Design evaluation
PK model user-defined
The equation corresponds to a one-compartment model with first-order
absorption, with parameters ka, V and Cl. The dose is passed via the
dose_RespPK keyword.
Model parameters
The model has three structural parameters, all log-normally distributed. Inter-individual variability (\(\omega\)) and inter-occasion variability (\(\gamma\)) are specified for each.
| Parameter | Description | \(\mu\) | \(\omega\) | \(\gamma\) | Fixed \(\mu\) | Fixed \(\omega\) |
|---|---|---|---|---|---|---|
| ka | Absorption rate constant (h\(^{-1}\)) | 1 | \(\sqrt{0.09} \approx 0.30\) | \(\sqrt{0.0225} = 0.15\) | No | No |
| V | Volume of distribution (L) | 3.5 | \(\sqrt{0.09} \approx 0.30\) | \(\sqrt{0.0225} = 0.15\) | No | No |
| Cl | Elimination clearance (L/h) | 2 | \(\sqrt{0.09} \approx 0.30\) | \(\sqrt{0.0225} = 0.15\) | No | No |
Define mu, omega and gamma for each parameter
modelParameters = list(
ModelParameter( name = "ka", distribution = LogNormal( mu = 1, omega = sqrt(0.09) ), gamma = sqrt(0.0225) ),
ModelParameter( name = "V", distribution = LogNormal( mu = 3.5, omega = sqrt(0.09) ), gamma = sqrt(0.0225) ),
ModelParameter( name = "Cl", distribution = LogNormal( mu = 2, omega = sqrt(0.09) ), gamma = sqrt(0.0225) )
)Residual error model
A constant (additive) residual error model is used, with
sigmaInter = 0.1 (variance = 0.01).
Covariates
Covariate effects are parameterised on the log scale
(modelCovariatesEquation = "exponential"), so each \(\beta\) coefficient represents the
log-ratio of the affected parameter between the non-reference and the
reference category.
| Covariate | Type | Categories | Proportions | Affected parameter | Effect (\(\beta\)) | Reference |
|---|---|---|---|---|---|---|
| Sex | Between-subject (fixed) | M / F | 50 % / 50 % | V | log(1.2) \(\approx\) 0.182 | M |
| Treatment | Within-subject (occasion) | R / T | 50 % / 50 % | Cl | log(1.1) \(\approx\) 0.095 | R |
Sex is a between-subject covariate with an
exponential effect on V. The log-ratio between female and
male typical values is log(1.2).
Define the between-subject covariate
sex = Covariate(
name = "Sex",
categories = c("M", "F"),
categoriesProportions = c(0.5, 0.5),
effects = list( "F" = c( "V" = log(1.2) ) )
)Treatment is a within-subject (occasion) covariate
following a two-sequence, two-period crossover design. The log-ratio of
clearance under treatment T relative to treatment R is
log(1.1).
Administration and sampling times
A single oral dose of 30 mg is administered at time 0. Five sampling times are specified to cover both the absorption and elimination phases.
Arm and design
A single arm of 40 subjects on the same regimen.
Population FIM evaluation
The covariate effects use an exponential parameterisation
(modelCovariatesEquation = "exponential"). The analytic
model does not require ODE solver parameters.
Evaluate the population FIM
evaluationPop = Evaluation(
name = "",
modelParameters = modelParameters,
modelCovariates = list( sex, treatment ),
modelCovariatesEquation = "exponential",
modelEquations = modelEquations,
modelError = modelError,
designs = list( design1 ),
fimType = "population",
outputs = list( "RespPK" = "RespPK" )
)
evaluationPopFIM = run( evaluationPop )Display the population FIM
***************************************
Population Fisher Matrix
***************************************
μ_ka μ_V μ_Cl β_V_Sex_F β_Cl_Treatment_T ω²_ka ω²_V ω²_Cl γ²_ka γ²_V γ²_Cl σ_inter_RespPK
μ_ka 339.888866 -12.1244029 2.2403232 -17.9762700 1.8294873 0.0000000 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.00000
μ_V -12.124403 29.3831440 0.5085979 52.3319426 0.4561115 0.0000000 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.00000
μ_Cl 2.240323 0.5085979 98.2790307 0.8587853 98.1435338 0.0000000 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.00000
β_V_Sex_F -17.976270 52.3319426 0.8587853 183.1617992 0.8270805 0.0000000 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.00000
β_Cl_Treatment_T 1.829487 0.4561115 98.1435338 0.8270805 953.4932901 0.0000000 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.00000
ω²_ka 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 1444.4800032 2.303490e+01 2.509671e-01 7.250348e+02 1.529776e+01 1.301725e-01 423.70188
ω²_V 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 23.0348957 1.620000e+03 1.586320e-01 1.385535e+01 8.131879e+02 8.052141e-02 269.48022
ω²_Cl 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.2509671 1.586320e-01 1.931754e+03 1.290869e-01 9.016087e-02 9.658788e+02 18.52264
γ²_ka 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 725.0347675 1.385535e+01 1.290869e-01 1.201459e+04 4.448247e+03 4.969634e+01 2591.78367
γ²_V 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 15.2977604 8.131879e+02 9.016087e-02 4.448247e+03 1.990898e+04 3.195887e+01 1814.53482
γ²_Cl 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.1301725 8.052141e-02 9.658788e+02 4.969634e+01 3.195887e+01 3.707617e+04 615.30442
σ_inter_RespPK 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 423.7018815 2.694802e+02 1.852264e+01 2.591784e+03 1.814535e+03 6.153044e+02 36667.05676
***************************************
Fixed effects (μ)
***************************************
μ_ka μ_V μ_Cl β_V_Sex_F β_Cl_Treatment_T
μ_ka 339.888866 -12.1244029 2.2403232 -17.9762700 1.8294873
μ_V -12.124403 29.3831440 0.5085979 52.3319426 0.4561115
μ_Cl 2.240323 0.5085979 98.2790307 0.8587853 98.1435338
β_V_Sex_F -17.976270 52.3319426 0.8587853 183.1617992 0.8270805
β_Cl_Treatment_T 1.829487 0.4561115 98.1435338 0.8270805 953.4932901
***************************************
Variance components (ω², γ², σ)
***************************************
ω²_ka ω²_V ω²_Cl γ²_ka γ²_V γ²_Cl σ_inter_RespPK
ω²_ka 1444.4800032 2.303490e+01 2.509671e-01 7.250348e+02 1.529776e+01 1.301725e-01 423.70188
ω²_V 23.0348957 1.620000e+03 1.586320e-01 1.385535e+01 8.131879e+02 8.052141e-02 269.48022
ω²_Cl 0.2509671 1.586320e-01 1.931754e+03 1.290869e-01 9.016087e-02 9.658788e+02 18.52264
γ²_ka 725.0347675 1.385535e+01 1.290869e-01 1.201459e+04 4.448247e+03 4.969634e+01 2591.78367
γ²_V 15.2977604 8.131879e+02 9.016087e-02 4.448247e+03 1.990898e+04 3.195887e+01 1814.53482
γ²_Cl 0.1301725 8.052141e-02 9.658788e+02 4.969634e+01 3.195887e+01 3.707617e+04 615.30442
σ_inter_RespPK 423.7018815 2.694802e+02 1.852264e+01 2.591784e+03 1.814535e+03 6.153044e+02 36667.05676
*********************************************
Determinant, condition numbers and D-criterion
***********************************************
Determinant: 9.179643e+37
D-criterion: 1457.367
Condition number (fixed effects): 73.40657
Condition number (variance components): 27.34871
***************************************
Parameters estimation
***************************************
Parameter Value SE RSE(%)
μ_ka 1.00000000 0.054662025 5.466202
μ_V 3.50000000 0.264574919 7.559283
μ_Cl 2.00000000 0.106506758 5.325338
β_V_Sex_F 0.18232156 0.105457033 57.841231
β_Cl_Treatment_T 0.09531018 0.034189595 35.871924
ω²_ka 0.09000000 0.026779506 29.755007
ω²_V 0.09000000 0.025141327 27.934808
ω²_Cl 0.09000000 0.022901883 25.446537
γ²_ka 0.02250000 0.009744927 43.310785
γ²_V 0.02250000 0.007496871 33.319425
γ²_Cl 0.02250000 0.005228287 23.236831
σ_inter_RespPK 0.10000000 0.005272574 5.272574
***************************************
Legend:
μ = fixed effects (population means)
β = covariate effects
ω² = inter-individual variability (IIV)
γ² = inter-occasion variability (IOV)
σ = residual error SD
***************************************
[1] 1457.367
plotsSI = plotSensitivityIndices( evaluationPopFIM, plotOptions )
print( plotsSI$design1$arm1$RespPK$V )
outputFile = "vignette3_evaluation_popFim_report.html"
Report(evaluationPopFIM, paths$reports, outputFile, plotOptions)Of note, we could also use these functions to extract specific results:
