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: 0.02156209945678711
1 2.343899e+02 1.747348e+03
* time: 0.8339729309082031
2 9.696232e+01 1.198088e+03
* time: 0.8372678756713867
3 -7.818699e+01 5.538151e+02
* time: 0.8398690223693848
4 -1.234803e+02 2.462514e+02
* time: 0.8425559997558594
5 -1.372888e+02 2.067458e+02
* time: 0.8452439308166504
6 -1.410579e+02 1.162950e+02
* time: 0.8478620052337646
7 -1.434754e+02 5.632816e+01
* time: 0.850471019744873
8 -1.453401e+02 7.859270e+01
* time: 0.8530569076538086
9 -1.498185e+02 1.455606e+02
* time: 0.8556530475616455
10 -1.534371e+02 1.303682e+02
* time: 0.8580820560455322
11 -1.563557e+02 5.975474e+01
* time: 0.8605079650878906
12 -1.575052e+02 9.308611e+00
* time: 0.8630430698394775
13 -1.579357e+02 1.234484e+01
* time: 0.8657219409942627
14 -1.581874e+02 7.478196e+00
* time: 0.8682129383087158
15 -1.582981e+02 2.027162e+00
* time: 0.870703935623169
16 -1.583375e+02 5.578262e+00
* time: 0.8732490539550781
17 -1.583556e+02 4.727050e+00
* time: 0.8757350444793701
18 -1.583644e+02 2.340173e+00
* time: 0.8782000541687012
19 -1.583680e+02 7.738100e-01
* time: 0.8806700706481934
20 -1.583696e+02 3.300689e-01
* time: 0.8832099437713623
21 -1.583704e+02 3.641985e-01
* time: 0.885685920715332
22 -1.583707e+02 4.365901e-01
* time: 0.8881509304046631
23 -1.583709e+02 3.887800e-01
* time: 0.890639066696167
24 -1.583710e+02 2.766977e-01
* time: 0.8931601047515869
25 -1.583710e+02 1.758029e-01
* time: 0.8955810070037842
26 -1.583710e+02 1.133947e-01
* time: 0.8981220722198486
27 -1.583710e+02 7.922544e-02
* time: 0.900709867477417
28 -1.583710e+02 5.954998e-02
* time: 0.9033980369567871
29 -1.583710e+02 4.157079e-02
* time: 0.906182050704956
30 -1.583710e+02 4.295447e-02
* time: 0.9088840484619141
31 -1.583710e+02 5.170753e-02
* time: 1.0704209804534912
32 -1.583710e+02 2.644383e-02
* time: 1.0737268924713135
33 -1.583710e+02 4.548993e-03
* time: 1.0765469074249268
34 -1.583710e+02 2.501804e-02
* time: 1.0791969299316406
35 -1.583710e+02 3.763440e-02
* time: 1.0811920166015625
36 -1.583710e+02 3.206026e-02
* time: 1.083172082901001
37 -1.583710e+02 1.003698e-02
* time: 1.0853180885314941
38 -1.583710e+02 2.209089e-02
* time: 1.0874629020690918
39 -1.583710e+02 4.954172e-03
* time: 1.0895729064941406
40 -1.583710e+02 1.609373e-02
* time: 1.0922091007232666
41 -1.583710e+02 1.579802e-02
* time: 1.094048023223877
42 -1.583710e+02 1.014113e-03
* time: 1.095870018005371
43 -1.583710e+02 6.050644e-03
* time: 1.0983729362487793
44 -1.583710e+02 1.354412e-02
* time: 1.1002819538116455
45 -1.583710e+02 4.473248e-03
* time: 1.1021230220794678
46 -1.583710e+02 4.644735e-03
* time: 1.1039209365844727
47 -1.583710e+02 9.829910e-03
* time: 1.1058390140533447
48 -1.583710e+02 1.047561e-03
* time: 1.107630968093872
49 -1.583710e+02 8.366895e-03
* time: 1.1096100807189941
50 -1.583710e+02 7.879055e-04
* time: 1.1113650798797607FittedPumasModel
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: 7.677078247070312e-5
1 -7.022088e+02 1.707063e+02
* time: 2.0021729469299316
2 -7.314067e+02 2.903269e+02
* time: 2.153850793838501
3 -8.520591e+02 2.285888e+02
* time: 2.319225788116455
4 -1.120191e+03 3.795410e+02
* time: 2.726613998413086
5 -1.178784e+03 2.323978e+02
* time: 2.872412919998169
6 -1.218320e+03 9.699907e+01
* time: 3.0526278018951416
7 -1.223641e+03 5.862105e+01
* time: 3.1764578819274902
8 -1.227620e+03 1.831403e+01
* time: 3.3014559745788574
9 -1.228381e+03 2.132323e+01
* time: 3.4573428630828857
10 -1.230098e+03 2.921228e+01
* time: 3.5831899642944336
11 -1.230854e+03 2.029662e+01
* time: 3.7117679119110107
12 -1.231116e+03 5.229097e+00
* time: 3.8451437950134277
13 -1.231179e+03 1.689232e+00
* time: 3.9541478157043457
14 -1.231187e+03 1.215379e+00
* time: 4.096207857131958
15 -1.231188e+03 2.770380e-01
* time: 4.189993858337402
16 -1.231188e+03 1.636653e-01
* time: 4.270173788070679
17 -1.231188e+03 2.701133e-01
* time: 4.361715793609619
18 -1.231188e+03 3.163363e-01
* time: 4.465590953826904
19 -1.231188e+03 1.505149e-01
* time: 4.549017906188965
20 -1.231188e+03 2.484999e-02
* time: 4.628176927566528
21 -1.231188e+03 8.446863e-04
* time: 4.72223687171936FittedPumasModel
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: 7.009506225585938e-5
1 -9.197817e+02 9.927951e+02
* time: 0.2710580825805664
2 -1.372640e+03 2.054986e+02
* time: 0.5651061534881592
3 -1.446326e+03 1.543987e+02
* time: 0.8720130920410156
4 -1.545570e+03 1.855028e+02
* time: 1.1435761451721191
5 -1.581449e+03 1.713157e+02
* time: 1.5802390575408936
6 -1.639433e+03 1.257382e+02
* time: 1.856180191040039
7 -1.695964e+03 7.450539e+01
* time: 2.132009983062744
8 -1.722243e+03 5.961044e+01
* time: 2.4169211387634277
9 -1.736883e+03 7.320921e+01
* time: 2.6618921756744385
10 -1.753547e+03 7.501938e+01
* time: 2.970576047897339
11 -1.764053e+03 6.185661e+01
* time: 3.2695841789245605
12 -1.778991e+03 4.831033e+01
* time: 3.542302131652832
13 -1.791492e+03 4.943278e+01
* time: 3.8526840209960938
14 -1.799847e+03 2.871410e+01
* time: 4.190831184387207
15 -1.805374e+03 7.520791e+01
* time: 4.528662204742432
16 -1.816260e+03 2.990621e+01
* time: 4.853327035903931
17 -1.818252e+03 2.401915e+01
* time: 5.109703063964844
18 -1.822988e+03 2.587225e+01
* time: 5.399553060531616
19 -1.824653e+03 1.550517e+01
* time: 5.709488153457642
20 -1.826074e+03 1.788927e+01
* time: 6.011183023452759
21 -1.826821e+03 1.888389e+01
* time: 6.269635200500488
22 -1.827900e+03 1.432840e+01
* time: 6.564840078353882
23 -1.828511e+03 9.422041e+00
* time: 6.859769105911255
24 -1.828754e+03 5.363442e+00
* time: 7.1353631019592285
25 -1.828862e+03 4.916159e+00
* time: 7.412130117416382
26 -1.829007e+03 4.695755e+00
* time: 7.702583074569702
27 -1.829358e+03 1.090249e+01
* time: 8.008338212966919
28 -1.829830e+03 1.451325e+01
* time: 8.272639989852905
29 -1.830201e+03 1.108715e+01
* time: 8.592926025390625
30 -1.830360e+03 2.891223e+00
* time: 8.900094032287598
31 -1.830390e+03 1.695557e+00
* time: 9.193721055984497
32 -1.830404e+03 1.601712e+00
* time: 9.435672998428345
33 -1.830432e+03 2.823385e+00
* time: 9.709078073501587
34 -1.830477e+03 4.060617e+00
* time: 9.999872207641602
35 -1.830528e+03 5.133499e+00
* time: 10.266378164291382
36 -1.830593e+03 2.830970e+00
* time: 10.555169105529785
37 -1.830616e+03 3.342835e+00
* time: 10.845858097076416
38 -1.830622e+03 3.708884e+00
* time: 11.140149116516113
39 -1.830625e+03 2.062934e+00
* time: 11.397561073303223
40 -1.830627e+03 1.278569e+00
* time: 11.669556140899658
41 -1.830628e+03 1.832895e+00
* time: 11.953103065490723
42 -1.830628e+03 3.768840e-01
* time: 12.185222148895264
43 -1.830629e+03 3.152895e-01
* time: 12.427147150039673
44 -1.830630e+03 4.871060e-01
* time: 12.699096202850342
45 -1.830630e+03 3.110627e-01
* time: 12.934239149093628
46 -1.830630e+03 2.687758e-02
* time: 13.179856061935425
47 -1.830630e+03 4.694018e-03
* time: 13.397958993911743
48 -1.830630e+03 8.272969e-03
* time: 13.570129156112671
49 -1.830630e+03 8.249151e-03
* time: 13.813857078552246
50 -1.830630e+03 8.245562e-03
* time: 14.130325078964233
51 -1.830630e+03 8.240030e-03
* time: 14.450873136520386
52 -1.830630e+03 8.240030e-03
* time: 14.760737180709839
53 -1.830630e+03 8.240030e-03
* time: 15.100085020065308FittedPumasModel
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 | 15.1 | 4.722 |
| 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: 7.295608520507812e-5
1 -8.682982e+02 1.000199e+03
* time: 0.35883092880249023
2 -1.381870e+03 5.008081e+02
* time: 0.6725928783416748
3 -1.551053e+03 6.833490e+02
* time: 1.0004088878631592
4 -1.680887e+03 1.834586e+02
* time: 1.3138949871063232
5 -1.726118e+03 8.870274e+01
* time: 1.6186330318450928
6 -1.761023e+03 1.162036e+02
* time: 1.8976669311523438
7 -1.786619e+03 1.114552e+02
* time: 2.2057838439941406
8 -1.863556e+03 9.914305e+01
* time: 2.5291078090667725
9 -1.882942e+03 5.342676e+01
* time: 2.8589258193969727
10 -1.888020e+03 2.010181e+01
* time: 3.1928110122680664
11 -1.889832e+03 1.867263e+01
* time: 3.5259838104248047
12 -1.891649e+03 1.668512e+01
* time: 3.8090100288391113
13 -1.892615e+03 1.820701e+01
* time: 4.118948936462402
14 -1.893453e+03 1.745195e+01
* time: 4.42402982711792
15 -1.894760e+03 1.850174e+01
* time: 4.7284300327301025
16 -1.895647e+03 1.773939e+01
* time: 5.038611888885498
17 -1.896597e+03 1.143462e+01
* time: 5.354621887207031
18 -1.897114e+03 9.720097e+00
* time: 5.642088890075684
19 -1.897373e+03 6.054321e+00
* time: 5.94549298286438
20 -1.897498e+03 3.985954e+00
* time: 6.249166965484619
21 -1.897571e+03 4.262464e+00
* time: 6.544780015945435
22 -1.897633e+03 4.010234e+00
* time: 6.867193937301636
23 -1.897714e+03 4.805375e+00
* time: 7.179778814315796
24 -1.897802e+03 3.508706e+00
* time: 7.4493608474731445
25 -1.897865e+03 3.691477e+00
* time: 7.743985891342163
26 -1.897900e+03 2.982720e+00
* time: 8.033143043518066
27 -1.897928e+03 2.563790e+00
* time: 8.323179960250854
28 -1.897968e+03 3.261485e+00
* time: 8.619223833084106
29 -1.898013e+03 3.064690e+00
* time: 8.879823923110962
30 -1.898040e+03 1.636525e+00
* time: 9.173550844192505
31 -1.898051e+03 1.439997e+00
* time: 9.464515924453735
32 -1.898057e+03 1.436504e+00
* time: 9.75806188583374
33 -1.898069e+03 1.881529e+00
* time: 10.054566860198975
34 -1.898095e+03 3.253165e+00
* time: 10.31788682937622
35 -1.898142e+03 4.257942e+00
* time: 10.608276844024658
36 -1.898199e+03 3.685241e+00
* time: 10.910993814468384
37 -1.898245e+03 2.567364e+00
* time: 11.233237028121948
38 -1.898246e+03 2.561591e+00
* time: 11.681350946426392
39 -1.898251e+03 2.530888e+00
* time: 12.079346895217896
40 -1.898298e+03 2.673696e+00
* time: 12.390853881835938
41 -1.898300e+03 2.794639e+00
* time: 12.763314008712769
42 -1.898337e+03 3.751590e+00
* time: 13.18537187576294
43 -1.898421e+03 4.878407e+00
* time: 13.503590822219849
44 -1.898433e+03 4.391719e+00
* time: 13.880887031555176
45 -1.898437e+03 4.216518e+00
* time: 14.34292483329773
46 -1.898442e+03 4.108397e+00
* time: 14.798146963119507
47 -1.898446e+03 3.934902e+00
* time: 15.273016929626465
48 -1.898449e+03 3.769838e+00
* time: 15.726444959640503
49 -1.898450e+03 3.739486e+00
* time: 16.170886039733887
50 -1.898450e+03 3.712049e+00
* time: 16.651981830596924
51 -1.898457e+03 3.623436e+00
* time: 17.063912868499756
52 -1.898471e+03 2.668312e+00
* time: 17.37425184249878
53 -1.898479e+03 2.302438e+00
* time: 17.680744886398315
54 -1.898480e+03 2.386566e-01
* time: 17.99843692779541
55 -1.898480e+03 7.802040e-01
* time: 18.292367935180664
56 -1.898480e+03 7.369786e-01
* time: 18.724740982055664
57 -1.898480e+03 5.113191e-01
* time: 19.09568691253662
58 -1.898480e+03 3.067709e-01
* time: 19.335248947143555
59 -1.898480e+03 3.076791e-01
* time: 19.61433982849121
60 -1.898480e+03 3.102066e-01
* time: 19.885987997055054
61 -1.898480e+03 3.102066e-01
* time: 20.249873876571655
62 -1.898480e+03 3.102069e-01
* time: 20.66685700416565
63 -1.898480e+03 3.102071e-01
* time: 21.30530881881714
64 -1.898480e+03 3.102074e-01
* time: 21.97941303253174
65 -1.898480e+03 3.102076e-01
* time: 22.615993976593018
66 -1.898480e+03 3.102079e-01
* time: 23.292935848236084
67 -1.898480e+03 3.102081e-01
* time: 23.923321962356567
68 -1.898480e+03 3.102081e-01
* time: 24.59940195083618
69 -1.898480e+03 3.102081e-01
* time: 25.22856092453003
70 -1.898480e+03 3.102082e-01
* time: 25.876300811767578
71 -1.898480e+03 3.102082e-01
* time: 26.543476819992065
72 -1.898480e+03 3.102082e-01
* time: 27.207408905029297
73 -1.898480e+03 3.102102e-01
* time: 27.861055850982666
74 -1.898480e+03 3.102102e-01
* time: 28.36388897895813
75 -1.898480e+03 3.102096e-01
* time: 28.816293954849243
76 -1.898480e+03 3.102096e-01
* time: 29.309242963790894
77 -1.898480e+03 3.125688e-01
* time: 29.76039981842041
78 -1.898480e+03 3.125640e-01
* time: 30.214128971099854
79 -1.898480e+03 3.125618e-01
* time: 30.662088871002197
80 -1.898480e+03 3.125615e-01
* time: 31.198736906051636
81 -1.898480e+03 3.125612e-01
* time: 31.683645963668823
82 -1.898480e+03 3.125610e-01
* time: 32.2076518535614
83 -1.898480e+03 3.125609e-01
* time: 32.78699803352356
84 -1.898480e+03 3.125604e-01
* time: 33.30263900756836
85 -1.898480e+03 3.125602e-01
* time: 33.802459955215454
86 -1.898480e+03 3.125602e-01
* time: 34.359484910964966
87 -1.898480e+03 3.125602e-01
* time: 35.126346826553345
88 -1.898480e+03 3.125602e-01
* time: 35.907317876815796
89 -1.898480e+03 3.125602e-01
* time: 36.69755983352661
90 -1.898480e+03 3.125602e-01
* time: 37.512157917022705
91 -1.898480e+03 3.125602e-01
* time: 38.274991035461426
92 -1.898480e+03 3.125602e-01
* time: 38.80830383300781
93 -1.898480e+03 3.125602e-01
* time: 39.366822957992554
94 -1.898480e+03 3.125602e-01
* time: 39.92230486869812
95 -1.898480e+03 1.387453e-01
* time: 40.26575183868408
96 -1.898480e+03 1.387453e-01
* time: 40.692107915878296
97 -1.898480e+03 1.387453e-01
* time: 41.26377081871033
98 -1.898480e+03 1.387453e-01
* time: 41.85437488555908
99 -1.898480e+03 1.387453e-01
* time: 42.26488995552063FittedPumasModel
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.