---
output:
  pdf_document:
    extra_dependencies: ["float"]
    number_sections: true
    toc: true
    toc_depth: 3
header_includes:
  -\usepackage{fancyheadr}
  -\setlength{\headheight}{22pt}%
  -\usepackage{lastpage}
  -\pagestyle{fancy}
  -\usepackage{pdflscape}
  -\usepackage{longtable}
  -\rhead{\includegraphics[width=.15\textwidth]{`r getwd()`/logo.png}}
params:
  FileName: NA
  newTitle: NA
  author: NA
  REP: NA
  coeffs: NA
author: "Author: `r params$author`"
title: |
    | ![](logo.png){width=1in}
    | 4PL bioassay evaluation
subtitle: |
  `r params$FileName`
  
  <left> Unique time: </left> <right> `r Sys.time()`</right>
date: "`r paste(params$Subway, params$Version)`"

---

<!-- \fancyfoot[C]{\thepage\ of \pageref{LastPage}} -->
<!-- \newpage -->

<!-- \newpage -->

```{r setup, include=FALSE}

knitr::opts_chunk$set(echo = TRUE)

library(knitr)
library(DT)

REP <- params$REP
author <- params$author
coeffs <- params$coeffs

all_l <- REP$all_l
ANOVAXLS <- REP$ANOVAXLS
DiagnTable <- REP$DiagnTable
UnRPLAausw <- REP$UnRPLAausw
UnRPLBend <- REP$UnRPLBend
PLAausw <- REP$PLAausw
PLBend <- REP$PLBend
LogPLAausw <- REP$LogPLAausw
LogUnrPLAausw <- REP$LogUnrPLAausw

XLdat2 <- REP$XLdat2

CIplot <- REP$CIplot
testsTab <- REP$testsTab
relpotTestPlot <- REP$relpotTestPlot



```


# Introduction

Bioassay potency estimation uses statistical methods to quantify the strength of a biological product or drug by comparing its response to that of a reference standard. Because biological responses are inherently variable, affected by assay conditions, cell systems or organisms, and measurement noise, the 4-parametric logistic regression is used to obtain reliable potency values. The variance for confidence interval calculation is coming from the regression procedure itself and is an excellent predictor for the variability of any future potency determinations.  
USP<1034> recommends calculation of standard errors of ratios of the parameters using Fieller's theorem [Finney D.J. 1978] or using the "delta" method (for a discussion about the "delta" method see [Ver Hoef 2012]). However, the presented gradient approach using the differences on the log-scale is methematically more stable und thus preferable compared to any ratio approach ([Franz, V.H. 2007]).

# Results

All data used for the 4PL evaluation is shown in table 1:

```{r alll, echo=FALSE, warning=FALSE, results='asis'}

kable(all_l, format = "markdown", caption= "Uploaded data (test and reference) in long format", digits=3)

```

The following 4 plots show all 4 models: restricted and unrestricted, and log transformed, respectively.

You can also embed plots, for example:

```{r XLplot, echo=FALSE, warning=FALSE, fig.height=4, fig.width=6, fig.cap="Plot of models", fig.align='left'}

plot_f <- function(dat, sigmoid,det_sig) {
  CORdat <- cor(dat[,1],dat[,ncol(dat)])

  all_l <- melt(data.frame(dat), id.vars="log_dose", variable.name="replname", value.name = "readout")
  isRef <- rep(c(1,0),1,each=nrow(all_l)/2)
  isSample <- rep(c(0,1),1,each=nrow(all_l)/2)
  all_l2 <- cbind(all_l, isRef, isSample)

  if(is.null(det_sig)) {
    if (CORdat<0) {
      startlist <- list(a=sigmoid[3], b=-sigmoid[5],cs=sigmoid[7],
                        d=sigmoid[1],r=sigmoid[8])
    } else {
      startlist <- list(a=sigmoid[3],b=sigmoid[5],cs=sigmoid[7],
                        d=sigmoid[1],r=sigmoid[8])
    }
  } else {
    startlist <- list(a=det_sig[5], b=det_sig[1],cs=det_sig[7],
                      d=det_sig[3],r=det_sig[7] - det_sig[8])
  }
  #browser()
  tryCatch({
    mr <- gsl_nls(fn = readout ~ a+(d-a)/(1+exp(b*(log_dose-(cs-r*isSample)))),
                  data=all_l2,
                  start=startlist,
                  control=gsl_nls_control(xtol=1e-6,ftol=1e-6, gtol=1e-6))
  },
  error = function(err) {
    err$message
  })
  s_mr <- summary(mr)
  a <- s_mr$coefficients[1,1]
  b <- s_mr$coefficients[2,1]
  cs <- s_mr$coefficients[3,1]
  d <- s_mr$coefficients[4,1]
  r <- s_mr$coefficients[5,1]

  log_dose <- unique(all_l$log_dose)
  seq_x <- seq(min(log_dose),max(log_dose),0.1)
  SAMPLE <- a+(d-a)/(1+exp(b*(seq_x-(cs-r))))
  REF <- a+(d-a)/(1+exp(b*(seq_x-(cs))))

  if (is.null(det_sig)) {
    SAMPLEtrue <- sigmoid[4] + (sigmoid[2] -sigmoid[4])/(1+exp(sigmoid[6]*(seq_x-(sigmoid[7]-sigmoid[8]))))
    REFtrue <- sigmoid[3] + (sigmoid[1] -sigmoid[3])/(1+exp(sigmoid[5]*(seq_x-(sigmoid[7]))))
  } else {
    SAMPLEtrue <- det_sig[4] + (det_sig[6] -det_sig[4])/(1+exp(-det_sig[2]*(seq_x-(det_sig[8]))))
    REFtrue <- det_sig[3] + (det_sig[5] -det_sig[3])/(1+exp(-det_sig[1]*(seq_x-(det_sig[7]))))
  }

  pl_df <- cbind(seq_x, SAMPLE, REF, SAMPLEtrue, REFtrue)
  all_l2$readout[all_l2$readout < 0] <- 0.01
  all_l2$readouttrans <- log(all_l2$readout)
  slopeEC50 <- b*(a-d)/4

  Xbendl3 <- cs-(1.31696/b)
  Xbendu3 <- cs+(1.31696/b)
  XbendlT <- cs-r-(1.31696/b)
  XbenduT <- cs-r+(1.31696/b)
  bendpoints <- c(bendREF_lower = round(Xbendl3,3), bendREF_upper=round(Xbendu3,3),
                  bendSAMPLE_lower = round(XbendlT,3), bendSAMPLE_upper=round(XbenduT,3))

  p <- ggplot(all_l2, aes(x=log_dose, y=readout, color=factor(isRef))) +
    geom_point(shape=factor(isRef), alpha=0.8) +
    labs(title = paste("restricted 4pl; bendp:", round(Xbendl3,3),round(Xbendu3,3),round(XbendlT,3),round(XbenduT,3)),
         color="product") +
    scale_color_manual(labels=c("test","reference"), values=c("red","blue")) +
    scale_shape_manual(labels=c("test","reference")) +
    theme_bw() +
    theme(axis.text = element_text(size=14))

  p2 <- p + geom_line(data=as.data.frame(pl_df), aes(x=seq_x, y=SAMPLE), color="red",
                      inherit.aes = F) +
    geom_line(data=as.data.frame(pl_df), aes(x=seq_x, y=REF), color="blue",
              inherit.aes = F) +
    geom_line(data=as.data.frame(pl_df), aes(x=seq_x, y=SAMPLEtrue), color="red", linetype=2, alpha=0.4,
              inherit.aes = F) +
    geom_line(data=as.data.frame(pl_df), aes(x=seq_x, y=REFtrue), color="blue", linetype=2, alpha=0.4,
                                          inherit.aes = F) +
    geom_vline(xintercept=c(Xbendl3, Xbendu3), col="blue",linetype=2) +
    geom_vline(xintercept=c(XbendlT, XbenduT), col="red",linetype=2) +
    annotate("text", x=cs, y=a+(d-a)/2, label="0", size=5) +
    theme(legend.position="none")


  # transformed plots
  p_rt <- ggplot(all_l2, aes(x=log_dose, y=readouttrans, color=factor(isRef))) +
    geom_point(shape=factor(isRef), alpha=0.8) +
    labs(title = paste("restricted transformed 4pl"), color="product") +
    scale_color_manual(labels=c("test","reference"), values=c("red","blue")) +
    theme_bw()

  mrt <- gsl_nls(fn = readouttrans ~ a+(d-a)/(1+exp(b*(log_dose-(cs-r*isSample)))),
                data=all_l2,
                start=startlist,
                control=gsl_nls_control(xtol=1e-6,ftol=1e-6, gtol=1e-6))
  s_mrt <- summary(mrt)
  a_trans <- s_mrt$coefficients[1,1]
  b_trans <- s_mrt$coefficients[2,1]
  cs_trans <- s_mrt$coefficients[3,1]
  d_trans <- s_mrt$coefficients[4,1]
  r_trans <- s_mrt$coefficients[5,1]

  XbendlTrans <- cs_trans-(1.31696/b_trans)
  XbenduTrans <- cs_trans+(1.31696/b_trans)
  XbendlTransT <- cs_trans-r_trans-(1.31696/b_trans)
  XbenduTransT <- cs_trans-r_trans+(1.31696/b_trans)
  bendpointsTRANS <- c(bendREF_lower = round(XbendlTrans,3), bendREF_upper=round(XbenduTrans,3),
                  bendSAMPLE_lower = round(XbendlTransT,3), bendSAMPLE_upper=round(XbenduTransT,3))

  SAMPLEtrans <- a_trans+(d_trans-a_trans)/(1+exp(b_trans*(seq_x-(cs_trans-r_trans))))
  REFtrans <- a_trans+(d_trans-a_trans)/(1+exp(b_trans*(seq_x-(cs_trans))))

  pl_df_trans <- cbind(seq_x, SAMPLEtrans, REFtrans)
  p_rt2 <- p_rt + geom_line(data=as.data.frame(pl_df_trans), aes(x=seq_x, y=SAMPLEtrans), color="red",
                            inherit.aes = F) +
    geom_line(data=as.data.frame(pl_df_trans), aes(x=seq_x, y=REFtrans), color="blue",
              inherit.aes = F) +
    geom_vline(xintercept=c(XbendlTrans, XbenduTrans), col="blue",linetype=2) +
    geom_vline(xintercept=c(XbendlTransT, XbenduTransT), col="red",linetype=2) +
    theme(legend.position = "none", axis.text=element_text(size=14))

  if (is.null(det_sig)) {
    unrestr <- drm(readout ~ exp(log_dose), isSample, data=all_l2, fct=LL.4(),
                   pmodels=data.frame(isSample, isSample,isSample,isSample))
    Sum_u <- summary(unrestr)
    ast <- Sum_u$coefficients[3,1]
    ate <- Sum_u$coefficients[4,1]
    bst <- Sum_u$coefficients[1,1]
    bte <- Sum_u$coefficients[2,1]
    cst <- log(Sum_u$coefficients[7,1])
    cte <- log(Sum_u$coefficients[8,1])
    dst <- Sum_u$coefficients[5,1]
    dte <- Sum_u$coefficients[6,1]
  } else {
    ast <- det_sig[5]
    ate <- det_sig[6]
    bst <- det_sig[1]
    bte <- det_sig[2]
    cst <- det_sig[7]
    cte <- det_sig[8]
    dst <- det_sig[3]
    dte <- det_sig[4]
  }
  REFu <- ast + (dst-ast)/(1+exp(bst*(seq_x-cst)))
  SAMPLEu <- ate + (dte-ate)/(1+exp(bte*(seq_x-cte)))
  pl_df2 <- cbind(seq_x, SAMPLEu, REFu)
#browser()
  pu <- ggplot(all_l2, aes(x=log_dose, y=readout, color=factor(isRef))) +
    geom_point() +
    labs(title="unrestricted 4_pl-Model", color="product") +
    scale_color_manual(labels = c("test","reference"), values=c("red","blue")) +
    theme_bw()
  pu2 <- pu + geom_line(data=as.data.frame(pl_df2), aes(x=seq_x, y=SAMPLEu),
                   color="red", inherit.aes = F) +
    geom_line(data=as.data.frame(pl_df2), aes(x=seq_x, y=REFu),
              color="blue", inherit.aes = F,
              show.legend = F)
    pu2_ <- pu2 +
    theme(legend.position = "none", axis.text = element_text(size=14))
  putrans <- ggplot(all_l2, aes(x=log_dose, y=readouttrans, color=factor(isRef))) +
    geom_point() +
    labs(title="unrestricted transformed 4_pl-Model", color="product") +
    scale_color_manual(labels = c("test","reference"), values=c("red","blue")) +
    theme_bw()

  unrestr_trans <- drm(readouttrans ~ exp(log_dose), isSample, data=all_l2, fct=LL.4(),
                 pmodels=data.frame(isSample, isSample,isSample,isSample))
  Sum_ut <- summary(unrestr_trans)
  ast_t <- Sum_ut$coefficients[3,1]
  ate_t <- Sum_ut$coefficients[4,1]
  bst_t <- Sum_ut$coefficients[1,1]
  bte_t <- Sum_ut$coefficients[2,1]
  cst_t <- log(Sum_ut$coefficients[7,1])
  cte_t <- log(Sum_ut$coefficients[8,1])
  dst_t <- Sum_ut$coefficients[5,1]
  dte_t <- Sum_ut$coefficients[6,1]

  REFu_trans <- ast_t + (dst_t-ast_t)/(1+exp(bst_t*(seq_x-cst_t)))
  SAMPLEu_trans <- ate_t + (dte_t-ate_t)/(1+exp(bte_t*(seq_x-cte_t)))
  pl_df2u_t <- cbind(seq_x, SAMPLEu_trans, REFu_trans)

  pu2_t <- putrans + geom_line(data=as.data.frame(pl_df2u_t), aes(x=seq_x, y=SAMPLEu_trans),
                               color="red", inherit.aes = F) +
    geom_line(data=as.data.frame(pl_df2u_t), aes(x=seq_x, y=REFu_trans),
              color="blue", inherit.aes = F,
              show.legend = F)
  pu3_t <- pu2_t
  grid.arrange(p2,p_rt2,pu2_,pu3_t, nrow=2)
}

plot_f(XLdat2, sigmoid=NULL, det_sig=coeffs)


```

The ANOVA of the unconstrained model is listed in table 2:

```{r anovaxls, echo=FALSE, warning=FALSE, results='asis'}

kable(ANOVAXLS, format = "markdown", caption= "ANOVA table unrestricted", digits=3)


```


```{r SST_ergebn, fig.align='center', fig.pos='htb!', echo=FALSE, cache=FALSE, warning=FALSE, message=FALSE, tidy=TRUE}

kable(testsTab[1:7,], row.names = F, format = "markdown", caption="SST results")


```


*...The estimate for F-test on regression and on non-linearity is the p-value
  F-test on regression passes if F-value > F-crit and thus p < 0.05
  F-test on non-linearity passes if F-value < F-crit and thus p > 0.05
  Test results outcome:

  0 ... test passed (for EQ tests: CI within limits);

  1 ... test failed (for EQ tests CI not within limits);

  -1 ... calculations unbound/denominator too close to 0


<!-- ```{r, label= 'CIplot', echo=FALSE, warning=FALSE, fig.width=100, fig.cap='Selected SSt confidence intervals with entered limits', fig.align='center'} -->

<!-- png("CIplot.png") -->
<!-- print(CIplot) -->
<!-- dev.off() -->


<!-- ``` -->


<!-- ![](CIplot.png){width=60%} -->


## Fitting results of the 4 models with bend points

The results of the non-linear fitting procedure for the restricted model (5 parameters) is listed in table 4:

```{r PLAausw, echo=FALSE, warning=FALSE, results='asis'}

kable(PLAausw, format = "markdown", caption= "Restricted 4PL evaluation", digits=3, row.names = F)


```


A depiction of the CI and corresponding limits of relative potency is shown here:

```{r, label='relpotPlot', echo=FALSE, warning=FALSE, fig.height=2, fig.width=3.5, fig.cap="Rel potency with CIs and limits", fig.align='left', results='asis'}

print(relpotTestPlot)


```


The bend points for test and reference sample are in table 5:

```{r PLBend, echo=FALSE, warning=FALSE, results='asis'}

kable(PLBend, format = "markdown", caption= "Bendpoints (Sebaugh) of restricted 4PL", digits=3)


```

The results of the non-linear fitting procedure for the unrestricted model (8 parameters) is listed in table 6:

```{r UnRPLAausw, echo=FALSE, warning=FALSE, results='asis'}

kable(UnRPLAausw, format = "markdown", caption= "UNrestricted 4PL evaluation", digits=3, row.names = F)


```


```{r UnRPLBend, echo=FALSE, warning=FALSE, results='asis'}

kable(UnRPLBend, format = "markdown", caption= "Bend points of 4PL unrestricted", digits=3, row.names = F)


```



```{r LogPLAausw, echo=FALSE, warning=FALSE, results='asis'}

kable(LogPLAausw, format = "markdown", caption= "Restricted 4PL evaluation with log-transformed response", digits=3)


```



```{r LogUnRPLAausw, echo=FALSE, warning=FALSE, results='asis'}

kable(LogUnrPLAausw, format = "markdown", caption= "Unrestricted 4PL evaluation with log-transformed response", digits=3)


```


# Appendix: Formulas

## 4PL regression

$$
Y = D + \frac{A-D} {1+(\frac{C} {x})^B } + \epsilon
$$


## log-logistic 4P regression

$$
Y = D + \frac{A-D} {1+e^{(B*(C - log(x))) }} + \epsilon
$$

where: x ... concentration of the analyte

A: upper asymptote

B: slope

D: lower asymptote

C ... EC50

# Literature

Finney, D.J.: (1978) Statistical Method in Biological Assay, London: Charles Griffin House, 3rd edition (pp. 80-82)

Franz, V.H.: Ratios: A short guide to confidence limits and proper use. arXiv:0710.2024v1, 10 Oct 2007

VerHoef, J.M.: Who invented the Delta Method? The American Statistician, 2012, 66:2, 124-127 DOI: 10.1080/00031305.2012.687494