Shock formation and non-linear dispersion in a microvascular capillary network

Pop, Serban R.; Richardson, Giles; Waters, Sarah L.; Jensen, Oliver E.

doi:10.1093/imammb/dqm007

Cited by 15 publications

(20 citation statements)

References 32 publications

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“…These solid EC cords initially develop a lumen, forming blind-ended vessels; the tips of these vessels may join with other segments, establishing blood flow through the loop, as recently shown in the zebrafish gastrula (Vogeli, 2006). The transport of oxygen and nutrients though these newly formed vessels is mediated by well-characterised non-linear rheological effects (such as the plasma-skimming, Fahraeus and FahraeusLindqvist effects (Popel and Johnson, 2005)), which influence both the spatial and temporal haematocrit distribution in microvascular networks (Pop, 2007). In particular, the phase-separation (plasmaskimming) effect acting at divergent bifurcations (Pries et al, 1996), in which plasma is drawn down low-flux branches in preference to suspended red blood cells, is likely to influence strongly the flow environment in BEVs.…”

Section: Introductionmentioning

confidence: 92%

Experimental and theoretical modelling of blind-ended vessels within a developing angiogenic plexus

Guerreiro-Lucas

Pop

Machado

et al. 2008

Microvascular Research

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Angiogenic sprouts at the leading edge of an expanding vascular plexus are recognised as major regulators of the structure of the developing network. Early in sprout development, a vascular lumen is often evident which communicates with the parent vessel while the distal tip is blind-ended. Here we describe the temporal evolution of blind-ended vessels (BEVs) in a small wound made in the panniculus carnosus muscle of a mouse viewed in a dorsal skin-fold window-chamber model with intra-vital microscopy during the most active period of angiogenesis (days 5-8 after injury). Although these structures have been mentioned anecdotally in previous studies, we observed BEVs to be frequent, albeit transient, features of plexus formation. Plasma leakage into the surrounding extracellular matrix occurring from these immature conduits could play an important role in preparing hypoxic tissue for vascular invasion. Although sprout growth is likely to be regulated by its flow environment, the parameters regulating flow into and through BEVs have not been characterised in situ. Longitudinal data from individual animals show that the number of BEVs filled with plasma alone peaks at day 7, when they can exceed 150 μm in length. Additionally, BEVs greater than 40 μm in length are more likely to be filled with stationary erythrocytes than with plasma alone. Using a mathematical model, we show how the flux of 150kD fluorinated (FITC-) dextran through an individual plasma-filled BEV is related to its geometry being determined primarily by its surface area; by fitting theoretical intensity values to experimental data we assess the permeability of the vessel to FITC-dextran. Plasma skimming provides a mechanistic explanation for the observation that BEVs with larger surface area are more likely to recruit erythrocytes.

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Section: Introductionmentioning

confidence: 92%

Experimental and theoretical modelling of blind-ended vessels within a developing angiogenic plexus

Guerreiro-Lucas

Pop

Machado

et al. 2008

Microvascular Research

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“…These constitutive equations are based upon the hypothesis that the flow occurs within high-aspect ratio tube-like channels. Furthermore, it will be assumed (following Pop et al 2007) that the particle suspension can be described by a continuous void fraction θ , so that the fluid has continuous properties everywhere. Let us consider the longitudinal velocity component u (r, z, t) and the associated lubricated pressure p(z, t), at radial and longitudinal positions r and z, in the cylindrical coordinate systems (r, φ, z).…”

Section: Lubrication Approximationmentioning

confidence: 99%

“…According to Pop et al (2007) and their associated notations, we consider the local relative volume of red blood cell θ(r, z, t) at location r and z, for which the dynamical behavior is related to the mass conservation laws given by the equation:…”

Section: Lubrication Approximationmentioning

confidence: 99%

“…Some efforts have been made to better quantify the origin of the phase separation phenomenon either from the analysis of numerical simulations (Enden and Popel 1994) or detailed fluid-mechanical interactions (El-Kareh and Secomb 2000) arising at the level of a single bifurcation. More recent investigations have addressed the influence of separation effect on the dynamical behavior of very simple networks (Carr and Lacoin 2000;Carr et al 2005;Pop et al 2007). Nevertheless, all the above mentioned contributions always consider that phase separation results from local mechanisms, and as such, should be modeled by local laws.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A New Approach to Model Confined Suspensions Flows in Complex Networks: Application to Blood Flow

2009

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The modeling of blood flows confined in micro-channels or micro-capillary beds depends on the interactions between the cell-phase, plasma and the complex geometry of the network. In the case of capillaries or channels having a high aspect ratio (their longitudinal size is much larger than their transverse one), this modeling is much simplified from the use of a continuous description of fluid viscosity as previously proposed in the literature. Phase separation or plasma skimming effect is a supplementary mechanism responsible for the relative distribution of the red blood cell's volume density in each branch of a given bifurcation. Different models have already been proposed to connect this effect to the various hydrodynamics and geometrical parameters at each bifurcation. We discuss the advantages and drawbacks of these models and compare them to an alternative approach for modeling phase distribution in complex channels networks. The main novelty of this new formulation is to show that albeit all the previous approaches seek for a local origin of the phase segregation phenomenon, it can arise from a global non-local and nonlinear structuration of the flow inside the network. This new approach describes how elementary conservation laws are sufficient principles (rather than the complex parametric models previously proposed) to provide non local phase separation. Spatial variations of the hematocrit field thus result from the topological complexity of the network as well as nonlinearities arising from solving a new free boundary problem associated with the flux and mass conservation. This network model approach could apply to model blood flow distribution either on artificial micro-models, micro-fluidic networks, or realistic reconstruction of biological micro-vascular networks.

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“…This omission simplifies the equations and allows us to set the divergence of all velocities equal to zero because of the incompressibility of blood. Pop et al 37 have included the Fahraeus effect in their computation on flow through microvessel networks. This inclusion changes the wave speed in the convection equations and leads to the formation of shock waves of red cell concentration as blood flows through the network.…”

Section: Discussionmentioning

confidence: 99%

Blood flow in microvascular networks: A study in nonlinear biology

Geddes

Carr

et al. 2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Plasma skimming and the Fahraeus-Lindqvist effect are well-known phenomena in blood rheology. By combining these peculiarities of blood flow in the microcirculation with simple topological models of microvascular networks, we have uncovered interesting nonlinear behavior regarding blood flow in networks. Nonlinearity manifests itself in the existence of multiple steady states. This is due to the nonlinear dependence of viscosity on blood cell concentration. Nonlinearity also appears in the form of spontaneous oscillations in limit cycles. These limit cycles arise from the fact that the physics of blood flow can be modeled in terms of state dependent delay equations with multiple interacting delay times. In this paper we extend our previous work on blood flow in a simple two node network and begin to explore how topological complexity influences the dynamics of network blood flow. In addition we present initial evidence that the nonlinear phenomena predicted by our model are observed experimentally. © 2010 American Institute of Physics. ͓doi:10.1063/1.3530122͔Nonlinear phenomena are very common in nature. Here we explore one example from biology, the flow of blood in small blood vessel networks. Nonlinearity is revealed in this case by the existence of multiple steady state solutions and by the evolution of spontaneous fluctuations in network flow. These nonlinear phenomena are not dependent on any biological control mechanisms but are solely due to the interaction of well-known blood rheological properties and network geometry. We examine these interactions through analysis and numerical computation. We also present our first in vitro experimental data on spontaneous oscillations in blood flow in this paper.

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Shock formation and non-linear dispersion in a microvascular capillary network

Cited by 15 publications

References 32 publications

Experimental and theoretical modelling of blind-ended vessels within a developing angiogenic plexus

Experimental and theoretical modelling of blind-ended vessels within a developing angiogenic plexus

A New Approach to Model Confined Suspensions Flows in Complex Networks: Application to Blood Flow

Blood flow in microvascular networks: A study in nonlinear biology

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