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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), 1304-1308. 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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