1. FRACTALS IN BIOLOGY ATTACKING THE ROOT OF CANCER http://www.uib.no/med/angiotargeting/research_activities/research_activities.html A Classroom Lesson From the MathScience Innovation Center

2. Fractals in Biology (c) MathScience Innovation Center 2007 HOW DO YOU DRAW LIFE? In straight lines? In circles? In squares?

3. Fractals in Biology (c) MathScience Innovation Center 2007 WHEN WAS THE LAST TIME YOU SAW A REAL TREE THAT LOOKED LIKE THIS?

4. Fractals in Biology (c) MathScience Innovation Center 2007 OR A FISH THAT LOOKED LIKE THIS?

5. Fractals in Biology (c) MathScience Innovation Center 2007 LIFE IS NOT MADE OF SIMPLE SHAPES

6. Fractals in Biology (c) MathScience Innovation Center 2007 AND WE’RE FINDING THAT OUR STUDY OF LIFE SHOULDN’T BE LIMITED TO THEM http://brain.mgh.harvard.edu/tumormodeling/models.htm

7. Fractals in Biology (c) MathScience Innovation Center 2007 IN FACT, THERE IS A WORD DESCRIBING THESE COMPLEX SHAPES

8. Fractals in Biology (c) MathScience Innovation Center 2007 WHAT IS A FRACTAL? Basically, it’s a shape that is defined by non- integer dimensions Like this satellite image of the Gulf Stream

9. Fractals in Biology (c) MathScience Innovation Center 2007 WHAT IS A FRACTAL? Or this model of the insulin molecule

10. Fractals in Biology (c) MathScience Innovation Center 2007 ALL OF THESE BEAUTIFUL DESIGNS ARE FRACTALS Images courtesy of www.wyomingwebdesign.com

11. Fractals in Biology (c) MathScience Innovation Center 2007 ALL FRACTALS ARE ROUGH Like this coastline

12. Fractals in Biology (c) MathScience Innovation Center 2007 ALL FRACTALS ARE SELF- SIMILAR Meaning that they are composed of smaller versions of themselves.

13. Fractals in Biology (c) MathScience Innovation Center 2007 A FRACTAL IS A SHAPE THAT CAN’T BE DESCRIBED BY THE USUAL GEOMETRIC TERMS For example, what shape is a plant’s root? More importantly, how do you measure a plant’s root? Let’s try it!

14. Fractals in Biology (c) MathScience Innovation Center 2007 WHAT PROBLEMS DID YOU HAVE MEASURING THE ROOT? –What part of the root did you measure? –Was a ruler adequate for the task? –Do you think your measurement would help you determine if a plant is growing properly?

15. Fractals in Biology (c) MathScience Innovation Center 2007 FRACTALS AREN’T MEASURED BY CONVENTIONAL METHODS We can use something called a “box count” We count how many squares in grids of different sizes the fractal occupies. As the grid gets smaller, the number of squares occupied gets bigger exponentially!

16. Fractals in Biology (c) MathScience Innovation Center 2007 TRY A BOX COUNT WITH A PLANT ROOT

17. Fractals in Biology (c) MathScience Innovation Center 2007 TRY A BOX COUNT WITH A PLANT ROOT

18. Fractals in Biology (c) MathScience Innovation Center 2007 TRY A BOX COUNT WITH A PLANT ROOT

19. Fractals in Biology (c) MathScience Innovation Center 2007 MEASURING A ROOT BY BOX COUNT GRID SIZE7.43.81.41.0.5 GRID SPACES OCCUPIED Fractal Dimension = After your BOX COUNT, use the graphing calculator to calculate the Fractal Dimension of your root sample. Data Table

20. Fractals in Biology (c) MathScience Innovation Center 2007 TRY A BOX COUNT WITH A PLANT ROOT It’s much easier to use a box counting program like Winfeed!Winfeed

21. Fractals in Biology (c) MathScience Innovation Center 2007 FRACTALS OCCUR EVERYWHERE IN NATURE The more we look, the more places we find fractals. Not just in the big, but also in the very small www.ghcc.msfc.nasa.gov

22. Fractals in Biology (c) MathScience Innovation Center 2007 FRACTALS IN MEDICINE Fractal shapes are everywhere in the human body © University of Alabama at Birmingham, Department of Pathology

23. Fractals in Biology (c) MathScience Innovation Center 2007 FRACTALS IN MEDICINE And there is hope that they will be useful in diagnosing and treating such things as cardiovascular disease and cancer. © University of Alabama at Birmingham, Department of Pathology

24. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS A tumor is a mass of cells that no longer recognize the growth limits of a normal cell. http://brain.mgh.harvard.edu/tumormodeling/models.htm

25. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS Normally, a cell stops growing when it contacts other cells or experiences a change in growth regulating genes Credit: Nicolle Rager, National Science Foundation

26. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS When a tumor first begins to grow, it is supplied nutrients and oxygen via diffusion from nearby blood vessels. www.cancerquest.org

27. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS As the tumor enlarges, it releases substances that stimulate branching of the blood vessels. www.cancerquest.org

28. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS Gradually, more branches are added and the tumor is able to enlarge. www.cancerquest.org

29. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS This process of blood vessel growth and branching is called “angiogenesis”. www.cancerquest.org

30. Fractals in Biology (c) MathScience Innovation Center 2007

35. TUMOR ANGIOGENESIS Once a tumor has commandeered blood vessels, tumor cells may break off and spread (termed “metastasize”) to other parts of the body. Illustration from: Horizons in Cancer Therapeutics: From Bench to Bedside

36. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS Some researchers are studying the possibility of using fractals to fight cancer.

37. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS One such study used a computer model to determine that blood vessels grow throughout a tumor the same way that a plant root penetrates the soil.

38. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS A plant root moves in response to water availability, but it still has to move around the less penetrable soil parts

39. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS Let’s try to recreate that computer model using a hands-on exercise.

40. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS This grid represents the extracellular matrix of a tumor mass. The dark spaces are resistant to blood vessel invasion

41. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS Beginning at one side of the tumor, draw a line through all available white spaces.

42. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS If you have no other white spaces to move into, you may “leap” one gray space into a white space and continue on. Use up every available white space before leaping a gray one.

43. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS When you are through, trace the shortest path from one side of the “tumor” to the other.

44. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS When you are through, trace the shortest path from one side of the “tumor” to the other. This is called the “minimum path” of vascular tissue.

45. Fractals in Biology (c) MathScience Innovation Center 2007 TUMOR ANGIOGENESIS Now find the fractal dimension of the minimum path Researchers have found that the minimum path in a tumor has a greater fractal dimension than that in a normal tissue

46. Fractals in Biology (c) MathScience Innovation Center 2007 SO WHAT! Researchers may be able to use the fractal dimension of tissue blood vessels to better diagnose the presence of tumors. Illustration from: Horizons in Cancer Therapeutics: From Bench to Bedside

47. Fractals in Biology (c) MathScience Innovation Center 2007 SO WHAT! A better understanding of the fractal nature of tumor angiogenesis is presenting new options for the delivery of chemotherapy. Illustration from: Horizons in Cancer Therapeutics: From Bench to Bedside

48. Fractals in Biology (c) MathScience Innovation Center 2007 SO WHAT! Fractal models of tumor growth are suggesting new ways of fighting tumors – such as shutting down tumor vasculature to “starve” the tumor. Illustration from: Horizons in Cancer Therapeutics: From Bench to Bedside

49. Fractals in Biology (c) MathScience Innovation Center 2007 IN CONCLUSION Fractals are providing a new way of studying biology. From the patterns of root growth in soil To the patterns of blood vessels in a tumor We are gaining new insights into the nature of growth and development.