Case Study III: Development of a population PKPD model

Author

Patrick Kofod Mogensen 

using Pumas
using PharmaDatasets
using DataFramesMeta
using PumasUtilities
using CSV

1 Introduction

This is the third tutorial in a series of case studies based on the tutorials found here . The third case study is about building a sequential PKPD model. It has IV infusion dosing, PK is governed by a simple one compartment model, and PD is an indirect response model (IDR) of histamine concentrations.

2 Data

The datasets are available from the PharmaDatasets package. One for the PK model (CS3_IVINFEST) and another for the PD model (CS3_IVINFPDEST). First, we read the data for the PK model. We define an :evid and a :cmt column and set all event row values of :CONC to missing.

pkdata = CSV.read(dataset("pumas/event_data/CS3_IVINFEST", String), DataFrame; header = 4)
@rtransform!(pkdata, :evid = :AMT == 0 ? 0 : 1)
@rtransform!(pkdata, :cmt = :AMT == 0 ? missing : Symbol("Central"))
@rtransform!(pkdata, :CONC = :evid == 1 ? missing : :CONC)

Then, we map the DataFrame to a population. We can omit specifying the cmt and evid keyword because we used the default value of lower case :cmt and :evid.

pk_population = read_pumas(
    pkdata;
    id = :CID,
    time = :TIME,
    observations = [:CONC],
    amt = :AMT,
    rate = :RATE,
)
Population
  Subjects: 20
  Observations: CONC

The data can be plotted using the observations_vs_time plot.

observations_vs_time(pk_population; axis = (title = "PK data plot",))

To emphasize individual trajectories, the sim_plot function also works on a population.

sim_plot(pk_population; axis = (title = "PK data plot",))

3 PK Model definition

The next step is to define the PK model. Since we have IV infusion we do not need a depot. The model does not contain any significant distribution phase. With just a Central compartment, we can use the Central1 predefined model and its associated closed form solution. This is equivalent to ADVAN1 in NONMEM.

inf1cmt = @model begin
    @param begin
        θcl ∈ RealDomain(lower = 0.0)
        θvc ∈ RealDomain(lower = 0.0)
        Ω ∈ PDiagDomain(2)
        σ_add ∈ RealDomain(lower = 0.0)
        σ_prop ∈ RealDomain(lower = 0.0)
    end
    @random begin
        η ~ MvNormal(Ω)
    end
    @pre begin
        CL = θcl * exp(η[1])
        Vc = θvc * exp(η[2])
    end
    @dynamics Central1
    @derived begin
        conc_model := @. Central / Vc
        CONC ~ @. Normal(conc_model, sqrt(σ_add^2 + (conc_model * σ_prop)^2))
    end
end
PumasModel
  Parameters: θcl, θvc, Ω, σ_add, σ_prop
  Random effects: η
  Covariates:
  Dynamical system variables: Central
  Dynamical system type: Closed form
  Derived: CONC
  Observed: CONC

To be able to fit the model we need to specify initial parameters.

initial_est_inf1cmt = (
    θcl = 1.0,
    θvc = 1.0,
    Ω = Diagonal([0.09, 0.09]),
    σ_add = sqrt(10.0),
    σ_prop = sqrt(0.01),
)
(θcl = 1.0,
 θvc = 1.0,
 Ω = [0.09 0.0; 0.0 0.09],
 σ_add = 3.1622776601683795,
 σ_prop = 0.1,)

And then we can fit the model to data.

inf1cmt_results = fit(inf1cmt, pk_population, initial_est_inf1cmt, FOCE())
[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
Iter     Function value   Gradient norm 
     0     3.174194e+03     1.463122e+03
 * time: 0.03469681739807129
     1     1.700194e+03     3.899473e+02
 * time: 1.7266778945922852
     2     1.372117e+03     2.133637e+02
 * time: 1.7342908382415771
     3     1.149219e+03     9.540101e+01
 * time: 1.742297887802124
     4     1.063179e+03     5.172865e+01
 * time: 1.7495667934417725
     5     1.025366e+03     2.646221e+01
 * time: 1.7566008567810059
     6     1.011715e+03     1.666561e+01
 * time: 1.7632169723510742
     7     1.007591e+03     1.182557e+01
 * time: 1.7694919109344482
     8     1.006663e+03     9.681515e+00
 * time: 1.7755868434906006
     9     1.006394e+03     8.838980e+00
 * time: 1.7815217971801758
    10     1.006061e+03     8.916639e+00
 * time: 1.78721284866333
    11     1.005287e+03     8.898501e+00
 * time: 1.7930629253387451
    12     1.003673e+03     1.594601e+01
 * time: 1.7990288734436035
    13     1.000433e+03     3.264510e+01
 * time: 1.8050928115844727
    14     9.939620e+02     5.680776e+01
 * time: 1.8112568855285645
    15     9.920492e+02     8.189008e+01
 * time: 1.8177649974822998
    16     9.853470e+02     4.792287e+01
 * time: 1.8244719505310059
    17     9.821580e+02     4.099167e+01
 * time: 1.8298218250274658
    18     9.784445e+02     4.359738e+01
 * time: 1.835981845855713
    19     9.753844e+02     1.715016e+01
 * time: 1.842167854309082
    20     9.742523e+02     1.244736e+01
 * time: 1.8471238613128662
    21     9.734762e+02     1.086879e+01
 * time: 1.8521909713745117
    22     9.729031e+02     1.529972e+01
 * time: 1.8571298122406006
    23     9.714972e+02     2.051054e+01
 * time: 1.8624320030212402
    24     9.702763e+02     1.703802e+01
 * time: 1.8676419258117676
    25     9.693945e+02     1.242588e+01
 * time: 1.8729209899902344
    26     9.690627e+02     1.193813e+01
 * time: 1.8779418468475342
    27     9.687959e+02     1.049254e+01
 * time: 1.8831830024719238
    28     9.683343e+02     7.801000e+00
 * time: 1.8882718086242676
    29     9.667001e+02     1.096197e+01
 * time: 1.8938570022583008
    30     9.660129e+02     9.792112e+00
 * time: 1.898535966873169
    31     9.655534e+02     9.823330e+00
 * time: 1.9033138751983643
    32     9.651971e+02     1.059550e+01
 * time: 1.9080917835235596
    33     9.636185e+02     1.011640e+01
 * time: 1.9135258197784424
    34     9.621257e+02     7.727968e+00
 * time: 1.9183719158172607
    35     9.603739e+02     9.857407e+00
 * time: 1.9234158992767334
    36     9.603277e+02     1.508837e+01
 * time: 1.928480863571167
    37     9.598745e+02     3.570285e-01
 * time: 1.933471918106079
    38     9.598694e+02     2.341098e-01
 * time: 1.9380009174346924
    39     9.598690e+02     9.077958e-02
 * time: 1.9424059391021729
    40     9.598690e+02     3.615731e-02
 * time: 1.9475657939910889
    41     9.598690e+02     3.631911e-03
 * time: 1.9527208805084229
    42     9.598690e+02     3.934849e-04
 * time: 1.9573948383331299

FittedPumasModel

Dynamical system type:                 Closed form

Number of subjects:                             20

Observation records:         Active        Missing
    CONC:                       220              0
    Total:                      220              0

Number of parameters:      Constant      Optimized
                                  0              6

Likelihood approximation:                     FOCE
Likelihood optimizer:                         BFGS

Termination Reason:                   GradientNorm
Log-likelihood value:                   -959.86896

------------------
         Estimate
------------------
θcl      0.024451
θvc      0.074289
Ω₁,₁     0.072037
Ω₂,₂     0.09384
σ_add    3.2726
σ_prop   0.10228
------------------

4 Model diagnostics

As usual, we use the inspect function to calculate all diagnostics.

inf1cmt_insp = inspect(inf1cmt_results)
[ Info: Calculating predictions.
[ Info: Calculating weighted residuals.
[ Info: Calculating empirical bayes.
[ Info: Evaluating dose control parameters.
[ Info: Evaluating individual parameters.
[ Info: Done.

FittedPumasModelInspection

Likelihood approximation used for weighted residuals: FOCE

These can be saved to a file by constructing a table representation of everything from predictions to weighted residuals and empirical bayes estimes and individual coefficients (POSTHOC in NONMEM).

df_inspect = DataFrame(inf1cmt_insp)
CSV.write("inspect_file.csv", df_inspect)
"inspect_file.csv"

Besides mean predictions, it is also simple to simulate from the estimated model using the empirical bayes estimates as the values for the random effects.

sim_plot(simobs(inf1cmt, pk_population, coef(inf1cmt_results)))

For the PD model we need individual CL and Vc values.

icoef_dataframe = unique(df_inspect[!, [:id, :time, :CL, :Vc]], :id)
rename!(icoef_dataframe, :CL => :CLi, :Vc => :Vci)
20×4 DataFrame
Row id time CLi Vci
String Float64 Float64? Float64?
1 1 0.0 0.0154794 0.0879781
2 10 0.0 0.0308652 0.0566826
3 11 0.0 0.028236 0.0814447
4 12 0.0 0.0397456 0.0597957
5 13 0.0 0.021964 0.0764398
6 14 0.0 0.0233953 0.115324
7 15 0.0 0.0270355 0.0653006
8 16 0.0 0.0253244 0.0579478
9 17 0.0 0.0278282 0.144549
10 18 0.0 0.0400327 0.0813417
11 19 0.0 0.0183681 0.0723819
12 2 0.0 0.0191042 0.0920158
13 20 0.0 0.0276561 0.0533032
14 3 0.0 0.0156637 0.0753704
15 4 0.0 0.0260312 0.0691899
16 5 0.0 0.0284244 0.103891
17 6 0.0 0.0245165 0.094743
18 7 0.0 0.0153045 0.0360822
19 8 0.0 0.0261443 0.0750091
20 9 0.0 0.0248482 0.0555691

5 Getting the data

The data file consists of data obtained from 10 individuals who were treated with 500mg dose TID (three times a day, every eight hours) for five days. The dataset exists in the PharmaDatasets package and we load it into memory as a DataFrame. We specify that the first column should be a String because we want to join the individual parameters from the PK step to the PD data, and the id column of the DataFrame from inspect is a String column. We use the first(input, number_of_elements) function to show the first 10 rows of the DataFrame.

pddata = CSV.read(
    dataset("pumas/event_data/CS3_IVINFPDEST", String),
    DataFrame;
    header = 5,
    types = Dict(1 => String),
)
rename!(pddata, :CID => :id, :TIME => :time, :AMT => :amt, :CMT => :cmt)
@rtransform!(pddata, :evid = :amt == 0 ? 0 : 1)
@rtransform!(pddata, :HIST = :evid == 1 ? missing : :HIST)
first(pddata, 10)
10×10 DataFrame
Row id time HIST amt cmt RATE MDV CLI VI evid
String Float64 Float64? Int64 Int64 Float64 Int64 Float64 Float64 Int64
1 1 0.0 missing 100 1 16.7 1 15.585 90.812 1
2 1 0.0 missing 1 2 0.0 1 15.585 90.812 1
3 1 0.0 13.008 0 2 0.0 0 15.585 90.812 0
4 1 0.5 13.808 0 2 0.0 0 15.585 90.812 0
5 1 1.0 8.6859 0 2 0.0 0 15.585 90.812 0
6 1 3.0 6.2601 0 2 0.0 0 15.585 90.812 0
7 1 5.0 4.0602 0 2 0.0 0 15.585 90.812 0
8 1 6.0 4.4985 0 2 0.0 0 15.585 90.812 0
9 1 8.0 3.2736 0 2 0.0 0 15.585 90.812 0
10 1 12.0 2.6026 0 2 0.0 0 15.585 90.812 0

The file contains 11 columns:

pd_dataframe = outerjoin(pddata, icoef_dataframe; on = [:id, :time])
sort!(pd_dataframe, [:id, :time])
280×12 DataFrame
255 rows omitted
Row id time HIST amt cmt RATE MDV CLI VI evid CLi Vci
String Float64 Float64? Int64? Int64? Float64? Int64? Float64? Float64? Int64? Float64? Float64?
1 1 0.0 missing 100 1 16.7 1 15.585 90.812 1 0.0154794 0.0879781
2 1 0.0 missing 1 2 0.0 1 15.585 90.812 1 0.0154794 0.0879781
3 1 0.0 13.008 0 2 0.0 0 15.585 90.812 0 0.0154794 0.0879781
4 1 0.5 13.808 0 2 0.0 0 15.585 90.812 0 missing missing
5 1 1.0 8.6859 0 2 0.0 0 15.585 90.812 0 missing missing
6 1 3.0 6.2601 0 2 0.0 0 15.585 90.812 0 missing missing
7 1 5.0 4.0602 0 2 0.0 0 15.585 90.812 0 missing missing
8 1 6.0 4.4985 0 2 0.0 0 15.585 90.812 0 missing missing
9 1 8.0 3.2736 0 2 0.0 0 15.585 90.812 0 missing missing
10 1 12.0 2.6026 0 2 0.0 0 15.585 90.812 0 missing missing
11 1 15.0 2.3422 0 2 0.0 0 15.585 90.812 0 missing missing
12 1 18.0 3.8008 0 2 0.0 0 15.585 90.812 0 missing missing
13 1 24.0 7.1397 0 2 0.0 0 15.585 90.812 0 missing missing
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
269 9 0.0 34.608 0 2 0.0 0 24.809 56.453 0 0.0248482 0.0555691
270 9 0.5 29.57 0 2 0.0 0 24.809 56.453 0 missing missing
271 9 1.0 27.68 0 2 0.0 0 24.809 56.453 0 missing missing
272 9 3.0 18.397 0 2 0.0 0 24.809 56.453 0 missing missing
273 9 5.0 14.908 0 2 0.0 0 24.809 56.453 0 missing missing
274 9 6.0 11.669 0 2 0.0 0 24.809 56.453 0 missing missing
275 9 8.0 8.5475 0 2 0.0 0 24.809 56.453 0 missing missing
276 9 12.0 16.895 0 2 0.0 0 24.809 56.453 0 missing missing
277 9 15.0 21.326 0 2 0.0 0 24.809 56.453 0 missing missing
278 9 18.0 26.131 0 2 0.0 0 24.809 56.453 0 missing missing
279 9 24.0 26.278 0 2 0.0 0 24.809 56.453 0 missing missing
280 9 30.0 39.643 0 2 0.0 0 24.809 56.453 0 missing missing

6 Converting the DataFrame to a collection of Subjects

population = read_pumas(pd_dataframe; observations = [:HIST], covariates = [:CLi, :Vci])
Population
  Subjects: 20
  Covariates: CLi, Vci
  Observations: HIST

7 IDR model

irm1 = @model begin
    @metadata begin
        desc = "POPULATION PK-PD MODELING"
        timeu = u"hr" # hour
    end
    @param begin
        tvkin ∈ RealDomain(lower = 0)
        tvkout ∈ RealDomain(lower = 0)
        tvic50 ∈ RealDomain(lower = 0)
        tvimax ∈ RealDomain(lower = 0)
        Ω ∈ PDiagDomain(3)
        σ_add_pd ∈ RealDomain(lower = 0)
        σ_prop_pd ∈ RealDomain(lower = 0)
    end

    @random begin
        η ~ MvNormal(Ω)
    end

    @covariates CLi Vci

    @pre begin
        kin = tvkin * exp(η[1])
        kout = tvkout * exp(η[2])
        bsl = kin / kout
        ic50 = tvic50 * exp(η[3])
        imax = tvimax
        CL = CLi
        Vc = Vci
    end

    @init begin
        Response = bsl
    end

    @dynamics begin
        Central' = -CL / Vc * Central
        Response' =
            kin * (1 - imax * (Central / Vc) / (ic50 + Central / Vc)) - kout * Response
    end

    @derived begin
        HIST ~ @. Normal(Response, sqrt(σ_add_pd^2 + (Response * σ_prop_pd)^2))
    end
end
PumasModel
  Parameters: tvkin, tvkout, tvic50, tvimax, Ω, σ_add_pd, σ_prop_pd
  Random effects: η
  Covariates: CLi, Vci
  Dynamical system variables: Central, Response
  Dynamical system type: Nonlinear ODE
  Derived: HIST
  Observed: HIST
init_θ = (
    tvkin = 5.4,
    tvkout = 0.3,
    tvic50 = 3.9,
    tvimax = 1.0,
    Ω = Diagonal([0.2, 0.2, 0.2]),
    σ_add_pd = 0.05,
    σ_prop_pd = 0.05,
)
(tvkin = 5.4,
 tvkout = 0.3,
 tvic50 = 3.9,
 tvimax = 1.0,
 Ω = [0.2 0.0 0.0; 0.0 0.2 0.0; 0.0 0.0 0.2],
 σ_add_pd = 0.05,
 σ_prop_pd = 0.05,)
irm1_results = fit(
    irm1,
    population,
    init_θ,
    Pumas.FOCE();
    constantcoef = (tvimax = 1.0,),
    optim_options = (show_every = 10,),
)
[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
Iter     Function value   Gradient norm 
     0     1.625453e+03     2.073489e+03
 * time: 1.811981201171875e-5
    10     5.899604e+02     6.134653e+00
 * time: 1.6497101783752441
    20     5.824375e+02     2.641109e+01
 * time: 2.58089017868042
    30     5.801204e+02     1.223989e+00
 * time: 3.4438860416412354
    40     5.798576e+02     5.986551e-01
 * time: 4.279842138290405
    50     5.668948e+02     6.290186e+00
 * time: 5.257589101791382
    60     5.591291e+02     5.129868e+00
 * time: 6.195143222808838
    70     5.559638e+02     3.219738e-04
 * time: 7.0542311668396

FittedPumasModel

Dynamical system type:               Nonlinear ODE
Solver(s): (OrdinaryDiffEqVerner.Vern7,OrdinaryDiffEqRosenbrock.Rodas5P)

Number of subjects:                             20

Observation records:         Active        Missing
    HIST:                       240              0
    Total:                      240              0

Number of parameters:      Constant      Optimized
                                  1              8

Likelihood approximation:                     FOCE
Likelihood optimizer:                         BFGS

Termination Reason:                   GradientNorm
Log-likelihood value:                   -555.96385

------------------------
              Estimate
------------------------
  tvkin        5.5533
  tvkout       0.27864
  tvic50      30.427
† tvimax       1.0
  Ω₁,₁         0.18348
  Ω₂,₂         0.067395
  Ω₃,₃         0.20683
  σ_add_pd     1.1094
  σ_prop_pd    0.1038
------------------------
† indicates constant parameters
irm1_insp = inspect(irm1_results)
goodness_of_fit(irm1_insp)
[ Info: Calculating predictions.
[ Info: Calculating weighted residuals.
[ Info: Calculating empirical bayes.
[ Info: Evaluating dose control parameters.
[ Info: Evaluating individual parameters.
[ Info: Done.

8 Conclusion

In this tutorial, we saw how to build on topics we learnt in the previous two case studies to build a sequential PKPD model. We built two different models and saw how to forward the results from the PK model to the PD model. This concludes the third of the three case studies.