using Pumas
using PumasUtilities
using NCA
using NCAUtilities
A Comprehensive Introduction to Pumas
This tutorial provides a comprehensive introduction to a modeling and simulation workflow in Pumas. The idea is not to get into the details of Pumas specifics, but instead provide a narrative on the lines of a regular workflow in our day-to-day work, with brevity where required to allow a broad overview. Wherever possible, cross-references will be provided to documentation and detailed examples that provide deeper insight into a particular topic.
As part of this workflow, you will be introduced to various aspects such as:
- Data wrangling in Julia
- Exploratory analysis in Julia
- Continuous data non-linear mixed effects modeling in Pumas
- Model comparison routines, post-processing, validation etc.
1 The Study and Design
CTMNopain is a novel anti-inflammatory agent under preliminary investigation. A dose-ranging trial was conducted comparing placebo with 3 doses of CTMNopain (5mg, 20mg and 80 mg QD). The maximum tolerated dose is 160 mg per day. Plasma concentrations (mg/L) of the drug were measured at 0, 0.5, 1, 1.5, 2, 2.5, 3-8 hours.
Pain score (0=no pain, 1=mild, 2=moderate, 3=severe) were obtained at time points when plasma concentration was collected. A pain score of 2 or more is considered as no pain relief.
The subjects can request for remedication if pain relief is not achieved after 2 hours post dose. Some subjects had remedication before 2 hours if they were not able to bear the pain. The time to remedication and the remedication status is available for subjects.
The pharmacokinetic dataset can be accessed using PharmaDatasets.jl.
2 Setup
2.1 Load libraries
These libraries provide the workhorse functionality in the Pumas ecosystem:
In addition, libraries below are good add-on’s that provide ancillary functionality:
using GLM: lm, @formula
using Random
using CSV
using DataFramesMeta
using CairoMakie
using PharmaDatasets2.2 Data Wrangling
We start by reading in the dataset and making some quick summaries.
Tip
If you want to learn more about data wrangling, don’t forget to check our Data Wrangling in Julia tutorials!
pkpain_df = dataset("pk_painrelief")
first(pkpain_df, 5)5×7 DataFrame
| Row | Subject | Time | Conc | PainRelief | PainScore | RemedStatus | Dose |
|---|---|---|---|---|---|---|---|
| Int64 | Float64 | Float64 | Int64 | Int64 | Int64 | String7 | |
| 1 | 1 | 0.0 | 0.0 | 0 | 3 | 1 | 20 mg |
| 2 | 1 | 0.5 | 1.15578 | 1 | 1 | 0 | 20 mg |
| 3 | 1 | 1.0 | 1.37211 | 1 | 0 | 0 | 20 mg |
| 4 | 1 | 1.5 | 1.30058 | 1 | 0 | 0 | 20 mg |
| 5 | 1 | 2.0 | 1.19195 | 1 | 1 | 0 | 20 mg |
Let’s filter out the placebo data as we don’t need that for the PK analysis.
pkpain_noplb_df = @rsubset pkpain_df :Dose != "Placebo";
first(pkpain_noplb_df, 5)5×7 DataFrame
| Row | Subject | Time | Conc | PainRelief | PainScore | RemedStatus | Dose |
|---|---|---|---|---|---|---|---|
| Int64 | Float64 | Float64 | Int64 | Int64 | Int64 | String7 | |
| 1 | 1 | 0.0 | 0.0 | 0 | 3 | 1 | 20 mg |
| 2 | 1 | 0.5 | 1.15578 | 1 | 1 | 0 | 20 mg |
| 3 | 1 | 1.0 | 1.37211 | 1 | 0 | 0 | 20 mg |
| 4 | 1 | 1.5 | 1.30058 | 1 | 0 | 0 | 20 mg |
| 5 | 1 | 2.0 | 1.19195 | 1 | 1 | 0 | 20 mg |
3 Analysis
3.1 Non-compartmental analysis
Let’s begin by performing a quick NCA of the concentration time profiles and view the exposure changes across doses. The input data specification for NCA analysis requires the presence of a :route column and an :amt column that specifies the dose. So, let’s add that in:
@rtransform! pkpain_noplb_df begin
:route = "ev"
:Dose = parse(Int, chop(:Dose; tail = 3))
endWe also need to create an :amt column:
@rtransform! pkpain_noplb_df :amt = :Time == 0 ? :Dose : missingNow, we map the data variables to the read_nca function that prepares the data for NCA analysis.
pkpain_nca = read_nca(
pkpain_noplb_df;
id = :Subject,
time = :Time,
amt = :amt,
observations = :Conc,
group = [:Dose],
route = :route,
)NCAPopulation (120 subjects):
Group: [["Dose" => 5], ["Dose" => 20], ["Dose" => 80]]
Number of missing observations: 0
Number of blq observations: 0Now that we mapped the data in, let’s visualize the concentration vs time plots for a few individuals. When paginate is set to true, a vector of plots are returned and below we display the first element with 9 individuals.
f = observations_vs_time(
pkpain_nca;
paginate = true,
axis = (; xlabel = "Time (hr)", ylabel = "CTMNoPain Concentration (ng/mL)"),
facet = (; combinelabels = true),
)
f[1]
or you can view the summary curves by dose group as passed in to the group argument in read_nca
summary_observations_vs_time(
pkpain_nca,
figure = (; fontsize = 22, size = (800, 1000)),
color = "black",
linewidth = 3,
axis = (; xlabel = "Time (hr)", ylabel = "CTMX Concentration (μg/mL)"),
facet = (; combinelabels = true, linkaxes = true),
)
A full NCA Report is now obtained for completeness purposes using the run_nca function, but later we will only extract a couple of key metrics of interest.
pk_nca = run_nca(pkpain_nca; sigdigits = 3)We can look at the NCA fits for some subjects. Here f is a vector or figures. We’ll showcase the first image by indexing f:
f = subject_fits(
pk_nca,
paginate = true,
axis = (; xlabel = "Time (hr)", ylabel = "CTMX Concentration (μg/mL)"),
facet = (; combinelabels = true, linkaxes = true),
)
f[1]
As CTMNopain’s effect maybe mainly related to maximum concentration (cmax) or area under the curve (auc), we present some summary statistics using the summarize function from NCA.
strata = [:Dose]1-element Vector{Symbol}:
:Doseparams = [:cmax, :aucinf_obs]2-element Vector{Symbol}:
:cmax
:aucinf_obsoutput = summarize(pk_nca; stratify_by = strata, parameters = params)6×10 DataFrame
| Row | Dose | parameters | numsamples | minimum | maximum | mean | std | geomean | geostd | geomeanCV |
|---|---|---|---|---|---|---|---|---|---|---|
| Int64 | String | Int64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | |
| 1 | 5 | cmax | 40 | 0.19 | 0.539 | 0.356075 | 0.0884129 | 0.345104 | 1.2932 | 26.1425 |
| 2 | 5 | aucinf_obs | 40 | 0.914 | 3.4 | 1.5979 | 0.490197 | 1.53373 | 1.32974 | 29.0868 |
| 3 | 20 | cmax | 40 | 0.933 | 2.7 | 1.4737 | 0.361871 | 1.43408 | 1.2633 | 23.6954 |
| 4 | 20 | aucinf_obs | 40 | 2.77 | 14.1 | 6.377 | 2.22239 | 6.02031 | 1.41363 | 35.6797 |
| 5 | 80 | cmax | 40 | 3.3 | 8.47 | 5.787 | 1.31957 | 5.64164 | 1.25757 | 23.2228 |
| 6 | 80 | aucinf_obs | 40 | 13.7 | 49.1 | 29.5 | 8.68984 | 28.2954 | 1.34152 | 30.0258 |
The statistics printed above are the default, but you can pass in your own statistics using the stats = [] argument to the summarize function.
We can look at a few parameter distribution plots.
parameters_vs_group(
pk_nca,
parameter = :cmax,
axis = (; xlabel = "Dose (mg)", ylabel = "Cₘₐₓ (ng/mL)"),
figure = (; fontsize = 18),
)
Dose normalized PK parameters, cmax and aucinf were essentially dose proportional between for 5 mg, 20 mg and 80 mg doses. You can perform a simple regression to check the impact of dose on cmax:
dp = NCA.DoseLinearityPowerModel(pk_nca, :cmax; level = 0.9)Dose Linearity Power Model
Variable: cmax
Model: log(cmax) ~ log(α) + β × log(dose)
────────────────────────────────────
Estimate low CI 90% high CI 90%
────────────────────────────────────
β 1.00775 0.97571 1.0398
────────────────────────────────────Here’s a visualization for the dose linearity using a power model for cmax:
power_model(dp)
We can also visualize a dose proportionality results with respect to a specific endpoint in a NCA Report; for example cmax and aucinf_obs:
dose_vs_dose_normalized(pk_nca, :cmax)
dose_vs_dose_normalized(pk_nca, :aucinf_obs)
Based on visual inspection of the concentration time profiles as seen earlier, CTMNopain exhibited monophasic decline, and perhaps a one compartment model best fits the PK data.
3.2 Pharmacokinetic modeling
As seen from the plots above, the concentrations decline monoexponentially. We will evaluate both one and two compartment structural models to assess best fit. Further, different residual error models will also be tested.
We will use the results from NCA to provide us good initial estimates.
3.2.1 Data preparation for modeling
PumasNDF requires the presence of :evid and :cmt columns in the dataset.
@rtransform! pkpain_noplb_df begin
:evid = :Time == 0 ? 1 : 0
:cmt = :Time == 0 ? 1 : 2
:cmt2 = 1 # for zero order absorption
endFurther, observations at time of dosing, i.e., when evid = 1 have to be missing
@rtransform! pkpain_noplb_df :Conc = :evid == 1 ? missing : :ConcThe dataframe will now be converted to a Population using read_pumas. Note that both observations and covariates are required to be an array even if it is one element.
pkpain_noplb = read_pumas(
pkpain_noplb_df;
id = :Subject,
time = :Time,
amt = :amt,
observations = [:Conc],
covariates = [:Dose],
evid = :evid,
cmt = :cmt,
)Population
Subjects: 120
Covariates: Dose
Observations: ConcNow that the data is transformed to a Population of subjects, we can explore different models.
3.2.2 One-compartment model
Note
If you are not familiar yet with the @model blocks and syntax, please check our documentation.
pk_1cmp = @model begin
@metadata begin
desc = "One Compartment Model"
timeu = u"hr"
end
@param begin
"""
Clearance (L/hr)
"""
tvcl ∈ RealDomain(; lower = 0, init = 3.2)
"""
Volume (L)
"""
tvv ∈ RealDomain(; lower = 0, init = 16.4)
"""
Absorption rate constant (h-1)
"""
tvka ∈ RealDomain(; lower = 0, init = 3.8)
"""
- ΩCL
- ΩVc
- ΩKa
"""
Ω ∈ PDiagDomain(init = [0.04, 0.04, 0.04])
"""
Proportional RUV
"""
σ_p ∈ RealDomain(; lower = 0.0001, init = 0.2)
end
@random begin
η ~ MvNormal(Ω)
end
@covariates begin
"""
Dose (mg)
"""
Dose
end
@pre begin
CL = tvcl * exp(η[1])
Vc = tvv * exp(η[2])
Ka = tvka * exp(η[3])
end
@dynamics Depots1Central1
@derived begin
cp := @. Central / Vc
"""
CTMx Concentration (ng/mL)
"""
Conc ~ @. Normal(cp, abs(cp) * σ_p)
end
end┌ Warning: Covariate Dose is not used in the model.
└ @ Pumas ~/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Pumas/aZRyj/src/dsl/model_macro.jl:2856PumasModel
Parameters: tvcl, tvv, tvka, Ω, σ_p
Random effects: η
Covariates: Dose
Dynamical system variables: Depot, Central
Dynamical system type: Closed form
Derived: Conc
Observed: Conc
Tip
Note that the local assignment := can be used to define intermediate statements that will not be carried outside of the block. This means that all the resulting data workflows from this model will not contain the intermediate variables defined with :=. We use this when we want to suppress the variable from any further output.
The idea behind := is for performance reasons. If you are not carrying the variable defined with := outside of the block, then it is not necessary to store it in the resulting data structures. Not only will your model run faster, but the resulting data structures will also be smaller.
Before going to fit the model, let’s evaluate some helpful steps via simulation to check appropriateness of data and model
# zero out the random effects
etas = zero_randeffs(pk_1cmp, init_params(pk_1cmp), pkpain_noplb)Above, we are generating a vector of η’s of the same length as the number of subjects to zero out the random effects. We do this as we are evaluating the trajectories of the concentrations at the initial set of parameters at a population level. Other helper functions here are sample_randeffs and init_randeffs. Please refer to the documentation.
simpk_iparams = simobs(pk_1cmp, pkpain_noplb, init_params(pk_1cmp), etas)Simulated population (Vector{<:Subject})
Simulated subjects: 120
Simulated variables: Concsim_plot(
pk_1cmp,
simpk_iparams;
observations = [:Conc],
figure = (; fontsize = 18),
axis = (;
xlabel = "Time (hr)",
ylabel = "Observed/Predicted \n CTMx Concentration (ng/mL)",
),
)
Our NCA based initial guess on the parameters seem to work well.
Lets change the initial estimate of a couple of the parameters to evaluate the sensitivity.
pkparam = (; init_params(pk_1cmp)..., tvka = 2, tvv = 10)(tvcl = 3.2,
tvv = 10,
tvka = 2,
Ω = [0.04 0.0 0.0; 0.0 0.04 0.0; 0.0 0.0 0.04],
σ_p = 0.2,)simpk_changedpars = simobs(pk_1cmp, pkpain_noplb, pkparam, etas)Simulated population (Vector{<:Subject})
Simulated subjects: 120
Simulated variables: Concsim_plot(
pk_1cmp,
simpk_changedpars;
observations = [:Conc],
figure = (; fontsize = 18),
axis = (
xlabel = "Time (hr)",
ylabel = "Observed/Predicted \n CTMx Concentration (ng/mL)",
),
)
Changing the tvka and decreasing the tvv seemed to make an impact and observations go through the simulated lines.
To get a quick ballpark estimate of your PK parameters, we can do a NaivePooled analysis.
3.2.2.1 NaivePooled
pkfit_np = fit(pk_1cmp, pkpain_noplb, init_params(pk_1cmp), NaivePooled(); omegas = (:Ω,))[ 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 7.744356e+02 3.715711e+03
* time: 0.022433996200561523
1 2.343899e+02 1.747348e+03
* time: 1.001481056213379
2 9.696232e+01 1.198088e+03
* time: 1.0043270587921143
3 -7.818699e+01 5.538151e+02
* time: 1.0062170028686523
4 -1.234803e+02 2.462514e+02
* time: 1.008086919784546
5 -1.372888e+02 2.067458e+02
* time: 1.010080099105835
6 -1.410579e+02 1.162950e+02
* time: 1.0121159553527832
7 -1.434754e+02 5.632816e+01
* time: 1.0140111446380615
8 -1.453401e+02 7.859270e+01
* time: 1.0158779621124268
9 -1.498185e+02 1.455606e+02
* time: 1.0177860260009766
10 -1.534371e+02 1.303682e+02
* time: 1.0197601318359375
11 -1.563557e+02 5.975474e+01
* time: 1.0218760967254639
12 -1.575052e+02 9.308611e+00
* time: 1.0238909721374512
13 -1.579357e+02 1.234484e+01
* time: 1.02577805519104
14 -1.581874e+02 7.478196e+00
* time: 1.0275850296020508
15 -1.582981e+02 2.027162e+00
* time: 1.0293891429901123
16 -1.583375e+02 5.578262e+00
* time: 1.031135082244873
17 -1.583556e+02 4.727050e+00
* time: 1.0328409671783447
18 -1.583644e+02 2.340173e+00
* time: 1.0348501205444336
19 -1.583680e+02 7.738100e-01
* time: 1.036916971206665
20 -1.583696e+02 3.300689e-01
* time: 1.039703130722046
21 -1.583704e+02 3.641985e-01
* time: 1.0413639545440674
22 -1.583707e+02 4.365901e-01
* time: 1.0434410572052002
23 -1.583709e+02 3.887800e-01
* time: 1.0451350212097168
24 -1.583710e+02 2.766977e-01
* time: 1.0471839904785156
25 -1.583710e+02 1.758029e-01
* time: 1.049255132675171
26 -1.583710e+02 1.133947e-01
* time: 1.0507850646972656
27 -1.583710e+02 7.922544e-02
* time: 1.0527489185333252
28 -1.583710e+02 5.954998e-02
* time: 1.0547010898590088
29 -1.583710e+02 4.157079e-02
* time: 1.0562341213226318
30 -1.583710e+02 4.295447e-02
* time: 1.058223009109497
31 -1.583710e+02 5.170753e-02
* time: 1.0602819919586182
32 -1.583710e+02 2.644383e-02
* time: 1.0624330043792725
33 -1.583710e+02 4.548993e-03
* time: 1.0650079250335693
34 -1.583710e+02 2.501804e-02
* time: 1.0675570964813232
35 -1.583710e+02 3.763440e-02
* time: 1.0695810317993164
36 -1.583710e+02 3.206026e-02
* time: 1.0711669921875
37 -1.583710e+02 1.003698e-02
* time: 1.0731761455535889
38 -1.583710e+02 2.209089e-02
* time: 1.0751590728759766
39 -1.583710e+02 4.954172e-03
* time: 1.0771400928497314
40 -1.583710e+02 1.609373e-02
* time: 1.0791940689086914
41 -1.583710e+02 1.579802e-02
* time: 1.0811691284179688
42 -1.583710e+02 1.014113e-03
* time: 1.08315110206604
43 -1.583710e+02 6.050644e-03
* time: 1.0856380462646484
44 -1.583710e+02 1.354412e-02
* time: 1.0872550010681152
45 -1.583710e+02 4.473248e-03
* time: 1.0892341136932373
46 -1.583710e+02 4.644735e-03
* time: 1.0912070274353027
47 -1.583710e+02 9.829910e-03
* time: 1.0931861400604248
48 -1.583710e+02 1.047561e-03
* time: 1.0947351455688477
49 -1.583710e+02 8.366895e-03
* time: 1.0967061519622803
50 -1.583710e+02 7.879055e-04
* time: 1.0986759662628174FittedPumasModel
Successful minimization: true
Likelihood approximation: NaivePooled
Likelihood Optimizer: BFGS
Dynamical system type: Closed form
Log-likelihood value: 158.37103
Number of subjects: 120
Number of parameters: Fixed Optimized
1 6
Observation records: Active Missing
Conc: 1320 0
Total: 1320 0
------------------
Estimate
------------------
tvcl 3.0054
tvv 14.089
tvka 44.228
Ω₁,₁ 0.0
Ω₂,₂ 0.0
Ω₃,₃ 0.0
σ_p 0.32999
------------------coefficients_table(pkfit_np)7×3 DataFrame
| Row | Parameter | Description | Estimate |
|---|---|---|---|
| String | Abstract… | Float64 | |
| 1 | tvcl | Clearance (L/hr)\n | 3.005 |
| 2 | tvv | Volume (L)\n | 14.089 |
| 3 | tvka | Absorption rate constant (h-1)\n | 44.228 |
| 4 | Ω₁,₁ | ΩCL | 0.0 |
| 5 | Ω₂,₂ | ΩVc | 0.0 |
| 6 | Ω₃,₃ | ΩKa | 0.0 |
| 7 | σ_p | Proportional RUV\n | 0.33 |
The final estimates from the NaivePooled approach seem reasonably close to our initial guess from NCA, except for the tvka parameter. We will stick with our initial guess.
One way to be cautious before going into a complete fitting routine is to evaluate the likelihood of the individual subjects given the initial parameter values and see if any subject(s) pops out as unreasonable. There are a few ways of doing this:
- check the
loglikelihoodsubject wise - check if there any influential subjects
Below, we are basically checking if the initial estimates for any subject are way off that we are unable to compute the initial loglikelihood.
lls = []
for subj in pkpain_noplb
push!(lls, loglikelihood(pk_1cmp, subj, pkparam, FOCE()))
end
# the plot below is using native CairoMakie `hist`
hist(lls; bins = 10, normalization = :none, color = (:black, 0.5))
The distribution of the loglikelihood’s suggest no extreme outliers.
A more convenient way is to use the findinfluential function that provides a list of k top influential subjects by showing the normalized (minus) loglikelihood for each subject. As you can see below, the minus loglikelihood in the range of 16 agrees with the histogram plotted above.
influential_subjects = findinfluential(pk_1cmp, pkpain_noplb, pkparam, FOCE())120-element Vector{@NamedTuple{id::String, nll::Float64}}:
(id = "148", nll = 16.65965885684477)
(id = "135", nll = 16.648985190076335)
(id = "156", nll = 15.959069556607496)
(id = "159", nll = 15.441218240496484)
(id = "149", nll = 14.71513464411951)
(id = "88", nll = 13.09709837464614)
(id = "16", nll = 12.98228052193144)
(id = "61", nll = 12.652182902303679)
(id = "71", nll = 12.500330088085505)
(id = "59", nll = 12.241510254805235)
⋮
(id = "57", nll = -22.79767423253431)
(id = "93", nll = -22.836900711478208)
(id = "12", nll = -23.007742339519247)
(id = "123", nll = -23.292751843079234)
(id = "41", nll = -23.425412534960515)
(id = "99", nll = -23.535214841901112)
(id = "29", nll = -24.025959868383083)
(id = "52", nll = -24.164757842493685)
(id = "24", nll = -25.57209232565845)3.2.2.2 FOCE
Now that we have a good handle on our data, lets go ahead and fit a population model with FOCE:
pkfit_1cmp = fit(pk_1cmp, pkpain_noplb, pkparam, FOCE(); constantcoef = (; tvka = 2))[ 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 -5.935351e+02 5.597318e+02
* time: 7.605552673339844e-5
1 -7.022088e+02 1.707063e+02
* time: 0.3131880760192871
2 -7.314067e+02 2.903269e+02
* time: 0.5381979942321777
3 -8.520591e+02 2.285888e+02
* time: 0.6782650947570801
4 -1.120191e+03 3.795410e+02
* time: 0.9890539646148682
5 -1.178784e+03 2.323978e+02
* time: 1.1938450336456299
6 -1.218320e+03 9.699907e+01
* time: 1.324463129043579
7 -1.223641e+03 5.862105e+01
* time: 1.465644121170044
8 -1.227620e+03 1.831403e+01
* time: 1.6084120273590088
9 -1.228381e+03 2.132323e+01
* time: 1.7682039737701416
10 -1.230098e+03 2.921228e+01
* time: 1.888746976852417
11 -1.230854e+03 2.029662e+01
* time: 2.0225961208343506
12 -1.231116e+03 5.229097e+00
* time: 2.150954008102417
13 -1.231179e+03 1.689232e+00
* time: 2.3053860664367676
14 -1.231187e+03 1.215379e+00
* time: 2.4097530841827393
15 -1.231188e+03 2.770380e-01
* time: 2.5148589611053467
16 -1.231188e+03 1.636653e-01
* time: 2.609405994415283
17 -1.231188e+03 2.701133e-01
* time: 2.745171070098877
18 -1.231188e+03 3.163363e-01
* time: 2.8302531242370605
19 -1.231188e+03 1.505149e-01
* time: 2.9132330417633057
20 -1.231188e+03 2.484999e-02
* time: 2.997634172439575
21 -1.231188e+03 8.446863e-04
* time: 3.0766701698303223FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Closed form
Log-likelihood value: 1231.188
Number of subjects: 120
Number of parameters: Fixed Optimized
1 6
Observation records: Active Missing
Conc: 1320 0
Total: 1320 0
-------------------
Estimate
-------------------
tvcl 3.1642
tvv 13.288
tvka 2.0
Ω₁,₁ 0.08494
Ω₂,₂ 0.048568
Ω₃,₃ 5.5811
σ_p 0.10093
-------------------infer(pkfit_1cmp)[ Info: Calculating: variance-covariance matrix.
[ Info: Done.Asymptotic inference results using sandwich estimator
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Closed form
Log-likelihood value: 1231.188
Number of subjects: 120
Number of parameters: Fixed Optimized
1 6
Observation records: Active Missing
Conc: 1320 0
Total: 1320 0
-------------------------------------------------------------------
Estimate SE 95.0% C.I.
-------------------------------------------------------------------
tvcl 3.1642 0.08662 [ 2.9944 ; 3.334 ]
tvv 13.288 0.27481 [12.749 ; 13.827 ]
tvka 2.0 NaN [ NaN ; NaN ]
Ω₁,₁ 0.08494 0.011022 [ 0.063338; 0.10654 ]
Ω₂,₂ 0.048568 0.0063502 [ 0.036122; 0.061014]
Ω₃,₃ 5.5811 1.2188 [ 3.1922 ; 7.97 ]
σ_p 0.10093 0.0057196 [ 0.089718; 0.11214 ]
-------------------------------------------------------------------Notice that tvka is fixed to 2 as we don’t have a lot of information before tmax. From the results above, we see that the parameter precision for this model is reasonable.
3.2.3 Two-compartment model
Just to be sure, let’s fit a 2-compartment model and evaluate:
pk_2cmp = @model begin
@param begin
"""
Clearance (L/hr)
"""
tvcl ∈ RealDomain(; lower = 0, init = 3.2)
"""
Central Volume (L)
"""
tvv ∈ RealDomain(; lower = 0, init = 16.4)
"""
Peripheral Volume (L)
"""
tvvp ∈ RealDomain(; lower = 0, init = 10)
"""
Distributional Clearance (L/hr)
"""
tvq ∈ RealDomain(; lower = 0, init = 2)
"""
Absorption rate constant (h-1)
"""
tvka ∈ RealDomain(; lower = 0, init = 1.3)
"""
- ΩCL
- ΩVc
- ΩKa
- ΩVp
- ΩQ
"""
Ω ∈ PDiagDomain(init = [0.04, 0.04, 0.04, 0.04, 0.04])
"""
Proportional RUV
"""
σ_p ∈ RealDomain(; lower = 0.0001, init = 0.2)
end
@random begin
η ~ MvNormal(Ω)
end
@covariates begin
"""
Dose (mg)
"""
Dose
end
@pre begin
CL = tvcl * exp(η[1])
Vc = tvv * exp(η[2])
Ka = tvka * exp(η[3])
Vp = tvvp * exp(η[4])
Q = tvq * exp(η[5])
end
@dynamics Depots1Central1Periph1
@derived begin
cp := @. Central / Vc
"""
CTMx Concentration (ng/mL)
"""
Conc ~ @. Normal(cp, cp * σ_p)
end
end┌ Warning: Covariate Dose is not used in the model.
└ @ Pumas ~/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Pumas/aZRyj/src/dsl/model_macro.jl:2856PumasModel
Parameters: tvcl, tvv, tvvp, tvq, tvka, Ω, σ_p
Random effects: η
Covariates: Dose
Dynamical system variables: Depot, Central, Peripheral
Dynamical system type: Closed form
Derived: Conc
Observed: Conc3.2.3.1 FOCE
pkfit_2cmp =
fit(pk_2cmp, pkpain_noplb, init_params(pk_2cmp), FOCE(); constantcoef = (; tvka = 2))[ 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 -6.302369e+02 1.021050e+03
* time: 9.608268737792969e-5
1 -9.197817e+02 9.927951e+02
* time: 0.4063079357147217
2 -1.372640e+03 2.054986e+02
* time: 0.7915420532226562
3 -1.446326e+03 1.543987e+02
* time: 1.1256020069122314
4 -1.545570e+03 1.855028e+02
* time: 1.5302700996398926
5 -1.581449e+03 1.713157e+02
* time: 2.047740936279297
6 -1.639433e+03 1.257382e+02
* time: 2.4201691150665283
7 -1.695964e+03 7.450539e+01
* time: 2.7611329555511475
8 -1.722243e+03 5.961044e+01
* time: 3.1375930309295654
9 -1.736883e+03 7.320921e+01
* time: 3.494723081588745
10 -1.753547e+03 7.501938e+01
* time: 3.8869619369506836
11 -1.764053e+03 6.185661e+01
* time: 4.272141933441162
12 -1.778991e+03 4.831033e+01
* time: 4.683020114898682
13 -1.791492e+03 4.943278e+01
* time: 5.12828803062439
14 -1.799847e+03 2.871410e+01
* time: 5.488668918609619
15 -1.805374e+03 7.520791e+01
* time: 5.909689903259277
16 -1.816260e+03 2.990621e+01
* time: 6.248203992843628
17 -1.818252e+03 2.401915e+01
* time: 6.598495960235596
18 -1.822988e+03 2.587225e+01
* time: 6.924659967422485
19 -1.824653e+03 1.550517e+01
* time: 7.265336990356445
20 -1.826074e+03 1.788927e+01
* time: 7.583723068237305
21 -1.826821e+03 1.888389e+01
* time: 7.921534061431885
22 -1.827900e+03 1.432840e+01
* time: 8.236469030380249
23 -1.828511e+03 9.422041e+00
* time: 8.578222036361694
24 -1.828754e+03 5.363442e+00
* time: 8.954122066497803
25 -1.828862e+03 4.916159e+00
* time: 9.241298913955688
26 -1.829007e+03 4.695755e+00
* time: 9.609203100204468
27 -1.829358e+03 1.090249e+01
* time: 9.915071964263916
28 -1.829830e+03 1.451325e+01
* time: 10.270534038543701
29 -1.830201e+03 1.108715e+01
* time: 10.58910608291626
30 -1.830360e+03 2.891223e+00
* time: 10.936537981033325
31 -1.830390e+03 1.695557e+00
* time: 11.241647005081177
32 -1.830404e+03 1.601712e+00
* time: 11.556674003601074
33 -1.830432e+03 2.823385e+00
* time: 11.851758003234863
34 -1.830477e+03 4.060617e+00
* time: 12.184998989105225
35 -1.830528e+03 5.133499e+00
* time: 12.503180980682373
36 -1.830593e+03 2.830970e+00
* time: 12.854108095169067
37 -1.830616e+03 3.342835e+00
* time: 13.17300009727478
38 -1.830622e+03 3.708884e+00
* time: 13.529642105102539
39 -1.830625e+03 2.062934e+00
* time: 13.867832899093628
40 -1.830627e+03 1.278569e+00
* time: 14.118606090545654
41 -1.830628e+03 1.832895e+00
* time: 14.451056957244873
42 -1.830628e+03 3.768840e-01
* time: 14.701940059661865
43 -1.830629e+03 3.152895e-01
* time: 14.998039960861206
44 -1.830630e+03 4.871060e-01
* time: 15.258908033370972
45 -1.830630e+03 3.110627e-01
* time: 15.54232406616211
46 -1.830630e+03 2.687758e-02
* time: 15.832128047943115
47 -1.830630e+03 4.694018e-03
* time: 16.060713052749634
48 -1.830630e+03 8.272969e-03
* time: 16.30059003829956
49 -1.830630e+03 8.249151e-03
* time: 16.561913013458252
50 -1.830630e+03 8.245562e-03
* time: 16.913420915603638
51 -1.830630e+03 8.240030e-03
* time: 17.246982097625732
52 -1.830630e+03 8.240030e-03
* time: 17.631911039352417
53 -1.830630e+03 8.240030e-03
* time: 18.02180790901184FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Closed form
Log-likelihood value: 1830.6304
Number of subjects: 120
Number of parameters: Fixed Optimized
1 10
Observation records: Active Missing
Conc: 1320 0
Total: 1320 0
-------------------
Estimate
-------------------
tvcl 2.8138
tvv 11.005
tvvp 5.54
tvq 1.5159
tvka 2.0
Ω₁,₁ 0.10267
Ω₂,₂ 0.060776
Ω₃,₃ 1.2012
Ω₄,₄ 0.4235
Ω₅,₅ 0.24473
σ_p 0.048405
-------------------3.3 Comparing One- versus Two-compartment models
The 2-compartment model has a much lower objective function compared to the 1-compartment. Let’s compare the estimates from the 2 models using the compare_estimates function.
compare_estimates(; pkfit_1cmp, pkfit_2cmp)11×3 DataFrame
| Row | parameter | pkfit_1cmp | pkfit_2cmp |
|---|---|---|---|
| String | Float64? | Float64? | |
| 1 | tvcl | 3.1642 | 2.81378 |
| 2 | tvv | 13.288 | 11.0046 |
| 3 | tvka | 2.0 | 2.0 |
| 4 | Ω₁,₁ | 0.0849405 | 0.102669 |
| 5 | Ω₂,₂ | 0.0485682 | 0.0607757 |
| 6 | Ω₃,₃ | 5.58107 | 1.20115 |
| 7 | σ_p | 0.100928 | 0.0484049 |
| 8 | tvvp | missing | 5.53998 |
| 9 | tvq | missing | 1.51591 |
| 10 | Ω₄,₄ | missing | 0.423495 |
| 11 | Ω₅,₅ | missing | 0.244731 |
We perform a likelihood ratio test to compare the two nested models. The test statistic and the \(p\)-value clearly indicate that a 2-compartment model should be preferred.
lrtest(pkfit_1cmp, pkfit_2cmp)Statistic: 1200.0
Degrees of freedom: 4
P-value: 0.0We should also compare the other metrics and statistics, such ηshrinkage, ϵshrinkage, aic, and bic using the metrics_table function.
@chain metrics_table(pkfit_2cmp) begin
leftjoin(metrics_table(pkfit_1cmp); on = :Metric, makeunique = true)
rename!(:Value => :pk2cmp, :Value_1 => :pk1cmp)
end20×3 DataFrame
| Row | Metric | pk2cmp | pk1cmp |
|---|---|---|---|
| String | Any | Any | |
| 1 | Successful | true | true |
| 2 | Estimation Time | 18.022 | 3.077 |
| 3 | Subjects | 120 | 120 |
| 4 | Fixed Parameters | 1 | 1 |
| 5 | Optimized Parameters | 10 | 6 |
| 6 | Conc Active Observations | 1320 | 1320 |
| 7 | Conc Missing Observations | 0 | 0 |
| 8 | Total Active Observations | 1320 | 1320 |
| 9 | Total Missing Observations | 0 | 0 |
| 10 | Likelihood Approximation | Pumas.FOCE{Optim.NewtonTrustRegion{Float64}, Optim.Options{Float64, Nothing}} | Pumas.FOCE{Optim.NewtonTrustRegion{Float64}, Optim.Options{Float64, Nothing}} |
| 11 | LogLikelihood (LL) | 1830.63 | 1231.19 |
| 12 | -2LL | -3661.26 | -2462.38 |
| 13 | AIC | -3641.26 | -2450.38 |
| 14 | BIC | -3589.41 | -2419.26 |
| 15 | (η-shrinkage) η₁ | 0.037 | 0.016 |
| 16 | (η-shrinkage) η₂ | 0.047 | 0.04 |
| 17 | (η-shrinkage) η₃ | 0.516 | 0.733 |
| 18 | (ϵ-shrinkage) Conc | 0.185 | 0.105 |
| 19 | (η-shrinkage) η₄ | 0.287 | missing |
| 20 | (η-shrinkage) η₅ | 0.154 | missing |
We next generate some goodness of fit plots to compare which model is performing better. To do this, we first inspect the diagnostics of our model fit.
res_inspect_1cmp = inspect(pkfit_1cmp)[ 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, )
)res_inspect_2cmp = inspect(pkfit_2cmp)[ 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, )
)gof_1cmp = goodness_of_fit(res_inspect_1cmp; figure = (; fontsize = 12))
gof_2cmp = goodness_of_fit(res_inspect_2cmp; figure = (; fontsize = 12))
These plots clearly indicate that the 2-compartment model is a better fit compared to the 1-compartment model.
We can look at selected sample of individual plots.
fig_subject_fits = subject_fits(
res_inspect_2cmp;
separate = true,
paginate = true,
facet = (; combinelabels = true),
figure = (; fontsize = 18),
axis = (; xlabel = "Time (hr)", ylabel = "CTMx Concentration (ng/mL)"),
)
fig_subject_fits[1]
There a lot of important plotting functions you can use for your standard model diagnostics. Please make sure to read the documentation for plotting. Below, we are checking the distribution of the empirical Bayes estimates.
empirical_bayes_dist(res_inspect_2cmp; zeroline_color = :red)
empirical_bayes_vs_covariates(
res_inspect_2cmp;
categorical = [:Dose],
figure = (; size = (600, 800)),
)
Clearly, our guess at tvka seems off-target. Let’s try and estimate tvka instead of fixing it to 2:
pkfit_2cmp_unfix_ka = fit(pk_2cmp, pkpain_noplb, init_params(pk_2cmp), 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 -3.200734e+02 1.272671e+03
* time: 6.29425048828125e-5
1 -8.682982e+02 1.000199e+03
* time: 0.3818359375
2 -1.381870e+03 5.008081e+02
* time: 0.7042748928070068
3 -1.551053e+03 6.833490e+02
* time: 1.055907964706421
4 -1.680887e+03 1.834586e+02
* time: 1.3785779476165771
5 -1.726118e+03 8.870274e+01
* time: 1.6552119255065918
6 -1.761023e+03 1.162036e+02
* time: 1.9778239727020264
7 -1.786619e+03 1.114552e+02
* time: 2.310001850128174
8 -1.863556e+03 9.914305e+01
* time: 2.666949987411499
9 -1.882942e+03 5.342676e+01
* time: 3.010939836502075
10 -1.888020e+03 2.010181e+01
* time: 3.3433988094329834
11 -1.889832e+03 1.867263e+01
* time: 3.657357931137085
12 -1.891649e+03 1.668512e+01
* time: 3.9899909496307373
13 -1.892615e+03 1.820701e+01
* time: 4.318882942199707
14 -1.893453e+03 1.745195e+01
* time: 4.641011953353882
15 -1.894760e+03 1.850174e+01
* time: 4.972251892089844
16 -1.895647e+03 1.773939e+01
* time: 5.299329996109009
17 -1.896597e+03 1.143462e+01
* time: 5.597846984863281
18 -1.897114e+03 9.720097e+00
* time: 5.921367883682251
19 -1.897373e+03 6.054321e+00
* time: 6.249831914901733
20 -1.897498e+03 3.985954e+00
* time: 6.568563938140869
21 -1.897571e+03 4.262464e+00
* time: 6.909961938858032
22 -1.897633e+03 4.010234e+00
* time: 7.228219032287598
23 -1.897714e+03 4.805375e+00
* time: 7.511170864105225
24 -1.897802e+03 3.508706e+00
* time: 7.826236009597778
25 -1.897865e+03 3.691477e+00
* time: 8.138185024261475
26 -1.897900e+03 2.982720e+00
* time: 8.483932971954346
27 -1.897928e+03 2.563790e+00
* time: 8.794867992401123
28 -1.897968e+03 3.261485e+00
* time: 9.068126916885376
29 -1.898013e+03 3.064690e+00
* time: 9.369340896606445
30 -1.898040e+03 1.636525e+00
* time: 9.676731824874878
31 -1.898051e+03 1.439997e+00
* time: 9.983341932296753
32 -1.898057e+03 1.436504e+00
* time: 10.290406942367554
33 -1.898069e+03 1.881529e+00
* time: 10.565873861312866
34 -1.898095e+03 3.253165e+00
* time: 10.87034797668457
35 -1.898142e+03 4.257942e+00
* time: 11.18009901046753
36 -1.898199e+03 3.685241e+00
* time: 11.495620965957642
37 -1.898245e+03 2.567364e+00
* time: 11.838673830032349
38 -1.898246e+03 2.561591e+00
* time: 12.307148933410645
39 -1.898251e+03 2.530888e+00
* time: 12.726053953170776
40 -1.898298e+03 2.673696e+00
* time: 13.06554889678955
41 -1.898300e+03 2.794639e+00
* time: 13.453397989273071
42 -1.898337e+03 3.751590e+00
* time: 13.89956283569336
43 -1.898421e+03 4.878407e+00
* time: 14.199062824249268
44 -1.898433e+03 4.391719e+00
* time: 14.603793859481812
45 -1.898437e+03 4.216518e+00
* time: 15.164515972137451
46 -1.898442e+03 4.108397e+00
* time: 15.744963884353638
47 -1.898446e+03 3.934902e+00
* time: 16.38485097885132
48 -1.898449e+03 3.769838e+00
* time: 17.09025287628174
49 -1.898450e+03 3.739486e+00
* time: 17.76390290260315
50 -1.898450e+03 3.712049e+00
* time: 18.54351782798767
51 -1.898457e+03 3.623436e+00
* time: 19.244205951690674
52 -1.898471e+03 2.668312e+00
* time: 19.66318392753601
53 -1.898479e+03 2.302438e+00
* time: 20.08320903778076
54 -1.898480e+03 2.386566e-01
* time: 20.497478008270264
55 -1.898480e+03 7.802040e-01
* time: 20.844905853271484
56 -1.898480e+03 7.369786e-01
* time: 21.422022819519043
57 -1.898480e+03 5.113191e-01
* time: 21.904071807861328
58 -1.898480e+03 3.067709e-01
* time: 22.259647846221924
59 -1.898480e+03 3.076791e-01
* time: 22.62203288078308
60 -1.898480e+03 3.102066e-01
* time: 22.98664093017578
61 -1.898480e+03 3.102066e-01
* time: 23.4666268825531
62 -1.898480e+03 3.102069e-01
* time: 24.017258882522583
63 -1.898480e+03 3.102071e-01
* time: 24.85649585723877
64 -1.898480e+03 3.102074e-01
* time: 25.555357933044434
65 -1.898480e+03 3.102076e-01
* time: 26.227010011672974
66 -1.898480e+03 3.102079e-01
* time: 26.928642988204956
67 -1.898480e+03 3.102081e-01
* time: 27.587378978729248
68 -1.898480e+03 3.102081e-01
* time: 28.27103090286255
69 -1.898480e+03 3.102081e-01
* time: 28.96690082550049
70 -1.898480e+03 3.102082e-01
* time: 29.654830932617188
71 -1.898480e+03 3.102082e-01
* time: 30.359181880950928
72 -1.898480e+03 3.102082e-01
* time: 31.04028081893921
73 -1.898480e+03 3.102102e-01
* time: 31.720247983932495
74 -1.898480e+03 3.102102e-01
* time: 32.2472620010376
75 -1.898480e+03 3.102096e-01
* time: 32.72541904449463
76 -1.898480e+03 3.102096e-01
* time: 33.22709393501282
77 -1.898480e+03 3.125688e-01
* time: 33.67114186286926
78 -1.898480e+03 3.125640e-01
* time: 34.127923011779785
79 -1.898480e+03 3.125618e-01
* time: 34.57367300987244
80 -1.898480e+03 3.125615e-01
* time: 35.06596398353577
81 -1.898480e+03 3.125612e-01
* time: 35.58578395843506
82 -1.898480e+03 3.125610e-01
* time: 36.074020862579346
83 -1.898480e+03 3.125609e-01
* time: 36.619977951049805
84 -1.898480e+03 3.125604e-01
* time: 37.10071086883545
85 -1.898480e+03 3.125602e-01
* time: 37.566136837005615
86 -1.898480e+03 3.125602e-01
* time: 38.07646989822388
87 -1.898480e+03 3.125602e-01
* time: 38.78900098800659
88 -1.898480e+03 3.125602e-01
* time: 39.5242018699646
89 -1.898480e+03 3.125602e-01
* time: 40.272294998168945
90 -1.898480e+03 3.125602e-01
* time: 40.997889041900635
91 -1.898480e+03 3.125602e-01
* time: 41.73891997337341
92 -1.898480e+03 3.125602e-01
* time: 42.23625087738037
93 -1.898480e+03 3.125602e-01
* time: 42.752264976501465
94 -1.898480e+03 3.125602e-01
* time: 43.26782703399658
95 -1.898480e+03 1.387453e-01
* time: 43.58665990829468
96 -1.898480e+03 1.387453e-01
* time: 43.98843693733215
97 -1.898480e+03 1.387453e-01
* time: 44.52016282081604
98 -1.898480e+03 1.387453e-01
* time: 45.06600093841553
99 -1.898480e+03 1.387453e-01
* time: 45.456135988235474FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Closed form
Log-likelihood value: 1898.4797
Number of subjects: 120
Number of parameters: Fixed Optimized
0 11
Observation records: Active Missing
Conc: 1320 0
Total: 1320 0
-------------------
Estimate
-------------------
tvcl 2.6191
tvv 11.378
tvvp 8.453
tvq 1.3164
tvka 4.8926
Ω₁,₁ 0.13243
Ω₂,₂ 0.059669
Ω₃,₃ 0.41581
Ω₄,₄ 0.080679
Ω₅,₅ 0.24996
σ_p 0.049097
-------------------compare_estimates(; pkfit_2cmp, pkfit_2cmp_unfix_ka)11×3 DataFrame
| Row | parameter | pkfit_2cmp | pkfit_2cmp_unfix_ka |
|---|---|---|---|
| String | Float64? | Float64? | |
| 1 | tvcl | 2.81378 | 2.6191 |
| 2 | tvv | 11.0046 | 11.3784 |
| 3 | tvvp | 5.53998 | 8.45297 |
| 4 | tvq | 1.51591 | 1.31637 |
| 5 | tvka | 2.0 | 4.89257 |
| 6 | Ω₁,₁ | 0.102669 | 0.132432 |
| 7 | Ω₂,₂ | 0.0607757 | 0.0596693 |
| 8 | Ω₃,₃ | 1.20115 | 0.415811 |
| 9 | Ω₄,₄ | 0.423495 | 0.0806789 |
| 10 | Ω₅,₅ | 0.244731 | 0.249956 |
| 11 | σ_p | 0.0484049 | 0.0490975 |
Let’s revaluate the goodness of fits and η distribution plots.
Not much change in the general gof plots
res_inspect_2cmp_unfix_ka = inspect(pkfit_2cmp_unfix_ka)[ 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, )
)goodness_of_fit(res_inspect_2cmp_unfix_ka; figure = (; fontsize = 12))
But you can see a huge improvement in the ηka, (η₃) distribution which is now centered around zero
empirical_bayes_vs_covariates(
res_inspect_2cmp_unfix_ka;
categorical = [:Dose],
ebes = [:η₃],
figure = (; size = (600, 800)),
)
Finally looking at some individual plots for the same subjects as earlier:
fig_subject_fits2 = subject_fits(
res_inspect_2cmp_unfix_ka;
separate = true,
paginate = true,
facet = (; combinelabels = true, linkyaxes = false),
figure = (; fontsize = 18),
axis = (; xlabel = "Time (hr)", ylabel = "CTMx Concentration (ng/mL)"),
)
fig_subject_fits2[6]
The randomly sampled individual fits don’t seem good in some individuals, but we can evaluate this via a vpc to see how to go about.
3.4 Visual Predictive Checks (VPC)
We can now perform a vpc to check. The default plots provide a 80% prediction interval and a 95% simulated CI (shaded area) around each of the quantiles
pk_vpc = vpc(
pkfit_2cmp_unfix_ka,
200;
observations = [:Conc],
stratify_by = [:Dose],
ensemblealg = EnsembleThreads(), # multi-threading
)[ Info: Continuous VPCVisual Predictive Check
Type of VPC: Continuous VPC
Simulated populations: 200
Subjects in data: 120
Stratification variable(s): [:Dose]
Confidence level: 0.95
VPC lines: quantiles ([0.1, 0.5, 0.9])vpc_plot(
pk_2cmp,
pk_vpc;
rows = 1,
columns = 3,
figure = (; size = (1400, 1000), fontsize = 22),
axis = (;
xlabel = "Time (hr)",
ylabel = "Observed/Predicted\n CTMx Concentration (ng/mL)",
),
facet = (; combinelabels = true),
)
The visual predictive check suggests that the model captures the data well across all dose levels.
4 Additional Help
If you have questions regarding this tutorial, please post them on our discourse site.