In cacha0227/ncappc: NCA Calculation and Population PK Model Diagnosis

Area Under the Curve
AUC can be estimated using either the linear trapezoidal method or the log-linear trapezoidal method. It is shown that linear approximation is a better method to estimate of the area under a curve at the positive or zero local slopes (increasing concentration or at the peak), while the log-linear approximation is the method of choice to estimate of the area under a curve at the negative local slope (decreasing concentration) assuming that the plasma concentration decline mono-exponentially in the elimination phase. There are three different options available in ncappc: “linear”, “log”, and “linearup-logdown”. The user can choose any of these three methods for the estimating the area under the zero-order moment (AUClast) and first-order moment (AUMClast) curves between the first and the last sampling time. If the lower and upper limit of the time range is provided, AUC is also calculated for that specified window of time (AUClower_upper).

The "linear" option under the argument, namely "method" allows the function to employ linear trapezoidal approximation (Equations 1 and 2) to estimate the area under the zero and first order moment curves.

$AUC|{i}^{i+1} = \frac{C{i+1}+C_{i}}{2}*\Delta{t}$, if $C_{i+1} \geq C_{i}$ (Equation 1)

$AUMC|{i}^{i+1} = \frac{(C{i+1}t_{i+1})+(C_{i}t_{i})}{2}*\Delta{t}$, if $C_{i+1} \geq C_{i}$ (Equation 2)

The “log” options employ the log trapezoidal approximation (Equations 3 and 4) to estimate the area under the zero and first order moment curves.

$AUC|{i}^{i+1} = \frac{C{i+1}-C_{i}}{ln(\frac{C_{i+1}}{C_{i}})}\Delta{t}$, if $C_{i+1} < C_{i}$ (Equation 3)
$AUMC|{i}^{i+1} = \frac{(C{i+1}t_{i+1})-(C_{i}t_{i})}{ln(\frac{C_{i+1}}{C_{i}})}\Delta{t}-\frac{C_{i+1}-C_{i}}{ln(\frac{C_{i+1}}{C_{i}})}*\Delta{t}^2$, if $ßC_{i+1} < C_{i}$ (Equation 4)

The “log-linear” option employs the linear trapezoidal approximation for the increasing concentration between two consecutive observations or at the peak concentration, while it uses the log trapezoidal approximation at the declining concentration between two consecutive observations.

\paragraph{} where, ${C_i}$ is the plasma concentration at time ${t_i}$, ${\Delta{t}}$ is ${t_{i+1}-t_{i}}$ and ${n}$ is the number of data points. Each estimated segment of AUC and AUMC is added to obtain AUClast and AUMClast, respectively.

\subsection{Extrapolation in the elimination phase} \label{subsection} The plasma concentration of the drug in the elimination phase is assumed to follow the mono-exponential decay with the rate of elimination, Lambda_z. The extrapolation of the plasma concentration from the last sampling time to infinity is performed based on the regression line estimated from the elimination phase data. All concentration data after Cmax are considered as the elimination phase data. The following steps are performed to determine the optimum regression line representing the plasma concentration in the elimination phase. First, a regression line is obtained from the last three non-zero concentrations in the elimination phase and the regression coefficient or ${R^2}$ (Rsq) and Adjusted ${R^2}$ (Rsq_adjusted) are calculated. Next, one data point at a time upto Cmax is added to the set and the corresponding Rsq and Rsq_adjusted are calculated each time. In order to extrapolate the NCA metrics from the last observed value to infinity in the terminal phase, the ncappc functionality requires the following two criteria to be satisfied.
* Minimum three non-zero concentrations in the elimination phase * At least one of the combinations of data points in the elimination phase yields negative slope for the corresponding regression line

If any individual in the data set does not satisfy the above mention criteria, the NCA metrics for that individual are not extrapolated to infinity for the elimination phase. Othrwise, the regression line yielding the highest Rsq_adjusted is used to estimate Lambda_z. If there is any other regression line that yields Rsq_adjustzed within 0.0001 of the highest Rsq_adjusted and includes larger number of data points, that regression line is used to estimate Lambda_z. Lambda_z is estimated as the negative of the slope of the selected regression line representing the elimination phase of the concentration profile (Equation 5). The number of points used to determine the Lambda_z is called No_points_Lambda_z. It is possible to specify the lower and upper limit on time for the calculation of the regression line in the elimination phase (Lambda_z_lower and Lambda_z_upper, respectively).

$Lambda_z = -(slope) = \frac{ln(Clast)-ln(Clast-k)}{Tlast-(Tlast-k)}$ (Equation 5)

where ${k+1}$ is the number of data points out of ${n}$ data samples that are used to calculate the regression line representing the elimination phase. Clast is the plasma concentration corresponding to the last sampled time Tlast. The elimination half-life (HL_Lambda_z) is calculated from Lambda_z using Equation 6.

$HL_Lambda_z = \frac{log2}{Lambda_z}$ (Equation 6)

cacha0227/ncappc documentation built on Nov. 14, 2020, 9:09 a.m.