---
title: "Design Evaluation and Optimization in Discrete Space"
date: ""
output:
  rmarkdown::html_vignette:
    toc: true
    toc_depth: 3
bibliography: references.bib
biblio-style: apalike
link-citations: yes
vignette: >
  %\VignetteIndexEntry{Design Evaluation and Optimization in Discrete Space}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
  
<style type="text/css">
/* Typography */
body       { font-size: 14px; line-height: 1.7; color: #2c3e50;
max-width: 1200px; }
h1, h2, h3 { font-weight: 600; margin-top: 1.8em; color: #2c3e50; border-bottom: none !important;}
p          { margin-bottom: 0.9em; }

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border: 1px solid #e0e0e0; padding: 10px; }
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</style>
  
```{r setup, include=FALSE}
backup_options = options()
options(width = 120)

knitr::opts_chunk$set(
echo       = TRUE,
eval       = FALSE,          # chunks do NOT execute by default (CRAN compliance)
warning    = FALSE,
message    = FALSE,
comment    = "#>",
cache      = FALSE,
fig.width  = 8,
fig.height = 5,
fig.align  = "center",
out.width  = "90%"
)

library(PFIM)

# paths$data    : pre-computed RDS files shipped in inst/extdata/
# paths$outputs : text outputs  -> tempdir() (CRAN-compliant)
# paths$reports : HTML reports  -> tempdir()
paths = list(
data    = system.file("extdata", package = "PFIM"),
outputs = file.path(tempdir(), "pfim_out"),
reports = file.path(tempdir(), "pfim_rep")
)
dir.create(paths$outputs, showWarnings = FALSE, recursive = TRUE)
dir.create(paths$reports, showWarnings = FALSE, recursive = TRUE)

# Shared plot options
plotOptions = list(unitTime = "hour", unitOutcomes = c("mcg/mL", "DI%"))
```

---

# Scientific Background

This example is based on a landmark PK/PD population study of **Tolmetin** (a
non-steroidal anti-inflammatory drug) in rats [@FloresMurrieta1998]. The integrated model couples a one-compartment PK model with a PD model to describe nociception, through the dysfunction index. The antinociceptive effect of Tolmetin was characterized using an indirect response model.

**Original experimental design.** Six parallel dose groups of rats (n=6 per group, total N=36) received single oral doses from 1 to 100 mg/kg. Blood samples and inflammation scores (DI) were collected at 10 time points: 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 2, 3, and 4 h.

**Objectives:**

1. **Evaluation** -- Compute the Population and Bayesian Fisher Information
Matrices (FIM) for the original design. Assess parameter precision via RSE%
and Shrinkage.
2. **Optimization** -- Find the D-optimal design for 30 rats by selecting doses
and sampling times from a discrete candidate set. Two algorithms are
compared: Fedorov-Wynn and Multiplicative.

---

# Design Evaluation

## Model Equations

### PK model: one-compartment, first-order oral absorption

$$\frac{dC_c}{dt} = \frac{D}{V}\,k_a\,e^{-k_a t} - \frac{Cl}{V}\,C_c$$

| Symbol | Description | Unit |
|:------:|:------------|:----:|
| $C_c$ | Plasma drug concentration (state variable) | mcg/mL |
| $V$   | Volume of distribution | L |
| $k_a$ | First-order absorption rate constant | h$^{-1}$ |
| $Cl$  | Total clearance | L/h |
| $D$   | Administered dose (`dose_RespPK`) | mg |

### PD model: indirect response, inhibition of production 

$$\frac{dE}{dt} = R_{in}\!\left(1 - \frac{I_{max}\,C_c^{\gamma}}{C_c^{\gamma} + IC_{50}^{\gamma}}\right) - k_{out}\,E$$

| Symbol | Description | Unit |
|:------:|:------------|:----:|
| $E$      | Dysfunction index $%$ | -- |
| $R_{in}$ | Zero-order input rate | $%/h$ |
| $I_{max}$| Maximum fractional inhibition | -- |
| $IC_{50}$| Concentration at 50% of $I_{max}$ | $mg/mL$ |
| $\gamma$ | Hill coefficient | -- |
| $k_{out}$| First-order dissipation rate | $/h$ |

> **Steady-state initial condition.** At $t = 0$ (i.e. $C_c = 0$), the PD system
> is at equilibrium: $E(0) = R_{in}/k_{out}.

**Naming conventions for ODE models:**

- The prefix `Deriv_` is mandatory; the suffix must match the state variable name (`Cc` or `E`).
- The dose is specified as `dose_` followed by the name of the PK response (e.g. `dose_RespPK`).
- Use `**` for exponentiation.
- Model outcomes are represented as a list associating each response with its state variable:
  `RespPK` $\leftrightarrow$ `Cc`, `RespPD` $\leftrightarrow$ `E`.

```{r model-equations, echo=TRUE, comment=''}
modelEquations = list(
  Deriv_Cc = "dose_RespPK/V*ka*exp(-ka*t) - Cl/V*Cc",
  Deriv_E  = "Rin*(1-Imax*(Cc**gamma)/(Cc**gamma + IC50**gamma)) - kout*E"
)
```

## Model Parameters

Parameters are specified via their population typical value (fixed effect $\mu$)
and inter-individual variability (IIV $\omega$). We assume a **log-normal**
distribution for all parameters.

The estimable parameter vector has dimension $p = 14$ (6 fixed effects + 8
variance components )

$$\theta = \bigl\{\mu_V,\,\mu_{Cl},\,\mu_{k_{out}},\,\mu_{I_{max}},\,
\mu_{IC_{50}},\,\mu_{\gamma},\;
\omega^2_V,\,\omega^2_{Cl},\,\omega^2_{k_{out}},\,\omega^2_{I_{max}},\,
\omega^2_{IC_{50}},\,\omega^2_{\gamma}, \sigma_{slope_{RespPK}}, \sigma_{inter_{RespPD}}\bigr\}$$

Setting `fixedMu = TRUE` excludes the fixed effect from the FIM. Setting
`omega = 0` implies no IIV; the corresponding variance is not estimated
(`fixedOmega = TRUE` is set automatically).

| Parameter  | Description                         |   $\mu$ | $\omega$ | Fixed $\mu$ | Fixed $\omega$ |
|:----------:|:------------------------------------|--------:|---------:|:-----------:|:--------------:|
| $V$        | Volume of distribution (L)          |    0.74 |    0.316 | No          | No             |
| $Cl$       | Total clearance (L/h)               |    0.28 |    0.456 | No          | No             |
| $k_a$      | Absorption rate constant (h$^{-1}$) |      10 |        0 | Yes         | Yes            |
| $k_{out}$  | Effect elimination rate (h$^{-1}$)  |    6.14 |    0.947 | No          | No             |
| $R_{in}$   | Baseline production rate (h$^{-1}$) |     614 |        0 | Yes         | Yes            |
| $I_{max}$  | Maximum inhibition (--)             |    0.76 |    0.439 | No          | No             |
| $IC_{50}$  | Potency (mcg/mL)                    |    9.22 |    0.452 | No          | No             |
| $\gamma$   | Hill coefficient (--)               |    2.77 |    1.761 | No          | No             |

```{r model-parameters, echo=TRUE, comment=''}
modelParameters = list(
  ModelParameter(name         = "V",
                 distribution = LogNormal(mu = 0.74,  omega = 0.316)),
  ModelParameter(name         = "Cl",
                 distribution = LogNormal(mu = 0.28,  omega = 0.456)),
  ModelParameter(name         = "ka",
                 distribution = LogNormal(mu = 10,    omega = 0),
                 fixedMu      = TRUE),
  ModelParameter(name         = "kout",
                 distribution = LogNormal(mu = 6.14,  omega = 0.947)),
  ModelParameter(name         = "Rin",
                 distribution = LogNormal(mu = 614,   omega = 0),
                 fixedMu      = TRUE),
  ModelParameter(name         = "Imax",
                 distribution = LogNormal(mu = 0.76,  omega = 0.439)),
  ModelParameter(name         = "IC50",
                 distribution = LogNormal(mu = 9.22,  omega = 0.452)),
  ModelParameter(name         = "gamma",
                 distribution = LogNormal(mu = 2.77,  omega = 1.761))
)
```

## Residual Error Models

Two residual error structures are used, one per response.
A proportional error model is considered for the PK (`Combined1` with $\sigma_{inter}$= 0, which is equivalent to `Proportional`). 
An additive error model is considered for the PD (Constant). 

| Response | Error model | $\sigma_{inter}$ | $\sigma_{slope}$ |
|:--------:|:-----------:|:----------------:|:----------------:|
| PK       | `Combined1` | 0.0              | 0.21             |
| PD       | `Constant`  | 9.6              | --               |

```{r error-models, echo=TRUE, comment=''}
errorModelRespPK = Combined1(output = "RespPK", sigmaInter = 0,   sigmaSlope = 0.21)
errorModelRespPD = Constant(output = "RespPD", sigmaInter = 9.6)
modelError       = list(errorModelRespPK, errorModelRespPD)
```

## Sampling Times

```{r sampling-times, echo = TRUE,   comment=''} 
samplingTimesRespPK = SamplingTimes(
  outcome   = "RespPK",
  samplings = c(0.25, 0.5, 0.75, 1, 1.25, 1.5, 2, 3, 4)
)
samplingTimesRespPD = SamplingTimes(
  outcome   = "RespPD",
  samplings = c(0.25, 0.5, 0.75, 1, 1.25, 1.5, 2, 3, 4)
)
```

## Arms 

Six arms correspond to the original dose levels converted to absolute doses for
a 200 g rat: $D_\text{mg} = D_\text{mg/kg} \times 0.200\,\text{kg}$.
**Total: 36 subjects** (6 arms x 6 subjects each).

| Arm | mg/kg | Absolute dose | Subjects |
|:---:|------:|:-------------:|:--------:|
| 1 |   1.0 | 0.20 mg  | 6 |
| 2 |   3.2 | 0.64 mg  | 6 |
| 3 |  10.0 | 2.00 mg  | 6 |
| 4 |  31.6 | 6.32 mg  | 6 |
| 5 |  56.2 | 11.24 mg | 6 |
| 6 | 100.0 | 20.00 mg | 6 |

Initial conditions: $C_c(0) = 0$ and $E(0) = 100 = R_{in}/k_{out}$ 

```{r arms, echo = TRUE,   comment=''} 
# Helper to avoid repeating the Arm() constructor six times
makeArm = function(armName, dose, size = 6) {
  admin = Administration(outcome = "RespPK", timeDose = 0, dose = dose)
  Arm(
    name             = armName,
    size             = size,
    administrations  = list(admin),
    samplingTimes    = list(samplingTimesRespPK, samplingTimesRespPD),
    initialCondition = list(Cc = 0, E = 100)
  )
}

arm1 = makeArm("0.20mg Arm",  0.20)
arm2 = makeArm("0.64mg Arm",  0.64)
arm3 = makeArm("2.00mg Arm",  2.00)
arm4 = makeArm("6.32mg Arm",  6.32)
arm5 = makeArm("11.24mg Arm", 11.24)
arm6 = makeArm("20.00mg Arm", 20.00)
```

## 1.6 Design Assembly

```{r design1, echo = TRUE,   comment=''} 
design1 = Design( name = "design1", 
                  arms = list( arm1, arm2, arm3, arm4, arm5, arm6 ) )
```

# Evaluation of the population and Bayesian FIM 

## Population FIM Evaluation

```{r evaluation-objects, echo = TRUE,   comment=''} 
evaluationPop = Evaluation(
  name                = "evaluationPop",
  modelEquations      = modelEquations,
  modelParameters     = modelParameters,
  modelError          = modelError,
  outputs             = list("RespPK" = "Cc", "RespPD" = "E"),
  designs             = list(design1),
  fimType             = "population",
  odeSolverParameters = list(atol = 1e-8, rtol = 1e-8)
)

evaluationPopResults = run (evaluationPop )
```

## Bayesian FIM Evaluation

```{r evaluation-run, echo=TRUE, comment=''}
evaluationBayesian = Evaluation(
  name                = "evaluationBayesian",
  modelEquations      = modelEquations,
  modelParameters     = modelParameters,
  modelError          = modelError,
  outputs             = list("RespPK" = "Cc", "RespPD" = "E"),
  designs             = list(design1),
  fimType             = "Bayesian",
  odeSolverParameters = list(atol = 1e-8, rtol = 1e-8)
)

evaluationBayesianResults = run( evaluationBayesian )
```

## Results 

### Population FIM

```{r results-pop-show, echo = TRUE, eval= FALSE, comment=''} 
show(evaluationPopResults)
```

```{r results-pop-show_from_outputs, echo = FALSE,  eval=TRUE, comment=''} 
lines = readLines("outputs/vignette1_evaluation_populationFIM_show.txt")
cat(paste(lines, collapse = "\n"))
```

All these elements can also be accessed using the following methods:

```{r results-pop-detail}
cat("Fisher Information Matrix") 
print(getFisherMatrix(evaluationPopResults))

cat("Correlation Matrix")
print(getCorrelationMatrix(evaluationPopResults))

cat("Standard Errors (SE)")
print(getSE(evaluationPopResults))

cat("Relative Standard Errors")
print(getRSE(evaluationPopResults))

cat("Shrinkage (%)")
print(getShrinkage(evaluationPopResults))

cat("Determinant")
print(getDeterminant(evaluationPopResults))

cat("D-Criterion")
print(getDcriterion(evaluationPopResults))
```

### Bayesian FIM

```{r results-bayesian-show, echo = TRUE, eval= FALSE, comment=''} 
show(evaluationBayesianResults)
```

```{r results-bayesian_from_outputs, echo = FALSE,  eval=TRUE, comment=''} 
lines = readLines("outputs/vignette1_evaluation_BayesianFIM_show.txt")
cat(paste(lines, collapse = "\n"))
```

```{r results-bayesian-detail}
cat("Fisher Information Matrix")
print(getFisherMatrix(evaluationBayesianResults))

cat("Correlation Matrix")
print(getCorrelationMatrix(evaluationBayesianResults))

cat("Standard Errors (SE)")
print(getSE(evaluationBayesianResults))

cat("Relative Standard Errorn")
print(getRSE(evaluationBayesianResults))

cat("Shrinkage (%)")
print(getShrinkage(evaluationBayesianResults))

cat("Determinant")
print(getDeterminant(evaluationBayesianResults))

cat("D-Criterion")
print(getDcriterion(evaluationBayesianResults))
```

## Diagnostic Plots

### Response profiles

```{r plot-eval-pk, echo = TRUE, eval=FALSE, comment=''}
# plot() is the unified OO entry point — dispatches on class, returns a named list:
#   $evaluation         -> nested [["design"]][["arm"]][["outcome"]]
#   $sensitivityIndices -> nested [["design"]][["arm"]][["outcome"]][["param"]]
#   $SE / $RSE          -> ggplot2 bar charts
# which = c(...) selects a subset; omitting it computes all plots for the class.
evalPlots = plot(evaluationPopResults,
                 plotOptions = plotOptions,
                 which       = c("evaluation", "sensitivityIndices", "SE", "RSE"))
print(evalPlots$evaluation[["design1"]][["20.00mg Arm"]][["RespPK"]])
```

```{r plot-eval-pk_from_figures, echo=FALSE, eval=TRUE, out.width="50%", fig.align="center", fig.cap="PK response profile -- 20.00 mg arm (population FIM)"}
knitr::include_graphics("figures/vignette1_evaluation_populationFim_design1_arm20mg_RespPK.png")
```

```{r plot-eval-pd, echo = TRUE, eval=FALSE,  comment=''}
print(evalPlots$evaluation[["design1"]][["20.00mg Arm"]][["RespPD"]])
```

```{r plot-eval-pd_from_figures, echo = FALSE,  eval=TRUE, out.width="50%",comment='', fig.align = "center", fig.cap="PD response profile -- 20.00 mg arm (population FIM)"} 
knitr::include_graphics("figures/vignette1_evaluation_populationFim_design1_arm20mg_RespPD.png")
```

### Sensitivity indices

```{r plot-si-cl, echo = TRUE, eval= FALSE,  comment=''}
# $sensitivityIndices is computed in the same plot() call above
print(evalPlots$sensitivityIndices[["design1"]][["20.00mg Arm"]][["RespPK"]][["Cl"]])
```

```{r plot-si-cl_from_figures, echo = FALSE,  eval=TRUE, out.width="50%", comment='', fig.align = "center", fig.cap="Sensitivity index for Cl -- RespPK, 20.00 mg arm"} 
knitr::include_graphics("figures/vignette1_evaluation_populationFim_design1_SI_RespPK_Cl.png")
```

### Standard errors and Relative Standard errors

```{r plot-se, echo = TRUE, eval= FALSE,   comment=''}
print(evalPlots$SE)
```

```{r plot-se_figures, out.width="33%", echo = FALSE, eval= TRUE,  comment='', fig.align = "center",fig.cap="Standard Errors (SE)"} 
knitr::include_graphics("figures/vignette1_evaluation_populationFim_design1_SE.png")
```

```{r plot-rse, fig.width=5, fig.width=15, echo = TRUE, eval= FALSE,  comment=''}
print(evalPlots$RSE)
```


```{r plot-rse_figures, out.width="33%", echo = FALSE, eval= TRUE,  comment='', fig.align = "center",fig.cap="Relative Standard Errors (RSE %)"} 
knitr::include_graphics("figures/vignette1_evaluation_populationFim_design1_RSE.png")
```

## HTML Report

```{r report-pop, eval=FALSE, echo=TRUE, comment=''}
#Define your path to save your report: pathsReports ="C:/..."
Report(evaluationPopResults, pathsReports, "vignette1_evaluation_popFIM.html", plotOptions)
```

---

# Design Optimization

**Objective.** Find a D-optimal design for a future study under practical
constraints:

- Total sample size fixed at **30 subjects**.
- Doses restricted to the discrete set: {0.2, 0.64, 2, 6.32, 11.24, 20} mg.
- **RespPK:** 2 mandatory times + 4 optimisable times (from 7 candidates).
- **RespPD:** 2 mandatory times + 4 optimisable times (from 8 candidates).

## Shared Optimization Setup

```{r optim-setup, echo = TRUE,  eval = FALSE, comment=''} 
#  Initial administration
# Starting dose; optimizer will reassign from the discrete set in Section 2.6.
adminRespPK = Administration(outcome = "RespPK", timeDose = 0, dose = 6.32)

# Candidate sampling grids
sampTimesOptPK = SamplingTimes( outcome = "RespPK", samplings = c(0.25, 0.75, 1, 1.5, 2, 4, 6) )
sampTimesOptPD = SamplingTimes( outcome   = "RespPD", samplings = c(0.25, 0.75, 1.5, 2, 3, 6, 8, 12) )

# Sampling constraints -- RespPK
# Fixed:       0.25 h (absorption/Cmax) and 4 h (late elimination).
# Optimisable: 4 times chosen from {0.75, 1, 1.5, 2, 6}.
sampConstraintsPK = SamplingTimeConstraints(
  outcome                      = "RespPK",
  initialSamplings             = c(0.25, 0.75, 1, 1.5, 2, 4, 6),
  fixedTimes                   = c(0.25, 4),
  numberOfsamplingsOptimisable = 4
)

# Sampling constraints -- RespPD
# Fixed:       2 h (near peak effect / IC50) and 6 h (mid-recovery / kout).
# Optimisable: 4 times chosen from {0.25, 0.75, 1.5, 3, 8, 12}.
sampConstraintsPD = SamplingTimeConstraints(
  outcome                      = "RespPD",
  initialSamplings             = c(0.25, 0.75, 1.5, 2, 3, 6, 8, 12),
  fixedTimes                   = c(2, 6),
  numberOfsamplingsOptimisable = 4
)

# 2.5 Initial elementary protocols (Fedorov-Wynn only)
initialElementaryProtocols = list(
  c(0.25, 0.75, 1, 4),  # PK-focused: absorption + elimination
  c(1.5,  2, 6, 12)     # PD-focused: near-peak + late recovery
)

# Dose constraints -- discrete set matching the original study
adminConstraintsPK = AdministrationConstraints(
  outcome = "RespPK",
  doses   = list(0.2, 0.64, 2, 6.32, 11.24, 20)
)

# Constrained arm and design
# E(0) = "Rin/kout" as a formula string: PFIM evaluates this at typical values,
# giving E0 = 614/6.14 = 100. This is preferable to hardcoding the numeric
# value because it stays consistent if parameter estimates are updated.
armConstraint = Arm(
  name                       = "armConstraint",
  size                       = 30,
  administrations            = list(adminRespPK),
  samplingTimes              = list(sampTimesOptPK, sampTimesOptPD),
  administrationsConstraints = list(adminConstraintsPK),
  samplingTimesConstraints   = list(sampConstraintsPK, sampConstraintsPD),
  initialCondition           = list(Cc = 0, E = "Rin/kout")
)

# numberOfArms: upper bound on distinct elementary protocols
designConstraint = Design(
  name         = "designConstraints",
  arms         = list(armConstraint),
  numberOfArms = 30
)

numberOfSubjects      = 30
proportionsOfSubjects = 1   # single group at init; optimizer redistributes
```

## Fedorov-Wynn Algorithm

```{r fw-object, echo = TRUE,  eval=FALSE, comment=''} 
optimizationFW = Optimization(
  name                = "FedorovWynn",
  modelEquations      = modelEquations,
  modelParameters     = modelParameters,
  modelError          = modelError,
  optimizer           = "FedorovWynnAlgorithm",
  optimizerParameters = list(
    elementaryProtocols   = initialElementaryProtocols,
    numberOfSubjects      = numberOfSubjects,
    proportionsOfSubjects = proportionsOfSubjects,
    showProcess           = TRUE
  ),
  designs             = list(designConstraint),
  fimType             = "population",
  outputs             = list("RespPK" = "Cc", "RespPD" = "E"),
  odeSolverParameters = list(atol = 1e-8, rtol = 1e-8)
)
```

```{r fw-run, echo=TRUE, eval=FALSE, comment=''}
optimizationFWResults = run(optimizationFW)
```

### Results

```{r fw-show, echo=TRUE, eval=FALSE, comment=''}
show(optimizationFWResults)
```

```{r results-optiFW-popFIM_from_outputs, echo = FALSE,  eval=TRUE, comment=''} 
lines = readLines("outputs/vignette1_optimization_FedorovWynn_populationFIM_show.txt")
cat(paste(lines, collapse = "\n"))
```

```{r fw-detail, echo=TRUE, eval=FALSE, comment=''}
cat("Fisher Information Matrix\n");  print(getFisherMatrix(optimizationFWResults))
cat("\nCorrelation Matrix\n");        print(getCorrelationMatrix(optimizationFWResults))
cat("\nStandard Errors (SE)\n");      print(getSE(optimizationFWResults))
cat("\nRelative Standard Errors\n");  print(getRSE(optimizationFWResults))
cat("\nShrinkage (%)\n");             print(getShrinkage(optimizationFWResults))
cat("\nDeterminant\n");               print(getDeterminant(optimizationFWResults))
cat("\nD-Criterion\n");               print(getDcriterion(optimizationFWResults))
```

### HTML Report

```{r fw-report, eval=FALSE, echo=TRUE, comment=''}
#Define your path to save your report: pathsReports ="C:/..."
Report(optimizationFWResults, pathsReports, "vignette1_optimization_FedorovWynn_populationFIM.html", plotOptions)
```

## Multiplicative Algorithm

Unlike Fedorov-Wynn algorithm, the Multiplicative Algorithm requires no initial protocols and may converge to a different local optimum -- comparing both confirms robustness.

```{r mult-object, echo=TRUE, eval=FALSE, comment=''}
optimizationMult = Optimization(
  name                = "Multiplicative",
  modelEquations      = modelEquations,
  modelParameters     = modelParameters,
  modelError          = modelError,
  optimizer           = "MultiplicativeAlgorithm",
  optimizerParameters = list(
    lambda             = 0.99,
    numberOfIterations = 1000,
    weightThreshold    = 0.01,
    delta              = 1e-4,
    showProcess        = TRUE
  ),
  designs             = list( designConstraint),
  fimType             = "population",
  outputs             = list( "RespPK" = "Cc", "RespPD" = "E" ),
  odeSolverParameters = list( atol = 1e-8, rtol = 1e-8 )
)
```

```{r mult-run, echo=TRUE, eval=FALSE, comment=''}
optimizationMultResults = run( optimizationMult )
```

### Results

```{r mult-show, echo=TRUE, eval=FALSE, comment=''}
show(optimizationMultResults)
```


```{r results-optiAlgoMult-popFIM_from_outputs, echo = FALSE,  eval=TRUE, comment=''} 
lines = readLines("outputs/vignette1_optimization_MultiplicativeAlgorithm_populationFIM_show.txt")
cat(paste(lines, collapse = "\n"))
```

```{r mult-detail}
cat("Fisher Information Matrix")
print(getFisherMatrix(optimizationMultResults))

cat("Correlation Matrix")
print(getCorrelationMatrix(optimizationMultResults))

cat("Standard Errors (SE)")
print(getSE(optimizationMultResults))

cat("Relative Standard Errors")
print(getRSE(optimizationMultResults))

cat("Shrinkage (%)")
print(getShrinkage(optimizationMultResults))

cat("Determinant")
print(getDeterminant(optimizationMultResults))

cat("D-Criterion")
print(getDcriterion(optimizationMultResults))
```

### Arm weights

```{r algoMul-show, echo=TRUE, eval=FALSE, comment=''}
# plot() on an Optimization object -- $weights is specific to MultiplicativeAlgorithm:
# final weight per candidate protocol; non-zero bars define the design support.
plotsMult = plot(optimizationMultResults,
                 plotOptions = plotOptions,
                 which       = c("evaluation", "SE", "RSE", "weights"))

print( plotsMult$weights )
```

```{r mult-plot, out.width="50%", echo = FALSE, eval= TRUE,  comment='', fig.align = "center",fig.cap="Weights"} 
knitr::include_graphics("figures/vignette1_optimization_MultiplicativeAlgorithm_populationFIM_weights.png")
```

### HTML Report

```{r mult-report, eval=FALSE, echo=TRUE, comment=''}
#Define your path to save your report: pathsReports ="C:/...
Report(optimizationMultResults,pathsReports ,"vignette1_optimization_Multiplicative_populationFIM.html",plotOptions)
```

---

# References

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