using Dates
using Pumas
using PumasUtilities
using DataFramesMeta
using PharmaDatasets
using CairoMakie
using AlgebraOfGraphics
using Random
Why are non-Gaussian random effects relevant?
1 Motivation - PK model
Why using a non-Gaussian distribution as the underlying distribution for the random effects? There are a couple of arguments.
First, the Gaussian distribution has unbounded support, i.e. it take any value in \((-\infty, \infty)\). While phamacokinetic parameters typically are (semi) bounded, e.g.:
- clearance and volumes, \((0, \infty)\)
- bioavailability, \([0, 1]\)
Additionally, in order for a Gaussian distribution to work as the underlying distribution, often we need to transform them (e.g. exponentiation and logistic transformation). But these transformations in some settings, when the random effects do not have a great impact, i.e. they do not have large values, may shift the mean of the typical values (\(\theta\)) so that the expectation of the typical values (\(\operatorname{E}\)) are not equal to the mean. For example, the following code block is a traditional 1-compartment PK model with a Gaussian random effect that needs to be constrained to positive values, \((0, \infty)\):
@random begin
ηCL ~ Normal(0.0, ωCL)
ηVc ~ Normal(0.0, ωVc)
end
@pre begin
CL = θCL * exp(ηCL)
Vc = θVc * exp(ηVc)
end
@dynamics begin
Central' = -CL / Vc * Central
endIf we recover the formula for the expectation of the log-normal distribution, we have that:
\[\operatorname{E}[CL] = \exp \left\{ \log(\theta_{CL}) + \frac{\omega^2_{CL}}{2} \right\} \approx \theta_{CL}\]
This approximation only holds for small \(\omega_{CL}\).
Hence, \(\theta_{CL}\) is only the typical value when \(\omega_{CL}\) is small.
Here is a small tabulation for \(\operatorname{E}[CL]\) when \(\theta_{CL} = 0.5\):
ωs = [0.1, 0.2, 0.4, 0.8, 1.6]
DataFrame(; ω_CL = ωs, E_CL = (ω -> exp(log(0.5) + ω^2 / 2)).(ωs))5×2 DataFrame
| Row | ω_CL | E_CL |
|---|---|---|
| Float64 | Float64 | |
| 1 | 0.1 | 0.502506 |
| 2 | 0.2 | 0.510101 |
| 3 | 0.4 | 0.541644 |
| 4 | 0.8 | 0.688564 |
| 5 | 1.6 | 1.79832 |
As you can see, the larger the \(\omega_{CL}\) the more \(\operatorname{E}[CL]\) deviates from \(\theta_{CL}\).
1.1 Gamma distribution for the rescue
We can use the gamma distribution which has the following parametrization:
\[\text{Gamma}(k, \theta)\]
where \(k\) is a shape parameter and \(\theta\) is a scale parameter.
Note
Shape parameters generally control the shape of the distribution rather than shifting it (as a location parameter) of stretching/shrinking it (as a scale parameter)
We can use an alternative parametrization where the mean-value appears directly a parameter:
\[\text{Gamma}(\mu, \sigma)\]
where:
- \(\mu = \theta k\)
- \(\sigma = k^{-\frac{1}{2}}\)
Tip
The \(\sigma\) parameter is the coefficient of variation, i.e.
\[\sigma = \frac{\operatorname{Var} X}{\operatorname{E} X},\]
because that mimics the role of \(\sigma\) in the LogNormal(log(μ), σ) where for small values of \(\sigma\)
\[\sigma \approx \\frac{\operatorname{Var} X}{\operatorname{E} X}.\]
So, our previous PK model now becomes:
@random begin
_CL ~ Gamma(1 / ωCL^2, θCL * ωCL^2)
_Vc ~ Gamma(1 / ωVc^2, θVc * ωVc^2)
end
@pre begin
CL = _CL
Vc = _Vc
end
@dynamics begin
Central' = -CL / Vc * Central
endAs you can see the mean from the gamma distribution becomes:
\[\operatorname{E}[CL] = \theta k = \frac{1}{\omega^2_{CL}} \theta_{CL} \omega^2_{CL} = \theta_{CL}\]
It does not dependent on the between-subject variability \(\omega\)!
Note
We are avoiding η notation here since we are modeling the subject-specific parameter directly.
1.2 Gamma versus Log-Nogmal Numerical Simulations
Before we dive into our PK examples, let us showcase the case for gamma versus log-normal with some numerical simulations.
First, let’s define a mean μ_PK value for a typical value along with an array of possible standard deviations σ values:
μ_PK = 1.0
σ = [0.1, 0.2, 0.5, 1.0, 1.5, 2.0]These will serve as the mean and standard deviations for our gamma and log-normal distributions.
Now let’s compare the coefficient of variation (CV) as a function of σ for LogNormal and Gamma:
num_df_gamma = DataFrame(;
μ = μ_PK,
σ = σ,
meanLogNormal = mean.(LogNormal.(log.(μ_PK), σ)),
cvLogNormal = std.(LogNormal.(log.(μ_PK), σ)) ./ mean.(LogNormal.(log.(μ_PK), σ)),
meanGamma = mean.(Gamma.(1 ./ σ .^ 2, μ_PK .* σ .^ 2)),
cvGamma = std.(Gamma.(1 ./ σ .^ 2, μ_PK .* σ .^ 2)) ./
mean.(Gamma.(1 ./ σ .^ 2, μ_PK .* σ .^ 2)),
)6×6 DataFrame
| Row | μ | σ | meanLogNormal | cvLogNormal | meanGamma | cvGamma |
|---|---|---|---|---|---|---|
| Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | |
| 1 | 1.0 | 0.1 | 1.00501 | 0.100251 | 1.0 | 0.1 |
| 2 | 1.0 | 0.2 | 1.0202 | 0.202017 | 1.0 | 0.2 |
| 3 | 1.0 | 0.5 | 1.13315 | 0.53294 | 1.0 | 0.5 |
| 4 | 1.0 | 1.0 | 1.64872 | 1.31083 | 1.0 | 1.0 |
| 5 | 1.0 | 1.5 | 3.08022 | 2.91337 | 1.0 | 1.5 |
| 6 | 1.0 | 2.0 | 7.38906 | 7.32108 | 1.0 | 2.0 |
f, ax, plotobj = lines(
num_df_gamma.σ,
num_df_gamma.meanLogNormal;
label = "μ - LogNormal",
linewidth = 3,
axis = (; xlabel = L"\sigma", ylabel = "value"),
)
lines!(
num_df_gamma.σ,
num_df_gamma.meanGamma;
label = "μ - Gamma",
linewidth = 3,
linestyle = :dashdot,
)
lines!(num_df_gamma.σ, num_df_gamma.cvLogNormal; label = "CV - LogNormal", linewidth = 3)
lines!(
num_df_gamma.σ,
num_df_gamma.cvGamma;
label = "CV - Gamma",
linewidth = 3,
linestyle = :dashdot,
)
axislegend(ax; position = :lt)
f
In the graph above, the dashed lines correspond to the mean and CV for the gamma distribution, whereas the solid lines correspond to the log-normal distribution.
There is clearly a bias in both the log-normal’s mean and CV that we don’t see in the gamma distribution.
2 Motivation - Bioavailability
Here is a very common model that can benefit from a non-Gaussian random effects distribution.
The model has one-compartment elimination and oral absorption with modeled bioavailability based on a crossover design.
The following code is a traditional PK model with a Gaussian random effect that needs to be constrained to the unit interval, \([0, 1]\):
@param begin
θF ∈ RealDomain(lower = 0.0, upper = 1.0)
ωF ∈ RealDomain(lower = 0.0)
end
@random begin
ηF ~ Normal(0.0, ωF)
end
@dosecontrol begin
bioav = (Depot = logistic(logit(θF) + ηF),)
endThe expectation \(\operatorname{E}[F]\) doesn’t have closed form and is generally different from \(\theta_F\). However, we have that:
\[\operatorname{E}[F] \approx \theta_F\]
when \(ωF\) is small. I.e. \(\theta_F\) is only the typical value when \(ωF\) is small.
2.1 Beta versus Logit-Normal Numerical Simulations
Let’s perform the same type of simulations we did before, but now we will be using the numerical integrator quadgk from the QuadGK.jl package. This is because we don’t have a closed form solution for \(\operatorname{E}[F]\) in the logit-normal parameterization.
using QuadGK: quadgkμ_bioav = 0.7We’ll also reuse the same σ values for the CVs.
num_df_beta = DataFrame(;
μ = μ_bioav,
σ = σ,
meanLogitNormal = map(
σ -> quadgk(
t -> logistic(t) * pdf(Normal(logit(μ_bioav), σ), t),
-100 * σ,
100 * σ,
)[1],
σ,
),
meanBeta = mean.(Beta.(μ_bioav ./ σ, (1 - μ_bioav) ./ σ)),
)6×4 DataFrame
| Row | μ | σ | meanLogitNormal | meanBeta |
|---|---|---|---|---|
| Float64 | Float64 | Float64 | Float64 | |
| 1 | 0.7 | 0.1 | 0.699582 | 0.7 |
| 2 | 0.7 | 0.2 | 0.698345 | 0.7 |
| 3 | 0.7 | 0.5 | 0.690393 | 0.7 |
| 4 | 0.7 | 1.0 | 0.668971 | 0.7 |
| 5 | 0.7 | 1.5 | 0.646064 | 0.7 |
| 6 | 0.7 | 2.0 | 0.626038 | 0.7 |
f, ax, plotobj = lines(
num_df_beta.σ,
num_df_beta.meanLogitNormal;
label = "μ - LogitNormal",
linewidth = 3,
axis = (; xlabel = L"\sigma", ylabel = "value"),
)
lines!(
num_df_beta.σ,
num_df_beta.meanBeta;
label = "μ - Beta",
linewidth = 3,
linestyle = :dashdot,
)
axislegend(ax; position = :lb)
f
In the graph above, the dashed lines correspond to the mean for the beta distribution, whereas the solid lines correspond to the logit-normal distribution.
As before, there is clearly a bias in the logit-normal’s mean that we don’t see in the beta distribution.
3 Warfarin data
We’ll demonstrate those intuitions using the Warfarin dataset.
pop = read_pumas(dataset("pumas/warfarin"))Population
Subjects: 32
Observations: dv4 Models and Simulations
Here we will provide a Gaussian and a non-Gaussian approach for:
- PK IV 1-compartment model fit for the Warfarin dataset
- Bioavaliability parallel absorption model simulation
4.1 Warfarin Gaussian and non-Gaussian PK model
The first model is a simple 1-compartment PK IV model with proportional error. This is for the Gaussian versus gamma random effects:
model_lognormal = @model begin
@param begin
θCL ∈ RealDomain(; lower = 0)
θVc ∈ RealDomain(; lower = 0)
ωCL ∈ RealDomain(; lower = 0)
ωVc ∈ RealDomain(; lower = 0)
σ ∈ RealDomain(; lower = 0)
end
@random begin
_CL ~ LogNormal(log(θCL), ωCL)
_Vc ~ LogNormal(log(θVc), ωVc)
end
# This is equivalent to defining
# CL = θCL*exp(ηCL)
# with
# ηCL = Normal(0, ωCL)
@pre begin
CL = _CL
Vc = _Vc
end
@dynamics Central1
@derived begin
μ := @. Central / Vc
dv ~ @. Normal(μ, μ * σ)
end
endPumasModel
Parameters: θCL, θVc, ωCL, ωVc, σ
Random effects: _CL, _Vc
Covariates:
Dynamical system variables: Central
Dynamical system type: Closed form
Derived: dv
Observed: dvmodel_gamma = @model begin
@param begin
θCL ∈ RealDomain(; lower = 0)
θVc ∈ RealDomain(; lower = 0)
ωCL ∈ RealDomain(; lower = 0)
ωVc ∈ RealDomain(; lower = 0)
σ ∈ RealDomain(; lower = 0)
end
@random begin
_CL ~ Gamma(1 / ωCL^2, θCL * ωCL^2)
_Vc ~ Gamma(1 / ωVc^2, θVc * ωVc^2)
end
@pre begin
CL = _CL
Vc = _Vc
end
@dynamics begin
Central' = -CL / Vc * Central
end
@derived begin
μ := @. Central / Vc
dv ~ @. Normal(μ, μ * σ)
end
endPumasModel
Parameters: θCL, θVc, ωCL, ωVc, σ
Random effects: _CL, _Vc
Covariates:
Dynamical system variables: Central
Dynamical system type: Matrix exponential
Derived: dv
Observed: dvWe also need some initial values for the fitting:
iparams_pk = (; θCL = 1.0, θVc = 5.0, ωCL = 0.1, ωVc = 0.1, σ = 0.2)(θCL = 1.0,
θVc = 5.0,
ωCL = 0.1,
ωVc = 0.1,
σ = 0.2,)We proceed by fitting both models:
fit_lognormal = fit(model_lognormal, pop, iparams_pk, FOCE())[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
Iter Function value Gradient norm
0 5.770212e+03 7.912060e+03
* time: 0.02602100372314453
1 9.433464e+02 6.079483e+02
* time: 1.0018219947814941
2 8.189627e+02 4.423725e+02
* time: 1.0778021812438965
3 5.917683e+02 1.819248e+02
* time: 1.1044740676879883
4 5.421783e+02 1.121313e+02
* time: 1.125730037689209
5 5.255651e+02 7.407230e+01
* time: 1.144388198852539
6 5.208427e+02 8.699271e+01
* time: 1.1616590023040771
7 5.174883e+02 8.974584e+01
* time: 1.1784040927886963
8 5.138523e+02 7.328235e+01
* time: 1.195033073425293
9 5.109883e+02 4.155805e+01
* time: 1.2110869884490967
10 5.094359e+02 3.170517e+01
* time: 1.225978136062622
11 5.086172e+02 3.327331e+01
* time: 1.2407701015472412
12 5.080941e+02 2.942077e+01
* time: 1.2558159828186035
13 5.074009e+02 2.839941e+01
* time: 1.2705881595611572
14 5.059302e+02 3.330093e+01
* time: 1.2850661277770996
15 5.036399e+02 3.172884e+01
* time: 1.30059814453125
16 5.017004e+02 3.160020e+01
* time: 1.3166542053222656
17 5.008553e+02 2.599524e+01
* time: 1.360314130783081
18 5.005913e+02 2.139314e+01
* time: 1.375521183013916
19 5.003573e+02 2.134778e+01
* time: 1.3901910781860352
20 4.997249e+02 2.069868e+01
* time: 1.405493974685669
21 4.984453e+02 1.859010e+01
* time: 1.4211821556091309
22 4.959584e+02 2.156209e+01
* time: 1.4371252059936523
23 4.923347e+02 3.030833e+01
* time: 1.4541740417480469
24 4.906916e+02 1.652278e+01
* time: 1.4722859859466553
25 4.902955e+02 6.360800e+00
* time: 1.4892420768737793
26 4.902870e+02 7.028603e+00
* time: 1.5114049911499023
27 4.902193e+02 1.176895e+00
* time: 1.5289530754089355
28 4.902189e+02 1.170642e+00
* time: 1.54256010055542
29 4.902186e+02 1.167624e+00
* time: 1.5540320873260498
30 4.902145e+02 1.110377e+00
* time: 1.5685679912567139
31 4.902079e+02 1.010507e+00
* time: 1.582970142364502
32 4.901917e+02 9.619218e-01
* time: 1.5974969863891602
33 4.901683e+02 1.001006e+00
* time: 1.6227531433105469
34 4.901473e+02 6.138233e-01
* time: 1.6379129886627197
35 4.901412e+02 1.754342e-01
* time: 1.6530921459197998
36 4.901406e+02 2.617009e-02
* time: 1.6666510105133057
37 4.901405e+02 4.585882e-03
* time: 1.6785120964050293
38 4.901405e+02 7.668184e-04
* time: 1.6881821155548096FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Closed form
Log-likelihood value: -490.14052
Number of subjects: 32
Number of parameters: Fixed Optimized
0 5
Observation records: Active Missing
dv: 256 0
Total: 256 0
-----------------
Estimate
-----------------
θCL 0.16025
θVc 10.262
ωCL 0.23505
ωVc 0.10449
σ 0.3582
-----------------fit_gamma = fit(model_gamma, pop, iparams_pk, FOCE())[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
Iter Function value Gradient norm
0 4.160707e+03 5.960578e+03
* time: 6.318092346191406e-5
1 9.056373e+02 5.541920e+02
* time: 0.03891611099243164
2 7.787963e+02 3.988904e+02
* time: 0.0725240707397461
3 5.915054e+02 1.495777e+02
* time: 0.09677004814147949
4 5.533120e+02 8.826583e+01
* time: 0.11900115013122559
5 5.389239e+02 9.144086e+01
* time: 0.14025497436523438
6 5.323499e+02 1.000933e+02
* time: 0.16268706321716309
7 5.270252e+02 8.423844e+01
* time: 0.18620514869689941
8 5.233813e+02 5.194402e+01
* time: 0.20844817161560059
9 5.213366e+02 3.461331e+01
* time: 0.2733621597290039
10 5.200972e+02 3.888113e+01
* time: 0.2935221195220947
11 5.191933e+02 3.556605e+01
* time: 0.31372809410095215
12 5.181335e+02 3.624436e+01
* time: 0.33366918563842773
13 5.161626e+02 4.322775e+01
* time: 0.3528480529785156
14 5.133202e+02 3.722515e+01
* time: 0.37081408500671387
15 5.107758e+02 3.401586e+01
* time: 0.38916015625
16 5.095157e+02 2.854997e+01
* time: 0.4082601070404053
17 5.090165e+02 2.644560e+01
* time: 0.42623114585876465
18 5.085184e+02 2.744429e+01
* time: 0.44341015815734863
19 5.074309e+02 2.793918e+01
* time: 0.4605841636657715
20 5.053757e+02 2.616169e+01
* time: 0.47840213775634766
21 5.018507e+02 2.257667e+01
* time: 0.4974551200866699
22 4.942495e+02 3.832878e+01
* time: 0.519212007522583
23 4.940229e+02 5.518159e+01
* time: 0.5466179847717285
24 4.909110e+02 3.042064e+01
* time: 0.5746350288391113
25 4.900234e+02 6.929306e+00
* time: 0.6005170345306396
26 4.897974e+02 1.087865e+00
* time: 0.6263010501861572
27 4.897942e+02 6.456402e-01
* time: 0.6489071846008301
28 4.897940e+02 6.467689e-01
* time: 0.6981141567230225
29 4.897939e+02 6.463480e-01
* time: 0.7166881561279297
30 4.897935e+02 6.408914e-01
* time: 0.7354199886322021
31 4.897924e+02 6.208208e-01
* time: 0.7547152042388916
32 4.897900e+02 1.035462e+00
* time: 0.7742931842803955
33 4.897850e+02 1.452099e+00
* time: 0.7939980030059814
34 4.897776e+02 1.482593e+00
* time: 0.8134751319885254
35 4.897718e+02 8.420646e-01
* time: 0.8339540958404541
36 4.897702e+02 2.023876e-01
* time: 0.8545091152191162
37 4.897700e+02 1.885486e-02
* time: 0.873035192489624
38 4.897700e+02 2.343932e-03
* time: 0.8896360397338867
39 4.897700e+02 4.417566e-04
* time: 0.9039561748504639FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Matrix exponential
Log-likelihood value: -489.77002
Number of subjects: 32
Number of parameters: Fixed Optimized
0 5
Observation records: Active Missing
dv: 256 0
Total: 256 0
-----------------
Estimate
-----------------
θCL 0.16466
θVc 10.329
ωCL 0.23348
ωVc 0.10661
σ 0.35767
-----------------Finally, let’s compare the estimates:
compare_estimates(; lognormal = fit_lognormal, gamma = fit_gamma)5×3 DataFrame
| Row | parameter | lognormal | gamma |
|---|---|---|---|
| String | Float64? | Float64? | |
| 1 | θCL | 0.160253 | 0.164658 |
| 2 | θVc | 10.2617 | 10.3288 |
| 3 | ωCL | 0.235046 | 0.233484 |
| 4 | ωVc | 0.10449 | 0.106611 |
| 5 | σ | 0.358205 | 0.357667 |
As mention above, the mean of a log-normal is \(\exp \left\{ \mu + \frac{\sigma^2}{2} \right\}\).
So let’s compare that with the gamma typical values:
DataFrame(;
parameter = ["θCL", "θVc"],
θLogNormal = [coef(fit_lognormal).θCL, coef(fit_lognormal).θVc],
ELogNormal = [
exp(log(coef(fit_lognormal).θCL) + coef(fit_lognormal).ωCL^2 / 2),
exp(log(coef(fit_lognormal).θVc) + coef(fit_lognormal).ωVc^2 / 2),
],
θGamma = [coef(fit_gamma).θCL, coef(fit_gamma).θVc],
)2×4 DataFrame
| Row | parameter | θLogNormal | ELogNormal | θGamma |
|---|---|---|---|---|
| String | Float64 | Float64 | Float64 | |
| 1 | θCL | 0.160253 | 0.164741 | 0.164658 |
| 2 | θVc | 10.2617 | 10.3178 | 10.3288 |
As you can see the Gaussian model has a slight bias in the estimation of both θCL and θVc.
Let’s also plot the two probability density functions (PDF) for θCL:
plotdataPK = @chain DataFrame(; x = range(0, 0.5; length = 1_000)) begin
@rtransform begin
:LogNormal =
pdf(LogNormal(log(coef(fit_lognormal).θCL), coef(fit_lognormal).ωCL), :x)
:Gamma = pdf(LogNormal(log(coef(fit_gamma).θCL), coef(fit_gamma).ωCL), :x)
end
end
first(plotdataPK, 5)5×3 DataFrame
| Row | x | LogNormal | Gamma |
|---|---|---|---|
| Float64 | Float64 | Float64 | |
| 1 | 0.0 | 0.0 | 0.0 |
| 2 | 0.000500501 | 5.27258e-128 | 5.2502e-131 |
| 3 | 0.001001 | 9.25648e-99 | 3.24235e-101 |
| 4 | 0.0015015 | 2.10132e-83 | 1.45529e-85 |
| 5 | 0.002002 | 2.71651e-73 | 2.9785e-75 |
data(stack(plotdataPK, [:LogNormal, :Gamma])) *
mapping(:x, :value; color = :variable) *
visual(Lines) |> draw
4.2 Bioavaliability Parallel Absorption Simulation
This is a parallel absorption model with bioavaliabity in both the “fast” as the “slow” depots.
First, the traditional approach with a logistic transformation of a Gaussian random variable. This makes the individual relative bioavailibility logit-normally distributed.
model_logitnormal = @model begin
@param begin
θkaFast ∈ RealDomain(; lower = 0)
θkaSlow ∈ RealDomain(; lower = 0)
θCL ∈ RealDomain(; lower = 0)
θVc ∈ RealDomain(; lower = 0)
θbioav ∈ RealDomain(; lower = 0.0, upper = 1.0)
ωCL ∈ RealDomain(; lower = 0)
ωVc ∈ RealDomain(; lower = 0)
# I call this one ξ to distinguish it from ω since the interpretation is NOT a relative error (coefficient of variation)
ξbioav ∈ RealDomain(; lower = 0, init = 0.1)
σ ∈ RealDomain(; lower = 0)
end
@random begin
_CL ~ Gamma(1 / ωCL^2, θCL * ωCL^2)
_Vc ~ Gamma(1 / ωVc^2, θVc * ωVc^2)
# define the latent Gaussian random effect. Notice the logit transform
ηbioavLogit ~ Normal(logit(θbioav), ξbioav)
end
@pre begin
kaFast = θkaFast
kaSlow = θkaSlow
CL = _CL
Vc = _Vc
end
@dosecontrol begin
# _bioav is LogitNormal distributed
_bioav = logistic(ηbioavLogit)
bioav = (; DepotFast = _bioav, DepotSlow = 1 - _bioav)
end
@dynamics begin
DepotFast' = -kaFast * DepotFast
DepotSlow' = -kaSlow * DepotSlow
Central' = kaFast * DepotFast + kaSlow * DepotSlow - CL / Vc * Central
end
@derived begin
μ := @. Central / Vc
dv ~ @. Normal(μ, abs(μ) * σ)
end
endPumasModel
Parameters: θkaFast, θkaSlow, θCL, θVc, θbioav, ωCL, ωVc, ξbioav, σ
Random effects: _CL, _Vc, ηbioavLogit
Covariates:
Dynamical system variables: DepotFast, DepotSlow, Central
Dynamical system type: Matrix exponential
Derived: dv
Observed: dvNow the same model but with the non-Gaussian random-effects using a beta distribution instead of the logit parameterization of the Gaussian distribution:
model_beta = @model begin
@param begin
θkaFast ∈ RealDomain(; lower = 0)
θkaSlow ∈ RealDomain(; lower = 0)
θCL ∈ RealDomain(; lower = 0)
θVc ∈ RealDomain(; lower = 0)
θbioav ∈ RealDomain(; lower = 0.0, upper = 1.0)
ωCL ∈ RealDomain(; lower = 0)
ωVc ∈ RealDomain(; lower = 0)
# We call this one n since the interpretation is like the length of a Binomial distribution
nbioav ∈ RealDomain(; lower = 0, init = 10)
σ ∈ RealDomain(; lower = 0)
end
@random begin
ηCL ~ Gamma(1 / ωCL^2, θCL * ωCL^2)
ηVc ~ Gamma(1 / ωVc^2, θVc * ωVc^2)
# The makes E(_bioav) = θbioav
# See https://en.wikipedia.org/wiki/Beta_distribution
ηbioav ~ Beta(θbioav * nbioav, (1 - θbioav) * nbioav)
end
@pre begin
kaFast = θkaFast
kaSlow = θkaSlow
CL = ηCL
Vc = ηVc
end
@dosecontrol begin
bioav = (; DepotFast = ηbioav, DepotSlow = 1 - ηbioav)
end
@dynamics begin
DepotFast' = -kaFast * DepotFast
DepotSlow' = -kaSlow * DepotSlow
Central' = kaFast * DepotFast + kaSlow * DepotSlow - CL / Vc * Central
end
@derived begin
μ := @. Central / Vc
dv ~ @. Normal(μ, abs(μ) * σ)
end
endPumasModel
Parameters: θkaFast, θkaSlow, θCL, θVc, θbioav, ωCL, ωVc, nbioav, σ
Random effects: ηCL, ηVc, ηbioav
Covariates:
Dynamical system variables: DepotFast, DepotSlow, Central
Dynamical system type: Matrix exponential
Derived: dv
Observed: dvWe have two types of random effects here.
First, as you are already familiar from the previous example, the clearance (CL), volume of concentration (Vc), and absorption rate (ka) have typical values (i.e. fixed effects) and between-subject variability (i.e. random effects) modelled as a gamma distribution.
Second, bioavailability Bioav is modelled as a beta distribution. Generally the beta distribution is parametrized as:
\[\text{Beta}(\alpha, \beta)\]
where both parameters \(\alpha\) and \(\beta\) are shape parameters.
One nice thing about the beta distribution is that it only takes values between and including 0 and 1, i.e. \([0, 1]\). This makes it the perfect candidate to model bioavailability parameters which are generally bounded in that interval. So, we don’t need to do a logistic transformation.
Another nice thing about the beta distribution is that we can use the alternative \((\mu, n)\)-parametrization with with \(\mu\) serving as a mean-value parameter:
\[\text{Beta}(\mu, n)\]
where in the original beta parametrization:
- \(\alpha = \mu n\)
- \(\beta = (1 - \mu) n\)
Hence, our mean is:
\[\operatorname{E}[F] = \mu = \theta_F\]
which, again, does not depend on any other parameters. The variance is
\[\operatorname{Var}(F) = \frac{\mu(1 - \mu)}{n}\]
so similar to the mean of Bernoulli trials.
Now let’s generate some data for the simulation:
dr = DosageRegimen(
DosageRegimen(100; cmt = :DepotFast),
DosageRegimen(100; cmt = :DepotSlow),
)2×10 DataFrame
| Row | time | cmt | amt | evid | ii | addl | rate | duration | ss | route |
|---|---|---|---|---|---|---|---|---|---|---|
| Float64 | Symbol | Float64 | Int8 | Float64 | Int64 | Float64 | Float64 | Int8 | NCA.Route | |
| 1 | 0.0 | DepotFast | 100.0 | 1 | 0.0 | 0 | 0.0 | 0.0 | 0 | NullRoute |
| 2 | 0.0 | DepotSlow | 100.0 | 1 | 0.0 | 0 | 0.0 | 0.0 | 0 | NullRoute |
simtimes = [0.5, 1.0, 2.0, 4.0, 8.0, 24.0]6-element Vector{Float64}:
0.5
1.0
2.0
4.0
8.0
24.0trueparam = (;
θkaFast = 0.9,
θkaSlow = 0.2,
θCL = 1.1,
θVc = 10.0,
θbioav = 0.7,
ωCL = 0.1,
ωVc = 0.1,
nbioav = 40,
σ = 0.1,
)(θkaFast = 0.9,
θkaSlow = 0.2,
θCL = 1.1,
θVc = 10.0,
θbioav = 0.7,
ωCL = 0.1,
ωVc = 0.1,
nbioav = 40,
σ = 0.1,)For simplicity, we just add 20% to the true values for initial values:
initparamBeta = map(t -> 1.2 * t, trueparam)(θkaFast = 1.08,
θkaSlow = 0.24,
θCL = 1.32,
θVc = 12.0,
θbioav = 0.84,
ωCL = 0.12,
ωVc = 0.12,
nbioav = 48.0,
σ = 0.12,)The initial values for the LogitNormal need to have ξbioav defined instead of nbioav:
initparamLogitNormal =
(Base.structdiff(initparamBeta, NamedTuple{(:nbioav,)})..., ξbioav = 0.1)(θkaFast = 1.08,
θkaSlow = 0.24,
θCL = 1.32,
θVc = 12.0,
θbioav = 0.84,
ωCL = 0.12,
ωVc = 0.12,
σ = 0.12,
ξbioav = 0.1,)Setup empty Subjects with the dose information:
skeletonpop = map(i -> Subject(; id = i, events = dr), 1:40)Population
Subjects: 40
Observations: Next, we simulate the data (while setting the seed for reprocibility):
Random.seed!(128)
simpop = Subject.(simobs(model_beta, skeletonpop, trueparam; obstimes = simtimes))Finally let’s fit both models:
fit_logitnormal = fit(model_logitnormal, simpop, initparamLogitNormal, FOCE())[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
Iter Function value Gradient norm
0 2.512964e+02 3.650794e+02
* time: 7.581710815429688e-5
1 2.005689e+02 1.252637e+02
* time: 0.3474099636077881
2 1.875728e+02 4.225518e+01
* time: 0.5161159038543701
3 1.863803e+02 3.429785e+01
* time: 0.6789588928222656
4 1.845581e+02 3.694229e+01
* time: 0.868034839630127
5 1.828416e+02 1.170915e+01
* time: 1.0379559993743896
6 1.823277e+02 1.004017e+01
* time: 1.1548748016357422
7 1.810629e+02 5.326780e+00
* time: 1.2712910175323486
8 1.810479e+02 1.767987e+00
* time: 1.385589838027954
9 1.810272e+02 3.852037e+00
* time: 1.509922981262207
10 1.809414e+02 1.196625e+01
* time: 1.6359548568725586
11 1.806489e+02 1.861440e+01
* time: 1.7637288570404053
12 1.801984e+02 1.361774e+01
* time: 2.254415988922119
13 1.800427e+02 2.177039e+01
* time: 2.3771588802337646
14 1.796554e+02 7.079039e+00
* time: 2.504149913787842
15 1.795832e+02 1.581499e+01
* time: 2.633692979812622
16 1.795220e+02 6.552120e+00
* time: 2.768831968307495
17 1.794621e+02 6.612645e+00
* time: 2.89694881439209
18 1.793643e+02 6.603903e+00
* time: 3.0205190181732178
19 1.793502e+02 1.025787e+01
* time: 3.165799856185913
20 1.793178e+02 1.202199e+01
* time: 3.2765138149261475
21 1.792424e+02 1.625661e+01
* time: 3.3882968425750732
22 1.791560e+02 1.128856e+01
* time: 3.5046188831329346
23 1.790991e+02 7.625637e+00
* time: 3.6228508949279785
24 1.790756e+02 8.557258e+00
* time: 3.740238904953003
25 1.790633e+02 4.848482e+00
* time: 3.859757900238037
26 1.790408e+02 5.859306e+00
* time: 3.978538990020752
27 1.789734e+02 1.106035e+01
* time: 4.121018886566162
28 1.789070e+02 1.143361e+01
* time: 4.231564998626709
29 1.788321e+02 7.098448e+00
* time: 4.341272830963135
30 1.787896e+02 5.709857e+00
* time: 4.454293966293335
31 1.787809e+02 7.456555e+00
* time: 4.573001861572266
32 1.787671e+02 7.320931e-01
* time: 4.692580938339233
33 1.787660e+02 5.023990e-01
* time: 4.810844898223877
34 1.787656e+02 3.813994e-01
* time: 4.925073862075806
35 1.787639e+02 8.608909e-01
* time: 5.069129943847656
36 1.787607e+02 2.047321e+00
* time: 5.17998480796814
37 1.787525e+02 3.859529e+00
* time: 5.289959907531738
38 1.787349e+02 5.864920e+00
* time: 5.413597822189331
39 1.787017e+02 6.966045e+00
* time: 5.548888921737671
40 1.786574e+02 5.348939e+00
* time: 5.685351848602295
41 1.786270e+02 1.642025e+00
* time: 5.811041831970215
42 1.786181e+02 5.211823e-01
* time: 5.934624910354614
43 1.786153e+02 1.186061e+00
* time: 6.062311887741089
44 1.786125e+02 1.292005e+00
* time: 6.205183982849121
45 1.786099e+02 7.814598e-01
* time: 6.315224885940552
46 1.786086e+02 1.369456e-01
* time: 6.42535400390625
47 1.786082e+02 1.912170e-01
* time: 6.535862922668457
48 1.786080e+02 2.670802e-01
* time: 6.654233932495117
49 1.786078e+02 1.979262e-01
* time: 6.768675804138184
50 1.786077e+02 5.177918e-02
* time: 6.885594844818115
51 1.786076e+02 2.998328e-02
* time: 7.003453969955444
52 1.786076e+02 5.799706e-02
* time: 7.1177818775177
53 1.786076e+02 4.280434e-02
* time: 7.259394884109497
54 1.786076e+02 1.467829e-02
* time: 7.3675549030303955
55 1.786076e+02 6.928178e-03
* time: 7.468969821929932
56 1.786076e+02 1.236015e-02
* time: 7.573523998260498
57 1.786076e+02 9.950826e-03
* time: 7.686716794967651
58 1.786076e+02 2.974370e-03
* time: 7.793972969055176
59 1.786076e+02 1.374576e-03
* time: 7.90009880065918
60 1.786076e+02 2.860078e-03
* time: 7.998617887496948
61 1.786076e+02 2.100127e-03
* time: 8.097692966461182
62 1.786076e+02 2.100127e-03
* time: 8.266541957855225
63 1.786076e+02 6.412834e-03
* time: 8.383128881454468
64 1.786076e+02 6.412834e-03
* time: 8.541384935379028
65 1.786076e+02 6.412834e-03
* time: 8.77962589263916
66 1.786076e+02 6.412834e-03
* time: 9.194366931915283
67 1.786076e+02 6.412834e-03
* time: 9.422312021255493FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Matrix exponential
Log-likelihood value: -178.60759
Number of subjects: 40
Number of parameters: Fixed Optimized
0 9
Observation records: Active Missing
dv: 240 0
Total: 240 0
-----------------------
Estimate
-----------------------
θkaFast 0.91097
θkaSlow 0.13112
θCL 1.0854
θVc 7.1008
θbioav 0.4802
ωCL 0.088113
ωVc 0.12133
ξbioav 1.8429e-5
σ 0.10545
-----------------------fit_beta = fit(model_beta, simpop, initparamBeta, FOCE())[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
Iter Function value Gradient norm
0 2.523111e+02 3.644346e+02
* time: 6.103515625e-5
1 2.014577e+02 1.265001e+02
* time: 0.3875720500946045
2 1.880885e+02 4.190708e+01
* time: 0.5650589466094971
3 1.870317e+02 8.825666e+01
* time: 0.7014479637145996
4 1.846027e+02 4.400156e+01
* time: 0.9199090003967285
5 1.834445e+02 1.906624e+01
* time: 1.0387461185455322
6 1.828599e+02 1.113882e+01
* time: 1.15706205368042
7 1.815719e+02 7.449355e+00
* time: 1.280817985534668
8 1.815131e+02 2.164678e+00
* time: 1.4090170860290527
9 1.814896e+02 2.167319e+00
* time: 1.5333330631256104
10 1.814458e+02 4.615738e+00
* time: 1.6596269607543945
11 1.813173e+02 9.576967e+00
* time: 1.7890710830688477
12 1.809756e+02 2.052077e+01
* time: 1.9889180660247803
13 1.807625e+02 4.553366e+01
* time: 2.1098780632019043
14 1.802224e+02 6.550892e+00
* time: 2.2313129901885986
15 1.800862e+02 2.865509e+00
* time: 2.350567102432251
16 1.800780e+02 1.164611e+00
* time: 2.4725489616394043
17 1.800737e+02 7.952462e-01
* time: 2.600260019302368
18 1.800352e+02 4.860618e+00
* time: 2.7309439182281494
19 1.800089e+02 5.176689e+00
* time: 2.863070011138916
20 1.799679e+02 4.303892e+00
* time: 2.990499973297119
21 1.799153e+02 4.612832e+00
* time: 3.1363110542297363
22 1.798423e+02 1.209387e+01
* time: 3.249553918838501
23 1.796821e+02 1.712256e+01
* time: 3.360459089279175
24 1.794275e+02 1.435100e+01
* time: 3.473684072494507
25 1.793773e+02 4.137313e+00
* time: 3.5931200981140137
26 1.793488e+02 1.846545e+00
* time: 3.745600938796997
27 1.793331e+02 5.502533e+00
* time: 3.865586996078491
28 1.793234e+02 2.894037e+00
* time: 3.9850361347198486
29 1.793119e+02 1.453372e+00
* time: 4.131155967712402
30 1.792879e+02 4.109884e+00
* time: 4.242964029312134
31 1.792599e+02 4.613173e+00
* time: 4.359745025634766
32 1.792384e+02 4.254549e+00
* time: 4.469506025314331
33 1.792251e+02 4.415024e+00
* time: 4.589903116226196
34 1.792026e+02 3.229145e+00
* time: 4.709100961685181
35 1.791841e+02 3.422094e+00
* time: 4.8633880615234375
36 1.791705e+02 1.621228e+00
* time: 5.015721082687378
37 1.791555e+02 3.462667e+00
* time: 5.143255949020386
38 1.791293e+02 5.586685e+00
* time: 5.28898811340332
39 1.790713e+02 9.927638e+00
* time: 5.483910083770752
40 1.789891e+02 1.133271e+01
* time: 5.598509073257446
41 1.788585e+02 1.279172e+01
* time: 5.716385126113892
42 1.787407e+02 5.291681e+00
* time: 5.8444530963897705
43 1.786633e+02 5.367971e+00
* time: 5.975500106811523
44 1.786405e+02 2.571548e+00
* time: 6.099683046340942
45 1.786339e+02 2.720314e+00
* time: 6.22140908241272
46 1.786252e+02 1.930583e+00
* time: 6.37310004234314
47 1.786182e+02 1.523164e+00
* time: 6.483062028884888
48 1.786126e+02 5.027920e-01
* time: 6.592617034912109
49 1.786100e+02 3.733023e-01
* time: 6.7067201137542725
50 1.786089e+02 2.749553e-01
* time: 6.823368072509766
51 1.786083e+02 1.981772e-01
* time: 6.9436609745025635
52 1.786079e+02 1.005262e-01
* time: 7.060589075088501
53 1.786078e+02 2.549641e-02
* time: 7.177897930145264
54 1.786077e+02 4.452931e-02
* time: 7.293743133544922
55 1.786076e+02 2.967125e-02
* time: 7.407896041870117
56 1.786076e+02 1.068274e-02
* time: 7.5468199253082275
57 1.786076e+02 5.355447e-03
* time: 7.65038800239563
58 1.786076e+02 8.180920e-03
* time: 7.752283096313477
59 1.786076e+02 4.935439e-03
* time: 7.855996131896973
60 1.786076e+02 4.387986e-03
* time: 7.992188930511475
61 1.786076e+02 4.388023e-03
* time: 8.174855947494507
62 1.786076e+02 4.387934e-03
* time: 8.339471101760864
63 1.786076e+02 4.387934e-03
* time: 8.506487131118774FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Matrix exponential
Log-likelihood value: -178.60759
Number of subjects: 40
Number of parameters: Fixed Optimized
0 9
Observation records: Active Missing
dv: 240 0
Total: 240 0
----------------------
Estimate
----------------------
θkaFast 0.91099
θkaSlow 0.13112
θCL 1.0854
θVc 7.1007
θbioav 0.48019
ωCL 0.088114
ωVc 0.12134
nbioav 2.7333e7
σ 0.10545
----------------------As before, let’s compare the estimates:
compare_estimates(; logitnormal = fit_logitnormal, beta = fit_beta)10×3 DataFrame
| Row | parameter | logitnormal | beta |
|---|---|---|---|
| String | Float64? | Float64? | |
| 1 | θkaFast | 0.910971 | 0.910988 |
| 2 | θkaSlow | 0.131115 | 0.131117 |
| 3 | θCL | 1.0854 | 1.0854 |
| 4 | θVc | 7.10076 | 7.10073 |
| 5 | θbioav | 0.480202 | 0.480191 |
| 6 | ωCL | 0.0881132 | 0.0881137 |
| 7 | ωVc | 0.121334 | 0.121335 |
| 8 | σ | 0.105448 | 0.105449 |
| 9 | ξbioav | 1.84292e-5 | missing |
| 10 | nbioav | missing | 2.73333e7 |
Again, we’ll both PDFs from the estimated values:
plotdatabioav = @chain DataFrame(; x = range(0, 1; length = 1_000)) begin
@rtransform begin
:logitnormal =
1 / coef(fit_logitnormal).ξbioav / √(2π) / (:x * (1 - :x)) * exp(
-(logit(:x) - logit(coef(fit_logitnormal).θbioav))^2 /
(2 * coef(fit_logitnormal).ξbioav^2),
)
:beta = pdf(
Beta(
coef(fit_beta).θbioav * coef(fit_beta).nbioav,
(1 - coef(fit_beta).θbioav) * coef(fit_beta).nbioav,
),
:x,
)
end
end
first(plotdatabioav, 5)5×3 DataFrame
| Row | x | logitnormal | beta |
|---|---|---|---|
| Float64 | Float64 | Float64 | |
| 1 | 0.0 | NaN | 0.0 |
| 2 | 0.001001 | 0.0 | 0.0 |
| 3 | 0.002002 | 0.0 | 0.0 |
| 4 | 0.003003 | 0.0 | 0.0 |
| 5 | 0.004004 | 0.0 | 0.0 |
plt_pdf_bioav =
data(stack(plotdatabioav, [:logitnormal, :beta])) *
mapping(:x, :value; color = :variable) *
visual(Lines);
draw(plt_pdf_bioav; axis = (; xticks = 0.1:0.1:1.0))
For this dataset, the two distributions differ significantly with the Beta model producing a distribution much closer to the truth but for other realizations of the simulated data they are closer to each other.