The immersed boundary (IB) method is a mathematical and numerical framework for problems of fluidstructure interaction, treating the particular case in which an elastic structure is immersed in a viscous incompressible fluid. The IB approach to such problems is to describe the elasticity of the immersed structure in Lagrangian form, and to describe the momentum, viscosity, and incompressibility of the coupled fluidstructure system in Eulerian form. Interaction between Lagrangian and Eulerian variables is mediated by integral equations with Dirac delta function kernels. The IB method provides a unified formulation for fluidstructure interaction models involving both thin elastic boundaries and also thick viscoelastic bodies. In this work, we describe the application of an adaptive, staggered-grid version of the IB method to the three-dimensional simulation of the fluid dynamics of the aortic heart valve. Our model describes the thin leaflets of the aortic valve as immersed elastic boundaries, and describes the wall of the aortic root as a thick, semi-rigid elastic structure. A physiological left-ventricular pressure waveform is used to drive flow through the model valve, and dynamic pressure loading conditions are provided by a reduced (zero-dimensional) circulation model that has been fit to clinical data. We use this model and method to simulate aortic valve dynamics over multiple cardiac cycles. The model is shown to approach rapidly a periodic steady state in which physiological cardiac output is obtained at physiological pressures. These realistic flow rates are not specified in the model, however. Instead, they emerge from the fluidstructure interaction simulation. Copyright (c) 2011 John Wiley & Sons, Ltd.
The purpose of this paper is to put in evidence that the fractional-step method (FSM) used to solve the incompressible transient Euler and Navier-Stokes equations for free-surface flows has a problem inherent to the method that may produce unacceptable variations of the domain volume. A simple modification of the free-surface boundary term is introduced in order to reduce considerably the volume loss and preserve the computational advantages of the FSM. Copyright (C) 2008 John Wiley & Sons, Ltd.
The immersed boundary method is an approach to fluid‐structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid‐structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach uses a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian‐Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach. We employ a nodal FE discretization of the structural equations while retaining a finite difference discretization of the Eulerian equations, and we apply this new scheme to benchmark problems involving elastic, rigid, and actively contracting structures of left ventricle. The primary contribution of this work is that it introduces a novel approach to coupling the Lagrangian and Eulerian variables that enables Independent Lagrangian and Eulerian spatial discretizations and thus the effective use of coarse structural meshes with the IB method.
In simulation of biomechanical structures the patient-specific geometry of the object of interest is very often reconstructed from in vivo medical imaging such as CT scans. Such geometries therefore represent a deformed configuration stressed by typical in vivo conditions. Commonly, such structures are considered stress free in simulation. In this contribution we present and compare two methods to introduce a physically meaningful stress/strain state to the obtained geometry for simulations in the finite strain regime and demonstrate the necessity of such prestressing techniques. One method is based on an inverse design analysis to calculate a stress-free reference configuration. The other method developed here is based on a modified updated Lagrangian formulation. The formulation of both methods is provided in detail and implementation issues are discussed. Applicability and accurateness of both approaches are compared and evaluated utilizing an analytical aorta model and fully three-dimensional patient-specific abdominal aortic aneurysm structures in the finite strain regime. Copyright (C) 2009 John Wiley & Sons, Ltd.
The coupling of lightweight and often thin-walled structures to fluids in an incompressible regime is a recurring theme in biomechanics. There are many fluid structure interaction (FSI) solution schemes to address these kinds of problem, each one with its costs and benefits. Here, we attempt a comparison of the most important FSI schemes in the context of biomechanical problems, that is a comparison of different fixed-point schemes and a block preconditioned monolithic scheme. The emphasis of this study is on the numerical behavior of these FSI schemes to gain an understanding of their effectiveness in comparison with each other. To this end a simplified benchmark problem is studied to show its applicability for more involved biomechanical problems. Two such examples with patient-specific geometries are also discussed. The monolithic scheme proved to be much more efficient than the partitioned schemes in biomechanical problems. Copyright (C) 2009 John Wiley & Sons, Ltd.
Mechanical ventilation is a key therapy for patients who cannot breathe adequately by themselves, and dynamics of mechanical ventilation system is of great significance for life support of patients. Recently, models of mechanical ventilated respiratory system with 1 lung are used to simulate the respiratory system of patients. However, humans have 2 lungs. When the respiratory characteristics of 2 lungs are different, a single‐lung model cannot reflect real respiratory system. In this paper, to illustrate dynamic characteristics of mechanical ventilated respiratory system with 2 different lungs, we propose a mathematical model of mechanical ventilated respiratory system with 2 different lungs and conduct experiments to verify the model. Furthermore, we study the dynamics of mechanical ventilated respiratory system with 2 different lungs. This research study can be used for improving the efficiency and safety of volume‐controlled mechanical ventilation system. Because coupling effects of 2 lungs has a significant influence on safety and efficiency of mechanical ventilation, a pneumatic model with 2 lungs has been built in this paper to study the coupling effects of 2 lungs in volume‐controlled ventilation. It can be concluded that a change of compliance or air resistance of one lung can affect both lungs and an unbalance of 2 lungs may result in overly high pressure in the trachea and overventilation.
Bronchial diameter is a key parameter that affects the respiratory treatment of mechanically ventilated patients. In this paper, to reveal the influence of bronchial diameter on the airflow dynamics of pressure‐controlled mechanically ventilated patients, a new respiratory system model is presented that combines multigeneration airways with lungs. Furthermore, experiments and simulation studies to verify the model are performed. Finally, through the simulation study, it can be determined that in airway generations 2 to 7, when the diameter is reduced to half of the original value, the maximum air pressure (maximum air pressure in lungs) decreases by nearly 16%, the maximum flow decreases by nearly 30%, and the total airway pressure loss (sum of each generation pressure drop) is more than 5 times the original value. Moreover, in airway generations 8 to 16, with increasing diameter, the maximum air pressure, maximum flow, and total airway pressure loss remain almost constant. When the diameter is reduced to half of the original value, the maximum air pressure decreases by 3%, the maximum flow decreases by nearly 5%, and the total airway pressure loss increases by 200%. The study creates a foundation for improvement in respiratory disease diagnosis and treatment. We propose a novel mathematical model of mechanical ventilated respiratory system with multigeneration airways. The mathematical model was verified through experimental study. Influence of bronchial diameter on airflow dynamics of pressure‐controlled ventilation system is illustrated. The bronchial diameter change in generations 2 to 7 has a greater impact on airflow dynamics than that in generations 8 to 16. The study lays a foundation for the improvement in respiratory disease diagnosis and treatment.
Protein‐ligand binding is a fundamental biological process that is paramount to many other biological processes, such as signal transduction, metabolic pathways, enzyme construction, cell secretion, and gene expression. Accurate prediction of protein‐ligand binding affinities is vital to rational drug design and the understanding of protein‐ligand binding and binding induced function. Existing binding affinity prediction methods are inundated with geometric detail and involve excessively high dimensions, which undermines their predictive power for massive binding data. Topology provides the ultimate level of abstraction and thus incurs too much reduction in geometric information. Persistent homology embeds geometric information into topological invariants and bridges the gap between complex geometry and abstract topology. However, it oversimplifies biological information. This work introduces element specific persistent homology (ESPH) or multicomponent persistent homology to retain crucial biological information during topological simplification. The combination of ESPH and machine learning gives rise to a powerful paradigm for macromolecular analysis. Tests on 2 large data sets indicate that the proposed topology‐based machine‐learning paradigm outperforms other existing methods in protein‐ligand binding affinity predictions. ESPH reveals protein‐ligand binding mechanism that can not be attained from other conventional techniques. The present approach reveals that protein‐ligand hydrophobic interactions are extended to 40Å away from the binding site, which has a significant ramification to drug and protein design. Most physical models are based on geometry, which leads to high number of degrees of freedom for biomolecular data sets. In contrast, topology is often too abstract to be practically useful. Persistent homology bridges the gap between geometry and topology but neglects biological information. This work introduces element‐specific persistent homology to retain essential biological information during topological simplification. The integration of element‐specific persistent homology and machine learning sheds light on the molecular mechanism of protein‐ligand binding that cannot obtain from other conventional techniques.
Image‐based noninvasive fractional flow reserve (FFR) is an emergent approach to determine the functional relevance of coronary stenoses. The present work aimed to determine the feasibility of using a method based on coronary computed tomography angiography (CCTA) and reduced‐order models (0D‐1D) for the evaluation of coronary stenoses. The reduced‐order methodology (cFFR RO ) was kept as simple as possible and did not include pressure drop or stenosis models. The geometry definition was incorporated into the physical model used to solve coronary flow and pressure. cFFRRO was assessed on a virtual cohort of 30 coronary artery stenoses in 25 vessels and compared with a standard approach based on 3D computational fluid dynamics (cFFR 3D ). In this proof‐of‐concept study, we sought to investigate the influence of geometry and boundary conditions on the agreement between both methods. Performance on a per‐vessel level showed a good correlation between both methods (Pearson's product‐moment R =0.885, P <0.01), when using cFFR 3D as the reference standard. The 95% limits of agreement were −0.116 and 0.08, and the mean bias was −0.018 (SD =0.05). Our results suggest no appreciable difference between cFFR RO and cFFR 3D with respect to lesion length and/or aspect ratio. At a fixed aspect ratio, however, stenosis severity and shape appeared to be the most critical factors accounting for differences in both methods. Despite the assumptions inherent to the 1D formulation, asymmetry did not seem to affect the agreement. The choice of boundary conditions is critical in obtaining a functionally significant drop in pressure. Our initial data suggest that this approach may be part of a broader risk assessment strategy aimed at increasing the diagnostic yield of cardiac catheterisation for in‐hospital evaluation of haemodynamically significant stenoses. In this proof‐of‐concept study, we proposed a novel method based on coronary computed tomography angiography and reduced‐order (0D‐1D) models for the evaluation of coronary stenoses, where the geometry definition is incorporated into the physical model via a reference area and material properties using high order polynomials. The method was assessed on a virtual cohort of 30 coronary artery stenoses in 25 vessels and compared with a standard approach based on 3D computational fluid dynamics. Performance on a per‐vessel level showed a good correlation between both methods with no significant bias.
The models used for modeling the airflow in the human airways are either 0‐dimensional compartmental or full 3‐dimensional (3D) computational fluid dynamics (CFD) models. In the former, airways are treated as compartments, and the computations are performed with several assumptions, thereby generating a low‐fidelity solution. The CFD method displays extremely high fidelity since the solution is obtained by solving the conservation equations in a physiologically consistent geometry. However, CFD models (1) require millions of degrees of freedom to accurately describe the geometry and to reduce the discretization errors, (2) have convergence problems, and (3) require several days to simulate a few breathing cycles. In this paper, we present a novel, fast‐running, and robust quasi‐3D wire model for modeling the airflow in the human lung airway. The wire mesh is obtained by contracting the high‐fidelity lung airway surface mesh to a system of connected wires, with well‐defined radii. The conservation equations are then solved in each wire. These wire meshes have around O (1000) degrees of freedom and hence are 3000 to 25 000 times faster than their CFD counterparts. The 3D spatial nature is also preserved since these wires are contracted out of the actual lung STL surface. The pressure readings between the 2 approaches showed minor difference (maximum error = 15%). In general, this formulation is fast and robust, allows geometric changes, and delivers high‐fidelity solutions. Hence, this approach has great potential for more complicated problems including modeling of constricted/diseased lung sections and for calibrating the lung flow resistances through parameter inversion. Schematic showing the CFD to Q3D conversion, pressure distribution for a sample simulation. Q3D is 3000–27000 times faster than the CFD approach.