Noncompartmental Analysis

NCA parameter formulas

Plasma or serum data
Urine data
Sparse sampling (pre-clinical) data
Drug effect data model 220
User defined parameters
References

See also “NCA models” and, for additional reading, see “References”.

Plasma or serum data

Data structure: NCA for blood concentration data requires the following input data:

•Time of each sample

•Plasma or serum concentrations

Output: Models 200–202 estimate the parameters in the following tables.

Plasma parameters that do not require Lambda Z estimation
Plasma parameters that are estimated when Lambda Z is estimated
Plasma parameters that are estimated when at steady-state

Plasma parameters that do not require Lambda Z estimation
Parameter	Definition
Dosing time	Available as ‘Time’ in the Dosing Used results. Time of last administered dose. It is assumed to be zero unless otherwise specified. This parameter is used mainly with steady-state data, where time may be coded as the time elapsed since the first dose, or the elapsed time since the time of the first dose.
N_Samples	This parameter reports the number of non-missing observations used in the analysis of the profile (time is at or after dosing time, the observation is numeric, and the volume is positive for urine models). It does not count points inserted by engine, e.g., inserted at dosing time.
Dose	Amount of last administered dose. This is assumed to be zero if not specified.
No_points_Lambda_z	Number of points used in computing Lambda Z. If Lambda Z is not estimable, zero.
Tlag	Time of observation prior to the first observation with a measurable (non-zero) concentration. For plasma models, Tlag is only computed when the dosing type is extravascular.
Tmax	Time of maximum observed concentration. For non-steady-state data, the entire curve is considered. If the maximum observed concentration is not unique, then the first maximum is used.
Cmax	Maximum observed concentration, occurring at time Tmax, as defined above.
Cmax_D	=
C0	Initial concentration. Given only for IV Bolus dosing. It is equal to the first observed concentration value if that value occurs at the dose time. Otherwise, it is estimated by back-extrapolating (see “AUC_%Back_Ext(_obs, _pred)” below).
Tlast	Time of last measurable (positive) observed concentration.
Clast	Observed concentration corresponding to Tlast.
AUClast	Area under the curve from the time of dosing to the time of the last measurable (positive) concentration (Tlast).
AUClast_D	=
AUCall	Area under the curve from the time of dosing to the time of the last observation. If the last concentration is positive, AUClast=AUCall. Otherwise, AUCall will not be equal to AUClast, as it includes the additional area from the last measurable (positive) concentration down to zero or negative observations.
AUMClast	Area under the moment curve from the time of dosing to the last measurable (positive) concentration.
MRTlast	Mean residence time from the time of dosing to the time of the last measurable concentration. For non-infusion models: = For infusion models: = where TI is the length of infusion.

The following table includes several parameters that are extrapolated to infinity. These parameters are calculated two ways: based on the last observed concentration (indicated by “_obs” appended to the parameter name), or based on the last predicted concentration (indicated by “_pred” appended to the parameter name), where the predicted value is based on the linear regression performed to estimate Lambda Z.

Plasma parameters that are estimated when Lambda Z is estimated
Parameter	Definition
Rsq	Goodness of fit statistic for the terminal elimination phase.
Rsq_adjusted	Goodness of fit statistic for the terminal elimination phase, adjusted for the number of points used in the estimation of Lambda Z.
Corr_XY	Correlation between time (X) and log concentration (Y) for the points used in the estimation of Lambda Z.
Lambda_z	First-order rate constant associated with the terminal (log-linear) portion of the curve. Estimated by linear regression of time vs. log concentration.
Lambda_z_intercept	Intercept on log scale estimated via linear regression of time vs. log concentration.
Lambda_z_lower	Lower limit on time for values to be included in the calculation of Lambda Z.
Lambda_z_upper	Upper limit on time for values to be included in the calculation of Lambda Z.
HL_Lambda_z	Terminal half-life=
Span	=
Clast_pred	Predicted concentration at Tlast: = exp(Lambda_z_intercept – Lambda_z*Tlast)
AUCINF(_obs, _pred)	AUC from time of dosing extrapolated to infinity, based on the last observed concentration (_obs) or last predicted concentration (_pred). =
AUCINF_D(_obs, _pred)	=
AUC_%Extrap(_obs, _pred)	Percentage of AUCINF(_obs, _pred) due to extrapolation from Tlast to infinity: =
AUC_%Back_Ext(_obs, _pred)	Computed for IV Bolus models. Percentage of AUCINF that was due to back extrapolation to estimate C0 when the first measured concentration is not at dosing time.
Vz(_obs, _pred) Vz_F(_obs, _pred)^a	Volume of distribution based on the terminal phase. For non-steady-state data: =
Cl(_obs, _pred) Cl_F(_obs, _pred)^a	Total body clearance for extravascular administration. =
AUMCINF(_obs, _pred)	Area under the first moment curve (AUMC) extrapolated to infinity, based on the last observed concentration (obs) or the last predicted concentration (pred). =
AUMC_%Extrap(_obs, _pred)	Percent of AUMCINF(_obs, _pred) that is extrapolated. =
MRTINF(_obs, _pred)	Mean residence time (MRT) extrapolated to infinity. For non-steady-state data: For non-infusion models: = For infusion models: = where TI is the length of infusion. (Note that, for extravascular dosing (oral model 200), MRTINF includes Mean Input Time as well as time in systemic circulation.)
Vss(_obs, _pred)	For non-steady-state data: An estimate of the volume of distribution at steady-state based on the last observed (obs) or last predicted (pred) concentration. = Computed for IV Bolus and Infusion dosing only. Not computed for extravascular dosing (oral model 200), as MRTINF for oral models includes Mean Input Time as well as time in systemic circulation and therefore is not appropriate to use in calculating Vss.

^aFor extravascular models (model 200), the fraction of dose absorbed cannot be estimated; therefore Volume and Clearance for these models are actually Volume/F or Clearance/F where F is the fraction of dose absorbed.

Plasma parameters that are estimated when at steady-state
Parameter	Definition
Tau	Available in the Dosing Used results worksheet for steady-state data. The (assumed equal) dosing interval for steady-state data.
Tmax	Time of maximum observed concentration. For steady-state data, based on observations collected during the dosing interval, that is, at or after the dosing time, but no later than the dosing time plus Tau, where Tau is the dosing interval. If the maximum observed concentration is not unique, then the first maximum is used.
Cmax	Maximum observed concentration, occurring at time Tmax, as defined above.
Tmin	Time of minimum observed concentration. For steady-state data, based on observations collected during the dosing interval (i.e., after the dosing time, but no later than dosing time plus Tau, where Tau is the dosing interval). If the minimum observed concentration is not unique, then the first minimum is used.
Cmin	Minimum observed concentration occurring at time Tmin as defined above. Note: Regulatory agencies differ on the definition of Cmin: some agencies define Cmin the same as Ctau is defined below. Both Cmin and Ctau are included in the output so that users can use the correct parameter for their situation.
Ctau	Concentration at dosing time plus Tau. Observed concentration if the value exists in the input data; otherwise, the predicted concentration value. Predicted concentrations are calculated following the same rules as for computing inserting missing endpoints needed for partial areas, see “Partial area calculation”.
Cavg	Average concentration, computed: =
Swing	=
Swing_Tau	=
Fluctuation%	= where Cmin and Cmax were obtained between dosing time and dosing time plus Tau.
Fluctuation%_Tau	=
CLss, CLss_F ^a	An estimate of the total body clearance, computed for IV Bolus and Infusion dosing only. =
MRTINF(_obs, _pred)	Mean residence time (MRT) extrapolated to infinity based on AUCINF(_obs, _pred). For non-infusion: For infusion: where TI represents infusion duration. (Note that, for oral model 200, MRTINF includes Mean Input Time as well as time in systemic circulation.)
Vz, Vz_F^a	=
Vss(_obs, _pred)	An estimate of the volume of distribution at steady-state based on the last observed (obs) or last predicted (pred) concentration. Computed for IV Bolus and infusion dosing only. = Not computed for extravascular dosing (oral model 200), as MRTINF for oral models includes Mean Input Time as well as time in systemic circulation and therefore is not appropriate to use in calculating Vss.
Accumulation Index	=
AUC_TAU	The partial area from dosing time to dosing time plus Tau. See “Partial area calculation” for information on how it is computed.
AUC_TAU_D	=
AUC_TAU_%Extrap	Percentage of AUC_TAU that is due to extrapolation from Tlast to infinity. = =
AUMC_TAU	Area under the first moment curve from dosing time to dosing time plus Tau. See “Partial area calculation” for information on how it is computed.
AUClower_upper	(Optional) User-requested area(s) under the curve from time “lower” to “upper”.

Urine data

Data structure: NCA for urine data requires the following input data:

•Starting and ending time of each urine collection interval

•Urine concentrations

•Urine volumes

From this data, models 210–212 compute the following for the analysis:

•Midpoint of each collection interval=(Starting time+Ending time)/2

•Excretion rate for each interval (amount eliminated per unit of time)=(Concentration*Volume)/(Ending time-Starting time)

Output: Models 210–212 estimate the following parameters.

The worksheet will include the Sort(s), Carry(ies), parameter names, units, and computed values. A User Defined Parameters Pivoted worksheet will include the pivoted form of the User Defined Parameters worksheet.

Urine parameters that do not depend on Lambda Z
Parameter	Definition
Dosing time	Available as ‘Time’ in the Dosing Used results. Time of last administered dose. It is assumed to be zero unless otherwise specified. This parameter is used mainly with steady-state data, where time may be coded as the time elapsed since the first dose, or the elapsed time since the time of the first dose.
Dose	Amount of last administered dose. This is assumed to be zero if not specified.
N_Samples	Number of non-missing observations in the analysis. Does not include missing or non-numeric observations, observations before dosing time, or urine observations where volume is zero. Does not include the point at dosing time if it was not observed but was inserted by the engine.
No_points_lambda_z	Number of points used in the computation of Lambda Z. If Lambda Z cannot be estimated, this is set to zero.
Tlag	Midpoint of the collection interval prior to the first collection interval with measurable (non-zero) rate. Computed for all urine models.
Tmax_Rate	Midpoint of the collection interval associated with the maximum observed excretion rate. If the maximum observed excretion rate is not unique, then the first maximum is used.
Max_Rate	Maximum observed excretion rate, at time Tmax_Rate as defined above.
Mid_Pt_last	Midpoint of collection interval associated with last measurable (positive) observed excretion rate.
Rate_last	Last observed measurable (positive) rate at time Mid_Pt_last.
AURC_last	Area under the urinary excretion rate curve from time of dosing to Mid_Pt_last.
AURC_last_D	=
Vol_UR	Sum of Urine Volumes (urine)
Amount_Recovered	Cumulative amount eliminated. =
Percent_Recovered	=
AURC_all	Area under the urinary excretion rate curve from the time of dosing to the midpoint of the interval with the last rate. If the last rate is positive, AURC_last=AURC_all.

The following table includes some parameters that are extrapolated to infinity. These parameters are calculated two ways: based on the last observed excretion rate: Rate_last (indicated by “_obs” appended to the parameter name), or based on the last predicted excretion rate: Rate_last_pred (indicated by “_pred” appended to the parameter name), where the predicted value is based on the linear regression performed to estimate Lambda Z.

Urine parameters that are estimated when Lambda Z is estimated
Parameter	Definition
Rsq	Goodness of fit statistic for the terminal elimination phase.
Rsq_adjusted	Goodness of fit statistic for the terminal elimination phase, adjusted for the number of points used in the estimation of Lambda Z.
Corr_XY	Correlation between midpoints and log excretion rates for the points used in the estimation of Lambda Z.
Lambda_z	First-order rate constant associated with the terminal (log-linear) portion of the curve. This is estimated via linear regression of midpoints vs. log excretion rates.
Lambda_z_intercept	Intercept on log scale estimated via linear regression of midpoints vs. log excretion rates.
Lambda_z_lower	Lower limit on midpoint for values to be included in Lambda Z estimation.
Lambda_z_upper	Upper limit on midpoint for values to be included in Lambda Z estimation.
HL_Lambda_z	Terminal half-life=ln(2)/Lambda Z
Span	=
Rate_last_pred	Predicted rate at Mid_Pt_last.
AURC_INF(_obs, _pred)	Area under the urinary excretion rate curve extrapolated to infinity, based on the last observed excretion rate (_obs) or the last predicted rate (_pred), i.e., the excretion rate at the final midpoint estimated using the linear regression for Lambda Z. Note that AURC_INF is theoretically equal to Amount_Recovered, but will differ due to experimental error.
AURC_%Extrap(_obs, _pred)	Percent of AURC_INF(_obs, _pred) that is extrapolated.

Sparse sampling (pre-clinical) data

When an NCA model is loaded with the Sparse Sampling option (see “Options tab”), the data are treated as a special case of plasma or urine concentration data. The NCA engine computes the mean concentration or rate at each unique time value or interval. Using the mean concentration curve across subjects, it estimates the same parameters normally calculated for plasma or urine data, plus those listed below. See “Sparse sampling calculation” for additional details of NCA computations with sparse data.

Note:The names of some of the output worksheets change when the data is sparse: Final Parameters becomes Mean Curve Final Parameters, and Summary Table becomes Mean Curve Summary Table.

Plasma or serum concentration parameters

Sparse sampling methods for plasma data (models 200–202) compute the following parameters in addition to those listed in “Plasma or serum data”.

Parameter	Definition
SE_Cmax	Standard error of data at Tmax (time of maximum mean concentration).
SE_AUClast	It is the standard error of the area under the mean concentration curve from dose time to Tlast, where Tlast is the time of last measurable (positive) mean concentration.
SE_AUCall	Standard error of the area under the mean concentration curve from dose time to the final observation time.

Note:SE_AUClast and SE_AUCall provide a measurement of the uncertainty for AUClast and AUCall, respectively, and are usually the same. Differences between these parameter values will only be observed if some of the measurements were flagged as BQL (below the quantitation limit).

Urine excretion rate parameters

Sparse sampling methods for urine data (models 210–212) compute the following parameters in addition to those listed under “Urine data”.

Parameter	Definition
SE_Max_Rate	Standard error of the data at the time of maximum mean rate.
SE_AURC_last	Standard error of the area under the mean urinary excretion rate curve from dose time through the last interval that has a measurable (positive) mean rate.
SE_AURC_all	Standard error of the area under the mean urinary excretion rate curve from dose time through the final interval.

Individuals by time

This sheet includes the individual subject data, along with N (number of non-missing observations), the mean and standard error for each unique time value (plasma data) or unique midpoint value (urine data).

References

Holder (2001). Comments on Nedelman and Jia's extension of Satterthwaite's approximation applied to pharmacokinetics. J Biopharm Stat 11(1-2):75–9.

Nedelman, Gibiansky and Lau (1995). Applying Bailer's method for AUC intervals to sparse sampling. Pharm Res 12:124–8.

Nedelman and Jia (1998). An extension of Satterthwaite's approximation applied to pharmacokinetics. J Biopharm Stat 8(2):317–28.

Yeh (1990). Estimation and Significant Tests of Area Under the Curve Derived from Incomplete Blood Sampling. ASA Proceedings of the Biopharmaceutical Section 74–81.

Drug effect data model 220

Illustration of a time-effect curve with AUCs highlighted

Output parameter names use the following conventions:

•B is the baseline effect value (discussed above).

•T is the user-supplied threshold effect value.

•“Above” means towards increasing Y values, even for inhibitory effects.

Estimated parameters for model 220
Parameter	Definition
N_Samples	This parameter reports the number of non-missing observations used in the analysis of the profile (time is at or after dosing time, the observation is numeric, and the volume is positive for urine models). It does not count points inserted by engine, e.g., inserted at dosing time.
Slope1 (or 2)	Slope of the first (or second) segment of the curve. See “Lambda Z or Slope Estimation settings”.
Rsq_Slope1 (or 2)	Goodness of fit statistic for slope 1 or 2.
Rsq_adj_Slope1 (or 2)	Goodness of fit statistic for slope 1 or 2, adjusted for the number of points used in the estimation.
Corr_XY_Slope1 (or 2)	Correlation between time (X) and effect (or log effect, for log regression) (Y) for the points used in the slope estimation.
No_points_Slope1 (or 2)	The number of data points included in calculation of slope 1 or 2.
Slope1_lower or Slope2_lower	Lower limit on Time for values to be included in the slope calculation.
Slope1_upper or Slope2_upper	Upper limit on Time for values to be included in the slope calculation.
Tmin	Time of minimum observed response value (Rmin).
Rmin	Minimum observed response value.
Tmax	Time of maximum observed response value (Rmax).
Rmax	Maximum observed response value.
Baseline	Baseline response (Y) value supplied by the user, (assumed to be zero if none is supplied) or, for Certara Integral studies with no user-supplied baseline value, effect value at dose time.
AUC_Above_B	Area under the response curve that is above the baseline (dark gray areas in the above diagram).
AUC_Below_B	Area that is below the baseline and above the response curve (combined blue and pink areas in the above diagram).
AUC_Net_B	= AUC_Above_B – AUC_Below_B This is likely to be a negative value for inhibitory effects.
Time_Above_B	Total time that Response >= Baseline.
Time_Below_B	Total time that Response < Baseline.
Time_%Below_B	= 100*Time_Below_B/(Tfinal – Tdose) where Tfinal is the final observation time and Tdose is dosing time.

When a threshold value is provided, model 220 also computes the following.

Additional parameters for model 220 with threshold
Parameter	Definition
Threshold	Threshold value used.
AUC_Above_T	Area under the response curve that is above the threshold value (combined light and dark gray areas in the above diagram).
AUC_Below_T	Area that is below the threshold and above the response curve (pink area in the above diagram).
AUC_Net_T	= AUC_Above_T – AUC_Below_T
Time_Above_T	Total time that Response >= Threshold.
Time_Below_T	Total time that Response < Threshold.
Time_%Below_T	= 100*Time_Below_T/(Tlast – Tdose) where Tlast is the final observation time and Tdose is dosing time.
Tonset	Time that the response first crosses the threshold coming from the direction of the baseline value, as shown in the above diagram. Tonset=Tdose if the first response value is across the threshold, relative to baseline. The time will be interpolated using the calculation method selected in the model options. (See “Options tab”.)^a
Toffset	Time greater than Tonset at which the curve first crosses back to the baseline side of threshold, as shown in the diagram above.^a
Diff_Toffset_Tonset	= Toffset – Tonset
Time_Between_BT	Total time spent between baseline and threshold (sum of length of green arrows in diagram).
AUClower_upper	(Optional) user-requested area(s) under the curve from time “lower” to time “upper”.

^aUse caution in interpreting Tonset and Toffset for noisy data if Baseline and Threshold are close together.

User defined parameters

Any user defined parameters will be computed and reported in the User Defined Parameters and User Defined Parameters Pivoted worksheets. The worksheets include entries for Sort, Carry, parameter name, unit, and estimated value. A User Defined Parameters Pivoted worksheet will include the pivoted form of the User Defined Parameters worksheet.

Parameter	Definition
C<time value>	For Plasma models with user-defined times only, one column per requested time value. (For plasma models, a computed concentration at time zero will be reported as the parameter C0_0, since C0 is already used for the initial concentration measurement.)
Y<x-value>	For Drug Effect models with user-defined x-values only, one column per requested x-value.
<user-defined name>	Parameter defined using the User Defined Parameters tab.

References

Holder (2001). “Comments on Nedelman and Jia's extension of Satterthwaite's approximation applied to pharmacokinetics.” J Biopharm Stat 11(1–2):75–9.

Nedelman and Jia (1998). “An extension of Satterthwaite's approximation applied to pharmacokinetics.” J Biopharm Stat 8(2):317–28.