Fig 1.
Schematic overview of descriptors of effective drug concentration and the used models.
(A). This graph shows the in vivo concentration profile after drug ingestion in arbitrary units. Three descriptors are commonly used to predict efficacy: The time above a specific threshold, commonly the MIC (dashed line, TC>MIC), the peak concentration (Cmax), or the area under the curve (AUC, hatched area). (B) Overview of the used models. Model 1 (upper panel) only follows the extracellular antibiotic concentration, bound and unbound target molecules and assumes that the antibiotic concentration remains constant over time during the period of administration. Model 2 (middle panel) allows for fluctuating antibiotic concentrations and a diffusion barrier between antibiotic molecules outside the cell and their intracellular targets. Model 3 (lower panel) incorporates the reproduction of target molecules through bacterial replication and considers unspecific binding of the antibiotic.
Fig 2.
Delay to onset of antibiotic action depends on turnover rate and target occupancy at MIC.
(A) A definition of the MIC based on physicochemical characteristics. This graph shows the expected MIC in mol/L (based on Eq (3), y-axis) as a function of target occupancy at MIC (fc, x-axis). The colors indicate different affinities of drug target binding (KD). Blue: KD = 10−7 M, yellow: KD = 10−6 M, green: KD = 10−5 M. (B) This graph illustrates the time course of drug-target reaction (based on Eq (4)) for various parameter sets and a fixed antibiotic concentration just above MIC (1.01 x MIC). Dotted lines: Slow turnover rate of antibiotic-target binding with half-life of drug-target complex tbound = 1h 55 min (kr = 10−4). Solid lines: Fast turnover rate of antibiotic-target binding with half-life of drug-target complex tbound = 11.5 min (kr = 10−3). The colors indicate different target occupancies at MIC. Red: fc = 90%, Dark blue: fc = 50%, Yellow: fc = 10%. The grey lines at 90%, 50% and 10% indicate the fc, the threshold of bound target required to kill/inactivate the cell. The light blue solid and dotted vertical lines indicate when the fast and slow reactions reach the fc, i.e. the time of onset of the antibiotic action (tonset).
Fig 3.
The benefit of high drug concentrations depends on the velocity with which the reaction reaches equilibrium.
These graphs show the dependence of the onset of antibiotic action as measured in time required for reaching the threshold fc on the drug concentration in fold-MIC. It is based on Eq (5). (A) Illustration of scenarios with different thresholds fc with a constant drug target half-life tbound of 36.5 min (kr = 10−3.5). Blue: fc = 10%, yellow: fc = 50%, green: fc = 90%. (B) Illustration of scenarios with different drug target half-lives tbound with a constant threshold fc = 50%. Blue: tbound = 11.5 min (kr = 10−3), yellow: tbound = 36.5 min (kr = 10−3.5), green: tbound = 1h 55min (kr = 10−4).
Fig 4.
Turnover rate and target occupancy at MIC determine end of antibiotic action.
This graph illustrates the time course of drug-target dissociation after 99.9% of the target was bound according to an exponential decay . Blue: tbound = 11.5 min (kr = 10−3), yellow: tbound = 36.5 min (kr = 10−3.5), green: tbound = 1h 55min (kr = 10−4). The grey lines at 90%, 50% and 10% illustrate when the fraction of bound target falls below a particular threshold fc.
Table 1.
Parameters and references.
Fig 5.
Biochemical properties shape antibiotic pharmacodynamics.
These graphs show the expected dynamics of antibiotic-target reaction according to model 2 with all parameters adapted to ampicillin except stated. The x-axes show the time after initiation of antibiotic therapy in hours, the y-axes the current antibiotic concentration in fold MIC (black, left side) and the % bound target (violet, right side). Note that the y-axis is on a logarithmic scale. The green line shows the antibiotic concentration outside and the red inside the cell (both refer to the y-axis on the left), the violet line shows the amount of bound target (refers to y-axis on the right). The grey area indicates that either the antibiotic concentration is below MIC or the fraction of bound target is below the inhibitory threshold fc. The dotted vertical lines indicate beginning and end of antibiotic action. Graphs in the first column depict bolus injections with an initial antibiotic concentration of 50MIC and a half-life of 1h. The second column shows a hypothetical dosing regimen with a constant concentration just above the MIC (1.01 MIC) that has the same TC>MIC as in the first column. The third column shows a hypothetical dosing regimen with a constant concentration just above the MIC (1.01 MIC) that has the equivalent area under the curve (AUC) as in the first column. Note the different timescale in the third column. (A) Biochemical properties are sufficient to explain time-dependent action of beta-lactams. The graphs show drug-target binding expected based on physicochemical characteristics of ampicillin drug-target binding from the literature (Table 1). (B) Area under the curve is best predictor of antibiotic action for equilibration times in the range of hours. We introduced a diffusion barrier of p = 10-4s-1 while all other parameters remain as in (A). (C) Peak concentration is best predictor of drug action when equilibration is slow. We introduced a stronger diffusion barrier of p = 10-5s-1 while all other parameters remain as in (A).
Fig 6.
Correlation between pharmacokinetic drivers and time until average cells are predicted to killed with isoniazid.
This graph shows the time until the threshold fc is reached after simulated bolus injections (model 2b) of 5-100x MIC isoniazid (typical Cmax values during therapy are at around 25x MIC [51, 52]) and a half-life of 0.5 – 4h. We used experimentally determined values for the MIC (0.1mg/l) for M. tuberculosis [53]. The different panels show the correlation between the time until the threshold is reached and different pharmacokinetic measures: (A) time above MIC; (B) area under the curve; (C) Cmax. We correlated the logarithm of tonset with the logarithm of the three pharmacokinetic indices in single linear regressions.
Fig 7.
Antibiotic action of tetracycline used in different concentrations.
These graphs show results of numerical simulations of Eq (10) parameterized with previously fitted values [23]. The x-axis shows the time in hours after antibiotic administration. The different colors indicate the initial antibiotic concentration in fold MIC (see legend). (A) Simplified pharmacokinetics (first-order clearance) of a tetracycline bolus injection with a half-life of 6h. The y-axis indicates the antibiotic concentration in fold MIC. The grey shaded area indicates an antibiotic concentration below MIC. The vertical dotted lines indicate when the antibiotic concentration for different dose levels falls below MIC. (B) Effect of supra-MIC doses on bacterial growth rate (y-axis). Again, the vertical dotted lines indicate when the antibiotic concentration for different dose levels falls below MIC. (C) Effect of sub-MIC concentrations of tetracycline on bacterial population size (y-axis).
Fig 8.
Time above MIC is insufficient predictor when sub-MIC concentrations are biologically active.
These graphs show results from the numerical simulations shown in Fig 7. Antibiotic efficacy is measured in fold reduction of total bacterial load during 24h normalized to a bacterial population growing in the absence of antibiotics. Blue = peak concentration < MIC, red = peak concentration > MIC. The lower panels in (A) and (B) show sub- and supra-MIC concentrations separately for clarity. (A) This graph shows the correlation between TC>MIC and antibiotic efficacy. (B) This graph shows the correlation between area under the curve and antibiotic efficacy.
Table 2.
Explanation of variables, constants and parameters.