively small 306 obstructions in critical topological sites cause a much larger decrease on flow than 307 larger obstructions in non-critical sites.

308
Despite the tendency shown in Fig. 8A that seems to imply that the larger the worth noticing that, as depicted in Fig. 8A, a minimum in the response can be 315 observed when occlusions take place at the same position that the resistance jump.

317
In this work we relate network structure to hemodynamics after occlusions. We study  For networks in which jumps in resistance between subsequent levels exist, we have 331 identified the sites of the jumps as critical for the overall capacity of the network to 332 supply blood to a tissue. We have also demonstrated that relatively small obstructions 333 in these critical topological sites, cause a much larger decrease on flow than larger 334 obstructions in non-critical sites. By simple observation of the structure of a vascular 335 network, these key sites could be readily identified and monitored in vivo. 336 We have derived analytical expressions for the dynamic response of the network to the effect of the addition of anastomotic vessels [14]. When in the former case  For our model, we have considered that vessels are rigid. However, real vessels are 360 elastic tubes. Important insight could be gained from the inclusion of elastic effects in 361 the model, especially for large arteries, where elastic effects are very important.

362
Regional tissue metabolism, such as the myogenic effect in arteries, can be considered, 363 PLOS 9/18 as was done in reference [14], where it was shown that the impact that is has on flow is 364 large, but tendencies with the geometrical place where anatomical variations occur, 365 are qualitatively unaltered.

366
A validation of our model with an in-vivo biological system is currently not possible, 367 since it would require data of flow measurements after a systematic variation of the        Fig. 1. 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 Fig.2. The pressure drop was set to 110 Pa.

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A B Figure 6. Dynamic response for a network with vessel radii that follow Murray's law. A: Real and imaginary parts of the response of the network (in m 4 ) as a function of the level n at which obstructions occur. B: Real and imaginary parts of the ratio of two sequential resistances a i = Ri Ri−1 as a function of the level i of the underlying network. Note that we use the subindex i, whenever we refer to a property of the underlying network, we use the subindex n whenever we refer to the response of the whole network when obstructions occur at level n.  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 Fig. 8. The total pressure drop was set to 600 Pa.