Purpose
To investigate the rapid growth of cancer cells
To generate a model from a problem situation
To communicate the rapid spread of a disease in an organism
Materials
Graph paper, graphing calculator, (Math)
Petri dish, organism (science)
Scenario I: Cancer Cells too Aggressive to be contained
The evidence of certain cancer has been found to be prevalent the population of mice. Researchers have found that the tumor is growing at a rapid rate when the growth rate is compared with the cells of the organ in which it is located. You have been asked to do a research on the growth of this cancer cell and compare it with the growth of other cells.
For your research you found the following description of cancer cell growth.
Cancer cells can arise in almost any location in the body or tumors or collection of cancer cells can be vastly different from organ to organ and person to person. A cell forming a tumor is different from normal cells within the organ; it has undergone a malignant transformation. The molecular event that occurs during this transformation are not completely understood, and there is more than one set of molecular changes that will cause a malignant transformation. All cancer cells share a number of characteristics. They proliferate or divide rapidly compared to normal signals. They generally do not respond normally to signals that are provided by neighboring cells. They do not differentiate normally, but tend to remain as immature dedifferentiated cells. They do not become specialized or die, even when they are moved to a part of the body that is different from their normal environment.
From your research project you are required to examine the growth of these cells in the lab.
Activities
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1. Make a table and generate a graph showing the number of cells created during a 3 hour period if there was 100 cancer cells present initially C = 10t
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2. Generate a table to show the rate of growth of the normal tissue cells assume that the initial population was 1,000 and these cells reproduce at
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N(t) = 2t.
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3. Compare the graph of both cell growths. Write a paragraph describing the behavior of the graph.
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4. a) If the cancer cells grow at the rate of Pn = Po ekt estimate the number of cells when t = 5 with the initial population of 1,000 cells and k = 0.05.
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b) If there were also 1000 normal cells in the tissue sample and the normal cells reproduce at Pn = Po ekt with k = 0.02. Graph the cancer cells and the normal cells on the same axis use ( 0 =t=20). Determine when the cancer cells will out number the normal cells.
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5. a) A group of cancer cells starts with 400 and grows at a constant rate The Amount after t hours is given by A(t) = (450.268)e1.125 t cells per day. How many cells will there be in 3 days
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b) Graph the growth of the cells from 0 =t=10 days.
For Calculus Students: Rate of change Application problem
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6. A spherical cell is growing at a constant rate of 4000 um3 / day (1um = 10 - 6 ). At what rate is its radius increasing when the radius is 10um?
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7. When the growth of a spherical cell depends on the flow of nutrients through the surface, the growth rate dv/dt is proportional to its surface area, s. Assume that for a particular cell dv/dt = 1/3 s. At what rate is its radius r increasing.
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8. The rate of growth of a tumor is proportional to the size of the tumor.
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a) Write a differential equation for S the size of the tumor, in mm, as a function of time.
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b) Find the general solution to the differential equation.
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c) If the tumor is 5 mm across at time t = 0 what does that mean?
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d) If the tumor is 8 mm across at t =3 what does the solution mean?