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.

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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

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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.

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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

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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>.

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Special Thanks

Alan Knoerr Angela GallegosRon Buckmire

Mathematics DepartmentFamilyFriends

“Mis Locas”♥

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