1 / 33

Algebraic-Maclaurin-Padè Solutions to the Three-Dimensional Thin-Walled Spherical Inflation Model Applied to Intracrani

Algebraic-Maclaurin-Padè Solutions to the Three-Dimensional Thin-Walled Spherical Inflation Model Applied to Intracranial Saccular Aneurysms . J. B. Collins II & Matthew Watts July 29, 2004 REU Symposium. OVERVIEW. MOTIVATION

donald
Download Presentation

Algebraic-Maclaurin-Padè Solutions to the Three-Dimensional Thin-Walled Spherical Inflation Model Applied to Intracrani

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Algebraic-Maclaurin-Padè Solutions to the Three-Dimensional Thin-Walled Spherical Inflation Model Applied to Intracranial Saccular Aneurysms. J. B. Collins II & Matthew Watts July 29, 2004 REU Symposium

  2. OVERVIEW MOTIVATION “It is only through biomechanics that we can understand, and thus address, many of the biophysical phenomena that occur at the molecular, cellular, tissue, organ, and organism levels”[4] METHODOLOGY Model intracranial saccular aneurysm as incompressible nonlinear thin-walled hollow sphere. Examine dynamics of spherical inflation caused by biological forcing function. Employ Algebraic-Maclaurin-Padé numerical method to solve constitutive equations.

  3. HISTOLOGY CELL BIOLOGY Cells and the ECM Collagen & Elastin[1] SOFT TISSUE MECHANICS Nonlinear Anisotropy Visco-Elasticity Incompressibility[2]

  4. The Arterial Wall THE ARTERIAL WALL[3] Structure – I, M, A Multi-Layer Material Model Vascular Disorders Hypertension, Artherosclerosis, Intracranial Saccular Aneurymsms,etc.

  5. Aneurysms MOTIVATION[4] Two to five percent of the general population in the Western world, and more so in other parts of the world, likely harbors a saccular aneurysm.[4] INTRACRANIAL SACCULAR ANEURYMS Pathogenesis; Enlargement; Rupture THE ANEURYSMAL WALL[5] Humphrey et al.’s vs. Three-Dimensional Membrane TheoryNonlinear Elasticty

  6. Modeling the Problem FULLY BLOWN THREE-DIMENSIONAL DEFORMATION SPHERICAL INFLATION

  7. Modeling the Problem[4] INNER PRESSURE - BLOOD OUTER PRESSURE – CEREBROSPINAL FLUID

  8. Governing Equations Dimensional Equation Non-dimensional change of variables Non-dimensional Equation

  9. Material Models Neo-Hookean Model Fung Isotropic Model Fung Anisotropic Model

  10. Model Dependent Term Neo-Hookean Model Fung Isotropic Model Fung Anisotropic Model

  11. Algebraic-Maclaurin-Padé MethodParker and Sochacki (1996 & 1999)

  12. Algebraic-Maclaurin Consider Substitute into

  13. STRAIGHTFORWARD 1st 2nd Calculate the coefficients, of of (Not DIFFICULT since RHS is POLYNOMIAL) So can iteratively determine :

  14. Programming Nuts & Bolts A) RHS f typically higher than 2nd degree in y B) Introduce dummy “product” variables C) Numerically, (FORTRAN), calculate coefficients of with a sequence of nested Cauchy Products & where

  15. Algebraic Maclaurin Padé • Determine the Maclaurin coefficients kjfor a solution y, to the 2Ndegree with the (AM) Method then the well known Padé approximation foryis

  16. Setb0 = 1, determine remainingbjusing Gaussian Elimination

  17. Determine theajby Cauchy Product ofkjand thebj • Then to approximateyat some valuet*, calculate

  18. Adaptive time-stepping • Determine the first Padé error term, using 2N+1 order term of MacLaurin series • Calculate the next time step

  19. Numerical Problem Differential equation for the Fung model Convert to system of polynomial equations…

  20. Recast as polynomial system:

  21. Results Forcing Pressures

  22. Fung Isotropic

  23. Neo-Hookean and Fung Isotropic

  24. Fung Anisotropic(k2 = 1, k2 = 43) and Fung Isotropic

  25. RELATIVE ERRORS CAVITY RADIUS(=1.5)

  26. Adaptive Step Size(n=12, n=24)

  27. Dynamic Animation Fung Model

  28. Dynamic Animation Neo-Hookean Model

  29. SUMMATION Solutions were produced from full three-dimensional nonlinear theory of elasticity analogous to Humphrey et al. without simplifications of membrane theory. Comparison of material models (neo-Hookean & Fung) reinforced continuum theory. Developed novel strain-energy function capturing anisotropy of radially fiber-reinforced composite materials.

  30. SUMMATION The AMP Method provides an algorithm for solving mathematical models, including singular complex IVPs, that is: Efficientfewer number of operations for a higher level of accuracy Adaptable“on the fly” control of order Accurateconvergence to within machine ε Quickerror of machine ε obtained with few time steps Potentialroom for improvement

  31. Acknowledgements National Science Foundation NSF REU DMS 0243845 Dr. Jay D. Humphrey – U. Texas A & M Dr. Paul G. Warne Dr. Debra Polignone Warne Adam Schweiger JMU Department of Mathematics & Statistics JMU College of Science and Mathematics

  32. References [1] Adams, Josephine Clare, 2000. Schematic view of an arterial wall in cross-section. Expert Reviews in Molecular Medicine, Cambridge University Press. http://www-rmm.cbcu.cam.ac.uk/02004064h.htm. Retrieved July 21, 2004. [2] Holzapfel, G.A., Gasser, T.C., Ogden, R.W., 2000. A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models. Journal of Elasticity 61, 1-48. [3] Fox, Stuart. Human Psychology 4th, Brown Publishers. http://www.sci.sdsu.edu/class/bio590/pictures/lect5/5.2.html. Retrieved July 25, 2004. [4] Humphrey, J.D., Cardiovascular Solid Mechanics: Cells, Tissues, and Organs. Springer New York, 2002.

  33. Questions?

More Related