時弘 哲治

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.

概要．血管新生を記述する単純な数理モデルを考察する．最近の血管新生における内皮細胞運動のタイムラプス撮影の実験を参考に，内皮細胞の新生血管先端での密度のみにより，新生血管の伸長と分岐が定まることを仮定すると，非線形連立常微分方程式によって記述されるモデルが導かれる．細胞分裂および VEGFなどの内皮細胞の運動を活性化する因子の影響も取り入れ，厳密解および数値シミュレーションの結果を示す．