Urszula Ledzewicz

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Lecture 3 - Part 1 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

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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
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Research Support
Research supported by NSF grants DMS 0205093,
DMS 0305965 and collaborative research grants
DMS 0405827/0405848 DMS 0707404/0707410
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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

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Synthesis of Optimal Controls for Hahnfeldt et
al.
ua
u0
p
q
Full synthesis 0asa0 typical synthesis - as0
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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
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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 p012,000 and q015,000
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Minimum tumor volumes
pmin
u

• full dose
• averaged optimal dose
• optimal control

q0
q0
Values of the minimum tumor volume for a fixed
initial tumor volume as functions of the
initial endothelial support
averaged optimal dose
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Minimum tumor volumes
pmin
u
half dose
full dose
averaged optimal dose
optimal control
q0
q0
Values of the minimum tumor volume for a fixed
initial tumor volume as functions of the
initial endothelial support
averaged optimal dose
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Minimum tumor volumes
pmin
u
full dose
half dose
averaged optimal dose
optimal control
q0
q0
Values of the minimum tumor volume for a fixed
initial tumor volume as functions of the
initial endothelial support
averaged optimal dose
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Comparison of Trajectories
0
full dose
half dose
optimal control
singular arc
averaged optimal dose
0
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Optimal Constant Dose Protocols
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Minimal Tumor Size
dosages from u10 to u100
blow-up of the value for dosages from u46 to
u47
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Optimal 2-Stage Protocols
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Cross-section of the Value
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Cross-section of the Value
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Optimal 1- and 2-Stage Controls
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Optimal Daily Dosages
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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
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Ergun, Camphausen and Wein, Bull. Math. Biol.,
2003

• For a free terminal time minimize
• over all measurable functions that satisfy
• subject to the dynamics

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Synthesis for Model by Ergun et al.

Full synthesis 0asa0, typical synthesis - as0
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Example of optimal control and corresponding
trajectory for Model by Ergun et al.
Initial condition p0 8,000 q0 10,000
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Value of tumor for one dose protocols
dosages from u0 to u15
blow-up of the value for dosages from u8 to u12
minimum at u10.37, p(T)2328.1
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Cross-section of the Value
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Optimal trajectory corresponding to 2-Stage
Protocol
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Optimal Daily Dosages
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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