A new model for the emergence of blood capillary networks

Pedro Aceves-Sanchez; Benjamin Aymard; Diane Peurichard; Pol Kennel; Anne Lorsignol; Franck Plouraboué; Louis Casteilla; Pierre Degond

doi:10.3934/nhm.2021001

Article Contents

2021, Volume 16, Issue 1: 91-138. Doi: 10.3934/nhm.2021001

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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

^* Corresponding author
^* Corresponding author

The first two authors contributed equally to this research

Received: February 2020

Revised: November 2020

Early access: December 2020

Published: March 2021

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 92C15, 76Z05; Secondary: 92C42, 76S05.

Citation:

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Figure 1. A capillary element of length $ L_c $ and width $ w_c $ with center at $ \textbf{X} $ and direction $ \boldsymbol{\omega} $

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Figure 2. (A) The function $ g \mapsto \psi_1 ( g) = \psi ( ( L_0^c g - 1)/h_c) $ where $ \psi $ is defined in (8) models an on/off switch. Its fuzziness region is shadowed in gray. On its left-hand-side the switch is off whereas on its right-hand-side it is on. (B) The function $ \rho \mapsto \psi_2 ( \rho) = \psi ( (1 - \rho / \rho_s) / h_s) $ with fuzzy region shadowed in gray. As opposed to (A) the switch is on at the left-hand-side of the shadowed region and it is off on the right-hand-side

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Figure 3. Given a point $ \textbf{X} $ in the tissue, the second term of the right-hand-side of the tensors $ \textbf{K} $ and $ \textbf{D} $ defined in (15) and (16) are computed by summing the tensors $ \boldsymbol{\omega}_k \otimes \boldsymbol{\omega}_k $ over all capillary elements $ k $ that contain $ \textbf{X} $ in their domain $ S_k $. For instance, in this sketch, only five (dark-shadowed rods) out of the nine capillary elements are combined to form tensors $ \textbf{K} $ and $ \textbf{D} $ at $ \textbf{X} $

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Figure 4. (A) Geometrical setting for $ \Omega_1 $, which mimics a cross-section of the tissue in the direction normal to a blood vessel. (B) Geometrical setting for $ \Omega_2 $ which mimics a cross-section in a plane containing the blood vessel. The dimensions of $ \Omega_1 $ and $ \Omega_2 $ are given in Table. 1

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Figure 5. Labeling of boundaries and boundary conditions for the pressure $ p $ and oxygen density $ \rho $ in $ \Omega_1 $

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Figure 6. Labeling of boundaries and boundary conditions for the pressure $ p $ and oxygen density $ \rho $ for $ \Omega_2 $

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Figure 7. Positions of oxygen particles (red spots) and of capillary elements (blue rods) in the rectangular domain $ \Omega_1 $ for a realization of the model. As red spots overlay the blue rods, capillary elements lying below the red oxygen particles are present although not seen. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameter used are those given in Tables 1 and 2. Pictures (A) to (F) give snapshots at increasing times: $ 2 $ min (A), $ 4 $ min (B), $ 6 $ min (C), $ 8 $ min (D), $ 10 $ min (E), $ 12 $ min (F) after initialization

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Figure 8. Isolines and heatmap of the pressure $ p $ in the rectangular domain $ \Omega_1 $ for the same realization of the model as in Fig. 7. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameters used are those given in Tables 1 and 2. Pictures (A) to (F) give snapshots at increasing times: $ 2 $ min (A), $ 4 $ min (B), $ 6 $ min (C), $ 8 $ min (D), $ 10 $ min (E), $ 12 $ min (F) after initialization. The units are given in mmHg

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Figure 9. Heatmap of the Frobenius norm $ \gamma $ of the hydraulic conductivity tensor $ \boldsymbol{K} $ in the rectangular domain $ \Omega_1 $ for the same realization of the model as in Fig. 7. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameter used are those given in Tables 1 and 2. Pictures (A) to (F) give snapshots at increasing times: $ 2 $ min (A), $ 4 $ min (B), $ 6 $ min (C), $ 8 $ min (D), $ 10 $ min (E), $ 12 $ min (F) after initialization. The units are given in $ 10^5 $ $ \mu $m$ ^2 / (mmHg \, \, min) $

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Figure 10. Two binary decision trees, placed back-to-back. The upper one (I) includes capillary creation by reinforcement while the lower one (II) excludes it. Each tree successively includes or excludes capillary creation by WSS and oxygen gradient (noted $ O2\nabla $). At the end of each branch, a typical realization of the model with corresponding inclusion/exclusion of the mechanism is shown. The picture shows the positions of the oxygen particles (red spots) and those of the capillary elements (tiny blue rods). The times for each of the snapshots are the following: 12 min (A), 12 min (B), 19.5 min (C), 30 min (D), 12 min (E), 12 min (F) and 19.5 min (G), after initialization

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Figure 11. Positions of the oxygen particles (red spots) and of the capillary elements (tiny blue rods) in the domain $ \Omega_1 $ (see caption of Fig. 7 for details) for a realization with mesh-size $ \Delta x = \Delta y = 5/8 $, the other parameters in Tables 1 and 2 being unchanged. Pictures (A) to (D) give snapshots at increasing times: $ 2.5 $ min (A), $ 5 $ min (B), $ 7.5 $ min (C), $ 10 $ min (D) after initialization

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Figure 12. influence of the pressure gradient. (B) is the same as Fig. 7 (E). (A) is for pressure gradient reduced by $ 10 \, \% $. (C) is for pressure gradient increased by $ 10 \, \% $. Positions of oxygen particles (red spots) and of capillary elements (blue rods) in the rectangular domain $ \Omega_1 $ are plotted at time $ 10 $ min after initialization. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameter used are those given in Tables 1 and 2 except for Figs (A) and (C) where the boundary conditions for the pressure are modified as detailed in the text

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Figure 13. influence of the capillary element size. (B) is the same as Fig. 7 (D). (A) is for capillary length $ L_c $ divided by $ 2 $. (C) is for capillary length $ L_c $ multiplied by $ 2 $. Positions of oxygen particles (red spots) and of capillary elements (blue rods) in the rectangular domain $ \Omega_1 $ are plotted at time $ 8 $ min after initialization. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameter used are those given in Tables 1 and 2 except for Figs (A) and (C) where the capillary length is modified as detailed in the text

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Figure 14. Positions of the oxygen particles (red spots) and of the capillary elements (tiny blue rods) in the domain $ \Omega_2 $ (see caption of Fig. 7 for details) for a realization of the model. All the creation/deletion mechanisms of capillary elements are turned 'On'. All the parameters used are those given in Tables 1 and 2. Pictures (A) to (F) give snapshots at increasing times: $ 6.8 $ min (A), $ 13.6 $ min (B), $ 20.4 $ min (C), $ 27.2 $ min (D), $ 34 $ min (E), $ 40.8 $ min (F) after initialization

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Table 1. Parameters of the model. In case of estimated parameters ("estim." in the last column), we refer to the corresponding section (indicated in the first column) for the details of this estimation

Quantity	Sym.	Value	Units	Source
Geometry 1 (Sect. 2.6)
Domain size in $ x $-direction	$ L_x $	$ 1000 $	$ \mu \mbox{m} $	estim.
Domain size in $ y $-direction	$ L_y $	$ 2000 $	$ \mu \mbox{m} $	estim.
Oxygen injection region:	$ L_{\min} $	$ 950 $	$ \mu \mbox{m} $	estim.
$ y $-coordinate of lower end
Oxygen injection region:	$ L_{\max} $	$ 1050 $	$ \mu \mbox{m} $	estim.
$ y $-coordinate of upper end
Geometry 2 (Sect. 2.6)
Domain size in $ x $-direction	$ L_x $	$ 2000 $	$ \mu \mbox{m} $	estim.
Domain size in $ y $-direction	$ L_y $	$ 1000 $	$ \mu \mbox{m} $	estim.
Oxygen injection region:	$ L_{\min} $	$ 450 $	$ \mu \mbox{m} $	estim.
$ y $-coordinate of lower end
Oxygen injection region:	$ L_{\max} $	$ 550 $	$ \mu \mbox{m} $	estim.
$ y $-coordinate of upper end
Blood (Sect. 2.2 & 2.4.4)
Pressure at high pressure boundary	$ p_0 $	$ 37.7 $	$ \mbox{mmHg} $	[81]
Pressure at low pressure boundary	$ p_1 $	$ 14.6 $	$ \mbox{mmHg} $	[81]
Dynamic viscosity	$ \mu $	$ 3.75 \times 10^{-7} $	$ \mbox{mmHg min} $	[21]
Oxygen and oxygen
dynamics (Sect. 2.3)
Concentration at injection boundary	$ \rho_0 $	$ 0.025 $	$ \mu \mbox{m}^{-2} $	estim.
Concentration for linear/nonlinear	$ \widetilde{\rho} $	$ 0.1 \times \rho_0 $	$ \mu \mbox{m}^{-2} $	estim.
diffusion shift
Maximum consumption rate	$ \beta_{\rm{sat}} $	$ 0.01 \times \rho_0 $	$ \mbox{min}^{-1}\mu \mbox{m}^{-2} $	estim.
				from [13,75]
Michaelis constant	$ K_m $	$ 0.5 \times \rho_0 $	$ \mu \mbox{m}^{-2} $	estim.
				from [13,75]
Capillary elements
(Sect. 2.4.1 & 2.5)
Length	$ L_c $	$ 15 $	$ \mu \mbox{m} $	[23]
Width	$ w_c $	$ 4 $	$ \mu \mbox{m} $	[23]
Hydraulic conductivity	$ \kappa $	$ 80000 $	$ \mu \mbox{m}^2 \mbox{min}^{-1}\mbox{mmHg}^{-1} $	estim.
Oxygen diffusivity	$ \Delta $	$ 200 $	$ \mu \mbox{m}^2\mbox{min}^{-1} $	estim.
Capillary creation:
oxygen gradient (Sect. 2.4.2)
Maximal creation rate	$ \nu_c^* $	$ 0.05 $	$ \mu \mbox{m}^{-2}\mbox{min}^{-1} $	estim.
Oxygen concentration gradient	$ L_0^c $	$ 8 $	$ \mu \mbox{m} $	estim.
length threshold
Concentration for regularization	$ \rho^* $	$ 0.1 \times \rho_0 $	$ \mu \mbox{m}^{-2} $	estim.
of logarithmic sensing
Width of sigmoid: oxygen gradient	$ h_c $	$ 0.1 $	$ - $	estim.
Oxygen concentration threshold	$ \rho_s $	$ \rho_0 $	$ \mu \mbox{m}^{-2} $	estim.
Width of sigmoid: oxygen concentration	$ h_s $	$ 0.1 $	$ - $	estim.
Capillary creation:
reinforcement (Sect. 2.4.3)
Maximal creation rate	$ \nu_f^* $	$ 0.01 $	$ \mu \mbox{m}^{-2}\mbox{min}^{-1} $	estim.
Blood velocity threshold	$ \bar{u} $	$ 20 $	$ \mu \mbox{m } \mbox{min}^{-1} $	estim.
Lower oxygen concentration threshold	$ \underline{\rho} $	$ 0.1 \times \rho_0 $	$ \mu \mbox{m}^{-2} $	estim.
Upper oxygen concentration threshold	$ \bar{\rho} $	$ 0.5 \times \rho_0 $	$ \mu \mbox{m}^{-2} $	estim.
Width of sigmoids	$ h_f $	$ 0.1 $	$ - $	estim.
Capillary creation:
WSS (Sect. 2.4.4)
Maximal creation rate	$ \nu_w^* $	$ 0.3 $	$ \mu \mbox{m}^{-2}\mbox{min}^{-1} $	estim.
Width of sigmoid	$ h_w $	$ 0.1 $	$ - $	estim.
WSS threshold	$ \lambda^* $	$ 3.75 \times 10^{-8} $	mmHg	estim.
				from [56]
Capillary removal (Sect. 2.4.5)
Removal rate at twice threshold	$ \nu_r^* $	$ 30.0 $	$ \mbox{min}^{-1} $	estim.
Hydraulic conductivity threshold	$ \gamma^* $	$ 400000 $	$ \mu \mbox{m}^2 \, \mbox{min}^{-1} \mbox{mmHg}^{-1} $	estim.
Tissue (Sect. 2.5)
Hydraulic conductivity	$ k_h $	$ 400 $	$ \mu \mbox{m}^2 \mbox{min}^{-1}\mbox{mmHg}^{-1} $	[74]
Oxygen diffusivity	$ \Delta_h $	$ 10 $	$ \mu \mbox{m}^2\mbox{min}^{-1} $	[76]

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Table 2. Numerical parameters

Quantity	Symbol	Value	Unit
Finite-element-method for blood flow
Mesh size in $ x $-direction	$ \Delta x $	$ 1.25 $	$ \mu $m
Mesh size in $ y $-direction	$ \Delta y $	$ 1.25 $	$ \mu $m
SPH particle method for oxygen concentration
Particle "mass"	$ m $	$ 1.0 $	$ - $
Smoothing parameter	$ \eta $	$ 5.0 $	$ \mu $m
CFL parameter	$ C $	$ 0.45 $	$ - $
Point Poisson process for capillary creation
Number of sample points per time step	$ N_c $	$ 10^5 $	$ - $

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