Algebraic-Maclaurin-Pad

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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
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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
  levels4
• 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.

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HISTOLOGY

• CELL BIOLOGY
• Cells and the ECM
• Collagen Elastin1
• SOFT TISSUE MECHANICS
• Nonlinear
• Anisotropy
• Visco-Elasticity
• Incompressibility2

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The Arterial Wall

• THE ARTERIAL WALL3
• Structure I, M, A
• Multi-Layer Material
• Model
• Vascular Disorders
• Hypertension, Artherosclerosis,
• Intracranial Saccular Aneurymsms,etc.

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Aneurysms

• MOTIVATION4
• 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 WALL5
• Humphrey et al.s vs. Three-Dimensional
• Membrane Theory Nonlinear Elasticty

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Modeling the Problem

• FULLY BLOWN THREE-DIMENSIONAL DEFORMATION
  SPHERICAL INFLATION

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Modeling the Problem4

• INNER PRESSURE - BLOOD
• OUTER PRESSURE CEREBROSPINAL FLUID

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Governing Equations
Dimensional Equation
Non-dimensional change of variables
Non-dimensional Equation
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Material Models
Neo-Hookean Model
Fung Isotropic Model
Fung Anisotropic Model
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Model Dependent Term
Neo-Hookean Model
Fung Isotropic Model
Fung Anisotropic Model
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Algebraic-Maclaurin-Padé MethodParker and
Sochacki (1996 1999)
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Algebraic-Maclaurin

• Consider

Substitute into
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STRAIGHTFORWARD

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

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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
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Algebraic Maclaurin Padé

• Determine the Maclaurin coefficients kj for a
  solution y, to the 2N degree with the (AM)
  Method

then the well known Padé approximation for y is
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1. Set b0 1, determine remaining bj using Gaussian
   Elimination

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1. Determine the aj by Cauchy Product of kj and the
   bj
2. Then to approximate y at some value t, calculate

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Adaptive time-stepping

1. Determine the first Padé error term, using 2N1
   order term of MacLaurin series
2. Calculate the next time step

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Numerical Problem
Differential equation for the Fung model
Convert to system of polynomial equations
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• Recast as polynomial system

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Results
Forcing Pressures
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Fung Isotropic
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Neo-Hookean and Fung Isotropic
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Fung Anisotropic(k2 1, k2 43) and Fung
Isotropic
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RELATIVE ERRORS CAVITY RADIUS (?1.5)
Order Step Runge-Kutta Taylor Series Padé
4 10 0.529 E-1 0.761 E-1 0.474
4 100 0.106 E-5 0.226 E-6 0.182 E-6
4 100,000 0.104 E-11 0.298 E-12 0.163 E-12

8 10 0.128 0.177
8 100 0.240 E-8 0.255 E-14

12 1 0.152 0.902 E-1
12 100 0.121 E-9 0.279 E-14

100 1 0.999 0.344 E-11
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Adaptive Step Size(n12, n24)
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Dynamic Animation
Fung Model
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Dynamic Animation
Neo-Hookean Model
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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.

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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 e
• Quick ? error of machine e obtained with few time
  steps
• Potential ? room for improvement

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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

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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/pic
  tures/lect5/5.2.html.
• Retrieved July 25, 2004.
• 4 Humphrey, J.D., Cardiovascular Solid
  Mechanics Cells, Tissues, and Organs.
• Springer New York, 2002.

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Questions?