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(A) One-compartment model of creatinine kinetics. (B) Two-compartment model of creatinine kinetics.

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The rise and fall of SCr after severe AKI and recovery. In this simulation, creatinine clearance dropped acutely by 90% at 8 h and then recovered acutely to baseline levels 7 d later. Results from one- and two-compartment models of creatinine kinetics are shown.

The rise in SCr after severe AKI, defined as a 90% reduction in CrCl from baseline, is shown in
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(two-compartment model), according to baseline kidney function (no CKD and stages 2, 3, and 4 CKD). At 24 h after severe AKI, the
*
absolute
*
increase in SCr is nearly identical (1.8 to 2.0 mg/dl) irrespective of whether CKD is present. By contrast, the
*
percentage
*
increases over baseline are 246% (no CKD), 173% (stage 2), 92% (stage 3), and 47% (stage 4). The results are similar for less severe AKI, defined as a 50% reduction in CrCl from baseline (
Figure 4
). After 24 h, the
*
absolute
*
increase in SCr ranges from 0.6 mg/dl (no CKD) to 0.9 mg/dl (stage 4), whereas the
*
percentage
*
increase in SCr ranges from 23% (stage 4) to 78% (no CKD). For both severe and less severe AKI, the time necessary to reach a 50% percent increase in SCr increases markedly with increasing stages of CKD, whereas the time necessary to reach an absolute 0.5-mg/dl increase remains relatively constant (
Figures 3
and
4
).
Table 2
illustrates the application of the RIFLE criteria to AKI in patients with and without CKD and how different AKI stages are reached despite an identical reduction in CrCl.

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SCr concentrations after an abrupt 90% reduction in CrCl, superimposed on four different levels of baseline kidney function (no CKD and stages 2 through 4 CKD). Solid squares show the point at which a 100% increase in SCr has occurred; open triangles show the point at which a 1.0-mg/dl increase in SCr has occurred.

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SCr concentrations after an abrupt 50% reduction in CrCl, superimposed on four different levels of baseline kidney function (no CKD and stages 2 through 4 CKD). Solid squares show the point at which a 100% increase in SCr has occurred; open triangles show the point at which a 1.0-mg/dl increase in SCr has occurred.

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Classification of AKI as proposed by the RIFLE criteria after severe (90% reduction in CrCl) and moderate (50% reduction in CrCl) AKI, according to baseline level of kidney function

The trajectory of the SCr increase differs according to the severity of AKI and baseline kidney function, as seen in
Figures 3
and
4
. The time to reach within 0.1 mg/dl of predicted steady-state SCr concentrations is shown in
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, according to baseline kidney function and percentage reduction in CrCl. At any given level of baseline kidney function, the more severe the AKI, the longer it takes to approach steady-state SCr concentrations. Similarly, at any given percentage reduction in CrCl, the higher the baseline SCr (
*
i.e.
*
, higher CKD stage), the longer it takes to approach steady-state SCr concentrations. Simple calculations from a single-compartment model also illustrate this point. A 100% increase in SCr from 2.0 to 4.0 mg/dl requires retention of 2.0 mg/dl creatinine throughout the volume of distribution (420 dl), or 840 mg of creatinine. By contrast, the same percentage increase from 1.0 to 2.0 mg/dl requires retention of 420 mg of creatinine. At a constant creatinine generation rate of 60 mg/h and complete cessation of CrCl, the time required to reach a 100% increase is 14 h, when baseline SCr is 2.0, and 7 h, when baseline SCr is 1.0 mg/dl.

In a first step, the above methodology is applied to investigate the difference between the biasing policies assuming the scenarios of Examples 1 and 2. We set the favoured treatment groups to be for biasing policy I and for biasing policy II. We assume an selection effect of
*
η
*
=
*
f
*
_{
4,3
}
= 1.07.
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shows the result of the comparison for the sample size
*
N
*
= 12 based on the distribution of the type I error probabilities following
Eq 11
. It can be seen that the distribution of the type I error probabilities is shifted away from the nominal significance level of 5% in all investigated settings. In case of a single block of length
*
N
*
(PBD(
*
N
*
)), the influence of the biasing policies was comparable. For smaller block sizes, biasing policy II leads to higher type I error probabilities than the biasing policy I.

Fig 1.
Distribution of the type I error probability under selection bias for different biasing policies.

Each scenario is based on a sample of
*
r
*
= 10,000 sequences, sample size
*
N
*
= 12 and number of treatment groups
*
K
*
= 3, assuming the selection effect
*
η
*
=
*
f
*
_{
4,3
}
= 1.07 for permuted block design (PBD). The red dashed line marks the 5% significance level.

https://doi.org/10.1371/journal.pone.0192065.g001

In the second step, we restricted our attention to the strict biasing policy with to investigate the impact of selection bias under variation of the number of groups, the sample size and the selection effect. To that aim, we varied the number of treatment groups
*
K
*
∈ {3, 4, 6} and the number of patients per group
*
m
*
=
*
N
*
/
*
K
*
∈ {4, 8, 32}, speaking of a small trial if
*
m
*
= 4, a medium trial if
*
m
*
= 8, and a large trial if
*
m
*
= 32. Figs
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and
3
show the proportion of sequences that lead to an inflation of the type I error probability as proposed in
Eq 12
. In
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we fixed the selection effect
*
η
*
=
*
f
*
_{
m
,
K
}
, but varied
*
K
*
and
*
m
*
. In
Fig 3
we fixed the number of groups at
*
K
*
= 3, but varied
*
η
*
=
*
ρ
*
⋅
*
f
*
_{
m
,
K
}
and
*
m
*
. In all scenarios we investigated, at least thirty percent of the sequences in the sample lead to an inflation of the type I error-probability. However, the maximum proportion of inflated sequences varied according to the randomization procedure. The permuted block design with block size
*
K
*
had up to 100% of inflated sequences in medium and large trials (middle and right hand panels of Figs
2
and
3
). For permuted block randomization with block length
*
N
*
/2 or
*
N
*
, the proportion of inflated sequences ranged up to 84% right hand panel of
Fig 3
and 76% middle panel of
Fig 3
and generally attained its maximum in large trials with
*
K
*
= 3 treatment groups. For all the randomization procedures we investigated, the proportion of inflated sequences grew when the number of treatment groups remained the same but the number of patients per group was increased. Consider for example the situation of
*
K
*
= 6 treatment groups and permuted block design with block length
*
K
*
shown in red in
Fig 2
. In a small trial, one third of the sequences had inflated type I error probability. This proportion was more than doubled in a medium trial (71%), and reached 100% in a large trial. Interestingly,
Fig 3
shows that the proportion of sequences with inflated type I error probability remained constant when the selection effect
*
η
*
=
*
ρ
*
⋅
*
f
*
_{
m
,
K
}
was varied with
*
ρ
*
∈ {0, 1/4, 1/2, 1} and the number of groups was fixed to
*
K
*
= 3. This means that already a relatively small bias can lead to the same proportion of sequences with inflated type I error probability as a large bias.
Table 3
shows that this is also true for
*
K
*
= 4 and
*
K
*
= 6. For
*
η
*
=
*
ρ
*
= 0, all sequences maintain the type I error in all investigated scenarios, as expected.

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