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February  2017, 14(1): 195-216. doi: 10.3934/mbe.2017013

Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays

1. 

Silesian University of Technology, Department of Automatic Control, Akademicka 16, 44101 Gliwice, Poland

2. 

University of Münster, Institute of Computational and Applied Mathematics, Einsteinstr. 62,48149 Münster, Germany

3. 

Silesian University of Technology, Department of Automatic Control, Akademicka 16, 44101 Gliwice, Poland

Received  October 06, 2015 Accepted  July 15, 2016 Published  October 2016

Fund Project: The research presented here was partially supported by the National Science Centre (NCN) in Poland grant DEC-2014/13/B/ST7/00755 (JK, AS). Some preliminary results were presented at the Conference: Micro and Macro Systems in Life Sciences, Bedlewo, 2015. HM is grateful to Urszula Ledzewicz who provided a grant for him to attend the Conference in Bedlewo

We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.

Citation: Jerzy Klamka, Helmut Maurer, Andrzej Swierniak. Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 195-216. doi: 10.3934/mbe.2017013
References:
[1]

G. Bergers and D. Hanahan, Modes of resistance to anti-angiogenic therapy, Nature Reviews Cancer, 8 (2008), 592-603. doi: 10.1038/nrc2442. Google Scholar

[2]

A. C. Billioux, U. Modlich and R, Bicknell, The Cancer Handbook: Angiogenesis, 2nd Edition, John Wiley & Sons, 2007.Google Scholar

[3]

R. F. Brammer, Controllability in linear autonomous systems with positive controllers, SIAM J. Control, 10 (1972), 339-353. doi: 10.1137/0310026. Google Scholar

[4]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen, PhD thesis, Institut für Numerische Mathematik, Universität Münster, Germany, 1998.Google Scholar

[5]

C. Büskens and H. Maurer, SQP methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real time control, J. Comput. Appl. Math., 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8. Google Scholar

[6]

J. M. CollinsD. S. ZaharkoR. L. Dedrick and B. A. Chabner, Potential roles for preclinical pharmacology in phase Ⅰ clinical trials, Cancer Treat. Rep., 70 (1986), 73-80. Google Scholar

[7]

V. T. Devita and J. Folkman, Cancer: Principles and Practice of Oncology, 6th edition, Lippincott Williams & Wilkins Publishers, 2001.Google Scholar

[8]

M. Dolbniak and A. Swierniak, Comparison of simple models of periodic protocols for combined anticancer therapy, Computational and Mathematical Methods in Medicine, 2013 (2013), Article ID 567213, 11pp. Google Scholar

[9]

A. D'Onofrio and A. Gandolfi, Tumor eradication by anti-angiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004), 159-184. doi: 10.1016/j.mbs.2004.06.003. Google Scholar

[10]

A. D'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Mathematical Medicine and Biology, 26 (2009), 63-95. doi: 10.1093/imammb/dqn024. Google Scholar

[11]

A. D'Onofrio and A. Gandolfi, Chemotherapy of vascularised tumors: Role of vessel density and the effect of vascular "pruning", Journal of Theoretical Biology, 264 (2010), 253-265. doi: 10.1016/j.jtbi.2010.01.023. Google Scholar

[12]

A. D'OnofrioU. LedzewiczH. Maurer and H. Schaettler, On optimal delivery of combination therapy for tumors, Math. Biosciences, 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004. Google Scholar

[13]

A. ErgunK. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424. doi: 10.1016/S0092-8240(03)00006-5. Google Scholar

[14]

J. Folkman, Anti-angiogenesis: New concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416. doi: 10.1097/00000658-197203000-00014. Google Scholar

[15]

J. Folkman, Tumor angiogenesis therapeutic implications, New England Journal of Medicine, 285 (1971), 1182-1186. Google Scholar

[16]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993.Google Scholar

[17]

G. GaspariniR. LongoM. Fanelli and B. A. Teicher, Combination of anti-angiogenic therapy with other anticancer therapies: Results, challenges, and open questions, Journal of Clinical Oncology, 23 (2005), 1295-1311. Google Scholar

[18]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Special Issue on Computational Methods for Optimization and Control, J. of Industrial and Management Optimization, 10 (2014), 413-441. Google Scholar

[19]

P. HahnfeldtD. PanigrahyJ. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775. Google Scholar

[20]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013. Google Scholar

[21]

R. S. Kerbel, Inhibition of tumor angiogenesis as a strategy to circumvent acquired resistance to anti-cancer therapeutic agents, BioEssays, 13 (1991), 31-36. doi: 10.1002/bies.950130106. Google Scholar

[22]

M. Kimmel and A. Swierniak, Control theory approach to cancer chemotherapy: Benefiting from phase dependence and overcoming drug resistance, Tutorials in Mathematical Biosciences Ⅲ: Cell Cycle, Proliferation, and Cancer (A. Friedman-Ed. ), Lecture Notes in Mathematics, Mathematical Biosciences Subseries, Springer, Heidelberg, 1872 (2006), 185-221. doi: 10.1007/11561606_5. Google Scholar

[23]

J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1991. Google Scholar

[24]

J. Klamka, Constrained controllability of nonlinear systems, J. Math. Anal. Appl., 201 (1996), 365-374. doi: 10.1006/jmaa.1996.0260. Google Scholar

[25]

U. Ledzewicz and H. Schaettler, Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM Journal on Control and Optimization, 46 (2007), 1052-1079. doi: 10.1137/060665294. Google Scholar

[26]

U. Ledzewicz and H. Schaettler, Analysis of optimal controls for a mathematical model of tumor anti-angiogenesis, Optimal Control Applications and Methods, 29 (2008), 41-57. doi: 10.1002/oca.814. Google Scholar

[27]

U. Ledzewicz and H. Schaettler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. Theor. Biol., 252 (2008), 295-312. doi: 10.1016/j.jtbi.2008.02.014. Google Scholar

[28]

U. Ledzewicz and H. Schaettler, On the optimality of singular controls for a class of mathematical models for tumor antiangiogenesis, Discrete and Continuous Dynamical Systems, Series B, 11 (2009), 691-715. doi: 10.3934/dcdsb.2009.11.691. Google Scholar

[29]

U. LedzewiczJ. MarriottH. Maurer and H. Schaettler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179. doi: 10.1093/imammb/dqp012. Google Scholar

[30]

U. Ledzewicz, H. Maurer and H. Schaettler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, in Recent Advances in Optimization and its Applications in Engineering, 267-276, Springer, 2010. doi: 10.1007/978-3-642-12598-0_23. Google Scholar

[31]

U. LedzewiczH. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor antiangiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323. doi: 10.3934/mbe.2011.8.307. Google Scholar

[32]

J. Ma and D. J. Waxman, Combination of antiangiogenesis with chemotherapy for more effective cancer treatment, Molecular Cancer Therapeutics, 7 (2008), 3670-3684. doi: 10.1158/1535-7163.MCT-08-0715. Google Scholar

[33]

H. MaurerC. BüskensJ. H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang control, Optimal Control Appl. Meth., 26 (2005), 129-156. doi: 10.1002/oca.756. Google Scholar

[34]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368. Google Scholar

[35]

M. J. Piotrowska and U. Forys, Analysis of the Hopf bifurcation for the family of angiogenesis models, Journal of Mathematical Analysis and Applications, 382 (2011), 180-203. doi: 10.1016/j.jmaa.2011.04.046. Google Scholar

[36]

R. K. SachcsL. R. Hlatky and P. Hahnfeldt, Simple ODE models of tumor growth and anti-angiogenic or radiation treatment, Math. Comput. Mod, 33 (2001), 1297-1305. doi: 10.1016/S0895-7177(00)00316-2. Google Scholar

[37]

H. Schättler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. Google Scholar

[38]

A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences: Technical Sciences, 56 (2008), 367-378. Google Scholar

[39]

A. Swierniak, Modelling combined anti-angiogenic and chemo-therapies, in: Proc. 14th National Conf. Appl. Math. Biol Medicine, Leszno, 2008,127-133.Google Scholar

[40]

A. Swierniak, Comparison of six models of anti-angiogenic therapy, Applicationes Mathematicae, 36 (2009), 333-348. doi: 10.4064/am36-3-6. Google Scholar

[41]

A. Swierniak, A. d'Onofrio and A. Gandolfi, Control problems related to tumor angiogenesis, Proc. of the 32nd Annual Conference on IEEE Industrial Electronics (IECON '06), Paris, 677-681, November 2006. doi: 10.1109/IECON.2006.347815. Google Scholar

[42]

A. Swierniak and J. Klamka, Local controllability of models of combined anticancer therapy with delays in control, Math. Model. Nat. Phenom., 9 (2014), 216-226. doi: 10.1051/mmnp/20149413. Google Scholar

[43]

L. S. TengK. T. JinK. F. HeH. H. WangJ. Cao and D. C. Yu, Advances in combination of anti-angiogenic agents targeting VEGF-binding and conventional chemotherapy and radiation for cancer treatment, Journal of the Chinese Medical Association, 73 (2010), 281-288. Google Scholar

[44]

The Internet Drug Index, (2015), http://www.rxlist.com/avastin-drug/clinical-pharmacology.htmlGoogle Scholar

[45]

US National Institutes of Health, Clinical Trials, (last updated June 2015), http://www.clinicaltrials.gov/ct2/show/NCT00520013Google Scholar

[46]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. Google Scholar

show all references

References:
[1]

G. Bergers and D. Hanahan, Modes of resistance to anti-angiogenic therapy, Nature Reviews Cancer, 8 (2008), 592-603. doi: 10.1038/nrc2442. Google Scholar

[2]

A. C. Billioux, U. Modlich and R, Bicknell, The Cancer Handbook: Angiogenesis, 2nd Edition, John Wiley & Sons, 2007.Google Scholar

[3]

R. F. Brammer, Controllability in linear autonomous systems with positive controllers, SIAM J. Control, 10 (1972), 339-353. doi: 10.1137/0310026. Google Scholar

[4]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen, PhD thesis, Institut für Numerische Mathematik, Universität Münster, Germany, 1998.Google Scholar

[5]

C. Büskens and H. Maurer, SQP methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real time control, J. Comput. Appl. Math., 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8. Google Scholar

[6]

J. M. CollinsD. S. ZaharkoR. L. Dedrick and B. A. Chabner, Potential roles for preclinical pharmacology in phase Ⅰ clinical trials, Cancer Treat. Rep., 70 (1986), 73-80. Google Scholar

[7]

V. T. Devita and J. Folkman, Cancer: Principles and Practice of Oncology, 6th edition, Lippincott Williams & Wilkins Publishers, 2001.Google Scholar

[8]

M. Dolbniak and A. Swierniak, Comparison of simple models of periodic protocols for combined anticancer therapy, Computational and Mathematical Methods in Medicine, 2013 (2013), Article ID 567213, 11pp. Google Scholar

[9]

A. D'Onofrio and A. Gandolfi, Tumor eradication by anti-angiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004), 159-184. doi: 10.1016/j.mbs.2004.06.003. Google Scholar

[10]

A. D'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Mathematical Medicine and Biology, 26 (2009), 63-95. doi: 10.1093/imammb/dqn024. Google Scholar

[11]

A. D'Onofrio and A. Gandolfi, Chemotherapy of vascularised tumors: Role of vessel density and the effect of vascular "pruning", Journal of Theoretical Biology, 264 (2010), 253-265. doi: 10.1016/j.jtbi.2010.01.023. Google Scholar

[12]

A. D'OnofrioU. LedzewiczH. Maurer and H. Schaettler, On optimal delivery of combination therapy for tumors, Math. Biosciences, 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004. Google Scholar

[13]

A. ErgunK. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424. doi: 10.1016/S0092-8240(03)00006-5. Google Scholar

[14]

J. Folkman, Anti-angiogenesis: New concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416. doi: 10.1097/00000658-197203000-00014. Google Scholar

[15]

J. Folkman, Tumor angiogenesis therapeutic implications, New England Journal of Medicine, 285 (1971), 1182-1186. Google Scholar

[16]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993.Google Scholar

[17]

G. GaspariniR. LongoM. Fanelli and B. A. Teicher, Combination of anti-angiogenic therapy with other anticancer therapies: Results, challenges, and open questions, Journal of Clinical Oncology, 23 (2005), 1295-1311. Google Scholar

[18]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Special Issue on Computational Methods for Optimization and Control, J. of Industrial and Management Optimization, 10 (2014), 413-441. Google Scholar

[19]

P. HahnfeldtD. PanigrahyJ. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775. Google Scholar

[20]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013. Google Scholar

[21]

R. S. Kerbel, Inhibition of tumor angiogenesis as a strategy to circumvent acquired resistance to anti-cancer therapeutic agents, BioEssays, 13 (1991), 31-36. doi: 10.1002/bies.950130106. Google Scholar

[22]

M. Kimmel and A. Swierniak, Control theory approach to cancer chemotherapy: Benefiting from phase dependence and overcoming drug resistance, Tutorials in Mathematical Biosciences Ⅲ: Cell Cycle, Proliferation, and Cancer (A. Friedman-Ed. ), Lecture Notes in Mathematics, Mathematical Biosciences Subseries, Springer, Heidelberg, 1872 (2006), 185-221. doi: 10.1007/11561606_5. Google Scholar

[23]

J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1991. Google Scholar

[24]

J. Klamka, Constrained controllability of nonlinear systems, J. Math. Anal. Appl., 201 (1996), 365-374. doi: 10.1006/jmaa.1996.0260. Google Scholar

[25]

U. Ledzewicz and H. Schaettler, Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM Journal on Control and Optimization, 46 (2007), 1052-1079. doi: 10.1137/060665294. Google Scholar

[26]

U. Ledzewicz and H. Schaettler, Analysis of optimal controls for a mathematical model of tumor anti-angiogenesis, Optimal Control Applications and Methods, 29 (2008), 41-57. doi: 10.1002/oca.814. Google Scholar

[27]

U. Ledzewicz and H. Schaettler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. Theor. Biol., 252 (2008), 295-312. doi: 10.1016/j.jtbi.2008.02.014. Google Scholar

[28]

U. Ledzewicz and H. Schaettler, On the optimality of singular controls for a class of mathematical models for tumor antiangiogenesis, Discrete and Continuous Dynamical Systems, Series B, 11 (2009), 691-715. doi: 10.3934/dcdsb.2009.11.691. Google Scholar

[29]

U. LedzewiczJ. MarriottH. Maurer and H. Schaettler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179. doi: 10.1093/imammb/dqp012. Google Scholar

[30]

U. Ledzewicz, H. Maurer and H. Schaettler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, in Recent Advances in Optimization and its Applications in Engineering, 267-276, Springer, 2010. doi: 10.1007/978-3-642-12598-0_23. Google Scholar

[31]

U. LedzewiczH. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor antiangiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323. doi: 10.3934/mbe.2011.8.307. Google Scholar

[32]

J. Ma and D. J. Waxman, Combination of antiangiogenesis with chemotherapy for more effective cancer treatment, Molecular Cancer Therapeutics, 7 (2008), 3670-3684. doi: 10.1158/1535-7163.MCT-08-0715. Google Scholar

[33]

H. MaurerC. BüskensJ. H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang control, Optimal Control Appl. Meth., 26 (2005), 129-156. doi: 10.1002/oca.756. Google Scholar

[34]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368. Google Scholar

[35]

M. J. Piotrowska and U. Forys, Analysis of the Hopf bifurcation for the family of angiogenesis models, Journal of Mathematical Analysis and Applications, 382 (2011), 180-203. doi: 10.1016/j.jmaa.2011.04.046. Google Scholar

[36]

R. K. SachcsL. R. Hlatky and P. Hahnfeldt, Simple ODE models of tumor growth and anti-angiogenic or radiation treatment, Math. Comput. Mod, 33 (2001), 1297-1305. doi: 10.1016/S0895-7177(00)00316-2. Google Scholar

[37]

H. Schättler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. Google Scholar

[38]

A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences: Technical Sciences, 56 (2008), 367-378. Google Scholar

[39]

A. Swierniak, Modelling combined anti-angiogenic and chemo-therapies, in: Proc. 14th National Conf. Appl. Math. Biol Medicine, Leszno, 2008,127-133.Google Scholar

[40]

A. Swierniak, Comparison of six models of anti-angiogenic therapy, Applicationes Mathematicae, 36 (2009), 333-348. doi: 10.4064/am36-3-6. Google Scholar

[41]

A. Swierniak, A. d'Onofrio and A. Gandolfi, Control problems related to tumor angiogenesis, Proc. of the 32nd Annual Conference on IEEE Industrial Electronics (IECON '06), Paris, 677-681, November 2006. doi: 10.1109/IECON.2006.347815. Google Scholar

[42]

A. Swierniak and J. Klamka, Local controllability of models of combined anticancer therapy with delays in control, Math. Model. Nat. Phenom., 9 (2014), 216-226. doi: 10.1051/mmnp/20149413. Google Scholar

[43]

L. S. TengK. T. JinK. F. HeH. H. WangJ. Cao and D. C. Yu, Advances in combination of anti-angiogenic agents targeting VEGF-binding and conventional chemotherapy and radiation for cancer treatment, Journal of the Chinese Medical Association, 73 (2010), 281-288. Google Scholar

[44]

The Internet Drug Index, (2015), http://www.rxlist.com/avastin-drug/clinical-pharmacology.htmlGoogle Scholar

[45]

US National Institutes of Health, Clinical Trials, (last updated June 2015), http://www.clinicaltrials.gov/ct2/show/NCT00520013Google Scholar

[46]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. Google Scholar

Figure 5.  Optimal solution for the Hahnfeldt model with logistic growth function and delays $h_1= 10.6$ in $u$ and $h=1.84 $ in $v$, objective $J(u) = p(T) + 0.5\, q(T)$ and fixed final time $T=16$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 40, w_{\rm max} = 320, v_{\rm max} = 2, \, z_{\rm max} = 10$. (a) control $u$, (b) control $v$, (c) tumor volume $p$ and vasculature $q$
Figure 1.  Optimal solution for the Hahnfeldt model with Gompertz-type growth function $f(p, q) = - \xi\, p \, \ln (p/q)$, objective $J(u) = p(T)$ and free terminal time $T$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 75, w_{\rm max} = 300, v_{\rm max} = 2, z_{\rm max} = 10$. (a) control $u$, (b) control $v$, (c) tumor volume $p$ and vasculature $q$
Figure 2.  Optimal solution for the Hahnfeldt model with logistic-type growth function $f(p, q)= \xi p (1-p/q)$, objective $J(u) = p(T) + 0.2\, q(T)$ and fixed final time $T=16$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 40, w_{\rm max} = 320, v_{\rm max} = 2, \, z_{\rm max} = 10$. (a) control $u$, (b) control $v$, (c) tumor volume $p$ and vasculature $q$
Figure 3.  Optimal solution for the Hahnfeldt model with logistic growth function$ f(p, q) = \xi p (1-p/q) $, objective $J(u) = p(T)$ and free terminal time $T$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 75, w_{\rm max} = 300, v_{\rm max} = 2, z_{\rm max} = 10$. Optimal control $v(t) \equiv 2$ and $T=5$. (a) control $u$ and switching function $\phi_u$ satisfying (41), (b) tumor volume $p$ and vasculature $q$
Figure 4.  Optimal solution for the Hahnfeldt model with logistic growth function $f(p, q) = \xi p (1-p/q) $, objective $J(u) = p(T) + 0.5 q(T)$ and free terminal time $T$. Initial conditions $p(0)=12000, \, q(0) = 15000$ and control bounds $u_{\rm max} = 75, w_{\rm max} = 300, v_{\rm max} = 2, z_{\rm max} = 10$. Optimal control $v(t) \equiv 2$ and $T=5$. (a) control $u$, (b) tumor volume $p$ and vasculature $q$
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