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

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] “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.

CELL BIOLOGY Cells and the ECM Collagen & Elastin [1] SOFT TISSUE MECHANICS NonlinearAnisotropy Visco-Elasticity Incompressibility [2]

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

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

Modeling the Problem FULLY BLOWN THREE-DIMENSIONAL DEFORMATION SPHERICAL INFLATION

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

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

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

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

Algebraic-Maclaurin-Padé Method Parker and Sochacki (1996 & 1999)

Algebraic-Maclaurin Substitute into Consider

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

Algebraic Maclaurin Padé 1)Determine the Maclaurin coefficients k j for a solution y, to the 2N degree with the (AM) Method y then the well known Padé approximation for y is

2)Set b 0 = 1, determine remaining b j using Gaussian Elimination

3)Determine the a j by Cauchy Product of k j and the b j 4)Then to approximate y at some value t*, calculate

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

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

Results Forcing Pressures

Fung Isotropic

Neo-Hookean and Fung Isotropic

Fung Anisotropic(k 2 = 1, k 2 = 43) and Fung Isotropic

OrderStepRunge-Kutta Taylor Series Padé E E E E E-6 100, E E E E E E E E E-11

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

Dynamic Animation Fung Model

Dynamic Animation Neo-Hookean Model

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.

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

Acknowledgements National Science Foundation NSF REU DMS 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

References [1] Adams, Josephine Clare, Schematic view of an arterial wall in cross-section. Expert Reviews in Molecular Medicine, Cambridge University Press. Expert Reviews in Molecular Medicine, Cambridge University Press. Retrieved July 21, Retrieved July 21, [2] Holzapfel, G.A., Gasser, T.C., Ogden, R.W., A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models. Journal for Arterial Wall Mechanics and a Comparative Study of Material Models. Journal of Elasticity 61, of Elasticity 61, [3] Fox, Stuart. Human Psychology 4 th, Brown Publishers. Retrieved July 25, Retrieved July 25, [4] Humphrey, J.D., Cardiovascular Solid Mechanics: Cells, Tissues, and Organs. Springer New York, Springer New York, 2002.

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