Tetsuji Tokihiro

Branching structures such as vascular networks are representative morphological patterns in living systems, and they often arise from collective cell migration. Angiogenesis, the sprouting of new blood vessels from pre-existing ones, is a fundamental process in development. Experimental and theoretical studies have demonstrated that sprout formation depends on the collective movements and shapes of endothelial cells, as well as the remodelling of the extracellular matrix. Many discrete models have been proposed to describe cell dynamics, successfully reproducing vascular patterns and collective behaviours. In this study, we present a two-dimensional mathematical model that represents each endothelial cell as an ellipse and incorporates the effects of the extracellular matrix. We performed computer simulations under two scenarios: invasion from a pre-formed sprout and collective advancement into an extracellular matrix region. The results show that the extracellular matrix helps maintain linear sprout extension and suppresses the formation of dispersed or curved branches, while elongated cell shapes promote sprouting more effectively than round cells. The model also reproduces experimentally observed behaviours such as tip-cell replacement and the mixing of cells within sprouts. These findings highlight the importance of integrating cell shape and extracellular matrix remodelling to understand early blood vessel formation.

We examine stochastic phase models for the community effect of cardiac muscle cells. Our model extends the stochastic integrate-and-fire model by incorporating irreversibility after beating, induced beating, and refractoriness. We focus on investigating the expectation and variance in the synchronized beating interval. Specifically, for a single isolated cell, we obtain the closed-form expectation and variance in the beating interval, discovering that the coefficient of variation has an upper limit of 2/3. For two coupled cells, we derive the partial differential equations for the expected synchronized beating intervals and the distribution density of phases. Furthermore, we consider the conventional Kuramoto model for both two- and N-cell models. We establish a new analysis using stochastic calculus to obtain the coefficient of variation in the synchronized beating interval, thereby improving upon existing literature.

Abstract We introduce a three-dimensional mathematical model for the dynamics of vascular endothelial cells during sprouting angiogenesis. Angiogenesis is the biological process by which new blood vessels form from existing ones. It has been the subject of numerous theoretical models. These models have successfully replicated various aspects of angiogenesis. 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. 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. 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. Numerical simulations demonstrate that our model reproduces the patterns of elongation and branching observed in the early stages of angiogenesis. We show that the pattern formation of the cell population is strongly dependent on the cell shape. Finally, we demonstrate that our current mathematical model reproduces the cell behaviours, specifically cell-mixing, observed in sprouts.

Abstract. We investigate a simple mathematical model for angiogenesis. From recent time-lapse imaging experiments on the dynamics of endothelial cells (ECs) in angiogenesis, we suppose that elongation and bifurcation of neogenetic vessel is determined by only the density of ECs near the tip, and introduce a model described by nonlinear simultaneous differential equations. We also incorporate proliferation of ECs and activation factor such as VEGF and show the exact solutions to that model and numerical simulations.