2012, 9(4): 899-914. doi: 10.3934/mbe.2012.9.899

Hybrid optimal control for HIV multi-drug therapies: A finite set control transcription approach

1. 

Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712, United States

2. 

School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, United States

Received  August 2011 Revised  March 2012 Published  October 2012

In this study, the treatment of Human Immunodeficiency Virus (HIV) infection is investigated through an optimal structured treatment interruption (STI) schedule of two classes of antiretroviral drugs, mainly, reverse transcriptase inhibitors and protease inhibitors. An STI treatment strategy may be beneficial in lowering the risk of HIV mutating to drug-resistant strains, and could provide patients with respite from toxic side effects of HAART. A shorter treatment period is considered compared to previous studies and the solution to the HIV STI problem is obtained via the Finite Set Control Transcription (FSCT) formulation. The FSCT formulation offers a unique approach for handling multiple independent decision variables simultaneously, and, as is shown by the results of this study, is well-suited for an effective treatment of the optimal STI problem. The results obtained in the present investigation demonstrate that immune boosting and subsequent natural suppression of the viral load are possible even when a reduced STI therapy treatment duration is in consideration.
Citation: Divya Thakur, Belinda Marchand. Hybrid optimal control for HIV multi-drug therapies: A finite set control transcription approach. Mathematical Biosciences & Engineering, 2012, 9 (4) : 899-914. doi: 10.3934/mbe.2012.9.899
References:
[1]

B. M. Adams, H. T. Banks, M. Davidian, H. Kwon, H. T. Tran and S. N. Wynne, HIV dynamics: Modeling, data analysis, and optimal treatment protocols,, Journal of Computational and Applied Mathematics, 184 (2005), 10.  doi: 10.1016/j.cam.2005.02.004.  Google Scholar

[2]

B. M. Adams, H. T. Banks, H. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: optimal and STI control approaches,, Mathematical Biosciences and Engineering, 1 (2004), 223.   Google Scholar

[3]

R. J. Allgor and P. I. Barton, Mixed-integer dynamic optimization I: Problem formulation,, Computers and Chemical Engineering, 23 (1999), 567.  doi: 10.1016/S0098-1354(98)00294-4.  Google Scholar

[4]

J. T. Betts, Survey of numerical methods for trajectory optimization,, Journal of Guidance, 21 (1998), 193.   Google Scholar

[5]

J. T. Betts, "Practical Methods for Optimal Control Using Nonlinear Quadratic Programming,", Society of Industrial and Applied Mathematics, (2001).   Google Scholar

[6]

S. Bonhoeffer, M. Rembiszewski, B. M. Ortiz and D. F. Nixon, Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infections,, AIDS, 14 (2000), 2313.  doi: 10.1097/00002030-200010200-00012.  Google Scholar

[7]

M. A. L. Caetano and T. Yoneyama, Short and long period optimization of drug doses in the treatment of AIDS,, Anais da Academia Brasileira de Ciéncias (Annals of the Brazilian Academy of Sciences), 74 (2002), 379.   Google Scholar

[8]

D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads,, Bulletin of Mathematical Biology, 238 (2001), 29.   Google Scholar

[9]

N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics,, Journal of Mathematical Analysis and Applications, 341 (2008), 1084.   Google Scholar

[10]

M. A. Duran and I. E. Grossmann, An outer-approximation algorithm for a class of mixed-integer nonlinear programs,, Mathematical Programming, 36 (1986), 307.  doi: 10.1007/BF02592064.  Google Scholar

[11]

K. R. Fister and J. C. Panetta, Optimal control applied to cell-cycle-specific cancer chemotherapy,, SIAM Journal on Applied Mathematics, 3 (2000), 1059.   Google Scholar

[12]

R. Fletcher and S. Leyffer, Solving mixed integer nonlinear programs by outer approximation,, Mathematical Programming, 66 (1994), 327.  doi: 10.1007/BF01581153.  Google Scholar

[13]

C. A. Floudas, "Nonlinear and Mixed-Integer Optimization,", Oxford University Press, (1995).   Google Scholar

[14]

A. M. Geoffrion, Generalized benders decomposition,, Journal of Optimization Theory and Applications, 10 (1972), 237.  doi: 10.1007/BF00934810.  Google Scholar

[15]

M. Gerdts, Solving mixed-integer optimal control problems by branch & bound: A case study from automobile test-driving with gear shift,, Optimal Control Applications and Methods, 26 (2005), 1.  doi: 10.1002/oca.751.  Google Scholar

[16]

M. Gerdts, A variable time transformation method for mixed-integer optimal control problems,, Optimal Control Applications and Methods, 27 (2006), 169.  doi: 10.1002/oca.778.  Google Scholar

[17]

C. R. Hargraves and S. W. Paris, Direct trajectory optimization using nonlinear Pprogramming and collocation,, Journal of Guidance, 10 (1987), 338.   Google Scholar

[18]

D. Kirschner, S. Lenhart and S. Serbin, A model for treatment strategy in the chemotherapy of AIDS,, Bulletin of Mathematical Biology, 58 (1996), 367.   Google Scholar

[19]

J. J. Kutch and P. Gurfil, Optimal control of HIV infection with a continuously mutating viral population,, Proceedings of the 2002 American Control Conference, 5 (2002), 4033.   Google Scholar

[20]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for optimal discrete-valued control problems,, Automatica, 35 (1999), 1401.  doi: 10.1016/S0005-1098(99)00050-3.  Google Scholar

[21]

F. Neri, J. Toivanen and R. A. E. Mäkinen, An adaptive evolutionary algorithm with intelligent mutation local searchers for designing multidrug therapies for HIV,, Applied Intelligence, 27 (2007), 219.   Google Scholar

[22]

F. Neri, J. Toivanen, G. L. Cascella and Y. Ong, An adaptive multimeme algorithm for designing HIV multidrug therapies,, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 4 (2007), 264.   Google Scholar

[23]

S. Sager, H. G. Bock, M. Diehl, G. Reinelt and Schlöder, A variable time transformation method for mixed-integer optimal control problems,, in, (2006), 269.   Google Scholar

[24]

S. A. Stanton and B. G. Marchand, Finite set control transcription for optimal control applications,, Journal of Spacecraft and Rockets, 47 (2010), 457.  doi: 10.2514/1.44056.  Google Scholar

[25]

S. A. Stanton, "Finite Set Control Transcription for Optimal Control Applications,", Ph.D thesis, (2009).   Google Scholar

[26]

O. Stryk and M. Glocker, Numerical mixed-integer optimal control and motorized traveling salesmen problems,, Journal Europeen des Systemes Automatises, 35 (2001), 519.   Google Scholar

[27]

W. Tan and Z. Xiang, Some state space models of HIV pathogenesis under treatment by anti-viral drug in HIV-infected individuals,, Mathematical Biosciences, 156 (1999), 69.   Google Scholar

[28]

H. C. Tuckwell and E. Le Corfec, A stochastic model for early HIV-1 population dynamics,, Mathematical Biosciences, 195 (1998), 451.   Google Scholar

[29]

S. Wei, K. Uthaichana, M. Žefran, R. A. DeCaralo and S. Bengea, Applications of numerical optimal control to nonlinear hybrid systems,, Nonlinear Analysis: Hybrid Systems, 1 (2007), 264.   Google Scholar

[30]

L. M. Wein, S. A. Zenios and M. A. Nowak, Dynamic multidrug therapies for HIV: a control theoretic approach,, Journal of Theoretical Biology, 185 (1997), 15.   Google Scholar

[31]

R. Zurakowski and A. R. Teel, A model predictive control based scheduling method for HIV therapy,, Journal of Theoretical Biology, 238 (2006), 368.   Google Scholar

show all references

References:
[1]

B. M. Adams, H. T. Banks, M. Davidian, H. Kwon, H. T. Tran and S. N. Wynne, HIV dynamics: Modeling, data analysis, and optimal treatment protocols,, Journal of Computational and Applied Mathematics, 184 (2005), 10.  doi: 10.1016/j.cam.2005.02.004.  Google Scholar

[2]

B. M. Adams, H. T. Banks, H. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: optimal and STI control approaches,, Mathematical Biosciences and Engineering, 1 (2004), 223.   Google Scholar

[3]

R. J. Allgor and P. I. Barton, Mixed-integer dynamic optimization I: Problem formulation,, Computers and Chemical Engineering, 23 (1999), 567.  doi: 10.1016/S0098-1354(98)00294-4.  Google Scholar

[4]

J. T. Betts, Survey of numerical methods for trajectory optimization,, Journal of Guidance, 21 (1998), 193.   Google Scholar

[5]

J. T. Betts, "Practical Methods for Optimal Control Using Nonlinear Quadratic Programming,", Society of Industrial and Applied Mathematics, (2001).   Google Scholar

[6]

S. Bonhoeffer, M. Rembiszewski, B. M. Ortiz and D. F. Nixon, Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infections,, AIDS, 14 (2000), 2313.  doi: 10.1097/00002030-200010200-00012.  Google Scholar

[7]

M. A. L. Caetano and T. Yoneyama, Short and long period optimization of drug doses in the treatment of AIDS,, Anais da Academia Brasileira de Ciéncias (Annals of the Brazilian Academy of Sciences), 74 (2002), 379.   Google Scholar

[8]

D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads,, Bulletin of Mathematical Biology, 238 (2001), 29.   Google Scholar

[9]

N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics,, Journal of Mathematical Analysis and Applications, 341 (2008), 1084.   Google Scholar

[10]

M. A. Duran and I. E. Grossmann, An outer-approximation algorithm for a class of mixed-integer nonlinear programs,, Mathematical Programming, 36 (1986), 307.  doi: 10.1007/BF02592064.  Google Scholar

[11]

K. R. Fister and J. C. Panetta, Optimal control applied to cell-cycle-specific cancer chemotherapy,, SIAM Journal on Applied Mathematics, 3 (2000), 1059.   Google Scholar

[12]

R. Fletcher and S. Leyffer, Solving mixed integer nonlinear programs by outer approximation,, Mathematical Programming, 66 (1994), 327.  doi: 10.1007/BF01581153.  Google Scholar

[13]

C. A. Floudas, "Nonlinear and Mixed-Integer Optimization,", Oxford University Press, (1995).   Google Scholar

[14]

A. M. Geoffrion, Generalized benders decomposition,, Journal of Optimization Theory and Applications, 10 (1972), 237.  doi: 10.1007/BF00934810.  Google Scholar

[15]

M. Gerdts, Solving mixed-integer optimal control problems by branch & bound: A case study from automobile test-driving with gear shift,, Optimal Control Applications and Methods, 26 (2005), 1.  doi: 10.1002/oca.751.  Google Scholar

[16]

M. Gerdts, A variable time transformation method for mixed-integer optimal control problems,, Optimal Control Applications and Methods, 27 (2006), 169.  doi: 10.1002/oca.778.  Google Scholar

[17]

C. R. Hargraves and S. W. Paris, Direct trajectory optimization using nonlinear Pprogramming and collocation,, Journal of Guidance, 10 (1987), 338.   Google Scholar

[18]

D. Kirschner, S. Lenhart and S. Serbin, A model for treatment strategy in the chemotherapy of AIDS,, Bulletin of Mathematical Biology, 58 (1996), 367.   Google Scholar

[19]

J. J. Kutch and P. Gurfil, Optimal control of HIV infection with a continuously mutating viral population,, Proceedings of the 2002 American Control Conference, 5 (2002), 4033.   Google Scholar

[20]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for optimal discrete-valued control problems,, Automatica, 35 (1999), 1401.  doi: 10.1016/S0005-1098(99)00050-3.  Google Scholar

[21]

F. Neri, J. Toivanen and R. A. E. Mäkinen, An adaptive evolutionary algorithm with intelligent mutation local searchers for designing multidrug therapies for HIV,, Applied Intelligence, 27 (2007), 219.   Google Scholar

[22]

F. Neri, J. Toivanen, G. L. Cascella and Y. Ong, An adaptive multimeme algorithm for designing HIV multidrug therapies,, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 4 (2007), 264.   Google Scholar

[23]

S. Sager, H. G. Bock, M. Diehl, G. Reinelt and Schlöder, A variable time transformation method for mixed-integer optimal control problems,, in, (2006), 269.   Google Scholar

[24]

S. A. Stanton and B. G. Marchand, Finite set control transcription for optimal control applications,, Journal of Spacecraft and Rockets, 47 (2010), 457.  doi: 10.2514/1.44056.  Google Scholar

[25]

S. A. Stanton, "Finite Set Control Transcription for Optimal Control Applications,", Ph.D thesis, (2009).   Google Scholar

[26]

O. Stryk and M. Glocker, Numerical mixed-integer optimal control and motorized traveling salesmen problems,, Journal Europeen des Systemes Automatises, 35 (2001), 519.   Google Scholar

[27]

W. Tan and Z. Xiang, Some state space models of HIV pathogenesis under treatment by anti-viral drug in HIV-infected individuals,, Mathematical Biosciences, 156 (1999), 69.   Google Scholar

[28]

H. C. Tuckwell and E. Le Corfec, A stochastic model for early HIV-1 population dynamics,, Mathematical Biosciences, 195 (1998), 451.   Google Scholar

[29]

S. Wei, K. Uthaichana, M. Žefran, R. A. DeCaralo and S. Bengea, Applications of numerical optimal control to nonlinear hybrid systems,, Nonlinear Analysis: Hybrid Systems, 1 (2007), 264.   Google Scholar

[30]

L. M. Wein, S. A. Zenios and M. A. Nowak, Dynamic multidrug therapies for HIV: a control theoretic approach,, Journal of Theoretical Biology, 185 (1997), 15.   Google Scholar

[31]

R. Zurakowski and A. R. Teel, A model predictive control based scheduling method for HIV therapy,, Journal of Theoretical Biology, 238 (2006), 368.   Google Scholar

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