diff --git a/posts/vascularNetworks/.ipynb_checkpoints/VascularNetworks-checkpoint.ipynb b/posts/vascularNetworks/.ipynb_checkpoints/VascularNetworks-checkpoint.ipynb
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+{
+ "cells": [
+ {
+ "cell_type": "raw",
+ "id": "683d2f78-2a94-4988-acc8-a3444522edc9",
+ "metadata": {},
+ "source": [
+ "---\n",
+ "title: \"Vascular Networks\"\n",
+ "author: \"Ali Fele Paranj\"\n",
+ "date: \"November 5 2024\"\n",
+ "categories: [MATH 612]\n",
+ "bibliography: references.bib\n",
+ "---"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "637dc720-8a97-46cb-986b-e240a88f0b7d",
+ "metadata": {},
+ "source": [
+ "## Abstract\n",
+ "I have introduced some basic concepts of micro-circulation and the vascular networks and how they get created (angiogenesis) in health and disease. Then I discuss some angiogenesis models (Anderson-Chaplain as well as BARW) and use the tools of the geomstats to analyze the loopy structure in these networks. I explained the characteristics of the loopy structures in the networks in terms of the parameters of the model. Furthermore, I consider the time evolution of the graphs created by these networks and how the characterization of the loopy structures change through time in these networks."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cf9a4a1c-1f0a-45f9-af4f-ae30efcee890",
+ "metadata": {},
+ "source": [
+ "## Introduction"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5c752ca0-fba7-4d58-beb4-f08d17d7dc13",
+ "metadata": {},
+ "source": [
+ "Vascular network in animals is a complex network of vessels to carry the nutrients to and wastes out from the tissue. Main parts of the blood circulation system is determined during the embryo development (like the location and characterization of the main veins and arteries). However, the exact positioning and hierarchy of the capillary networks is not predetermined genetically, but is an adaptive process. A tissue under hypoxia (experiencing low oxygen) will initiate biochemical signals to recruit new vessels. One of the main signaling molecules is the tumor angiogenic factor (TAF) and as the name suggests it is highly expressed by tumor under oxygen stress (hypoxia). The TAF molecules will diffuse in the interstitial space. Upon interaction with pre-existing vessels, they will digest through the basal membrane of the pre-existing vessels and will activate the endothelial cells (that are lining the inner part lumen of all vessels in the body). The endothelial cells activated by TAF will turn into tip cells which will move up in the gradient of TAF, and they will make branches of vessels which will grow as the tip cells are moving. See figure below for details."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "38259f5c-e17c-4489-b165-61128f37b56e",
+ "metadata": {},
+ "source": [
+ "\n",
+ "![Steps of angiogenesis process. The endothelial cells are \"activated\" by the TAF molecules and turn into tip cells that move towards the source of TAF (up the gradient) and produce new blood vessel segments along their way. This is called tip-induced angiogenesis. This figure is taken from [@Atale2020]](images/Angiogenesis.png){width=60%}\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "96403519-ee11-45ca-8f63-0816dc457885",
+ "metadata": {},
+ "source": [
+ "There are also other factors important in the process of tip cell movement among which is the concentration of fibronectin in the environment. Fibronectin is a protein that acts like a glue and attaches the cells to intracellular matrix, which mediates the cell movement. In the section related to the angiogenesis models we will discuss how this protein is considered in the models."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c418cf59-8b3d-4792-91d6-db2c36acf075",
+ "metadata": {},
+ "source": [
+ "## Anderson-Chaplain Model of Angiogenesis"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "db04f88b-25a4-482d-b062-3de44f34fd65",
+ "metadata": {},
+ "source": [
+ "Anderson-Chaplain model of angiogenesis \\cite{Anderson1998} describes the angiogenesis process considering the factors like TAF and fibronectin. This model contains three variables\n",
+ "$\\newcommand{\\R}{\\mathbb{R}}$\n",
+ "$\\newcommand{\\abs}[1]{|#1|}$ \n",
+ "\n",
+ "* $n = n(X,t): \\Omega \\times \\R \\to \\R$: the endothelial-cell density (per unit area).\n",
+ "* $c = c(X,t): \\Omega \\times \\R \\to \\R$: the tumor angiogenic factor (TAF) concentration (nmol per unit area).\n",
+ "* $f = f(X,t): \\Omega \\times \\R \\to \\R$: the fibronectin concentration (nmol per unit area).\n",
+ " \n",
+ "and the time evolution is governed by the following system of PDEs"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3aadfa90-6fbe-4946-9f84-0eb173791927",
+ "metadata": {},
+ "source": [
+ "\\begin{align*}\n",
+ "\t\t&\\frac{\\partial n}{\\partial t} = D_n\\nabla^2 n - \\nabla\\cdot(\\chi n\\nabla c) - \\nabla\\cdot(\\rho n \\nabla f), \\\\\n",
+ "\t\t&\\frac{\\partial c}{\\partial t} = -\\lambda n c, \\\\\n",
+ "\t\t&\\frac{\\partial f}{\\partial t} = \\omega n - \\mu n f,\n",
+ "\t\\end{align*}"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "59abe1de-231b-459f-b5b3-375a43cc866f",
+ "metadata": {},
+ "source": [
+ "where $D_n$ is a diffusion constant taking the random movement of tip cells into account, $\\chi, \\rho$ reflects the strength of the chemotaxis of tip cells due to the gradient of TAF, and fibronectin respectively. Furthermore, $\\lambda, \\mu$ is the rate at which tip cells consume the TAF and fibronectin respectively, and $\\omega$ denotes the production of fibronectin by the tip cells. Note that we assume at the start of the angiogenesis process, we have a steady state distribution of fibronectin and TAF and is not diffusing. This assumption is not entirely true and can be enhanced."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e6a427c7-189e-42c8-844a-5a25275765dd",
+ "metadata": {},
+ "source": [
+ "Here in this report, we will be using the discrete and stochastic variation of this model. For more detail see [@Anderson1998]. See figure below for some example outputs of the model."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4c974f74-ac03-4136-b3a2-768e8c11d3d5",
+ "metadata": {},
+ "source": [
+ "\n",
+ "![Some example output of the Anderson-Chaplain model of angiogenesis using the implementation of the model shared by [@Nardini2021]. We have assumed the source of TAF molecules is located at the right edge of the domain, while the pre-existing parent vessels is located at the left edge of the domain. The strength of the chemotaxis and haptotactic (due to fibronectin) signaling is set to be $\\chi = 0.4$, and $\\rho = 0.4$.](images/Anderson.png){width=60%,fig-align=\"left\",#fig:anderson}\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "49e671ed-6694-4176-8886-0474f7620583",
+ "metadata": {},
+ "source": [
+ "## Branching-Annihilating Random Walker\n",
+ "The Anderson-Chaplain model of angiogenesis is not the only formulation of this phenomena. A popular alternative formulation is using the notion of branching annihilating random walkers for the to explain the branching morphogenesis of vascular networks. A very detailed discussion on this formulation can be found in [@Hannezo2017,@Ucar2021]. This formulation has been also successful to models a vast variety of tip-driven morphogenesis in mammary-glands, prostate, kidney [@Hannezo2017], lymphatic system [@Ucar2023], neural branching [@Ucar2021], and etc.\n",
+ "\t\n",
+ "The core idea behind this formulation is to assume that the tip cells undergo a branching-annihilating random walk, i.e. they move randomly in the space, turn into pairs randomly (branching), and as they move they produce new cells (stalk) behind their trails, and finally annihilate if they encounter any of the stalk cells. See figure below:"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b036fbe6-4a90-4131-a09b-2907b6d9e3e4",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "{width=50%,fig-align=\"center\",#BARW}\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "34ab7e20-4aa1-4e5a-8571-535f3e99518d",
+ "metadata": {},
+ "source": [
+ "## Data Pre-Processing\n",
+ "The data generated by the Anderson-Chaplain model or BARW model are typically matrices (i.e. images) (however, there are certain ways to construct a graph representing the final structure during the simulation). To be able to do any form of computation with the simulated data (like finding the loops, etc) it is convenient to turn the images into a graph.\n",
+ "\t\n",
+ "For the outputs of the Anderson-Chaplain model , since the vessels are more crowded at some regions, we first did a dilation followed by a erosion (using OpenCV) to enhance the images for further processing. We then applied a connectivity-8 kernel to the enhanced images to find the nodes, and finally used the processes information to turn the vascular network in the image into a graph in python (using NetworkX library). See figure below for more details. We also applied the same processing pipeline on the data generated by the BARW model which is shown in the following figure"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f1e13e4a-22f8-4b29-9a61-50e3b055d555",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "{width=50%, fig-align=\"center\"}\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "49ef607d-c494-4fdd-b2d8-6f98b426e208",
+ "metadata": {},
+ "source": [
+ "\n",
+ "{width=50%,fig-align=\"center\"}\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "51eb6cfb-275a-4fe6-be0a-8e2070a6e140",
+ "metadata": {},
+ "source": [
+ "The new data structure that we store the synthetic data (i.e. as graphs in NetworkX) gives us a tremendous flexibility in working and analyzing the shape of each individual loops. See the following figure for further details on this. This data structure will also come in handy when we study the time evolution of the graphs."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a25f7530-9b44-471c-8fc4-9c5ba90449f0",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "{width=50%,fig-align=\"center\"}\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9e6ce76d-c876-4ddc-81f3-cc32ac5b499b",
+ "metadata": {},
+ "source": [
+ "**Study Plan**: In this section we will use the notions of Kendall shape space to study the structure of the loops that appear under different parameters of the model."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0d4cf518-ee34-492b-ad32-7c3b8966d402",
+ "metadata": {},
+ "source": [
+ "## Time Evolution Of Networks"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ca9cc283-bfc6-44eb-aace-82990ae746e3",
+ "metadata": {},
+ "source": [
+ "Vascular networks are not static structure, but rather the evolve in time in response to the changing metabolic demand of the underlying tissue, as well as the metabolic cost of the network itself, and the overall energy required to pump the fluid through the network (See [@Pries2010,Secomb2012,@Pries2014,Ouarne2021] for more discussion). To put this in different words, the role of vascular networks is to deliver nutrients to the tissue and remove the wastes. To do this, it needs to have a space filling configuration with lots of branches. However, due to the Poiseuille law for the flow of fluids in a tube, the power needed to pump the fluid through the tube scales with $r^{-4}$ where $r$ is the radius of the tube. I.e. smaller vessel segments needs a huge power to pump the blood through them. Thus have a massively branched structure is not an optimal solution. On the other hand, the vascular network consists of cells which requires maintenance as well. Thus the optimized vascular network should have a low volume as well. Because of these dynamics in action, in the angiogenesis process first a mesh of new blood vessels form which later evolve to a more ordered and hierarchical structure in a self-organization process."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b04b2843-013e-4644-a1ff-76588501f916",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "![Remodeling of vascular network of chick chorioallantoic membrane. Initially (sub-figure 1) a mesh of vascular networks form. Then (sub-figures 2,3,4), through the remodeling dynamics, a more ordered and hierarchical structure emerges. Images are taken from [@Richard2018].](images/remodeling2.png){width=50%,fig-align=\"center\"}\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cf2eb844-82eb-4805-9c8e-43fbbdef012a",
+ "metadata": {},
+ "source": [
+ "To determine the time evolution of the vascular network we first need to formulate the problem in an appropriate way. First, we represent a given vascular network with a multi-weighted graph $G=(\\mathcal{V},\\mathcal{E})$ where $V$ is the set of vertices and $E$ is the edge set. We define the pressure $\\mathbf{P}$ on the nodes, the flow $ \\mathbf{Q} $ on the edges, and let $C_{i,j}, L_{i,j}$ denote the conductivity of an edge, and $L_{i,j}$ denote the length of the same edge. Given the source and sink terms on the nodes $ \\mathbf{q} $, the flow in the edges can be determined by\n",
+ "\t$$\\mathcal{L} \\mathbf{P} = \\mathbf{q},$$\n",
+ "\twhere $\\mathcal{L}$ is the Laplacian matrix of the graph. For more details on this see \\autoref{appndix}. Once we know the pressures on the nodes, we can easily calculate the flow through the edges by \n",
+ "\t$$\\bf{Q} = \\bf{C} L^{-1} \\bf{\\Delta} \\bf{P}, \\tag{2}$$\n",
+ "\twhere $C$ is a diagonal matrix of the conductance of the edges, $L$ is the diagonal matrix of the length of each edge, $ \\Delta $ is the transpose of the incidence matrix, and $ P $ is the pressure on the nodes. $Q$ is the flow of the edges. Once we know the flow in the edges, we can design evolution law to describe the time evolution of the weights of the edges (which by Poiseuille's is a function of the radius of the vessel segment). The evolution law can be derived by defining an energy functional and moving down the gradient of the energy functional to minimize it, or we can take an ad-hoc method and write a mechanistic ODE for time evolution of the conductances. For the energy functional one can write\n",
+ "\t$$ E(\\mathbf{C}) = \\frac{1}{2} \\sum_{e\\in \\mathcal{E}}(\\frac{Q_e^2}{C_e} + \\nu C_e^\\gamma), $$\n",
+ "\twhere $ \\mathcal{E} $ is the edge set of the graph, $ Q_e, C_e $ is the flow and conductance of the edge $ e $, and $ \\nu,\\gamma $ are parameters. The first term in the sum is of the form ``power=current$ \\times $potential'' and reflects the power required to pump the flow, and the second term can be shown that reflects the volume of the total network. We can set\n",
+ "\t$$ \\frac{d \\mathbf{C}}{dt} = -\\nabla E, $$\n",
+ "\twhich determines the time evolution of the weights in a direction that reduces the total energy. The steady-state solution of this ODE system is precisely the Euler-Lagrange formulation of the least action principle. Alternatively, one can come up with carefully designed ODEs for the time evolution of the conductances that represents certain biological facts. In particular\n",
+ "\t$$ \\frac{d C_e}{dt} = \\alpha |Q_e|^{2\\sigma} - b C_e + g $$\n",
+ "\tproposed by \\cite{Ronellenfitsch2016}, and \n",
+ "\t$$ \\frac{d}{dt} \\sqrt{C_e} = F(Q_e) - c\\sqrt{C_e}, $$\n",
+ "\tproposed by \\cite{Almeida2022} has been popular choices. See \\autoref{fig:GraphEvolution} for more details. It is important to note that in the simulations shown here, the initial network is a toy network. This can be improved by using any of the vascular network generated by any of the angiogenesis models discussed before."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f6834722-7ae8-46e1-8af6-56ad9ccfa3ec",
+ "metadata": {},
+ "source": [
+ "\n",
+ "{width=50%,fig-align=\"center\"}\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "99d20ad1-ec5f-452c-806d-850c75bbbf3e",
+ "metadata": {},
+ "source": [
+ "## Appendix"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1318c753-cd56-4470-a8d8-a2a6bc8ab5ad",
+ "metadata": {},
+ "source": [
+ "For a graph, the Laplacian matrix contains the information on the in/out flow of stuff into the nodes.\n",
+ "\n",
+ " ![\"BARW\"](\"images/graph1.png\")
\n",
+ "
\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3227cc0f-18a3-483a-9cdc-8afab3f42e39",
+ "metadata": {},
+ "source": [
+ "Then the Laplacian matrix is given by\n",
+ "\t$$ D = \\begin{pmatrix}\n",
+ "\t\t2 & 0 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & 4 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & 0 & 2 & 0 & 0 \\\\\n",
+ "\t\t0 & 0 & 0 & 2 & 0 \\\\\n",
+ "\t\t0 & 0 & 0 & 0 & 2\n",
+ "\t\\end{pmatrix}, $$\n",
+ "\tand the adjacency matrix is given by\n",
+ "\t$$ A = \\begin{pmatrix}\n",
+ "\t\t0 & 1 & 1 & 0 & 0 \\\\\n",
+ "\t\t1 & 0 & 1 & 1 & 1 \\\\\n",
+ "\t\t1 & 1 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & 1 & 0 & 0 & 1 \\\\\n",
+ "\t\t0 & 1 & 0 & 1 & 0\n",
+ "\t\\end{pmatrix}, $$\n",
+ "\tand the Laplacian matrix is given by\n",
+ "\t$$ L = D -A = \n",
+ "\t\\begin{pmatrix}\n",
+ "\t\t2 & -1 & -1 & 0 & 0 \\\\\n",
+ "\t\t-1 & 4 & -1 & -1 & -1 \\\\\n",
+ "\t\t-1 & -1 & 2 & 0 & 0 \\\\\n",
+ "\t\t0 & -1 & 0 & 2 & -1 \\\\\n",
+ "\t\t0 & -1 & 0 & -1 & 2\n",
+ "\t\\end{pmatrix}.\n",
+ "\t$$\n",
+ "It is straight forward to generalize the notion of Laplacian matrix to the weighed graphs, where the degree matrix $ D $, the diagonal entries will be the sum of all weights of the edges connected to that node, and for the adjacency matrix, instead of zeros and ones, we will have the weights of the connections..\n",
+ "\t\n",
+ "There is also another way of finding the Laplacian matrix by using the notion of incidence matrix. To do so, we first need to make our graph to be directed. Any combination of the direction on the edges will do the job and will yield in a correct answer. For instance, consider the following directed graph"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "193df266-5f85-4659-9d11-48a9e73011bc",
+ "metadata": {},
+ "source": [
+ "For a graph, the Laplacian matrix contains the information on the in/out flow of stuff into the nodes.\n",
+ "\n",
+ " ![\"BARW\"](\"images/graph2.png\")
\n",
+ "
\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "099eb981-35c7-4091-adfb-b3ea3451677a",
+ "metadata": {},
+ "source": [
+ "Its incidence matrix will be\n",
+ "\t$$\n",
+ "\tM = \\begin{pmatrix}\n",
+ "\t\t-1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & -1 & 1 & -1 & 0 & -1 \\\\\n",
+ "\t\t1 & 0 & -1 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & 0 & 0 & 1 & 1 & 0 \\\\\n",
+ "\t\t0 & 0 & 0 & 0 & -1 & 1 \\\\\n",
+ "\t\\end{pmatrix}\n",
+ "\t$$\n",
+ "\tThe Laplacian matrix can be written as\n",
+ "\t$$ \\mathcal{L} = M M^T. $$\n",
+ "\tNote that in the case of the weighed graphs, we will have\n",
+ "\t$$ \\mathcal{L} = M W M^T \\tag{1}$$\n",
+ "\twhere $ W $ is a diagonal matrix containing the weights. These computations can be done easily on the NetworkX. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4b73b031-c340-4d88-be86-d4f5c909f634",
+ "metadata": {},
+ "source": [
+ "The incidence matrix is also very useful in calculating the pressure difference between nodes of a particular edge. Let $\\Delta = M^T$. Then given the vector $P$ that contains the pressures on the vertices, then the pressure difference on the edges will be given by $\\Delta P$, where $\\Delta$ is the transpose of the incidence matrix. This comes in handy when we want to calculate the flow of the edges which will be given by\n",
+ "$$ \\bf{Q} = \\bf{C} L^{-1} \\bf{\\Delta} \\bf{P}, \\tag{2} $$\n",
+ "where $ C $ is a diagonal matrix of the conductance of the edges, $L$ is the diagonal matrix of the ``length'' of each edge, $\\Delta$ is the transpose of the incidence matrix, and $P$ is the pressure on the nodes. $Q$ is the flow of the edges. In this particular example we are assuming that the relation between flow and the pressure difference is $Q_e = C_e (p_i - p_j)/L$. But we can have many other choices.\n",
+ "\t\n",
+ "Knowing the sources and sinks on the nodes, the pressure can be determined by the Kirchhoff law\n",
+ "\t$$ \\mathcal{L} \\bf{P} = \\bf{q}, $$\n",
+ "where the vector $ q $ is the sources and the sinks values for each node. This is the same as solving the \\textbf{Poisson equation}. This can also be written in terms of the flow, i.e.\n",
+ "\t$$ \\Delta^T \\bf{Q} = \\bf{q}. $$\n",
+ "\tBy $ (2) $ we can write\n",
+ "\t$$ (\\bf{\\Delta}^T \\bf{C}\\bf{L}^{-1}\\Delta) \\bf{P} = \\bf{q}. $$\n",
+ "\tSince $ \\Delta = M^T $, the expression inside the parentheses is clearly Equation (1).\n",
+ "\t\n",
+ "Similar to the Poisson equation on the graph which is equivalent Kirchhoff's law, we can solve other types of heat and wave equations on the graph as well. The Laplacian matrix play a key role.\n",
+ "$$ \\frac{\\partial p}{\\partial t} = - \\mathcal{L} p + q, $$\n",
+ "\tfor the heat equation, and\n",
+ "$$ \\frac{\\partial^2 p}{\\partial t^2} = -\\mathcal{L}p + q, $$\n",
+ "\tfor the wave equation."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6bc26e96-34af-431c-84f3-7b535f68bd15",
+ "metadata": {},
+ "source": [
+ "## References"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1aa75011-0b16-4f25-b32c-79efe74c5f41",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.10.12"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/posts/vascularNetworks/VascularNetworks.ipynb b/posts/vascularNetworks/VascularNetworks.ipynb
new file mode 100644
index 0000000..4fb0f64
--- /dev/null
+++ b/posts/vascularNetworks/VascularNetworks.ipynb
@@ -0,0 +1,409 @@
+{
+ "cells": [
+ {
+ "cell_type": "raw",
+ "id": "683d2f78-2a94-4988-acc8-a3444522edc9",
+ "metadata": {},
+ "source": [
+ "---\n",
+ "title: \"Vascular Networks\"\n",
+ "author: \"Ali Fele Paranj\"\n",
+ "date: \"November 5 2024\"\n",
+ "categories: [MATH 612]\n",
+ "bibliography: references.bib\n",
+ "---"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "637dc720-8a97-46cb-986b-e240a88f0b7d",
+ "metadata": {},
+ "source": [
+ "## Abstract\n",
+ "I have introduced some basic concepts of micro-circulation and the vascular networks and how they get created (angiogenesis) in health and disease. Then I discuss some angiogenesis models (Anderson-Chaplain as well as BARW) and use the tools of the geomstats to analyze the loopy structure in these networks. I explained the characteristics of the loopy structures in the networks in terms of the parameters of the model. Furthermore, I consider the time evolution of the graphs created by these networks and how the characterization of the loopy structures change through time in these networks."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cf9a4a1c-1f0a-45f9-af4f-ae30efcee890",
+ "metadata": {},
+ "source": [
+ "## Introduction"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5c752ca0-fba7-4d58-beb4-f08d17d7dc13",
+ "metadata": {},
+ "source": [
+ "Vascular network in animals is a complex network of vessels to carry the nutrients to and wastes out from the tissue. Main parts of the blood circulation system is determined during the embryo development (like the location and characterization of the main veins and arteries). However, the exact positioning and hierarchy of the capillary networks is not predetermined genetically, but is an adaptive process. A tissue under hypoxia (experiencing low oxygen) will initiate biochemical signals to recruit new vessels. One of the main signaling molecules is the tumor angiogenic factor (TAF) and as the name suggests it is highly expressed by tumor under oxygen stress (hypoxia). The TAF molecules will diffuse in the interstitial space. Upon interaction with pre-existing vessels, they will digest through the basal membrane of the pre-existing vessels and will activate the endothelial cells (that are lining the inner part lumen of all vessels in the body). The endothelial cells activated by TAF will turn into tip cells which will move up in the gradient of TAF, and they will make branches of vessels which will grow as the tip cells are moving. See figure below for details."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "38259f5c-e17c-4489-b165-61128f37b56e",
+ "metadata": {},
+ "source": [
+ "\n",
+ "![Steps of angiogenesis process. The endothelial cells are \"activated\" by the TAF molecules and turn into tip cells that move towards the source of TAF (up the gradient) and produce new blood vessel segments along their way. This is called tip-induced angiogenesis. This figure is taken from [@Atale2020]](images/Angiogenesis.png){width=60%}\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "96403519-ee11-45ca-8f63-0816dc457885",
+ "metadata": {},
+ "source": [
+ "There are also other factors important in the process of tip cell movement among which is the concentration of fibronectin in the environment. Fibronectin is a protein that acts like a glue and attaches the cells to intracellular matrix, which mediates the cell movement. In the section related to the angiogenesis models we will discuss how this protein is considered in the models."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c418cf59-8b3d-4792-91d6-db2c36acf075",
+ "metadata": {},
+ "source": [
+ "## Anderson-Chaplain Model of Angiogenesis"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "db04f88b-25a4-482d-b062-3de44f34fd65",
+ "metadata": {},
+ "source": [
+ "Anderson-Chaplain model of angiogenesis \\cite{Anderson1998} describes the angiogenesis process considering the factors like TAF and fibronectin. This model contains three variables\n",
+ "$\\newcommand{\\R}{\\mathbb{R}}$\n",
+ "$\\newcommand{\\abs}[1]{|#1|}$ \n",
+ "\n",
+ "* $n = n(X,t): \\Omega \\times \\R \\to \\R$: the endothelial-cell density (per unit area).\n",
+ "* $c = c(X,t): \\Omega \\times \\R \\to \\R$: the tumor angiogenic factor (TAF) concentration (nmol per unit area).\n",
+ "* $f = f(X,t): \\Omega \\times \\R \\to \\R$: the fibronectin concentration (nmol per unit area).\n",
+ " \n",
+ "and the time evolution is governed by the following system of PDEs"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3aadfa90-6fbe-4946-9f84-0eb173791927",
+ "metadata": {},
+ "source": [
+ "\\begin{align*}\n",
+ "\t\t&\\frac{\\partial n}{\\partial t} = D_n\\nabla^2 n - \\nabla\\cdot(\\chi n\\nabla c) - \\nabla\\cdot(\\rho n \\nabla f), \\\\\n",
+ "\t\t&\\frac{\\partial c}{\\partial t} = -\\lambda n c, \\\\\n",
+ "\t\t&\\frac{\\partial f}{\\partial t} = \\omega n - \\mu n f,\n",
+ "\t\\end{align*}"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "59abe1de-231b-459f-b5b3-375a43cc866f",
+ "metadata": {},
+ "source": [
+ "where $D_n$ is a diffusion constant taking the random movement of tip cells into account, $\\chi, \\rho$ reflects the strength of the chemotaxis of tip cells due to the gradient of TAF, and fibronectin respectively. Furthermore, $\\lambda, \\mu$ is the rate at which tip cells consume the TAF and fibronectin respectively, and $\\omega$ denotes the production of fibronectin by the tip cells. Note that we assume at the start of the angiogenesis process, we have a steady state distribution of fibronectin and TAF and is not diffusing. This assumption is not entirely true and can be enhanced."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e6a427c7-189e-42c8-844a-5a25275765dd",
+ "metadata": {},
+ "source": [
+ "Here in this report, we will be using the discrete and stochastic variation of this model. For more detail see [@Anderson1998]. See figure below for some example outputs of the model."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4c974f74-ac03-4136-b3a2-768e8c11d3d5",
+ "metadata": {},
+ "source": [
+ "\n",
+ "![Some example output of the Anderson-Chaplain model of angiogenesis using the implementation of the model shared by [@Nardini2021]. We have assumed the source of TAF molecules is located at the right edge of the domain, while the pre-existing parent vessels is located at the left edge of the domain. The strength of the chemotaxis and haptotactic (due to fibronectin) signaling is set to be $\\chi = 0.4$, and $\\rho = 0.4$.](images/Anderson.png){width=60%,fig-align=\"left\",#fig:anderson}\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "49e671ed-6694-4176-8886-0474f7620583",
+ "metadata": {},
+ "source": [
+ "## Branching-Annihilating Random Walker\n",
+ "The Anderson-Chaplain model of angiogenesis is not the only formulation of this phenomena. A popular alternative formulation is using the notion of branching annihilating random walkers for the to explain the branching morphogenesis of vascular networks. A very detailed discussion on this formulation can be found in [@Hannezo2017,@Ucar2021]. This formulation has been also successful to models a vast variety of tip-driven morphogenesis in mammary-glands, prostate, kidney [@Hannezo2017], lymphatic system [@Ucar2023], neural branching [@Ucar2021], and etc.\n",
+ "\t\n",
+ "The core idea behind this formulation is to assume that the tip cells undergo a branching-annihilating random walk, i.e. they move randomly in the space, turn into pairs randomly (branching), and as they move they produce new cells (stalk) behind their trails, and finally annihilate if they encounter any of the stalk cells. See figure below:"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b036fbe6-4a90-4131-a09b-2907b6d9e3e4",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "{width=50%,fig-align=\"center\",#BARW}\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "34ab7e20-4aa1-4e5a-8571-535f3e99518d",
+ "metadata": {},
+ "source": [
+ "## Data Pre-Processing\n",
+ "The data generated by the Anderson-Chaplain model or BARW model are typically matrices (i.e. images) (however, there are certain ways to construct a graph representing the final structure during the simulation). To be able to do any form of computation with the simulated data (like finding the loops, etc) it is convenient to turn the images into a graph.\n",
+ "\t\n",
+ "For the outputs of the Anderson-Chaplain model , since the vessels are more crowded at some regions, we first did a dilation followed by a erosion (using OpenCV) to enhance the images for further processing. We then applied a connectivity-8 kernel to the enhanced images to find the nodes, and finally used the processes information to turn the vascular network in the image into a graph in python (using NetworkX library). See figure below for more details. We also applied the same processing pipeline on the data generated by the BARW model which is shown in the following figure"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f1e13e4a-22f8-4b29-9a61-50e3b055d555",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "{width=50%, fig-align=\"center\"}\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "49ef607d-c494-4fdd-b2d8-6f98b426e208",
+ "metadata": {},
+ "source": [
+ "\n",
+ "{width=50%,fig-align=\"center\"}\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "51eb6cfb-275a-4fe6-be0a-8e2070a6e140",
+ "metadata": {},
+ "source": [
+ "The new data structure that we store the synthetic data (i.e. as graphs in NetworkX) gives us a tremendous flexibility in working and analyzing the shape of each individual loops. See the following figure for further details on this. This data structure will also come in handy when we study the time evolution of the graphs."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a25f7530-9b44-471c-8fc4-9c5ba90449f0",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "{width=50%,fig-align=\"center\"}\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9e6ce76d-c876-4ddc-81f3-cc32ac5b499b",
+ "metadata": {},
+ "source": [
+ "**Study Plan**: In this section we will use the notions of Kendall shape space to study the structure of the loops that appear under different parameters of the model."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0d4cf518-ee34-492b-ad32-7c3b8966d402",
+ "metadata": {},
+ "source": [
+ "## Time Evolution Of Networks"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ca9cc283-bfc6-44eb-aace-82990ae746e3",
+ "metadata": {},
+ "source": [
+ "Vascular networks are not static structure, but rather the evolve in time in response to the changing metabolic demand of the underlying tissue, as well as the metabolic cost of the network itself, and the overall energy required to pump the fluid through the network (See [@Pries2010,Secomb2012,@Pries2014,Ouarne2021] for more discussion). To put this in different words, the role of vascular networks is to deliver nutrients to the tissue and remove the wastes. To do this, it needs to have a space filling configuration with lots of branches. However, due to the Poiseuille law for the flow of fluids in a tube, the power needed to pump the fluid through the tube scales with $r^{-4}$ where $r$ is the radius of the tube. I.e. smaller vessel segments needs a huge power to pump the blood through them. Thus have a massively branched structure is not an optimal solution. On the other hand, the vascular network consists of cells which requires maintenance as well. Thus the optimized vascular network should have a low volume as well. Because of these dynamics in action, in the angiogenesis process first a mesh of new blood vessels form which later evolve to a more ordered and hierarchical structure in a self-organization process."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b04b2843-013e-4644-a1ff-76588501f916",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "![Remodeling of vascular network of chick chorioallantoic membrane. Initially (sub-figure 1) a mesh of vascular networks form. Then (sub-figures 2,3,4), through the remodeling dynamics, a more ordered and hierarchical structure emerges. Images are taken from [@Richard2018].](images/remodeling2.png){width=50%,fig-align=\"center\"}\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cf2eb844-82eb-4805-9c8e-43fbbdef012a",
+ "metadata": {},
+ "source": [
+ "To determine the time evolution of the vascular network we first need to formulate the problem in an appropriate way. First, we represent a given vascular network with a multi-weighted graph $G=(\\mathcal{V},\\mathcal{E})$ where $V$ is the set of vertices and $E$ is the edge set. We define the pressure $\\mathbf{P}$ on the nodes, the flow $ \\mathbf{Q} $ on the edges, and let $C_{i,j}, L_{i,j}$ denote the conductivity of an edge, and $L_{i,j}$ denote the length of the same edge. Given the source and sink terms on the nodes $ \\mathbf{q} $, the flow in the edges can be determined by\n",
+ "\t$$\\mathcal{L} \\mathbf{P} = \\mathbf{q},$$\n",
+ "\twhere $\\mathcal{L}$ is the Laplacian matrix of the graph. For more details on this see \\autoref{appndix}. Once we know the pressures on the nodes, we can easily calculate the flow through the edges by \n",
+ "\t$$\\bf{Q} = \\bf{C} L^{-1} \\bf{\\Delta} \\bf{P}, \\tag{2}$$\n",
+ "\twhere $C$ is a diagonal matrix of the conductance of the edges, $L$ is the diagonal matrix of the length of each edge, $ \\Delta $ is the transpose of the incidence matrix, and $ P $ is the pressure on the nodes. $Q$ is the flow of the edges. Once we know the flow in the edges, we can design evolution law to describe the time evolution of the weights of the edges (which by Poiseuille's is a function of the radius of the vessel segment). The evolution law can be derived by defining an energy functional and moving down the gradient of the energy functional to minimize it, or we can take an ad-hoc method and write a mechanistic ODE for time evolution of the conductances. For the energy functional one can write\n",
+ "\t$$ E(\\mathbf{C}) = \\frac{1}{2} \\sum_{e\\in \\mathcal{E}}(\\frac{Q_e^2}{C_e} + \\nu C_e^\\gamma), $$\n",
+ "\twhere $ \\mathcal{E} $ is the edge set of the graph, $ Q_e, C_e $ is the flow and conductance of the edge $ e $, and $ \\nu,\\gamma $ are parameters. The first term in the sum is of the form ``power=current$ \\times $potential'' and reflects the power required to pump the flow, and the second term can be shown that reflects the volume of the total network. We can set\n",
+ "\t$$ \\frac{d \\mathbf{C}}{dt} = -\\nabla E, $$\n",
+ "\twhich determines the time evolution of the weights in a direction that reduces the total energy. The steady-state solution of this ODE system is precisely the Euler-Lagrange formulation of the least action principle. Alternatively, one can come up with carefully designed ODEs for the time evolution of the conductances that represents certain biological facts. In particular\n",
+ "\t$$ \\frac{d C_e}{dt} = \\alpha |Q_e|^{2\\sigma} - b C_e + g $$\n",
+ "\tproposed by \\cite{Ronellenfitsch2016}, and \n",
+ "\t$$ \\frac{d}{dt} \\sqrt{C_e} = F(Q_e) - c\\sqrt{C_e}, $$\n",
+ "\tproposed by \\cite{Almeida2022} has been popular choices. See \\autoref{fig:GraphEvolution} for more details. It is important to note that in the simulations shown here, the initial network is a toy network. This can be improved by using any of the vascular network generated by any of the angiogenesis models discussed before."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f6834722-7ae8-46e1-8af6-56ad9ccfa3ec",
+ "metadata": {},
+ "source": [
+ "\n",
+ "{width=50%,fig-align=\"center\"}\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "99d20ad1-ec5f-452c-806d-850c75bbbf3e",
+ "metadata": {},
+ "source": [
+ "## Appendix"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1318c753-cd56-4470-a8d8-a2a6bc8ab5ad",
+ "metadata": {},
+ "source": [
+ "For a graph, the Laplacian matrix contains the information on the in/out flow of stuff into the nodes.\n",
+ "\n",
+ " \n",
+ "
\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3227cc0f-18a3-483a-9cdc-8afab3f42e39",
+ "metadata": {},
+ "source": [
+ "Then the Laplacian matrix is given by\n",
+ "\t$$ D = \\begin{pmatrix}\n",
+ "\t\t2 & 0 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & 4 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & 0 & 2 & 0 & 0 \\\\\n",
+ "\t\t0 & 0 & 0 & 2 & 0 \\\\\n",
+ "\t\t0 & 0 & 0 & 0 & 2\n",
+ "\t\\end{pmatrix}, $$\n",
+ "\tand the adjacency matrix is given by\n",
+ "\t$$ A = \\begin{pmatrix}\n",
+ "\t\t0 & 1 & 1 & 0 & 0 \\\\\n",
+ "\t\t1 & 0 & 1 & 1 & 1 \\\\\n",
+ "\t\t1 & 1 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & 1 & 0 & 0 & 1 \\\\\n",
+ "\t\t0 & 1 & 0 & 1 & 0\n",
+ "\t\\end{pmatrix}, $$\n",
+ "\tand the Laplacian matrix is given by\n",
+ "\t$$ L = D -A = \n",
+ "\t\\begin{pmatrix}\n",
+ "\t\t2 & -1 & -1 & 0 & 0 \\\\\n",
+ "\t\t-1 & 4 & -1 & -1 & -1 \\\\\n",
+ "\t\t-1 & -1 & 2 & 0 & 0 \\\\\n",
+ "\t\t0 & -1 & 0 & 2 & -1 \\\\\n",
+ "\t\t0 & -1 & 0 & -1 & 2\n",
+ "\t\\end{pmatrix}.\n",
+ "\t$$\n",
+ "It is straight forward to generalize the notion of Laplacian matrix to the weighed graphs, where the degree matrix $ D $, the diagonal entries will be the sum of all weights of the edges connected to that node, and for the adjacency matrix, instead of zeros and ones, we will have the weights of the connections..\n",
+ "\t\n",
+ "There is also another way of finding the Laplacian matrix by using the notion of incidence matrix. To do so, we first need to make our graph to be directed. Any combination of the direction on the edges will do the job and will yield in a correct answer. For instance, consider the following directed graph"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "193df266-5f85-4659-9d11-48a9e73011bc",
+ "metadata": {},
+ "source": [
+ "For a graph, the Laplacian matrix contains the information on the in/out flow of stuff into the nodes.\n",
+ "\n",
+ " \n",
+ "
\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "099eb981-35c7-4091-adfb-b3ea3451677a",
+ "metadata": {},
+ "source": [
+ "Its incidence matrix will be\n",
+ "\t$$\n",
+ "\tM = \\begin{pmatrix}\n",
+ "\t\t-1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & -1 & 1 & -1 & 0 & -1 \\\\\n",
+ "\t\t1 & 0 & -1 & 0 & 0 & 0 \\\\\n",
+ "\t\t0 & 0 & 0 & 1 & 1 & 0 \\\\\n",
+ "\t\t0 & 0 & 0 & 0 & -1 & 1 \\\\\n",
+ "\t\\end{pmatrix}\n",
+ "\t$$\n",
+ "\tThe Laplacian matrix can be written as\n",
+ "\t$$ \\mathcal{L} = M M^T. $$\n",
+ "\tNote that in the case of the weighed graphs, we will have\n",
+ "\t$$ \\mathcal{L} = M W M^T \\tag{1}$$\n",
+ "\twhere $ W $ is a diagonal matrix containing the weights. These computations can be done easily on the NetworkX. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4b73b031-c340-4d88-be86-d4f5c909f634",
+ "metadata": {},
+ "source": [
+ "The incidence matrix is also very useful in calculating the pressure difference between nodes of a particular edge. Let $\\Delta = M^T$. Then given the vector $P$ that contains the pressures on the vertices, then the pressure difference on the edges will be given by $\\Delta P$, where $\\Delta$ is the transpose of the incidence matrix. This comes in handy when we want to calculate the flow of the edges which will be given by\n",
+ "$$ \\bf{Q} = \\bf{C} L^{-1} \\bf{\\Delta} \\bf{P}, \\tag{2} $$\n",
+ "where $ C $ is a diagonal matrix of the conductance of the edges, $L$ is the diagonal matrix of the ``length'' of each edge, $\\Delta$ is the transpose of the incidence matrix, and $P$ is the pressure on the nodes. $Q$ is the flow of the edges. In this particular example we are assuming that the relation between flow and the pressure difference is $Q_e = C_e (p_i - p_j)/L$. But we can have many other choices.\n",
+ "\t\n",
+ "Knowing the sources and sinks on the nodes, the pressure can be determined by the Kirchhoff law\n",
+ "\t$$ \\mathcal{L} \\bf{P} = \\bf{q}, $$\n",
+ "where the vector $ q $ is the sources and the sinks values for each node. This is the same as solving the \\textbf{Poisson equation}. This can also be written in terms of the flow, i.e.\n",
+ "\t$$ \\Delta^T \\bf{Q} = \\bf{q}. $$\n",
+ "\tBy $ (2) $ we can write\n",
+ "\t$$ (\\bf{\\Delta}^T \\bf{C}\\bf{L}^{-1}\\Delta) \\bf{P} = \\bf{q}. $$\n",
+ "\tSince $ \\Delta = M^T $, the expression inside the parentheses is clearly Equation (1).\n",
+ "\t\n",
+ "Similar to the Poisson equation on the graph which is equivalent Kirchhoff's law, we can solve other types of heat and wave equations on the graph as well. The Laplacian matrix play a key role.\n",
+ "$$ \\frac{\\partial p}{\\partial t} = - \\mathcal{L} p + q, $$\n",
+ "\tfor the heat equation, and\n",
+ "$$ \\frac{\\partial^2 p}{\\partial t^2} = -\\mathcal{L}p + q, $$\n",
+ "\tfor the wave equation."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6bc26e96-34af-431c-84f3-7b535f68bd15",
+ "metadata": {},
+ "source": [
+ "## References"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1aa75011-0b16-4f25-b32c-79efe74c5f41",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.10.12"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
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+@InCollection{Atale2020,
+ author = {Atale, Neha and Rani, Vibha},
+ publisher = {Springer},
+ title = {Angiogenesis: {A} {Therapeutic} {Target} for {Cancer}},
+ year = {2020},
+ address = {Singapore},
+ editor = {Tuli, Hardeep Singh},
+ isbn = {9789811575860},
+ pages = {165--183},
+ abstract = {The proliferation and metastatic spread of tumor cells depend on the newly developed blood vessels. Vasculature not only provides an adequate supply of oxygen and nutrients but also removes waste products or gas exchange. The process of angiogenesis is controlled by various transcriptional factors and growth factors. It has been observed that the discovery of angiogenic inhibitors can help to reduce carcinomas growth. Presently, chemotherapeutic drugs mediated inhibition of hypoxia-inducible factor (HIF-1), which initiates neovascularization under hypoxic conditions in the tumor, is being investigated. Vascular endothelial growth factor (VEGF) and receptor VEGFR mediated activation of endothelial cells are also inhibited by chemotherapeutic drugs. Furthermore, chemotherapeutic drugs inhibit the PI3K/AKT/mTOR signaling pathways mediated growth of new blood vessels. The aim of this chapter would be to highlight the role of angiogenesis in cancer progression. Furthermore, various anti-cancer therapeutic strategies/trials based upon inhibition of blood vessels would also be discussed.},
+ doi = {10.1007/978-981-15-7586-0_9},
+ keywords = {Tumor angiogenesis, Endothelial cell, Vascular endothelial growth factor, Hypoxia-inducible factor, Anti-angiogenesis drugs},
+ language = {en},
+ shorttitle = {Angiogenesis},
+ url = {https://doi.org/10.1007/978-981-15-7586-0_9},
+ urldate = {2024-09-19},
+}
+
+@Article{Anderson1998,
+ author = {Anderson, A. R. and Chaplain, M. A.},
+ journal = {Bulletin of Mathematical Biology},
+ title = {Continuous and discrete mathematical models of tumor-induced angiogenesis},
+ year = {1998},
+ issn = {0092-8240},
+ month = sep,
+ number = {5},
+ pages = {857--899},
+ volume = {60},
+ abstract = {Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial-cell migration, and organize themselves into a dendritic structure. Subsequent cell proliferation near the sprout tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumors. In this paper we present both continuous and discrete mathematical models which describe the formation of the capillary sprout network in response to chemical stimuli (tumor angiogenic factors, TAF) supplied by a solid tumor. The models also take into account essential endothelial cell-extracellular matrix interactions via the inclusion of the matrix macromolecule fibronectin. The continuous model consists of a system of nonlinear partial differential equations describing the initial migratory response of endothelial cells to the TAF and the fibronectin. Numerical simulations of the system, using parameter values based on experimental data, are presented and compared qualitatively with in vivo experiments. We then use a discretized form of the partial differential equations to develop a biased random-walk model which enables us to track individual endothelial cells at the sprout tips and incorporate anastomosis, mitosis and branching explicitly into the model. The theoretical capillary networks generated by computer simulations of the discrete model are compared with the morphology of capillary networks observed in in vivo experiments.},
+ doi = {10.1006/bulm.1998.0042},
+ keywords = {Angiogenesis Inducing Agents, Animals, Arteriovenous Anastomosis, Capillaries, Cell Communication, Cell Division, Computer Simulation, Endothelium, Vascular, Fibronectins, Humans, Mitosis, Models, Biological, Neoplasms, Experimental, Neovascularization, Pathologic, Numerical Analysis, Computer-Assisted, Stochastic Processes},
+ language = {eng},
+ pmid = {9739618},
+}
+
+@Article{Nardini2021,
+ author = {Nardini, John T. and Stolz, Bernadette J. and Flores, Kevin B. and Harrington, Heather A. and Byrne, Helen M.},
+ journal = {PLoS Computational Biology},
+ title = {Topological data analysis distinguishes parameter regimes in the {Anderson}-{Chaplain} model of angiogenesis},
+ year = {2021},
+ issn = {1553-734X},
+ month = jun,
+ number = {6},
+ pages = {e1009094},
+ volume = {17},
+ abstract = {Angiogenesis is the process by which blood vessels form from pre-existing vessels. It plays a key role in many biological processes, including embryonic development and wound healing, and contributes to many diseases including cancer and rheumatoid arthritis. The structure of the resulting vessel networks determines their ability to deliver nutrients and remove waste products from biological tissues. Here we simulate the Anderson-Chaplain model of angiogenesis at different parameter values and quantify the vessel architectures of the resulting synthetic data. Specifically, we propose a topological data analysis (TDA) pipeline for systematic analysis of the model. TDA is a vibrant and relatively new field of computational mathematics for studying the shape of data. We compute topological and standard descriptors of model simulations generated by different parameter values. We show that TDA of model simulation data stratifies parameter space into regions with similar vessel morphology. The methodologies proposed here are widely applicable to other synthetic and experimental data including wound healing, development, and plant biology., Vascular networks play a key role in many physiological processes, by delivering nutrition to, and removing waste from, biological tissues. In cancer, tumors stimulate the growth of new blood vessels, via a process called angiogenesis. The resulting vascular structure comprises many inter-connected vessels which lead to emergent morphologies that influence the rate of tumor growth and treatment efficacy. In this work, we consider several approaches to summarize the morphology of synthetic vascular networks generated from a mathematical model of tumor-induced angiogenesis. We find that a topological approach can be used quantify vascular morphology of model simulations and group the simulations into biologically interpretable clusters. This methodology may be useful for the diagnosis of abnormal blood vessel networks and quantifying the efficacy of vascular-targeting treatments.},
+ doi = {10.1371/journal.pcbi.1009094},
+ file = {PubMed Central Full Text PDF:Nardini2021 - Topological Data Analysis Distinguishes Parameter Regimes in the Anderson Chaplain Model of Angiogenesis.pdf:PDF:https\://www.ncbi.nlm.nih.gov/pmc/articles/PMC8270459/pdf/pcbi.1009094.pdf},
+ pmcid = {PMC8270459},
+ pmid = {34181657},
+ url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8270459/},
+ urldate = {2024-09-19},
+}
+
+@Article{Hannezo2017,
+ author = {Hannezo, Edouard and Scheele, Colinda L. G. J. and Moad, Mohammad and Drogo, Nicholas and Heer, Rakesh and Sampogna, Rosemary V. and van Rheenen, Jacco and Simons, Benjamin D.},
+ journal = {Cell},
+ title = {A {Unifying} {Theory} of {Branching} {Morphogenesis}},
+ year = {2017},
+ issn = {0092-8674},
+ month = sep,
+ number = {1},
+ pages = {242--255.e27},
+ volume = {171},
+ abstract = {The morphogenesis of branched organs remains a subject of abiding interest. Although much is known about the underlying signaling pathways, it remains unclear how macroscopic features of branched organs, including their size, network topology, and spatial patterning, are encoded. Here, we show that, in mouse mammary gland, kidney, and human prostate, these features can be explained quantitatively within a single unifying framework of branching and annihilating random walks. Based on quantitative analyses of large-scale organ reconstructions and proliferation kinetics measurements, we propose that morphogenesis follows from the proliferative activity of equipotent tips that stochastically branch and randomly explore their environment but compete neutrally for space, becoming proliferatively inactive when in proximity with neighboring ducts. These results show that complex branched epithelial structures develop as a self-organized process, reliant upon a strikingly simple but generic rule, without recourse to a rigid and deterministic sequence of genetically programmed events.},
+ doi = {10.1016/j.cell.2017.08.026},
+ file = {ScienceDirect Full Text PDF:https\://www.sciencedirect.com/science/article/pii/S0092867417309510/pdfft?md5=a53bc8b7423aa0fc223341aeb0553091&pid=1-s2.0-S0092867417309510-main.pdf&isDTMRedir=Y:application/pdf},
+ keywords = {branching morphogenesis, mammary gland, kidney, prostate, mathematical modeling, branching and annihilating random walks, self-organization},
+ url = {https://www.sciencedirect.com/science/article/pii/S0092867417309510},
+ urldate = {2024-09-19},
+}
+
+@Article{Ucar2021,
+ author = {Uçar, Mehmet Can and Kamenev, Dmitrii and Sunadome, Kazunori and Fachet, Dominik and Lallemend, Francois and Adameyko, Igor and Hadjab, Saida and Hannezo, Edouard},
+ journal = {Nature Communications},
+ title = {Theory of branching morphogenesis by local interactions and global guidance},
+ year = {2021},
+ issn = {2041-1723},
+ month = nov,
+ number = {1},
+ pages = {6830},
+ volume = {12},
+ abstract = {Branching morphogenesis governs the formation of many organs such as lung, kidney, and the neurovascular system. Many studies have explored system-specific molecular and cellular regulatory mechanisms, as well as self-organizing rules underlying branching morphogenesis. However, in addition to local cues, branched tissue growth can also be influenced by global guidance. Here, we develop a theoretical framework for a stochastic self-organized branching process in the presence of external cues. Combining analytical theory with numerical simulations, we predict differential signatures of global vs. local regulatory mechanisms on the branching pattern, such as angle distributions, domain size, and space-filling efficiency. We find that branch alignment follows a generic scaling law determined by the strength of global guidance, while local interactions influence the tissue density but not its overall territory. Finally, using zebrafish innervation as a model system, we test these key features of the model experimentally. Our work thus provides quantitative predictions to disentangle the role of different types of cues in shaping branched structures across scales.},
+ doi = {10.1038/s41467-021-27135-5},
+ keywords = {Animals, Animals, Genetically Modified, Genes, Reporter, Intravital Microscopy, Models, Animal, Models, Biological, Morphogenesis, Sensory Receptor Cells, Stochastic Processes, Zebrafish},
+ language = {eng},
+ pmcid = {PMC8613190},
+ pmid = {34819507},
+}
+
+@Article{Ucar2023,
+ author = {Uçar, Mehmet Can and Hannezo, Edouard and Tiilikainen, Emmi and Liaqat, Inam and Jakobsson, Emma and Nurmi, Harri and Vaahtomeri, Kari},
+ journal = {Nature Communications},
+ title = {Self-organized and directed branching results in optimal coverage in developing dermal lymphatic networks},
+ year = {2023},
+ issn = {2041-1723},
+ month = sep,
+ pages = {5878},
+ volume = {14},
+ abstract = {Branching morphogenesis is a ubiquitous process that gives rise to high exchange surfaces in the vasculature and epithelial organs. Lymphatic capillaries form branched networks, which play a key role in the circulation of tissue fluid and immune cells. Although mouse models and correlative patient data indicate that the lymphatic capillary density directly correlates with functional output, i.e., tissue fluid drainage and trafficking efficiency of dendritic cells, the mechanisms ensuring efficient tissue coverage remain poorly understood. Here, we use the mouse ear pinna lymphatic vessel network as a model system and combine lineage-tracing, genetic perturbations, whole-organ reconstructions and theoretical modeling to show that the dermal lymphatic capillaries tile space in an optimal, space-filling manner. This coverage is achieved by two complementary mechanisms: initial tissue invasion provides a non-optimal global scaffold via self-organized branching morphogenesis, while VEGF-C dependent side-branching from existing capillaries rapidly optimizes local coverage by directionally targeting low-density regions. With these two ingredients, we show that a minimal biophysical model can reproduce quantitatively whole-network reconstructions, across development and perturbations. Our results show that lymphatic capillary networks can exploit local self-organizing mechanisms to achieve tissue-scale optimization., The lymphatic system is a transport network that controls immune response and tissue fluid circulation in the body. Here the authors combine experiment and theory to reveal that developing lymphatic capillary networks exploit complementary branching strategies to optimize tissue coverage.},
+ doi = {10.1038/s41467-023-41456-7},
+ file = {PubMed Central Full Text PDF:Ucar2023 - Self Organized and Directed Branching Results in Optimal Coverage in Developing Dermal Lymphatic Networks.pdf:PDF:https\://www.ncbi.nlm.nih.gov/pmc/articles/PMC10514270/pdf/41467_2023_Article_41456.pdf},
+ pmcid = {PMC10514270},
+ pmid = {37735168},
+ url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10514270/},
+ urldate = {2024-09-19},
+}
+
+@Article{Pries2010,
+ author = {Pries, Axel R. and Höpfner, Michael and le Noble, Ferdinand and Dewhirst, Mark W. and Secomb, Timothy W.},
+ journal = {Nature Reviews Cancer},
+ title = {The shunt problem: control of functional shunting in normal and tumour vasculature},
+ year = {2010},
+ issn = {1474-1768},
+ month = aug,
+ number = {8},
+ pages = {587--593},
+ volume = {10},
+ abstract = {This Opinion article discusses how aberrant vascular remodelling might lead to tumour hypoxia. What controls this process, and could this be reversed — or normalized — by anti-angiogenic therapies?},
+ copyright = {2010 Springer Nature Limited},
+ doi = {10.1038/nrc2895},
+ keywords = {Mechanisms of disease, Targeted therapies, Tumour angiogenesis},
+ language = {en},
+ publisher = {Nature Publishing Group},
+ shorttitle = {The shunt problem},
+ url = {https://www.nature.com/articles/nrc2895},
+ urldate = {2024-09-19},
+}
+
+@Article{Secomb2012,
+ author = {Secomb, Timothy W. and Dewhirst, Mark W. and Pries, Axel R.},
+ journal = {Basic \& Clinical Pharmacology \& Toxicology},
+ title = {Structural {Adaptation} of {Normal} and {Tumour} {Vascular} {Networks}},
+ year = {2012},
+ issn = {1742-7843},
+ number = {1},
+ pages = {63--69},
+ volume = {110},
+ abstract = {Abstract: Vascular networks are dynamic structures, adapting to changing conditions by structural remodelling of vessel diameters and by growth of new vessels and regression of existing vessels. The vast number of blood vessels in the circulatory system, more than 109, implies that vessels’ arrangement and structure are not under individual genetic control but emerge as a result of generic responses of each segment to the various stimuli that it experiences. To obtain insight into the types of response that are needed, a network-oriented approach has been used, in which theoretical models are used to simulate structural adaptation in vascular networks, and the results are compared with experimental observations. With regard to the structural control of vessel diameters, this approach shows that responses to both haemodynamic and metabolic stimuli are needed for the formation of functionally adequate and efficient network structures. Furthermore, information transfer in both upstream and downstream directions is essential for balancing flows between long and short flow pathways. Otherwise, functional shunting occurs, that is, short pathways become enlarged and flow bypasses longer pathways. Information transfer in the upstream direction is achieved by conducted responses communicated along vessel walls. Simulations of structural adaptation in tumour microvascular networks indicate that impaired vascular communication, resulting in functional shunting, may be an important factor causing the dysfunctional microcirculation and local hypoxia typically observed in tumours. Anti-angiogenic treatment of tumours may restore vascular communication and thereby improve or normalize flow distribution in tumour vasculature.},
+ copyright = {© 2011 The Authors. Basic \& Clinical Pharmacology \& Toxicology © 2011 Nordic Pharmacological Society},
+ doi = {10.1111/j.1742-7843.2011.00815.x},
+ file = {Full Text PDF:https\://onlinelibrary.wiley.com/doi/pdfdirect/10.1111/j.1742-7843.2011.00815.x:application/pdf},
+ language = {en},
+ url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1742-7843.2011.00815.x},
+ urldate = {2024-09-19},
+}
+
+@Article{Pries2014,
+ author = {Pries, Axel R. and Secomb, Timothy W.},
+ journal = {Physiology},
+ title = {Making {Microvascular} {Networks} {Work}: {Angiogenesis}, {Remodeling}, and {Pruning}},
+ year = {2014},
+ issn = {1548-9213},
+ month = nov,
+ number = {6},
+ pages = {446--455},
+ volume = {29},
+ abstract = {The adequate and efficient functioning of the microcirculation requires not only numerous vessels providing a large surface area for transport but also a structure that provides short diffusion distances from capillaries to tissue and efficient distribution of convective blood flow. Theoretical models show how a combination of angiogenesis, remodeling, and pruning in response to hemodynamic and metabolic stimuli, termed “angioadaptation,” generates well organized, functional networks.},
+ doi = {10.1152/physiol.00012.2014},
+ pmcid = {PMC4280154},
+ pmid = {25362638},
+ shorttitle = {Making {Microvascular} {Networks} {Work}},
+ url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4280154/},
+ urldate = {2024-09-19},
+}
+
+@Article{Ouarne2021,
+ author = {Ouarné, Marie and Pena, Andreia and Franco, Cláudio Areias},
+ journal = {Cells \& Development},
+ title = {From remodeling to quiescence: {The} transformation of the vascular network},
+ year = {2021},
+ issn = {2667-2901},
+ month = dec,
+ pages = {203735},
+ volume = {168},
+ abstract = {The vascular system is essential for embryogenesis, healing, and homeostasis. Dysfunction or deregulated blood vessel function contributes to multiple diseases, including diabetic retinopathy, cancer, hypertension, or vascular malformations. A balance between the formation of new blood vessels, vascular remodeling, and vessel quiescence is fundamental for tissue growth and function. Whilst the major mechanisms contributing to the formation of new blood vessels have been well explored in recent years, vascular remodeling and quiescence remain poorly understood. In this review, we highlight the cellular and molecular mechanisms responsible for vessel remodeling and quiescence during angiogenesis. We further underline how impaired remodeling and/or destabilization of vessel networks can contribute to vascular pathologies. Finally, we speculate how addressing the molecular mechanisms of vascular remodeling and stabilization could help to treat vascular-related disorders.},
+ doi = {10.1016/j.cdev.2021.203735},
+ keywords = {Humans, Neovascularization, Pathologic, Vascular Remodeling, Angiogenesis, Endothelial cell, Vascular disease, Vascular quiescence, Vascular remodeling},
+ language = {eng},
+ pmid = {34425253},
+ shorttitle = {From remodeling to quiescence},
+}
+
+@Article{Richard2018,
+ author = {Richard, Sophie and Brun, Amanda and Tedesco, Antonio and Gallois, Benjamin and Taghi, Naoual and Dantan, Philippe and Seguin, Johanne and Fleury, Vincent},
+ journal = {Communications Biology},
+ title = {Direct imaging of capillaries reveals the mechanism of arteriovenous interlacing in the chick chorioallantoic membrane},
+ year = {2018},
+ issn = {2399-3642},
+ month = dec,
+ number = {1},
+ pages = {1--11},
+ volume = {1},
+ abstract = {Understanding vascular development in vertebrates is an important scientific endeavor. Normal vasculatures generally start off as a disorganized capillary lattice which progressively matures into a well-organized vascular loop comprising a hierarchy of arteries and veins. One striking feature of vascular development is the interlacing of arteries and veins. How arteries and veins manage to avoid themselves and interlace with such a perfect architecture is not understood. Here we present a detailed view of the development of the vasculature in the chorioallantoic membrane of the chicken embryo. We find that the origin of arteriovenous interlacing lies in the presence of an increased hemodynamic resistance at the distal part of the arteries due to vascular flattening onto the ectodermal surface. This reduces the vascular conductance distally, thus repelling veins away. In more proximal parts, vessels round off into cylinders and the increased flow attracts veins.},
+ copyright = {2018 The Author(s)},
+ doi = {10.1038/s42003-018-0229-x},
+ file = {Full Text PDF:Richard2018 - Direct Imaging of Capillaries Reveals the Mechanism of Arteriovenous Interlacing in the Chick Chorioallantoic Membrane.pdf:PDF:https\://www.nature.com/articles/s42003-018-0229-x.pdf},
+ keywords = {Angiogenesis, Cardiovascular biology, Time-lapse imaging},
+ language = {en},
+ publisher = {Nature Publishing Group},
+ url = {https://www.nature.com/articles/s42003-018-0229-x},
+ urldate = {2024-09-19},
+}
+
+@Article{Ronellenfitsch2016,
+ author = {Ronellenfitsch, Henrik and Katifori, Eleni},
+ journal = {Physical Review Letters},
+ title = {Global {Optimization}, {Local} {Adaptation}, and the {Role} of {Growth} in {Distribution} {Networks}},
+ year = {2016},
+ month = sep,
+ number = {13},
+ pages = {138301},
+ volume = {117},
+ abstract = {Highly optimized complex transport networks serve crucial functions in many man-made and natural systems such as power grids and plant or animal vasculature. Often, the relevant optimization functional is nonconvex and characterized by many local extrema. In general, finding the global, or nearly global optimum is difficult. In biological systems, it is believed that such an optimal state is slowly achieved through natural selection. However, general coarse grained models for flow networks with local positive feedback rules for the vessel conductivity typically get trapped in low efficiency, local minima. In this work we show how the growth of the underlying tissue, coupled to the dynamical equations for network development, can drive the system to a dramatically improved optimal state. This general model provides a surprisingly simple explanation for the appearance of highly optimized transport networks in biology such as leaf and animal vasculature.},
+ doi = {10.1103/PhysRevLett.117.138301},
+ file = {Full Text PDF:https\://journals.aps.org/prl/pdf/10.1103/PhysRevLett.117.138301:application/pdf},
+ publisher = {American Physical Society},
+ url = {https://link.aps.org/doi/10.1103/PhysRevLett.117.138301},
+ urldate = {2024-09-19},
+}
+
+@Article{Almeida2022,
+ author = {Almeida, Rodrigo and Dilão, Rui},
+ journal = {Physica D: Nonlinear Phenomena},
+ title = {Adaptive {Hagen}–{Poiseuille} flows on graphs},
+ year = {2022},
+ issn = {0167-2789},
+ month = aug,
+ pages = {133322},
+ volume = {436},
+ abstract = {We derive a class of equations describing low Reynolds number steady flows of incompressible and viscous fluids in networks made of straight channels, with several sources and sinks and adaptive conductivities. A graph represents the network, and the fluxes at sources and sinks control the flow. The adaptive conductivities describe the transverse channel elasticities, mirroring several network structures found in physics and biology. Minimising the dissipated energy per unit of time, we have found an explicit form for the adaptation equations and, asymptotically in time, a steady state tree geometry for the graph connecting sources and sinks is reached. A phase transition tuned by an order parameter for the adapted steady state graph has been found.},
+ doi = {10.1016/j.physd.2022.133322},
+ file = {ScienceDirect Full Text PDF:https\://www.sciencedirect.com/science/article/pii/S0167278922001014/pdfft?md5=ea8ba7d5cfac168b6d6c0f287f798189&pid=1-s2.0-S0167278922001014-main.pdf&isDTMRedir=Y:application/pdf},
+ keywords = {Hagen–Poiseuille flows in networks, Biological networks, Flows in microchannels, Networks},
+ url = {https://www.sciencedirect.com/science/article/pii/S0167278922001014},
+ urldate = {2024-09-19},
+}
+
+@Comment{jabref-meta: databaseType:bibtex;}