Model development

We construct a mathematical model of an infected burn wound, focussing upon the bacterial population and the treatment strategies employed to clear an infection. For our purposes a burn wound consists of a layer of host cells, over which lies a fluid layer, exuded by the host cells, called the exudate. The exudate is partially covered by a layer of necrotic tissue, except in the region of a surgical excision, where it is exposed to the air and from which fluid may leak. If left undisturbed, a scab forms across the excision after 24 hr, preventing further fluid loss (see Fig 1(a)). The environment-dependent parameters used in this paper were fitted to an in vivo rat model with the bacterial species P. aeruginosa in [26]; however, this model is also of relevance to burn wounds in humans and for any bacterial species for which host cell attachment is partly mediated by MAM7. See the ‘Experimental set-up’ and ‘Model formulation’ sections of [26] for more details.

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Fig 1. Wound geometry and model structure.

(a) The wound is pictured in the transverse plane. The (liquid) exudate overlies the host cells, while a layer of necrotic tissue covers the wound, except at the excision where the exudate is exposed to the air. (b) The processes described in the mathematical model as they relate to the wound geometry (the diagram does not distinguish between susceptible and resistant bacteria, nor does it depict conjugation or segregation). (c) A representation of the model topology (Eqs 1–11). : free susceptible bacteria, : free resistant bacteria, : bound susceptible bacteria, : bound resistant bacteria, I_F: free inhibitors, I_B: bound inhibitors, and A: antibiotic. Antibiotic directly inhibits bacteria by killing them, while inhibitors indirectly inhibit bacteria by limiting their ability to bind to host cells. The intrinsic growth rate of susceptible bacteria is greater than that of resistant bacteria, while the maximum killing rate of susceptible bacteria by antibiotic is greater than that for resistant bacteria. This is represented by thicker arrows/inhibition symbols in each case. See Table 1 for variable descriptions and units, and Tables 2–4 for parameter descriptions and units.

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Our model considers three types of species: bacteria, inhibitors and antibiotics. Both bacteria and inhibitors may exist in one of two physical states, either swimming/floating freely in the exudate or bound to the host cells, while antibiotics remain in solution in the exudate at all times. Further, bacteria come in two varieties: those which are more vulnerable to antibiotic (susceptible bacteria) and those which have developed resistance to the antibiotic (resistant bacteria). In this study we model a bactericidal antibiotic, employing parameter values fitted to newly measured in vitro kill curves for P. aeruginosa using the antibiotic meropenem (see Parameter fitting and justification and S1 Text for more details). Meropenem is commonly used to treat P. aeruginosa burn wound infections [49] and can be administered intravenously; thus, it is a natural choice for this study. Inhibitors are applied directly to the exudate, whereas antibiotics are applied systemically, entering the wound through the host cell layer, having reached the wound via the bloodstream.

Assuming, as in [26], that the system is well-mixed, we define an ordinary differential equation (ODE) model for free susceptible bacteria density, (cells cm⁻³), free resistant bacteria density, (cells cm⁻³), bound susceptible bacteria density, (cells cm⁻²), bound resistant bacteria density, (cells cm⁻²), free inhibitor concentration, I_F(t) (inhib. cm⁻³), bound inhibitor concentration, I_B(t) (inhib. cm⁻²), and antibiotic concentration A(t) (μg cm⁻³), over time, t (hr) (the dependent and independent variables are summarised in Table 1). It is assumed that the total binding site density on the host cells, consisting of both free and occupied sites, is conserved, such that the free binding site density (sites cm⁻²), where E_total (sites cm⁻²) is the total density of binding sites (both free and bound), and ϕ_Bac (sites cell⁻¹) and ϕ_I (sites inhib.⁻¹) are the number of binding sites occupied by a bacterium or an inhibitor respectively.

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Table 1. Dependent and independent variables for Eqs 1–11.

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The model, summarised in Fig 1(b) and 1(c), is described by the following governing equations (1) (2) (3) (4) (5) (6) (7) where parameter definitions and values are given in Tables 2–4. See Parameter fitting and justification for details on how the parameter values were obtained.

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Table 2. Parameter values fitted to in vivo data for Eqs 1–11.

https://doi.org/10.1371/journal.pcbi.1007211.t002

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Table 3. Parameter values fitted to in vitro data for Eqs 1–11.

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Table 4. Measured, calculated, literature-derived and estimated parameter values for Eqs 1–11.

https://doi.org/10.1371/journal.pcbi.1007211.t004

We note that this model differs from that presented in [26] in the following respects:

• it includes antibiotic concentration, resulting in an additional ODE (Eq 7);
• both the free and bound bacterial compartments are split into susceptible and resistant sub-compartments (having been tacitly susceptible in [26]), such that the two ODEs describing free and bound bacterial dynamics are replaced with four ODEs for free susceptible, free resistant, bound susceptible and bound resistant bacteria (Eqs 1–4);
• it includes parameters and terms for fitness cost to resistant bacteria, conjugation, segregation, killing of bacteria by antibiotic and the factor difference in antibiotic potency against bound bacteria compared with free bacteria (see below for more details);
• it includes terms to prevent the regrowth of bacteria once their population size goes beneath one (see below).

These key extensions to our model in [26] facilitate investigation into how the combination therapies presented here can best be employed to tackle an otherwise untreatable antibiotic resistant infection.

Both free and bound bacteria are assumed to grow logistically with carrying capacities K_F (cells cm⁻³) and K_B (cells cm⁻²) respectively. In our model, the carrying capacities represent the maximum number of bacteria that can be sustained by available nutrients and are such that bacterial division is negligible when or (see [62, 63]). It is important to note that the number of bacteria that can be supported by nutrients near the host cells is not in general equal to the number of available binding sites on the host cells (K_B ≠ E_total/ϕ_Bac), indeed K_B < E_total/ϕ_Bac for all parameter sets considered here (see Tables 2 and 4).

Susceptible bacteria have intrinsic growth rates (hr⁻¹) (free) and (hr⁻¹) (bound), while resistant bacteria incur a fitness cost, 0 < c < 1 (dimensionless), such that their intrinsic growth rates are (hr⁻¹) (free) and (hr⁻¹) (bound). This fitness cost only operates when the logistic terms represent bacterial growth. If the density of free cells, , exceeds the free carrying capacity, K_F, then the free logistic growth term becomes a death term, and likewise for bound bacteria. In this case the intrinsic growth rates of resistant bacteria revert to those of susceptible bacteria, since resistant bacteria are assumed to die at the same rate as susceptible bacteria. This is achieved through the use of Heaviside step functions, and , in Eqs 2 and 4, where (8) Further, the growth of any bacterial subtype (free-susceptible/free-resistant/bound-susceptible/bound-resistant) ceases once the number of bacteria in that subtype falls beneath one, since at least one cell is required in order for division to be possible. This is achieved using the Heaviside step functions , , and , in Eqs 1–4, where H is defined in Eq 8.

Daughter cells derived from bound bacteria may enter either the bound compartment (in the proportion 0 ≤ η(E(t)) ≤ 1 (dimensionless)) or the free compartment (1 − η(E(t))), the proportion entering the bound compartment increasing as the density of free binding sites, E(t), increases. We model this dependence using a Hill function as follows (9) where η_max (dimensionless) is the maximum proportion of daughter cells which may remain bound to the surface, γ (sites cm⁻²) is the binding site density at which η(E) = η_max/2 and n (dimensionless) is the Hill coefficient. We use a Heaviside step function, , in Eqs 1–4 to restrict cell death due to the bound logistic growth term to the bound compartment when , where H is defined in Eq 8.

The resistant strain of P. aeruginosa used in our in vitro experiments, PA1004 Evo10, transfers resistance genes vertically, but not horizontally. Therefore, throughout most of this study we neglect horizontal gene transfer and segregation. We include conjugation and segregation terms in Eqs 1–11 so as to make our model relevant to a wider class of infections, performing a sensitivity analysis on these parameters in Sensitivity analysis. In those cases where horizontal gene transfer does occur, resistant bacteria transfer plasmids conferring resistance to susceptible bacteria via conjugation at a rate λ (cm³cell⁻¹hr⁻¹). It is assumed that this process occurs within the free compartment, but not within the bound compartment or between the two compartments, since bound bacteria are typically physically separated from each other and free bacteria are unlikely to interact with bound bacteria. Horizontal gene transfer can also occur via transformation and transduction; however, we consider only conjugation here since it is the most common of the three mechanisms [10]. When a bacterium divides, its plasmids are segregated (divided) between the resulting daughter cells. A portion of the daughter cells of resistant bacteria produced upon division fail to inherit the resistance plasmid, leading to the production of susceptible offspring (by resistant bacteria) at a rate ρ (hr⁻¹) (see [53] for an example). Similarly to the processes described above, segregation only occurs where the number of free or bound bacteria are below carrying capacity and where the number of free or bound resistant bacteria is greater than one. This is achieved through the use of Heaviside step functions, , , and , in Eqs 1–4, where H is defined in Eq 8.

Bacteria and inhibitors bind to and unbind from the host cells with respective binding rates α_Bac (hr⁻¹ sites⁻¹) and α_I (hr⁻¹ sites⁻¹), and unbinding rates β_Bac (hr⁻¹) and β_I (hr⁻¹), in accordance with the law of mass action.

Neutrophils are present only on the surface of the host cells and are fully upregulated throughout an infection, such that bound bacteria can be assumed to decay exponentially at rate δ_B (hr⁻¹), where δ_B accounts for neutrophil density.

We use Michaelis-Menten terms for the killing rates of susceptible and resistant bacteria by antibiotics to capture the saturating effects of increased antibiotic concentration. The maximum killing rates are given by (hr⁻¹) (susceptible) and (hr⁻¹) (resistant), where , while the Michaelis constants (μg cm⁻³) (susceptible) and (μg cm⁻³) (resistant) give the antibiotic concentrations at which the killing rate is half-maximal, where (see Table 3). We multiply the bound bacteria antibiotic killing terms by a factor ω (dimensionless), to account for the potential difference in the antibiotic potency against bound bacteria as compared with free bacteria. Bound bacteria may be less vulnerable to antibiotic than free bacteria, in which case ω < 1; however, they may also be exposed to higher concentrations of antibiotic, which enters the wound through the host cell layer, in which case they may be more vulnerable, such that ω > 1. If bound bacteria are equally as vulnerable to antibiotic as free bacteria then ω = 1.

The clearance of bacteria and inhibitors (ψ_Bac(t) (hr⁻¹) and ψ_I(t) (hr⁻¹)) is assumed to occur at a constant rate for the first 24 hours, after which it ceases when a scab forms over the excision. Therefore, clearance occurs at rates (10) where (hr⁻¹) and (hr⁻¹) are the constant clearance rates which apply in the first 24 hours, and H is a Heaviside step function (see Eq 8).

Antibiotics may either be administered in discrete doses or applied continuously, such that the antibiotic concentration remains fixed. In the former case, antibiotic is assumed to be eliminated from the system (e.g. through degradation and clearance into the bloodstream and surrounding tissues) at a rate δ_A (hr⁻¹) following a dosing event. It is assumed that the loss of antibiotic through its interaction with bacteria is negligible in comparison to its elimination rate, and hence we do not include it in the model. Further, we do not include an antibiotic clearance term, similar to those given in Eq 10 for bacteria and inhibitors, since, while some antibiotic will leave the wound within the leaking exudate, this will not affect the antibiotic concentration in the remaining exudate (which is replenished via passage cross the host cell layer). It is assumed that inhibitor degradation, if it occurs, is sufficiently gradual that it can be neglected.

Several of the terms in Eqs 1–7 contain the exudate height, h, or volume, V, or the wound area, A_r, as a factor in order to ensure dimensional consistency. We retain them in explicit form in the interests of clarity, though we note they could have been combined with their multipliers to create new parameters.

Bacteria are applied to the burn wound, following the excision, at time t = 0 (hr). This is also the first occasion upon which inhibitor or antibiotic treatment may be applied. Therefore, initially (11) where , , and A_init are constants. The bound compartments are empty initially, since bacteria and inhibitors have not had an opportunity to bind to the host cells. See Tables 2–4 for parameter values. We note that we retain equations in dimensional form to ease biological interpretation.

Treatment types

Previously we considered a susceptible only bacterial population, treated using inhibitors and debridement [26]. There the focus was upon optimising inhibitor properties to improve treatment. Here we consider how to optimally combine antibiotic, inhibitor and debridement therapies so as to eliminate a mixed susceptible-resistant population of bacteria.

Antibiotics are applied systemically and may be administered either in discrete doses (e.g. administered orally as tablets) or continuously (e.g. administered intravenously via a drip). In the continuous case the antibiotic concentration is held at a constant value such that A ≡ A_init. Hellinger et al. [59] have shown that meropenem dosages as high as 6 g day⁻¹ can be used in humans without increasing the frequency of adverse effects, a result which has been confirmed by other groups [60, 61]. Furthermore, Roberts et al. [60] found that continuous dosing of meropenem at 3 g day⁻¹ in humans resulted in subcutaneous tissue concentrations of 4 μg cm⁻³. Therefore, since the daily dosage could be up to twice this value, we can infer (assuming a linear scaling) that subcutaneous tissue (and hence burn wound) concentrations up to 8 μg cm⁻³ are achievable. In the discrete dosing case, antibiotic degrades and is cleared from the body following each dosing event. We assume that discrete doses may not exceed tissue concentrations of 8 μg cm⁻³, consistent with the continuous case. We take the dosing frequency to be once a day, at the same times at which inhibitors are applied (see below), thus ensuring that our treatment regimens are feasible to implement clinically.

Inhibitors are applied topically to the wound. There is no hard limit on how frequently inhibitors may be applied; however, twice daily is a reasonable upper limit (that is, at 0, 12, 24, 36, … hr), fixing the frequency at daily dosing in the present study for simplicity. We take the dose used by Huebinger et al. [18] in their experiments, that is 3×10⁸ inhibitors (which, when added to the exudate, corresponds to a concentration of 6.12×10⁷ inhib. cm⁻³), as standard. The total number of inhibitors in the system (free and bound) is conserved in the absence of debridement, except during the first 24 hr (after the necrotic tissue is first excised), when free inhibitors are lost through leakage of the exudate.

Debridement involves the mechanical cleansing of a wound, for example with a cloth. In our model this corresponds to the instantaneous removal of the exudate and with it all of the free bacteria and inhibitors. The exudate is quickly replenished (on the timescale of a few minutes) such that its volume fluctuation can be neglected. Debridement can be administered at most once daily, starting from the first day after the excision is made (that is at 24, 48, 72, … hr). In those cases where debridement and dosing with inhibitors coincides, debridement is performed first, to avoid immediately removing the newly administered inhibitors. Since debridement involves the removal of the scab that forms over the wound, clearance of bacteria and inhibitors is re-established in the first 24 hr after each debridement event.

Parameter fitting and justification

The parameters in Table 3 were fitted to newly gathered in vitro data. Susceptible, PA1004 WT, and resistant, PA1004 Evo10, strains of P. aeruginosa were grown both in the absence of antibiotic and in the presence of a range of concentrations of meropenem. Simplified equations, containing only logistic growth and antibiotic killing terms were then fitted to the data using the Matlab routine fminsearch, providing fits for c, , , and . See S1 Text for further details.

The parameters in Table 2 come from [26] where they were fitted to in vivo data from the rat burn wound model described in [18]. Twelve valid parameter sets were identified, which were grouped into four qualitatively distinct cases (Case A–Case D). Treatment with inhibitors is effective in Cases A and B, worsens an infection in Case C and has little effect in Case D. Each parameter set gave an equally good fit to the data, while insufficient experimental data is currently available to distinguish between them. In the present work we use a single parameter set from each case, Set 2 from Case A, Set 6 from Case B, Set 10 from Case C and Set 12 from Case D. Set 2 was chosen as it is the most biologically realistic parameter set in Case A, Sets 6 and 10 were chosen since they are the most resistant to treatment, allowing us to consider the worst-case-scenario, and Set 12 was chosen since it is the only parameter set in Case D. We note that we used Sets 3 and 8, rather than Sets 6 and 10, in the main text of [26] in Cases B and C respectively.

The combination of the parameters fitted to in vitro data in Table 3 with the parameters fitted to in vivo data in Table 2 is valid, both since the effects and processes with which each set of parameters are associated are independent from each other, and because the in vitro data were gathered using the same bacterial species (P. aeruginosa) as the in vivo data and using a growth medium which replicates the nutrient levels in a burn wound exudate (see S1 Text).

Each of the parameters in Table 4 were either measured, calculated, derived from the literature or estimated, as indicated in the fourth column. The parameters ϕ_Bac, ϕ_I, V, A_r, h, n and E_total are justified in [26] (where E_total is written as E_init), while and A_init are justified above in Treatment types.

We set the conjugation and segregation rates, λ and ρ, to zero unless otherwise stated. This is because the resistance genes to meropenem in the PA1004 Evo10 strain of P. aeruginosa under consideration are chromosomal and hence cannot be transferred by conjugation or lost through segregation, which requires the resistance gene to be carried on a plasmid. While we have the PA1004 Evo10 strain in mind throughout this study, we have included terms for conjugation and segregation in order to make our model sufficiently general to account for other bacterial strains. In S3 Text we perform a sensitivity analysis to investigate the effect of these parameters on the bacterial dynamics, using values informed by the literature as described below.

Hall et al. [50] measured the intraspecific conjugation rates of P. fluorescens and P. putida to be 10^−11±0.2 cell⁻¹hr⁻¹ and 10^−14±0.4 cell⁻¹hr⁻¹ respectively (these values must be multiplied by V = 4.9 cm³ to make them dimensionally consistent with our model), which fall within the range of values measured by [51–53], while Simonsen et al. [51] have measured conjugation rates as high as 10⁻⁹ cm³ cell⁻¹hr⁻¹ in E. coli.

Smets et al. [52] measured a plasmid loss rate of 2.52×10⁻⁴hr⁻¹ which informed the value of 1×10⁻⁴hr⁻¹ used in [50, 53] and falls within the range of values measured by [54].

The antibiotic elimination rate, δ_A, has been measured to lie in the range 0.62–1.72 hr⁻¹ for meropenem in both humans and pigs [55–57], thus we choose δ_A = 1 hr⁻¹ as a typical value.

The factor difference in antibiotic potency against bound bacteria ω is assumed to be one (i.e. no difference) by default and in the absence of further information. We perform a sensitivity analysis on ω in Sensitivity analysis, varying it within the range ω ∈ [0.5, 2].

The initial bacterial burden is taken to be 5×10⁶ CFU (colony-forming units), corresponding to an initial free density (all bacteria are free initially) of 1.02×10⁶ cells cm⁻³, in accordance with the in vivo model in [18]. The initial ratio of susceptible to resistant bacteria may vary; however, susceptible bacteria will be in the majority prior to treatment with antibiotic due to the resistance-associated fitness cost. Therefore, we assume that only 2% (2.04×10⁴ cells cm⁻³) of the initial bacterial population exhibits the resistant phenotype, the remaining 98% (1.00×10⁶ cells cm⁻³) being susceptible.

Steady-state analysis

We begin by considering a steady-state analysis of Eqs 1–10, performed using Maple, to determine the number of steady-states exhibited by the system under various conditions, together with their stability properties. While the system may take a long while to approach steady-state in practice, depending upon the choice of parameters and initial conditions, this analysis is instructive for at least two reasons. Firstly, it allows us to ensure that we are not overlooking any potential stable steady-state solutions in the time-dependent simulations presented below. Secondly, it allows us to make more clear-cut comparisons between different scenarios, looking beyond the transient dynamics resulting from the choice of initial conditions.

We consider four scenarios: untreated, antibiotic treatment only, inhibitor treatment only, and treatment with both antibiotics and inhibitors, comparing Cases A–D in each scenario. We use the maximum continuous concentration (8 μg cm⁻³) for antibiotic treatment and a single standard dose (6.12×10⁷ inhib. cm⁻³) for inhibitor treatment (see Treatment types). We set the clearance terms to zero ( hr⁻¹ and hr⁻¹) since fluid only leaks from the wound in the first 24 hr. Further, we neglect conjugation and segregation (λ = 0 cm³cell⁻¹hr⁻¹ and ρ = 0 hr⁻¹), and assume that there is no difference in the potency of antibiotics against bound bacteria as compared with free bacteria (ω = 1) (see Parameter fitting and justification). We also remove the Heaviside step functions preventing the logistic growth of bacteria when their population size goes beneath one (, , and ) since these are only required in dynamic simulations to prevent biologically unrealistic regrowth when bacteria have been eliminated. Lastly, where antibiotics are applied, we assume a constant dose, since with a discrete dose the antibiotic concentration is zero at steady-state, being identical to the equivalent scenario without antibiotic treatment. Following these simplifications, the governing equations reduce to Eqs A–E in S2 Text. All remaining parameter values are as given in Tables 2–4.

The results of the steady-state analysis are described in detail in S2 Text and summarised here, in Table 5 and in Fig 2. In all cases except Case A under the inhibitor only treatment the system is monostable, the number of steady-states (stable plus unstable) varying between one and three depending upon the treatment scenario and the parameter set. In the absence of antibiotics, resistant bacteria go extinct at the stable steady-state, while free and bound susceptible bacteria survive. The situation is reversed in the presence of antibiotics, with susceptible bacteria going extinct at the stable steady-state, while free and bound resistant bacteria survive. There are three exceptions to this rule. The first two are for the scenario in which treatment with both antibiotics and inhibitors is applied, in Cases B and C, for which all bacteria go extinct at the stable steady-state. The third is Case A under the inhibitor only treatment (noted above), for which there exist no isolated stable steady-states. Instead, there exists a region of non-isolated steady-states [64], in which susceptible and resistant bacteria may coexist, including the extremes (unstable steady-states) at which only one of these subtypes survives. As such, the state to which the system settles depends upon the initial conditions. For simplicity of exposition, we plot the unstable steady-state solution in which only susceptible bacteria survive in Fig 2, this being the typical state in the absence of antibiotics under most parameter sets.

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Table 5. Steady-states and their stability properties.

https://doi.org/10.1371/journal.pcbi.1007211.t005

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Fig 2. Steady-state solutions.

Four stacked bars are plotted for each case: the first bar shows the number of free and bound susceptible bacteria, and , at steady-state in the untreated scenario; the second bar shows the number of free and bound resistant bacteria, and , at steady-state in the antibiotic only scenario; the third bar shows the number of free and bound susceptible bacteria at steady-state in the inhibitor only scenario, and the fourth bar shows the number of free and bound resistant bacteria at steady-state in the antibiotic and inhibitor scenario. There are no resistant bacteria at steady-state in the absence of antibiotic, nor are there any susceptible bacteria at steady-state in the presence of antibiotic. Thus, the combined height of the bars in each stack gives the total number of bacteria, B_T. Treatment with antibiotics alone reduces the total bacterial burden in Cases A, C and D, and increases it in Case B. Treatment with inhibitors alone reduces the total bacterial burden in Cases A and B, increases it in Case C and has a negligible effect in Case D. Treatment with both antibiotics and inhibitors eliminates the bacterial burden in Cases B and C, and reduces it more than either treatment in isolation in Cases A and D. Steady-state solutions to Eqs 1–10 were calculated using Maple, neglecting the Heaviside step functions which prevent the logistic growth of bacteria when their population size goes beneath one. Parameter values: hr⁻¹, hr⁻¹, λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹, ω = 1, A = 0 or 8 μg cm⁻³, and or 6.12×10⁷ inhib. cm⁻³. See Tables 2–4 for the remaining parameter values.

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Treatment with antibiotics alone reduces the total number of bacteria, B_T, in Cases A, C and D; however, it slightly increases the bacterial burden in Case B (this counter-intuitive result is discussed below in Sensitivity analysis). As was the case in [26], treatment with inhibitors reduces the total number of bacteria in Cases A and B, increases the bacterial burden in Case C and has little effect in Case D (see Sensitivity analysis and [26] for discussion). Treatment with antibiotics and inhibitors in combination is more effective than treatment with either therapy in isolation, eliminating the bacterial burden in Cases B and C and greatly reducing it in Cases A and D. It is evident from these results that antibiotics and inhibitors are predicted to work together in a synergistic manner, as opposed to an additive one, their combined effect reducing the total bacterial burden by a greater quantity in Cases B–D than the sum of the reductions when applied in isolation, and by a smaller quantity in Case A.

Dynamic simulations

Having examined the behaviour of the system at steady-state, we consider the bacterial population dynamics over time in response to treatment. We use the Matlab routine ode15s, a variable-step, variable-order solver based upon numerical differentiation formulas, to solve the time-dependent problem (Eqs 1–11) both here and throughout the paper. The untreated scenario is compared with four treatment scenarios: regular antibiotic and inhibitor dosing with and without regular debridement, and constant antibiotic concentration with regular inhibitor dosing, with and without regular debridement (see Fig 3), for Cases A–D. We note that while only the total number of bacteria, B_T, is plotted for clarity, the simulations include susceptible/resistant and free/bound bacteria. Regular antibiotic/inhibitor/debridement treatments are performed every 24 hr, antibiotic/inhibitor dosing occurring for the first time at t = 0 hr and debridement being performed for the first time at t = 24 hr. Antibiotic doses of 8 μg cm⁻³ and standard inhibitor doses of 6.12×10⁷ inhib. cm⁻³ are used in all cases, while each debridement event results in the removal of all free bacteria and inhibitors. The antibiotic concentration is held fixed at A = 8 μg cm⁻³ in the constant antibiotic scenarios (see Treatment types for more details).

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Fig 3. Dynamic simulations.

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 log₁₀ scale on the y-axis. Inset graphs show the number of resistant bacteria . 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 1–11 were solved using ode15s. Parameter values: λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹ and ω = 1. Antibiotics doses: 8 μg cm⁻³, inhibitor doses: 6.12×10⁷ inhib. cm⁻³, constant antibiotic scenarios: A = 8 μg cm⁻³. See Tables 2–4 for the remaining parameter values.

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Constant antibiotic concentration with regular inhibitor dosing and debridement is the most effective treatment, eliminating the bacterial burden in all cases and doing so more rapidly than the other strategies. Constant antibiotic concentration with regular inhibitor dosing and no debridement eliminates the bacterial population in Cases B and C, but has a more modest effect in Cases A and D. Regular antibiotic and inhibitor dosing with debridement eliminates all bacteria in Cases A and B, but is ineffective in Cases C and D. Lastly, regular antibiotic and inhibitor dosing without debridement is the least effective strategy, having little effect in Cases A–D. We note that while constant antibiotic concentration with regular inhibitor dosing and debridement reduces the number of antibiotic resistant bacteria, , in all cases (see inset graphs, Fig 3), the remaining strategies increase B_R above untreated levels in some cases; indeed, regular antibiotic and inhibitor dosing without debridement does so in all cases.

A comparison between regular inhibitor and debridement treatment, which is the most effective therapy in the absence of antibiotics (see [26]), and treatment which combines regular inhibitor and debridement therapy with a constant antibiotic dose, shows that the combined therapy is significantly more effective (see Fig. A in S3 Text). Inhibitors and debridement alone eliminate the bacterial burden in Case A only, whereas, when combined with antibiotics, the bacterial burden is eliminated in all four cases (A–D).

Sensitivity analysis

In the results that follow we consider the effect of varying the antibiotic and inhibitor doses, and other key parameters, upon the bacterial population dynamics and their steady-state values. In the cases where time-dependent simulations are employed, the solutions are shown at 4 weeks (672 hr), with solutions at 1 week (168 hr) and 1 year (365 days = 8760 hr) provided in S3 Text. Results are given at 1 week since ideally we would like to clear an infection within this time, while results are shown at 4 weeks and 1 year to demonstrate the dynamics of more persistent infections and since the sensitivity of the system to changes in parameter values varies over time.

Antibiotic and inhibitor doses.

Fig 4 shows the effect of varying the antibiotic concentration, A, upon the total number of bacteria, B_T, the number of free bacteria, B_F, and the number of bound bacteria, B_B, at steady-state, in the absence of inhibitors (we note that, except in a very narrow region around A = 0 μg ml⁻¹, all bacteria are resistant at steady-state in the presence of antibiotic, see Steady-state analysis). Eqs 1–10 were solved using the Matlab routine fsolve, employing the Trust-Region-Dogleg algorithm, in the same form and with the same parameter values as in Steady-state analysis, but with A ∈ [0, 50] μg ml⁻¹. This range of antibiotic values goes well above the maximum concentration discussed in Treatment types. We use a wider range here in order to elucidate the theoretical effect of increased antibiotic concentrations in isolation.

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Fig 4. Steady-state sensitivity analysis for antibiotic concentration.

The number of bound bacteria, B_B, is a monotone decreasing function of the antibiotic concentration, A, in all cases, as are the number of free bacteria, B_F, and the total number of bacteria, B_T, in Case A. In Cases B–D the free and total bacterial burdens increase initially, before decreasing with increasing antibiotic concentration. (Note that, except near A = 0 μg ml⁻¹, all bacteria are resistant at steady-state in the presence of antibiotic, see Steady-state analysis). Steady-state solutions to Eqs 1–10 were calculated using fsolve, neglecting the Heaviside step functions which prevent the logistic growth of bacteria when their population size goes beneath one. Parameter values: hr⁻¹, hr⁻¹, λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹, ω = 1 and inhib. cm⁻³. See Tables 2–4 for the remaining parameter values.

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It can be seen that B_B is a monotone decreasing function of A in all cases (A–D), as are B_F and B_T in Case A. However, in Cases B–D, we have the counter-intuitive result that B_F increases initially as A is increased above zero, before reaching a maximum and decreasing, causing a similar behaviour in B_T. The increase in B_F, and hence in B_T, is caused by an increase in the logistic growth rate of bound bacteria, occurring as a result of a decrease in the number of bound bacteria, the majority of the daughter cells from bound bacteria entering the exudate since η_max ≪ 1 in all cases. Given the quadratic dependence of the bound logistic growth rate upon the number of bound bacteria, a change to the system resulting in a reduction in the number of bound bacteria has the potential to increase the bound growth rate provided the original bound bacterial population is greater than half the bound carrying capacity, K_B/2 (see [26] for more details). Since the increase in the number of free bacteria outweighs the decrease in bound bacteria, this also increases the total bacterial burden. All bacteria are eliminated for sufficiently high antibiotic concentrations in Case A–C, while the antibiotic killing effect saturates in Case D such that B_T does not drop far below 2 × 10⁷ as the antibiotic concentration is increased beyond 50 μg ml⁻¹.

Fig 5 shows the effect of varying the (single) inhibitor dose, , upon the total number of bacteria, the number of free bacteria and the number of bound bacteria at steady-state, in the absence of antibiotic (we note that all bacteria are susceptible at steady-state in the absence of antibiotic, see Steady-state analysis). As discussed in Steady-state analysis there are inhibitor concentrations at which the system does not possess an isolated steady-state in Case A; therefore, in this case, we plot the steady-state for which only susceptible bacteria survive across all values of for consistency. The Matlab routine ode15s was used to solve the time-dependent problem (Eqs 1–11), 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). Aside from using the time-dependent form of the equations, the same assumptions were made as in Steady-state analysis, but with inhib. cm⁻³, the maximum dose being the standard dose as discussed in Treatment types.

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Fig 5. Steady-state sensitivity analysis for inhibitor doses.

Note that inhibitor treatments are plotted as multiples of the standard dose (6.12×10⁷ inhib. cm⁻³). In Cases A and B the number of bound bacteria, B_B, is a monotone decreasing function of the inhibitor dose, , 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 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 1–11 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: hr⁻¹, hr⁻¹, λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹, ω = 1 and A = 0 μg cm⁻³. See Tables 2–4 for the remaining parameter values.

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It can be seen that B_B is a monotone decreasing function of in Cases A and B, while the number of free bacteria increases initially as is increased above zero, before reaching a maximum and decreasing. This rather surprising increase in the number of free bacteria results from the same phenomenon as that described above for antibiotic treatment, whereby a decrease in the number of bound bacteria results in an increase in their growth rate and hence in the rate of their contribution of daughter cells to the exudate. In Case B this causes the total number of bacteria to increase for intermediate vales of , while in Case A the reduction in bound bacteria is greater than the increase in free bacteria such that the total number of bacteria is a monotone decreasing function of . In both Cases A and B, further increases in above about half the standard dose have relatively little effect on the steady-state bacterial population.

Case C gives the highly counter-intuitive prediction that the addition of inhibitors will increase both the number of free and bound bacteria, and, therefore, the total bacterial burden, within the range inhib. cm⁻³. This cannot be due to the phenomenon described above since B_B is close to its carrying capacity, K_B, in the absence of inhibitors and comes to slightly exceed it as is increased. Rather, the addition of inhibitors greatly decreases the free binding site density, E, such that the number of free bacteria can greatly increase before the bacterial binding terms in Eqs 1–2, which act as a sink on free bacteria, achieve a similar magnitude to that in the inhibitor free case. Indeed, since the increase in free bacterial numbers results in an increase in the growth rate of free bacteria, the magnitude of the bacterial binding terms can exceed that in the untreated case before the growth and binding terms balance in Eqs 1–2. Consequently, the bacterial binding terms, which act as a source in Eqs 3–4, make a greater contribution to the number of bound bacteria, increasing the bound bacterial burden beyond that in the untreated case. Lastly, in Case D, the addition of inhibitors has a negligible effect within the range inhib. cm⁻³. The inset graphs for Cases C and D show that at higher inhibitor doses B_B falls, while B_F and B_T rise and fall. We note that this behaviour only occurs for unrealistically high doses, thousands of times higher than the standard dose.

Fig 6 shows the effect of varying the antibiotic concentration and inhibitor dose upon the total number of bacteria at t = 672 hr (4 weeks). The full model (Eqs 1–11) was solved with a constant antibiotic concentration and with hr⁻¹, hr⁻¹, λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹ and ω = 1, for A ∈ [0, 16] μg ml⁻¹ and inhib. cm⁻³, where the antibiotic upper bound is twice the maximum concentration that could be applied constantly, while the inhibitor upper bound is fourteen times the standard dose. The white curves are the contours along which B_T(672) = 1, separating regions in which bacteria survive from those in which they are eliminated. Treatment has a relatively minor effect in Cases A and D; however, the bacterial burden can be eliminated in Cases B and C, provided the antibiotic concentration and inhibitor dose are high enough, corresponding to the region above-right of the white contours. Note that the use of a modest dose of inhibitors is sufficient to dramatically reduce the antibiotic concentration required to eliminate the bacterial burden in these cases. The equivalent results at 1 week and 1 year are given in Figs B and C in S3 Text. Treatment is less effective after 1 week than after 4 weeks in Cases B and C, but more effective in Cases A and D, while treatment is more effective in Cases B and C after 1 year than after 4 weeks and there is little change in Cases A and D over this period.

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Fig 6. Sensitivity analysis for antibiotic concentration and inhibitor dose—4 weeks.

The log₁₀ 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 (6.12×10⁷ 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 1–11 were solved using ode15s and with a constant antibiotic concentration. Parameter values: hr⁻¹, hr⁻¹, λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹ and ω = 1. See Tables 2–4 for the remaining parameter values.

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Fig 7 shows the effect of varying the antibiotic concentration and inhibitor dose upon the total number of bacteria at t = 672 hr (4 weeks) when daily debridement is included. The model details are the same as for Fig 6, except for the inclusion of debridement and the clearance of free bacteria and inhibitors ( and ). Treatment is more effective in Cases A and B than in the antibiotic and inhibitor only scenario (Fig 6) and less effective in Cases C and D. The equivalent results at 1 week and 1 year are given in Figs D and E in S3 Text. Treatment is less effective after 1 week than after 4 weeks in Cases A–C, but more effective in Case D. Comparing the results at 4 weeks with those at 1 year, bacteria are always eliminated in Case A at both times, while there is little change in Case D over this period. In Cases B and C, bacteria are eliminated for a larger range of inhibitor doses and antibiotic concentrations at 1 year; however, in those regions of parameter space where bacteria are not eliminated, B_T increases over this period.

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Fig 7. Sensitivity analysis for antibiotic concentration and inhibitor dose with debridement—4 weeks.

The log₁₀ 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 (6.12×10⁷ 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 1–11 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 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹ and ω = 1. See Tables 2–4 for the remaining parameter values.

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Factor difference in antibiotic potency against bound bacteria compared with free bacteria.

We have assumed until now that antibiotics are equally potent against bound and free bacteria, that is ω = 1. Fig 8 shows the effect of varying ω upon the total number of bacteria, B_T, the number of free bacteria, B_F, and the number of bound bacteria, B_B, at t = 672 hr (4 weeks) (we note that the majority of bacteria are resistant in all cases, such that a plot of the susceptible bacterial population size would almost always lie along the x-axis at this scale). The full model (Eqs 1–11) was solved with a constant antibiotic dose of A = 8 μg ml⁻¹ and without inhibitors, where hr⁻¹, hr⁻¹, λ = 0 cm³cell⁻¹hr⁻¹ and ρ = 0 hr⁻¹. The factor difference in antibiotic potency was varied in the range ω ∈ [0.5, 2]. Both B_B and B_T are monotone decreasing functions of ω in all cases within the range examined, while B_F is monotone decreasing in Cases A–C and increases initially, before decreasing, in Case D. The system is sensitive in all cases, and particularly in Case C, for which B_F, B_B and B_T are O(10) when ω = 2. Figs L and M in S3 Text show the equivalent results at 1 week and 1 year. The behaviour is similar to that at 4 weeks, with B_T, B_F and B_B monotone decreasing in most cases.

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Fig 8. Sensitivity analysis for the factor difference in antibiotic potency against bound bacteria compared with free bacteria—4 weeks.

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 log₂ 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 1–11 were solved using ode15s, with a constant antibiotic dose and without inhibitors. Parameter values: A = 8 μg ml⁻¹, hr⁻¹, hr⁻¹, λ = 0 cm³cell⁻¹hr⁻¹ and ρ = 0 hr⁻¹. See Tables 2–4 for the remaining parameter values.

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Optimising treatment

Informed by the preceding sensitivity analyses, we used our mathematical model to predict the optimum treatment regimens in Cases A–D under certain constraints.

Two sets of initial conditions were considered, each consisting of a mixture of susceptible and antibiotic resistant bacteria. The first set is the standard initial conditions given in Table 4, which corresponds to a new infection in which bacteria have not yet had time to bind to host cells. The second set corresponds to an established infection. Here we chose the initial conditions to be the untreated steady-states for each parameter set (Cases A–D), in which all surviving bacteria are susceptible, modified so that 2% of the free and bound bacteria are resistant.

We chose to optimise the treatment over the period of a week—this being a standard period over which to treat a bacterial infection and also reducing the number of regimens over which to search compared with longer periods—exploring combination therapies including antibiotics, inhibitors and debridement. We assume continuous dosing with antibiotics, fixing the concentration at its maximum value of A = 8 μg ml⁻¹ since this was found to have the greatest effect against bound bacteria (see Fig 4), while free bacteria can be removed using debridement. Debridement may be applied at the beginning of days 2–7, but not at the start of the first day (see Treatment types), giving 2⁶ = 64 possible treatment regimens. Lastly, we assume that inhibitors may be applied in multiples of the standard dose (6.12×10⁷ inhib. cm⁻³), using exactly seven standard doses worth of inhibitors over the week (4.28×10⁸ inhib. cm⁻³), and that inhibitors may only be applied at t = 0 hr and immediately following a debridement event. This brings the total number of possible treatment regimens to 14,407. We note that in preliminary work we used a genetic algorithm approach to investigate optimum solutions; however, there is no guarantee of identifying the global optimum via this method. Rather, by accounting for the clinical constraints on the treatment regimen (as described above), we sample the complete space of possible treatment regimens, enabling us to identify the global optimum, subject to these constraints. While the clinical constraints imposed on our optimisation limit the options to a discrete set of points in decision space, we note that the theoretical range of treatment options lies on a continuum (e.g. the timing and concentration of inhibitor doses).

Four separate optimisations were performed for each of Cases A–D and for each set of initial conditions, each using a different objective function which we sought to minimise. The first objective function gives the final number of bound bacteria, B_B(168), the second gives the final total number of bacteria, B_T(168), the third gives the mean number of bound bacteria over the week, 〈B_B〉, while the fourth gives the mean total number of bacteria over the week, 〈B_T〉. We seek to optimise for each objective function individually, rather than performing a multi-objective optimisation, since we wish to find the regimens which fully-optimise each criteria and to compare between these. A unique optimal regimen can always be found under the first two optimality criteria. In those cases where multiple treatment regimens are equally optimal under the third and fourth criteria, we designate that regimen which gives the lowest final value of B_B (for the 〈B_B〉 criterion) or B_T (for the 〈B_T〉 criterion) as being optimal. We performed separate optimisations for the bound bacterial burden since it is bound bacteria, rather than free bacteria, that actively damage host tissue. Therefore, it may be more important to remove bound bacteria than free bacteria. Further, we performed separate optimisations for the final and mean number of bacteria since we aim both to eliminate the bacterial burden as rapidly as possible (final), while also keeping the bacterial burden low during treatment (mean). In each case we search through the full set of 14,407 possible treatment regimens. We note that, unlike in the steady-state and sensitivity analyses above (with the exception of Fig 7), clearance of free bacteria and free inhibitors is included in these simulations, occurring both in the first 24 hr and in the first 24 hr after each debridement event.

Fig 9 shows the optimum treatment regimens for the B_B(168) and B_T(168) objective functions (columns) and for each parameter set (rows) in the new infection scenario. It is predicted to be optimal to apply all of the inhibitors at the start of the first day under both optimality conditions in Cases A–C and to distribute inhibitors more evenly across the week in Case D. Further, it is predicted to be optimal to debride every day (days 2–7) under both optimality conditions in Cases A and D, and to debride only on some of the later days in Cases B and C. The results under the 〈B_B〉 and 〈B_T〉 criteria are similar (see Fig 10).

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Fig 9. Optimum treatment regimens—minimising final bacterial burdens with a new infection.

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 (6.12×10⁷ 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 1–11 were solved using ode15s, with a constant antibiotic dose. Parameter values: A = 8 μg ml⁻¹, λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹ and ω = 1. See Tables 2–4 for the remaining parameter values.

https://doi.org/10.1371/journal.pcbi.1007211.g009

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Fig 10. Optimum treatment regimens—minimising mean bacterial burdens with a new infection.

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 (6.12×10⁷ 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 1–11 were solved using ode15s, with a constant antibiotic dose. Parameter values: A = 8 μg ml⁻¹, λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹ and ω = 1. See Tables 2–4 for the remaining parameter values.

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Fig 11 shows the dynamics of B_B and B_T under the optimal B_B(168) and B_T(168) treatment regimens in the new infection scenario. The total bacterial burden is eliminated by the end of the week in Case A, is reduced to O(10) in Case B, to O(10²) in Case C and to just below 10³ in Case D (where B_T(168) = O(10⁷) to O(10⁸) in the untreated scenario in Cases A–D). The results under the 〈B_B〉 and 〈B_T〉 criteria are presented in Fig 12. The difference in the bacterial dynamics between the different optimisation regimens is minor. Both here and in Figs 15 and 16 we plot just the total number of bacteria and the number of bound bacteria for clarity. In both cases the majority of bacteria are susceptible for approximately the first 2 days, after which antibiotic resistant bacteria dominate.

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Fig 11. Optimum treatment outcomes—minimising final bacterial burdens with a new infection.

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 Fig 9) for each parameter set (rows). Note the log₁₀ 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(10²) in Case C and just below 10³ 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 12, 15 and 16, see Treatment types). Eqs 1–11 were solved using ode15s, with a constant antibiotic dose. Parameter values: A = 8 μg ml⁻¹, λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹ and ω = 1. See Tables 2–4 for the remaining parameter values.

https://doi.org/10.1371/journal.pcbi.1007211.g011

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Fig 12. Optimum treatment outcomes—minimising mean bacterial burdens with a new infection.

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 Fig 10) for each parameter set (rows). Note the log₁₀ 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(10²) respectively, while B_T(168) = O(10⁴) in Case D. There is little difference in the effects of the optimum regimens under the different optimality conditions for any given case. Eqs 1–11 were solved using ode15s, with a constant antibiotic dose. Parameter values: A = 8 μg ml⁻¹, λ = 0 cm³cell⁻¹hr⁻¹, ρ = 0 hr⁻¹ and ω = 1. See Tables 2–4 for the remaining parameter values.

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Fig 13 shows the optimum treatment regimens for the B_B(168) and B_T(168) objective functions (columns) and for each parameter set (rows) in the case of an established infection. It is predicted to be optimal to apply all or most of the inhibitors at the start of the first day under both optimality conditions in Cases B–D and to distribute inhibitors more evenly across the week in Case A. In this respect the optimal debridement is similar to the new infection scenario for Cases B and C, differing in Case A, where all inhibitors were used on the first day, and Case D, where inhibitor doses were distributed throughout the week. The predicted optimal debridement patterns differ markedly from the new infection scenario, with debridement being less frequent in all cases and entirely absent in Cases C and D under the B_B(168) criterion. The results under the 〈B_B〉 and 〈B_T〉 criteria differ from all of those discussed above (see Fig 14). Here, it is predicted to be optimal to use all inhibitors at the start of day 1 for Cases A–D under the 〈B_B〉 criterion and in Case B under the 〈B_T〉 criterion, while it is better to distribute inhibitors across multiple days in Cases A, C and D under the 〈B_T〉 criterion. Further it is predicted to be optimal to debride every day in Cases A–D under the 〈B_T〉 criterion, not at all in Cases A and D under the 〈B_B〉 criterion and only on some later days in Cases B and C under the 〈B_B〉 criterion.

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Fig 13. Optimum treatment regimens—minimising final bacterial burdens with an established infection.

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 (6.12×10⁷ 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 1–11 were solved using ode15s, with a constant antibiotic dose. Parameter values: A = 8 μg ml⁻¹, λ = 0 cm³cell⁻¹hr⁻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 2–4 for the remaining parameter values.

https://doi.org/10.1371/journal.pcbi.1007211.g013

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Fig 14. Optimum treatment regimens—minimising mean bacterial burdens with an established infection.

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 (6.12×10⁷ 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 1–11 were solved using ode15s, with a constant antibiotic dose. Parameter values: A = 8 μg ml⁻¹, λ = 0 cm³cell⁻¹hr⁻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 2–4 for the remaining parameter values.

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Fig 15 shows the dynamics of B_B and B_T under the optimal B_B(168) and B_T(168) treatment regimens in the case of an established infection. Here the efficacy is more modest in comparison with the new infection scenario, as would be expected. The total bacterial burden is eliminated by the end of the week in Case A, is reduced to O(10²) in Case B, to O(10⁴)–O(10⁵) in Case C and to O(10⁵)–O(10⁶) in Case D (where B_T(168) = O(10⁷) to O(10⁸) in the untreated scenario in Cases A–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. The results under the 〈B_B〉 and 〈B_T〉 criteria are presented in Fig 16. The difference in the bacterial dynamics between the different optimisation regimens is minor, except under the 〈B_B〉 optimum regimen in Case A, for which the total bacterial burden is not eliminated since debridement is not employed.

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Fig 15. Optimum treatment outcomes—minimising final bacterial burdens with an established infection.

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 Fig 13) for each parameter set (rows). Note the log₁₀ 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(10²) in Case B, O(10⁴)–O(10⁵) in Case C and O(10⁵)–O(10⁶) 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 1–11 were solved using ode15s, with a constant antibiotic dose. Parameter values: A = 8 μg ml⁻¹, λ = 0 cm³cell⁻¹hr⁻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 2–4 for the remaining parameter values.

https://doi.org/10.1371/journal.pcbi.1007211.g015

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Fig 16. Optimum treatment outcomes—minimising mean bacterial burdens with an established infection.

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 Fig 14) for each parameter set (rows). Note the log₁₀ 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(10⁶) under the 〈B_B〉 regimen. Both regimens achieve B_T(168) = O(10⁴) in Cases B and C, and B_T(168) = O(10⁶) in Case D. There is little difference in the effects of the optimum regimens under the different optimality conditions for Cases B–D. Eqs 1–11 were solved using ode15s, with a constant antibiotic dose. Parameter values: A = 8 μg ml⁻¹, λ = 0 cm³cell⁻¹hr⁻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 2–4 for the remaining parameter values.

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