---
title: "Fraction Unbound in Plasma - Ultracentrifugation (f~up~ UC)"
author: "US EPA's Center for Computational Toxicology and Exposure ccte@epa.gov"
output:
  rmarkdown::html_vignette
params:
  css: css/rmdformats.css
bibliography: '`r system.file("REFERENCES.bib", package="invitroTKstats")`'
vignette: >
  %\VignetteIndexEntry{Fup UC}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{css,code = readLines(params$css),hide=TRUE,echo=FALSE}
```

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.align = 'center'
)
```

## Introduction 

This vignette guides users on how to estimate fraction unbound in plasma (f~up~) from mass spectrometry data using ultracentrifugation (UC). Fraction unbound in plasma is a chemical specific parameter that describes the amount of free chemical in the plasma that is usually responsible for pharmacological effects (@redgrave1975separation). 

The mass spectrometry data should be collected from an assay that uses ultracentrifugation as seen in Figure 1 (@smeltz2023plasma, @kreutz2023category). 

```{r, echo = FALSE, out.width = "100%", fig.cap = "Fig 1: f~up~ UC experimental set up", fig.topcaption = TRUE, fig.align = "center"}
knitr::include_graphics("img/fupUC_diagram.png")
```

### Suggested packages for use with this vignette 

```{r setup, message = FALSE, warning = FALSE}
# Primary package 
library(invitroTKstats)
# Data manipulation package 
library(dplyr)
# Table formatting package 
library(flextable)
```

## Load Data

First, we load in the example dataset from `invitroTKstats`. 

```{r Load example data}
# Load example fup UC data 
data("Fup-UC-example")
```

Many datasets are loaded in: `fup_uc_L0`, `fup_uc_L1`, `fup_uc_L2`, `fup_uc_L3`, and `fup_uc_L4`. These datasets are f~up~ data at Level 0, 1, 2, 3, and 4 respectively. Additional datasets associated with Level 4 processing are also loaded in: `fup_uc_L2_heldout` and `fup_uc_PREJAGS`. These will be described later in the "Level 4 processing" section. Lastly, a `fup_uc_cheminfo` dataset is loaded in that contains chemical information necessary for identification mapping; it is used to create Level 0 data. For the purpose of this vignette, we'll start with `fup_uc_L0`, the Level 0 data, to demonstrate the complete pipelining process. 

`fup_uc_L0` is the output from the `merge_level0` function which compiles raw lab data from specified Excel files into a singular data frame. The data frame contains exactly one row per sample with information obtained from the mass spectrometer. For more details on curating raw lab data to a singular Level 0 data frame, see the "Data Guide Creation and Level-0 Data Compilation" vignette.

The following table displays the first three rows of `fup_uc_L0`, our Level 0 data. 

```{r, echo = FALSE, warning = FALSE}
head(fup_uc_L0, n = 3) %>% 
  flextable() %>% 
  bg(bg = "#DDDDDD", part = "header") %>% 
  autofit() %>% 
  set_table_properties(
    opts_html = list(
      scroll = list(
        
      )
    )
  ) %>% 
  set_caption(caption = "Table 1: Level 0 data", 
              align_with_table = FALSE) %>% 
  fontsize(size = 10, part = "all") %>% 
  theme_vanilla()
```

## Level 1 processing 

`format_fup_uc` is the Level 1 function used to create a standardized data frame. This level of processing is necessary because naming conventions or formatting can differ across data sets. 

If the Level 0 data already contains the required column, then the existing column name can be specified. For example, `fup_uc_L0` already contains a column specifying the sample name called "Sample". However, the default column name for sample name is "Lab.Sample.Name". Therefore, we specify the correct column with `sample.col = "Sample"`. In general, to specify an already existing column that differs from the default, the user must use the parameter with the `.col` suffix. 

If the Level 0 data does not already contain the required column, then the entire column can be populated with a single value. For example, `fup_uc_L0` does not contain a column specifying biological replicates. Therefore, we populate the required column with `biological.replicates = 1`. In general, to specify a single value for an entire column, the user must use the parameter without the `.col` suffix. 

Users should be mindful if they choose to specify a single value for all of their samples; they should verify this action is one they wish to take. 

Some columns must be present in the Level 0 data while others can be filled with a single value. At minimum, the following columns must be present in the Level 0 data and specification with a single entry is not permitted: `sample.col`, `lab.compound.col`, `dtxsid.col`, `compound.col`, `area.col`, `type.col`, and `istd.col`.

If there is no additional `note.col` in the Level 0 data, users should use `note.col = NULL` to fill the column with "Note".

The rest of the following columns may either be specified from the Level 0 data or filled with a single value: `date.col` or `date`, `test.conc.col` or `test.conc`, `cal.col` or `cal`, `dilution.col` or `dilution`, `istd.name.col` or `istd.name`, `istd.conc.col` or `istd.conc`, `uc.assay.conc.col` or `uc.assay.conc`, `biological.replicates.col` or `biological.replicates`, `technical.replicates.col` or `technical.replicates`, `analysis.method.col` or `analysis.method`, `analysis.instrument.col` or `analysis.instrument`, `analysis.parameters.col` or `analysis.parameters`, `level0.file.col` or `level0.file`, and `level0.sheet.col` or `level0.sheet`. 

```{r required cols, echo = FALSE}
# Create table of required arguments for Level 1 

req_cols <- data.frame(matrix(nrow = 29, ncol = 5))
vars <- c("Argument", "Default", "Required in L0?", "Corresp. single-entry Argument", "Descr.")
colnames(req_cols) <- vars

# Argument names 
arguments <- c("FILENAME", "data.in", "sample.col", "lab.compound.col", "dtxsid.col", 
               "date.col", "compound.col", "area.col", "type.col", "test.conc.col", 
               "cal.col", "dilution.col", "istd.col", "istd.name.col", "istd.conc.col", 
               "uc.assay.conc.col", "biological.replicates.col",
               "technical.replicates.col", "analysis.method.col", 
               "analysis.instrument.col", "analysis.parameters.col", "note.col", 
               "level0.file.col", "level0.sheet.col", "output.res", "save.bad.types", 
               "sig.figs", "INPUT.DIR", "OUTPUT.DIR"
               )
req_cols[, "Argument"] <- arguments 

# Default arguments 
defaults <- c("MYDATA", NA, "Lab.Sample.Name", "Lab.Compound.Name", "DTXSID", 
              "Date", "Compound.Name", "Area", "Sample.Type", "Test.Compound.Conc", 
              "Cal", "Dilution.Factor", "ISTD.Area", "ISTD.Name", "ISTD.Conc", 
              "UC.Assay.Conc", "Biological.Replicates", "Technical.Replicates", 
              "Analysis.Method", "Analysis.Instrument", "Analysis.Parameters", 
              "Note", "Level0.File", "Level0.Sheet", FALSE, FALSE, 5, NA, NA)
req_cols[, "Default"] <- defaults 

# Argument required in L0?
req_cols <- req_cols %>% 
  mutate("Required in L0?" = case_when(
    Argument %in% c("sample.col", "lab.compound.col", "dtxsid.col",
                    "compound.col", "area.col", "type.col", 
                    "istd.col") ~ "Y", 
    Argument %in% c("FILENAME", "data.in", "output.res", "save.bad.types", 
                    "sig.figs", "INPUT.DIR", "OUTPUT.DIR") ~ "N/A",
    .default = "N"
  ))

# Corresponding single-entry Argument 
req_cols <- req_cols %>% 
  mutate("Corresp. single-entry Argument" = ifelse(.data[[vars[3]]] == "N" & .data[[vars[[1]]]] != "note.col",
                                                   gsub(".col", "", Argument), NA))

# Brief description 
description <- c("Output and input filename", 
                 "Level 0 data frame", 
                 "Lab sample name", 
                 "Lab test compound name (abbr.)", 
                 "EPA's DSSTox Structure ID",
                 "Lab measurement date", 
                 "Formal test compound name", 
                 "Target analyte peak area", 
                 "Sample type (CC/AF/T1/T5)",
                 "Standard test chemical concentration", 
                 "MS calibration", 
                 "Number of times sample was diluted", 
                 "Internal standard peak area", 
                 "Internal standard name", 
                 "Internal standard concentration", 
                 "Intended initial test concentration",
                 "Replicates with the same analyte", 
                 "Repeated measurements from one sample", 
                 "Analytical chemistry analysis method", 
                 "Analytical chemistry analysis instrument", 
                 "Analytical chemistry analysis parameters", 
                 "Additional notes", 
                 "Raw data filename", 
                 "Raw data sheet name", 
                 "Export results (TSV)?", 
                 "Export bad data (TSV)?",
                 "Number of significant figures",
                 "Input directory of Level 0 file", 
                 "Export directory to save Level 1 files")
req_cols[, "Descr."] <- description
```

```{r, echo = FALSE, warning = FALSE}
req_cols %>% 
  flextable() %>% 
  bg(bg = "#DDDDDD", part = "header") %>% 
  autofit() %>% 
  set_table_properties(
    opts_html = list(
      scroll = list(
        height = 200
      )
    )
  ) %>% 
  set_caption(caption = "Table 2: Level 1 `format_fup_uc` function arguments", align_with_table = FALSE) %>% 
  fontsize(size = 10, part = "all") %>% 
  theme_vanilla()
```

A TSV file containing the level-1 data can be exported to the user's per-session temporary directory. This temporary directory is a per-session directory whose path can be found with the following code: `tempdir()`. For more details, see 
[https://www.collinberke.com/til/posts/2023-10-24-temp-directories/].

To avoid exporting to this temporary directory, an `OUTPUT.DIR` must be specified. We have omitted this export entirely with `output.res = FALSE` (the default). The option to omit exporting a TSV file is also available at levels 2 and 3 and will be used from this point forward. 

```{r L1 processing}
fup_uc_L1_curated <- format_fup_uc(FILENAME = "Fup_UC_vignette",
                                   data.in = fup_uc_L0,
                                   # columns present in L0 data
                                   sample.col = "Sample",
                                   lab.compound.col = "Lab.Compound.ID",
                                   compound.col = "Compound",
                                   area.col = "Peak.Area", 
                                   test.conc.col = "Compound.Conc",
                                   cal.col = "Date", 
                                   istd.col = "ISTD.Peak.Area",
                                   technical.replicates.col = "Replicate",
                                   analysis.parameters.col = "Analysis.Params",
                                   # columns not present in L0 data 
                                   istd.conc = 1, 
                                   test.nominal.conc = 10, 
                                   biological.replicates = 1, 
                                   analysis.method = "UPLC-MS/MS",
                                   analysis.instrument = "Waters Xevo TQ-S micro(QEB0036)",
                                   note.col = NULL,
                                   # don't export output TSV file
                                   output.res = FALSE
                                   )
```

All of our samples are successfully formatted and returned in `fup_uc_L1_curated`, our Level 1 data produced from `format_fup_uc`. Each sample has one of the following sample types

1. Calibration Curve (CC)
2. Ultracentrifugation Aqueous Fraction (AF)
3. Whole Plasma T1h Sample (T1)
4. Whole Plasma T5h Sample (T5)

If any samples had a different sample type, they would have been removed and reported to the user. If the user wants to export the removed samples as a TSV, the user should set the parameter `save.bad.types = TRUE`. 

The following table displays the first three rows of `fup_uc_L1_curated`. In addition to the columns specified by the user, there is an additional column called `Response`. This column is the test compound concentration and is calculated as $\textrm{Response} = \frac{\textrm{Analyte Area}}{\textrm{ISTD Area}} * \textrm{ISTD Conc}$ where $\textrm{Analyte Area}$ is defined by the `Area` column, $\textrm{ISTD Area}$ is defined by the `ISTD.Area` column, and $\textrm{ISTD Conc}$ is defined by the `ISTD.Conc` column. 

```{r, echo = FALSE, warning = FALSE}
fup_uc_L1_curated %>% 
  head(n = 3) %>% 
  flextable() %>% 
  bg(bg = "#DDDDDD", part = "header") %>% 
  autofit() %>% 
  set_table_properties(
    opts_html = list(
      scroll = list(
      )
      )
  ) %>% 
  set_caption(caption = "Table 3: Level 1 data",
              align_with_table = FALSE) %>% 
  fontsize(size = 10, part = "all") %>% 
  theme_vanilla()
```

## Level 2 processing 

`sample_verification` is the Level 2 function used to add a verification column. The verification column indicates whether a sample should be included in the point estimation (Level 3) and credible interval (Level 4) processing. This column allows users to keep all samples in their data but only utilize the reliable samples for f~up~ estimation. All of the data in Level 2 is identical to the data in Level 1 with the exception of the additional `Verified` column.

To determine whether a sample should be included, the user should consult the wet-lab scientists from where their data originates or a chemist who may be able to provide reliable rationale for samples that should not be verified. This level of processing allows the user to receive feedback from the wet-lab scientists, exclude erroneous or unreliable samples, and produce new f~up~ estimates. Thus, there is an open channel of communication between the user and the wet-lab scientists or chemists. 

We will use the already processed Level 2 data frame, `fup_uc_L2`, to regenerate our exclusion data. Note, all of our samples are verified but we are explaining how to create an exclusion list for learning purposes. In general, the user would not have access to the exclusion information *a priori*. 

The exclusion data frame must include the following columns: `Variables`, `Values`, and `Message`. The `Variables` column contains the variable names used to filter the excluded rows. Here, we are using `Lab.Sample.Name` and `DTXSID` to identify the excluded rows separated by a "|". The `Values` column contains the values of the variables, as a character, also separated by a "|". The `Message` column contains the reason for exclusion. Here we are using the reasons listed in the `Verified` column in `fup_uc_L2`. The user should refrain from using "|" in any of their descriptions to avoid conflicts with the `sample_verifiation` function. 

```{r}
# Use verification data from loaded in `fup_uc_L2` data frame 
exclusion <- fup_uc_L2 %>% 
  filter(Verified != "Y") %>% 
  mutate("Variables" = "Lab.Sample.Name|DTXSID") %>% 
  mutate("Values" = paste(Lab.Sample.Name, DTXSID, sep = "|")) %>% 
  mutate("Message" = Verified) %>% 
  select(Variables, Values, Message)
```

```{r, echo = FALSE}
exclusion %>% 
  flextable() %>% 
  bg(bg = "#DDDDDD", part = "header") %>% 
  autofit() %>% 
  set_table_properties(
    opts_html = list(
      scroll = list()
      )
  ) %>% 
  set_caption("Table 4: Exclusion data", align_with_table = FALSE) %>% 
  fontsize(size = 10, part = "body") %>% 
  theme_vanilla()
```

As expected, our exclusion data frame is empty because all of our samples are verified. If all of the user's samples are verified, they simply do not provide an `exclusion.info` data frame in `sample_verification`. 

```{r L2 processing}
fup_uc_L2_curated <- sample_verification(FILENAME = "fup_UC_vignette", 
                                         data.in = fup_uc_L1_curated,
                                         assay = "fup-UC",
                                         # don't export output TSV file
                                         output.res = FALSE)
```

Our Level 2 data now contains a `Verified` column. If the sample should be included, the column contains a "Y" for yes. If the sample should should be excluded, the column contains the reason for exclusion. 

The following table displays some rows of the Level 2 data. 

```{r, echo = FALSE}
fup_uc_L2_curated %>% 
  head(n = 3) %>% 
  flextable() %>% 
  bg(bg = "#DDDDDD", part = "header") %>% 
  autofit() %>% 
  set_table_properties(
    opts_html = list(
      scroll = list()
      )
  ) %>% 
  set_caption("Table 5: Level 2 data", align_with_table = FALSE) %>% 
  fontsize(size = 10, part = "body") %>% 
  theme_vanilla()
```

## Level 3 processing 

`calc_fup_uc` is the Level 3 function used to calculate the f~up~ point estimate from ultracentrifugation for each test compound using a Frequentist framework. 

Mathematically, f~up~ is the ratio of the compound concentration in the aqueous fraction after centrifugation to the compound concentration in the sample after incubation for 5 hours. It can be expressed as $$f_{up} = \frac{C_{\textrm{aqueous fraction}}}{C_{\textrm{incubation}}}$$ where $C_{\textrm{aqueous fraction}}$ is the compound concentration in the aqueous fraction and $C_{\textrm{incubation}}$ is the compound concentration in the incubated sample. 

The concentrations are defined as the mean AF or T5 response multiplied by its dilution factor. 

```{r L3 processing}
fup_uc_L3_curated <- calc_fup_uc_point(FILENAME = "Fup_UC_vignette", 
                                 data.in = fup_uc_L2_curated,
                                 # don't export output TSV file 
                                 output.res = FALSE)
```

```{r, echo = FALSE}
fup_uc_L3_curated %>% 
  flextable() %>% 
  bg(bg = "#DDDDDD", part = "header") %>% 
  autofit() %>% 
  set_caption("Table 6: Level 3 data", align_with_table = FALSE) %>% 
  fontsize(size = 10, part = "body") %>% 
  theme_vanilla()
```

Our Level 3 data contains a `Fup` estimate and a `Calibration` column that details which data was used to calculate the point estimate. 

## Level 4 processing 

`calc_fup_uc` is the Level 4 function used to calculate f~up~ point estimates and credible intervals from ultracentrifugation using a Bayesian framework. Markov chain Monte Carlo (MCMC) simulations are used to randomly sample from the posterior distribution with a uniform prior. 

To run Level 4, one needs to have JAGS installed on their machine. To determine the correct path, the user may use the `runjags::findjags()` or may need to specify the JAGS installation location for the `JAGS.PATH` argument. 

:::{.noticebox}

JAGS is a software used for conducting Bayesian hierarchical modeling using Markov Chain Monte Carlo (MCMC) simulation. `invitroTKstats` contains the JAGS models for each of the applicable assays and utilizes the JAGS model under the hood to run the MCMC simulations via the `runjags` dependency to obtain the level-4 Bayesian estimates.

  1.  Download and install the "Just Another Gibbs Sampler" (JAGS) software, which is freely available [here](https://mcmc-jags.sourceforge.io/).
  2. Once JAGS is installed one will need to identify the installation location, which is likely to be in a system location (e.g. "Program Files"). Users may need this location to specify the `JAGS.PATH` argument within the level-4 function.
  3. Finally, if not already installed, the `rjags`, `runjags`, and `coda` R packages need to be installed via the R console using the `install.packages` function.

:::

We pass `fup_uc_L2_curated`, the Level 2 data frame, and **not** `fup_uc_L3_curated`, the Level 3 data frame, into `calc_fup_uc`. This is because Level 3 and Level 4 processing are not sequential; they are methods that calculate different statistical quantities. The following code chunk takes a while to run; previous runtimes are around 10 minutes.

```{r L4 processing, message = FALSE, eval = FALSE}
fup_uc_L4_curated <- calc_fup_uc(FILENAME = "Fup_UC_vignette",
                                 data.in = fup_uc_L2_curated,
                                 JAGS.PATH = runjags::findjags()
                                 )
```

```{r, echo = FALSE}
fup_uc_L4_curated <- fup_uc_L4
```

The f~up~ intervals are returned to the user's R session, in an exported TSV file, and in an exported .RData file. There is no parameter to prevent the TSV or RData files from being exported because of the potential for the simulations to crash. If there are no crashes, then the exported TSV file is identical to the user's R session and the exported .RData file. `fup_uc_L4` is an example exported .RData file.  

Additionally, intermediate files are saved to the user's current working directory if `TEMP.DIR = NULL`. These include a Level 2 heldout set, `fup_uc_L2_heldout`, containing unverified samples and a Level 4 PREJAGS list, `fup_uc_PREJAGS`, containing arguments provided to JAGS. Because the PREJAGS list is overwritten with each compound, `fup_uc_PREJAGS` only contains information relevant to the last tested compound, K-PFBS in this case.

Our Level 4 data contains a credible interval for f~up~ and for f~stable~ which measures the compound's stability. Mathematically, $f_{stable} = \frac{\textrm{T5}}{\textrm{T1}}$ where a value of $1$ indicates no compound breakdown. 

```{r, echo = FALSE}
fup_uc_L4_curated %>% 
  flextable() %>% 
  bg(bg = "#DDDDDD", part = "header") %>% 
  autofit() %>% 
  set_table_properties(
    opts_html = list(
      scroll = list(
      )
      )
  ) %>% 
  set_caption(caption = "Table 7: Level 4 data",
              align_with_table = FALSE) %>% 
  fontsize(size = 10, part = "all") %>% 
  theme_vanilla()
```

## Best Practices: Food for thought 

Generally, data processing pipelines should include minimal to no manual coding. It is best to keep clean code that is easily reproducible and transferable. The user should aim to have all the required data and meta-data files properly formatted to avoid further modifications throughout the pipeline. 

## Appendix

### L4 Bayesian Modeling Information

In this section, we provide the equations for the Bayesian model (i.e., priors and likelihoods) used to estimate the fraction unbound in plasma ($f_{up}$), from the ultracentrifugation assay, and the uncertainty about that estimate. The following sub-sections are organized such that:

* Each section contains the relevant hyper-parameters and their priors followed by the variable likelihoods
* Sections are ordered from the most fundamental variables/parameters (i.e., those used in later equations) to the likelihood for the observations used to estimate the fraction unbound in plasma ($f_{up}$)

Some of the indices are reused between sections (e.g. $i$, $w^*_i$, etc.). However, it should be noted that these are not meant to be understood across sub-sections; rather, only understood within the context of the section they are in.

*NOTE: Readers unfamiliar with JAGS should be aware that JAGS software uses precision ($\tau$) rather than variance ($\sigma^2$) (i.e., $\tau = \sigma^2$).*

#### Measurement Model

Each chemical may have more than one day of experimentation, and thus multiple calibrations. Suppose for the chemical of interest there are a total of $n_{cal}$ calibrations (`Num.cal`). For a particular calibration $w \in (1,\ldots,n_{cal})$ we assume the following priors for our Bayesian model:

**Prior for log-scale constant analytic standard deviation (`log.const.analytic.sd`):**

$$ log(\sigma_a)_w \sim \textrm{Unif} \left( a = -6,b = 1 \right); \space \sigma_{a,w} = 10^{log(\sigma_a)_w} $$

where $a$ and $b$ are the minimum and maximum, respectively, and $\sigma_{a,w}$ is the converted parameter used in later equations (`const.analytic.sd`).

**Prior for the log-scale heteroscedastic analytic slope (`log.hetero.analytic.slope`):**

$$ log(m_h)_w \sim \textrm{Unif} \left( a = -6,b = 1 \right) ; \space m_{h,w} = 10^{log(m_h)_w} $$

where $a$ and $b$ are the minimum and maximum, respectively, and $m_{h,w}$ is the converted parameter used in later equations (`hetero.analytic.slope`).

**Prior for the threshold concentration (`C.thresh`):**

$$ C_{thresh,w} \sim \textrm{Unif} \left( a = 0, b = \frac{conc_{target,w}}{10} \right)$$

where $a$ and $b$ are the minimum and maximum, respectively, and $conc_{target,w}$ is the expected initial concentration (`Test.Nominal.Conc`) for calibration index $w$.

**Prior for the log-scale calibration (`log.calibration`):**

$$ log(cal)_w \sim \textrm{N} \left( \mu = 0, \tau = 0.01 \right); \space cal_w = 10^{log(cal)_w} $$

where $\mu$ and $\tau$ are the mean and precision, respectively, and $cal_w$ is the converted parameter used in later equations (`calibration`).

**Prior for the background (`background`):**

$$ \gamma_w \sim \textrm{Exp} \left( \lambda = 100 \right) $$

where $\lambda$ is the rate parameter.

#### Estimation of Response Observations

Suppose $n$ is the total number of response observations (`Num.obs`), and $y_i$ indicates the $i^{th}$ observation, for all sample types. For each observation we obtain a posterior MCMC estimate for the observations with the following:

Calibration curve slope for the $i^{th}$ observation (`slope`) is assumed to be the calibration estimate corresponding to the calibration index ($w^*_i$):

$$ m_i = cal_{w^*_i}$$

where $w^*_i$ is the calibration index (`obs.cal`) for the $i^{th}$ observation, such that $w^* = (w^*_1,...,w^*_n)$ and $w^*_i \in w$ (i.e. $w^*_i$ indicates one of the $n_{cal}$ calibrations).

Calibration curve intercept for the $i^{th}$ observation (`intercept`) is assumed to be the background estimate corresponding to the calibration index ($w^*_i$):

$$ \alpha_{CC,i} = \gamma_{w^*_i} $$

Estimation for the predicted values for response observations (`Response.pred`):

$$ x_i = \frac{m_i*((C_{c^*_i}- C_{thresh,w^*_i}) * \beta_i + \alpha_i)}{df_i} $$

where

* $c^*_i$ is the index for the standard test chemical concentration (`obs.conc`) for the $i^{th}$ observation
* $df_i$ is the dilution factor (`Dilution.Factor`) for the $i^{th}$ observation
* $\beta_i$ is the JAGS step function for the $i^{th}$ observation, such that

$$ \beta_i = 
\begin{cases}
1 & if \space (C_{c^*_i}- C_{thresh,w^*_i}) \ge 0\\
0 & o.w.
\end{cases}
$$

Estimation of the precision for observations (`Response.prec`):

$$  \tau^*_i = \frac{1}{(\sigma_{a,w^*_i}+m_{h,w^*_i}*x_i)^2} $$

**Likelihood for the observations (`Response.obs`):**

$$ y_i \sim N(\mu = x_i,\tau = \tau^*_i) $$

#### Binding Model

**Prior for the log-scale fraction unbound in plasma (`log.Fup`):**

$$ log(f_{up}) \sim \textrm{Unif} \left( a = -15, b = 0 \right) ; \space f_{up} = 10^{log(f_{up})} $$

where $a$ and $b$ are the minimum and maximum, respectively, and $f_{up}$ is the converted parameter used in later equations (`Fup`).

**Prior for the log-scale chemical loss fraction (`log.Floss`):** 

$$ log(f_{loss}) \sim \textrm{Unif} \left( a = -6,b = 0 \right); \space f_{stable} = 1 - 10^{log(f_{loss})} $$

where $a$ and $b$ are the minimum and maximum, respectively, and $f_{stable}$ is the converted parameter used in later equations (`Fstable`) estimating the fraction of stable chemical in the assay and available for plasma binding.

Suppose $n_s$ is the total number of series run (i.e. biological replicates) (`Num.series`) and $C_i$ indicates the $i^{th}$ series. Each series has a total of 3 observations, including the 1 and 5 hour whole plasma and aqueous fraction observations (i.e. sample types = `T1` or `T5` or `AF`, respectively), such that there are a total of $n*$ observations (i.e., $n^* = 3*n_s$).  For each series obtain the following:

**Prior for the whole plasma 1 hour sample (`T1`) concentration (`Conc`):** 

$$ C_i \sim \textrm{N} \left( \mu = conc_{target,w^*_i}, \tau = 100 \right) $$

where

* $\mu$ and $\tau$ are the mean and precision, respectively
* $i$ indicates the $i^{th}$ series run (biological replicate)
* $w^*_i$ represents the calibration index (`obs.cal`) for the $i^{th}$ series $w^* = (w^*_1, \ldots,w^*_n)$ and $w^*_i \in w$ (i.e. $w^*_i$ indicates one of the $n_{cal}$ calibrations)
* $conc_{target,w^*_i}$ is the expected initial concentration (`Test.Nominal.Conc`) corresponding to the $i^{th}$ series, given the calibration index $w^*_i$

Posterior estimate for the whole plasma 5 hour sample (`T5`) concentrations (after potential breakdown):  

$$ C_{T5,i} = f_{stable} * C_i $$

Posterior estimate for the aqueous fraction sample (`AF`) concentrations for the stable chemical at `T5`: 

$$ C_{AF,i} = f_{up}*C_{T5,i} $$

### Post-MCMC Estimates

#### Fraction of Chemical Stability

Posterior estimates for the fraction of stable chemical in the assay, including the median (`Fstable.Med`) and $95\%$ credible interval (`Fstable.Low` and `Fstable.High`):

$$ \textrm{Fstable.Med} = f_{stable,0.5} $$

$$ \textrm{Fstable.CI} = (\textrm{Fstable.Low},\textrm{Fstable.High}) = (f_{stable,0.025},f_{stable,0.975}) $$

where $f_{stable,p}$ indicates the percentile ($p$) for the posterior distribution of the chemical stability fraction, ($p = 0.5$ indicates the $50\%$ percentile, i.e. median, for the posterior distribution).

#### Fraction of Chemical Unbound in Plasma (Bayesian Estimates)

Posterior estimates for the chemical fraction unbound in plasma, including the median (`Fup.Med`) and $95\%$ credible interval (`Fup.Low` and `Fup.High`):

$$ \textrm{Fup.Med} = f_{up,0.5} $$

$$ \textrm{Fup.CI} = (\textrm{Fup.Low},\textrm{Fup.High}) = (f_{up,0.025},f_{up,0.975}) $$

where $f_{up,p}$ indicates the percentile ($p$) for the posterior distribution of the fraction of the chemical unbound in plasma, ($p = 0.5$ indicates the $50\%$ percentile, i.e. median, for the posterior distribution).

#### Fraction Unbound in Plasma (Empirical Point Estimate)

Point estimate for the chemical fraction unbound in plasma (`Fup.point`):

Suppose there are a total of $n_{AF}$ observed aqueous fraction (`AF`) responses, then estimate the mean response:

$$ \hat{\mu_{AF}} = \frac{\sum_{j = 1}^{n_{AF}}(y_{AF,j} * df_{AF,j})}{n_{AF}} $$

where $y_{AF,j}$ is the response and $df_{AF,j}$ is the dilution factor for the $j^{th}$ aqueous fraction observation, sample type = `AF`.

Suppose there are a total of $n_{T5}$ observed whole plasma 5 hour (`T5`) responses, then estimate the mean response:

$$ \hat{\mu_{T5}} = \frac{\sum_{j = 1}^{n_{T5}}(y_{T5,j} * df_{T5})}{n_{T5}} $$

where $y_{T5,j}$ is the response and $df_{T5,j}$ is the dilution factor for the $j^{th}$ whole plasma 5 hour observation, sample type = `T5`.

Then the fraction unbound in plasma point estimate can be estimated as follows:

$$ \textrm{Fup.point} = \frac{\hat{\mu_{AF}}}{\hat{\mu_{T5}}} $$


### JAGS to L4 Equation Notation Tables

Data passed to JAGS as part of the `PREJAGS` object:

|       Data        |   Notation    | Description |
|-------------------|---------------|-------------|
| `Test.Nominal.Conc` | $conc_{target}$ | expected initial concentration |
| `Num.cal` | $n_{cal}$ | total number of calibrations |
| `Num.obs` | $n$ | total number of responses |
| `Response.obs` | $y$ | all sample responses |
| `obs.conc` | $c^*$ | concentration indices for all samples |
| `obs.cal` | $w^*$ | calibration index for all samples |
| `Dilution.Factor` | $df$ | dilution factors for all samples |
| `Conc` | $C$ | the standard test chemical concentration of `CC` samples (and `NA` placeholders for `T1`, `T5`, and `AF` samples) |
| `Num.cc.obs` | $n_{CC}$ | total number of `CC` samples |
| `Num.series` | $n_s$ | number of biological replicates (series) |

MCMC Parameters in JAGS:

| JAGS Parameter Name | Parameter | Distribution | Prior/Posterior/Calculated |
|---------------------|-----------|--------------|----------------------------|
| log.const.analytic.sd | $log(\sigma_a)$ | Uniform | Prior |
| log.hetero.analytic.slope | $log(m_h)$ | Uniform | Prior |
| C.thresh | $C_{thresh}$ | Uniform | Prior |
| log.calibration | $log(cal)$ | Normal | Prior |
| background | $\gamma$ | Exponential | Prior |
| const.analytic.sd | $\sigma_a$ | | Calculated |
| hetero.analytic.slope | $m_h$ | | Calculated |
| calibration | $cal$ | | Calculated |
| slope | $m$ | | Calculated |
| intercept | $\alpha$ | | Calculated |
| Response.pred | $x$ | | Calculated |
| Response.prec | $\tau^*$ | | Calculated |
| Response.obs | $y$ | Normal | Posterior |
| `log.Fup` | $log(f_{up})$ | Uniform | Prior |
| `Fup` | $f_{up}$ | | Calculated |
| `log.Floss` | $log(f_{loss})$| Uniform | Prior |
| `Fstable` | $f_{stable}$| | Calculated |
| `Conc` | $C$ | Normal (`T1`) / - (`T5`, `AF`) | Prior (`T1`) / Calculated (`T5`, `AF`) |

## References 

\references{
\insertRef{kreutz2023category}{invitroTKstats}
\insertRef{redgrave1975separation}{invitroTKstats}
\insertRef{smeltz2023plasma}{invitroTKstats}
}
