![]() ![]() They are described by PDEs for the fields of cell density (mass conservation law and the momentum balance equation) and chemotactic factor concentration (reaction diffusion equation).Ĭell-based models have also been proposed for vascular network construction. Mathematical models are proposed in which the cells are accelerated by gradients of chemoattractants released by the cells, diffuse and degrade in finite time 9, 10. Before mesenchymal movement is activated, cells undergo a faster amoeboid-type migration driven by chemical factors such as Vascular Endothelial Growth Factor (VEGF). Another important factor in cell movement is the effect of chemoattraction. According to these hypotheses, the model is described by nonlinear partial differential equations (PDEs). Cell movement is assumed to mimic a random walk with a bias towards areas of maximum strain, and cell locomotion is modelled as a diffusive movement with diffusion dependent on the local strain. For example, a mechanical model is based on the hypothesis that the extracellular matrix (ECM) is reorganised and the cellular networks are formed as a result of the traction forces exerted by the cells on the matrix and the elasticity of the matrix 8. It is also interesting to explore which properties of endothelial cells are associated with the vascular network.Īngiogenesis and vasculogenesis, the latter of wihch refers to the de novo formation of endothelial cells from mesodermal precursors, have inspired numerous mathematical models that focus on the formation of the initial embryonic vascular network 5, 6, 7. Understanding endothelial cell behaviour during sprouting, such as ’cell-mixing’, is of considerable theoretical and experimental interest. Particularly in the early stages of angiogenesis, endothelial cells exhibit complex behaviours such as overtaking each other, making U-turns, and intermixing, which are referred to as ’cell-mixing’ 3, 4. Over time, these sprouts are lumenized, culminating in the formation of mature blood vessels. This triggers the degradation of the underlying basement membrane, enabling endothelial cells to form new sprouts. When cells surrounding an existing blood vessel become hypoxic or inflamed, they secrete vascular endothelial growth factor (VEGF) among other factors. The general process of angiogenesis unfolds as follows. Consequently, understanding the mechanisms of angiogenesis has become a crucial topic in modern medicine. It is also implicated in the progression of pathological conditions such as malignant tumours and retinopathy 2. Similar content being viewed by othersĪngiogenesis, a biological phenomenon in which new vascular networks are constructed from existing blood vessels, plays a significant role in development, wound healing, menstruation, and pregnancy 1. Finally, we demonstrate that our current mathematical model reproduces the cell behaviours, specifically cell-mixing, observed in sprouts. We show that the pattern formation of the cell population is strongly dependent on the cell shape. Numerical simulations demonstrate that our model reproduces the patterns of elongation and branching observed in the early stages of angiogenesis. In our model, the shape of a vascular endothelial cell is represented as a spheroid, and a discrete dynamical system is constructed based on the simple assumption of two-body interactions. While most mathematical models are two-dimensional, we aim to investigate whether ellipsoids also form branch-like structures and how their shape affects the pattern. Recent studies using particle-based models have highlighted the significant influence of cell shape on network formation, with elongated cells contributing to the formation of branching structures. These models have successfully replicated various aspects of angiogenesis. ![]() It has been the subject of numerous theoretical models. Angiogenesis is the biological process by which new blood vessels form from existing ones. ![]() We introduce a three-dimensional mathematical model for the dynamics of vascular endothelial cells during sprouting angiogenesis. ![]()
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