A model of optimal dosing of antibiotic treatment in biofilm

Mudassar Imran; Hal L. Smith

doi:10.3934/mbe.2014.11.547

Article Contents

2014, Volume 11, Issue 3: 547-571. Doi: 10.3934/mbe.2014.11.547

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A model of optimal dosing of antibiotic treatment in biofilm

Mudassar Imran^1, and
Hal L. Smith^2,

1.
Department of Mathematics, Syed Babar Ali School of Science and Engineering, Lahore University of Management Sciences, Lahore
2.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804

Received: November 30, 2012

Revised: May 31, 2013

Published: December 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 49J15, 92C50; Secondary: 92C45, 34D15.

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