FRACTALS IN BIOLOGY ATTACKING THE ROOT OF CANCER A Classroom Lesson From

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?

WHEN WAS THE LAST TIME YOU SAW A REAL TREE THAT

LOOKED LIKE THIS?

OR A FISH THAT LOOKED LIKE THIS?

LIFE IS NOT MADE OF SIMPLE SHAPES

AND WE’RE FINDING THAT OUR STUDY OF LIFE SHOULDN’T BE

LIMITED TO THEM

http://brain.mgh.harvard.edu/tumormodeling/models.htm

IN FACT, THERE IS A WORD DESCRIBING THESE COMPLEX

SHAPES

WHAT IS A FRACTAL?

• Basically, it’s a shape that is defined by non-integer dimensions

• Like this satellite image of the Gulf Stream

WHAT IS A FRACTAL?

• Or this model of the insulin molecule

ALL OF THESE BEAUTIFUL DESIGNS ARE FRACTALS

Images courtesy of www.wyomingwebdesign.com

ALL FRACTALS ARE ROUGH

• Like this coastline

ALL FRACTALS ARE SELF-SIMILAR

• Meaning that they are composed of smaller versions of themselves.

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!

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 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!

TRY A BOX COUNT WITH A PLANT ROOT

MEASURING A ROOTBY BOX COUNT

GRID SIZE 7.4 3.8 1.4 1.0 .5

SPACES

OCCUPIED

Fractal Dimension =

After your BOX COUNT, use the graphing calculator to calculate the Fractal Dimension of your root sample.

Data Table

• It’s much easier to use a box counting program like Winfeed!

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 MEDICINE

• Fractal shapes are everywhere in the human body

FRACTALS IN MEDICINE

• And there is hope that they will be useful in diagnosing and treating such things as cardiovascular disease and cancer.

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

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

TUMOR ANGIOGENESIS

• When a tumor first begins to grow, it is supplied nutrients and oxygen via diffusion from nearby blood vessels.

www.cancerquest.org

TUMOR ANGIOGENESIS

• As the tumor enlarges, it releases substances that stimulate branching of the blood vessels.

www.cancerquest.org

TUMOR ANGIOGENESIS

• Gradually, more branches are added and the tumor is able to enlarge.

www.cancerquest.org

TUMOR ANGIOGENESIS

• This process of blood vessel growth and branching is called “angiogenesis”.

www.cancerquest.org

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

TUMOR ANGIOGENESIS

• Some researchers are studying the possibility of using fractals to fight cancer.

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.

TUMOR ANGIOGENESIS

• A plant root moves in response to water availability, but it still has to move around the less penetrable soil parts

TUMOR ANGIOGENESIS

Let’s try to recreate that computer model using a hands-on exercise.

TUMOR ANGIOGENESIS

This grid represents the extracellular matrix of a tumor mass.

The dark spaces are resistant to blood vessel invasion

TUMOR ANGIOGENESIS

Beginning at one side of the tumor, draw a line through all available white spaces.

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.

TUMOR ANGIOGENESIS

When you are through, trace the shortest path from one side of the “tumor” to the other.

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.

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

SO WHAT!

• Researchers may be able to use the fractal dimension of tissue blood vessels to better diagnose the presence of tumors.

SO WHAT!

• A better understanding of the fractal nature of tumor angiogenesis is presenting new options for the delivery of chemotherapy.

SO WHAT!• Fractal models of tumor growth are suggesting

new ways of fighting tumors – such as shutting down tumor vasculature to “starve” the tumor.

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.

FRACTALS IN BIOLOGY ATTACKING THE ROOT OF CANCER A Classroom Lesson From

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