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
“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.
HISTOLOGY
CELL BIOLOGY
Cells and the ECM
Collagen & Elastin[1]
SOFT TISSUE MECHANICS
Nonlinear
Anisotropy
Visco-Elasticity
Incompressibility[2]
The Arterial Wall
THE ARTERIAL WALL[3]
Structure – I, M, A
Multi-Layer Material
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
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.
Membrane Theory
Three-Dimensional
Nonlinear Elasticty
Modeling the Problem
FULLY BLOWN THREE-DIMENSIONAL
DEFORMATION SPHERICAL INFLATION
r  R, t     R, t  R
   
   
Modeling the Problem
[4]
INNER PRESSURE - BLOOD
10
Pi  Pm    An cos(n t )  Bn sin(n t ) 
n 1
OUTER PRESSURE – CEREBROSPINAL FLUID
Po (t )  p   csf
 d  3  d 
A 
 

2
 dt
2  dt 

2
2
2



Governing Equations
Dimensional Equation
 ( R, t ) R 2
 T   ( R, t )   pR  R, t   2 ( R, t ) R

Non-dimensional change of variables
 
c
R
c
t, R  ,  
2
 HA
A
H
Non-dimensional Equation
A
Acsf
    T        p 
c
H


2
2


d 3  d   A
    2      Pi   

d

2
d

c





Material Models
Neo-Hookean Model

W  I1    I1  3
2
Fung Isotropic Model
W  I1    e
Fung Anisotropic Model
W  I1 , I 4    e
 I1 3


k1 k2  I4 1

e
1
k2
 I1 3
Model Dependent Term
Neo-Hookean Model
TH    
2   3  1
5
Fung Isotropic Model
TFI    
2
2
4    2    2     1      1 e
4
 2  1  1   1
2
3
2
2
2
4
9
Fung Anisotropic Model
 
TFA     TFI    
e
2
2
2 2

   1   1  1 


8




17 k2
 4k  1 2 
1
14
 13  12   11  7  10   9   8   8   6   5   4   4   2   2  1

Algebraic-Maclaurin-Padé Method
Parker and Sochacki (1996 & 1999)
y  f  y , t  y (a)  y 0
A) Autonomous: f  y , t   f  y 

B) Initial Condition set at a  0
C) f y is polynomial in terms of the y
 
i

y  f  y  y (0)  y 0
Algebraic-Maclaurin
Consider
y (t )  k 0  k1t  k 2 t  k 3t 
2
3

y (t )  k1  2 k 2 t  3k 3t 2  4k 4 t 3 
Substitute into
y   f  y  , y (0)  y 0
2

y (t )  k1  2 k 2t  3k 3t 
 f(y)

 f k 0  k1t  k 2t 2  k 3t 3 
Need only to determine the k j

STRAIGHTFORWARD
1st
2nd
y (0)  k 0  y 0
2
f
(
k

k
t

k
t
 ...)
Calculate the coefficients, of t of
0
1
2
j
(Not DIFFICULT since RHS is POLYNOMIAL)
So can iteratively determine :
 1
 coefficient  f , t  2
 coefficient  f , t  3
k 1  coefficient f , t 0
k2
k3
depends on k 0
1
depends on k 0 , k 1
2
depends on k 0 , k 1 , k 2
Programming Nuts & Bolts
A) RHS f typically higher than 2nd degree in y
B) Introduce dummy “product” variables
C) Numerically, (FORTRAN), calculate coefficients of
j
t with a sequence of nested Cauchy Products

a   ai t i
i 0

&
b   bi t i
i 0
n

 d  ab   di t i
i 0
where
d n   ai bn i
i 0
Algebraic Maclaurin Padé
1)
Determine the Maclaurin coefficients kj for a solution y, to the 2N
degree with the (AM) Method
2N
y(t )   k j t j
j 0
then the well known Padé approximation for y is
N
PN (t ) 
j
a
t
 j
j 0
N
j
b
t
 j

  k j t j to O (t 2 N 1 )
j 0
j 0


N
N
j 0
j 0
j 0
j
j
j
2 N 1
k
t
b
t

a
t

0
to
O
(
t
)
 j  j  j
2)
Set b0 = 1, determine remaining bj using Gaussian Elimination
Aij  k N i  j
A N N
 kN
k
 N 1


 k2 N 1
k N 1
kN
k2 N  2
k1 

k2 


k N 
 b1   k N 1 
 b   k 
 2    N 2 

  

  
bN   k2 N 
3)
Determine the aj by Cauchy Product of kj and the bj

N
N
j 0
j 0
j 0
j
j
j
2 N 1
k
t
b
t

a
t

0
to
O
(
t
)
 j  j  j

n
a n   k j bn j for n  0, 1, ..., N
j 0
4)
Then to approximate y at some value t*, calculate
N
PN (t ) 
*
a t
*j
b t
*j
j 0
N
j 0
j
j
Adaptive time-stepping
1)
Determine the first Padé error term, using 2N+1 order term
of MacLaurin series
N
2 N 1

j 0
k jt j 
j
a
t
 j
j 0
N
j
b
t
 j
 p2 N 1t 2 N 1  O (2 N  2)
j 0
p2 N 1  k 2 N 1  k2 N b1  k 2 N 1b2 
2)
Calculate the next time step
1

 2N 
h
q
 w w 

i 1 
 i 1

 p
 2 N 1
  
 qh  h  

p
 2 N 1 
1

2 N 1 

h

h
2N
 k N 1bN
1
2N
Numerical Problem
Differential equation for the Fung model
 2  2 1  12   12
2
2
d
4
2
4
3
2
4



2


2


1




1
e

  

d 2  9 1   3 


 d 
[





   cos     sin     cos     sin  
3
 d 
1   
2
2
  cos      sin      cos      sin      cos   
  sin      cos      sin      cos      sin   
  cos      sin      cos       sin       cos   
  sin   ]
Convert to system of polynomial equations…
Recast as polynomial system:
y   y
1
2
y2    4 p46   p17   p19   p20   p21   p22
  p23  p24  p25   p26   p27   p28  p29
  p30  p31  p32  p33  p34  p35  p36
 p37  p38  p39
y3   p2
y4  3 p9
y5  4 p10  6 p8
y6  y2
y7  2 p11  y2
y8  4 p10  2 p11
  4 p16
y10
   y12
y11
   y11
y12
   y14
y13
   y13
y14
    y16
y15
   y15
y16
    y18
y17
   y17
y18
  y22
y21
  y21
y22
  y24
y23
  y23
y24
   y26
y25
   y25
y26
   y28
y27
   y27
y28
  y30
y29
    y20
y19
  y29
y30
   y19
y20
y9  2 p11
 (0)  1  (0)  0
Results
Forcing Pressures
Fung Isotropic
Neo-Hookean and Fung Isotropic
Fung Anisotropic(k2 = 1, k2 = 43) and Fung Isotropic
RELATIVE ERRORS CAVITY RADIUS
(=1.5)
Order
4
8
Step
Runge-Kutta
Taylor
Series
Padé
10
0.529 E-1
0.761 E-1
0.474
100
0.106 E-5
0.226 E-6
0.182 E-6
100,000
0.104 E-11
0.298 E-12
0.163 E-12
10
0.128
0.177
100
0.240 E-8
0.255 E-14
1
0.152
0.902 E-1
100
0.121 E-9
0.279 E-14
1
0.999
0.344 E-11
12
100
Adaptive Step Size(n=12, n=24)
Dynamic Animation
Fung Model
Dynamic Animation
Neo-Hookean Model
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.
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
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
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.
Questions?