Fitting and Inferring a Compartmental PK Model

Author

Vijay Ivaturi

1 Introduction

A typical workflow for fitting a Pumas model and deriving parameter precision typically involves:

  1. Preparing the data and the model.
  2. Checking model-data compatibility.
  3. Obtaining initial parameter estimates.
  4. Fitting the model via a chosen estimation method.
  5. Interpreting the fit results.
  6. Computing parameter precision.
  7. (Optionally) proceeding with more advanced techniques like bootstrapping or SIR for robust uncertainty quantification.

The following sections will walk through these steps using a one-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_tabs, pop_lag
  • Inter-individual variability (IIV) components: pk_Ω
  • Residual error model parameters: σ_prop, σ_add
  • Covariates for scaling: FSZCL and FSZV
  • Differential equations describing the PK behavior in the compartments.
using Pumas
using PharmaDatasets
using DataFramesMeta
using CairoMakie
using AlgebraOfGraphics
using PumasUtilities
using Markdown
warfarin_pk_model = @model begin

    @metadata begin
        desc = "Warfarin 1-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)
        """
        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])

        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
    end

    @derived begin
        """
        Concentration (ng/mL)
        """
        conc ~ @. Normal(cp, sqrt((σ_prop * cp)^2 + σ_add^2))
    end
end
PumasModel
  Parameters: pop_CL, pop_Vc, pop_tabs, pop_lag, pk_Ω, σ_prop, σ_add
  Random effects: pk_η
  Covariates: FSZCL, FSZV
  Dynamical system variables: Depot, Central
  Dynamical system type: Matrix exponential
  Derived: conc
  Observed: conc

2.2 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)
end
317×13 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 include FOCE(), 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_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())
-11626.613157863481

The initial loglikelihood of the warfarin PK model given the data and parameter values is `LL = -11626.613157863481 `.

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 = "8", nll = 2609.1127547470246)
 (id = "14", nll = 2113.146717564112)
 (id = "12", nll = 1928.5840732530462)
 (id = "13", nll = 1627.4380743700433)
 (id = "2", nll = 780.9109417396234)
 (id = "7", nll = 512.7488860179473)
 (id = "11", nll = 470.0423353858021)
 (id = "6", nll = 311.9666456226985)
 (id = "3", nll = 192.83907815907168)
 (id = "22", nll = 187.1431308413218)
 ⋮
 (id = "16", nll = 22.434033734725986)
 (id = "5", nll = 18.475981181576064)
 (id = "32", nll = 17.87557264312157)
 (id = "17", nll = 17.341505598982042)
 (id = "26", nll = 8.191625356662307)
 (id = "24", nll = 6.4001549755272755)
 (id = "31", nll = 6.363630970170725)
 (id = "23", nll = 5.48597914877157)
 (id = "29", nll = 4.343343958293616)

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     5.070332e+04     5.741309e+04
 * time: 0.020396947860717773
     1     1.990291e+04     3.525936e+04
 * time: 0.764348030090332
     2     3.850566e+03     7.927044e+03
 * time: 0.7657740116119385
     3     2.326447e+03     4.958575e+03
 * time: 0.7671420574188232
     4     1.147033e+03     2.356630e+03
 * time: 0.7685039043426514
     5     7.221759e+02     1.181586e+03
 * time: 0.7698509693145752
     6     5.432686e+02     5.064601e+02
 * time: 0.7712290287017822
     7     4.864460e+02     1.677683e+02
 * time: 0.7726130485534668
     8     4.724264e+02     7.050247e+01
 * time: 0.7739808559417725
     9     4.702526e+02     6.322024e+01
 * time: 0.7753548622131348
    10     4.699078e+02     6.864635e+01
 * time: 0.7767288684844971
    11     4.695689e+02     7.516235e+01
 * time: 0.7781078815460205
    12     4.688608e+02     7.294852e+01
 * time: 0.7795019149780273
    13     4.679633e+02     5.227783e+01
 * time: 0.7809040546417236
    14     4.672220e+02     2.711848e+01
 * time: 0.782310962677002
    15     4.669432e+02     2.373879e+01
 * time: 0.7836978435516357
    16     4.668886e+02     2.300102e+01
 * time: 0.7852139472961426
    17     4.668459e+02     2.284229e+01
 * time: 0.7866089344024658
    18     4.667377e+02     2.378758e+01
 * time: 0.7879898548126221
    19     4.665009e+02     2.684403e+01
 * time: 0.7893760204315186
    20     4.659955e+02     2.835852e+01
 * time: 0.7907688617706299
    21     4.652873e+02     2.371497e+01
 * time: 0.7921669483184814
    22     4.647592e+02     3.073448e+01
 * time: 0.7935428619384766
    23     4.645466e+02     3.812126e+01
 * time: 0.794921875
    24     4.645013e+02     3.839585e+01
 * time: 0.7963018417358398
    25     4.644583e+02     3.795849e+01
 * time: 0.7976789474487305
    26     4.643176e+02     3.599161e+01
 * time: 0.7992238998413086
    27     4.640337e+02     3.141265e+01
 * time: 0.8007998466491699
    28     4.634991e+02     2.152192e+01
 * time: 0.8023660182952881
    29     4.629314e+02     1.579662e+01
 * time: 0.803912878036499
    30     4.626793e+02     8.229231e+00
 * time: 0.8054709434509277
    31     4.626228e+02     4.593455e+00
 * time: 0.8070578575134277
    32     4.626174e+02     4.703436e+00
 * time: 0.8086240291595459
    33     4.626166e+02     4.711371e+00
 * time: 0.81020188331604
    34     4.626100e+02     4.754289e+00
 * time: 0.8117749691009521
    35     4.625971e+02     4.818111e+00
 * time: 0.8133590221405029
    36     4.625584e+02     7.283490e+00
 * time: 0.8149549961090088
    37     4.624573e+02     1.322306e+01
 * time: 0.8165109157562256
    38     4.621496e+02     2.504876e+01
 * time: 0.8180489540100098
    39     4.606593e+02     6.055247e+01
 * time: 0.8195939064025879
    40     4.574668e+02     9.046732e+01
 * time: 0.8211250305175781
    41     4.493736e+02     1.320107e+02
 * time: 0.8226809501647949
    42     4.476881e+02     1.201131e+02
 * time: 0.8246438503265381
    43     4.399545e+02     6.343714e+01
 * time: 0.8262569904327393
    44     4.308404e+02     1.074644e+02
 * time: 0.8278639316558838
    45     4.279440e+02     4.992808e+01
 * time: 0.8294780254364014
    46     4.269382e+02     2.089849e+01
 * time: 0.8310039043426514
    47     4.259042e+02     2.284828e+01
 * time: 0.8325319290161133
    48     4.250072e+02     3.457276e+01
 * time: 0.8340668678283691
    49     4.225016e+02     5.670587e+01
 * time: 0.8356039524078369
    50     4.218595e+02     8.075863e+01
 * time: 0.8371620178222656
    51     4.156464e+02     5.623544e+01
 * time: 0.8386929035186768
    52     4.144306e+02     1.062421e+01
 * time: 0.8402559757232666
    53     4.139388e+02     9.442026e+00
 * time: 0.8417849540710449
    54     4.133251e+02     9.168600e+00
 * time: 0.8433070182800293
    55     4.125579e+02     1.007895e+01
 * time: 0.8448410034179688
    56     4.116527e+02     1.162167e+01
 * time: 0.8464038372039795
    57     4.106767e+02     1.382208e+01
 * time: 0.8479280471801758
    58     4.092564e+02     2.392522e+01
 * time: 0.8505289554595947
    59     4.088745e+02     1.991206e+01
 * time: 0.8520748615264893
    60     4.084320e+02     1.729494e+01
 * time: 0.8536410331726074
    61     4.082868e+02     2.139214e+01
 * time: 0.8555450439453125
    62     4.078151e+02     2.066959e+01
 * time: 0.8570849895477295
    63     4.075154e+02     1.547607e+01
 * time: 0.8586528301239014
    64     4.074365e+02     2.050007e+01
 * time: 0.8606259822845459
    65     4.073436e+02     2.087001e+01
 * time: 0.8626999855041504
    66     4.072557e+02     2.244000e+01
 * time: 0.8648040294647217
    67     4.071114e+02     2.538153e+01
 * time: 0.8669328689575195
    68     4.069218e+02     3.482650e+01
 * time: 0.8690638542175293
    69     4.067389e+02     4.414308e+01
 * time: 0.8711628913879395
    70     4.066095e+02     4.254209e+01
 * time: 0.8732619285583496
    71     4.065063e+02     4.513420e+01
 * time: 0.8753950595855713
    72     4.063491e+02     3.571645e+01
 * time: 0.8775360584259033
    73     4.061858e+02     6.436632e+01
 * time: 0.8796968460083008
    74     4.060687e+02     6.862397e+01
 * time: 0.8818600177764893
    75     4.056652e+02     7.310609e+01
 * time: 0.8840219974517822
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   139     3.891230e+02     1.971994e+06
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   141     3.885389e+02     1.421010e+07
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   142     3.876703e+02     1.828068e+07
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   143     3.872001e+02     1.936569e+07
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   144     3.864437e+02     7.177840e+06
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   145     3.862728e+02     2.351227e+07
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   146     3.861345e+02     1.476714e+07
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   149     3.850837e+02     5.358988e+07
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   150     3.846268e+02     5.942891e+07
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   152     3.835490e+02     1.465098e+07
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   153     3.832654e+02     1.633041e+08
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   154     3.829528e+02     5.727909e+07
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   157     3.803226e+02     1.187691e+08
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   158     3.798624e+02     3.217154e+08
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   159     3.797492e+02     4.467329e+08
 * time: 1.156074047088623
   160     3.795408e+02     4.886666e+08
 * time: 1.158642053604126
   161     3.792743e+02     5.037401e+08
 * time: 1.1611878871917725
   162     3.789825e+02     5.111849e+08
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 * time: 1.1662938594818115
   164     3.783367e+02     5.171820e+08
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   165     3.779772e+02     5.263783e+08
 * time: 1.1714420318603516
   166     3.775719e+02     5.000187e+08
 * time: 1.174025058746338
   167     3.771293e+02     8.335380e+08
 * time: 1.176602840423584
   168     3.759346e+02     1.046396e+09
 * time: 1.1792850494384766
   169     3.755284e+02     1.160020e+09
 * time: 1.1819159984588623
   170     3.749055e+02     4.168182e+08
 * time: 1.184556007385254
   171     3.743422e+02     2.645442e+09
 * time: 1.1882920265197754
   172     3.740647e+02     5.457301e+09
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   173     3.737548e+02     6.572679e+09
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   174     3.734376e+02     6.309385e+09
 * time: 1.1961839199066162
   175     3.731891e+02     5.923826e+09
 * time: 1.1988458633422852
   176     3.729631e+02     5.750542e+09
 * time: 1.2014939785003662
   177     3.724857e+02     5.555957e+09
 * time: 1.2041609287261963
   178     3.722050e+02     5.222174e+09
 * time: 1.2068979740142822
   179     3.717245e+02     6.984869e+09
 * time: 1.209709882736206
   180     3.706393e+02     1.976407e+09
 * time: 1.2125558853149414
   181     3.705300e+02     2.033361e+10
 * time: 1.2159898281097412
   182     3.702465e+02     2.437418e+10
 * time: 1.2188339233398438
   183     3.700494e+02     2.195495e+10
 * time: 1.2216570377349854
   184     3.694023e+02     2.298811e+10
 * time: 1.224519968032837
   185     3.692452e+02     2.149516e+10
 * time: 1.2275609970092773
   186     3.687094e+02     2.166279e+10
 * time: 1.2303948402404785
   187     3.685261e+02     2.143947e+10
 * time: 1.2332818508148193
   188     3.673781e+02     1.883299e+10
 * time: 1.2361578941345215
   189     3.670650e+02     1.930317e+10
 * time: 1.2406609058380127
   190     3.670156e+02     1.930877e+10
 * time: 1.2457129955291748
   191     3.668443e+02     1.930877e+10
 * time: 1.2566540241241455
   192     3.668443e+02     1.930877e+10
 * time: 1.2669758796691895
   193     3.668404e+02     1.930877e+10
 * time: 1.2743568420410156
   194     3.668404e+02     1.930877e+10
 * time: 1.2845280170440674
   195     3.668404e+02     1.930877e+10
 * time: 1.2942860126495361
FittedPumasModel

Successful minimization:                      true

Likelihood approximation:              NaivePooled
Likelihood Optimizer:                         BFGS
Dynamical system type:          Matrix exponential

Log-likelihood value:                   -366.84038
Number of subjects:                             31
Number of parameters:         Fixed      Optimized
                                  1              8
Observation records:         Active        Missing
    conc:                       239             47
    Total:                      239             47

-------------------------
             Estimate
-------------------------
pop_CL        0.12562
pop_Vc        8.5443
pop_tabs      9.8269e-11
pop_lag       1.0
pk_Ω₁,₁       0.01
pk_Ω₂,₂       0.01
pk_Ω₃,₃       0.01
σ_prop        0.24564
σ_add         2.1685e-8
-------------------------

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_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_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_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_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_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_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,
)
9×6 DataFrame
Row parameter p1 p2 p3 p4 original
String Float64? Float64? Float64? Float64? Float64?
1 pop_CL 0.12727 0.119671 0.12727 0.12694 0.125617
2 pop_Vc 8.34567 7.65772 8.34567 8.49744 8.54432
3 pop_tabs 0.371906 0.737079 0.371904 2.83552e-11 9.82686e-11
4 pop_lag 0.764775 0.5 0.764777 1.0 1.0
5 pk_Ω₁,₁ 0.009 0.0095 0.0105 0.011 0.01
6 pk_Ω₂,₂ 0.009 0.0095 0.0105 0.011 0.01
7 pk_Ω₃,₃ 0.009 0.0095 0.0105 0.011 0.01
8 σ_prop 0.228485 0.318661 0.228485 0.252298 0.245643
9 σ_add 0.380457 4.0701e-24 0.38046 8.61276e-9 2.16855e-8

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 p3 results seem to be reasonable amongst all the fits and these results can 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 using the initial parameter values specified in the model definition:

foce_fit = fit(warfarin_pk_model, pop, init_params(warfarin_pk_model), FOCE())

The init_params function is a convenience function that extracts the initial parameter values from the model definition.

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 tabs at a particular value:

foce_fit_fixedtabs = fit(
    warfarin_pk_model,
    pop,
    (; param_vals..., pop_tabs = 0.5),
    FOCE();
    constantcoef = (:pop_tabs,),
);   # This fixes the pop_tabs parameter
foce_fit_fixedtabs
FittedPumasModel

Successful minimization:                      true

Likelihood approximation:                     FOCE
Likelihood Optimizer:                         BFGS
Dynamical system type:          Matrix exponential

Log-likelihood value:                   -320.22372
Number of subjects:                             31
Number of parameters:         Fixed      Optimized
                                  1              8
Observation records:         Active        Missing
    conc:                       239             47
    Total:                      239             47

-----------------------
             Estimate
-----------------------
pop_CL        0.13091
pop_Vc        8.0799
pop_tabs      0.5
pop_lag       0.91828
pk_Ω₁,₁       0.051096
pk_Ω₂,₂       0.019074
pk_Ω₃,₃       1.0492
σ_prop        0.090995
σ_add         0.33185
-----------------------

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_fixedtabs = fit(
    warfarin_pk_model,
    pop,
    (; param_vals..., pop_tabs = 0.5),
    FOCE();
    constantcoef = (:pop_tabs,),
    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:

  1. Objective Function Value (OFV): The minimized negative log-likelihood (or similar objective). Lower is better.
  2. Convergence Status: An indication of whether the solver found a local optimum or if it ran into issues (e.g., maximum iterations reached).
  3. Parameter Estimates: A table listing final estimates of fixed effects, random effects (variances or standard deviations), and residual variability components.
  4. 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_fixedtabs
FittedPumasModel

Successful minimization:                      true

Likelihood approximation:                     FOCE
Likelihood Optimizer:                         BFGS
Dynamical system type:          Matrix exponential

Log-likelihood value:                    -320.2235
Number of subjects:                             31
Number of parameters:         Fixed      Optimized
                                  1              8
Observation records:         Active        Missing
    conc:                       239             47
    Total:                      239             47

-----------------------
             Estimate
-----------------------
pop_CL        0.13091
pop_Vc        8.08
pop_tabs      0.5
pop_lag       0.91842
pk_Ω₁,₁       0.050929
pk_Ω₂,₂       0.019161
pk_Ω₃,₃       1.056
σ_prop        0.090974
σ_add         0.33189
-----------------------

We can break down the output of the fit function into the following components:

2.9.1 FittedPumasModel

FittedPumasModel

The FittedPumasModel object contains the following fields:

  • model: The original model definition
foce_fit_fixedtabs.model
PumasModel
  Parameters: pop_CL, pop_Vc, pop_tabs, pop_lag, pk_Ω, σ_prop, σ_add
  Random effects: pk_η
  Covariates: FSZCL, FSZV
  Dynamical system variables: Depot, Central
  Dynamical system type: Matrix exponential
  Derived: conc
  Observed: conc
  • data: The population data
foce_fit_fixedtabs.data
Population
  Subjects: 31
  Covariates: SEX, WEIGHT, FSZV, FSZCL
  Observations: conc
  • optim: The optimization results
foce_fit_fixedtabs.optim
 * Status: success

 * Candidate solution
    Final objective value:     3.202237e+02

 * Found with
    Algorithm:     BFGS

 * Convergence measures
    |x - x'|               = 6.79e-08 ≰ 1.0e-08
    |x - x'|/|x'|          = 1.72e-08 ≤ 1.0e-06
    |f(x) - f(x')|         = 6.12e-07 ≰ 0.0e+00
    |f(x) - f(x')|/|f(x')| = 1.91e-09 ≰ 0.0e+00
    |g(x)|                 = 4.41e-02 ≰ 1.0e-03

 * Work counters
    Seconds run:   3  (vs limit Inf)
    Iterations:    70
    f(x) calls:    79
    ∇f(x) calls:   72
  • approx: The likelihood approximation method
foce_fit_fixedtabs.approx
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, )
)
  • kwargs: Additional keyword arguments
foce_fit_fixedtabs.kwargs
  • fixedparamset: The fixed parameter set
foce_fit_fixedtabs.fixedparamset
  • optim_state: The optimization state
foce_fit_fixedtabs.optim_state

2.9.2 Minimization Status

Successful minimization:                      true

The 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:                     FOCE

The Likelihood approximation field indicates the likelihood approximation method used.

2.9.4 Likelihood Optimizer

  Likelihood Optimizer:                         BFGS

The 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 exponential

The 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: foce_fit.model.prob which results in this case to Pumas.LinearODE(). 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
end

The 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:                    -320.2235

The 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:                             31

The 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              8

The 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             47

The 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        0.13091
pop_Vc        8.08
pop_tabs      0.5
pop_lag       0.91842
pk_Ω₁,₁       0.050929
pk_Ω₂,₂       0.019161
pk_Ω₃,₃       1.056
σ_prop        0.090974
σ_add         0.33189
-----------------------

The Parameter estimates field indicates the final parameter estimates. The Estimate column indicates the maximum likelihood estimates 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 of fit (e.g., foce_fit).
  • level: The confidence interval level (e.g., 0.95). The confidence intervals are calculated as the (1-level)/2 and (1+level)/2 quantiles of the estimated parameters
  • rethrow_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 to true (the default value), the sandwich estimator will be used. If set to false, the standard error will be calculated using the inverse of the Hessian matrix.

An example usage:

inference_results = infer(foce_fit_fixedtabs; 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:                   -320.22372
Number of subjects:                             31
Number of parameters:         Fixed      Optimized
                                  1              8
Observation records:         Active        Missing
    conc:                       239             47
    Total:                      239             47

-----------------------------------------------------------------------
            Estimate           SE                      95.0% C.I.
-----------------------------------------------------------------------
pop_CL       0.13091         0.0055387        [ 0.12006  ; 0.14177 ]
pop_Vc       8.0799          0.23677          [ 7.6158   ; 8.5439  ]
pop_tabs     0.5             NaN              [  NaN     ;  NaN      ]
pop_lag      0.91828         0.042229         [ 0.83551  ; 1.001   ]
pk_Ω₁,₁      0.051096        0.017609         [ 0.016584 ; 0.085608]
pk_Ω₂,₂      0.019074        0.0056721        [ 0.0079565; 0.030191]
pk_Ω₃,₃      1.0492          0.56299          [-0.054208 ; 2.1527  ]
σ_prop       0.090995        0.014419         [ 0.062735 ; 0.11926 ]
σ_add        0.33185         0.087824         [ 0.15972  ; 0.50398 ]
-----------------------------------------------------------------------

We can use the sandwich_estimator argument to get a more robust estimate of the standard errors.

inference_results = infer(foce_fit_fixedtabs; 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:                   -320.22372
Number of subjects:                             31
Number of parameters:         Fixed      Optimized
                                  1              8
Observation records:         Active        Missing
    conc:                       239             47
    Total:                      239             47

---------------------------------------------------------------------
            Estimate           SE                     95.0% C.I.
---------------------------------------------------------------------
pop_CL       0.13091         0.0054659       [ 0.1202   ; 0.14162]
pop_Vc       8.0799          0.23392         [ 7.6214   ; 8.5384 ]
pop_tabs     0.5             NaN             [  NaN     ;  NaN     ]
pop_lag      0.91828         0.029387        [ 0.86068  ; 0.97588]
pk_Ω₁,₁      0.051096        0.013885        [ 0.023883 ; 0.07831]
pk_Ω₂,₂      0.019074        0.006447        [ 0.0064378; 0.03171]
pk_Ω₃,₃      1.0492          0.55326         [-0.035156 ; 2.1336 ]
σ_prop       0.090995        0.0086472       [ 0.074047 ; 0.10794]
σ_add        0.33185         0.048238        [ 0.2373   ; 0.42639]
---------------------------------------------------------------------

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:

  1. Model Definition and Data Preparation.
  2. Checking Compatibility via loglikelihood and findinfluential.
  3. Initial Parameter Estimation using Naive Pooled.
  4. Fitting with FOCE, demonstrating how to fix parameters or change solver tolerances.
  5. Interpreting Fit Results, exploring the components of a fittedpumasmodel.
  6. Computing Precision of estimates with infer using 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.”