Mathematics in Pink

AwardHonorable Mention

SchoolKenston High School

TeacherGreg Koltas

Selected Research

"Pilot Study to Identify Women at High Risk for Hereditary Breast Cancer in a Community Mammography Clinic Population," Randi Michel

Selected Art*Optimism in DNA,* Grace Kelemen

Selected Language

"Biology," Margo Uhrman

*At first glance, the scientific research done by Randi Michel seems confusing and only able to be interpreted by someone in the medical field. However, when analyzed closely, the mathematics is more evident. The aspects that are most useful are the charts and graphs that anyone can interpret. From there we decided to also focus on Margo Uhrman's piece "Biology" and Grace Kelemen's piece Optimism in DNA. The research combined with the pieces of art created interesting mathematical connections that at first glance were overlooked even by us.*

-Rachel Bullock, Alex Deuley, Stephanie Dueley, Kate Kupiec

- MATH IN SCIENCE
Analysis of Research by Randi Michel

In Randi Michel's study, patients filled out an anonymous family history collection tool which would be used to calculate a Pedigree Assessment Tool (PAT) score. After analyzing the family history of each patient, a case of breast or ovarian cancer in a patient's history was then given a point value using a scoring system. After all points had been assigned, the risk was assigned. Having five points indicated a low risk, six to seven points represented a moderate risk, and eight or more points indicated a high risk.

The only information obtained from the study that was put into the pie chart was the risk level of the patients who filled out the PAT. However, in order to compare the risk level to the actual number of cases in an individual's family history, we made a chart that put them right next to each other (titled "Family History of Breast/Ovarian Cancer"). Also, instead of only comparing the patients with any number of cases of breast or ovarian cancer in family history, we also took into account the patients with no family history of breast or ovarian cancer.

With this study, the only assessment of the risk of the patient getting breast cancer was the number of cases in family history. Therefore, having no cases in family history indicated no risk of having breast or ovarian cancer. This made up 47%, or 110 out of the 234 family histories. This stayed the same for both of the values (number of cases and risk level) being studied. Low risk made up 29%, or 67 out of the 234 family histories. Moderate risk was 12%, or 28 out of 234 family histories. High risk was about 12%, or 29 out of 234 family histories. There were 67 patients, or 29%, who had only one case in family history. There were 40 patients, or 17%, who had two cases in family history. There were 17 patients, or 7%, who had three or more cases in family history.

By looking at this chart, it is a lot easier to compare the number of cases of a patient's history with the risk level of that patient. It can be concluded that most patients with three cases of breast or ovarian cancer in family history had a high risk. However, patients could have only two or one case in family history and have a high risk of getting breast or ovarian cancer. Also, patients with one case in an individual's family history most likely had a low risk of getting breast or ovarian cancer. This is known because the only point assigned to a case in an individual's family history that was higher than five was for male breast cancer, which gave that case eight points (high risk).

The methodology of Randi Michel's research, titled "Pilot Study to Identify Women at High Risk for Hereditary Breast Cancer in a Community Mammography Clinic Population,"has many important connections to statistics. Foremost, her research outlines an observational study in which the patients were asked to fill out the Pedigree Assessment Tool (PAT) upon checking into Hillcrest Hospital for their mammogram. These tests were then utilized in assigning risk level for breast and ovarian cancer among the population. In an observational study, individuals are observed and the variables are measured without any attempt to influence the response. This cannot be described as an experiment because there was no control group and no treatment implemented. Since other variables are not controlled for, there may be multiple confounding variables in this set-up. The patients' willingness to come to Hillcrest Hospital on one's own initiative for a mammogram may have created bias. Due to the voluntary response for collecting this data, older women and those who believe ovarian and breast cancer run in their family are more likely to get mammograms in attempts to protect themselves from these cancers. This means that there is a lack of this sample capturing the entire population, so age and hereditary factors may have been key confounding variables in this study. Additionally, only certain insurance plans may cover getting the procedure done at Hillcrest. The factors involved in personal coverage and location could attract different types of subjects than at other offices. For example, it is not plausible to conclude that all Ashkenazi Jewish women will get breast or ovarian cancer as given in Michel's data with a 100% rating. Alone, this characteristic, being only 4 pts on the PAT itself, would only cause low risk. Therefore, these individuals must also have had initiative to go get a mammogram because of family reasons or another characteristic accounted for on the PAT, explaining why all Ashkenazi Jewish ancestry women were in the high risk interval. Also, only 4 individuals in this sample had such a characteristic, which is not very assuring because of the low proportion of Ashkenazi Jewish women represented. Therefore, for Randi Michel to conclude that all women of Ashkenazi Jewish ancestry are at high risk would be incorrect because of the multiple lurking variables in her sample.

The questionability of this research can again arise from the mean and standard deviation of PAT scores from the high risk cohort. Mathematically, this scientific analysis was done well; however, during the discussion of results Michel states that using the PAT provides an overestimation of cancer risk. If this is an overestimation, then it is expected that individuals would be flagged as high risk using the PAT, when instead they are at moderate or low risk level for breast and ovarian cancer. Mathematically, Michel provides data in opposition to this idea. Of those in the high risk cohort, it is said that the average was 12.2, but there was a very large standard deviation of 5.6. A score of 8 or higher flagged an individual to belong to high risk. Just one standard deviation out, the score of this cohort would already be marked as a 6.6, which is categorized as just moderately increased hereditary risk and nearly the mean of those in the moderately increased risk cohort (mean = 6.4). To find the proportion of individuals defined as high risk upon the test, one should take the normal curve and apply the given standard deviation and mean values. First, one must standardize the value of 8 at which being marked as high risk begins (See Figure 2). This equals -.75, which is named the z-score. Therefore, the P (z ≥ -.75) is .7734. However, one also needs to take into consideration that 24 is the highest value attainable on the PAT. Taking into account that 24 is the greatest possible score for the PAT with the given characteristics and their values, then the true proportion of high risk individuals would be marked as those who lie within scoring an 8 and a 24. The z-score for 24, using the same procedure as before, is 2.11. The P (-.75 ≤ z ≤ 2.11), is equal to just .756 (See Figure 3).

This yields evidence that approximately 75% of individuals who are genetically at high risk are recognized based upon the PAT, leaving 25% unaccounted for. To determine that risk is overestimated by such a tool is statistically invalid and may then give hope to those scoring below an 8 who should in fact be considered high risk (23%). Such results could cause those individuals to pay less attention to regular check-ups for these types of cancer when they instead should be extremely careful due to their high risk. This can conclude that the PAT may not be as reliable as Michel is giving it credit for during her presentation.

Further, and more controlled, research would need to be implemented in order to effectively declare the PAT a successful tool for breast and ovarian cancer risk screening. Moreover, it may be beneficial to not only expand the collection pool, but to randomly choose women from the population instead of waiting for them to arrive at the hospital based upon personal initiative and doctor or facility choice.

- MATH IN LANGUAGE
Analysis of "Biology" by Margo Uhrman

Mathematical aspects were found in Margo Uhrman's piece of writing "Biology." She based her writing off of Randi Michel's research of women who were at a particularly high risk for developing breast cancer. Her writing relates the physical self to the psychological and spiritual self in terms of the cancer. Though the focus was mainly on the biological aspects of the disease, there was an evident presence of mathematics.

The first mathematical point came in the second stanza of the piece. It states, "A series of interconnected molecules. Water. 73.8 percent water. 213 bones and 600 skeletal muscles" (line 3). Therefore, if her facts are correct, then our bodies are not only 73.8 percent water, but also 73.8 percent skeletal muscle when compared with bones and muscles combined (See Figure 4 for more information). This proves to be an interesting mathematical connection with the information provided in the writing.

The next point of interest came when the author states, "Fact: Each and every one of us have different fingerprints. Identical twins disprove this fact" (Lines 13-14). This is an example of a paradox. Paradoxes in mathematics occur very often and definitely have their place in this piece of writing. Paradoxes disprove a previously known fact. In this case, the fact that each human has their own distinct fingerprint is disproved by the identical fingerprints of identical twins. Paradoxes have a rich history in mathematics. Some famous paradoxes include Zeno's paradoxes, Cantor's paradox, Russell's paradox, and Einstein and Twin's paradox. Specifically, Zeno proposed four paradoxes (between 495 and 480 B.C.) in order to challenge the accepted notions of space and time. Zeno focused on the fact that space was infinitely divisible, and that motion was, therefore, continuous. The most famous of his paradoxes is the "Tortoise and Achilles." Basically, the paradox begins with the tale of a tortoise and Achilles preparing to have a race. Achilles agrees to give the tortoise a head start of ten meters (thinking that he will catch him in a short amount of time). However, as Zeno states, as Achilles progresses to catch up the ten meters, the tortoise has also progressed and has pushed himself farther ahead. Therefore, every time Achilles catches up to the tortoise's lead, the tortoise has moved forward and is still ahead.

If one were to rephrase this paradox it could be compared to crossing a room. In order to cross a room one must first cover half the distance. Then after that he or she must cover half of the distance remaining. After covering that distance, they must cover half of the distance remaining again. This pattern would continue on forever, and the person would never reach the other side of the room. Zeno's paradox disproves a previously know fact regarding time and distance just like the paradox that disproves the fact that each human has their own distinct fingerprint in Margo Uhrman's piece of writing "Biology." Although an answer was eventually reached to solve "Tortoise and Achilles," his paradoxes baffled mathematicians for centuries, and aided in the development of future paradoxes.

Finally, one of the most notable mathematical influences is the presence of an exponential growth in human body cells and cancer cells. The final line of the poem states, "For half an hour, you existed as one single cell. One single cell." One views that the accumulation of growth of cells occurs exponentially and an equation can be found that includes the information given in the last line. After half an hour the number of cells would total two illuminating the fact that it is an exponential growth (See Figures 5 and 6 for more information).

This results in an incredibly fast growth of cells. The metastasizing of a cancerous cell, therefore, can be seen to be detrimental and to occur very rapidly. From the information given in the writing, the amount of cells would total four after one hour. After one day they would total around 300 trillion. If this fact were true, the growth of a cancer cell would be extremely quick and almost impossible to stop. However, since the number of cells in the human body totals around 100 trillion her information is either incorrect, or the rate of death of cells also occurs exponentially.

These various pieces of Margo Urhman's writing stood out because of their connections to mathematics. They proved to have not only a meaning in the world of writing, but also in the world of mathematics.

- MATH IN ART
Analysis of

*Optimism in DNA*by Grace KelemenThe sculpture

*Optimism in DNA*by Grace Kelemen of Kirtland High School represents a breast cancer ribbon and DNA. It can be modeled as part of a five petal rose using polar equations and parametric equations (See figure 8). The equation in polar is r=sin50– and the parametric equation is XT=sin(5T)cosT,YT=sin(5T)sin(T). Both equations represent the ribbon over 0–∈[1.08, 2.07] radians or T∈[1.08, 2.07]radians.If the ribbon is inscribed in a unit circle the unwound ribbon will be approximately 3.683 units in length. The area bounded by the ribbon is approximately .157 units2. Both of these values can be found using Arc Length and Area formulas (See Figure 11). The approximate lengths (as seen in a two-dimensional projection) of the bars or DNA base pairs can also easily be found using a calculator. Bar 1 (See Figure 9) is approximately .176 units long, Bar 2 is .222 units, Bar 3 is .016 units, Bar 4 is .196, and Bar 5 is .498 units long. Based on the bars' locations on the unit circle, they also seem to be evenly placed. Bar 1 is located at about =1.47 R, Bar 2 is at =1.39 R, Bar 3 is at =1.32 R, Bar 4 is at =1.94 R, and Bar 5 is at =2.01 R. It seems that there is about a .7 difference between each of the bars on the ribbon (For all mathematical equations regarding calculations, see Figure 11).

Rose curves, as seen above, are any type of curves that consist of loops all coming from a single point, the origin. They can be graphed in both parametric and polar form, and may contain any number of petals. The entire rose curve pictured above is drawn on the interval 0– ∈[0 π], but in order to trace only the breast cancer ribbon, the graph 0– ∈[1.08, 2.07] is necessary.

Rose curves produce a very interesting graph because they fail to pass the Vertical Line Test (when a vertical line is drawn down the graph, it crosses more than one point), yet they are still functions. It is apparent that they are functions because for every 0– there is only one r value (polar), and for every T value, there is exactly one (x,y) coordinate (See Figure 10 for more information).

Randi Michel

Pilot Study to Identify Women at High Risk for Hereditary Breast Cancer in a Community Mammography Clinic Population