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 3.130150e+06 5.915753e+06 * time: 0.04027104377746582 1 5.185877e+05 8.742010e+05 * time: 3.254331111907959 2 3.866957e+05 6.366584e+05 * time: 4.136708974838257 3 1.795019e+05 2.835377e+05 * time: 5.034359931945801 4 9.682619e+04 1.546512e+05 * time: 5.952147960662842 5 4.791898e+04 6.820022e+04 * time: 6.856826066970825 6 2.907369e+04 3.509683e+04 * time: 7.77540397644043 7 1.827377e+04 1.713122e+04 * time: 8.689090967178345 8 1.260634e+04 9.563328e+03 * time: 9.626774072647095 9 9.403430e+03 8.611643e+03 * time: 10.584319114685059 10 7.325839e+03 7.634814e+03 * time: 11.471596002578735 11 5.926864e+03 6.625143e+03 * time: 12.36168098449707 12 4.942277e+03 5.562818e+03 * time: 13.218662023544312 13 4.139082e+03 4.326560e+03 * time: 14.079061031341553 14 3.563490e+03 3.065255e+03 * time: 14.944560050964355 15 3.296158e+03 2.170246e+03 * time: 15.800405025482178 16 3.216281e+03 1.669628e+03 * time: 16.65265989303589 17 3.205943e+03 1.487868e+03 * time: 17.505381107330322 18 3.204981e+03 1.444253e+03 * time: 18.361736059188843 19 3.204107e+03 1.417661e+03 * time: 19.199899911880493 20 3.201145e+03 1.361617e+03 * time: 20.05643105506897 21 3.194218e+03 1.282918e+03 * time: 20.894515991210938 22 3.175800e+03 1.158036e+03 * time: 21.7493839263916 23 3.130001e+03 9.774722e+02 * time: 22.586612939834595 24 3.016853e+03 7.265878e+02 * time: 23.43086004257202 25 2.749857e+03 4.107604e+02 * time: 24.231614112854004 26 2.137213e+03 2.318395e+02 * time: 25.03315806388855 27 1.756667e+03 2.281481e+02 * time: 25.83544898033142 28 1.380674e+03 1.613750e+02 * time: 27.559083938598633 29 1.328451e+03 1.298786e+02 * time: 28.279186964035034 30 1.287368e+03 2.435584e+02 * time: 28.91658091545105 31 1.263071e+03 1.616830e+02 * time: 29.56758189201355 32 1.254713e+03 1.796575e+02 * time: 30.216628074645996 33 1.247205e+03 1.996163e+02 * time: 30.87345004081726 34 1.243765e+03 1.938233e+02 * time: 31.543585062026978 35 1.240829e+03 1.739536e+02 * time: 32.23684000968933 36 1.240788e+03 1.744026e+02 * time: 32.91271710395813 37 1.240776e+03 1.743249e+02 * time: 33.58285689353943 38 1.240093e+03 1.687609e+02 * time: 34.25345706939697 39 1.239010e+03 1.586880e+02 * time: 34.941107988357544 40 1.236253e+03 1.305344e+02 * time: 35.61800408363342 41 1.232129e+03 8.229765e+01 * time: 36.283889055252075 42 1.228151e+03 3.735853e+01 * time: 36.94801902770996 43 1.226337e+03 5.238455e+01 * time: 37.6187949180603 44 1.226025e+03 4.868891e+01 * time: 38.28610706329346 45 1.226011e+03 4.583321e+01 * time: 38.95399808883667 46 1.226010e+03 4.536409e+01 * time: 39.616377115249634 47 1.226008e+03 4.466374e+01 * time: 40.284754037857056 48 1.226002e+03 4.336404e+01 * time: 40.95414209365845 49 1.225988e+03 4.102589e+01 * time: 41.628408908843994 50 1.225951e+03 3.656316e+01 * time: 42.30089592933655 51 1.225858e+03 2.865017e+01 * time: 42.96701192855835 52 1.225646e+03 2.863718e+01 * time: 43.66602301597595 53 1.225243e+03 3.015482e+01 * time: 44.38126492500305 54 1.224739e+03 4.057249e+01 * time: 45.04973602294922 55 1.224438e+03 5.072112e+01 * time: 45.71704411506653 56 1.224368e+03 4.561597e+01 * time: 46.38602805137634 57 1.224362e+03 4.124712e+01 * time: 47.05290603637695 58 1.224360e+03 4.008302e+01 * time: 47.714771032333374 59 1.224357e+03 3.807314e+01 * time: 48.381216049194336 60 1.224349e+03 3.507776e+01 * time: 49.06674790382385 61 1.224328e+03 3.152674e+01 * time: 49.75519299507141 62 1.224274e+03 2.940406e+01 * time: 50.44197106361389 63 1.224133e+03 2.814507e+01 * time: 51.121541023254395 64 1.223771e+03 2.816089e+01 * time: 51.80075001716614 65 1.222880e+03 4.493376e+01 * time: 52.46784591674805 66 1.220919e+03 8.628440e+01 * time: 53.13873291015625 67 1.217683e+03 1.137013e+02 * time: 53.79784393310547 68 1.214493e+03 9.066643e+01 * time: 54.44950604438782 69 1.212820e+03 8.381730e+01 * time: 55.104881048202515 70 1.212582e+03 8.589809e+01 * time: 55.750256061553955 71 1.212574e+03 8.557503e+01 * time: 56.381829023361206 72 1.212569e+03 8.522123e+01 * time: 57.00205898284912 73 1.212552e+03 8.416230e+01 * time: 57.63934803009033 74 1.212516e+03 8.236943e+01 * time: 58.26696491241455 75 1.212415e+03 7.841502e+01 * time: 58.9020049571991 76 1.212165e+03 7.017293e+01 * time: 59.539676904678345 77 1.211553e+03 5.231397e+01 * time: 60.176669120788574 78 1.210236e+03 7.254841e+01 * time: 60.817163944244385 79 1.208052e+03 8.652477e+01 * time: 61.472455978393555 80 1.205979e+03 9.811697e+01 * time: 62.12971210479736 81 1.205158e+03 1.085054e+02 * time: 62.76586198806763 82 1.205038e+03 9.706902e+01 * time: 63.39622092247009 83 1.205030e+03 9.297153e+01 * time: 64.02270197868347 84 1.205025e+03 9.142094e+01 * time: 64.6404800415039 85 1.205007e+03 8.794517e+01 * time: 65.27202892303467 86 1.204968e+03 8.329872e+01 * time: 65.90216994285583 87 1.204860e+03 7.519252e+01 * time: 66.53303098678589 88 1.204586e+03 6.676934e+01 * time: 67.16360902786255 89 1.203889e+03 5.914964e+01 * time: 67.8032910823822 90 1.202275e+03 5.659520e+01 * time: 68.47505211830139 91 1.199277e+03 5.457734e+01 * time: 69.11474800109863 92 1.195791e+03 6.434155e+01 * time: 69.74380302429199 93 1.193281e+03 4.404749e+01 * time: 70.38005805015564 94 1.192403e+03 4.696509e+01 * time: 71.00876903533936 95 1.192344e+03 4.826237e+01 * time: 71.64119791984558 96 1.192341e+03 4.836344e+01 * time: 72.26130700111389 97 1.192340e+03 4.843132e+01 * time: 72.8712809085846 98 1.192336e+03 4.854903e+01 * time: 73.52037906646729 99 1.192325e+03 4.873523e+01 * time: 74.13997507095337 100 1.192297e+03 4.901991e+01 * time: 74.75897407531738 101 1.192226e+03 4.941566e+01 * time: 75.3819830417633 102 1.192043e+03 4.985207e+01 * time: 76.020742893219 103 1.191593e+03 5.001010e+01 * time: 76.63857507705688 104 1.190562e+03 4.892721e+01 * time: 77.27579307556152 105 1.188621e+03 4.482422e+01 * time: 77.92431807518005 106 1.186171e+03 5.565271e+01 * time: 78.58953189849854 107 1.184411e+03 7.195174e+01 * time: 79.2555661201477 108 1.183640e+03 7.480916e+01 * time: 79.93634104728699 109 1.183456e+03 7.309340e+01 * time: 80.6063449382782 110 1.183420e+03 7.276592e+01 * time: 81.27220106124878 111 1.183410e+03 7.363636e+01 * time: 81.92923593521118 112 1.183407e+03 7.448587e+01 * time: 82.59276103973389 113 1.183403e+03 7.577977e+01 * time: 83.32104301452637 114 1.183397e+03 7.712665e+01 * time: 83.98159289360046 115 1.183377e+03 7.953569e+01 * time: 84.64769196510315 116 1.183329e+03 8.300886e+01 * time: 85.31324100494385 117 1.183203e+03 8.805324e+01 * time: 85.98515295982361 118 1.182889e+03 9.423587e+01 * time: 86.66518092155457 119 1.182143e+03 9.919012e+01 * time: 87.36795210838318 120 1.180607e+03 9.577137e+01 * time: 88.07200407981873 121 1.178255e+03 7.337667e+01 * time: 88.79630899429321 122 1.176240e+03 3.404242e+01 * time: 89.54150009155273 123 1.175531e+03 1.922941e+01 * time: 90.29272294044495 124 1.175410e+03 1.933202e+01 * time: 91.05586504936218 125 1.175396e+03 1.964875e+01 * time: 91.78286099433899 126 1.175395e+03 1.966955e+01 * time: 92.49878406524658 127 1.175394e+03 1.960524e+01 * time: 93.2031900882721 128 1.175394e+03 1.957692e+01 * time: 93.91531801223755 129 1.175392e+03 1.945994e+01 * time: 94.62854790687561 130 1.175387e+03 1.938255e+01 * time: 95.38211512565613 131 1.175374e+03 1.915648e+01 * time: 96.12786197662354 132 1.175342e+03 1.892418e+01 * time: 96.85899305343628 133 1.175256e+03 1.840923e+01 * time: 97.62305498123169 134 1.175030e+03 1.936471e+01 * time: 98.36383509635925 135 1.174436e+03 2.656839e+01 * time: 99.16814994812012 136 1.172888e+03 3.906048e+01 * time: 99.94532608985901 137 1.169012e+03 5.824526e+01 * time: 100.78434205055237 138 1.160857e+03 7.724845e+01 * time: 101.57162809371948 139 1.151984e+03 5.979641e+01 * time: 102.37913393974304 140 1.150239e+03 2.132792e+02 * time: 103.22348308563232 141 1.147461e+03 1.267625e+02 * time: 104.07426309585571 142 1.143928e+03 2.373475e+01 * time: 104.90579509735107 143 1.142853e+03 2.225755e+01 * time: 105.68621301651001 144 1.141788e+03 3.915063e+01 * time: 106.41290593147278 145 1.141298e+03 2.253375e+01 * time: 107.13300395011902 146 1.140996e+03 2.098981e+01 * time: 107.87195205688477 147 1.140981e+03 2.094768e+01 * time: 108.59949398040771 148 1.140978e+03 2.098189e+01 * time: 109.3288209438324 149 1.140977e+03 2.101185e+01 * time: 110.05342602729797 150 1.140977e+03 2.100810e+01 * time: 110.83264398574829 151 1.140977e+03 2.097542e+01 * time: 111.54763698577881 152 1.140976e+03 2.094477e+01 * time: 112.25739312171936 153 1.140974e+03 2.089057e+01 * time: 112.97688388824463 154 1.140970e+03 2.083453e+01 * time: 113.6947660446167 155 1.140963e+03 2.076320e+01 * time: 114.4404981136322 156 1.140944e+03 2.068383e+01 * time: 115.24259090423584 157 1.140896e+03 2.060212e+01 * time: 116.00710391998291 158 1.140770e+03 2.053302e+01 * time: 116.73308992385864 159 1.140444e+03 2.051790e+01 * time: 117.43134903907776 160 1.139610e+03 2.179094e+01 * time: 118.14153909683228 161 1.138122e+03 3.277638e+01 * time: 118.87747001647949 162 1.135109e+03 4.596766e+01 * time: 119.61206603050232 163 1.132075e+03 5.671178e+01 * time: 120.59048199653625 164 1.130256e+03 6.299917e+01 * time: 121.59097290039062 165 1.126132e+03 1.302587e+02 * time: 122.41041493415833 166 1.116038e+03 4.101274e+01 * time: 123.46991395950317 167 1.114892e+03 3.653675e+01 * time: 124.40893912315369 168 1.112342e+03 3.434411e+01 * time: 125.33218693733215 169 1.111707e+03 1.638796e+01 * time: 126.27798008918762 170 1.111291e+03 9.906829e+00 * time: 127.2151141166687 171 1.111206e+03 1.001220e+01 * time: 128.18087697029114 172 1.111191e+03 1.040896e+01 * time: 129.1692659854889
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: -1111.1911
-----------------------
Estimate
-----------------------
pop_CL 0.13526
pop_V 7.9651
pop_tabs 0.61588
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 0.034279
pk_Ω₃,₃ 0.24152
pd_Ω₁,₁ 0.0029639
pd_Ω₂,₂ 0.0022285
pd_Ω₃,₃ 0.02459
pd_Ω₄,₄ 0.015792
σ_prop 0.013052
σ_add 0.81409
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: 2.002716064453125e-5 1 5.174461e+05 8.708698e+05 * time: 2.038646936416626 2 3.865265e+05 6.344302e+05 * time: 3.8881800174713135 3 1.804274e+05 2.829723e+05 * time: 5.742444038391113 4 9.706640e+04 1.550547e+05 * time: 9.669104814529419 5 4.769637e+04 6.778818e+04 * time: 12.429404973983765 6 2.902319e+04 3.499747e+04 * time: 15.156613826751709 7 1.823472e+04 1.705751e+04 * time: 17.338454008102417 8 1.258819e+04 9.569381e+03 * time: 19.28728699684143 9 9.389984e+03 8.615851e+03 * time: 21.25524401664734 10 7.314702e+03 7.636883e+03 * time: 23.31206488609314 11 5.916029e+03 6.624325e+03 * time: 25.357045888900757 12 4.930519e+03 5.558140e+03 * time: 27.271600008010864 13 4.125060e+03 4.315759e+03 * time: 29.145792961120605 14 3.549280e+03 3.051093e+03 * time: 31.05953598022461 15 3.283489e+03 2.157292e+03 * time: 32.917186975479126 16 3.204886e+03 1.659798e+03 * time: 34.786945819854736 17 3.194875e+03 1.480528e+03 * time: 36.69229698181152 18 3.193944e+03 1.437921e+03 * time: 38.53877401351929 19 3.193070e+03 1.411186e+03 * time: 40.36453700065613 20 3.190129e+03 1.355327e+03 * time: 42.223654985427856 21 3.183228e+03 1.276603e+03 * time: 44.02393698692322 22 3.164897e+03 1.151838e+03 * time: 45.82613682746887 23 3.119250e+03 9.712651e+02 * time: 47.655014991760254 24 3.006297e+03 7.204342e+02 * time: 49.364956855773926 25 2.738913e+03 4.050545e+02 * time: 51.03362703323364 26 2.123834e+03 2.318194e+02 * time: 52.541319847106934 27 1.789138e+03 2.290465e+02 * time: 54.08745503425598 28 1.396455e+03 1.683969e+02 * time: 57.74677896499634 29 1.333545e+03 1.336195e+02 * time: 59.37213397026062 30 1.297771e+03 2.452189e+02 * time: 60.75433802604675 31 1.266002e+03 1.523968e+02 * time: 62.121484994888306 32 1.255506e+03 1.733993e+02 * time: 63.48962688446045 33 1.247789e+03 1.971624e+02 * time: 64.88391089439392 34 1.244490e+03 1.915728e+02 * time: 66.28190684318542 35 1.240568e+03 1.704250e+02 * time: 67.67508792877197 36 1.240503e+03 1.711787e+02 * time: 69.0648980140686 37 1.240492e+03 1.711657e+02 * time: 70.43627595901489 38 1.239992e+03 1.687239e+02 * time: 71.82237195968628 39 1.239199e+03 1.624610e+02 * time: 73.19775295257568 40 1.236971e+03 1.400045e+02 * time: 74.60032796859741 41 1.233203e+03 9.442285e+01 * time: 76.0111289024353 42 1.228682e+03 3.273926e+01 * time: 77.41088581085205 43 1.226466e+03 4.997777e+01 * time: 78.80167293548584 44 1.226104e+03 4.904407e+01 * time: 80.19011282920837 45 1.226088e+03 4.675833e+01 * time: 81.56534481048584 46 1.226088e+03 4.628496e+01 * time: 82.94446802139282 47 1.226085e+03 4.541356e+01 * time: 84.32142090797424 48 1.226080e+03 4.402741e+01 * time: 85.69129180908203 49 1.226064e+03 4.142300e+01 * time: 87.0664758682251 50 1.226026e+03 3.663066e+01 * time: 88.44160294532776 51 1.225931e+03 2.851375e+01 * time: 89.83773398399353 52 1.225713e+03 2.844436e+01 * time: 91.2216329574585 53 1.225303e+03 3.026439e+01 * time: 92.60339403152466 54 1.224791e+03 4.157121e+01 * time: 93.98989796638489 55 1.224489e+03 5.047475e+01 * time: 95.36107683181763 56 1.224420e+03 4.451066e+01 * time: 96.743332862854 57 1.224413e+03 3.994027e+01 * time: 98.11266493797302 58 1.224412e+03 3.879446e+01 * time: 99.82611203193665 59 1.224408e+03 3.677251e+01 * time: 101.80664587020874 60 1.224400e+03 3.377994e+01 * time: 103.83263802528381 61 1.224379e+03 3.158188e+01 * time: 105.4466118812561 62 1.224324e+03 2.925156e+01 * time: 106.84423303604126 63 1.224180e+03 2.815886e+01 * time: 108.25090980529785 64 1.223813e+03 2.818297e+01 * time: 109.6441638469696 65 1.222906e+03 4.598739e+01 * time: 111.09390592575073 66 1.220910e+03 8.692122e+01 * time: 112.60115599632263 67 1.217609e+03 1.136529e+02 * time: 113.9562828540802 68 1.214313e+03 9.030296e+01 * time: 115.3127498626709 69 1.212533e+03 8.868945e+01 * time: 116.67022681236267 70 1.212272e+03 8.970280e+01 * time: 118.04214000701904 71 1.212264e+03 8.921923e+01 * time: 119.36563491821289 72 1.212259e+03 8.882367e+01 * time: 120.69239687919617 73 1.212242e+03 8.761510e+01 * time: 122.01685190200806 74 1.212205e+03 8.561996e+01 * time: 123.33133697509766 75 1.212103e+03 8.125316e+01 * time: 124.66439580917358 76 1.211850e+03 7.229635e+01 * time: 126.01195001602173 77 1.211238e+03 5.321841e+01 * time: 127.35932397842407 78 1.209945e+03 7.377542e+01 * time: 128.70038485527039 79 1.207890e+03 8.404090e+01 * time: 130.0276129245758 80 1.206058e+03 9.148111e+01 * time: 131.33733987808228 81 1.205389e+03 9.689402e+01 * time: 132.64629101753235 82 1.205303e+03 8.594691e+01 * time: 134.08903288841248 83 1.205297e+03 8.262969e+01 * time: 135.50547885894775 84 1.205293e+03 8.096366e+01 * time: 136.85238599777222 85 1.205277e+03 7.747617e+01 * time: 138.31153988838196 86 1.205241e+03 7.259257e+01 * time: 139.62329983711243 87 1.205143e+03 6.664833e+01 * time: 140.9456329345703 88 1.204895e+03 6.374646e+01 * time: 142.2852029800415 89 1.204268e+03 5.753767e+01 * time: 143.68840193748474 90 1.202818e+03 4.866325e+01 * time: 145.12029886245728 91 1.200116e+03 5.085841e+01 * time: 146.54891800880432 92 1.196895e+03 7.139478e+01 * time: 148.08338499069214 93 1.194610e+03 4.693643e+01 * time: 149.54172492027283 94 1.193880e+03 4.874637e+01 * time: 150.95180487632751 95 1.193833e+03 4.997341e+01 * time: 152.3726568222046 96 1.193831e+03 5.008752e+01 * time: 153.77960300445557 97 1.193829e+03 5.014860e+01 * time: 155.0893018245697 98 1.193824e+03 5.025195e+01 * time: 156.4021818637848 99 1.193812e+03 5.040244e+01 * time: 157.74737286567688 100 1.193778e+03 5.061928e+01 * time: 159.10484385490417 101 1.193693e+03 5.087910e+01 * time: 160.50053000450134 102 1.193472e+03 5.105104e+01 * time: 161.84537601470947 103 1.192926e+03 5.068579e+01 * time: 163.17244601249695 104 1.191681e+03 4.863476e+01 * time: 164.4832408428192 105 1.189351e+03 4.892981e+01 * time: 165.81381487846375 106 1.186420e+03 7.777396e+01 * time: 167.1595778465271 107 1.184527e+03 8.874388e+01 * time: 168.51327085494995 108 1.184034e+03 8.377338e+01 * time: 169.87817001342773 109 1.183978e+03 7.932701e+01 * time: 171.22644686698914 110 1.183969e+03 7.794331e+01 * time: 172.57162594795227 111 1.183966e+03 7.787273e+01 * time: 173.91549396514893 112 1.183962e+03 7.833691e+01 * time: 175.2577519416809 113 1.183955e+03 7.929110e+01 * time: 176.61764693260193 114 1.183940e+03 8.084135e+01 * time: 177.95211696624756 115 1.183904e+03 8.331063e+01 * time: 179.29603385925293 116 1.183812e+03 8.698432e+01 * time: 180.6378858089447 117 1.183576e+03 9.188955e+01 * time: 181.98618483543396 118 1.182997e+03 9.668923e+01 * time: 183.35552287101746 119 1.181702e+03 9.635664e+01 * time: 184.735200881958 120 1.179436e+03 8.061105e+01 * time: 186.1466670036316 121 1.177026e+03 4.571925e+01 * time: 187.60149097442627 122 1.175794e+03 1.863482e+01 * time: 189.0432448387146 123 1.175474e+03 1.968114e+01 * time: 190.49391794204712 124 1.175429e+03 1.958620e+01 * time: 191.9702558517456 125 1.175427e+03 1.993702e+01 * time: 193.3754370212555 126 1.175426e+03 1.962020e+01 * time: 194.78105401992798 127 1.175426e+03 1.965822e+01 * time: 196.14109802246094 128 1.175424e+03 1.984458e+01 * time: 197.47322297096252 129 1.175421e+03 2.001151e+01 * time: 198.81580996513367 130 1.175413e+03 2.035203e+01 * time: 200.1918499469757 131 1.175392e+03 2.084295e+01 * time: 201.59662985801697 132 1.175336e+03 2.164587e+01 * time: 202.99810791015625 133 1.175189e+03 2.287862e+01 * time: 204.3963119983673 134 1.174801e+03 2.475389e+01 * time: 205.80923986434937 135 1.173792e+03 4.143547e+01 * time: 207.23343086242676 136 1.171220e+03 7.151619e+01 * time: 208.68525981903076 137 1.165144e+03 1.078851e+02 * time: 210.25424790382385 138 1.155079e+03 7.750674e+01 * time: 211.96672296524048 139 1.149212e+03 6.869309e+01 * time: 213.66858196258545 140 1.146920e+03 6.356778e+01 * time: 215.3637239933014 141 1.144663e+03 4.948700e+01 * time: 217.0967400074005 142 1.143084e+03 2.539148e+01 * time: 218.70491480827332 143 1.141829e+03 1.979199e+01 * time: 220.24662899971008 144 1.141293e+03 2.050876e+01 * time: 221.7892849445343 145 1.141052e+03 2.076835e+01 * time: 223.33905601501465 146 1.140979e+03 2.090550e+01 * time: 224.88066792488098 147 1.140976e+03 2.094541e+01 * time: 226.4251070022583 148 1.140975e+03 2.099654e+01 * time: 227.99200701713562 149 1.140974e+03 2.099916e+01 * time: 229.55026602745056 150 1.140974e+03 2.099880e+01 * time: 231.10132694244385 151 1.140973e+03 2.099210e+01 * time: 232.67069792747498 152 1.140972e+03 2.098164e+01 * time: 234.23330688476562 153 1.140970e+03 2.096558e+01 * time: 235.760803937912 154 1.140963e+03 2.094348e+01 * time: 237.30035495758057 155 1.140947e+03 2.091734e+01 * time: 238.86845588684082 156 1.140905e+03 2.089968e+01 * time: 240.44372081756592 157 1.140795e+03 2.093513e+01 * time: 242.0844488143921 158 1.140505e+03 2.116001e+01 * time: 243.67987895011902 159 1.139717e+03 2.228769e+01 * time: 245.2926390171051 160 1.137490e+03 3.498391e+01 * time: 246.9752378463745 161 1.132263e+03 6.110620e+01 * time: 248.828519821167 162 1.129968e+03 7.039255e+01 * time: 251.1916048526764 163 1.128181e+03 7.684085e+01 * time: 253.99378180503845 164 1.125090e+03 1.650673e+02 * time: 257.0761368274689 165 1.124218e+03 1.513316e+02 * time: 259.51599502563477 166 1.117007e+03 4.305495e+01 * time: 261.64677691459656 167 1.113027e+03 1.731902e+01 * time: 263.6966679096222 168 1.111575e+03 1.058505e+01 * time: 265.7290539741516 169 1.111230e+03 1.232050e+01 * time: 267.7804398536682 170 1.111199e+03 1.227441e+01 * time: 269.8051929473877 171 1.111189e+03 1.013419e+01 * time: 272.03501200675964 172 1.111188e+03 1.117708e+01 * time: 274.6148009300232 173 1.111187e+03 1.066970e+01 * time: 276.74446392059326 174 1.111187e+03 1.057794e+01 * time: 278.8028919696808 175 1.111187e+03 1.040593e+01 * time: 280.87565302848816 176 1.111186e+03 1.023758e+01 * time: 282.96445393562317 177 1.111185e+03 1.018183e+01 * time: 285.27909088134766 178 1.111181e+03 1.018006e+01 * time: 288.0036129951477 179 1.111172e+03 1.017403e+01 * time: 290.24673080444336 180 1.111148e+03 1.230488e+01 * time: 292.8202049732208 181 1.111084e+03 1.995781e+01 * time: 295.4611089229584 182 1.110919e+03 3.222739e+01 * time: 297.8854749202728 183 1.110490e+03 5.141692e+01 * time: 300.00062680244446 184 1.109411e+03 7.920072e+01 * time: 302.4842610359192 185 1.106922e+03 1.111906e+02 * time: 304.7299280166626 186 1.102355e+03 1.240590e+02 * time: 306.9949929714203 187 1.097127e+03 8.551331e+01 * time: 309.4299988746643 188 1.093684e+03 3.313309e+01 * time: 312.21650886535645 189 1.092621e+03 1.147153e+01 * time: 315.2738208770752 190 1.092198e+03 1.049704e+01 * time: 318.093279838562 191 1.092058e+03 4.261193e+00 * time: 320.78474497795105 192 1.091989e+03 4.455297e+00 * time: 323.38252782821655 193 1.091944e+03 4.499157e+00 * time: 325.8687698841095 194 1.091907e+03 4.410964e+00 * time: 328.2382218837738 195 1.091904e+03 4.346219e+00 * time: 330.576740026474 196 1.091904e+03 4.341837e+00 * time: 332.8840010166168 197 1.091904e+03 4.341552e+00 * time: 335.28278493881226 198 1.091904e+03 4.341455e+00 * time: 337.65420603752136 199 1.091904e+03 4.341307e+00 * time: 340.04330492019653 200 1.091904e+03 4.340490e+00 * time: 342.3786869049072 201 1.091903e+03 4.338928e+00 * time: 344.66221594810486 202 1.091901e+03 4.334976e+00 * time: 346.9737708568573 203 1.091896e+03 4.326218e+00 * time: 349.2848799228668 204 1.091882e+03 4.384246e+00 * time: 351.67737197875977 205 1.091845e+03 7.392590e+00 * time: 354.14512491226196 206 1.091752e+03 1.220967e+01 * time: 356.7521288394928 207 1.091516e+03 1.961825e+01 * time: 359.332692861557 208 1.090962e+03 2.954677e+01 * time: 361.98716282844543 209 1.089857e+03 3.695896e+01 * time: 364.6618688106537 210 1.088317e+03 2.850061e+01 * time: 367.20686292648315 211 1.087314e+03 7.728854e+00 * time: 369.53150701522827 212 1.087074e+03 5.430740e+00 * time: 371.8809428215027 213 1.087047e+03 5.450106e+00 * time: 374.2029678821564 214 1.087044e+03 5.402925e+00 * time: 376.42341685295105 215 1.087043e+03 5.377293e+00 * time: 378.6237208843231 216 1.087043e+03 5.379011e+00 * time: 380.82324481010437 217 1.087043e+03 5.383711e+00 * time: 383.0670759677887 218 1.087043e+03 5.386507e+00 * time: 385.35623383522034 219 1.087043e+03 5.393833e+00 * time: 387.64855003356934 220 1.087043e+03 5.405582e+00 * time: 389.9288568496704 221 1.087042e+03 5.425634e+00 * time: 392.18535900115967 222 1.087042e+03 5.459261e+00 * time: 394.44627594947815 223 1.087039e+03 5.517515e+00 * time: 396.718209028244 224 1.087033e+03 5.621798e+00 * time: 398.9919250011444 225 1.087018e+03 5.819426e+00 * time: 401.2257788181305 226 1.086976e+03 6.228064e+00 * time: 403.485524892807 227 1.086857e+03 7.213421e+00 * time: 405.64953994750977 228 1.086375e+03 1.055856e+01 * time: 407.7489798069 229 1.085683e+03 1.335167e+01 * time: 409.8204028606415 230 1.084499e+03 1.995178e+01 * time: 411.88772892951965 231 1.084114e+03 2.279485e+01 * time: 414.60697388648987 232 1.083303e+03 2.490520e+01 * time: 416.71446800231934 233 1.081442e+03 2.096658e+01 * time: 418.85954689979553 234 1.071363e+03 4.643961e+01 * time: 420.9830539226532 235 1.070356e+03 1.244811e+01 * time: 423.15628600120544 236 1.070177e+03 1.063405e+01 * time: 425.3080048561096 237 1.069824e+03 1.309356e+01 * time: 427.3964898586273 238 1.069578e+03 6.660870e+00 * time: 429.519348859787 239 1.069490e+03 2.255943e+00 * time: 431.6236460208893 240 1.069488e+03 2.247488e+00 * time: 433.7520749568939 241 1.069488e+03 2.235872e+00 * time: 435.86096000671387 242 1.069488e+03 2.235347e+00 * time: 437.9629259109497 243 1.069488e+03 2.235455e+00 * time: 440.0918040275574 244 1.069488e+03 2.235483e+00 * time: 442.22585582733154 245 1.069488e+03 2.235692e+00 * time: 444.4057319164276 246 1.069488e+03 2.235754e+00 * time: 446.56019401550293 247 1.069488e+03 2.235730e+00 * time: 448.6959578990936 248 1.069487e+03 2.234852e+00 * time: 450.8306088447571 249 1.069486e+03 2.231694e+00 * time: 453.00720381736755 250 1.069483e+03 3.351708e+00 * time: 455.14236402511597 251 1.069476e+03 5.437839e+00 * time: 457.27106499671936 252 1.069457e+03 8.668871e+00 * time: 460.2361190319061 253 1.069412e+03 1.327559e+01 * time: 463.2661349773407 254 1.069313e+03 1.834554e+01 * time: 466.44025683403015 255 1.069150e+03 1.978067e+01 * time: 468.6938989162445 256 1.068997e+03 1.293343e+01 * time: 470.89009284973145 257 1.068938e+03 4.185093e+00 * time: 473.1128509044647 258 1.068930e+03 8.685190e-01 * time: 475.2997579574585 259 1.068929e+03 8.687490e-01 * time: 477.4497938156128 260 1.068929e+03 8.689190e-01 * time: 479.57472801208496 261 1.068929e+03 8.689190e-01 * time: 481.9438099861145 262 1.068929e+03 8.692069e-01 * time: 484.10817885398865 263 1.068929e+03 8.690477e-01 * time: 486.2577679157257 264 1.068929e+03 8.689800e-01 * time: 488.3981719017029 265 1.068929e+03 8.688207e-01 * time: 490.5732879638672 266 1.068929e+03 8.686832e-01 * time: 492.728178024292 267 1.068929e+03 8.685443e-01 * time: 494.9002959728241 268 1.068929e+03 8.684483e-01 * time: 497.0671570301056 269 1.068929e+03 8.683904e-01 * time: 499.23548889160156 270 1.068929e+03 8.682908e-01 * time: 501.4175820350647 271 1.068928e+03 8.926147e-01 * time: 503.58963084220886 272 1.068925e+03 1.338379e+00 * time: 505.76021003723145 273 1.068919e+03 1.995877e+00 * time: 507.9361038208008 274 1.068904e+03 2.943299e+00 * time: 510.1244640350342 275 1.068866e+03 4.196484e+00 * time: 512.3166220188141 276 1.068785e+03 5.459732e+00 * time: 514.4768488407135 277 1.068639e+03 5.733599e+00 * time: 516.6196799278259 278 1.068466e+03 3.827168e+00 * time: 518.7567918300629 279 1.068385e+03 1.244218e+00 * time: 520.9281458854675 280 1.068375e+03 1.320170e+00 * time: 523.1134889125824 281 1.068375e+03 1.304430e+00 * time: 525.2812678813934 282 1.068374e+03 1.287290e+00 * time: 527.4261898994446 283 1.068374e+03 1.283980e+00 * time: 529.5560688972473 284 1.068374e+03 1.283336e+00 * time: 531.6768720149994 285 1.068374e+03 1.283336e+00 * time: 534.1037268638611 286 1.068374e+03 1.283336e+00 * time: 536.542582988739 287 1.068374e+03 1.281411e+00 * time: 538.7809548377991 288 1.068374e+03 1.281745e+00 * time: 540.9885489940643 289 1.068374e+03 1.281770e+00 * time: 543.2421689033508 290 1.068374e+03 1.281857e+00 * time: 545.5166230201721 291 1.068374e+03 1.281872e+00 * time: 547.8256750106812 292 1.068374e+03 1.281875e+00 * time: 550.201318025589 293 1.068374e+03 1.281875e+00 * time: 552.6846868991852 294 1.068374e+03 1.292555e+00 * time: 554.8408138751984 295 1.068374e+03 1.294426e+00 * time: 557.0669808387756 296 1.068374e+03 1.307443e+00 * time: 559.2217988967896 297 1.068374e+03 1.316687e+00 * time: 562.0229518413544 298 1.068372e+03 1.322745e+00 * time: 565.0892608165741 299 1.068371e+03 1.310269e+00 * time: 567.3231840133667 300 1.068369e+03 1.271371e+00 * time: 569.5327208042145 301 1.068367e+03 1.636139e+00 * time: 571.7091119289398 302 1.068365e+03 2.209215e+00 * time: 573.8983788490295 303 1.068362e+03 2.798144e+00 * time: 576.0553689002991 304 1.068355e+03 3.580926e+00 * time: 578.1741440296173 305 1.068337e+03 4.603098e+00 * time: 580.298094034195 306 1.068297e+03 5.633992e+00 * time: 582.4164199829102 307 1.068227e+03 5.816034e+00 * time: 584.5466828346252 308 1.068141e+03 4.303758e+00 * time: 586.7451088428497 309 1.068067e+03 1.730033e+00 * time: 588.9244909286499 310 1.068026e+03 3.589916e-01 * time: 591.0922138690948 311 1.068010e+03 5.810053e-01 * time: 593.2428879737854 312 1.068002e+03 4.592203e-01 * time: 595.4031209945679 313 1.067996e+03 1.569451e-01 * time: 597.5500328540802 314 1.067993e+03 7.120208e-02 * time: 599.6918709278107 315 1.067992e+03 8.724747e-02 * time: 601.8393540382385 316 1.067992e+03 7.564415e-02 * time: 603.9808459281921 317 1.067991e+03 8.648572e-02 * time: 606.0968029499054 318 1.067991e+03 9.291430e-02 * time: 608.2179310321808 319 1.067991e+03 9.291506e-02 * time: 610.5130839347839 320 1.067991e+03 9.521700e-02 * time: 612.6549849510193 321 1.067991e+03 9.526222e-02 * time: 614.7766978740692 322 1.067991e+03 9.528281e-02 * time: 616.9025318622589 323 1.067991e+03 9.528473e-02 * time: 619.0175158977509 324 1.067991e+03 9.529642e-02 * time: 621.1245529651642 325 1.067991e+03 9.529646e-02 * time: 623.3140358924866 326 1.067991e+03 9.529643e-02 * time: 625.5763688087463 327 1.067991e+03 9.529642e-02 * time: 627.8732299804688 328 1.067991e+03 9.529645e-02 * time: 630.1866278648376 329 1.067991e+03 9.529645e-02 * time: 632.5435259342194 330 1.067991e+03 9.529645e-02 * time: 634.8885200023651 331 1.067991e+03 9.529645e-02 * time: 637.2058849334717 332 1.067991e+03 9.529645e-02 * time: 639.4884510040283 333 1.067991e+03 9.529645e-02 * time: 641.7723948955536 334 1.067991e+03 9.529645e-02 * time: 644.0616388320923 335 1.067991e+03 9.529645e-02 * time: 646.3328399658203 336 1.067991e+03 9.529645e-02 * time: 648.654935836792 337 1.067991e+03 9.529645e-02 * time: 650.9889409542084 338 1.067991e+03 9.529645e-02 * time: 653.3579878807068 339 1.067991e+03 9.531647e-02 * time: 655.4713048934937 340 1.067991e+03 9.531313e-02 * time: 657.7392148971558 341 1.067991e+03 9.530784e-02 * time: 659.9957098960876 342 1.067991e+03 9.530712e-02 * time: 662.297798871994 343 1.067991e+03 9.530562e-02 * time: 664.5318348407745 344 1.067991e+03 9.530550e-02 * time: 666.7757248878479 345 1.067991e+03 9.530550e-02 * time: 669.1192598342896 346 1.067991e+03 9.530550e-02 * time: 671.4657700061798 347 1.067991e+03 9.530549e-02 * time: 673.8027739524841 348 1.067991e+03 9.530549e-02 * time: 676.1358139514923 349 1.067991e+03 9.530549e-02 * time: 678.4601030349731 350 1.067991e+03 9.530549e-02 * time: 680.8360228538513 351 1.067991e+03 9.530549e-02 * time: 683.3559148311615 352 1.067991e+03 9.530549e-02 * time: 685.7374889850616 353 1.067991e+03 9.530549e-02 * time: 688.1872899532318 354 1.067991e+03 9.530537e-02 * time: 690.4277949333191 355 1.067991e+03 9.530506e-02 * time: 692.6699759960175 356 1.067991e+03 9.530469e-02 * time: 694.9139759540558 357 1.067991e+03 9.530403e-02 * time: 697.1973028182983 358 1.067991e+03 9.530394e-02 * time: 699.4866459369659 359 1.067991e+03 9.530380e-02 * time: 701.8161430358887 360 1.067991e+03 9.530372e-02 * time: 704.14919090271 361 1.067991e+03 9.530371e-02 * time: 706.4999399185181 362 1.067991e+03 9.530374e-02 * time: 708.8377358913422 363 1.067991e+03 9.530373e-02 * time: 711.2627718448639 364 1.067991e+03 9.530373e-02 * time: 713.6875009536743 365 1.067991e+03 9.530373e-02 * time: 716.1032629013062 366 1.067991e+03 9.530372e-02 * time: 718.4884829521179 367 1.067991e+03 9.530372e-02 * time: 720.8648128509521 368 1.067991e+03 9.530372e-02 * time: 723.2424669265747 369 1.067991e+03 9.530372e-02 * time: 725.662328004837 370 1.067991e+03 9.530372e-02 * time: 728.0895459651947 371 1.067991e+03 9.530372e-02 * time: 730.5171709060669 372 1.067991e+03 9.530372e-02 * time: 733.0118279457092 373 1.067991e+03 9.530372e-02 * time: 735.5368738174438 374 1.067991e+03 9.530361e-02 * time: 737.9585168361664 375 1.067991e+03 9.530337e-02 * time: 740.2525029182434 376 1.067991e+03 9.530331e-02 * time: 742.5948770046234 377 1.067991e+03 9.530325e-02 * time: 744.9344499111176 378 1.067991e+03 9.530318e-02 * time: 747.3146350383759 379 1.067991e+03 9.530309e-02 * time: 749.7848110198975 380 1.067991e+03 9.530308e-02 * time: 752.2446880340576 381 1.067991e+03 9.530307e-02 * time: 754.644889831543 382 1.067991e+03 9.530306e-02 * time: 757.3667068481445 383 1.067991e+03 9.530305e-02 * time: 759.8309888839722 384 1.067991e+03 9.530304e-02 * time: 762.3057758808136 385 1.067991e+03 9.530304e-02 * time: 764.7751150131226 386 1.067991e+03 9.530304e-02 * time: 767.2786228656769 387 1.067991e+03 1.898051e-01 * time: 769.2326810359955 388 1.067991e+03 1.898051e-01 * time: 771.5718259811401 389 1.067991e+03 2.257344e-01 * time: 773.6951439380646 390 1.067991e+03 1.949550e-01 * time: 775.8358688354492 391 1.067991e+03 1.707081e-01 * time: 777.9725089073181 392 1.067991e+03 1.354320e-01 * time: 780.1233768463135 393 1.067991e+03 1.140622e-01 * time: 782.2818429470062 394 1.067991e+03 2.216227e-01 * time: 784.4352428913116 395 1.067991e+03 3.226737e-01 * time: 786.5787470340729 396 1.067991e+03 3.200315e-01 * time: 788.7049069404602 397 1.067990e+03 3.343502e-01 * time: 790.8311100006104 398 1.067990e+03 2.993885e-01 * time: 792.964262008667 399 1.067990e+03 2.003437e-01 * time: 795.092887878418 400 1.067990e+03 1.236484e-01 * time: 797.2002890110016 401 1.067990e+03 1.020407e-01 * time: 799.3552238941193 402 1.067990e+03 5.335225e-02 * time: 801.5091309547424 403 1.067990e+03 2.405451e-02 * time: 803.6540598869324 404 1.067990e+03 2.683056e-02 * time: 805.8660428524017 405 1.067990e+03 2.680502e-02 * time: 807.9754259586334 406 1.067990e+03 2.294308e-02 * time: 810.0784168243408 407 1.067990e+03 2.294308e-02 * time: 812.4120860099792 408 1.067990e+03 2.677630e-02 * time: 814.6323609352112 409 1.067990e+03 2.941466e-02 * time: 816.7543158531189 410 1.067990e+03 4.804222e-02 * time: 818.8137040138245 411 1.067990e+03 5.249561e-02 * time: 820.7885839939117 412 1.067990e+03 5.258353e-02 * time: 822.7769198417664 413 1.067990e+03 5.250493e-02 * time: 824.8119459152222 414 1.067990e+03 5.250493e-02 * time: 827.3216829299927 415 1.067990e+03 5.250493e-02 * time: 829.8326139450073 416 1.067990e+03 5.250493e-02 * time: 832.5518579483032 417 1.067990e+03 5.250493e-02 * time: 835.3408398628235 418 1.067990e+03 5.250493e-02 * time: 837.9799158573151
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: NoObjectiveChange
Log-likelihood value: -1067.9901
------------------------
Estimate
------------------------
pop_CL 0.1352
pop_V 8.0132
pop_tabs 0.57169
pop_lag 0.87545
pop_e0 96.401
pop_emax -1.0614
pop_c50 1.4902
pop_tover 14.047
pk_Ω₁,₁ 0.067979
pk_Ω₂,₂ 0.021023
pk_Ω₃,₃ 0.85609
pd_Ω₁,₁ 0.0029369
pd_Ω₂,₂ 9.7306e-10
pd_Ω₃,₃ 0.14607
pd_Ω₄,₄ 0.015382
σ_prop 0.088486
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: 2.3126602172851562e-5 1 5.174461e+05 8.708699e+05 * time: 13.337918996810913 2 3.865265e+05 6.344302e+05 * time: 24.32446599006653 3 1.804274e+05 2.829723e+05 * time: 35.46498703956604 4 9.706641e+04 1.550547e+05 * time: 46.42454814910889 5 4.769637e+04 6.778818e+04 * time: 57.39849805831909 6 2.902319e+04 3.499747e+04 * time: 67.98965311050415 7 1.823472e+04 1.705751e+04 * time: 78.81974697113037 8 1.258819e+04 9.569382e+03 * time: 89.67385601997375 9 9.389985e+03 8.615853e+03 * time: 100.52160310745239 10 7.314703e+03 7.636886e+03 * time: 111.20888710021973 11 5.916030e+03 6.624328e+03 * time: 121.87213706970215 12 4.930520e+03 5.558143e+03 * time: 132.52129697799683 13 4.125062e+03 4.315761e+03 * time: 143.41537404060364 14 3.549280e+03 3.051094e+03 * time: 154.3607370853424 15 3.283490e+03 2.157293e+03 * time: 165.0271611213684 16 3.204886e+03 1.659798e+03 * time: 175.65110802650452 17 3.194875e+03 1.480528e+03 * time: 186.2473030090332 18 3.193944e+03 1.437922e+03 * time: 196.78797507286072 19 3.193070e+03 1.411186e+03 * time: 207.2668001651764 20 3.190129e+03 1.355327e+03 * time: 217.77894616127014 21 3.183228e+03 1.276603e+03 * time: 228.7139961719513 22 3.164897e+03 1.151838e+03 * time: 239.17675805091858 23 3.119250e+03 9.712652e+02 * time: 249.65389800071716 24 3.006297e+03 7.204344e+02 * time: 260.20063614845276 25 2.738913e+03 4.050546e+02 * time: 270.7286970615387 26 2.123834e+03 2.318194e+02 * time: 280.9914619922638 27 1.789142e+03 2.290466e+02 * time: 292.0883800983429 28 1.396456e+03 1.683975e+02 * time: 314.63406896591187 29 1.333545e+03 1.336198e+02 * time: 326.2059690952301 30 1.297773e+03 2.452181e+02 * time: 336.37269401550293 31 1.266002e+03 1.523972e+02 * time: 346.8023011684418 32 1.255505e+03 1.734000e+02 * time: 356.8953640460968 33 1.247788e+03 1.971637e+02 * time: 367.06115913391113 34 1.244490e+03 1.915744e+02 * time: 377.22988295555115 35 1.240568e+03 1.704280e+02 * time: 387.60817313194275 36 1.240502e+03 1.711813e+02 * time: 398.0118761062622 37 1.240491e+03 1.711683e+02 * time: 408.0859651565552 38 1.239992e+03 1.687296e+02 * time: 418.31823110580444 39 1.239200e+03 1.624744e+02 * time: 428.5529611110687 40 1.236975e+03 1.400443e+02 * time: 438.82662296295166 41 1.233209e+03 9.449763e+01 * time: 449.0860559940338 42 1.228687e+03 3.276923e+01 * time: 459.7660491466522 43 1.226467e+03 4.997214e+01 * time: 470.032044172287 44 1.226104e+03 4.905105e+01 * time: 480.56359100341797 45 1.226088e+03 4.676122e+01 * time: 491.2116219997406 46 1.226087e+03 4.628674e+01 * time: 501.71351504325867 47 1.226085e+03 4.541753e+01 * time: 511.8243420124054 48 1.226080e+03 4.403288e+01 * time: 521.9873280525208 49 1.226064e+03 4.143315e+01 * time: 532.2489449977875 50 1.226026e+03 3.664876e+01 * time: 542.4695291519165 51 1.225931e+03 2.851383e+01 * time: 552.7979409694672 52 1.225714e+03 2.844460e+01 * time: 562.9916400909424 53 1.225304e+03 3.025732e+01 * time: 573.3238830566406 54 1.224793e+03 4.151448e+01 * time: 583.5820870399475 55 1.224489e+03 5.047755e+01 * time: 594.115228176117 56 1.224420e+03 4.452753e+01 * time: 604.6760830879211 57 1.224413e+03 3.994380e+01 * time: 614.7549519538879 58 1.224412e+03 3.879465e+01 * time: 624.8423960208893 59 1.224408e+03 3.677922e+01 * time: 634.9104659557343 60 1.224400e+03 3.379126e+01 * time: 645.0253009796143 61 1.224379e+03 3.158821e+01 * time: 656.0217611789703 62 1.224324e+03 2.926306e+01 * time: 666.2277359962463 63 1.224181e+03 2.815869e+01 * time: 676.760584115982 64 1.223815e+03 2.818300e+01 * time: 687.0316009521484 65 1.222911e+03 4.583041e+01 * time: 699.0480971336365 66 1.220921e+03 8.675281e+01 * time: 709.711531162262 67 1.217625e+03 1.136294e+02 * time: 720.1392350196838 68 1.214324e+03 9.050244e+01 * time: 730.7749500274658 69 1.212536e+03 8.867180e+01 * time: 741.2235260009766 70 1.212272e+03 8.970655e+01 * time: 752.0565390586853 71 1.212263e+03 8.922293e+01 * time: 762.291855096817 72 1.212259e+03 8.882951e+01 * time: 772.8572201728821 73 1.212242e+03 8.761908e+01 * time: 783.674674987793 74 1.212205e+03 8.562573e+01 * time: 794.4898781776428 75 1.212102e+03 8.125881e+01 * time: 805.0866811275482 76 1.211850e+03 7.230599e+01 * time: 815.723207950592 77 1.211238e+03 5.323492e+01 * time: 826.1147119998932 78 1.209946e+03 7.373618e+01 * time: 836.2689340114594 79 1.207891e+03 8.401772e+01 * time: 846.6160750389099 80 1.206059e+03 9.146801e+01 * time: 856.9359421730042 81 1.205389e+03 9.690592e+01 * time: 867.1562969684601 82 1.205302e+03 8.595286e+01 * time: 877.3749930858612 83 1.205296e+03 8.263040e+01 * time: 887.4145851135254 84 1.205292e+03 8.096547e+01 * time: 897.2020440101624 85 1.205276e+03 7.747632e+01 * time: 907.23273396492 86 1.205240e+03 7.259211e+01 * time: 917.1031501293182 87 1.205143e+03 6.664635e+01 * time: 926.9798741340637 88 1.204894e+03 6.374389e+01 * time: 936.8805840015411 89 1.204267e+03 5.753484e+01 * time: 946.8237261772156 90 1.202818e+03 4.869938e+01 * time: 956.7559349536896 91 1.200117e+03 5.087313e+01 * time: 966.607861995697 92 1.196896e+03 7.141162e+01 * time: 976.3480701446533 93 1.194612e+03 4.695316e+01 * time: 986.0859911441803 94 1.193881e+03 4.874366e+01 * time: 995.7820200920105 95 1.193834e+03 4.997038e+01 * time: 1005.4474351406097 96 1.193832e+03 5.008442e+01 * time: 1015.044764995575 97 1.193830e+03 5.014549e+01 * time: 1024.616954088211 98 1.193825e+03 5.024884e+01 * time: 1034.2359759807587 99 1.193813e+03 5.039938e+01 * time: 1043.8869471549988 100 1.193779e+03 5.061632e+01 * time: 1053.5854880809784 101 1.193694e+03 5.087641e+01 * time: 1063.3008320331573 102 1.193473e+03 5.104895e+01 * time: 1073.010362148285 103 1.192927e+03 5.068500e+01 * time: 1082.7447531223297 104 1.191682e+03 4.863684e+01 * time: 1092.4907369613647 105 1.189353e+03 4.890365e+01 * time: 1102.1895530223846 106 1.186424e+03 7.778082e+01 * time: 1112.1221361160278 107 1.184533e+03 8.873607e+01 * time: 1121.7144150733948 108 1.184042e+03 8.376145e+01 * time: 1131.0230450630188 109 1.183986e+03 7.931813e+01 * time: 1140.6123020648956 110 1.183978e+03 7.793645e+01 * time: 1150.1174330711365 111 1.183974e+03 7.786675e+01 * time: 1159.8720819950104 112 1.183970e+03 7.833168e+01 * time: 1169.9709169864655 113 1.183963e+03 7.928653e+01 * time: 1179.86359500885 114 1.183949e+03 8.083870e+01 * time: 1189.7006499767303 115 1.183913e+03 8.331100e+01 * time: 1199.6437740325928 116 1.183820e+03 8.698958e+01 * time: 1209.8645210266113 117 1.183584e+03 9.190093e+01 * time: 1219.7203271389008 118 1.183003e+03 9.670514e+01 * time: 1229.5921511650085 119 1.181706e+03 9.636548e+01 * time: 1239.4470210075378 120 1.179435e+03 8.059042e+01 * time: 1249.3455369472504 121 1.177024e+03 4.567337e+01 * time: 1261.533800125122 122 1.175793e+03 1.863452e+01 * time: 1271.3185861110687 123 1.175474e+03 1.967979e+01 * time: 1282.1732051372528 124 1.175430e+03 1.958939e+01 * time: 1292.9334950447083 125 1.175427e+03 1.993415e+01 * time: 1304.6089370250702 126 1.175427e+03 1.962015e+01 * time: 1315.2274100780487 127 1.175426e+03 1.965822e+01 * time: 1325.6836471557617 128 1.175424e+03 1.984190e+01 * time: 1336.0319740772247 129 1.175422e+03 2.000843e+01 * time: 1346.5157301425934 130 1.175413e+03 2.034664e+01 * time: 1357.2343580722809 131 1.175392e+03 2.083524e+01 * time: 1367.8236570358276 132 1.175336e+03 2.163395e+01 * time: 1378.1501109600067 133 1.175190e+03 2.286075e+01 * time: 1388.5123620033264 134 1.174803e+03 2.472708e+01 * time: 1398.9679470062256 135 1.173797e+03 4.124367e+01 * time: 1410.3830981254578 136 1.171232e+03 7.123154e+01 * time: 1420.9448730945587 137 1.165167e+03 1.075848e+02 * time: 1432.1197810173035 138 1.155096e+03 7.751300e+01 * time: 1443.59268116951 139 1.149210e+03 6.815707e+01 * time: 1455.0884139537811 140 1.146904e+03 6.218499e+01 * time: 1466.1711280345917 141 1.144688e+03 5.158716e+01 * time: 1477.0096261501312 142 1.143100e+03 2.555104e+01 * time: 1487.9836900234222 143 1.141816e+03 1.970245e+01 * time: 1498.9004549980164 144 1.141294e+03 2.048648e+01 * time: 1509.8919229507446 145 1.141044e+03 2.076701e+01 * time: 1520.6147570610046 146 1.140979e+03 2.090339e+01 * time: 1531.2721049785614 147 1.140976e+03 2.094515e+01 * time: 1541.9833180904388 148 1.140975e+03 2.099576e+01 * time: 1552.650936126709 149 1.140974e+03 2.099888e+01 * time: 1563.2025661468506 150 1.140974e+03 2.099931e+01 * time: 1573.7318141460419 151 1.140973e+03 2.099295e+01 * time: 1584.708328962326 152 1.140972e+03 2.098305e+01 * time: 1595.5327661037445 153 1.140970e+03 2.096756e+01 * time: 1606.1853470802307 154 1.140963e+03 2.094619e+01 * time: 1616.803719997406 155 1.140947e+03 2.092087e+01 * time: 1627.5186021327972 156 1.140906e+03 2.090400e+01 * time: 1638.2148411273956 157 1.140797e+03 2.093964e+01 * time: 1649.3110120296478 158 1.140511e+03 2.116231e+01 * time: 1660.3002080917358 159 1.139732e+03 2.193009e+01 * time: 1672.013345003128 160 1.137531e+03 3.209812e+01 * time: 1684.9339451789856 161 1.132353e+03 5.451900e+01 * time: 1698.5651080608368 162 1.130098e+03 6.354058e+01 * time: 1714.4340951442719 163 1.128599e+03 6.862520e+01 * time: 1727.3912091255188 164 1.127623e+03 8.385757e+01 * time: 1739.4436609745026 165 1.122228e+03 1.204824e+02 * time: 1750.4380781650543 166 1.118774e+03 1.225695e+02 * time: 1761.3137819766998 167 1.115679e+03 4.611265e+01 * time: 1772.127161026001 168 1.113979e+03 3.434831e+01 * time: 1782.8834521770477 169 1.112521e+03 3.523199e+01 * time: 1794.202516078949 170 1.111267e+03 1.228404e+01 * time: 1805.1972861289978 171 1.111189e+03 1.078351e+01 * time: 1816.2691061496735 172 1.111187e+03 1.048243e+01 * time: 1827.2123341560364 173 1.111187e+03 1.070654e+01 * time: 1837.696886062622 174 1.111187e+03 1.057812e+01 * time: 1847.862864971161 175 1.111187e+03 1.055136e+01 * time: 1857.8691821098328 176 1.111186e+03 1.049713e+01 * time: 1868.4386510849 177 1.111185e+03 1.042509e+01 * time: 1878.7995371818542 178 1.111181e+03 1.030466e+01 * time: 1888.7547221183777 179 1.111173e+03 1.018029e+01 * time: 1898.7038359642029 180 1.111150e+03 1.017084e+01 * time: 1908.8663339614868 181 1.111091e+03 1.547995e+01 * time: 1920.9656391143799 182 1.110938e+03 2.528675e+01 * time: 1932.9952969551086 183 1.110546e+03 4.010074e+01 * time: 1945.0814290046692 184 1.109576e+03 6.035772e+01 * time: 1956.6388659477234 185 1.107416e+03 8.168845e+01 * time: 1968.1647210121155 186 1.103899e+03 9.586417e+01 * time: 1979.5202331542969 187 1.100518e+03 1.082022e+02 * time: 1990.3253750801086 188 1.096703e+03 1.051879e+02 * time: 2001.2265009880066 189 1.092614e+03 3.372346e+01 * time: 2012.550920009613 190 1.092108e+03 2.854387e+01 * time: 2024.0932149887085 191 1.091924e+03 8.267860e+00 * time: 2035.0896060466766 192 1.091910e+03 4.388023e+00 * time: 2045.7833721637726 193 1.091908e+03 4.381249e+00 * time: 2056.913866996765 194 1.091906e+03 4.351858e+00 * time: 2067.831127166748 195 1.091905e+03 4.347982e+00 * time: 2078.381769180298 196 1.091904e+03 4.344943e+00 * time: 2088.6248710155487 197 1.091904e+03 4.343567e+00 * time: 2098.625465154648 198 1.091904e+03 4.341811e+00 * time: 2108.603266954422 199 1.091904e+03 4.338160e+00 * time: 2118.571408033371 200 1.091904e+03 4.331287e+00 * time: 2128.466262102127 201 1.091904e+03 4.319915e+00 * time: 2138.352035045624 202 1.091903e+03 4.299874e+00 * time: 2148.2937140464783 203 1.091901e+03 4.267809e+00 * time: 2158.244080066681 204 1.091897e+03 4.211172e+00 * time: 2168.215698003769 205 1.091886e+03 4.118019e+00 * time: 2178.214600086212 206 1.091856e+03 4.885696e+00 * time: 2188.2280299663544 207 1.091780e+03 7.437907e+00 * time: 2198.2812271118164 208 1.091587e+03 1.171705e+01 * time: 2208.373736143112 209 1.091126e+03 1.721239e+01 * time: 2218.519726037979 210 1.090066e+03 2.181183e+01 * time: 2228.663264989853 211 1.088567e+03 2.176703e+01 * time: 2238.776967048645 212 1.087529e+03 2.216618e+01 * time: 2248.818817138672 213 1.087131e+03 6.874410e+00 * time: 2258.875859975815 214 1.087053e+03 5.487630e+00 * time: 2268.90096116066 215 1.087044e+03 5.346752e+00 * time: 2278.889070034027 216 1.087043e+03 5.364345e+00 * time: 2288.882313013077 217 1.087043e+03 5.380873e+00 * time: 2299.643881082535 218 1.087043e+03 5.385018e+00 * time: 2310.507280111313 219 1.087043e+03 5.387075e+00 * time: 2320.429589986801 220 1.087043e+03 5.393036e+00 * time: 2331.0308001041412 221 1.087043e+03 5.401736e+00 * time: 2341.443987131119 222 1.087042e+03 5.417218e+00 * time: 2351.347270011902 223 1.087041e+03 5.443368e+00 * time: 2361.2895460128784 224 1.087039e+03 5.489834e+00 * time: 2371.231119155884 225 1.087032e+03 5.575603e+00 * time: 2381.200968980789 226 1.087015e+03 5.744372e+00 * time: 2391.1848261356354 227 1.086967e+03 8.460880e+00 * time: 2401.1890890598297 228 1.086833e+03 1.435286e+01 * time: 2411.2058911323547 229 1.086295e+03 2.780785e+01 * time: 2421.2722210884094 230 1.086212e+03 6.228588e+01 * time: 2434.341662168503 231 1.085721e+03 4.234553e+01 * time: 2444.4111671447754 232 1.085189e+03 5.048698e+01 * time: 2454.4381790161133 233 1.084656e+03 5.746214e+01 * time: 2464.4458589553833 234 1.084415e+03 6.141722e+01 * time: 2474.9956760406494 235 1.084038e+03 6.809867e+01 * time: 2484.9938490390778 236 1.083436e+03 6.850144e+01 * time: 2494.9156250953674 237 1.081492e+03 5.978072e+01 * time: 2504.8726799488068 238 1.080761e+03 6.149768e+01 * time: 2514.7910001277924 239 1.078780e+03 5.347455e+01 * time: 2524.722897052765 240 1.076968e+03 4.156009e+01 * time: 2534.6588430404663 241 1.073814e+03 2.237215e+01 * time: 2544.5471551418304 242 1.070458e+03 1.333345e+01 * time: 2554.4871599674225 243 1.069603e+03 5.047246e+00 * time: 2564.3617491722107 244 1.069495e+03 2.297975e+00 * time: 2574.7633969783783 245 1.069488e+03 2.249209e+00 * time: 2584.6452651023865 246 1.069488e+03 2.237153e+00 * time: 2594.9734921455383 247 1.069488e+03 2.235110e+00 * time: 2605.290740966797 248 1.069488e+03 2.235110e+00 * time: 2616.3480060100555 249 1.069488e+03 2.236833e+00 * time: 2627.020509958267 250 1.069488e+03 2.238073e+00 * time: 2636.9210550785065 251 1.069488e+03 2.240833e+00 * time: 2646.8046560287476 252 1.069487e+03 2.244456e+00 * time: 2656.670403957367 253 1.069487e+03 2.249819e+00 * time: 2666.5069210529327 254 1.069486e+03 2.256246e+00 * time: 2676.401690006256 255 1.069483e+03 2.946324e+00 * time: 2686.296406030655 256 1.069474e+03 4.816137e+00 * time: 2697.0503561496735 257 1.069454e+03 7.706597e+00 * time: 2708.102396965027 258 1.069404e+03 1.178904e+01 * time: 2719.6368520259857 259 1.069297e+03 1.616397e+01 * time: 2731.5308520793915 260 1.069127e+03 1.694616e+01 * time: 2743.0238020420074 261 1.068983e+03 1.008907e+01 * time: 2754.285213947296 262 1.068935e+03 2.703971e+00 * time: 2765.5764620304108 263 1.068930e+03 8.679397e-01 * time: 2776.8269081115723 264 1.068929e+03 8.687530e-01 * time: 2788.098988056183 265 1.068929e+03 8.688140e-01 * time: 2799.040897130966 266 1.068929e+03 8.688140e-01 * time: 2810.6895451545715 267 1.068929e+03 8.688140e-01 * time: 2822.746241092682 268 1.068929e+03 8.689166e-01 * time: 2833.8097569942474 269 1.068929e+03 8.689166e-01 * time: 2845.965171098709 270 1.068929e+03 8.689166e-01 * time: 2859.3691029548645 271 1.068929e+03 8.689166e-01 * time: 2872.8940510749817
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: -1068.9294
------------------------
Estimate
------------------------
pop_CL 0.13521
pop_V 8.0111
pop_tabs 0.56615
pop_lag 0.87615
pop_e0 96.395
pop_emax -1.0613
pop_c50 1.4884
pop_tover 14.053
pk_Ω₁,₁ 0.069288
pk_Ω₂,₂ 0.020318
pk_Ω₃,₃ 0.89964
pd_Ω₁,₁ 0.0028776
pd_Ω₂,₂ 0.00044801
pd_Ω₃,₃ 0.15375
pd_Ω₄,₄ 0.015014
σ_prop 0.088936
σ_add 0.41486
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: 1.0967254638671875e-5 1 5.174461e+05 8.708698e+05 * time: 2.3504390716552734 2 3.865265e+05 6.344302e+05 * time: 3.763947010040283 3 1.804274e+05 2.829723e+05 * time: 5.1270551681518555 4 9.706640e+04 1.550547e+05 * time: 6.4688560962677 5 4.769637e+04 6.778818e+04 * time: 7.773163080215454 6 2.902319e+04 3.499747e+04 * time: 9.06303596496582 7 1.823472e+04 1.705751e+04 * time: 10.36414909362793 8 1.258819e+04 9.569381e+03 * time: 11.654446125030518 9 9.389984e+03 8.615851e+03 * time: 12.928815126419067 10 7.314702e+03 7.636883e+03 * time: 14.210600137710571 11 5.916029e+03 6.624325e+03 * time: 15.488165140151978 12 4.930519e+03 5.558140e+03 * time: 16.77163314819336 13 4.125060e+03 4.315759e+03 * time: 18.043378114700317 14 3.549280e+03 3.051093e+03 * time: 19.3059720993042 15 3.283489e+03 2.157292e+03 * time: 20.57533097267151 16 3.204886e+03 1.659798e+03 * time: 21.83706307411194 17 3.194875e+03 1.480528e+03 * time: 23.111901998519897 18 3.193944e+03 1.437921e+03 * time: 24.362020015716553 19 3.193070e+03 1.411186e+03 * time: 25.616013050079346 20 3.190129e+03 1.355327e+03 * time: 26.918020963668823 21 3.183228e+03 1.276603e+03 * time: 28.251516103744507 22 3.164897e+03 1.151838e+03 * time: 29.53156805038452 23 3.119250e+03 9.712651e+02 * time: 30.810620069503784 24 3.006297e+03 7.204342e+02 * time: 32.11590504646301 25 2.738913e+03 4.050545e+02 * time: 33.4037070274353 26 2.123834e+03 2.318194e+02 * time: 34.5408980846405 27 1.789139e+03 2.290465e+02 * time: 35.77036905288696 28 1.396455e+03 1.683969e+02 * time: 176.12951111793518 29 1.333545e+03 1.336195e+02 * time: 177.50298595428467 30 1.297771e+03 2.452175e+02 * time: 178.71981716156006 31 1.266002e+03 1.523957e+02 * time: 179.9164800643921 32 1.255506e+03 1.733982e+02 * time: 181.1105351448059 33 1.247789e+03 1.971613e+02 * time: 182.33535313606262 34 1.244490e+03 1.915717e+02 * time: 183.54821109771729 35 1.240568e+03 1.704241e+02 * time: 184.78742909431458 36 1.240503e+03 1.711777e+02 * time: 185.9951159954071 37 1.240492e+03 1.711647e+02 * time: 187.2159321308136 38 1.239992e+03 1.687233e+02 * time: 188.4315869808197 39 1.239199e+03 1.624614e+02 * time: 189.6709690093994 40 1.236972e+03 1.400083e+02 * time: 190.904559135437 41 1.233204e+03 9.443139e+01 * time: 192.15486216545105 42 1.228683e+03 3.274305e+01 * time: 193.37648701667786 43 1.226466e+03 4.997666e+01 * time: 194.60761499404907 44 1.226104e+03 4.904455e+01 * time: 195.8263680934906 45 1.226088e+03 4.675831e+01 * time: 197.0472161769867 46 1.226088e+03 4.628481e+01 * time: 198.24502801895142 47 1.226085e+03 4.541370e+01 * time: 199.43380498886108 48 1.226080e+03 4.402775e+01 * time: 200.543475151062 49 1.226064e+03 4.142396e+01 * time: 201.6574580669403 50 1.226026e+03 3.663267e+01 * time: 202.7688591480255 51 1.225931e+03 2.851376e+01 * time: 203.8795771598816 52 1.225714e+03 2.844439e+01 * time: 204.99360394477844 53 1.225303e+03 3.026353e+01 * time: 206.1157729625702 54 1.224791e+03 4.156415e+01 * time: 207.23835110664368 55 1.224489e+03 5.047549e+01 * time: 208.34930801391602 56 1.224420e+03 4.451326e+01 * time: 209.45680713653564 57 1.224413e+03 3.994115e+01 * time: 210.55235409736633 58 1.224412e+03 3.879491e+01 * time: 211.6535120010376 59 1.224408e+03 3.677379e+01 * time: 212.74576711654663 60 1.224400e+03 3.378181e+01 * time: 213.84569907188416 61 1.224379e+03 3.158275e+01 * time: 214.950786113739 62 1.224324e+03 2.925310e+01 * time: 216.0745611190796 63 1.224181e+03 2.815884e+01 * time: 217.20979714393616 64 1.223813e+03 2.818297e+01 * time: 218.3489270210266 65 1.222906e+03 4.596708e+01 * time: 219.47388100624084 66 1.220911e+03 8.689986e+01 * time: 220.6734640598297 67 1.217611e+03 1.136509e+02 * time: 221.78291606903076 68 1.214314e+03 9.033016e+01 * time: 222.90369606018066 69 1.212533e+03 8.868756e+01 * time: 224.11485195159912 70 1.212272e+03 8.970398e+01 * time: 225.29155707359314 71 1.212263e+03 8.922046e+01 * time: 226.41549396514893 72 1.212259e+03 8.882520e+01 * time: 227.64228916168213 73 1.212242e+03 8.761642e+01 * time: 228.85575914382935 74 1.212205e+03 8.562156e+01 * time: 230.07331204414368 75 1.212102e+03 8.125479e+01 * time: 231.28080701828003 76 1.211850e+03 7.229855e+01 * time: 232.51611399650574 77 1.211238e+03 5.322149e+01 * time: 233.73439812660217 78 1.209945e+03 7.377053e+01 * time: 234.91851115226746 79 1.207890e+03 8.403807e+01 * time: 236.12126517295837 80 1.206058e+03 9.147814e+01 * time: 237.35737013816833 81 1.205389e+03 9.689402e+01 * time: 238.53633999824524 82 1.205303e+03 8.594641e+01 * time: 239.70233297348022 83 1.205297e+03 8.262863e+01 * time: 240.94235515594482 84 1.205292e+03 8.096275e+01 * time: 242.13172006607056 85 1.205277e+03 7.747512e+01 * time: 243.24207711219788 86 1.205241e+03 7.259152e+01 * time: 244.33572006225586 87 1.205143e+03 6.664782e+01 * time: 245.4345030784607 88 1.204895e+03 6.374591e+01 * time: 246.52581596374512 89 1.204267e+03 5.753713e+01 * time: 247.6290590763092 90 1.202818e+03 4.866654e+01 * time: 248.75005102157593 91 1.200116e+03 5.085943e+01 * time: 249.90765500068665 92 1.196895e+03 7.139637e+01 * time: 251.04908299446106 93 1.194610e+03 4.693841e+01 * time: 252.1901659965515 94 1.193880e+03 4.874622e+01 * time: 253.34200596809387 95 1.193833e+03 4.997328e+01 * time: 254.50410509109497 96 1.193831e+03 5.008739e+01 * time: 255.73292803764343 97 1.193829e+03 5.014847e+01 * time: 256.850711107254 98 1.193824e+03 5.025182e+01 * time: 257.9943540096283 99 1.193812e+03 5.040232e+01 * time: 259.1261031627655 100 1.193778e+03 5.061917e+01 * time: 260.2526640892029 101 1.193693e+03 5.087902e+01 * time: 261.35335516929626 102 1.193472e+03 5.105102e+01 * time: 262.43938302993774 103 1.192926e+03 5.068588e+01 * time: 263.5303280353546 104 1.191681e+03 4.863506e+01 * time: 264.6128661632538 105 1.189351e+03 4.892935e+01 * time: 265.69260716438293 106 1.186420e+03 7.777156e+01 * time: 266.7764849662781 107 1.184527e+03 8.874260e+01 * time: 267.89406299591064 108 1.184034e+03 8.377283e+01 * time: 268.9875030517578 109 1.183978e+03 7.932657e+01 * time: 270.08589005470276 110 1.183969e+03 7.794290e+01 * time: 271.220587015152 111 1.183966e+03 7.787253e+01 * time: 272.36798310279846 112 1.183962e+03 7.833696e+01 * time: 273.50313806533813 113 1.183955e+03 7.929137e+01 * time: 274.61871314048767 114 1.183940e+03 8.084179e+01 * time: 275.7783260345459 115 1.183904e+03 8.331129e+01 * time: 276.9091911315918 116 1.183812e+03 8.698526e+01 * time: 278.0200231075287 117 1.183576e+03 9.189102e+01 * time: 279.13590908050537 118 1.182997e+03 9.669181e+01 * time: 280.3037209510803 119 1.181703e+03 9.636170e+01 * time: 281.4303891658783 120 1.179436e+03 8.062012e+01 * time: 282.5633010864258 121 1.177026e+03 4.572956e+01 * time: 283.69928097724915 122 1.175794e+03 1.863503e+01 * time: 284.853404045105 123 1.175474e+03 1.968076e+01 * time: 286.02651715278625 124 1.175429e+03 1.958612e+01 * time: 287.1737389564514 125 1.175427e+03 1.993705e+01 * time: 288.28662300109863 126 1.175426e+03 1.962015e+01 * time: 289.4238030910492 127 1.175426e+03 1.965820e+01 * time: 290.5202660560608 128 1.175424e+03 1.984461e+01 * time: 291.62093210220337 129 1.175421e+03 2.001158e+01 * time: 292.7491171360016 130 1.175413e+03 2.035215e+01 * time: 293.8511140346527 131 1.175392e+03 2.084316e+01 * time: 294.9866211414337 132 1.175336e+03 2.164622e+01 * time: 296.11107897758484 133 1.175189e+03 2.287922e+01 * time: 297.238440990448 134 1.174802e+03 2.475497e+01 * time: 298.36141705513 135 1.173793e+03 4.143173e+01 * time: 299.51229214668274 136 1.171222e+03 7.150973e+01 * time: 300.6576600074768 137 1.165149e+03 1.078889e+02 * time: 301.8359501361847 138 1.155085e+03 7.759079e+01 * time: 303.05647706985474 139 1.149213e+03 6.862147e+01 * time: 304.3083870410919 140 1.146917e+03 6.326661e+01 * time: 305.5527141094208 141 1.144667e+03 4.997041e+01 * time: 306.820307970047 142 1.143086e+03 2.541688e+01 * time: 308.03285098075867 143 1.141825e+03 1.975967e+01 * time: 309.22648906707764 144 1.141293e+03 2.050313e+01 * time: 310.41299510002136 145 1.141050e+03 2.076852e+01 * time: 311.61468505859375 146 1.140979e+03 2.090534e+01 * time: 312.81991696357727 147 1.140976e+03 2.094562e+01 * time: 314.0004451274872 148 1.140975e+03 2.099640e+01 * time: 315.1859140396118 149 1.140974e+03 2.099904e+01 * time: 316.36300706863403 150 1.140974e+03 2.099888e+01 * time: 317.5467550754547 151 1.140973e+03 2.099225e+01 * time: 318.7398490905762 152 1.140972e+03 2.098193e+01 * time: 319.91878604888916 153 1.140970e+03 2.096599e+01 * time: 321.09982800483704 154 1.140963e+03 2.094403e+01 * time: 322.2816390991211 155 1.140947e+03 2.091801e+01 * time: 323.47728204727173 156 1.140905e+03 2.090038e+01 * time: 324.66283798217773 157 1.140796e+03 2.093557e+01 * time: 325.8692581653595 158 1.140506e+03 2.115931e+01 * time: 327.0862030982971 159 1.139721e+03 2.222025e+01 * time: 328.2945501804352 160 1.137501e+03 3.402239e+01 * time: 329.53060698509216 161 1.132295e+03 5.964366e+01 * time: 330.8257510662079 162 1.130015e+03 6.884181e+01 * time: 332.2783901691437 163 1.128409e+03 7.427004e+01 * time: 333.7853031158447 164 1.126419e+03 1.090627e+02 * time: 335.28106117248535 165 1.120677e+03 1.367968e+02 * time: 336.6449489593506 166 1.116325e+03 7.465640e+01 * time: 337.9958710670471 167 1.115223e+03 4.224641e+01 * time: 339.27745509147644 168 1.112608e+03 3.602555e+01 * time: 340.59220695495605 169 1.111487e+03 1.611035e+01 * time: 341.92792797088623 170 1.111244e+03 1.730205e+01 * time: 343.2664830684662 171 1.111202e+03 1.118481e+01 * time: 344.5985691547394 172 1.111188e+03 1.043851e+01 * time: 345.93772196769714 173 1.111187e+03 1.048990e+01 * time: 347.27853298187256 174 1.111187e+03 1.060111e+01 * time: 348.59413599967957 175 1.111187e+03 1.073269e+01 * time: 349.9134330749512 176 1.111186e+03 1.091744e+01 * time: 351.23578095436096 177 1.111185e+03 1.120428e+01 * time: 352.5651080608368 178 1.111182e+03 1.164828e+01 * time: 353.8924310207367 179 1.111175e+03 1.234981e+01 * time: 355.2222900390625 180 1.111156e+03 1.346222e+01 * time: 356.54751896858215 181 1.111106e+03 1.523323e+01 * time: 357.9040219783783 182 1.110975e+03 2.097544e+01 * time: 359.26089811325073 183 1.110638e+03 3.408181e+01 * time: 360.6080651283264 184 1.109785e+03 5.427198e+01 * time: 361.9711651802063 185 1.107762e+03 8.193896e+01 * time: 363.34891295433044 186 1.103637e+03 1.077201e+02 * time: 364.7835991382599 187 1.097569e+03 1.003652e+02 * time: 366.2988181114197 188 1.092967e+03 5.096083e+01 * time: 367.8593349456787 189 1.091994e+03 1.584738e+01 * time: 369.39310812950134 190 1.091933e+03 5.113684e+00 * time: 370.92089200019836 191 1.091918e+03 4.300736e+00 * time: 372.4406681060791 192 1.091912e+03 4.320276e+00 * time: 373.94323205947876 193 1.091907e+03 4.348644e+00 * time: 375.4366309642792 194 1.091905e+03 4.353250e+00 * time: 376.93143010139465 195 1.091904e+03 4.346010e+00 * time: 378.42278003692627 196 1.091904e+03 4.343141e+00 * time: 379.9144389629364 197 1.091904e+03 4.341661e+00 * time: 381.38539600372314 198 1.091904e+03 4.340192e+00 * time: 382.856281042099 199 1.091904e+03 4.336230e+00 * time: 384.32773208618164 200 1.091904e+03 4.330351e+00 * time: 385.8459801673889 201 1.091903e+03 4.319693e+00 * time: 387.43713212013245 202 1.091901e+03 4.301698e+00 * time: 389.0251021385193 203 1.091897e+03 4.269750e+00 * time: 390.6275780200958 204 1.091884e+03 4.211883e+00 * time: 392.1785089969635 205 1.091853e+03 4.102286e+00 * time: 393.73617696762085 206 1.091772e+03 5.258973e+00 * time: 395.41366600990295 207 1.091574e+03 8.172454e+00 * time: 397.1466181278229 208 1.091123e+03 1.178724e+01 * time: 398.82024216651917 209 1.090245e+03 1.457136e+01 * time: 400.6152379512787 210 1.088868e+03 1.369431e+01 * time: 402.35442209243774 211 1.087425e+03 8.299086e+00 * time: 403.88132405281067 212 1.087110e+03 5.812640e+00 * time: 405.35406017303467 213 1.087046e+03 5.485431e+00 * time: 406.8415801525116 214 1.087043e+03 5.388012e+00 * time: 408.3091480731964 215 1.087043e+03 5.384068e+00 * time: 409.77692794799805 216 1.087043e+03 5.384641e+00 * time: 411.2282061576843 217 1.087043e+03 5.384014e+00 * time: 412.6743321418762 218 1.087043e+03 5.383782e+00 * time: 414.13456106185913 219 1.087043e+03 5.383930e+00 * time: 415.58486914634705 220 1.087043e+03 5.384549e+00 * time: 417.04563212394714 221 1.087042e+03 5.386342e+00 * time: 418.6253011226654 222 1.087042e+03 5.390485e+00 * time: 420.10045099258423 223 1.087039e+03 5.400620e+00 * time: 421.52663803100586 224 1.087034e+03 5.425468e+00 * time: 422.95523500442505 225 1.087019e+03 5.488586e+00 * time: 424.38521003723145 226 1.086979e+03 6.874461e+00 * time: 425.825021982193 227 1.086868e+03 1.129603e+01 * time: 427.2687990665436 228 1.086507e+03 1.863818e+01 * time: 428.6943140029907 229 1.085639e+03 2.222381e+01 * time: 430.2790491580963 230 1.085276e+03 2.099129e+01 * time: 431.79383301734924 231 1.084589e+03 2.368163e+01 * time: 433.5643050670624 232 1.083748e+03 2.730613e+01 * time: 435.073832988739 233 1.080654e+03 6.007070e+01 * time: 436.61378502845764 234 1.072587e+03 5.434177e+01 * time: 438.1010501384735 235 1.070694e+03 1.598435e+01 * time: 439.57811212539673 236 1.070017e+03 1.161830e+01 * time: 441.10464906692505 237 1.069752e+03 1.918608e+01 * time: 442.5689220428467 238 1.069640e+03 9.742903e+00 * time: 444.0264241695404 239 1.069517e+03 4.099627e+00 * time: 445.5099730491638 240 1.069495e+03 3.815363e+00 * time: 446.9579231739044 241 1.069488e+03 2.216629e+00 * time: 448.33486795425415 242 1.069488e+03 2.233237e+00 * time: 449.73813915252686 243 1.069488e+03 2.235862e+00 * time: 451.1525731086731 244 1.069488e+03 2.235888e+00 * time: 452.57002902030945 245 1.069488e+03 2.235786e+00 * time: 454.0303370952606 246 1.069488e+03 2.235560e+00 * time: 455.52723598480225 247 1.069488e+03 2.235068e+00 * time: 456.97886395454407 248 1.069488e+03 2.234047e+00 * time: 458.32697010040283 249 1.069487e+03 2.231804e+00 * time: 459.66279315948486 250 1.069486e+03 2.226682e+00 * time: 461.01168417930603 251 1.069484e+03 2.218474e+00 * time: 462.35030698776245 252 1.069478e+03 3.597494e+00 * time: 463.69751811027527 253 1.069463e+03 5.725019e+00 * time: 465.05444717407227 254 1.069426e+03 8.773863e+00 * time: 466.4594180583954 255 1.069343e+03 1.233147e+01 * time: 467.9462101459503 256 1.069199e+03 1.423932e+01 * time: 469.40702295303345 257 1.069038e+03 1.107739e+01 * time: 470.8811790943146 258 1.068948e+03 4.305391e+00 * time: 472.360888004303 259 1.068930e+03 8.661019e-01 * time: 473.8403341770172 260 1.068929e+03 8.685835e-01 * time: 475.27386808395386 261 1.068929e+03 8.689206e-01 * time: 476.6756081581116 262 1.068929e+03 8.689472e-01 * time: 478.1250250339508 263 1.068929e+03 8.689831e-01 * time: 479.6149401664734 264 1.068929e+03 8.690444e-01 * time: 481.17668199539185 265 1.068929e+03 8.691394e-01 * time: 482.595253944397 266 1.068929e+03 8.692847e-01 * time: 484.0207941532135 267 1.068929e+03 8.694972e-01 * time: 485.3888599872589 268 1.068929e+03 8.697830e-01 * time: 486.75368309020996 269 1.068929e+03 8.700931e-01 * time: 488.12266516685486 270 1.068928e+03 1.196930e+00 * time: 489.4835801124573 271 1.068927e+03 1.942720e+00 * time: 490.8559470176697 272 1.068923e+03 3.132419e+00 * time: 492.2287311553955 273 1.068912e+03 4.986894e+00 * time: 493.61767411231995 274 1.068885e+03 7.704936e+00 * time: 495.0121121406555 275 1.068825e+03 1.106840e+01 * time: 496.38876008987427 276 1.068706e+03 1.347148e+01 * time: 497.7670180797577 277 1.068534e+03 1.175424e+01 * time: 499.13039898872375 278 1.068403e+03 5.825066e+00 * time: 500.50286316871643 279 1.068376e+03 1.542530e+00 * time: 501.8714060783386 280 1.068375e+03 1.320283e+00 * time: 503.2480499744415 281 1.068375e+03 1.291873e+00 * time: 504.6058940887451 282 1.068374e+03 1.284133e+00 * time: 505.9587559700012 283 1.068374e+03 1.283996e+00 * time: 507.35157012939453 284 1.068374e+03 1.283987e+00 * time: 508.7587330341339 285 1.068374e+03 1.283973e+00 * time: 510.17090702056885 286 1.068374e+03 1.283948e+00 * time: 511.5888640880585 287 1.068374e+03 1.283944e+00 * time: 513.0333151817322 288 1.068374e+03 1.283939e+00 * time: 514.4798879623413 289 1.068374e+03 1.283931e+00 * time: 515.9274799823761 290 1.068374e+03 1.283931e+00 * time: 517.4443221092224 291 1.068374e+03 1.283931e+00 * time: 518.9739661216736 292 1.068374e+03 1.283931e+00 * time: 520.5062050819397 293 1.068374e+03 1.283931e+00 * time: 522.0958471298218 294 1.068374e+03 1.283931e+00 * time: 523.7067620754242 295 1.068374e+03 1.283931e+00 * time: 525.3156690597534 296 1.068374e+03 1.277366e+00 * time: 526.6698000431061 297 1.068374e+03 1.276676e+00 * time: 528.0194549560547 298 1.068374e+03 1.273787e+00 * time: 529.367115020752 299 1.068374e+03 1.270733e+00 * time: 530.7121839523315 300 1.068374e+03 1.264549e+00 * time: 532.0575060844421 301 1.068374e+03 1.254567e+00 * time: 533.4073350429535 302 1.068373e+03 1.236379e+00 * time: 534.7559909820557 303 1.068370e+03 1.889956e+00 * time: 536.1147711277008 304 1.068364e+03 2.910582e+00 * time: 537.4767069816589 305 1.068348e+03 4.391420e+00 * time: 538.8366780281067 306 1.068313e+03 6.164560e+00 * time: 540.1913051605225 307 1.068246e+03 7.233428e+00 * time: 541.5575680732727 308 1.068154e+03 6.055187e+00 * time: 542.9138519763947 309 1.068069e+03 3.059256e+00 * time: 544.2669241428375 310 1.068025e+03 5.434011e-01 * time: 545.6382501125336 311 1.068010e+03 6.146314e-01 * time: 547.0022530555725 312 1.068003e+03 8.114366e-01 * time: 548.3549439907074 313 1.067997e+03 5.072251e-01 * time: 549.7095420360565 314 1.067993e+03 7.983059e-02 * time: 551.0575511455536 315 1.067992e+03 1.029348e-01 * time: 552.399670124054 316 1.067992e+03 1.192737e-01 * time: 553.7479090690613 317 1.067991e+03 8.345899e-02 * time: 555.1091101169586 318 1.067991e+03 9.188838e-02 * time: 556.4687490463257 319 1.067991e+03 9.481172e-02 * time: 557.821161031723 320 1.067991e+03 9.481216e-02 * time: 559.2196459770203 321 1.067991e+03 9.481276e-02 * time: 560.6228411197662 322 1.067991e+03 9.481276e-02 * time: 562.1291840076447 323 1.067991e+03 9.523276e-02 * time: 563.5955350399017 324 1.067991e+03 9.527485e-02 * time: 564.9410741329193 325 1.067991e+03 9.527975e-02 * time: 566.319216966629 326 1.067991e+03 9.528102e-02 * time: 567.7668011188507 327 1.067991e+03 9.529977e-02 * time: 569.1490941047668 328 1.067991e+03 9.554032e-02 * time: 570.5199360847473 329 1.067991e+03 9.554111e-02 * time: 572.6347720623016 330 1.067991e+03 9.554114e-02 * time: 575.2774939537048 331 1.067991e+03 9.554121e-02 * time: 576.9334809780121 332 1.067991e+03 9.554285e-02 * time: 578.4859280586243 333 1.067991e+03 9.554319e-02 * time: 580.0383989810944 334 1.067991e+03 9.554319e-02 * time: 581.8009359836578 335 1.067991e+03 9.554319e-02 * time: 583.5525670051575
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: NoObjectiveChange
Log-likelihood value: -1067.9914
-----------------------
Estimate
-----------------------
pop_CL 0.13521
pop_V 8.0133
pop_tabs 0.57108
pop_lag 0.87562
pop_e0 96.399
pop_emax -1.0615
pop_c50 1.4912
pop_tover 14.05
pk_Ω₁,₁ 0.068013
pk_Ω₂,₂ 0.021048
pk_Ω₃,₃ 0.86272
pd_Ω₁,₁ 0.0029774
pd_Ω₂,₂ 1.9312e-7
pd_Ω₃,₃ 0.14561
pd_Ω₄,₄ 0.015351
σ_prop 0.088484
σ_add 0.41685
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.