Cancer Tumor Kinetics

Gretchen A. Koch-NobleGoucher CollegePEER UTK 2012

Special Thanks To:Dr. Claudia Neuhauser

University of Minnesota – Rochester Author and creator of modules

Learning ObjectivesAfter completion of this module, the student will be able to:

1. Build a data‐driven phenomenological model of tumor growth with a minimal number of parameters

2. Make predictions about the kinetic behavior of a tumor based on a mathematical model

3. Define growth rate and exponential growth

4. Develop a differential equations describing tumor growth

5. Use WolframAlpha to solve algebraic equations and take limits

Prerequisites1. Volume of a sphere

2. Straight lines

3. Natural logarithm and exponential functions

4. Graphing in Excel

5. Logarithmic transformation

6. Fitting a straight line to data points in Excel and displaying the equation

Knowledge Gained1. Continuous time population models

2. Fitting a straight line to data

3. Doubling time of an exponentially growing population

4. Growth rate and exponential growth

Concept Map

New Cases of Cancer 2011

Map from American Cancer Society. Cancer Facts & Figures 2011. Atlanta: American Cancer Society; 2011.

Cancer Tumor KineticsThe growth and spread of the cancer tumor

Tumor metastasis and survival rates

Table from American Cancer Society. Cancer Facts & Figures 2011. Atlanta: American Cancer Society; 2011.

Why model cancer tumor kinetics?

Case StudyPatient with breast cancer tumor and growth of

(untreated) tumor over time

Diameter (mm)

Measurement Date D1 D2 D3

1 06/26/69 4 4 4

2 11/27/69 5 4 6

3 11/24/70 7 8 9

4 07/06/71 11 12 14

5 08/17/73 29 33 31

6 09/18/73 32 36 34

D. v. Fournier, E. Weber, W. Hoeffken, M. Bauer, F. Kubli, and V. Barth. 1980. Growth rate of 147 mammary carcinoma. Cancer 8: 2198‐2207.

Questions to AnswerWhen will the patient die?

Lethal burden of tumor

When did the cancer start?Depends on growth rate (doubling time)

Model Assumptions1. The shape of a tumor is a sphere

2. A tumor is a solid mass of tumor cells

3. An individual tumor cell is a sphere with diameter

4. 1 gram of tumor cells corresponds to 109 cells

Create the Model: Background Information

Volume of a sphere with radius, r:

Create the Model: Background Information

Volume of a sphere with radius, r:

Create the Model: Background Information

Volume of a sphere with radius, r:

Relationship between diameter, d, and radius, r:

Create the Model: Background Information

Volume of a sphere with radius, r:

Relationship between diameter, d, and radius, r:

Create the ModelVolume of a cancer tumor, VT, with diameter, D:

Volume of individual cancer tumor cell, VC, with diameter, d:

Think, Pair, Share:Create the Model

Given the two volumes, find the number of tumor cells in any given cancer tumor.

5:00

Think, Pair, Share:Create the Model

Given the two volumes, find the number of tumor cells in any given cancer tumor.

4:00

Think, Pair, Share:Create the Model

Given the two volumes, find the number of tumor cells in any given cancer tumor.

3:00

Think, Pair, Share:Create the Model

Given the two volumes, find the number of tumor cells in any given cancer tumor.

2:00

Think, Pair, Share:Create the Model

Given the two volumes, find the number of tumor cells in any given cancer tumor.

1:00

Think, Pair, Share:Create the Model

Given the two volumes, find the number of tumor cells in any given cancer tumor.

Time to Share!

Create the ModelThe number of cells in any tumor is

Create the ModelThe number of cells in any tumor is

Create the ModelSince 109 tumor cells weigh 1 gram, the weight

of the tumor is

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

10:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

9:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

8:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

7:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

6:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

5:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

4:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

3:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

2:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

1:00

Think, Pair, Share:Create the Model

1. Download the cancer data set from the Schedule webpage.

2. Under the Patient 1 tab, calculate each of the following

a. Column G: Average diameter for the tumor of the patient

b. Column H: Volume of the tumor based on the average diameter

c. Column I: Number of cells in the tumor

d. Column J: Weight of the tumor

Time to Share!

Create the ModelExcel Time!

Think, Pair, Share:Kinetics Model

1. Under the Patient 1 tab, calculate each of the following

a. Column C – Days between observations: Excel can calculate the number of days between observations by using simple subtraction. Set the date of the first observation to be day 0, and calculate the days between subsequent observations.

b. Plot the Number of Tumor Cells (Column I) as a function of time (Column C).

c. Determine if transforming either or both axes logarithmically gives a straight line fit.

d. What type of function should we use to fit our data?

5:00

Think, Pair, Share:Kinetics Model

1. Under the Patient 1 tab, calculate each of the following

a. Column C – Days between observations: Excel can calculate the number of days between observations by using simple subtraction. Set the date of the first observation to be day 0, and calculate the days between subsequent observations.

b. Plot the Number of Tumor Cells (Column I) as a function of time (Column C).

c. Determine if transforming either or both axes logarithmically gives a straight line fit.

d. What type of function should we use to fit our data?

4:00

Think, Pair, Share:Kinetics Model

1. Under the Patient 1 tab, calculate each of the following

a. Column C – Days between observations: Excel can calculate the number of days between observations by using simple subtraction. Set the date of the first observation to be day 0, and calculate the days between subsequent observations.

b. Plot the Number of Tumor Cells (Column I) as a function of time (Column C).

c. Determine if transforming either or both axes logarithmically gives a straight line fit.

d. What type of function should we use to fit our data?

3:00

Think, Pair, Share:Kinetics Model

1. Under the Patient 1 tab, calculate each of the following

a. Column C – Days between observations: Excel can calculate the number of days between observations by using simple subtraction. Set the date of the first observation to be day 0, and calculate the days between subsequent observations.

b. Plot the Number of Tumor Cells (Column I) as a function of time (Column C).

c. Determine if transforming either or both axes logarithmically gives a straight line fit.

d. What type of function should we use to fit our data?

2:00

Think, Pair, Share:Kinetics Model

1. Under the Patient 1 tab, calculate each of the following

a. Column C – Days between observations: Excel can calculate the number of days between observations by using simple subtraction. Set the date of the first observation to be day 0, and calculate the days between subsequent observations.

b. Plot the Number of Tumor Cells (Column I) as a function of time (Column C).

c. Determine if transforming either or both axes logarithmically gives a straight line fit.

d. What type of function should we use to fit our data?

1:00

Think, Pair, Share:Kinetics Model

1. Under the Patient 1 tab, calculate each of the following

a. Column C – Days between observations: Excel can calculate the number of days between observations by using simple subtraction. Set the date of the first observation to be day 0, and calculate the days between subsequent observations.

b. Plot the Number of Tumor Cells (Column I) as a function of time (Column C).

c. Determine if transforming either or both axes logarithmically gives a straight line fit.

d. What type of function should we use to fit our data?

Time to Share!

Kinetics ModelExcel Time!

Think, Pair, Share:Kinetics Model

1. Use the Trendline option to fit an exponential function to the data and on the graph, display the equation of the form

2. Determine and record the values of a and c.

5:00

Think, Pair, Share:Kinetics Model

1. Use the Trendline option to fit an exponential function to the data and on the graph, display the equation of the form

2. Determine and record the values of a and c.

4:00

Think, Pair, Share:Kinetics Model

1. Use the Trendline option to fit an exponential function to the data and on the graph, display the equation of the form

2. Determine and record the values of a and c.

3:00

Think, Pair, Share:Kinetics Model

1. Use the Trendline option to fit an exponential function to the data and on the graph, display the equation of the form

2. Determine and record the values of a and c.

2:00

Think, Pair, Share:Kinetics Model

1. Use the Trendline option to fit an exponential function to the data and on the graph, display the equation of the form

2. Determine and record the values of a and c.

1:00

Think, Pair, Share:Kinetics Model

1. Use the Trendline option to fit an exponential function to the data and on the graph, display the equation of the form

2. Determine and record the values of a and c.

Time to Share!

Kinetics ModelExcel Time!

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

10:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

9:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

8:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

7:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

6:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

5:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

4:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

3:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

2:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

1:00

Think, Pair, Share:Kinetics Model

1. A number of studies have shown that a primary tumor starts from a single cell. Use the model equation to predict the date when the tumor started.

2. Tumors can be detected by palpitation when their size is about 107 to 109 cells. Tumors become lethal when their size is about 1012 to 1013 cells. This size is called the lethal burden. Based on the model equation, determine when the tumor was detectable and when the tumor reached the lethal burden?

Time to Share!

Kinetics ModelExcel Time!

Doubling Time

Doubling Time

Doubling Time

Doubling Time

Doubling Time

Doubling Time

Doubling Time

Doubling TimeThen, the doubling time does not depend on the

number of cells present.

Doubling TimeI forget how to do this!

WolframAlpha

Think, Pair, Share:Doubling Time

1. Use Excel to find the doubling time for our tumor kinetics model.

3:00

Think, Pair, Share:Doubling Time

1. Use Excel to find the doubling time for our tumor kinetics model.

2:00

Think, Pair, Share:Doubling Time

1. Use Excel to find the doubling time for our tumor kinetics model.

1:00

Think, Pair, Share:Doubling Time

1. Use Excel to find the doubling time for our tumor kinetics model.

Time to Share!

Think, Pair, Share:Doubling Time

1. Excel time!

Learning ObjectivesAfter completion of this module, the student will be able

to:

1. Build a data‐driven phenomenological model of tumor growth with a minimal number of parameters

2. Make predictions about the kinetic behavior of a tumor based on a mathematical model

3. Define growth rate and exponential growth

4. Develop a differential equations describing tumor growth

5. Use WolframAlpha to solve algebraic equations and take limits

Putting it all together Complete the group project on page 7 of the Cancer Tumor

Kinetics pdf to find the time to lethal burden and detection time for:

Primary Cancer Doubling Time (days)

Number of Cases

Malignant Melanoma

48 10

Colon 109 10

116 25

Kidney 66 5

132 8

Thyroid, anaplastic

29 7

Data Source: Table III from Friberg, S. and S. Mattson. 1997. On the growth rates of human malignant tumors: Implications for medical decision making. Journal of Surgical Oncology 65: 284‐297