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Robust Increase in Supply by Vessel Dilation in Globally Coupled Microvasculature

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Robust Increase in Supply by Vessel Dilation in Globally Coupled Microvasculature Robust increase in supply by vessel dilation in globally coupled microvasculature Felix J. Meigel,1 Peter Cha,2 Michael P. Brenner,2 and Karen Alim1, 3 1Max Planck Institute for Dynamics and Self-Organization, 37077 G¨ottingen,Germany 2John A. Paulson School of Engineering and Applied Sciences and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138, USA 3 Physik-Department, Technische Universit¨atM¨unchen,85748 Garching, Germany∗ Neuronal activity induces changes in blood flow by locally dilating vessels in the brain microvas- culature. How can the local dilation of a single vessel increase flow-based metabolite supply, given that flows are globally coupled within microvasculature? Solving the supply dynamics for rat brain microvasculature, we find one parameter regime to dominate physiologically. This regime allows for robust increase in supply independent of the position in the network, which we explain analytically. We show that local coupling of vessels promotes spatially correlated increased supply by dilation. Vascular networks pervade all organs of animals tures [21, 22], including solute absorption [23]. The im- and are the paradigm of adaptive transport networks. pact of a dilating vessel is hard to estimate since not only Their self-organized architecture continuously inspires the absorption dynamics on the level of single vessels is the search for their underlying physical principles [1{4] changed, but also solute flux throughout the network is and at the same time serves as a template for design- rerouted since fluid flow and thus solute flux are glob- ing efficient networks in engineering [5]. The blood flow- ally coupled. However, to connect fMRI, which relies ing through vessels transports nutrients, hormones, and on a fluid dynamic signal [9, 24, 25], and the change in metabolites to adjacent tissues. Metabolite exchange pri- blood flow with neuronal activity [7, 11, 26{28], we need marily occurs within the fine vessel meshwork formed by to understand how vessel dilations affect the supply with microvasculature. In the brain, local metabolite demand metabolites. can abruptly rise due to an increase in neural activity [6], In this letter, we present a theoretical model to deter- altering blood flow [7, 8] in the same brain region, observ- mine the change in supply resulting from the dilation of a able in fMRI [9]. During the process of increased neu- single vessel. On the level of an individual vessel, we an- ronal activity, neurons signal their increased demand to alytically identify three regimes, each yielding a different adjacent astrocyte cells, which in turn trigger small ring functional dependence of the overall supply by absorp- muscles surrounding blood vessels to relax [10]. Thus, tion along the vessel wall on vessel geometry, blood flow, neural activity drives local dilation of a vessel [11, 12], and blood flow based solute flux. Numerically analyz- and hence regulates metabolite supply [7, 13]. However, ing supply dynamics in a microvasculature excerpt of a from a fluid dynamics perspective there is a mystery: rat brain supplied from the Kleinfeld laboratory [8], we blood vessels form a highly interconnected network in find that a single regime dominates. This regime has the the microvasculature [8], resulting in a global coupling of important property that dilating a single vessel robustly blood flow. A single dilating vessel can potentially change increases the supply along the dilated vessel independent the metabolite supply in a broad region of the network - of the exact location of the vessel in the network. We ex- and thus the local increase due to dilation is a function plain analytically how a single vessel can buffer the global of specific network topology. Quantitatively, how much coupling of solute fluxes within the network and yield a control over changes in blood-based supply resides in a robust local increase independent of network topology. single dilating vessel? We further discuss how a single dilating vessel impacts Models considering metabolite spread in tissue date the solute flux downstream and thereby induces spatial back more than a hundred years to A. Krogh [14]. correlations in supply increase. Krogh's model estimates the supply pattern in a tissue To understand how a change in flow induces changes in enclosed by vessels assuming that supply is constant on solute flux and supply dynamics, we first focus on a single arXiv:1911.13197v1 [physics.bio-ph] 29 Nov 2019 all vessel walls. Yet, on a larger tissue scale, supply spa- vessel. We assume that the flow is laminar with longi- tially varies along the vasculature since resources sup- tudinal velocity profile U(r) = 2U¯(1 (r=R)2) [29, 30], − plied upstream are not available downstream. Alterna- where U¯ denotes the cross-sectional averaged longitudi- tive models consider vessel-based transport [15], yet only nal flow velocity. The dispersion of soluble molecules of diffusive transport is taken into account. The combined concentration C by the fluid flow within a tubular vessel importance of advection and diffusion for transporting of radius R and length L is then given by solutes in a single tube was discovered by G.I. Taylor @C @C [16, 17], with subsequent work outlining modifications + U(r) = κ 2C; (1) due to solute absorption at the tube boundary [18{20]. @t @z r Yet, there has been much less work capturing the cou- where κ denotes the molecular diffusivity of the solute, pling of advection and diffusion in tubular network struc- and r and z parameterize the radial and longitudinal 2 where Pe = UL/κ¯ is the P´ecletnumber, α = γL, and 103 0 S = κγL=RU¯ measures the ratio of absorption rate to ad- vection rate. Note, that the concentration decays along 2 -0.5 10 the vessel starting from an initial concentration C0 that Vessels itself is determined by the solute flux entering a vessel 1 -1 10 50 % all J0. Also for the solute influx into a vessel advective and ) 90 % 0 diffusive transport contribute, /J 0 -1.5 ( 10 99 % 10 2 κβ 2 β log J = πR C U¯ + = πR C U¯ 1 + : (5) 0 0 L 0 Pe -1 advective 10 -2 We define as supply of a vessel φ the integrated diffusive -2 diffusive 10 -2.5 flux through the entire vessel surface of the cylindrical Pe (advection rate / diffusion rate) vessel, S R >1 10-3 -3 @C 10-2 100 102 φ = κ 2πRdz: (6) S (absorption rate / advection rate) − @r r=R ZS FIG. 1. Supply φ by a single vessel can be partitioned into resulting in, three distinctive regimes as a function of dimensionless pa- 2 ¯ α S UL 1 12SPe + 2 β rameters characterizing flow and absorption, Pe = κ and φ =J0 2 (1 exp ( β)) : (7) S = κγL . Dotted lines indicate separation of regimes. Re- β · α · − − RU¯ 1 + Pe 1 + 4SPe ! maining non-dimensional parameter fixed at α = 0:001. Error ellipsoids contain the annotated percentage of vessels of the For physical intuition on how flow and vessel proper- here considered rat brain microvasculature [8] with physiolog- ties affect supply, we partition the phase space of sup- ical parameters for γ and κ, see main text. ply dynamics spanned by Pe and S into three regimes, keeping α fixed, see Fig. 1. At large values of S 1 and S 1=Pe the solute decays very quickly along the component of the vessel. The soluble molecule is ab- vessel. Here, all solute that flows into the vessel of cross- sorbed at the vessel boundary, following sectional area πR2 is absorbed at the wall, here denoted all-absorbing regime @C κ + κγC(R) = 0; (2) @r β r=R φ J = πR2C U¯ 1 + : (8) all 0 0 ≈ Pe with absorption parameter γ. In analogy to the deriva- tion of Taylor Dispersion [16, 17, 23], we simplify the For a network this implies that after a vessel in this multidimensional diffusion-advection for C = C¯+C~ to an regime, no solute for further absorption downstream of equation for the cross-sectionally averaged concentration this vessel is available, which indeed is physiologically C¯ if the cross-sectional variations of the concentration C~ rare, 1:0% in the rat brain microvasculature considered are much smaller than the averaged concentration itself. here. A second regime occurs at Pe 1=S, Pe S This is true if the time scale to diffuse radially within the where diffusive transport dominates, here denoted diffu- vessel is much shorter than the time scale of advection sive regime. We distinguish a third regime, which we along the vessel, R2/κ L=U¯, if the vessel itself can be denote advective regime where advective transport dom- characterized as a long, slender vessel, R L, and if the inates, defined by S 1 and S Pe. In both cases absorption parameter is small enough to keep a shallow the solute decay is very shallow, β 1 in Eqs. (3), (6), gradient in concentration across the vessel's cross-section resulting in supply independent of flow velocity, except γR 1, which states that the length scale of absorption for the dependence on the initial concentration C0 is much bigger than the vessel radius. All these approx- φ φ 2πRL κγ C : (9) imations are valid for the rat brain microvasculature ex- advective ≈ diffusive ≈ · · 0 ample considered here [8]. With these assumptions, the Yet, note that the reason for the solute decay, i.e. β being concentration profile along the vessel approaches a steady small, arises from entirely different transport dynamics, state over a timescale L=U¯ given by (see the Supplemen- see Fig. 1. This is reflected in the very different relation tal Material S1 for derivation) between initial solute concentration at the start of the vessel C and the solute influx J for the two regimes z 0 0 C¯(z) = C exp β(Pe; S; α) ; (3) (see the Supplemental Material S1 for derivation) 0 − L 2 ¯ 24 Pe 8S α2 J0;advective C0 πR U; (10) ≈ · · β(Pe; S; α) = · α2 1 + + 1 ; (4) 3 Pe 6PeS − 2 48 + S2 r ! J0;diffusive C0 πR κ 2γ: (11) ≈ · · p 3 Hence, under constant solute influx J0 the diffusive and the advective regime show a fundamentally different, yet both non-linear dependence on the vessel radius, 2γκL φ J ; (12) advective ≈ 0 RU¯ p2γL φdiffusive J0 : (13) ≈ pR Based on these results for a single vessel we expect largely varying increase in supply in response to vessel dilation.
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