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.

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

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.

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

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