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.
6.1 Model Simulation
- Default Solvers: Rosenbrock23 (stiff) and Tsit5 (non-stiff)
- Tolerances: Relative tolerance \(1 \times 10^{-3}\) and absolute tolerance \(1 \times 10^{-6}\)
- Rationale: These higher (less stringent) tolerances permit faster simulations while maintaining sufficient accuracy for exploratory and predictive modeling.
6.2 Model Fitting
- Default Solvers: Rodas5P (stiff) and Vern7 (non-stiff)
- Tolerances: Relative tolerance \(10^{-8}\) and absolute tolerance \(10^{-12}\)
- Rationale: These lower (more stringent) tolerances ensure high precision during parameter estimation, which is critical for 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.
6.3 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: 0.0444488525390625 1 5.185877e+05 8.742010e+05 * time: 3.0191988945007324 2 3.866957e+05 6.366584e+05 * time: 4.067097902297974 3 1.795019e+05 2.835377e+05 * time: 5.092190980911255 4 9.682619e+04 1.546512e+05 * time: 6.1065168380737305 5 4.791898e+04 6.820022e+04 * time: 7.071563959121704 6 2.907369e+04 3.509683e+04 * time: 8.029161930084229 7 1.827377e+04 1.713122e+04 * time: 8.97847294807434 8 1.260634e+04 9.563328e+03 * time: 9.94101095199585 9 9.403430e+03 8.611643e+03 * time: 10.879899024963379 10 7.325839e+03 7.634814e+03 * time: 11.808027029037476 11 5.926864e+03 6.625143e+03 * time: 12.735996961593628 12 4.942276e+03 5.562817e+03 * time: 13.637043952941895 13 4.139082e+03 4.326560e+03 * time: 14.539255857467651 14 3.563490e+03 3.065255e+03 * time: 15.447733879089355 15 3.296158e+03 2.170246e+03 * time: 16.335453987121582 16 3.216281e+03 1.669628e+03 * time: 17.210973024368286 17 3.205943e+03 1.487868e+03 * time: 18.08922004699707 18 3.204981e+03 1.444253e+03 * time: 18.967967987060547 19 3.204107e+03 1.417661e+03 * time: 19.84506392478943 20 3.201145e+03 1.361617e+03 * time: 20.752863883972168 21 3.194218e+03 1.282918e+03 * time: 21.656405925750732 22 3.175799e+03 1.158036e+03 * time: 22.590276956558228 23 3.130001e+03 9.774722e+02 * time: 23.483536958694458 24 3.016853e+03 7.265878e+02 * time: 24.35519003868103 25 2.749857e+03 4.107604e+02 * time: 25.20760202407837 26 2.137213e+03 2.318395e+02 * time: 26.058462858200073 27 1.756667e+03 2.281482e+02 * time: 26.99558687210083 28 1.380674e+03 1.613751e+02 * time: 29.494125843048096 29 1.328451e+03 1.298787e+02 * time: 30.345643043518066 30 1.287368e+03 2.435586e+02 * time: 31.080091953277588 31 1.263071e+03 1.616831e+02 * time: 31.819727897644043 32 1.254713e+03 1.796576e+02 * time: 32.584303855895996 33 1.247205e+03 1.996165e+02 * time: 33.35411095619202 34 1.243765e+03 1.938235e+02 * time: 34.118725061416626 35 1.240829e+03 1.739538e+02 * time: 34.9112708568573 36 1.240788e+03 1.744028e+02 * time: 35.67048096656799 37 1.240776e+03 1.743250e+02 * time: 36.4374418258667 38 1.240093e+03 1.687609e+02 * time: 37.245545864105225 39 1.239010e+03 1.586878e+02 * time: 38.0805549621582 40 1.236253e+03 1.305335e+02 * time: 38.85179805755615 41 1.232129e+03 8.229601e+01 * time: 39.6273238658905 42 1.228151e+03 3.735809e+01 * time: 40.40343189239502 43 1.226337e+03 5.238476e+01 * time: 41.179723024368286 44 1.226025e+03 4.868885e+01 * time: 41.97121286392212 45 1.226011e+03 4.583328e+01 * time: 42.739656925201416 46 1.226010e+03 4.536419e+01 * time: 43.41727304458618 47 1.226008e+03 4.466378e+01 * time: 44.113921880722046 48 1.226002e+03 4.336406e+01 * time: 44.78502893447876 49 1.225988e+03 4.102579e+01 * time: 45.48313093185425 50 1.225951e+03 3.656289e+01 * time: 46.19092583656311 51 1.225858e+03 2.865017e+01 * time: 46.8719379901886 52 1.225645e+03 2.863717e+01 * time: 47.54545497894287 53 1.225243e+03 3.015495e+01 * time: 48.22722005844116 54 1.224739e+03 4.057375e+01 * time: 48.960087060928345 55 1.224438e+03 5.072098e+01 * time: 49.649543046951294 56 1.224368e+03 4.561543e+01 * time: 50.33049988746643 57 1.224362e+03 4.124686e+01 * time: 51.016664028167725 58 1.224360e+03 4.008283e+01 * time: 51.7152738571167 59 1.224357e+03 3.807281e+01 * time: 52.39108991622925 60 1.224349e+03 3.507733e+01 * time: 53.070642948150635 61 1.224328e+03 3.152658e+01 * time: 53.75957989692688 62 1.224274e+03 2.940378e+01 * time: 54.45708703994751 63 1.224133e+03 2.814508e+01 * time: 55.15429091453552 64 1.223771e+03 2.816089e+01 * time: 55.86393404006958 65 1.222880e+03 4.493737e+01 * time: 56.572295904159546 66 1.220919e+03 8.628817e+01 * time: 57.27905201911926 67 1.217683e+03 1.137018e+02 * time: 58.03884196281433 68 1.214493e+03 9.066168e+01 * time: 58.79125905036926 69 1.212820e+03 8.381803e+01 * time: 59.53952693939209 70 1.212582e+03 8.589808e+01 * time: 60.31252694129944 71 1.212574e+03 8.557497e+01 * time: 61.037479877471924 72 1.212569e+03 8.522111e+01 * time: 61.78223705291748 73 1.212552e+03 8.416219e+01 * time: 62.55606389045715 74 1.212516e+03 8.236925e+01 * time: 63.3102068901062 75 1.212415e+03 7.841477e+01 * time: 64.07686305046082 76 1.212165e+03 7.017247e+01 * time: 64.86221385002136 77 1.211553e+03 5.231312e+01 * time: 65.6284658908844 78 1.210236e+03 7.254984e+01 * time: 66.38427400588989 79 1.208052e+03 8.652548e+01 * time: 67.12837886810303 80 1.205978e+03 9.811810e+01 * time: 67.81938791275024 81 1.205158e+03 1.085052e+02 * time: 68.50058102607727 82 1.205038e+03 9.706885e+01 * time: 69.14765286445618 83 1.205030e+03 9.297156e+01 * time: 69.79661893844604 84 1.205025e+03 9.142093e+01 * time: 70.46677684783936 85 1.205007e+03 8.794517e+01 * time: 71.12442684173584 86 1.204968e+03 8.329868e+01 * time: 71.82089400291443 87 1.204860e+03 7.519245e+01 * time: 72.48261094093323 88 1.204586e+03 6.676950e+01 * time: 73.18347597122192 89 1.203889e+03 5.914985e+01 * time: 73.88511204719543 90 1.202275e+03 5.659425e+01 * time: 74.57274389266968 91 1.199277e+03 5.457598e+01 * time: 75.27330994606018 92 1.195791e+03 6.434187e+01 * time: 76.00647902488708 93 1.193281e+03 4.404726e+01 * time: 76.70292091369629 94 1.192403e+03 4.696513e+01 * time: 77.39984083175659 95 1.192344e+03 4.826238e+01 * time: 78.09739804267883 96 1.192341e+03 4.836346e+01 * time: 78.77679491043091 97 1.192340e+03 4.843134e+01 * time: 79.45247602462769 98 1.192336e+03 4.854905e+01 * time: 80.1184389591217 99 1.192325e+03 4.873525e+01 * time: 80.76821899414062 100 1.192297e+03 4.901993e+01 * time: 81.40853905677795 101 1.192226e+03 4.941568e+01 * time: 82.04220700263977 102 1.192043e+03 4.985206e+01 * time: 82.73750901222229 103 1.191593e+03 5.001007e+01 * time: 83.43123388290405 104 1.190562e+03 4.892711e+01 * time: 84.15450882911682 105 1.188621e+03 4.482401e+01 * time: 84.83892583847046 106 1.186171e+03 5.565401e+01 * time: 85.5174469947815 107 1.184411e+03 7.195267e+01 * time: 86.19085884094238 108 1.183640e+03 7.480949e+01 * time: 86.8679530620575 109 1.183456e+03 7.309347e+01 * time: 87.5378429889679 110 1.183420e+03 7.276593e+01 * time: 88.22886896133423 111 1.183410e+03 7.363637e+01 * time: 88.90527296066284 112 1.183407e+03 7.448590e+01 * time: 89.61400699615479 113 1.183403e+03 7.577977e+01 * time: 90.35319304466248 114 1.183397e+03 7.712667e+01 * time: 91.04749488830566 115 1.183377e+03 7.953569e+01 * time: 91.74914693832397 116 1.183329e+03 8.300885e+01 * time: 92.40900087356567 117 1.183203e+03 8.805322e+01 * time: 93.10949492454529 118 1.182889e+03 9.423584e+01 * time: 93.77632689476013 119 1.182143e+03 9.919011e+01 * time: 94.46516394615173 120 1.180607e+03 9.577147e+01 * time: 95.17568302154541 121 1.178255e+03 7.337696e+01 * time: 95.89746904373169 122 1.176240e+03 3.404284e+01 * time: 96.66613698005676 123 1.175531e+03 1.922941e+01 * time: 97.46626400947571 124 1.175410e+03 1.933203e+01 * time: 98.16923785209656 125 1.175396e+03 1.964875e+01 * time: 98.87998104095459 126 1.175395e+03 1.966957e+01 * time: 99.59074592590332 127 1.175394e+03 1.960524e+01 * time: 100.29266405105591 128 1.175394e+03 1.957692e+01 * time: 100.98692584037781 129 1.175392e+03 1.945993e+01 * time: 101.69481086730957 130 1.175387e+03 1.938252e+01 * time: 102.39847898483276 131 1.175374e+03 1.915643e+01 * time: 103.13967704772949 132 1.175342e+03 1.892407e+01 * time: 103.85099291801453 133 1.175256e+03 1.840907e+01 * time: 104.59721398353577 134 1.175030e+03 1.936487e+01 * time: 105.36507987976074 135 1.174436e+03 2.656836e+01 * time: 106.09787702560425 136 1.172888e+03 3.906036e+01 * time: 106.83714294433594 137 1.169012e+03 5.824500e+01 * time: 107.57362198829651 138 1.160858e+03 7.724789e+01 * time: 108.34717988967896 139 1.151984e+03 5.979641e+01 * time: 109.13442587852478 140 1.150239e+03 2.132816e+02 * time: 109.90594291687012 141 1.147461e+03 1.267649e+02 * time: 110.68494486808777 142 1.143928e+03 2.373475e+01 * time: 111.44041895866394 143 1.142853e+03 2.225755e+01 * time: 112.18134593963623 144 1.141788e+03 3.914831e+01 * time: 112.89826202392578 145 1.141298e+03 2.253303e+01 * time: 113.60847997665405 146 1.140996e+03 2.098978e+01 * time: 114.3228690624237 147 1.140981e+03 2.094768e+01 * time: 115.02130484580994 148 1.140978e+03 2.098189e+01 * time: 115.74608087539673 149 1.140977e+03 2.101184e+01 * time: 116.43827795982361 150 1.140977e+03 2.100810e+01 * time: 117.16852688789368 151 1.140977e+03 2.097542e+01 * time: 117.85696792602539 152 1.140976e+03 2.094478e+01 * time: 118.56182193756104 153 1.140974e+03 2.089055e+01 * time: 119.25551891326904 154 1.140970e+03 2.083449e+01 * time: 119.95206689834595 155 1.140963e+03 2.076312e+01 * time: 120.65537190437317 156 1.140944e+03 2.068373e+01 * time: 121.35652089118958 157 1.140896e+03 2.060201e+01 * time: 122.08560085296631 158 1.140770e+03 2.053295e+01 * time: 122.80344605445862 159 1.140444e+03 2.051792e+01 * time: 123.5052740573883 160 1.139610e+03 2.178464e+01 * time: 124.2343099117279 161 1.138121e+03 3.276681e+01 * time: 124.94513201713562 162 1.135107e+03 4.596086e+01 * time: 125.66745901107788 163 1.132128e+03 5.673428e+01 * time: 126.56104898452759 164 1.130311e+03 6.297114e+01 * time: 127.43138599395752 165 1.126112e+03 1.303349e+02 * time: 128.25241684913635 166 1.116080e+03 4.102784e+01 * time: 129.22865104675293 167 1.114892e+03 3.653943e+01 * time: 130.19353604316711 168 1.112342e+03 3.437496e+01 * time: 131.1588659286499 169 1.111707e+03 1.636383e+01 * time: 132.1578860282898 170 1.111291e+03 9.909471e+00 * time: 133.14245891571045 171 1.111206e+03 1.001147e+01 * time: 134.16351699829102 172 1.111191e+03 1.040660e+01 * time: 135.16239190101624
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: 2.384185791015625e-5 1 5.174461e+05 8.708698e+05 * time: 2.234921932220459 2 3.865265e+05 6.344302e+05 * time: 4.1615400314331055 3 1.804274e+05 2.829723e+05 * time: 5.9937028884887695 4 9.706640e+04 1.550547e+05 * time: 7.820833921432495 5 4.769637e+04 6.778818e+04 * time: 9.510324954986572 6 2.902319e+04 3.499747e+04 * time: 11.189435005187988 7 1.823472e+04 1.705751e+04 * time: 12.910559892654419 8 1.258819e+04 9.569381e+03 * time: 14.624377965927124 9 9.389984e+03 8.615851e+03 * time: 16.45893883705139 10 7.314702e+03 7.636883e+03 * time: 18.36890983581543 11 5.916029e+03 6.624325e+03 * time: 20.273990869522095 12 4.930519e+03 5.558140e+03 * time: 22.15579581260681 13 4.125060e+03 4.315759e+03 * time: 24.039878845214844 14 3.549280e+03 3.051093e+03 * time: 25.816280841827393 15 3.283489e+03 2.157292e+03 * time: 27.508412837982178 16 3.204886e+03 1.659798e+03 * time: 29.139386892318726 17 3.194875e+03 1.480528e+03 * time: 30.704494953155518 18 3.193944e+03 1.437921e+03 * time: 32.51869487762451 19 3.193070e+03 1.411186e+03 * time: 34.46353888511658 20 3.190129e+03 1.355327e+03 * time: 36.18931198120117 21 3.183228e+03 1.276603e+03 * time: 37.83227181434631 22 3.164897e+03 1.151838e+03 * time: 39.528690814971924 23 3.119250e+03 9.712651e+02 * time: 41.13296699523926 24 3.006297e+03 7.204342e+02 * time: 42.6832389831543 25 2.738913e+03 4.050545e+02 * time: 44.27626395225525 26 2.123834e+03 2.318194e+02 * time: 45.736785888671875 27 1.789138e+03 2.290465e+02 * time: 47.28368592262268 28 1.396455e+03 1.683969e+02 * time: 51.55344295501709 29 1.333545e+03 1.336195e+02 * time: 53.195307970047 30 1.297771e+03 2.452189e+02 * time: 54.557859897613525 31 1.266002e+03 1.523968e+02 * time: 55.84942603111267 32 1.255506e+03 1.733993e+02 * time: 57.17598581314087 33 1.247789e+03 1.971624e+02 * time: 58.48611903190613 34 1.244490e+03 1.915728e+02 * time: 59.784265995025635 35 1.240568e+03 1.704250e+02 * time: 61.10211801528931 36 1.240503e+03 1.711787e+02 * time: 62.396113872528076 37 1.240492e+03 1.711657e+02 * time: 63.65488600730896 38 1.239992e+03 1.687240e+02 * time: 64.9413628578186 39 1.239199e+03 1.624610e+02 * time: 66.23924899101257 40 1.236971e+03 1.400044e+02 * time: 67.58775091171265 41 1.233203e+03 9.442277e+01 * time: 68.88796091079712 42 1.228682e+03 3.273922e+01 * time: 70.19587302207947 43 1.226466e+03 4.997778e+01 * time: 71.47292399406433 44 1.226104e+03 4.904407e+01 * time: 72.7358968257904 45 1.226088e+03 4.675833e+01 * time: 74.03177094459534 46 1.226088e+03 4.628497e+01 * time: 75.37533783912659 47 1.226085e+03 4.541356e+01 * time: 76.7222650051117 48 1.226080e+03 4.402741e+01 * time: 78.1068480014801 49 1.226064e+03 4.142299e+01 * time: 79.38507890701294 50 1.226026e+03 3.663064e+01 * time: 80.71971797943115 51 1.225931e+03 2.851375e+01 * time: 82.01586198806763 52 1.225713e+03 2.844436e+01 * time: 83.3860399723053 53 1.225303e+03 3.026440e+01 * time: 84.67640399932861 54 1.224791e+03 4.157127e+01 * time: 85.95411586761475 55 1.224489e+03 5.047474e+01 * time: 87.23834800720215 56 1.224420e+03 4.451064e+01 * time: 88.51272892951965 57 1.224413e+03 3.994026e+01 * time: 89.77175188064575 58 1.224412e+03 3.879446e+01 * time: 91.03320789337158 59 1.224408e+03 3.677250e+01 * time: 92.77796792984009 60 1.224400e+03 3.377993e+01 * time: 95.81799602508545 61 1.224379e+03 3.158187e+01 * time: 98.62408781051636 62 1.224324e+03 2.925155e+01 * time: 100.26513481140137 63 1.224180e+03 2.815886e+01 * time: 101.59188389778137 64 1.223813e+03 2.818297e+01 * time: 102.93980383872986 65 1.222906e+03 4.598758e+01 * time: 104.22323489189148 66 1.220910e+03 8.692142e+01 * time: 105.58843088150024 67 1.217609e+03 1.136529e+02 * time: 106.87528681755066 68 1.214313e+03 9.030271e+01 * time: 108.14076781272888 69 1.212533e+03 8.868947e+01 * time: 109.39353895187378 70 1.212272e+03 8.970279e+01 * time: 110.63007998466492 71 1.212264e+03 8.921922e+01 * time: 111.87661981582642 72 1.212259e+03 8.882366e+01 * time: 113.11237597465515 73 1.212242e+03 8.761509e+01 * time: 114.32794380187988 74 1.212205e+03 8.561995e+01 * time: 115.56921195983887 75 1.212103e+03 8.125315e+01 * time: 116.80078601837158 76 1.211850e+03 7.229633e+01 * time: 118.02375483512878 77 1.211238e+03 5.321838e+01 * time: 119.26348185539246 78 1.209945e+03 7.377546e+01 * time: 120.4910659790039 79 1.207890e+03 8.404092e+01 * time: 121.72184586524963 80 1.206058e+03 9.148114e+01 * time: 122.94782781600952 81 1.205389e+03 9.689402e+01 * time: 124.1878650188446 82 1.205303e+03 8.594692e+01 * time: 125.40383100509644 83 1.205297e+03 8.262969e+01 * time: 127.99246883392334 84 1.205293e+03 8.096366e+01 * time: 129.37879180908203 85 1.205277e+03 7.747618e+01 * time: 131.24889588356018 86 1.205241e+03 7.259258e+01 * time: 133.57813787460327 87 1.205143e+03 6.664833e+01 * time: 134.95729780197144 88 1.204895e+03 6.374646e+01 * time: 136.21653580665588 89 1.204268e+03 5.753767e+01 * time: 137.4621238708496 90 1.202818e+03 4.866322e+01 * time: 138.73243689537048 91 1.200116e+03 5.085841e+01 * time: 140.02400493621826 92 1.196895e+03 7.139477e+01 * time: 141.28587293624878 93 1.194610e+03 4.693642e+01 * time: 142.5424349308014 94 1.193880e+03 4.874637e+01 * time: 143.8006730079651 95 1.193833e+03 4.997341e+01 * time: 145.09883189201355 96 1.193831e+03 5.008752e+01 * time: 146.29489398002625 97 1.193829e+03 5.014860e+01 * time: 147.4955849647522 98 1.193824e+03 5.025195e+01 * time: 148.68387699127197 99 1.193812e+03 5.040245e+01 * time: 149.8904058933258 100 1.193778e+03 5.061928e+01 * time: 151.152037858963 101 1.193693e+03 5.087910e+01 * time: 152.38475799560547 102 1.193472e+03 5.105104e+01 * time: 153.66539692878723 103 1.192926e+03 5.068579e+01 * time: 154.980397939682 104 1.191681e+03 4.863476e+01 * time: 156.2937068939209 105 1.189351e+03 4.892982e+01 * time: 157.65465593338013 106 1.186420e+03 7.777398e+01 * time: 158.93764901161194 107 1.184527e+03 8.874390e+01 * time: 160.26991391181946 108 1.184034e+03 8.377339e+01 * time: 161.5546998977661 109 1.183978e+03 7.932702e+01 * time: 162.8830428123474 110 1.183969e+03 7.794331e+01 * time: 164.18269085884094 111 1.183966e+03 7.787273e+01 * time: 165.5228488445282 112 1.183962e+03 7.833691e+01 * time: 166.86115097999573 113 1.183955e+03 7.929110e+01 * time: 168.21114492416382 114 1.183940e+03 8.084135e+01 * time: 169.53141593933105 115 1.183904e+03 8.331062e+01 * time: 170.79682898521423 116 1.183812e+03 8.698431e+01 * time: 172.0329999923706 117 1.183576e+03 9.188953e+01 * time: 173.27450299263 118 1.182997e+03 9.668921e+01 * time: 174.51346802711487 119 1.181702e+03 9.635659e+01 * time: 175.77193880081177 120 1.179436e+03 8.061098e+01 * time: 177.0483798980713 121 1.177026e+03 4.571918e+01 * time: 178.35170602798462 122 1.175794e+03 1.863482e+01 * time: 179.6647388935089 123 1.175474e+03 1.968114e+01 * time: 181.02070784568787 124 1.175429e+03 1.958620e+01 * time: 182.84070301055908 125 1.175427e+03 1.993702e+01 * time: 185.18890500068665 126 1.175426e+03 1.962020e+01 * time: 187.00636887550354 127 1.175426e+03 1.965822e+01 * time: 188.46444988250732 128 1.175424e+03 1.984458e+01 * time: 189.91618490219116 129 1.175421e+03 2.001151e+01 * time: 191.3489909172058 130 1.175413e+03 2.035203e+01 * time: 192.80149698257446 131 1.175392e+03 2.084295e+01 * time: 194.18834495544434 132 1.175336e+03 2.164587e+01 * time: 195.5569658279419 133 1.175189e+03 2.287862e+01 * time: 196.99633502960205 134 1.174801e+03 2.475388e+01 * time: 198.44568300247192 135 1.173792e+03 4.143552e+01 * time: 199.92840385437012 136 1.171220e+03 7.151629e+01 * time: 201.83148980140686 137 1.165144e+03 1.078851e+02 * time: 203.6147379875183 138 1.155079e+03 7.750603e+01 * time: 205.42039895057678 139 1.149212e+03 6.869378e+01 * time: 207.469664812088 140 1.146920e+03 6.357052e+01 * time: 209.12604689598083 141 1.144663e+03 4.948266e+01 * time: 210.73121094703674 142 1.143084e+03 2.539126e+01 * time: 212.29919981956482 143 1.141829e+03 1.979229e+01 * time: 213.93441200256348 144 1.141293e+03 2.050881e+01 * time: 215.57394695281982 145 1.141052e+03 2.076835e+01 * time: 217.20781993865967 146 1.140979e+03 2.090550e+01 * time: 218.8016698360443 147 1.140976e+03 2.094541e+01 * time: 220.34354496002197 148 1.140975e+03 2.099654e+01 * time: 221.93538403511047 149 1.140974e+03 2.099916e+01 * time: 223.43596982955933 150 1.140974e+03 2.099880e+01 * time: 224.92119693756104 151 1.140973e+03 2.099210e+01 * time: 226.38576793670654 152 1.140972e+03 2.098163e+01 * time: 227.86579084396362 153 1.140970e+03 2.096557e+01 * time: 229.34565782546997 154 1.140963e+03 2.094347e+01 * time: 230.87114691734314 155 1.140947e+03 2.091734e+01 * time: 232.39119291305542 156 1.140905e+03 2.089968e+01 * time: 233.90505981445312 157 1.140795e+03 2.093513e+01 * time: 235.42123889923096 158 1.140505e+03 2.116003e+01 * time: 236.95499181747437 159 1.139717e+03 2.228845e+01 * time: 238.49653482437134 160 1.137490e+03 3.499275e+01 * time: 240.15833282470703 161 1.132263e+03 6.112003e+01 * time: 241.7646279335022 162 1.129968e+03 7.040706e+01 * time: 243.8004858493805 163 1.128183e+03 7.684081e+01 * time: 245.96678280830383 164 1.125121e+03 1.651402e+02 * time: 247.9547770023346 165 1.124367e+03 1.524469e+02 * time: 249.7699489593506 166 1.117089e+03 4.431295e+01 * time: 251.54892086982727 167 1.113092e+03 1.609702e+01 * time: 253.38566994667053 168 1.111588e+03 1.011714e+01 * time: 255.19143795967102 169 1.111231e+03 1.280570e+01 * time: 256.9553678035736 170 1.111200e+03 1.186560e+01 * time: 258.79461789131165 171 1.111188e+03 1.091436e+01 * time: 260.5425100326538 172 1.111187e+03 1.052729e+01 * time: 262.31232500076294 173 1.111187e+03 1.067420e+01 * time: 264.0367410182953 174 1.111187e+03 1.055472e+01 * time: 265.75105381011963 175 1.111187e+03 1.036659e+01 * time: 267.5291819572449 176 1.111185e+03 1.017962e+01 * time: 269.26445984840393 177 1.111183e+03 1.017379e+01 * time: 271.0845229625702 178 1.111176e+03 1.016131e+01 * time: 273.6650218963623 179 1.111157e+03 1.013351e+01 * time: 275.45040798187256 180 1.111109e+03 1.006842e+01 * time: 277.2688579559326 181 1.110986e+03 1.142083e+01 * time: 279.0684599876404 182 1.110675e+03 1.751466e+01 * time: 280.8874189853668 183 1.109918e+03 2.591233e+01 * time: 282.6962468624115 184 1.108234e+03 3.515773e+01 * time: 284.40694999694824 185 1.105216e+03 3.902425e+01 * time: 286.0809099674225 186 1.101588e+03 3.080377e+01 * time: 287.83286690711975 187 1.098345e+03 3.607134e+01 * time: 289.5950508117676 188 1.094148e+03 3.626407e+01 * time: 291.43521881103516 189 1.093478e+03 3.314569e+01 * time: 293.4693748950958 190 1.092996e+03 2.938811e+01 * time: 295.5720899105072 191 1.092406e+03 2.138312e+01 * time: 297.6074719429016 192 1.092297e+03 2.975339e+01 * time: 299.5973129272461 193 1.091949e+03 7.639324e+00 * time: 301.6387119293213 194 1.091909e+03 4.276154e+00 * time: 303.63990592956543 195 1.091904e+03 4.346069e+00 * time: 305.62014985084534 196 1.091904e+03 4.337170e+00 * time: 307.55117988586426 197 1.091904e+03 4.342274e+00 * time: 309.45277881622314 198 1.091904e+03 4.344184e+00 * time: 311.4497790336609 199 1.091904e+03 4.346951e+00 * time: 313.4514739513397 200 1.091904e+03 4.349040e+00 * time: 315.48049783706665 201 1.091904e+03 4.348757e+00 * time: 317.4290268421173 202 1.091904e+03 4.343922e+00 * time: 319.3945939540863 203 1.091903e+03 4.333452e+00 * time: 321.34094285964966 204 1.091902e+03 4.317872e+00 * time: 323.2677118778229 205 1.091901e+03 4.295486e+00 * time: 325.2726049423218 206 1.091897e+03 4.261316e+00 * time: 327.2876088619232 207 1.091889e+03 5.125139e+00 * time: 329.29788303375244 208 1.091867e+03 9.013047e+00 * time: 331.37801480293274 209 1.091811e+03 1.513012e+01 * time: 333.5072338581085 210 1.091668e+03 2.427311e+01 * time: 335.6348948478699 211 1.091325e+03 3.642351e+01 * time: 337.6991198062897 212 1.090580e+03 4.801992e+01 * time: 339.8031129837036 213 1.089340e+03 5.062255e+01 * time: 341.93780183792114 214 1.088109e+03 4.437004e+01 * time: 343.9912078380585 215 1.087458e+03 3.115088e+01 * time: 345.8685438632965 216 1.087119e+03 1.183083e+01 * time: 347.708860874176 217 1.087046e+03 5.394070e+00 * time: 349.56295680999756 218 1.087043e+03 5.390830e+00 * time: 351.4101219177246 219 1.087043e+03 5.383737e+00 * time: 353.2492938041687 220 1.087043e+03 5.384236e+00 * time: 355.1112868785858 221 1.087043e+03 5.384636e+00 * time: 356.91737699508667 222 1.087043e+03 5.385864e+00 * time: 358.7464590072632 223 1.087043e+03 5.387179e+00 * time: 360.56298089027405 224 1.087042e+03 5.389948e+00 * time: 362.38911390304565 225 1.087041e+03 5.395184e+00 * time: 364.2165608406067 226 1.087039e+03 5.407029e+00 * time: 366.05086493492126 227 1.087033e+03 5.434556e+00 * time: 367.8843078613281 228 1.087016e+03 5.502571e+00 * time: 369.7469799518585 229 1.086972e+03 5.677167e+00 * time: 371.5855460166931 230 1.086849e+03 8.425733e+00 * time: 373.43654680252075 231 1.086459e+03 1.475408e+01 * time: 375.27375984191895 232 1.086446e+03 4.299557e+01 * time: 377.01288080215454 233 1.085092e+03 2.387717e+01 * time: 378.8178608417511 234 1.083538e+03 2.726422e+01 * time: 380.5686848163605 235 1.081685e+03 2.845459e+01 * time: 382.3016378879547 236 1.080396e+03 2.805425e+01 * time: 384.1446828842163 237 1.077803e+03 2.873905e+01 * time: 385.8347508907318 238 1.073139e+03 2.780340e+01 * time: 387.5888638496399 239 1.069921e+03 1.583445e+01 * time: 389.39954590797424 240 1.069621e+03 7.856085e+00 * time: 391.18015694618225 241 1.069531e+03 2.311783e+00 * time: 393.02971482276917 242 1.069509e+03 2.021453e+00 * time: 394.86608695983887 243 1.069491e+03 2.157500e+00 * time: 396.5996639728546 244 1.069488e+03 2.225422e+00 * time: 398.35064601898193 245 1.069488e+03 2.236782e+00 * time: 400.08276200294495 246 1.069488e+03 2.235390e+00 * time: 401.8094379901886 247 1.069488e+03 2.235490e+00 * time: 403.5200560092926 248 1.069488e+03 2.235491e+00 * time: 405.4117829799652 249 1.069488e+03 2.235259e+00 * time: 407.25454592704773 250 1.069488e+03 2.235150e+00 * time: 409.0679090023041 251 1.069488e+03 2.234587e+00 * time: 410.8860869407654 252 1.069487e+03 2.233171e+00 * time: 412.7629108428955 253 1.069486e+03 2.229015e+00 * time: 414.6252658367157 254 1.069484e+03 2.218016e+00 * time: 416.4838709831238 255 1.069477e+03 2.188614e+00 * time: 418.36225390434265 256 1.069460e+03 2.790832e+00 * time: 420.23526883125305 257 1.069418e+03 4.259790e+00 * time: 422.1472179889679 258 1.069326e+03 5.893844e+00 * time: 424.0497839450836 259 1.069168e+03 6.471844e+00 * time: 425.9710228443146 260 1.069008e+03 4.528747e+00 * time: 427.8906879425049 261 1.068940e+03 1.710450e+00 * time: 429.79487586021423 262 1.068930e+03 8.661473e-01 * time: 431.690495967865 263 1.068929e+03 8.687543e-01 * time: 433.56481099128723 264 1.068929e+03 8.689456e-01 * time: 435.39104890823364
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: 1.6927719116210938e-5 1 5.174461e+05 8.708699e+05 * time: 9.564247131347656 2 3.865265e+05 6.344302e+05 * time: 17.227109909057617 3 1.804274e+05 2.829723e+05 * time: 25.345355987548828 4 9.706641e+04 1.550547e+05 * time: 34.17463707923889 5 4.769637e+04 6.778818e+04 * time: 41.24401807785034 6 2.902319e+04 3.499747e+04 * time: 48.099628925323486 7 1.823472e+04 1.705751e+04 * time: 55.15830993652344 8 1.258819e+04 9.569382e+03 * time: 62.04436898231506 9 9.389985e+03 8.615853e+03 * time: 68.90869998931885 10 7.314703e+03 7.636886e+03 * time: 75.98258805274963 11 5.916030e+03 6.624328e+03 * time: 83.26359510421753 12 4.930520e+03 5.558143e+03 * time: 90.3055169582367 13 4.125062e+03 4.315761e+03 * time: 96.95682096481323 14 3.549280e+03 3.051094e+03 * time: 103.967453956604 15 3.283490e+03 2.157293e+03 * time: 111.1126549243927 16 3.204886e+03 1.659798e+03 * time: 118.12255311012268 17 3.194875e+03 1.480528e+03 * time: 125.21693396568298 18 3.193944e+03 1.437922e+03 * time: 132.4953489303589 19 3.193070e+03 1.411186e+03 * time: 139.7904829978943 20 3.190129e+03 1.355327e+03 * time: 147.26393294334412 21 3.183228e+03 1.276603e+03 * time: 154.29383301734924 22 3.164897e+03 1.151838e+03 * time: 161.56506395339966 23 3.119250e+03 9.712652e+02 * time: 168.8311960697174 24 3.006297e+03 7.204344e+02 * time: 176.17929792404175 25 2.738913e+03 4.050546e+02 * time: 183.48810005187988 26 2.123834e+03 2.318194e+02 * time: 190.31236791610718 27 1.789142e+03 2.290466e+02 * time: 199.42411994934082 28 1.396456e+03 1.683975e+02 * time: 220.02368593215942 29 1.333545e+03 1.336198e+02 * time: 229.02982902526855 30 1.297773e+03 2.452181e+02 * time: 237.8410611152649 31 1.266002e+03 1.523972e+02 * time: 245.64952397346497 32 1.255505e+03 1.734000e+02 * time: 253.5784809589386 33 1.247788e+03 1.971637e+02 * time: 261.512412071228 34 1.244490e+03 1.915744e+02 * time: 269.08797693252563 35 1.240568e+03 1.704280e+02 * time: 276.64229106903076 36 1.240502e+03 1.711813e+02 * time: 283.97336602211 37 1.240491e+03 1.711683e+02 * time: 290.9710500240326 38 1.239992e+03 1.687296e+02 * time: 298.1024169921875 39 1.239200e+03 1.624744e+02 * time: 305.15316009521484 40 1.236975e+03 1.400443e+02 * time: 312.2441580295563 41 1.233209e+03 9.449763e+01 * time: 319.597275018692 42 1.228687e+03 3.276923e+01 * time: 326.78851795196533 43 1.226467e+03 4.997214e+01 * time: 333.74038910865784 44 1.226104e+03 4.905105e+01 * time: 340.7573299407959 45 1.226088e+03 4.676122e+01 * time: 347.6935420036316 46 1.226087e+03 4.628674e+01 * time: 354.536376953125 47 1.226085e+03 4.541753e+01 * time: 361.41191697120667 48 1.226080e+03 4.403288e+01 * time: 368.18379306793213 49 1.226064e+03 4.143315e+01 * time: 374.916463136673 50 1.226026e+03 3.664876e+01 * time: 382.5936470031738 51 1.225931e+03 2.851383e+01 * time: 389.60882592201233 52 1.225714e+03 2.844460e+01 * time: 396.3549530506134 53 1.225304e+03 3.025732e+01 * time: 403.13479709625244 54 1.224793e+03 4.151448e+01 * time: 409.96794295310974 55 1.224489e+03 5.047755e+01 * time: 416.54454803466797 56 1.224420e+03 4.452753e+01 * time: 423.30796003341675 57 1.224413e+03 3.994380e+01 * time: 429.9929530620575 58 1.224412e+03 3.879465e+01 * time: 436.99037194252014 59 1.224408e+03 3.677922e+01 * time: 443.7821590900421 60 1.224400e+03 3.379126e+01 * time: 450.470477104187 61 1.224379e+03 3.158821e+01 * time: 457.0885920524597 62 1.224324e+03 2.926306e+01 * time: 463.8231029510498 63 1.224181e+03 2.815869e+01 * time: 470.557902097702 64 1.223815e+03 2.818300e+01 * time: 477.3216440677643 65 1.222911e+03 4.583041e+01 * time: 484.0668399333954 66 1.220921e+03 8.675281e+01 * time: 490.8283860683441 67 1.217625e+03 1.136294e+02 * time: 497.7941861152649 68 1.214324e+03 9.050244e+01 * time: 504.77468395233154 69 1.212536e+03 8.867180e+01 * time: 511.6411371231079 70 1.212272e+03 8.970655e+01 * time: 518.4601669311523 71 1.212263e+03 8.922293e+01 * time: 525.782112121582 72 1.212259e+03 8.882951e+01 * time: 532.7552311420441 73 1.212242e+03 8.761908e+01 * time: 539.622740983963 74 1.212205e+03 8.562573e+01 * time: 546.4403150081635 75 1.212102e+03 8.125881e+01 * time: 553.223384141922 76 1.211850e+03 7.230599e+01 * time: 559.7855930328369 77 1.211238e+03 5.323492e+01 * time: 566.4570059776306 78 1.209946e+03 7.373618e+01 * time: 572.8393499851227 79 1.207891e+03 8.401772e+01 * time: 579.4438970088959 80 1.206059e+03 9.146801e+01 * time: 586.1869020462036 81 1.205389e+03 9.690592e+01 * time: 592.9429619312286 82 1.205302e+03 8.595286e+01 * time: 599.6726109981537 83 1.205296e+03 8.263040e+01 * time: 606.5938370227814 84 1.205292e+03 8.096547e+01 * time: 613.7157459259033 85 1.205276e+03 7.747632e+01 * time: 620.398411989212 86 1.205240e+03 7.259211e+01 * time: 627.3925409317017 87 1.205143e+03 6.664635e+01 * time: 634.3302760124207 88 1.204894e+03 6.374389e+01 * time: 641.1192979812622 89 1.204267e+03 5.753484e+01 * time: 647.9874129295349 90 1.202818e+03 4.869938e+01 * time: 654.5895130634308 91 1.200117e+03 5.087313e+01 * time: 660.8868651390076 92 1.196896e+03 7.141162e+01 * time: 667.2440299987793 93 1.194612e+03 4.695316e+01 * time: 673.7502119541168 94 1.193881e+03 4.874366e+01 * time: 679.8972790241241 95 1.193834e+03 4.997038e+01 * time: 686.088268995285 96 1.193832e+03 5.008442e+01 * time: 692.1031010150909 97 1.193830e+03 5.014549e+01 * time: 698.178906917572 98 1.193825e+03 5.024884e+01 * time: 704.2186529636383 99 1.193813e+03 5.039938e+01 * time: 710.3019299507141 100 1.193779e+03 5.061632e+01 * time: 716.7501790523529 101 1.193694e+03 5.087641e+01 * time: 723.5223860740662 102 1.193473e+03 5.104895e+01 * time: 729.612270116806 103 1.192927e+03 5.068500e+01 * time: 735.7495429515839 104 1.191682e+03 4.863684e+01 * time: 741.8964419364929 105 1.189353e+03 4.890365e+01 * time: 748.0847179889679 106 1.186424e+03 7.778082e+01 * time: 754.2442560195923 107 1.184533e+03 8.873607e+01 * time: 760.4056119918823 108 1.184042e+03 8.376145e+01 * time: 766.8266670703888 109 1.183986e+03 7.931813e+01 * time: 774.3262801170349 110 1.183978e+03 7.793645e+01 * time: 781.0351390838623 111 1.183974e+03 7.786675e+01 * time: 787.6291790008545 112 1.183970e+03 7.833168e+01 * time: 794.148992061615 113 1.183963e+03 7.928653e+01 * time: 800.490788936615 114 1.183949e+03 8.083870e+01 * time: 806.875657081604 115 1.183913e+03 8.331100e+01 * time: 813.2905900478363 116 1.183820e+03 8.698958e+01 * time: 819.7315990924835 117 1.183584e+03 9.190093e+01 * time: 826.1487169265747 118 1.183003e+03 9.670514e+01 * time: 832.6487619876862 119 1.181706e+03 9.636548e+01 * time: 839.0561239719391 120 1.179435e+03 8.059042e+01 * time: 845.3655459880829 121 1.177024e+03 4.567337e+01 * time: 851.7256000041962 122 1.175793e+03 1.863452e+01 * time: 858.1589479446411 123 1.175474e+03 1.967979e+01 * time: 864.3829779624939 124 1.175430e+03 1.958939e+01 * time: 870.673730134964 125 1.175427e+03 1.993415e+01 * time: 876.8294229507446 126 1.175427e+03 1.962015e+01 * time: 883.2819080352783 127 1.175426e+03 1.965822e+01 * time: 890.2989299297333 128 1.175424e+03 1.984190e+01 * time: 896.939444065094 129 1.175422e+03 2.000843e+01 * time: 903.6549799442291 130 1.175413e+03 2.034664e+01 * time: 910.2151279449463 131 1.175392e+03 2.083524e+01 * time: 916.7032480239868 132 1.175336e+03 2.163395e+01 * time: 923.1635639667511 133 1.175190e+03 2.286075e+01 * time: 929.634603023529 134 1.174803e+03 2.472708e+01 * time: 936.2510330677032 135 1.173797e+03 4.124367e+01 * time: 942.8426921367645 136 1.171232e+03 7.123154e+01 * time: 949.5383780002594 137 1.165167e+03 1.075848e+02 * time: 957.3163070678711 138 1.155096e+03 7.751300e+01 * time: 964.7580709457397 139 1.149210e+03 6.815707e+01 * time: 972.0830600261688 140 1.146904e+03 6.218499e+01 * time: 979.3557410240173 141 1.144688e+03 5.158716e+01 * time: 986.6182990074158 142 1.143100e+03 2.555104e+01 * time: 993.6984441280365 143 1.141816e+03 1.970245e+01 * time: 1000.8065099716187 144 1.141294e+03 2.048648e+01 * time: 1007.906336069107 145 1.141044e+03 2.076701e+01 * time: 1014.9837980270386 146 1.140979e+03 2.090339e+01 * time: 1022.0832829475403 147 1.140976e+03 2.094515e+01 * time: 1029.0339241027832 148 1.140975e+03 2.099576e+01 * time: 1036.0739920139313 149 1.140974e+03 2.099888e+01 * time: 1043.1141419410706 150 1.140974e+03 2.099931e+01 * time: 1050.135108947754 151 1.140973e+03 2.099295e+01 * time: 1057.1122570037842 152 1.140972e+03 2.098305e+01 * time: 1064.1448090076447 153 1.140970e+03 2.096756e+01 * time: 1071.2270259857178 154 1.140963e+03 2.094619e+01 * time: 1079.986871957779 155 1.140947e+03 2.092087e+01 * time: 1089.2260010242462 156 1.140906e+03 2.090400e+01 * time: 1096.2722990512848 157 1.140797e+03 2.093964e+01 * time: 1106.8431010246277 158 1.140511e+03 2.116231e+01 * time: 1115.0258779525757 159 1.139732e+03 2.193009e+01 * time: 1121.851938009262 160 1.137531e+03 3.209812e+01 * time: 1128.6939299106598 161 1.132353e+03 5.451900e+01 * time: 1135.6939430236816 162 1.130098e+03 6.354058e+01 * time: 1143.1910309791565 ┌ 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: 1150.8915359973907 164 1.127623e+03 8.385757e+01 * time: 1158.3381469249725 165 1.122228e+03 1.204824e+02 * time: 1165.3083460330963 166 1.118774e+03 1.225695e+02 * time: 1172.1833729743958 167 1.115679e+03 4.611268e+01 * time: 1178.971202135086 168 1.113979e+03 3.434831e+01 * time: 1185.7568929195404 169 1.112521e+03 3.523199e+01 * time: 1192.562714099884 170 1.111267e+03 1.228403e+01 * time: 1199.4687149524689 171 1.111189e+03 1.078353e+01 * time: 1206.3638129234314 172 1.111187e+03 1.048242e+01 * time: 1213.0903940200806 173 1.111187e+03 1.070654e+01 * time: 1219.8539700508118 174 1.111187e+03 1.057812e+01 * time: 1226.4931271076202 175 1.111187e+03 1.055136e+01 * time: 1233.2575120925903 176 1.111186e+03 1.049712e+01 * time: 1241.0216619968414 177 1.111185e+03 1.042508e+01 * time: 1247.8158819675446 178 1.111181e+03 1.030463e+01 * time: 1254.5020790100098 179 1.111173e+03 1.018028e+01 * time: 1261.5860290527344 180 1.111150e+03 1.017084e+01 * time: 1268.9996869564056 181 1.111091e+03 1.547976e+01 * time: 1276.3248529434204 182 1.110938e+03 2.528645e+01 * time: 1283.7103209495544 183 1.110546e+03 4.010028e+01 * time: 1291.0377659797668 184 1.109576e+03 6.035708e+01 * time: 1298.412682056427 185 1.107416e+03 8.168777e+01 * time: 1305.7562301158905 186 1.103899e+03 9.586365e+01 * time: 1313.0045490264893 187 1.100518e+03 1.082019e+02 * time: 1320.2475769519806 188 1.096703e+03 1.051881e+02 * time: 1327.4893231391907 189 1.092614e+03 3.371950e+01 * time: 1334.9684300422668 190 1.092108e+03 2.854225e+01 * time: 1342.3084149360657 191 1.091924e+03 8.267155e+00 * time: 1349.379548072815 192 1.091910e+03 4.388021e+00 * time: 1359.1478741168976 193 1.091908e+03 4.381246e+00 * time: 1366.608146905899 194 1.091906e+03 4.351855e+00 * time: 1374.6234240531921 195 1.091905e+03 4.347981e+00 * time: 1382.7338030338287 196 1.091904e+03 4.344942e+00 * time: 1391.199077129364 197 1.091904e+03 4.343567e+00 * time: 1398.7811980247498 198 1.091904e+03 4.341811e+00 * time: 1407.1851029396057 199 1.091904e+03 4.338160e+00 * time: 1414.9101159572601 200 1.091904e+03 4.331286e+00 * time: 1423.3089709281921 201 1.091904e+03 4.319913e+00 * time: 1431.365534067154 202 1.091903e+03 4.299872e+00 * time: 1439.0423829555511 203 1.091901e+03 4.267805e+00 * time: 1446.0664689540863 204 1.091897e+03 4.211166e+00 * time: 1452.9909870624542 205 1.091886e+03 4.118011e+00 * time: 1461.526076078415 206 1.091856e+03 4.884541e+00 * time: 1469.0294330120087 207 1.091780e+03 7.435767e+00 * time: 1476.3803629875183 208 1.091587e+03 1.171416e+01 * time: 1484.5305740833282 209 1.091126e+03 1.720748e+01 * time: 1492.0590479373932 210 1.090066e+03 2.180540e+01 * time: 1499.547033071518 211 1.088566e+03 2.176055e+01 * time: 1506.8953120708466 212 1.087529e+03 2.216379e+01 * time: 1514.4804999828339 213 1.087131e+03 6.871069e+00 * time: 1521.8601710796356 214 1.087053e+03 5.487618e+00 * time: 1529.0764410495758 215 1.087044e+03 5.346767e+00 * time: 1536.4355199337006 216 1.087043e+03 5.364351e+00 * time: 1543.4889059066772 217 1.087043e+03 5.380873e+00 * time: 1550.9210081100464 218 1.087043e+03 5.385017e+00 * time: 1557.981153011322 219 1.087043e+03 5.387076e+00 * time: 1565.2259891033173 220 1.087043e+03 5.393037e+00 * time: 1572.5008299350739 221 1.087043e+03 5.401737e+00 * time: 1579.8430199623108 222 1.087042e+03 5.417220e+00 * time: 1587.0042719841003 223 1.087041e+03 5.443370e+00 * time: 1594.2000019550323 224 1.087039e+03 5.489836e+00 * time: 1601.4302699565887 225 1.087032e+03 5.575602e+00 * time: 1609.1601889133453 226 1.087015e+03 5.744360e+00 * time: 1616.7147190570831 227 1.086967e+03 8.458307e+00 * time: 1624.278650045395 228 1.086833e+03 1.434823e+01 * time: 1631.8946211338043 229 1.086295e+03 2.779692e+01 * time: 1639.6357259750366 ┌ 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: 1650.7464311122894 231 1.085699e+03 4.269436e+01 * time: 1660.8280310630798 232 1.085153e+03 5.101086e+01 * time: 1676.3455090522766 233 1.084625e+03 5.803468e+01 * time: 1687.4738569259644 234 1.084371e+03 6.244198e+01 * time: 1699.0825290679932 235 1.083958e+03 6.785742e+01 * time: 1710.8908619880676 236 1.083163e+03 6.963805e+01 * time: 1720.6300101280212 237 1.081667e+03 6.332470e+01 * time: 1731.2140791416168 238 1.080941e+03 5.916175e+01 * time: 1750.4753930568695 239 1.079241e+03 5.431208e+01 * time: 1761.100774049759 240 1.076096e+03 3.465388e+01 * time: 1771.687518119812 241 1.072649e+03 1.670281e+01 * time: 1780.8012299537659 242 1.069992e+03 1.935661e+01 * time: 1788.6073529720306 243 1.069558e+03 3.678375e+00 * time: 1796.4083559513092 244 1.069496e+03 2.270102e+00 * time: 1804.2123279571533 245 1.069488e+03 2.221349e+00 * time: 1811.8281841278076 246 1.069488e+03 2.237998e+00 * time: 1820.3182721138 247 1.069488e+03 2.236062e+00 * time: 1827.980148077011 248 1.069488e+03 2.235507e+00 * time: 1835.7463490962982 249 1.069488e+03 2.234690e+00 * time: 1843.6164569854736 250 1.069488e+03 2.233625e+00 * time: 1851.464313030243 251 1.069488e+03 2.231719e+00 * time: 1859.7571630477905 252 1.069488e+03 2.228573e+00 * time: 1867.4882979393005 253 1.069487e+03 2.223013e+00 * time: 1875.2372579574585 254 1.069487e+03 2.212970e+00 * time: 1885.185487985611 255 1.069485e+03 2.193875e+00 * time: 1892.7460520267487 256 1.069481e+03 2.558103e+00 * time: 1900.3854820728302 257 1.069470e+03 4.099866e+00 * time: 1909.4473600387573 258 1.069441e+03 6.445399e+00 * time: 1917.183618068695 259 1.069374e+03 9.551161e+00 * time: 1925.2555899620056 260 1.069241e+03 1.204286e+01 * time: 1935.4242289066315 261 1.069065e+03 1.050115e+01 * time: 1943.2474761009216 262 1.068958e+03 5.184001e+00 * time: 1950.9922270774841 263 1.068932e+03 1.288840e+00 * time: 1958.7259039878845 264 1.068929e+03 8.682448e-01 * time: 1966.2240579128265 265 1.068929e+03 8.688495e-01 * time: 1973.8010659217834 266 1.068929e+03 8.688500e-01 * time: 1981.6565670967102 267 1.068929e+03 8.689152e-01 * time: 1989.2063081264496 268 1.068929e+03 8.689157e-01 * time: 1996.95889210701 269 1.068929e+03 8.690133e-01 * time: 2004.4847071170807 270 1.068929e+03 8.690133e-01 * time: 2013.5737841129303 271 1.068929e+03 8.693257e-01 * time: 2022.082277059555 272 1.068929e+03 8.693100e-01 * time: 2029.2785460948944 273 1.068919e+03 9.277276e-01 * time: 2037.3966550827026 274 1.068907e+03 1.626480e+00 * time: 2046.6672599315643 275 1.068835e+03 4.106347e+00 * time: 2057.245882987976 276 1.068726e+03 6.157731e+00 * time: 2068.4691281318665 277 1.068548e+03 7.201969e+00 * time: 2080.8171451091766 278 1.068415e+03 5.646200e+00 * time: 2091.022810935974 279 1.068386e+03 3.803550e+00 * time: 2100.4798350334167 280 1.068384e+03 3.184721e+00 * time: 2112.120849132538 281 1.068384e+03 2.938192e+00 * time: 2123.9794409275055 282 1.068384e+03 2.942962e+00 * time: 2137.305746078491 283 1.068384e+03 2.943184e+00 * time: 2148.965430021286 284 1.068384e+03 2.943716e+00 * time: 2164.621393918991 285 1.068384e+03 2.944531e+00 * time: 2179.9104170799255 286 1.068384e+03 2.944646e+00 * time: 2198.838725090027 287 1.068384e+03 2.944770e+00 * time: 2211.9734830856323 288 1.068384e+03 2.944962e+00 * time: 2227.798793077469 289 1.068384e+03 2.944984e+00 * time: 2240.173574924469 290 1.068384e+03 2.945026e+00 * time: 2252.1921689510345 291 1.068384e+03 2.945033e+00 * time: 2264.480735063553 292 1.068384e+03 2.945039e+00 * time: 2278.1172411441803 293 1.068384e+03 2.945044e+00 * time: 2293.1370990276337 294 1.068384e+03 2.945044e+00 * time: 2304.5833909511566 295 1.068384e+03 2.945044e+00 * time: 2316.9158329963684 296 1.068384e+03 2.945044e+00 * time: 2328.5985469818115 297 1.068384e+03 2.945044e+00 * time: 2337.980791091919 298 1.068384e+03 2.945044e+00 * time: 2346.998826980591 299 1.068384e+03 2.945044e+00 * time: 2356.8434779644012 300 1.068384e+03 2.945044e+00 * time: 2366.515361070633 301 1.068384e+03 2.945044e+00 * time: 2376.1823360919952 302 1.068384e+03 2.945044e+00 * time: 2385.078877925873 303 1.068384e+03 2.945044e+00 * time: 2394.3149909973145 304 1.068384e+03 2.945044e+00 * time: 2409.1285910606384 305 1.068384e+03 3.074478e+00 * time: 2418.3943049907684 306 1.068384e+03 3.057700e+00 * time: 2426.3685591220856 307 1.068384e+03 3.030813e+00 * time: 2434.2940471172333 308 1.068384e+03 3.005745e+00 * time: 2442.023374080658 309 1.068384e+03 2.974044e+00 * time: 2450.4006791114807 310 1.068384e+03 2.945854e+00 * time: 2460.219269990921 311 1.068383e+03 2.905870e+00 * time: 2469.0537810325623 312 1.068382e+03 2.819225e+00 * time: 2484.123179912567 313 1.068381e+03 2.581914e+00 * time: 2495.0115010738373 314 1.068376e+03 1.940126e+00 * time: 2502.665009021759 315 1.068367e+03 1.682057e+00 * time: 2510.273323059082 316 1.068350e+03 2.787006e+00 * time: 2517.8747720718384 317 1.068331e+03 6.870737e+00 * time: 2525.4204199314117 318 1.068322e+03 9.136899e+00 * time: 2532.6881170272827 319 1.068319e+03 9.093888e+00 * time: 2541.096415042877 320 1.068318e+03 8.702840e+00 * time: 2548.1413190364838 321 1.068317e+03 8.067498e+00 * time: 2555.060490131378 322 1.068313e+03 7.024566e+00 * time: 2562.017143011093 323 1.068305e+03 5.566463e+00 * time: 2568.9350731372833 324 1.068294e+03 4.311204e+00 * time: 2576.0539610385895 325 1.068285e+03 4.366941e+00 * time: 2583.1316571235657 326 1.068282e+03 5.297662e+00 * time: 2590.9156980514526 327 1.068282e+03 5.794863e+00 * time: 2600.8535120487213 328 1.068282e+03 5.858759e+00 * time: 2608.492994070053 329 1.068282e+03 5.858759e+00 * time: 2617.479418992996 330 1.068282e+03 5.890965e+00 * time: 2624.998419046402 331 1.068282e+03 5.889642e+00 * time: 2632.6605911254883 332 1.068282e+03 5.755927e+00 * time: 2640.347897052765 333 1.068282e+03 5.654191e+00 * time: 2648.6581149101257 334 1.068281e+03 5.457610e+00 * time: 2655.9265460968018 335 1.068280e+03 5.325147e+00 * time: 2663.280911922455 336 1.068280e+03 5.359845e+00 * time: 2672.979134082794 337 1.068279e+03 5.558458e+00 * time: 2680.3815290927887 338 1.068278e+03 5.765445e+00 * time: 2687.827162027359 339 1.068278e+03 5.998734e+00 * time: 2695.183377981186 340 1.068276e+03 6.340911e+00 * time: 2702.45348405838 341 1.068272e+03 6.826786e+00 * time: 2709.8414130210876 342 1.068262e+03 7.444708e+00 * time: 2717.178615093231 343 1.068237e+03 7.966869e+00 * time: 2724.5161480903625 344 1.068187e+03 7.662772e+00 * time: 2732.061765909195 345 1.068105e+03 5.453572e+00 * time: 2740.41206908226 346 1.068030e+03 2.116637e+00 * time: 2747.8398559093475 347 1.068004e+03 2.672269e-01 * time: 2755.5182321071625 348 1.068001e+03 4.076140e-01 * time: 2763.127718925476 349 1.067999e+03 5.992600e-01 * time: 2770.8160939216614 350 1.067997e+03 5.810130e-01 * time: 2778.35297703743 351 1.067994e+03 3.112045e-01 * time: 2786.5953829288483 352 1.067992e+03 7.689723e-02 * time: 2796.2118039131165 353 1.067992e+03 9.091969e-02 * time: 2806.3966660499573 354 1.067992e+03 9.785125e-02 * time: 2818.222280025482 355 1.067992e+03 1.024830e-01 * time: 2829.2268121242523 356 1.067992e+03 1.051058e-01 * time: 2839.9516320228577 357 1.067992e+03 1.057116e-01 * time: 2848.9413590431213 358 1.067992e+03 1.057207e-01 * time: 2858.4513239860535 359 1.067992e+03 1.057207e-01 * time: 2872.8086800575256 360 1.067992e+03 1.057044e-01 * time: 2888.176573038101 361 1.067992e+03 1.056554e-01 * time: 2899.0949490070343 362 1.067992e+03 1.056554e-01 * time: 2910.5532641410828 363 1.067992e+03 1.052666e-01 * time: 2923.9838371276855 364 1.067992e+03 1.052895e-01 * time: 2940.477483034134 365 1.067992e+03 1.052895e-01 * time: 2959.7988369464874 366 1.067992e+03 1.052895e-01 * time: 2977.7432510852814
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: 4.9114227294921875e-5 1 5.174461e+05 8.708698e+05 * time: 3.28534197807312 2 3.865265e+05 6.344302e+05 * time: 6.624495983123779 3 1.804274e+05 2.829723e+05 * time: 9.015691995620728 4 9.706640e+04 1.550547e+05 * time: 12.425055027008057 5 4.769637e+04 6.778818e+04 * time: 14.47299599647522 6 2.902319e+04 3.499747e+04 * time: 17.126474142074585 7 1.823472e+04 1.705751e+04 * time: 18.949612140655518 8 1.258819e+04 9.569381e+03 * time: 21.18776297569275 9 9.389984e+03 8.615851e+03 * time: 24.231640100479126 10 7.314702e+03 7.636883e+03 * time: 27.641842126846313 11 5.916029e+03 6.624325e+03 * time: 30.89994215965271 12 4.930519e+03 5.558140e+03 * time: 34.41958713531494 13 4.125060e+03 4.315759e+03 * time: 37.01551699638367 14 3.549280e+03 3.051093e+03 * time: 38.910706996917725 15 3.283489e+03 2.157292e+03 * time: 40.843899965286255 16 3.204886e+03 1.659798e+03 * time: 42.58100509643555 17 3.194875e+03 1.480528e+03 * time: 44.34722900390625 18 3.193944e+03 1.437921e+03 * time: 46.056424140930176 19 3.193070e+03 1.411186e+03 * time: 47.82005500793457 20 3.190129e+03 1.355327e+03 * time: 49.63676309585571 21 3.183228e+03 1.276603e+03 * time: 51.85410499572754 22 3.164897e+03 1.151838e+03 * time: 53.608317136764526 23 3.119250e+03 9.712651e+02 * time: 55.51257801055908 24 3.006297e+03 7.204342e+02 * time: 58.33063817024231 25 2.738913e+03 4.050545e+02 * time: 61.28430700302124 26 2.123834e+03 2.318194e+02 * time: 63.04982113838196 27 1.789139e+03 2.290465e+02 * time: 64.76023316383362 ┌ 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: 75.8752830028534 29 1.380652e+03 1.621863e+02 * time: 82.60585713386536 30 1.298107e+03 1.130479e+02 * time: 84.21117305755615 31 1.271740e+03 2.292541e+02 * time: 86.05455613136292 32 1.255835e+03 1.721669e+02 * time: 87.56385016441345 33 1.251089e+03 1.813354e+02 * time: 88.9577100276947 34 1.243725e+03 1.920537e+02 * time: 90.26416206359863 35 1.241416e+03 1.835909e+02 * time: 91.55497717857361 36 1.240510e+03 1.715569e+02 * time: 92.83500409126282 37 1.240496e+03 1.710861e+02 * time: 94.10288500785828 38 1.240467e+03 1.703009e+02 * time: 95.39114499092102 39 1.240383e+03 1.685321e+02 * time: 96.67335915565491 40 1.240176e+03 1.649610e+02 * time: 97.94923400878906 41 1.239647e+03 1.571549e+02 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307 1.068239e+03 8.485885e+00 * time: 499.50799107551575 308 1.068153e+03 6.787305e+00 * time: 501.37801599502563 309 1.068076e+03 2.962239e+00 * time: 503.20950198173523 310 1.068028e+03 3.648163e-01 * time: 504.9974420070648 311 1.068010e+03 6.696155e-01 * time: 507.3346540927887 312 1.068002e+03 6.065508e-01 * time: 509.784912109375 313 1.067996e+03 2.826797e-01 * time: 511.76007294654846 314 1.067993e+03 2.585896e-01 * time: 514.159383058548 315 1.067992e+03 2.949494e-01 * time: 516.0277111530304 316 1.067992e+03 3.200153e-01 * time: 517.8984501361847 317 1.067992e+03 3.366476e-01 * time: 519.843670129776 318 1.067992e+03 3.446883e-01 * time: 523.0763230323792 319 1.067992e+03 3.468853e-01 * time: 525.1016550064087 320 1.067992e+03 3.472675e-01 * time: 527.3784439563751 321 1.067992e+03 3.472738e-01 * time: 529.6928460597992 322 1.067992e+03 3.473614e-01 * time: 531.9515571594238 323 1.067992e+03 3.473715e-01 * time: 534.5297420024872 324 1.067992e+03 3.473714e-01 * time: 536.4936339855194 325 1.067992e+03 3.473224e-01 * time: 539.5539650917053 326 1.067992e+03 3.473228e-01 * time: 541.7922141551971 327 1.067992e+03 3.473221e-01 * time: 543.9502620697021 328 1.067992e+03 3.399790e-01 * time: 546.0242841243744 329 1.067992e+03 3.362705e-01 * time: 547.9090449810028 330 1.067992e+03 3.005757e-01 * time: 549.7174179553986 331 1.067992e+03 2.429240e-01 * time: 551.5789210796356 332 1.067991e+03 1.361280e-01 * time: 553.4982681274414 333 1.067991e+03 1.071745e-01 * time: 555.3499269485474 334 1.067991e+03 9.741600e-02 * time: 556.9606239795685 335 1.067991e+03 9.434333e-02 * time: 559.1467590332031 336 1.067991e+03 9.456144e-02 * time: 561.0986039638519 337 1.067991e+03 9.456602e-02 * time: 563.0062980651855 338 1.067991e+03 9.457219e-02 * time: 564.9513821601868 339 1.067991e+03 9.457802e-02 * time: 567.0127429962158 340 1.067991e+03 9.458467e-02 * time: 568.8960750102997 341 1.067991e+03 9.459192e-02 * time: 570.8002710342407 342 1.067991e+03 9.459264e-02 * time: 572.6477589607239 343 1.067991e+03 9.459354e-02 * time: 575.0698771476746 344 1.067991e+03 9.459363e-02 * time: 577.1101710796356 345 1.067991e+03 9.459380e-02 * time: 580.0148050785065 346 1.067991e+03 9.459391e-02 * time: 583.4739320278168 347 1.067991e+03 9.459411e-02 * time: 585.7895300388336 348 1.067991e+03 9.459412e-02 * time: 587.7760901451111 349 1.067991e+03 9.459414e-02 * time: 589.9174671173096 350 1.067991e+03 9.459420e-02 * time: 591.9095711708069 351 1.067991e+03 9.459421e-02 * time: 593.8837850093842 352 1.067991e+03 9.459423e-02 * time: 595.8362641334534 353 1.067991e+03 9.459423e-02 * time: 598.1158490180969 354 1.067991e+03 9.459424e-02 * time: 602.1133010387421 355 1.067991e+03 9.459424e-02 * time: 605.0288171768188 356 1.067991e+03 9.459424e-02 * time: 607.6077451705933 357 1.067991e+03 9.459424e-02 * time: 609.6037640571594 358 1.067991e+03 9.459425e-02 * time: 611.6056251525879 359 1.067991e+03 9.459425e-02 * time: 613.6342561244965 360 1.067991e+03 9.459425e-02 * time: 615.6277520656586 361 1.067991e+03 9.459425e-02 * time: 618.2075991630554 362 1.067991e+03 9.459426e-02 * time: 620.5964889526367 363 1.067991e+03 9.459426e-02 * time: 623.0132110118866 364 1.067991e+03 9.459426e-02 * time: 625.4707181453705 365 1.067991e+03 9.459426e-02 * time: 627.5397300720215 366 1.067991e+03 9.459426e-02 * time: 629.6662740707397 367 1.067991e+03 9.459426e-02 * time: 631.7079780101776 368 1.067991e+03 9.459426e-02 * time: 633.9353420734406 369 1.067991e+03 9.459426e-02 * time: 637.1764481067657 370 1.067991e+03 9.459426e-02 * time: 640.0038540363312 371 1.067991e+03 9.459426e-02 * time: 642.1507339477539 372 1.067991e+03 9.459426e-02 * time: 642.6700010299683
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.