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nca.tss

Time to steady state from the trough concentrations of the dosing intervals.

Repeated dosing approaches a plateau: the trough of every dosing interval rises until the amount eliminated over an interval equals the amount given, and the time at which that plateau is practically reached is what a study design and a dose escalation decision need. ICH M13A asks for the evidence outright ("applicants should document appropriate dosage administration and sampling to demonstrate the attainment of steady-state").

A multiple dose analysis already reports the trough of every dosing interval (interval_ctrough, the value at the end of the interval, and interval_cmin, its smallest value), so the estimate is a curve through those troughs. The trough of an interval is the value at its end, so interval_end is the time it was taken at and the time the troughs are read against; interval_start is what the stepwise estimate reports, the start of the interval from which the troughs no longer rise. time_to_steady_state offers the two estimators of PKNCA (pk.tss.monoexponential, pk.tss.stepwise.linear):

  • monoexponential: the troughs approach the plateau as \(C_\mathrm{trough}(t) = C_\mathrm{ss}\left(1 - e^{-k t}\right)\), which is the accumulation of a one compartment drug, and the time to reach a fraction \(f\) of \(C_\mathrm{ss}\) is \(t_\mathrm{ss} = -\ln(1 - f) / k\). It is a smooth estimate which uses every interval and reports the plateau itself.
  • stepwise: no model. The troughs from interval \(i\) on are regressed linearly against time and the slope is tested against 0; the first interval from which the trend is no longer significant at alpha is where the plateau starts, and its start is the estimate. It is the conservative estimate of a study report, since it asks only that the troughs stop rising.

Both are estimates of a design quantity and not of a parameter of the drug: a study which stops before the plateau reports a time to steady state which is its own last interval, and a study with two intervals reports nothing.

TSSResult dataclass 

TSSResult(tss, method, fraction, c_ss)

Time to steady state of every sample of a multiple dose analysis.

Attributes:

Name Type Description
tss DataArray

the time to steady state per sample, in the time unit of the analysis, over the sample dimensions of the result; NaN for a sample whose troughs do not carry the estimate
method TSSMethod

the estimator, "monoexponential" or "stepwise"
fraction float

the fraction of the plateau the monoexponential estimate reports the time to; it does not apply to the stepwise estimate
c_ss DataArray | None

the estimated plateau per sample of the monoexponential estimate, None for the stepwise one

to_dataframe 

to_dataframe()

The estimate as one row per sample.

Returns:

Type Description
DataFrame

The sample coordinates, tss and, for the monoexponential
DataFrame

estimate, c_ss; the single row of a result of one curve
DataFrame

(nca_single, no sample dimensions) with its scalar coordinates.

time_to_steady_state 

time_to_steady_state(
    result,
    *,
    method="monoexponential",
    fraction=0.9,
    alpha=0.05,
)

Time to steady state from the troughs of the dosing intervals.

The troughs of every sample (interval_ctrough, interval_cmin when the analysis reports no trough) are read against the end of their interval (interval_end), the time the trough was taken at, and the plateau is estimated with one of the two methods of the module, which are those of PKNCA (pk.tss):

  • "monoexponential" fits \(C_\mathrm{trough}(t) = C_\mathrm{ss}\left(1 - e^{-k t}\right)\) by least squares and reports \(t_\mathrm{ss} = -\ln(1 - f) / k\), the time to the fraction \(f\) of the plateau, together with the plateau; \(t\) is measured from the start of the first dosing interval and the estimate is reported in the times of the analysis;
  • "stepwise" regresses the troughs from every interval on linearly and reports interval_start of the first interval from which the slope is no longer significant at alpha.

Parameters:

Name Type Description Default
result NCAResult

the result of a multiple dose analysis with per-interval parameters (NCAOptions(intervals=True), the default)

 required

Other Parameters:

Name Type Description
method TSSMethod

the estimator
fraction float

the fraction of the plateau of the monoexponential estimate, 0.9 by default (ninety percent of steady state)
alpha float

significance level of the trend test of the stepwise estimate

Returns:

Type Description
TSSResult

The estimate per sample.

Raises:

Type Description
ValueError

if the result carries no per-interval troughs, if fraction is not in (0, 1) or if alpha is not in (0, 1).