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A model of optimal dosing of antibiotic treatment in biofilm
1. | Department of Mathematics, Syed Babar Ali School of Science and Engineering, Lahore University of Management Sciences, Lahore, Pakistan |
2. | School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804 |
References:
[1] |
N. Abramzon, C. Joaquin, J. D. Bray and G. Brelles-Mario, Biofilm Destruction by RF High-Pressure Cold Plasma Jet, IEEE Trans. Plasma Science, 34 (2006), 1304-1308.
doi: 10.1109/TPS.2006.877515. |
[2] |
J. N. Anderl, M. J. Franklin and P. S. Stewart, Role of antibiotic penetration limitation in Klebsiella pneumoniae biofilm resistance to ampicillin and ciprofloxacin, Antimicrob Agents Chemotherapy, 44 (2000), 1818-1824.
doi: 10.1128/AAC.44.7.1818-1824.2000. |
[3] |
D. J. Austin, N. J. White and R. M. Anderson, The dynamics of drug action on the within-host population growth of infectious agents: melding pharmacokinetics with pathogen population dynamics, J. Theor. Biol., 194 (1998), 313-339.
doi: 10.1006/jtbi.1997.0438. |
[4] |
N. G. Cogan, R. Cortez and L. Fauci, Modeling physiological resistance in bacterial biofilms, B. Math. Biol., 67 (2005), 831-853.
doi: 10.1016/j.bulm.2004.11.001. |
[5] |
N. G. Cogan, Effects of persister formation on bacterial response to dosing, J. Theor. Biol., 238 (2006), 694-703.
doi: 10.1016/j.jtbi.2005.06.017. |
[6] |
N. G. Cogan, Incorporating toxin hypothesis into a mathematical model of persister formation and dynamics, J. Theor. Biol., 248(2) (2007), 340-349.
doi: 10.1016/j.jtbi.2007.05.021. |
[7] |
N. G. Cogan, J. S. Gunn and J. W. Daniel, Biofilms and infectious diseases: biology to mathematics and back again, EMS Microbiol. Lett., 322 (2011), 1-7.
doi: 10.1111/j.1574-6968.2011.02314.x. |
[8] |
N. G. Cogan, J. S. Gunn and J. W. Daniel, Optimal control strategies for disinfection of bacterial populations with persister/susceptible dynamics, Antimicrob Agents Chemotherapy, 248 (2012), 4816-4826.
doi: 10.1128/AAC.00675-12. |
[9] |
D. E. Corpet, S. Lumeau and F. Corpet, Minimum antibiotics levels for selecting a resistance plasmid in a gnotobiotic animal model, Antimicrob Agents Chemotherapy, 33 (1989), 535-540.
doi: 10.1128/AAC.33.4.535. |
[10] |
R. M. Cozens, E. Tuomanen, W. Tosch, O. Zak, J. Suter and A. Tomasz, Evaluation of the bactericidal activity of beta-lactam antibiotics on slowly growing bacteria cultured in the chemostat, Antimicrob Agents Chemotherapy, 29 (1986), 797-802.
doi: 10.1128/AAC.29.5.797. |
[11] |
W. A. Craig, Pharmacokinetics/pharmacodynamic parameters: rationale for antibacterial dosing of mice and men, Clinical Infectious Diseases, 26 (1998), 1-12.
doi: 10.1086/516284. |
[12] |
P. De Leenheer and N. G. Cogan, Failure of antibiotic treatment in microbial populations, J. Math. Biol., 59 (2009), 563-579.
doi: 10.1007/s00285-008-0243-6. |
[13] |
R. M. Donlan and J. W. Costerton, Biofilms: Survival mechanisms of clinically relevant microorganisms, Clin. Microbiol. Rev., 15(2) (2002), 167-193.
doi: 10.1128/CMR.15.2.167-193.2002. |
[14] |
G. D. Ehrlich, P. Stoodley, S. Kathju, S. Zhao, B. R. McLeod, N. Balaban, F. Z. Hu, G. N. Sotereanos, J. W. Costerton, P. S. Stewart and Q. Lin, Engineering approaches for the detection and control of orthopaedic biofilm infections, Clin. Orthop Relat. Res., 437 (2005), 59-66.
doi: 10.1097/00003086-200508000-00011. |
[15] |
K. Fister, S. Lenhart and J. McNally, Optimizing chemotherapy in an HIV model, E. J. Differential Equations, 32 (1998), 1-12. |
[16] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975. |
[17] |
E. L. Gillespie, J. L. Kuti, and D. P. Nicolau, Pharmacodynamics of antimicrobials: treatment optimisation, Expert Opin. Drug Metabolism and Toxi., 1 (2005), 351-361.
doi: 10.1517/17425255.1.3.351. |
[18] |
L. Hall-Stoodley, J. W. Costerton and P. Stoodley, Bacterial biofilms: From the environment to infectious disease, Nature Review Microbiology, 2 (2004), 95-108.
doi: 10.1038/nrmicro821. |
[19] |
J. Hofbauer and J. W.-H. So, Uniform persistence and repellors for maps, Proc. Amer. Math. Soc., 107 (1989), 1137-1142.
doi: 10.1090/S0002-9939-1989-0984816-4. |
[20] |
N. G. Holford and L. B. Sheiner, Kinetics of pharmacologic response, Pharmac. Ther., 16 (1982), 143-166.
doi: 10.1016/0163-7258(82)90051-1. |
[21] |
S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Mathematical Biosciences, 187 (2004), 53-91.
doi: 10.1016/j.mbs.2003.07.004. |
[22] |
M. Imran and H. L Smith, The pharmacodynamics of antibiotic treatment, Computational and Mathematical Methods in Medicine, 7 (2006), 229-263.
doi: 10.1080/10273660601122773. |
[23] |
M. Imran and H. L. Smith, A Mathematical Model of Gene Transfer in a Biofilm, Mathematics for Ecology and Environmental Sciences, Springer-Verlag, New York, 2007.
doi: 10.1007/978-3-540-34428-5_6. |
[24] |
M. Imran and H. L Smith, The dynamics of bacterial infection, innate immune, response and antibiotic treatmnet, Discrete and continous dynamical systems-series B, 8 (2007), 127-143.
doi: 10.3934/dcdsb.2007.8.127. |
[25] |
E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Sustems, 2 (2002), 473-482.
doi: 10.3934/dcdsb.2002.2.473. |
[26] |
D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792.
doi: 10.1007/s002850050076. |
[27] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995. |
[28] |
S. Lenhart and J. T. Workman, Forward-Backward Sweep Method, Chapman & Hall/CRC, Taylor & Francis Group, 2007 |
[29] |
R. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor, J. Theor. Biology, 122 (1986), 83-92.
doi: 10.1016/S0022-5193(86)80226-0. |
[30] |
B. R. Levin and K. I. Udekwu, Population Dynamics of Antibiotic treatment: Mathematical model and hypotheses for time-kill and continous culture experiments, Antimicrob. Agents Chemother., 54 (2010), 3414-3426.
doi: 10.1128/AAC.00381-10. |
[31] |
K. Lewis, Riddle of biofilm resistence, Antimicrob. Agents Chemother., 45 (2001), 999-1007.
doi: 10.1128/AAC.45.4.999-1007.2001. |
[32] |
D. M. Livermore, Antibiotic uptake and transport by bacteria, Scand. J. Infect. Dis. Suppl., 74 (1990), 15-22. |
[33] |
C. T. Mascio, J. D. Alder and J. A. Silverman, Bactericidal Action of Daptomycin against Stationary-Phase and Nondividing Staphylococcus aureus Cells, Antimicrob Agents Chemother., 51(12) (2007), 4255-4260.
doi: 10.1128/AAC.00824-07. |
[34] |
R. Pena-Miller, D. Laehnemann, H. Schulenburg, M. Ackermann and R. Beardmore, Selecting against drug-resistant pathogens: Optimal treatments in the presence of commensal bacteria, Bull. Math. Biol., 74 (2012), 908-934.
doi: 10.1007/s11538-011-9698-5. |
[35] |
R. Regoes, C. Wiuff, R. M. Zappala, N. Garner, F. Baquero and B. R. Levin, Pharmacodynamic functions: A multiparameter approach to the design of antibiotic treatment regimens, Antimicrob. Agents Chemother., 48 (2004), 3670-3676.
doi: 10.1128/AAC.48.10.3670-3676.2004. |
[36] |
M. Robert and P. S. Stewart, Modeling antibiotic tolerance in biofilms by accounting for nutrient limitation, Antimicrob. Agents Chemother., 48 (2004), 48-52.
doi: 10.1128/AAC.48.1.48-52.2004. |
[37] |
M. A. Ryder, Catheter-related infections: It's all about biofilm, Topics in Advanced Practice Nursing eJournal, 5 (2005). |
[38] |
H. L. Smith, On the existence and stability of bounded almost periodic and periodic solutions of a singularly perturbed nonautonomous system, Diff. and Integ. Equations, 8 (1995), 2125-2144. |
[39] |
P. S. Stewart, Biofilm accumulation model that predicts antibiotic resistance of Pseudomonas aeruginosa biofilms, Antimicrob Agents Chemotherapy, 38 (1994), 1052-1058.
doi: 10.1128/AAC.38.5.1052. |
[40] |
P. S. Stewart, Theoretical aspects of antibiotic diffusion into microbial biofilms, Antimicrob Agents Chemotherapy, 40 (1996), 2517-2522. |
[41] |
H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an epidemic model), SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[42] |
E. Tuomanen, Phenotypic tolerance: The search for beta-lactam antibiotics that kill nongrowing bacteria, Reviews of Infectious Disease, 8 (1986), 279-291. |
[43] |
E. Tuomanen, R. Cozens, W. Tosch, O. Zak and A. Tomasz, The rate of killing of Escherichia coli by beta-lactam antibiotics is strictly proportional to the rate of bacterial growth, Journal of General Microbiology, 132 (1986), 1297-1304. |
[44] |
C. Wiuff, R. M. Zappala, R. Regoes, K. Garner, F. Baquero and B. R. Levin, Phenotypic tolerance: antibiotic enrichment of noninherited resistance in bacterial populations, Antimicrob. Agents Chemotherapy, 49 (2005), 775-792.
doi: 10.1128/AAC.49.4.1483-1494.2005. |
[45] |
X. Yan and Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, World Journal of Modelling and Simulation, 47 (2008), 235-245.
doi: 10.1016/j.mcm.2007.04.003. |
[46] |
P. J. Yeh, M. J. Hegreness, A. P. Aiden and R. Kishony, Drug interactions and the evolution of antibiotic resistance, Nat. Rev., Microbiol., 7 (2009), 460-466.
doi: 10.1038/nrmicro2133. |
show all references
References:
[1] |
N. Abramzon, C. Joaquin, J. D. Bray and G. Brelles-Mario, Biofilm Destruction by RF High-Pressure Cold Plasma Jet, IEEE Trans. Plasma Science, 34 (2006), 1304-1308.
doi: 10.1109/TPS.2006.877515. |
[2] |
J. N. Anderl, M. J. Franklin and P. S. Stewart, Role of antibiotic penetration limitation in Klebsiella pneumoniae biofilm resistance to ampicillin and ciprofloxacin, Antimicrob Agents Chemotherapy, 44 (2000), 1818-1824.
doi: 10.1128/AAC.44.7.1818-1824.2000. |
[3] |
D. J. Austin, N. J. White and R. M. Anderson, The dynamics of drug action on the within-host population growth of infectious agents: melding pharmacokinetics with pathogen population dynamics, J. Theor. Biol., 194 (1998), 313-339.
doi: 10.1006/jtbi.1997.0438. |
[4] |
N. G. Cogan, R. Cortez and L. Fauci, Modeling physiological resistance in bacterial biofilms, B. Math. Biol., 67 (2005), 831-853.
doi: 10.1016/j.bulm.2004.11.001. |
[5] |
N. G. Cogan, Effects of persister formation on bacterial response to dosing, J. Theor. Biol., 238 (2006), 694-703.
doi: 10.1016/j.jtbi.2005.06.017. |
[6] |
N. G. Cogan, Incorporating toxin hypothesis into a mathematical model of persister formation and dynamics, J. Theor. Biol., 248(2) (2007), 340-349.
doi: 10.1016/j.jtbi.2007.05.021. |
[7] |
N. G. Cogan, J. S. Gunn and J. W. Daniel, Biofilms and infectious diseases: biology to mathematics and back again, EMS Microbiol. Lett., 322 (2011), 1-7.
doi: 10.1111/j.1574-6968.2011.02314.x. |
[8] |
N. G. Cogan, J. S. Gunn and J. W. Daniel, Optimal control strategies for disinfection of bacterial populations with persister/susceptible dynamics, Antimicrob Agents Chemotherapy, 248 (2012), 4816-4826.
doi: 10.1128/AAC.00675-12. |
[9] |
D. E. Corpet, S. Lumeau and F. Corpet, Minimum antibiotics levels for selecting a resistance plasmid in a gnotobiotic animal model, Antimicrob Agents Chemotherapy, 33 (1989), 535-540.
doi: 10.1128/AAC.33.4.535. |
[10] |
R. M. Cozens, E. Tuomanen, W. Tosch, O. Zak, J. Suter and A. Tomasz, Evaluation of the bactericidal activity of beta-lactam antibiotics on slowly growing bacteria cultured in the chemostat, Antimicrob Agents Chemotherapy, 29 (1986), 797-802.
doi: 10.1128/AAC.29.5.797. |
[11] |
W. A. Craig, Pharmacokinetics/pharmacodynamic parameters: rationale for antibacterial dosing of mice and men, Clinical Infectious Diseases, 26 (1998), 1-12.
doi: 10.1086/516284. |
[12] |
P. De Leenheer and N. G. Cogan, Failure of antibiotic treatment in microbial populations, J. Math. Biol., 59 (2009), 563-579.
doi: 10.1007/s00285-008-0243-6. |
[13] |
R. M. Donlan and J. W. Costerton, Biofilms: Survival mechanisms of clinically relevant microorganisms, Clin. Microbiol. Rev., 15(2) (2002), 167-193.
doi: 10.1128/CMR.15.2.167-193.2002. |
[14] |
G. D. Ehrlich, P. Stoodley, S. Kathju, S. Zhao, B. R. McLeod, N. Balaban, F. Z. Hu, G. N. Sotereanos, J. W. Costerton, P. S. Stewart and Q. Lin, Engineering approaches for the detection and control of orthopaedic biofilm infections, Clin. Orthop Relat. Res., 437 (2005), 59-66.
doi: 10.1097/00003086-200508000-00011. |
[15] |
K. Fister, S. Lenhart and J. McNally, Optimizing chemotherapy in an HIV model, E. J. Differential Equations, 32 (1998), 1-12. |
[16] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975. |
[17] |
E. L. Gillespie, J. L. Kuti, and D. P. Nicolau, Pharmacodynamics of antimicrobials: treatment optimisation, Expert Opin. Drug Metabolism and Toxi., 1 (2005), 351-361.
doi: 10.1517/17425255.1.3.351. |
[18] |
L. Hall-Stoodley, J. W. Costerton and P. Stoodley, Bacterial biofilms: From the environment to infectious disease, Nature Review Microbiology, 2 (2004), 95-108.
doi: 10.1038/nrmicro821. |
[19] |
J. Hofbauer and J. W.-H. So, Uniform persistence and repellors for maps, Proc. Amer. Math. Soc., 107 (1989), 1137-1142.
doi: 10.1090/S0002-9939-1989-0984816-4. |
[20] |
N. G. Holford and L. B. Sheiner, Kinetics of pharmacologic response, Pharmac. Ther., 16 (1982), 143-166.
doi: 10.1016/0163-7258(82)90051-1. |
[21] |
S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Mathematical Biosciences, 187 (2004), 53-91.
doi: 10.1016/j.mbs.2003.07.004. |
[22] |
M. Imran and H. L Smith, The pharmacodynamics of antibiotic treatment, Computational and Mathematical Methods in Medicine, 7 (2006), 229-263.
doi: 10.1080/10273660601122773. |
[23] |
M. Imran and H. L. Smith, A Mathematical Model of Gene Transfer in a Biofilm, Mathematics for Ecology and Environmental Sciences, Springer-Verlag, New York, 2007.
doi: 10.1007/978-3-540-34428-5_6. |
[24] |
M. Imran and H. L Smith, The dynamics of bacterial infection, innate immune, response and antibiotic treatmnet, Discrete and continous dynamical systems-series B, 8 (2007), 127-143.
doi: 10.3934/dcdsb.2007.8.127. |
[25] |
E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Sustems, 2 (2002), 473-482.
doi: 10.3934/dcdsb.2002.2.473. |
[26] |
D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792.
doi: 10.1007/s002850050076. |
[27] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995. |
[28] |
S. Lenhart and J. T. Workman, Forward-Backward Sweep Method, Chapman & Hall/CRC, Taylor & Francis Group, 2007 |
[29] |
R. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor, J. Theor. Biology, 122 (1986), 83-92.
doi: 10.1016/S0022-5193(86)80226-0. |
[30] |
B. R. Levin and K. I. Udekwu, Population Dynamics of Antibiotic treatment: Mathematical model and hypotheses for time-kill and continous culture experiments, Antimicrob. Agents Chemother., 54 (2010), 3414-3426.
doi: 10.1128/AAC.00381-10. |
[31] |
K. Lewis, Riddle of biofilm resistence, Antimicrob. Agents Chemother., 45 (2001), 999-1007.
doi: 10.1128/AAC.45.4.999-1007.2001. |
[32] |
D. M. Livermore, Antibiotic uptake and transport by bacteria, Scand. J. Infect. Dis. Suppl., 74 (1990), 15-22. |
[33] |
C. T. Mascio, J. D. Alder and J. A. Silverman, Bactericidal Action of Daptomycin against Stationary-Phase and Nondividing Staphylococcus aureus Cells, Antimicrob Agents Chemother., 51(12) (2007), 4255-4260.
doi: 10.1128/AAC.00824-07. |
[34] |
R. Pena-Miller, D. Laehnemann, H. Schulenburg, M. Ackermann and R. Beardmore, Selecting against drug-resistant pathogens: Optimal treatments in the presence of commensal bacteria, Bull. Math. Biol., 74 (2012), 908-934.
doi: 10.1007/s11538-011-9698-5. |
[35] |
R. Regoes, C. Wiuff, R. M. Zappala, N. Garner, F. Baquero and B. R. Levin, Pharmacodynamic functions: A multiparameter approach to the design of antibiotic treatment regimens, Antimicrob. Agents Chemother., 48 (2004), 3670-3676.
doi: 10.1128/AAC.48.10.3670-3676.2004. |
[36] |
M. Robert and P. S. Stewart, Modeling antibiotic tolerance in biofilms by accounting for nutrient limitation, Antimicrob. Agents Chemother., 48 (2004), 48-52.
doi: 10.1128/AAC.48.1.48-52.2004. |
[37] |
M. A. Ryder, Catheter-related infections: It's all about biofilm, Topics in Advanced Practice Nursing eJournal, 5 (2005). |
[38] |
H. L. Smith, On the existence and stability of bounded almost periodic and periodic solutions of a singularly perturbed nonautonomous system, Diff. and Integ. Equations, 8 (1995), 2125-2144. |
[39] |
P. S. Stewart, Biofilm accumulation model that predicts antibiotic resistance of Pseudomonas aeruginosa biofilms, Antimicrob Agents Chemotherapy, 38 (1994), 1052-1058.
doi: 10.1128/AAC.38.5.1052. |
[40] |
P. S. Stewart, Theoretical aspects of antibiotic diffusion into microbial biofilms, Antimicrob Agents Chemotherapy, 40 (1996), 2517-2522. |
[41] |
H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an epidemic model), SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[42] |
E. Tuomanen, Phenotypic tolerance: The search for beta-lactam antibiotics that kill nongrowing bacteria, Reviews of Infectious Disease, 8 (1986), 279-291. |
[43] |
E. Tuomanen, R. Cozens, W. Tosch, O. Zak and A. Tomasz, The rate of killing of Escherichia coli by beta-lactam antibiotics is strictly proportional to the rate of bacterial growth, Journal of General Microbiology, 132 (1986), 1297-1304. |
[44] |
C. Wiuff, R. M. Zappala, R. Regoes, K. Garner, F. Baquero and B. R. Levin, Phenotypic tolerance: antibiotic enrichment of noninherited resistance in bacterial populations, Antimicrob. Agents Chemotherapy, 49 (2005), 775-792.
doi: 10.1128/AAC.49.4.1483-1494.2005. |
[45] |
X. Yan and Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, World Journal of Modelling and Simulation, 47 (2008), 235-245.
doi: 10.1016/j.mcm.2007.04.003. |
[46] |
P. J. Yeh, M. J. Hegreness, A. P. Aiden and R. Kishony, Drug interactions and the evolution of antibiotic resistance, Nat. Rev., Microbiol., 7 (2009), 460-466.
doi: 10.1038/nrmicro2133. |
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