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A new model for the emergence of blood capillary networks

  • * Corresponding author

    * Corresponding author

The first two authors contributed equally to this research

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  • 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.

    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} $

    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

    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} $

    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

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

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

    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

    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

    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) $

    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

    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

    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

    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

    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

    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]
     | Show Table
    DownLoad: CSV

    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 $ $ - $
     | Show Table
    DownLoad: CSV
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