Mathematical Modeling and Computer Simulation of Moving Boundary Problems in Biofluids

研究人员开发了数学模型和计算方法，用于研究生物流体中两个特定的移动边界问题：真核纤毛和鞭毛的运动以及脊椎动物胚胎肢芽生长的数学建模和数值模拟。纤毛和鞭毛轴突模型结合了单个动力臂的离散表示，轴突弹性结构（如微管和内链）的力学模型，以及周围流体力学的连续描述。肢芽生长和模式模型包括肢芽组织的流体力学描述，反应-扩散-平流成分，控制生长因子和形态因子在特定区域（如顶端外胚层和极化活动区）的时空分布，以及代表肢芽外胚层粘弹性特性的移动弹性边界。在三维模型中，肢体芽外胚层被建模为一个由线性弹性元件连接的节点网络。研究者还开发了一种算法来施加适当的边界条件来求解反应-扩散-平流方程。因为每个这些模型包括流体力学表示，研究者使用类似的计算方法，基于浸入边界法和计算机代码共享许多主要子程序。本研究项目的目的是发展研究复杂生物系统中移动边界问题的数学模型和计算方法。研究者开发了两种类型的模型：一种用于真核纤毛和鞭毛的运动，第二种用于多细胞生物系统的生长动力学。这两种模型都涉及到粘弹性材料在生物学中的建模方法，并共享一个共同的计算观点。第一个项目的目标是为单个纤毛或鞭毛建立一个三维模型。虽然人们对真核生物纤毛和鞭毛的生物化学、超微结构和运动已经有了大量的了解，但对其振荡运动和弯曲的控制机制还没有很好的了解。纤毛和鞭毛建模的目的是建立一个三维模型，可以作为研究控制波形的机械化学机制的平台。生长对多细胞生物和单细胞生物群落都起着重要的作用。研究者开发了一个脊椎动物肢体发育模型，该模型结合了发育中的肢体芽内细胞和组织的生长和运动，生长因子在肢体芽内的运输和产生，以及代表肢体芽外胚层机械和生化特性的移动边界。该模型的发展和求解相关数学方程的计算方法已应用于各种多细胞系统的研究，其中生长是一个基本特征，包括发育生物学中的生长和模式形成，细菌菌落的生长以及肿瘤的生长。该项目的一个重要方面是发展研究模型的数值方法；这一领域的进展可用于计算生物流体领域的各种问题。

Dillon9805501 The investigator develops mathematical models andcomputational methods for studying two specific moving boundaryproblems in biofluids: the motion of eukaryotic cilia andflagella and the mathematical modeling and numerical simulationof the growing embryonic vertebrate limb bud. The model for thecilia and flagella axoneme incorporates discrete representationsof the individual dynein arms, a mechanical model of theaxoneme's elastic structures such as the microtubules and nexinlinks, and a continuous description of the surrounding fluidmechanics. The model for limb bud outgrowth and patterningconsists of a fluid-mechanical description of the limb budtissue, a reaction-diffusion-advection component that governs thespatio-temporal distribution of growth factors and morphogensproduced in specialized regions such as the apical ectodermalridge and the zone of polarizing activity, and a moving elasticboundary that represents the limb bud ectoderm's viscoelasticproperties. In the three-dimensional model, the limb bud ectodermis modeled as a network of nodes connected by linear elasticelements. The investigator also develops an algorithm forimposing the appropriate boundary conditions in solving thereaction-diffusion-advection equations. Because each of thesemodels includes a fluid-mechanical representation, theinvestigator uses a similar computational approach based on theimmersed boundary method and the computer codes share many of themajor subroutines. The purpose of this research project is the development ofmathematical models and computational methods for studying movingboundary problems in complex biological systems. The investigatordevelops two types of models: one for the motion of eukaryoticcilia and flagella and the second for the growth dynamics ofmulticellular biological systems. Both of these models involvenew approaches to the modeling of viscoelastic materials inbiology and share a common computational point of view. Theobject in the first project is to develop a three-dimensionalmodel for the individual cilium or flagellum. Although a greatdeal has been discovered regarding the biochemistry,ultrastructure and movement of eukaryotic cilia and flagella, themechanisms governing the control of oscillatory motion andbending are not well understood. The goal of the cilia andflagella modeling is to develop a three-dimensional model thatcan serve as a platform for studying the mechanochemicalmechanisms that control the waveforms. Growth plays an importantprocess for multicellular organisms as well as communities ofsingle cell organisms. The investigator develops a model ofvertebrate limb development that incorporates the growth andmovement of cells and tissue within the developing limb bud, thetransport and production of growth factors within the limb bud,and a moving boundary that represents the mechanical andbiochemical properties of the limb bud ectoderm. The developmentof this model and the computational methods for solving theassociated mathematical equations has application to the study ofa variety of multicellular systems where growth is an essentialfeature, including growth and pattern formation in developmentalbiology, the growth of bacterial colonies, as well as tumorgrowth. An important aspect of this project is the development ofnumerical methods for studying the models; advances in this areacan be used in a variety of problems in the field ofcomputational biofluids.