Wanho Lee

LUMPED PARAMETER MODELS OF CARDIOVASCULAR CIRCULATION IN NORMAL AND ARRHYTHMIA CASES

A new mathematical model of pumping heart coupled to lumped compartments of blood circulation is presented. This lumped pulsatile cardiovascular model consists of eight compart- ments of the body that include pumping heart, the systemic circu- lation, and the pulmonary circulation. The governing equations for the pressure and volume in each vascular compartment are derived from the following equations: Ohms law, conservation of volume, and the deflnition of compliances. The pumping heart is modeled by the time-dependent linear curves of compliances in the heart. We show that the numerical results in normal case are in agree- ment with corresponding data found in the literature. We extend the developed lumped model of circulation in normal case into a speciflc model for arrhythmia. These models provide valuable tools in examining and understanding cardiovascular diseases.

An Immersed Boundary Heart Model Coupled with a Multicompartment Lumped Model of the Circulatory System

A new computational model of the circulatory system is developed to investigate the intracardiac blood flow patterns and the motion of the mitral valve. In this work, we couple an existing two-dimensional model for the left heart with a multicompartment lumped model for the whole circulation to analyze the hemodynamics of the normal circulation in humans. The two-dimensional left-heart model is based on the immersed boundary method, and the lumped circulation model is governed by a system of ordinary differential equations. We investigate the intraventricular velocity field and the velocity curves over the mitral ring, between the valve leaflets, and across the aortic outflow tract. The flow and pressure curves are also measured in the left and right hearts and the systemic and pulmonary arteries. Our simulation results are in reasonably good agreement with ones in the literature and comparable to the existing magnetic resonance data, despite the inherent deficiency of a two-dimensional modeling of the left heart.

Simulations of Valveless Pumping in an Open Elastic Tube

Mathematical models and numerical simulations of flows driven by pumping without valves (valveless pumping) are presented. This work has been originally motivated by a biomedical objective to explain the complicated valveless blood flow mechanism in the circulation. For instance, the fetal circulation before the formation of valves and blood circulation during cardiopulmonary resuscitation. A mathematical model of valveless pumping either in a closed loop system or in an open system consists of a couple of tubes with different elasticities or radii. In this work, we develop new models which are not necessarily connected by two different materials: only one open elastic tube and an open elastic tube contained in a closed (almost) rigid tank. Although only one soft material is used or the soft and rigid materials are not connected, we have observed the existence of a net flow driven by the periodic compress-and-release action and the important features of valveless pumping that have been observed in earlier models or experiments. We have also shown that there exist mostly one-directional net flows inside the elastic tube if a long and thin tube is considered and the pumping is applied to the short portion near the edge of the tube. This might explain why most earlier reports on the physical experiments could not easily observe the existence of both directional net flows depending on the parameters, such as frequency. Another important result is that the direction and the magnitude of a net flow can be explained by the sign and the amount of power, which is work done on the fluid by the fluid pressure and the elastic wall over one period, respectively. A new feature in this work is that only one elastic material is sufficient to show the existence of a net flow in a valveless pump system. Because of a simple structure in our new model, it is much easier to construct a valveless pump system applicable to real world applications, such as a microelectromechanical system device.

The role of myosin II in glioma invasion: A mathematical model

Gliomas are malignant tumors that are commonly observed in primary brain cancer. Glioma cells migrate through a dense network of normal cells in microenvironment and spread long distances within brain. In this paper we present a two-dimensional multiscale model in which a glioma cell is surrounded by normal cells and its migration is controlled by cell-mechanical components in the microenvironment via the regulation of myosin II in response to chemoattractants. Our simulation results show that the myosin II plays a key role in the deformation of the cell nucleus as the glioma cell passes through the narrow intercellular space smaller than its nuclear diameter. We also demonstrate that the coordination of biochemical and mechanical components within the cell enables a glioma cell to take the mode of amoeboid migration. This study sheds lights on the understanding of glioma infiltration through the narrow intercellular spaces and may provide a potential approach for the development of anti-invasion strategies via the injection of chemoattractants for localization.

Modeling polymorphic transformation of rotating bacterial flagella in a viscous fluid

The helical flagella that are attached to the cell body of bacteria such as Escherichia coli and Salmonella typhimurium allow the cell to swim in a fluid environment. These flagella are capable of polymorphic transformation in that they take on various helical shapes that differ in helical pitch, radius, and chirality. We present a mathematical model of a single flagellum described by Kirchhoff rod theory that is immersed in a fluid governed by Stokes equations. We perform numerical simulations to demonstrate two mechanisms by which polymorphic transformation can occur, as observed in experiments. First, we consider a flagellar filament attached to a rotary motor in which transformations are triggered by a reversal of the direction of motor rotation [L. Turner et al., J. Bacteriol. 182, 2793 (2000)10.1128/JB.182.10.2793-2801.2000]. We then consider a filament that is fixed on one end and immersed in an external fluid flow [H. Hotani, J. Mol. Biol. 156, 791 (1982)10.1016/0022-2836(82)90142-5]. The detailed dynamics of the helical flagellum interacting with a viscous fluid is discussed and comparisons with experimental and theoretical results are provided.

An Immersed Boundary Method for a Contractile Elastic Ring in a Three-Dimensional Newtonian Fluid

In this paper, we present an immersed boundary method for modeling a contractile elastic ring in a three-dimensional Newtonian fluid. The governing equations are the modified Navier–Stokes equations with an elastic force from the contractile ring. The length of the elastic ring is time dependent and the ring shrinks with time because of its elastic nature in our proposed model. We dynamically reduce the number of Lagrangian boundary points when the distance between adjacent points is too small. This point-deleting algorithm helps keep the number of immersed boundary points in a single Cartesian mesh grid from becoming too high. We perform numerical experiments with various initial configurations of the contractile elastic ring, and numerical simulations to investigate the effects of the parameters are also conducted. The numerical results show that the proposed method can model and simulate the time-dependent contractile elastic ring in a three-dimensional Newtonian fluid.

A Multiscale Model of Cardiovascular System Including an Immersed Whole Heart in the Cases of Normal and Ventricular Septal Defect (VSD)

A mathematical and computational model combining the heart and circulatory system has been developed to understand the hemodynamics of circulation under normal conditions and ventricular septal defect (VSD). The immersed boundary method has been introduced to describe the interaction between the moving two-dimensional heart and intracardiac blood flow. The whole-heart model is governed by the Navier–Stokes system; this system is combined with a multi-compartment model of circulation using pressure–flow relations and the linearity of the discretized Navier–Stokes system. We investigate the velocity field, flowmeters, and pressure–volume loop in both normal and VSD cases. Simulation results show qualitatively good agreements with others found in the literature. This model, combining the heart and circulation, is useful for understanding the complex, hemodynamic mechanisms involved in normal circulation and cardiac diseases.

Axial Green Function Method for Axisymmetric Electromagnetic Field Computation

1D Green functions are applied to efficiently calculate the electrostatic field axisymmetric in 3D. The point of this paper is laid on the computation of the axisymmetric numerical solution which may have derivative discontinuity due to the transmission condition across the interface between different kinds of matter. A combined axial Green function method(AGM) is proposed to accurately solve this axisymmetric problem.

Axial green function method for axisymmetric electromagnetic field computation

Only with 1-D Green function for 1-D elliptic differential operator, we can solve 2-D/3-D general elliptic problems by applying the axial Green function method (AGM). An extension of AGM is proposed to enforce Neumann boundary conditions. This extension is directly available for 2-D problems with straight boundaries parallel to axes on which Neumann boundary conditions are assigned. It is thoroughly attributed to the specific axial Green functions associated with the Neumann conditions. Moreover, since this extended AGM (XAGM) in 1-D satisfies the transmission condition across an interface along which the permittivity is discontinuous, it can be applied to 2-D problems with interfaces parallel to axes without loss of accuracy. Finally, we apply the XAGM in 2-D to 3-D axisymmetric electric potential problems with variable and/or even discontinuous permittivities along interfaces. Owing to the cylindrical coordinate transform, the transformed problem is tractable to solve using this XAGM. Arbitrary distribution of axial lines is available, which must be a marked advantage of XAGM compared to other methods.

MULTIDIMENSIONAL OPEN SYSTEM FOR VALVELESS PUMPING

In this study, we present a multidimensional open system for valveless pumping (VP). This system consists of an elastic tube connected to two open tanks filled with a fluid under gravity. The two-dimensional elastic tube model is constructed based on the immersed boundary method, and the tank model is governed by a system of ordinary differential equations based on the work-energy principle. The flows into and out of the elastic tube are modeled in terms of the source/sink patches inside the tube. The fluid dynamics of this system is generated by the periodic compress-and-release action applied to an asymmetric region of the elastic tube. We have developed an algorithm to couple these partial differential equations and ordinary differential equations using the pressure-flow relationship and the linearity of the discretized Navier-Stokes equations. We have observed the most important feature of VP, namely, the existence of a unidirectional net flow in the system. Our computations are focused on the factors that strongly influence the occurrence of unidirectional flows, for example, the frequency, compression duration, and location of pumping. Based on these investigations, some case studies are performed to observe the details of the ow features.