Fractal Dimension and Roughness Measurement in Biological Cell Boundaries
This content discusses the concept of fractal dimension in cell colony boundaries, focusing on measuring roughness using the Box-Counting method. It delves into the practical estimation of fractal dimension, covering definitions, preliminary concepts, and the application to tumor boundaries. Additionally, it explains the covering dimension, compact sets, and open covers in relation to fractal analysis.
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Fractal Dimension of Cell Colony Boundaries Gabriela Rodriguez April 15, 2010
Tumor Boundaries Isolated tumor growing in a Petri dish Interested in roughness of boundary in 2-D How can roughness be measured? *10 2
Fractal Dimension Measure of roughness (Mandelbrot ): a boundary is a fractal if its covering dimension fractal dimension Practical method of estimating fractal dimension: Box-counting 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
2 Limit Points in R p 2 An -neighborhood of R ( ) p O is an open disk , with p radius , centered at p. 0 X 2 R is a limit point of p ( ) p O iff for all X 0 . 6
2 R Compact Sets in is closed if it contains all its limit R X 2 points. 2 R X is bounded if it lies in a finite region of . 2 X is compact in if it is closed and bounded. R 7
2 R Open Covers of Compact Sets in X 2 An open cover of a compact set is a collection of neighborhoods of points in X whose union containsX. R Heine-Borel Theorem Every open cover of a compact set contains a finite sub-cover. 8
Covering Dimension X 2 R 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. The covering dimension of the curve is n = 1 because some points of the curve must lie in 2 =1+1 open disks. 9
Another View of Dimension *6 = 1 = 1 2 KEY : section size N: # of sections D: dimension = 1 3 ln N ( ) ( ) D = = = 1 ln ln 1 N N D D ( ) ln 1 10
2 R Closed Covers of Compact Sets in X 2 R A closed cover of a compact set is a collection of closed disks centered at points in X whose union containsX. 11
Fractal Dimension 2 Let X be a compact subset of . R The fractal dimension DofXis defined as ( ) ( ) 1 ln ( ) , X N ln , N X = lim D (if this limit exists), 0 where is the smallest number of closed disks of radius needed to cover X. 0 12
Box-Counting Method Cover 2 R with a grid, = k 1 2 whose squares have side length ( ) X Bk . Let be the number of grid squares (boxes) that intersect X. , the fractal dimension of X. D ( ) ( ) Bk ln ln 1 Plot vs. ) ) . 2 ln ( ( = ln ln 1 k B k Slope of plot D. 13
2= = 2 40 1 2 B , *5 16
Estimating Fractal Dimension by Box Counting *5 4 3.5 3 2.5 ln(B_k) . 1 161 D 2 1.5 (slope) 1 0.5 0 0 0.5 1 1.5 Ln(2^k) 17
Box-counting Theorem 2 R Let Xbe a compact subset of , be the box-count for X ( ) X Bk let 1 using boxes of side ,and k 2 ( ) k ln B X = lim k L suppose exists. ( ) 2 ln k Then L = D, the fractal dimension of X. 18
Outline of Proof Let be the smallest number of closed disks of ( k X N 2 , ) 1 radius needed to cover X. 1 k 2 Step 1: ( ) ( ) X ( ) X k , B N X B 1 1 1 k k 4 2 Step 2: ( ln ) ( ( ) 2 ln N ln ) ln B ln B 1 = = 1 k k lim k lim k 4 L ) ( ) 2 k k Step 3: , since ( ( ) 2 ln , X 1 D = k = L lim 2 D k k 19
( ) ( ) X ( ) X k , B N X B 1 1 Step 1: 1 k k 4 2 ( 1 ) = 1 k k 1 2 2 A closed disk of radius can 1 2 1 k 1 2 intersect at most 4 grid boxes of side . ( ) ( ) X , Bk N X 1 1 Therefore . 1 4 k 2 20
( ) ( ) X ( ) X k , B N X B 1 1 Step 1: 1 k k 4 2 A square box of side s can fit inside a ball of ( 2s r radius r iff . 2 2 ) 2 Pythagoras: s s + 2 2 2 ( ) ( ) r 2 2 Therefore every disk intersects at least 1 box: N . B ( ) ( ) X k , X 1 k 2 21
Step 2: ( ln ) ( ( ) ( ) ( ) 2 ln B ln ln 4 B 1 = 1 k 1 lim k lim k k 4 ( ) 2 ) + 1 1 k k ln 2 ln ( ( ) ) ( ( ) 2 ln ) ln ln B 2 B = = = 1 lim lim k k L 1 k k ln k k 22
D = L ) ) Step 3: Prove that ( ) X B k 1 4 . ( ( ) X k , N X B 1 1 k 2 ( ( ( ) 2 ln ) ( ) ( ) X ( ln ( ) ( ) 2 ) ln , N X 1 ln B ln B X 1 k 1 k k 4 2 ( ) 2 k k k ln As , L D L k ( ( ) 2 ln ) ln , N X 1 since k = lim 2 D k k 23
Estimating Fractal Dimension by Box Counting 8 7 6 5 Ln(B_K) 4 . 1 (slope) 273 D 3 2 1 0 0 1 2 3 4 5 6 Ln(2^k) *2 25
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
References 1. 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>. Bauer, Wolfgang. "Cancer Detection via Determination of Fractal Cell Dimension." 1-5. Web. 15 Mar. 2010. Barnsley, M. F. Fractals Everywhere. Boston: Academic, 1988. Print. Bru, Antonio. "The Universal Dynamics of Tumor Growth." Biophysical Journal 85 (2003): 2948-961. Print. Baish, James W. "Fractals and Cancer." Cancer Research 60 (2000): 3683-688. Print. 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>. "Fractal Dimension." OSU Mathematics. Web. 15 Mar. 2010. <http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node37.html>. Izquierdo-Kulich, Elena. "Morphogenesis of the Tumor Patterns." Mathematical Biosciences and Engineering 5.2 (2008): 299-313. Print. 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>. 2. 3. 4. 5. 6. 7. 8. 9. 27
Special Thanks Alan Knoerr Angela Gallegos Ron Buckmire Mathematics Department Family Friends Mis Locas 28
