Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: The total number of bacteria, B T , is shown over time for the untreated scenario and for four treatment strategies: regular antibiotic and inhibitor dosing with and without regular debridement, and constant antibiotic concentration with regular inhibitor dosing, with and without regular debridement. Note the log 10 scale on the y -axis. Inset graphs show the number of resistant bacteria B R = V B F R + A r B B R . Treatment with constant antibiotics together with regular inhibitor dosing and debridement is most effective, eliminating the bacterial population in all cases (A–D), while regular antibiotic and inhibitor dosing without debridement is least effective, failing to eliminate the bacterial burden in all cases. The remaining two treatment strategies are intermediate in their efficacy, eliminating the bacterial population in some, but not all cases. Eqs – were solved using ode15s . Parameter values: λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. Antibiotics doses: 8 μ g cm −3 , inhibitor doses: 6.12×10 7 inhib. cm −3 , constant antibiotic scenarios: A = 8 μ g cm −3 . See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Bacteria, Concentration Assay, Inhibition

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: Note that inhibitor treatments are plotted as multiples of the standard dose (6.12×10 7 inhib. cm −3 ). In Cases A and B the number of bound bacteria, B B , is a monotone decreasing function of the inhibitor dose, I F i n i t , while the number of free bacteria, B F , increases initially, before decreasing with increasing inhibitor dose. The total number of bacteria, B T , is a monotone decreasing function of the inhibitor dose in Case A, while it increases initially, before decreasing with increasing inhibitor dose, in Case B. In Case C both free and bound bacterial numbers, and hence the total number of bacteria, are increasing functions of the inhibitor dose within the range I F i n i t ∈ [ 0 , 6 . 12 × 10 7 ] inhib. cm −3 , while in Case D inhibitors have little effect on the steady-state bacterial numbers within this range. The insets in the lower panels show the steady-state behaviour for higher inhibitor doses. Eqs – were solved using ode15s , allowing the system to evolve until it reached steady-state and neglecting the Heaviside step functions which prevent the logistic growth of bacteria when their population size goes beneath one. Parameter values: ψ ˜ B a c = 0 hr −1 , ψ ˜ I = 0 hr −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 , ω = 1 and A = 0 μ g cm −3 . See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Inhibition, Bacteria

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: The log 10 of the total number of bacteria at 4 weeks (672 hr), B T (672), is plotted for a range of antibiotic concentrations and inhibitor doses. Note that inhibitor treatments are plotted as multiples of the standard dose (6.12×10 7 inhib. cm −3 ) and that values of B T (672) < 1 are plotted as B T (672) = 1 to maximise visual clarity. The white curves are the contours along which B T (672) = 1; hence, B T (672) < 1 above-right of these contours. The effect of treatment is relatively minor in Cases A and D; however, the bacterial burden may be eliminated for sufficiently high antibiotic concentrations and inhibitor doses in Cases B and C. Eqs – were solved using ode15s and with a constant antibiotic concentration. Parameter values: ψ ˜ B a c = 0 hr −1 , ψ ˜ I = 0 hr −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Bacteria, Inhibition, Concentration Assay

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: The log 10 of the total number of bacteria at 4 weeks (672 hr), B T (672), is plotted for a range of antibiotic concentrations and inhibitor doses. Note that inhibitor treatments are plotted as multiples of the standard dose (6.12×10 7 inhib. cm −3 ) and that values of B T (672) < 1 are plotted as B T (672) = 1 to maximise visual clarity. The white curves are the contours along which B T (672) = 1; hence, B T (672) < 1 above-right of these contours. All bacteria are eliminated for all antibiotic concentrations and inhibitor doses tested in Case A, except where they are both absent. The bacterial burden can also be eliminated in Cases B and C for sufficiently high antibiotic concentrations and inhibitor doses. Treatment has relatively little effect in Case D, reducing the bacterial population by no more than a factor of five. Eqs – were solved using ode15s and with a constant antibiotic concentration. Debridement takes place at the start of each day, occurring for the first time at t = 24 hr, effecting the removal of all free bacteria and inhibitors. Parameter values: λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Bacteria, Inhibition, Concentration Assay

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: The total number of bacteria and the numbers of free and bound bacteria at 4 weeks (672 hr), B T (672), B F (672) and B B (672) respectively, are plotted for a range of values of the potency factor, ω . Note the log scale on the x -axis. In all cases both B T (672) and B B (672) decrease monotonically with increasing ω , while B F (672) is monotone decreasing in Cases A–C and increases before decreasing in Case D. The effect is particularly pronounced in Case C, where the bacterial burden is almost eliminated ( B T = O (10)) as ω approaches 2. Eqs – were solved using ode15s , with a constant antibiotic dose and without inhibitors. Parameter values: A = 8 μ g ml −1 , ψ ˜ B a c = 0 hr −1 , ψ ˜ I = 0 hr −1 , λ = 0 cm 3 cell −1 hr −1 and ρ = 0 hr −1 . See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Bacteria

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: For each parameter set (rows) the optimum regimen of inhibitor doses and debridement is shown, where we minimise either the final number of bound bacteria, B B (168) (left-hand column), or the final total number of bacteria, B T (168) (right-hand column). Note that inhibitor treatments are plotted as multiples of the standard dose (6.12×10 7 inhib. cm −3 ). Using all of the inhibitors on the first day is optimal in Cases A–C under both optimality conditions, whereas inhibitor doses should be distributed across the week in Case D. It is optimal to debride every day in Cases A and D, and only on some of the later days in Case B and C. Eqs – were solved using ode15s , with a constant antibiotic dose. Parameter values: A = 8 μ g ml −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Bacteria, Inhibition

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: For each parameter set (rows) the optimum regimen of inhibitor doses and debridement is shown, where we minimise either the mean number of bound bacteria, 〈 B B 〉 (left-hand column), or the mean total number of bacteria, 〈 B T 〉 (right-hand column). Note that inhibitor treatments are plotted as multiples of the standard dose (6.12×10 7 inhib. cm −3 ). Using most of the inhibitors on the first day is optimal in all instances. It is optimal to debride every day in all cases (A–D), except in Case C under the 〈 B B 〉 criterion, where it is optimal to debride on days 6 and 7 only. Eqs – were solved using ode15s , with a constant antibiotic dose. Parameter values: A = 8 μ g ml −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Bacteria, Inhibition

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: Graphs show the dynamics of the bound and total bacterial burdens, B B and B T respectively (columns), in the untreated case and under the optimal treatment regimens (see ) for each parameter set (rows). Note the log 10 scale on the y -axis. The bacterial burden is eliminated ( B T (168) < 1) under the optimal treatment regimens in Case A and is significantly reduced in Cases B–D (to O (10) in Case B, O (10 ) in Case C and just below 10 3 in Case D). There is little difference in the effects of the optimum treatments under the different optimality conditions for any given case. Note that the discontinuities in B T are caused by the instantaneous removal of free bacteria upon debridement (similarly in Figs , and , see Treatment types). Eqs – were solved using ode15s , with a constant antibiotic dose. Parameter values: A = 8 μ g ml −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Bacteria

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: Graphs show the dynamics of the bound and total bacterial burdens, B B and B T respectively (columns), in the untreated case and under the optimal treatment regimens (see ) for each parameter set (rows). Note the log 10 scale on the y -axis. The bacterial burden is eliminated ( B T (168) < 1) under the optimal treatment regimen in Case A and almost eliminated in Cases B and C, where B T (168) = O (10) and O (10 ) respectively, while B T (168) = O (10 ) in Case D. There is little difference in the effects of the optimum regimens under the different optimality conditions for any given case. Eqs – were solved using ode15s , with a constant antibiotic dose. Parameter values: A = 8 μ g ml −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques:

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: Graphs show the dynamics of the bound and total bacterial burdens, B B and B T respectively (columns), in the untreated case and under the optimal treatment regimens (see ) for each parameter set (rows). Note the log 10 scale on the y -axis. The bacterial burden is eliminated ( B T (168) < 1) under the optimal treatment regimens in Case A and is significantly reduced in Cases B–D (to O (10 ) in Case B, O (10 )– O (10 5 ) in Case C and O (10 5 )– O (10 6 ) in Case D, lower values corresponding to the B T (168) optimality condition and higher values to the B B (168) optimality condition where ranges are given). Eqs – were solved using ode15s , with a constant antibiotic dose. Parameter values: A = 8 μ g ml −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. The initial conditions are the untreated steady-states for each parameter set, modified so that 2% of the free and bound bacteria are resistant. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Modification, Bacteria

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: Graphs show the dynamics of the bound and total bacterial burdens, B B and B T respectively (columns), in the untreated case and under the optimal treatment regimens (see ) for each parameter set (rows). Note the log 10 scale on the y -axis. The bacterial burden is eliminated ( B T (168) < 1) under the 〈 B T 〉 optimal treatment regimen in Case A, wile B T (168) = O (10 6 ) under the 〈 B B 〉 regimen. Both regimens achieve B T (168) = O (10 ) in Cases B and C, and B T (168) = O (10 6 ) in Case D. There is little difference in the effects of the optimum regimens under the different optimality conditions for Cases B–D. Eqs – were solved using ode15s , with a constant antibiotic dose. Parameter values: A = 8 μ g ml −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. The initial conditions are the untreated steady-states for each parameter set, modified so that 2% of the free and bound bacteria are resistant. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Modification, Bacteria

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: For each parameter set (rows) the optimum regimen of inhibitor doses and debridement is shown, where we minimise either the final number of bound bacteria, B B (168) (left-hand column), or the final total number of bacteria, B T (168) (right-hand column). Note that inhibitor treatments are plotted as multiples of the standard dose (6.12×10 7 inhib. cm −3 ). Using most or all of the inhibitors on the first day is optimal in Cases B–D, whereas inhibitor doses should be distributed across the week in Case A. It is optimal to debride on most days in Case A, on the last three days in Case B and either once on the last day ( B T (168) optimum) or not at all ( B B (168) optimum) in Cases C and D. Eqs – were solved using ode15s , with a constant antibiotic dose. Parameter values: A = 8 μ g ml −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. The initial conditions are the untreated steady-states for each parameter set, modified so that 2% of the free and bound bacteria are resistant. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Bacteria, Inhibition, Modification

Journal: PLoS Computational Biology

Article Title: Mathematical model predicts anti-adhesion–antibiotic–debridement combination therapies can clear an antibiotic resistant infection

doi: 10.1371/journal.pcbi.1007211

Figure Lengend Snippet: For each parameter set (rows) the optimum regimen of inhibitor doses and debridement is shown, where we minimise either the mean number of bound bacteria, 〈 B B 〉 (left-hand column), or the mean total number of bacteria, 〈 B T 〉 (right-hand column). Note that inhibitor treatments are plotted as multiples of the standard dose (6.12×10 7 inhib. cm −3 ). Using all of the inhibitors on the first day is optimal for all cases under the 〈 B B 〉 criterion and in Case B under the 〈 B T 〉 criterion, while it is optimal to distribute inhibitor treatment across more of the week in Cases A, C and D under the 〈 B T 〉 criterion. It is optimal to debride every day in all cases under the 〈 B T 〉 criterion, while under the 〈 B B 〉 criterion it is optimal to debride less frequently or not at all. Eqs – were solved using ode15s , with a constant antibiotic dose. Parameter values: A = 8 μ g ml −1 , λ = 0 cm 3 cell −1 hr −1 , ρ = 0 hr −1 and ω = 1. The initial conditions are the untreated steady-states for each parameter set, modified so that 2% of the free and bound bacteria are resistant. See Tables – for the remaining parameter values.

Article Snippet: The Matlab routine ode15s was used to solve the time-dependent problem (Eqs – ), allowing the system to evolve until it reached steady-state ( fsolve struggles to find the steady-state solution when inhibitors are included due to the difficulty in choosing an initial guess that will converge to the desired steady-state).

Techniques: Bacteria, Inhibition, Modification