Cancer Tumor Kinetics
Gretchen A. Koch
Goucher College
PEER UTK 2011
Special Thanks To:
 Dr. Claudia Neuhauser
 University of Minnesota – Rochester
 Author and creator of modules
Learning Objectives
 After 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
Prerequisites
1. 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 Gained
1. 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 2010
Map from American Cancer Society. Cancer Facts & Figures 2010. Atlanta: American
Society; 2010.
Cancer
Cancer Tumor Kinetics
 The growth and spread of the cancer tumor
 Tumor metastasis and survival rates
Table from American Cancer Society. Cancer Facts & Figures 2010. Atlanta: American
Society; 2010.
Cancer
Why model cancer tumor kinetics?
Case Study
 Patient 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 Answer
 When will the patient die?
 Lethal burden of tumor
 When did the cancer start?
 Depends on growth rate (doubling time)
Model Assumptions
1. 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
d = 10 mm
4. 1 gram of tumor cells corresponds to 109 cells
Create the Model: Background
Information
 Volume of a sphere with radius, r:
4 3
V = pr
3
 Relationship between diameter, d, and radius, r:
d = 2r
Create the Model
 Volume of a cancer tumor, VT, with diameter, D:
4 æ Dö
VT = p ç ÷
3 è 2ø
3
 Volume of individual cancer tumor cell, VC, with
diameter, d:
4 æ dö
VC = p ç ÷
3 è 2ø
3
Think, Pair, Share:
Create the Model
 Given the two volumes, find the number of tumor cells
in any given cancer tumor.
4 æ Dö
VT = p ç ÷
3 è 2ø
3
4 æ dö
VC = p ç ÷
3 è 2ø
Time to Share!
3
Create the Model
 The number of cells in any tumor is
3
4 æ Dö
pç ÷
3
VT 3 è 2 ø
æ Dö
=
÷
3 =ç
è dø
VC 4 æ d ö
pç ÷
3 è 2ø
Create the Model
 The number of cells in any tumor is
3
4 æ Dö
pç ÷
3
VT 3 è 2 ø
æ Dö
Number of tumor cells= =
÷ø
3 =ç
è
VC 4 æ d ö
d
pç ÷
3 è 2ø
Create the Model
 Since 109 tumor cells weigh 1 gram, the weight of the
tumor is
æ 1g ö
Weight = Number of tumor cells ç 9
÷
è 10 cells ø
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 Model
 Excel 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?
Time to Share!
Kinetics Model
 Excel 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
N(t) = ae
ct
2. Determine and record the values of a and c.
Time to Share!
Kinetics Model
 Excel 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?
Time to Share!
Kinetics Model
 Excel Time!
Doubling Time
N(t) = ae
ct
N(T2 ) = 2N(0)
Doubling Time
N(t) = ae
N(T2 ) = 2N(0)
ct
ae
cT2
= 2ae
c0
Doubling Time
N(t) = aect
N(T2 ) = 2N(0)
ae
cT2
= 2ae
aecT2 = 2a
c0
Doubling Time
N(t) = aect
N(T2 ) = 2N(0)
aecT2 = 2aec0
aecT2 = 2a
ecT2 = 2
Doubling Time
N(t) = aect
N(T2 ) = 2N(0)
ae
cT2
= 2ae
c0
aecT2 = 2a
ecT2 = 2
ln ( ecT2 ) = ln ( 2 )
Doubling Time
N(t) = aect
N(T2 ) = 2N(0)
aecT2 = 2aec0
aecT2 = 2a
e
cT2
=2
ln ( ecT2 ) = ln ( 2 )
cT2 = ln ( 2 )
Doubling Time
N(t) = aect
N(T2 ) = 2N(0)
ae
cT2
= 2ae
c0
aecT2 = 2a
e
cT2
=2
ln ( ecT2 ) = ln ( 2 )
cT2 = ln ( 2 )
ln ( 2 )
T2 =
c
Doubling Time
 Then, the doubling time does not depend on the
number of cells present.
ln ( 2 )
T2 =
c
Doubling Time
 I forget how to do this!
e
 WolframAlpha
cT2
=2
Think, Pair, Share:
Doubling Time
1. Use Excel to find the doubling time for our tumor
kinetics model.
e
cT2
=2
Time to Share!
Think, Pair, Share:
Doubling Time
1. Excel time!
Learning Objectives
 After 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
66
5
132
8
29
7
Kidney
Thyroid, anaplastic
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