Generating and Simulating Populations

using Dates
using Random
using Pumas
using PumasUtilities

Random.seed!(1234)

1 Introduction

In this tutorial, we will cover the fundamentals of generating populations to simulate with Pumas. We will demonstrate how to specify dosage regimens and covariates, and then how to piece these together to form a population to simulate.

2 The model

Below is a Pumas model that specifies a 1-compartment oral absorption system with between-subject variability on all the parameters. Details of the model specification are provided in the documentation.

model = @model begin
    @param begin
        tvcl ∈ RealDomain(; lower = 0)
        tvvc ∈ RealDomain(; lower = 0)
        tvka ∈ RealDomain(; lower = 0)
        Ω ∈ PSDDomain(3)
        σ_prop ∈ RealDomain(; lower = 0)
    end

    @random begin
        η ~ MvNormal(Ω)
    end

    @covariates Wt

    @pre begin
        CL = tvcl * (Wt / 70)^0.75 * exp(η[1])
        Vc = tvvc * (Wt / 70) * exp(η[2])
        Ka = tvka * exp(η[3])
    end

    @dynamics begin
        Depot' = -Ka * Depot
        Central' = Ka * Depot - Central * CL / Vc
    end

    @derived begin
        conc = @. Central / Vc
        dv ~ @. Normal(conc, conc * σ_prop)
    end

end

PumasModel
  Parameters: tvcl, tvvc, tvka, Ω, σ_prop
  Random effects: η
  Covariates: Wt
  Dynamical variables: Depot, Central
  Derived: conc, dv
  Observed: conc, dv

3 Setting up parameters

Next we provide the initial estimates of the parameters to simulate from. The fixed effects are provided in the typical values tv parameters (tvcl, tvvc, tvka) and the between-subject variability parameters are provided in the Ω matrix as a covariance matrix. So, 0.04 variance on Ω₁,₁ suggests a 20% coefficient of variation (\(\sqrt{0.04} = 0.2\)). Similarly, σ_prop has a 20% proportional residual error.

fixeffs =
    (; tvcl = 0.4, tvvc = 20, tvka = 1.1, Ω = Diagonal([0.04, 0.04, 0.04]), σ_prop = 0.2)

(tvcl = 0.4,
 tvvc = 20,
 tvka = 1.1,
 Ω = [0.04 0.0 0.0; 0.0 0.04 0.0; 0.0 0.0 0.04],
 σ_prop = 0.2,)

4 Single dose example

DosageRegimen() is the function that lets you construct a dosing regimen. The first argument of the DosageRegimen is amt and is a positional argument and not a keyword argument. All subsequent arguments need to be named, since they are keyword arguments. Let’s try a simple example where you provide a 100 mg dose at time = 0.

ev = DosageRegimen(100; time = 0)

1×10 DataFrame

Row	time	cmt	amt	evid	ii	addl	rate	duration	ss	route
	Float64	Int64	Float64	Int8	Float64	Int64	Float64	Float64	Int8	NCA.Route
1	0.0	1	100.0	1	0.0	0	0.0	0.0	0	NullRoute

As you can see above, we provided a single 100 mg dose. DosageRegimen provides some defaults when it creates the dosage regimen dataset, time = 0, evid = 1, cmt = 1, rate = 0, ii = 0 and addl = 0.

Note that ev is of type DosageRegimen. Specified like above, DosageRegimen is one of the four fundamental building block of a Subject (more on Subject below).

4.1 Building Subjects

Let’s create a single subject

s1 = Subject(;
    id = 1,
    events = ev,
    covariates = (; Wt = 70),
    observations = (; dv = nothing),
)

Subject
  ID: 1
  Events: 1
  Observations: dv: (n=0)
  Covariates: Wt

Note that the s1 Subject created above is composed of:

• id: an unique identifier
• observations: observations, represented by a NamedTuple
• covariates: covariates, represented by a NamedTuple
• events: events, represented by a DosageRegimen

DataFrame(s1.events)

1×9 DataFrame

Row	time	amt	evid	cmt	rate	duration	ss	ii	route
	Float64	Float64	Int8	Int64	Float64	Float64	Int8	Float64	NCA.Route
1	0.0	100.0	1	1	0.0	0.0	0	0.0	NullRoute

The events are presented by basic information such as the dose of drug, the time of dose administration, the compartment number for administration, and whether the dose is an instantaneous input or an infusion.

Below is how the covariates are represented, as a NamedTuple.

s1.covariates

Pumas.ConstantCovar{NamedTuple{(:Wt,), Tuple{Int64}}}((Wt = 70,))

(Note: defining distributions for covariates will be discussed in detail later.)

Using this one subject, s1, let us simulate a simple concentration time profile using the model above, and plot the result:

sims1 = simobs(model, s1, fixeffs; obstimes = 0:0.1:120)

sim_plot(
    model,
    sims1;
    observations = [:conc],
    figure = (; fontsize = 18),
    axis = (; xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)"),
)

4.2 Building Populations

Now, lets create one more subject, s2.

s2 = Subject(;
    id = 2,
    events = ev,
    covariates = (; Wt = 70),
    observations = (; dv = nothing),
)

Subject
  ID: 2
  Events: 1
  Observations: dv: (n=0)
  Covariates: Wt

If we want to simulate both s1 and s2 together, we need to bring these subjects together to form a Population. A Population is essentially a collection (vector) of subjects.

twosubjs = [s1, s2]

Population
  Subjects: 2
  Covariates: Wt
  Observations: dv

Let’s see the details of the first and the second subject:

twosubjs[1]

Subject
  ID: 1
  Events: 1
  Observations: dv: (n=0)
  Covariates: Wt

twosubjs[2]

Subject
  ID: 2
  Events: 1
  Observations: dv: (n=0)
  Covariates: Wt

Now, we can simulate this Population of 2 subjects, and plot the result:

sims2 = simobs(model, twosubjs, fixeffs; obstimes = 0:0.1:120)

sim_plot(
    model,
    sims2;
    observations = [:conc],
    figure = (; fontsize = 18),
    axis = (; xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)"),
)

Similarly, we can build a population of any number of subjects. But before we do that, let’s dive into covariate generation.

4.3 Covariates

As was discussed earlier, a Subject can also be provided details regarding covariates. In the model above, a continuous covariate body weight Wt impacts both CL and Vc. Let us now specify covariates to a population of 10 subjects:

choose_covariates() = (; Wt = rand(55:80))

choose_covariates (generic function with 1 method)

choose_covariates will randomly choose a Wt between 55-80 kgs

We can make a vector with covariates for ten subjects through an array comprehension:

cvs = [choose_covariates() for i = 1:10]

Now, we add these covariates to the population as below. The map(f, xs) will return the result of f on each element of the collection xs. Let’s map a function that build’s a subject with the randomly chosen covariates in order to build a Population:

pop_with_covariates = map(
    i -> Subject(;
        id = i,
        events = ev,
        covariates = choose_covariates(),
        observations = (; dv = nothing),
    ),
    1:10,
)

Population
  Subjects: 10
  Covariates: Wt
  Observations: dv

Simulate into the population, and visualize the output:

sims3 = simobs(model, pop_with_covariates, fixeffs; obstimes = 0:0.1:120)

sim_plot(
    model,
    sims3;
    observations = [:conc],
    figure = (; fontsize = 18),
    axis = (; xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)"),
)

5 Multiple dose example

DosageRegimen provides additional controls over the specification of the dosage regimen. For example, ii defines the “interdose interval”, or the time difference between two doses, while addl defines how many additional times to repeat a dose. Thus, let’s define a dose of 100 that’s repeated 7 times at 24 hour intervals:

md = DosageRegimen(100; ii = 24, addl = 6)

1×10 DataFrame

Row	time	cmt	amt	evid	ii	addl	rate	duration	ss	route
	Float64	Int64	Float64	Int8	Float64	Int64	Float64	Float64	Int8	NCA.Route
1	0.0	1	100.0	1	24.0	6	0.0	0.0	0	NullRoute

Let’s create a new subject, s3 with this dosage regimen, and see the results:

s3 = Subject(;
    id = 3,
    events = md,
    covariates = (; Wt = 70),
    observations = (; dv = nothing),
)

sims4 = simobs(model, s3, fixeffs; obstimes = 0:0.1:240)

sim_plot(
    model,
    [sims4];
    observations = [:conc],
    figure = (; fontsize = 18),
    axis = (; xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)"),
)

6 Combining dosage regimens

We can also combine dosage regimens to build a more complex regimen. Recall from the introduction that using arrays will build the element-wise combinations. Thus, let’s build a dose of 500 into compartment 1 at time 0, and 7 doses into compartment 1 of 100 spaced by 24 hours:

ldmd = DosageRegimen([500, 100]; time = [0, 24], cmt = 1, addl = [0, 6], ii = [0, 24])

2×10 DataFrame

Row	time	cmt	amt	evid	ii	addl	rate	duration	ss	route
	Float64	Int64	Float64	Int8	Float64	Int64	Float64	Float64	Int8	NCA.Route
1	0.0	1	500.0	1	0.0	0	0.0	0.0	0	NullRoute
2	24.0	1	100.0	1	24.0	6	0.0	0.0	0	NullRoute

Let’s see if this result matches our intuition:

s4 = Subject(;
    id = 4,
    events = ldmd,
    covariates = (; Wt = 70),
    observations = (; dv = nothing),
)

sims5 = simobs(model, s4, fixeffs; obstimes = 0:0.1:120)

sim_plot(
    model,
    [sims5];
    observations = [:conc],
    figure = (; fontsize = 18),
    axis = (; xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)"),
)

Another way to build complex dosage regiments is to combine previously constructed regimens into a single regimen. For example:

e1 = DosageRegimen(500; cmt = 1, time = 0, addl = 0, ii = 0)
e2 = DosageRegimen(100; cmt = 1, time = 24, addl = 6, ii = 24)
evs = DosageRegimen(e1, e2)

2×10 DataFrame

Row	time	cmt	amt	evid	ii	addl	rate	duration	ss	route
	Float64	Int64	Float64	Int8	Float64	Int64	Float64	Float64	Int8	NCA.Route
1	0.0	1	500.0	1	0.0	0	0.0	0.0	0	NullRoute
2	24.0	1	100.0	1	24.0	6	0.0	0.0	0	NullRoute

s5 = Subject(;
    id = 5,
    events = evs,
    covariates = (; Wt = 70),
    observations = (; dv = nothing),
)

sims6 = simobs(model, s5, fixeffs; obstimes = 0:0.1:120)

sim_plot(
    model,
    [sims6];
    observations = [:conc],
    figure = (; fontsize = 18),
    axis = (; xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)"),
)

is the same regimen as before.

Putting these ideas together, we can define a Population where individuals with different covariates undergo different regimens, and simulate them all together:

e3 = DosageRegimen(100; ii = 24, addl = 6)
e4 = DosageRegimen(50; ii = 12, addl = 13)
e5 = DosageRegimen(200; ii = 24, addl = 2)

pop1 = map(
    i -> Subject(;
        id = i,
        events = e3,
        covariates = choose_covariates(),
        observations = (; dv = nothing),
    ),
    1:5,
)

pop2 = map(
    i -> Subject(;
        id = i,
        events = e4,
        covariates = choose_covariates(),
        observations = (; dv = nothing),
    ),
    6:8,
)

pop3 = map(
    i -> Subject(;
        id = i,
        events = e5,
        covariates = choose_covariates(),
        observations = (; dv = nothing),
    ),
    9:10,
)

pop = vcat(pop1, pop2, pop3)

sims7 = simobs(model, pop, fixeffs; obstimes = 0:0.1:120)

sim_plot(
    model,
    sims7;
    observations = [:conc],
    figure = (; fontsize = 18),
    axis = (; xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)"),
)

7 Defining Infusions

An infusion is a dosage which is defined as having a non-zero positive rate at which the drug enters the system. Let’s define a single infusion dose of total amount 100 with a rate of 3 unit amount per time into the compartment 2:

inf = DosageRegimen(100; rate = 3, cmt = 2)

1×10 DataFrame

Row	time	cmt	amt	evid	ii	addl	rate	duration	ss	route
	Float64	Int64	Float64	Int8	Float64	Int64	Float64	Float64	Int8	NCA.Route
1	0.0	2	100.0	1	0.0	0	3.0	33.3333	0	NullRoute

Now let’s simulate a subject undergoing this treatment strategy:

s6 = Subject(;
    id = 5,
    events = inf,
    covariates = (; Wt = 70),
    observations = (; dv = nothing),
)

sims8 = simobs(model, s6, fixeffs; obstimes = 0:0.1:120)

sim_plot(
    model,
    [sims8];
    observations = [:conc],
    figure = (; fontsize = 18),
    axis = (; xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)"),
)

8 Final Note on Julia Programming

Note that all of these functions are standard Julia functions, and thus standard Julia programming constructions can be utilized to simplify the construction of large populations. We already demonstrated the use of map and array comprehension, but we can also make use of constructs like for loops and if-else statements.

9 Conclusion

This tutorial shows the tools for generating populations of infinite complexity, defining covariates and dosage regimens on the fly and simulating the results of the model.