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
Fractals in Biology (c) MathScience Innovation Center 2007
HOW DO YOU DRAW LIFE?
• In straight lines? • In circles?• In squares?
Fractals in Biology (c) MathScience Innovation Center 2007
WHEN WAS THE LAST TIME YOU SAW A REAL TREE THAT
LOOKED LIKE THIS?
Fractals in Biology (c) MathScience Innovation Center 2007
OR A FISH THAT LOOKED LIKE THIS?
Fractals in Biology (c) MathScience Innovation Center 2007
LIFE IS NOT MADE OF SIMPLE SHAPES
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
Fractals in Biology (c) MathScience Innovation Center 2007
IN FACT, THERE IS A WORD DESCRIBING THESE COMPLEX
SHAPES
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
Fractals in Biology (c) MathScience Innovation Center 2007
WHAT IS A FRACTAL?
• Or this model of the insulin molecule
Fractals in Biology (c) MathScience Innovation Center 2007
ALL OF THESE BEAUTIFUL DESIGNS ARE FRACTALS
Images courtesy of www.wyomingwebdesign.com
Fractals in Biology (c) MathScience Innovation Center 2007
ALL FRACTALS ARE ROUGH
• Like this coastline
Fractals in Biology (c) MathScience Innovation Center 2007
ALL FRACTALS ARE SELF-SIMILAR
• Meaning that they are composed of smaller versions of themselves.
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!
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?
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!
Fractals in Biology (c) MathScience Innovation Center 2007
TRY A BOX COUNT WITH A PLANT ROOT
Fractals in Biology (c) MathScience Innovation Center 2007
TRY A BOX COUNT WITH A PLANT ROOT
Fractals in Biology (c) MathScience Innovation Center 2007
TRY A BOX COUNT WITH A PLANT ROOT
Fractals in Biology (c) MathScience Innovation Center 2007
MEASURING A ROOTBY BOX COUNT
GRID SIZE 7.4 3.8 1.4 1.0 .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
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!
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
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
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
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
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
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
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
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
Fractals in Biology (c) MathScience Innovation Center 2007
TUMOR ANGIOGENESIS
• This process of blood vessel growth and branching is called “angiogenesis”.
www.cancerquest.org
Fractals in Biology (c) MathScience Innovation Center 2007
Fractals in Biology (c) MathScience Innovation Center 2007
Fractals in Biology (c) MathScience Innovation Center 2007
Fractals in Biology (c) MathScience Innovation Center 2007
Fractals in Biology (c) MathScience Innovation Center 2007
Fractals in Biology (c) MathScience Innovation Center 2007
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
Fractals in Biology (c) MathScience Innovation Center 2007
TUMOR ANGIOGENESIS
• Some researchers are studying the possibility of using fractals to fight cancer.
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.
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
Fractals in Biology (c) MathScience Innovation Center 2007
TUMOR ANGIOGENESIS
Let’s try to recreate that computer model using a hands-on exercise.
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
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.
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.
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.
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.
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
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
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
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
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.