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4PL report update

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This commit is contained in:
Franz Innerbichler
2026-05-14 18:38:27 +02:00
parent 9861af5fba
commit 9422490f25
4 changed files with 338 additions and 455 deletions
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Doc_BioassayReport.Rmd
+105 -238
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@@ -42,6 +42,7 @@ knitr::opts_chunk$set(echo = TRUE)
library(knitr)
library(DT)
library(kableExtra)
REP <- params$REP
author <- params$author
@@ -49,13 +50,14 @@ coeffs <- params$coeffs
all_l <- REP$all_l
ANOVAXLS <- REP$ANOVAXLS
XLplot4pl <- REP$XLplot4pl
DiagnTable <- REP$DiagnTable
UnRPLAausw <- REP$UnRPLAausw
UnRPLBend <- REP$UnRPLBend
PLAausw <- REP$PLAausw
PLBend <- REP$PLBend
LogPLAausw <- REP$LogPLAausw
LogUnrPLAausw <- REP$LogUnrPLAausw
pottab4plXL <- REP$pottab4plXL
Lim <- REP$Lim
XLdat2 <- REP$XLdat2
@@ -70,262 +72,144 @@ relpotTestPlot <- REP$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]).
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.
USP<1034> recommends calculation of standard errors of ratios of the parameters using Fieller's theorem [Finney DJ 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 mathematically more stable und thus preferable compared to a ratio approach ([Franz VH 2007]).
# Results
# Raw data
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)
kable(XLdat2, format = "markdown", caption= "Uploaded data (test and reference) ", digits=3)
```
The following 4 plots show all 4 models: restricted and unrestricted, and log transformed, respectively.
You can also embed plots, for example:
# Results
```{r XLplot, echo=FALSE, warning=FALSE, fig.height=4, fig.width=6, fig.cap="Plot of models", fig.align='left'}
## Overall result
plot_f <- function(dat, sigmoid,det_sig) {
CORdat <- cor(dat[,1],dat[,ncol(dat)])
```{r Over_all, echo=FALSE, comment=NA, warning=NA, message=NA}
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)
potFlag <- 0
if (pottab4plXL["test_result"][[1]][1]==1) potFlag <- 1
AnalysisFlag <- FALSE
if (potFlag==1 | sum(testsTab$test_results)>0) AnalysisFlag <- TRUE
if(is.null(det_sig)) {
if (CORdat<0) {
startlist <- list(a=sigmoid[3], b=-sigmoid[5],cs=sigmoid[7],
d=sigmoid[1],r=sigmoid[8])
} else {
startlist <- list(a=sigmoid[3],b=sigmoid[5],cs=sigmoid[7],
d=sigmoid[1],r=sigmoid[8])
}
} else {
startlist <- list(a=det_sig[5], b=det_sig[1],cs=det_sig[7],
d=det_sig[3],r=det_sig[7] - det_sig[8])
}
#browser()
tryCatch({
mr <- gsl_nls(fn = readout ~ a+(d-a)/(1+exp(b*(log_dose-(cs-r*isSample)))),
data=all_l2,
start=startlist,
control=gsl_nls_control(xtol=1e-6,ftol=1e-6, gtol=1e-6))
},
error = function(err) {
err$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$coefficients[6,1]
REFu_trans <- ast_t + (dst_t-ast_t)/(1+exp(bst_t*(seq_x-cst_t)))
SAMPLEu_trans <- ate_t + (dte_t-ate_t)/(1+exp(bte_t*(seq_x-cte_t)))
pl_df2u_t <- cbind(seq_x, SAMPLEu_trans, REFu_trans)
pu2_t <- putrans + geom_line(data=as.data.frame(pl_df2u_t), aes(x=seq_x, y=SAMPLEu_trans),
color="red", inherit.aes = F) +
geom_line(data=as.data.frame(pl_df2u_t), aes(x=seq_x, y=REFu_trans),
color="blue", inherit.aes = F,
show.legend = F)
pu3_t <- pu2_t
grid.arrange(p2,p_rt2,pu2_,pu3_t, nrow=2)
colFmt <- function() {
outputFormat <- knitr::opts_knit$get("rmarkdown.pandoc.to")
if(AnalysisFlag) {
text <- paste("\\textcolor{red}{Analysis failed}",sep="")
} else {
text <- paste("\\textcolor{black}{Analysis succeeded}>",sep="")
}
return(text)
}
plot_f(XLdat2, sigmoid=NULL, det_sig=coeffs)
```
`r colFmt()`
## 4pl-regression
Relative potency (absolute and relative confidence limits) are shown in Table 2:
```{r Pot_tab4pl, echo=FALSE, comment=NA, warning=NA, message=NA}
#browser()
if (pottab4plXL["test_result"][[1]][1]==1) { cat(paste("FAILED: relative potency CL result of restricted model outside limits: ", Lim[[9]], "and" ,Lim[[10]] ))}
kable(pottab4plXL, format = "markdown", caption= "Relative potency with absolute and relative CLs ", digits=3, row.names = F) %>%
kable_styling(latex_options = "hold_position")
```
## Plot of the data and models
The following plots show the restricted and unrestricted model, respectively.
```{r XLplot, echo=FALSE, warning=FALSE, fig.height=4, fig.width=6, fig.cap="Plot of models", fig.align='left', comment=F, message=F, results='asis', fig.pos='H'}
library(cowplot)
plot_grid(XLplot4pl)
```
## ANOVA table
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)
kable(ANOVAXLS, format = "markdown", caption= "ANOVA table unrestricted", digits=3) %>%
kable_styling(latex_options = "hold_position")
```
## Analysis suitability tests
The following table lists the chosen suitabilit test results with confidence limits, where applicable:
```{r SST_ergebn, echo=FALSE, cache=FALSE, warning=FALSE, message=FALSE, tidy=TRUE}
kable(testsTab, row.names = F, format = "markdown", caption="SST results")
```
```{r SST_ergebn, fig.align='center', fig.pos='htb!', echo=FALSE, cache=FALSE, warning=FALSE, message=FALSE, tidy=TRUE}
\footnotesize
kable(testsTab[1:7,], row.names = F, format = "markdown", caption="SST results")
```{r Fussnote, echo=F, comment=NA}
cat("*...The estimate for F-test on regression and on non-linearity is the p-value")
cat( "F-test on regression passes if F-value > F-crit and thus p < 0.05")
cat( "F-test on non-linearity passes if F-value < F-crit and thus p > 0.05")
cat( "Test results outcome:")
cat(" 0 ... test passed (for EQ tests: CL within limits);")
cat(" 1 ... test failed (for EQ tests: CL not within limits);")
```
\normalsize
```{r AST_Ergebn, echo=FALSE, cache=FALSE, warning=FALSE, message=FALSE, tidy=TRUE}
TestsTabFlag <- FALSE
if (sum(testsTab$test_results)>0) TestsTabFlag <- TRUE
colFmt2 <- function() {
outputFormat <- knitr::opts_knit$get("rmarkdown.pandoc.to")
if(TestsTabFlag) {
text <- paste("\\textcolor{red}{Analysis suitability tests failed}",sep="")
} else {
text <- paste("\\textcolor{black}{Analysis suitability tests succeeded}",sep="")
}
return(text)
}
```
*...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 colFmt2()`
<!-- ```{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
@@ -377,23 +261,6 @@ kable(UnRPLBend, format = "markdown", caption= "Bend points of 4PL unrestricted"
```{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