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: 1.713428020477295
1 2.343899e+02 1.747348e+03
* time: 2.442026138305664
2 9.696232e+01 1.198088e+03
* time: 2.4444739818573
3 -7.818699e+01 5.538151e+02
* time: 2.446171998977661
4 -1.234803e+02 2.462514e+02
* time: 2.447812080383301
5 -1.372888e+02 2.067458e+02
* time: 2.4494149684906006
6 -1.410579e+02 1.162950e+02
* time: 2.4510200023651123
7 -1.434754e+02 5.632816e+01
* time: 2.4526379108428955
8 -1.453401e+02 7.859270e+01
* time: 2.4542629718780518
9 -1.498185e+02 1.455606e+02
* time: 2.4559760093688965
10 -1.534371e+02 1.303682e+02
* time: 2.4577670097351074
11 -1.563557e+02 5.975474e+01
* time: 2.459545135498047
12 -1.575052e+02 9.308611e+00
* time: 2.4613120555877686
13 -1.579357e+02 1.234484e+01
* time: 2.4630699157714844
14 -1.581874e+02 7.478196e+00
* time: 2.464935064315796
15 -1.582981e+02 2.027162e+00
* time: 2.4668431282043457
16 -1.583375e+02 5.578262e+00
* time: 2.4688150882720947
17 -1.583556e+02 4.727050e+00
* time: 2.470684051513672
18 -1.583644e+02 2.340173e+00
* time: 2.472554922103882
19 -1.583680e+02 7.738100e-01
* time: 2.474376916885376
20 -1.583696e+02 3.300689e-01
* time: 2.476161003112793
21 -1.583704e+02 3.641985e-01
* time: 2.4778950214385986
22 -1.583707e+02 4.365901e-01
* time: 2.479696035385132
23 -1.583709e+02 3.887800e-01
* time: 2.481490135192871
24 -1.583710e+02 2.766977e-01
* time: 2.4845259189605713
25 -1.583710e+02 1.758029e-01
* time: 2.4865050315856934
26 -1.583710e+02 1.133947e-01
* time: 2.488286018371582
27 -1.583710e+02 7.922544e-02
* time: 2.4899449348449707
28 -1.583710e+02 5.954998e-02
* time: 2.4916179180145264
29 -1.583710e+02 4.157079e-02
* time: 2.493360996246338
30 -1.583710e+02 4.295447e-02
* time: 2.4951469898223877
31 -1.583710e+02 5.170753e-02
* time: 2.4969370365142822
32 -1.583710e+02 2.644383e-02
* time: 2.4993410110473633
33 -1.583710e+02 4.548993e-03
* time: 2.5019009113311768
34 -1.583710e+02 2.501804e-02
* time: 2.504662036895752
35 -1.583710e+02 3.763440e-02
* time: 2.506448984146118
36 -1.583710e+02 3.206026e-02
* time: 2.508155107498169
37 -1.583710e+02 1.003698e-02
* time: 2.5097649097442627
38 -1.583710e+02 2.209089e-02
* time: 2.5112879276275635
39 -1.583710e+02 4.954172e-03
* time: 2.5127999782562256
40 -1.583710e+02 1.609373e-02
* time: 2.5153090953826904
41 -1.583710e+02 1.579802e-02
* time: 2.516775131225586
42 -1.583710e+02 1.014113e-03
* time: 2.5185680389404297
43 -1.583710e+02 6.050644e-03
* time: 2.5210371017456055
44 -1.583710e+02 1.354412e-02
* time: 2.5228641033172607
45 -1.583710e+02 4.473248e-03
* time: 2.524730920791626
46 -1.583710e+02 4.644735e-03
* time: 2.5261189937591553
47 -1.583710e+02 9.829910e-03
* time: 2.527967929840088
48 -1.583710e+02 1.047561e-03
* time: 2.529910087585449
49 -1.583710e+02 8.366895e-03
* time: 2.531398057937622
50 -1.583710e+02 7.879055e-04
* time: 2.533268928527832FittedPumasModel
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: 7.581710815429688e-5
1 -7.022088e+02 1.707063e+02
* time: 0.31902599334716797
2 -7.314067e+02 2.903269e+02
* time: 0.5302929878234863
3 -8.520591e+02 2.285888e+02
* time: 0.6740000247955322
4 -1.120191e+03 3.795410e+02
* time: 1.0613529682159424
5 -1.178784e+03 2.323978e+02
* time: 1.197326898574829
6 -1.218320e+03 9.699907e+01
* time: 1.3325469493865967
7 -1.223641e+03 5.862105e+01
* time: 1.4668078422546387
8 -1.227620e+03 1.831403e+01
* time: 1.628661870956421
9 -1.228381e+03 2.132323e+01
* time: 1.7356178760528564
10 -1.230098e+03 2.921228e+01
* time: 1.861311912536621
11 -1.230854e+03 2.029662e+01
* time: 1.9884798526763916
12 -1.231116e+03 5.229097e+00
* time: 2.132563829421997
13 -1.231179e+03 1.689232e+00
* time: 2.230721950531006
14 -1.231187e+03 1.215379e+00
* time: 2.3364429473876953
15 -1.231188e+03 2.770380e-01
* time: 2.43475079536438
16 -1.231188e+03 1.636653e-01
* time: 2.5534520149230957
17 -1.231188e+03 2.701133e-01
* time: 2.6290578842163086
18 -1.231188e+03 3.163363e-01
* time: 2.706772804260254
19 -1.231188e+03 1.505149e-01
* time: 2.793069839477539
20 -1.231188e+03 2.484999e-02
* time: 2.8723859786987305
21 -1.231188e+03 8.446863e-04
* time: 2.978125810623169FittedPumasModel
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: 6.794929504394531e-5
1 -9.197817e+02 9.927951e+02
* time: 0.30261683464050293
2 -1.372640e+03 2.054986e+02
* time: 0.5687398910522461
3 -1.446326e+03 1.543987e+02
* time: 0.8557100296020508
4 -1.545570e+03 1.855028e+02
* time: 1.1449289321899414
5 -1.581449e+03 1.713157e+02
* time: 1.572429895401001
6 -1.639433e+03 1.257382e+02
* time: 1.8320250511169434
7 -1.695964e+03 7.450539e+01
* time: 2.093984842300415
8 -1.722243e+03 5.961044e+01
* time: 2.3543899059295654
9 -1.736883e+03 7.320921e+01
* time: 2.6266160011291504
10 -1.753547e+03 7.501938e+01
* time: 2.8995280265808105
11 -1.764053e+03 6.185661e+01
* time: 3.180608034133911
12 -1.778991e+03 4.831033e+01
* time: 3.4747869968414307
13 -1.791492e+03 4.943278e+01
* time: 3.785245895385742
14 -1.799847e+03 2.871410e+01
* time: 4.112076044082642
15 -1.805374e+03 7.520791e+01
* time: 4.435018062591553
16 -1.816260e+03 2.990621e+01
* time: 4.746316909790039
17 -1.818252e+03 2.401915e+01
* time: 5.0217649936676025
18 -1.822988e+03 2.587225e+01
* time: 5.310658931732178
19 -1.824653e+03 1.550517e+01
* time: 5.57533597946167
20 -1.826074e+03 1.788927e+01
* time: 5.853234052658081
21 -1.826821e+03 1.888389e+01
* time: 6.1313090324401855
22 -1.827900e+03 1.432840e+01
* time: 6.407709836959839
23 -1.828511e+03 9.422041e+00
* time: 6.692572832107544
24 -1.828754e+03 5.363442e+00
* time: 6.984563827514648
25 -1.828862e+03 4.916159e+00
* time: 7.260490894317627
26 -1.829007e+03 4.695755e+00
* time: 7.536607027053833
27 -1.829358e+03 1.090249e+01
* time: 7.844152927398682
28 -1.829830e+03 1.451325e+01
* time: 8.135375022888184
29 -1.830201e+03 1.108715e+01
* time: 8.42812204360962
30 -1.830360e+03 2.891223e+00
* time: 8.720229864120483
31 -1.830390e+03 1.695557e+00
* time: 8.996635913848877
32 -1.830404e+03 1.601712e+00
* time: 9.272814989089966
33 -1.830432e+03 2.823385e+00
* time: 9.54322600364685
34 -1.830477e+03 4.060617e+00
* time: 9.825235843658447
35 -1.830528e+03 5.133499e+00
* time: 10.111971855163574
36 -1.830593e+03 2.830970e+00
* time: 10.401591062545776
37 -1.830616e+03 3.342835e+00
* time: 10.683997869491577
38 -1.830622e+03 3.708884e+00
* time: 10.96933889389038
39 -1.830625e+03 2.062934e+00
* time: 11.237452030181885
40 -1.830627e+03 1.278569e+00
* time: 11.486078023910522
41 -1.830628e+03 1.832895e+00
* time: 11.753257036209106
42 -1.830628e+03 3.768840e-01
* time: 12.002631902694702
43 -1.830629e+03 3.152895e-01
* time: 12.23666501045227
44 -1.830630e+03 4.871060e-01
* time: 12.495190858840942
45 -1.830630e+03 3.110627e-01
* time: 12.7477388381958
46 -1.830630e+03 2.687758e-02
* time: 12.99698805809021
47 -1.830630e+03 4.694018e-03
* time: 13.185861825942993
48 -1.830630e+03 8.272969e-03
* time: 13.386101007461548
49 -1.830630e+03 8.249151e-03
* time: 13.643970966339111
50 -1.830630e+03 8.245562e-03
* time: 13.954442977905273
51 -1.830630e+03 8.240030e-03
* time: 14.268861055374146
52 -1.830630e+03 8.240030e-03
* time: 14.629942893981934
53 -1.830630e+03 8.240030e-03
* time: 14.964189052581787FittedPumasModel
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 | 14.964 | 2.978 |
| 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: 7.200241088867188e-5
1 -8.682982e+02 1.000199e+03
* time: 0.334136962890625
2 -1.381870e+03 5.008081e+02
* time: 0.6585738658905029
3 -1.551053e+03 6.833490e+02
* time: 1.004654884338379
4 -1.680887e+03 1.834586e+02
* time: 1.3426499366760254
5 -1.726118e+03 8.870274e+01
* time: 1.672989845275879
6 -1.761023e+03 1.162036e+02
* time: 1.9619579315185547
7 -1.786619e+03 1.114552e+02
* time: 2.284024953842163
8 -1.863556e+03 9.914305e+01
* time: 2.6183228492736816
9 -1.882942e+03 5.342676e+01
* time: 2.9576239585876465
10 -1.888020e+03 2.010181e+01
* time: 3.290156841278076
11 -1.889832e+03 1.867263e+01
* time: 3.5988688468933105
12 -1.891649e+03 1.668512e+01
* time: 3.922437906265259
13 -1.892615e+03 1.820701e+01
* time: 4.245962858200073
14 -1.893453e+03 1.745195e+01
* time: 4.573210000991821
15 -1.894760e+03 1.850174e+01
* time: 4.907455921173096
16 -1.895647e+03 1.773939e+01
* time: 5.196255922317505
17 -1.896597e+03 1.143462e+01
* time: 5.523356914520264
18 -1.897114e+03 9.720097e+00
* time: 5.844550848007202
19 -1.897373e+03 6.054321e+00
* time: 6.17809796333313
20 -1.897498e+03 3.985954e+00
* time: 6.47012996673584
21 -1.897571e+03 4.262464e+00
* time: 6.77438497543335
22 -1.897633e+03 4.010234e+00
* time: 7.079877853393555
23 -1.897714e+03 4.805375e+00
* time: 7.390949964523315
24 -1.897802e+03 3.508706e+00
* time: 7.71281886100769
25 -1.897865e+03 3.691477e+00
* time: 7.990954875946045
26 -1.897900e+03 2.982720e+00
* time: 8.293629884719849
27 -1.897928e+03 2.563790e+00
* time: 8.590051889419556
28 -1.897968e+03 3.261485e+00
* time: 8.899805784225464
29 -1.898013e+03 3.064690e+00
* time: 9.173626899719238
30 -1.898040e+03 1.636525e+00
* time: 9.478863954544067
31 -1.898051e+03 1.439997e+00
* time: 9.785226821899414
32 -1.898057e+03 1.436504e+00
* time: 10.089764833450317
33 -1.898069e+03 1.881529e+00
* time: 10.363648891448975
34 -1.898095e+03 3.253165e+00
* time: 10.662033796310425
35 -1.898142e+03 4.257942e+00
* time: 10.963122844696045
36 -1.898199e+03 3.685241e+00
* time: 11.275807857513428
37 -1.898245e+03 2.567364e+00
* time: 11.60645580291748
38 -1.898246e+03 2.561591e+00
* time: 12.055300951004028
39 -1.898251e+03 2.530888e+00
* time: 12.457650899887085
40 -1.898298e+03 2.673696e+00
* time: 12.74339485168457
41 -1.898300e+03 2.794639e+00
* time: 13.127835988998413
42 -1.898337e+03 3.751590e+00
* time: 13.574963808059692
43 -1.898421e+03 4.878407e+00
* time: 13.898703813552856
44 -1.898433e+03 4.391719e+00
* time: 14.29328179359436
45 -1.898437e+03 4.216518e+00
* time: 14.771390914916992
46 -1.898442e+03 4.108397e+00
* time: 15.238492012023926
47 -1.898446e+03 3.934902e+00
* time: 15.715795993804932
48 -1.898449e+03 3.769838e+00
* time: 16.18308401107788
49 -1.898450e+03 3.739486e+00
* time: 16.652670860290527
50 -1.898450e+03 3.712049e+00
* time: 17.13361382484436
51 -1.898457e+03 3.623436e+00
* time: 17.548874855041504
52 -1.898471e+03 2.668312e+00
* time: 17.873390913009644
53 -1.898479e+03 2.302438e+00
* time: 18.195745944976807
54 -1.898480e+03 2.386566e-01
* time: 18.51852798461914
55 -1.898480e+03 7.802040e-01
* time: 18.783370971679688
56 -1.898480e+03 7.369786e-01
* time: 19.225034952163696
57 -1.898480e+03 5.113191e-01
* time: 19.603256940841675
58 -1.898480e+03 3.067709e-01
* time: 19.88420581817627
59 -1.898480e+03 3.076791e-01
* time: 20.178939819335938
60 -1.898480e+03 3.102066e-01
* time: 20.444341897964478
61 -1.898480e+03 3.102066e-01
* time: 20.824752807617188
62 -1.898480e+03 3.102069e-01
* time: 21.268054962158203
63 -1.898480e+03 3.102071e-01
* time: 21.931060791015625
64 -1.898480e+03 3.102074e-01
* time: 22.568915843963623
65 -1.898480e+03 3.102076e-01
* time: 23.217228889465332
66 -1.898480e+03 3.102079e-01
* time: 23.872769832611084
67 -1.898480e+03 3.102081e-01
* time: 24.534252882003784
68 -1.898480e+03 3.102081e-01
* time: 25.236604928970337
69 -1.898480e+03 3.102081e-01
* time: 25.886016845703125
70 -1.898480e+03 3.102082e-01
* time: 26.524367809295654
71 -1.898480e+03 3.102082e-01
* time: 27.200507879257202
72 -1.898480e+03 3.102082e-01
* time: 27.857213973999023
73 -1.898480e+03 3.102102e-01
* time: 28.491069793701172
74 -1.898480e+03 3.102102e-01
* time: 29.05099391937256
75 -1.898480e+03 3.102096e-01
* time: 29.514575004577637
76 -1.898480e+03 3.102096e-01
* time: 29.990442991256714
77 -1.898480e+03 3.125688e-01
* time: 30.432599782943726
78 -1.898480e+03 3.125640e-01
* time: 30.88490390777588
79 -1.898480e+03 3.125618e-01
* time: 31.324219942092896
80 -1.898480e+03 3.125615e-01
* time: 31.807673931121826
81 -1.898480e+03 3.125612e-01
* time: 32.29063391685486
82 -1.898480e+03 3.125610e-01
* time: 32.785194873809814
83 -1.898480e+03 3.125609e-01
* time: 33.3365797996521
84 -1.898480e+03 3.125604e-01
* time: 33.816112995147705
85 -1.898480e+03 3.125602e-01
* time: 34.27131390571594
86 -1.898480e+03 3.125602e-01
* time: 34.777485847473145
87 -1.898480e+03 3.125602e-01
* time: 35.48393678665161
88 -1.898480e+03 3.125602e-01
* time: 36.17909598350525
89 -1.898480e+03 3.125602e-01
* time: 36.90882682800293
90 -1.898480e+03 3.125602e-01
* time: 37.60477900505066
91 -1.898480e+03 3.125602e-01
* time: 38.33198380470276
92 -1.898480e+03 3.125602e-01
* time: 38.81040978431702
93 -1.898480e+03 3.125602e-01
* time: 39.30093693733215
94 -1.898480e+03 3.125602e-01
* time: 39.77143383026123
95 -1.898480e+03 1.387453e-01
* time: 40.117082834243774
96 -1.898480e+03 1.387453e-01
* time: 40.50619196891785
97 -1.898480e+03 1.387453e-01
* time: 40.99233078956604
98 -1.898480e+03 1.387453e-01
* time: 41.53955292701721
99 -1.898480e+03 1.387453e-01
* time: 41.907896995544434FittedPumasModel
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.