Chapter 4 Equations

For the parameter table below, concentrations measured below the limit of concentration are assumed to be written as zero. Also, mathematical terminology differs in many references; throughout this document, \(ln\) is used to indicate the natural logarithm (also known as the logarithm base \(e\), \(log_e\), and \({}^elog\)).

4.1 Defined Parameters

The following parameter definitions are used throughout the parameter table, and these parameters are defined parameter (not calculated).

Symbol Units Definition
C concentration Concentration
D amount Dose
t time Time
\(\tau\) time Dosing interval

4.2 Parameter Table

symbol units definition cdisc single multiple intravenous extravascular equation notes reference
\(C_{max}\) concentration The maximum concentration occurring at \(t_{max}\). CMAX TRUE TRUE TRUE TRUE Observed
\(t_{max}\) time The time of \(C_{max}\). TMAX TRUE TRUE TRUE TRUE Observed
\(C_{min}\) concentration The minimum concentration occurring between dose time and dose time plus \(\tau\) (at \(t_{min}\)). CMIN TRUE TRUE TRUE TRUE Observed 1
\(t_{min}\) time The time of \(C_{min}\). TMIN TRUE TRUE TRUE TRUE Observed
\(C_{trough}\) concentration Concentration at end of dosing interval. CTROUGH TRUE TRUE TRUE TRUE Observed 2
\(t_{lag}\) time The time prior to the first increase in concentration. TLAG TRUE TRUE TRUE TRUE Observed (, 2013)
\(C_{last,obs}\) concentration The last observed concentration above the limit of quantification. Equivalently, the concentration corresponding to \(t_{last}\). CLST TRUE TRUE TRUE TRUE Observed (, 2013)
\(C_{last,pred}\) concentration The concentration at \(t_{last}\) predicted from the log-linear regression of the terminal part of the concentration-time curve (as estimated for half-life). TRUE TRUE TRUE TRUE \[C_{last,pred} = \lambda_z \times t_{last} + A\] 3 (, 2013)
\(t_{last}\) time The time of the last measurable (positive) concentration. TLST TRUE TRUE TRUE TRUE Observed (, 2013)
\(AUC_{a,b,linear}\) concentration*time Area under the concentration time curve from measurement at time \(a\) to time \(b\) using the linear trapezoidal rule. TRUE TRUE TRUE TRUE \[AUC_{t_i,t_{i+1}} = \frac{1}{2} \left(C_i + C_{i+1}\right) \left(t_{i+1} - t_i\right)\] (, 2013)
\(AUC_{a,b,log-linear}\) concentration*time Area under the concentration time curve from measurement at time \(a\) to time \(b\) using the log-linear trapezoidal rule. TRUE TRUE TRUE TRUE \[AUC_{t_i,t_{i+1}} = \frac{\left(C_i - C_{i+1}\right) \left(t_{i+1} - t_i\right)}{ln{C_i} - ln{C_{i+1}}}\] (, 2013)
\(AUC_{0,t}\), \(AUC_{last}\), \(AUC_{\tau}\) concentration*time Area under the concentration time curve during a defined interval using only samples above the limit of quantification AUCINT, AUCLST, AUCTAU TRUE TRUE TRUE TRUE \[AUC = \sum_{t=a}^{t=min(t_{last}, b)} AUC_{t_i,t_{i+1}}\] 4 (, 2013)
\(AUC_{t_{last}-\infty,pred}\), \(AUC_{t_{last}-\infty,obs}\) concentration*time Area under the concentration-time curve from \(t_{last}\) to \(\infty\) TRUE TRUE TRUE TRUE \[\frac{C_{last}}{\lambda_z}\] 5 (, 2013)
\(AUC_{t_{last}-\infty,all}\) concentration*time Area under the concentration-time curve from \(t_{last}\) to \(\infty\) as used for \(AUC_{all}\). TRUE TRUE TRUE TRUE \[AUC_{t_{last},t_{last+1},linear}\] 6 (, 2013)
\(AUC_{0-\infty,pred}\), \(AUC_{0-\infty,obs}\) concentration*time The area under the curve (AUC) extrapolated to \(\infty\), calculated using the observed or predicted value of the last non-zero concentration. AUCIFP, AUCIFO TRUE FALSE TRUE TRUE \[AUC_{last} + AUC_{t_{last}-\infty}\] 7 (, 2013)
\(AUC_{0,all}\) concentration*time The area under the curve (AUC) from the time of dosing to the time of the last observation, regardless of whether the last concentration is measurable or not. AUCALL TRUE TRUE TRUE TRUE \[AUC_{last} + AUC_{t_{last}-\infty,all}\] 8 (, 2013)
\(AUC_{\%extrap,obs}\), \(AUC_{\%extrap,pred}\) % The area under the curve (AUC) from the last observed non-zero concentration value to infinity as a percentage of the area under the curve extrapolated to infinity using either the observed or predicted \(C_{last}\). AUCPEO, AUCPEP TRUE FALSE TRUE TRUE \[\frac{AUC_{t_{last},\infty}}{AUC_{0,\infty}} \times 100\] 9 (, 2013)
\(F\) fraction Bioavailability TRUE FALSE TRUE TRUE \[\frac{AUC_{0,\infty,ev}}{AUC_{0,\infty,iv}} \frac{D_{iv}}{D_{ev}}\] \[\frac{AUC_{0,\infty,test}}{AUC_{0,\infty,reference}} \frac{D_{reference}}{D_{test}}\] 10 (, 2013)
\(AUMC_{a,b,linear}\) concentration*time2 Area under the first moment of the concentration time curve from measurement at time \(a\) to time \(b\) using the linear trapezoidal rule. TRUE TRUE TRUE TRUE \[AUMC_{t_i,t_{i+1},linear} = \frac{1}{2} \left(t_{i+1} - t_{i}\right) \left(C_{i+1} t_{i+1} + C_i t_i\right)\] (, 2013)
\(AUMC_{a,b,log-linear}\) concentration*time2 Area under the first moment of the concentration time curve from measurement at time \(a\) to time \(b\) using the log-linear trapezoidal rule. TRUE TRUE TRUE TRUE \[AUMC_{t_i,t_{i+1},log-linear} = \frac{t_{i+1} - t_{i}}{ln\left(C_i\right) - ln\left(C_{i+1}\right)} \left(C_{i+1} \times t_{i+1} + C_i \times t_i\right) + \\ \left(\frac{t_{i+1} - t_{i}}{ln\left(C_i\right) - ln\left(C_{i+1}\right)}\right)^2 \left(C_i - C_{i+1}\right)\] (, 2013)
\(AUMC_{0,t}\), \(AUMC_{last}\), \(AUMC_{\tau}\) concentration*time2 Area under the first moment of the concentration time curve during a defined interval using only samples above the limit of quantification AUMCLST, AUMCTAU TRUE TRUE TRUE TRUE \[AUMC_{last} = \sum_{t=a}^{t=min(t_{last}, b)} AUMC_{t_i,t_{i+1}}\] 11 (, 2013)
\(AUMC_{t_{last}-\infty,obs}\), \(AUMC_{t_{last}-\infty,pred}\) concentration*time2 Area under the first moment of the concentration-time curve from \(t_{last}\) to \(\infty\) TRUE TRUE TRUE TRUE \[\frac{C_{last} t_{last}}{\lambda_z} + \frac{C_{last}}{\lambda_z^2}\] 12 (, 2013)
\(AUMC_{t_{last}-\infty,all}\) concentration*time2 Area under the first moment of the concentration-time curve from \(t_{last}\) to \(\infty\) as used for \(AUC_{all}\). TRUE TRUE TRUE TRUE \[AUC_{t_{last},t_{last+1},linear}\] 13 (, 2013)
\(AUMC_{0-\infty,obs}\), \(AUMC_{0-\infty,pred}\) concentration*time2 Area under the first moment of the concentration-time curve extrapolated to \(\infty\), calculated using the observed or predicted value of the last non-zero concentration. AUMCIFP, AUMCIFO TRUE FALSE TRUE TRUE \[AUMC_{last} + AUMC_{t_{last}-\infty}\] 14 (, 2013)
\(AUMC_{0,all}\) concentration*time2 Area under the first moment of the concentration-time curve from the time of dosing to the time of the last observation, regardless of whether the last concentration is measurable or not. TRUE TRUE TRUE TRUE \[AUMC_{last} + AUMC_{t_{last}-\infty,all}\] 15 (, 2013)
\(AUMC_{\%extrap,obs}\), \(AUMC_{\%extrap,pred}\) % Area under the first moment of the concentration-time curve from the last observed non-zero concentration value to infinity as a percentage of the area under the curve extrapolated to infinity using either the observed or predicted \(C_{last}\). AUMCPEO, AUMCPEP TRUE FALSE TRUE TRUE \[\frac{AUMC_{t_{last}-\infty}}{AUMC_{0,\infty}} \times 100\] 16 (, 2013)
\(t_{1/2}\) time Terminal half-life LAMZHL TRUE TRUE TRUE TRUE \[\frac{ln 2}{\lambda_z}\] (, 2013)
\(\lambda_z\) 1/time The first order rate constant associated with the terminal (log-linear) portion of the curve. LAMZ TRUE TRUE TRUE TRUE \[ln C = A - \lambda_z t\] 17 (, 2013)
\(t_{first,\lambda_z}\), \(t_{last,\lambda_z}\) time The first and last time (lower and upper limit on time) for values to be included in the calculation of \(\lambda_z\). LAMZLL, LAMZUL TRUE TRUE TRUE TRUE (, 2013)
\(N_{\lambda_z}\) count The number of time points used in computing \(\lambda_z\). LAMZNPT TRUE TRUE TRUE TRUE 18
\(r^2\) fraction The goodness of fit statistic for the terminal elimination phase. R2 TRUE TRUE TRUE TRUE \[1 - \frac{\sum_i \left(C_i - \hat{C_i}\right)^2}{\sum_i \left(C_i - \overline{C_i}\right)^2}\] 19
\(r^2_{adj}\) fraction The goodness of fit statistic for the terminal elimination phase, adjusted for the number of time points used in the estimation of \(\lambda_z\). R2ADJ TRUE TRUE TRUE TRUE \[1 - \left(1 - r^2\right) \frac{1}{N_{\lambda_z}-2}\] 20 (, 2013)
\(MRT_{ev,last}\), \(MRT_{ev,\infty,obs}\), \(MRT_{ev,\infty,pred}\), \(MRT_{iv,last}\), \(MRT_{iv,\infty,obs}\), \(MRT_{iv,\infty,pred}\) time Mean residence time (MRT) from the time of dosing to the time of the last measurable concentration or infinity, for a substance administered by intravascular or extravascular dosing. MRTEVLST, MRTEVIFO, MRTEVIFP, MRTIVLST, MRTIVIFO, MRTIVIFP TRUE TRUE TRUE TRUE \[\frac{AUMC}{AUC}\] 21 (, 2013)
\(MAT_{last}\), \(MAT_{\infty,obs}\), \(MAT_{\infty,pred}\) time Mean absorption time of a substance administered by extravascular dosing. MAT TRUE TRUE FALSE TRUE \[MRT_{ev} - MRT{iv}\] \[t_{lag} + \frac{1}{K_a}\] 22 (, 2013)
\(PTR\) ratio The maximum concentration during a dosing interval divided by the concentration at the end of the dosing interval. PTROUGHR FALSE TRUE TRUE TRUE \[\frac{C_{max}}{C_{trough}}\] (, 2013)
\(TPR\) ratio The concentration at the start of a dosing interval divided by the maximum concentration during the dosing interval. TROUGHPR FALSE TRUE TRUE TRUE \[\frac{C_{trough}}{C_{max}}\]
\(PTF\) % The difference between Cmin and Cmax standardized to Cavg, between dose time and \(\tau\). FLUCP FALSE TRUE TRUE TRUE \[100 \times \frac{C_{max} - C_{min}}{C_{avg}}\] (, 2013)
\(C_{avg}\), \(C_{av}\) concentration Average concentration CAVG FALSE TRUE TRUE TRUE \[\frac{AUC_{0,\tau}}{\tau}\] (, 2013)
\(R_{A,AUC}\) fraction The area under the curve at steady state divided by the area under the curve over the initial dosing interval. ARAUC FALSE TRUE TRUE TRUE \[R_{A,AUC} = \frac{AUC_{\tau,ss}}{AUC_{\tau,sd}}\] 23 (, 2013)
\(R_{A,C_{max}}\) fraction The maximum concentration at steady state divided by the maximum concentration during the initial dosing interval. ARCMAX FALSE TRUE TRUE TRUE \[R_{A,C_{max}} = \frac{C_{max,ss}}{C_{max,sd}}\] 24 (, 2013)
\(R_{A,C_{min}}\) fraction The minimum concentration at steady state divided by the minimum concentration during the initial dosing interval. ARCMIN FALSE TRUE TRUE TRUE \[R_{A,C_{min}} = \frac{C_{min,ss}}{C_{min,sd}}\] 25 (, 2013)
\(R_{A,C_{trough}}\) fraction The trough concentration at steady state divided by the trough concentration during the initial dosing interval. ARCTROUG FALSE TRUE TRUE TRUE \[R_{A,C_{trough}} = \frac{C_{trough,ss}}{C_{trough,sd}}\] 26 (, 2013)
\(CL_{ev,obs}\), \(CL_{ev,pred}\), \(CL_{ev,\tau}\), \(CL_{iv,obs}\), \(CL_{iv,pred}\), \(CL_{iv,\tau}\) volume/time The total body clearance for extravascular or intravascular administration (divided by the fraction of dose absorbed for extravascular), calculated using the equivalent \(AUC\). CLFO, CLFP, CLFTAU, CLO, CLP, CLTAU TRUE TRUE TRUE TRUE \[\frac{F \times D}{AUC}\] 27 (, 2013)
\(V_{ss,obs}\), \(V_{ss,pred}\) volume The volume of distribution at steady state based on the equivalent for a substance administered by intravascular dosing. VSSO, VSSP TRUE FALSE TRUE FALSE \[CL \times MRT\] \[\frac{F \times D \times AUMC}{AUC^2}\] 28 (, 2013)
\(V_{z,ev,obs}\), \(V_{z,ev,pred}\), \(V_{z,ev,\tau}\), \(V_{z,iv,obs}\), \(V_{z,iv,pred}\), \(V_{z,iv,\tau}\) volume The volume of distribution associated with the terminal slope following extravascular or intravascular administration (divided by the fraction of dose absorbed for extravascular), calculated using equivalent \(AUC\). VZFO, VZFP, VZFTAU, VZO, VZP, VZTAU TRUE TRUE TRUE TRUE \[\frac{F \times D}{AUC \times \lambda_z}\] (, 2013)
\(C_0\) concentration Initial concentration C0 TRUE TRUE TRUE FALSE 29
\(V_0\) volume The initial volume of distribution for a substance administered by bolus intravascular dosing. V0 TRUE FALSE TRUE FALSE \[\frac{D}{C_0}\] (, 2013)
\(AUC_{\%backextrap,obs}\), \(AUC_{\%backextrap,pred}\) concentration*time TRUE TRUE TRUE FALSE \[\frac{AUC_{0,first}}{AUC_{0,\infty}}\] 30

4.3 Notes

4.4 \(C_{min}\)

Compare with \(C_{trough}\).

4.5 \(C_{trough}\)

Compare with \(C_{min}\).

4.6 \(C_{last,pred}\)

Parameters in the equation are fit during half-life fitting.

4.7 \(AUC_{0,t}\), \(AUC_{last}\), \(AUC_{\tau}\)

Often, the \(AUC_{a,b,linear}\) and \(AUC_{a,b,log-linear}\) are combined where the linear trapezoidal rule is used for ascending concentrations (\(C_{i+1} >= C_i\)) or when the next concentration is below the limit of quantification (\(C_{i+1} > 0\), if applicable), and the log-linear trapezoidal rule is used for descending concentrations \(C_{i+1} < C_i\).

4.8 \(AUC_{t_{last}-\infty,pred}\), \(AUC_{t_{last}-\infty,obs}\)

\(C_{last}\) should be \(C_{last,pred}\) or \(C_{last,obs}\) depending on which version of the parameter is desired.

4.9 \(AUC_{t_{last}-\infty,all}\)

\(t_{last+1}\) is the first time point below the limit of quantification.

4.10 \(AUC_{0-\infty,pred}\), \(AUC_{0-\infty,obs}\)

Use the ‘pred’ or ‘obs’ version of \(AUC_{t_{last}-\infty}\) for the equivalent version of \(AUC_{0-\infty}\).

4.11 \(AUC_{0,all}\)

If all concentrations are above the limit of quantification, then \(AUC_{0,all}=AUC_{last}\).

4.12 \(AUC_{\%extrap,obs}\), \(AUC_{\%extrap,pred}\)

Use the ‘pred’ or ‘obs’ version of \(AUC_{t_{last},\infty}\) and \(AUC_{0,\infty}\) for the equivalent version of \(AUC_{\%extrap}\).

4.13 \(F\)

Bioavailability is the ratio of two AUC values. In addition to the comparison of extravascular (\(ev\)) to intravascular (\(iv\)) for absolute bioavailability, relative bioavailability compares any two \(AUC_{0,\infty} values, a 'test' to a 'reference'\). In some cases, bioavailability may be tested at steady-state using \(AUC_{0,\tau}\) instead of \(AUC_{0,\infty}\).

4.14 \(AUMC_{0,t}\), \(AUMC_{last}\), \(AUMC_{\tau}\)

See \(AUC_{last}\) notes for suggested calculation options.

4.15 \(AUMC_{t_{last}-\infty,obs}\), \(AUMC_{t_{last}-\infty,pred}\)

\(C_{last}\) should be \(C_{last,pred}\) or \(C_{last,obs}\) depending on which version of the parameter is desired.

4.16 \(AUMC_{t_{last}-\infty,all}\)

\(t_{last+1}\) is the first time point below the limit of quantification.

4.17 \(AUMC_{0-\infty,obs}\), \(AUMC_{0-\infty,pred}\)

Use the ‘obs’ or ‘pred’ version of \(AUC_{t_{last}-\infty}\) for the equivalent version of \(AUC_{0-\infty}\).

4.18 \(AUMC_{0,all}\)

If all concentrations are above the limit of quantification, then \(AUMC_{0,all}=AUMC_{last}\).

4.19 \(AUMC_{\%extrap,obs}\), \(AUMC_{\%extrap,pred}\)

Use the ‘pred’ or ‘obs’ version of \(AUMC_{t_{last}-\infty}\) for the equivalent version of \(AUMC_{0-\infty}\).

4.20 \(\lambda_z\)

Fit the equation using observed concentration (\(C\)) and time (\(t\)) from the terminal slope of the concentration-time curve. Selection of points for inclusion may be either manual or regression-quality based.

4.21 \(N_{\lambda_z}\)

\(N_{\lambda_z}\) should always be greater than or equal to 3.

4.22 \(r^2\)

\(r^2\) is typically directly available from line-fitting functions and rarely requires direct calculation. \(\overline{C_i}\) is the mean concentration of values used; \(\hat{C_i}\) is the estimated concentation.

4.23 \(r^2_{adj}\)

The canonical form of \(r^2_{adj}\) has the fraction on the right of the equation as \(\frac{p}{n-p-1}\), but for fitting terminal regression \(p=1\).

4.24 \(MRT_{ev,last}\), \(MRT_{ev,\infty,obs}\), \(MRT_{ev,\infty,pred}\), \(MRT_{iv,last}\), \(MRT_{iv,\infty,obs}\), \(MRT_{iv,\infty,pred}\)

The equation shown here includes the mean absorption time for extravascular administration aligning with the SDTM definition. The \(AUMC\) and \(AUC\) should be chosen to match the parameter type desired (e.g. use \(AUMC_{last}\) and \(AUC_{last}\) for \(MRT_{last}\)).

4.25 \(MAT_{last}\), \(MAT_{\infty,obs}\), \(MAT_{\infty,pred}\)

\(K_a\) is the absorption rate from multi-exponential compartmental analysis curve fitting (and is not typically applicable for noncompartmental analysis).

4.26 \(R_{A,AUC}\)

\(ss\) indicates that value is at steady-state while \(sd\) indicated that value is from the first (single) dose.

4.27 \(R_{A,C_{max}}\)

\(ss\) indicates that value is at steady-state while \(sd\) indicated that value is from the first (single) dose.

4.28 \(R_{A,C_{min}}\)

\(ss\) indicates that value is at steady-state while \(sd\) indicated that value is from the first (single) dose.

4.29 \(R_{A,C_{trough}}\)

\(ss\) indicates that value is at steady-state while \(sd\) indicated that value is from the first (single) dose.

4.30 \(CL_{ev,obs}\), \(CL_{ev,pred}\), \(CL_{ev,\tau}\), \(CL_{iv,obs}\), \(CL_{iv,pred}\), \(CL_{iv,\tau}\)

For each of the symbols, select the equivalent \(AUC\). Such as, \(CL_{ev,pred}\) uses \(AUC = AUC_{0,\infty,pred}\)

4.31 \(V_{ss,obs}\), \(V_{ss,pred}\)

SDTM parameters are specific to intravascular dosing. For each of the symbols, select the equivalent \(AUMC\) and \(AUC\). When extravascular dosing is used, it is referred to as the observed or apparent volume of distribution at steady-state. CL and MRT may be given either for single dosing with \(AUC_{0,\infty}\) or steady-state with \(AUC_{0,\tau}\).

4.32 \(C_0\)

IV bolus only. \(C_0\) is back-extrapolated from the first two data points after the infusion. See data cleaning rules SDC-4 and MDC-4 for more details on the calculation and cleaning associated with \(C_0\).

4.33 \(AUC_{\%backextrap,obs}\), \(AUC_{\%backextrap,pred}\)

IV bolus only. \(first\) is the first time point after 0, and both \(AUC\) values should be calculated with \(C_0\) included at time 0.