Design Evaluation and Optimization in Discrete Space

Scientific Background

This example is based on a landmark PK/PD population study of Tolmetin (a non-steroidal anti-inflammatory drug) in rats (Flores-Murrieta et al. 1998). 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.
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 
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 – 
errorModelRespPK = Combined1(output = "RespPK", sigmaInter = 0,   sigmaSlope = 0.21)
errorModelRespPD = Constant(output = "RespPD", sigmaInter = 9.6)
modelError       = list(errorModelRespPK, errorModelRespPD)

Sampling Times

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 \(E(0) = 100 = R_{in}/k_{out}\)

# 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

design1 = Design( name = "design1", 
                  arms = list( arm1, arm2, arm3, arm4, arm5, arm6 ) )

Evaluation of the population and Bayesian FIM

Population FIM Evaluation

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

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

show(evaluationPopResults)

*************************************** 
 Population Fisher Matrix 
*************************************** 

                        μ_V          μ_Cl      μ_kout      μ_Imax      μ_IC50     μ_gamma         ω²_V        ω²_Cl      ω²_kout      ω²_Imax    ω²_IC50   ω²_gamma σ_slope_RespPK σ_inter_RespPD
μ_V            583.89694102    9.19036099 -0.02315464 -3.35612090  0.32079096 -0.24507445 0.000000e+00 0.000000e+00 0.000000e+00  0.000000000  0.0000000 0.00000000   0.000000e+00    0.000000000
μ_Cl             9.19036099 2110.23408215 -0.01715815 -0.03198433  0.52533166  0.35319017 0.000000e+00 0.000000e+00 0.000000e+00  0.000000000  0.0000000 0.00000000   0.000000e+00    0.000000000
μ_kout          -0.02315464   -0.01715815  1.04306085  0.63336569 -0.03141002 -0.04209541 0.000000e+00 0.000000e+00 0.000000e+00  0.000000000  0.0000000 0.00000000   0.000000e+00    0.000000000
μ_Imax          -3.35612090   -0.03198433  0.63336569 60.63827631 -2.70317529  1.54206818 0.000000e+00 0.000000e+00 0.000000e+00  0.000000000  0.0000000 0.00000000   0.000000e+00    0.000000000
μ_IC50           0.32079096    0.52533166 -0.03141002 -2.70317529  0.65184243  0.07044445 0.000000e+00 0.000000e+00 0.000000e+00  0.000000000  0.0000000 0.00000000   0.000000e+00    0.000000000
μ_gamma         -0.24507445    0.35319017 -0.04209541  1.54206818  0.07044445  0.74226627 0.000000e+00 0.000000e+00 0.000000e+00  0.000000000  0.0000000 0.00000000   0.000000e+00    0.000000000
ω²_V             0.00000000    0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 1.419936e+03 5.071130e-02 5.312959e-04  0.174193007  0.4767726 0.01121281   1.694963e+02    0.022484025
ω²_Cl            0.00000000    0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 5.071130e-02 3.801555e+02 9.340249e-05  0.003376237  0.1153787 0.00215635   3.442924e+01    0.004590882
ω²_kout          0.00000000    0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 5.312959e-04 9.340249e-05 2.149292e+01  0.386744465  0.1693550 0.01150713   8.462700e-03    0.050138553
ω²_Imax          0.00000000    0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 1.741930e-01 3.376237e-03 3.867445e-01 41.669312017 16.1932689 0.55767351   1.413532e+00    0.818888973
ω²_IC50          0.00000000    0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 4.767726e-01 1.153787e-01 1.693550e-01 16.193268935 83.8259799 0.15388072   8.155428e+00    1.229640291
ω²_gamma         0.00000000    0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 1.121281e-02 2.156350e-03 1.150713e-02  0.557673507  0.1538807 0.69850714   1.549203e-01    0.107175844
σ_slope_RespPK   0.00000000    0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 1.694963e+02 3.442924e+01 8.462700e-03  1.413531724  8.1554276 0.15492032   1.146177e+04    0.328767259
σ_inter_RespPD   0.00000000    0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 2.248402e-02 4.590882e-03 5.013855e-02  0.818888973  1.2296403 0.10717584   3.287673e-01    5.297969806

*************************************** 
 Fixed effects 
*************************************** 

                 μ_V          μ_Cl      μ_kout      μ_Imax      μ_IC50     μ_gamma
μ_V     583.89694102    9.19036099 -0.02315464 -3.35612090  0.32079096 -0.24507445
μ_Cl      9.19036099 2110.23408215 -0.01715815 -0.03198433  0.52533166  0.35319017
μ_kout   -0.02315464   -0.01715815  1.04306085  0.63336569 -0.03141002 -0.04209541
μ_Imax   -3.35612090   -0.03198433  0.63336569 60.63827631 -2.70317529  1.54206818
μ_IC50    0.32079096    0.52533166 -0.03141002 -2.70317529  0.65184243  0.07044445
μ_gamma  -0.24507445    0.35319017 -0.04209541  1.54206818  0.07044445  0.74226627

*************************************** 
 Variance components 
*************************************** 

                       ω²_V        ω²_Cl      ω²_kout      ω²_Imax    ω²_IC50   ω²_gamma σ_slope_RespPK σ_inter_RespPD
ω²_V           1.419936e+03 5.071130e-02 5.312959e-04  0.174193007  0.4767726 0.01121281   1.694963e+02    0.022484025
ω²_Cl          5.071130e-02 3.801555e+02 9.340249e-05  0.003376237  0.1153787 0.00215635   3.442924e+01    0.004590882
ω²_kout        5.312959e-04 9.340249e-05 2.149292e+01  0.386744465  0.1693550 0.01150713   8.462700e-03    0.050138553
ω²_Imax        1.741930e-01 3.376237e-03 3.867445e-01 41.669312017 16.1932689 0.55767351   1.413532e+00    0.818888973
ω²_IC50        4.767726e-01 1.153787e-01 1.693550e-01 16.193268935 83.8259799 0.15388072   8.155428e+00    1.229640291
ω²_gamma       1.121281e-02 2.156350e-03 1.150713e-02  0.557673507  0.1538807 0.69850714   1.549203e-01    0.107175844
σ_slope_RespPK 1.694963e+02 3.442924e+01 8.462700e-03  1.413531724  8.1554276 0.15492032   1.146177e+04    0.328767259
σ_inter_RespPD 2.248402e-02 4.590882e-03 5.013855e-02  0.818888973  1.2296403 0.10717584   3.287673e-01    5.297969806

********************************************* 
 Determinant, condition numbers and D-criterion  
*********************************************** 

Determinant: 4.24689711411024e+22 
D-criterion: 41.332420450232 
Condition number of the fixed effects: 4677.28366293334 
Condition number of the random effects: 16644.2310751313 

*************************************** 
 Parameters estimation 
*************************************** 

               parametersValues          SE       RSE
μ_V                    0.740000 0.041396101  5.594068
μ_Cl                   0.280000 0.021772569  7.775917
μ_kout                 6.140000 0.984617978 16.036123
μ_Imax                 0.760000 0.149908136 19.724755
μ_IC50                 9.220000 1.409222187 15.284406
μ_gamma                2.770000 1.227830929 44.326026
ω²_V                   0.099856 0.026561314 26.599617
ω²_Cl                  0.207936 0.051295421 24.668851
ω²_kout                0.896809 0.215721035 24.054290
ω²_Imax                0.192721 0.162032545 84.076227
ω²_IC50                0.204304 0.113693973 55.649411
ω²_gamma               3.101121 1.204551613 38.842458
σ_slope_RespPK         0.210000 0.009350449  4.452595
σ_inter_RespPD         9.600000 0.436125367  4.542973

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

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

show(evaluationBayesianResults)

*************************************** 
 Bayesian Fisher Matrix 
*************************************** 

                μ_V       μ_Cl    μ_kout    μ_Imax      μ_IC50     μ_gamma
μ_V      732.518441  108.32260 -314.8704 -202.6583  193.580756   -9.312555
μ_Cl     108.322602 1119.42551 -300.7189 -139.9528  172.742664   17.150077
μ_kout  -314.870389 -300.71887 4069.5078  539.1362 -615.589231 -108.750485
μ_Imax  -202.658307 -139.95280  539.1362  552.9046 -342.611104   66.745296
μ_IC50   193.580756  172.74266 -615.5892 -342.6111  366.668883    7.837506
μ_gamma   -9.312555   17.15008 -108.7505   66.7453    7.837506   46.714749

*************************************** 
 Fixed effects 
*************************************** 

                μ_V       μ_Cl    μ_kout    μ_Imax      μ_IC50     μ_gamma
μ_V      732.518441  108.32260 -314.8704 -202.6583  193.580756   -9.312555
μ_Cl     108.322602 1119.42551 -300.7189 -139.9528  172.742664   17.150077
μ_kout  -314.870389 -300.71887 4069.5078  539.1362 -615.589231 -108.750485
μ_Imax  -202.658307 -139.95280  539.1362  552.9046 -342.611104   66.745296
μ_IC50   193.580756  172.74266 -615.5892 -342.6111  366.668883    7.837506
μ_gamma   -9.312555   17.15008 -108.7505   66.7453    7.837506   46.714749

*********************************************** 
 Determinant, condition numbers and D-criterion 
*********************************************** 

Determinant: 3455702611551547 
D-criterion: 388.82668275192 
Condition number of the fixed effects: 255.011768254495 

*************************************** 
 Shrinkage 
*************************************** 

           Shrinkage
μ_V     1.905362e+01
μ_Cl    3.753865e+01
μ_kout  3.208707e-03
μ_Imax  1.000000e+02
μ_IC50  9.999902e+01
μ_gamma 9.998219e+01

*************************************** 
 Parameters estimation 
*************************************** 

        parametersValues         SE        RSE
μ_V                 0.74 0.03997149  5.4015531
μ_Cl                0.28 0.03110851 11.1101808
μ_kout              6.14 0.01916800  0.3121824
μ_Imax              0.76 0.09527373 12.5360173
μ_IC50              9.22 0.10815373  1.1730339
μ_gamma             2.77 0.22066132  7.9661127
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

# 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"]])
PK response profile -- 20.00 mg arm (population FIM)

PK response profile – 20.00 mg arm (population FIM) 

print(evalPlots$evaluation[["design1"]][["20.00mg Arm"]][["RespPD"]])
PD response profile -- 20.00 mg arm (population FIM)

PD response profile – 20.00 mg arm (population FIM)

Sensitivity indices

# $sensitivityIndices is computed in the same plot() call above
print(evalPlots$sensitivityIndices[["design1"]][["20.00mg Arm"]][["RespPK"]][["Cl"]])
Sensitivity index for Cl -- RespPK, 20.00 mg arm

Sensitivity index for Cl – RespPK, 20.00 mg arm

Standard errors and Relative Standard errors

print(evalPlots$SE)
Standard Errors (SE)

Standard Errors (SE) 

print(evalPlots$RSE)
Relative Standard Errors (RSE %)

Relative Standard Errors (RSE %)

HTML Report

#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:

Shared Optimization Setup

#  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

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)
)
optimizationFWResults = run(optimizationFW)

Results

show(optimizationFWResults)

===================================== 
  Initial design 
===================================== 

      Arms name Number of subjects Outcome Dose                    Sampling times
1 armConstraint                 30  RespPK 6.32     (0.25, 0.75, 1, 1.5, 2, 4, 6)
2 armConstraint                 30  RespPD    . (0.25, 0.75, 1.5, 2, 3, 6, 8, 12)

*************************************** 
 Population Fisher Matrix 
*************************************** 

                         μ_V          μ_Cl        μ_kout        μ_Imax       μ_IC50     μ_gamma         ω²_V        ω²_Cl      ω²_kout      ω²_Imax      ω²_IC50     ω²_gamma σ_slope_RespPK
μ_V            483.943136181 -2.515935e+01 -0.0021215242 -10.359410838  1.436756981  0.01963642 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_Cl           -25.159351796  1.783803e+03 -0.0006308506  -3.299783812  0.494965586  0.28030930 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_kout          -0.002121524 -6.308506e-04  0.8845243295   0.009428532 -0.001235346 -0.01067545 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_Imax         -10.359410838 -3.299784e+00  0.0094285322  44.462907428 -6.214531153 -0.71598124 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_IC50           1.436756981  4.949656e-01 -0.0012353465  -6.214531153  0.892950381  0.23982726 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_gamma          0.019636419  2.803093e-01 -0.0106754464  -0.715981240  0.239827262  1.05954052 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
ω²_V             0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 1.170481e+03 4.529259e-01 1.548620e-06 5.657304e-01 1.601551e+00 2.700202e-05   1.414551e+02
ω²_Cl            0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 4.529259e-01 3.259676e+02 1.960445e-08 8.217940e-03 2.721302e-02 7.877698e-04   1.987900e+01
ω²_kout          0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 1.548620e-06 1.960445e-08 1.853283e+01 3.226271e-05 8.151263e-05 5.494360e-04   1.063131e-05
ω²_Imax          0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 5.657304e-01 8.217940e-03 3.226271e-05 1.099256e+01 3.160491e+01 3.786507e-02   3.861180e+00
ω²_IC50          0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 1.601551e+00 2.721302e-02 8.151263e-05 3.160491e+01 9.603446e+01 6.252695e-01   1.087403e+01
ω²_gamma         0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 2.700202e-05 7.877698e-04 5.494360e-04 3.786507e-02 6.252695e-01 1.101547e+00   5.667870e-02
σ_slope_RespPK   0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 1.414551e+02 1.987900e+01 1.063131e-05 3.861180e+00 1.087403e+01 5.667870e-02   6.839433e+03
σ_inter_RespPD   0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 5.017060e-02 5.310514e-04 1.061828e-02 7.675634e-01 1.874276e+00 1.056648e-01   4.046508e-01
               σ_inter_RespPD
μ_V              0.0000000000
μ_Cl             0.0000000000
μ_kout           0.0000000000
μ_Imax           0.0000000000
μ_IC50           0.0000000000
μ_gamma          0.0000000000
ω²_V             0.0501706038
ω²_Cl            0.0005310514
ω²_kout          0.0106182763
ω²_Imax          0.7675634046
ω²_IC50          1.8742757046
ω²_gamma         0.1056648445
σ_slope_RespPK   0.4046508170
σ_inter_RespPD   3.3502933639

*************************************** 
 Fixed effects 
*************************************** 

                  μ_V          μ_Cl        μ_kout        μ_Imax       μ_IC50     μ_gamma
μ_V     483.943136181 -2.515935e+01 -0.0021215242 -10.359410838  1.436756981  0.01963642
μ_Cl    -25.159351796  1.783803e+03 -0.0006308506  -3.299783812  0.494965586  0.28030930
μ_kout   -0.002121524 -6.308506e-04  0.8845243295   0.009428532 -0.001235346 -0.01067545
μ_Imax  -10.359410838 -3.299784e+00  0.0094285322  44.462907428 -6.214531153 -0.71598124
μ_IC50    1.436756981  4.949656e-01 -0.0012353465  -6.214531153  0.892950381  0.23982726
μ_gamma   0.019636419  2.803093e-01 -0.0106754464  -0.715981240  0.239827262  1.05954052

*************************************** 
 Variance components 
*************************************** 

                       ω²_V        ω²_Cl      ω²_kout      ω²_Imax      ω²_IC50     ω²_gamma σ_slope_RespPK σ_inter_RespPD
ω²_V           1.170481e+03 4.529259e-01 1.548620e-06 5.657304e-01 1.601551e+00 2.700202e-05   1.414551e+02   0.0501706038
ω²_Cl          4.529259e-01 3.259676e+02 1.960445e-08 8.217940e-03 2.721302e-02 7.877698e-04   1.987900e+01   0.0005310514
ω²_kout        1.548620e-06 1.960445e-08 1.853283e+01 3.226271e-05 8.151263e-05 5.494360e-04   1.063131e-05   0.0106182763
ω²_Imax        5.657304e-01 8.217940e-03 3.226271e-05 1.099256e+01 3.160491e+01 3.786507e-02   3.861180e+00   0.7675634046
ω²_IC50        1.601551e+00 2.721302e-02 8.151263e-05 3.160491e+01 9.603446e+01 6.252695e-01   1.087403e+01   1.8742757046
ω²_gamma       2.700202e-05 7.877698e-04 5.494360e-04 3.786507e-02 6.252695e-01 1.101547e+00   5.667870e-02   0.1056648445
σ_slope_RespPK 1.414551e+02 1.987900e+01 1.063131e-05 3.861180e+00 1.087403e+01 5.667870e-02   6.839433e+03   0.4046508170
σ_inter_RespPD 5.017060e-02 5.310514e-04 1.061828e-02 7.675634e-01 1.874276e+00 1.056648e-01   4.046508e-01   3.3502933639

********************************************* 
 Determinant, condition numbers and D-criterion  
*********************************************** 

Determinant: 1892186514468350720 
D-criterion: 20.2067944869215 
Condition number of the fixed effects: 323832.174335363 
Condition number of the random effects: 14200.0226338072 

*************************************** 
 Parameters estimation 
*************************************** 

               parametersValues          SE        RSE
μ_V                    0.740000  0.04559038   6.160862
μ_Cl                   0.280000  0.02368822   8.460077
μ_kout                 6.140000  1.06357081  17.322000
μ_Imax                 0.760000  1.82700380 240.395237
μ_IC50                 9.220000 13.23044351 143.497218
μ_gamma                2.770000  2.01694542  72.813914
ω²_V                   0.099856  0.02926613  29.308334
ω²_Cl                  0.207936  0.05539254  26.639224
ω²_kout                0.896809  0.23228946  25.901776
ω²_Imax                0.192721  1.33935341 694.970141
ω²_IC50                0.204304  0.45266182 221.562877
ω²_gamma               3.101121  0.97863817  31.557562
σ_slope_RespPK         0.210000  0.01210916   5.766266
σ_inter_RespPD         9.600000  0.55405294   5.771385

===================================== 
  Optimal design 
===================================== 

  Arms name Number of subjects Outcome  Dose     Sampling times
1      Arm1               9.97  RespPK    20 (0.25, 0.75, 4, 6)
2      Arm1               9.97  RespPD     .   (0.75, 2, 6, 12)
3      Arm2               13.3  RespPK    20 (0.25, 0.75, 4, 6)
4      Arm2               13.3  RespPD     .    (0.75, 2, 3, 6)
5      Arm3               6.73  RespPK 11.24 (0.25, 0.75, 4, 6)
6      Arm3               6.73  RespPD     .    (0.75, 2, 3, 6)

*************************************** 
 Population Fisher Matrix 
*************************************** 

                        μ_V          μ_Cl      μ_kout      μ_Imax      μ_IC50      μ_gamma         ω²_V        ω²_Cl      ω²_kout      ω²_Imax      ω²_IC50     ω²_gamma σ_slope_RespPK σ_inter_RespPD
μ_V            442.42767946 -5.722735e+01 -0.01283957  -6.7730365  1.15398862 -0.405928724 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_Cl           -57.22735145  1.757798e+03 -0.04702067  -4.7644365  1.54590129  0.005111016 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_kout          -0.01283957 -4.702067e-02  0.87179742   0.7062229 -0.02945391 -0.044150390 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_Imax          -6.77303648 -4.764437e+00  0.70622291 151.9673571 -3.40928835  4.462377068 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_IC50           1.15398862  1.545901e+00 -0.02945391  -3.4092884  1.02517615 -0.028420509 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_gamma         -0.40592872  5.111016e-03 -0.04415039   4.4623771 -0.02842051  0.955210290 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
ω²_V             0.00000000  0.000000e+00  0.00000000   0.0000000  0.00000000  0.000000000 9.782766e+02 2.343473e+00 6.785969e-05   0.36834443   1.11109049 0.0120839481   2.147356e+02    0.016637130
ω²_Cl            0.00000000  0.000000e+00  0.00000000   0.0000000  0.00000000  0.000000000 2.343473e+00 3.165327e+02 1.231163e-04   0.02799261   0.27418887 0.0001660741   2.632674e+01    0.005574841
ω²_kout          0.00000000  0.000000e+00  0.00000000   0.0000000  0.00000000  0.000000000 6.785969e-05 1.231163e-04 1.800493e+01   0.24293648   0.05155276 0.0124677850   6.828079e-03    0.039470891
ω²_Imax          0.00000000  0.000000e+00  0.00000000   0.0000000  0.00000000  0.000000000 3.683444e-01 2.799261e-02 2.429365e-01 134.21483469  15.34076803 1.6048142318   1.525019e+00    1.977219903
ω²_IC50          0.00000000  0.000000e+00  0.00000000   0.0000000  0.00000000  0.000000000 1.111090e+00 2.741889e-01 5.155276e-02  15.34076803 130.36343842 0.0863674220   1.088414e+01    2.574826139
ω²_gamma         0.00000000  0.000000e+00  0.00000000   0.0000000  0.00000000  0.000000000 1.208395e-02 1.660741e-04 1.246779e-02   1.60481423   0.08636742 0.9052593492   8.729990e-02    0.105996297
σ_slope_RespPK   0.00000000  0.000000e+00  0.00000000   0.0000000  0.00000000  0.000000000 2.147356e+02 2.632674e+01 6.828079e-03   1.52501867  10.88414156 0.0872999001   2.795737e+03    0.232692783
σ_inter_RespPD   0.00000000  0.000000e+00  0.00000000   0.0000000  0.00000000  0.000000000 1.663713e-02 5.574841e-03 3.947089e-02   1.97721990   2.57482614 0.1059962969   2.326928e-01    0.426249789

*************************************** 
 Fixed effects 
*************************************** 

                 μ_V          μ_Cl      μ_kout      μ_Imax      μ_IC50      μ_gamma
μ_V     442.42767946 -5.722735e+01 -0.01283957  -6.7730365  1.15398862 -0.405928724
μ_Cl    -57.22735145  1.757798e+03 -0.04702067  -4.7644365  1.54590129  0.005111016
μ_kout   -0.01283957 -4.702067e-02  0.87179742   0.7062229 -0.02945391 -0.044150390
μ_Imax   -6.77303648 -4.764437e+00  0.70622291 151.9673571 -3.40928835  4.462377068
μ_IC50    1.15398862  1.545901e+00 -0.02945391  -3.4092884  1.02517615 -0.028420509
μ_gamma  -0.40592872  5.111016e-03 -0.04415039   4.4623771 -0.02842051  0.955210290

*************************************** 
 Variance components 
*************************************** 

                       ω²_V        ω²_Cl      ω²_kout      ω²_Imax      ω²_IC50     ω²_gamma σ_slope_RespPK σ_inter_RespPD
ω²_V           9.782766e+02 2.343473e+00 6.785969e-05   0.36834443   1.11109049 0.0120839481   2.147356e+02    0.016637130
ω²_Cl          2.343473e+00 3.165327e+02 1.231163e-04   0.02799261   0.27418887 0.0001660741   2.632674e+01    0.005574841
ω²_kout        6.785969e-05 1.231163e-04 1.800493e+01   0.24293648   0.05155276 0.0124677850   6.828079e-03    0.039470891
ω²_Imax        3.683444e-01 2.799261e-02 2.429365e-01 134.21483469  15.34076803 1.6048142318   1.525019e+00    1.977219903
ω²_IC50        1.111090e+00 2.741889e-01 5.155276e-02  15.34076803 130.36343842 0.0863674220   1.088414e+01    2.574826139
ω²_gamma       1.208395e-02 1.660741e-04 1.246779e-02   1.60481423   0.08636742 0.9052593492   8.729990e-02    0.105996297
σ_slope_RespPK 2.147356e+02 2.632674e+01 6.828079e-03   1.52501867  10.88414156 0.0872999001   2.795737e+03    0.232692783
σ_inter_RespPD 1.663713e-02 5.574841e-03 3.947089e-02   1.97721990   2.57482614 0.1059962969   2.326928e-01    0.426249789

********************************************* 
 Determinant, condition numbers and D-criterion  
*********************************************** 

Determinant: 6.36420159165367e+21 
D-criterion: 36.0919444747579 
Condition number of the fixed effects: 2312.37488361135 
Condition number of the random effects: 8272.56086431556 

*************************************** 
 Parameters estimation 
*************************************** 

               parametersValues         SE       RSE
μ_V                    0.740000 0.04772793  6.449720
μ_Cl                   0.280000 0.02392109  8.543246
μ_kout                 6.140000 1.07624035 17.528345
μ_Imax                 0.760000 0.09133501 12.017765
μ_IC50                 9.220000 1.03216841 11.194885
μ_gamma                2.770000 1.10845621 40.016470
ω²_V                   0.099856 0.03224496 32.291455
ω²_Cl                  0.207936 0.05622909 27.041537
ω²_kout                0.896809 0.23569615 26.281644
ω²_Imax                0.192721 0.08998922 46.694038
ω²_IC50                0.204304 0.09353945 45.784443
ω²_gamma               3.101121 1.07457623 34.651219
σ_slope_RespPK         0.210000 0.01908444  9.087828
σ_inter_RespPD         9.600000 1.69963286 17.704509
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

#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.

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 )
)
optimizationMultResults = run( optimizationMult )

Results

show(optimizationMultResults)

===================================== 
  Initial design 
===================================== 

      Arms name Number of subjects Outcome Dose                    Sampling times
1 armConstraint                 30  RespPK 6.32     (0.25, 0.75, 1, 1.5, 2, 4, 6)
2 armConstraint                 30  RespPD    . (0.25, 0.75, 1.5, 2, 3, 6, 8, 12)

*************************************** 
 Population Fisher Matrix 
*************************************** 

                         μ_V          μ_Cl        μ_kout        μ_Imax       μ_IC50     μ_gamma         ω²_V        ω²_Cl      ω²_kout      ω²_Imax      ω²_IC50     ω²_gamma σ_slope_RespPK
μ_V            483.943136181 -2.515935e+01 -0.0021215242 -10.359410838  1.436756981  0.01963642 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_Cl           -25.159351796  1.783803e+03 -0.0006308506  -3.299783812  0.494965586  0.28030930 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_kout          -0.002121524 -6.308506e-04  0.8845243295   0.009428532 -0.001235346 -0.01067545 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_Imax         -10.359410838 -3.299784e+00  0.0094285322  44.462907428 -6.214531153 -0.71598124 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_IC50           1.436756981  4.949656e-01 -0.0012353465  -6.214531153  0.892950381  0.23982726 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
μ_gamma          0.019636419  2.803093e-01 -0.0106754464  -0.715981240  0.239827262  1.05954052 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00   0.000000e+00
ω²_V             0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 1.170481e+03 4.529259e-01 1.548620e-06 5.657304e-01 1.601551e+00 2.700202e-05   1.414551e+02
ω²_Cl            0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 4.529259e-01 3.259676e+02 1.960445e-08 8.217940e-03 2.721302e-02 7.877698e-04   1.987900e+01
ω²_kout          0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 1.548620e-06 1.960445e-08 1.853283e+01 3.226271e-05 8.151263e-05 5.494360e-04   1.063131e-05
ω²_Imax          0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 5.657304e-01 8.217940e-03 3.226271e-05 1.099256e+01 3.160491e+01 3.786507e-02   3.861180e+00
ω²_IC50          0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 1.601551e+00 2.721302e-02 8.151263e-05 3.160491e+01 9.603446e+01 6.252695e-01   1.087403e+01
ω²_gamma         0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 2.700202e-05 7.877698e-04 5.494360e-04 3.786507e-02 6.252695e-01 1.101547e+00   5.667870e-02
σ_slope_RespPK   0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 1.414551e+02 1.987900e+01 1.063131e-05 3.861180e+00 1.087403e+01 5.667870e-02   6.839433e+03
σ_inter_RespPD   0.000000000  0.000000e+00  0.0000000000   0.000000000  0.000000000  0.00000000 5.017060e-02 5.310514e-04 1.061828e-02 7.675634e-01 1.874276e+00 1.056648e-01   4.046508e-01
               σ_inter_RespPD
μ_V              0.0000000000
μ_Cl             0.0000000000
μ_kout           0.0000000000
μ_Imax           0.0000000000
μ_IC50           0.0000000000
μ_gamma          0.0000000000
ω²_V             0.0501706038
ω²_Cl            0.0005310514
ω²_kout          0.0106182763
ω²_Imax          0.7675634046
ω²_IC50          1.8742757046
ω²_gamma         0.1056648445
σ_slope_RespPK   0.4046508170
σ_inter_RespPD   3.3502933639

*************************************** 
 Fixed effects 
*************************************** 

                  μ_V          μ_Cl        μ_kout        μ_Imax       μ_IC50     μ_gamma
μ_V     483.943136181 -2.515935e+01 -0.0021215242 -10.359410838  1.436756981  0.01963642
μ_Cl    -25.159351796  1.783803e+03 -0.0006308506  -3.299783812  0.494965586  0.28030930
μ_kout   -0.002121524 -6.308506e-04  0.8845243295   0.009428532 -0.001235346 -0.01067545
μ_Imax  -10.359410838 -3.299784e+00  0.0094285322  44.462907428 -6.214531153 -0.71598124
μ_IC50    1.436756981  4.949656e-01 -0.0012353465  -6.214531153  0.892950381  0.23982726
μ_gamma   0.019636419  2.803093e-01 -0.0106754464  -0.715981240  0.239827262  1.05954052

*************************************** 
 Variance components 
*************************************** 

                       ω²_V        ω²_Cl      ω²_kout      ω²_Imax      ω²_IC50     ω²_gamma σ_slope_RespPK σ_inter_RespPD
ω²_V           1.170481e+03 4.529259e-01 1.548620e-06 5.657304e-01 1.601551e+00 2.700202e-05   1.414551e+02   0.0501706038
ω²_Cl          4.529259e-01 3.259676e+02 1.960445e-08 8.217940e-03 2.721302e-02 7.877698e-04   1.987900e+01   0.0005310514
ω²_kout        1.548620e-06 1.960445e-08 1.853283e+01 3.226271e-05 8.151263e-05 5.494360e-04   1.063131e-05   0.0106182763
ω²_Imax        5.657304e-01 8.217940e-03 3.226271e-05 1.099256e+01 3.160491e+01 3.786507e-02   3.861180e+00   0.7675634046
ω²_IC50        1.601551e+00 2.721302e-02 8.151263e-05 3.160491e+01 9.603446e+01 6.252695e-01   1.087403e+01   1.8742757046
ω²_gamma       2.700202e-05 7.877698e-04 5.494360e-04 3.786507e-02 6.252695e-01 1.101547e+00   5.667870e-02   0.1056648445
σ_slope_RespPK 1.414551e+02 1.987900e+01 1.063131e-05 3.861180e+00 1.087403e+01 5.667870e-02   6.839433e+03   0.4046508170
σ_inter_RespPD 5.017060e-02 5.310514e-04 1.061828e-02 7.675634e-01 1.874276e+00 1.056648e-01   4.046508e-01   3.3502933639

********************************************* 
 Determinant, condition numbers and D-criterion  
*********************************************** 

Determinant: 1892186514468350720 
D-criterion: 20.2067944869215 
Condition number of the fixed effects: 323832.174335363 
Condition number of the random effects: 14200.0226338072 

*************************************** 
 Parameters estimation 
*************************************** 

               parametersValues          SE        RSE
μ_V                    0.740000  0.04559038   6.160862
μ_Cl                   0.280000  0.02368822   8.460077
μ_kout                 6.140000  1.06357081  17.322000
μ_Imax                 0.760000  1.82700380 240.395237
μ_IC50                 9.220000 13.23044351 143.497218
μ_gamma                2.770000  2.01694542  72.813914
ω²_V                   0.099856  0.02926613  29.308334
ω²_Cl                  0.207936  0.05539254  26.639224
ω²_kout                0.896809  0.23228946  25.901776
ω²_Imax                0.192721  1.33935341 694.970141
ω²_IC50                0.204304  0.45266182 221.562877
ω²_gamma               3.101121  0.97863817  31.557562
σ_slope_RespPK         0.210000  0.01210916   5.766266
σ_inter_RespPD         9.600000  0.55405294   5.771385

===================================== 
  Optimal design 
===================================== 

   Arms name Number of subjects Outcome  Dose     Sampling times
1     Arm667               0.64  RespPK 11.24    (0.25, 1, 4, 6)
2     Arm667               0.64  RespPD     .    (0.75, 2, 3, 6)
3     Arm837               0.88  RespPK    20    (0.25, 1, 4, 6)
4     Arm837               0.88  RespPD     .   (0.75, 2, 6, 12)
5     Arm817               1.77  RespPK    20    (0.25, 1, 4, 6)
6     Arm817               1.77  RespPD     .    (0.75, 2, 3, 6)
7     Arm664               5.82  RespPK 11.24 (0.25, 0.75, 4, 6)
8     Arm664               5.82  RespPD     .    (0.75, 2, 3, 6)
9     Arm834               7.18  RespPK    20 (0.25, 0.75, 4, 6)
10    Arm834               7.18  RespPD     .   (0.75, 2, 6, 12)
11    Arm814              13.71  RespPK    20 (0.25, 0.75, 4, 6)
12    Arm814              13.71  RespPD     .    (0.75, 2, 3, 6)

*************************************** 
 Population Fisher Matrix 
*************************************** 

                        μ_V          μ_Cl      μ_kout     μ_Imax      μ_IC50      μ_gamma         ω²_V        ω²_Cl      ω²_kout      ω²_Imax      ω²_IC50     ω²_gamma σ_slope_RespPK σ_inter_RespPD
μ_V            441.94766215 -5.733100e+01 -0.01236585  -6.688188  1.17545168 -0.403675305 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_Cl           -57.33099676  1.757901e+03 -0.04910061  -4.563042  1.58876877  0.005979464 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_kout          -0.01236585 -4.910061e-02  0.87055103   0.767240 -0.03072718 -0.047741108 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_Imax          -6.68818843 -4.563042e+00  0.76723997 149.841076 -3.27482827  4.643952232 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_IC50           1.17545168  1.588769e+00 -0.03072718  -3.274828  1.05291300 -0.026930370 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
μ_gamma         -0.40367531  5.979464e-03 -0.04774111   4.643952 -0.02693037  0.945572176 0.000000e+00 0.000000e+00 0.000000e+00   0.00000000   0.00000000 0.0000000000   0.000000e+00    0.000000000
ω²_V             0.00000000  0.000000e+00  0.00000000   0.000000  0.00000000  0.000000000 9.761645e+02 2.352061e+00 6.426557e-05   0.35962926   1.14226753 0.0119071393   2.154440e+02    0.016465616
ω²_Cl            0.00000000  0.000000e+00  0.00000000   0.000000  0.00000000  0.000000000 2.352061e+00 3.165700e+02 1.333508e-04   0.02662137   0.28893220 0.0001551839   2.626014e+01    0.005516857
ω²_kout          0.00000000  0.000000e+00  0.00000000   0.000000  0.00000000  0.000000000 6.426557e-05 1.333508e-04 1.795348e+01   0.27612949   0.05581461 0.0141175668   7.505416e-03    0.041594169
ω²_Imax          0.00000000  0.000000e+00  0.00000000   0.000000  0.00000000  0.000000000 3.596293e-01 2.662137e-02 2.761295e-01 129.99710162  14.59817254 1.7306834484   1.474160e+00    1.964070533
ω²_IC50          0.00000000  0.000000e+00  0.00000000   0.000000  0.00000000  0.000000000 1.142268e+00 2.889322e-01 5.581461e-02  14.59817254 137.15736498 0.0800695183   1.144645e+01    2.550765126
ω²_gamma         0.00000000  0.000000e+00  0.00000000   0.000000  0.00000000  0.000000000 1.190714e-02 1.551839e-04 1.411757e-02   1.73068345   0.08006952 0.8875950952   8.590151e-02    0.104646111
σ_slope_RespPK   0.00000000  0.000000e+00  0.00000000   0.000000  0.00000000  0.000000000 2.154440e+02 2.626014e+01 7.505416e-03   1.47416000  11.44645305 0.0859015137   2.796442e+03    0.230205164
σ_inter_RespPD   0.00000000  0.000000e+00  0.00000000   0.000000  0.00000000  0.000000000 1.646562e-02 5.516857e-03 4.159417e-02   1.96407053   2.55076513 0.1046461108   2.302052e-01    0.428440823

*************************************** 
 Fixed effects 
*************************************** 

                 μ_V          μ_Cl      μ_kout     μ_Imax      μ_IC50      μ_gamma
μ_V     441.94766215 -5.733100e+01 -0.01236585  -6.688188  1.17545168 -0.403675305
μ_Cl    -57.33099676  1.757901e+03 -0.04910061  -4.563042  1.58876877  0.005979464
μ_kout   -0.01236585 -4.910061e-02  0.87055103   0.767240 -0.03072718 -0.047741108
μ_Imax   -6.68818843 -4.563042e+00  0.76723997 149.841076 -3.27482827  4.643952232
μ_IC50    1.17545168  1.588769e+00 -0.03072718  -3.274828  1.05291300 -0.026930370
μ_gamma  -0.40367531  5.979464e-03 -0.04774111   4.643952 -0.02693037  0.945572176

*************************************** 
 Variance components 
*************************************** 

                       ω²_V        ω²_Cl      ω²_kout      ω²_Imax      ω²_IC50     ω²_gamma σ_slope_RespPK σ_inter_RespPD
ω²_V           9.761645e+02 2.352061e+00 6.426557e-05   0.35962926   1.14226753 0.0119071393   2.154440e+02    0.016465616
ω²_Cl          2.352061e+00 3.165700e+02 1.333508e-04   0.02662137   0.28893220 0.0001551839   2.626014e+01    0.005516857
ω²_kout        6.426557e-05 1.333508e-04 1.795348e+01   0.27612949   0.05581461 0.0141175668   7.505416e-03    0.041594169
ω²_Imax        3.596293e-01 2.662137e-02 2.761295e-01 129.99710162  14.59817254 1.7306834484   1.474160e+00    1.964070533
ω²_IC50        1.142268e+00 2.889322e-01 5.581461e-02  14.59817254 137.15736498 0.0800695183   1.144645e+01    2.550765126
ω²_gamma       1.190714e-02 1.551839e-04 1.411757e-02   1.73068345   0.08006952 0.8875950952   8.590151e-02    0.104646111
σ_slope_RespPK 2.154440e+02 2.626014e+01 7.505416e-03   1.47416000  11.44645305 0.0859015137   2.796442e+03    0.230205164
σ_inter_RespPD 1.646562e-02 5.516857e-03 4.159417e-02   1.96407053   2.55076513 0.1046461108   2.302052e-01    0.428440823

********************************************* 
 Determinant, condition numbers and D-criterion  
*********************************************** 

Determinant: 6.31315138109825e+21 
D-criterion: 36.0711877593656 
Condition number of the fixed effects: 2376.71661961602 
Condition number of the random effects: 8155.99219628966 

*************************************** 
 Parameters estimation 
*************************************** 

               parametersValues         SE       RSE
μ_V                    0.740000 0.04775536  6.453427
μ_Cl                   0.280000 0.02392115  8.543267
μ_kout                 6.140000 1.07821075 17.560436
μ_Imax                 0.760000 0.09256982 12.180240
μ_IC50                 9.220000 1.01515063 11.010311
μ_gamma                2.770000 1.12502992 40.614799
ω²_V                   0.099856 0.03228218 32.328733
ω²_Cl                  0.207936 0.05622566 27.039889
ω²_kout                0.896809 0.23603646 26.319591
ω²_Imax                0.192721 0.09164467 47.553026
ω²_IC50                0.204304 0.09073074 44.409674
ω²_gamma               3.101121 1.08695256 35.050311
σ_slope_RespPK         0.210000 0.01908354  9.087401
σ_inter_RespPD         9.600000 1.68756908 17.578845
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

# 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 )
Weights

Weights

HTML Report

#Define your path to save your report: pathsReports ="C:/...
Report(optimizationMultResults,pathsReports ,"vignette1_optimization_Multiplicative_populationFIM.html",plotOptions)

References

Flores-Murrieta, Francisco, Holly Kimko, Dora Flores-Acevedo, Francisco López-Muñoz, William Jusko, Mark Sale, and Gilberto Castañeda-Hernández. 1998. “Pharmacokinetic–Pharmacodynamic Modeling of Tolmetin Antinociceptive Effect in the Rat Using an Indirect Response Model: A Population Approach.” Journal of Pharmacokinetics and Biopharmaceutics 26 (November): 547–57.