warfarin_pkpd_model = @model begin
...
@dynamics begin
Depot' = -Ka * Depot
Central' = Ka * Depot - CL / Vc * Central
Turnover' =
rin * (1 + emax * (Central / Vc) / (cp50 + Central / Vc)) - kout * Turnover
end
...
end
Differential Equations in Pumas
1 Introduction
Pumas automatically chooses a differential equation solver that is suitable for the simulation or estimation of the dynamical system of the NLME (Nonlinear Mixed Effects) model at hand. This default solver is the preferred choice and optimized for most users and use cases. Nevertheless, in some cases the performance-accuracy trade-off can be improved by adjusting the tolerances, or possibly even the algorithm, of the differential equation solver.
In this tutorial, the Warfarin PK/PD model is used to demonstrate how to configure the differential equation solver.
2 Learning Goals
- Observe the utility of the
@varsblock of a Pumas model with respect to storing dynamic variables associated with differential equations - Understand the main differences between common differential equation solvers for nonlinear dynamical systems
- Learn how to adjust the algorithm and the tolerances of the differential equation solver
3 Warfarin PK/PD Model
We return to the Warfarin PK/PD model. Its dynamical system consists of three states, \(\operatorname{Depot}\), \(\operatorname{Central}\), and \(\operatorname{Turnover}\), whose dynamics are governed by the ordinary differential equations:
\[ \begin{aligned} \operatorname{Depot}'(t) &= - \operatorname{Ka} \operatorname{Depot}(t),\\ \operatorname{Central}'(t) &= \operatorname{Ka} \operatorname{Depot}(t) - \frac{\operatorname{CL}}{\operatorname{Vc}} \operatorname{Central}(t),\\ \operatorname{Turnover}'(t) &= \operatorname{rin} (1 + \operatorname{emax} \frac{\operatorname{Central}(t) / \operatorname{Vc}}{\operatorname{c50} + \operatorname{Central}(t)/\operatorname{Vc}}) - \operatorname{kout} \operatorname{Turnover}(t) \end{aligned} \]
with PK parameters \(\operatorname{Ka}\) (absorption rate), \(\operatorname{CL}\) (clearance), and \(\operatorname{Vc}\) (volume of distribution) and PD parameters \(\operatorname{rin}\), \(\operatorname{emax}\), \(\operatorname{c50}\), and \(\operatorname{kout}\).
The dynamical system can be written more concisely by introducing auxiliary variables for repeated expressions:
\[ \begin{aligned} \operatorname{Depot}'(t) &= -\operatorname{ratein}(t),\\ \operatorname{Central}'(t) &= \operatorname{ratein}(t) - \operatorname{CL} \operatorname{cp}(t),\\ \operatorname{Turnover}'(t) &= \operatorname{rin} \operatorname{pd}(t) - \operatorname{kout} \operatorname{Turnover}(t) \end{aligned} \]
with influx rate \(\operatorname{ratein}(t) := \operatorname{Ka} \operatorname{Depot}(t)\), concentration \(\operatorname{cp}(t) := \operatorname{Central}(t) / \operatorname{Vc}\), and \(\operatorname{pd}(t) := 1 + \operatorname{emax} \frac{\operatorname{cp}(t)}{\operatorname{c50} + \operatorname{cp}(t)}\).
4 Auxiliary Variables in @vars
In Pumas, dynamical systems are defined in the @dynamics block inside of the @model definition. For instance, the dynamical system of the Warfarin PK/PD model can be implemented as follows:
The same concise rewriting can be applied in a Pumas @model by defining auxiliary variables (“aliases”) in the @vars block:
warfarin_pkpd_model = @model begin
...
@vars begin
cp := Central / Vc
ratein := Ka * Depot
pd := 1 + emax * cp / (c50 + cp)
end
@dynamics begin
Depot' = -ratein
Central' = ratein - CL * cp
Turnover' = rin * pd - kout * Turnover
end
...
end
Tip
The walrus operator (:=) ensures that the aliases do not show up in the simulation output of the model. However, if you would like to access an alias in the simulation output, you should define the alias with =. For instance, if you want to obtain concentration cp as part of the simulation output, you can change the @vars block to
@vars begin
cp = Central / Vc
ratein := Ka * Depot
pd := 1 + emax * cp / (c50 + cp)
end5 Differential Equation Solvers
The differential equation in the Warfarin model is non-linear, as detected by Pumas (“Dynamical system type: Nonlinear ODE”):
using Pumas
warfarin_pkpd_model = @model begin
@param begin
# PK parameters
"""
Clearance (L/h/70kg)
"""
pop_CL ∈ RealDomain(lower = 0.0, init = 0.134)
"""
Central Volume L/70kg
"""
pop_V ∈ RealDomain(lower = 0.0, init = 8.11)
"""
Absorption time (h)
"""
pop_tabs ∈ RealDomain(lower = 0.0, init = 0.523)
"""
Lag time (h)
"""
pop_lag ∈ RealDomain(lower = 0.0, init = 0.1)
# PD parameters
"""
Baseline
"""
pop_e0 ∈ RealDomain(lower = 0.0, init = 100.0)
"""
Emax
"""
pop_emax ∈ RealDomain(init = -1.0)
"""
EC50
"""
pop_c50 ∈ RealDomain(lower = 0.0, init = 1.0)
"""
Turnover
"""
pop_tover ∈ RealDomain(lower = 0.0, init = 14.0)
# Inter-individual variability
"""
- ΩCL
- ΩVc
- ΩTabs
"""
pk_Ω ∈ PDiagDomain([0.01, 0.01, 0.01])
"""
- Ωe0
- Ωemax
- Ωec50
- Ωturn
"""
pd_Ω ∈ PDiagDomain([0.01, 0.01, 0.01, 0.01])
# Residual variability
"""
Proportional residual error for drug concentration
"""
σ_prop ∈ RealDomain(lower = 0.0, init = 0.00752)
"""
Additive residual error for drug concentration (mg/L)
"""
σ_add ∈ RealDomain(lower = 0.0, init = 0.0661)
"""
Additive error for PCA
"""
σ_fx ∈ RealDomain(lower = 0.0, init = 0.01)
end
@random begin
# mean = 0, covariance = pk_Ω
pk_η ~ MvNormal(pk_Ω)
# mean = 0, covariance = pd_Ω
pd_η ~ MvNormal(pd_Ω)
end
@covariates FSZV FSZCL
@pre begin
# PK
CL = FSZCL * pop_CL * exp(pk_η[1])
Vc = FSZV * pop_V * exp(pk_η[2])
tabs = pop_tabs * exp(pk_η[3])
Ka = log(2) / tabs
# PD
e0 = pop_e0 * exp(pd_η[1])
emax = pop_emax * exp(pd_η[2])
c50 = pop_c50 * exp(pd_η[3])
tover = pop_tover * exp(pd_η[4])
kout = log(2) / tover
rin = e0 * kout
time = t
end
@dosecontrol begin
lags = (Depot = pop_lag,)
end
@init begin
Turnover = e0
end
# aliases for use in @dynamics and @derived
@vars begin
cp := Central / Vc
ratein := Ka * Depot
pd := 1 + emax * cp / (c50 + cp)
end
@dynamics begin
Depot' = -ratein
Central' = ratein - CL * cp
Turnover' = rin * pd - kout * Turnover
end
@derived begin
"""
Warfarin Concentration (mg/L)
"""
conc ~ @. Normal(cp, sqrt((σ_prop * cp)^2 + σ_add^2))
"""
PCA
"""
pca ~ @. Normal(Turnover, σ_fx)
end
endPumasModel
Parameters: pop_CL, pop_V, pop_tabs, pop_lag, pop_e0, pop_emax, pop_c50, pop_tover, pk_Ω, pd_Ω, σ_prop, σ_add, σ_fx
Random effects: pk_η, pd_η
Covariates: FSZV, FSZCL
Dynamical system variables: Depot, Central, Turnover
Dynamical system type: Nonlinear ODE
Derived: conc, pca
Observed: conc, pcaPumas approximates the solution of the differential equation with a numerical differential equation solver. Generally, one distinguishes between solvers for stiff and non-stiff differential equations.
5.1 Stiff vs. Non-Stiff Systems
A key distinction among numerical solvers is whether they are designed for stiff or non-stiff differential equations:
Non-Stiff Differential Equations: These systems exhibit relatively moderate changes in their variables. Standard non-stiff solvers can efficiently approximate solutions of these systems.
Stiff Differential Equations: These systems contain rapidly changing components alongside more slowly varying dynamics. Non-stiff solvers typically perform poorly on stiff systems, as they may require exceedingly small step sizes to maintain numerical stability. Specialized stiff solvers are therefore employed to handle the sharp gradients and large timescale differences without compromising accuracy.
6 Pumas’s Automatic Solver Selection
By default, Pumas adopts a hybrid approach with automatic stiffness detection to switch between stiff and non-stiff solvers as needed.
- Default Solvers: Rodas5P (stiff) and Vern7 (non-stiff)
- Tolerances: Relative tolerance \(10^{-8}\) and absolute tolerance \(10^{-12}\)
- Rationale: These tolerances ensure high precision during both simulation and parameter estimation, which is critical for accurate exploratory and predictive modeling and matching the model’s predictions to observed data.
The default solvers and tolerances are recommended for most users in most instances. If desired, however, it is possible to adjust these settings with the diffeq_options keyword argument.
Important
Computation time decreases as tolerances are increased. However, higher tolerances come at the cost of a less strict error control, and hence generally a less accurate solution.
6.1 Adjusting the Tolerances
The absolute and relative tolerance of the solver can be specified with abstol and reltol.
7 Absolute and Relative Tolerances
When employing a numerical solver, it is necessary to specify how accurately the solution should be computed. This precision is controlled by two key parameters:
Absolute Tolerance \((\text{abstol})\)
- Interpreted as the maximum allowable error when the solution values are near zero.
- Ensures that numerical approximations stay within a reasonable bound, preventing physically impossible outcomes (e.g., negative concentrations) or excessive drift at small scales.
- For instance, an absolute tolerance of \(10^{-6}\) means the solver attempts to keep the absolute error below \(10^{-6}\) whenever the solution magnitude is close to zero.
Relative Tolerance \((\text{reltol})\)
- Enforces the number of correct digits throughout the simulation, effectively controlling error relative to the current scale of the solution.
- For example, a relative tolerance of \(10^{-3}\) implies the solver aims for three correct decimal places (i.e., the solution is accurate to within 0.1% of its current magnitude).
- As the solution grows or shrinks, the solver adjusts its time-step size and internal computations to maintain this relative accuracy.
Sometimes decreasing tolerances can help to reduce numerical problems, e.g. to keep solutions non-negative that are mathematically guaranteed to be non-negative. Additionally, the choice of tolerances can be motivated by the application of the numerical solution: For plotting a less accurate solution, and hence larger tolerances, might be tolerable, whereas typically for model fitting a more accurate solution, and hence smaller tolerances, are beneficial.
This can be demonstrated when fitting the Warfarin model with an example dataset: Optimization fails with large tolerances of 1e-3 (relative) and 1e-6 (absolute)
fit(
warfarin_pkpd_model,
pop,
init_params(warfarin_pkpd_model),
FOCE();
diffeq_options = (; reltol = 1e-3, abstol = 1e-6),
)[ 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 3.130181e+06 5.915753e+06 * time: 0.04384589195251465 1 5.185877e+05 8.742010e+05 * time: 4.800498008728027 2 3.866957e+05 6.366584e+05 * time: 5.858506917953491 3 1.795019e+05 2.835377e+05 * time: 6.889288902282715 4 9.682619e+04 1.546512e+05 * time: 7.811128854751587 5 4.791898e+04 6.820022e+04 * time: 8.696491956710815 6 2.907369e+04 3.509683e+04 * time: 9.584167003631592 7 1.827377e+04 1.713122e+04 * time: 10.537323951721191 8 1.260634e+04 9.563328e+03 * time: 11.51173996925354 9 9.403430e+03 8.611643e+03 * time: 12.472425937652588 10 7.325839e+03 7.634814e+03 * time: 13.432734966278076 11 5.926864e+03 6.625143e+03 * time: 14.379194021224976 12 4.942276e+03 5.562817e+03 * time: 15.294863939285278 13 4.139082e+03 4.326560e+03 * time: 16.210917949676514 14 3.563490e+03 3.065255e+03 * time: 17.130348920822144 15 3.296158e+03 2.170246e+03 * time: 18.04499387741089 16 3.216281e+03 1.669628e+03 * time: 18.94379496574402 17 3.205943e+03 1.487868e+03 * time: 19.84991693496704 18 3.204981e+03 1.444253e+03 * time: 20.740763902664185 19 3.204107e+03 1.417661e+03 * time: 21.6385760307312 20 3.201145e+03 1.361617e+03 * time: 22.526020050048828 21 3.194218e+03 1.282918e+03 * time: 23.41675901412964 22 3.175799e+03 1.158036e+03 * time: 24.306293964385986 23 3.130001e+03 9.774722e+02 * time: 25.199190855026245 24 3.016853e+03 7.265878e+02 * time: 26.08452796936035 25 2.749857e+03 4.107604e+02 * time: 26.950739860534668 26 2.137213e+03 2.318395e+02 * time: 27.82140588760376 27 1.756667e+03 2.281482e+02 * time: 28.753108978271484 28 1.380674e+03 1.613751e+02 * time: 31.269019842147827 29 1.328451e+03 1.298787e+02 * time: 32.153708934783936 30 1.287368e+03 2.435586e+02 * time: 32.927605867385864 31 1.263071e+03 1.616831e+02 * time: 33.70522093772888 32 1.254713e+03 1.796576e+02 * time: 34.47108602523804 33 1.247205e+03 1.996165e+02 * time: 35.249754905700684 34 1.243765e+03 1.938235e+02 * time: 36.01481890678406 35 1.240829e+03 1.739538e+02 * time: 36.821568965911865 36 1.240788e+03 1.744028e+02 * time: 37.61355495452881 37 1.240776e+03 1.743250e+02 * time: 38.374022006988525 38 1.240093e+03 1.687609e+02 * time: 39.156182050704956 39 1.239010e+03 1.586878e+02 * time: 39.94395089149475 40 1.236253e+03 1.305335e+02 * time: 40.725817918777466 41 1.232129e+03 8.229601e+01 * time: 41.49223184585571 42 1.228151e+03 3.735809e+01 * time: 42.286757946014404 43 1.226337e+03 5.238476e+01 * time: 43.06374502182007 44 1.226025e+03 4.868885e+01 * time: 43.839536905288696 45 1.226011e+03 4.583328e+01 * time: 44.60988783836365 46 1.226010e+03 4.536419e+01 * time: 45.36117887496948 47 1.226008e+03 4.466378e+01 * time: 46.093708992004395 48 1.226002e+03 4.336406e+01 * time: 46.8202919960022 49 1.225988e+03 4.102579e+01 * time: 47.591376066207886 50 1.225951e+03 3.656289e+01 * time: 48.337640047073364 51 1.225858e+03 2.865017e+01 * time: 49.08858799934387 52 1.225645e+03 2.863717e+01 * time: 49.82658004760742 53 1.225243e+03 3.015495e+01 * time: 50.56348991394043 54 1.224739e+03 4.057375e+01 * time: 51.30551099777222 55 1.224438e+03 5.072098e+01 * time: 52.047581911087036 56 1.224368e+03 4.561543e+01 * time: 52.78227496147156 57 1.224362e+03 4.124686e+01 * time: 53.51751685142517 58 1.224360e+03 4.008283e+01 * time: 54.23265790939331 59 1.224357e+03 3.807281e+01 * time: 54.962846994400024 60 1.224349e+03 3.507733e+01 * time: 55.69300293922424 61 1.224328e+03 3.152658e+01 * time: 56.422492027282715 62 1.224274e+03 2.940378e+01 * time: 57.15742087364197 63 1.224133e+03 2.814508e+01 * time: 57.90731406211853 64 1.223771e+03 2.816089e+01 * time: 58.659225940704346 65 1.222880e+03 4.493737e+01 * time: 59.386653900146484 66 1.220919e+03 8.628817e+01 * time: 60.11530685424805 67 1.217683e+03 1.137018e+02 * time: 60.822242975234985 68 1.214493e+03 9.066168e+01 * time: 61.523202896118164 69 1.212820e+03 8.381803e+01 * time: 62.23194599151611 70 1.212582e+03 8.589808e+01 * time: 62.942939043045044 71 1.212574e+03 8.557497e+01 * time: 63.65787601470947 72 1.212569e+03 8.522111e+01 * time: 64.3703088760376 73 1.212552e+03 8.416219e+01 * time: 65.09645986557007 74 1.212516e+03 8.236925e+01 * time: 65.78960490226746 75 1.212415e+03 7.841477e+01 * time: 66.49769997596741 76 1.212165e+03 7.017247e+01 * time: 67.2089569568634 77 1.211553e+03 5.231312e+01 * time: 67.90825605392456 78 1.210236e+03 7.254984e+01 * time: 68.65167498588562 79 1.208052e+03 8.652548e+01 * time: 69.39358687400818 80 1.205978e+03 9.811810e+01 * time: 70.15770292282104 81 1.205158e+03 1.085052e+02 * time: 70.88729596138 82 1.205038e+03 9.706885e+01 * time: 71.62432098388672 83 1.205030e+03 9.297156e+01 * time: 72.3486750125885 84 1.205025e+03 9.142093e+01 * time: 73.07483005523682 85 1.205007e+03 8.794517e+01 * time: 73.78387498855591 86 1.204968e+03 8.329868e+01 * time: 74.49967193603516 87 1.204860e+03 7.519245e+01 * time: 75.22060203552246 88 1.204586e+03 6.676950e+01 * time: 75.9117419719696 89 1.203889e+03 5.914985e+01 * time: 76.60696792602539 90 1.202275e+03 5.659425e+01 * time: 77.29526901245117 91 1.199277e+03 5.457598e+01 * time: 77.99242401123047 92 1.195791e+03 6.434187e+01 * time: 78.654217004776 93 1.193281e+03 4.404726e+01 * time: 79.30896401405334 94 1.192403e+03 4.696513e+01 * time: 79.99026799201965 95 1.192344e+03 4.826238e+01 * time: 80.6446099281311 96 1.192341e+03 4.836346e+01 * time: 81.28319597244263 97 1.192340e+03 4.843134e+01 * time: 81.91030597686768 98 1.192336e+03 4.854905e+01 * time: 82.53473091125488 99 1.192325e+03 4.873525e+01 * time: 83.16785407066345 100 1.192297e+03 4.901993e+01 * time: 83.80186891555786 101 1.192226e+03 4.941568e+01 * time: 84.45286893844604 102 1.192043e+03 4.985206e+01 * time: 85.13519406318665 103 1.191593e+03 5.001007e+01 * time: 85.81375885009766 104 1.190562e+03 4.892711e+01 * time: 86.50051784515381 105 1.188621e+03 4.482401e+01 * time: 87.18688797950745 106 1.186171e+03 5.565401e+01 * time: 87.89409184455872 107 1.184411e+03 7.195267e+01 * time: 88.60426092147827 108 1.183640e+03 7.480949e+01 * time: 89.31642889976501 109 1.183456e+03 7.309347e+01 * time: 90.02860593795776 110 1.183420e+03 7.276593e+01 * time: 90.69497489929199 111 1.183410e+03 7.363637e+01 * time: 91.36429691314697 112 1.183407e+03 7.448590e+01 * time: 92.02231287956238 113 1.183403e+03 7.577977e+01 * time: 92.6972029209137 114 1.183397e+03 7.712667e+01 * time: 93.35962700843811 115 1.183377e+03 7.953569e+01 * time: 94.04420304298401 116 1.183329e+03 8.300885e+01 * time: 94.7211389541626 117 1.183203e+03 8.805322e+01 * time: 95.40114188194275 118 1.182889e+03 9.423584e+01 * time: 96.0783679485321 119 1.182143e+03 9.919011e+01 * time: 96.78499484062195 120 1.180607e+03 9.577147e+01 * time: 97.48721385002136 121 1.178255e+03 7.337696e+01 * time: 98.20086002349854 122 1.176240e+03 3.404284e+01 * time: 98.91404485702515 123 1.175531e+03 1.922941e+01 * time: 99.66240191459656 124 1.175410e+03 1.933203e+01 * time: 100.41189885139465 125 1.175396e+03 1.964875e+01 * time: 101.1775050163269 126 1.175395e+03 1.966957e+01 * time: 101.91872787475586 127 1.175394e+03 1.960524e+01 * time: 102.67614102363586 128 1.175394e+03 1.957692e+01 * time: 103.43248701095581 129 1.175392e+03 1.945993e+01 * time: 104.1843490600586 130 1.175387e+03 1.938252e+01 * time: 104.9326388835907 131 1.175374e+03 1.915643e+01 * time: 105.6806709766388 132 1.175342e+03 1.892407e+01 * time: 106.40796899795532 133 1.175256e+03 1.840907e+01 * time: 107.15405106544495 134 1.175030e+03 1.936487e+01 * time: 107.86935901641846 135 1.174436e+03 2.656836e+01 * time: 108.62122201919556 136 1.172888e+03 3.906036e+01 * time: 109.38319206237793 137 1.169012e+03 5.824500e+01 * time: 110.14776802062988 138 1.160858e+03 7.724789e+01 * time: 110.91178393363953 139 1.151984e+03 5.979641e+01 * time: 111.69499707221985 140 1.150239e+03 2.132816e+02 * time: 112.49608302116394 141 1.147461e+03 1.267649e+02 * time: 113.29408288002014 142 1.143928e+03 2.373475e+01 * time: 114.07671189308167 143 1.142853e+03 2.225755e+01 * time: 114.83840084075928 144 1.141788e+03 3.914831e+01 * time: 115.57437086105347 145 1.141298e+03 2.253303e+01 * time: 116.30886507034302 146 1.140996e+03 2.098978e+01 * time: 117.05825805664062 147 1.140981e+03 2.094768e+01 * time: 117.79100894927979 148 1.140978e+03 2.098189e+01 * time: 118.50924491882324 149 1.140977e+03 2.101184e+01 * time: 119.22071695327759 150 1.140977e+03 2.100810e+01 * time: 119.98072695732117 151 1.140977e+03 2.097542e+01 * time: 120.68533205986023 152 1.140976e+03 2.094478e+01 * time: 121.36990284919739 153 1.140974e+03 2.089055e+01 * time: 122.10394096374512 154 1.140970e+03 2.083449e+01 * time: 122.86092400550842 155 1.140963e+03 2.076312e+01 * time: 123.57725095748901 156 1.140944e+03 2.068373e+01 * time: 124.29382300376892 157 1.140896e+03 2.060201e+01 * time: 125.06949806213379 158 1.140770e+03 2.053295e+01 * time: 125.81379389762878 159 1.140444e+03 2.051792e+01 * time: 126.77110385894775 160 1.139610e+03 2.178464e+01 * time: 127.88144898414612 161 1.138121e+03 3.276681e+01 * time: 129.08301901817322 162 1.135107e+03 4.596086e+01 * time: 130.29972791671753 163 1.132128e+03 5.673428e+01 * time: 131.31528687477112 164 1.130311e+03 6.297114e+01 * time: 132.17991590499878 165 1.126112e+03 1.303349e+02 * time: 132.98048901557922 166 1.116080e+03 4.102784e+01 * time: 133.94178700447083 167 1.114892e+03 3.653943e+01 * time: 134.82184386253357 168 1.112342e+03 3.437496e+01 * time: 135.6621880531311 169 1.111707e+03 1.636383e+01 * time: 136.52938985824585 170 1.111291e+03 9.909471e+00 * time: 137.40249705314636 171 1.111206e+03 1.001147e+01 * time: 138.2972309589386 172 1.111191e+03 1.040660e+01 * time: 139.18353605270386
FittedPumasModel
Dynamical system type: Nonlinear ODE
Solver(s): (OrdinaryDiffEqVerner.Vern7,OrdinaryDiffEqRosenbrock.Rodas5P)
Number of subjects: 32
Observation records: Active Missing
conc: 251 47
pca: 232 66
Total: 483 113
Number of parameters: Constant Optimized
0 18
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: NoXChange
Log-likelihood value: -1111.1914
-----------------------
Estimate
-----------------------
pop_CL 0.13526
pop_V 7.9651
pop_tabs 0.61589
pop_lag 0.86532
pop_e0 96.343
pop_emax -1.085
pop_c50 1.6306
pop_tover 14.55
pk_Ω₁,₁ 0.11739
pk_Ω₂,₂ 0.03428
pk_Ω₃,₃ 0.24153
pd_Ω₁,₁ 0.0029639
pd_Ω₂,₂ 0.0022284
pd_Ω₃,₃ 0.02459
pd_Ω₄,₄ 0.015792
σ_prop 0.013052
σ_add 0.81408
σ_fx 3.5163
-----------------------but fares much better with lower tolerances of 1e-8 (relative) and 1e-12 (absolute):
fit(
warfarin_pkpd_model,
pop,
init_params(warfarin_pkpd_model),
FOCE();
diffeq_options = (; reltol = 1e-8, abstol = 1e-12),
)[ 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 3.125741e+06 5.911802e+06 * time: 2.5987625122070312e-5 1 5.174461e+05 8.708698e+05 * time: 2.2354819774627686 2 3.865265e+05 6.344302e+05 * time: 4.204480886459351 3 1.804274e+05 2.829723e+05 * time: 6.008337020874023 4 9.706640e+04 1.550547e+05 * time: 7.893967866897583 5 4.769637e+04 6.778818e+04 * time: 9.764966011047363 6 2.902319e+04 3.499747e+04 * time: 11.628520011901855 7 1.823472e+04 1.705751e+04 * time: 13.471780061721802 8 1.258819e+04 9.569381e+03 * time: 15.302196025848389 9 9.389984e+03 8.615851e+03 * time: 17.16966485977173 10 7.314702e+03 7.636883e+03 * time: 19.061110019683838 11 5.916029e+03 6.624325e+03 * time: 20.895668983459473 12 4.930519e+03 5.558140e+03 * time: 22.727866888046265 13 4.125060e+03 4.315759e+03 * time: 24.570483922958374 14 3.549280e+03 3.051093e+03 * time: 26.38404893875122 15 3.283489e+03 2.157292e+03 * time: 28.173277854919434 16 3.204886e+03 1.659798e+03 * time: 29.878135919570923 17 3.194875e+03 1.480528e+03 * time: 31.56321406364441 18 3.193944e+03 1.437921e+03 * time: 33.2578330039978 19 3.193070e+03 1.411186e+03 * time: 34.94093894958496 20 3.190129e+03 1.355327e+03 * time: 36.62148404121399 21 3.183228e+03 1.276603e+03 * time: 38.30476403236389 22 3.164897e+03 1.151838e+03 * time: 39.96260905265808 23 3.119250e+03 9.712651e+02 * time: 41.63005089759827 24 3.006297e+03 7.204342e+02 * time: 43.24989295005798 25 2.738913e+03 4.050545e+02 * time: 44.8462860584259 26 2.123834e+03 2.318194e+02 * time: 46.306967973709106 27 1.789138e+03 2.290465e+02 * time: 47.85612893104553 28 1.396455e+03 1.683969e+02 * time: 51.94181489944458 29 1.333545e+03 1.336195e+02 * time: 53.567774057388306 30 1.297771e+03 2.452189e+02 * time: 54.927698850631714 31 1.266002e+03 1.523968e+02 * time: 56.28854584693909 32 1.255506e+03 1.733993e+02 * time: 57.656753063201904 33 1.247789e+03 1.971624e+02 * time: 59.02541399002075 34 1.244490e+03 1.915728e+02 * time: 60.38836693763733 35 1.240568e+03 1.704250e+02 * time: 61.776084899902344 36 1.240503e+03 1.711787e+02 * time: 63.15067791938782 37 1.240492e+03 1.711657e+02 * time: 64.49610495567322 38 1.239992e+03 1.687240e+02 * time: 65.876620054245 39 1.239199e+03 1.624610e+02 * time: 67.28721785545349 40 1.236971e+03 1.400044e+02 * time: 68.68989896774292 41 1.233203e+03 9.442277e+01 * time: 70.0905499458313 42 1.228682e+03 3.273922e+01 * time: 71.4747359752655 43 1.226466e+03 4.997778e+01 * time: 72.85972690582275 44 1.226104e+03 4.904407e+01 * time: 74.24275588989258 45 1.226088e+03 4.675833e+01 * time: 75.61262202262878 46 1.226088e+03 4.628497e+01 * time: 76.96887493133545 47 1.226085e+03 4.541356e+01 * time: 78.32595992088318 48 1.226080e+03 4.402741e+01 * time: 79.6824209690094 49 1.226064e+03 4.142299e+01 * time: 81.04272484779358 50 1.226026e+03 3.663064e+01 * time: 82.41365385055542 51 1.225931e+03 2.851375e+01 * time: 83.77667593955994 52 1.225713e+03 2.844436e+01 * time: 85.19013500213623 53 1.225303e+03 3.026440e+01 * time: 86.57701802253723 54 1.224791e+03 4.157127e+01 * time: 87.94020295143127 55 1.224489e+03 5.047474e+01 * time: 89.31216096878052 56 1.224420e+03 4.451064e+01 * time: 90.74336194992065 57 1.224413e+03 3.994026e+01 * time: 92.09214901924133 58 1.224412e+03 3.879446e+01 * time: 93.44346594810486 59 1.224408e+03 3.677250e+01 * time: 94.80109596252441 60 1.224400e+03 3.377993e+01 * time: 96.1508629322052 61 1.224379e+03 3.158187e+01 * time: 97.52853894233704 62 1.224324e+03 2.925155e+01 * time: 98.8903980255127 63 1.224180e+03 2.815886e+01 * time: 100.25663805007935 64 1.223813e+03 2.818297e+01 * time: 101.62795686721802 65 1.222906e+03 4.598758e+01 * time: 103.03600597381592 66 1.220910e+03 8.692142e+01 * time: 104.40268206596375 67 1.217609e+03 1.136529e+02 * time: 105.77456092834473 68 1.214313e+03 9.030271e+01 * time: 107.12677597999573 69 1.212533e+03 8.868947e+01 * time: 108.51322889328003 70 1.212272e+03 8.970279e+01 * time: 109.87358784675598 71 1.212264e+03 8.921922e+01 * time: 111.22936296463013 72 1.212259e+03 8.882366e+01 * time: 112.58034586906433 73 1.212242e+03 8.761509e+01 * time: 113.92434787750244 74 1.212205e+03 8.561995e+01 * time: 115.25925087928772 75 1.212103e+03 8.125315e+01 * time: 116.57272100448608 76 1.211850e+03 7.229633e+01 * time: 117.8692479133606 77 1.211238e+03 5.321838e+01 * time: 119.19737792015076 78 1.209945e+03 7.377546e+01 * time: 120.49001383781433 79 1.207890e+03 8.404092e+01 * time: 121.82098293304443 80 1.206058e+03 9.148114e+01 * time: 123.1292769908905 81 1.205389e+03 9.689402e+01 * time: 124.40561604499817 82 1.205303e+03 8.594692e+01 * time: 125.70369505882263 83 1.205297e+03 8.262969e+01 * time: 126.99411392211914 84 1.205293e+03 8.096366e+01 * time: 128.2679100036621 85 1.205277e+03 7.747618e+01 * time: 129.56158185005188 86 1.205241e+03 7.259258e+01 * time: 130.8484559059143 87 1.205143e+03 6.664833e+01 * time: 132.1497540473938 88 1.204895e+03 6.374646e+01 * time: 133.49039387702942 89 1.204268e+03 5.753767e+01 * time: 134.84536290168762 90 1.202818e+03 4.866322e+01 * time: 136.17918395996094 91 1.200116e+03 5.085841e+01 * time: 137.4999668598175 92 1.196895e+03 7.139477e+01 * time: 138.83528184890747 93 1.194610e+03 4.693642e+01 * time: 140.16371488571167 94 1.193880e+03 4.874637e+01 * time: 141.5410978794098 95 1.193833e+03 4.997341e+01 * time: 142.8707640171051 96 1.193831e+03 5.008752e+01 * time: 144.2117109298706 97 1.193829e+03 5.014860e+01 * time: 145.62985587120056 98 1.193824e+03 5.025195e+01 * time: 147.03822684288025 99 1.193812e+03 5.040245e+01 * time: 148.44854187965393 100 1.193778e+03 5.061928e+01 * time: 149.8715488910675 101 1.193693e+03 5.087910e+01 * time: 151.29440999031067 102 1.193472e+03 5.105104e+01 * time: 152.64622688293457 103 1.192926e+03 5.068579e+01 * time: 153.9582118988037 104 1.191681e+03 4.863476e+01 * time: 155.22451186180115 105 1.189351e+03 4.892982e+01 * time: 156.47253489494324 106 1.186420e+03 7.777398e+01 * time: 157.7377049922943 107 1.184527e+03 8.874390e+01 * time: 158.99657487869263 108 1.184034e+03 8.377339e+01 * time: 160.26790690422058 109 1.183978e+03 7.932702e+01 * time: 161.5211079120636 110 1.183969e+03 7.794331e+01 * time: 162.77012705802917 111 1.183966e+03 7.787273e+01 * time: 164.0265028476715 112 1.183962e+03 7.833691e+01 * time: 165.33348989486694 113 1.183955e+03 7.929110e+01 * time: 166.65898990631104 114 1.183940e+03 8.084135e+01 * time: 167.98653388023376 115 1.183904e+03 8.331062e+01 * time: 169.27821493148804 116 1.183812e+03 8.698431e+01 * time: 170.5929639339447 117 1.183576e+03 9.188953e+01 * time: 171.95075392723083 118 1.182997e+03 9.668921e+01 * time: 173.3104648590088 119 1.181702e+03 9.635659e+01 * time: 174.66923999786377 120 1.179436e+03 8.061098e+01 * time: 176.04682183265686 121 1.177026e+03 4.571918e+01 * time: 177.44839096069336 122 1.175794e+03 1.863482e+01 * time: 178.88221788406372 123 1.175474e+03 1.968114e+01 * time: 180.25503396987915 124 1.175429e+03 1.958620e+01 * time: 181.59373784065247 125 1.175427e+03 1.993702e+01 * time: 183.041002035141 126 1.175426e+03 1.962020e+01 * time: 184.42174696922302 127 1.175426e+03 1.965822e+01 * time: 185.77183604240417 128 1.175424e+03 1.984458e+01 * time: 187.16407585144043 129 1.175421e+03 2.001151e+01 * time: 188.5387978553772 130 1.175413e+03 2.035203e+01 * time: 189.9340078830719 131 1.175392e+03 2.084295e+01 * time: 191.32540798187256 132 1.175336e+03 2.164587e+01 * time: 192.70958995819092 133 1.175189e+03 2.287862e+01 * time: 194.05450201034546 134 1.174801e+03 2.475388e+01 * time: 195.38134384155273 135 1.173792e+03 4.143552e+01 * time: 196.71206498146057 136 1.171220e+03 7.151629e+01 * time: 198.07408094406128 137 1.165144e+03 1.078851e+02 * time: 199.5346188545227 138 1.155079e+03 7.750603e+01 * time: 201.10307598114014 139 1.149212e+03 6.869378e+01 * time: 202.70089983940125 140 1.146920e+03 6.357052e+01 * time: 204.32509899139404 141 1.144663e+03 4.948266e+01 * time: 205.94313097000122 142 1.143084e+03 2.539126e+01 * time: 207.44376301765442 143 1.141829e+03 1.979229e+01 * time: 208.8740930557251 144 1.141293e+03 2.050881e+01 * time: 210.38423895835876 145 1.141052e+03 2.076835e+01 * time: 211.83070588111877 146 1.140979e+03 2.090550e+01 * time: 213.26495504379272 147 1.140976e+03 2.094541e+01 * time: 214.67772793769836 148 1.140975e+03 2.099654e+01 * time: 216.10724592208862 149 1.140974e+03 2.099916e+01 * time: 217.57909893989563 150 1.140974e+03 2.099880e+01 * time: 219.03184604644775 151 1.140973e+03 2.099210e+01 * time: 220.53654289245605 152 1.140972e+03 2.098163e+01 * time: 222.0392780303955 153 1.140970e+03 2.096557e+01 * time: 223.5425808429718 154 1.140963e+03 2.094347e+01 * time: 225.09362387657166 155 1.140947e+03 2.091734e+01 * time: 226.66516304016113 156 1.140905e+03 2.089968e+01 * time: 228.26662492752075 157 1.140795e+03 2.093513e+01 * time: 229.84285187721252 158 1.140505e+03 2.116003e+01 * time: 231.38762092590332 159 1.139717e+03 2.228845e+01 * time: 232.948224067688 160 1.137490e+03 3.499275e+01 * time: 234.59126806259155 161 1.132263e+03 6.112003e+01 * time: 236.28288888931274 162 1.129968e+03 7.040706e+01 * time: 238.39471101760864 163 1.128183e+03 7.684081e+01 * time: 240.6832139492035 164 1.125121e+03 1.651402e+02 * time: 242.86091995239258 165 1.124367e+03 1.524469e+02 * time: 244.81681895256042 166 1.117089e+03 4.431295e+01 * time: 246.7511019706726 167 1.113092e+03 1.609702e+01 * time: 248.7464849948883 168 1.111588e+03 1.011714e+01 * time: 250.70158100128174 169 1.111231e+03 1.280570e+01 * time: 252.61687397956848 170 1.111200e+03 1.186560e+01 * time: 254.48356986045837 171 1.111188e+03 1.091436e+01 * time: 256.39161586761475 172 1.111187e+03 1.052729e+01 * time: 258.21701288223267 173 1.111187e+03 1.067420e+01 * time: 260.07829689979553 174 1.111187e+03 1.055472e+01 * time: 261.97083806991577 175 1.111187e+03 1.036659e+01 * time: 263.8777709007263 176 1.111185e+03 1.017962e+01 * time: 265.8946440219879 177 1.111183e+03 1.017379e+01 * time: 267.8660509586334 178 1.111176e+03 1.016131e+01 * time: 269.82666397094727 179 1.111157e+03 1.013351e+01 * time: 271.78626585006714 180 1.111109e+03 1.006842e+01 * time: 273.73224401474 181 1.110986e+03 1.142083e+01 * time: 275.63618898391724 182 1.110675e+03 1.751466e+01 * time: 277.5228068828583 183 1.109918e+03 2.591233e+01 * time: 279.4431529045105 184 1.108234e+03 3.515773e+01 * time: 281.4320569038391 185 1.105216e+03 3.902425e+01 * time: 283.41052293777466 186 1.101588e+03 3.080377e+01 * time: 285.3537268638611 187 1.098345e+03 3.607134e+01 * time: 287.35308599472046 188 1.094148e+03 3.626407e+01 * time: 289.42933106422424 189 1.093478e+03 3.314569e+01 * time: 291.7464950084686 190 1.092996e+03 2.938811e+01 * time: 294.1085638999939 191 1.092406e+03 2.138312e+01 * time: 296.55131793022156 192 1.092297e+03 2.975339e+01 * time: 298.87501883506775 193 1.091949e+03 7.639324e+00 * time: 301.11450004577637 194 1.091909e+03 4.276154e+00 * time: 303.3737909793854 195 1.091904e+03 4.346069e+00 * time: 305.6366620063782 196 1.091904e+03 4.337170e+00 * time: 307.8746359348297 197 1.091904e+03 4.342274e+00 * time: 310.1256539821625 198 1.091904e+03 4.344184e+00 * time: 312.3570020198822 199 1.091904e+03 4.346951e+00 * time: 314.59187483787537 200 1.091904e+03 4.349040e+00 * time: 316.83250188827515 201 1.091904e+03 4.348757e+00 * time: 319.0764379501343 202 1.091904e+03 4.343922e+00 * time: 321.3279929161072 203 1.091903e+03 4.333452e+00 * time: 323.57368683815 204 1.091902e+03 4.317872e+00 * time: 325.8067078590393 205 1.091901e+03 4.295486e+00 * time: 328.03695583343506 206 1.091897e+03 4.261316e+00 * time: 330.2564649581909 207 1.091889e+03 5.125139e+00 * time: 332.51975083351135 208 1.091867e+03 9.013047e+00 * time: 335.7650909423828 209 1.091811e+03 1.513012e+01 * time: 338.71681904792786 210 1.091668e+03 2.427311e+01 * time: 341.3882648944855 211 1.091325e+03 3.642351e+01 * time: 343.70767188072205 212 1.090580e+03 4.801992e+01 * time: 346.15532398223877 213 1.089340e+03 5.062255e+01 * time: 348.5607559680939 214 1.088109e+03 4.437004e+01 * time: 350.8699269294739 215 1.087458e+03 3.115088e+01 * time: 353.07188391685486 216 1.087119e+03 1.183083e+01 * time: 355.26864194869995 217 1.087046e+03 5.394070e+00 * time: 357.46981596946716 218 1.087043e+03 5.390830e+00 * time: 359.64521884918213 219 1.087043e+03 5.383737e+00 * time: 361.888552904129 220 1.087043e+03 5.384236e+00 * time: 364.52495884895325 221 1.087043e+03 5.384636e+00 * time: 366.9340159893036 222 1.087043e+03 5.385864e+00 * time: 369.1040370464325 223 1.087043e+03 5.387179e+00 * time: 371.28600883483887 224 1.087042e+03 5.389948e+00 * time: 373.53688406944275 225 1.087041e+03 5.395184e+00 * time: 375.70870995521545 226 1.087039e+03 5.407029e+00 * time: 377.9040639400482 227 1.087033e+03 5.434556e+00 * time: 380.13645601272583 228 1.087016e+03 5.502571e+00 * time: 382.36235785484314 229 1.086972e+03 5.677167e+00 * time: 384.5774199962616 230 1.086849e+03 8.425733e+00 * time: 386.8095920085907 231 1.086459e+03 1.475408e+01 * time: 389.0079228878021 232 1.086446e+03 4.299557e+01 * time: 391.10253500938416 233 1.085092e+03 2.387717e+01 * time: 393.2545289993286 234 1.083538e+03 2.726422e+01 * time: 395.3614890575409 235 1.081685e+03 2.845459e+01 * time: 397.4341070652008 236 1.080396e+03 2.805425e+01 * time: 399.6169328689575 237 1.077803e+03 2.873905e+01 * time: 401.59939193725586 238 1.073139e+03 2.780340e+01 * time: 403.61367297172546 239 1.069921e+03 1.583445e+01 * time: 405.68462586402893 240 1.069621e+03 7.856085e+00 * time: 407.71548295021057 241 1.069531e+03 2.311783e+00 * time: 409.7601869106293 242 1.069509e+03 2.021453e+00 * time: 411.8024249076843 243 1.069491e+03 2.157500e+00 * time: 413.86292600631714 244 1.069488e+03 2.225422e+00 * time: 415.91369700431824 245 1.069488e+03 2.236782e+00 * time: 417.9519028663635 246 1.069488e+03 2.235390e+00 * time: 419.98416805267334 247 1.069488e+03 2.235490e+00 * time: 422.0035319328308 248 1.069488e+03 2.235491e+00 * time: 424.1308250427246 249 1.069488e+03 2.235259e+00 * time: 426.1524498462677 250 1.069488e+03 2.235150e+00 * time: 428.1738200187683 251 1.069488e+03 2.234587e+00 * time: 430.2406680583954 252 1.069487e+03 2.233171e+00 * time: 432.2974829673767 253 1.069486e+03 2.229015e+00 * time: 434.35016894340515 254 1.069484e+03 2.218016e+00 * time: 436.41360998153687 255 1.069477e+03 2.188614e+00 * time: 438.4706959724426 256 1.069460e+03 2.790832e+00 * time: 440.52958583831787 257 1.069418e+03 4.259790e+00 * time: 442.59208393096924 258 1.069326e+03 5.893844e+00 * time: 444.66876792907715 259 1.069168e+03 6.471844e+00 * time: 446.62949991226196 260 1.069008e+03 4.528747e+00 * time: 448.6796419620514 261 1.068940e+03 1.710450e+00 * time: 450.71741795539856 262 1.068930e+03 8.661473e-01 * time: 452.80285000801086 263 1.068929e+03 8.687543e-01 * time: 454.8346788883209 264 1.068929e+03 8.689456e-01 * time: 456.8873689174652
FittedPumasModel
Dynamical system type: Nonlinear ODE
Solver(s): (OrdinaryDiffEqVerner.Vern7,OrdinaryDiffEqRosenbrock.Rodas5P)
Number of subjects: 32
Observation records: Active Missing
conc: 251 47
pca: 232 66
Total: 483 113
Number of parameters: Constant Optimized
0 18
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: NoXChange
Log-likelihood value: -1068.9294
------------------------
Estimate
------------------------
pop_CL 0.13521
pop_V 8.0112
pop_tabs 0.56615
pop_lag 0.87614
pop_e0 96.395
pop_emax -1.0613
pop_c50 1.4884
pop_tover 14.053
pk_Ω₁,₁ 0.06929
pk_Ω₂,₂ 0.020318
pk_Ω₃,₃ 0.89963
pd_Ω₁,₁ 0.0028776
pd_Ω₂,₂ 0.00044803
pd_Ω₃,₃ 0.15375
pd_Ω₄,₄ 0.015014
σ_prop 0.088936
σ_add 0.41486
σ_fx 3.5814
------------------------
Tip
It is not recommended to decrease tolerances below 1e-14.
7.1 Changing the Algorithm
Usually, it should not be necessary to adjust the differential equation solver. If you change the solver, you should follow the guidelines in the SciML documentation that explains which solvers are the most efficient at the desired tolerance level.
For instance, if it is known that a differential equation is stiff, a stiff solver such as Rosenbrock23 at high tolerances or Rodas5P at low tolerances could be a possible alternative to the default auto-switching solver:
# Fitting with stiff solver Rodas5P at low tolerances (relative: 1e-8, absolute: 1e-12)
fit(
warfarin_pkpd_model,
pop,
init_params(warfarin_pkpd_model),
FOCE();
diffeq_options = (; alg = Rodas5P(), reltol = 1e-8, abstol = 1e-12),
)[ 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 3.125741e+06 5.911803e+06 * time: 3.0994415283203125e-5 1 5.174461e+05 8.708699e+05 * time: 10.08549189567566 2 3.865265e+05 6.344302e+05 * time: 18.191150903701782 3 1.804274e+05 2.829723e+05 * time: 26.057904958724976 4 9.706641e+04 1.550547e+05 * time: 34.09296798706055 5 4.769637e+04 6.778818e+04 * time: 42.12146806716919 6 2.902319e+04 3.499747e+04 * time: 50.21349096298218 7 1.823472e+04 1.705751e+04 * time: 57.94804787635803 8 1.258819e+04 9.569382e+03 * time: 65.69003200531006 9 9.389985e+03 8.615853e+03 * time: 73.36286687850952 10 7.314703e+03 7.636886e+03 * time: 80.92074704170227 11 5.916030e+03 6.624328e+03 * time: 88.43818092346191 12 4.930520e+03 5.558143e+03 * time: 95.92720603942871 13 4.125062e+03 4.315761e+03 * time: 104.45348691940308 14 3.549280e+03 3.051094e+03 * time: 112.81527400016785 15 3.283490e+03 2.157293e+03 * time: 120.09153389930725 16 3.204886e+03 1.659798e+03 * time: 127.57366108894348 17 3.194875e+03 1.480528e+03 * time: 135.08616495132446 18 3.193944e+03 1.437922e+03 * time: 142.59069108963013 19 3.193070e+03 1.411186e+03 * time: 149.92508792877197 20 3.190129e+03 1.355327e+03 * time: 157.59825587272644 21 3.183228e+03 1.276603e+03 * time: 165.28803896903992 22 3.164897e+03 1.151838e+03 * time: 173.19753289222717 23 3.119250e+03 9.712652e+02 * time: 182.10435605049133 24 3.006297e+03 7.204344e+02 * time: 190.0906000137329 25 2.738913e+03 4.050546e+02 * time: 197.79244709014893 26 2.123834e+03 2.318194e+02 * time: 205.34040999412537 27 1.789142e+03 2.290466e+02 * time: 213.61461091041565 28 1.396456e+03 1.683975e+02 * time: 230.96926093101501 29 1.333545e+03 1.336198e+02 * time: 240.36588501930237 30 1.297773e+03 2.452181e+02 * time: 247.83066201210022 31 1.266002e+03 1.523972e+02 * time: 255.29916787147522 32 1.255505e+03 1.734000e+02 * time: 262.897696018219 33 1.247788e+03 1.971637e+02 * time: 270.64445090293884 34 1.244490e+03 1.915744e+02 * time: 278.2859649658203 35 1.240568e+03 1.704280e+02 * time: 285.8861758708954 36 1.240502e+03 1.711813e+02 * time: 293.3388168811798 37 1.240491e+03 1.711683e+02 * time: 300.796128988266 38 1.239992e+03 1.687296e+02 * time: 308.377436876297 39 1.239200e+03 1.624744e+02 * time: 316.07769894599915 40 1.236975e+03 1.400443e+02 * time: 323.73160099983215 41 1.233209e+03 9.449763e+01 * time: 333.49364709854126 42 1.228687e+03 3.276923e+01 * time: 341.31556391716003 43 1.226467e+03 4.997214e+01 * time: 348.94008588790894 44 1.226104e+03 4.905105e+01 * time: 356.6202130317688 45 1.226088e+03 4.676122e+01 * time: 364.2222430706024 46 1.226087e+03 4.628674e+01 * time: 371.62658405303955 47 1.226085e+03 4.541753e+01 * time: 379.11325907707214 48 1.226080e+03 4.403288e+01 * time: 386.51107597351074 49 1.226064e+03 4.143315e+01 * time: 394.0518629550934 50 1.226026e+03 3.664876e+01 * time: 401.6194040775299 51 1.225931e+03 2.851383e+01 * time: 409.1823420524597 52 1.225714e+03 2.844460e+01 * time: 418.55847096443176 53 1.225304e+03 3.025732e+01 * time: 427.0618488788605 54 1.224793e+03 4.151448e+01 * time: 434.8933370113373 55 1.224489e+03 5.047755e+01 * time: 442.6357419490814 56 1.224420e+03 4.452753e+01 * time: 450.3093030452728 57 1.224413e+03 3.994380e+01 * time: 457.8525040149689 58 1.224412e+03 3.879465e+01 * time: 465.3744649887085 59 1.224408e+03 3.677922e+01 * time: 472.9092769622803 60 1.224400e+03 3.379126e+01 * time: 480.3521909713745 61 1.224379e+03 3.158821e+01 * time: 487.8674170970917 62 1.224324e+03 2.926306e+01 * time: 495.3346300125122 63 1.224181e+03 2.815869e+01 * time: 502.81563997268677 64 1.223815e+03 2.818300e+01 * time: 510.3729269504547 65 1.222911e+03 4.583041e+01 * time: 517.923749923706 66 1.220921e+03 8.675281e+01 * time: 525.4331960678101 67 1.217625e+03 1.136294e+02 * time: 532.9681289196014 68 1.214324e+03 9.050244e+01 * time: 540.432126045227 69 1.212536e+03 8.867180e+01 * time: 547.8575918674469 70 1.212272e+03 8.970655e+01 * time: 555.2410318851471 71 1.212263e+03 8.922293e+01 * time: 562.5489919185638 72 1.212259e+03 8.882951e+01 * time: 569.8617570400238 73 1.212242e+03 8.761908e+01 * time: 577.1773819923401 74 1.212205e+03 8.562573e+01 * time: 584.5113599300385 75 1.212102e+03 8.125881e+01 * time: 591.9010560512543 76 1.211850e+03 7.230599e+01 * time: 599.3711540699005 77 1.211238e+03 5.323492e+01 * time: 606.7648060321808 78 1.209946e+03 7.373618e+01 * time: 614.2040779590607 79 1.207891e+03 8.401772e+01 * time: 621.5980889797211 80 1.206059e+03 9.146801e+01 * time: 630.1043679714203 81 1.205389e+03 9.690592e+01 * time: 637.5221469402313 82 1.205302e+03 8.595286e+01 * time: 644.908117055893 83 1.205296e+03 8.263040e+01 * time: 652.1829519271851 84 1.205292e+03 8.096547e+01 * time: 659.593190908432 85 1.205276e+03 7.747632e+01 * time: 666.9177839756012 86 1.205240e+03 7.259211e+01 * time: 674.340658903122 87 1.205143e+03 6.664635e+01 * time: 681.7046210765839 88 1.204894e+03 6.374389e+01 * time: 689.219358921051 89 1.204267e+03 5.753484e+01 * time: 696.5747950077057 90 1.202818e+03 4.869938e+01 * time: 703.9712300300598 91 1.200117e+03 5.087313e+01 * time: 711.3690550327301 92 1.196896e+03 7.141162e+01 * time: 718.7447528839111 93 1.194612e+03 4.695316e+01 * time: 726.0801439285278 94 1.193881e+03 4.874366e+01 * time: 733.4677369594574 95 1.193834e+03 4.997038e+01 * time: 740.7661969661713 96 1.193832e+03 5.008442e+01 * time: 747.9423789978027 97 1.193830e+03 5.014549e+01 * time: 755.1373920440674 98 1.193825e+03 5.024884e+01 * time: 765.5117809772491 99 1.193813e+03 5.039938e+01 * time: 772.9342908859253 100 1.193779e+03 5.061632e+01 * time: 780.0720548629761 101 1.193694e+03 5.087641e+01 * time: 787.2071130275726 102 1.193473e+03 5.104895e+01 * time: 794.3223650455475 103 1.192927e+03 5.068500e+01 * time: 801.4699790477753 104 1.191682e+03 4.863684e+01 * time: 808.5548150539398 105 1.189353e+03 4.890365e+01 * time: 815.4794020652771 106 1.186424e+03 7.778082e+01 * time: 822.5067028999329 107 1.184533e+03 8.873607e+01 * time: 829.4352869987488 108 1.184042e+03 8.376145e+01 * time: 836.4146299362183 109 1.183986e+03 7.931813e+01 * time: 843.435338973999 110 1.183978e+03 7.793645e+01 * time: 850.5612668991089 111 1.183974e+03 7.786675e+01 * time: 858.3795380592346 112 1.183970e+03 7.833168e+01 * time: 865.5764429569244 113 1.183963e+03 7.928653e+01 * time: 872.7332870960236 114 1.183949e+03 8.083870e+01 * time: 879.8570690155029 115 1.183913e+03 8.331100e+01 * time: 887.1313829421997 116 1.183820e+03 8.698958e+01 * time: 894.4087228775024 117 1.183584e+03 9.190093e+01 * time: 903.6383080482483 118 1.183003e+03 9.670514e+01 * time: 910.9472780227661 119 1.181706e+03 9.636548e+01 * time: 918.2859179973602 120 1.179435e+03 8.059042e+01 * time: 925.5823578834534 121 1.177024e+03 4.567337e+01 * time: 932.9123959541321 122 1.175793e+03 1.863452e+01 * time: 940.3037650585175 123 1.175474e+03 1.967979e+01 * time: 947.5520920753479 124 1.175430e+03 1.958939e+01 * time: 954.8643209934235 125 1.175427e+03 1.993415e+01 * time: 962.1931190490723 126 1.175427e+03 1.962015e+01 * time: 969.4152438640594 127 1.175426e+03 1.965822e+01 * time: 976.5550479888916 128 1.175424e+03 1.984190e+01 * time: 983.8179590702057 129 1.175422e+03 2.000843e+01 * time: 991.1798169612885 130 1.175413e+03 2.034664e+01 * time: 998.3535859584808 131 1.175392e+03 2.083524e+01 * time: 1005.4594168663025 132 1.175336e+03 2.163395e+01 * time: 1012.8585200309753 133 1.175190e+03 2.286075e+01 * time: 1020.8742668628693 134 1.174803e+03 2.472708e+01 * time: 1028.3417830467224 135 1.173797e+03 4.124367e+01 * time: 1035.883635044098 136 1.171232e+03 7.123154e+01 * time: 1043.5812079906464 137 1.165167e+03 1.075848e+02 * time: 1051.492215871811 138 1.155096e+03 7.751300e+01 * time: 1059.4278349876404 139 1.149210e+03 6.815707e+01 * time: 1067.51943898201 140 1.146904e+03 6.218499e+01 * time: 1075.4210000038147 141 1.144688e+03 5.158716e+01 * time: 1083.3619079589844 142 1.143100e+03 2.555104e+01 * time: 1091.3381559848785 143 1.141816e+03 1.970245e+01 * time: 1099.2035639286041 144 1.141294e+03 2.048648e+01 * time: 1108.420124053955 145 1.141044e+03 2.076701e+01 * time: 1116.7233469486237 146 1.140979e+03 2.090339e+01 * time: 1124.4975650310516 147 1.140976e+03 2.094515e+01 * time: 1132.246516942978 148 1.140975e+03 2.099576e+01 * time: 1140.047329902649 149 1.140974e+03 2.099888e+01 * time: 1147.8510870933533 150 1.140974e+03 2.099931e+01 * time: 1155.7224719524384 151 1.140973e+03 2.099295e+01 * time: 1163.7300870418549 152 1.140972e+03 2.098305e+01 * time: 1171.5957670211792 153 1.140970e+03 2.096756e+01 * time: 1179.4612119197845 154 1.140963e+03 2.094619e+01 * time: 1187.4338519573212 155 1.140947e+03 2.092087e+01 * time: 1195.2383868694305 156 1.140906e+03 2.090400e+01 * time: 1202.9390060901642 157 1.140797e+03 2.093964e+01 * time: 1210.8675019741058 158 1.140511e+03 2.116231e+01 * time: 1219.024020910263 159 1.139732e+03 2.193009e+01 * time: 1227.0754880905151 160 1.137531e+03 3.209812e+01 * time: 1235.1209509372711 161 1.132353e+03 5.451900e+01 * time: 1243.2788889408112 162 1.130098e+03 6.354058e+01 * time: 1251.7811479568481 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 163 1.128599e+03 6.862520e+01 * time: 1260.6219720840454 164 1.127623e+03 8.385757e+01 * time: 1269.4574558734894 165 1.122228e+03 1.204824e+02 * time: 1277.8529529571533 166 1.118774e+03 1.225695e+02 * time: 1286.062891960144 167 1.115679e+03 4.611268e+01 * time: 1294.1571969985962 168 1.113979e+03 3.434831e+01 * time: 1302.3025388717651 169 1.112521e+03 3.523199e+01 * time: 1310.587739944458 170 1.111267e+03 1.228403e+01 * time: 1318.7387800216675 171 1.111189e+03 1.078353e+01 * time: 1326.9373590946198 172 1.111187e+03 1.048242e+01 * time: 1335.3299369812012 173 1.111187e+03 1.070654e+01 * time: 1345.7729210853577 174 1.111187e+03 1.057812e+01 * time: 1353.7987558841705 175 1.111187e+03 1.055136e+01 * time: 1361.8972699642181 176 1.111186e+03 1.049712e+01 * time: 1370.0765290260315 177 1.111185e+03 1.042508e+01 * time: 1379.687306880951 178 1.111181e+03 1.030463e+01 * time: 1387.9390530586243 179 1.111173e+03 1.018028e+01 * time: 1396.2664229869843 180 1.111150e+03 1.017084e+01 * time: 1404.479413986206 181 1.111091e+03 1.547976e+01 * time: 1412.65838098526 182 1.110938e+03 2.528645e+01 * time: 1420.8021450042725 183 1.110546e+03 4.010028e+01 * time: 1428.996183872223 184 1.109576e+03 6.035708e+01 * time: 1437.0932619571686 185 1.107416e+03 8.168777e+01 * time: 1445.2246930599213 186 1.103899e+03 9.586365e+01 * time: 1453.3344569206238 187 1.100518e+03 1.082019e+02 * time: 1461.4210278987885 188 1.096703e+03 1.051881e+02 * time: 1469.6246409416199 189 1.092614e+03 3.371950e+01 * time: 1478.0849659442902 190 1.092108e+03 2.854225e+01 * time: 1486.5472779273987 191 1.091924e+03 8.267155e+00 * time: 1498.0392289161682 192 1.091910e+03 4.388021e+00 * time: 1506.301176071167 193 1.091908e+03 4.381246e+00 * time: 1514.5959470272064 194 1.091906e+03 4.351855e+00 * time: 1523.017145872116 195 1.091905e+03 4.347981e+00 * time: 1531.4109258651733 196 1.091904e+03 4.344942e+00 * time: 1539.6831409931183 197 1.091904e+03 4.343567e+00 * time: 1547.753278017044 198 1.091904e+03 4.341811e+00 * time: 1555.9967720508575 199 1.091904e+03 4.338160e+00 * time: 1564.1387119293213 200 1.091904e+03 4.331286e+00 * time: 1572.2593879699707 201 1.091904e+03 4.319913e+00 * time: 1580.3326919078827 202 1.091903e+03 4.299872e+00 * time: 1588.4866309165955 203 1.091901e+03 4.267805e+00 * time: 1597.5446979999542 204 1.091897e+03 4.211166e+00 * time: 1605.81454205513 205 1.091886e+03 4.118011e+00 * time: 1616.958899974823 206 1.091856e+03 4.884541e+00 * time: 1625.9775259494781 207 1.091780e+03 7.435767e+00 * time: 1634.2627580165863 208 1.091587e+03 1.171416e+01 * time: 1642.5469369888306 209 1.091126e+03 1.720748e+01 * time: 1650.9167029857635 210 1.090066e+03 2.180540e+01 * time: 1659.2485930919647 211 1.088566e+03 2.176055e+01 * time: 1667.6395409107208 212 1.087529e+03 2.216379e+01 * time: 1677.723247051239 213 1.087131e+03 6.871069e+00 * time: 1688.5370450019836 214 1.087053e+03 5.487618e+00 * time: 1703.0826480388641 215 1.087044e+03 5.346767e+00 * time: 1715.1214179992676 216 1.087043e+03 5.364351e+00 * time: 1727.4196119308472 217 1.087043e+03 5.380873e+00 * time: 1737.2756400108337 218 1.087043e+03 5.385017e+00 * time: 1748.9424018859863 219 1.087043e+03 5.387076e+00 * time: 1760.6793599128723 220 1.087043e+03 5.393037e+00 * time: 1770.4370019435883 221 1.087043e+03 5.401737e+00 * time: 1780.3918149471283 222 1.087042e+03 5.417220e+00 * time: 1792.3762118816376 223 1.087041e+03 5.443370e+00 * time: 1801.5814599990845 224 1.087039e+03 5.489836e+00 * time: 1810.3141560554504 225 1.087032e+03 5.575602e+00 * time: 1822.68763589859 226 1.087015e+03 5.744360e+00 * time: 1833.023402929306 227 1.086967e+03 8.458307e+00 * time: 1841.150279045105 228 1.086833e+03 1.434823e+01 * time: 1849.2932028770447 229 1.086295e+03 2.779692e+01 * time: 1857.9522750377655 ┌ Warning: First function call produced NaNs. Exiting. Double check that none of the initial conditions, parameters, or timespan values are NaN. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/initdt.jl:253 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: First function call produced NaNs. Exiting. Double check that none of the initial conditions, parameters, or timespan values are NaN. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/initdt.jl:253 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: First function call produced NaNs. Exiting. Double check that none of the initial conditions, parameters, or timespan values are NaN. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/initdt.jl:253 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: First function call produced NaNs. Exiting. Double check that none of the initial conditions, parameters, or timespan values are NaN. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/initdt.jl:253 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: First function call produced NaNs. Exiting. Double check that none of the initial conditions, parameters, or timespan values are NaN. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/initdt.jl:253 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: First function call produced NaNs. Exiting. Double check that none of the initial conditions, parameters, or timespan values are NaN. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/initdt.jl:253 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: First function call produced NaNs. Exiting. Double check that none of the initial conditions, parameters, or timespan values are NaN. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/initdt.jl:253 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: First function call produced NaNs. Exiting. Double check that none of the initial conditions, parameters, or timespan values are NaN. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/initdt.jl:253 ┌ Warning: Automatic dt set the starting dt as NaN, causing instability. Exiting. └ @ OrdinaryDiffEqCore ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/OrdinaryDiffEqCore/LG1u6/src/solve.jl:654 ┌ Warning: NaN dt detected. Likely a NaN value in the state, parameters, or derivative value caused this outcome. └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:583 ┌ Warning: At t=0.4500886809099674, dt was forced below floating point epsilon 5.551115123125783e-17, and step error estimate = 4.8708450078891075e20. Aborting. There is either an error in your model specification or the true solution is unstable (or the true solution can not be represented in the precision of ForwardDiff.Dual{ForwardDiff.Tag{Pumas.Tag, Float64}, Float64, 7}). └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:623 ┌ Warning: At t=0.4500886809099674, dt was forced below floating point epsilon 5.551115123125783e-17, and step error estimate = 4.8708450078891075e20. Aborting. There is either an error in your model specification or the true solution is unstable (or the true solution can not be represented in the precision of ForwardDiff.Dual{ForwardDiff.Tag{Pumas.Tag, Float64}, Float64, 7}). └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:623 230 1.086121e+03 6.179899e+01 * time: 1868.2224929332733 231 1.085699e+03 4.269436e+01 * time: 1876.4216940402985 232 1.085153e+03 5.101086e+01 * time: 1884.5097138881683 233 1.084625e+03 5.803468e+01 * time: 1892.5904490947723 234 1.084371e+03 6.244198e+01 * time: 1901.1619610786438 235 1.083958e+03 6.785742e+01 * time: 1909.2674760818481 236 1.083163e+03 6.963805e+01 * time: 1917.3103029727936 237 1.081667e+03 6.332470e+01 * time: 1925.3473188877106 238 1.080941e+03 5.916175e+01 * time: 1933.1295099258423 239 1.079241e+03 5.431208e+01 * time: 1940.9890339374542 240 1.076096e+03 3.465388e+01 * time: 1951.40802693367 241 1.072649e+03 1.670281e+01 * time: 1959.6910519599915 242 1.069992e+03 1.935661e+01 * time: 1967.6729390621185 243 1.069558e+03 3.678375e+00 * time: 1975.3966999053955 244 1.069496e+03 2.270102e+00 * time: 1986.0689570903778 245 1.069488e+03 2.221349e+00 * time: 1994.4514379501343 246 1.069488e+03 2.237998e+00 * time: 2002.0855770111084 247 1.069488e+03 2.236062e+00 * time: 2009.7287380695343 248 1.069488e+03 2.235507e+00 * time: 2019.4155640602112 249 1.069488e+03 2.234690e+00 * time: 2027.027909040451 250 1.069488e+03 2.233625e+00 * time: 2034.334240913391 251 1.069488e+03 2.231719e+00 * time: 2041.6176300048828 252 1.069488e+03 2.228573e+00 * time: 2048.89395904541 253 1.069487e+03 2.223013e+00 * time: 2056.1925899982452 254 1.069487e+03 2.212970e+00 * time: 2063.4897220134735 255 1.069485e+03 2.193875e+00 * time: 2070.8365230560303 256 1.069481e+03 2.558103e+00 * time: 2078.186975955963 257 1.069470e+03 4.099866e+00 * time: 2085.768434047699 258 1.069441e+03 6.445399e+00 * time: 2093.7621150016785 259 1.069374e+03 9.551161e+00 * time: 2100.850672006607 260 1.069241e+03 1.204286e+01 * time: 2107.8825080394745 261 1.069065e+03 1.050115e+01 * time: 2115.268637895584 262 1.068958e+03 5.184001e+00 * time: 2122.4012138843536 263 1.068932e+03 1.288840e+00 * time: 2129.3792209625244 264 1.068929e+03 8.682448e-01 * time: 2136.677870988846 265 1.068929e+03 8.688495e-01 * time: 2143.8077199459076 266 1.068929e+03 8.688500e-01 * time: 2151.2941660881042 267 1.068929e+03 8.689152e-01 * time: 2158.743479013443 268 1.068929e+03 8.689157e-01 * time: 2166.0727310180664 269 1.068929e+03 8.690133e-01 * time: 2173.9399960041046 270 1.068929e+03 8.690133e-01 * time: 2183.5664670467377 271 1.068929e+03 8.693257e-01 * time: 2193.099380970001 272 1.068929e+03 8.693100e-01 * time: 2201.1834840774536 273 1.068919e+03 9.277276e-01 * time: 2208.997227907181 274 1.068907e+03 1.626480e+00 * time: 2216.510533094406 275 1.068835e+03 4.106347e+00 * time: 2223.9146480560303 276 1.068726e+03 6.157731e+00 * time: 2231.6306250095367 277 1.068548e+03 7.201969e+00 * time: 2239.875617980957 278 1.068415e+03 5.646200e+00 * time: 2247.4018890857697 279 1.068386e+03 3.803550e+00 * time: 2254.8616259098053 280 1.068384e+03 3.184721e+00 * time: 2262.28914809227 281 1.068384e+03 2.938192e+00 * time: 2269.742798089981 282 1.068384e+03 2.942962e+00 * time: 2277.0469329357147 283 1.068384e+03 2.943184e+00 * time: 2284.4481179714203 284 1.068384e+03 2.943716e+00 * time: 2291.9473390579224 285 1.068384e+03 2.944531e+00 * time: 2299.438082933426 286 1.068384e+03 2.944646e+00 * time: 2307.0241260528564 287 1.068384e+03 2.944770e+00 * time: 2315.892790079117 288 1.068384e+03 2.944962e+00 * time: 2323.6786000728607 289 1.068384e+03 2.944984e+00 * time: 2332.1880509853363 290 1.068384e+03 2.945026e+00 * time: 2340.5053429603577 291 1.068384e+03 2.945033e+00 * time: 2348.7700939178467 292 1.068384e+03 2.945039e+00 * time: 2357.4254760742188 293 1.068384e+03 2.945044e+00 * time: 2366.2432069778442 294 1.068384e+03 2.945044e+00 * time: 2374.7894558906555 295 1.068384e+03 2.945044e+00 * time: 2383.451961994171 296 1.068384e+03 2.945044e+00 * time: 2391.987457036972 297 1.068384e+03 2.945044e+00 * time: 2400.915272951126 298 1.068384e+03 2.945044e+00 * time: 2410.190519094467 299 1.068384e+03 2.945044e+00 * time: 2419.0747430324554 300 1.068384e+03 2.945044e+00 * time: 2428.338721036911 301 1.068384e+03 2.945044e+00 * time: 2438.0237119197845 302 1.068384e+03 2.945044e+00 * time: 2446.6547570228577 303 1.068384e+03 2.945044e+00 * time: 2455.5501260757446 304 1.068384e+03 2.945044e+00 * time: 2463.875478029251 305 1.068384e+03 3.074478e+00 * time: 2470.6915628910065 306 1.068384e+03 3.057700e+00 * time: 2477.4511439800262 307 1.068384e+03 3.030813e+00 * time: 2484.2298719882965 308 1.068384e+03 3.005745e+00 * time: 2491.0156400203705 309 1.068384e+03 2.974044e+00 * time: 2497.9032928943634 310 1.068384e+03 2.945854e+00 * time: 2504.6720700263977 311 1.068383e+03 2.905870e+00 * time: 2511.4514129161835 312 1.068382e+03 2.819225e+00 * time: 2518.258208990097 313 1.068381e+03 2.581914e+00 * time: 2525.096179008484 314 1.068376e+03 1.940126e+00 * time: 2531.968715906143 315 1.068367e+03 1.682057e+00 * time: 2538.83180809021 316 1.068350e+03 2.787006e+00 * time: 2545.6913990974426 317 1.068331e+03 6.870737e+00 * time: 2552.5707199573517 318 1.068322e+03 9.136899e+00 * time: 2559.435222864151 319 1.068319e+03 9.093888e+00 * time: 2566.283092021942 320 1.068318e+03 8.702840e+00 * time: 2573.127382040024 321 1.068317e+03 8.067498e+00 * time: 2579.9573950767517 322 1.068313e+03 7.024566e+00 * time: 2586.776678085327 323 1.068305e+03 5.566463e+00 * time: 2593.6040070056915 324 1.068294e+03 4.311204e+00 * time: 2600.419604063034 325 1.068285e+03 4.366941e+00 * time: 2607.4299490451813 326 1.068282e+03 5.297662e+00 * time: 2614.3870170116425 327 1.068282e+03 5.794863e+00 * time: 2621.324644088745 328 1.068282e+03 5.858759e+00 * time: 2628.2105309963226 329 1.068282e+03 5.858759e+00 * time: 2635.8782579898834 330 1.068282e+03 5.890965e+00 * time: 2642.7700119018555 331 1.068282e+03 5.889642e+00 * time: 2649.921215057373 332 1.068282e+03 5.755927e+00 * time: 2657.00225687027 333 1.068282e+03 5.654191e+00 * time: 2664.053260087967 334 1.068281e+03 5.457610e+00 * time: 2671.0283830165863 335 1.068280e+03 5.325147e+00 * time: 2678.223057985306 336 1.068280e+03 5.359845e+00 * time: 2685.3982009887695 337 1.068279e+03 5.558458e+00 * time: 2692.336699962616 338 1.068278e+03 5.765445e+00 * time: 2699.8987300395966 339 1.068278e+03 5.998734e+00 * time: 2707.4405229091644 340 1.068276e+03 6.340911e+00 * time: 2714.4940259456635 341 1.068272e+03 6.826786e+00 * time: 2721.5697689056396 342 1.068262e+03 7.444708e+00 * time: 2728.8970019817352 343 1.068237e+03 7.966869e+00 * time: 2736.2187399864197 344 1.068187e+03 7.662772e+00 * time: 2743.5834410190582 345 1.068105e+03 5.453572e+00 * time: 2750.612259864807 346 1.068030e+03 2.116637e+00 * time: 2757.680116891861 347 1.068004e+03 2.672269e-01 * time: 2764.9063789844513 348 1.068001e+03 4.076140e-01 * time: 2771.929300069809 349 1.067999e+03 5.992600e-01 * time: 2778.8596448898315 350 1.067997e+03 5.810130e-01 * time: 2785.7961809635162 351 1.067994e+03 3.112045e-01 * time: 2795.1923899650574 352 1.067992e+03 7.689723e-02 * time: 2802.5668959617615 353 1.067992e+03 9.091969e-02 * time: 2810.21625995636 354 1.067992e+03 9.785125e-02 * time: 2817.8544459342957 355 1.067992e+03 1.024830e-01 * time: 2825.4843440055847 356 1.067992e+03 1.051058e-01 * time: 2833.2433738708496 357 1.067992e+03 1.057116e-01 * time: 2840.4835329055786 358 1.067992e+03 1.057207e-01 * time: 2848.0463030338287 359 1.067992e+03 1.057207e-01 * time: 2856.4472219944 360 1.067992e+03 1.057044e-01 * time: 2864.2110888957977 361 1.067992e+03 1.056554e-01 * time: 2871.9153439998627 362 1.067992e+03 1.056554e-01 * time: 2880.530816078186 363 1.067992e+03 1.052666e-01 * time: 2889.360293865204 364 1.067992e+03 1.052895e-01 * time: 2898.2155590057373 365 1.067992e+03 1.052895e-01 * time: 2907.4585468769073 366 1.067992e+03 1.052895e-01 * time: 2916.890515089035
FittedPumasModel
Dynamical system type: Nonlinear ODE
Solver(s): OrdinaryDiffEqRosenbrock.Rodas5P
Number of subjects: 32
Observation records: Active Missing
conc: 251 47
pca: 232 66
Total: 483 113
Number of parameters: Constant Optimized
0 18
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: NoObjectiveChange
Log-likelihood value: -1067.9916
-----------------------
Estimate
-----------------------
pop_CL 0.13521
pop_V 8.0132
pop_tabs 0.57101
pop_lag 0.87564
pop_e0 96.399
pop_emax -1.0615
pop_c50 1.4912
pop_tover 14.05
pk_Ω₁,₁ 0.068018
pk_Ω₂,₂ 0.02105
pk_Ω₃,₃ 0.86339
pd_Ω₁,₁ 0.0029815
pd_Ω₂,₂ 2.3946e-7
pd_Ω₃,₃ 0.14556
pd_Ω₄,₄ 0.015351
σ_prop 0.088484
σ_add 0.41684
σ_fx 3.5802
-----------------------On the other hand, if it is known that a differential equation is non-stiff (this might be difficult to guarantee for all admissible parameter values), a non-stiff solver such as Tsit5 at high tolerances or Vern7 at low tolerances could be an alternative to the default solver:
# Fitting with the non-stiff solver Vern7 at low tolerances (relative: 1e-8, absolute: 1e-12)
fit(
warfarin_pkpd_model,
pop,
init_params(warfarin_pkpd_model),
FOCE();
diffeq_options = (; alg = Vern7(), reltol = 1e-8, abstol = 1e-12),
)[ 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 3.125741e+06 5.911802e+06 * time: 2.8848648071289062e-5 1 5.174461e+05 8.708698e+05 * time: 2.2333338260650635 2 3.865265e+05 6.344302e+05 * time: 3.731431007385254 3 1.804274e+05 2.829723e+05 * time: 5.179621934890747 4 9.706640e+04 1.550547e+05 * time: 6.615917921066284 5 4.769637e+04 6.778818e+04 * time: 8.036172866821289 6 2.902319e+04 3.499747e+04 * time: 9.408237934112549 7 1.823472e+04 1.705751e+04 * time: 10.888195991516113 8 1.258819e+04 9.569381e+03 * time: 12.373081922531128 9 9.389984e+03 8.615851e+03 * time: 13.837177991867065 10 7.314702e+03 7.636883e+03 * time: 15.313628911972046 11 5.916029e+03 6.624325e+03 * time: 16.786343812942505 12 4.930519e+03 5.558140e+03 * time: 18.250086784362793 13 4.125060e+03 4.315759e+03 * time: 19.702292919158936 14 3.549280e+03 3.051093e+03 * time: 21.151182889938354 15 3.283489e+03 2.157292e+03 * time: 22.602295875549316 16 3.204886e+03 1.659798e+03 * time: 24.07211184501648 17 3.194875e+03 1.480528e+03 * time: 25.452760934829712 18 3.193944e+03 1.437921e+03 * time: 26.891008853912354 19 3.193070e+03 1.411186e+03 * time: 28.31724786758423 20 3.190129e+03 1.355327e+03 * time: 29.68156599998474 21 3.183228e+03 1.276603e+03 * time: 31.041510820388794 22 3.164897e+03 1.151838e+03 * time: 32.43494486808777 23 3.119250e+03 9.712651e+02 * time: 33.801060914993286 24 3.006297e+03 7.204342e+02 * time: 35.13729381561279 25 2.738913e+03 4.050545e+02 * time: 36.43084478378296 26 2.123834e+03 2.318194e+02 * time: 37.63926887512207 27 1.789139e+03 2.290465e+02 * time: 38.911783933639526 ┌ Warning: Interrupted. Larger maxiters is needed. If you are using an integrator for non-stiff ODEs or an automatic switching algorithm (the default), you may want to consider using a method for stiff equations. See the solver pages for more details (e.g. https://docs.sciml.ai/DiffEqDocs/stable/solvers/ode_solve/#Stiff-Problems). └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:589 ┌ Warning: Interrupted. Larger maxiters is needed. If you are using an integrator for non-stiff ODEs or an automatic switching algorithm (the default), you may want to consider using a method for stiff equations. See the solver pages for more details (e.g. https://docs.sciml.ai/DiffEqDocs/stable/solvers/ode_solve/#Stiff-Problems). └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:589 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117 ┌ Warning: Interrupted. Larger maxiters is needed. If you are using an integrator for non-stiff ODEs or an automatic switching algorithm (the default), you may want to consider using a method for stiff equations. See the solver pages for more details (e.g. https://docs.sciml.ai/DiffEqDocs/stable/solvers/ode_solve/#Stiff-Problems). └ @ SciMLBase ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/SciMLBase/JKXkh/src/integrator_interface.jl:589 28 1.576318e+03 2.147821e+02 * time: 48.31771779060364 29 1.380652e+03 1.621863e+02 * time: 54.60399580001831 30 1.298107e+03 1.130479e+02 * time: 55.961283922195435 31 1.271740e+03 2.292541e+02 * time: 57.23762583732605 32 1.255835e+03 1.721669e+02 * time: 58.50220584869385 33 1.251089e+03 1.813354e+02 * time: 59.77135181427002 34 1.243725e+03 1.920537e+02 * time: 61.05448389053345 35 1.241416e+03 1.835909e+02 * time: 62.328558921813965 36 1.240510e+03 1.715569e+02 * time: 63.60202097892761 37 1.240496e+03 1.710861e+02 * time: 64.8607850074768 38 1.240467e+03 1.703009e+02 * time: 66.13109183311462 39 1.240383e+03 1.685321e+02 * time: 67.3918068408966 40 1.240176e+03 1.649610e+02 * time: 68.6569299697876 41 1.239647e+03 1.571549e+02 * time: 69.92803978919983 42 1.238394e+03 1.407042e+02 * time: 71.19908690452576 43 1.235743e+03 1.084408e+02 * time: 72.44106483459473 44 1.231513e+03 5.668753e+01 * time: 73.66167998313904 45 1.227771e+03 4.125472e+01 * time: 74.88389182090759 46 1.226339e+03 5.546989e+01 * time: 76.11234188079834 47 1.226095e+03 4.927853e+01 * time: 77.3901309967041 48 1.226086e+03 4.642266e+01 * time: 78.65242981910706 49 1.226086e+03 4.611503e+01 * time: 79.87912797927856 50 1.226081e+03 4.478422e+01 * time: 81.13784980773926 51 1.226071e+03 4.294093e+01 * time: 82.41126084327698 52 1.226042e+03 3.911828e+01 * time: 83.66753101348877 53 1.225973e+03 3.210937e+01 * time: 84.90588688850403 54 1.225806e+03 2.839654e+01 * time: 86.15470290184021 55 1.225466e+03 2.859127e+01 * time: 87.4140989780426 56 1.224959e+03 3.226886e+01 * time: 88.6843159198761 57 1.224556e+03 4.925686e+01 * time: 89.9598228931427 58 1.224428e+03 4.689186e+01 * time: 91.23084998130798 59 1.224411e+03 4.101617e+01 * time: 92.47948694229126 60 1.224410e+03 3.916788e+01 * time: 93.74462485313416 61 1.224408e+03 3.787270e+01 * time: 94.98624992370605 62 1.224402e+03 3.519636e+01 * time: 96.24421691894531 63 1.224389e+03 3.233949e+01 * time: 97.49690794944763 64 1.224351e+03 3.054354e+01 * time: 98.76662087440491 65 1.224256e+03 2.816053e+01 * time: 100.01584196090698 66 1.224007e+03 2.821380e+01 * time: 101.27148079872131 67 1.223382e+03 3.119472e+01 * time: 102.53297185897827 68 1.221919e+03 6.976985e+01 * time: 103.81634092330933 69 1.219110e+03 1.073633e+02 * time: 105.01742792129517 70 1.215576e+03 1.072842e+02 * time: 106.19050884246826 71 1.213034e+03 8.400960e+01 * time: 107.36192297935486 72 1.212294e+03 8.951044e+01 * time: 108.56186079978943 73 1.212259e+03 8.933198e+01 * time: 109.74496579170227 74 1.212255e+03 8.895637e+01 * time: 110.9097728729248 75 1.212241e+03 8.777217e+01 * time: 112.10962581634521 76 1.212214e+03 8.600288e+01 * time: 113.32086777687073 77 1.212135e+03 8.213714e+01 * time: 114.5501389503479 78 1.211943e+03 7.451404e+01 * time: 115.76547384262085 79 1.211464e+03 5.844663e+01 * time: 116.94675087928772 80 1.210406e+03 6.363959e+01 * time: 118.1483428478241 81 1.208527e+03 7.916604e+01 * time: 119.35058283805847 82 1.206464e+03 8.366506e+01 * time: 120.55367088317871 83 1.205476e+03 1.005964e+02 * time: 121.73480582237244 84 1.205305e+03 8.882827e+01 * time: 122.92152285575867 85 1.205293e+03 8.309057e+01 * time: 124.08534288406372 86 1.205290e+03 8.172481e+01 * time: 125.25852084159851 87 1.205275e+03 7.814716e+01 * time: 126.47384786605835 88 1.205243e+03 7.364643e+01 * time: 127.66153693199158 89 1.205155e+03 6.632764e+01 * time: 128.84665989875793 90 1.204934e+03 6.325952e+01 * time: 130.03138279914856 91 1.204369e+03 5.693461e+01 * time: 131.2556118965149 92 1.203050e+03 5.234111e+01 * time: 132.44794082641602 93 1.200504e+03 5.128249e+01 * time: 133.64694786071777 94 1.197282e+03 7.013671e+01 * time: 134.83573079109192 95 1.194826e+03 5.064375e+01 * time: 136.01982378959656 96 1.193905e+03 4.851059e+01 * time: 137.2034158706665 97 1.193833e+03 4.996663e+01 * time: 138.4188048839569 98 1.193831e+03 5.008119e+01 * time: 139.59921789169312 99 1.193829e+03 5.013190e+01 * time: 140.7463297843933 100 1.193824e+03 5.022328e+01 * time: 141.91279578208923 101 1.193812e+03 5.034993e+01 * time: 143.0969319343567 102 1.193779e+03 5.053003e+01 * time: 144.26484394073486 103 1.193695e+03 5.072909e+01 * time: 145.46637678146362 104 1.193477e+03 5.080775e+01 * time: 146.66769790649414 105 1.192938e+03 5.030899e+01 * time: 147.87584781646729 106 1.191706e+03 4.810245e+01 * time: 149.0541808605194 107 1.189391e+03 4.971073e+01 * time: 150.26439690589905 108 1.186449e+03 8.056474e+01 * time: 151.4451789855957 109 1.184524e+03 9.072066e+01 * time: 152.6276638507843 110 1.184030e+03 8.460397e+01 * time: 153.79987692832947 111 1.183978e+03 7.969454e+01 * time: 154.9771909713745 112 1.183972e+03 7.815446e+01 * time: 156.18289399147034 113 1.183968e+03 7.789814e+01 * time: 157.41558194160461 114 1.183964e+03 7.808136e+01 * time: 158.8024377822876 115 1.183956e+03 7.875580e+01 * time: 160.03819298744202 116 1.183938e+03 8.003236e+01 * time: 161.26422500610352 117 1.183894e+03 8.217667e+01 * time: 162.4765019416809 118 1.183783e+03 8.541976e+01 * time: 163.6682767868042 119 1.183501e+03 8.965327e+01 * time: 164.83293294906616 120 1.182810e+03 9.316549e+01 * time: 165.99618482589722 121 1.181298e+03 9.013778e+01 * time: 167.14797496795654 122 1.178841e+03 7.025076e+01 * time: 168.31451082229614 123 1.176625e+03 3.473146e+01 * time: 169.505872964859 124 1.175679e+03 1.951082e+01 * time: 170.7292709350586 125 1.175454e+03 2.004503e+01 * time: 171.97003698349 126 1.175428e+03 1.978517e+01 * time: 173.19885277748108 127 1.175426e+03 1.976376e+01 * time: 174.4319019317627 128 1.175426e+03 1.969368e+01 * time: 175.66282296180725 129 1.175425e+03 1.972651e+01 * time: 176.89209294319153 130 1.175424e+03 1.966406e+01 * time: 178.12023282051086 131 1.175420e+03 1.973019e+01 * time: 179.65336179733276 132 1.175410e+03 1.971215e+01 * time: 182.48550081253052 133 1.175383e+03 1.983589e+01 * time: 183.9717698097229 134 1.175313e+03 1.991370e+01 * time: 185.20777583122253 135 1.175129e+03 2.013321e+01 * time: 186.37510991096497 136 1.174641e+03 2.030513e+01 * time: 187.64069199562073 137 1.173354e+03 2.816243e+01 * time: 189.62395787239075 138 1.170026e+03 4.486231e+01 * time: 191.50137281417847 139 1.162269e+03 5.857731e+01 * time: 193.12811589241028 140 1.152132e+03 5.506464e+01 * time: 194.74579191207886 141 1.149729e+03 1.693931e+02 * time: 196.44204998016357 142 1.146433e+03 7.151290e+01 * time: 198.10267281532288 143 1.144677e+03 2.342636e+01 * time: 199.42189693450928 144 1.142987e+03 4.134765e+01 * time: 200.7526228427887 145 1.142072e+03 4.624246e+01 * time: 202.09439992904663 146 1.141319e+03 2.544630e+01 * time: 203.43123483657837 147 1.141015e+03 2.115539e+01 * time: 204.8118658065796 148 1.140986e+03 2.102616e+01 * time: 206.18677377700806 149 1.140977e+03 2.108739e+01 * time: 207.53756189346313 150 1.140976e+03 2.105936e+01 * time: 208.8918149471283 151 1.140975e+03 2.101748e+01 * time: 210.24441194534302 152 1.140974e+03 2.098036e+01 * time: 211.5923728942871 153 1.140974e+03 2.096305e+01 * time: 212.93098187446594 154 1.140973e+03 2.094761e+01 * time: 214.28020977973938 155 1.140972e+03 2.092782e+01 * time: 215.64678597450256 156 1.140969e+03 2.089927e+01 * time: 217.04898500442505 157 1.140961e+03 2.085855e+01 * time: 218.41135787963867 158 1.140941e+03 2.080395e+01 * time: 219.76034879684448 159 1.140887e+03 2.074229e+01 * time: 221.11157083511353 160 1.140748e+03 2.070368e+01 * time: 222.46634483337402 161 1.140385e+03 2.968484e+01 * time: 223.82905197143555 162 1.139445e+03 4.800342e+01 * time: 225.20662593841553 163 1.137434e+03 7.357687e+01 * time: 226.59376192092896 164 1.134445e+03 9.639494e+01 * time: 227.9870889186859 165 1.131010e+03 1.115418e+02 * time: 229.51169991493225 166 1.127363e+03 1.263037e+02 * time: 231.23877882957458 167 1.125050e+03 1.311065e+02 * time: 232.7852268218994 168 1.115334e+03 5.808827e+01 * time: 234.31652688980103 169 1.112799e+03 3.748309e+01 * time: 236.5046579837799 170 1.111761e+03 2.320620e+01 * time: 238.99316096305847 171 1.111313e+03 1.021803e+01 * time: 240.78599095344543 172 1.111234e+03 1.165118e+01 * time: 242.60841393470764 173 1.111194e+03 1.100397e+01 * time: 244.35175800323486 174 1.111188e+03 1.096404e+01 * time: 245.869295835495 175 1.111187e+03 1.052785e+01 * time: 247.38061499595642 176 1.111187e+03 1.058517e+01 * time: 248.88084387779236 177 1.111187e+03 1.071182e+01 * time: 250.39083981513977 178 1.111186e+03 1.088955e+01 * time: 251.9052929878235 179 1.111184e+03 1.118993e+01 * time: 253.44240880012512 180 1.111179e+03 1.166631e+01 * time: 255.39467692375183 181 1.111167e+03 1.244302e+01 * time: 257.94237089157104 182 1.111134e+03 1.369715e+01 * time: 259.4918758869171 183 1.111047e+03 1.572515e+01 * time: 261.0670328140259 184 1.110820e+03 1.903860e+01 * time: 262.6113049983978 185 1.110234e+03 3.096405e+01 * time: 264.16577196121216 186 1.108788e+03 4.898071e+01 * time: 265.74148893356323 187 1.105699e+03 7.038625e+01 * time: 267.38957500457764 188 1.100930e+03 7.685637e+01 * time: 269.11721992492676 189 1.095734e+03 4.447770e+01 * time: 270.91539001464844 190 1.092518e+03 7.975607e+00 * time: 272.7608118057251 191 1.092216e+03 1.029575e+01 * time: 274.5689058303833 192 1.092108e+03 4.169550e+00 * time: 276.3573088645935 193 1.091973e+03 9.176673e+00 * time: 278.13406896591187 194 1.091920e+03 7.552603e+00 * time: 279.8633267879486 195 1.091907e+03 4.322688e+00 * time: 281.57999181747437 196 1.091904e+03 4.363464e+00 * time: 283.28416180610657 197 1.091904e+03 4.353570e+00 * time: 284.972797870636 198 1.091904e+03 4.343995e+00 * time: 286.6579518318176 199 1.091904e+03 4.339119e+00 * time: 288.34594798088074 200 1.091904e+03 4.332190e+00 * time: 290.02867579460144 201 1.091904e+03 4.318726e+00 * time: 291.73756289482117 202 1.091904e+03 4.297989e+00 * time: 293.4409899711609 203 1.091903e+03 4.263272e+00 * time: 295.12430787086487 204 1.091901e+03 4.206755e+00 * time: 296.81512999534607 205 1.091895e+03 4.112804e+00 * time: 298.540452003479 206 1.091882e+03 3.955547e+00 * time: 300.301864862442 207 1.091845e+03 4.211209e+00 * time: 301.9976170063019 208 1.091753e+03 6.664228e+00 * time: 303.75610280036926 209 1.091523e+03 1.044273e+01 * time: 305.4288549423218 210 1.090981e+03 1.598948e+01 * time: 307.0988988876343 211 1.089797e+03 1.981501e+01 * time: 308.81782579421997 212 1.088403e+03 1.645660e+01 * time: 310.41896295547485 213 1.087420e+03 9.435957e+00 * time: 312.03738498687744 214 1.087072e+03 5.707506e+00 * time: 313.711678981781 215 1.087045e+03 5.468838e+00 * time: 315.35236382484436 216 1.087043e+03 5.373236e+00 * time: 316.98988580703735 217 1.087043e+03 5.381042e+00 * time: 318.62679982185364 218 1.087043e+03 5.384621e+00 * time: 320.24694991111755 219 1.087043e+03 5.385776e+00 * time: 321.86213183403015 220 1.087043e+03 5.389036e+00 * time: 323.4170649051666 221 1.087043e+03 5.392164e+00 * time: 324.962073802948 222 1.087042e+03 5.398604e+00 * time: 326.53738498687744 223 1.087042e+03 5.409320e+00 * time: 328.078067779541 224 1.087040e+03 5.429855e+00 * time: 329.6481599807739 225 1.087035e+03 5.470145e+00 * time: 331.2582697868347 226 1.087023e+03 7.328713e+00 * time: 332.87839698791504 227 1.086989e+03 1.212423e+01 * time: 334.520956993103 228 1.086898e+03 2.015768e+01 * time: 336.15204882621765 229 1.086600e+03 3.491298e+01 * time: 337.7881889343262 230 1.085509e+03 5.565514e+01 * time: 339.35071897506714 231 1.085364e+03 6.302934e+01 * time: 340.99111795425415 232 1.083398e+03 7.339780e+01 * time: 342.5948488712311 233 1.081123e+03 8.030617e+01 * time: 344.188982963562 234 1.079382e+03 7.453755e+01 * time: 345.7607488632202 235 1.076762e+03 3.790063e+01 * time: 347.31511187553406 236 1.074930e+03 6.335405e+01 * time: 348.8751769065857 237 1.072774e+03 5.513460e+01 * time: 350.41324400901794 238 1.071755e+03 4.021548e+01 * time: 351.9602839946747 239 1.070070e+03 3.590102e+01 * time: 353.5386109352112 240 1.069558e+03 3.932180e+00 * time: 355.1027297973633 241 1.069494e+03 2.133214e+00 * time: 356.6439299583435 242 1.069489e+03 2.261641e+00 * time: 358.1981358528137 243 1.069488e+03 2.240254e+00 * time: 359.7385218143463 244 1.069488e+03 2.235314e+00 * time: 361.2339859008789 245 1.069488e+03 2.235235e+00 * time: 362.7281918525696 246 1.069488e+03 2.235056e+00 * time: 364.20541882514954 247 1.069488e+03 2.234829e+00 * time: 365.69299578666687 248 1.069488e+03 2.234354e+00 * time: 367.1695239543915 249 1.069488e+03 2.233506e+00 * time: 368.67799282073975 250 1.069487e+03 2.231791e+00 * time: 370.1735680103302 251 1.069487e+03 2.228226e+00 * time: 371.7147707939148 252 1.069486e+03 2.220326e+00 * time: 373.32600593566895 253 1.069483e+03 2.442988e+00 * time: 374.8686308860779 254 1.069475e+03 3.940528e+00 * time: 376.4121689796448 255 1.069454e+03 6.247492e+00 * time: 377.97104597091675 256 1.069405e+03 9.508044e+00 * time: 379.5714638233185 257 1.069300e+03 1.307650e+01 * time: 381.0915439128876 258 1.069131e+03 1.408019e+01 * time: 382.63350081443787 259 1.068981e+03 8.693949e+00 * time: 384.23002195358276 260 1.068935e+03 2.128772e+00 * time: 385.8037898540497 261 1.068930e+03 8.672316e-01 * time: 387.36090993881226 262 1.068929e+03 8.688287e-01 * time: 388.9138457775116 263 1.068929e+03 8.689320e-01 * time: 390.3955738544464 264 1.068929e+03 8.689320e-01 * time: 392.00895500183105 265 1.068929e+03 8.689320e-01 * time: 393.64708399772644 266 1.068929e+03 8.689151e-01 * time: 395.1534798145294 267 1.068929e+03 8.689249e-01 * time: 396.65899085998535 268 1.068929e+03 8.689538e-01 * time: 398.1391317844391 269 1.068929e+03 8.690025e-01 * time: 399.62266278266907 270 1.068929e+03 8.691057e-01 * time: 401.1464807987213 271 1.068929e+03 8.693123e-01 * time: 402.6340899467468 272 1.068929e+03 8.697336e-01 * time: 404.1389877796173 273 1.068928e+03 8.705109e-01 * time: 405.64310598373413 274 1.068927e+03 8.716320e-01 * time: 407.13754391670227 275 1.068926e+03 8.725278e-01 * time: 408.6096889972687 276 1.068926e+03 8.725828e-01 * time: 410.0848557949066 277 1.068926e+03 8.721383e-01 * time: 411.56991600990295 278 1.068925e+03 8.713478e-01 * time: 413.06353998184204 279 1.068924e+03 9.225718e-01 * time: 414.54989099502563 280 1.068922e+03 1.485996e+00 * time: 416.02474880218506 281 1.068915e+03 2.429714e+00 * time: 417.5075488090515 282 1.068899e+03 3.861471e+00 * time: 418.98104786872864 283 1.068860e+03 5.845992e+00 * time: 420.42924189567566 284 1.068776e+03 7.973861e+00 * time: 421.87981486320496 285 1.068627e+03 8.779578e+00 * time: 423.3428008556366 286 1.068457e+03 6.374437e+00 * time: 424.7874789237976 287 1.068383e+03 2.378837e+00 * time: 426.2387948036194 288 1.068376e+03 1.345639e+00 * time: 427.70383381843567 289 1.068375e+03 1.310336e+00 * time: 429.1694509983063 290 1.068374e+03 1.285895e+00 * time: 430.60361886024475 291 1.068374e+03 1.284951e+00 * time: 432.0822649002075 292 1.068374e+03 1.284951e+00 * time: 433.7251977920532 293 1.068374e+03 1.283462e+00 * time: 435.4100069999695 294 1.068374e+03 1.283455e+00 * time: 436.8802058696747 295 1.068374e+03 1.282722e+00 * time: 438.39613795280457 296 1.068374e+03 1.280664e+00 * time: 439.8759219646454 297 1.068374e+03 1.278888e+00 * time: 441.3187267780304 298 1.068374e+03 1.275000e+00 * time: 442.8133850097656 299 1.068374e+03 1.269230e+00 * time: 444.2982997894287 300 1.068374e+03 1.259257e+00 * time: 445.8136258125305 301 1.068374e+03 1.242726e+00 * time: 447.3267250061035 302 1.068373e+03 1.260054e+00 * time: 448.84030294418335 303 1.068370e+03 2.043047e+00 * time: 450.3476388454437 304 1.068363e+03 3.272705e+00 * time: 451.8810977935791 305 1.068346e+03 5.080497e+00 * time: 453.38277792930603 306 1.068308e+03 7.259240e+00 * time: 454.8907299041748 307 1.068239e+03 8.485885e+00 * time: 456.3600687980652 308 1.068153e+03 6.787305e+00 * time: 457.8610498905182 309 1.068076e+03 2.962239e+00 * time: 459.38173389434814 310 1.068028e+03 3.648163e-01 * time: 460.9150619506836 311 1.068010e+03 6.696155e-01 * time: 462.4024279117584 312 1.068002e+03 6.065508e-01 * time: 463.92383193969727 313 1.067996e+03 2.826797e-01 * time: 465.3901858329773 314 1.067993e+03 2.585896e-01 * time: 466.88756799697876 315 1.067992e+03 2.949494e-01 * time: 468.3804018497467 316 1.067992e+03 3.200153e-01 * time: 469.8789517879486 317 1.067992e+03 3.366476e-01 * time: 471.3481378555298 318 1.067992e+03 3.446883e-01 * time: 472.83821392059326 319 1.067992e+03 3.468853e-01 * time: 474.3045427799225 320 1.067992e+03 3.472675e-01 * time: 475.77925086021423 321 1.067992e+03 3.472738e-01 * time: 477.2796947956085 322 1.067992e+03 3.473614e-01 * time: 478.80944085121155 323 1.067992e+03 3.473715e-01 * time: 480.3207628726959 324 1.067992e+03 3.473714e-01 * time: 481.8991389274597 325 1.067992e+03 3.473224e-01 * time: 483.4537320137024 326 1.067992e+03 3.473228e-01 * time: 485.08415699005127 327 1.067992e+03 3.473221e-01 * time: 486.6752779483795 328 1.067992e+03 3.399790e-01 * time: 488.1113429069519 329 1.067992e+03 3.362705e-01 * time: 489.56944394111633 330 1.067992e+03 3.005757e-01 * time: 491.04181599617004 331 1.067992e+03 2.429240e-01 * time: 492.47851300239563 332 1.067991e+03 1.361280e-01 * time: 493.92034697532654 333 1.067991e+03 1.071745e-01 * time: 495.4468548297882 334 1.067991e+03 9.741600e-02 * time: 496.87898778915405 335 1.067991e+03 9.434333e-02 * time: 498.30126094818115 336 1.067991e+03 9.456144e-02 * time: 499.7226629257202 337 1.067991e+03 9.456602e-02 * time: 501.1835608482361 338 1.067991e+03 9.457219e-02 * time: 502.6515967845917 339 1.067991e+03 9.457802e-02 * time: 504.09935092926025 340 1.067991e+03 9.458467e-02 * time: 505.5463619232178 341 1.067991e+03 9.459192e-02 * time: 507.01256489753723 342 1.067991e+03 9.459264e-02 * time: 508.4890458583832 343 1.067991e+03 9.459354e-02 * time: 509.9607138633728 344 1.067991e+03 9.459363e-02 * time: 511.4626088142395 345 1.067991e+03 9.459380e-02 * time: 512.9669988155365 346 1.067991e+03 9.459391e-02 * time: 514.4617698192596 347 1.067991e+03 9.459411e-02 * time: 515.9668538570404 348 1.067991e+03 9.459412e-02 * time: 517.4879670143127 349 1.067991e+03 9.459414e-02 * time: 519.0075309276581 350 1.067991e+03 9.459420e-02 * time: 520.5319309234619 351 1.067991e+03 9.459421e-02 * time: 522.0506639480591 352 1.067991e+03 9.459423e-02 * time: 523.5863728523254 353 1.067991e+03 9.459423e-02 * time: 525.1306939125061 354 1.067991e+03 9.459424e-02 * time: 526.7076969146729 355 1.067991e+03 9.459424e-02 * time: 528.3091359138489 356 1.067991e+03 9.459424e-02 * time: 529.8898119926453 357 1.067991e+03 9.459424e-02 * time: 531.4563229084015 358 1.067991e+03 9.459425e-02 * time: 533.0241839885712 359 1.067991e+03 9.459425e-02 * time: 534.627956867218 360 1.067991e+03 9.459425e-02 * time: 536.2101719379425 361 1.067991e+03 9.459425e-02 * time: 537.8283488750458 362 1.067991e+03 9.459426e-02 * time: 539.4456617832184 363 1.067991e+03 9.459426e-02 * time: 541.1266949176788 364 1.067991e+03 9.459426e-02 * time: 542.7916629314423 365 1.067991e+03 9.459426e-02 * time: 544.4413290023804 366 1.067991e+03 9.459426e-02 * time: 546.1449677944183 367 1.067991e+03 9.459426e-02 * time: 547.6939618587494 368 1.067991e+03 9.459426e-02 * time: 549.3151547908783 369 1.067991e+03 9.459426e-02 * time: 550.929016828537 370 1.067991e+03 9.459426e-02 * time: 552.5268139839172 371 1.067991e+03 9.459426e-02 * time: 554.219260931015 372 1.067991e+03 9.459426e-02 * time: 554.7838389873505
FittedPumasModel
Dynamical system type: Nonlinear ODE
Solver(s): OrdinaryDiffEqVerner.Vern7
Number of subjects: 32
Observation records: Active Missing
conc: 251 47
pca: 232 66
Total: 483 113
Number of parameters: Constant Optimized
0 18
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: NoXChange
Log-likelihood value: -1067.9913
-----------------------
Estimate
-----------------------
pop_CL 0.13521
pop_V 8.0133
pop_tabs 0.57114
pop_lag 0.87561
pop_e0 96.399
pop_emax -1.0615
pop_c50 1.4912
pop_tover 14.05
pk_Ω₁,₁ 0.068012
pk_Ω₂,₂ 0.021048
pk_Ω₃,₃ 0.86273
pd_Ω₁,₁ 0.002977
pd_Ω₂,₂ 1.9398e-7
pd_Ω₃,₃ 0.14561
pd_Ω₄,₄ 0.015351
σ_prop 0.088485
σ_add 0.41684
σ_fx 3.5803
-----------------------8 Concluding Remarks
In this tutorial, you have seen how to adjust the tolerances and the algorithm of the differential solver. Usually, the default differential equation solver in Pumas is an efficient choice. To reduce numerical issues, sometimes it can be helpful to decrease the default tolerances.