Noncompartmental Analysis
Kyun-Seop Bae MD PhD
2022-03-11
Introduction
Noncompartmental Analysis(NCA) is relatively robust and used frequently for regulatory purpose.
Individual NCA parameters
Preparation
This is just for the prepartion of data for the subsequent R scripts.
Adm = c("BOLUS", "INFUSION", "EXTRAVASCULAR")[3] # Drug Administration Method
Dose = 320 # mg
x = x0 = Theoph[Theoph$Subject==1, "Time"] # h
y = y0 = Theoph[Theoph$Subject==1, "conc"] # ug/L
# For the calculation of AUClast
iLastNonZero = max(which(y > 0)) # index of last non-zero concentration
x1 = x0[1:iLastNonZero]
y1 = y0[1:iLastNonZero]
# For the log-concentration vs. time regression
x2 = x0[y0 > 0]
y2 = y0[y0 > 0]
# Print data
cbind(Time=x0, Conc=y0)## Time Conc
## [1,] 0.00 0.74
## [2,] 0.25 2.84
## [3,] 0.57 6.57
## [4,] 1.12 10.50
## [5,] 2.02 9.66
## [6,] 3.82 8.58
## [7,] 5.10 8.36
## [8,] 7.03 7.47
## [9,] 9.05 6.89
## [10,] 12.12 5.94
## [11,] 24.37 3.28Plot of raw data
Cmax (CMAX)
Maximum concentration.
CMAX = max(y)
CMAX## [1] 10.5Cmax_D (CMAXD)
Dose normalized Cmax.
CMAXD = NA
if (Dose > 0) CMAXD = CMAX/Dose
CMAXD## [1] 0.0328125Tmax (TMAX)
Time of maximum concentration.
TMAX = NA
if (CMAX > 0) TMAX = x[which.max(y)]
TMAX## [1] 1.12Tlag (TLAG)
Time until first non-zero concentration.
TLAG = NA
if (CMAX > 0) TLAG = x[max(1, min(which(y > 0)) - 1)]
TLAG## [1] 0Clast (CLST)
Last non-zero concentration.
CLST = y[iLastNonZero]
CLST## [1] 3.28Tlast (TLST)
Time of last non-zero concentration.
TLST = x[iLastNonZero]
TLST## [1] 24.37Rsq (R2)
R-squared value from the log concentration and time regression.
Rsq_adjusted (R2ADJ)
Adujsted R-squared value.
\(R_{adj}^2 = 1 - (1 - R^2)\frac{n - 1}{n - 2}\)
Corr_XY (CORRXY)
Correlation value of the regression
b0 (b0)
Intercept of the simple linear regression of log(y) vs x
Lambda_z (LAMZ)
Terminal slope as a positive value
No_points_Lambda_z (LAMZNPT)
Number of points used for the regression
Lambda_z_lower (LAMZLL)
First time point used for the regression
Lambda_z_upper (LAMZUL)
Last time point used for the regression
Calculation of R2, R2ADJ, CORRXY, b0, LAMZ, LAMZNPT, LAMZLL, and LAMZUL
Only positive concentrations are used. In case of oral administration, the first possible point is next to Tmax point. In case of intravascular administration, the first point can be Tmax point. If the difference of R2ADJ (R2-squared adjusted) is less than 1e-4, the longer slope is chosen. Regression points should be at least 3.
x = x2
y = y2
if (toupper(Adm) == "EXTRAVASCULAR") {
iFirst = which.max(y) + 1 # for oral administration
} else {
iFirst = which.max(y) # for intravenous administration
}
iLast = iLastNonZero
ColNames = c("R2", "R2ADJ", "CORRXY", "b0", "LAMZ", "LAMZNPT", "LAMZLL", "LAMZUL")
mRes = matrix(nrow = iLast - iFirst + 1 - 2, ncol=length(ColNames))
colnames(mRes) = ColNames
for (i in iFirst:(iLast - 2)) {
Res = lm(log(y[i:iLast]) ~ x[i:iLast])
mRes[i - iFirst + 1, "R2"] = summary(Res)$r.squared
mRes[i - iFirst + 1, "R2ADJ"] = summary(Res)$adj.r.squared
mRes[i - iFirst + 1, "CORRXY"] = cor(log(y[i:iLast]), x[i:iLast])
mRes[i - iFirst + 1, "b0"] = Res$coefficients[1]
mRes[i - iFirst + 1, "LAMZ"] = -Res$coefficients[2]
mRes[i - iFirst + 1, "LAMZNPT"] = iLast - i + 1
mRes[i - iFirst + 1, "LAMZLL"] = x[i]
mRes[i - iFirst + 1, "LAMZUL"] = x[iLast]
}
mRes## R2 R2ADJ CORRXY b0 LAMZ LAMZNPT LAMZLL LAMZUL
## [1,] 0.9988013 0.9985615 -0.9994005 2.355187 0.04778625 7 2.02 24.37
## [2,] 0.9987305 0.9984131 -0.9993650 2.350845 0.04751440 6 3.82 24.37
## [3,] 0.9995671 0.9994229 -0.9997836 2.362429 0.04817356 5 5.10 24.37
## [4,] 0.9996109 0.9994164 -0.9998054 2.356834 0.04787556 4 7.03 24.37
## [5,] 0.9999997 0.9999995 -0.9999999 2.368785 0.04845700 3 9.05 24.37OKs = abs(max(mRes[,"R2ADJ"]) - mRes[,"R2ADJ"]) < 1e-4
resNCA = as.data.frame(mRes[which(OKs)[1],,drop=FALSE])
resNCA## R2 R2ADJ CORRXY b0 LAMZ LAMZNPT LAMZLL LAMZUL
## 1 0.9999997 0.9999995 -0.9999999 2.368785 0.048457 3 9.05 24.37attach(resNCA, warn.conflicts=FALSE)If you want to manually omit some points, use an R package for convenience.
HL_Lambda_z (LAMZHL)
Terminal half-life calculated by ln(2)/LAMZ
LAMZHL = NA
if (LAMZ > 0) LAMZHL = log(2)/LAMZ
LAMZHL## [1] 14.30438Clast_pred (CLSTP)
Predicted Clast, predicted concentration at Tlast.
\(C_{last,pred}=exp(\beta_0 - \lambda \cdot T_{last})\)
CLSTP = NA
if (LAMZ > 0) CLSTP = exp(b0 - LAMZ*x[iLast])
CLSTP## [1] 3.280146C0 (C0)
Concentration at time 0, intial concentration. This is calculated only BOLUS administration.
\(C_0 = exp(log(c_1) - t_1 \frac{log(c_2) - log(c_1)}{t_2 - t_1})\)
x = x0
y = y0
if (toupper(Adm) == "BOLUS") {
if (y[1] > y[2] & y[2] > 0) {
C0 = exp(log(y[1]) - x[1]*(log(y[2]) - log(y[1]))/(x[2] - x[1]))
} else {
C0 = y[x==min(x[y > 0])]
}
} else {
C0 = NA
}AUClast (AUCLST)
Area under the time-concentration curve from dosing to the last positive concentration.
For linear trapezoidal method,
\(AUC_{last} = \sum_{i = 2} \frac{(t_i - t_{i-1}) \times (c_i - c_{i-1})}{2}\)
n = length(x1)
AUCLST = sum((y1[-1] + y1[-n]) * (x1[-1] - x1[-n]))/2
AUCLST## [1] 148.923For ‘linear-up log-down’ method,
AUCLST = 0
for (i in 2:n) {
if (y1[i] < y1[i-1] & y1[i] > 0) {
k = (log(y1[i - 1]) - log(y1[i]))/(x1[i] - x1[i - 1]) # slope in log
AUCLST = AUCLST + (y1[i - 1] - y1[i])/k
} else {
AUCLST = AUCLST + (x1[i] - x1[i - 1])*(y1[i] + y1[i - 1])/2
}
}
AUCLSTAUCall (AUCALL)
AUC values including all zero concentrations.
For linear trapezoidal method,
AUCALL = sum((y0[-1] + y0[-n]) * (x0[-1] - x0[-n]))/2
AUCALL## [1] 148.923For ‘linear-up log-down’ method,
AUCALL = 0
for (i in 2:n) {
if (y0[i] < y0[i-1] & y0[i] > 0) {
k = (log(y0[i - 1]) - log(y0[i]))/(x0[i] - x0[i - 1]) # slope in log
AUCALL = AUCALL + (y0[i - 1] - y0[i])/k
} else {
AUCALL = AUCALL + (x0[i] - x0[i - 1])*(y0[i] + y0[i - 1])/2
}
}
AUCALLZero concentrations to be log-transformed need not be removed, because R can handle infinity value.
AUCinf_obs (AUCIFO)
AUCinf observed.
\(AUC_{inf,obs} = AUC_{last} + \frac{C_{last}}{\lambda_z}\)
AUC_%Extrap_obs (AUCPEO)
AUC percent extrapolated observed.
\(AUC_{\%Extrap,obs} = (1 - \frac{AUC_{last}}{AUC_{inf,obs}}) \times 100\)
AUCinf_D_obs (AUCIFOD)
Dose normalized AUCinf observed.
\(AUC_{dose,inf,obs} = frac{AUC_{inf,obs}}{Dose}\)
AUCinf_pred (AUCIFP)
AUCinf predicted.
\(AUC_{inf,pred} = AUC_{last} + \frac{C_{last,pred}}{\lambda_z}\)
AUC_%Extrap_pred (AUCPEP)
AUC percent extrapolated predicted.
\(AUC_{\%Extrap,pred} = (1 - \frac{AUC_{last}}{AUC_{inf,pred}}) \times 100\)
AUCinf_D_pred (AUCIFPD)
Dose normalized AUCinf predicted.
\(AUC_{dose,inf,pred} = frac{AUC_{inf,pred}}{Dose}\)
AUMClast (AUMCLST)
\(AUMC_{last} = \int_{0}^{t_{last}} tC(t)dt\)
For linear trapezoidal method;
\(AUMC_{last} \approx \sum_{i=2} \frac{(t_i - t_{i-1})(t_i c_i + t_{i-1} c_{i-1})}{2}\)
AUMCLST = sum((x1[-1] - x1[-n])*(x1[-1]*y1[-1] + x1[-n]*y1[-n]))/2For ‘linear-up log-down’ method;
AUMCLST = 0
for (i in 2:n) {
if (y1[i] < y1[i-1] & y1[i] > 0) {
k = (log(y1[i-1]) - log(y1[i]))/(x1[i] - x1[i-1]) # slope in log
AUMCLST = AUMCLST + (x1[i-1]*y1[i-1] - x1[i]*y1[i])/k + (y1[i-1] - y1[i])/k/k
} else {
AUMCLST = AUMCLST + (x1[i] - x1[i-1])*(x1[i]*y1[i] + x1[i-1]*y1[i-1])/2
}
}AUMCinf_obs (AUMCIFO)
AUMC infinity observed.
\(AUMC_{inf,obs} = AUMC_{last} + \frac{C_{last}T_{last}}{\lambda_z} + \frac{C_{last}}{(\lambda_z)^2}\)
AUMC_%Extrap_obs (AUMCPEO)
AUMC percent extrapolated observed.
\(AUMC_{\%Extrap,obs} = (1 - \frac{AUMC_{last}}{AUMC_{inf,obs}}) \times 100\)
AUMCinf_pred (AUMCIFP)
AUMC infinity predicted.
\(AUMC_{inf,pred} = AUMC_{last} + \frac{C_{last,pred}T_{last}}{\lambda_z} + \frac{C_{last,pred}}{(\lambda_z)^2}\)
AUMC_%Extrap_pred (AUMCPEP)
AUMC percent extrapolated predicted.
\(AUMC_{\%Extrap,pred} = (1 - \frac{AUMC_{last}}{AUMC_{inf,pred}}) \times 100\)
AUC_Back_Ext_obs (AUCBEO)
AUC back extrapolated observed. This is only for BOLUS administration.
For trapezoidal method;
\(AUC_{back extrap} = \frac{t_1 \times (C_0 + C_1)}{2}\)
For log-down method;
\(AUC_{back extrap} = \frac{t_1 \times (C_0 + C_1)}{log(C_0) - log(C_1)}\)
AUC_%Back_Ext_obs (AUCPBEO)
AUC percent back extrapolated observed. This is only for BOLUS administration.
\(AUC_{\%back extrap,obs} = \frac{AUC_{back_extrap}}{AUC_{inf,obs}} \times 100\)
AUC_%Back_Ext_pred (AUCPBEP)
AUC percent back extrapolated predicted. This is only for BOLUS administration.
\(AUC_{\%back extrap,pred} = \frac{AUC_{back_extrap}}{AUC_{inf,pred}} \times 100\)
MRTlast (MRTEVLST, MRTIVLST)
Mean Residence Time (MRT) from 0 to Tlast.
\(MRT_{EV,last} = \frac{AUMC_{last}}{AUC_{last}}\)
\(MRT_{IV,last} = \frac{AUMC_{last}}{AUC_{last}} - \frac{Dur}{2}\)
Here ‘Dur’ is infusion duration.
MRTinf_obs (MRTEVIFO, MRTIVIFO)
MRT infinity observed.
\(MRT_{EV,inf,obs} = \frac{AUMC_{inf,obs}}{AUC_{inf,obs}}\)
\(MRT_{IV,inf,obs} = \frac{AUMC_{inf,obs}}{AUC_{inf,obs}} - \frac{Dur}{2}\)
MRTinf_pred (MRTEVIFP, MRTIVIFP)
MRT infinity predicted.
\(MRT_{EV,inf,pred} = \frac{AUMC_{inf,pred}}{AUC_{inf,pred}}\)
\(MRT_{IV,inf,pred} = \frac{AUMC_{inf,pred}}{AUC_{inf,pred}} - \frac{Dur}{2}\)
Vz_obs (VZO) or Vz_F_obs (VZFO)
Volume of distribution by terminal slope observed. VZO is for intravascular administration and VZFO for extravascular administration.
\(V_{z,obs} = \frac{Dose}{AUC_{inf,obs} \times \lambda_z}\)
Vz_pred (VZP) or Vz_F_pred (VZFP)
Volume of distribution by terminal slope predicted. VZP is for intravascular administration and VZFP for extravascular administration.
\(V_{z,pred} = \frac{Dose}{AUC_{inf,pred} \times \lambda_z}\)
CL_obs (CLO) or CL_F_obs (CLFO)
Clearance observed. CLO is for intravascular administration and CLFO for extravascular administration.
\(CL_{obs} = \frac{Dose}{AUC_{inf,obs}}\)
CL_pred (CLP) or CL_F_pred (CLFP)
Clearance predicted. CLP is for intravascular administration and CLFP for extravascular administration.
\(CL_{pred} = \frac{Dose}{AUC_{inf,pred}}\)
Vss_obs (VSSO)
Volume of distribution at steady state, observed. This is for intravascular administration only.
\(V_{ss,obs} = MRT_{IV,inf,obs} \times CL_{obs}\)
Vss_pred (VSSP)
Volume of distribution at stead state, predicted. This is for intravascular administration only.
\(V_{ss,pred} = MRT_{IV,inf,pred} \times CL_{pred}\)
AmtRecLast (RCAMLST)
Amount recovered in urine.
\(Ae_last = \sum_{i=1} Vol_i Conc_i\)
CLrenal (RENALCL)
Renal clearance.
\(CL_R = \frac{Ae_{last}}{AUC_{last}}\)
fe (FE)
Fraction excreted unchanged.
\(fe = \frac{CL_R}{CL_{obs}} = \frac{{Ae_{last}}/{AUC_{last}}}{{Dose}/{AUC_{inf,obs}}} = \frac{Ae_{last}}{Dose} \times \frac{AUC_{inf,obs}}{AUC_{last}}\)
AI (ARCMAX, ARAUC, ARCMIN)
Accumulation index, accumulation ratio.
\(R_{ac} = \frac{C_{max,ss}}{C_{max,1}} = \frac{AUC_{tau,ss}}{AUC_{tau,1}} = \frac{AUC_{inf,ss}}{AUC_{inf,1}} = \frac{C_{min,ss}}{C_{min,1}}\)