Chapter_03-NCA_Equations.Rmd

---
title: "NCA Equations"
author: "William Denney"
date: "October 21, 2016"
output:
  html_document:
    toc: 1
---

```{r setup, include=FALSE}
library(pander)
knitr::opts_chunk$set(echo = TRUE)
source("report.helpers.R")
library(knitcitations)
pander::panderOptions("keep.line.breaks", TRUE)
cleanbib()
agah <- citep("http://www.agah.eu/fileadmin/_migrated/content_uploads/PK-glossary_PK_working_group_2004.pdf")
```

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 ${}^elog$).

# Defined Parameters

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

```{r defined, echo=FALSE}
defined <- data.frame(
  Symbol=c("C", "D", "t", "$\\tau$"),
  Units=c("concentration", "amount", "time", "time"),
  Definition=c("Concentration", "Dose", "Time", "Dosing interval"))
pander(defined)
```


# Parameter Table

```{r paramdeffun, echo=FALSE}
#' Add a parameter to the parameter list
#'
#' @param symbol is the NCA abbreviation as it would typically be used in the
#'   text of a paper.  It may use LaTex inline formula notation with dollar
#'   signs around the value.  It may also be a vector of symbols if more than
#'   one have nearly identical definitions.
#' @param units A text description of the units like "concentration" or
#'   "concentration*time".
#' @param definition A text definition of the parameter.
#' @param cdisc The CDISC PKPARMCD value for the term (all capital letters
#'   <= 8 characters).
#' @param single Can the parameter be used with single dosing? (logical)
#' @param multiple Can the parameter be used with steady-state multiple
#'   dosing? (logical)
#' @param iv Can the parameter be used with intravenous dosing? (logical)
#' @param po Can the parameter be used with oral dosing? (logical)
#' @param equation The LaTeX-formatted equation for the parameter (without
#'   the dollar signs).
#' @param notes Notes on the use of the parameter.  This will often define
#'   the differences between variants of the parameter.
addparam <- local({
  paramlist <- data.frame()
  function(symbol, units, definition, cdisc,
           single, multiple,
           iv, po, equation, notes="",
           reference="") {
    if (all(equation %in% "Observed")) {
      ## Do nothing
    } else if (!all(equation %in% "")) {
      equation <-
        paste0("$$", equation, "$$",
               collapse="\n")
    }
    paramlist <<-
      rbind(paramlist,
            data.frame(
              symbol=paste(symbol, collapse=", "),
              units=units,
              definition=definition,
              cdisc=paste(cdisc, collapse=", "),
              single=single,
              multiple=multiple,
              intravenous=iv,
              extravascular=po,
              equation=equation,
              notes=paste(notes, collapse="\n"),
              reference=reference,
              stringsAsFactors=FALSE))
  }
})
```

```{r parameterdefs, echo=FALSE}
## Common, observed parameters
addparam(symbol="$C_{max}$",
         units="concentration",
         definition="The maximum concentration occurring at $t_{max}$.",
         cdisc=c("CMAX"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="Observed",
         notes="",
         reference="")

addparam(symbol="$t_{max}$",
         units="time",
         definition="The time of $C_{max}$.",
         cdisc=c("TMAX"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="Observed",
         notes="",
         reference="")

addparam(symbol="$C_{min}$",
         units="concentration",
         definition="The minimum concentration occurring between dose time and dose time plus $\\tau$ (at $t_{min}$).",
         cdisc=c("CMIN"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="Observed",
         notes="Compare with $C_{trough}$.",
         reference="")

addparam(symbol="$t_{min}$",
         units="time",
         definition="The time of $C_{min}$.",
         cdisc=c("TMIN"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="Observed",
         notes="",
         reference="")

addparam(symbol="$C_{trough}$",
         units="concentration",
         definition="Concentration at end of dosing interval.",
         cdisc=c("CTROUGH"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="Observed",
         notes="Compare with $C_{min}$.",
         reference="")

addparam(symbol="$t_{lag}$",
         units="time",
         definition="The time prior to the first increase in concentration.",
         cdisc=c("TLAG"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="Observed",
         notes="",
         reference=agah)


## Clast-related parameters
addparam(symbol="$C_{last,obs}$",
         units="concentration",
         definition="The last observed concentration above the limit of quantification.  Equivalently, the concentration corresponding to $t_{last}$.",
         cdisc=c("CLST"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="Observed",
         notes="",
         reference=agah)

addparam(symbol="$C_{last,pred}$",
         units="concentration",
         definition="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).",
         cdisc=c(),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="C_{last,pred} = \\lambda_z \\times t_{last} + A",
         notes="Parameters in the equation are fit during half-life fitting.",
         reference=agah)

addparam(symbol="$t_{last}$",
         units="time",
         definition="The time of the last measurable (positive) concentration.",
         cdisc=c("TLST"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="Observed",
         notes="",
         reference=agah)

# AUC-related parameters
addparam(symbol="$AUC_{a,b,linear}$",
         units="concentration*time",
         definition="Area under the concentration time curve from measurement at time $a$ to time $b$ using the linear trapezoidal rule.",
         cdisc="",
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="AUC_{t_i,t_{i+1}} = \\frac{1}{2} \\left(C_i + C_{i+1}\\right) \\left(t_{i+1} - t_i\\right)",
         reference=agah)

addparam(symbol="$AUC_{a,b,log-linear}$",
         units="concentration*time",
         definition="Area under the concentration time curve from measurement at time $a$ to time $b$ using the log-linear trapezoidal rule.",
         cdisc="",
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="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}}}",
         reference=agah)

addparam(symbol=c("$AUC_{0,t}$",
                  "$AUC_{last}$",
                  "$AUC_{\\tau}$"),
         units="concentration*time",
         definition="Area under the concentration time curve during a defined interval using only samples above the limit of quantification",
         cdisc=c("AUCINT",
                 "AUCLST",
                 "AUCTAU"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="AUC = \\sum_{t=a}^{t=min(t_{last}, b)} AUC_{t_i,t_{i+1}}",
         notes="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 $C_{i+1} < C_i$.",
         reference=agah)

addparam(symbol=c("$AUC_{t_{last}-\\infty,pred}$",
                  "$AUC_{t_{last}-\\infty,obs}$"),
         units="concentration*time",
         definition="Area under the concentration-time curve from $t_{last}$ to $\\infty$",
         cdisc=c(),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{C_{last}}{\\lambda_z}",
         notes="$C_{last}$ should be $C_{last,pred}$ or $C_{last,obs}$ depending on which version of the parameter is desired.",
         reference=agah)

addparam(symbol="$AUC_{t_{last}-\\infty,all}$",
         units="concentration*time",
         definition="Area under the concentration-time curve from $t_{last}$ to $\\infty$ as used for $AUC_{all}$.",
         cdisc=c(),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="AUC_{t_{last},t_{last+1},linear}",
         notes="$t_{last+1}$ is the first time point below the limit of quantification.",
         reference=agah)

addparam(c("$AUC_{0-\\infty,pred}$",
           "$AUC_{0-\\infty,obs}$"),
         units="concentration*time",
         definition="The area under the curve (AUC) extrapolated to $\\infty$, calculated using the observed or predicted value of the last non-zero concentration.",
         cdisc=c("AUCIFP",
                 "AUCIFO"),
         single=TRUE,
         multiple=FALSE,
         iv=TRUE,
         po=TRUE,
         equation="AUC_{last} + AUC_{t_{last}-\\infty}",
         notes="Use the 'pred' or 'obs' version of $AUC_{t_{last}-\\infty}$ for the equivalent version of $AUC_{0-\\infty}$.",
         reference=agah)

addparam(symbol="$AUC_{0,all}$",
         units="concentration*time",
         definition="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.",
         cdisc=c("AUCALL"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="AUC_{last} + AUC_{t_{last}-\\infty,all}",
         notes="If all concentrations are above the limit of quantification, then $AUC_{0,all}=AUC_{last}$.",
         reference=agah)

addparam(symbol=c("$AUC_{\\%extrap,obs}$",
                  "$AUC_{\\%extrap,pred}$"),
         units="%",
         definition="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 $C_{last}$.",
         cdisc=c("AUCPEO", "AUCPEP"),
         single=TRUE,
         multiple=FALSE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{AUC_{t_{last},\\infty}}{AUC_{0,\\infty}} \\times 100",
         notes="Use the 'pred' or 'obs' version of $AUC_{t_{last},\\infty}$ and $AUC_{0,\\infty}$ for the equivalent version of $AUC_{\\%extrap}$.",
         reference=agah)

addparam(symbol="$F$",
         units="fraction",
         definition="Bioavailability",
         cdisc=c(),
         single=TRUE,
         multiple=FALSE,
         iv=TRUE,
         po=TRUE,
         equation=c("\\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}}"),
         notes="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 $AUC_{0,\\infty}$.",
         reference=agah)

# AUMC parameters

addparam(symbol="$AUMC_{a,b,linear}$",
         units="concentration*time^2^",
         definition="Area under the first moment of the concentration time curve from measurement at time $a$ to time $b$ using the linear trapezoidal rule.",
         cdisc="",
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="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)",
         reference=agah)

addparam(symbol="$AUMC_{a,b,log-linear}$",
         units="concentration*time^2^",
         definition="Area under the first moment of the concentration time curve from measurement at time $a$ to time $b$ using the log-linear trapezoidal rule.",
         cdisc="",
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="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)",
         reference=agah)

addparam(symbol=c("$AUMC_{0,t}$",
                  "$AUMC_{last}$",
                  "$AUMC_{\\tau}$"),
         units="concentration*time^2^",
         definition="Area under the first moment of the concentration time curve during a defined interval using only samples above the limit of quantification",
         cdisc=c("AUMCLST",
                 "AUMCTAU"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="AUMC_{last} = \\sum_{t=a}^{t=min(t_{last}, b)} AUMC_{t_i,t_{i+1}}",
         notes="See $AUC_{last}$ notes for suggested calculation options.",
         reference=agah)

addparam(symbol=c("$AUMC_{t_{last}-\\infty,obs}$",
                  "$AUMC_{t_{last}-\\infty,pred}$"),
         units="concentration*time^2^",
         definition="Area under the first moment of the concentration-time curve from $t_{last}$ to $\\infty$",
         cdisc=c(),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{C_{last} t_{last}}{\\lambda_z} + \\frac{C_{last}}{\\lambda_z^2}",
         notes="$C_{last}$ should be $C_{last,pred}$ or $C_{last,obs}$ depending on which version of the parameter is desired.",
         reference=agah)

addparam(symbol="$AUMC_{t_{last}-\\infty,all}$",
         units="concentration*time^2^",
         definition="Area under the first moment of the concentration-time curve from $t_{last}$ to $\\infty$ as used for $AUC_{all}$.",
         cdisc=c(),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="AUC_{t_{last},t_{last+1},linear}",
         notes="$t_{last+1}$ is the first time point below the limit of quantification.",
         reference=agah)

addparam(c("$AUMC_{0-\\infty,obs}$",
           "$AUMC_{0-\\infty,pred}$"),
         units="concentration*time^2^",
         definition="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.",
         cdisc=c("AUMCIFP",
                 "AUMCIFO"),
         single=TRUE,
         multiple=FALSE,
         iv=TRUE,
         po=TRUE,
         equation="AUMC_{last} + AUMC_{t_{last}-\\infty}",
         notes="Use the 'obs' or 'pred' version of $AUC_{t_{last}-\\infty}$ for the equivalent version of $AUC_{0-\\infty}$.",
         reference=agah)

addparam(symbol="$AUMC_{0,all}$",
         units="concentration*time^2^",
         definition="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.",
         cdisc=c(),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="AUMC_{last} + AUMC_{t_{last}-\\infty,all}",
         notes="If all concentrations are above the limit of quantification, then $AUMC_{0,all}=AUMC_{last}$.",
         reference=agah)

addparam(symbol=c("$AUMC_{\\%extrap,obs}$",
                  "$AUMC_{\\%extrap,pred}$"),
         units="%",
         definition="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 $C_{last}$.",
         cdisc=c("AUMCPEO", "AUMCPEP"),
         single=TRUE,
         multiple=FALSE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{AUMC_{t_{last}-\\infty}}{AUMC_{0,\\infty}} \\times 100",
         notes="Use the 'pred' or 'obs' version of $AUMC_{t_{last}-\\infty}$ for the equivalent version of $AUMC_{0-\\infty}$.",
         reference=agah)

## Half-life related parameters
addparam(symbol="$t_{1/2}$",
         units="time",
         definition="Terminal half-life",
         cdisc=c("LAMZHL"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{ln 2}{\\lambda_z}",
         notes="",
         reference=agah)

addparam(symbol="$\\lambda_z$",
         units="1/time",
         definition="The first order rate constant associated with the terminal (log-linear) portion of the curve.",
         cdisc=c("LAMZ"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="ln C = A - \\lambda_z t",
         notes="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.",
         reference=agah)

addparam(symbol=c("$t_{first,\\lambda_z}$",
                  "$t_{last,\\lambda_z}$"),
         units="time",
         definition="The first and last time (lower and upper limit on time) for values to be included in the calculation of $\\lambda_z$.",
         cdisc=c("LAMZLL",
                 "LAMZUL"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="",
         notes="",
         reference=agah)

addparam(symbol="$N_{\\lambda_z}$",
         units="count",
         definition="The number of time points used in computing $\\lambda_z$.",
         cdisc=c("LAMZNPT"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="",
         notes="$N_{\\lambda_z}$ should always be greater than or equal to 3.",
         reference="")

addparam(symbol="$r^2$",
         units="fraction",
         definition="The goodness of fit statistic for the terminal elimination phase.",
         cdisc=c("R2"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="1 - \\frac{\\sum_i \\left(C_i - \\hat{C_i}\\right)^2}{\\sum_i \\left(C_i - \\overline{C_i}\\right)^2}",
         notes="$r^2$ 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.",
         reference="")

addparam(symbol="$r^2_{adj}$",
         units="fraction",
         definition="The goodness of fit statistic for the terminal elimination phase, adjusted for the number of time points used in the estimation of $\\lambda_z$.",
         cdisc=c("R2ADJ"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="1 - \\left(1 - r^2\\right) \\frac{1}{N_{\\lambda_z}-2}",
         notes="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 $p=1$.",
         reference=agah)

## Mean Residence Time

addparam(symbol=c("$MRT_{ev,last}$",
                  "$MRT_{ev,\\infty,obs}$",
                  "$MRT_{ev,\\infty,pred}$",
                  "$MRT_{iv,last}$",
                  "$MRT_{iv,\\infty,obs}$",
                  "$MRT_{iv,\\infty,pred}$"),
         units="time",
         definition="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.",
         cdisc=c("MRTEVLST",
                 "MRTEVIFO",
                 "MRTEVIFP",
                 "MRTIVLST",
                 "MRTIVIFO",
                 "MRTIVIFP"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{AUMC}{AUC}",
         notes="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 $MRT_{last}$).",
         reference=agah)

addparam(symbol=c("$MAT_{last}$",
                  "$MAT_{\\infty,obs}$",
                  "$MAT_{\\infty,pred}$"),
         units="time",
         definition="Mean absorption time of a substance administered by extravascular dosing.",
         cdisc=c("MAT"),
         single=TRUE,
         multiple=TRUE,
         iv=FALSE,
         po=TRUE,
         equation=c("MRT_{ev} - MRT{iv}",
                    "t_{lag} + \\frac{1}{K_a}"),
         notes="$K_a$ is the absorption rate from multi-exponential compartmental analysis curve fitting (and is not typically applicable for noncompartmental analysis).",
         reference=agah)

## Peak to trough and similar

addparam(symbol="$PTR$",
         units="ratio",
         definition="The maximum concentration during a dosing interval divided by the concentration at the end of the dosing interval.",
         cdisc=c("PTROUGHR"),
         single=FALSE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{C_{max}}{C_{trough}}",
         notes="",
         reference=agah)

addparam(symbol="$TPR$",
         units="ratio",
         definition="The concentration at the start of a dosing interval divided by the maximum concentration during the dosing interval.",
         cdisc=c("TROUGHPR"),
         single=FALSE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{C_{trough}}{C_{max}}",
         notes="",
         reference="")

addparam(symbol="$PTF$",
         units="%",
         definition="The difference between Cmin and Cmax standardized to Cavg, between dose time and $\\tau$.",
         cdisc=c("FLUCP"),
         single=FALSE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="100 \\times \\frac{C_{max} - C_{min}}{C_{avg}}",
         notes="",
         reference=agah)

addparam(symbol=c("$C_{avg}$", "$C_{av}$"),
         units="concentration",
         definition="Average concentration",
         cdisc=c("CAVG"),
         single=FALSE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{AUC_{0,\\tau}}{\\tau}",
         notes="",
         reference=agah)

## Accumulation ratio
addparam(symbol="$R_{A,AUC}$",
         units="fraction",
         definition="The area under the curve at steady state divided by the area under the curve over the initial dosing interval.",
         cdisc=c("ARAUC"),
         single=FALSE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="R_{A,AUC} = \\frac{AUC_{\\tau,ss}}{AUC_{\\tau,sd}}",
         notes="$ss$ indicates that value is at steady-state while $sd$ indicated that value is from the first (single) dose.",
         reference=agah)

addparam(symbol="$R_{A,C_{max}}$",
         units="fraction",
         definition="The maximum concentration at steady state divided by the maximum concentration during the initial dosing interval.",
         cdisc=c("ARCMAX"),
         single=FALSE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="R_{A,C_{max}} = \\frac{C_{max,ss}}{C_{max,sd}}",
         notes="$ss$ indicates that value is at steady-state while $sd$ indicated that value is from the first (single) dose.",
         reference=agah)

addparam(symbol="$R_{A,C_{min}}$",
         units="fraction",
         definition="The minimum concentration at steady state divided by the minimum concentration during the initial dosing interval.",
         cdisc=c("ARCMIN"),
         single=FALSE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="R_{A,C_{min}} = \\frac{C_{min,ss}}{C_{min,sd}}",
         notes="$ss$ indicates that value is at steady-state while $sd$ indicated that value is from the first (single) dose.",
         reference=agah)

addparam(symbol="$R_{A,C_{trough}}$",
         units="fraction",
         definition="The trough concentration at steady state divided by the trough concentration during the initial dosing interval.",
         cdisc=c("ARCTROUG"),
         single=FALSE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="R_{A,C_{trough}} = \\frac{C_{trough,ss}}{C_{trough,sd}}",
         notes="$ss$ indicates that value is at steady-state while $sd$ indicated that value is from the first (single) dose.",
         reference=agah)

## Clearance
addparam(symbol=c("$CL_{ev,obs}$",
                  "$CL_{ev,pred}$",
                  "$CL_{ev,\\tau}$",
                  "$CL_{iv,obs}$",
                  "$CL_{iv,pred}$",
                  "$CL_{iv,\\tau}$"),
         units="volume/time",
         definition="The total body clearance for extravascular or intravascular administration (divided by the fraction of dose absorbed for extravascular), calculated using the equivalent $AUC$.",
         cdisc=c("CLFO",
                 "CLFP",
                 "CLFTAU",
                 "CLO",
                 "CLP",
                 "CLTAU"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{F \\times D}{AUC}",
         notes="For each of the symbols, select the equivalent $AUC$.  Such as, $CL_{ev,pred}$ uses $AUC = AUC_{0,\\infty,pred}$",
         reference=agah)

## Volumes
addparam(symbol=c("$V_{ss,obs}$",
                  "$V_{ss,pred}$"),
         units="volume",
         definition="The volume of distribution at steady state based on the equivalent  for a substance administered by intravascular dosing.",
         cdisc=c("VSSO", "VSSP"),
         single=TRUE,
         multiple=FALSE,
         iv=TRUE,
         po=FALSE,
         equation=c("CL \\times MRT",
                    "\\frac{F \\times D \\times AUMC}{{AUC}^2}"),
         notes="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 $AUC_{0,\\tau}$.",
         reference=agah)

addparam(symbol=c("$V_{z,ev,obs}$",
                  "$V_{z,ev,pred}$",
                  "$V_{z,ev,\\tau}$",
                  "$V_{z,iv,obs}$",
                  "$V_{z,iv,pred}$",
                  "$V_{z,iv,\\tau}$"),
         units="volume",
         definition="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 $AUC$.",
         cdisc=c("VZFO",
                 "VZFP",
                 "VZFTAU",
                 "VZO",
                 "VZP",
                 "VZTAU"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=TRUE,
         equation="\\frac{F \\times D}{AUC \\times \\lambda_z}",
         notes="",
         reference=agah)

## IV dosing parameters
addparam(symbol="$C_0$",
         units="concentration",
         definition="Initial concentration",
         cdisc=c("C0"),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=FALSE,
         equation="",
         notes="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 $C_0$.",
         reference="")

addparam(symbol="$V_0$",
         units="volume",
         definition="The initial volume of distribution for a substance administered by bolus intravascular dosing.",
         cdisc=c("V0"),
         single=TRUE,
         multiple=FALSE,
         iv=TRUE,
         po=FALSE,
         equation="\\frac{D}{C_0}",
         notes="",
         reference=agah)

addparam(symbol=c("$AUC_{\\%backextrap,obs}$",
                  "$AUC_{\\%backextrap,pred}$"),
         units="concentration*time",
         definition="",
         cdisc=c(),
         single=TRUE,
         multiple=TRUE,
         iv=TRUE,
         po=FALSE,
         equation="\\frac{AUC_{0,first}}{AUC_{0,\\infty}}",
         notes="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.",
         reference="")


## Convert the notes to references for later use
paramtable <- environment(addparam)$paramlist
mask.note <- !(paramtable$notes %in% c(NA, ""))
notetable <- paramtable[mask.note,
                        c("symbol", "notes")]
idx.note <- 1:sum(mask.note)
paramtable$notes[mask.note] <-
  sprintf("[%d][%s]", idx.note, paramtable$symbol[mask.note])
```

```{r maketable, echo=FALSE}
pander(paramtable, split.tables=Inf)
```

# Notes

```{r notelist, results="asis", echo=FALSE}
for (i in seq_len(nrow(notetable))) {
  ## The link to the section header is automatic
  cat(sprintf("## %s\n", notetable$symbol[i]))
  cat(sprintf("%s\n\n", notetable$note[i]))
}
```

# References

```{r bibliography, echo=FALSE, results="asis"}
knitcitations::bibliography()
```