Introduction to Pumas

This is an introduction to Pumas, a software for pharmaceutical modeling and simulation.

The basic workflow of Pumas is:

  1. Build a model.
  2. Define subjects or populations to simulate or estimate.
  3. Analyze the results with post-processing and plots.

We will show how to build a multiple-response PK/PD model via the @model macro, define a subject with multiple doses, analyze the results of the simulation, roudtrip the data for estimation and fit the data using maximum likelihood methods. This tutorial is made to be a broad overview of the workflow and more in-depth treatment of each section can be found in the subsequent tutorials and documentation.

Working Example

Let's start by showing a complete simulation code, and then break down how it works.

using Pumas
using Random
using Plots
######### Turnover model

inf_2cmt_lin_turnover = @model begin
    @param   begin
        tvcl ∈ RealDomain(lower=0)
        tvvc ∈ RealDomain(lower=0)
        tvq ∈ RealDomain(lower=0)
        tvvp ∈ RealDomain(lower=0)
        Ω_pk ∈ PDiagDomain(4)
        σ_prop_pk ∈ RealDomain(lower=0)
        # PD parameters
        tvturn ∈ RealDomain(lower=0)
        tvebase ∈ RealDomain(lower=0)
        tvec50 ∈ RealDomain(lower=0)
        Ω_pd ∈ PDiagDomain(1)
        σ_add_pd ∈ RealDomain(lower=0)
    end

    @random begin
      ηpk ~ MvNormal(Ω_pk)
      ηpd ~ MvNormal(Ω_pd)
    end

    @pre begin
        CL = tvcl * exp(ηpk[1])
        Vc = tvvc * exp(ηpk[2])
        Q = tvq * exp(ηpk[3])
        Vp = tvvp * exp(ηpk[4])

        ebase = tvebase*exp(ηpd[1])
        ec50 = tvec50
        emax = 1
        turn = tvturn
        kout = 1/turn
        kin0 = ebase*kout
    end

    @init begin
        Resp = ebase
    end

    @vars begin
        conc := Central/Vc
        edrug := emax*conc/(ec50 + conc)
        kin := kin0*(1-edrug)
    end

     @dynamics begin
         Central' =  - (CL/Vc)*Central + (Q/Vp)*Peripheral - (Q/Vc)*Central
         Peripheral' = (Q/Vc)*Central - (Q/Vp)*Peripheral
         Resp' = kin - kout*Resp
     end

    @derived begin
        dv ~ @. Normal(conc, sqrt(conc^2*σ_prop_pk))
        resp ~ @. Normal(Resp, sqrt(σ_add_pd))
    end
end

turnover_params = (tvcl = 1.5,
                   tvvc = 25.0,
                   tvq = 5.0,
                   tvvp = 150.0,
                   tvturn = 10,
                   tvebase = 10,
                   tvec50 = 0.3,
                   Ω_pk = Diagonal([0.05,0.05,0.05,0.05]),
                   Ω_pd = Diagonal([0.05]),
                   σ_prop_pk = 0.02,
                   σ_add_pd = 0.2)
regimen = DosageRegimen(150, rate = 10, cmt = 1)
pop = Population(map(i -> Subject(id = i,events = regimen), 1:10))
sd_obstimes = [0.25, 0.5, 0.75, 1, 2, 4, 8,
                12, 16, 20, 24, 36, 48, 60, 71.9] # single dose observation times
Random.seed!(123)
sims = simobs(inf_2cmt_lin_turnover,
               pop,
               turnover_params,
               obstimes = sd_obstimes)
plot(sims, obsnames=[:dv, :resp], title = "",  guidefontsize = 25,
           linewidth = 5, xtickfont = font(25), ytickfont = font(25),
           size=(1000, 800), xlabel = "Time (hours)")

Plot sim

In this code, we defined a nonlinear mixed effects model by describing the parameters, the random effects, the dynamical model, and the derived (result) values. Then we generated a population of 10 subjects who received a single dose of 150mg, specified parameter values, simulated the model, and generated a plot of the results. Now let's walk through this process!

Using the Model Macro

First we define the model. The simplest way to do is via the @model DSL. Inside of this block we have a few subsections. The first of which is @param. In here we define what kind of parameters we have. For this model we will define structural model parameters of PK and PD and their corresponding variances where applicable:

@param   begin
    tvcl ∈ RealDomain(lower=0)
    tvvc ∈ RealDomain(lower=0)
    tvq ∈ RealDomain(lower=0)
    tvvp ∈ RealDomain(lower=0)
    Ω_pk ∈ PDiagDomain(4)
    σ_prop_pk ∈ RealDomain(lower=0)
    # PD parameters
    tvturn ∈ RealDomain(lower=0)
    tvebase ∈ RealDomain(lower=0)
    tvec50 ∈ RealDomain(lower=0)
    Ω_pd ∈ PDiagDomain(1)
    σ_add_pd ∈ RealDomain(lower=0)
end

Next we define our random effects. The random effects are defined by a distribution from Distributions.jl. For more information on defining distributions, please see the Distributions.jl documentation. For this tutorial, we wish to have a multivariate normal of uncorrelated random effects, one for PK and another for PD so we utilize the syntax:

ηpk ~ MvNormal(Ω_pk)
ηpd ~ MvNormal(Ω_pd)

Now we define our pre-processing step in @pre. This is where we choose how the parameters, random effects, and the covariates collate. We define the values and give them a name as follows:

@pre begin
    CL = tvcl * exp(ηpk[1])
    Vc = tvvc * exp(ηpk[2])
    Q = tvq * exp(ηpk[3])
    Vp = tvvp * exp(ηpk[4])

    ebase = tvebase*exp(ηpd[1])
    ec50 = tvec50
    emax = 1
    turn = tvturn
    kout = 1/turn
    kin0 = ebase*kout
end

Next we define the @init block which gives the initial values for our differential equations. Any variable not mentioned in this block is assumed to have a zero for its starting value. We wish to only set the starting value for Resp, and thus we use:

@init begin
    Resp = ebase
end

Now we define our dynamics. We do this via the @dynamics block. Differential variables are declared by having a line defining their derivative. For our model, we use:

@dynamics begin
    Central' =  - (CL/Vc)*Central + (Q/Vp)*Peripheral - (Q/Vc)*Central
    Peripheral' = (Q/Vc)*Central - (Q/Vp)*Peripheral
    Resp' = kin - kout*Resp
end

Next we setup alias variables that can be used later in the code. Such alias code can be setup in the @vars block

@vars begin
    conc := Central/Vc
    edrug := emax*conc/(ec50 + conc)
    kin := kin0*(1-edrug)
end

Lastly we utilize the @derived macro to define our post-processing. We can output values using the following:

@derived begin
    dv ~ @. Normal(conc, sqrt(conc^2*σ_prop_pk))
    resp ~ @. Normal(Resp, sqrt(σ_add_pd))
end

Building a Subject

Now let's build a subject to simulate the model with. A subject is defined by the following components:

  1. An identifier - id
  2. The dosage regimen - events
  3. The covariates of the individual - covariates
  4. Observations associated with the individual - observations
  5. The timing of the sampling - time
  6. A vector of times if the covariates are time-varying - covariate_time
  7. Interpolation direction for covariates - covariate_direction

Our model did not make use of covariates so we will ignore (3, 6 and 7) for now, and (4) is only necessary for fitting parameters to data which will not be covered in this tutorial. Thus our subject will be defined simply by its dosage regimen.

To do this, we use the DosageRegimen constructor. The first value is always the dosing amount. Then there are optional arguments, the most important of which is time which specifies the time that the dosing occurs. For example,

DosageRegimen(150, time=0)

is a dosage regimen which simply does a single dose at time t=0 of amount 150. Let's assume the dose is an infusion administered at the rate of 10 mg/hour into the first compartment

regimen = DosageRegimen(150, rate=10, cmt=1)

Let's define our subject to have id=1 and this multiple dosing regimen:

subject = Subject(id = 1, events = regimen)

You can also create a collection of subjects which becomes a Population.

pop = Population(map(i -> Subject(id= i, events = regimen), 1:10))

Running a Simulation

The main function for running a simulation is simobs. simobs on a population simulates all of the population, while simobs on a subject simulates just that subject. If we wish to change the parameters from the initialized values, then we pass them in. Let's simulate subject 1 with a set of chosen parameters:

turnover_params = (tvcl = 1.5,
                   tvvc = 25.0,
                   tvq = 5.0,
                   tvvp = 150.0,
                   tvturn = 10,
                   tvebase = 10,
                   tvec50 = 0.3,
                   Ω_pk = Diagonal([0.05,0.05,0.05,0.05]),
                   Ω_pd = Diagonal([0.05]),
                   σ_prop_pk = 0.02,
                   σ_add_pd = 0.2)

Random.seed!(123)
sims = simobs(inf_2cmt_lin_turnover,
               pop,
               turnover_params,
               obstimes = sd_obstimes)

We can then plot the simulated observations by using the plot command:

using Plots
plot(sim)

simple plot

Note that we can use the attributes from Plots.jl to further modify the plot. For example,

plot(sims, obsnames=[:dv, :resp], title = "",  guidefontsize = 25,
           linewidth = 5, xtickfont = font(25), ytickfont = font(25),
           size=(1000, 800), xlabel = "Time (hours)")

Plot sim

When we only give the parameters, the random effects are automatically sampled from their distributions. If we wish to prescribe a value for the random effects, we pass initial values similarly:

randeffs = (ηpk = rand(4), ηpd = rand(1),)

If a population simulation is required with no random effects, then the values of the η's can be set to zero that will result in a simulation only at the mean level:

randeffs = (ηpk = zeros(4), ηpd = zeros(1))
Random.seed!(123)
sims = simobs(inf_2cmt_lin_turnover,
               pop,
               turnover_params,
               [randeffs for i in 1:length(pop)];
               obstimes = sd_obstimes)
plot(sims, title = "",  guidefontsize = 25,
           linewidth = 5, xtickfont = font(25), ytickfont = font(25),
           size=(1000, 800), xlabel = "Time (hours)")

Notice that since we are simulating a population, the randeffs have to be the same lenght of the population, i.e., one set of random effects for each subject.

Plot sim

You still see variability in the plot above, mainly due to the residual variability components in the model. It is quite trivial to change the parameter estimates of only a subset of parameters as below

turnover_params_wo_sigma  = (turnover_params..., σ_prop_pk = 0.0, σ_add_pd = 0.0)

and now if we perform the simulation again without random effects to get only the population mean,

randeffs = (ηpk = zeros(4), ηpd = zeros(1))
Random.seed!(123)
sims = simobs(inf_2cmt_lin_turnover,
               pop,
               turnover_params_wo_sigma,
               [randeffs for i in 1:length(pop)];
               obstimes = sd_obstimes)
plot(sims, title = "",  guidefontsize = 25, color=:blue,
           linewidth = 5, xtickfont = font(25), ytickfont = font(25),
           size=(1000, 800), xlabel = "Time (hours)")

Plot sim

Handling the SimulatedObservations

The resulting SimulatedObservations type has two fields for each subject. sim[1].time is an array of time points for which the data was saved. sim[1].observations is the result of the derived variables. From there, the derived variables are accessed by name. For example,

sims[1].observations[:dv]

is the array of dv values at the associated time points for subject 1. We can turn this into a DataFrame via using the DataFrame command:

DataFrame(sims)

From there, any Julia tools can be used to analyze these arrays and DataFrames.

Using these commands, a Julia program can be written to post-process the program however you like!

Conclusion

This tutorial covered basic workflow for how to build a model and simulate results from it. The subsequent tutorials will go into more detail in the components, such as:

  1. More detailed treatment of specifying populations, dosage regimens, and covariates.
  2. Reading in dosage regimens and observations from standard input data.
  3. Fitting models