David M. McQueen

A three-dimensional computational method for blood flow in the heart. 1. Immersed elastic fibers in a viscous incompressible fluid

Abstract This paper describes the numerical solution of the 3-dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers. Implementation details such as vectorization and the efficient use of external memory are discussed. The method is applied to the damped vibrations of a fiber-wound toroidal tube, and empirical evidence of convergence is presented.

An adaptive, formally second order accurate version of the immersed boundary method

Like many problems in biofluid mechanics, cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid (the blood) and a (visco-)elastic structure (the muscular walls and the valves of the heart). The immersed boundary method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves, and extensions of this work have yielded a three-dimensional model of the heart and great vessels. In the present work, we introduce a new adaptive version of the immersed boundary method. This adaptive scheme employs the same hierarchical structured grid approach (but a different numerical scheme) as the two-dimensional adaptive immersed boundary method of Roma et al. [A multilevel self adaptive version of the immersed boundary method, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 1996; An adaptive version of the immersed boundary method, J. Comput. Phys. 153 (2) (1999) 509–534] and is based on a formally second order accurate (i.e., second order accurate for problems with sufficiently smooth solutions) version of the immersed boundary method that we have recently described [B.E. Griffith, C.S. Peskin, On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems, J. Comput. Phys. 208 (1) (2005) 75–105]. Actual second order convergence rates are obtained for both the uniform and adaptive methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell. We also present initial results from the application of this methodology to the three-dimensional simulation of blood flow in the heart and great vessels. The results obtained by the adaptive method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the method, but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution. Differences are also observed in the flow about the mitral valve leaflets.

Modeling prosthetic heart valves for numerical analysis of blood flow in the heart

This paper extends our previous work on numerical analysis of blood flow in the heart. In that work the boundary forces were evaluated by solving a fixed-point problem, which we now reformulate as a problem in optimization. This optimization problem, which involves the energy function from which the boundary forces are derived, is solved by Murrays modification of Newtons method. The energy function turns out to be an extremely useful tool in modeling prosthetic heart valves. To enforce a constraint on the valve, we use an energy function which is zero when the constraint is satisfied and positive otherwise. The energy function must be invariant under translation and rotation so that conservation of momentum and angular momentum will be satisfied. We use this technique to construct computer models of several prosthetic valves, and we study the flow patterns of these valves in our computer test chamber.

A three-dimensional computational method for blood flow in the heart. II. contractile fibers

Abstract This paper is the second in a series that describes the development of a 3-dimensional computer model of the heart. The problem studied here is that of a contractile fiber-wound toroidal tube immersed in a viscous incompressible fluid. A wave of contraction propagates around the tube, and this results in peristaltic pumping of the internal fluid in the direction of the wave. When the contraction is sufficiently strong, there is a small region of entrained fluid that is convected along at the speed of the wave.

SIMULATING THE FLUID DYNAMICS OF NATURAL AND PROSTHETIC HEART VALVES USING THE IMMERSED BOUNDARY METHOD

The immersed boundary method is both a general mathematical framework and a particular numerical approach to problems of fluid-structure interaction. In the present work, we describe the application of the immersed boundary method to the simulation of the fluid dynamics of heart valves, including a model of a natural aortic valve and a model of a chorded prosthetic mitral valve. Each valve is mounted in a semi-rigid flow chamber. In the case of the mitral valve, the flow chamber is a circular pipe, and in the case of the aortic valve, the flow chamber is a model of the aortic root. The model valves and flow chambers are immersed in a viscous incompressible fluid, and realistic fluid boundary conditions are prescribed at the upstream and downstream ends of the chambers. To connect the immersed boundary models to the boundaries of the fluid domain, we introduce a novel modification of the standard immersed boundary scheme. In particular, near the outer boundaries of the fluid domain, we modify the construction of the regularized delta function which mediates fluid-structure coupling in the immersed boundary method, whereas in the interior of the fluid domain, we employ a standard four-point delta function which is frequently used with the immersed boundary method. The standard delta function is used wherever possible, and the modified delta function continuously transitions to the standard delta function away from the outer boundaries of the fluid domain. Three-dimensional computational results are presented to demonstrate the capabilities of our immersed boundary approach to simulating the fluid dynamics of heart valves.

Shared-Memory Parallel Vector Implementation of the Immersed Boundary Method for the Computation of Blood Flow in the Beating Mammalian Heart

This paper describes the parallel implementation of the immersed boundary method on a shared-memory machine such as the Cray C-90 computer. In this implementation, outer loops are parallelized and inner loops are vectorized. The sustained computation rates achieved are 0.258 Gflops with a single processor, 1.89 Gflops with 8 processors, and 2.50 Gflops with 16 processors. An application to the computer simulation of blood flow in the heart is presented.

Heart Simulation by an Immersed Boundary Method with Formal Second-order Accuracy and Reduced Numerical Viscosity

This paper describes a formally second-order accurate version of the immersed boundary method and its application to the computer simulation of blood flow in a three-dimensional model of the human heart.

Computer-Assisted Design of Butterfly Bileaflet Valves for the Mitral Position

This paper describes the application of computer testing to a design study of butterfly bileaflet mitral prostheses having flat or curved leaflets. The curvature is in the plane normal to the pivot axes and is such that the convex sides of the leaflets face each other when the valve is open. The design parameters considered are the curvature of the leaflets and the location of the pivot points. In this study, stagnation is assessed by computing the smallest value (over the three openings of the valve) of the peak velocity, and hemodynamic performance is judged by a benefit/cost ratio: the net stroke volume divided by the mean transvalvular pressure difference. Unlike the case of a pivoting single-disc valve, the inclusion of a constraint on the maximum angle of opening of the leaflets is found to be essential for adequate, competent performance. Results are presented with both 85 degrees and 90 degrees constraints, since best performance is achieved with the opening-angle constraint in this range. Asymmetry of leaflet motion which is observed with flat leaflets in the mitral position is reduced with modest leaflet curvature. Leaflet curvature also ameliorates central orifice stagnation, which is observed with flat leaflets. Curvature of the valve produces the following improvements in comparison with the best flat valve when the opening-angle constraint is 85 degrees: a 38% increase in the minimum peak velocity and a 16% increase in the hemodynamic benefit/cost ratio. With a 90 degrees constraint the corresponding improvements are 34% and 20%, respectively.

Effect of bundle branch block on cardiac output: A whole heart simulation study

The heart is an electrically controlled fluid pump which operates by mechanical contraction. Whole heart modelling is a computationally daunting task which must incorporate several subsystems: mechanical, electrical, and fluidic. Numerous feedback mechanisms on many levels, and operating at different scales, exist to finely control behaviour. Understanding these interactions is necessary to understand heart operation, as well as pathologies and therapies. A review of the components in such a model is given. The authors then present a framework for their electro-mechano-fluidic whole heart model based on cable methods. The model incorporates atria and ventricles, and has functioning valves with papillary muscles. The effect of altered propagation due to left and right bundle branch block on cardiac output is examined using the cable-based model. Results are compared to clinically observed phenomena. Good agreement was obtained, but tighter coupling of mechanical and electrical events is needed to fully account for behaviour. Cable-based models offer an alternative to continuum models.

The influence of volume exclusion by chromatin on the time required to find specific DNA binding sites by diffusion

Within the nuclei of eukaryotic cells, the density of chromatin is nonuniform. We study the influence of this nonuniform density, which we derive from microscopic images [Schermelleh L, et al. (2008) Science 320:1332–1336], on the diffusion of proteins within the nucleus, under the hypothesis that chromatin density is proportional to an effective potential that tends to exclude the diffusing protein from regions of high chromatin density. The constant of proportionality, which we call the volume exclusivity of chromatin, is a model parameter that we can tune to study the influence of such volume exclusivity on the random time required for a diffusing particle to find its target. We consider randomly chosen binding sites located in regions of low (20th–30th percentile) chromatin density, and we compute the median time to find such a binding site by a protein that enters the nucleus at a randomly chosen nuclear pore. As the volume exclusivity of chromatin increases from zero, we find that the median time needed to reach the target binding site at first decreases to a minimum, and then increases again as the volume exclusivity of chromatin increases further. Random permutation of the voxel values of chromatin density abolishes the minimum, thus demonstrating that the speedup seen with increasing volume exclusivity at low to moderate volume exclusivity is dependent upon the spatial structure of chromatin within the nucleus.