mathematical modeling
DESCRIPTION
Mathematical modeling. And its possible applications in the diagnosis of cancer. Researched by: Mrs. Himani Asija PGT Mathematics (Delhi Public School Vasant Kunj New Delhi). The Problem. We have the solution to cancer but is it the cure? - PowerPoint PPT PresentationTRANSCRIPT
And its possible applications in the diagnosis of cancer
Researched by:Mrs. Himani AsijaPGT Mathematics
(Delhi Public School Vasant KunjNew Delhi)
We have the solution to cancer but is it the cure?
Patient specific titration of the dosage of cancer related drugs is at best a vague field.
Behavior of cancerous cells is seemingly random and unpredictable. It is little understood and therefore, at present inadequately treated.
The Problem
The SolutionTo try and understand how the cogs of a tumor turn; to try and predict how it’ll react to change in it’s environment (the body of the patient).
This allows us to maximize tumor damage and minimize patient damage.
WHERE AND HOW IT STARTED….A holiday homework assignment given to
children where they had to draw fractals figures of Koch snowflake, Sierpinski’s carpet and Sierpinski’s triangle and find their areas and perimeters at different stages
Generalise the above to the nth stageKoch snowflakeLINK : koch snowlfake.gspSerspenki’s carpetLINK : serspenki.gsp
ABOUT THE PROJECT…THOSE WHO SAY THAT MATH HOMEWORKS ARE
BORING AND FAR FROM REAL WORLD
BEWARE!!!
This project is an endeavor not only to talk, discuss, and research about cancer cells, but also correlate the biology of the cells with the mathematics in it.
The project is based on two hypotheses. In the first hypotheses, a dynamic software called the geometer’s sketchpad has been used and for the second hypothesis, MS EXCEL and a freeware Graphmatica has been used.
OBSERVATION -1
ITERATION
PERIMETER
AREA ENCLOSED
0 3x (√3/4)x²
1 3(4x)/3=4x (√3/4)x²+ 3(√3/4)(x/3) ²= (√3/4)x² (1+3/3²)
2 16x/3 (√3/4)x²+ 3(√3/4)(x/3) ²+ 12(√3/4)(x/9) ²= (√3/4)x² (1+3/3²+12/9²)
3 64x/9 (√3/4)x²+ 3(√3/4)(x/3) ²+ 12(√3/4)(x/9) ² +48(√3/4)(x/27) ² = (√3/4)x² (1+3/3²+12/9²+48/27²)
4 192x/27 (√3/4)x²+ 3(√3/4)(x/3) ²+ 12(√3/4)(x/9) ² +48(√3/4)(x/27) ² +192(√3/4)(x/81) ² = (√3/4)x² (1+3/3²+12/9²+48/27² +192/81²)
Make a Koch snowflake with an equilateral triangle of side x cm. We obtain the following table LINK : KOCH SNOWFLAKE
PERIMETERThe perimeters form a geometric progression with common ratio 4/3, which is greater than one3x, 4x, 16x/3, 48x/9, 192x/27, …………So, the nth term Tn = 3x(4/3)n-1 which increases infinitely as n increases infinitely.
Conclusion: The perimeter of the polygon approaches infinity as n approaches infinity
AREA ENCLOSEDThe area enclosed by the polygon forms a geometric progression with common ratio 4/9, which is less than one (√3/4)x² (1+3/3²+12/9²+48/27² +192/81²+……….)= (√3/4)x² (1+ ) = (√3/4)x² ( 8/5) = (√3/5)2x²
= 8/5 times the area of the original triangle
Conclusion: The area enclosed by the polygon is finite even when n approaches infinity
THE RATIO OF THE PERMETER SQUARED AND AREA INCREASES INFINITELY AS THE NO. OF SIDES OF THE POLYGON INCREASES INFINITELY
NOTE The perimeter has been squared to produce a dimensionless quantity in the ratio
OBSERVATION 2 THE RATIO OF PERIMETER SQUARED AND
AREA IS MAXIMUM WHEN THE NO. OF SIDES OF A FIGURE IS MINIMUM; IT BEING MAXIMUM FOR A TRIANGULAR FIGURE AND MINIMUM FOR A CIRCLE ( THE NO. OF SIDES OF A CIRCLE IS INFINITE)
TO BEGIN WITH THE CANCER CELLS…HYPOTHESIS – 1Cancer cells follow the fractal figure, the Koch Snowflake.
The ratio of the square of the perimeter and the area of a normal cell is the least and that of the cell at the advanced stage is the maximum; it increases with the increase in the stage of malignancy.
Normal (non cancerous) human cellPerimeter Perimeter2 Area Perimeter2
Area
22.84 521.56 35.19 14.82
Gsp snapshot..
P1
Perimeter P1 2
Area P1 = 14.82
Area P1 = 35.19 cm2
Perimeter P1 = 22.84 cm
Cancer cell in preliminary stagePerimeter Perimeter2 Area Perimeter2
Area
28.80 829.28 34.64 23.94
Gsp snapshot….
P1
perimeter2
area = 23.94
perimeter = 28.80 cm
area = 34.64 cm2
B
T
S
R Q
P
O
N
M
L
K
J
I
HG
F
E
D
C
A
Cancer cell in intermediate stagePerimeter Perimeter2 Area Perimeter2
Area
82.23 6761.31 56.65 119.36
Gsp snapshot..
P2
Perimeter P2 2
Area P2 = 119.36
Perimeter P2 = 82.23 cm
Area P2 = 56.65 cm2
Cancer cell in advanced stageS.No
Perimeter Perimeter2 Area Perimeter2
Area
1 335.22 112375.34 60.48 1858.05
2 315.36 99451.84 54.95 1809.84
3 396.60 157289.45 100.95 1558.10
1.
2.
3 .
Gsp snapshot…P1
Perimeter P1 2
Area P1 = 1858.05
Perimeter P1 = 335.22 cm
Area P1 = 60.48 cm2
Gsp snapshot …
P1
Perimeter P1 2
Area P1 = 1809.84Perimeter P1 = 315.36 cm
Area P1 = 54.95 cm2
Gsp snapshot
P1
Perimeter P1 2
Area P1 = 1558.10
Area P1 = 100.95 cm2
Perimeter P1 = 396.60 cm
Hypothesis 2FRACTAL DIMENSIONS BY BOX COUNTING METHOD:
•Fractal dimension by box method is calculated as the slope of the line of best fit obtained by plotting the points ( Ln(S), Ln (Ne )) where S is the dimension of the square grid required to cover the picture and Ne is the no. of boxes of the grid required to cover the picture. Ln is the natural log of the respective values
HIGHER THE STAGE OF MALIGNANCY, LESS IS THE FRACTAL DIMENSION
OUR HYPOTHESIS IS:
THE FRACTAL DIMENSION OF THE NORMAL CELL IS MAXIMUM AND IT REDUCES AS THE STAGE ADVANCES SO
Normal Cell (non cancerous)
1 1 1 1 1 1 6 1/s=11
1 1 1 1 1 1 1 1 1 9 covered= 78
1 1 1 1 1 1 1 1 1 9 4.356708827
1 1 1 1 1 1 1 1 1 1 10 2.397895273
1 1 1 1 1 1 1 1 1 1 10
1 1 1 1 1 1 1 1 1 1 10
1 1 1 1 1 1 1 1 1 1 10
1 1 1 1 1 1 1 1 8
1 1 1 1 1 5
1 1
78LINK: EXCEL SHEET SCALES
OBTAINING THE LINE OF BEST FIT AND ITS SLOPE BY PLOTING THE POINTS ON THE X-Y AXIS
Ln(1/s)
Ln(Ne )
Ln (7) = 1.95
Ln(37) = 3.6
Ln(11) = 2.4
Ln(78) = 4.36
Ln(16) = 2.77
Ln(187) = 5.23
Ln(21) = 3.04
Ln(314) = 5.78
FRACTAL DIMENSION = 2.03 LINK: GRAPHMATICA FILE
Cancer cell in preliminary stage
1 1 1 1 1 1 1 1 1 91 1 1 1 1 1 1 1 1 1 10
1 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 1 1 1 1 1 1 141 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15
1 1 1 1 1 1 1 1 1 1 1 1 1 1 141 1 1 1 1 1 1 1 1 1 1 1 1 1 141 1 1 1 1 1 1 1 1 1 1 1 1 1 141 1 1 1 1 1 1 1 1 1 1 1 1 1 14
1 1 1 1 1 1 1 1 1 1 1 1 12 1/S = 291 1 1 1 1 1 1 1 1 1 10
1 1 1 1 1 1 1 1 8 COVERED = 3151 1 1 1 1 51 1 1 1 1 5 5.752572639
1 1 1 1 1 1 61 1 1 1 1 1 6
1 1 1 1 1 1 1 1 8 3.367295831 1 1 1 1 1 1 1 1 1 101 1 1 1 1 1 1 1 1 1 1 11
1 1 1 1 1 1 1 1 1 1 1 1 121 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 1 1 1 1 12
1 1 1 1 1 1 1 1 1 1 1 111 1 1 1 1 1 1 1 1 9
1 1 1 1 1 1 1 1 8
315
LINK ; EXCEL SHEET SCALES
OBTAINING THE LINE OF BEST FIT AND ITS SLOPE BY PLOTING THE POINTS ON THE X-Y AXIS
Ln(1/s)
Ln(Ne )
Ln (9) = 2.2
Ln(38) = 3.64
Ln(14) = 2.64
Ln(76) = 4.36
Ln(22) = 3.09
Ln(185) = 5.22
Ln(29) = 3.37
Ln(31) = 5.75
FRACTAL DIMENSION = 1.82 LINK: GRAPHMATICA FILE
Cancer cell in intermediate stage1 1
1 1 1 31 1 1 1 41 1 1 1 1 1 6
1 1 1 1 1 1 61 1 1 1 1 1 1 1 1 1 101 1 1 1 1 1 1 1 1 1 1 1 12
1 1 1 1 1 1 1 1 1 1 1 111 1 1 1 1 1 1 1 1 1 1 1 121 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 161 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 161 1 1 1 1 1 1 1 1 1 1 1 1 1 1 151 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 1 1 1 1 1 13
1 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 1 1 1 1 12
1 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 1 1 1 1 1 1 14
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 161 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 171 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 18 3.8286413961 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 19 6.381816017
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 211 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 23
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 241 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 26
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 281 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 271 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 211 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 161 1 1 1 1 1 1 1 1 1 1 1 121 1 1 1 1 1 1 1 1 1 101 1 1 1 1 1 1 1 1 91 1 1 1 1 1 1 1 1 9
1 1 1 1 1 1 1 1 81 1 1 1 1 1 1 1 8
1 1 1 1 1 1 1 1 81 1 1 1 1 1 1 1 1 91 1 1 1 1 1 1 1 1 91 1 1 1 1 1 61 1 1 1 1 5
1 1 1 1 1 1 1 71 1 1 1 1 5
1 1 1 3
591
LINK :EXCEL SHEET SCALES
OBTAINING THE LINE OF BEST FIT AND ITS SLOPE BY PLOTING THE POINTS ON THE X-Y AXIS
FRACTAL DIMENSION = 1.73
Ln(1/s)
Ln(Ne )
Ln (14) = 2.64
Ln(76) = 4.33
Ln(22) = 3.09
Ln(157) = 5.06
Ln(35) = 3.56
Ln(359) = 5.88
Ln(46) = 3.83
Ln(591) = 6.38
LINK: GRAPHMATICA FILE
Cancer cell in advanced stage -1
1 11 1 1 1 1 1 1 1 1 1 1 11
1 1 1 1 1 1 1 1 1 1 1 1 1 1 141 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 201 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 24
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 251 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 28
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 321 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 35
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 421 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 44
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 461 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 44
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 421 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49 1/S = 751 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 58 COVERED = 14131 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 61
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 59 7.2534703831 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 52
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 4.3174881141 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 48
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 621 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 451 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 44
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 441 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 42
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 381 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 30
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 271 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 27
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 261 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 22
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 241 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17
1 1 1 1 1 1 1 1 1 1 1 1 1 131 1 1 1 1 1 1 1 8
1 1 21413
LINK: EXCEL SHEET SCALE
OBTAINING THE LINE OF BEST FIT AND ITS SLOPE BY PLOTING THE POINTS ON THE X-Y AXIS
Ln(1/s)
Ln(Ne )
Ln (23) = 3.13
Ln(192) = 5.26
Ln(35) = 3.56
Ln(381) = 5.94
Ln(56) = 4.03
Ln(876) = 6.78
Ln(75) = 4.32
Ln(1413) = 7.25
FRACTAL DIMENSION = 1.7 LINK : GRAPHMATICA FILE
Cancer cell in advanced stage-21 1 21 1 1 1 1 1 1 1 8
1 1 1 1 1 1 1 71 1 1 1 1 1 1 1 1 1 1 1 12
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 191 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 241 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 27
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 27 1/s=341 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 30
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 27 6.3647507571 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 30
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 26 3.5263605251 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 321 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 26
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 301 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 25
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 191 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 22
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 251 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 24
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 221 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 21
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 211 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 221 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 18
1 1 1 1 1 1 1 1 1 91 1 1 1 1 1 1 1 1 91 1 1 1 1 1 1 1 8
1 1 1 1 1 1 1 71 1 2
581
OBTAINING THE LINE OF BEST FIT AND ITS SLOPE BY PLOTING THE POINTS ON THE X-Y AXIS
Ln(1/s)
Ln(Ne )
Ln (23) = 2.64
Ln(192) = 4.84
Ln(35) = 3.04
Ln(381) = 5.58
Ln(56) = 3.53
Ln(876) = 6.36
Ln(75) = 3.81
Ln(1413) = 6.8FRACTAL DIMENSION = 1.68 LINK : GRAPHMATICA FILES
1 1 1 1 1 1 1 1 1 1 10
1 1 1 1 1 1 1 1 1 1 10
1 1 1 1 1 1 1 1 1 1 1 1 12
1 1 1 1 1 1 1 1 1 1 1 11 2.564949357
1 1 1 1 1 1 1 1 1 1 1 1 12 4.762173935
1 1 1 1 1 1 1 1 1 1 1 1 12
1 1 1 1 1 1 1 1 1 1 1 1 1 13
1 1 1 1 1 1 1 1 1 1 1 1 1 13
1 1 1 1 1 1 1 1 1 1 10
1 1 1 1 1 1 1 1 1 9
1 1 1 1 1 5
117
FRACTAL DIMENSION = 1.61
Ln(1/s)
Ln(Ne )
Ln (13) = 3
Ln(117) = 5.52
Ln(35) = 3.47
Ln(381) = 6.33
Ln(56) = 3.78
Ln(876) = 6.78
LINK : GRAPHMATICA FILE
STAGE R = P^2/A FRACTAL DIMENSION
NORMAL (SLIDE 8) 14.82 2.03
PRELIMINARY STAGE
23.94 1.82
INTERMIDIATE STAGE
119.36 1.73
ADVANCED STAGE - 1
1558 1.7
ADVANCED STAGE - 2
1809.84 1.68
ADVANCED STAGE- 3
1858.05 1.61
INCREASING
DECREASINGOther Links for fractal dimensions excel files1.Fractal dimension of a rectangle (same as to the topological dimension=2)
1.Fractal dimension remains same even if the size of the figure under study is reduced/increased.
Microsoft Office Excel Worksheet
Microsoft Office Excel Worksheet
DIMENIOSN OF RECTANGLE BY BOX COUNTING
FD OF SMALLER SIZE CELL STAGE 1
Limitations of the approach Making the equations that govern the model is a slow, hit and trial process.
There are always biological fudge factors which are almost impossible to predict and
even harder to stimulate. For example: A particular patient’s tumor could develop a
unique adaptation mechanism to counter the effects of the chemotherapy and
radiation.
Each individual’s body is unique in its own way and it is very difficult to define a
model that would give results to the same degree of accuracy for all the patients.
For example, a fat person will have a slower and more restricted blood flow, which in
turn affects the oxygen supply to the tumor and even how effective a given dosage
of drugs will be and whether or not the drugs will reach the intended site in the
intended concentration.
Our Resource Limitations We couldn’t get access to hands-on pictures of cancer cells from doctors and certified
hospitals due to the patient privacy policy. So, we had to rely on pictures found on the
internet.
Due to our limited knowledge in the field of cellular structure and medicine, we couldn’t
satisfactorily explore the biological depth of the subject.
The pictures were converted into polygons, which gave a very approximate shape.
For clinical purposes, approximations do not give accurate results. So a better
software is needed to apply the concept.
To find the fractal dimension by box counting method was done by using
MS EXCEL which does not give very accurate result to be used for diagnostic
researches. Software giving more accurate fractal dimension shall be needed to
conclude the actual stage of a patient. This was just an attempt to make the students
realize how the mathematics they study is so closely related to their lives.
REFERENCES1. http://mste.illinois.edu/dildine/cancer/2. Fratals for the classroom , by Evan
Maletsky, Terry Perciantae, Lee Yunker3. http://cancerres.aacrjournals.org/content/
60/14/3683.full.pdf4. A free trial version of the soft wares used
can be downloaded from the following links: http://www.keypress.com/x24795.xml
(geometer’s sketchpad) http://www8.pair.com/ksoft/ (graphmatica)
CLINICAL JUSTIFICATIONS OF THE HYPOTHESIS
http://www.newscientist.com/article/mg15721182.100-fractal-cancer.html
Personal details of the researcherNAME: Himani AsijaSCHOOL: Delhi Public School Vasant KunjADDRESS: B 804, NPSC Apts., Plot no. 5,
Sector-2, Dwarka New Delhi - 110075CONTACT NO.: 9717160042 E MAIL [email protected] http://mathemagic-
himani.spaces.live.com/