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## Augmenting the Immersed Boundary Method with Radial Basis Functions (RBFs) for the Modeling of Platelets in Hemodynamic Flows (2014)

### Citations

331 |
Numerical analysis of blood flow in the heart
- Peskin
- 1977
(Show Context)
Citation Context ...u (Robert M. Kirby), fogelson@math.utah.edu (Aaron L. Fogelson). 1 Corresponding Author Preprint submitted to JCP Resubmitted April 4, 2014 massless, elastic surfaces or objects immersed in the fluid =-=[24]-=-. The IB method was originally developed to model blood flow in the heart and through heart valves [24, 26, 27], but has since been used in a wide variety of other applications, particularly in bioflu... |

246 | Scattered Data Approximation - Wendland - 2005 |

94 |
A three-dimensional computational method for blood flow in the heart I. Immersed elastic fibers in a viscous incompressible fluid
- Peskin, McQueen
- 1989
(Show Context)
Citation Context ...ted to JCP Resubmitted April 4, 2014 massless, elastic surfaces or objects immersed in the fluid [24]. The IB method was originally developed to model blood flow in the heart and through heart valves =-=[24, 26, 27]-=-, but has since been used in a wide variety of other applications, particularly in biofluid dynamics problems where complex geometries and immersed elastic membranes or structures are present and make... |

72 |
A computational model of aquatic animal locomotion
- Fauci, Peskin
- 1988
(Show Context)
Citation Context ...uid dynamics problems where complex geometries and immersed elastic membranes or structures are present and make traditional computational approaches difficult. Examples include swimming of organisms =-=[7, 8]-=-, biofilm processes [4], mechanical properties of cells [1], cochlear dynamics [2], and insect flight [20, 21]. In this work, we are motivated by the application of the IB method to platelet aggregati... |

59 |
The immersed boundary method," Acta Numerica
- Peskin
- 2002
(Show Context)
Citation Context ...tions, we assume that the IB objects are neutrally buoyant; the IB membrane itself 4 carries no mass and each object’s mass is attributed to the fluid in which it sits. For more on the IB method, see =-=[25]-=-. 3 Geometric modeling of platelets In this section, we review the two geometric modeling strategies to be compared in the context of the Immersed Boundary method applied to platelet aggregation. For ... |

54 |
A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting.
- FOGELSON
- 1984
(Show Context)
Citation Context ... 2D models, while activated platelets are approximately circular. Piecewise linear approximations of platelets are currently used in IB methods applied to the simulation of platelet aggregation (e.g. =-=[7, 12, 13]-=-). In previous work [30], we found that interpolation with radial basis functions restricted to the circle (or sphere in 3D) offered accuracy and computational cost comparable to that offered by Fouri... |

53 |
A computational model of the cochlea using the immersed boundary method.
- BEYER
- 1992
(Show Context)
Citation Context ...ot of a platelet aggregation simulation achieved with the RBF-IB method with a time-step of ∆t = 10−4. The snapshot was taken at simulation time t = 2.4. The simulation was run on a 128× 64 grid on a =-=[0, 2]-=-× [0, 1] domain. The arrows show the magnitude and direction of the velocity field. is a consequence (and function) of the stiffness of each platelet, the shear rate of the flow and the number of link... |

52 |
Initiation of platelet adhesion by arrest onto fibrinogen or translocation on von Willebrand factor.
- Savage, Saldivar, et al.
- 1996
(Show Context)
Citation Context ...lecules on the platelets’ surfaces. In addition to slowing or stopping platelet motion over the subendothelium, this binding triggers intracellular signaling pathways that lead to platelet activation =-=[18, 29]-=-. Our platelet aggregation models [7, 11–13, 33] track the motion and behavior of a collection of individual platelets as they interact with the suspending fluid, one another, and the vessel walls. We... |

50 | Scattered data fitting on the sphere - Fasshauer, Schumaker - 1998 |

49 | Error estimates for scattered data interpolation on spheres.
- Jetter, Stockler, et al.
- 1999
(Show Context)
Citation Context ... as the zonal basis function (ZBF) or spherical basis function (SBF) method [31, Ch. 17]. Several studies have provided error estimates for RBF interpolation on circles and spheres; see, for example, =-=[19, 22]-=-. In [19], it was shown that these interpolants can provide spectral accuracy provided the underlying target function is sufficiently smooth, while in [22], error estimates are given in the case that ... |

43 | C.: A stable algorithm for flat radial basis functions on a sphere
- Fornberg, Piret
- 2007
(Show Context)
Citation Context ...ed in our previous work, lower values of ε can make the RBF interpolation matrix more ill-conditioned. While methods (such as RBF-QR and RBF-RA) have been developed to overcome this poor conditioning =-=[14]-=-, they are much more expensive than forming and inverting the standard RBF interpolation matrix. We thus choose a small value of ε = 1.2 for all our tests. When using Nd = 100, we use ε = 2.0. These a... |

42 |
Truncated Newton Method and the Modeling of Complex Immersed Elastic Structures
- Fauci, Fogelson
- 1993
(Show Context)
Citation Context ...uid dynamics problems where complex geometries and immersed elastic membranes or structures are present and make traditional computational approaches difficult. Examples include swimming of organisms =-=[7, 8]-=-, biofilm processes [4], mechanical properties of cells [1], cochlear dynamics [2], and insect flight [20, 21]. In this work, we are motivated by the application of the IB method to platelet aggregati... |

40 | 2007] Unconditionally stable discretizations of the immersed boundary equations
- Newren, Fogelson, et al.
(Show Context)
Citation Context ...id-structure interaction problem: We describe a standard fluid-structure interaction problem on which we test both versions of the IB method. This problem is commonly used in the IB literature (e.g., =-=[23]-=-). The problem involves placing an elliptical object with its center of mass at the center of the [0, 1]2 physical domain. The elliptical object has a circle of the same area as its rest configuration... |

38 |
Modeling biofilm processes using the immersed boundary method.
- DILLON, FAUCI, et al.
- 1996
(Show Context)
Citation Context ...e complex geometries and immersed elastic membranes or structures are present and make traditional computational approaches difficult. Examples include swimming of organisms [7, 8], biofilm processes =-=[4]-=-, mechanical properties of cells [1], cochlear dynamics [2], and insect flight [20, 21]. In this work, we are motivated by the application of the IB method to platelet aggregation in blood clotting, b... |

36 |
Modeling prosthetic heart valves for numerical analysis of blood flow in the heart
- Peskin, McQueen
- 1980
(Show Context)
Citation Context ...ted to JCP Resubmitted April 4, 2014 massless, elastic surfaces or objects immersed in the fluid [24]. The IB method was originally developed to model blood flow in the heart and through heart valves =-=[24, 26, 27]-=-, but has since been used in a wide variety of other applications, particularly in biofluid dynamics problems where complex geometries and immersed elastic membranes or structures are present and make... |

31 | Transport schemes on a sphere using radial basis functions
- Flyer, Wright
(Show Context)
Citation Context ...ven in the case that the target function belongs to a Sobolev space. The RBF method has also been used successfully for numerically solving partial differential equations on the surface of the sphere =-=[9, 10]-=-, as well as more general surfaces [15, 28]. 5 Here, we present the RBF model developed in our earlier work [30]. It is based on explicit parametric representations of the objects. Since our target ob... |

30 |
A computational fluid dynamics of ’clap and fling’ in the smallest insects
- Miller, Peskin
- 2005
(Show Context)
Citation Context ...make traditional computational approaches difficult. Examples include swimming of organisms [7, 8], biofilm processes [4], mechanical properties of cells [1], cochlear dynamics [2], and insect flight =-=[20, 21]-=-. In this work, we are motivated by the application of the IB method to platelet aggregation in blood clotting, but expect our method to be useful in other applications as well. Intravascular blood cl... |

28 |
Meshfree Approximation Methods with
- Fasshauer
- 2007
(Show Context)
Citation Context ...model The RBF method is a popular tool for approximating multidimensional scattered data. An excellent overview of the theory and application of this method can be found in the two books by Fasshauer =-=[5]-=- and Wendland [31]. The restriction of the RBF method to interpolation on a circle and/or sphere began to receive considerable attention from a theoretical standpoint in the mid 1990s (see [6, §6] for... |

27 |
An Adaptive, Cartesian, Front-Tracking Method for the Motion, Deformation and Adhesion of Circulating Cells,’’
- Agresar, Linderman, et al.
- 1998
(Show Context)
Citation Context ...astic membranes or structures are present and make traditional computational approaches difficult. Examples include swimming of organisms [7, 8], biofilm processes [4], mechanical properties of cells =-=[1]-=-, cochlear dynamics [2], and insect flight [20, 21]. In this work, we are motivated by the application of the IB method to platelet aggregation in blood clotting, but expect our method to be useful in... |

26 | When vortices stick: and aerodynamic transition in tiny insects
- Miller, Peskin
(Show Context)
Citation Context ...make traditional computational approaches difficult. Examples include swimming of organisms [7, 8], biofilm processes [4], mechanical properties of cells [1], cochlear dynamics [2], and insect flight =-=[20, 21]-=-. In this work, we are motivated by the application of the IB method to platelet aggregation in blood clotting, but expect our method to be useful in other applications as well. Intravascular blood cl... |

25 | A radial basis function method for shallow water equations on a sphere
- Flyer, Wright
(Show Context)
Citation Context ...ven in the case that the target function belongs to a Sobolev space. The RBF method has also been used successfully for numerically solving partial differential equations on the surface of the sphere =-=[9, 10]-=-, as well as more general surfaces [15, 28]. 5 Here, we present the RBF model developed in our earlier work [30]. It is based on explicit parametric representations of the objects. Since our target ob... |

18 | Simulating the fluid dynamics of natural and prosthetic heart valves using the immersed boundary method
- Griffith, Luo, et al.
(Show Context)
Citation Context ...) is the fiber tension and kt > 0 is constant. We set l0,i = ‖τ‖|λi,t0 , where t0 is the initial time of the simulation. For a bending-resistant force, we use a linear variant of the force defined in =-=[16]-=- and define the elastic force density at X(λi, tk) due to how much the platelet surface there is bent to be F B(λi, tk) = − kb ( ∂4X ∂λ4 − ∂4X0 ∂λ4 )∣∣∣∣∣ λi,tk . (16) HereX0 =X(λi, t0) is the initial... |

17 |
Signaling events underlying thrombus formation,”
- Jackson, Nesbitt, et al.
- 2003
(Show Context)
Citation Context ...lecules on the platelets’ surfaces. In addition to slowing or stopping platelet motion over the subendothelium, this binding triggers intracellular signaling pathways that lead to platelet activation =-=[18, 29]-=-. Our platelet aggregation models [7, 11–13, 33] track the motion and behavior of a collection of individual platelets as they interact with the suspending fluid, one another, and the vessel walls. We... |

17 |
Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions
- Narcowich, Sun, et al.
(Show Context)
Citation Context ... as the zonal basis function (ZBF) or spherical basis function (SBF) method [31, Ch. 17]. Several studies have provided error estimates for RBF interpolation on circles and spheres; see, for example, =-=[19, 22]-=-. In [19], it was shown that these interpolants can provide spectral accuracy provided the underlying target function is sufficiently smooth, while in [22], error estimates are given in the case that ... |

10 |
The orthogonal gradients method: a radial basis functions method for solving partial differential equations on arbitrary surfaces
- Piret
- 2012
(Show Context)
Citation Context ...elongs to a Sobolev space. The RBF method has also been used successfully for numerically solving partial differential equations on the surface of the sphere [9, 10], as well as more general surfaces =-=[15, 28]-=-. 5 Here, we present the RBF model developed in our earlier work [30]. It is based on explicit parametric representations of the objects. Since our target objects are platelets, which in 2D models are... |

9 | A high-order kernel method for diffusion and reactiondiffusion equations on surfaces,
- Fuselier, Wright
- 2013
(Show Context)
Citation Context ...elongs to a Sobolev space. The RBF method has also been used successfully for numerically solving partial differential equations on the surface of the sphere [9, 10], as well as more general surfaces =-=[15, 28]-=-. 5 Here, we present the RBF model developed in our earlier work [30]. It is based on explicit parametric representations of the objects. Since our target objects are platelets, which in 2D models are... |

9 |
R.M.: A study of different modeling choices for simulating platelets within the immersed boundary method
- Shankar, Wright, et al.
- 2013
(Show Context)
Citation Context ...telets are approximately circular. Piecewise linear approximations of platelets are currently used in IB methods applied to the simulation of platelet aggregation (e.g. [7, 12, 13]). In previous work =-=[30]-=-, we found that interpolation with radial basis functions restricted to the circle (or sphere in 3D) offered accuracy and computational cost comparable to that offered by Fourier-based methods in mode... |

8 | Computational Modeling of Blood Clotting: Coagulation and Three-Dimensional Platelet Aggregation." Polymer and Cell Dynamics (2003 - Fogelson, Yu, et al. |

7 |
2005] Stability of approximate projection methods on cell-centered grids
- Guy, Fogelson
(Show Context)
Citation Context ...r. As in [3], we use a fractional-step projection method. First, a backward Euler discretization of the momentum equations is used. The pressure that enforces discrete incompressibility is determined =-=[17]-=-. This gives us the velocity field un+1/2g , the mid-step approximation required in an RK2 method. (5) Using the mid-step fluid velocity un+1/2g and the mid-step IB point positions X n+1/2 q , update ... |

6 |
Simulations of chemical transport and reaction in a suspension of cells i: an augmented forcing point method for the stationary case
- Yao, Fogelson
(Show Context)
Citation Context ...such as normal vectors) is needed at the sample points, an approximation to the surface represented by the sample sites may be formed by a piecewise quadratic interpolation of the sample sites (e.g., =-=[32]-=-). After the incompressible Navier Stokes equations are solved, velocities from the portions of the Eulerian grid surrounding the sample sites are interpolated to the sample sites using a discretizati... |

5 |
An immersed boundary energy-based method for incompressible viscoelasticity
- Devendran, Peskin
- 2012
(Show Context)
Citation Context ... a restriction that the IB point spacing on the surface be no greater than 0.5h, where h is the Eulerian grid cell width. We use the formally second-order Runge-Kutta time-stepping scheme outlined in =-=[3]-=-. This time-stepping scheme appears to show second-order convergence in time for a smooth forcing function, or for an elastic material that fills the whole domain, as demonstrated in [3]. However, as ... |

5 | Immersed-boundary-type models of intravascular platelet aggregation
- Fogelson, Guy
- 2008
(Show Context)
Citation Context ... 2D models, while activated platelets are approximately circular. Piecewise linear approximations of platelets are currently used in IB methods applied to the simulation of platelet aggregation (e.g. =-=[7, 12, 13]-=-). In previous work [30], we found that interpolation with radial basis functions restricted to the circle (or sphere in 3D) offered accuracy and computational cost comparable to that offered by Fouri... |

5 | Three-dimensional computational modeling and simulation of platelet aggregation on parallel computers - Yu - 2000 |