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A two-dimensional multi-class traffic flow model
A new model for the emergence of blood capillary networks
1. | Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA |
2. | MathNeuro Team, Inria Sophia-Antipolis Mediterrannee, 2004 Routes des Lucioles, BP93, 06902 Valbonne Cedex, France |
3. | INRIA Paris, 2, rue Simone Iff, 75589 Paris Cedex 12, France |
4. | Université de Toulouse-INPT-UPS, Institut de Mécanique des Fluides, 31000 Toulouse, France |
5. | STROMALab, Université de Toulouse, Inserm U1031, EFS, INP-ENVT, UPS, CNRS ERL5311, Toulouse, France |
6. | Department of Mathematics, Imperial College London, London SW7 2AZ, UK |
We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and interstitial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper.
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show all references
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[4] |
C. Bardos and E. Tadmor,
Stability and spectral convergence of fourier method for nonlinear problems: On the shortcomings of the $2/3$ de-aliasing method, Numer. Math., 129 (2015), 749-782.
doi: 10.1007/s00211-014-0652-y. |
[5] |
A. L. Bauer, T. L. Jackson and Y. Jiang, Topography of extracellular matrix mediates vascular morphogenesis and migration speeds in angiogenesis, PLoS Computational Biology, 5 (2009), e1000445, 18pp.
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[6] |
E. Boissard, P. Degond and S. Motsch,
Trail formation based on directed pheromone deposition, J. Math. Biol., 66 (2013), 1267-1301.
doi: 10.1007/s00285-012-0529-6. |
[7] |
S. C. Brenner and R. L. Scott, The Mathematical Theory of Finite Element Methods, vol. 15, Springer Science & Business Media, 2008.
doi: 10.1007/978-0-387-75934-0. |
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T. Büscher, A. L. Diez, G. Gompper and J. Elgeti, Instability and fingering of interfaces in growing tissue, New J. Phys., 22 (2020), 083005, 11pp.
doi: 10.1088/1367-2630/ab9e88. |
[9] |
H. Byrne and M. Chaplain,
Mathematical models for tumour angiogenesis: Numerical simulations and nonlinear wave solutions, Bull. Math. Biol., 57 (1995), 461-486.
doi: 10.1007/BF02460635. |
[10] |
P. Carmeliet and R. K. Jain,
Angiogenesis in cancer and other diseases, Nature, 407 (2000), 249-257.
doi: 10.1038/35025220. |
[11] |
A. Chen, J. Darbon, G. Buttazzo, F. Santambrogio and J.-M. Morel,
On the equations of landscape formation, Interfaces Free Bound., 16 (2014), 105-136.
doi: 10.4171/IFB/315. |
[12] |
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Landscape evolution models: A review of their fundamental equations, Geomorphology, 219 (2014), 68-86.
doi: 10.1016/j.geomorph.2014.04.037. |
[13] |
E. Curcio, A. Piscioneri, S. Morelli, S. Salerno, P. Macchiarini and L. De Bartolo,
Kinetics of oxygen uptake by cells potentially used in a tissue engineered trachea, Biomaterials, 35 (2014), 6829-6837.
doi: 10.1016/j.biomaterials.2014.04.100. |
[14] |
G. Dahlquist and Å. Björck, Numerical Methods in Scientific Computing, Volume i, Society for Industrial and Applied Mathematics, 2008.
doi: 10.1137/1.9780898717785. |
[15] |
J. T. Daub and R. M. H. Merks,
A cell-based model of extracellular-matrix-guided endothelial cell migration during angiogenesis, Bull. Math. Biol., 75 (2013), 1377-1399.
doi: 10.1007/s11538-013-9826-5. |
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P. Degond and S. Mas-Gallic,
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Quantity | Sym. | Value | Units | Source |
Geometry 1 (Sect. 2.6) | ||||
Domain size in |
estim. | |||
Domain size in |
estim. | |||
Oxygen injection region: | estim. | |||
Oxygen injection region: | estim. | |||
Geometry 2 (Sect. 2.6) | ||||
Domain size in |
estim. | |||
Domain size in |
estim. | |||
Oxygen injection region: | estim. | |||
Oxygen injection region: | estim. | |||
Blood (Sect. 2.2 & 2.4.4) | ||||
Pressure at high pressure boundary | [81] | |||
Pressure at low pressure boundary | [81] | |||
Dynamic viscosity | [21] | |||
Oxygen and oxygen | ||||
dynamics (Sect. 2.3) | ||||
Concentration at injection boundary | estim. | |||
Concentration for linear/nonlinear | estim. | |||
diffusion shift | ||||
Maximum consumption rate | estim. | |||
from [13,75] | ||||
Michaelis constant | estim. | |||
from [13,75] | ||||
Capillary elements | ||||
(Sect. 2.4.1 & 2.5) | ||||
Length | [23] | |||
Width | [23] | |||
Hydraulic conductivity | estim. | |||
Oxygen diffusivity | estim. | |||
Capillary creation: | ||||
oxygen gradient (Sect. 2.4.2) | ||||
Maximal creation rate | estim. | |||
Oxygen concentration gradient | estim. | |||
length threshold | ||||
Concentration for regularization | estim. | |||
of logarithmic sensing | ||||
Width of sigmoid: oxygen gradient | estim. | |||
Oxygen concentration threshold | estim. | |||
Width of sigmoid: oxygen concentration | estim. | |||
Capillary creation: | ||||
reinforcement (Sect. 2.4.3) | ||||
Maximal creation rate | estim. | |||
Blood velocity threshold | estim. | |||
Lower oxygen concentration threshold | estim. | |||
Upper oxygen concentration threshold | estim. | |||
Width of sigmoids | estim. | |||
Capillary creation: | ||||
WSS (Sect. 2.4.4) | ||||
Maximal creation rate | estim. | |||
Width of sigmoid | estim. | |||
WSS threshold | mmHg | estim. | ||
from [56] | ||||
Capillary removal (Sect. 2.4.5) | ||||
Removal rate at twice threshold | estim. | |||
Hydraulic conductivity threshold | estim. | |||
Tissue (Sect. 2.5) | ||||
Hydraulic conductivity | [74] | |||
Oxygen diffusivity | [76] |
Quantity | Sym. | Value | Units | Source |
Geometry 1 (Sect. 2.6) | ||||
Domain size in |
estim. | |||
Domain size in |
estim. | |||
Oxygen injection region: | estim. | |||
Oxygen injection region: | estim. | |||
Geometry 2 (Sect. 2.6) | ||||
Domain size in |
estim. | |||
Domain size in |
estim. | |||
Oxygen injection region: | estim. | |||
Oxygen injection region: | estim. | |||
Blood (Sect. 2.2 & 2.4.4) | ||||
Pressure at high pressure boundary | [81] | |||
Pressure at low pressure boundary | [81] | |||
Dynamic viscosity | [21] | |||
Oxygen and oxygen | ||||
dynamics (Sect. 2.3) | ||||
Concentration at injection boundary | estim. | |||
Concentration for linear/nonlinear | estim. | |||
diffusion shift | ||||
Maximum consumption rate | estim. | |||
from [13,75] | ||||
Michaelis constant | estim. | |||
from [13,75] | ||||
Capillary elements | ||||
(Sect. 2.4.1 & 2.5) | ||||
Length | [23] | |||
Width | [23] | |||
Hydraulic conductivity | estim. | |||
Oxygen diffusivity | estim. | |||
Capillary creation: | ||||
oxygen gradient (Sect. 2.4.2) | ||||
Maximal creation rate | estim. | |||
Oxygen concentration gradient | estim. | |||
length threshold | ||||
Concentration for regularization | estim. | |||
of logarithmic sensing | ||||
Width of sigmoid: oxygen gradient | estim. | |||
Oxygen concentration threshold | estim. | |||
Width of sigmoid: oxygen concentration | estim. | |||
Capillary creation: | ||||
reinforcement (Sect. 2.4.3) | ||||
Maximal creation rate | estim. | |||
Blood velocity threshold | estim. | |||
Lower oxygen concentration threshold | estim. | |||
Upper oxygen concentration threshold | estim. | |||
Width of sigmoids | estim. | |||
Capillary creation: | ||||
WSS (Sect. 2.4.4) | ||||
Maximal creation rate | estim. | |||
Width of sigmoid | estim. | |||
WSS threshold | mmHg | estim. | ||
from [56] | ||||
Capillary removal (Sect. 2.4.5) | ||||
Removal rate at twice threshold | estim. | |||
Hydraulic conductivity threshold | estim. | |||
Tissue (Sect. 2.5) | ||||
Hydraulic conductivity | [74] | |||
Oxygen diffusivity | [76] |
Quantity | Symbol | Value | Unit |
Finite-element-method for blood flow | |||
Mesh size in |
|||
Mesh size in |
|||
SPH particle method for oxygen concentration | |||
Particle "mass" | |||
Smoothing parameter | |||
CFL parameter | |||
Point Poisson process for capillary creation | |||
Number of sample points per time step |
Quantity | Symbol | Value | Unit |
Finite-element-method for blood flow | |||
Mesh size in |
|||
Mesh size in |
|||
SPH particle method for oxygen concentration | |||
Particle "mass" | |||
Smoothing parameter | |||
CFL parameter | |||
Point Poisson process for capillary creation | |||
Number of sample points per time step |
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