Modeling of Soft Tissues Interacting with Fluid (Blood or Air) Using the Immersed Finite Element Method

Lucy T. Zhang

JEC 2049, Department of Mechanical, Aerospace, and Nuclear Engineering, Troy, USA.

DOI: 10.4236/jbise.2014.73018                

Abstract

This paper presents some biomedical applications that involve fluid-structure interactions which are simulated using the Immersed Finite Element Method (IFEM). Here, we first review the original and enhanced IFEM methods that are suitable to model incompressible or compressible fluid that can have densities that are significantly lower than the solid, such as air. Then, three biomedical applications are studied using the IFEM. Each of the applications may require a specific set of IFEM formulation for its respective numerical stability and accuracy due to the disparities between the fluid and the solid. We show that these biomedical applications require a fully-coupled and stable numerical technique in order to produce meaningful results.

Keywords

Biomechanics; Fluid-Structure Interactions; Biomechanics

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Peskin, C. (1977) Numerical analysis of blood flow in the heart. Journal of Computational Physics, 25, 220-252. http://dx.doi.org/10.1016/0021-9991(77)90100-0
[2]	McCracken, M. and Peskin, C. (1980) A vortex method for blood flow through heart valves. Journal of Computational Physics, 35, 183-205. http://dx.doi.org/10.1016/0021-9991(80)90085-6
[3]	McQueen, D. and Peskin, C. (1983) Computer-assisted design of pivoting-disc prosthetic mitral valves. Journal of Computational Physics, 86, 126-135.
[4]	Peskin, C. and McQueen, D. (1989) A three-dimensional computational method for blood flow in the heart. I. Immersed elastic fibers in a viscous incompressible fluid. Journal of Computational Physics, 81, 372-405. http://dx.doi.org/10.1016/0021-9991(89)90213-1
[5]	Peskin, C. and McQueen, D. (1992) Cardiac fluid dynamics. Critical Reviews in Biomedical Engineering, SIAM Journal on Scientific and Statistical Computing, 20, 451-459.
[6]	Peskin, C. and McQueen, D. (1994) Me-chanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets. American Journal of Physiology, 266, H319-H328.
[7]	Peskin, C. and McQueen, D. (1996) Case studies in mathematical modeling-ecology, physiology, and cell biology. Prentice-Hall, Upper Saddle River.
[8]	LeVeque, R.J. and Li, Z. (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM Journal on Numerical Analysis, 31, 1091-1044. http://dx.doi.org/10.1137/0731054
[9]	LeVeque, R.J. and Li, Z. (1997) Immersed interface methods for stokes flow with elastic boundries or surface tension. SIAM Journal on Scientific Computing, 18, 709-735. http://dx.doi.org/10.1137/S1064827595282532
[10]	Fogelson, A. and Keener, J. (2000) Immersed interface method for Neumann and related problems in two and three dimensions. SIAM Journal on Scientific Computing, 22, 1630-1654. http://dx.doi.org/10.1137/S1064827597327541
[11]	Lee, L. and LeVeque, R. (2003) An immersed interface method for incompressible Navier-Stokes equations, SIAM Journal on Scientific Computing, 25, 832-856. http://dx.doi.org/10.1137/S1064827502414060
[12]	Li, Z. and Lai, M. (2001) The immersed interface method for the Navier-Stokes equations with singular forces. Journal of Computational Physics, 171, 822-842. http://dx.doi.org/10.1006/jcph.2001.6813
[13]	Wiegmann, A. and Bube, K.P. (1998) The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources. SIAM Journal on Numerical Analysis, 35, 177-200. http://dx.doi.org/10.1137/S003614299529378X
[14]	Wiegmann, A. and Bube, K.P. (2000) The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions. SIAM Journal on Numerical Analysis, 37, 827-862. http://dx.doi.org/10.1137/S0036142997328664
[15]	Wang, X. and Liu, W. (2004) Extended immersed boundary method using FEM and RKPM. Computer Methods in Applied Mechanics and Engineering, 193, 1305-1321. http://dx.doi.org/10.1016/j.cma.2003.12.024
[16]	Boffi, D. and Gastaldi, L. (2003) A finite element approach for the immersed boundary method. Computers and Structures, 81, 491-501. http://dx.doi.org/10.1016/S0045-7949(02)00404-2
[17]	Boffi, D., Gastaldi, L. and Heltai, L. (2007) On the CFL condition for the finite element immersed boundary method. Computers and Structures, 85, 775-783. http://dx.doi.org/10.1016/j.compstruc.2007.01.009
[18]	Wang, S.X., Zhang, L.T. and Liu, W.K. (2009) Finite element formulations for immersed methods: Explicit and implicit approaches. Journal of Computational Physics, 228, 2535-2551. http://dx.doi.org/10.1016/j.jcp.2008.12.012
[19]	Zhang, L., Gerstenberger, A., Wang, X. and Liu, W. (2004) Immersed finite element method. Computer Methods in Applied Mechanics and Engineering, 193, 2051-2067. http://dx.doi.org/10.1016/j.cma.2003.12.044
[20]	Zhang, L. and Gay, M. (2007) Immersed finite element method for fluid-structure interactions. Journal of Fluids and Structures, 23, 839-857. http://dx.doi.org/10.1016/j.jfluidstructs.2007.01.001
[21]	Zhang, L.T. and Gay, M. (2008) Imposing rigidity constraints on immersed objects in unsteady fluid flows. Computational Mechanics, 42, 357-370. http://dx.doi.org/10.1007/s00466-008-0244-8
[22]	Wang, X. and Zhang, L.T. (2010) Interpolation functions in the immersed boundary and finite element methods. Computational Mechanics, 45, 321-334.
[23]	Liu, W., Liu, Y., Farrell, D., Zhang, L., Wang, S., Fukui, Y., Patankar, N., Zhang, Y., Bajaj, C., Lee, J., Hong, J., Chen, X. and Hsu, H. (2006) Immersed finite element method and its applications to biological systems. Computer Methods in Applied Mechanics and Engineering, 195, 1722-1749. http://dx.doi.org/10.1016/j.cma.2005.05.049
[24]	Liu, W., Liu, Y., Zhang, L., Wang, X., Gerstenberger, A. and Farrell, D. (2004) Immersed finite element method and applications to biological systems. Finite Element Methods: 1970’s and Beyond. International Center for Numerical Methods and Engineering.
[25]	Liu, Y. and Liu, W. (2006) Rheology of red blood cell aggregation in capillary by computer simulation. Journal of Computational Physics, 220, 139-154. http://dx.doi.org/10.1016/j.jcp.2006.05.010
[26]	Liu, Y., Zhang, L., Wang, X. and Liu, W. (2004) Coupling of Navier-Stokes equations with protein molecular dynamics and its application to hemodynamics. International Journal for Numerical Methods in Fluids, 46, 1237-1252. http://dx.doi.org/10.1002/fld.798
[27]	Gay, M., Zhang, L. and Liu, W. (2006) Stent modeling using immersed finite element method. Computer Methods in Applied Mechanics and Engineering, 195, 4358-4370. http://dx.doi.org/10.1016/j.cma.2005.09.012
[28]	Zhang, L.T. (2008) Shear stress and shear-induced particle residence in stenosed blood vessels, International Journal of Multiscale Computational Engineering, 6, 141-152. http://dx.doi.org/10.1615/IntJMultCompEng.v6.i2.30
[29]	Zhang, L.T. and Gay, M. (2008) Characterizing left atrial appendage functions in sinus rhythm and atrial fibrillation using computational models. Journal of Biomechanics, 41, 2515-2523. http://dx.doi.org/10.1016/j.jbiomech.2008.05.012
[30]	Gay, M. and Zhang, L.T. (2009) Numerical studies of healthy, stenosed, and stented coronary arteries. International Journal of Numerical Methods in Fluids, 61, 453-472. http://dx.doi.org/10.1002/fld.1966
[31]	M. Gay, and L. T. Zhang, (2009) Numerical studies on fluid-structure interactions of stent deployment and stented arteries. Engineering with Computers, 25, 61-72. http://dx.doi.org/10.1007/s00366-008-0105-2
[32]	Tezduyar, T. (1992) Stabilized finite element formulations for incompressible-flow computations. Advanced Application Mechanics, 28, 1-44.
[33]	Tezduyar, T. (2001) Finite element methods for flow problems with moving boundaries and interfaces. Archives of Computational Methods in Engineering, 8, 83-130. http://dx.doi.org/10.1007/BF02897870
[34]	Hughes, T., Franca, L. and Balestra, M. (1986) A new finite element formulation for computational fluid dynamics: V. Circumventing the babuska-brezzi condition: A stable Petrov-Galerkin formulation of the stokes problem accommodating equal-order interpolations. Computer Methods in Applied Mechanics and Engineering, 59, 85-99. http://dx.doi.org/10.1016/0045-7825(86)90025-3
[35]	Wang, X., Wang, C. and Zhang, L.T. (2011) Semi-implicit formulation of the immersed finite element method. Computational Mechanics, 49, 421-430. http://dx.doi.org/10.1007/s00466-011-0652-z
[36]	Wang, X. and Zhang, L.T. (2013) Modified immersed finite element method for solid-dominated fully-coupled fluid-structure interactions. Computer Methods in Computer Methods in Applied Mechanics and Engineering, 267, 150-169. http://dx.doi.org/10.1016/j.cma.2013.07.019
[37]	Peskin, C. (2002) The immersed boundary method. Acta Numerica, 11, 479-517. http://dx.doi.org/10.1017/S0962492902000077
[38]	Torres, D. and Brackbill, J. (2000) The point-set method: front-tracking without connectivity. Journal of Computational Physics, 165, 620-644. http://dx.doi.org/10.1006/jcph.2000.6635
[39]	Colombo, A., Hall, P., Nakamura, S., Almagor, Y., Maiello, L., Martini, G., Gaglione, A., Goldberg, S. and Tobis, J. (1995) Intracoronary stenting without anticoagulation accomplished with intravascular ultrasound guidance. Circulation, 91, 1676-1688. http://dx.doi.org/10.1161/01.CIR.91.6.1676
[40]	Goldberg, S., Colombo, A., Nakamura, S., Almagor, Y., Maiello, L. and Tobis, J. (1994) Benefit of intracoronary ultrasound in the deployment of Palmaz-Schatz stents. Journal of the American College of Cardiology, 24, 996-1003. http://dx.doi.org/10.1016/0735-1097(94)90861-3
[41]	Segers, P., Kostopoulos, K., Scheerder, I.D. and Verdonck, P. (1998) Biomechanical aspects of intracoronary stents. In: Verdonck, P., Ed., Intra and Extracorporeal Cardiovascular Fluid Dynamics, Vol. 1, General Principles in Application, WIT Press, Ashurst, 203-232.
[42]	Russo, R., Schatz, R., Sklar, M., Johnson, A., Tobis, J. and Teirstein, P. (1995) Ultrasound guided coronary stent placement without prolonged systemic anticoagulation. Journal of the American College of Cardiology, 25, 50A. http://dx.doi.org/10.1016/0735-1097(95)91662-H
[43]	SolidWorks (2004) Sp03.1. SolidWorks Corporation, Concord.
[44]	Serruys, P. and Rensing, B. (2002) Handbook of coronary stents. 4th Edition, Martin Dunitz, London.
[45]	Saab, M. (1999) Applications of high-pressure balloons in the medical device industry. Advanced Polymers, Inc. Salem.
[46]	Murphy, B., Savage, P., McHugh, P. and Quinn, D. (2003) The stress-strain behavior of coronary stent struts is size dependent. Annals of Biomedical Engineering, 31, 686-691. http://dx.doi.org/10.1114/1.1569268
[47]	Dumoulin, C. and Cochelin, B. (2000) Mechanical behavior modeling of balloon-expandable stents. Journal of Biomechanics, 33, 1461-1470. http://dx.doi.org/10.1016/S0021-9290(00)00098-1
[48]	Moore Jr., J.E. and Berry, J. (2002) Fluid and solid mechanical implications of vascular stenting. Annals of Biomedical Engineering, 30, 498-508. http://dx.doi.org/10.1114/1.1458594
[49]	Tao, C., Zhang, Y., Hottinger, D.G. and Jiang, J.J. (2007) Asymmetric airflow and vibration induced by the Coanda effect in a symmetric model of the vocal folds. Journal of the Acoustical Society of America, 122, 2270-2278. http://dx.doi.org/10.1121/1.2773960
[50]	Drechsel, J.S. and Thomson, S.L. (2008) Influence of supraglottal structures on the glottal jet exiting a two-layer synthetic, self-oscillating vocal fold model. Journal of the Acoustical Society of America, 123, 4434-4445. http://dx.doi.org/10.1121/1.2897040