A Biomathematical Model of Pneumococcal Lung Infection and Antibiotic Treatment in Mice

Abstract

Pneumonia is considered to be one of the leading causes of death worldwide. The outcome depends on both, proper antibiotic treatment and the effectivity of the immune response of the host. However, due to the complexity of the immunologic cascade initiated during infection, the latter cannot be predicted easily. We construct a biomathematical model of the murine immune response during infection with pneumococcus aiming at predicting the outcome of antibiotic treatment. The model consists of a number of non-linear ordinary differential equations describing dynamics of pneumococcal population, the inflammatory cytokine IL-6, neutrophils and macrophages fighting the infection and destruction of alveolar tissue due to pneumococcus. Equations were derived by translating known biological mechanisms and assuming certain response kinetics. Antibiotic therapy is modelled by a transient depletion of bacteria. Unknown model parameters were determined by fitting the predictions of the model to data sets derived from mice experiments of pneumococcal lung infection with and without antibiotic treatment. Time series of pneumococcal population, debris, neutrophils, activated epithelial cells, macrophages, monocytes and IL-6 serum concentrations were available for this purpose. The antibiotics Ampicillin and Moxifloxacin were considered. Parameter fittings resulted in a good agreement of model and data for all experimental scenarios. Identifiability of parameters is also estimated. The model can be used to predict the performance of alternative schedules of antibiotic treatment. We conclude that we established a biomathematical model of pneumococcal lung infection in mice allowing predictions regarding the outcome of different schedules of antibiotic treatment. We aim at translating the model to the human situation in the near future.

Fig 1. Structure of the model.

Compartments: pneumococcal population (P), IL-6, debris (D), unaffected epithelial cells (EU), epithelial cells with pneumococci attached to cell surface receptors (EA) and macrophages/neutrophils (M, N).

Fig 2. Antibiotic therapy.

The resulting antibiotic effect function A(t) of Ampicillin (0.02mg/g) and Moxifloxacin (0.1 mg/g) is shown. Antibiotics were given every 12 hours, starting 24 hours after infection.

Fig 3. Modelling different initial doses of pneumococci without treatment.

Time series of pneumococcal population, neutrophils, IL-6, debris, EA and macrophages in BALF are presented. The solid black curves represent simulation results with an initial bacterial load of 2.5 ⋅ 10⁵, the dashed lines with 2.4 ⋅ 10⁵. As one can see, the smaller initial dose is successfully removed by the immune system, while larger doses result in disseminated infection.

Fig 4. Bacterial growth and elimination by the immune system.

We simulated different initial dosages of bacteria without antibiotic treatment. A: In dependance on the inhalated dose (x axis), we calculate the total amount of bacteria produced over a period of 120 hours per μl of BALF (blue line), the total amount of bacteria eliminated by neutrophils over this period (black dashed line), and the total amount of bacteria eliminated by macrophages over the same period (black solid line). The green dashed line represents the sum of bacteria eliminated by neutrophils and macrophages over the period. As one can see, immune system is unable to remove all bacteria if the initial dose exceeds about 2.5 ⋅ 10⁵. B: In dependance on the initial dose, the percentage of bacteria eliminated by neutrophils respectively macrophages over a period of 120 hours is depicted. After infection with lower doses, 100% of bacteria were eliminated by macrophages (solid line) over a period of 120 hours. For higher doses, neutrophils are also involved (dashed line).

Fig 5. Sensitivity of model parameters.

Single parameter values were changed by ±2.5% while the other parameters were kept constant. Corresponding percental deterioration of the fitness function Eq (17) was evaluated as a measure of sensitivity of the considered parameter. Longer bars correspond to more sensitive parameters, i.e. better identifiability.

Fig 6. Modelling infection without treatment.

We compare results of model simulation with data from [9] without antibiotic treatment. Time series of pneumococcal population, neutrophils, IL-6, debris, EU, EA and macrophages in BALF are displayed. Solid black curve represents simulation results. Circles represent data points. The green dashed lines represent minimal and maximal observed values.

Fig 7. Ampicillin treatment.

We compare results of model simulation with data from [9] under Ampicillin treatment. Therapy starts 24 hours after infection and is continued every 12h. Time series of pneumococcal population, neutrophils, IL-6, debris, EU, EA and macrophages in BALF are displayed. Solid black curve represents simulation results. Circles represent data points. The green dashed lines represent minimal and maximal observed values.

Fig 8. Moxifloxacin treatment.

We compare results of model simulation with data from [9] under Moxifloxacin treatment. Therapy starts 24 hours after infection and is continued every 12h. Time series of pneumococcal population, neutrophils, IL-6, debris, EU, EA and macrophages in BALF are displayed. Solid black curve represents simulation results. Circles represent data points. The green dashed lines represent minimal and maximal observed values.

Fig 9. Prediction: Moxifloxacin treatment.

Treatment with different doses of Moxifloxacin after infection with 5 ⋅ 10⁶ Streptococcus pneumoniae are simulated. Moxifloxacin is given every 12 hours starting 24 hours after infection. Time series of pneumococcal population, neutrophils, IL-6, debris, EU, EA and macrophages in BALF are presented. Solid red curves represent the predicted time course with lower dosed Moxifloxacin (0.01 mg/g). The black dashed line represents the simulation results of the higher dose (0.1 mg/g). Circles represent time points of antibiotic treatment.

Fig 10. Prediction: Different schedules of antibiotic treatment.

Treatment with different schedules of Ampicillin or Moxifloxacin after infection with 5 ⋅ 10⁶ Streptococcus pneumoniae are simulated. Time series of pneumococcal population, neutrophils, IL-6, debris, EU, EA and macrophages in BALF are presented. Ampicillin 0.02 mg/g is given every 12 hours starting 24 hours after infection (dashed blue line) or every 48 hours starting 24 hours after infection (red line). Moxifloxacin 0.1 mg/g is given every 12 hours starting 24 hours after infection (dashed black line) or every 48 hours starting 24 hours after infection (green line).

Fig 11. Prediction: Dose reduction in antibiotic treatment.

Treatment with modified Ampicillin schedule after infection with 5 ⋅ 10⁶ Streptococcus pneumoniae are simulated. Time series of pneumococcal population, neutrophils, IL-6, debris, EU, EA and macrophages in BALF are presented. Therapy starts at 24 h after infection with 0.02 mg/g. Then, dose is reduced to 0.01 mg/g given every 12 hours. Ampicillin treatment is stopped after 48 h (red line) or 60 h (green line). The black dashed line shows the standard treatment with Ampicillin 0.02 mg/g every 12 hours, stopped after 108 hours.