The hardware and bandwidth for this mirror is donated by dogado GmbH, the Webhosting and Full Service-Cloud Provider. Check out our Wordpress Tutorial.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]dogado.de.

G-CSF PK-PD design evaluation

Overview

This example evaluates a G-CSF / filgrastim population design with PFIM, based on the PK-PD model of Krzyzanski et al. (Krzyzanski et al. 2010).

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

Evaluate the population FIM of Design 1 (ODE model, two responses, Combined2 residual error).
Report relative standard errors (RSE %) for fixed effects, IIV and residual error.
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, example04_execute.R reuses vignettes/data/vignette4_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 example04_execute.R. The evaluation only needs the equation list, the algebraic PK output, and baseline initial conditions .pfimGcsfBaselineICs().

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.

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 example04_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

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

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

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)

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

                        μ_KA        μ_KEL          μ_VD          μ_KD        μ_KINT         μ_KSI         μ_KMT        μ_KTT         μ_NB0         μ_SC1         μ_SM1        μ_SM2       ω²_KEL
μ_KA            3971.1311156 1186.9218298 -2.036910e+01   -43.0557970   13963.30995   -244.505941 -6.798335e+02    7174.6916   -7.09980650   12.44065292  1.916216e+00    0.1939741  0.000000000
μ_KEL           1186.9218298 3276.6125964  6.240750e+00  -156.3236244    9250.01901     -4.066780  2.179500e+03    8740.5620   -2.35299025    0.73943343  4.034423e-01    1.2497026  0.000000000
μ_VD             -20.3691010    6.2407503  1.143137e+01     3.8378322     -94.80479     46.860295 -3.208658e+00    -728.5192    0.47053982    0.09993536  2.053618e-04   -0.1446296  0.000000000
μ_KD             -43.0557970 -156.3236244  3.837832e+00   284.9748591   -6639.55813   -193.952738 -1.759877e+03    6329.5127   -1.59457642   -2.93350653 -6.571671e-01    1.2214591  0.000000000
μ_KINT         13963.3099523 9250.0190055 -9.480479e+01 -6639.5581293  280625.07795   2536.417058  7.140062e+04 -162853.1163  -13.80764317  128.70795076  4.117431e+01  -40.5458885  0.000000000
μ_KSI           -244.5059408   -4.0667795  4.686030e+01  -193.9527380    2536.41706   1757.093492  1.291056e+03  -35042.1221    7.95346612    9.43280439  2.382649e+00   -9.9768049  0.000000000
μ_KMT           -679.8334795 2179.5003307 -3.208658e+00 -1759.8765917   71400.62066   1291.055650  2.516249e+05 -102117.7445  402.89285135  277.64187376  2.988309e+01  110.7877980  0.000000000
μ_KTT           7174.6915719 8740.5619687 -7.285192e+02  6329.5127120 -162853.11626 -35042.122084 -1.021177e+05 1371025.1260 -261.47968877 -581.86254459  4.025250e+02 -125.4816343  0.000000000
μ_NB0             -7.0998065   -2.3529903  4.705398e-01    -1.5945764     -13.80764      7.953466  4.028929e+02    -261.4797   35.37313289   -0.50505033 -8.216462e-02    1.0780692  0.000000000
μ_SC1             12.4406529    0.7394334  9.993536e-02    -2.9335065     128.70795      9.432804  2.776419e+02    -581.8625   -0.50505033    3.20793111 -6.164331e-02   -0.1954826  0.000000000
μ_SM1              1.9162163    0.4034423  2.053618e-04    -0.6571671      41.17431      2.382649  2.988309e+01     402.5250   -0.08216462   -0.06164331  1.947301e+00   -1.9885161  0.000000000
μ_SM2              0.1939741    1.2497026 -1.446296e-01     1.2214591     -40.54589     -9.976805  1.107878e+02    -125.4816    1.07806922   -0.19548259 -1.988516e+00    2.4205231  0.000000000
ω²_KEL             0.0000000    0.0000000  0.000000e+00     0.0000000       0.00000      0.000000  0.000000e+00       0.0000    0.00000000    0.00000000  0.000000e+00    0.0000000 96.431122784
ω²_VD              0.0000000    0.0000000  0.000000e+00     0.0000000       0.00000      0.000000  0.000000e+00       0.0000    0.00000000    0.00000000  0.000000e+00    0.0000000  0.121067757
ω²_KSI             0.0000000    0.0000000  0.000000e+00     0.0000000       0.00000      0.000000  0.000000e+00       0.0000    0.00000000    0.00000000  0.000000e+00    0.0000000  0.001812156
ω²_NB0             0.0000000    0.0000000  0.000000e+00     0.0000000       0.00000      0.000000  0.000000e+00       0.0000    0.00000000    0.00000000  0.000000e+00    0.0000000  0.007300912
ω²_SC1             0.0000000    0.0000000  0.000000e+00     0.0000000       0.00000      0.000000  0.000000e+00       0.0000    0.00000000    0.00000000  0.000000e+00    0.0000000  0.008543976
ω²_SM1             0.0000000    0.0000000  0.000000e+00     0.0000000       0.00000      0.000000  0.000000e+00       0.0000    0.00000000    0.00000000  0.000000e+00    0.0000000  0.779939847
σ_slope_RespPK     0.0000000    0.0000000  0.000000e+00     0.0000000       0.00000      0.000000  0.000000e+00       0.0000    0.00000000    0.00000000  0.000000e+00    0.0000000 17.867806893
σ_inter_RespPD     0.0000000    0.0000000  0.000000e+00     0.0000000       0.00000      0.000000  0.000000e+00       0.0000    0.00000000    0.00000000  0.000000e+00    0.0000000  0.868053723
σ_slope_RespPD     0.0000000    0.0000000  0.000000e+00     0.0000000       0.00000      0.000000  0.000000e+00       0.0000    0.00000000    0.00000000  0.000000e+00    0.0000000  8.192692601
                    ω²_VD       ω²_KSI       ω²_NB0       ω²_SC1       ω²_SM1 σ_slope_RespPK σ_inter_RespPD σ_slope_RespPD
μ_KA            0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_KEL           0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_VD            0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_KD            0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_KINT          0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_KSI           0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_KMT           0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_KTT           0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_NB0           0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_SC1           0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_SM1           0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
μ_SM2           0.0000000 0.000000e+00 0.000000e+00  0.000000000 0.000000e+00       0.000000      0.0000000       0.000000
ω²_KEL          0.1210678 1.812156e-03 7.300912e-03  0.008543976 7.799398e-01      17.867807      0.8680537       8.192693
ω²_VD          96.4111185 1.501672e+01 2.001819e-01  0.149257544 3.307127e-01       6.457008      0.6128612       2.418923
ω²_KSI         15.0167242 1.157098e+02 3.188220e-01  0.696509059 6.474389e+00      21.993134      1.1966022      15.044794
ω²_NB0          0.2001819 3.188220e-01 1.549261e+02  0.335554086 1.096557e+00       1.147716      1.2844870       4.533607
ω²_SC1          0.1492575 6.965091e-01 3.355541e-01 18.249985669 1.096434e+00       1.315882      0.8195170       8.313513
ω²_SM1          0.3307127 6.474389e+00 1.096557e+00  1.096434311 8.753110e+04      50.382840      8.4710336      43.458001
σ_slope_RespPK  6.4570079 2.199313e+01 1.147716e+00  1.315881531 5.038284e+01   10401.414719      5.7904536     104.906026
σ_inter_RespPD  0.6128612 1.196602e+00 1.284487e+00  0.819517001 8.471034e+00       5.790454    380.5603736    1719.216620
σ_slope_RespPD  2.4189227 1.504479e+01 4.533607e+00  8.313512679 4.345800e+01     104.906026   1719.2166196   48953.726924

*************************************** 
  Fixed effects (μ) 
*************************************** 

                μ_KA        μ_KEL          μ_VD          μ_KD        μ_KINT         μ_KSI         μ_KMT        μ_KTT         μ_NB0         μ_SC1         μ_SM1        μ_SM2
μ_KA    3971.1311156 1186.9218298 -2.036910e+01   -43.0557970   13963.30995   -244.505941 -6.798335e+02    7174.6916   -7.09980650   12.44065292  1.916216e+00    0.1939741
μ_KEL   1186.9218298 3276.6125964  6.240750e+00  -156.3236244    9250.01901     -4.066780  2.179500e+03    8740.5620   -2.35299025    0.73943343  4.034423e-01    1.2497026
μ_VD     -20.3691010    6.2407503  1.143137e+01     3.8378322     -94.80479     46.860295 -3.208658e+00    -728.5192    0.47053982    0.09993536  2.053618e-04   -0.1446296
μ_KD     -43.0557970 -156.3236244  3.837832e+00   284.9748591   -6639.55813   -193.952738 -1.759877e+03    6329.5127   -1.59457642   -2.93350653 -6.571671e-01    1.2214591
μ_KINT 13963.3099523 9250.0190055 -9.480479e+01 -6639.5581293  280625.07795   2536.417058  7.140062e+04 -162853.1163  -13.80764317  128.70795076  4.117431e+01  -40.5458885
μ_KSI   -244.5059408   -4.0667795  4.686030e+01  -193.9527380    2536.41706   1757.093492  1.291056e+03  -35042.1221    7.95346612    9.43280439  2.382649e+00   -9.9768049
μ_KMT   -679.8334795 2179.5003307 -3.208658e+00 -1759.8765917   71400.62066   1291.055650  2.516249e+05 -102117.7445  402.89285135  277.64187376  2.988309e+01  110.7877980
μ_KTT   7174.6915719 8740.5619687 -7.285192e+02  6329.5127120 -162853.11626 -35042.122084 -1.021177e+05 1371025.1260 -261.47968877 -581.86254459  4.025250e+02 -125.4816343
μ_NB0     -7.0998065   -2.3529903  4.705398e-01    -1.5945764     -13.80764      7.953466  4.028929e+02    -261.4797   35.37313289   -0.50505033 -8.216462e-02    1.0780692
μ_SC1     12.4406529    0.7394334  9.993536e-02    -2.9335065     128.70795      9.432804  2.776419e+02    -581.8625   -0.50505033    3.20793111 -6.164331e-02   -0.1954826
μ_SM1      1.9162163    0.4034423  2.053618e-04    -0.6571671      41.17431      2.382649  2.988309e+01     402.5250   -0.08216462   -0.06164331  1.947301e+00   -1.9885161
μ_SM2      0.1939741    1.2497026 -1.446296e-01     1.2214591     -40.54589     -9.976805  1.107878e+02    -125.4816    1.07806922   -0.19548259 -1.988516e+00    2.4205231

*************************************** 
  Variance components (ω², γ², σ) 
*************************************** 

                     ω²_KEL      ω²_VD       ω²_KSI       ω²_NB0       ω²_SC1       ω²_SM1 σ_slope_RespPK σ_inter_RespPD σ_slope_RespPD
ω²_KEL         96.431122784  0.1210678 1.812156e-03 7.300912e-03  0.008543976 7.799398e-01      17.867807      0.8680537       8.192693
ω²_VD           0.121067757 96.4111185 1.501672e+01 2.001819e-01  0.149257544 3.307127e-01       6.457008      0.6128612       2.418923
ω²_KSI          0.001812156 15.0167242 1.157098e+02 3.188220e-01  0.696509059 6.474389e+00      21.993134      1.1966022      15.044794
ω²_NB0          0.007300912  0.2001819 3.188220e-01 1.549261e+02  0.335554086 1.096557e+00       1.147716      1.2844870       4.533607
ω²_SC1          0.008543976  0.1492575 6.965091e-01 3.355541e-01 18.249985669 1.096434e+00       1.315882      0.8195170       8.313513
ω²_SM1          0.779939847  0.3307127 6.474389e+00 1.096557e+00  1.096434311 8.753110e+04      50.382840      8.4710336      43.458001
σ_slope_RespPK 17.867806893  6.4570079 2.199313e+01 1.147716e+00  1.315881531 5.038284e+01   10401.414719      5.7904536     104.906026
σ_inter_RespPD  0.868053723  0.6128612 1.196602e+00 1.284487e+00  0.819517001 8.471034e+00       5.790454    380.5603736    1719.216620
σ_slope_RespPD  8.192692601  2.4189227 1.504479e+01 4.533607e+00  8.313512679 4.345800e+01     104.906026   1719.2166196   48953.726924

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

Determinant: 4.833743e+56 
D-criterion: 500.3243 
Condition number (fixed effects): 19670405 
Condition number (variance components): 4798.393 

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

Parameter               Value           SE     RSE(%)
μ_KA               0.6420000  0.021116261   3.289137
μ_KEL              0.1480000  0.020007804  13.518786
μ_VD               2.5600000  0.322665143  12.604107
μ_KD               1.2700000  0.104970938   8.265428
μ_KINT             0.1010000  0.003860719   3.822495
μ_KSI              0.2110000  0.041203487  19.527719
μ_KMT              0.0723000  0.002940078   4.066498
μ_KTT              0.0102000  0.001735777  17.017423
μ_NB0              1.6500000  0.177727482  10.771363
μ_SC1              3.2100000  0.669325581  20.851264
μ_SM1             34.3000000  2.843452279   8.289948
μ_SM2             32.3000000  2.475776746   7.664943
ω²_KEL            0.3120000  0.101851203  32.644616
ω²_VD             0.3280000  0.102890719  31.369122
ω²_KSI            0.2240000  0.093948373  41.941238
ω²_NB0            0.2980000  0.080344111  26.961111
ω²_SC1            0.8030000  0.234129307  29.156825
ω²_SM1            0.0128000  0.003380033  26.406507
σ_slope_RespPK      0.5029911  0.009808890   1.950112
σ_inter_RespPD      1.4491377  0.055888640   3.856683
σ_slope_RespPD      0.1506652  0.004927586   3.270554

[1] 500.3243

Relative standard errors

RSE (%) reported by PFIM for Design 1.

Fixed effects (\(\mu\))

Fixed-effect RSE (%) for Design 1.
Parameter	Description	RSE (%)
\(\mu_{\mathrm{KA}}\)	Absorption rate	3.289
\(\mu_{\mathrm{KEL}}\)	Elimination rate of free G-CSF	13.519
\(\mu_{\mathrm{VD}}\)	Central volume	12.604
\(\mu_{\mathrm{KD}}\)	Equilibrium dissociation constant	8.265
\(\mu_{\mathrm{KINT}}\)	Internalization rate	3.822
\(\mu_{\mathrm{KSI}}\)	Binding capacity (Rmax-related)	19.528
\(\mu_{\mathrm{KMT}}\)	Neutrophil elimination from blood	4.066
\(\mu_{\mathrm{KTT}}\)	Myeloid transit rate	17.017
\(\mu_{\mathrm{NB0}}\)	Baseline circulating ANC	10.771
\(\mu_{\mathrm{SC1}}\)	G-CSF EC50 for stimulation	20.851
\(\mu_{\mathrm{SM1}}\)	Max. stimulation of production	8.290
\(\mu_{\mathrm{SM2}}\)	Max. stimulation of maturation	7.665

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

IIV variance (\(\omega^2\)) RSE (%) for Design 1.
Parameter	Description	RSE (%)
\(\omega^2_{\mathrm{NB0}}\)	Baseline circulating ANC	26.961
\(\omega^2_{\mathrm{KEL}}\)	Elimination rate of free G-CSF	32.645
\(\omega^2_{\mathrm{VD}}\)	Central volume	31.369
\(\omega^2_{\mathrm{KSI}}\)	Binding capacity (Rmax-related)	41.941
\(\omega^2_{\mathrm{SC1}}\)	G-CSF EC50 for stimulation	29.157
\(\omega^2_{\mathrm{SM1}}\)	Max. stimulation of production	26.407

Residual error (\(\sigma\))

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

Residual-error SD and RSE (%) for Design 1.
Parameter	Description	Value (SD)	RSE (%)
\(\sigma_{\mathrm{slope,PK}}\)	PK proportional (SD)	0.5030	1.950
\(\sigma_{\mathrm{slope,ANC}}\)	ANC proportional (SD)	0.1507	3.271
\(\sigma_{\mathrm{inter,ANC}}\)	ANC additive (SD)	1.4491	3.857

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.

plotOutcomesEvaluationModel
PFIM::plotSE(evaluationPop)
PFIM::plotRSE(evaluationPop)

References

Krzyzanski, Wojciech, Paweł Wiczling, Peter J. Lowe, Etienne Pigeolet, Markus Fink, Amin Berghout, and Stephan Balser. 2010. “Population Modeling of Filgrastim PK-PD in Healthy Adults Following Intravenous and Subcutaneous Administrations.” Journal of Clinical Pharmacology 50 (9 Suppl): 101S–112S. https://doi.org/10.1177/0091270010376966.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.