Resumen de Cellular dynamics models of angiogenesis
Rocío Vega Martínez
INTRODUCTION AND MOTIVATION Angiogenesis is a process by which new blood vessels sprout and grow from existing ones. This process occurs in normal and vital processes such as wound healing or the growth of a fetus. However, it also plays a crucial role in the development of several pathological processes, such as cancer, age-related macular degeneration or diabetes.
The difference between physiological and pathological angiogenesis is a matter of balance. In a healthy process, angiogenesis develops to its proper extent and then stops, while in pathological processes angiogenesis does not stop or it does not develop sufficiently. Angiogenesis keeps the number of blood vessels needed in balance: few blood vessels cause tissue death, while uncontrolled vascular proliferation can lead to cancer, macular degeneration and other diseases.
Cancer kills 26.4% of Spanish people. It is the second cause of death, just behind diseases of the circulatory system, 28.3%. On the other hand, age-related macular degeneration may cause severe loss of vision or blindness in many people, particularly the elderly. It is projected that 196 million people will be affected by age-related macular degeneration in 2020, increasing to 288 million by 2040. Therefore, the study of angiogenesis is essential for the treatment of certain diseases that are very present in our society.
Angiogenesis is a complex, multistep and well regulated process where biochemistry and physics are intertwined. The process entails signaling in vessel cells being driven by both chemical and mechanical mechanisms that result in vascular cell movement, deformation and proliferation. Cells migrates following increasing gradients of Vascular Endothelial Growth Factors (VEGF) secreted from the hypoxic region (chemotaxis), of adhesion to substrate (haptotaxis) and of substrate stiffness (durotaxis). In a later stage of angiogenesis, vessel cells rearrange to form lumen and allow the perfusion of the blood inside the sprout. Depending on what induces the angiogenesis, different environments and cells should be considered, for instance in the retina.
PREVIOUS MATHEMATICAL MODELS OF ANGIOGENESIS Beyond experimental investigations, mathematical models of angiogenesis try to help in understanding the process and how the relevant mechanisms of angiogenesis interact. A crucial question about modeling is how to integrate the multiple scales and mechanisms present in angiogenesis in a mathematical model. A model is expected to be useful to explore methods for promoting and inhibiting angiogenesis. However, answering this question with this expectation is not a simple task. Early models in angiogenesis described Endothelial Cells (ECs) and the blood vessel network as continuous density fields, thus omitting the structural details of the network that the sprouts form. Since these models, the field has evolved significantly. Currently, we could classify the models that describe evolving angiogenesis networks into three large groups: Tip cell (agent based) models, phase field models and cellular Potts models.
Tip cell models of angiogenesis assume that the tip cell establishes the path that the stalk cells of the sprout should follow. These models ignore length scales smaller than a capillary and consider tip cells to be point particles. Then blood vessels advancing toward the hypoxic region are simply tip cell trajectories. Similar but more complex agent models incorporate tip-stalk cell dynamics, chemotaxis and haptotaxis.
A phase field model is a continuum model that is able to describe vascular networks by introducing an extra phase field that adopts different values inside and outside vessels and marks the boundaries thereof. Phase field models can include other features such as elasticity, a force at the vessel tip and haptotaxis.
The models of angiogenesis of the first two groups do not describe the shape of cells or how it changes during vessel formation. However, we need to consider models that capture dynamics of the cell for a more precise analysis of the role of cell mechanics and adhesion in angiogenesis, since endothelial cells migrate by durotaxis and haptotaxis too. Thus, treating cells as point particles of tip cells model or a limited distinction between stalk and tip cells of phase field models are not enough and we need to add extra dynamics for them. Assembling all the processes involved with their different time and length scales requires to develop a cellular dynamics model combined with models for the continuum fields. A Cellular Potts Model (CPM) is particularly useful at these scales. CPM uses a Monte Carlo dynamics coupled to continuum fields (elastic fields, VEGF, …) to be capable of dynamically capturing the shape of the cell. Based on the Metropolis algorithm, this lattice-based computational modeling method allows to simulate the collective behavior of cells and Extracellular Matrix (ECM). Despite all the advantages of the CPM, no CPM presented to date has considered durotaxis, chemotaxis and cell signaling together.
MATHEMATICAL MODEL OF EARLY STAGE ANGIOGENESIS For this reason, the first mathematical model that we propose in the thesis is a hybrid cellular Potts model of early stage angiogenesis. The energy functional of the CPM takes into account terms to control the area, perimeter and length of the endothelial cells, as well as adhesion between ECs or ECs with ECM.
To account for chemotaxis, we include a term in the energy functional that favors cell motion towards the hypoxic region, where the VEGF concentration is higher. The VEGF concentration obeys an initial-boundary value problem for a partial differential equation containing diffusion, decay and a consumption term.
The term added to the energy functional related to durotaxis needs the strains of the extracellular matrix. We calculate the ECM strains by using the finite element method to solve the stationary Navier equations of linear elasticity. We consider a stiffness matrix, an array of the displacements of all nodes and an array of the traction forces per unit substrate thickness exerted by the cells. The displacements at each pixel are used to obtain the strain tensor.
The most novel part of our CPM is the inclusion of a system of ordinary differential equations that describes the Notch Signaling Pathway for a given cell. The solution of the system gives the number of Notch, Delta and Jagged proteins and the number of Notch intracellular domain (NICD), VEGF receptor (VEGFR) and VEGF molecules for each cell. These equations are coupled with those of other neighboring cells through ‘external’ terms that include the numbers of some neighbors’ proteins.
The Notch signaling pathway is used to distinguish the phenotype of endothelial cells. They may be on a tip, hybrid or stalk cell phenotype depending on the number of VEGF molecules they possess.
Endothelial cells also split and proliferate in our model. For each sprouting vessel, only one stalk cell in contact with a tip cell is randomly chosen to undergo proliferation. We use the unsupervised machine learning algorithm K-means clustering to split the cell. This algorithm calculates the Euclidean distance of each pixel in the cell to the centroid of two groups of pixels and corrects the centroids until the two groups are balanced. These two groups become the new cells.
Additionally, when a stalk cell acquires the tip cell or the hybrid tip/stalk cell phenotype, they may lead a new active sprouting branch depending on their localization within the existing branch and on their moving direction. The ECM strains are considered to decide whether the new tip cell branches out.
This hybrid cellular Potts model is used to unravel the regulating role of Jagged, Notch and Delta dynamics in vascular cells. These membrane proteins have an important part in determining the leading cell in each neovascular sprout and also in branching, anastomosis and speed of angiogenesis. Simulations of this model show that although anastomosis is driven by chemotaxis, it may be favored or impeded depending on the mechanical configuration of the strains near the tip cells. Regarding to the Notch signaling pathway, we have found that increasing the production rate of Jagged produce a thinner vasculature that advances faster due to the larger number of cells with hybrid phenotype. On the other hand, increasing the production rate of Delta lowers the number of tip cells therefore there are less sprouts and anastomosis is less frequent. Quantitative results have confirmed that an imbalance of the Jagged production results in pathological angiogenesis that can be corrected by increasing the Delta production rate to diminish the number of tips and slow down somewhat angiogenesis. These results have been obtained for two types of cell dynamics: rounder and insensitive to chemical and mechanical cues stalk cells and more elongate and motile stalk cells. In contrast to recent models, the mathematical model of angiogenesis presented in this thesis illustrates the relative importance of mechanical, chemical and cellular cues when they are all considered simultaneously.
MATHEMATICAL MODEL OF ANGIOGENESIS IN THE RETINA The advantages of discovering the reasons why angiogenesis starts in the retina or inhibitory mechanisms are innumerable. Unraveling the causes of Choroidal Neovascularization (CNV) in the retina and giving possible solutions for age-related macular degeneration are our motivation to adapt the angiogenesis model described above to the retina in this thesis.
We consider the simple geometry: a square domain in which the Bruch’s Membrane (BM) separates the choroid crisscrossed by blood vessels, which may issue angiogenic sprouts, from retinal pigment epithelium (RPE) cells, eventual drusen and a subretinal space on top of which there are photoreceptors. The choroid vessels may issue sprouts at randomly chosen points provided the VEGF concentration surpasses some threshold in those points. The growth of drusen above RPE cells turns on VEGF sources that attract the sprouts issued from the choroid vessels to them. Once ECs have crossed the BM, they either form subRPE type 1 CNV or subretinal type 2 CNV. Type 1 CNV occurs if the sprouts form a network between the BM and the RPE cells, whereas type 2 CNV occurs if the sprouts succeed moving beyond the RPE layer and towards the VEGF emitting photoreceptors.
Besides ECs and the ECM, our CPM also considers the new objects: BM, RPE cells and drusen. Constraints of area and perimeter now affect all these objects, as well as the adhesion between the different types is considered. We ignore the outer segments of photoreceptors and their dynamics. Notch signaling, chemotaxis, durotaxis, cell proliferation, branching and anastomosis are also part of this model.
Our results confirm the often considered relationship between adhesion and type of CNV. Given enough VEGF concentration in the choroidal space, CNV occurs and its type is affected by adhesion defects. Impaired adhesion between the basement membrane of the RPE and the BM allows the cells to move easily in this space. Our simulations also show that a reduced lateral adhesion between RPE cells facilitates type 2 CNV. Furthermore, we studied adhesion between ECs realizing that whenever it is weakened, the ECs are able to intersperse RPE cells and drusen to change quickly from type 1 to type 2 CNV.
It is also known that high levels of VEGF concentration are needed for angiogenesis to begin. However, under the same parameter values and conditions, the value of the VEGF concentration at the point where the sprout tries to cross the RPE determines the sprout chances of starting type 2 CNV.
Additionally, our results show an extraordinary relationship between Notch signaling and age-related macular degeneration. Reducing the production rate of Jagged in a subretinal CNV decreases the number of blood vessel in this area and slows the speed of angiogenesis. Finally, we are able to recognize the type of CNV that develops during age-related macular degeneration looking and some Notch signaling proteins that work as markers.
MODELS OF LUMEN FORMATION IN SPROUTING ANGIOGENESIS Lumen weakening in capillaries may cause blood leaking from the sprouts in the retina. Therefore, due to its biological implications, lumen formation is an important feature to consider in mathematical and computational models of angiogenesis. However, the work that has been done in this field is limited. In this thesis, we present the two models of lumen formation we have developed.
In the first model, we start with a preformed sprout simulated with our angiogenesis model. We assume that the lumen formation occurs as a result of cell repulsion, due to the polarization of cells membranes, and vacuolization, that is the secretion of vacuoles filled with ECM fluids that form inside the cells. This assumption involves considering compartments within cells in the CPM, such as apical and basolateral membranes, vesicles, vacuoles, cytoplasm and luminal fluid. The adhesion between these compartments is the key of this model, but it is not enough to form the lumen and we have to fixed the cell’s membranes that are in contact to the ECM in order to achieve our goal. Although some restrictions in the model make its applications and possibilities limited, its study is convenient to establish the basis of the second proposed model.
The second and promising model is a hybrid CPM and it is inspired by some experiments of lumen formation in angiogenesis. The experiments suggest that lumen expansion during sprouting angiogenesis in vivo is motivated by blood flow through a process termed inverse blebbing. During inverse blebbing, ECs react to high external pressure by inducing spherical deformations of the apical membrane of ECs. The CPM includes haptotaxis, durotaxis, chemotaxis and constraints for area, perimeter and length of ECs and it oversees moving ECs and change their size. Meanwhile the unsupervised K-means algorithm is used to implement proliferation of ECs. Notch signaling pathway is incorporated to the model and it decides the phenotype of the cells of the sprout. However, only one sprout is considered without branching nor anastomosis in order to simplify at this point of model development. During the evolution of the sprout, the formation of the lumen is included through two extra terms in the energy functional. One of them favors the movement of ECs in the direction of some forces. These forces represent the pressure of the blood from the parent vessel and the mentioned unsupervised K-means algorithm is used again to set the forces. Since it is a 2D model and we consider the cross section of the vessel, ECs will be separated in two sides after the lumen formation. ECs except the tip cell are previously classified in one of these sides to add cell polarization in the other extra term of the energy functional for the adhesion between sides. This model is a work in progress, but the preliminary results show the capacity of the endothelial cells to rearrange and form the lumen in a sprout using the mechanism of inverse blebbing.
GPU-BASED PARALLEL IMPLEMENTATION OF CELLULAR DYNAMICS MODELS OF ANGIOGENESIS A critical step in the development of a mathematical and computational model is to analyze the viability of its simulations. Finally, we highlight the role that the simulation of these models has had in this thesis. We have carried out a novel parallel computing implementation of the models that uses C-CUDA on Graphics Processing Units (GPUs) to reduce the computational times as much as possible. The parallelization of numerical codes has allowed us to solve in an efficient, local and controlled way the complex problems that appear when coupling the multiple mechanisms and scales of angiogenesis. Facing the complexity of accomplish a GPU-based parallel implementation has been necessary to obtain the results of our models. This thesis includes a chapter with detailed information on the numerical codes that have been created to simulate our models.
