A general model-based design of experiments approach to achieve practical identifiability of pharmacokinetic and pharmacodynamic models

F. Galvanin, C.C. Ballan, M. Barolo, F. Bezzo

Online Supplemental Material 4

Case study II

In Case study II, a realistic measurements noise level has been considered by assuming a standard deviation of 0.2 (on log10 basis). The experiment design results provided by Configuration 1 and Configuration 2 for the first experiment,providing the data forNp = 4 kill curves simultaneously, are illustrated in Table S1.

Table S1.Case study II: experimental design results for the first experiment.

Configuration 1 / Configuration 2
C1 / Growth control
(no administration) / D-OPTC1 / d1= [6.5 77.0 28.2 19.5 19.3 19.8 21.6 25.5]


C2 / Continuous infusion of 12.5 mg/h (equivalent to 300 mg/day) / D-OPTC2 / d2 = 0

= 30.15 mg/h
C3 / 400 mg every 8 h / D-OPTC3 / d3 = [375.4 375.3 375.3 375.2 2199.0 2166.0 2182.0 2182.0]


C4 / 2800 mg every 12 h / D-OPTC4 / d4 = [17.4 33.7 17.2 17.4 17.4 17.4 17.4 17.4]


Configuration 1 suggests the utilisation of a growth control (C1), a kill curve at constant antimicrobial administration rate (C2) and two curveswhere the ciprofloxacin doses are evenly distributed along the experiment horizon (C3-C4); C4 is a nearly maximum-killing curve which is obtainedthanks to high ciprofloxacin doses (Figure S1), but only the first samples can be effectively used for parameter estimation. The experimental settings determined by Configuration 1 have been used as initial guess for MBDoE optimisation in Configuration 2.

(a) (b)

Figure S1. Case study II: configuration1 (first experiment).(a) Ciprofloxacin concentration profiles obtained by design. (b) Bacterial concentration profiles obtained after parameter identification. The thin short dashed line represents the detectability threshold. The experimental samples are indicated by symbols; samples taken below the detectability threshold are indicated with empty symbols.

Results for Configuration 2 suggest a different administration of ciprofloxacin doses along the experiment duration in order to maximise the information level of experiment (Figure S2). In D-OPTC1, D-OPTC3 and D-OPTC4Np = 8 different ciprofloxacin doses are optimally allocated in time, while D-OPTC2 suggests a constant antibioticadministration rate, which is higher than the one realised by Configuration 1. The concentration of the bacterial population is always kept well above the detectability threshold not only during the design phase, but also during the execution of the experiment (Figure S3). In this specific case study, the adoption of a backoff strategy is therefore unnecessary.

Figure S2. Case study II: configuration 2 (first experiment). Ciprofloxacin concentration profiles obtained by design.

Figure S3. Case study II: Configuration 2 (first experiment). Bacterial concentration profiles predicted by the model during the experiment design (dashed lines), after parameter identification (solid lines) and experimental samples (diamonds) for D-OPTC1, D-OPTC2,D-OPTC3 and D-OPTC4.The thin short dashed line represents the detectability threshold.

The parameter estimation results in terms of a posteriori statistics for the two experiment design configurations after the execution of the first experiment is shown in Table S2 (model parametersare scaled to unity using the “true” values reported in Table 1, in such a way that the unit vector represents the true vector defining the system). The parameter estimation obtained from Configuration 1 is unsatisfactory, as it fails to provide a precise description of parameters kr, EC50r and R0. Configuration 2 allows a precise and accurate description of the entire set of model parameters excepts R0, which is therefore the most critical parameter to be estimated.

Table S2. Case study II. Parameter estimation after the design and execution of the first experiment. Estimated value and a posteriori statistics (95% confidence intervals, t-values and standard deviations) obtained from Configuration 1 (tref = 1.667, Figure S1) and Configuration 2 (tref = 1.663, Figure S2 and Figure S3). Superscript (*) denotes t-values failing the t-test.

gs
[hr-] / ks
[hr-1] / EC50s
[μg/mL] / gr
[hr-1] / kr
[hr-1] / EC50r
[μg/mL] / R0
[cfu·mL-1] / Nmax
[cfu·mL-1]
Configuration 1 / Est. / 0.807 / 0.884 / 1.392 / 1.069 / 0.961 / 0.832 / 0.478 / 0.970
95% c.i. / 0.267 / 0.168 / 0.414 / 0.257 / 1.060 / 2.097 / 0.513 / 0.179
t-value / 3.0 / 5.3 / 3.3 / 4.1 / 0.9* / 0.4* / 0.9* / 5.4
st. dev. / 0.1338 / 0.0841 / 0.2079 / 0.1289 / 0.5317 / 1.052 / 0.2573 / 0.0900
Configuration
2 / Est. / 1.030 / 1.003 / 0.859 / 0.911 / 1.000 / 1.287 / 1.243 / 1.099
95% c.i. / 0.218 / 0.130 / 0.266 / 0.101 / 0.118 / 0.476 / 1.371 / 0.190
t-value / 4.7 / 7.7 / 3.2 / 9.0 / 8.5 / 2.7 / 0.9* / 5.8
st. dev. / 0.1101 / 0.0650 / 0.1339 / 0.0509 / 0.0594 / 0.2399 / 0.6899 / 0.0958

As illustrated in [1], if a traditional design approach is followed, a second experiment constituted by (at least) four kill curves (whose settings are illustrated in Table S3) is required to achieve a reliable estimation of the full set of model parameters. Conversely, adopting Configuration 2, a statistically sound estimation of the entire set of model parameters can be obtained by adding a single additional kill curve only (D-OPTC5). Results from parameter estimation (Table S4) underline that Configuration 1 tends to provide better results on the estimation of R0, while Configuration 2 allows to estimate parameters kr and EC50r in a more precise way with a reduced experimental effort.

Note that, for experiments over multiple days, additional biological mechanisms (such as those related to emergence of resistance) may manifest. If these mechanisms are not part of the original model structure, additional optimal design analyses will become warranted and are expected to be beneficial to elucidate the complex biology involved.

Table S3.Case study II: experimental design results for the second experiment.

Configuration 1 / Configuration 2
C5 / Replicate growth control / D-OPTC5 / d1 = [557.4 557.2 557.0 556.7 556.7 556.7 556.7 556.7]

dc = 0
C6 / 400 mg every 12 h / -
C7 / 750 mg every 12 h / -
C8 / 6250 mg every 12 h / -

Table S4. Case study II. Parameter estimation after the design and execution of the second experiment. Estimated value and a posteriori statistics (95% confidence intervals, t-values and standard deviations) obtained from Configuration 1 (tref = 1.658) and Configuration 2 (tref = 1.663 ).

gs
[hr-] / ks
[hr-1] / EC50s
[μg/mL] / gr
[hr-1] / kr
[hr-1] / EC50r
[μg/mL] / R0
[cfu·mL-1] / Nmax
[cfu·mL-1]
Configuration
1 / Est. / 0.993 / 0.999 / 1.115 / 1.046 / 0.989 / 0.898 / 0.681 / 0.953
95% c.i. / 0.240 / 0.151 / 0.251 / 0.071 / 0.106 / 0.279 / 0.349 / 0.104
t-value / 4.1 / 6.5 / 4.4 / 14.8 / 9.3 / 3.2 / 1.9 / 9.1
st. dev. / 0.1214 / 0.0767 / 0.1269 / 0.0357 / 0.0538 / 0.1416 / 0.1765 / 0.0529
Configuration
2 / Est. / 1.017 / 1.003 / 0.923 / 0.947 / 0.956 / 1.041 / 1.575 / 1.023
95% c.i. / 0.214 / 0.128 / 0.259 / 0.061 / 0.056 / 0.137 / 0.947 / 0.146
t-value / 4.7 / 7.8 / 3.6 / 15.4 / 16.8 / 7.6 / 1.7 / 6.9
st. dev. / 0.1101 / 0.0648 / 0.1307 / 0.0310 / 0.0287 / 0.0694 / 0.4781 / 0.0730

10E9m population size t = 0 hwhich 50% of the maximal killing occurs original paper [2] and kept constant during the design foReferences

  1. Campion J J, McNamara P J, Evans M E (2005) Pharmacodynamic modeling of ciprofloxacin resistance in Staphylococcus aureus. Antimicrob. Agents Chemother. 49:209-219.