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 ~ @. CombinedNormal(cp, σ_add, σ_prop)
"""
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 3.130049e+06 5.915673e+06 * time: 0.10085296630859375 1 5.185947e+05 8.741866e+05 * time: 31.95722794532776 2 3.866970e+05 6.366598e+05 * time: 33.38027906417847 3 1.795038e+05 2.835250e+05 * time: 33.979835987091064 4 9.682944e+04 1.546505e+05 * time: 34.59925889968872 5 4.791895e+04 6.819657e+04 * time: 35.20427703857422 6 2.907473e+04 3.509578e+04 * time: 35.8135130405426 7 1.827466e+04 1.713041e+04 * time: 36.413214921951294 8 1.260726e+04 9.564194e+03 * time: 36.99838900566101 9 9.403896e+03 8.612114e+03 * time: 37.54739999771118 10 7.326086e+03 7.634997e+03 * time: 38.088762044906616 11 5.927094e+03 6.625137e+03 * time: 38.59825801849365 12 4.942762e+03 5.562953e+03 * time: 39.14595890045166 13 4.139610e+03 4.326715e+03 * time: 39.62843990325928 14 3.563998e+03 3.065221e+03 * time: 40.12851095199585 15 3.296712e+03 2.170265e+03 * time: 40.61025810241699 16 3.216830e+03 1.669547e+03 * time: 41.10696196556091 17 3.206499e+03 1.487841e+03 * time: 41.588062047958374 18 3.205537e+03 1.444242e+03 * time: 42.09421110153198 19 3.204662e+03 1.417622e+03 * time: 42.61991596221924 20 3.201697e+03 1.361560e+03 * time: 43.163373947143555 21 3.194761e+03 1.282787e+03 * time: 43.667803049087524 22 3.176327e+03 1.157868e+03 * time: 44.181999921798706 23 3.130481e+03 9.772233e+02 * time: 44.680757999420166 24 3.017242e+03 7.263321e+02 * time: 45.15919804573059 25 2.750133e+03 4.106853e+02 * time: 45.6402530670166 26 2.137975e+03 2.318410e+02 * time: 46.0722930431366 27 1.357961e+03 1.422407e+02 * time: 46.774665117263794 28 1.296792e+03 2.619606e+02 * time: 47.43133997917175 29 1.276960e+03 1.838772e+02 * time: 47.817694902420044 30 1.267251e+03 1.950227e+02 * time: 48.1839759349823 31 1.246464e+03 2.121277e+02 * time: 48.58478903770447 32 1.244107e+03 1.960492e+02 * time: 48.979257106781006 33 1.243941e+03 1.908395e+02 * time: 49.3779571056366 34 1.243927e+03 1.905050e+02 * time: 49.78055000305176 35 1.243788e+03 1.876624e+02 * time: 50.16678595542908 36 1.243545e+03 1.837093e+02 * time: 50.56037497520447 37 1.242827e+03 1.738853e+02 * time: 50.97185707092285 38 1.241221e+03 1.547074e+02 * time: 51.38754391670227 39 1.237772e+03 1.170472e+02 * time: 51.78935503959656 40 1.232408e+03 5.798690e+01 * time: 52.191704988479614 41 1.227950e+03 4.597807e+01 * time: 52.5912880897522 42 1.226363e+03 5.777629e+01 * time: 52.99941802024841 43 1.226084e+03 5.151307e+01 * time: 53.43212008476257 44 1.226077e+03 4.917111e+01 * time: 53.868854999542236 45 1.226076e+03 4.892725e+01 * time: 54.283987045288086 46 1.226069e+03 4.758509e+01 * time: 54.69982099533081 47 1.226057e+03 4.572770e+01 * time: 55.11691999435425 48 1.226020e+03 4.166826e+01 * time: 55.53787112236023 49 1.225933e+03 3.401203e+01 * time: 55.959665060043335 50 1.225728e+03 2.835333e+01 * time: 56.358500957489014 51 1.225334e+03 2.817036e+01 * time: 56.75296401977539 52 1.224813e+03 3.424252e+01 * time: 57.15521311759949 53 1.224477e+03 4.798431e+01 * time: 57.5795259475708 54 1.224395e+03 4.437361e+01 * time: 57.97052597999573 55 1.224386e+03 3.979127e+01 * time: 58.35424304008484 56 1.224385e+03 3.851708e+01 * time: 58.73781895637512 57 1.224383e+03 3.680673e+01 * time: 59.11666703224182 58 1.224376e+03 3.402533e+01 * time: 59.51831793785095 59 1.224359e+03 3.268119e+01 * time: 59.90696310997009 60 1.224314e+03 3.090784e+01 * time: 60.28792691230774 61 1.224196e+03 2.816919e+01 * time: 60.67724895477295 62 1.223896e+03 2.822262e+01 * time: 61.06475806236267 63 1.223162e+03 3.872359e+01 * time: 61.46628403663635 64 1.221566e+03 7.595349e+01 * time: 61.8562068939209 65 1.219004e+03 9.939356e+01 * time: 62.26878595352173 66 1.216625e+03 7.736576e+01 * time: 62.69918394088745 67 1.215517e+03 7.064562e+01 * time: 63.108803033828735 68 1.215381e+03 6.708583e+01 * time: 63.509095907211304 69 1.215375e+03 6.675536e+01 * time: 63.888076066970825 70 1.215371e+03 6.640207e+01 * time: 64.2688980102539 71 1.215359e+03 6.551217e+01 * time: 64.65693402290344 72 1.215331e+03 6.395606e+01 * time: 65.0343689918518 73 1.215255e+03 6.066088e+01 * time: 65.41432094573975 74 1.215062e+03 5.381643e+01 * time: 65.81467700004578 75 1.214585e+03 3.902384e+01 * time: 66.19644403457642 76 1.213513e+03 5.729613e+01 * time: 66.5833740234375 77 1.211585e+03 7.543450e+01 * time: 66.9708399772644 78 1.209446e+03 1.047478e+02 * time: 67.35754299163818 79 1.208363e+03 1.260683e+02 * time: 67.76032209396362 80 1.208147e+03 1.160237e+02 * time: 68.1766710281372 81 1.208132e+03 1.100932e+02 * time: 68.55112290382385 82 1.208127e+03 1.084514e+02 * time: 68.92857193946838 83 1.208106e+03 1.039380e+02 * time: 69.30905795097351 84 1.208062e+03 9.804794e+01 * time: 69.70384097099304 85 1.207936e+03 8.751468e+01 * time: 70.08549308776855 86 1.207621e+03 7.057805e+01 * time: 70.4692919254303 87 1.206819e+03 6.533132e+01 * time: 70.8565559387207 88 1.204953e+03 5.687190e+01 * time: 71.26108407974243 89 1.201401e+03 5.997970e+01 * time: 71.66377401351929 90 1.197045e+03 8.595365e+01 * time: 72.04711508750916 91 1.193720e+03 5.814563e+01 * time: 72.44187903404236 92 1.192424e+03 4.918237e+01 * time: 72.82445502281189 93 1.192332e+03 5.056415e+01 * time: 73.20462012290955 94 1.192328e+03 5.064817e+01 * time: 73.64135003089905 95 1.192328e+03 5.069435e+01 * time: 74.0106770992279 96 1.192324e+03 5.077239e+01 * time: 74.39533710479736 97 1.192315e+03 5.089603e+01 * time: 74.79334998130798 98 1.192292e+03 5.107354e+01 * time: 75.18372201919556 99 1.192233e+03 5.128045e+01 * time: 75.58056402206421 100 1.192079e+03 5.138224e+01 * time: 75.99807810783386 101 1.191690e+03 5.093178e+01 * time: 76.38790512084961 102 1.190747e+03 6.233310e+01 * time: 76.76628303527832 103 1.188711e+03 8.393857e+01 * time: 77.1613199710846 104 1.185262e+03 8.919115e+01 * time: 77.57585096359253 105 1.181507e+03 6.478905e+01 * time: 78.0280430316925 106 1.179507e+03 6.342536e+01 * time: 78.47319197654724 107 1.179066e+03 5.651175e+01 * time: 78.92499899864197 108 1.179007e+03 5.355351e+01 * time: 79.39091396331787 109 1.178989e+03 5.342382e+01 * time: 79.82625102996826 110 1.178981e+03 5.452378e+01 * time: 80.24451088905334 111 1.178978e+03 5.571710e+01 * time: 80.65409207344055 112 1.178976e+03 5.645145e+01 * time: 81.0625069141388 113 1.178974e+03 5.703539e+01 * time: 81.486407995224 114 1.178968e+03 5.789071e+01 * time: 81.88936305046082 115 1.178951e+03 5.907442e+01 * time: 82.2967050075531 116 1.178908e+03 6.064947e+01 * time: 82.71039199829102 117 1.178799e+03 6.229592e+01 * time: 83.13639211654663 118 1.178533e+03 6.273385e+01 * time: 83.57649397850037 119 1.177947e+03 5.849461e+01 * time: 83.99810194969177 120 1.176912e+03 4.388558e+01 * time: 84.42919611930847 121 1.175763e+03 2.091799e+01 * time: 84.88180804252625 122 1.175177e+03 2.044403e+01 * time: 85.3150749206543 123 1.175045e+03 2.016046e+01 * time: 85.76054501533508 124 1.175028e+03 2.019076e+01 * time: 86.20812606811523 125 1.175028e+03 2.022824e+01 * time: 86.62758994102478 126 1.175027e+03 2.024135e+01 * time: 87.06428790092468 127 1.175026e+03 2.027385e+01 * time: 87.47796702384949 128 1.175023e+03 2.030820e+01 * time: 87.88658905029297 129 1.175015e+03 2.036436e+01 * time: 88.31808805465698 130 1.174993e+03 2.043864e+01 * time: 88.73513007164001 131 1.174937e+03 2.054254e+01 * time: 89.15437889099121 132 1.174789e+03 2.067532e+01 * time: 89.59600496292114 133 1.174399e+03 2.080795e+01 * time: 90.01402711868286 134 1.173370e+03 2.144892e+01 * time: 90.45199704170227 135 1.170694e+03 3.548471e+01 * time: 90.90973806381226 136 1.164233e+03 5.629167e+01 * time: 91.37341809272766 137 1.154010e+03 6.151737e+01 * time: 91.89041304588318 138 1.150573e+03 1.639707e+02 * time: 92.41977190971375 139 1.147394e+03 2.917842e+01 * time: 93.13551592826843 140 1.145496e+03 2.863275e+01 * time: 93.65010809898376 141 1.143192e+03 3.094888e+01 * time: 94.13974595069885 142 1.142078e+03 2.111828e+01 * time: 94.60862302780151 143 1.141269e+03 2.280164e+01 * time: 95.10577297210693 144 1.140892e+03 2.102961e+01 * time: 95.5939519405365 145 1.140884e+03 2.116517e+01 * time: 96.05637502670288 146 1.140881e+03 2.120892e+01 * time: 96.53991889953613 147 1.140875e+03 2.117831e+01 * time: 97.62418508529663 148 1.140876e+03 2.117287e+01 * time: 98.84433007240295 149 1.140877e+03 2.116565e+01 * time: 100.18232703208923 150 1.140878e+03 2.115760e+01 * time: 101.80237102508545 151 1.140878e+03 2.108868e+01 * time: 102.33382201194763 152 1.140879e+03 2.106545e+01 * time: 102.84422707557678 153 1.140877e+03 2.099005e+01 * time: 103.39663600921631 154 1.140869e+03 2.088107e+01 * time: 103.90508198738098 155 1.140851e+03 2.072350e+01 * time: 104.40935897827148 156 1.140805e+03 2.050201e+01 * time: 104.9587140083313 157 1.140672e+03 2.020905e+01 * time: 105.4999930858612 158 1.140327e+03 1.993126e+01 * time: 106.06782007217407 159 1.139343e+03 2.220619e+01 * time: 106.60322999954224 160 1.136580e+03 3.909688e+01 * time: 107.16822910308838 161 1.131533e+03 6.943445e+01 * time: 107.72238206863403 162 1.128306e+03 8.886539e+01 * time: 108.7072479724884 163 1.119536e+03 1.154839e+02 * time: 109.25799202919006 164 1.114376e+03 8.035326e+01 * time: 110.09401106834412 165 1.112204e+03 6.285092e+01 * time: 110.6754059791565 166 1.111325e+03 9.747779e+00 * time: 111.29452800750732 167 1.111283e+03 1.107469e+01 * time: 112.06412696838379 168 1.111254e+03 1.023420e+01 * time: 112.6079330444336 169 1.111253e+03 1.079507e+01 * time: 113.19580912590027 170 1.111254e+03 1.073945e+01 * time: 114.42395901679993 171 1.111255e+03 1.068732e+01 * time: 123.97819805145264 172 1.111252e+03 1.060491e+01 * time: 133.29102110862732 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/z6105/src/multivariate/optimize/optimize.jl:153 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/z6105/src/multivariate/optimize/optimize.jl:153 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/z6105/src/multivariate/optimize/optimize.jl:153 173 1.110664e+03 3.083628e+01 * time: 175.7226219177246 174 1.110332e+03 4.730705e+01 * time: 176.67596793174744 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/z6105/src/multivariate/optimize/optimize.jl:153 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/z6105/src/multivariate/optimize/optimize.jl:153 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/z6105/src/multivariate/optimize/optimize.jl:153 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/z6105/src/multivariate/optimize/optimize.jl:153 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/z6105/src/multivariate/optimize/optimize.jl:153 175 1.106751e+03 7.819364e+01 * time: 200.38744401931763 176 1.105947e+03 9.045065e+01 * time: 201.3515899181366 177 1.101707e+03 3.557938e+01 * time: 202.21357893943787 178 1.099097e+03 2.521659e+01 * time: 207.54717803001404 179 1.097610e+03 2.247338e+01 * time: 209.39091897010803 180 1.096552e+03 2.038806e+01 * time: 210.36381793022156 181 1.095186e+03 1.994325e+01 * time: 211.8545560836792 182 1.094889e+03 1.775126e+01 * time: 212.7726559638977 183 1.090935e+03 1.213116e+02 * time: 213.61496806144714 184 1.088311e+03 6.837435e+01 * time: 214.4737319946289 185 1.080704e+03 3.224134e+01 * time: 215.0809919834137 186 1.079956e+03 3.674062e+01 * time: 216.03220200538635 187 1.076150e+03 8.858208e+01 * time: 216.7093780040741 188 1.072169e+03 6.023963e+01 * time: 217.36745405197144 189 1.068995e+03 7.602314e+00 * time: 218.03085803985596 190 1.068636e+03 1.100790e+01 * time: 218.87113404273987 191 1.068473e+03 8.239482e+00 * time: 219.57624697685242 192 1.068406e+03 2.015959e+00 * time: 220.3769989013672 193 1.068361e+03 3.813509e+00 * time: 221.10253810882568 194 1.068328e+03 6.558561e+00 * time: 222.1658890247345 195 1.068241e+03 8.551467e+00 * time: 223.0452959537506 196 1.068151e+03 5.769737e+00 * time: 223.7558469772339 197 1.068071e+03 4.657982e+00 * time: 224.59823489189148 198 1.068036e+03 3.841275e+00 * time: 225.41302394866943 199 1.068021e+03 2.649143e+00 * time: 226.112459897995 200 1.068001e+03 3.885827e+00 * time: 226.94122910499573 201 1.067999e+03 5.022667e-01 * time: 227.59934401512146 202 1.068000e+03 6.120586e-01 * time: 239.20092797279358 203 1.067998e+03 6.225937e-01 * time: 251.6999180316925
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: -1067.9978
-----------------------
Estimate
-----------------------
pop_CL 0.13522
pop_V 8.0114
pop_tabs 0.57138
pop_lag 0.87464
pop_e0 96.409
pop_emax -1.06
pop_c50 1.4826
pop_tover 14.008
pk_Ω₁,₁ 0.067892
pk_Ω₂,₂ 0.021036
pk_Ω₃,₃ 0.84945
pd_Ω₁,₁ 0.0029348
pd_Ω₂,₂ 2.5044e-6
pd_Ω₃,₃ 0.14671
pd_Ω₄,₄ 0.015478
σ_prop 0.088415
σ_add 0.41724
σ_fx 3.581
-----------------------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: 2.5033950805664062e-5 1 5.174461e+05 8.708698e+05 * time: 1.5654969215393066 2 3.865265e+05 6.344302e+05 * time: 2.9129581451416016 3 1.804274e+05 2.829723e+05 * time: 4.104030132293701 4 9.706640e+04 1.550547e+05 * time: 5.3237950801849365 5 4.769637e+04 6.778817e+04 * time: 6.516402959823608 6 2.902319e+04 3.499747e+04 * time: 7.609928131103516 7 1.823472e+04 1.705751e+04 * time: 8.730916976928711 8 1.258819e+04 9.569381e+03 * time: 9.939985036849976 9 9.389984e+03 8.615851e+03 * time: 11.066648006439209 10 7.314702e+03 7.636883e+03 * time: 12.179367065429688 11 5.916029e+03 6.624325e+03 * time: 13.29023003578186 12 4.930519e+03 5.558140e+03 * time: 14.400984048843384 13 4.125060e+03 4.315759e+03 * time: 15.518592119216919 14 3.549280e+03 3.051093e+03 * time: 16.719094038009644 15 3.283489e+03 2.157292e+03 * time: 17.804913997650146 16 3.204886e+03 1.659798e+03 * time: 18.85201406478882 17 3.194875e+03 1.480528e+03 * time: 19.96847891807556 18 3.193944e+03 1.437921e+03 * time: 21.102582931518555 19 3.193070e+03 1.411186e+03 * time: 22.187807083129883 20 3.190129e+03 1.355327e+03 * time: 23.34429097175598 21 3.183228e+03 1.276603e+03 * time: 24.608520030975342 22 3.164897e+03 1.151838e+03 * time: 25.74379801750183 23 3.119250e+03 9.712651e+02 * time: 26.949666023254395 24 3.006297e+03 7.204342e+02 * time: 28.059625148773193 25 2.738913e+03 4.050545e+02 * time: 29.104063034057617 26 2.123834e+03 2.318194e+02 * time: 30.008044004440308 27 1.350274e+03 1.386111e+02 * time: 31.375566005706787 28 1.292769e+03 2.316810e+02 * time: 32.86082100868225 29 1.273651e+03 1.664584e+02 * time: 33.74436593055725 30 1.263845e+03 1.764773e+02 * time: 34.633508920669556 31 1.243264e+03 1.918458e+02 * time: 35.485023975372314 32 1.240721e+03 1.767778e+02 * time: 36.34144401550293 33 1.240528e+03 1.714649e+02 * time: 37.18366312980652 34 1.240516e+03 1.711394e+02 * time: 38.01062202453613 35 1.240469e+03 1.701233e+02 * time: 38.876343965530396 36 1.240367e+03 1.683191e+02 * time: 39.74546813964844 37 1.240086e+03 1.640882e+02 * time: 40.62927198410034 38 1.239406e+03 1.549688e+02 * time: 41.578848123550415 39 1.237810e+03 1.353098e+02 * time: 42.48259496688843 40 1.234651e+03 9.788573e+01 * time: 43.38705611228943 41 1.230250e+03 4.266570e+01 * time: 44.23580002784729 42 1.227165e+03 4.913304e+01 * time: 45.08145308494568 43 1.226203e+03 5.435433e+01 * time: 45.91524910926819 44 1.226090e+03 4.781720e+01 * time: 46.74526596069336 45 1.226088e+03 4.634324e+01 * time: 47.59111404418945 46 1.226087e+03 4.601513e+01 * time: 48.48559093475342 47 1.226081e+03 4.453279e+01 * time: 49.32145094871521 48 1.226069e+03 4.248078e+01 * time: 50.185007095336914 49 1.226037e+03 3.820894e+01 * time: 51.03336501121521 50 1.225957e+03 3.039316e+01 * time: 51.90381908416748 51 1.225769e+03 2.845915e+01 * time: 52.74452209472656 52 1.225398e+03 2.944618e+01 * time: 53.57852792739868 53 1.224884e+03 3.649371e+01 * time: 54.43102502822876 54 1.224523e+03 5.005295e+01 * time: 55.277178049087524 55 1.224424e+03 4.581771e+01 * time: 56.18644714355469 56 1.224413e+03 4.045804e+01 * time: 57.069414138793945 57 1.224411e+03 3.898760e+01 * time: 57.899819135665894 58 1.224409e+03 3.740817e+01 * time: 58.73792099952698 59 1.224403e+03 3.465864e+01 * time: 59.59060096740723 60 1.224387e+03 3.207203e+01 * time: 60.44272494316101 61 1.224344e+03 3.010607e+01 * time: 61.33860206604004 62 1.224234e+03 2.815612e+01 * time: 62.187644958496094 63 1.223949e+03 2.819773e+01 * time: 63.06050205230713 64 1.223237e+03 3.622961e+01 * time: 63.90958094596863 65 1.221600e+03 7.594055e+01 * time: 64.75888705253601 66 1.218599e+03 1.106912e+02 * time: 65.6617419719696 67 1.215109e+03 1.024965e+02 * time: 66.52719807624817 68 1.212822e+03 8.616050e+01 * time: 67.39629292488098 69 1.212285e+03 8.967803e+01 * time: 68.28398609161377 70 1.212264e+03 8.932132e+01 * time: 69.12013006210327 71 1.212260e+03 8.896522e+01 * time: 69.93823003768921 72 1.212244e+03 8.763366e+01 * time: 70.78216600418091 73 1.212211e+03 8.567001e+01 * time: 71.64305806159973 74 1.212117e+03 8.128684e+01 * time: 72.47710013389587 75 1.211887e+03 7.260216e+01 * time: 73.34248805046082 76 1.211324e+03 5.425869e+01 * time: 74.20944595336914 77 1.210115e+03 6.776024e+01 * time: 75.08428597450256 78 1.208108e+03 8.001108e+01 * time: 75.93232011795044 79 1.206180e+03 9.050256e+01 * time: 76.79371905326843 80 1.205414e+03 9.882265e+01 * time: 77.66950798034668 81 1.205304e+03 8.686897e+01 * time: 78.5058810710907 82 1.205297e+03 8.274517e+01 * time: 79.3553159236908 83 1.205293e+03 8.123277e+01 * time: 80.25385904312134 84 1.205277e+03 7.756801e+01 * time: 81.13073396682739 85 1.205243e+03 7.268799e+01 * time: 81.9905469417572 86 1.205146e+03 6.634996e+01 * time: 82.84709310531616 87 1.204904e+03 6.323779e+01 * time: 83.68327307701111 88 1.204289e+03 5.674357e+01 * time: 84.55984306335449 89 1.202870e+03 5.241928e+01 * time: 85.51994514465332 90 1.200212e+03 5.141237e+01 * time: 86.36468696594238 91 1.197000e+03 7.220230e+01 * time: 87.3042299747467 92 1.194663e+03 4.802449e+01 * time: 88.24461197853088 93 1.193885e+03 4.877522e+01 * time: 89.12818098068237 94 1.193833e+03 4.999065e+01 * time: 89.92944312095642 95 1.193831e+03 5.008260e+01 * time: 90.7244040966034 96 1.193830e+03 5.013799e+01 * time: 91.51061010360718 97 1.193824e+03 5.023264e+01 * time: 92.30193710327148 98 1.193812e+03 5.037089e+01 * time: 93.1235089302063 99 1.193778e+03 5.056811e+01 * time: 94.02658104896545 100 1.193690e+03 5.079476e+01 * time: 94.90433192253113 101 1.193465e+03 5.091049e+01 * time: 95.75796914100647 102 1.192908e+03 5.045173e+01 * time: 96.64663410186768 103 1.191643e+03 4.826733e+01 * time: 97.51607203483582 104 1.189287e+03 4.966444e+01 * time: 98.47149705886841 105 1.186354e+03 8.019652e+01 * time: 99.41906809806824 106 1.184499e+03 8.997361e+01 * time: 100.25601291656494 107 1.184032e+03 8.409142e+01 * time: 101.09798407554626 108 1.183980e+03 7.941840e+01 * time: 101.92823100090027 109 1.183972e+03 7.800909e+01 * time: 102.81652903556824 110 1.183969e+03 7.789073e+01 * time: 103.6474039554596 111 1.183965e+03 7.827288e+01 * time: 104.47205209732056 112 1.183957e+03 7.914384e+01 * time: 105.33942008018494 113 1.183941e+03 8.062266e+01 * time: 106.26173400878906 114 1.183902e+03 8.300645e+01 * time: 107.15159201622009 115 1.183800e+03 8.655837e+01 * time: 108.02165603637695 116 1.183541e+03 9.123920e+01 * time: 108.88705611228943 117 1.182906e+03 9.549885e+01 * time: 109.7315719127655 118 1.181503e+03 9.386248e+01 * time: 110.57409310340881 119 1.179134e+03 7.587261e+01 * time: 111.6832001209259 120 1.176807e+03 4.015019e+01 * time: 112.56940412521362 121 1.175726e+03 1.893891e+01 * time: 113.48626708984375 122 1.175462e+03 1.993836e+01 * time: 114.40321803092957 123 1.175429e+03 1.972240e+01 * time: 115.3362820148468 124 1.175427e+03 1.981999e+01 * time: 116.25580310821533 125 1.175426e+03 1.966086e+01 * time: 117.1625280380249 126 1.175426e+03 1.968277e+01 * time: 118.0519289970398 127 1.175424e+03 1.968319e+01 * time: 118.94418907165527 128 1.175421e+03 1.974292e+01 * time: 119.79079699516296 129 1.175412e+03 1.984730e+01 * time: 120.65187406539917 130 1.175390e+03 2.002879e+01 * time: 121.51699209213257 131 1.175331e+03 2.031908e+01 * time: 122.40993809700012 132 1.175176e+03 2.075996e+01 * time: 123.28043508529663 133 1.174766e+03 2.137371e+01 * time: 124.17863512039185 134 1.173682e+03 2.425284e+01 * time: 125.13869595527649 135 1.170870e+03 4.399728e+01 * time: 126.10479998588562 136 1.164080e+03 6.850329e+01 * time: 127.13975405693054 137 1.153404e+03 5.019046e+01 * time: 128.2863941192627 138 1.148838e+03 1.017073e+02 * time: 129.42170095443726 139 1.146757e+03 4.472171e+01 * time: 130.733225107193 140 1.145077e+03 2.419303e+01 * time: 131.72443008422852 141 1.142868e+03 4.737016e+01 * time: 132.65195202827454 142 1.142013e+03 3.798302e+01 * time: 133.6248381137848 143 1.141124e+03 2.039545e+01 * time: 134.65999698638916 144 1.140996e+03 2.088957e+01 * time: 135.68250703811646 145 1.140977e+03 2.088137e+01 * time: 136.67269802093506 146 1.140975e+03 2.100483e+01 * time: 137.61280608177185 147 1.140975e+03 2.099463e+01 * time: 138.50802397727966 148 1.140975e+03 2.098670e+01 * time: 139.40089797973633 149 1.140974e+03 2.098287e+01 * time: 140.32732105255127 150 1.140973e+03 2.097843e+01 * time: 141.23241209983826 151 1.140970e+03 2.097297e+01 * time: 142.12967705726624 152 1.140964e+03 2.096408e+01 * time: 143.02989411354065 153 1.140946e+03 2.094990e+01 * time: 143.96359992027283 154 1.140902e+03 2.092943e+01 * time: 144.87747311592102 155 1.140790e+03 2.090361e+01 * time: 145.78244996070862 156 1.140511e+03 2.083496e+01 * time: 146.72119808197021 157 1.139843e+03 2.015856e+01 * time: 147.6427960395813 158 1.138652e+03 1.563386e+01 * time: 148.59687614440918 159 1.136642e+03 1.940663e+01 * time: 149.52811908721924 160 1.130881e+03 3.438761e+01 * time: 151.6082580089569 161 1.127116e+03 2.732724e+01 * time: 152.66782999038696 162 1.121172e+03 2.329341e+01 * time: 153.8684220314026 163 1.114032e+03 3.693136e+01 * time: 155.1689190864563 164 1.113083e+03 1.625969e+01 * time: 156.4665710926056 165 1.111657e+03 9.839735e+00 * time: 157.758455991745 166 1.111245e+03 1.672316e+01 * time: 159.03842306137085 167 1.111200e+03 1.011289e+01 * time: 160.18154311180115 168 1.111189e+03 1.126428e+01 * time: 161.3076159954071 169 1.111187e+03 1.072074e+01 * time: 162.4282250404358 170 1.111187e+03 1.058154e+01 * time: 163.54794192314148 171 1.111187e+03 1.059935e+01 * time: 164.66437292099 172 1.111186e+03 1.066755e+01 * time: 165.82640600204468 173 1.111185e+03 1.073422e+01 * time: 166.95971202850342 174 1.111182e+03 1.084102e+01 * time: 168.12472796440125 175 1.111174e+03 1.099326e+01 * time: 169.257089138031 176 1.111153e+03 1.122224e+01 * time: 170.4001181125641 177 1.111099e+03 1.155538e+01 * time: 171.63402605056763 178 1.110959e+03 1.202824e+01 * time: 172.85179805755615 179 1.110607e+03 1.265924e+01 * time: 174.02826595306396 180 1.109748e+03 1.342521e+01 * time: 175.20587706565857 181 1.107812e+03 1.843518e+01 * time: 176.4319930076599 182 1.104084e+03 2.482762e+01 * time: 177.68730807304382 183 1.098538e+03 3.156038e+01 * time: 179.08151292800903 184 1.093041e+03 2.102741e+01 * time: 180.59333205223083 185 1.092139e+03 4.376057e+00 * time: 181.9588029384613 186 1.091949e+03 4.214735e+00 * time: 183.36548399925232 187 1.091924e+03 4.423897e+00 * time: 184.66636610031128 188 1.091915e+03 4.379812e+00 * time: 185.97601509094238 189 1.091911e+03 4.325356e+00 * time: 187.26099801063538 190 1.091907e+03 4.301718e+00 * time: 188.55438494682312 191 1.091905e+03 4.312479e+00 * time: 189.8496561050415 192 1.091904e+03 4.333231e+00 * time: 191.1568100452423 193 1.091904e+03 4.341267e+00 * time: 192.42251896858215 194 1.091904e+03 4.344148e+00 * time: 193.677325963974 195 1.091904e+03 4.347377e+00 * time: 194.90643906593323 196 1.091904e+03 4.350432e+00 * time: 196.14116597175598 197 1.091904e+03 4.354607e+00 * time: 197.4068751335144 198 1.091903e+03 4.360357e+00 * time: 198.78310108184814 199 1.091902e+03 4.368853e+00 * time: 200.09028506278992 200 1.091899e+03 4.728219e+00 * time: 201.343768119812 201 1.091891e+03 7.600915e+00 * time: 203.73512601852417 202 1.091869e+03 1.221674e+01 * time: 205.05579805374146 203 1.091814e+03 1.955276e+01 * time: 206.3858199119568 204 1.091671e+03 3.083649e+01 * time: 207.74562311172485 205 1.091313e+03 4.663244e+01 * time: 209.08951807022095 206 1.090503e+03 6.282148e+01 * time: 210.4137270450592 207 1.089183e+03 6.388542e+01 * time: 211.85959100723267 208 1.088151e+03 4.909379e+01 * time: 213.2343361377716 209 1.087509e+03 3.494681e+01 * time: 214.62998700141907 210 1.087080e+03 7.718616e+00 * time: 216.17087697982788 211 1.087045e+03 5.434022e+00 * time: 217.84268593788147 212 1.087043e+03 5.392509e+00 * time: 219.2256419658661 213 1.087043e+03 5.381737e+00 * time: 220.46033692359924 214 1.087043e+03 5.383544e+00 * time: 221.7077419757843 215 1.087043e+03 5.384736e+00 * time: 223.2341480255127 216 1.087043e+03 5.389168e+00 * time: 224.5976960659027 217 1.087043e+03 5.395864e+00 * time: 225.90008807182312 218 1.087042e+03 5.407975e+00 * time: 227.17838406562805 219 1.087041e+03 5.428555e+00 * time: 228.43401098251343 220 1.087039e+03 5.465713e+00 * time: 229.69690108299255 221 1.087033e+03 5.535143e+00 * time: 230.94412302970886 222 1.087016e+03 5.674084e+00 * time: 232.2751989364624 223 1.086972e+03 8.532540e+00 * time: 233.55292391777039 224 1.086848e+03 1.461753e+01 * time: 234.83977603912354 225 1.086412e+03 2.842371e+01 * time: 236.07178807258606 226 1.085222e+03 6.994075e+01 * time: 238.6683099269867 227 1.083851e+03 6.134735e+01 * time: 239.84890294075012 228 1.082442e+03 6.257404e+01 * time: 241.45631408691406 229 1.077492e+03 7.243971e+01 * time: 243.0501389503479 230 1.075473e+03 4.639961e+01 * time: 244.23957204818726 231 1.073275e+03 2.264633e+01 * time: 245.36646795272827 232 1.071725e+03 1.828391e+01 * time: 246.52636098861694 233 1.070530e+03 1.749938e+01 * time: 247.70466208457947 234 1.069856e+03 1.401865e+01 * time: 248.92396306991577 235 1.069530e+03 2.540004e+00 * time: 250.1129469871521 236 1.069495e+03 2.324898e+00 * time: 251.29384994506836 237 1.069488e+03 2.238028e+00 * time: 252.45547914505005 238 1.069488e+03 2.234884e+00 * time: 253.6480131149292 239 1.069488e+03 2.235545e+00 * time: 254.81162810325623 240 1.069488e+03 2.235598e+00 * time: 255.95970010757446 241 1.069488e+03 2.235952e+00 * time: 257.0792820453644 242 1.069488e+03 2.236398e+00 * time: 258.218337059021 243 1.069488e+03 2.237191e+00 * time: 259.4100661277771 244 1.069488e+03 2.238247e+00 * time: 260.61712098121643 245 1.069487e+03 2.239436e+00 * time: 261.8789939880371 246 1.069486e+03 2.239775e+00 * time: 263.1123399734497 247 1.069484e+03 2.236150e+00 * time: 264.31440711021423 248 1.069478e+03 2.819848e+00 * time: 265.4916410446167 249 1.069462e+03 4.569597e+00 * time: 266.82227396965027 250 1.069423e+03 7.151931e+00 * time: 268.10915207862854 251 1.069337e+03 1.033894e+01 * time: 269.37299513816833 252 1.069182e+03 1.222813e+01 * time: 270.57503294944763 253 1.069012e+03 8.750861e+00 * time: 271.7598991394043 254 1.068940e+03 2.347851e+00 * time: 272.936674118042 255 1.068930e+03 8.660232e-01 * time: 274.0925679206848 256 1.068929e+03 8.684927e-01 * time: 275.25603103637695 257 1.068929e+03 8.688773e-01 * time: 276.40380597114563 258 1.068929e+03 8.689364e-01 * time: 277.5343520641327 259 1.068929e+03 8.689906e-01 * time: 278.7022521495819 260 1.068929e+03 8.690639e-01 * time: 279.82327008247375 261 1.068929e+03 8.691862e-01 * time: 280.9646430015564 262 1.068929e+03 8.693731e-01 * time: 282.1010820865631 263 1.068929e+03 8.696634e-01 * time: 283.2346260547638 264 1.068929e+03 8.700941e-01 * time: 284.38801312446594 265 1.068929e+03 8.706948e-01 * time: 285.52480006217957 266 1.068929e+03 8.714109e-01 * time: 286.66775012016296 267 1.068928e+03 1.135042e+00 * time: 287.85991191864014 268 1.068925e+03 1.834310e+00 * time: 289.1886429786682 269 1.068918e+03 2.941732e+00 * time: 290.7323660850525 270 1.068901e+03 4.631750e+00 * time: 292.5980670452118 271 1.068860e+03 6.959147e+00 * time: 294.00758695602417 272 1.068771e+03 9.324762e+00 * time: 295.2287321090698 273 1.068620e+03 9.719454e+00 * time: 296.4024999141693 274 1.068450e+03 6.253902e+00 * time: 297.56492805480957 275 1.068381e+03 2.060774e+00 * time: 298.7944140434265 276 1.068375e+03 1.329067e+00 * time: 300.1364951133728 277 1.068375e+03 1.303384e+00 * time: 301.3344449996948 278 1.068374e+03 1.285284e+00 * time: 302.48182010650635 279 1.068374e+03 1.283587e+00 * time: 303.61964106559753 280 1.068374e+03 1.283262e+00 * time: 304.73621010780334 281 1.068374e+03 1.282394e+00 * time: 305.8754241466522 282 1.068374e+03 1.281210e+00 * time: 307.0003590583801 283 1.068374e+03 1.279149e+00 * time: 308.15380001068115 284 1.068374e+03 1.275842e+00 * time: 309.3928129673004 285 1.068374e+03 1.270320e+00 * time: 310.68015003204346 286 1.068374e+03 1.261096e+00 * time: 311.92820501327515 287 1.068374e+03 1.245320e+00 * time: 313.0929801464081 288 1.068372e+03 1.217687e+00 * time: 314.23217391967773 289 1.068369e+03 1.880347e+00 * time: 315.3807189464569 290 1.068360e+03 2.989937e+00 * time: 316.5441589355469 291 1.068339e+03 4.564004e+00 * time: 317.70784997940063 292 1.068294e+03 6.255183e+00 * time: 318.9276020526886 293 1.068218e+03 6.693297e+00 * time: 320.1140160560608 294 1.068129e+03 4.691438e+00 * time: 321.27065801620483 295 1.068058e+03 1.778432e+00 * time: 322.48358702659607 296 1.068021e+03 1.060128e-01 * time: 323.6565909385681 297 1.068007e+03 5.380572e-01 * time: 324.8908619880676 298 1.068000e+03 4.880990e-01 * time: 326.1059651374817 299 1.067995e+03 2.128989e-01 * time: 327.32036995887756 300 1.067993e+03 5.531827e-02 * time: 328.57333302497864 301 1.067992e+03 8.171941e-02 * time: 329.7879719734192 302 1.067992e+03 7.780559e-02 * time: 331.01171112060547 303 1.067991e+03 8.837140e-02 * time: 332.4144170284271 304 1.067991e+03 9.376056e-02 * time: 333.7503011226654 305 1.067991e+03 9.513439e-02 * time: 334.92954993247986 306 1.067991e+03 9.526533e-02 * time: 336.09844493865967 307 1.067991e+03 9.529304e-02 * time: 337.2295660972595 308 1.067991e+03 9.528758e-02 * time: 338.3707821369171 309 1.067991e+03 9.527374e-02 * time: 339.55013704299927 310 1.067991e+03 9.524600e-02 * time: 340.70316910743713 311 1.067991e+03 9.519336e-02 * time: 341.9980230331421 312 1.067991e+03 9.511341e-02 * time: 343.1991379261017 313 1.067991e+03 9.495795e-02 * time: 344.45249009132385 314 1.067991e+03 9.467216e-02 * time: 345.5718550682068 315 1.067991e+03 9.410027e-02 * time: 346.6997411251068 316 1.067991e+03 9.294065e-02 * time: 347.8513159751892 317 1.067991e+03 1.029917e-01 * time: 348.9983811378479 318 1.067991e+03 1.610004e-01 * time: 350.15007996559143 319 1.067991e+03 2.392391e-01 * time: 351.36393213272095 320 1.067991e+03 3.102433e-01 * time: 352.6082510948181 321 1.067991e+03 3.039617e-01 * time: 353.76532006263733 322 1.067990e+03 1.709451e-01 * time: 354.9542829990387 323 1.067990e+03 3.340140e-02 * time: 356.10167598724365 324 1.067990e+03 1.340511e-02 * time: 357.2319369316101 325 1.067990e+03 2.463356e-02 * time: 358.380912065506 326 1.067990e+03 2.689690e-02 * time: 359.5507061481476 327 1.067990e+03 1.754625e-02 * time: 360.7076749801636 328 1.067990e+03 4.559747e-03 * time: 361.8265149593353 329 1.067990e+03 2.117925e-03 * time: 362.96642994880676 330 1.067990e+03 3.040111e-03 * time: 364.09634494781494 331 1.067990e+03 2.450615e-03 * time: 365.1975030899048 332 1.067990e+03 6.637513e-04 * time: 366.32954692840576
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: GradientNorm
Log-likelihood value: -1067.9901
------------------------
Estimate
------------------------
pop_CL 0.13521
pop_V 8.0131
pop_tabs 0.57176
pop_lag 0.87544
pop_e0 96.401
pop_emax -1.0614
pop_c50 1.4902
pop_tover 14.047
pk_Ω₁,₁ 0.067971
pk_Ω₂,₂ 0.021022
pk_Ω₃,₃ 0.85621
pd_Ω₁,₁ 0.0029367
pd_Ω₂,₂ 1.9257e-10
pd_Ω₃,₃ 0.14607
pd_Ω₄,₄ 0.01538
σ_prop 0.088485
σ_add 0.41684
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: 0.00016999244689941406 1 5.174461e+05 8.708699e+05 * time: 14.550684928894043 2 3.865265e+05 6.344302e+05 * time: 20.07855486869812 3 1.804274e+05 2.829723e+05 * time: 25.854876041412354 4 9.706641e+04 1.550547e+05 * time: 30.83341383934021 5 4.769637e+04 6.778818e+04 * time: 38.68268084526062 6 2.902319e+04 3.499747e+04 * time: 43.69065594673157 7 1.823472e+04 1.705751e+04 * time: 47.99008393287659 8 1.258819e+04 9.569382e+03 * time: 52.87374496459961 9 9.389985e+03 8.615853e+03 * time: 57.113117933273315 10 7.314703e+03 7.636886e+03 * time: 61.480814933776855 11 5.916030e+03 6.624328e+03 * time: 66.43713188171387 12 4.930520e+03 5.558143e+03 * time: 70.61531090736389 13 4.125062e+03 4.315761e+03 * time: 74.82996487617493 14 3.549280e+03 3.051094e+03 * time: 79.23385405540466 15 3.283490e+03 2.157293e+03 * time: 83.44006395339966 16 3.204886e+03 1.659798e+03 * time: 87.89565205574036 17 3.194875e+03 1.480528e+03 * time: 92.04142189025879 18 3.193944e+03 1.437922e+03 * time: 96.44697093963623 19 3.193070e+03 1.411186e+03 * time: 100.71159887313843 20 3.190129e+03 1.355327e+03 * time: 105.06885695457458 21 3.183228e+03 1.276603e+03 * time: 109.23470187187195 22 3.164897e+03 1.151838e+03 * time: 113.48449397087097 23 3.119250e+03 9.712652e+02 * time: 117.74986791610718 24 3.006297e+03 7.204344e+02 * time: 122.58709406852722 25 2.738913e+03 4.050546e+02 * time: 126.72298693656921 26 2.123834e+03 2.318194e+02 * time: 130.802307844162 27 1.350274e+03 1.386114e+02 * time: 137.1480839252472 28 1.292769e+03 2.316788e+02 * time: 143.45056796073914 29 1.273650e+03 1.664576e+02 * time: 147.45281791687012 30 1.263845e+03 1.764780e+02 * time: 151.3947229385376 31 1.243263e+03 1.918515e+02 * time: 155.83680391311646 32 1.240721e+03 1.767819e+02 * time: 159.86959600448608 33 1.240527e+03 1.714670e+02 * time: 163.8618688583374 34 1.240515e+03 1.711414e+02 * time: 167.92909789085388 35 1.240468e+03 1.701273e+02 * time: 171.8130588531494 36 1.240366e+03 1.683252e+02 * time: 175.90008997917175 37 1.240086e+03 1.641003e+02 * time: 180.07083988189697 38 1.239407e+03 1.549923e+02 * time: 184.10015892982483 39 1.237812e+03 1.353555e+02 * time: 188.2578890323639 40 1.234656e+03 9.796266e+01 * time: 192.1853370666504 41 1.230256e+03 4.275685e+01 * time: 198.36414003372192 42 1.227168e+03 4.910425e+01 * time: 204.07284593582153 43 1.226203e+03 5.436423e+01 * time: 208.11162304878235 44 1.226090e+03 4.782434e+01 * time: 211.95224905014038 45 1.226088e+03 4.634508e+01 * time: 215.79323196411133 46 1.226087e+03 4.601737e+01 * time: 219.61934399604797 47 1.226081e+03 4.453430e+01 * time: 223.39968705177307 48 1.226069e+03 4.248196e+01 * time: 227.3222758769989 49 1.226036e+03 3.820877e+01 * time: 231.1504988670349 50 1.225957e+03 3.039100e+01 * time: 234.98302483558655 51 1.225769e+03 2.845924e+01 * time: 238.83892393112183 52 1.225398e+03 2.944735e+01 * time: 242.76048183441162 53 1.224884e+03 3.650133e+01 * time: 246.86232686042786 54 1.224523e+03 5.005193e+01 * time: 250.76804995536804 55 1.224424e+03 4.581270e+01 * time: 254.95302605628967 56 1.224413e+03 4.045440e+01 * time: 258.80726289749146 57 1.224411e+03 3.898473e+01 * time: 262.58337593078613 58 1.224409e+03 3.740456e+01 * time: 266.5492489337921 59 1.224403e+03 3.465479e+01 * time: 270.30845403671265 60 1.224387e+03 3.207084e+01 * time: 274.1657409667969 61 1.224344e+03 3.010429e+01 * time: 278.01020884513855 62 1.224234e+03 2.815615e+01 * time: 281.9327800273895 63 1.223949e+03 2.819773e+01 * time: 285.7356538772583 64 1.223237e+03 3.624525e+01 * time: 289.8103530406952 65 1.221599e+03 7.595776e+01 * time: 293.62435388565063 66 1.218598e+03 1.106972e+02 * time: 297.48278284072876 67 1.215107e+03 1.024784e+02 * time: 301.46459007263184 68 1.212822e+03 8.617173e+01 * time: 305.5966670513153 69 1.212285e+03 8.968159e+01 * time: 309.9504599571228 70 1.212264e+03 8.932390e+01 * time: 315.3000738620758 71 1.212260e+03 8.896775e+01 * time: 319.6141788959503 72 1.212243e+03 8.763569e+01 * time: 323.4746370315552 73 1.212211e+03 8.567133e+01 * time: 327.6819980144501 74 1.212116e+03 8.128655e+01 * time: 331.47232484817505 75 1.211887e+03 7.259875e+01 * time: 335.47084188461304 76 1.211323e+03 5.424907e+01 * time: 339.3401520252228 77 1.210114e+03 6.777926e+01 * time: 343.3141338825226 78 1.208107e+03 8.001941e+01 * time: 347.57910895347595 79 1.206178e+03 9.051730e+01 * time: 351.46726989746094 80 1.205413e+03 9.881818e+01 * time: 355.6359848976135 81 1.205304e+03 8.686459e+01 * time: 359.50358390808105 82 1.205297e+03 8.274385e+01 * time: 363.35196900367737 83 1.205293e+03 8.123098e+01 * time: 368.33727502822876 84 1.205277e+03 7.756664e+01 * time: 372.4802930355072 85 1.205242e+03 7.268640e+01 * time: 376.32764506340027 86 1.205146e+03 6.634851e+01 * time: 380.27435088157654 87 1.204903e+03 6.323589e+01 * time: 383.931911945343 88 1.204289e+03 5.674088e+01 * time: 387.67822098731995 89 1.202869e+03 5.242005e+01 * 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2.828484e-01 * time: 1373.9364738464355 320 1.067991e+03 3.568974e-01 * time: 1378.1144468784332 321 1.067990e+03 3.280426e-01 * time: 1384.0172319412231 322 1.067990e+03 1.629326e-01 * time: 1387.9558148384094 323 1.067990e+03 2.262745e-02 * time: 1392.605318069458 324 1.067990e+03 1.806059e-02 * time: 1396.585224866867 325 1.067990e+03 2.892375e-02 * time: 1400.499295949936 326 1.067990e+03 2.939738e-02 * time: 1404.3604400157928 327 1.067990e+03 1.819943e-02 * time: 1408.2225549221039 328 1.067990e+03 3.826783e-03 * time: 1412.0905890464783 329 1.067990e+03 2.380260e-03 * time: 1415.9229929447174 330 1.067990e+03 3.364719e-03 * time: 1419.940201997757 331 1.067990e+03 2.287408e-03 * time: 1424.975431919098 332 1.067990e+03 7.907173e-04 * time: 1430.074373960495
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: GradientNorm
Log-likelihood value: -1067.9901
------------------------
Estimate
------------------------
pop_CL 0.13521
pop_V 8.0131
pop_tabs 0.57176
pop_lag 0.87544
pop_e0 96.401
pop_emax -1.0614
pop_c50 1.4903
pop_tover 14.047
pk_Ω₁,₁ 0.067971
pk_Ω₂,₂ 0.021022
pk_Ω₃,₃ 0.85621
pd_Ω₁,₁ 0.0029368
pd_Ω₂,₂ 1.7943e-10
pd_Ω₃,₃ 0.14607
pd_Ω₄,₄ 0.01538
σ_prop 0.088485
σ_add 0.41684
σ_fx 3.5814
------------------------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: 2.5033950805664062e-5 1 5.174461e+05 8.708698e+05 * time: 5.074887990951538 2 3.865265e+05 6.344302e+05 * time: 6.147215127944946 3 1.804274e+05 2.829723e+05 * time: 7.145032167434692 4 9.706640e+04 1.550547e+05 * time: 8.020995140075684 5 4.769637e+04 6.778817e+04 * time: 8.883856058120728 6 2.902319e+04 3.499747e+04 * time: 9.730170011520386 7 1.823472e+04 1.705751e+04 * time: 10.857994079589844 8 1.258819e+04 9.569381e+03 * time: 11.885348081588745 9 9.389984e+03 8.615851e+03 * time: 12.760809183120728 10 7.314702e+03 7.636883e+03 * time: 13.60620903968811 11 5.916029e+03 6.624325e+03 * time: 14.44742202758789 12 4.930519e+03 5.558140e+03 * time: 15.27165699005127 13 4.125060e+03 4.315759e+03 * time: 16.078763961791992 14 3.549280e+03 3.051093e+03 * time: 16.91045117378235 15 3.283489e+03 2.157292e+03 * time: 17.71856713294983 16 3.204886e+03 1.659798e+03 * time: 18.523853063583374 17 3.194875e+03 1.480528e+03 * time: 19.3107750415802 18 3.193944e+03 1.437921e+03 * time: 20.120126008987427 19 3.193070e+03 1.411186e+03 * time: 20.91841197013855 20 3.190129e+03 1.355327e+03 * time: 21.703907012939453 21 3.183228e+03 1.276603e+03 * time: 22.504170179367065 22 3.164897e+03 1.151838e+03 * time: 23.31883716583252 23 3.119250e+03 9.712651e+02 * time: 24.155182123184204 24 3.006297e+03 7.204342e+02 * time: 24.94045114517212 25 2.738913e+03 4.050545e+02 * time: 25.705734968185425 26 2.123834e+03 2.318194e+02 * time: 26.4302020072937 27 1.350274e+03 1.386112e+02 * time: 27.56884002685547 28 1.292769e+03 2.316796e+02 * time: 28.720519065856934 29 1.273651e+03 1.664572e+02 * time: 29.43631601333618 30 1.263845e+03 1.764763e+02 * time: 30.162296056747437 31 1.243263e+03 1.918451e+02 * time: 30.889852046966553 32 1.240721e+03 1.767770e+02 * time: 31.606821060180664 33 1.240528e+03 1.714639e+02 * time: 32.33573913574219 34 1.240516e+03 1.711384e+02 * time: 33.035346031188965 35 1.240469e+03 1.701225e+02 * time: 33.765491008758545 36 1.240367e+03 1.683186e+02 * time: 34.48118710517883 37 1.240086e+03 1.640884e+02 * time: 35.18774700164795 38 1.239406e+03 1.549705e+02 * time: 35.90207815170288 39 1.237810e+03 1.353142e+02 * time: 36.638734102249146 40 1.234651e+03 9.789409e+01 * time: 37.40723896026611 41 1.230250e+03 4.267605e+01 * time: 38.14742398262024 42 1.227165e+03 4.912917e+01 * time: 38.86614108085632 43 1.226203e+03 5.435504e+01 * time: 39.57666611671448 44 1.226090e+03 4.781765e+01 * time: 40.316648960113525 45 1.226088e+03 4.634307e+01 * time: 41.01972198486328 46 1.226087e+03 4.601501e+01 * time: 41.71532607078552 47 1.226081e+03 4.453260e+01 * time: 42.42231202125549 48 1.226069e+03 4.248058e+01 * time: 43.121623039245605 49 1.226037e+03 3.820863e+01 * time: 43.85700011253357 50 1.225957e+03 3.039268e+01 * time: 44.57602095603943 51 1.225769e+03 2.845915e+01 * time: 45.3095760345459 52 1.225398e+03 2.944634e+01 * time: 46.015511989593506 53 1.224884e+03 3.649468e+01 * time: 46.724472999572754 54 1.224523e+03 5.005317e+01 * time: 47.46618700027466 55 1.224424e+03 4.581761e+01 * time: 48.17306900024414 56 1.224413e+03 4.045809e+01 * time: 48.88673114776611 57 1.224411e+03 3.898771e+01 * time: 49.5840220451355 58 1.224409e+03 3.740820e+01 * time: 50.284790992736816 59 1.224403e+03 3.465864e+01 * time: 51.01718497276306 60 1.224387e+03 3.207196e+01 * time: 51.72339701652527 61 1.224344e+03 3.010594e+01 * time: 52.42959713935852 62 1.224234e+03 2.815612e+01 * time: 53.133602142333984 63 1.223949e+03 2.819773e+01 * time: 53.83869910240173 64 1.223237e+03 3.623110e+01 * time: 54.57143807411194 65 1.221600e+03 7.594252e+01 * time: 55.29539513587952 66 1.218599e+03 1.106926e+02 * time: 56.004554986953735 67 1.215108e+03 1.024959e+02 * time: 56.70123815536499 68 1.212822e+03 8.616184e+01 * time: 57.42044711112976 69 1.212285e+03 8.967904e+01 * time: 58.11575508117676 70 1.212264e+03 8.932231e+01 * time: 58.80633211135864 71 1.212260e+03 8.896622e+01 * time: 59.50137901306152 72 1.212243e+03 8.763463e+01 * time: 60.189088106155396 73 1.212211e+03 8.567093e+01 * time: 60.90437197685242 74 1.212117e+03 8.128762e+01 * time: 61.62605118751526 75 1.211887e+03 7.260263e+01 * time: 62.318788051605225 76 1.211323e+03 5.425847e+01 * time: 63.00772404670715 77 1.210115e+03 6.776239e+01 * time: 63.71281599998474 78 1.208108e+03 8.001211e+01 * time: 64.4258291721344 79 1.206179e+03 9.050308e+01 * time: 65.11869215965271 80 1.205413e+03 9.882075e+01 * time: 65.81401300430298 81 1.205304e+03 8.686728e+01 * time: 66.49158096313477 82 1.205297e+03 8.274394e+01 * time: 67.18838500976562 83 1.205293e+03 8.123149e+01 * time: 67.91744804382324 84 1.205277e+03 7.756682e+01 * time: 68.61448812484741 85 1.205243e+03 7.268681e+01 * time: 69.3054780960083 86 1.205146e+03 6.634955e+01 * time: 69.9879560470581 87 1.204904e+03 6.323739e+01 * time: 70.6698579788208 88 1.204289e+03 5.674317e+01 * time: 71.37492609024048 89 1.202870e+03 5.241823e+01 * time: 72.06206202507019 90 1.200212e+03 5.141235e+01 * time: 72.74678015708923 91 1.197000e+03 7.220159e+01 * time: 73.43803000450134 92 1.194663e+03 4.802355e+01 * time: 74.13504815101624 93 1.193885e+03 4.877511e+01 * time: 74.83901810646057 94 1.193833e+03 4.999053e+01 * time: 75.52200508117676 95 1.193831e+03 5.008249e+01 * time: 76.1981680393219 96 1.193830e+03 5.013788e+01 * time: 76.8628900051117 97 1.193824e+03 5.023254e+01 * time: 77.54531002044678 98 1.193812e+03 5.037081e+01 * time: 78.23246097564697 99 1.193778e+03 5.056804e+01 * time: 78.9149911403656 100 1.193690e+03 5.079472e+01 * time: 79.58702802658081 101 1.193465e+03 5.091051e+01 * time: 80.26062417030334 102 1.192908e+03 5.045181e+01 * time: 80.97579908370972 103 1.191643e+03 4.826748e+01 * time: 81.67998504638672 104 1.189287e+03 4.966410e+01 * time: 82.4994101524353 105 1.186354e+03 8.019602e+01 * time: 83.4256420135498 106 1.184499e+03 8.997306e+01 * time: 84.36496114730835 107 1.184032e+03 8.409107e+01 * time: 85.41715812683105 108 1.183980e+03 7.941815e+01 * time: 86.40492916107178 109 1.183972e+03 7.800888e+01 * time: 87.33815908432007 110 1.183969e+03 7.789060e+01 * time: 88.18999600410461 111 1.183965e+03 7.827285e+01 * time: 88.99979496002197 112 1.183957e+03 7.914388e+01 * time: 89.83023810386658 113 1.183941e+03 8.062278e+01 * time: 90.6430561542511 114 1.183901e+03 8.300665e+01 * time: 91.48837304115295 115 1.183800e+03 8.655866e+01 * time: 92.29888010025024 116 1.183541e+03 9.123963e+01 * time: 93.13248205184937 117 1.182906e+03 9.549957e+01 * time: 93.95252704620361 118 1.181504e+03 9.386385e+01 * time: 94.77458596229553 119 1.179134e+03 7.587507e+01 * time: 95.49980807304382 120 1.176807e+03 4.015289e+01 * time: 96.20320105552673 121 1.175726e+03 1.893886e+01 * time: 96.92298698425293 122 1.175462e+03 1.993822e+01 * time: 97.64284706115723 123 1.175429e+03 1.972233e+01 * time: 98.52489399909973 124 1.175427e+03 1.982000e+01 * time: 99.51714205741882 125 1.175426e+03 1.966081e+01 * time: 100.49192500114441 126 1.175426e+03 1.968272e+01 * time: 101.5032811164856 127 1.175424e+03 1.968319e+01 * time: 102.41827416419983 128 1.175421e+03 1.974296e+01 * time: 103.36392498016357 129 1.175412e+03 1.984742e+01 * time: 104.22106313705444 130 1.175390e+03 2.002902e+01 * time: 105.05401396751404 131 1.175331e+03 2.031948e+01 * time: 105.90818309783936 132 1.175176e+03 2.076063e+01 * time: 106.76003909111023 133 1.174766e+03 2.137481e+01 * time: 107.60595297813416 134 1.173683e+03 2.425753e+01 * time: 108.4549331665039 135 1.170871e+03 4.400799e+01 * time: 109.38272500038147 136 1.164081e+03 6.851982e+01 * time: 110.27794313430786 137 1.153405e+03 5.020585e+01 * time: 111.04883408546448 138 1.148838e+03 1.016698e+02 * time: 111.89185309410095 139 1.146758e+03 4.473168e+01 * time: 112.99575901031494 140 1.145077e+03 2.419327e+01 * time: 113.76883602142334 141 1.142868e+03 4.735929e+01 * time: 114.52840518951416 142 1.142013e+03 3.798978e+01 * time: 115.30032110214233 143 1.141124e+03 2.039426e+01 * time: 116.05578207969666 144 1.140996e+03 2.088977e+01 * time: 116.80515503883362 145 1.140977e+03 2.088121e+01 * time: 117.54671502113342 146 1.140975e+03 2.100463e+01 * time: 118.28562712669373 147 1.140975e+03 2.099458e+01 * time: 119.03301405906677 148 1.140975e+03 2.098676e+01 * time: 119.76730298995972 149 1.140974e+03 2.098300e+01 * time: 120.5019371509552 150 1.140973e+03 2.097870e+01 * time: 121.25429511070251 151 1.140970e+03 2.097346e+01 * time: 122.02808904647827 152 1.140964e+03 2.096496e+01 * time: 122.76924109458923 153 1.140946e+03 2.095149e+01 * time: 123.5110981464386 154 1.140902e+03 2.093231e+01 * time: 124.25476002693176 155 1.140789e+03 2.090863e+01 * time: 125.0073561668396 156 1.140509e+03 2.084238e+01 * time: 125.77425813674927 157 1.139838e+03 2.015651e+01 * time: 126.5304319858551 158 1.138645e+03 1.562309e+01 * time: 127.31028914451599 159 1.136609e+03 1.936256e+01 * time: 128.07906198501587 160 1.130573e+03 3.336898e+01 * time: 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6.476647e+00 * time: 210.15239715576172 249 1.069110e+03 6.416702e+00 * time: 211.20450806617737 250 1.068975e+03 3.200148e+00 * time: 212.23416709899902 251 1.068934e+03 8.643648e-01 * time: 213.3029429912567 252 1.068930e+03 8.679826e-01 * time: 214.30638003349304 253 1.068929e+03 8.688201e-01 * time: 215.1902461051941 254 1.068929e+03 8.689251e-01 * time: 216.02453303337097 255 1.068929e+03 8.689498e-01 * time: 216.8728790283203 256 1.068929e+03 8.690232e-01 * time: 217.7128300666809 257 1.068929e+03 8.691127e-01 * time: 218.55410718917847 258 1.068929e+03 8.692685e-01 * time: 219.4085991382599 259 1.068929e+03 8.694978e-01 * time: 220.3910710811615 260 1.068929e+03 8.698396e-01 * time: 221.6115870475769 261 1.068929e+03 8.703020e-01 * time: 222.8625111579895 262 1.068929e+03 8.708198e-01 * time: 223.9311511516571 263 1.068928e+03 8.710499e-01 * time: 224.93901300430298 264 1.068925e+03 8.698335e-01 * time: 226.3638231754303 265 1.068919e+03 8.637767e-01 * time: 227.58704614639282 266 1.068903e+03 1.045517e+00 * time: 228.7748420238495 267 1.068864e+03 1.614828e+00 * time: 229.7796630859375 268 1.068779e+03 2.454711e+00 * time: 230.79736614227295 269 1.068628e+03 2.890169e+00 * time: 232.49964499473572 270 1.068456e+03 2.132359e+00 * time: 234.0608880519867 271 1.068383e+03 1.268143e+00 * time: 235.77049708366394 272 1.068375e+03 1.324055e+00 * time: 236.814355134964 273 1.068375e+03 1.304950e+00 * time: 237.68660497665405 274 1.068374e+03 1.286707e+00 * time: 238.5176031589508 275 1.068374e+03 1.283712e+00 * time: 239.34532618522644 276 1.068374e+03 1.283322e+00 * time: 240.2671821117401 277 1.068374e+03 1.282423e+00 * time: 241.14922714233398 278 1.068374e+03 1.281178e+00 * time: 242.1193151473999 279 1.068374e+03 1.279004e+00 * time: 243.10952806472778 280 1.068374e+03 1.275539e+00 * time: 243.95770406723022 281 1.068374e+03 1.269765e+00 * time: 245.04884600639343 282 1.068374e+03 1.260184e+00 * time: 246.40816402435303 283 1.068374e+03 1.243935e+00 * time: 247.81953811645508 284 1.068373e+03 1.215825e+00 * time: 249.2323100566864 285 1.068370e+03 1.165758e+00 * time: 250.65496611595154 286 1.068362e+03 1.074004e+00 * time: 252.11688899993896 287 1.068344e+03 1.165966e+00 * time: 253.58591604232788 288 1.068303e+03 1.641074e+00 * time: 255.0280261039734 289 1.068231e+03 1.855932e+00 * time: 256.3729040622711 290 1.068141e+03 1.431911e+00 * time: 257.39462518692017 291 1.068066e+03 6.141979e-01 * time: 258.44076895713806 292 1.068024e+03 1.146750e-01 * time: 259.4944791793823 293 1.068008e+03 1.760569e-01 * time: 260.5264241695404 294 1.068001e+03 1.619067e-01 * time: 261.5859751701355 295 1.067996e+03 8.232183e-02 * time: 262.6700701713562 296 1.067993e+03 5.385855e-02 * time: 263.7463221549988 297 1.067992e+03 6.528291e-02 * time: 264.80877113342285 298 1.067992e+03 7.684781e-02 * time: 265.8038899898529 299 1.067991e+03 8.733236e-02 * time: 266.79483103752136 300 1.067991e+03 9.329030e-02 * time: 267.8054540157318 301 1.067991e+03 9.501668e-02 * time: 268.80701518058777 302 1.067991e+03 9.523337e-02 * time: 269.8569631576538 303 1.067991e+03 9.527627e-02 * time: 270.91905403137207 304 1.067991e+03 9.528664e-02 * time: 271.94447898864746 305 1.067991e+03 9.529179e-02 * time: 272.99578404426575 306 1.067991e+03 9.529006e-02 * time: 274.06499004364014 307 1.067991e+03 9.528355e-02 * time: 275.1493020057678 308 1.067991e+03 9.527921e-02 * time: 276.09671998023987 309 1.067991e+03 9.525610e-02 * time: 276.9616231918335 310 1.067991e+03 9.517680e-02 * time: 277.9384980201721 311 1.067991e+03 9.497497e-02 * time: 279.1590120792389 312 1.067991e+03 9.442299e-02 * time: 280.2498199939728 313 1.067991e+03 9.297764e-02 * time: 281.3006081581116 314 1.067991e+03 8.928662e-02 * time: 282.3620641231537 315 1.067991e+03 8.037982e-02 * time: 283.3913700580597 316 1.067991e+03 7.638856e-02 * time: 284.42135310173035 317 1.067991e+03 8.359979e-02 * time: 285.478392124176 318 1.067990e+03 5.553555e-02 * time: 286.5992820262909 319 1.067990e+03 1.547449e-02 * time: 287.66377806663513 320 1.067990e+03 4.389889e-03 * time: 288.7101080417633 321 1.067990e+03 4.769016e-03 * time: 289.81074118614197 322 1.067990e+03 6.694099e-03 * time: 290.9175319671631 323 1.067990e+03 6.056627e-03 * time: 291.9200611114502 324 1.067990e+03 3.066253e-03 * time: 292.91921496391296 325 1.067990e+03 8.765056e-04 * time: 293.956915140152
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: GradientNorm
Log-likelihood value: -1067.9901
-----------------------
Estimate
-----------------------
pop_CL 0.13521
pop_V 8.0131
pop_tabs 0.57175
pop_lag 0.87544
pop_e0 96.401
pop_emax -1.0614
pop_c50 1.4903
pop_tover 14.047
pk_Ω₁,₁ 0.067972
pk_Ω₂,₂ 0.021022
pk_Ω₃,₃ 0.8563
pd_Ω₁,₁ 0.0029372
pd_Ω₂,₂ 1.4671e-9
pd_Ω₃,₃ 0.14607
pd_Ω₄,₄ 0.01538
σ_prop 0.088485
σ_add 0.41684
σ_fx 3.5814
-----------------------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.