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: 0.03019094467163086
1 9.433464e+02 6.079483e+02
* time: 1.353376865386963
2 8.189627e+02 4.423725e+02
* time: 1.3939549922943115
3 5.917683e+02 1.819248e+02
* time: 1.4251210689544678
4 5.421783e+02 1.121313e+02
* time: 1.4504680633544922
5 5.255651e+02 7.407230e+01
* time: 1.4725499153137207
6 5.208427e+02 8.699271e+01
* time: 1.492473840713501
7 5.174883e+02 8.974584e+01
* time: 1.509335994720459
8 5.138523e+02 7.328235e+01
* time: 1.526608943939209
9 5.109883e+02 4.155805e+01
* time: 1.542827844619751
10 5.094359e+02 3.170517e+01
* time: 1.5589170455932617
11 5.086172e+02 3.327331e+01
* time: 1.5766479969024658
12 5.080941e+02 2.942077e+01
* time: 1.5954580307006836
13 5.074009e+02 2.839941e+01
* time: 1.614182949066162
14 5.059302e+02 3.330093e+01
* time: 1.7384769916534424
15 5.036399e+02 3.172884e+01
* time: 1.7559590339660645
16 5.017004e+02 3.160020e+01
* time: 1.7732889652252197
17 5.008553e+02 2.599524e+01
* time: 1.7898850440979004
18 5.005913e+02 2.139314e+01
* time: 1.8053419589996338
19 5.003573e+02 2.134778e+01
* time: 1.820483922958374
20 4.997249e+02 2.069868e+01
* time: 1.8363189697265625
21 4.984453e+02 1.859010e+01
* time: 1.8533010482788086
22 4.959584e+02 2.156209e+01
* time: 1.870826005935669
23 4.923347e+02 3.030833e+01
* time: 1.888974905014038
24 4.906916e+02 1.652278e+01
* time: 1.9124529361724854
25 4.902955e+02 6.360800e+00
* time: 1.9307138919830322
26 4.902870e+02 7.028603e+00
* time: 1.9501080513000488
27 4.902193e+02 1.176895e+00
* time: 1.9682848453521729
28 4.902189e+02 1.170642e+00
* time: 1.9828920364379883
29 4.902186e+02 1.167624e+00
* time: 1.9954710006713867
30 4.902145e+02 1.110377e+00
* time: 2.010745048522949
31 4.902079e+02 1.010507e+00
* time: 2.0260698795318604
32 4.901917e+02 9.619218e-01
* time: 2.041718006134033
33 4.901683e+02 1.001006e+00
* time: 2.0571420192718506
34 4.901473e+02 6.138233e-01
* time: 2.0761020183563232
35 4.901412e+02 1.754342e-01
* time: 2.095212936401367
36 4.901406e+02 2.617009e-02
* time: 2.1125588417053223
37 4.901405e+02 4.585882e-03
* time: 2.1276509761810303
38 4.901405e+02 7.668184e-04
* time: 2.1401848793029785FittedPumasModel
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: 7.605552673339844e-5
1 9.056373e+02 5.541920e+02
* time: 0.04127001762390137
2 7.787963e+02 3.988904e+02
* time: 0.0713341236114502
3 5.915054e+02 1.495777e+02
* time: 0.09560513496398926
4 5.533120e+02 8.826583e+01
* time: 0.11838912963867188
5 5.389239e+02 9.144086e+01
* time: 0.14207100868225098
6 5.323499e+02 1.000933e+02
* time: 0.16589617729187012
7 5.270252e+02 8.423844e+01
* time: 0.18761801719665527
8 5.233813e+02 5.194402e+01
* time: 0.20717597007751465
9 5.213366e+02 3.461331e+01
* time: 0.22741103172302246
10 5.200972e+02 3.888113e+01
* time: 0.24939513206481934
11 5.191933e+02 3.556605e+01
* time: 0.2716820240020752
12 5.181335e+02 3.624436e+01
* time: 0.29453206062316895
13 5.161626e+02 4.322775e+01
* time: 0.3178591728210449
14 5.133202e+02 3.722515e+01
* time: 0.3403491973876953
15 5.107758e+02 3.401586e+01
* time: 0.36373114585876465
16 5.095157e+02 2.854997e+01
* time: 0.3889780044555664
17 5.090165e+02 2.644560e+01
* time: 0.414959192276001
18 5.085184e+02 2.744429e+01
* time: 0.4393889904022217
19 5.074309e+02 2.793918e+01
* time: 0.4628779888153076
20 5.053757e+02 2.616169e+01
* time: 0.48591017723083496
21 5.018507e+02 2.257667e+01
* time: 0.5973429679870605
22 4.942495e+02 3.832878e+01
* time: 0.6196150779724121
23 4.940229e+02 5.518159e+01
* time: 0.6472940444946289
24 4.909110e+02 3.042064e+01
* time: 0.6733942031860352
25 4.900234e+02 6.929306e+00
* time: 0.6974291801452637
26 4.897974e+02 1.087865e+00
* time: 0.7206649780273438
27 4.897942e+02 6.456402e-01
* time: 0.7410562038421631
28 4.897940e+02 6.467689e-01
* time: 0.7590069770812988
29 4.897939e+02 6.463480e-01
* time: 0.7779541015625
30 4.897935e+02 6.408914e-01
* time: 0.7976460456848145
31 4.897924e+02 6.208208e-01
* time: 0.816892147064209
32 4.897900e+02 1.035462e+00
* time: 0.8364500999450684
33 4.897850e+02 1.452099e+00
* time: 0.8562309741973877
34 4.897776e+02 1.482593e+00
* time: 0.8750159740447998
35 4.897718e+02 8.420646e-01
* time: 0.8949220180511475
36 4.897702e+02 2.023876e-01
* time: 0.9152951240539551
37 4.897700e+02 1.885486e-02
* time: 0.9348080158233643
38 4.897700e+02 2.343932e-03
* time: 0.9532670974731445
39 4.897700e+02 4.417566e-04
* time: 0.9695911407470703FittedPumasModel
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: 7.414817810058594e-5
1 2.005689e+02 1.252637e+02
* time: 0.4500119686126709
2 1.875728e+02 4.225518e+01
* time: 0.5945520401000977
3 1.863803e+02 3.429785e+01
* time: 0.7343041896820068
4 1.845581e+02 3.694229e+01
* time: 0.8472030162811279
5 1.828416e+02 1.170915e+01
* time: 1.0117571353912354
6 1.823277e+02 1.004017e+01
* time: 1.1256740093231201
7 1.810629e+02 5.326780e+00
* time: 1.2429859638214111
8 1.810479e+02 1.767987e+00
* time: 1.3514091968536377
9 1.810272e+02 3.852037e+00
* time: 1.458204984664917
10 1.809414e+02 1.196625e+01
* time: 1.5720210075378418
11 1.806489e+02 1.861440e+01
* time: 1.708198070526123
12 1.801984e+02 1.361774e+01
* time: 2.1012110710144043
13 1.800427e+02 2.177039e+01
* time: 2.2236781120300293
14 1.796554e+02 7.079039e+00
* time: 2.3736460208892822
15 1.795832e+02 1.581499e+01
* time: 2.4939141273498535
16 1.795220e+02 6.552120e+00
* time: 2.6164820194244385
17 1.794621e+02 6.612645e+00
* time: 2.7344610691070557
18 1.793643e+02 6.603903e+00
* time: 2.852046012878418
19 1.793502e+02 1.025787e+01
* time: 2.973180055618286
20 1.793178e+02 1.202199e+01
* time: 3.1122090816497803
21 1.792424e+02 1.625661e+01
* time: 3.227485179901123
22 1.791560e+02 1.128856e+01
* time: 3.3419041633605957
23 1.790991e+02 7.625637e+00
* time: 3.456753969192505
24 1.790756e+02 8.557258e+00
* time: 3.574429988861084
25 1.790633e+02 4.848482e+00
* time: 3.7008211612701416
26 1.790408e+02 5.859306e+00
* time: 3.842241048812866
27 1.789734e+02 1.106035e+01
* time: 3.970172166824341
28 1.789070e+02 1.143361e+01
* time: 4.086485147476196
29 1.788321e+02 7.098448e+00
* time: 4.204787015914917
30 1.787896e+02 5.709857e+00
* time: 4.33295202255249
31 1.787809e+02 7.456555e+00
* time: 4.457854986190796
32 1.787671e+02 7.320931e-01
* time: 4.583729028701782
33 1.787660e+02 5.023990e-01
* time: 4.6835291385650635
34 1.787656e+02 3.813994e-01
* time: 4.789961099624634
35 1.787639e+02 8.608909e-01
* time: 4.897998094558716
36 1.787607e+02 2.047321e+00
* time: 5.011054992675781
37 1.787525e+02 3.859529e+00
* time: 5.121412038803101
38 1.787349e+02 5.864920e+00
* time: 5.248653173446655
39 1.787017e+02 6.966045e+00
* time: 5.35902214050293
40 1.786574e+02 5.348939e+00
* time: 5.476291179656982
41 1.786270e+02 1.642025e+00
* time: 5.591240167617798
42 1.786181e+02 5.211823e-01
* time: 5.711748123168945
43 1.786153e+02 1.186061e+00
* time: 5.831022024154663
44 1.786125e+02 1.292005e+00
* time: 5.949275970458984
45 1.786099e+02 7.814598e-01
* time: 6.046231031417847
46 1.786086e+02 1.369456e-01
* time: 6.159016132354736
47 1.786082e+02 1.912170e-01
* time: 6.269965171813965
48 1.786080e+02 2.670802e-01
* time: 6.385892152786255
49 1.786078e+02 1.979262e-01
* time: 6.5037829875946045
50 1.786077e+02 5.177918e-02
* time: 6.645445108413696
51 1.786076e+02 2.998328e-02
* time: 6.7523791790008545
52 1.786076e+02 5.799706e-02
* time: 6.868740081787109
53 1.786076e+02 4.280434e-02
* time: 6.984434127807617
54 1.786076e+02 1.467829e-02
* time: 7.107870101928711
55 1.786076e+02 6.928178e-03
* time: 7.227673053741455
56 1.786076e+02 1.236015e-02
* time: 7.3375630378723145
57 1.786076e+02 9.950826e-03
* time: 7.4569830894470215
58 1.786076e+02 2.974370e-03
* time: 7.552105188369751
59 1.786076e+02 1.374576e-03
* time: 7.650382041931152
60 1.786076e+02 2.860078e-03
* time: 7.738620042800903
61 1.786076e+02 2.100127e-03
* time: 7.834052085876465
62 1.786076e+02 2.100127e-03
* time: 7.984111070632935
63 1.786076e+02 6.412834e-03
* time: 8.117985010147095
64 1.786076e+02 6.412834e-03
* time: 8.299153089523315
65 1.786076e+02 6.412834e-03
* time: 8.533292055130005
66 1.786076e+02 6.412834e-03
* time: 8.871594190597534
67 1.786076e+02 6.412834e-03
* time: 9.074562072753906FittedPumasModel
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: 0.00010704994201660156
1 2.014577e+02 1.265001e+02
* time: 0.7635269165039062
2 1.880885e+02 4.190708e+01
* time: 0.9961369037628174
3 1.870317e+02 8.825666e+01
* time: 1.1609859466552734
4 1.846027e+02 4.400156e+01
* time: 1.3571081161499023
5 1.834445e+02 1.906624e+01
* time: 1.531541109085083
6 1.828599e+02 1.113882e+01
* time: 1.6907451152801514
7 1.815719e+02 7.449355e+00
* time: 1.9133079051971436
8 1.815131e+02 2.164678e+00
* time: 2.1093130111694336
9 1.814896e+02 2.167319e+00
* time: 2.335047960281372
10 1.814458e+02 4.615738e+00
* time: 2.4730210304260254
11 1.813173e+02 9.576967e+00
* time: 2.6187009811401367
12 1.809756e+02 2.052077e+01
* time: 2.795315980911255
13 1.807625e+02 4.553366e+01
* time: 2.9571480751037598
14 1.802224e+02 6.550892e+00
* time: 3.1346209049224854
15 1.800862e+02 2.865509e+00
* time: 3.2632651329040527
16 1.800780e+02 1.164611e+00
* time: 3.3947110176086426
17 1.800737e+02 7.952462e-01
* time: 3.5190060138702393
18 1.800352e+02 4.860618e+00
* time: 3.662774085998535
19 1.800089e+02 5.176689e+00
* time: 3.8033759593963623
20 1.799679e+02 4.303892e+00
* time: 3.9650189876556396
21 1.799153e+02 4.612832e+00
* time: 4.096179008483887
22 1.798423e+02 1.209387e+01
* time: 4.224668979644775
23 1.796821e+02 1.712256e+01
* time: 4.354512929916382
24 1.794275e+02 1.435100e+01
* time: 4.488548994064331
25 1.793773e+02 4.137313e+00
* time: 4.630553960800171
26 1.793488e+02 1.846545e+00
* time: 4.8373329639434814
27 1.793331e+02 5.502533e+00
* time: 4.971108913421631
28 1.793234e+02 2.894037e+00
* time: 5.108158111572266
29 1.793119e+02 1.453372e+00
* time: 5.237026929855347
30 1.792879e+02 4.109884e+00
* time: 5.377150058746338
31 1.792599e+02 4.613173e+00
* time: 5.529743909835815
32 1.792384e+02 4.254549e+00
* time: 5.6778481006622314
33 1.792251e+02 4.415024e+00
* time: 5.839111089706421
34 1.792026e+02 3.229145e+00
* time: 5.9711339473724365
35 1.791841e+02 3.422094e+00
* time: 6.096899032592773
36 1.791705e+02 1.621228e+00
* time: 6.22212290763855
37 1.791555e+02 3.462667e+00
* time: 6.356561899185181
38 1.791293e+02 5.586685e+00
* time: 6.4936981201171875
39 1.790713e+02 9.927638e+00
* time: 6.648448944091797
40 1.789891e+02 1.133271e+01
* time: 6.787213087081909
41 1.788585e+02 1.279172e+01
* time: 6.924952030181885
42 1.787407e+02 5.291681e+00
* time: 7.057162046432495
43 1.786633e+02 5.367971e+00
* time: 7.19856595993042
44 1.786405e+02 2.571548e+00
* time: 7.342253923416138
45 1.786339e+02 2.720314e+00
* time: 7.494275093078613
46 1.786252e+02 1.930583e+00
* time: 7.671706914901733
47 1.786182e+02 1.523164e+00
* time: 7.812493085861206
48 1.786126e+02 5.027920e-01
* time: 7.9629480838775635
49 1.786100e+02 3.733023e-01
* time: 8.127568006515503
50 1.786089e+02 2.749553e-01
* time: 8.292948007583618
51 1.786083e+02 1.981772e-01
* time: 8.450971126556396
52 1.786079e+02 1.005262e-01
* time: 8.604620933532715
53 1.786078e+02 2.549641e-02
* time: 8.77908992767334
54 1.786077e+02 4.452931e-02
* time: 8.92313289642334
55 1.786076e+02 2.967125e-02
* time: 9.064459085464478
56 1.786076e+02 1.068274e-02
* time: 9.207327127456665
57 1.786076e+02 5.355447e-03
* time: 9.332042932510376
58 1.786076e+02 8.180920e-03
* time: 9.45642900466919
59 1.786076e+02 4.935439e-03
* time: 9.589504957199097
60 1.786076e+02 4.387986e-03
* time: 9.769643068313599
61 1.786076e+02 4.388023e-03
* time: 9.946478128433228
62 1.786076e+02 4.387934e-03
* time: 10.146687030792236
63 1.786076e+02 4.387934e-03
* time: 10.300440073013306FittedPumasModel
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.