using Pumas
using PharmaDatasets
using DataFramesMeta
using CairoMakie
using AlgebraOfGraphics
using PumasUtilities
using Markdown
Fitting and Inferring a Compartmental PK Model
1 Introduction
A typical workflow for fitting a Pumas model and deriving parameter precision typically involves:
- Preparing the data and the model.
- Checking model-data compatibility.
- Obtaining initial parameter estimates.
- Fitting the model via a chosen estimation method.
- Interpreting the fit results.
- Computing parameter precision.
- (Optionally) proceeding with more advanced techniques like bootstrapping or SIR for robust uncertainty quantification.
The following sections will walk through these steps using a two-compartment PK model for Warfarin, focusing on the PK aspects only. Exploratory data analysis (EDA), although extremely important, is out of scope here. Readers interested in EDA are encouraged to consult other tutorials.
2 Model and Data
2.1 Model Definition
Below is the PK model, named warfarin_pk_model, defined in Pumas. This model contains:
- Fixed effects (population parameters):
pop_CL, pop_Vc, pop_Q, pop_Vp, pop_tabs, pop_lag - Inter-individual variability (IIV) components:
pk_Ω, lag_ω - Residual error model parameters:
σ_prop, σ_add - Covariates for scaling:
FSZCLandFSZV - Differential equations describing the PK behavior in the compartments.
warfarin_pk_model = @model begin
@metadata begin
desc = "Warfarin 2-compartment PK model (PD removed)"
timeu = u"hr"
end
@param begin
# PK parameters
"""
Clearance (L/hr)
"""
pop_CL ∈ RealDomain(lower = 0.0, init = 0.134)
"""
Central volume (L)
"""
pop_Vc ∈ RealDomain(lower = 0.0, init = 8.11)
"""
Inter-compartmental clearance (L/hr)
"""
pop_Q ∈ RealDomain(lower = 0.0, init = 0.5)
"""
Peripheral volume (L)
"""
pop_Vp ∈ RealDomain(lower = 0.0, init = 20.0)
"""
Absorption lag time (hr)
"""
pop_tabs ∈ RealDomain(lower = 0.0, init = 0.523)
"""
Lag time (hr)
"""
pop_lag ∈ RealDomain(lower = 0.0, init = 0.1)
# Inter-individual variability
"""
- ΩCL: Clearance
- ΩVc: Central volume
- Ωtabs: Absorption lag time
"""
pk_Ω ∈ PDiagDomain([0.01, 0.01, 0.01])
# Residual variability
"""
σ_prop: Proportional error
"""
σ_prop ∈ RealDomain(lower = 0.0, init = 0.00752)
"""
σ_add: Additive error
"""
σ_add ∈ RealDomain(lower = 0.0, init = 0.0661)
end
@random begin
pk_η ~ MvNormal(pk_Ω) # mean = 0, covariance = pk_Ω
end
@covariates begin
"""
FSZCL: Clearance scaling factor
"""
FSZCL
"""
FSZV: Volume scaling factor
"""
FSZV
end
@pre begin
CL = FSZCL * pop_CL * exp(pk_η[1])
Vc = FSZV * pop_Vc * exp(pk_η[2])
Q = FSZCL * pop_Q
Vp = FSZV * pop_Vp
tabs = pop_tabs * exp(pk_η[3])
Ka = log(2) / tabs
end
@dosecontrol begin
lags = (Depot = pop_lag,)
end
@vars begin
cp := Central / Vc
end
@dynamics begin
Depot' = -Ka * Depot
Central' =
Ka * Depot - (CL / Vc) * Central - (Q / Vc) * Central + (Q / Vp) * Peripheral
Peripheral' = (Q / Vc) * Central - (Q / Vp) * Peripheral
end
@derived begin
"""
Concentration (ng/mL)
"""
conc ~ @. Normal(cp, sqrt((σ_prop * cp)^2 + σ_add^2))
end
endPumasModel
Parameters: pop_CL, pop_Vc, pop_Q, pop_Vp, pop_tabs, pop_lag, pk_Ω, σ_prop, σ_add
Random effects: pk_η
Covariates: FSZCL, FSZV
Dynamical system variables: Depot, Central, Peripheral
Dynamical system type: Matrix exponential
Derived: conc
Observed: conc2.2 Data Preparation
The Warfarin data used in this tutorial is pulled from PharmaDatasets for demonstration purposes. Note how the code reshapes and prepares the data in “wide” format for reading into Pumas. Only the conc column is treated as observations for the PK model.
warfarin_data = dataset("paganz2024/warfarin_long")
# Step 2: Fix Duplicate Time Points
# -------------------------------
# Some subjects have duplicate time points for DVID = 1
# For this dataset, the triple (ID, TIME, DVID) should define
# a row uniquely, but
nrow(warfarin_data)
nrow(unique(warfarin_data, ["ID", "TIME", "DVID"]))
# We can identify the problematic rows by grouping on the index variables
@chain warfarin_data begin
@groupby :ID :TIME :DVID
@transform :tmp = length(:ID)
@rsubset :tmp > 1
end
# It is important to understand the reason for the duplicate values.
# Sometimes the duplication is caused by recording errors, sometimes
# it is a data processing error, e.g. when joining tables, or it can
# be genuine records, e.g. when samples have been analyzed in multiple
# labs. The next step depends on which of the causes are behind the
# duplications.
#
# In this case, we will assume that both values are informative and
# we will therefore just adjust the time stamp a bit for the second
# observation.
warfarin_data_processed = @chain warfarin_data begin
@groupby :ID :TIME :DVID
@transform :tmp = 1:length(:ID)
@rtransform :TIME = :tmp == 1 ? :TIME : :TIME + 1e-6
@select Not(:tmp)
end
# Transform the data in a single chain of operations
warfarin_data_wide = @chain warfarin_data_processed begin
@rsubset !contains(:ID, "#")
@rtransform begin
# Scaling factors
:FSZV = :WEIGHT / 70 # volume scaling
:FSZCL = (:WEIGHT / 70)^0.75 # clearance scaling (allometric)
# Column name for the DV
:DVNAME = "DV$(:DVID)"
# Dosing indicator columns
:CMT = ismissing(:AMOUNT) ? missing : 1
:EVID = ismissing(:AMOUNT) ? 0 : 1
end
unstack(Not([:DVID, :DVNAME, :DV]), :DVNAME, :DV)
rename!(:DV1 => :conc, :DV2 => :pca)
end317×13 DataFrame
292 rows omitted
| Row | ID | TIME | WEIGHT | AGE | SEX | AMOUNT | FSZV | FSZCL | CMT | EVID | DV0 | pca | conc |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| String3 | Float64 | Float64 | Int64 | Int64 | Float64? | Float64 | Float64 | Int64? | Int64 | Float64? | Float64? | Float64? | |
| 1 | 1 | 0.0 | 66.7 | 50 | 1 | 100.0 | 0.952857 | 0.96443 | 1 | 1 | missing | missing | missing |
| 2 | 1 | 0.0 | 66.7 | 50 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | 100.0 | missing |
| 3 | 1 | 24.0 | 66.7 | 50 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | 49.0 | 9.2 |
| 4 | 1 | 36.0 | 66.7 | 50 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | 32.0 | 8.5 |
| 5 | 1 | 48.0 | 66.7 | 50 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | 26.0 | 6.4 |
| 6 | 1 | 72.0 | 66.7 | 50 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | 22.0 | 4.8 |
| 7 | 1 | 96.0 | 66.7 | 50 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | 28.0 | 3.1 |
| 8 | 1 | 120.0 | 66.7 | 50 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | 33.0 | 2.5 |
| 9 | 2 | 0.0 | 66.7 | 31 | 1 | 100.0 | 0.952857 | 0.96443 | 1 | 1 | missing | missing | missing |
| 10 | 2 | 0.0 | 66.7 | 31 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | 100.0 | missing |
| 11 | 2 | 0.5 | 66.7 | 31 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | missing | 0.0 |
| 12 | 2 | 2.0 | 66.7 | 31 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | missing | 8.4 |
| 13 | 2 | 3.0 | 66.7 | 31 | 1 | missing | 0.952857 | 0.96443 | missing | 0 | missing | missing | 9.7 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
| 306 | 31 | 48.0 | 83.3 | 24 | 1 | missing | 1.19 | 1.13936 | missing | 0 | missing | 24.0 | 6.4 |
| 307 | 31 | 72.0 | 83.3 | 24 | 1 | missing | 1.19 | 1.13936 | missing | 0 | missing | 22.0 | 4.5 |
| 308 | 31 | 96.0 | 83.3 | 24 | 1 | missing | 1.19 | 1.13936 | missing | 0 | missing | 28.0 | 3.4 |
| 309 | 31 | 120.0 | 83.3 | 24 | 1 | missing | 1.19 | 1.13936 | missing | 0 | missing | 42.0 | 2.5 |
| 310 | 32 | 0.0 | 62.0 | 21 | 1 | 93.0 | 0.885714 | 0.912999 | 1 | 1 | missing | missing | missing |
| 311 | 32 | 0.0 | 62.0 | 21 | 1 | missing | 0.885714 | 0.912999 | missing | 0 | missing | 100.0 | missing |
| 312 | 32 | 24.0 | 62.0 | 21 | 1 | missing | 0.885714 | 0.912999 | missing | 0 | missing | 36.0 | 8.9 |
| 313 | 32 | 36.0 | 62.0 | 21 | 1 | missing | 0.885714 | 0.912999 | missing | 0 | missing | 27.0 | 7.7 |
| 314 | 32 | 48.0 | 62.0 | 21 | 1 | missing | 0.885714 | 0.912999 | missing | 0 | missing | 24.0 | 6.9 |
| 315 | 32 | 72.0 | 62.0 | 21 | 1 | missing | 0.885714 | 0.912999 | missing | 0 | missing | 23.0 | 4.4 |
| 316 | 32 | 96.0 | 62.0 | 21 | 1 | missing | 0.885714 | 0.912999 | missing | 0 | missing | 20.0 | 3.5 |
| 317 | 32 | 120.0 | 62.0 | 21 | 1 | missing | 0.885714 | 0.912999 | missing | 0 | missing | 22.0 | 2.5 |
2.3 Creating a Pumas Population
Below is the creation of a population object in Pumas using read_pumas. Only the conc data are treated as the observation variable:
pop = read_pumas(
warfarin_data_wide;
id = :ID,
time = :TIME,
amt = :AMOUNT,
cmt = :CMT,
evid = :EVID,
covariates = [:SEX, :WEIGHT, :FSZV, :FSZCL],
observations = [:conc],
)Population
Subjects: 31
Covariates: SEX, WEIGHT, FSZV, FSZCL
Observations: conc
Note
The same data can contain multiple endpoints or PD observations. In this tutorial, the focus is solely on PK fitting. PKPD modeling on this warfarin dataset will be introduced later.
2.4 Checking Model-Data Compatibility
Before performing any fit, it is recommended to verify whether the defined model is consistent with the provided dataset. Pumas offers functions such as loglikelihood and findinfluential for these checks.
2.4.1 The loglikelihood Function
The loglikelihood function computes the log-likelihood of the model given data and parameters. In Pumas, the signature typically looks like:
loglikelihood(model, population, params, approx)where:
model: The Pumas model definition (e.g.,warfarin_pk_model).population: A Pumas population object (e.g.,pop).params: A named tuple or dictionary containing parameter values.approx: The approximation method to use. Common options includeFOCE(),FO(),LaplaceI(), etc.
For example, one might write:
# A named tuple of parameter values
param_vals = (
pop_CL = 0.134,
pop_Vc = 8.11,
pop_Q = 0.5,
pop_Vp = 20.0,
pop_tabs = 0.523,
pop_lag = 0.1,
pk_Ω = Diagonal([0.01, 0.01, 0.01]),
σ_prop = 0.00752,
σ_add = 0.0661,
)
ll_value = loglikelihood(warfarin_pk_model, pop, param_vals, FOCE())-91123.78983612436The initial loglikelihood of the warfarin PK model given the data and parameter values is `LL = -91123.78983612436 `.
If the model and data are incompatible (e.g., missing doses for compartments, or out-of-range parameter values), loglikelihood might return an error or produce a warning. A successful computation is a good sign that the model can handle the data.
2.4.2 The findinfluential Function
The findinfluential function helps identify observations that disproportionately influence the fit. It can be used before or after fitting, but even with initial guesses, it can highlight potentially problematic data points. The most notable case is when the loglikelihood cannot be evaluated in which case the data returned for the problematic subject does not return a finite value.
Typical usage is:
findinfluential(warfarin_pk_model, pop, params; approx)This will list the individual loglikelihoods for each subject. Readers can then inspect these observations further, potentially re-checking the data quality or adjusting the model assumptions if needed for each subject.
fi = findinfluential(warfarin_pk_model, pop, param_vals, FOCE())31-element Vector{@NamedTuple{id::String, nll::Float64}}:
(id = "12", nll = 7035.6308301243125)
(id = "8", nll = 6811.854099061583)
(id = "25", nll = 5140.036812994075)
(id = "27", nll = 4720.678341527095)
(id = "11", nll = 3856.5709209433408)
(id = "16", nll = 3853.3494408795573)
(id = "22", nll = 3616.0674596218623)
(id = "7", nll = 3592.014215601264)
(id = "3", nll = 3434.8872372376954)
(id = "26", nll = 3358.499378697223)
⋮
(id = "18", nll = 2084.1283016469442)
(id = "9", nll = 1950.7850340954703)
(id = "24", nll = 1717.087929029517)
(id = "29", nll = 1381.3498671515454)
(id = "17", nll = 1350.9070098668137)
(id = "6", nll = 1319.2234098688702)
(id = "21", nll = 1072.85052503673)
(id = "19", nll = 989.3840625094336)
(id = "5", nll = 984.368482951681)The default output is a vector of NamedTuples, which can easily be converted into a DataFrame for further analysis. One can then plot the distribution of the individual loglikelihoods to identify any potential problematic subjects.
fidf = DataFrame(fi)
hist(
fidf.nll,
axis = (xlabel = "Log-likelihood", ylabel = "Frequency"),
color = (:blue, 0.5),
)
The plot above shows that there are some subjects with very high initial log-likelihoods, which might be worth investigating. Notice, that a high initial log-likelihood does not necessarily imply a model or data problem. It can simply be that for the chosen initial population parameters the model does not fit the observations well for the subject in question.
2.5 Getting Good Initial Estimates Using Naive Pooled
A reliable starting point for nonlinear mixed-effects modeling is to get reasonable initial parameter estimates. Pumas provides approaches such as:
- Naive Pooled: Treats all data as if there were no inter-individual variability.
- NCA (Non-compartmental Analysis): Uses PK metrics (CL, V, etc.) from a non-compartmental approach to seed the parameter values (not showcased in this tutorial).
Here, the Naive Pooled approach is demonstrated. This method estimates population parameters by ignoring inter-individual differences, effectively pooling data as if from a single “super-subject.”
Below is the signature and documentation for the Pumas fit function. For more details, see the Fitting in Pumas Documentation.
fit(
model::PumasModel,
population::Population,
param::NamedTuple,
approx::Union{LikelihoodApproximation, MAP};
optim_alg = Optim.BFGS(linesearch = Optim.LineSearches.BackTracking(), initial_stepnorm = 1.0),
optim_options::NamedTuple = NamedTuple(),
optimize_fn = nothing,
constantcoef::Tuple = (),
ensemblealg::SciMLBase.EnsembleAlgorithm = EnsembleThreads(),
checkidentification = true,
diffeq_options = NamedTuple(),
verbose = true,
ignore_numerical_error = true,
)
Fit the Pumas model model to the dataset population with starting values param using the estimation method approx. Currently supported values for the `approx` argument are `FO`, `FOCE`,
`LaplaceI`, `NaivePooled`, and `BayesMCMC`. See the online documentation for more details about the different methods.
To control the optimization procedure for finding population parameters (fixed effects), use the `optim_alg` and `optim_options` keyword arguments. In previous versions of Pumas the argument
`optimize_fn` was used, but is now discouraged and will be removed in a later version of Pumas. These options control the optimization of all `approx` methods except BayesMCMC. The default
optimization function uses the quasi-Newton routine BFGS method from the `Optim` package. It can be changed by setting the `optim_alg` to an algorithm implemented in Optim.jl as long as it
does not use second order derivatives. Optimization specific options can be passed in using the `optim_options` keyword and has to be a `NamedTuple` with names and values that match the
`Optim.Options` type. For example, the optimization trace can be disabled and the algorithm can be changed to L-BFGS by passing `optim_alg=Optim.LBFGS(), optim_options = ;(show_trace=false)`
to fit. See [Optim](https://docs.pumas.ai/stable/basics/estimation/#Optimization-option) for more defails.
It is possible to fix one or more parameters of the fit by passing a Tuple of Symbols as the `constantcoef` argument with elements corresponding to the names of the fixed parameters, e.g.
`constantcoef=(:σ,)`.
When models include an `@random` block and fitting with NaivePooled is requested, it is required that the user sets all variability parameters to zero with `constantcoef` such that these can
be ignored in the optimization, e.g. `constantcoef = (:Ω,)` while overwriting the corresponding values in `param` with `(; init_params..., Ω = zeros(3, 3))`.
Parallelization of the optimization is supported for most estimation methods via the ensemble interface of DifferentialEquations.jl. Currently, the only supported options are:
• `EnsembleThreads()`: the default. Accelerate fits by using multiple threads.
• `EnsembleSerial()`: fit using a single thread.
• `EnsembleDistributed()`: fit by using multiple worker processes.
The `fit` function will check if any gradients and throw an exception if any of the elements are exactly zero unless `checkidentification` is set to false.
Further keyword arguments can be passed via the `diffeq_options` argument. This allows for passing arguments to the differential equations solver such as `alg`, `abstol`, and `reltol`. The
default values for these are `AutoVern7(Rodas5P(autodiff=true)), 1e-12, and 1e-8` respectively. See the [documentation](https://docs.pumas.ai/stable/basics/simulation/#Options-and-settings-for-simobs) for more details.
The keyword `verbose` controls if info statements about initial evaluations of the loglikelihood function and gradient should be printed or not. Defaults to true.
Since the numerical optimization routine can sometimes visit extreme regions of the parameter space during it's exploration we automatically handle the situation where the input
parameters result in errors when solving the dynamical system. This allows the algorithm to recover and continue in many situations that would have otherwise stalled early. Sometimes, it
is useful to turn this error handling off when debugging a model fit, and this can be done by setting `ignore_numerical_error = false`.
Below is how to run a Naive Pooled analysis:
naive_fit = fit(warfarin_pk_model, pop, param_vals, NaivePooled(); constantcoef = (:pk_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 3.188875e+05 5.217691e+05
* time: 0.026215076446533203
1 3.248274e+04 5.854052e+04
* time: 1.0637309551239014
2 2.414204e+04 4.403968e+04
* time: 1.0672879219055176
3 9.800450e+03 1.834949e+04
* time: 1.0707130432128906
4 5.218319e+03 9.776442e+03
* time: 1.0741140842437744
5 2.657496e+03 4.805323e+03
* time: 1.0775010585784912
6 1.508416e+03 2.469328e+03
* time: 1.0810329914093018
7 9.524014e+02 1.261463e+03
* time: 1.0844430923461914
8 6.999017e+02 6.470385e+02
* time: 1.0878310203552246
9 5.922734e+02 3.262900e+02
* time: 1.091162919998169
10 5.535691e+02 2.252527e+02
* time: 1.094351053237915
11 5.434957e+02 1.979047e+02
* time: 1.0975310802459717
12 5.419714e+02 1.852856e+02
* time: 1.1006379127502441
13 5.417601e+02 1.809508e+02
* time: 1.103679895401001
14 5.415227e+02 1.766418e+02
* time: 1.1067509651184082
15 5.408349e+02 1.663266e+02
* time: 1.1099491119384766
16 5.392823e+02 1.469356e+02
* time: 1.1131150722503662
17 5.357939e+02 1.101891e+02
* time: 1.1163370609283447
18 5.295274e+02 6.101717e+01
* time: 1.119739055633545
19 5.227089e+02 4.376424e+01
* time: 1.1231749057769775
20 5.205584e+02 6.762423e+01
* time: 1.126629114151001
21 5.200667e+02 6.460758e+01
* time: 1.1302299499511719
22 5.198970e+02 5.159635e+01
* time: 1.1338250637054443
23 5.198793e+02 5.168473e+01
* time: 1.1372630596160889
24 5.182416e+02 5.493697e+01
* time: 1.1407029628753662
25 5.154632e+02 5.617036e+01
* time: 1.144129991531372
26 5.085379e+02 6.965501e+01
* time: 1.1477079391479492
27 4.992897e+02 8.613012e+01
* time: 1.1510200500488281
28 4.952284e+02 6.666806e+01
* time: 1.154336929321289
29 4.937816e+02 4.845720e+01
* time: 1.1576409339904785
30 4.933235e+02 4.622865e+01
* time: 1.1609349250793457
31 4.932968e+02 4.584617e+01
* time: 1.1642329692840576
32 4.932703e+02 4.565222e+01
* time: 1.167543888092041
33 4.931155e+02 4.490937e+01
* time: 1.1709809303283691
34 4.927970e+02 4.552021e+01
* time: 1.1743929386138916
35 4.919145e+02 5.125228e+01
* time: 1.1777949333190918
36 4.898542e+02 5.869445e+01
* time: 1.1814560890197754
37 4.854151e+02 6.533002e+01
* time: 1.1850640773773193
38 4.786263e+02 5.894401e+01
* time: 1.1885490417480469
39 4.723401e+02 3.679497e+01
* time: 1.1920900344848633
40 4.676138e+02 2.454358e+01
* time: 1.1956191062927246
41 4.667006e+02 1.298513e+01
* time: 1.1990959644317627
42 4.666338e+02 1.353668e+01
* time: 1.2026050090789795
43 4.666323e+02 1.384287e+01
* time: 1.206115961074829
44 4.666314e+02 1.374470e+01
* time: 1.2096099853515625
45 4.666256e+02 1.344668e+01
* time: 1.213109016418457
46 4.666098e+02 1.292659e+01
* time: 1.2166039943695068
47 4.665692e+02 1.205145e+01
* time: 1.220120906829834
48 4.664639e+02 1.210731e+01
* time: 1.223634958267212
49 4.662037e+02 1.193827e+01
* time: 1.2271909713745117
50 4.656294e+02 1.246811e+01
* time: 1.2307629585266113
51 4.646738e+02 1.521581e+01
* time: 1.234287977218628
52 4.635500e+02 1.353950e+01
* time: 1.237696886062622
53 4.625753e+02 6.648229e+00
* time: 1.2410330772399902
54 4.622478e+02 3.557188e+00
* time: 1.2443010807037354
55 4.622081e+02 3.487277e+00
* time: 1.247499942779541
56 4.622033e+02 3.503679e+00
* time: 1.250627040863037
57 4.622028e+02 3.522469e+00
* time: 1.2539091110229492
58 4.622027e+02 3.528199e+00
* time: 1.257232904434204
59 4.622027e+02 3.533686e+00
* time: 1.2606050968170166
60 4.622025e+02 3.543319e+00
* time: 1.264112949371338
61 4.622021e+02 3.558678e+00
* time: 1.2678909301757812
62 4.622011e+02 3.584789e+00
* time: 1.2715730667114258
63 4.621983e+02 3.629313e+00
* time: 1.2753760814666748
64 4.621910e+02 3.708490e+00
* time: 1.2795109748840332
65 4.621716e+02 3.857090e+00
* time: 1.283452033996582
66 4.621190e+02 5.146880e+00
* time: 1.28751802444458
67 4.619639e+02 8.785594e+00
* time: 1.2915899753570557
68 4.611900e+02 1.480755e+01
* time: 1.2956769466400146
69 4.590238e+02 2.147025e+01
* time: 1.2997329235076904
70 4.586338e+02 1.072817e+02
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* time: 2.2105350494384766FittedPumasModel
Successful minimization: true
Likelihood approximation: NaivePooled
Likelihood Optimizer: BFGS
Dynamical system type: Matrix exponential
Log-likelihood value: 216.75859
Number of subjects: 31
Number of parameters: Fixed Optimized
1 10
Observation records: Active Missing
conc: 239 47
Total: 239 47
-------------------------
Estimate
-------------------------
pop_CL 0.12098
pop_Vc 8.2019
pop_Q 0.013114
pop_Vp 4.6242e12
pop_tabs 2.3621e-12
pop_lag 1.0
pk_Ω₁,₁ 0.01
pk_Ω₂,₂ 0.01
pk_Ω₃,₃ 0.01
σ_prop 0.24211
σ_add 3.0263e-96
-------------------------As you can see, the constantcoef argument is used to fix the inter-individual variability parameters to zero. The parameter estimates from the naive pooled fit seem out of place for some parameters such as pop_Vp and pop_tabs. Perhaps the initial parameter values are not close to the true values and some adjustments are needed. In Pumas, you can rerun the fit with new initial parameter values very easily. Suppose, we want to run the naive pooled fit again with 4 new initial parameter values that are within 10% of the original values.
We can use the power of the Julia language to generate 4 new parameter values that are within 10% of the original values.
# Generate parameter sets by scaling original values by ±10%
param_sets = [
NamedTuple(k => v * scale for (k, v) in pairs(param_vals)) for
scale in [0.9, 0.95, 1.05, 1.1]
]4-element Vector{@NamedTuple{pop_CL::Float64, pop_Vc::Float64, pop_Q::Float64, pop_Vp::Float64, pop_tabs::Float64, pop_lag::Float64, pk_Ω::Diagonal{Float64, Vector{Float64}}, σ_prop::Float64, σ_add::Float64}}:
(pop_CL = 0.12060000000000001, pop_Vc = 7.2989999999999995, pop_Q = 0.45, pop_Vp = 18.0, pop_tabs = 0.4707, pop_lag = 0.09000000000000001, pk_Ω = [0.009000000000000001 0.0 0.0; 0.0 0.009000000000000001 0.0; 0.0 0.0 0.009000000000000001], σ_prop = 0.006768, σ_add = 0.05949000000000001)
(pop_CL = 0.1273, pop_Vc = 7.7044999999999995, pop_Q = 0.475, pop_Vp = 19.0, pop_tabs = 0.49685, pop_lag = 0.095, pk_Ω = [0.0095 0.0 0.0; 0.0 0.0095 0.0; 0.0 0.0 0.0095], σ_prop = 0.007143999999999999, σ_add = 0.062795)
(pop_CL = 0.14070000000000002, pop_Vc = 8.5155, pop_Q = 0.525, pop_Vp = 21.0, pop_tabs = 0.54915, pop_lag = 0.10500000000000001, pk_Ω = [0.0105 0.0 0.0; 0.0 0.0105 0.0; 0.0 0.0 0.0105], σ_prop = 0.007896, σ_add = 0.06940500000000001)
(pop_CL = 0.14740000000000003, pop_Vc = 8.921, pop_Q = 0.55, pop_Vp = 22.0, pop_tabs = 0.5753, pop_lag = 0.11000000000000001, pk_Ω = [0.011000000000000001 0.0 0.0; 0.0 0.011000000000000001 0.0; 0.0 0.0 0.011000000000000001], σ_prop = 0.008272, σ_add = 0.07271000000000001)Now, we can run the fit again with the new parameter values.
new_fits = [
fit(warfarin_pk_model, pop, param_set, NaivePooled(), constantcoef = (:pk_Ω,)) for
param_set in param_sets
];You can compare the results of the new fits with perturbed initial parameter values.
compare_estimates(;
p1 = new_fits[1],
p2 = new_fits[2],
p3 = new_fits[3],
p4 = new_fits[4],
original = naive_fit,
)11×6 DataFrame
| Row | parameter | p1 | p2 | p3 | p4 | original |
|---|---|---|---|---|---|---|
| String | Float64? | Float64? | Float64? | Float64? | Float64? | |
| 1 | pop_CL | 0.0436272 | 0.000370378 | 0.0470208 | 0.127227 | 0.12098 |
| 2 | pop_Vc | 7.95126 | 14.3362 | 9.34242 | 8.34565 | 8.20193 |
| 3 | pop_Q | 0.0812342 | 0.179464 | 0.0867748 | 4.31655e-5 | 0.0131141 |
| 4 | pop_Vp | 181.671 | 1.92817e16 | 148.995 | 1.75913 | 4.62419e12 |
| 5 | pop_tabs | 1.09629 | 2.35217e-12 | 0.0381164 | 0.37191 | 2.36208e-12 |
| 6 | pop_lag | 0.5 | 0.5 | 0.5 | 0.764776 | 1.0 |
| 7 | pk_Ω₁,₁ | 0.009 | 0.0095 | 0.0105 | 0.011 | 0.01 |
| 8 | pk_Ω₂,₂ | 0.009 | 0.0095 | 0.0105 | 0.011 | 0.01 |
| 9 | pk_Ω₃,₃ | 0.009 | 0.0095 | 0.0105 | 0.011 | 0.01 |
| 10 | σ_prop | 0.310588 | 0.463829 | 0.256087 | 0.228482 | 0.242111 |
| 11 | σ_add | 1.48972e-17 | 1.39471e-161 | 6.52968e-11 | 0.380485 | 3.0263e-96 |
The compare_estimates function is a convenience function that is part of the PumasUtilities package. It is used to compare the estimates of the new fits with the original fit. The p4 results seem to be reasonable amongst all the fits and these results will be taken forward for the FOCE() fit.
2.6 Estimating Parameters via Maximum Likelihood (FOCE)
After obtaining initial estimates, the next step is fitting the model to data using a maximum likelihood approach. One of the most commonly used methods in Pumas is the First-Order Conditional Estimation (FOCE) method. This is invoked by specifying FOCE() as the estimation algorithm:
foce_fit = fit(warfarin_pk_model, pop, coef(new_fits[4]), FOCE())The coef(new_fits[4]) function extracts the parameter estimates from the new_fits[4] fit and passes them to the fit function.
Note
By default, the fit function will use all the cores of your machine to parallelize the optimization. You can control this using the ensemblealg argument. For more details, see the fit options section in the Pumas Documentation.
2.7 Fixing Parameters with constantcoef
Sometimes, it is desirable to hold certain parameters constant during estimation. For instance, suppose one wants to fix pop_Q at a particular value:
foce_fit_fixedQ = fit(
warfarin_pk_model,
pop,
(; param_vals..., pop_Q = 0.5),
FOCE();
constantcoef = (:pop_Q,),
); # This fixes the pop_Q parameterfoce_fit_fixedQFittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Matrix exponential
Log-likelihood value: -321.9637
Number of subjects: 31
Number of parameters: Fixed Optimized
1 10
Observation records: Active Missing
conc: 239 47
Total: 239 47
------------------------
Estimate
------------------------
pop_CL 0.13128
pop_Vc 8.0921
pop_Q 0.5
pop_Vp 0.0040524
pop_tabs 0.44966
pop_lag 0.92947
pk_Ω₁,₁ 0.049875
pk_Ω₂,₂ 0.018981
pk_Ω₃,₃ 1.7058
σ_prop 0.090433
σ_add 0.33064
------------------------This approach is useful for sensitivity analysis, exploring different values, or simplifying the model.
2.8 Changing Tolerance During Fitting
The fit function accepts additional keyword arguments to tweak the fitting process, such as tolerances for convergence:
foce_fit_fixedQ = fit(
warfarin_pk_model,
pop,
(; param_vals..., pop_Q = 0.5),
FOCE();
constantcoef = (pop_Q = 0.5,),
optim_options = (; x_reltol = 1e-6, x_abstol = 1e-8, iterations = 200),
);These arguments (x_reltol, x_abstol, iterations) help fine-tune the solver’s stopping criteria. For more details, see the Optimization Options during Fitting.
2.9 Interpreting the Printed Results
Once the model fit is complete, Pumas will print a summary of the fit to the console. This typically includes:
- Objective Function Value (OFV): The minimized negative log-likelihood (or similar objective). Lower is better.
- Convergence Status: An indication of whether the solver found a local optimum or if it ran into issues (e.g., maximum iterations reached).
- Parameter Estimates: A table listing final estimates of fixed effects, random effects (variances or standard deviations), and residual variability components.
- Standard Errors (if computed): By default, these might not be computed until using
infer(discussed in the next section).
The printed output might look like:
julia> foce_fit_fixedQ
FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Matrix exponential
Log-likelihood value: -321.9637
Number of subjects: 31
Number of parameters: Fixed Optimized
1 10
Observation records: Active Missing
conc: 239 47
Total: 239 47
------------------------
Estimate
------------------------
pop_CL 0.13128
pop_Vc 8.0919
pop_Q 0.5
pop_Vp 0.0042035
pop_tabs 0.44966
pop_lag 0.92947
pk_Ω₁,₁ 0.049875
pk_Ω₂,₂ 0.018981
pk_Ω₃,₃ 1.7058
σ_prop 0.090433
σ_add 0.33064
------------------------We can break down the output of the fit function into the following components:
2.9.1 FittedPumasModel
FittedPumasModelThe FittedPumasModel object contains the following fields:
model: The original model definition
foce_fit_fixedQ.modelPumasModel
Parameters: pop_CL, pop_Vc, pop_Q, pop_Vp, pop_tabs, pop_lag, pk_Ω, σ_prop, σ_add
Random effects: pk_η
Covariates: FSZCL, FSZV
Dynamical system variables: Depot, Central, Peripheral
Dynamical system type: Matrix exponential
Derived: conc
Observed: concdata: The population data
foce_fit_fixedQ.dataPopulation
Subjects: 31
Covariates: SEX, WEIGHT, FSZV, FSZCL
Observations: concoptim: The optimization results
foce_fit_fixedQ.optim * Status: success
* Candidate solution
Final objective value: 3.219637e+02
* Found with
Algorithm: BFGS
* Convergence measures
|x - x'| = 0.00e+00 ≤ 1.0e-08
|x - x'|/|x'| = 0.00e+00 ≤ 1.0e-06
|f(x) - f(x')| = 0.00e+00 ≤ 0.0e+00
|f(x) - f(x')|/|f(x')| = 0.00e+00 ≤ 0.0e+00
|g(x)| = 1.49e-03 ≰ 1.0e-03
* Work counters
Seconds run: 28 (vs limit Inf)
Iterations: 172
f(x) calls: 206
∇f(x) calls: 173approx: The likelihood approximation method
foce_fit_fixedQ.approxFOCE{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, )
)kwargs: Additional keyword arguments
foce_fit_fixedQ.kwargsfixedparamset: The fixed parameter set
foce_fit_fixedQ.fixedparamsetoptim_state: The optimization state
foce_fit_fixedQ.optim_state2.9.2 Minimization Status
Successful minimization: trueThe Successful minimization field indicates whether the minimization was successful. It depends on the convergence of the optimizer and rules set in the optim_options argument, for example x_reltol and x_abstol.
2.9.3 Likelihood Approximation
Likelihood approximation: FOCEThe Likelihood approximation field indicates the likelihood approximation method used.
2.9.4 Likelihood Optimizer
Likelihood Optimizer: BFGSThe Likelihood Optimizer field indicates the optimizer used which by default is BFGS from the Optim package. A user can change the optimizer by setting the optim_alg argument in the fit function.
2.9.5 Dynamical system type
Dynamical system type: Matrix exponentialThe Dynamical system type field indicates the type of dynamical system used. In the example here, even though the model is a differential equation model, the Matrix exponential indicates that the model is solved using a matrix exponential solver as the system has been deemed linear. The user can check the linear nature of the system as follows: This example model is: {julia} foce_fit.model.prob. If the user does not want the automatic check for linearity, they can turn it off in the @options block of the model definition.
@model begin
@param begin
...
end
@random begin
...
end
@dynamics begin
...
end
@pre begin
...
end
@options begin
checklinear = false
end
endThe checklinear option is a boolean that determines whether the solver should check if the system defined in the @dynamics block is linear. If the system is linear, setting this option to true (default) enables calculating the solution through matrix exponentials. If it is set to false or time (t) appears in the @pre block, this optimization is disabled. This option can be useful when the matrix exponential solver is not superior to general numerical integrators or for debugging purposes.
2.9.6 Log-likelihood value
Log-likelihood value: -321.9637The Log-likelihood value field indicates the value of the log-likelihood function at the maximum likelihood estimate.
2.9.7 Number of subjects
Number of subjects: 31The Number of subjects field indicates the number of subjects from the population data used in the fit.
2.9.8 Number of parameters
Number of parameters: Fixed Optimized
1 10The Number of parameters field indicates the number of fixed and optimized parameters. The Fixed column indicates the number of parameters that were fixed using the constantcoef argument during the fit and the Optimized column indicates the number of parameters that were optimized.
2.9.9 Observation records
Observation records: Active Missing
conc: 239 47
Total: 239 47The Observation records field indicates the number of active and missing observations in the population data. The Active column indicates the number of observations that were used in the fit and the Missing column indicates the number of observations that were missing and not used in the fit. missing in this case is due to the missingness of the conc observations, which is the collection of all the conc observations from all the subjects.
2.9.10 Parameter estimates
------------------------
Estimate
------------------------
pop_CL 0.13128
pop_Vc 8.0919
pop_Q 0.5
pop_Vp 0.0042035
pop_tabs 0.44966
pop_lag 0.92947
pk_Ω₁,₁ 0.049875
pk_Ω₂,₂ 0.018981
pk_Ω₃,₃ 1.7058
σ_prop 0.090433
σ_add 0.33064
------------------------The Parameter estimates field indicates the final parameter estimates. The Estimate column indicates the estimated value of the parameter.
2.10 Computing Parameter Precision with infer
The infer function in Pumas estimates the uncertainty (precision) of parameter estimates. Depending on the chosen method, infer can provide standard errors, confidence intervals, and correlation matrices.
The signature for infer often looks like:
infer(
fpm::FittedPumasModel;
level = 0.95,
rethrow_error::Bool = false,
sandwich_estimator::Bool = true,
)where:
fpm::FittedPumasModel: The result offit(e.g.,foce_fit).level: The confidence interval level (e.g., 0.95). The confidence intervals are calculated as the(1-level)/2and(1+level)/2quantiles of the estimated parametersrethrow_error: If rethrow_error is false (the default value), no error will be thrown if the variance-covariance matrix estimator fails. If it is true, an error will be thrown if the estimator fails.sandwich_estimator: Whether to use the sandwich estimator. If set totrue(the default value), the sandwich estimator will be used. If set tofalse, the standard error will be calculated using the inverse of the Hessian matrix.
An example usage:
inference_results = infer(foce_fit_fixedQ; level = 0.95)[ 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: Matrix exponential
Log-likelihood value: -321.9637
Number of subjects: 31
Number of parameters: Fixed Optimized
1 10
Observation records: Active Missing
conc: 239 47
Total: 239 47
-------------------------------------------------------------------------
Estimate SE 95.0% C.I.
-------------------------------------------------------------------------
pop_CL 0.13128 0.005417 [ 0.12066 ; 0.14189 ]
pop_Vc 8.0921 0.23906 [ 7.6235 ; 8.5606 ]
pop_Q 0.5 NaN [ NaN ; NaN ]
pop_Vp 0.0040524 0.0098746 [-0.015301 ; 0.023406]
pop_tabs 0.44966 0.22631 [ 0.0060942; 0.89322 ]
pop_lag 0.92947 0.052185 [ 0.82719 ; 1.0318 ]
pk_Ω₁,₁ 0.049875 0.01679 [ 0.016967 ; 0.082782]
pk_Ω₂,₂ 0.018981 0.0057796 [ 0.0076528; 0.030308]
pk_Ω₃,₃ 1.7058 1.1673 [-0.58197 ; 3.9936 ]
σ_prop 0.090433 0.014936 [ 0.061159 ; 0.11971 ]
σ_add 0.33064 0.086237 [ 0.16162 ; 0.49967 ]
-------------------------------------------------------------------------We can use the sandwich_estimator argument to get a more robust estimate of the standard errors.
inference_results = infer(foce_fit_fixedQ; level = 0.95, sandwich_estimator = false)[ Info: Calculating: variance-covariance matrix.
[ Info: Done.Asymptotic inference results using negative inverse Hessian
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Matrix exponential
Log-likelihood value: -321.9637
Number of subjects: 31
Number of parameters: Fixed Optimized
1 10
Observation records: Active Missing
conc: 239 47
Total: 239 47
------------------------------------------------------------------------
Estimate SE 95.0% C.I.
------------------------------------------------------------------------
pop_CL 0.13128 0.0054515 [ 0.12059 ; 0.14196 ]
pop_Vc 8.0921 0.47585 [ 7.1594 ; 9.0247 ]
pop_Q 0.5 NaN [ NaN ; NaN ]
pop_Vp 0.0040524 0.41066 [-0.80083 ; 0.80894 ]
pop_tabs 0.44966 0.21485 [ 0.028563 ; 0.87075 ]
pop_lag 0.92947 0.032557 [ 0.86566 ; 0.99328 ]
pk_Ω₁,₁ 0.049875 0.013572 [ 0.023273 ; 0.076476]
pk_Ω₂,₂ 0.018981 0.0067243 [ 0.0058011; 0.03216 ]
pk_Ω₃,₃ 1.7058 0.97858 [-0.21215 ; 3.6238 ]
σ_prop 0.090433 0.0086792 [ 0.073422 ; 0.10744 ]
σ_add 0.33064 0.047455 [ 0.23763 ; 0.42366 ]
------------------------------------------------------------------------This result above indicates that both with the sandwich estimator and the inverse of the Hessian matrix on the fixed parameter, the inference worked. We will pursue other methods to obtain parameter precision in later tutorials, such as bootstrap and SIR.
3 Concluding Remarks
This tutorial showcased a typical Pumas workflow for PK model fitting using a Warfarin dataset:
- Model Definition and Data Preparation.
- Checking Compatibility via
loglikelihoodandfindinfluential. - Initial Parameter Estimation using Naive Pooled.
- Fitting with FOCE, demonstrating how to fix parameters or change solver tolerances.
- Interpreting Fit Results, exploring the components of a
fittedpumasmodel. - Computing Precision of estimates with
inferusing different methods.
Readers are encouraged to refine their understanding by:
- Performing thorough Exploratory Data Analysis prior to modeling.
- Exploring Bootstrap or SIR methods for deeper uncertainty quantification (covered in later tutorials).
- Validating the final model through visual diagnostics (e.g., VPCs, GOF plots).
More advanced topics can be found throughout the Pumas Documentation and Pumas Tutorials.
Tip
Real-world workflows are iterative. Analysts often revisit model assumptions, re-check data, and explore alternative parameterizations before finalizing any one model as “best.”