1026lab5

Math 1026: Lab 5 Introduction Remember to print out this Lab Document to use as you work through the assignment. As you complete your work in Excel, don’t forget to SAVE EARLY and SAVE OFTEN! Last week you analyzed tumor growth models and (hopefully!) you found some ways to save your digital patient. In an ideal world, if someone were diagnosed with cancer, you could plug the tumor size, type, and location into an equation to find the optimal treatment plan. Unfortunately, due to the extreme complexity of the disease as well as the ethical difficulties of generating reliable data, it’s still a long way away from this becoming a reality. Right now, mathematicians studying cancer typically try to find tumor growth equations by “reverse-engineering” data that experimental scientists generate. In this lab, you’re going to do just this by building and investigating different tumor growth models from a recently-published research paper. Exponential Models and Corresponding Linear Models In this Lab Assignment, you will use Excel to find mathematical models representing data for tumor growth over time. In order to understand and use them to make predictions, let’s review some of the models we have been working with in this course. For example, here are 4 simple data points ( t, y ), first listed in a table and then plotted twice. Notice the label on the vertical axis for each plot above. One is a linear equation and the other is an exponential equation. Consider the general shape for a linear and for an exponential model (review your Lab 1 Document if you need help). Then, in the first plot above, add a linear graph that you think best fits the data points. In the second plot, add an exponential graph that you think best fits the data points. Which model is best for this data? The graphs that you sketched by hand on the above plots may resemble the ones shown in Figure 1 (for the linear graph) and Figure 2 (for the exponential graph) in Lab 5 Samples document. For this sample data set, it is not difficult to see that one model is better than another. However, since you will be analyzing results obtained from a real experiment, you need a better way to determine how well a model represents data. You have studied a way to do this for linear models in Lesson 4.1.3 in your online textbook, by calculating r 2 . For this lab, you will be looking for r 2 values that are close to 1, indicating that the model fits the data well. How- ever, it’s important to remember that r 2 values only make sense for linear models . While programs like Excel will give you an r 2 value for all kinds of different models, comparing an r 2 value of a linear model to an r 2 value of an exponential model is like comparing apples to oranges. So what can you do for exponential models? There is a way to create a linear equation from an exponential one. If you took Math 1025, you learned the process in Lesson 1.1.3. You can see more details in the online text for Math 1025, but the most important results will be reviewed here. Remember that a linear function can be written as y = mx + b (where m and b are numbers) and an exponential function as y = ce ax (where a and c are numbers). Let’s start with the formula y = ce ax and take the natural logarithm of both sides. 1

y = ce ax ln( y ) = ln ( ce ax ) = ln( c ) + ln ( e ax ) because ln( uv ) = ln( u ) + ln( v ) ln( y ) = ln( c ) + ax because ln ( e u ) = u ln( y ) = ax + ln( c ) reversing the terms on the right This shows that if you have an exponential model y = ce ax , then ln( y ) is a corresponding linear model: ln( y ) = mx + b with m = and b = m = a and b = ln ( c ) To see this result for our sample data, first redraw your exponential curve through the dots on the first graph below. Then use a calculator or Excel (whichever you prefer) to find the values for g = ln( y ), rounded to one decimal place and fill in the chart. Check your answers against Figure 3 in the Lab 5 Samples . Finally, plot the 4 points ( t, g ) on the second graph and draw the line through them. Compare your solution to Figure 4 in the Lab 5 Samples . Earlier in this Lab Document, you learned about changing an exponential equation into a linear form so that you can find a usable r 2 value. But eventually you want to go back to the exponential form for the actual model. So how do you go backwards? Just apply the natural exponential function to both sides of the equation ln( y ) = mx + b. ln( y ) = mx + b e ln( y ) = e mx + b = e mx · e b because e u + v = e u · e v y = e mx · e b because e ln( u ) = u y = e b · e mx reversing the factors on the right This shows that if you have a linear model ln( y ) = mx + b , then y is a corresponding exponential model: y = ce ax with c = and a = . c = e b and a = m For example, suppose you have the linear model g = ln( y ) = 2 x + 3. Then m = and b = . m = 2 and b = 3 For the corresponding exponential equation, y = ce ax , c = (calculate to one decimal place) and a = and the equation is y = . c = e 3 = 20 . 1 , a = 2 and y = 20 . 1 ( e 2 x ) . 2

The first sheet ( Graph B Data ) shows the data points from Graph B, along with the plots of the 4 different diet regimens from the article. We will examine the datasets for the control group SD and treatments CR-SD and CR-KD. If you click on the tabs at the bottom of the Excel workbook, you will see the datasets for the various treatments. Linear Models Linear models arise frequently in experimental data. Often times, this is related to the fact that many physical laws are linear in nature. To begin your investigation, you will use Excel’s Trendline function to create linear models for the data. You learned how to do this in your online text, but here is a repeat of the basic steps: 1. To create a graph: Highlight the data, including the labels. Click on Insert, then Chart, then XY(Scatter). 2. To create the linear trendline: Click once on any data point and make sure the Chart Design tab is selected. Then click Add Chart Element, then Trendline, then More Trendline Options. A Format Trendline box will open. 3. Select 3 things under Trendline Options: Linear, Display equation on chart, and Display R-squared value on chart. 4. If you want to enlarge the equation and R 2 , highlight the text, click on the Home tab, then increase the font size. Click on the appropriate tabs to find the 3 datasets (SD Linear Model, CR-SD Linear Model, CR-KD Linear Model). For each one, plot the points and find the equation and r 2 value for the linear model. Don’t just use x and y in your equations. Think about what the variables represent and use appropriate variable names. Round all numbers to 2 decimal places. Dataset Equation for Linear Model R 2 SD CR-SD CR-KD Based only on the r 2 values, which of these linear models fits its dataset best? Assume for the sake of argument that all of the datasets do follow a linear model. What would this say about how the tumor grows? (Hint: Think about the rate of change for a linear function.) Exponential Models Maybe an exponential model is better than a linear one. Click the Graph B Data tab to see the original graphs again and look at the two treatment options, CR-SD and CR-KD. Which one appears to fit an exponential model better? Excel has an exponential trendline option, too, but remember that r 2 values are only valid for linear equations . If you want to decide on a best fit, you need a way to compare the options. In the beginning of this lab, you saw that you can transform an exponential equation into a linear one using natural logarithms. Try that with your datasets. Click on the tab LN(SD) . You will see another copy of the original SD data in columns A and B. In cell C1 type LN of SD , and in cell C2 type =LN(B2) . Drag this formula down to C11. Now use Excel to fit a LINEAR model to the data in columns A and C (not B). Be sure to display the equation and R 2 on your graph. NOTE: To select two non-adjacent columns , hightlight the data in column A first, then hold down the com- mand key while you highlight column C. Repeat the process for the other two datasets ( LN(CR-SD) and LN(CR-KD) ) and then fill in the Equation for ln(y) and R 2 columns in the chart below. Again, round all numbers to 2 decimal places. 4

Dataset Equation for ln(y) R 2 Equation for Exponential Model SD ln(y)= CR-SD CR-KD In order to complete the last column of the chart, you need to convert from the linear model that Excel gave you to the corresponding exponential model. If you need help, go back to page 2 to review the work that you completed earlier. Be sure to develop your exponential models carefully and round the numbers to 2 decimals, as instructed. You will be using these models in the next section. Do you think the exponential models fit the data better than the linear models? Describe how the r 2 values and the graphs of the actual data support your answer. Assume for the sake of argument that all of the datasets do follow exponential models. What would this say about how the tumor grows? (Hint: Think about the rate of change for an exponential function.) Compare all of the information you have for CR-SD and CR-KD, including the graph. Which dataset appears to follow the exponential model more closely? Suggest a different model for the other dataset that might be better fit. Look at the shape and recall other models we have used in this course. Describe your reasoning . Application of the Tumor Model Now that you have calculated different models for the tumor growth, let’s see how we might use them. You will need to define formulas in Excel for this last part. You can work in the current sheets or copy the data into new sheets. Just be sure to label everything carefully so that you can follow your work. 1. In this experiment, the data was gathered from mice, where a tumor volume of 3 , 500 mm 3 meant probable death. For humans though, this threshold is likely much higher. Determine approximately how many days it would take the SD tumor to reach 10 , 000 mm 3 using both your linear model and the exponential model you developed in the previous section . Note: For the SD models only, you are given Excel check values to verify your Excel formulas. For the other methods, since you find your equations in the same way and enter them into Excel in the same way, check points will not be provided. For the SD model, you should have the following results: SDLinear(5) = 125 . 68 and SDExponential(5) = 259 . 82, rounded to 2 decimal places. If your calculations do not give these results, check your work and try again. SD Linear Model SD Exponential Model 2. Repeat problem 1 for the CR-SD tumor. CR-SD Linear Model CR-SD Exponential Model 5