1. And its possible applications
in the diagnosis of cancer
Researched by:
Mrs. Himani Asija
PGT Mathematics
(Delhi Public School Vasant Kunj
New Delhi)
2. The Problem
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.
3. The Solution
To 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.
4. 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
Generalize the above to the nth stage
Koch snowflake Sierpinski’s carpet
next 2
KOCH SNOWFLAKE SIERPENSKI'S
Open GSP file Open GSP fileSIERPENSKI'S
CARPET
CARPET
GSP snapshot GSP snapshot
J I
G H
E
B
5. 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.
6. OBSERVATION -1
Make a Koch snowflake with an equilateral
triangle of side x cm. We obtain the following
table LINK : KOCH SNOWFLAKE
ITERATIO PERIMETER AREA ENCLOSED
N
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)
7. PERIMETER
The perimeters form a geometric progression with common ratio 4/3, which is
greater than one
3x, 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 ENCLOSED
THE RATIO OF THE PERMETER SQUARED AND
The area enclosed by the polygon forms a geometric progression with common
AREA INCREASES INFINITELY AS THE NO. OF
ratio 4/9, which is less than one
SIDES OF THE POLYGON INCREASES INFINITELY
(√3/4)x² (1+3/3²+12/9²+48/27² +192/81²+……….)
= (√3/4)x² (1+ ) = (√3/4)x² ( 8/5) = (√3/5)2x²
NOTE The perimeter has been squared to produce area of the original
= 8/5 times the a dimensionless
quantity in the ratio triangle
Conclusion: The area enclosed by the polygon is finite
even when n approaches infinity
8. 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)
9. TO BEGIN WITH THE CANCER
CELLS…
HYPOTHESIS – 1
Cancer 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.
10. Normal (non cancerous) human
cell
Perimeter Perimeter2 Area Perimeter2
Area
22.84 521.56 35.19 14.82
Open GSP file
NORMAL CELL
11. GSP snapshot…
Perimeter P1 = 22.84 cm
Area P1 = 35.19 cm2
(Perimeter P1)2
= 14.82
(Area P1)
P1
12. Cancer cell in preliminary stage
Perimeter Perimeter2 Area Perimeter2
Area
28.80 829.28 34.64 23.94
Open GSP file PRELIMINARY
13. GSP snapshot…
area = 34.64 cm2
perimeter = 28.80 cm
perimeter2
R Q = 23.94
area
S P
T O
N
B
M
A L
C
K
D
J
E
I
F
P1
G H
14. Cancer cell in intermediate stage
Perimeter Perimeter2 Area Perimeter2
Area
82.23 6761.31 56.65 119.36
Open GSP file
INTERMIDIATE
15. GSP snapshot…
Ar ea P2 = 56.65 cm2
Perimeter P2 = 82.23 cm
(Perimeter P2)2
= 119.36
(Area P2)
P2
16. Cancer cell in advanced stage
S.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. Open GSP file
ADVANCED STAGE
3 .Open GSP file
3 ADVANCED STAGE
1
2. Open GSP file
ADVANCED STAGE
2
17. GSP snapshot…
Area P1 = 60.48 cm2
P1 Perimeter P1 = 335.22 cm
(Perimeter P1)2
= 1858.05
(Area P1)
19. GSP snapshot…
Perimeter P1 = 396.60 cm (Perimeter P1)2
= 1558.10
Area P1 = 100.95 cm2 (Area P1)
P1
20. Hypothesis 2
FRACTAL 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
OUR HYPOTHESIS IS: square grid required to cover the
S is the dimension of the
picture and
THE FRACTALboxes of the grid required to cover the picture.
Ne is the no. of DIMENSION OF THE NORMAL CELL IS
MAXIMUM AND IT REDUCES AS THE STAGE ADVANCES
Ln is the natural log of the respective values
SO
HIGHER THE STAGE OF MALIGNANCY, LESS IS THE
FRACTAL DIMENSION
22. NORMAL CELL
OBTAINING THE LINE OF BEST FIT AND ITS SLOPE
BY PLOTING THE POINTS ON THE X-Y AXIS
Graphmatica snapshot
Ln(1/s) Ln(Ne )
Ln (7) Ln(37)
= 1.95 = 3.6
Ln(11) = Ln(78)
2.4 = 4.36
Ln(16) Ln(187)
= 2.77 = 5.23
Ln(21) Ln(314)
= 3.04 = 5.78
OPEN GRAPHMATICA FILE
NORMAL CELL
FRACTAL DIMENSION = 2.03
24. PRELIMINARY STAGE
OBTAINING THE LINE OF BEST FIT AND ITS SLOPE
BY PLOTING THE POINTS ON THE X-Y AXIS
Graphmatica snapshot
Ln(1/s) Ln(Ne )
Ln (9) = Ln(38)
2.2 = 3.64
Ln(14) = Ln(76)
2.64 = 4.33
Ln(22) Ln(185)
= 3.09 = 5.22
Ln(29) Ln(31) =
= 3.37 5.75
OPEN GRAPHMATICA FILE
FRACTAL DIMENSION = 1.82 NORMAL CELL
26. intermediate stage
OBTAINING THE LINE OF BEST FIT AND ITS SLOPE
BY PLOTING THE POINTS ON THE X-Y AXIS
Graphmatica snapshot
Ln(1/s) Ln(Ne )
Ln (14) Ln(76)
= 2.64 = 4.33
Ln(22) Ln(157)
= 3.09 = 5.06
Ln(35) Ln(359)
= 3.56 = 5.88
Ln(46) Ln(591)
= 3.83 = 6.38
OPEN GRAPHMATICA FILE
FRACTAL DIMENSION = 1.73 INTERMIDIATE
STAGE
33. SUMMARY
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 DECREASING
Other Links for fractal dimensions excel files
1. Fractal dimension of a rectangle (same as to the topological dimension=2)
OPEN
OPEN EXEL FILE
GRAPHMATICA FILE RECTANGLE
RECTANGLE
2. Fractal dimension remains same even if the size of the figure under study is
reduced/increased. OPEN
GRAPHMATICA FILE FD OF SMALLER
OPEN EXEL FILE SIZE CELL STAGE 1
ADVANCED STAGE
1
34. 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.
35. 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.
36. REFERENCES
1. http://mste.illinois.edu/dildine/cancer/
2. Fractals for the classroom , by Evan Maletsky, Terry Perciantae, Lee
Yunker
3. http://cancerres.aacrjournals.org/content/60/14/3683.full.pdf
4. 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
37. Personal details of the researcher
NAME: Himani Asija
SCHOOL: Delhi Public School Vasant Kunj
ADDRESS: B 804, NPSC Apts., Plot no. 5,
Sector-2, Dwarka New Delhi - 110075
CONTACT NO.: 97171-60042
E MAIL himaniasija@hotmail.com
Blog http://mathemagic-himani.spaces.live.com/
