Example System – Tumor Growth
System
Tumor
Model

p
p   p 1  
 q
q  bq  dp2/3q  Guq
p(t) is the tumor volume in mm3
q(t) is the carrying capacity of the endothelial cells in mm3,
α ,B,d,G are constants.
u(t) is the angiogenic dose (input)
dp2/3q models endogenous inhibition of the tumor, 2/3
exponent represents the conversion of the tumor volume
into a tumor surface area
Example System – Tumor Growth
Control
Overall goal is to develop tools to show that a differential equation has a
solution, i.e. we are working towards Theorem 2 in this chapter.
Note: We are not saying that we can find this solution.
The errata for the book is on the class website (tinyurl.com/ece874)
such that
The set of all x such that x is an
element of set A or x is an
element of set B (or both)
The set of all pairs (a,b) such that a is an
element of set A and b is an element of
set B
Example : A  T using t   0,  
where t is time.

Example:
 non-negative real numbers
= set of integers
 set of all real numbers
mxn
Example:
 set of all complex numbers
Triangle inequality
The 2-norm
= set of real matrices with
m rows and n columns
d is a function that assigns each
ordered pair (x,y), where x X
and yX, to a unique element
d(x,y) [0,)
will be widely used
Example: ( 2 , d ) where d ( x, y )  ( y1  x1 ) 2  ( y2  x2 ) 2
Note that 2 is the 2-dimensional Euclidean space
n
is the n-dimensional Euclidean space
x1
x3
x2
? Is the d above a “distance” ?



Normed Vector Spaces (Vector space, distance measure)
Vector is a geometric entity with length and direction.
Definition 2.8: A normed vector space is a pair (X,
 x1 
x 
 2
 x3 
Starts at 0
) consisting
of a vector space X and a norm such that
(i ) x  0 iff x  0
Norm is the length
of the vector
(ii )  x   x for  
(iii ) x  y  x  y
x y  xzz y  xz  z y
Typical norms for x 
x
p

p
Distance is then defined as d ( x1 , x2 )  x  y
n
p
 x1  x2  ...  xn
p
specifically
x 1  x1  x2  ...  xn
x2
x

2
2
x1  x2  ...  xn
 max xi
i
2

1
p
which means this is a Metric Space
n1
Example: Neighborhood
of a point in
2
Example: Neighborhood
of a point in
3
Example: Set in
Both sets above are bounded
"Not Open" - require
r  0 for point on the
boundary of the set
2
Typically the Euclidian-norm for control discussions
Not Open
x2
Example:
2
Outside A
Needed in adaptive control
xn
Q
n and m are positive integers > 1 (since the set is open at 1)
Ex : If   .17 then above estimate provides N=12
Getting more dense
(
(we can see that N=2 would actually be ok)
11 1 1 1
98 7 6 5
1
4
1
3
1
2
)
Example:
with d= x  y
( No " holes "in the space)
Expand to n-dimensions
Will be used to analyze the existence and
uniqueness of solutions to some nonlinear
differential equations
)


[
[
f()
1
0
]
S
1
Need only one  for any x,y, i.e.
doesn’t have to hold for every  less
than 1.
Fixed point
Extend the local theorem
Global
and f ( x, t )  
 n
then (8) has a unique solution in [t0 , t1 ]
Chapter 2 Conclusions
• Can talk about the solutions of a differential
equation without actually solving.
– This will be the basis for the rest of the class
Homework 2
A. Chapter 2 - Problems 2.8, 2.13
B. Analyze the Tumor Growth Model:

p
p   p 1  
 q
q  bq  dp2/3q  Guq
Parameter values:  =1.08,b=0.243,d=3.63*.0001,G=1.3
1. For u=0
a) Find the equilibrium points
b) Plot phase portrait for u=0 (plot -2 to 20000 on both axes)
2. For u= .09 (constant dose)
a) Find the equilibrium points
b) Plot phase portrait for u=0 (plot -2 to 20000 on both axes)
3. For u=kq with k=10 (linear, proportional control)
a) Find the equilibrium points
b) Plot phase portrait for u=0 (plot -2 to 20000 on both axes)
c) What happens for other values of k?
Homework 2
C. Introduction to Simulink
Go to Controls Tutorial at:
http://ctms.engin.umich.edu/CTMS/index.php?aux=Home
Follow the instructions to use Simulink (not MATLAB) to simulate
motor speed and motor position models. Record model file and
output plots.
Homework – Solution
Homework – Solution
Homework – Solution
.
• u=0

p
0   p 1    p  q
q

0  bq  dp 2/3 q  Guq
• p0 = q0 =0
3/2
 b  Gu0 
p0  q0  

 d

• p0 = q0 = 17320 mm3
Tumor will reach a maximum size
Homework – Solution
• u= .09
• p0 = q0 =0
3/2
 b  Gu0 
p0  q0  

 d

• p0 = q0 = 6,466 mm3
• Tumor will shrink but not
disappear.
• If therapy is stopped,
tumor will grow to the
original equilibrium size.
Homework – Solution
• For u=kq with k=10 (linear, proportional
control)

p
0   p 1  
q


0  bp  dp 2/3 p  Gkqq  p b  dp 2/3  Gkp
• Linear control appears to
work well.
• Result is somewhat
misleading because we
assumed that the tumor had
grown to a certain size before
the angiogenic model
becomes valid (can’t really
show it goes to zero).
