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2014, 11(3): 547-571. doi: 10.3934/mbe.2014.11.547

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

Received  December 2012 Revised  June 2013 Published  January 2014

Biofilms are heterogeneous matrix enclosed micro-colonies of bacteria mostly found on moist surfaces. Biofilm formation is the primary cause of several persistent infections found in humans. We derive a mathematical model of biofilm and surrounding fluid dynamics to investigate the effect of a periodic dose of antibiotic on elimination of microbial population from biofilm. The growth rate of bacteria in biofilm is taken as Monod type for the limiting nutrient. The pharmacodynamics function is taken to be dependent both on limiting nutrient and antibiotic concentration. Assuming that flow rate of fluid compartment is large enough, we reduce the six dimensional model to a three dimensional model. Mathematically rigorous results are derived providing sufficient conditions for treatment success. Persistence theory is used to derive conditions under which the periodic solution for treatment failure is obtained. We also discuss the phenomenon of bi-stability where both infection-free state and infection state are locally stable when antibiotic dosing is marginal. In addition, we derive the optimal antibiotic application protocols for different scenarios using control theory and show that such treatments ensure bacteria elimination for a wide variety of cases. The results show that bacteria are successfully eliminated if the discrete treatment is given at an early stage in the infection or if the optimal protocol is adopted. Finally, we examine factors which if changed can result in treatment success of the previously treatment failure cases for the non-optimal technique.
Citation: Mudassar Imran, Hal L. Smith. A model of optimal dosing of antibiotic treatment in biofilm. Mathematical Biosciences & Engineering, 2014, 11 (3) : 547-571. doi: 10.3934/mbe.2014.11.547
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. doi: 10.1109/TPS.2006.877515. Google Scholar

[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. doi: 10.1128/AAC.44.7.1818-1824.2000. Google Scholar

[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. doi: 10.1006/jtbi.1997.0438. Google Scholar

[4]

N. G. Cogan, R. Cortez and L. Fauci, Modeling physiological resistance in bacterial biofilms,, B. Math. Biol., 67 (2005), 831. doi: 10.1016/j.bulm.2004.11.001. Google Scholar

[5]

N. G. Cogan, Effects of persister formation on bacterial response to dosing,, J. Theor. Biol., 238 (2006), 694. doi: 10.1016/j.jtbi.2005.06.017. Google Scholar

[6]

N. G. Cogan, Incorporating toxin hypothesis into a mathematical model of persister formation and dynamics,, J. Theor. Biol., 248 (2007), 340. doi: 10.1016/j.jtbi.2007.05.021. Google Scholar

[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. doi: 10.1111/j.1574-6968.2011.02314.x. Google Scholar

[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. doi: 10.1128/AAC.00675-12. Google Scholar

[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. doi: 10.1128/AAC.33.4.535. Google Scholar

[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. doi: 10.1128/AAC.29.5.797. Google Scholar

[11]

W. A. Craig, Pharmacokinetics/pharmacodynamic parameters: rationale for antibacterial dosing of mice and men,, Clinical Infectious Diseases, 26 (1998), 1. doi: 10.1086/516284. Google Scholar

[12]

P. De Leenheer and N. G. Cogan, Failure of antibiotic treatment in microbial populations,, J. Math. Biol., 59 (2009), 563. doi: 10.1007/s00285-008-0243-6. Google Scholar

[13]

R. M. Donlan and J. W. Costerton, Biofilms: Survival mechanisms of clinically relevant microorganisms,, Clin. Microbiol. Rev., 15 (2002), 167. doi: 10.1128/CMR.15.2.167-193.2002. Google Scholar

[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. doi: 10.1097/00003086-200508000-00011. Google Scholar

[15]

K. Fister, S. Lenhart and J. McNally, Optimizing chemotherapy in an HIV model,, E. J. Differential Equations, 32 (1998), 1. Google Scholar

[16]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Springer-Verlag, (1975). Google Scholar

[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. doi: 10.1517/17425255.1.3.351. Google Scholar

[18]

L. Hall-Stoodley, J. W. Costerton and P. Stoodley, Bacterial biofilms: From the environment to infectious disease,, Nature Review Microbiology, 2 (2004), 95. doi: 10.1038/nrmicro821. Google Scholar

[19]

J. Hofbauer and J. W.-H. So, Uniform persistence and repellors for maps,, Proc. Amer. Math. Soc., 107 (1989), 1137. doi: 10.1090/S0002-9939-1989-0984816-4. Google Scholar

[20]

N. G. Holford and L. B. Sheiner, Kinetics of pharmacologic response,, Pharmac. Ther., 16 (1982), 143. doi: 10.1016/0163-7258(82)90051-1. Google Scholar

[21]

S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor,, Mathematical Biosciences, 187 (2004), 53. doi: 10.1016/j.mbs.2003.07.004. Google Scholar

[22]

M. Imran and H. L Smith, The pharmacodynamics of antibiotic treatment,, Computational and Mathematical Methods in Medicine, 7 (2006), 229. doi: 10.1080/10273660601122773. Google Scholar

[23]

M. Imran and H. L. Smith, A Mathematical Model of Gene Transfer in a Biofilm,, Mathematics for Ecology and Environmental Sciences, (2007). doi: 10.1007/978-3-540-34428-5_6. Google Scholar

[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. doi: 10.3934/dcdsb.2007.8.127. Google Scholar

[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. doi: 10.3934/dcdsb.2002.2.473. Google Scholar

[26]

D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV,, J. Math. Biol., 35 (1997), 775. doi: 10.1007/s002850050076. Google Scholar

[27]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory,, Springer-Verlag, (1995). Google Scholar

[28]

S. Lenhart and J. T. Workman, Forward-Backward Sweep Method,, Chapman & Hall/CRC, (2007). Google Scholar

[29]

R. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor,, J. Theor. Biology, 122 (1986), 83. doi: 10.1016/S0022-5193(86)80226-0. Google Scholar

[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. doi: 10.1128/AAC.00381-10. Google Scholar

[31]

K. Lewis, Riddle of biofilm resistence,, Antimicrob. Agents Chemother., 45 (2001), 999. doi: 10.1128/AAC.45.4.999-1007.2001. Google Scholar

[32]

D. M. Livermore, Antibiotic uptake and transport by bacteria,, Scand. J. Infect. Dis. Suppl., 74 (1990), 15. Google Scholar

[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 (2007), 4255. doi: 10.1128/AAC.00824-07. Google Scholar

[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. doi: 10.1007/s11538-011-9698-5. Google Scholar

[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. doi: 10.1128/AAC.48.10.3670-3676.2004. Google Scholar

[36]

M. Robert and P. S. Stewart, Modeling antibiotic tolerance in biofilms by accounting for nutrient limitation,, Antimicrob. Agents Chemother., 48 (2004), 48. doi: 10.1128/AAC.48.1.48-52.2004. Google Scholar

[37]

M. A. Ryder, Catheter-related infections: It's all about biofilm,, Topics in Advanced Practice Nursing eJournal, 5 (2005). Google Scholar

[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. Google Scholar

[39]

P. S. Stewart, Biofilm accumulation model that predicts antibiotic resistance of Pseudomonas aeruginosa biofilms,, Antimicrob Agents Chemotherapy, 38 (1994), 1052. doi: 10.1128/AAC.38.5.1052. Google Scholar

[40]

P. S. Stewart, Theoretical aspects of antibiotic diffusion into microbial biofilms,, Antimicrob Agents Chemotherapy, 40 (1996), 2517. Google Scholar

[41]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an epidemic model),, SIAM J. Math. Anal., 24 (1993), 407. doi: 10.1137/0524026. Google Scholar

[42]

E. Tuomanen, Phenotypic tolerance: The search for beta-lactam antibiotics that kill nongrowing bacteria,, Reviews of Infectious Disease, 8 (1986), 279. Google Scholar

[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. Google Scholar

[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. doi: 10.1128/AAC.49.4.1483-1494.2005. Google Scholar

[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. doi: 10.1016/j.mcm.2007.04.003. Google Scholar

[46]

P. J. Yeh, M. J. Hegreness, A. P. Aiden and R. Kishony, Drug interactions and the evolution of antibiotic resistance,, Nat. Rev., 7 (2009), 460. doi: 10.1038/nrmicro2133. Google Scholar

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. doi: 10.1109/TPS.2006.877515. Google Scholar

[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. doi: 10.1128/AAC.44.7.1818-1824.2000. Google Scholar

[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. doi: 10.1006/jtbi.1997.0438. Google Scholar

[4]

N. G. Cogan, R. Cortez and L. Fauci, Modeling physiological resistance in bacterial biofilms,, B. Math. Biol., 67 (2005), 831. doi: 10.1016/j.bulm.2004.11.001. Google Scholar

[5]

N. G. Cogan, Effects of persister formation on bacterial response to dosing,, J. Theor. Biol., 238 (2006), 694. doi: 10.1016/j.jtbi.2005.06.017. Google Scholar

[6]

N. G. Cogan, Incorporating toxin hypothesis into a mathematical model of persister formation and dynamics,, J. Theor. Biol., 248 (2007), 340. doi: 10.1016/j.jtbi.2007.05.021. Google Scholar

[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. doi: 10.1111/j.1574-6968.2011.02314.x. Google Scholar

[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. doi: 10.1128/AAC.00675-12. Google Scholar

[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. doi: 10.1128/AAC.33.4.535. Google Scholar

[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. doi: 10.1128/AAC.29.5.797. Google Scholar

[11]

W. A. Craig, Pharmacokinetics/pharmacodynamic parameters: rationale for antibacterial dosing of mice and men,, Clinical Infectious Diseases, 26 (1998), 1. doi: 10.1086/516284. Google Scholar

[12]

P. De Leenheer and N. G. Cogan, Failure of antibiotic treatment in microbial populations,, J. Math. Biol., 59 (2009), 563. doi: 10.1007/s00285-008-0243-6. Google Scholar

[13]

R. M. Donlan and J. W. Costerton, Biofilms: Survival mechanisms of clinically relevant microorganisms,, Clin. Microbiol. Rev., 15 (2002), 167. doi: 10.1128/CMR.15.2.167-193.2002. Google Scholar

[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. doi: 10.1097/00003086-200508000-00011. Google Scholar

[15]

K. Fister, S. Lenhart and J. McNally, Optimizing chemotherapy in an HIV model,, E. J. Differential Equations, 32 (1998), 1. Google Scholar

[16]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Springer-Verlag, (1975). Google Scholar

[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. doi: 10.1517/17425255.1.3.351. Google Scholar

[18]

L. Hall-Stoodley, J. W. Costerton and P. Stoodley, Bacterial biofilms: From the environment to infectious disease,, Nature Review Microbiology, 2 (2004), 95. doi: 10.1038/nrmicro821. Google Scholar

[19]

J. Hofbauer and J. W.-H. So, Uniform persistence and repellors for maps,, Proc. Amer. Math. Soc., 107 (1989), 1137. doi: 10.1090/S0002-9939-1989-0984816-4. Google Scholar

[20]

N. G. Holford and L. B. Sheiner, Kinetics of pharmacologic response,, Pharmac. Ther., 16 (1982), 143. doi: 10.1016/0163-7258(82)90051-1. Google Scholar

[21]

S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor,, Mathematical Biosciences, 187 (2004), 53. doi: 10.1016/j.mbs.2003.07.004. Google Scholar

[22]

M. Imran and H. L Smith, The pharmacodynamics of antibiotic treatment,, Computational and Mathematical Methods in Medicine, 7 (2006), 229. doi: 10.1080/10273660601122773. Google Scholar

[23]

M. Imran and H. L. Smith, A Mathematical Model of Gene Transfer in a Biofilm,, Mathematics for Ecology and Environmental Sciences, (2007). doi: 10.1007/978-3-540-34428-5_6. Google Scholar

[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. doi: 10.3934/dcdsb.2007.8.127. Google Scholar

[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. doi: 10.3934/dcdsb.2002.2.473. Google Scholar

[26]

D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV,, J. Math. Biol., 35 (1997), 775. doi: 10.1007/s002850050076. Google Scholar

[27]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory,, Springer-Verlag, (1995). Google Scholar

[28]

S. Lenhart and J. T. Workman, Forward-Backward Sweep Method,, Chapman & Hall/CRC, (2007). Google Scholar

[29]

R. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor,, J. Theor. Biology, 122 (1986), 83. doi: 10.1016/S0022-5193(86)80226-0. Google Scholar

[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. doi: 10.1128/AAC.00381-10. Google Scholar

[31]

K. Lewis, Riddle of biofilm resistence,, Antimicrob. Agents Chemother., 45 (2001), 999. doi: 10.1128/AAC.45.4.999-1007.2001. Google Scholar

[32]

D. M. Livermore, Antibiotic uptake and transport by bacteria,, Scand. J. Infect. Dis. Suppl., 74 (1990), 15. Google Scholar

[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 (2007), 4255. doi: 10.1128/AAC.00824-07. Google Scholar

[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. doi: 10.1007/s11538-011-9698-5. Google Scholar

[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. doi: 10.1128/AAC.48.10.3670-3676.2004. Google Scholar

[36]

M. Robert and P. S. Stewart, Modeling antibiotic tolerance in biofilms by accounting for nutrient limitation,, Antimicrob. Agents Chemother., 48 (2004), 48. doi: 10.1128/AAC.48.1.48-52.2004. Google Scholar

[37]

M. A. Ryder, Catheter-related infections: It's all about biofilm,, Topics in Advanced Practice Nursing eJournal, 5 (2005). Google Scholar

[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. Google Scholar

[39]

P. S. Stewart, Biofilm accumulation model that predicts antibiotic resistance of Pseudomonas aeruginosa biofilms,, Antimicrob Agents Chemotherapy, 38 (1994), 1052. doi: 10.1128/AAC.38.5.1052. Google Scholar

[40]

P. S. Stewart, Theoretical aspects of antibiotic diffusion into microbial biofilms,, Antimicrob Agents Chemotherapy, 40 (1996), 2517. Google Scholar

[41]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an epidemic model),, SIAM J. Math. Anal., 24 (1993), 407. doi: 10.1137/0524026. Google Scholar

[42]

E. Tuomanen, Phenotypic tolerance: The search for beta-lactam antibiotics that kill nongrowing bacteria,, Reviews of Infectious Disease, 8 (1986), 279. Google Scholar

[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. Google Scholar

[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. doi: 10.1128/AAC.49.4.1483-1494.2005. Google Scholar

[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. doi: 10.1016/j.mcm.2007.04.003. Google Scholar

[46]

P. J. Yeh, M. J. Hegreness, A. P. Aiden and R. Kishony, Drug interactions and the evolution of antibiotic resistance,, Nat. Rev., 7 (2009), 460. doi: 10.1038/nrmicro2133. Google Scholar

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