Algebraic-Maclaurin-Padè Solutions to the Algebraic-Maclaurin-Padè Solutions to the Three-Dimensional Thin-Walled Spherical Three-Dimensional Thin-Walled Spherical

Inflation Model Applied to Inflation Model Applied to Intracranial Saccular AneurysmsIntracranial Saccular Aneurysms..

J. B. Collins II & Matthew Watts

July 29, 2004 REU Symposium

OVERVIEWOVERVIEWMOTIVATIONMOTIVATION ““It is only through biomechanics that we can It is only through biomechanics that we can

understand, and thus address, many of the understand, and thus address, many of the biophysical phenomena that occur at the molecular, biophysical phenomena that occur at the molecular, cellular, tissue, organ, and organism levels”cellular, tissue, organ, and organism levels” [4][4]

METHODOLOGYMETHODOLOGYModel intracranial saccular aneurysm as Model intracranial saccular aneurysm as

incompressible nonlinear thin-walled hollow incompressible nonlinear thin-walled hollow sphere.sphere.

Examine dynamics of spherical inflation caused by Examine dynamics of spherical inflation caused by biological forcing function.biological forcing function.

Employ Algebraic-Maclaurin-PadEmploy Algebraic-Maclaurin-Padéé numerical numerical method to solve constitutive equations.method to solve constitutive equations.

CELL BIOLOGYCELL BIOLOGYCells and the ECMCells and the ECMCollagen & ElastinCollagen & Elastin[1][1]

SOFT TISSUE SOFT TISSUE MECHANICSMECHANICS NonlinearNonlinearAnisotropyAnisotropyViscoVisco-Elasticity-ElasticityIncompressibilityIncompressibility[2][2]

HISTOLOGYHISTOLOGY

The Arterial WallThe Arterial Wall

THE ARTERIAL WALLTHE ARTERIAL WALL[3][3]

Structure – I, M, AStructure – I, M, A

Multi-Layer Material Multi-Layer Material ModelModel

Vascular DisordersVascular DisordersHypertension, Artherosclerosis, Hypertension, Artherosclerosis, Intracranial Saccular Intracranial Saccular

AneurymsmsAneurymsms,etc. ,etc.

AneurysmsAneurysmsMOTIVATIONMOTIVATION[4][4]

Two to five percent of the general populationTwo 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.parts of the world, likely harbors a saccular aneurysm.[4][4] INTRACRANIAL SACCULAR ANEURYMSINTRACRANIAL SACCULAR ANEURYMS Pathogenesis;Pathogenesis; Enlargement;Enlargement;

Rupture Rupture THE ANEURYSMAL WALLTHE ANEURYSMAL WALL[5][5]

Humphrey Humphrey et al.et al.’s’s vs. vs. Three-DimensionalThree-Dimensional Membrane TheoryMembrane Theory Nonlinear ElastictyNonlinear Elasticty

Modeling the ProblemModeling the ProblemFULLY BLOWN THREE-DIMENSIONAL FULLY BLOWN THREE-DIMENSIONAL DEFORMATION SPHERICAL INFLATIONDEFORMATION SPHERICAL INFLATION

, ,r R t R t R

Modeling the ProblemModeling the Problem [4][4]

INNER PRESSURE - INNER PRESSURE - BLOODBLOOD

OUTER PRESSURE – OUTER PRESSURE – CEREBROSPINAL CEREBROSPINAL FLUIDFLUID

10

i1

cos( ) sin( )m n nn

P P A n t B n t

222

2

3( )2o csf

d dP t p Adt dt

Governing EquationsGoverning Equations

2

2( , ) ( , ) , ( , )RR t R T R t p R t R t R

Dimensional Equation

Non-dimensional change of variables

2 , ,c R ct RHA A H

Non-dimensional Equation

22

2

32

csfi

AA d d AT p Pc H d d c

Material ModelsMaterial ModelsNeo-Hookean Model 1 1 3

2W I I

Fung Isotropic Model 1 31

IW I e

Fung Anisotropic Model

1 2 43 111 4

2

, 1I k IkW I I e ek

Model Dependent TermModel Dependent TermNeo-Hookean Model

3

5

2 1HT

Fung Isotropic Model

2 22

4

2 1 1 1224 3 2

9

4 2 2 1 1FI

eT

Fung Anisotropic Model

22 2 2

8

1 1 1

14 13 12 11 10 9 8 6 5 4 2 2117

2

4 1 2 7 8 4 1FA FIeT T k

k

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

0, ( )t a y f y y y

A) Autonomous: ,

B) Initial Condition set at 0

C) is polynomial in terms of the i

t

a

y

f y f y

f y

0(0) y f y y y

Algebraic-MaclaurinAlgebraic-Maclaurin2 3

0 1 2 3( )t t t t y k k k k

2 31 2 3 4( ) 2 3 4t t t t y k k k k

Substitute intoSubstitute into 0, (0) y f y y y

2

1 2 3

2 30 1 2 3

( ) 2 3t t t

t t t

y k k k f(y)

f k k k k

ConsiderConsider

Need only to determine the jk

11st st

22ndnd Calculate the coefficients, of of Calculate the coefficients, of of

(Not DIFFICULT since RHS is (Not DIFFICULT since RHS is POLYNOMIALPOLYNOMIAL))

So can iteratively determine :So can iteratively determine :

0 0(0) y k y2

0 1 2( ...)t t f k k k

01 0

12 0 1

23 0 1 2

coefficient , 1 depends on

coefficient , 2 depends on ,

coefficient , 3 depends on , ,

t

t

t

k f k

k f k k

k f k k k

jt

STRAIGHTFORWARDSTRAIGHTFORWARD

A)A) RHS RHS f f typically higher than 2typically higher than 2ndnd degree in degree in yy

B)B) Introduce dummy Introduce dummy ““productproduct”” variables variables

C)C) Numerically, Numerically, (FORTRAN),(FORTRAN), calculate coefficients of calculate coefficients of with a sequence of nested with a sequence of nested Cauchy ProductsCauchy Products

0

ii

i

a a t

0

ii

i

b b t

0

ii

i

d ab d t

&&

where0

n

n i n ii

d a b

jt

Programming Nuts & Programming Nuts & BoltsBolts

Algebraic Maclaurin PadAlgebraic Maclaurin Padéé1)1) Determine the Maclaurin coefficients Determine the Maclaurin coefficients kkjj for a solution for a solution yy,, to the to the 2N2N

degree with the degree with the (AM)(AM) Method Method

then the well known Padé approximation for yy is

2

0

( )N

jj

j

y t k t

0 2 1

0

0

( ) to ( )

Nj

jj j N

N jNj j

jj

a tP t k t O t

b t

2 1

0 0 0

0 to ( ) N N

j j j Nj j j

j j j

k t b t a t O t

2)2) SetSet bb00 = = 11, , determine remainingdetermine remaining bbjj using Gaussian Eliminationusing Gaussian Elimination

1 1 1 1

1 2 2 2

2 1 2 2 2

A

ij N i j

N N

N N N

N N N

N N N N N

A k

k k k b kk k k b k

k k k b k

2 1

0 0 0

0 to ( ) N N

j j j Nj j j

j j j

k t b t a t O t

0

a for 0, 1, ..., Nn

n j n jj

k b n

*

0*

*

0

( )

N jj

jN N j

jj

a tP t

b t

3)3) Determine theDetermine the aajj by by Cauchy Product ofCauchy Product of kkjj and theand the bbjj

4)4) Then to approximateThen to approximate yy at some valueat some value t*t*,, calculate calculate

Adaptive time-steppingAdaptive time-stepping1) Determine the first Padé error term, using 2N+1 order term

of MacLaurin series

2) Calculate the next time step

2 10 2 1

2 10

0

2 1 2 1 2 1 2 1 2 1

(2 2)

Nj

jNjj N

j NNjj

jj

N N N N N N

a tk t p t O N

b t

p k k b k b k b

1 12 2

2 11 1 2 1

N N

Ni i N

h hqw w p h

12

2 1

N

N

qh hp

Numerical ProblemNumerical ProblemDifferential equation for the Fung model

2 22

4

2 1 1 12 224 3 22 9 3

22

3

4 2 2 1 11

cos sin cos sin1

cos sin cos sin cos

sin cos sin cos sin

cos sin cos sin

[

d ed

dd

cos

sin ]

Convert to system of polynomial equations…

Recast as polynomial system:Recast as polynomial system:1 2

2 46 17 19 20 21 22

23 24 25 26 27 28 29

30 31 32 33 34 35 36

37 38 39

3 2

4 9

5 10 8

6 2

7 11 2

8 10 11

9 11

4

3

4 6

2

4 2

2

y y

y p p p p p pp p p p p p pp p p p p p pp p p

y p

y p

y p p

y y

y p y

y p p

y p

(0) 1 (0) 0

10 16

11 12

12 11

13 14

14 13

15 16

16 15

17 18

18 17

19 20

20 19

4y py yy yy yy yy yy yy yy yy yy y

21 22

22 21

23 24

24 23

25 26

26 25

27 28

28 27

29 30

30 29

y yy yy yy yy yy yy yy yy yy y

ResultsResultsForcing Pressures

Fung Isotropic

Neo-Hookean and Fung Isotropic

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

RELATIVE ERRORS CAVITY RADIUSRELATIVE ERRORS CAVITY RADIUS ((=1.5)=1.5)

OrderOrder StepStep Runge-KuttaRunge-Kutta Taylor Taylor SeriesSeries PadéPadé

441010 0.529 0.529 E-1E-1 0.761 0.761 E-1E-1 0.4740.474

100100 0.106 0.106 E-5E-5 0.226 0.226 E-6E-6 0.182 0.182 E-6E-6

100,000100,000 0.104 0.104 E-11E-11 0.298 0.298 E-12E-12 0.163 0.163 E-12E-12

881010 0.1280.128 0.1770.177

100100 0.240 0.240 E-8E-8 0.255 0.255 E-14E-14

121211 0.1520.152 0.902 0.902 E-1E-1

100100 0.121 0.121 E-9E-9 0.279 0.279 E-14E-14

100100 11 0.9990.999 0.344 0.344 E-11E-11

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

Dynamic AnimationDynamic Animation

Fung Model

Dynamic AnimationDynamic Animation

Neo-Hookean Model

SUMMATIONSUMMATIONSolutions were produced from full three-dimensional Solutions were produced from full three-dimensional

nonlinear theory of elasticity analogous to nonlinear theory of elasticity analogous to Humphrey Humphrey et al.et al. without simplifications of without simplifications of membrane theory.membrane theory.

Comparison of material models (neo-Hookean & Fung) Comparison of material models (neo-Hookean & Fung) reinforced continuum theory.reinforced continuum theory.

Developed novel strain-energy function capturing Developed novel strain-energy function capturing anisotropy of radially fiber-reinforced composite anisotropy of radially fiber-reinforced composite materials.materials.

SUMMATIONSUMMATIONThe The AMPAMP Method Method provides an algorithm for solving provides an algorithm for solving mathematical models, including singular complex mathematical models, including singular complex IVPs, that is:IVPs, that is:EfficientEfficient fewer number of operations for a higher level of accuracyfewer number of operations for a higher level of accuracy

AdaptableAdaptable “on the fly” control of order“on the fly” control of order

AccurateAccurate convergence to within machine convergence to within machine εε

QuickQuick error of machine error of machine εε obtained with few time steps obtained with few time steps PotentialPotential room for improvementroom for improvement

AcknowledgementsAcknowledgementsNational Science FoundationNational Science FoundationNSF REU DMS 0243845NSF REU DMS 0243845

Dr. Jay D. Humphrey – U. Texas A & MDr. Jay D. Humphrey – U. Texas A & MDr. Paul G. WarneDr. Paul G. WarneDr. Debra Polignone Warne Dr. Debra Polignone Warne Adam SchweigerAdam SchweigerJMU Department of Mathematics & StatisticsJMU Department of Mathematics & StatisticsJMU College of Science and MathematicsJMU College of Science and Mathematics

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

Questions?Questions?