Case Study III: Development of a population PKPD model

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 (inf_sd_4_pk) and another for the PD model (inf_sd_4_idr). 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 = dataset("inf_sd_4_pk")
@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 ~ @. CombinedNormal(conc_model, σ_add, σ_prop)
    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 = Diagonal([0.09, 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: 0.032286882400512695
     1     1.700194e+03     3.899473e+02
 * time: 4.653555870056152
     2     1.372117e+03     2.133637e+02
 * time: 4.657172918319702
     3     1.149219e+03     9.540101e+01
 * time: 4.661175012588501
     4     1.063179e+03     5.172865e+01
 * time: 4.66471791267395
     5     1.025366e+03     2.646221e+01
 * time: 4.668039083480835
     6     1.011715e+03     1.666561e+01
 * time: 4.671140909194946
     7     1.007591e+03     1.182557e+01
 * time: 4.6741108894348145
     8     1.006663e+03     9.681515e+00
 * time: 4.677121877670288
     9     1.006394e+03     8.838980e+00
 * time: 4.679976940155029
    10     1.006061e+03     8.916639e+00
 * time: 4.68265700340271
    11     1.005287e+03     8.898501e+00
 * time: 4.685354948043823
    12     1.003673e+03     1.594601e+01
 * time: 4.688066005706787
    13     1.000433e+03     3.264510e+01
 * time: 4.690990924835205
    14     9.939620e+02     5.680776e+01
 * time: 4.693846940994263
    15     9.920492e+02     8.189008e+01
 * time: 4.6968910694122314
    16     9.853470e+02     4.792287e+01
 * time: 4.700089931488037
    17     9.821580e+02     4.099167e+01
 * time: 4.702653884887695
    18     9.784445e+02     4.359738e+01
 * time: 4.705549955368042
    19     9.753844e+02     1.715016e+01
 * time: 4.708440065383911
    20     9.742523e+02     1.244736e+01
 * time: 4.7109270095825195
    21     9.734762e+02     1.086879e+01
 * time: 4.713479042053223
    22     9.729031e+02     1.529972e+01
 * time: 4.715928077697754
    23     9.714972e+02     2.051054e+01
 * time: 4.718554973602295
    24     9.702763e+02     1.703802e+01
 * time: 4.721091985702515
    25     9.693945e+02     1.242588e+01
 * time: 4.723649978637695
    26     9.690627e+02     1.193813e+01
 * time: 4.726101875305176
    27     9.687959e+02     1.049254e+01
 * time: 4.728684902191162
    28     9.683343e+02     7.801000e+00
 * time: 4.731173992156982
    29     9.667001e+02     1.096197e+01
 * time: 4.733782052993774
    30     9.660129e+02     9.792112e+00
 * time: 4.736250877380371
    31     9.655534e+02     9.823330e+00
 * time: 4.738784074783325
    32     9.651971e+02     1.059550e+01
 * time: 4.741294860839844
    33     9.636185e+02     1.011640e+01
 * time: 4.7439329624176025
    34     9.621257e+02     7.727968e+00
 * time: 4.746442079544067
    35     9.603739e+02     9.857407e+00
 * time: 4.7491419315338135
    36     9.599796e+02     3.356725e+00
 * time: 4.753638982772827
    37     9.598710e+02     4.280571e-01
 * time: 4.756067991256714
    38     9.598690e+02     1.553105e-01
 * time: 4.758415937423706
    39     9.598690e+02     6.159239e-02
 * time: 4.76058292388916
    40     9.598690e+02     1.194842e-02
 * time: 4.762799024581909
    41     9.598690e+02     1.228497e-03
 * time: 4.764816999435425
    42     9.598690e+02     2.242646e-04
 * time: 4.766718864440918

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     0.093838
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 convert the subject identification column to 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 = dataset("inf_sd_4_idr")
pddata.CID = string.(pddata.CID)
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 = "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 ~ @. CombinedNormal(Response, σ_add_pd, σ_prop_pd)
    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 = Diagonal([0.2, 0.2, 0.2]), σ_add_pd = 0.05, σ_prop_pd = 0.05)

irm1_results = fit(
    irm1,
    population,
    init_θ,
    Pumas.FOCE();
    constantcoef = (:tvimax,),
    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: 5.793571472167969e-5
    10     5.899604e+02     6.134653e+00
 * time: 16.470717906951904
    20     5.824248e+02     2.631300e+01
 * time: 17.518249988555908
    30     5.801204e+02     1.282116e+00
 * time: 19.308505058288574
    40     5.798576e+02     6.318885e-01
 * time: 19.715764045715332
    50     5.657245e+02     5.833103e+00
 * time: 20.380743980407715
    60     5.559678e+02     7.714349e-01
 * time: 20.874746084213257

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:                   -555.96384

------------------------
              Estimate
------------------------
  tvkin        5.5533
  tvkout       0.27864
  tvic50      30.427
† tvimax       1.0
  Ω₁,₁         0.18348
  Ω₂,₂         0.067396
  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.