Case Study III - Development of a population PKPD model using Pumas

using Pumas
using DataFramesMeta
using PumasUtilities
using CSV

1 Third of three case studies

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

2 Developing a population PKPD model using Pumas

In this tutorial, we will show how to build a PKPD model using a sequential approach. The PK model is a simple one compartment model with an IV infusion. The PD model is an indirect response model (IDR model) of histamine concentrations.

3 Data

Please download the PK dataset and PD dataset place them in the folder where this tutorial is placed First, we read the data from the csv file. We define an :evid and a :cmt column and set all event row values of :CONC to missing.

pkdata = CSV.read("CS3_IVINFEST.csv", 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",))

4 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 individual 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: 0.06037116050720215
     1     1.700194e+03     3.899473e+02
 * time: 0.7177481651306152
     2     1.372117e+03     2.133637e+02
 * time: 0.7240111827850342
     3     1.149219e+03     9.540101e+01
 * time: 0.7312791347503662
     4     1.063179e+03     5.172865e+01
 * time: 0.7382290363311768
     5     1.025366e+03     2.646221e+01
 * time: 0.7448410987854004
     6     1.011715e+03     1.666561e+01
 * time: 0.7507350444793701
     7     1.007591e+03     1.182557e+01
 * time: 0.756472110748291
     8     1.006663e+03     9.681515e+00
 * time: 0.7619431018829346
     9     1.006394e+03     8.838980e+00
 * time: 0.7676701545715332
    10     1.006061e+03     8.916639e+00
 * time: 0.7744929790496826
    11     1.005287e+03     8.898501e+00
 * time: 0.779555082321167
    12     1.003673e+03     1.594601e+01
 * time: 0.7845211029052734
    13     1.000433e+03     3.264510e+01
 * time: 0.8371751308441162
    14     9.939620e+02     5.680776e+01
 * time: 0.8425650596618652
    15     9.920492e+02     8.189008e+01
 * time: 0.848581075668335
    16     9.853470e+02     4.792287e+01
 * time: 0.854741096496582
    17     9.821580e+02     4.099167e+01
 * time: 0.8597002029418945
    18     9.784445e+02     4.359738e+01
 * time: 0.8656260967254639
    19     9.753844e+02     1.715016e+01
 * time: 0.8710691928863525
    20     9.742523e+02     1.244736e+01
 * time: 0.8754100799560547
    21     9.734762e+02     1.086879e+01
 * time: 0.8799381256103516
    22     9.729031e+02     1.529972e+01
 * time: 0.88425612449646
    23     9.714972e+02     2.051054e+01
 * time: 0.8889269828796387
    24     9.702763e+02     1.703802e+01
 * time: 0.8938281536102295
    25     9.693945e+02     1.242588e+01
 * time: 0.8987832069396973
    26     9.690627e+02     1.193813e+01
 * time: 0.9039239883422852
    27     9.687959e+02     1.049254e+01
 * time: 0.9403860569000244
    28     9.683343e+02     7.801000e+00
 * time: 0.9453761577606201
    29     9.667001e+02     1.096197e+01
 * time: 0.9503440856933594
    30     9.660129e+02     9.792112e+00
 * time: 0.9548861980438232
    31     9.655534e+02     9.823330e+00
 * time: 0.9594850540161133
    32     9.651971e+02     1.059550e+01
 * time: 0.9639551639556885
    33     9.636185e+02     1.011640e+01
 * time: 0.9689781665802002
    34     9.621257e+02     7.727968e+00
 * time: 0.973783016204834
    35     9.603739e+02     9.857407e+00
 * time: 0.9785830974578857
    36     9.603277e+02     1.508837e+01
 * time: 0.983422040939331
    37     9.598745e+02     3.570285e-01
 * time: 0.9881680011749268
    38     9.598694e+02     2.341098e-01
 * time: 0.9924931526184082
    39     9.598690e+02     9.077958e-02
 * time: 0.9964611530303955
    40     9.598690e+02     3.615731e-02
 * time: 1.0003631114959717
    41     9.598690e+02     3.631911e-03
 * time: 1.004249095916748
    42     9.598690e+02     3.934849e-04
 * time: 1.0081419944763184

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

5 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_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

6 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("CS3_IVINFPDEST.csv", 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

7 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

8 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: 9.107589721679688e-5
    10     5.899604e+02     6.134653e+00
 * time: 1.5002310276031494
    20     5.824375e+02     2.641101e+01
 * time: 2.2593960762023926
    30     5.801204e+02     1.223992e+00
 * time: 2.895132064819336
    40     5.798576e+02     5.986550e-01
 * time: 4.063896894454956
    50     5.668945e+02     6.290122e+00
 * time: 4.853703022003174
    60     5.591273e+02     5.076031e+00
 * time: 5.582738876342773
    70     5.559638e+02     3.255237e-04
 * time: 6.644665956497192

FittedPumasModel

Successful minimization:                      true

Likelihood approximation:                     FOCE
Likelihood Optimizer:                         BFGS
Dynamical system type:               Nonlinear ODE
Solver(s):(OrdinaryDiffEq.Vern7,OrdinaryDiffEq.Rodas5)

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.

9 Conclusion

In this tutorial, we saw how to map from data to a Population, how to formulate a model of oral administrationand fit it, and how to use built-in functionality to assess the quality of the formulated model. We also saw some advanced features such as bootstrap and SIR inference, VPC plots, and simulations.