using Pumas
using DataFramesMeta
using PumasUtilities
using CSV
Case Study III - Development of a population PKPD model using Pumas
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
.
= CSV.read("CS3_IVINFEST.csv", DataFrame; header = 4)
pkdata @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
.
= read_pumas(
pk_population
pkdata;= :CID,
id = :TIME,
time = [:CONC],
observations = :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.
= @model begin
inf1cmt @param begin
∈ RealDomain(lower = 0.0)
θcl ∈ RealDomain(lower = 0.0)
θvc ∈ PDiagDomain(2)
Ω ∈ RealDomain(lower = 0.0)
σ_add ∈ RealDomain(lower = 0.0)
σ_prop end
@random begin
~ MvNormal(Ω)
η end
@pre begin
= θcl * exp(η[1])
CL = θvc * exp(η[2])
Vc end
@dynamics Central1
@derived begin
:= @. Central / Vc
conc_model ~ @. Normal(conc_model, sqrt(σ_add^2 + (conc_model * σ_prop)^2))
CONC 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 = 1.0,
θcl = 1.0,
θvc = Diagonal([0.09, 0.09]),
Ω = sqrt(10.0),
σ_add = sqrt(0.01),
σ_prop )
(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.
= fit(inf1cmt, pk_population, initial_est_inf1cmt, FOCE()) inf1cmt_results
[ 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: 0.02028489112854004
1 1.700194e+03 3.899473e+02
* time: 0.47061896324157715
2 1.372117e+03 2.133637e+02
* time: 0.47582101821899414
3 1.149219e+03 9.540101e+01
* time: 0.48154187202453613
4 1.063179e+03 5.172865e+01
* time: 0.4863548278808594
5 1.025366e+03 2.646221e+01
* time: 0.49088096618652344
6 1.011715e+03 1.666561e+01
* time: 0.4952998161315918
7 1.007591e+03 1.182557e+01
* time: 0.4991018772125244
8 1.006663e+03 9.681515e+00
* time: 0.5033619403839111
9 1.006394e+03 8.838980e+00
* time: 0.50736403465271
10 1.006061e+03 8.916639e+00
* time: 0.5564849376678467
11 1.005287e+03 8.898501e+00
* time: 0.5603599548339844
12 1.003673e+03 1.594601e+01
* time: 0.5641708374023438
13 1.000433e+03 3.264510e+01
* time: 0.5680558681488037
14 9.939620e+02 5.680776e+01
* time: 0.5722169876098633
15 9.920492e+02 8.189008e+01
* time: 0.576685905456543
16 9.853470e+02 4.792287e+01
* time: 0.5812950134277344
17 9.821580e+02 4.099167e+01
* time: 0.5848639011383057
18 9.784445e+02 4.359738e+01
* time: 0.5891108512878418
19 9.753844e+02 1.715016e+01
* time: 0.593580961227417
20 9.742523e+02 1.244736e+01
* time: 0.5968108177185059
21 9.734762e+02 1.086879e+01
* time: 0.6004559993743896
22 9.729031e+02 1.529972e+01
* time: 0.6033468246459961
23 9.714972e+02 2.051054e+01
* time: 0.6350619792938232
24 9.702763e+02 1.703802e+01
* time: 0.6389219760894775
25 9.693945e+02 1.242588e+01
* time: 0.6424579620361328
26 9.690627e+02 1.193813e+01
* time: 0.6457438468933105
27 9.687959e+02 1.049254e+01
* time: 0.6491608619689941
28 9.683343e+02 7.801000e+00
* time: 0.6526038646697998
29 9.667001e+02 1.096197e+01
* time: 0.656203031539917
30 9.660129e+02 9.792112e+00
* time: 0.6595869064331055
31 9.655534e+02 9.823330e+00
* time: 0.6630709171295166
32 9.651971e+02 1.059550e+01
* time: 0.6663739681243896
33 9.636185e+02 1.011640e+01
* time: 0.6699228286743164
34 9.621257e+02 7.727968e+00
* time: 0.6733508110046387
35 9.603739e+02 9.857407e+00
* time: 0.6769578456878662
36 9.603277e+02 1.508837e+01
* time: 0.680668830871582
37 9.598745e+02 3.570285e-01
* time: 0.6843478679656982
38 9.598694e+02 2.341098e-01
* time: 0.7095229625701904
39 9.598690e+02 9.077958e-02
* time: 0.712709903717041
40 9.598690e+02 3.615731e-02
* time: 0.715569019317627
41 9.598690e+02 3.631911e-03
* time: 0.7184388637542725
42 9.598690e+02 3.934849e-04
* time: 0.7210068702697754
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.
= inspect(inf1cmt_results) inf1cmt_insp
[ 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).
= DataFrame(inf1cmt_insp)
df_inspect write("inspect_file.csv", df_inspect) CSV.
"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.
= unique(df_inspect[!, [:id, :time, :CL, :Vc]], :id)
icoef_dataframe rename!(icoef_dataframe, :CL => :CLi, :Vc => :Vci)
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
.
= CSV.read("CS3_IVINFPDEST.csv", DataFrame; header = 5, types = Dict(1 => String))
pddata 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)
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:
= outerjoin(pddata, icoef_dataframe; on = [:id, :time])
pd_dataframe sort!(pd_dataframe, [:id, :time])
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
= read_pumas(pd_dataframe; observations = [:HIST], covariates = [:CLi, :Vci]) population
Population
Subjects: 20
Covariates: CLi, Vci
Observations: HIST
8 IDR model
= @model begin
irm1 @metadata begin
= "POPULATION PK-PD MODELING"
desc = u"hr" # hour
timeu end
@param begin
∈ RealDomain(lower = 0)
tvkin ∈ RealDomain(lower = 0)
tvkout ∈ RealDomain(lower = 0)
tvic50 ∈ RealDomain(lower = 0)
tvimax ∈ PDiagDomain(3)
Ω ∈ RealDomain(lower = 0)
σ_add_pd ∈ RealDomain(lower = 0)
σ_prop_pd end
@random begin
~ MvNormal(Ω)
η end
@covariates CLi Vci
@pre begin
= tvkin * exp(η[1])
kin = tvkout * exp(η[2])
kout = kin / kout
bsl = tvic50 * exp(η[3])
ic50 = tvimax
imax = CLi
CL = Vci
Vc end
@init begin
= bsl
Response end
@dynamics begin
' = -CL / Vc * Central
Central' =
Response* (1 - imax * (Central / Vc) / (ic50 + Central / Vc)) - kout * Response
kin end
@derived begin
~ @. Normal(Response, sqrt(σ_add_pd^2 + (Response * σ_prop_pd)^2))
HIST 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 = 5.4,
tvkin = 0.3,
tvkout = 3.9,
tvic50 = 1.0,
tvimax = Diagonal([0.2, 0.2, 0.2]),
Ω = 0.05,
σ_add_pd = 0.05,
σ_prop_pd )
(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,)
= fit(
irm1_results
irm1,
population,
init_θ,FOCE();
Pumas.= (tvimax = 1.0,),
constantcoef = (show_every = 10,),
optim_options )
[ 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: 7.700920104980469e-5
10 5.899604e+02 6.134653e+00
* time: 0.7545020580291748
20 5.824375e+02 2.641101e+01
* time: 1.4005329608917236
30 5.801204e+02 1.223992e+00
* time: 2.055785894393921
40 5.798576e+02 5.986550e-01
* time: 2.621187925338745
50 5.668945e+02 6.290122e+00
* time: 3.201838970184326
60 5.591273e+02 5.076031e+00
* time: 3.814880847930908
70 5.559638e+02 3.255237e-04
* time: 4.427348852157593
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
------------------------
= inspect(irm1_results)
irm1_insp 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.