Lecture 3 - Part 1:
3
Realizable Suboptimal Protocols
for Tumor Anti-Angiogenesis
May 11-15, 2009
Department of Automatic Control
Silesian University of Technology, Gliwice
Urszula Ledzewicz
Department of Mathematics and Statistics
Southern Illinois University, Edwardsville, USA
Collaborators
Heinz Schättler
Dept. of Electrical and Systems Engineering
Washington University, St. Louis, Missouri, USA
Helmut Maurer
Rheinisch Westfälische Wilhelms-Universität Münster,
Münster, Germany
John Marriott
Dept. of Mathematics and Statistics,
Southern Illinois University, Edwardsville, USA
Research Support
Research supported by NSF grants
DMS 0205093,
DMS 0305965
and collaborative research grants
DMS 0405827/0405848
DMS 0707404/0707410
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.]
18000
p
u=a
16000
u=0
14000
tumor cells
12000
10000
8000
an optimal trajectory
end of “therapy”
begin of
therapy
6000
4000
final point – minimum of p
2000
0
0
2000
4000
6000
8000
10000
endothelial cells
12000
14000
16000
q 18000
Full synthesis 0asa0 typical synthesis - as0
An Optimal Controlled Trajectory
for [Hahnfeldt et al.]
1.3
70
x 10
4
singular arc
full dose
1.2
60
1.1
tumor volume, p
optimal control u
50
40
partial dose - singular
30
1
0.9
20
no dose
10
0.8
0
0
1
2
3
4
5
6
7
0.7
2000
4000
6000
time
8000
10000
12000
14000
16000
carrying capacity, q
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
Suboptimal Protocols for [Hahnfeldt et al.]
• full dose protocol:
give
over time
• half dose protocol:
give
over time
• averaged 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
pmin
Minimum tumor volumes
u
• full dose
• averaged optimal dose
• optimal control
q0
Values of the minimum tumor volume
for a fixed initial tumor volume
as
functions of the initial endothelial
support
q0
averaged optimal dose
Minimum tumor volumes
pmin
u
half dose
full dose
averaged optimal dose
optimal control
q0
Values of the minimum tumor volume
for a fixed initial tumor volume
as
functions of the initial endothelial
support
q0
averaged optimal dose
Minimum tumor volumes
pmin
u
full dose
half dose
averaged optimal dose
optimal control
Values of the minimum tumor volume
for a fixed initial tumor volume
as
functions of the initial endothelial
support
q0
q0
averaged optimal dose
Comparison of Trajectories
0
full dose
half dose
optimal control
singular arc
averaged optimal dose
0
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
1.3
70
x 10
4
singular arc
full dose
1.2
60
1.1
tumor volume, p
optimal control u
50
40
partial dose - singular
30
1
0.9
20
no dose
10
0.8
0
0
1
2
3
4
5
6
7
0.7
2000
4000
6000
time
8000
10000
12000
14000
16000
carrying capacity, q
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
[Ergun, Camphausen and Wein],
Bull. Math. Biol., 2003
For a free terminal time
over all measurable functions
subject to the dynamics
minimize
that satisfy
Synthesis for Model by [Ergun et al.]
tumor volume, p
18000
16000
14000
12000
full dose
beginning of therapy
10000
partial dose, singular control
8000
6000
no dose
4000
(q(T),p(T)), point where minimum is realized
2000
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
carrying capacity of the vasculature, q
Full synthesis 0asa0, typical synthesis - as0
1.6
1.8
2
4
x 10
Example of optimal control and corresponding
trajectory for Model by Ergun et al.
20
9000
18
8000
16
14
7000
12
6000
10
5000
8
4000
6
4
3000
2
2000
0
1000
-2
0
2
4
6
8
10
12
Initial condition: p0 = 8,000 q0 = 10,000
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
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