---
title: "G-CSF PK-PD design evaluation"
author: "Romain Leroux"
classoption: openany
output:
  rmarkdown::html_vignette:
    toc: true
bibliography: references.bib
biblio-style: apalike
link-citations: yes
linkcolor: blue
urlcolor: green
vignette: >
  %\VignetteIndexEntry{G-CSF PK-PD design evaluation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  %\VignetteAuthor{Romain Leroux}
---

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```{r global_options, echo = FALSE, include = FALSE}
knitr::opts_knit$set(tangle = FALSE)
backup_options = options()
library(PFIM)
set.seed(42)
options(width = 200)
utils = system.file("vignette-scripts", "pfim-vignette-utils.R", package = "PFIM")
if (!nzchar(utils)) stop("pfim-vignette-utils.R not found.", call. = FALSE)
source(utils, local = knitr::knit_global())
paths = pfimVignetteSetupPaths()
plotOptions = list(unitTime = c("hour"), unitOutcomes = c("ng/mL", "10^3/uL"))
.pfimVignetteHas = function( name ) {
  exists( name, inherits = TRUE ) && {
    val = get( name, inherits = TRUE )
    !is.null( val ) && ( !is.character( val ) || any( nzchar( val ) ) )
  }
}
knitr::opts_chunk$set(eval = !nzchar(Sys.getenv("_R_CHECK_PACKAGE_NAME_")),
                      purl = FALSE, collapse = TRUE,
                      comment = "#>", echo = FALSE, warning = FALSE, message = FALSE,
                      cache = FALSE, tidy = FALSE,
                      fig.align = "center", out.width = "100%", dpi = 160,
                      fig.width = 7, fig.height = 4, dev = "png",
                      dev.args = if (isTRUE(capabilities("cairo")))
                        list(png = list(type = "cairo", antialias = "default")) else list())
```

# Overview

This example evaluates a G-CSF / filgrastim population design with PFIM, based on the PK-PD model of Krzyzanski *et al.* [@Krzyzanski2010].

The model describes subcutaneous filgrastim with quasi-steady-state target-mediated drug disposition (TMDD) and a myelopoiesis cascade for absolute neutrophil count (ANC). The population Fisher information matrix (FIM) is evaluated for Design 1: three parallel arms (1, 3 and 10 µg/kg), 10 subjects per arm, dense PK and ANC sampling on days 1 and 7.

## Objectives

1. Evaluate the population FIM of Design 1 (ODE model, two responses, `Combined2` residual error).
2. Report relative standard errors (RSE %) for fixed effects, IIV and residual error.
3. Display typical PK and ANC predictions (dense ODE re-simulation) together with SE / RSE bar charts.

The 11-state ODE FIM is expensive. During rendering, `vignette_05_gcse_pkpd_execute.R` reuses `vignettes/data/vignette5_evaluation_populationFIM.RDS` when present; otherwise it runs the evaluation. Set `PFIM_GCSF_FORCE_RUN=true` to ignore the cache. HTML `Report()` is rebuilt only when regenerating the FIM or when `PFIM_VIGNETTE_REPORT=true`.

# Experimental design

Design 1 is a three-arm parallel study. Body weight is fixed at 75 kg so the administered amount is $\mathrm{DOSE}\times\mathrm{WT}$. Seven daily subcutaneous doses are given at times $0, 24, \ldots, 144$ h. Bioavailability `FF` enters the depot initial condition (`ABS = FF * dose_ABS`).

+------------------+--------+---------------------------+------------------------------------------+
| Arm              | $n$    | Dose                      | Sampling                                 |
+==================+========+===========================+==========================================+
| `dose_1ugkg`     | 10     | 1 µg/kg × 75 kg           | PK and ANC dense on days 1 and 7         |
+------------------+--------+---------------------------+------------------------------------------+
| `dose_3ugkg`     | 10     | 3 µg/kg × 75 kg           | same grid                                |
+------------------+--------+---------------------------+------------------------------------------+
| `dose_10ugkg`    | 10     | 10 µg/kg × 75 kg          | same grid                                |
+------------------+--------+---------------------------+------------------------------------------+

PK samples on day 1: 0.167–24 h (22 points), repeated on day 7, plus 172, 192 and 216 h. ANC adds daily troughs on days 2–6 and a 240 h point. Later PK peaks are lower than the day-1 peak because of TMDD feedback: higher ANC clears G-CSF faster.

# PK-PD model

Two observed responses:

- RespPK — serum G-CSF (ng/mL), algebraic quasi-steady-state free concentration from the central amount `CENT`.
- RespPD — circulating ANC ($10^3$/µL), ODE state `NB`.

Eleven ODE states: depot `ABS`, central `CENT`, nine bone-marrow transit compartments `B1`–`B9`, and blood neutrophils `NB`. Prefix `Deriv_` identifies each right-hand side; the suffix must match the state name. Operators follow R (`**` for exponentiation).

The full right-hand sides are built by `.pfimGcsfModelEquations()` in `vignette_05_gcse_pkpd_execute.R`. The evaluation only needs the equation list, the algebraic PK output, and baseline initial conditions `.pfimGcsfBaselineICs()`.

```{r, echo = TRUE, eval = FALSE, comment=''}
modelEquations = .pfimGcsfModelEquations()
outputs = list(RespPK = .pfimGcsfCp(), RespPD = "NB")
```

# Model parameters

Inter-individual variability ($\omega$) is set only for the parameters that are estimated (nonzero $\omega$). Fixed $\mu$ flags remove parameters from the FIM.

+------------+----------------------------------------------+-----------+------------------+----------+
| Parameter  | Description                                  | $\mu$     | $\omega$         | Fixed μ  |
+============+==============================================+===========+==================+==========+
| *FF*       | Bioavailability                              | 0.626     | 0                | Yes      |
+------------+----------------------------------------------+-----------+------------------+----------+
| *KA*       | Absorption rate (h⁻¹)                        | 0.642     | 0                | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *KEL*      | Elimination rate of free G-CSF (h⁻¹)         | 0.148     | √0.312           | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *VD*       | Central volume (L)                           | 2.56      | √0.328           | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *KD*       | Equilibrium dissociation constant (ng/mL)    | 1.27      | 0                | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *KINT*     | Internalization rate (h⁻¹)                   | 0.101     | 0                | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *KSI*      | Binding capacity (Rmax-related)              | 0.211     | √0.224           | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *KMT*      | Neutrophil elimination from blood (h⁻¹)      | 0.0723    | 0                | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *KTT*      | Myeloid transit rate (h⁻¹)                   | 0.0102    | 0                | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *NB0*      | Baseline circulating ANC (10³/µL)            | 1.65      | √0.298           | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *SC1*      | G-CSF EC₅₀ for stimulation (ng/mL)           | 3.21      | √0.803           | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *SM1*      | Max. stimulation of production               | 34.3      | √0.0128          | No       |
+------------+----------------------------------------------+-----------+------------------+----------+
| *SM2*      | Max. stimulation of maturation               | 32.3      | 0                | No       |
+------------+----------------------------------------------+-----------+------------------+----------+

`FR`, `D2`, `KOFF`, `KBB1`, `SM3` and `BAS` are fixed and do not appear in the FIM.

```{r, echo = TRUE, eval = FALSE, comment=''}
modelParameters = list(
  ModelParameter(name = "KA",  distribution = LogNormal(mu = 0.642, omega = 0)),
  ModelParameter(name = "KEL", distribution = LogNormal(mu = 0.148, omega = sqrt(0.312))),
  ModelParameter(name = "VD",  distribution = LogNormal(mu = 2.56,  omega = sqrt(0.328))),
  # ... remaining parameters as in vignette_05_gcse_pkpd_execute.R
)
```

# Residual error model

PFIM `Combined2` stores residual standard deviations (`sigmaInter`, `sigmaSlope`). The variance form is

$$
V = \sigma_{\mathrm{inter}}^2 + (\sigma_{\mathrm{slope}}\, f)^2.
$$

+-----------+------------------+------------------+
| Response  | Term             | PFIM SD          |
+===========+==================+==================+
| RespPK    | proportional     | √0.253           |
+-----------+------------------+------------------+
| RespPK    | additive         | 0, fixed         |
+-----------+------------------+------------------+
| RespPD    | proportional     | √0.0227          |
+-----------+------------------+------------------+
| RespPD    | additive         | √2.10            |
+-----------+------------------+------------------+

```{r, echo = TRUE, eval = FALSE, comment=''}
modelError = list(
  Combined2(output = "RespPK", sigmaInter = 0, sigmaSlope = sqrt(2.53e-01),
            sigmaInterFixed = TRUE),
  Combined2(output = "RespPD", sigmaInter = sqrt(2.10e+00),
            sigmaSlope = sqrt(2.27e-02))
)
```

# Administration, sampling times, arms

```{r, echo = TRUE, eval = FALSE, comment=''}
WT = 75
dose_times = seq(0, 6 * 24, by = 24)

mk_arm = function(name, dose_ug_per_kg) {
  Arm(
    name = name, size = 10,
    administrations = list(Administration(
      outcome = "ABS", timeDose = dose_times,
      dose = rep(dose_ug_per_kg * WT, length(dose_times))
    )),
    samplingTimes = list(samplingPK, samplingPD),
    initialConditions = .pfimGcsfBaselineICs()
  )
}

design1 = Design(
  name = "gcsf_design1",
  arms = list(
    mk_arm("dose_1ugkg", 1),
    mk_arm("dose_3ugkg", 3),
    mk_arm("dose_10ugkg", 10)
  )
)
```

# Population FIM evaluation

```{r, echo = TRUE, eval = FALSE, comment=''}
pfim_set_option(perf.fdLinearOnly = TRUE)

evaluationPop = Evaluation(
  name = "gcsf_design1",
  modelEquations = modelEquations,
  modelParameters = modelParameters,
  modelError = modelError,
  outputs = list(RespPK = .pfimGcsfCp(), RespPD = "NB"),
  designs = list(design1),
  fimType = "population",
  odeSolverParameters = list(atol = 1e-8, rtol = 1e-8)
)

evaluationPop = run(evaluationPop)
show(evaluationPop)
getRSE(evaluationPop)
```

```{r ex04_run, include = FALSE}
script = file.path("..", "inst", "vignette-scripts", "vignette_05_gcse_pkpd_execute.R")
if (!file.exists(script))
  script = system.file("vignette-scripts", "vignette_05_gcse_pkpd_execute.R", package = "PFIM")
if (!nzchar(script) || !file.exists(script))
  stop("vignette_05_gcse_pkpd_execute.R not found; reinstall PFIM or rebuild vignettes.")
source(script, local = knitr::knit_global())
```

```{r ex04_show_pop, echo = FALSE, results = "asis"}
if (.pfimVignetteHas("showOutputEvaluationPop")) {
  cat("<pre class=\"pfim-show\">", showOutputEvaluationPop, "</pre>\n", sep = "")
}
```

# PFIM, PopED, and cross-tool equivalence

This vignette compares PFIM to PopED on G-CSF Design 1. Many PFIM examples (workshop material, Tim Waterhouse oral PK) were originally built in PopED. The two tools do not share syntax; FIMs and RSE are comparable only when the statistical model, design, and variance versus SD conventions are aligned.

This section has three parts:

- General rules, valid for any PopED model translated into PFIM;
- Tim Waterhouse oral PK, the reference PopED layout (`optimal-design-2023`), implemented in `vignette("Vignette_04_tim_waterhouse_pediatric", package = "PFIM")`;
- G-CSF Design 1, the ODE model of this vignette.

Readers interested only in the G-CSF example can go directly to [G-CSF Design 1](#gcsf-poped) and the [relative standard errors](#gcsf-rse).

## General rules (all vignettes)

### Variance or SD? One conversion rule

PopED stores inter-individual `d` and residual `sigma` as variances. PFIM stores `omega` (IIV) and `sigmaInter` / `sigmaSlope` (residuals) as standard deviations on the same scales. When moving from PopED, NONMEM, or Tim's files to PFIM, apply `sqrt` to each variance separately (IIV and each residual component), never to the combined expression $\sigma_{\mathrm{add}}^2 + (\sigma_{\mathrm{prop}} f)^2$.

From PopED or NONMEM to PFIM: `omega = sqrt(d)` on each `LogNormal`; `sigmaSlope = sqrt(sigma[1])`, `sigmaInter = sqrt(sigma[2])` for `Combined2` (e.g. $\sqrt{1} = 1$).

PFIM only: type SDs directly; outputs label them $\sigma$, not $\sigma^2$. Use `Constant` or `Proportional` when a single error term is enough; `Combined2` matches PopED's two-term residual model.

Sanity check: at a typical $f_0$, compare $\sigma_{\mathrm{inter}}^2 + (\sigma_{\mathrm{slope}} f_0)^2$ in PFIM to `sigma[2] + sigma[1] * f_0^2` in PopED. Mismatch in FIM, RSE, or optimal times often comes from skipping the square root on `d` or `sigma`.

### Naming

Different names are expected. For peer review or cross-checking with PopED, align the equivalence table and the variance → SD rule, not a literal copy of PopED numbers into PFIM fields.

## Tim Waterhouse oral PK (Vignette 04)

The tables below refer to the pediatric oral PK example of `vignette("Vignette_04_tim_waterhouse_pediatric", package = "PFIM")`.

### What plays the same role

| Topic | Tim / PopED | PFIM (Vignette 04) |
|-------|-------------|----------------------|
| Structural PK | `ff()`, `fg()`; scaling in `ff` with `WT`, `WT_CL`, `WT_V` | One analytic `modelEquations` string; `(WT/70)^beta_*` inside the formula |
| Typical clearance, volume, absorption | `bpop`: `CL`, `V`, `KA` | `ModelParameter`: `Cl`, `V`, `ka` (`LogNormal`, μ on the natural scale, e.g. `V = 100` L) |
| Weight exponents (fixed) | `bpop`: `WT_CL = 0.75`, `WT_V = 1`; `notfixed_bpop = 0` for those | `beta_Cl_WT`, `beta_V_WT` with `fixedMu = TRUE` |
| Pediatric weight | Covariate `a` / `fg`: `WT = 32` | Global symbol `WT = 32` read from the R environment (not `Covariate()`) |
| IIV on CL, V, KA | `d`: `0.08`, `0.1`, `0.2` | `omega = sqrt(0.08)`, `sqrt(0.1)`, `sqrt(0.2)` on each `LogNormal` |
| Residual error | `feps.add.prop`; `sigma = c(0.05, 1)` | `Combined2`; `sigmaSlope = sqrt(0.05)`, `sigmaInter = 1` |
| Dose | `a`: `DOSE = 10` (mg); `fg` uses `DOSE * 1000` | `Administration`: `dose = 10 * 1000` at each `timeDose` |
| Sample size | `groupsize = 12` | `Arm(..., size = 12)` |
| Objective | D-optimal FIM on `bpop` (estimated thetas) | D-optimal population FIM on estimable `μ` (`fimType = "population"`) |

### Design and time scale

| PopED | PFIM (Vignette 04) |
|-------|----------------------|
| `xt`, `minxt`, `maxxt` often relative to dosing (e.g. hours after dose); steady-state samples use `model_switch = 2` and a dedicated formula | Calendar time from study start (hours); repeated doses via `timeDose` + superposition of the same oral formula |
| Five occasions in Tim: `xt = c(5, 24, 24, 24, 168)` in the SS frame | `5`, predose weeks 8/12/24, `lastDose + 168` in calendar hours |

Optimization windows in PFIM are chosen to mirror Tim's protocol; optimal times need not match PopED numerically (different time origin and PK engine).

### Structural model implementation

PopED switches between first-dose and steady-state concentration formulas (`model_switch`). PFIM uses one oral absorption expression and superposes daily doses. For a short half-life and once-daily dosing, profiles are usually similar but not identical.

### Naming only (no change to the model)

| PopED / Tim | PFIM (Vignette 04) |
|-------------|---------------------|
| `CL`, `KA` | `Cl`, `ka` (case) |
| `WT_CL`, `WT_V` | `beta_Cl_WT`, `beta_V_WT` |
| `bpop`, `d`, `sigma`, `xt` | `modelParameters`, `modelError`, `SamplingTimes` / constraints |

## G-CSF Design 1 (this vignette, Vignette 05) {#gcsf-poped}

### Variance → SD for this model

| Term | PopED (variance) | PFIM (SD) |
|------|------------------|-----------|
| IIV, e.g. `KEL` | `d = 0.312` | `omega = sqrt(0.312)` |
| PK proportional | `0.253` | `sigmaSlope = sqrt(0.253)` |
| ANC proportional | `0.0227` | `sigmaSlope = sqrt(0.0227)` |
| ANC additive | `2.10` | `sigmaInter = sqrt(2.10)` |

### Comparing RSE with PopED

- Fixed effects ($\mu$) and IIV ($\omega^2$): both tools report RSE on the same scale; values are directly comparable (see tables below).
- Residual error: PopED reports the RSE of the variance, PFIM the RSE of the SD. By the delta method, $\mathrm{RSE}(\sigma^2) \approx 2\,\mathrm{RSE}(\sigma)$.
- Finite differences: `perf.fdLinearOnly = TRUE` uses central differences, as PopED does, so that gradients are computed the same way.

# Relative standard errors {#gcsf-rse}

RSE (%) for Design 1. The PopED column is `get_rse(FIM_gcsf, poped_db_gcsf)` for this design, stored as `gcsf_design1` in `inst/fixtures/gold-fim.R`. That block is a PopED reference, not a PFIM result. The other entries in the same file (`cas1`–`cas6`) are frozen PFIM numbers and are not used here. Residual RSE in `gcsf_design1` (`rse_sigma_var`) is on the variance scale.

## Fixed effects ($\mu$)

```{r ex04_rse_mu, echo = FALSE, results = "asis"}
mu_desc = c(
  KA = "Absorption rate",
  KEL = "Elimination rate of free G-CSF",
  VD = "Central volume",
  KD = "Equilibrium dissociation constant",
  KINT = "Internalization rate",
  KSI = "Binding capacity (Rmax-related)",
  KMT = "Neutrophil elimination from blood",
  KTT = "Myeloid transit rate",
  NB0 = "Baseline circulating ANC",
  SC1 = "G-CSF EC50 for stimulation",
  SM1 = "Max. stimulation of production",
  SM2 = "Max. stimulation of maturation"
)
mu_tab = data.frame(
  Parameter = paste0("$\\mu_{\\mathrm{", cmp_mu$param, "}}$"),
  Description = unname(mu_desc[ cmp_mu$param ]),
  `PopED RSE (%)` = .gcsfFmtRse(cmp_mu$poped),
  `PFIM RSE (%)` = .gcsfFmtRse(cmp_mu$pfim),
  check.names = FALSE
)
.gcsfKbl(mu_tab, "Fixed-effect RSE (%) for Design 1 (PFIM vs PopED).")
```

## Inter-individual variances ($\omega^2$)

```{r ex04_rse_d, echo = FALSE, results = "asis"}
d_desc = c(
  NB0 = "Baseline circulating ANC",
  KEL = "Elimination rate of free G-CSF",
  VD = "Central volume",
  KSI = "Binding capacity (Rmax-related)",
  SC1 = "G-CSF EC50 for stimulation",
  SM1 = "Max. stimulation of production"
)
d_tab = data.frame(
  Parameter = paste0("$\\omega^2_{\\mathrm{", cmp_d$param, "}}$"),
  Description = unname(d_desc[ cmp_d$param ]),
  `PopED RSE (%)` = .gcsfFmtRse(cmp_d$poped),
  `PFIM RSE (%)` = .gcsfFmtRse(cmp_d$pfim),
  check.names = FALSE
)
.gcsfKbl(d_tab, "IIV variance ($\\omega^2$) RSE (%) for Design 1 (PFIM vs PopED).")
```

## Residual error ($\sigma$)

Console and report rows are labelled $\sigma_{\mathrm{slope/inter}}$ — the Value column is the SD, not the variance.

PFIM estimates the residual SD, PopED the residual variance, so their RSE are not on the same scale. By the delta method, $\mathrm{RSE}(\sigma^2) \approx 2\,\mathrm{RSE}(\sigma)$: the column $2 \times$ PFIM RSE is the one to compare with PopED.

```{r ex04_rse_sigma, echo = FALSE, results = "asis"}
sig_desc = c(
  slope_RespPK = "PK proportional (SD)",
  slope_RespPD = "ANC proportional (SD)",
  inter_RespPD = "ANC additive (SD)"
)
sig_tab = data.frame(
  Parameter = c(
    "$\\sigma_{\\mathrm{slope,PK}}$",
    "$\\sigma_{\\mathrm{slope,ANC}}$",
    "$\\sigma_{\\mathrm{inter,ANC}}$"
  ),
  Description = unname(sig_desc[ cmp_sigma$param ]),
  `Value PFIM (SD)` = .gcsfFmtRse(cmp_sigma$pfim_sd, 4),
  `PFIM RSE (SD, %)` = .gcsfFmtRse(cmp_sigma$pfim_rse_sd),
  `2 × PFIM RSE (variance scale, %)` = .gcsfFmtRse(cmp_sigma$pfim_rse_var_equiv),
  `PopED RSE (variance, %)` = .gcsfFmtRse(cmp_sigma$poped_rse_var),
  check.names = FALSE
)
.gcsfKbl(
  sig_tab,
  paste0(
    "Residual error for Design 1. Compare the last two columns: ",
    "$2 \\times$ PFIM RSE(SD) $\\approx$ RSE(variance), the scale used by PopED."
  )
)
```

# Diagnostic plots

Overlay of the three dose groups (PK | ANC): ODE re-simulation on a dense $0..t_{\max}$ grid. Sampling times are shown as points. SE and RSE bar charts follow.

```{r, echo = TRUE, eval = FALSE, comment=''}
plotOutcomesEvaluationModel
PFIM::plotSE(evaluationPop)
PFIM::plotRSE(evaluationPop)
```

```{r ex04_plot_model, echo = FALSE, fig.width = 10.5, fig.height = 4.4, out.width = "100%"}
if (.pfimVignetteHas("plotOutcomesEvaluationModel")) plotOutcomesEvaluationModel
```

```{r ex04_plot_se, echo = FALSE, fig.width = 11, fig.height = 4.4, out.width = "100%"}
if (.pfimVignetteHas("plotEval_SE")) plotEval_SE
```

```{r ex04_plot_rse, echo = FALSE, fig.width = 11, fig.height = 4.4, out.width = "100%"}
if (.pfimVignetteHas("plotEval_RSE")) plotEval_RSE
```

```{r cleanup, echo = FALSE, include = FALSE}
options(backup_options)
```

# References