Bone structure adaptation as a cellular automaton optimization process

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Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

Bone structure adaptation as a cellular
automaton optimization process

Andrés Tovar, Glen L. Niebur, Mihir Sen and John E. Renaud

Department of Aerospace and Mechanical Engineering

University of Notre Dame, Indiana


Brian Sanders

Air Force Research Laboratory

Wright
-
Patterson AFB, Ohio




Presentation at General Motors Corporation

Detroit, Michigan


6 May 2004



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

[
Meyer and Culmann, 1867; Wolff, 1892
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

Cellular Automata

(CAs)


Biological dynamics

Finite Element Method

(FEM)


Bone static models

Hybrid Cellular Automata

(HCA)


Bone adaptation dynamic model



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

Content

1.

Bone Adaptation


2.

Cellular Automata (CAs)


3.

The Hybrid Cellular Automaton (HCA) method


Local Control Rule


Performance


4.

Examples


5.

Final remarks



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

1. Bone Adaptation



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

100
m
m



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

Osteoclasts



resorb bone

Oscteoblasts



form bone

BMU

Basic

multi
-
cellular

unit

[
Frost, 1964, 1969
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

10
m
m



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

Oscteocytes



sense

mechanical stimuli

[
Skerry et al., 1989; Cowin et al., 1991; Lanyon, 1993; Klein
-
Nulend et al., 1995
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

[
Mullender et al. 1994, Mullender and Huiskes, 1995
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

[
Ott, 2001
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

The average density of osteocytes is

12,000 ~ 20,000 cells/mm3

[
Frost, 1960; Bodyne, 1972
]

0.1mm

2. Cellular Automata



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

2. Cellular Automata

CAs are dynamical systems that are
discrete in space and time

and

operate on a
uniform,

regular lattice
.

0.5mm



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders


CAs are characterized by
local

interactions.

Local rule

Neighborhood

Von Neumann

N
= 4

Moore

N
= 8

Expanded Moore

N
= 24

Empty

N
= 0

Boundary

Adiabatic

a

a

. . .

Periodic

z

a

. . .

z

Fixed

0

a

. . .

Reflecting

b

a

b

. . .



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

[
Conway, 1970
]

[
Tovar, 2003
]

[
Chopard and Droz, 1998
]

[
Wolfram, 2002
]


CAs have been used to simulate physical and biological phenomena since their
creation by von Neumann in 1940s.



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3. Hybrid Cellular Automaton Model

Mechanical

set point

Mechanical

signal

U*

U

[
Hajela and Kim, 2001;


Abdalla and Gürdal, 2002
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3. Hybrid Cellular Automaton Model

U*

U

FEM



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3. Hybrid Cellular Automaton Model

U

FEM

U*

[
Carter, 1977;


Beaupre, 1990
]

[
Bendsøe, 1989;


Sigmund, 2001
]

Local control



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3. Hybrid Cellular Automaton Model

U

FEM

U*

?

yes

no

End

Local control

Start



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

a) Two
-
position control

b) Proportional control

c) Integral control

d) Derivative control

3.1 Local control strategy



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.1 Local control strategy

Two
-
position control

0
2.5
5
7.5
10
0
5
10
15
20
t
U
0
0.25
0.5
0.75
1
M
U
M
t:21 U:6.7170 M:0.539

[c.f.
Sauter, 1992
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.1 Local control strategy

Proportional control

0
2.5
5
7.5
10
0
5
10
15
20
t
U
0
0.25
0.5
0.75
1
M
U
M
t:23 U:6.3265 M:0.581

[c.f.
Martin et al., 1998
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.1 Local control strategy

Proportional
-
Integral control

0
2.5
5
7.5
10
0
5
10
15
t
U
0
0.25
0.5
0.75
1
M
U
M
t:16 U:6.4576 M:0.568

[c.f.
Hazelwood et al., 2001
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.1 Local control strategy

Proportional
-
Derivative control

0
2.5
5
7.5
10
0
5
10
15
20
t
U
0
0.25
0.5
0.75
1
M
U
M
t:23 U:6.2938 M:0.585

[c.f.
Fyhrie and Schaffler, 1995
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.1 Local control strategy


Proportional
-
Integral
-
Derivative control

0
2.5
5
7.5
10
0
5
10
15
t
U
0
0.25
0.5
0.75
1
M
U
M
t:15 U:6.4338 M:0.569

[c.f.
Davidson et al., 2004
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.2 Performance

Initial design

M
= 1.0

M
= 0.5

M

= 0.0

M


0.5

t:21 U:6.4502 M:0.568

t:21 U:6.4668 M:0.568

t:17 U:6.4350 M:0.568

t:15 U:6.4338 M:0.569



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.2 Performance

Neighborhood

t:15 U:6.4338 M:0.569

t:16 U:6.8073 M:0.529

t:13 U:6.3511 M:0.574

t:16 U:6.2062 M:0.592



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.2 Performance

Boundary conditions

Periodic

z

a

. . .

z

Fixed

0

a

. . .

t:15 U:6.4338 M:0.569

t:13 U:6.3905 M:0.584



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.2 Performance

Size of the Design Domain

10x10

30x30

60x60

90x90

t:15 U:6.4338 M:0.569

t:16 U:6.9805 M:0.533

t:18 U:5.9944 M:0.598

t:20 U:6.8146 M:0.540

120x120

t:17 U:7.1222 M:0.526



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.2 Performance

Target mechanical stimulus
U
*

U
* =
U
0
/5

U
* =
U
0

U
* = 5
U
0

U
* = 10
U
0

U
0

0.005

t:15 U:6.4338 M:0.569

t:6 U:4.4033 M:0.920

t:19 U:13.1251 M:0.274

t:18 U:18.5379 M:0.193



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.2 Performance

The trade
-
off curve

1
10
100
1000
10000
0
0.2
0.4
0.6
0.8
1
M
U
HCA


Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

3.2 Performance

The trade
-
off curve

1
10
100
1000
10000
0
0.2
0.4
0.6
0.8
1
M
U
HCA
OC
[
Sigmund, 2001
]



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

4. Examples

Structures in cantilever

1:1

3:1

2:1

4:1



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

4. Examples

Structures in cantilever

t:10 U:10.3091 M:0.483

t:18 U:11.0233 M:0.551

t:10 U:12.9910 M:0.189



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

5. Examples

Trabecular bone (one
-
load case)



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

5. Examples

Trabecular bone (two
-
load case)



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

6. Final Remarks

1)
HCA = CA + FEM, using local control rules.


2)
HCA models are suitable to simulate biological structural optimization
process.


3)
HCA local control rules need to be “tuned” according to biological evidence.


4)
Time effects, like mineralization of bone tissue, can be included in the
model.


5)
A probabilistic HCA model can be implemented to simulate non
-
deterministic process in bone remodeling.


6)
The time scales are still a concern for HCA model.



Bone adaptation as a CA process



Tovar, Niebur, Sen, Renaud, Sanders

Thanks