This article was published online on [26 February 2013]. Errors were subsequently identified. This notice is included in the online and print versions to indicate that both have been corrected [09 April 2013].

Erratum

# Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions^{†}

Version of Record online: 26 FEB 2013

DOI: 10.1002/cnm.2543

Copyright © 2013 John Wiley & Sons, Ltd.

Issue

## International Journal for Numerical Methods in Biomedical Engineering

Volume 29, Issue 5, pages 641–643, May 2013

Additional Information

#### How to Cite

Griffith, B. E. (2013), Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions. Int. J. Numer. Meth. Biomed. Engng., 29: 641–643. doi: 10.1002/cnm.2543

^{†}

#### Publication History

- Issue online: 3 MAY 2013
- Version of Record online: 26 FEB 2013

Vol. 28, Issue 3, 317–345, Version of Record online: 16 MAY 2011

The aforementioned article (DOI: 10.1002/cnm.1445) was published online on May 16, 2011 and in print in the *International Journal for Numerical Methods in Biomedical Engineering* 2012; **28**(3):317–345. This article appeared with errors that were introduced by the publisher after the proofing process. Specifically, several colons appearing in mathematical expressions were omitted in the final published version of this paper, and all brackets were missing, including in all citations and in all formulae.

The following errors were subsequently identified:

Equations 5, 7, 13, 35, 36, and 37 were incorrect and should have read as follows:

- (5)

- (7)

- (13)

- (35)

- (36)

- (37)

On Page 319:

in which

**u**(**x**,*t*) = (*u*_{1}(**x**,*t*),*u*_{2}(**x**,*t*),*u*_{3}(**x**,*t*)) is the Eulerian velocity field,*p*(**x**,*t*) is the Eulerian pressure,**f**(**x**,*t*) = (*f*_{1}(**x**,*t*),*f*_{2}(**x**,*t*),*f*_{3}(**x**,*t*)) is the Eulerian elastic force density (i.e. the elastic force density with respect to the physical coordinate system, so that**f**(**x**,*t*) d**x**has units of force),**F**(**s**,*t*) = (*F*_{1}(**s**,*t*),*F*_{2}(**s**,*t*),*F*_{3}(**s**,*t*)) is the Lagrangian elastic force density (i.e the elastic force density with respect to the material coordinate system, so that**F**(**s**,*t*) d**s**has units of force), is a functional that specifies the Lagrangian elastic force density in terms of the deformation of the …The text was incorrect and should have read as follows:

in which

**u**(**x**,*t*) = (*u*_{1}(**x**,*t*),*u*_{2}(**x**,*t*),*u*_{3}(**x**,*t*)) is the Eulerian velocity field,*p*(**x**,*t*) is the Eulerian pressure,**f**(**x**,*t*) = (*f*_{1}(**x**,*t*),*f*_{2}(**x**,*t*),*f*_{3}(**x**,*t*)) is the Eulerian elastic force density (i.e., the elastic force density with respect to the physical coordinate system, so that**f**(**x**,*t*) d**x**has units of force),**F**(**s**,*t*) = (*F*_{1}(**s**,*t*),*F*_{2}(**s**,*t*),*F*_{3}(**s**,*t*)) is the Lagrangian elastic force density (i.e., the elastic force density with respect to the material coordinate system, so that**F**(**s**,) d*t***s**has units of force), is a functional that specifies the Lagrangian elastic force density in terms of the deformation of the …

On Page 320:

We next describe the form of the Lagrangian elastic force density functional used in our model. Like our earlier study of cardiac valve dynamics 11, we model the flexible leaflets of the aortic valve as thin elastic boundaries, and we model the vessel wall as a thick, semi-rigid elastic structure…

As is frequently done in IB models 3, we define the fiber elasticity in terms of a strain-energyfunctional

*E*=*E***X**( ⋅ ,*t*).The text was incorrect and should have read as follows:

We next describe the form of the Lagrangian elastic force density functional used in our model. Like our earlier study of cardiac valve dynamics [11], we model the flexible leaflets of the aortic valve as thin elastic boundaries, and we model the vessel wall as a thick, semi-rigid elastic structure…

As is frequently done in IB models [3], we define the fiber elasticity in terms of a strain-energy functional

*E*=*E*[**X**( ⋅ ,*t*)].

On Page 324:

To discretize the Eulerian incompressible Navier–Stokes equations in space, we employ a locally refined version of a three-dimensional staggered-grid finite difference scheme; see Figure 3. The computational domain Ω is a rectangular box, Ω = 0,

*L*_{1}× 0,*L*_{2}× 0,*L*_{3}, and the coarsest level of the locally refined Cartesian grid is a uniform discretization of Ω, so that the union of the level*ℓ*= 0 grid patches form a regular*N*_{1}×*N*_{2}×*N*_{3}Cartesian grid with grid spacings Δ*x*_{1}=*L*_{1}∕*N*_{1},Δ*x*_{2}=*L*_{2}∕*N*_{2}, and Δ*x*_{3}=*L*_{3}∕*N*_{3}.The text was incorrect and should have read as follows:

To discretize the Eulerian incompressible Navier–Stokes equations in space, we employ a locally refined version of a three-dimensional staggered-grid finite difference scheme; see Figure 3. The computational domain Ω is a rectangular box, Ω = [0,

*L*_{1}] × [0,*L*_{2}] × [0,*L*_{3}], and the coarsest level of the locally refined Cartesian grid is a uniform discretization of Ω, so that the union of the level*ℓ*= 0 grid patches form a regular*N*_{1}×*N*_{2}×*N*_{3}Cartesian grid with grid spacings Δ*x*_{1}=*L*_{1}∕*N*_{1},Δ*x*_{2}=*L*_{2}∕*N*_{2}, and Δ*x*_{3}=*L*_{3}∕*N*_{3}.

On Page 327:

in which we again consider only Cartesian grid cells on the finest level of the hierarchical grid. For those curvilinear mesh nodes that are in the vicinity of physical boundaries, we use the modified regularized delta function formulation of Griffith

*et al.*11. This approach ensures that force and torque are conserved during Lagrangian–Eulerian interaction, even near*∂*Ω. To simplify the description of our timestepping algorithm, we use the shorthand and (d/d*t*) , in which the force-spreading and velocity-interpolation operators, and , are implicitly defined by Equations (27)–(29) and (30)–(32), respectively.The text was incorrect and should have read as follows:

in which we again consider only Cartesian grid cells on the finest level of the hierarchical grid. For those curvilinear mesh nodes that are in the vicinity of physical boundaries, we use the modified regularized delta function formulation of Griffith

*et al.*[11]. This approach ensures that force and torque are conserved during Lagrangian–Eulerian interaction, even near*∂*Ω. To simplify the description of our timestepping algorithm, we use the shorthand and (d/d*t*) , in which the force-spreading and velocity-interpolation operators, and , are implicitly defined by Equations (27)–(29) and (30)–(32), respectively.

On Page 328:

in which

**N**^{n + 1 ∕ 2,k}≈**u**⋅ ∇**u**^{n + 1 ∕ 2}is an explicit approximation to the advection term that uses the xsPPM7 variant 24 of the piecewise parabolic method (PPM) 25 to discretize the nonlinear advection terms; see Griffith 23 for details.The text was incorrect and should have read as follows:

in which

**N**^{n + 1 ∕ 2,k}≈ [**u**⋅ ∇**u**]^{n + 1 ∕ 2}is an explicit approximation to the advection term that uses the xsPPM7 variant [24] of the piecewise parabolic method (PPM) [25] to discretize the nonlinear advection terms; see Griffith [23] for details.

The publishers wish to apologize for these errors.