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Fractal Dimension of
Cell Colony Boundaries
Gabriela RodriguezApril 15, 2010
Tumor Boundaries
• Isolated tumor growing in a Petri dish
• Interested in roughness of boundary in 2-D
• How can roughnessbe measured?
2*10
Fractal Dimension
• Measure of “roughness”
• (Mandelbrot ): a boundary is a fractal if its
• Practical method of estimating fractal dimension: Box-counting
covering dimension fractal dimension
3
Outline
• Definitions:– Preliminary concepts– Covering dimension – Fractal dimension
• Box-Counting method• Box-Counting Theorem• Application to Tumor Boundaries • Biological Significance
4
Preliminary Concepts
• Neighborhood• Limit point• Closed set• Bounded set• Compact set• Open cover
5
Limit Points in
• An ε-neighborhood of
is an open disk , with
radius , centered at p.
• is a limit point of
iff for all .
6
2Rp
2RX p
0 XpO
ε
p pO
0
2R
Compact Sets in
• is closed if it contains all its limit
points.
• X is bounded if it lies in a finite region of .
• X is compact in if it is closed and bounded.
7
2RX
2R
2R
2R
Open Covers of Compact Sets in• An open cover of a compact set is a collection of neighborhoods
of points in X whose union contains X.
• Heine-Borel TheoremEvery open cover of a compact set contains a finite sub-cover.
8
2RX
2R
Covering Dimension
The covering dimension of a compact is the smallest integer n for which there is an open cover of X such that no point of X lies in more than n+1 open disks.
9
The covering dimension of the curve is n = 1 because some points of the curve must lie in 2 =1+1 open disks.
2RX
Another View of Dimension
10
KEYε: section sizeN: # of sectionsD: dimension
1
21
31
DDNN D 1lnln1 1ln
ln N
*6
Closed Covers of Compact Sets in
A closed cover of a compact set is a collection of closed disks centered at points in X whose union contains X.
11
2RX
2R
Fractal Dimension
• Let X be a compact subset of .
• The fractal dimension D of X is defined as
(if this limit exists),
where is the smallest number of closed disks of radius
needed to cover X.
12
0
1ln
,lnlim
0
XND
,XN
2R
Box-Counting Method
• Cover with a grid,
whose squares have side length .
• Let be the number of grid squares (boxes) that
intersect X.
• , the fractal dimension of X.
• Plot vs. .
• Slope of plot D.
13
k21
XBk
2ln1ln k kBln
DBk 1lnln
2R
,
14
021 80 B
*5
,
15
121 181 B
*5
,221
16
402 B
*5
17
Estimating Fractal Dimension by Box Counting
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5
Ln(2^k)
ln(B
_k
)
161.1D(slope)
*5
Box-counting Theorem
Let X be a compact subset of ,
let be the “box-count” for X
using boxes of side , and
suppose exists.
Then L = D, the fractal dimension of X.
18
kk
k
XBL
2ln
lnlim
k21
XBk
2R
Outline of ProofLet be the smallest number of closed disks of
radius needed to cover X.
Step 1:
Step 2:
Step 3: , since
19
kXN21,
k21
XBXNXB kk k 21
141 ,
LBB
kk
kkk
k
2ln
lnlim
2ln
lnlim 14
1
LD kk
kXND
2ln
,lnlim 2
1
Step 1:
• A closed disk of radius can
intersect at most 4 grid boxes of side .
• Therefore .
20
XBXNXB kk k 21
141 ,
121 k
121 2121 kk
kXNXBk 21
141 ,
Step 1:
• A square box of side s can fit inside a ball of
radius r iff .
Pythagoras:
• Therefore every disk intersects at least 1 box:
. 21
XBXNXB kk k 21
141 ,
22 )2
(2s
r
222 )2
()2
(ss
r
XBXN kk 21,
Step 2:
22
11
1141
2ln2ln
4lnlnlim
2ln
lnlim
kk
kkk
k
BB
LBB
kk
kkk
k
2ln
lnlim
2ln
lnlim
11
Step 3: Prove that .
As ,
since
23
LD
kk
kXND
2ln
,lnlim 2
1
XBXNXB kk k 21
141 ,
kk
kkk XBXNXB k
2ln
ln
2ln
,ln
2ln
ln 21
141
LDL k
Boundary of Human Lymphocyte
24*2
25*2
Estimating Fractal Dimension by Box Counting
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6
Ln(2^k)
Ln
(B_K
)
273.1D(slope)
Biological Significance
• Bru (2003) and Izquierdo (2008) have shown
that fractal dimension and related critical
exponents can be used to classify growth
dynamics of a cell colony.
• A model of growth dynamics can potentially
predict tumor stages.26
References1. Aker, Eyvind. "The Box Counting Method." Fysisk Institutt, Universitetet I Oslo. 10 Feb. 1997. Web. 15
Mar. 2010. <http://www.fys.uio.no/~eaker/thesis/node55.html>.2. Bauer, Wolfgang. "Cancer Detection via Determination of Fractal Cell Dimension." 1-5. Web. 15 Mar.
2010.3. Barnsley, M. F. Fractals Everywhere. Boston: Academic, 1988. Print.4. Bru, Antonio. "The Universal Dynamics of Tumor Growth." Biophysical Journal 85 (2003): 2948-961. Print. 5. Baish, James W. "Fractals and Cancer." Cancer Research 60 (2000): 3683-688. Print.6. Clayton, Keith. "Fractals & the Fractal Dimension." Vanderbilt University | Nashville, Tennessee. Web. 15
Mar. 2010. <http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html>.7. "Fractal Dimension." OSU Mathematics. Web. 15 Mar. 2010.
<http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node37.html>.8. Izquierdo-Kulich, Elena. "Morphogenesis of the Tumor Patterns." Mathematical Biosciences and
Engineering 5.2 (2008): 299-313. Print. 9. Keefer, Tim. "American Metereological Society." Web. 20 Nov. 2009. 10. Lenkiewicz, Monika. "Culture and Isolation of Brain Tumor Initiating Cells | Current Protocols." Current
Protocols | The Fine Art of Experimentation. Dec. 2009. Web. 15 Mar. 2010. <http://www.currentprotocols.com/protocol/sc0303>.
11. Slice, Dennis E. "A Glossary for Geometric Morphometrics." Web. 20 Nov. 2009. 12. "Topological Dimension." OSU Mathematics. Web. 15 Mar. 2010.
<http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node36.html>.
27
Special Thanks
Alan Knoerr Angela GallegosRon Buckmire
Mathematics DepartmentFamilyFriends
“Mis Locas”♥
28