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3. Lecture 3 - Part 1: Realizable Suboptimal Protocols for Tumor Anti-Angiogenesis. Urszula Ledzewicz Department of Mathematics and Statistics Southern Illinois University, Edwardsville, USA. May 11-15, 2009 Department of Automatic Control Silesian University of Technology, Gliwice. - PowerPoint PPT Presentation
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3Lecture 3 - Part 1: Realizable Suboptimal Protocols for Tumor Anti-Angiogenesis
Urszula LedzewiczUrszula LedzewiczDepartment of Mathematics and StatisticsDepartment of Mathematics and StatisticsSouthern Illinois University, Edwardsville, USASouthern Illinois University, Edwardsville, USA
May 11-15, 2009
Department of Automatic Control
Silesian University of Technology, Gliwice
Heinz SchättlerDept. of Electrical and Systems EngineeringWashington University, St. Louis, Missouri, USA
Collaborators
Helmut MaurerRheinisch Westfälische Wilhelms-Universität Münster,Münster, Germany
John MarriottDept. of Mathematics and Statistics,
Southern Illinois University, Edwardsville, USA
Research supported by NSF grantsResearch supported by NSF grants
DMS 0205093, DMS 0305965DMS 0205093, DMS 0305965
and collaborative research grants and collaborative research grants
DMS 0405827/0405848DMS 0405827/0405848
DMS 0707404/0707410DMS 0707404/0707410
Research Support
References
• U. Ledzewicz and H. Schättler, Optimal and Suboptimal Protocols for Tumor Anti-Angiogenesis, J. of
Theoretical Biology, 252, (2008), pp. 295-312,
• U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, The scheduling of angiogenic inhibitors minimizing tumor volume, J. of Medical Informatics and Technologies, 12, (2008), pp. 23-28
• U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Math. Med. And Biology, (2009), to appear
Synthesis of Optimal Controls for [Hahnfeldt et al.]
Full synthesis 0asa0 0asa0 typical synthesis - as0as0
0 2000 4000 6000 8000 10000 12000 14000 16000 180000
2000
4000
6000
8000
10000
12000
14000
16000
18000
endothelial cells
tum
or
cells
an optimal trajectorybegin of therapy
final point – minimum of p
end of “therapy”
p
q
u=au=0
An Optimal Controlled Trajectory for [Hahnfeldt et al.]
Initial condition: p0 = 12,000 q0 = 15,000
Optimal terminal value: 8533.4 time: 6.7221
Terminal value for a0-trajectory: 8707.4 time: 5.1934
0 1 2 3 4 5 6 7
0
10
20
30
40
50
60
70
time
opt
imal
con
tro
l u
full dose
no dose
partial dose - singular
2000 4000 6000 8000 10000 12000 14000 160000.7
0.8
0.9
1
1.1
1.2
1.3x 10
4
carrying capacity, q
tum
or v
olum
e, p
singular arc
• full dose protocolfull dose protocol:
give over time
Suboptimal Protocols for [Hahnfeldt et al.]
• half dose protocolhalf dose protocol: give over time
• averaged optimal dose protocolaveraged optimal dose protocol: give over time where is the time when inhibitors are exhausted along the optimal solution and e.g., for p0=12,000
and q0=15,000
Minimum tumor volumes
Values of the minimum tumor volume for a fixed initial tumor volume as functions of the initial endothelial support
• full dose
• averaged optimal dose
• optimal control
pmin
q0
averaged optimal dose
u
q0
Minimum tumor volumes
Values of the minimum tumor volume for a fixed initial tumor volume as functions of the initial endothelial support
full dose
averaged optimal dose
u
optimal control
averaged optimal dose
half dose
pmin
q0 q0
Minimum tumor volumes
Values of the minimum tumor volume for a fixed initial tumor volume as functions of the initial endothelial support
full dose
averaged optimal dose
u
optimal control
averaged optimal dose
half dose
pmin
q0
q0
Comparison of Trajectories
averaged optimal dose
full dose
optimal control
0
0
singular arcsingular arc
half dose
Optimal Constant Dose Protocols
Minimal Tumor Size
dosages from u=10 to u=100
blow-up of the value for dosages from u=46 to u=47
Optimal 2-Stage Protocols
Cross-section of the Value
Cross-section of the Value
Optimal 1- and 2-Stage Controls
Optimal Daily Dosages
An Optimal Controlled Trajectory
Initial condition: p0 = 12,000 q0 = 15,000
Optimal terminal value: 8533.4 time: 6.7221
Terminal value for a0-trajectory: 8707.4 time: 5.1934
0 1 2 3 4 5 6 7
0
10
20
30
40
50
60
70
time
opt
imal
con
tro
l u
full dose
no dose
partial dose - singular
2000 4000 6000 8000 10000 12000 14000 160000.7
0.8
0.9
1
1.1
1.2
1.3x 10
4
carrying capacity, q
tum
or v
olum
e, p
singular arc
For a free terminal time minimizeminimize
over all measurable functions that satisfy
subject to the dynamics
[Ergun, Camphausen and Wein], Bull. Math. Biol., 2003
Synthesis for Model by [Ergun et al.]
Full synthesis 0asa0, typical synthesis - as0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
carrying capacity of the vasculature, q
tum
or
volu
me,
p
beginning of therapy
(q(T),p(T)), point where minimum is realized
full dose
partial dose, singular control
no dose
Example of optimal control and corresponding trajectory for Model by Ergun et al.
Initial condition: p0 = 8,000 q0 = 10,000
0 2 4 6 8 10 12
-2
0
2
4
6
8
10
12
14
16
18
20
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
1000
2000
3000
4000
5000
6000
7000
8000
9000
Value of tumor for one dose protocols
dosages from u=0 to u=15
blow-up of the value for dosages from u=8 to u=12
minimum at u=10.37, p(T)=2328.1
Cross-section of the Value
Optimal trajectory corresponding to 2-Stage Protocol
Optimal Daily Dosages
Conclusions
• The optimal control which has a singular piece is not medically
realizable (feedback), but it provides benchmark values and can
become the basis for the design of suboptimal, but realistic protocols.
• The averaged optimal dose protocol gives an excellent sub-optimal
protocol, generally within 1% of the optimal value. The averaged
optimal dose decreases with increasing initial tumor volume and is
very robust with respect to the endothelial support for fixed initial
tumor volume
• Optimal piecewise constant protocols can be constructed that
essentially reproduce the performance of the optimal controls
