Analyzing the effectiveness of different treatments for cancer typically involves collecting data from large clinical studies or animal testing. Such testing is always needed as a final validation for establishing treatment safety, but can be very costly and labor-intensive. Developing alternative testing approaches to be used in preliminary stages of evaluating new treatment strategies would be a great aid in speeding up research and development. We consider the use of mathematical models to describe the progression of cancer and how the influence of anti-cancer drugs can be incorporated into these models. There are many different forms of cancer, but several types share similar mechanisms for how they start and spread. The basic understanding of metastatic cancer consists of the following general stages: 1. The disease starts from a single primary tumor which grows in one location. 2. The primary tumor will start to shed cancer cells which get carried to other parts of the body by the circulatory or lymphatic systems. 3. These cells will attach to other organs and start new secondary tumors, called metastases (or meta-static tumors). 4. The metastases grow and will shed cancer cells to produce more tumors. Such rapid spreading of cancer (also called “progression”) typically leads to multiple organ failure and fatality. While there are many different types of clinical studies of cancer, there have been standards (RECIST) defined for many aspects of studies - including how to measure tumors and what 1  Figure 1: Two schematic representations of the spread of metastatic cancer: (left) from trialx.com (right) from www.cancer8.com. data should be recorded. These articles are also good sources for descriptions of the stages of progression of cancer and practical limitations in what data can be collected in clinical trials. An important question about the RECIST standards is whether collecting more clinical data can provide better assessments of progression and lead to more accurate models and better treatment protocols. Statistics and mathematical analysis can be applied to address these issues. Practical factors (effort, expense, record keeping, intrusiveness) have shaped the current standards and limited the amount and type of data that is collected in current studies. If benefits of increased data collection for guiding treatments could be demonstrated, this might lead to valuable improvements in the standards. Studies differ in conclusions about probability of fatality correlating with growth of tumors, but for our work, we will focus on increase in total tumor mass as a general descriptor of the progression of the disease.

1. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Dynamic Tumor Growth Modelling
Prof. Thomas Witelski,
Matthew Tanzy, Ben Owens, Oleksiy Varfolomiyev
NJIT, June 2011

2. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Cancer background
Current medical understanding of some types of cancer
1 Disease starts from a primary tumor which grows in one
location
2 The primary tumor may shed cancer cells which get carried to
other parts of the body by the circulatory system

3. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Colony size distribution of metastatic tumors with cell number x at
time t is governed by von Foerster Equation
∂ρ ∂
+ (g (x) ρ) = 0 (1)
∂t ∂x
g (x) is tumor growth rate determined by
dx
= g (x) (2)
dt
Initially no metastatic tumor exists
ρ(x, 0) = 0 (3)
Intitial tumor birth rate
∞
g (1)ρ(1, t) = β(x)ρ(x, t)dx + β(xp (t)), (4)
1
β(x) tumor birth rate

4. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Linear Growth Function: g (x) = k − Ex
Assuming that in the beginning the main contribution to the tumor
growth is given by primary tumor (i.e. neglecting the integral term
in the BC) we have
∂ρ(x, t) ∂ (g (x)ρ(x, t))
+ =0 (5)
∂t ∂x
ρ(x, 0) = 0 (6)
g (1)ρ(1, t) = β(xp ), β(x) = mx α (7)
dx
= k − Ex = g (x) (8)
dt

5. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Solution by the Method of Characteristics
dρ
= Eρ (9)
dt
along the characteristics given by
dx
= k − Ex (10)
dt
Therefore
ρ(x, t) = ρ0 (x(0)) e Et (11)
along the characteristics
K
x(t) = x(0)e −Et + 1 − e −Et (12)
E

6. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Solution by the Method of Characteristics
First we use the BC to solve for ρ0 (x(0)), i.e.
α
K K
(K − E ) ρ0 e Et + 1 − e Et e Et = m e −Et + 1 − e −Et
E E
Finally, the colony size distribution of metastatic tumors with cell
number x at time t is
α
mE −α e −Et (E −k)2
ρ (x, t) = k−Ex k+ Ex−k

7. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Growth function: g (x) = Ex
Figure: ρ(x) at t = 10 with all parameters at 1

8. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Growth function: g (x) = k − Ex
Figure: ρ(x) at t = 10 with E = 0.2, k = 0.3, and m = 0.1

9. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Single primary tumor size growth (no drug is active)
Growth function: g (x) = (k − E (t)e −rt )x
Here no drug is active (i.e. g (x) = kx)
Figure: ρ(x) at t = 3650 with k = 0.006, α = 0.663, and m = 5.3 × 10−8

10. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Single primary tumor size growth (drug effect)
At time t1 drug is activated
−r (t−t1 )
In a region of constant E : x (t) = Ae kt+Ee /r
Figure: ρ(x) at t = 3650 with k = 0.006, α = 0.663, E = 0.0083,
r = 1 × 10−5 and m = 5.3 × 10−8

11. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Linear Separable Growth Rate Solution
Now we have Linear Separable Growth Rate
dx
= [k − E (t)] x
dt
Colony size distribution
m m t
m −α−1− k−E (t) α+ k−E (t) ( k−E (s)ds )t
ρ(x, t) = k−E (t) x e 0

12. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Logistic Tumor Growth Rate Solution
Now we have Tumor Growth Rate given by Logistic Equation
dx
= (k − Ex) x
dt
Resulting colony size distribution
e −kt
ρ(x, t) = 2
(E x0 (e kt −1)+k )

13. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Future Work
Fitting data to models, parameter estimation for multiple
tumor and multiple patient data
Incorporating limitations on measurable clinical data
Comparing PDE, discrete population and polymerization
models
Solution of inverse problems for the birth rate
Probability and statistics of different forms of clinical data