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 (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: 0.02215409278869629
     1     1.700194e+03     3.899473e+02
 * time: 0.8523080348968506
     2     1.372117e+03     2.133637e+02
 * time: 0.8653719425201416
     3     1.149219e+03     9.540101e+01
 * time: 0.8800621032714844
     4     1.063179e+03     5.172865e+01
 * time: 0.8930540084838867
     5     1.025366e+03     2.646221e+01
 * time: 0.9466369152069092
     6     1.011715e+03     1.666561e+01
 * time: 0.9582979679107666
     7     1.007591e+03     1.182557e+01
 * time: 0.9693219661712646
     8     1.006663e+03     9.681515e+00
 * time: 0.9798901081085205
     9     1.006394e+03     8.838980e+00
 * time: 0.9896430969238281
    10     1.006061e+03     8.916639e+00
 * time: 0.9993569850921631
    11     1.005287e+03     8.898501e+00
 * time: 1.0092709064483643
    12     1.003673e+03     1.594601e+01
 * time: 1.0194060802459717
    13     1.000433e+03     3.264510e+01
 * time: 1.0296909809112549
    14     9.939620e+02     5.680776e+01
 * time: 1.0404210090637207
    15     9.920492e+02     8.189008e+01
 * time: 1.0518600940704346
    16     9.853470e+02     4.792287e+01
 * time: 1.0637140274047852
    17     9.821580e+02     4.099167e+01
 * time: 1.073099136352539
    18     9.784445e+02     4.359738e+01
 * time: 1.0842339992523193
    19     9.753844e+02     1.715016e+01
 * time: 1.0948340892791748
    20     9.742523e+02     1.244736e+01
 * time: 1.1033539772033691
    21     9.734762e+02     1.086879e+01
 * time: 1.1121289730072021
    22     9.729031e+02     1.529972e+01
 * time: 1.1207809448242188
    23     9.714972e+02     2.051054e+01
 * time: 1.1302599906921387
    24     9.702763e+02     1.703802e+01
 * time: 1.1395931243896484
    25     9.693945e+02     1.242588e+01
 * time: 1.1728579998016357
    26     9.690627e+02     1.193813e+01
 * time: 1.1819450855255127
    27     9.687959e+02     1.049254e+01
 * time: 1.1913480758666992
    28     9.683343e+02     7.801000e+00
 * time: 1.2005929946899414
    29     9.667001e+02     1.096197e+01
 * time: 1.210524082183838
    30     9.660129e+02     9.792112e+00
 * time: 1.2194669246673584
    31     9.655534e+02     9.823330e+00
 * time: 1.2284700870513916
    32     9.651971e+02     1.059550e+01
 * time: 1.2373039722442627
    33     9.636185e+02     1.011640e+01
 * time: 1.2465300559997559
    34     9.621257e+02     7.727968e+00
 * time: 1.2555060386657715
    35     9.603739e+02     9.857407e+00
 * time: 1.2648329734802246
    36     9.603277e+02     1.508837e+01
 * time: 1.2742259502410889
    37     9.598745e+02     3.570285e-01
 * time: 1.283505916595459
    38     9.598694e+02     2.341098e-01
 * time: 1.2919349670410156
    39     9.598690e+02     9.077958e-02
 * time: 1.2997701168060303
    40     9.598690e+02     3.615731e-02
 * time: 1.3080880641937256
    41     9.598690e+02     3.631911e-03
 * time: 1.3154199123382568
    42     9.598690e+02     3.934849e-04
 * time: 1.3221960067749023

FittedPumasModel

Successful minimization:                      true

Likelihood approximation:                     FOCE
Likelihood Optimizer:                         BFGS
Dynamical system type:                 Closed form

Log-likelihood value:                   -959.86896
Number of subjects:                             20
Number of parameters:         Fixed      Optimized
                                  0              6
Observation records:         Active        Missing
    CONC:                       220              0
    Total:                      220              0

---------------------
           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 individual parameters.
[ Info: Evaluating dose control parameters.
[ Info: Done.

FittedPumasModelInspection

Fitting was successful: true
Likehood approximation used for weighted residuals : Pumas.FOCE{Optim.NewtonTrustRegion{Float64}, Optim.Options{Float64, Nothing}}(Optim.NewtonTrustRegion{Float64}(1.0, 100.0, 1.4901161193847656e-8, 0.1, 0.25, 0.75, false), Optim.Options(x_abstol = 0.0, x_reltol = 0.0, f_abstol = 0.0, f_reltol = 0.0, g_abstol = 1.0e-5, g_reltol = 1.0e-8, outer_x_abstol = 0.0, outer_x_reltol = 0.0, outer_f_abstol = 0.0, outer_f_reltol = 0.0, outer_g_abstol = 1.0e-8, outer_g_reltol = 1.0e-8, f_calls_limit = 0, g_calls_limit = 0, h_calls_limit = 0, allow_f_increases = false, allow_outer_f_increases = true, successive_f_tol = 1, iterations = 1000, outer_iterations = 1000, store_trace = false, trace_simplex = false, show_trace = false, extended_trace = false, show_warnings = true, show_every = 1, time_limit = NaN, )
)

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: 7.009506225585938e-5
    10     5.899604e+02     6.134653e+00
 * time: 1.091763973236084
    20     5.824375e+02     2.641108e+01
 * time: 2.0990400314331055
    30     5.801204e+02     1.223991e+00
 * time: 3.0178050994873047
    40     5.798576e+02     5.986551e-01
 * time: 3.930305004119873
    50     5.668947e+02     6.290151e+00
 * time: 4.904735088348389
    60     5.591284e+02     5.109298e+00
 * time: 5.811630010604858
    70     5.559638e+02     3.233225e-04
 * time: 6.684618949890137

FittedPumasModel

Successful minimization:                      true

Likelihood approximation:                     FOCE
Likelihood Optimizer:                         BFGS
Dynamical system type:               Nonlinear ODE
Solver(s):(OrdinaryDiffEq.Vern7,OrdinaryDiffEq.Rodas5P)

Log-likelihood value:                   -555.96385
Number of subjects:                             20
Number of parameters:         Fixed      Optimized
                                  1              8
Observation records:         Active        Missing
    HIST:                       240              0
    Total:                      240              0

------------------------
              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
------------------------

irm1_insp = inspect(irm1_results)
goodness_of_fit(irm1_insp)

[ Info: Calculating predictions.
[ Info: Calculating weighted residuals.
[ Info: Calculating empirical bayes.
[ Info: Evaluating individual parameters.
[ Info: Evaluating dose control 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.