# Pharmacodynamic Model of the Dynamic Response of Pseudomonas aeruginosa Biofilms to Antibacterial Treatments

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{max}) or drug area under the curve (AUC) for the pharmacokinetics [8]. The physical barriers posed and community nature of a bacterial biofilm are such that it may be necessary to incorporate additional factors, such as the dynamics of drug transport and the delayed, cooperative effects of drugs on biofilm bacteria, in order to better describe drug response. The better experimental quantification of the dynamics of biofilm response to various drug treatments and their incorporation into pharmacodynamic (PD) models are crucial in understanding and incorporating the concentration-dependent and dynamic effects involved in overcoming biofilm infections. For example, recent developments in the application of confocal laser scanning microscopy with flow chambers has enabled the monitoring of the real-time killing of bacterial biofilms [9,10,11].

## 2. Materials and Methods

#### 2.1. Experimental Dataset

#### 2.2. Mathematical Model

_{1}, D

_{2}, …) in which the cell membrane integrity is maintained (i.e., they do not stain with propidium iodide) but the cells are no longer able to divide. Progression from the last transit state produces dead cells (X), corresponding experimentally to the nonviable biovolume. Mass balances were used to derive kinetic equations describing the populations of healthy biofilm cells, the respective transit compartments, and dead cells. For tobramycin administration, the number of transit compartments was determined via optimization to be five, leading to the following set of balance equations:

_{s}, and the bulk concentration, C

_{0}, raised to a cooperativity factor, γ (Equation (2)). The rate constant is proportional to the diffusive flux (Appendix S1) and can be expressed in terms of two model parameters, α and β, each of which is a grouping of physical constants, to give:

## 3. Results

## 4. Discussion

_{B}represents the specific growth rate of the biofilm. As little cell growth is observed during the time course of the experiments being modeled, its value is low, and no finer detail needs to be incorporated. The inclusion of drug diffusion results in two lumped parameters, $\alpha $ and $\beta $ (Equation (6) and Supplementary Information Appendix S1). The $\beta $ parameter is the value of $\frac{\pi D}{{H}^{2}}$, where D is the diffusion coefficient, and H is the thickness of the diffusion layer, which is a combination of the hydrodynamic layer resulting from the experimental setup in a flow cell, as well as the physical barrier imposed by the biofilm itself. The lumped parameter β results from the scaling of the diffusion problem and is the inverse of the characteristic time for diffusion. Using the biofilm thickness of ~20 μm, the fit values of β would correspond to diffusion coefficients (1.3–5.2 × 10

^{−5}mm

^{2}/h). These values are several orders of magnitude lower than those for typical drugs in water [31], suggesting that the diffusivity of the drugs is reduced in the biofilm and/or there is also a mass transfer boundary layer [32]. For this reason, β was retained as a fit, rather than fixed, parameter. Since the boundary layer thickness should be the same for both drugs, the slightly higher value fit for CST than for TOB can be interpreted as a higher effective diffusion coefficient for the former compound. Although CST has a higher molecular weight than TOB, it has biosurfactant properties that may allow it to diffuse (penetrate) more rapidly in the biofilm barrier [33].

_{t}, corresponds to the intercompartmental transit rate of the drug. This is not typically found in other PD models that are designed to capture data at one time point; however, it is critical in capturing the overall dynamic behavior of the drug and its effect on biofilm killing.

_{t}has the greatest impact on the model overall. Conceptually, this is expected as the intercompartmental transit rate dictates progress through the “death cascade” as well as the cellular response dynamics to the drug. The value of β dictates the delay due to diffusion between drug administration and cellular effects. It couples with k

_{t}and has a strong influence on the output. The pharmacologic rate constant, α, has a modest effect on the output, while the cooperativity, γ, exerts a stronger influence.

_{t}increases in order to mimic the effect of one transit compartment (Tables S1 and S2).

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Pharmacodynamic model structure for tobramycin. The pharmacodynamic model for response to tobramycin (subscript ‘t’) tracks the transit of biofilm cells going from a viable (B) to nonviable state (X

_{t}) following administration of tobramycin at bulk concentration C

_{0t}. There is a flux, J(D) of drug from the bulk to the biofilm cells, where the local concentration is C

_{t}. For tobramycin, there are five transit compartments (D

_{1t}, D

_{2t}, D

_{3t}, D

_{4t}, and D

_{5t}) mediating the cellular response to drug. Growth is governed by a specific growth rate, μ; the coupled diffusion and pharmacodynamic response are subject to parameters α, β, and γ; and the transit rate to subsequent compartments is given by k

_{tt}.

**Figure 2.**Dynamics of Pseudomonas aeruginosa killing in response to tobramycin at 20 µg/mL. The left panel shows the model comparison of the dead cells, shown in red, with the experimental data, shown in blue. The right panel shows the populations in each of the cellular compartments over the same time course.

**Figure 3.**Tobramycin model fits across a ten-fold concentration range. The PD model was used to simulate the treatment of biofilms treated with tobramycin at drug concentrations of 5, 20, and 50 µg/mL. The model was fit to the ensemble data of all three concentrations. The experimental data are shown in blue, and the model is shown in red.

**Figure 4.**Transient exposure of biofilms to tobramycin. The same set of parameters for the continuous data was used to simulate the treatment of biofilms using tobramycin transiently for four hours at the same drug concentrations of 5, 20, and 50 µg/mL. The experimental data are shown in blue, and the model is shown in red.

**Figure 5.**The pharmacodynamic model for colistin. (

**A**) Model structure with a single transit compartment. The symbols have the same meanings as in Figure 1, with subscripts changed from t (tobramycin) to c (colistin). (

**B**) Data and model fit for colistin killing of P. aeruginosa biofilms at 10 µg/mL. The model fit uses parameters determined by fitting to the ensemble of colistin continuous response data.

**Figure 6.**Colistin model fits across a ten-fold concentration range. The PD model was used to simulate the treatment of biofilms using colistin concentrations of 2.5, 10, and 25 µg/mL. The data of all three concentrations were used to fit one set of parameters, which was used to model the various concentrations shown above. The experimental data are shown in blue, and the model is shown in red.

**Figure 7.**Transient exposure of biofilms to colistin. The same set of parameters for the continuous data was used to simulate the treatment of biofilms using colistin for four hours at concentrations of 2.5, 10, and 25 µg/mL. The experimental data are shown in blue, and the model is shown in red.

**Figure 8.**Treatment with drug combinations. (

**A**) A schematic model for the combination treatment with two drugs. Transit compartments resulting from exposure to tobramycin and colistin are considered as parallel death pathways, with the possibility for crossover from the slow-acting tobramycin to fast-acting colistin pathway. The same parameters that were derived earlier for each respective drug were used to predict outcomes of combination treatments. (

**B**) Combination treatments were tested using various dose combinations of tobramycin and colistin over the course of 24 h. The mathematical model is shown in red, and the experimental data are shown in blue.

**Figure 9.**Impact of model simplifications on performance. The TOB continuous treatment was used to analyze the parameters within the model. Each TOB concentration was fit to the data without one indicated parameter. The ‘Full Model’ column shows the fit with the original parameter values found in Table 1.

**Figure 10.**Parametric sensitivity analysis. The TOB 20 µg/mL continuous data were used as the base case. The time course of dead biovolume (X*) production was compared for values of each parameter, varied one at a time. The black line represents the output when the parameter is at its best fit value (Table 1), the red and pink lines show the effect of higher values, and the green and blue lines illustrate the effect of lower values of each parameter.

**Figure 11.**Optimal number of transit compartments. The TOB 20 µg/mL and CST 10 µg/mL continuous data were used to determine the ideal number of compartments for the model. On the left graph, the model with three compartments gave the least error between the model and data for tobramycin. The graph on the right shows the model with a differing number of compartments for colistin, and as shown, only one compartment produced the least error between the model and data. The parameter values for each compartment can be found in the tables below.

Parameter | Description | Units | TOB Value | CST Value |
---|---|---|---|---|

μ_{B} | Growth rate of biofilm population | h^{−1} | 0.0321 | 0.0001 |

α | Rate constant for drug effect on biofilm | (μg/mL)^{γ−1} h^{−1} | 0.0002 | 0.0082 |

β | Normalized drug diffusivity | h^{−1} | 0.2088 | 0.3986 |

γ | Cooperativity in drug effect on biofilm | 3.5330 | 4.4313 | |

k_{t} | Intercompartmental transit rate of drug | h^{−1} | 0.5424 | 1.8924 |

Parameter | Full Model | Without μ_{B} | Without β | Without γ | Without k_{t} |
---|---|---|---|---|---|

μ_{B} | 0.0321 | - | 0.5372 | 0.0289 | 0.0000 |

α | 0.0002 | 0.0012 | 0.0072 | 0.0094 | 0.0351 |

β | 0.2088 | 0.0815 | - | 0.2480 | 0.0200 |

γ | 3.5330 | 3.2862 | 1.0705 | - | 3.4315 |

k_{t} | 0.5424 | 0.4370 | 0.4228 | 0.8053 | - |

Error | 0.0561 | 0.0971 | 0.4337 | 0.7883 | 1.6449 |

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**MDPI and ACS Style**

Roychowdhury, S.; Roth, C.M.
Pharmacodynamic Model of the Dynamic Response of *Pseudomonas aeruginosa* Biofilms to Antibacterial Treatments. *Biomedicines* **2023**, *11*, 2316.
https://doi.org/10.3390/biomedicines11082316

**AMA Style**

Roychowdhury S, Roth CM.
Pharmacodynamic Model of the Dynamic Response of *Pseudomonas aeruginosa* Biofilms to Antibacterial Treatments. *Biomedicines*. 2023; 11(8):2316.
https://doi.org/10.3390/biomedicines11082316

**Chicago/Turabian Style**

Roychowdhury, Swarnima, and Charles M. Roth.
2023. "Pharmacodynamic Model of the Dynamic Response of *Pseudomonas aeruginosa* Biofilms to Antibacterial Treatments" *Biomedicines* 11, no. 8: 2316.
https://doi.org/10.3390/biomedicines11082316