Flow and anastomosis in vascular networks.

We analyze the effect that the geometrical place of anastomosis in the circulatory tree has on blood flow. We introduce an idealized model that consists of a symmetric network for the arterial and venous vascular trees. We consider that the network contains a viscoelastic fluid with the rheological characteristics of blood, and analyze the network hydrodynamic response to a time-dependent periodic pressure gradient. This response is a measurement of the resistance to flow: the larger the response, the smaller the resistance to flow. We find that for networks whose vessels have the same radius and length, the outer the level of the branching tree in which anastomosis occurs, the larger the network response. Moreover, when anastomosis is incorporated in the form of bypasses that bridge vessels at different bifurcation levels, the further apart are the levels bridged by the bypass, the larger the response is. Furthermore, we apply the model to the available information for the dog circulatory system and find that the effect that anastomosis causes at different bifurcation levels is strongly determined by the structure of the underlying network without anastomosis. We rationalize our results by introducing two idealized models and approximated analytical expressions that allow us to argue that, to a large extent, the response of the network with anastomosis is determined locally. We have also considered the influence of the myogenic effect. This one has a large quantitative impact on the network response. However, the qualitative behavior of the network response with anastomosis is the same with or without consideration of the myogenic effect. That is, it depends on the structure that the underlying vessel network has in a small neighborhood around the place where anastomosis occurs. This implies that whenever there is an underlying tree-like network in an in vivo vasculature, our model is able to interpret the anastomotic effect.

Estimating Central Pulse Pressure From Blood Flow by Identifying the Main Physical Determinants of Pulse Pressure Amplification.

Several studies suggest that central (aortic) blood pressure (cBP) is a better marker of cardiovascular disease risk than peripheral blood pressure (pBP). The morphology of the pBP wave, usually assessed non-invasively in the arm, differs significantly from the cBP wave, whose direct measurement is highly invasive. In particular, pulse pressure, PP (the amplitude of the pressure wave), increases from central to peripheral arteries, leading to the so-called pulse pressure amplification (ΔPP). The main purpose of this study was to develop a methodology for estimating central PP (cPP) from non-invasive measurements of aortic flow and peripheral PP. Our novel approach is based on a comprehensive understanding of the main cardiovascular properties that determine ΔPP along the aortic-brachial arterial path, namely brachial flow wave morphology in late systole, and vessel radius and distance along this arterial path. This understanding was achieved by using a blood flow model which allows for workable analytical solutions in the frequency domain that can be decoupled and simplified for each arterial segment. Results show the ability of our methodology to (i) capture changes in cPP and ΔPP produced by variations in cardiovascular properties and (ii) estimate cPP with mean differences smaller than 3.3 ± 2.8 mmHg on in silico data for different age groups (25-75 years old) and 5.1 ± 6.9 mmHg on in vivo data for normotensive and hypertensive subjects. Our approach could improve cardiovascular function assessment in clinical cohorts for which aortic flow wave data is available.

Resonances in the response of fluidic networks inherent to the cooperation between elasticity and bifurcations.

A global response function (GRF) of an elastic network is introduced as a generalization of the response function (RF) of a rigid network, relating the average flow along the network with the pressure difference at its extremes. The GRF can be used to explore the frequency behaviour of a fluid confined in a tree-like symmetric elastic network in which vessels bifurcate into identical vessels. We study such dynamic response for elastic vessel networks containing viscous fluids. We find that the bifurcation structure, inherent to tree-like networks, qualitatively changes the dynamic response of a single elastic vessel, and gives resonances at certain frequencies. This implies that the average flow throughout the network could be enhanced if the pulsatile forcing at the network's inlet were imposed at the resonant frequencies. The resonant behaviour comes from the cooperation between the bifurcation structure and the elasticity of the network, since the GRF has no resonances either for a single elastic vessel or for a rigid network. We have found that resonances shift to high frequencies as the system becomes more rigid. We have studied two different symmetric tree-like network morphologies and found that, while many features are independent of network morphology, particular details of the response are morphology dependent. Our results could have applications to some biophysical networks, for which the morphology could be approximated to a tree-like symmetric structure and a constant pressure at the outlet. The GRF for these networks is a characteristic of the system fluid-network, being independent of the dynamic flow (or pressure) at the network's inlet. It might therefore represent a good quantity to differentiate healthy vasculatures from those with a medical condition. Our results could also be experimentally relevant in the design of networks engraved in microdevices, since the limit of the rigid case is almost impossible to attain with the materials used in microfluidics and the condition of constant pressure at the outlet is often given by the atmospheric pressure.

A Novel Analytical Approach to Pulsatile Blood Flow in the Arterial Network.

Haemodynamic simulations using one-dimensional (1-D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. We propose a novel linear 1-D dynamical theory of blood flow in networks of flexible vessels that is based on a generalized Darcy's model and for which a full analytical solution exists in frequency domain. We assess the accuracy of this formulation in a series of benchmark test cases for which computational 1-D and 3-D solutions are available. Accordingly, we calculate blood flow and pressure waves, and velocity profiles in the human common carotid artery, upper thoracic aorta, aortic bifurcation, and a 20-artery model of the aorta and its larger branches. Our analytical solution is in good agreement with the available solutions and reproduces the main features of pulse waveforms in networks of large arteries under normal physiological conditions. Our model reduces computational time and provides a new approach for studying arterial pulse wave mechanics; e.g.,  the analyticity of our model allows for a direct identification of the role played by physical properties of the cardiovascular system on the pressure waves.

Obstructions in Vascular Networks: Relation Between Network Morphology and 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, 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 in vivo vasculature, our model is able to interpret how important obstructions are for the irrigation of such tissue.

Tumor angiogenesis and vascular patterning: a mathematical model.

Understanding tumor induced angiogenesis is a challenging problem with important consequences for diagnosis and treatment of cancer. Recently, strong evidences suggest the dual role of endothelial cells on the migrating tips and on the proliferating body of blood vessels, in consonance with further events behind lumen formation and vascular patterning. In this paper we present a multi-scale phase-field model that combines the benefits of continuum physics description and the capability of tracking individual cells. The model allows us to discuss the role of the endothelial cells' chemotactic response and proliferation rate as key factors that tailor the neovascular network. Importantly, we also test the predictions of our theoretical model against relevant experimental approaches in mice that displayed distinctive vascular patterns. The model reproduces the in vivo patterns of newly formed vascular networks, providing quantitative and qualitative results for branch density and vessel diameter on the order of the ones measured experimentally in mouse retinas. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of different parameters in this process, hence underlining the necessary collaboration between mathematical modeling, in vivo imaging and molecular biology techniques to improve current diagnostic and therapeutic tools.