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Travasso RDM, Coelho-Santos V. Image-based angio-adaptation modelling: a playground to study cerebrovascular development. Front Physiol 2023; 14:1223308. [PMID: 37565149 PMCID: PMC10411953 DOI: 10.3389/fphys.2023.1223308] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2023] [Accepted: 07/12/2023] [Indexed: 08/12/2023] Open
Affiliation(s)
- Rui D. M. Travasso
- Department of Physics, Center for Physics of the University of Coimbra (CFisUC), University of Coimbra, Coimbra, Portugal
- Coimbra Institute for Biomedical Imaging and Translational Research (CIBIT), University of Coimbra, Coimbra, Portugal
| | - Vanessa Coelho-Santos
- Coimbra Institute for Biomedical Imaging and Translational Research (CIBIT), University of Coimbra, Coimbra, Portugal
- Institute for Nuclear Sciences Applied to Health (ICNAS), University of Coimbra, Coimbra, Portugal
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Qiu J, Chen X, Wu D, Zhang X, Cheng D. One-dimensional analysis method of pulsatile blood flow in arterial network for REBOA operations. Comput Biol Med 2023; 159:106898. [PMID: 37062253 DOI: 10.1016/j.compbiomed.2023.106898] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2022] [Revised: 03/07/2023] [Accepted: 04/09/2023] [Indexed: 04/18/2023]
Abstract
Based on the generalized Darcy model, here we develop a linear one-dimensional (1D) composite model to predict the effects of the inserted balloon under REBOA operations on the dynamic characteristics of blood flow in flexible arterial networks. We first consider the effect of the decrease of cardiac output under different degrees of blood loss through employing the fourth-order lumped parameter model of cardiovascular system. Then, the effect of the inserted balloon is included by developing the relation between flow resistance and occlusion ratio with the neural network approach. Finally, the accuracy of the developed 1D composite model for REBOA operations, which can be solved analytically in the frequency domain, is verified by comparing to computational fluid dynamics (CFD) simulations. It is demonstrated that the 1D model is able to reproduce main features of the systemic circulation under balloon occlusion of the aorta during REBOA surgery. The 1D composite model could substantially reduce the computational time, which makes it possible to give the instant prediction of the working parameters during RABOA operations.
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Affiliation(s)
- Jiade Qiu
- College of Chemical Engineering, Beijing University of Chemical Technology, Beijing, 100029, China
| | - Xin Chen
- College of Chemical Engineering, Beijing University of Chemical Technology, Beijing, 100029, China
| | - Dengfeng Wu
- College of Chemical Engineering, Beijing University of Chemical Technology, Beijing, 100029, China.
| | - Xianren Zhang
- College of Chemical Engineering, Beijing University of Chemical Technology, Beijing, 100029, China
| | - Daojian Cheng
- College of Chemical Engineering, Beijing University of Chemical Technology, Beijing, 100029, China.
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Notch signaling and taxis mechanisms regulate early stage angiogenesis: A mathematical and computational model. PLoS Comput Biol 2020; 16:e1006919. [PMID: 31986145 PMCID: PMC7021322 DOI: 10.1371/journal.pcbi.1006919] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2019] [Revised: 02/14/2020] [Accepted: 10/16/2019] [Indexed: 12/20/2022] Open
Abstract
During angiogenesis, new blood vessels sprout and grow from existing ones. This process plays a crucial role in organ development and repair, in wound healing and in numerous pathological processes such as cancer progression or diabetes. Here, we present a mathematical model of early stage angiogenesis that permits exploration of the relative importance of mechanical, chemical and cellular cues. Endothelial cells proliferate and move over an extracellular matrix by following external gradients of Vessel Endothelial Growth Factor, adhesion and stiffness, which are incorporated to a Cellular Potts model with a finite element description of elasticity. The dynamics of Notch signaling involving Delta-4 and Jagged-1 ligands determines tip cell selection and vessel branching. Through their production rates, competing Jagged-Notch and Delta-Notch dynamics determine the influence of lateral inhibition and lateral induction on the selection of cellular phenotypes, branching of blood vessels, anastomosis (fusion of blood vessels) and angiogenesis velocity. Anastomosis may be favored or impeded depending on the mechanical configuration of strain vectors in the ECM near tip cells. Numerical simulations demonstrate that increasing Jagged production results in pathological vasculatures with thinner and more abundant vessels, which can be compensated by augmenting the production of Delta ligands. Angiogenesis is the process by which new blood vessels grow from existing ones. This process plays a crucial role in organ development, in wound healing and in numerous pathological processes such as cancer growth or in diabetes. Angiogenesis is a complex, multi-step and well regulated process where biochemistry and physics are intertwined. The process entails signaling in vessel cells being driven by both chemical and mechanical mechanisms that result in vascular cell movement, deformation and proliferation. Mathematical models have the ability to bring together these mechanisms in order to explore their relative relevance in vessel growth. Here, we present a mathematical model of early stage angiogenesis that is able to explore the role of biochemical signaling and tissue mechanics. We use this model to unravel the regulating role of Jagged, Notch and Delta dynamics in vascular cells. These membrane proteins have an important part in determining the leading cell in each neo-vascular sprout. Numerical simulations demonstrate that increasing Jagged production results in pathological vasculatures with thinner and more abundant vessels, which can be compensated by augmenting the production of Delta ligands.
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Yáñez D, Travasso RDM, Corvera Poiré E. Resonances in the response of fluidic networks inherent to the cooperation between elasticity and bifurcations. ROYAL SOCIETY OPEN SCIENCE 2019; 6:190661. [PMID: 31598300 PMCID: PMC6774981 DOI: 10.1098/rsos.190661] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2019] [Accepted: 08/27/2019] [Indexed: 05/31/2023]
Abstract
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.
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Affiliation(s)
- Diana Yáñez
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, Ciudad de México 04510, Mexico
| | - Rui D. M. Travasso
- CFisUC, Department of Physics, University of Coimbra, Rua Larga, Coimbra, 3004-516Portugal
| | - Eugenia Corvera Poiré
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, Ciudad de México 04510, Mexico
- CFisUC, Department of Physics, University of Coimbra, Rua Larga, Coimbra, 3004-516Portugal
- Imaging Sciences and Biomedical Engineering Division, King’s College, St Thomas’ Hospital, London, UK
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Ramos JRD, Travasso R, Carvalho J. Capillary network formation from dispersed endothelial cells: Influence of cell traction, cell adhesion, and extracellular matrix rigidity. Phys Rev E 2018; 97:012408. [PMID: 29448490 DOI: 10.1103/physreve.97.012408] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2017] [Indexed: 11/07/2022]
Abstract
The formation of a functional vascular network depends on biological, chemical, and physical processes being extremely well coordinated. Among them, the mechanical properties of the extracellular matrix and cell adhesion are fundamental to achieve a functional network of endothelial cells, able to fully cover a required domain. By the use of a Cellular Potts Model and Finite Element Method it is shown that there exists a range of values of endothelial traction forces, cell-cell adhesion, and matrix rigidities where the network can spontaneously be formed, and its properties are characterized. We obtain the analytical relation that the minimum traction force required for cell network formation must obey. This minimum value for the traction force is approximately independent on the considered cell number and cell-cell adhesion. We quantify how these two parameters influence the morphology of the resulting networks (size and number of meshes).
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Affiliation(s)
- João R D Ramos
- Centro de Física da Universidade de Coimbra, CFisUC, 3007-516 Coimbra, Portugal.,Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
| | - Rui Travasso
- Centro de Física da Universidade de Coimbra, CFisUC, 3007-516 Coimbra, Portugal
| | - João Carvalho
- Centro de Física da Universidade de Coimbra, CFisUC, 3007-516 Coimbra, Portugal
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A Novel Analytical Approach to Pulsatile Blood Flow in the Arterial Network. Ann Biomed Eng 2016; 44:3047-3068. [PMID: 27138525 DOI: 10.1007/s10439-016-1625-3] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2016] [Accepted: 04/20/2016] [Indexed: 10/21/2022]
Abstract
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.
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Toward an optimal design principle in symmetric and asymmetric tree flow networks. J Theor Biol 2015; 389:101-9. [PMID: 26555845 DOI: 10.1016/j.jtbi.2015.10.027] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2015] [Revised: 10/03/2015] [Accepted: 10/19/2015] [Indexed: 11/22/2022]
Abstract
Fluid flow in tree-shaped networks plays an important role in both natural and engineered systems. This paper focuses on laminar flows of Newtonian and non-Newtonian power law fluids in symmetric and asymmetric bifurcating trees. Based on the constructal law, we predict the tree-shaped architecture that provides greater access to the flow subjected to the total network volume constraint. The relationships between the sizes of parent and daughter tubes are presented both for symmetric and asymmetric branching tubes. We also approach the wall-shear stresses and the flow resistance in terms of first tube size, degree of asymmetry between daughter branches, and rheological behavior of the fluid. The influence of tubes obstructing the fluid flow is also accounted for. The predictions obtained by our theory-driven approach find clear support in the findings of previous experimental studies.
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Torres Rojas AM, Meza Romero A, Pagonabarraga I, Travasso RDM, Corvera Poiré E. Obstructions in Vascular Networks: Relation Between Network Morphology and Blood Supply. PLoS One 2015; 10:e0128111. [PMID: 26086774 PMCID: PMC4472785 DOI: 10.1371/journal.pone.0128111] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2014] [Accepted: 04/22/2015] [Indexed: 11/19/2022] Open
Abstract
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.
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Affiliation(s)
- Aimee M. Torres Rojas
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, México D.F., Mexico
| | - Alejandro Meza Romero
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, México D.F., Mexico
| | | | - Rui D. M. Travasso
- Centro de Física da Universidade de Coimbra, Departamento de Física, Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Coimbra, Portugal
| | - Eugenia Corvera Poiré
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, México D.F., Mexico
- Departament de Física Fonamental, Universitat de Barcelona, Barcelona, Spain
- * E-mail:
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Gandica Y, Schwarz T, Oliveira O, Travasso RDM. Hypoxia in vascular networks: a complex system approach to unravel the diabetic paradox. PLoS One 2014; 9:e113165. [PMID: 25409306 PMCID: PMC4237512 DOI: 10.1371/journal.pone.0113165] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2014] [Accepted: 10/20/2014] [Indexed: 01/30/2023] Open
Abstract
In this work we model the extent of hypoxia in the diabetic retina as a function of the area affected by vessel disruption. We find two regimes that differ on the ratio between the area of disrupted vasculature and the area of tissue in hypoxia. In the first regime the hypoxia is localized in the vicinity of the vascular disruption, while in the second regime there is a generalized hypoxia in the affected tissue. The transition between these two regimes occurs when the tissue area affected by individual sites of vessel damage is on the order of the square of the characteristic irrigation length in the tissue (the maximum distance that an irrigated point in the tissue is from an existing vessel). We observe that very high levels of hypoxia are correlated with the rupture of larger vessels in the retina, and with smaller radii of individual sites of vessel damage. Based on this property of vascular networks, we propose a novel mechanism for the transition between the nonproliferative and the proliferative stages in diabetic retinopathy.
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Affiliation(s)
- Yérali Gandica
- Center for Computational Physics, Department of Physics, University of Coimbra, Coimbra, Portugal
| | - Tobias Schwarz
- Center for Computational Physics, Department of Physics, University of Coimbra, Coimbra, Portugal
- Heinz-Brandt-Schule, Berlin, Germany
| | - Orlando Oliveira
- Center for Computational Physics, Department of Physics, University of Coimbra, Coimbra, Portugal
| | - Rui D. M. Travasso
- Center for Computational Physics, Department of Physics, University of Coimbra, Coimbra, Portugal
- Center of Ophthalmology and Vision Sciences(COCV), Institute for Biomedical Imaging and Life Sciences (IBILI), Faculty of Medicine, University of Coimbra, Coimbra, Portugal
- * E-mail:
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