The Use of Fuzzy Optimization Methods for Radiation.ppt

1
Linear Optimization Under
Uncertainty: Comparisons
Weldon A. Lodwick

2
1. Introduction to Optimization Under Uncertainty
Part 1 of this presentation focuses on
relationships among some fuzzy,
possibilistic, stochastic, and deterministic
optimization methods for solving linear
programming problems. In particular, we
look at several methods to solve one
problem as a means of comparison and
interpretation of the solutions among the
methods.

3
OUTLINE: Part 1
1. Deterministic problem
2. Stochastic problem, stochastic optimization
3. Fuzzy problem – flexible constraints/goals,
flexible programming
4. Fuzzy problem – fuzzy coefficients, possibilistic
optimization
5. Fuzzy problem – Jamison&Lodwick approach

4
Definitions
Types of uncertainty
1. Deterministic – error which is a number
2. Interval – error which is an interval
3. Probabilistic – error which is a distribution, better
yet are distribution bounds (see recent research of
Lodwick&Jamison and Jamison&Lodwick
4. Possibilistic – error which is a possibility
distribution, better are necessity/possibility bounds
(see Jamison&Lodwick)
5. Fuzzy – errors which are membership function

5
Axioms
Confidence measures
( ) 0
( ) 1
The weakest axiom that one could conceive to insure
that the set function has a minimum of conherence
A B g(A) g(B)
When the reference set is infinite, we impose continuity
For every neste
g
g
g
 
 
  

0 1
0 1
d sequence: or
lim ( ) ( )
n
n
n n n n
A A A
A A A
g A g A
 
   
   


6
Measures of Possibility and of Necessity
Consequences of the axioms:
Thus we find, as the limiting case of confidence measure
union is called (by Zadeh) possibility measure
The limiting case of confidence measure intersection is
called necessity measure
, , ( ) max( ( ), ( ))
( ) min( ( ), ( ))
A B g A B g A g B
g A B g A g B
  

, , ( ) max( ( ), ( ))
A B A B A B
     
, , N( ) min( ( ), ( ))
A B A B N A N B
  

7
Observations
• When A and B are disjoint
• When E is a sure event such that:
( ) max( ( ), ( )).
Probability for this case is:
( ) ( ) ( )
Thus, probability and possibility are different. In particular,
suppose , ( )= ( )
A B A B A B
p A B p A p B
A B A B
   
 
   
=max( ( ), ( ))
=1
( ) 1 or ( ) 1 or both.
A B
A B
 
   
, we can define a function for which
( ) 1 if
( ) 0 otherwise
Then is a possibility measure.
E
A A E
A
 
  
 


8
Observations
• A function N can be constructed with values {0,1}
from a sure event E, by:
( ) 1 if
( ) 0 otherwise. Note that ( ) 1 means that
is sure (necessarily true).
The relation between necessity and possibility measures is:
, ( ) 1 ( )
( ) 1 ( )
min( ( ), (
N A E A
N A N A
A
A A N A
N A A
N A N
 
 
    
 
)) 0
, ( ) ( )
( ) 0 ( ) 1
( ) 1 ( ) 0
A
A A N A
N A A
A N A

   
  
   

9
Possibility distribution
Possibility measures are set functions. We also need
functions to act on individual elements (“points”).
Thus,
Necessity distributions are defined in the same way.
( ) ({ }) when is defined.
( ) sup{ ( ) | } when is defined.
is a mapping of into [0,1] called a
possibility distribution.
It is normalized in the sense that
, ( )=1 since ( )=1.
A A
  
   

  
 
  

  

10
The Deterministic Optimization Problem
The problem we consider is derived from the
deterministic LP
numbers.
real
are
and
where
,
0
,
1
,
:
subject to
max
,
,
1
1
j
i
ij
N
j
i
j
ij
N
j
j
j
c
b
a
x
M
i
b
x
a
x
c



 




11
Uncertainty and LP Models
Sources of uncertainty
1. The inequalities – flexible goals, vague goal, flexible
programming, vagueness
2. The coefficients – possibilistic optimization, ambiguity
3. Both in the inequalities and coefficients

12
Optimization in a Fuzzy Environment – Bellman &
Zadeh, “Decision making in a fuzzy environment,”
Management Science, 1970.
Let X be the set of alternatives that contain the
solution of a given optimization problem; that is,
the problem is feasible.
Let Ci be the fuzzy domain defined by the ith
constraint (i=1,…,m). For example, “United
Airline pilots must have good vision.” In this case
“good vision” is the associated fuzzy domain.
Let Gj be the fuzzy domain of the jth goal (j=1,…,J).
For example, “Profits must be high.” In this case
“high” is the associated fuzzy domain.

13
Bellman & Zadeh called a fuzzy decision, the fuzzy set D
on X
<figure next>
1 1
( ) ( )
For example, Air Canada wants pilots with good vision
and wants its profits high. Let denote "membership
function."
, ( ) min{ ( ), ( ), 1... , 1... }
The final decisi
i j
m J
i j
D C G
i j
D C G
m
x X m x m x m x i m j J
 

    
on, , is chosen from the maximal
decision set:
{ | ( ) ( )}
f
f f D f D
x
M x m x m x
 

14
When goals & constraints have unequal importance,
membership functions can be weighted by x
dependent coefficients as follows:
1 1
1 1
1 1 1 1
( ) ( ) 1 ; that is, the weights are a convex combo
( ) ( ) ( ) ( ) ( )
The fuzzy decision set has the property that:
( ) ( ) ( ) ( )
m J
i j
i j
m J
D C G j
i j
i j
i
m J m J
i j i i
i j i j
x x
m x x m x x m x
D
C G D C G
 
 
 
 
   
 
 
 
 
 

15
The definition of optimal decision as given by
Zadeh & Bellman is not always satisfactory
especially when mD(xf ) is very small (the graph is
close to the x-axis). When this occurs goals and
constraints are close to being contradictory (empty
intersections). This issue is addressed in the sequel.

16
An Example Optimization Problem
We will use a simple example from Birge and Louveaux, page 4. A
farmer has 500 acres on which to plant corn, sugar beets and wheat.
The decision as to how many acres to plant of each crop must be made
in the winter and implemented in the spring. Corn, sugar beets and
wheat have an average yield of 3.0, 20 and 2.5 tons per acre
respectively with a +/- 20% variation in the yields uniformly
distributed. The planting costs of these crops are, respectively, 150,
230, and 260 dollars per acre and the selling prices are, respectively,
170, 150, and 36 dollars per ton. However, there is a less favorable
selling price for sugar beets of 10 dollars per ton for any production
over 6,000 tons. The farmer also has cattle that require a minimum of
240 and 200 tons of corn and wheat, respectively. The farmer can buy
corn and wheat for 210 and 238 dollars per ton. The objective is to
minimize costs. It is assumed that the costs and prices are crisp.

17
The Deterministic Model
negative.
-
non
are
variables
all
and
i
crop
to
ing
correspond
high
and
average
low,
are
1,2,3
j
,
yields
and
wheat
and
beet,
sugar
corn,
be
to
1,2,3
i
crop
take
We
.
200
3
3
3
3
6000
2
1
0
2
2
2
1
2
2
240
1
1
1
1
500
3
2
1
:
subject to
3
238
3
170
2
2
10
2
1
36
1
210
1
150
3
150
2
260
1
230
min























ij
y
p
s
x
j
y
s
s
s
x
j
y
p
s
x
j
y
x
x
x
p
s
s
s
p
s
x
x
x

18
The Stochastic Model
matrix.
recourse
fixed
the
is
,
respect to
n with
expectatio
the
is
,
and
,
,
of
components
th
vector wi
the
is
},
0
,
|
min{
)
,
(
,
0
,
:
subject to
)
,
(
min
W
E
T
h
q
y
Ty
h
Wy
y
t
q
x
Q
x
b
Ax
x
Q
E
x
t
c














19
The Stochastic Model - Continued
For our problem we have:
.
200
6
5
3
)
(
3
0
4
3
2
)
(
2
240
2
1
1
)
(
1
:
subject to
}
6
238
5
170
4
10
3
36
2
210
1
150
min{
)
,
(

















y
y
x
s
t
y
y
x
s
t
y
y
x
s
t
y
y
y
y
y
y
y
t
q
s
x
Q

20
Fuzzy LP – Tanaka (1974), Zimmermann (1976, 78)
0
Consider the standard LP:
min
subject to: , 0
The standard "flexible" fuzzy LP is:
, 0
Let , be "crisp" coefficients and define me
T
T
ij i
z c x
Ax b x
c x z
A x b x
a b

 

 
1
1
-
1 1
1
>
1
mbership
functions , 0, ..., as follows:
1 for
( ) 1 - ( ) for
0 for
i
n
ij i i
j
n n n
i d
ij i ij i i i ij i i
j j
j i
n
ij i i i
j
m i M
a x b
m a x a x b b a x b
a x b d



 
 
 







i
d















21
Fuzzy LP – Tanaka and Zimmermann’s approach
where
0 0 0
0 0
,
are subjectively chosen (see radiation therapy problem)
These represent the amount of acceptable violation of each of the
constraints. The initial constraint is
j j
j
b z a c
d
b z
 

0
often determined by solving
the standard LP and obtaining the optimal value and use this for z
figure
 

22
Fuzzy LP – Tanaka and Zimmermann’s approach
A fuzzy decision for the fuzzy LP is D such that:
( ) min{ ( )}
n
i ij j
i j
m x m a x
 

23
The maximization of mD(x) is the equivalent “crisp” LP:
***
1
1
0 0
0
max this is the maximization of
( ) 0, ,
0 1, , 1
The constant is determined by solving
the above without constraint 0 ; let be its
solution, then
n
n
n i ij j
j
j
x m
x m a x i M
x j n
z d
i x
z


 

  


 0 0
= where is the
optimal solution of the standard LP with
replaced by , 1, , .
n
j j
j
i
i i
d c x z
b
b d i M

 

24
Fuzzy LP - Tanaka, et.al., fuzzy in coefficients,
possibilistic programming
max
1 1
subject to: [(1 )( ) ( )]
2 2
1
1 1
(1 )( ) ( ), 1
2 2
1 1 1 1
[ ( ) (1 )( )] ( ) (1 )( ), 1
2 2 2 2
1
is the level or degre
t
c x
N
h a a h a a x
ij
ij ij ij j
j
h b b h b b i M
i
i i i
N
h a a h a a x h b b h b b i M
ij i
ij ij ij j i i i
j
h
    


     
          


e of optimism
for the satisfaction of the constraint.

25
Fuzzy LP - Tanaka, et.al. continued, possibilistic
programming
Here aij and bi are triangular fuzzy numbers
Below h = 0.00, 0.25, 0.50, 0.75 and 1.00
is used.
].
1
,
0
[
and
crisp,
,
/
/
,
/
/ 
h
c
i
b
i
b
i
b
ij
a
ij
a
ij
a

26
Fuzzy LP – Inuiguchi, et. al., fuzzy coefficients,
possibilistic programming
Necessity measure for constraint satisfaction
satisfied.
is
constraint
the
of
necessity
the
which
to
degree
the
is
[0,1]
h
and
/
/
~
where
1
,
1
)
(
1
)
~
(














ij
a
ij
a
ij
a
ij
a
N
j i
b
N
j j
x
ij
a
ij
a
h
j
x
ij
a
N
j
h
i
b
j
x
ij
a
Nec

27
Fuzzy LP – Inuiguchi, et. al. continued, possibilistic
programming
Possibility measure for constraints
satisfied.
is
constraint
the
of
necessity
the
which
to
degree
the
is
[0,1]
h
where
1
,
1
)
(
)
1
(
1
)
~
(














N
j i
b
N
j j
x
ij
a
ij
a
h
j
x
ij
a
N
j
h
i
b
j
x
ij
a
Pos

28
Fuzzy LP – Jamison & Lodwick
Jamison&Lodwick consider the fuzzy LP
constraints a penalty on the objective as
follows:
,
0
,
:
subject to
},
~
~
,
0
max{
~
~
)
(
~





x
d
Bx
x
A
b
t
d
x
t
c
x
f

29
Fuzzy LP – Jamison & Lodwick, continued 2
The constraints are considered hard and the
uncertainty is contained in the objective
function. The expected average of this
objective is minimized; that is,
.
0
,
:
subject to
1
0
)}
(
)
(
{
2
1
)
(
min







x
d
Bx
d
x
f
x
f
x
F 



30
Fuzzy LP – Jamison & Lodwick, continued 3
• F(x) is convex
• Maximization is not differentiable
• Integration over the maximization is differentiable
• We can make the integrand differentiable by
transforming a max as follows:
small
0
,
)
2
(
2
1
}
,
0
max{ 



 x
x
x

31
Table 1: Computational Results – Stochastic and
Deterministic Cases
Corn Sugar Beets Wheat Profit ($)
LOW yield:
Acres planted – det.
25.0 375.0 100.0 $59,950
AVERAGE yield:
80.0 300.0 120.0 $118,600
HIGH yield:
66.7 250.0 183.3 $167,670
Prob. of 1/3 for each yld.
– discrete stochastic
80.0 250.0 170.0 $108,390
Recourse model –
continuous stochastic
85.1 279.1 135.8 $111,250

32
Table 2: Computational Results – Tanaka, Ochihashi,
and Asai
Fuzzy LP
Acres planted, h = 0
72.8 272.7 154.5 $143,430
Fuzzy LP
Acres planted, h = 0.25
71.8 269.2 159.0 $146,930
Fuzzy LP
70.6 264.7 164.7 $151,570
Fuzzy LP
69.0 258.6 172.4 $158,030
Fuzzy LP
66.7 250.0 183.3 $167,670

33
Table 3: Computational Results – Necessity,
Inuiguchi, et. al.
Fuzzy LP, necessity > h
h = 0
80.0 300.0 120.0 $118,600
h = 0.25
76.2 285.7 138.1 $131,100
h = 0.50
72.8 272.7 154.6 $143,430
h = 0.75
69.6 260.9 169.5 $155,610
h = 1.00
66.7 250.0 183.3 $167,667

34
Table 4: Computational Results – Possibility,
Inuiguchi, et. al.
Fuzzy LP, possibility > h
h = 0.00
100.0 300.0 100.0 $100,000
h = 0.25
94.1 302.8 103.1 $103,380
h = 0.50
88.9 303.8 107.3 $107,520
h = 0.75
84.2 302.8 113.0 $112,550
h = 1.00
80.0 300.0 120.0 $118,600

35
Table 5: Computational Results – Jamison and
Lodwick
Fuzzy LP
Jamison & Lodwick
85.1 280.4 134.5 $111,240
Recourse Model
Continuous Stochastic
85.1 279.1 135.8 $111,250

36
Analysis of Numerical Results
• The extreme of the necessity measure, h=0, and
the extreme of the possibility measure, h=1,
generate the same solution which is the average
yield scenario.
• Tanaka with h=0 (total lack of optimism)
corresponds to the necessity h=0.5 model.
• Tanaka starts with a solution halfway between the
deterministic average and high yield and ends up
at the high yield solution.

37
Analysis of Numerical Results
• Possibility measure starts with a solution half way between
the low and average yield deterministic and ends at the
deterministic average yield solution.
• Necessity measure starts with the solution corresponding
to average yield deterministic model and ends at the high
yield solution.
• Lodwick & Jamison is most similar to the stochastic
recourse optimization model yielding virtually identical
solutions

38
• Complexity of the fuzzy LP using triangular or
trapezoidal numbers corresponds to that of the
deterministic LP.
• There is an overhead in the data structure
conversion.
• The Lodwick&Jamison penalty approach is more
complex than other fuzzy linear programming
problems, especially since an integration rule must
be used to evaluate the expected average.

39
• Complexity of Jamison & Lodwick corresponds to
that of the recourse model with the addition of the
evaluation of one integral per iteration.
• The penalty approach is simpler than stochastic
programming in its modeling structure; that is, it
can be modeled more simply. The transformation
into a NLP using triangular or trapezoidal fuzzy
numbers is straight forward.
• Used MATLAB and a 21-point Simpson’s
integration rule.

40
2. Optimization Under Uncertainty -Methods and
Applications in Radiation Therapy
The extension of flexible programming problems in
order to allow for large “industrial strength”
optimization is given.
Methods to handle large optimization under
uncertainty problems and an application of these
methods of to radiation therapy planning is
presented. Two themes are developed in this
study: (1) the modeling of inherent uncertainty of
the problems and (2) the application of uncertainty
optimization

41
Objectives of part 2 of this presentation
1. To demonstrate that fuzzy mathematical
programming (fmp) is useful in solving
large, “industrial-strength” problems
2. To demonstrate the usefulness and
tractability of the Jamison & Lodwick and
surprise approaches to fuzzy linear
programming in solving large problems

42
OUTLINE – Part 2
I. Introduction: The radiation therapy
treatment problem (RTP)
II. Modeling of uncertainty in the RTP
III. Optimization under uncertainty
A. Zimmermann
B. Inuiguchi, Tanaka, Ichihashi, Ramik,
and others
C. Jamison & Lodwick
D. Surprise functions
IV. Numerical results – A, C and D

43
I. The Radiation Therapy Problem
• The radiation therapy problem (RTP) is
to obtain, for a given radiation machine, a
set of beam angles and beam intensities at
these angles so that the delivered dosage
destroys the tumor while sparing
surrounding healthy tissue through which
radiation must travel to intersect at the
tumor.

44
I. Why Use a Fuzzy Approach?
• Boundary between tumor and healthy tissue
• Minimum radiation value for tumor a range of
values
• Maximum values for healthy tissue a range of
values
• The calculation of delivered dosage at a particular
pixel is derived from a mathematical model
• Alignment of the patient at the time of radiation
• Position of the tumor at the time of radiation

45
I. CT Scan - Pixels and Pencils

46
I. ATTENUATION MATRIX
























































































m
b
b
n
x
n
m
a
x
m
a
x
m
a
n
x
n
a
x
a
x
a
n
x
x
x
n
m
a
m
a
m
a
n
a
a
a
Ax












1
m
pixel
at
radiation
attenuated
total
1
pixel
at
radiation
attenuated
total
,
2
2
,
1
1
,
,
1
2
2
,
1
1
1
,
1
2
1
,
2
,
1
,
,
1
2
,
1
1
,
1

47
I. EXAMPLE - Attenuation Matrix
• Suppose there are two pencils per beams
and two voxels





















































2
1
3
5
.
0
2
1
4
5
.
0
3
2
4
3
2
1
0
5
.
0
1
1
5
.
0
1
1
0
b
b
x
x
x
x
x
x
x
x
x
x
Ax

48
I. Constraint Inequalities
 
 
 
tumor
T
Ax
x
x
x
x
x x x z



















    
1 0
0 1 1 5
1 1 5 0
5
1
2
3
4
2 3 4 1
.
.
.
radiation at voxel 1

49
I. Objective Functions
f c x c x c x
c x
j j
j
J
j j
( , ) ,
,
   
  
 

0 0
Minimize total weighted radiation

50
I. The Fuzzy Optimization Model
min ( )
, ,
, ,
, ,
min
max
max
f x x
Ax T p p t
Ax T q q
Ax d r r
j
j
J
t t
t t
c
T
c c
k k k

    
    
    


  
  
  

 (minimize total radiation)
subject to:
indices of tumor voxels
c indices of critical tissue
x 0
T
T
k



0
0
0

51
II. Modeling of uncertainty in the RTP
Four sources of uncertainty and fuzziness in the
RTP:
1. Delineation of tumors and critical tissue
2. Radiation tolerances or critical dose levels for
each tissue type or tumor
3. Model for the delivered dose, that is the dose
transfer matrix
4. The location of tissue at the time of radiation

52
II. The RTP process – in practice
1. The oncologist delineates the tumor and critical
structures
2. A candidate set of beam intensities is obtained;
for example by linear programming, fuzzy
mathematical programming, simulated
annealing, or purely human choice.
3. These beam intensities are used as inputs to a
FDA (Federal Drug Administration) approved
dose calculator computer program to produce the
graphical depiction of the dose deposition of
each pixel (as color scales and dose-volume
histograms, DVH’s – see Figure 1).

53
II. Example Dose Volume Histogram (DVH)

54
III. Optimization Under Uncertainty
The general fuzzy/possibilistic model
considered here is:
intensity)
beam
(max
0
~
:
subject to
min
x
x
b
Bx
x
T
c




55
III. Zimmermann’s approach
Translate to a real-value linear program
membership
-
trapezoid
,
0
0
number
fuzzy
the
is
~
Where
0
1
,
:
subject to
max
p
/b/b
/
b
x
x
m
i
i
p
i
b
x
i
B
i
p






 




56
III. Jamison & Lodwick approach
Translate
into the nonlinear programming problem
x
x
to
subject
b
Bx
T
p
x
f
x
F





0
:
}
~
,
0
max{
~
)
(
)
(
~
:
obj
x
x
d
x
F
x
F
F
EA
f(x)






 
0
:
subject to
}
))
(
1
0
)
(
(
2
1
)
~
(
{
max 



57
III. Advantages to the J&L approach
1. If f(x) is convex, then the problem is a
convex nlp with simple bound constraints
2. It optimizes over all alpha-levels; that is, it
does not force each constraint to be at the
same alpha-level
3. Large problems can be solved quickly;
that is, it is tractable for large problems

58
III. Surprise function approach
Each fuzzy constraint
2
1 )
1
))
(
i
((
)
(
:
function
surprise
a
into
function
membership
each
Translate
).
~
(
)
(
i
function
membership
the
where
~
,
constraint
equality
fuzzy
a
into
d
translate
is
~
,














i
s
i
b
Pos
i
x
i
B
i
b
x
i
B



59
III. Surprise function approach - continued
The fuzzy problem is translated into the
nonlinear programming problem
This is a convex nlp with simple bound
constraints.
x
x
x
i
B



0
:
subject to
i
)
,
(
i
s
min 

60
III. Why use the surprise function approach?
1. It is a convex nlp with simple bound
constraints
2. It optimizes over all the alpha-levels; that
is, it does not force each constraint to be a
the same alpha-level
3. Large problems can be solved quickly;
that is, it is tractable for large problems

61
IV. Surprise – problem: Black is out of body, blue
is critical organ, yellow/green is other critical organs,
red is tumor – 32x32 image, 8 angles
Set-up time =5.4580
Optimization time= 1.7130

62
IV. Surprise 32x32 with 8 angles – delivered dosage
10
20
30
40
50
60
70

63
IV. Surprise 32x32 with 8 angles - Tumor dvh
0 10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy
Tumor DVH

64
IV. Surprise 32x32 with 8 angles – Critical dvh
0 10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy
Critical Structure DVH

65
IV. Surprise 64x64 with 8 angles – delivered dosage
Set-up time = 11.0160, optimization time = 2.2930
10
20
30
40
50
60
70

66
IV. Surprise 64x64 with 8 angles – Tumor dvh
0 10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy
Tumor DVH

67
IV. Surprise 64x64 with 8 angles – Critical dvh
0 10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy

68
IV. Zimmermann – 32x32 with 8 angles
Set-up time = 4.6060
Opt time =171.1060
10
20
30
40
50
60
70

69
IV. Zimmermann 32x32 with 8 angles – tumor dvh
0 10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy
Tumor DVH

70
IV. Zimmermann 32x32 with 8 angles – critical dvh
0 10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy

71
IV. Zimmermann – 64x64 with 8 angles
Set-up time = 8.8930, Optimization time =125.1100
10
20
30
40
50
60
70

72
IV. Zimmermann: 64x64 with 8 angles – Tumor dvh
0 10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy
Tumor DVH

73
IV. Zimmermann: 64x64 with 8 angles – Critical dvh
0 10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy

74
IV. J & L – 32x32 with 8 angles
Setup time - 5.3070
Optimization time - 7.3410
10
20
30
40
50
60
70
80
90
100

75
IV. J & L 32x32 with 8 angles – tumor dvh
0 10 20 30 40 50 60 70 80 90 100 110
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy
Tumor DVH

76
IV. J & L 32x32 with 8 angles – critical dvh
0 10 20 30 40 50 60 70 80 90 100 110
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy

77
IV. J & L – 64x64 with 8 angles
Set-up time=13.0290, optimization time = 3.145
10
20
30
40
50
60
70
80
90
100

78
IV. J & L - 64x64 with 8 angles
Tumor dvh
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy
Tumor DVH

79
IV. J & L - 64x64 with 8 angles
Critical structure dvh
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction
of
Volume
Dose in Gy

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