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: 0.04304909706115723 1 5.185876e+05 8.742030e+05 * time: 4.374430894851685 2 3.866954e+05 6.366579e+05 * time: 5.301578998565674 3 1.795029e+05 2.835394e+05 * time: 6.1889190673828125 4 9.682657e+04 1.546520e+05 * time: 7.1284019947052 5 4.791913e+04 6.820049e+04 * time: 8.015537023544312 6 2.907378e+04 3.509701e+04 * time: 8.934067010879517 7 1.827380e+04 1.713127e+04 * time: 9.839226961135864 8 1.260636e+04 9.563331e+03 * time: 10.720464944839478 9 9.403443e+03 8.611646e+03 * time: 11.638986110687256 10 7.325848e+03 7.634817e+03 * time: 12.494635105133057 11 5.926870e+03 6.625147e+03 * time: 13.357939004898071 12 4.942281e+03 5.562822e+03 * time: 14.23684310913086 13 4.139086e+03 4.326566e+03 * time: 15.086654901504517 14 3.563492e+03 3.065260e+03 * time: 15.918452024459839 15 3.296158e+03 2.170249e+03 * time: 16.76041889190674 16 3.216280e+03 1.669629e+03 * time: 17.605499982833862 17 3.205942e+03 1.487868e+03 * time: 18.42154312133789 18 3.204980e+03 1.444252e+03 * time: 19.253813982009888 19 3.204106e+03 1.417661e+03 * time: 20.075377941131592 20 3.201144e+03 1.361617e+03 * time: 20.88825011253357 21 3.194217e+03 1.282919e+03 * time: 21.702234983444214 22 3.175799e+03 1.158037e+03 * time: 22.50766897201538 23 3.130001e+03 9.774739e+02 * time: 23.33515191078186 24 3.016853e+03 7.265903e+02 * time: 24.17748212814331 25 2.749859e+03 4.107629e+02 * time: 24.988327980041504 26 2.137217e+03 2.318395e+02 * time: 25.770344018936157 27 1.756675e+03 2.281485e+02 * time: 26.606022119522095 28 1.380669e+03 1.613775e+02 * time: 28.67688798904419 29 1.328440e+03 1.298802e+02 * time: 29.417633056640625 30 1.287369e+03 2.434675e+02 * time: 30.063885927200317 31 1.263062e+03 1.616159e+02 * time: 30.702299118041992 32 1.254700e+03 1.795912e+02 * time: 31.325376987457275 33 1.247195e+03 1.995476e+02 * time: 31.960274934768677 34 1.243756e+03 1.937611e+02 * time: 32.59869599342346 35 1.240818e+03 1.738948e+02 * time: 33.2686231136322 36 1.240776e+03 1.743455e+02 * time: 33.90410590171814 37 1.240765e+03 1.742681e+02 * time: 34.563947916030884 38 1.240074e+03 1.686540e+02 * time: 35.19507098197937 39 1.238981e+03 1.584905e+02 * time: 35.837584018707275 40 1.236205e+03 1.301147e+02 * time: 36.494946002960205 41 1.232071e+03 8.166215e+01 * time: 37.151511907577515 42 1.228115e+03 3.717443e+01 * time: 37.80254411697388 43 1.226327e+03 5.239396e+01 * time: 38.451884031295776 44 1.226024e+03 4.861318e+01 * time: 39.10285997390747 45 1.226011e+03 4.580542e+01 * time: 39.762574911117554 46 1.226010e+03 4.534695e+01 * time: 40.41445302963257 47 1.226008e+03 4.462779e+01 * time: 41.08552002906799 48 1.226002e+03 4.331824e+01 * time: 41.74311900138855 49 1.225987e+03 4.094327e+01 * time: 42.39648103713989 50 1.225949e+03 3.642019e+01 * time: 43.080918073654175 51 1.225855e+03 2.864881e+01 * time: 43.74250912666321 52 1.225639e+03 2.863354e+01 * time: 44.409523010253906 53 1.225232e+03 3.020554e+01 * time: 45.06808304786682 54 1.224729e+03 4.103561e+01 * time: 45.725642919540405 55 1.224434e+03 5.071657e+01 * time: 46.39027690887451 56 1.224368e+03 4.548491e+01 * time: 47.02430295944214 57 1.224361e+03 4.120954e+01 * time: 47.674741983413696 58 1.224360e+03 4.006985e+01 * time: 48.327943086624146 59 1.224357e+03 3.801109e+01 * time: 48.98255896568298 60 1.224349e+03 3.498015e+01 * time: 49.65358805656433 61 1.224327e+03 3.147637e+01 * time: 50.28519797325134 62 1.224271e+03 2.931465e+01 * time: 50.93686890602112 63 1.224126e+03 2.814671e+01 * time: 51.59349489212036 64 1.223753e+03 2.816101e+01 * time: 52.25110197067261 65 1.222837e+03 4.614252e+01 * time: 52.9343740940094 66 1.220832e+03 8.757572e+01 * time: 53.58036398887634 67 1.217567e+03 1.138929e+02 * time: 54.23774695396423 68 1.214404e+03 8.919354e+01 * time: 54.885634899139404 69 1.212787e+03 8.410207e+01 * time: 55.51684808731079 70 1.212569e+03 8.598204e+01 * time: 56.137048959732056 71 1.212562e+03 8.564567e+01 * time: 56.74971008300781 72 1.212557e+03 8.527574e+01 * time: 57.37280893325806 73 1.212540e+03 8.422259e+01 * time: 57.99724006652832 74 1.212503e+03 8.240816e+01 * time: 58.61121392250061 75 1.212401e+03 7.843332e+01 * time: 59.24232292175293 76 1.212148e+03 7.012172e+01 * time: 59.90290689468384 77 1.211531e+03 5.213081e+01 * time: 60.53248310089111 78 1.210204e+03 7.305043e+01 * time: 61.187443017959595 79 1.208013e+03 8.679967e+01 * time: 61.79960608482361 80 1.205949e+03 9.839154e+01 * time: 62.422693967819214 81 1.205142e+03 1.083109e+02 * time: 63.04945492744446 82 1.205024e+03 9.689232e+01 * time: 63.674262046813965 83 1.205016e+03 9.287604e+01 * time: 64.29216194152832 84 1.205012e+03 9.131434e+01 * time: 64.91503095626831 85 1.204994e+03 8.785522e+01 * time: 65.51611804962158 86 1.204955e+03 8.321345e+01 * time: 66.1206750869751 87 1.204846e+03 7.512847e+01 * time: 66.69163703918457 88 1.204571e+03 6.677960e+01 * time: 67.31184101104736 89 1.203874e+03 5.917097e+01 * time: 67.93780899047852 90 1.202256e+03 5.625398e+01 * time: 68.55029201507568 91 1.199255e+03 5.407775e+01 * time: 69.15749001502991 92 1.195772e+03 6.418968e+01 * time: 69.77275490760803 93 1.193272e+03 4.385779e+01 * time: 70.37839889526367 94 1.192403e+03 4.695234e+01 * time: 70.9763240814209 95 1.192345e+03 4.824611e+01 * time: 71.54971098899841 96 1.192343e+03 4.834837e+01 * time: 72.15219306945801 97 1.192341e+03 4.841664e+01 * time: 72.7602128982544 98 1.192337e+03 4.853485e+01 * time: 73.34136509895325 99 1.192326e+03 4.872157e+01 * time: 73.9321780204773 100 1.192298e+03 4.900641e+01 * time: 74.54442811012268 101 1.192226e+03 4.940187e+01 * time: 75.15099310874939 102 1.192043e+03 4.983597e+01 * time: 75.75468301773071 103 1.191590e+03 4.998730e+01 * time: 76.33803391456604 104 1.190555e+03 4.888824e+01 * time: 76.9432909488678 105 1.188610e+03 4.476259e+01 * time: 77.5702760219574 106 1.186163e+03 5.610726e+01 * time: 78.21266603469849 107 1.184413e+03 7.224666e+01 * time: 78.88323402404785 108 1.183656e+03 7.490583e+01 * time: 79.53874802589417 109 1.183479e+03 7.310985e+01 * time: 80.18608212471008 110 1.183445e+03 7.276225e+01 * time: 80.85224199295044 111 1.183435e+03 7.362156e+01 * time: 81.49638104438782 112 1.183432e+03 7.447416e+01 * time: 82.15064811706543 113 1.183428e+03 7.576303e+01 * time: 82.80592393875122 114 1.183422e+03 7.712921e+01 * time: 83.4615490436554 115 1.183402e+03 7.954539e+01 * time: 84.10541200637817 116 1.183354e+03 8.304573e+01 * time: 84.72451710700989 117 1.183227e+03 8.811035e+01 * time: 85.35203289985657 118 1.182910e+03 9.433287e+01 * time: 85.9739100933075 119 1.182159e+03 9.929986e+01 * time: 86.66459608078003 120 1.180613e+03 9.583938e+01 * time: 87.37240099906921 121 1.178253e+03 7.334167e+01 * time: 88.12905406951904 122 1.176240e+03 3.397205e+01 * time: 88.89573311805725 123 1.175532e+03 1.920932e+01 * time: 89.6529049873352 124 1.175411e+03 1.932720e+01 * time: 90.40943789482117 125 1.175397e+03 1.965053e+01 * time: 91.13668489456177 126 1.175396e+03 1.966443e+01 * time: 91.86466789245605 127 1.175395e+03 1.960396e+01 * time: 92.58368992805481 128 1.175395e+03 1.957495e+01 * time: 93.3163890838623 129 1.175393e+03 1.946094e+01 * time: 94.0526270866394 130 1.175388e+03 1.938728e+01 * time: 94.80828309059143 131 1.175375e+03 1.916215e+01 * time: 95.54454493522644 132 1.175343e+03 1.894152e+01 * time: 96.29219794273376 133 1.175257e+03 1.842699e+01 * time: 97.03752589225769 134 1.175031e+03 1.935678e+01 * time: 97.81835508346558 135 1.174436e+03 2.681160e+01 * time: 98.62505197525024 136 1.172888e+03 3.945075e+01 * time: 99.43357110023499 137 1.169017e+03 5.876234e+01 * time: 100.22343802452087 138 1.160885e+03 7.779933e+01 * time: 101.01137089729309 139 1.152001e+03 6.044664e+01 * time: 101.81194806098938 140 1.150076e+03 2.093881e+02 * time: 102.63891005516052 141 1.147572e+03 1.330305e+02 * time: 103.44896912574768 142 1.143865e+03 2.372423e+01 * time: 104.25883507728577 143 1.142815e+03 2.227954e+01 * time: 105.01806092262268 144 1.141776e+03 3.847481e+01 * time: 105.71958708763123 145 1.141295e+03 2.197611e+01 * time: 106.40395903587341 146 1.140996e+03 2.098791e+01 * time: 107.1034779548645 147 1.140982e+03 2.095235e+01 * time: 107.79986906051636 148 1.140979e+03 2.098518e+01 * time: 108.48278093338013 149 1.140978e+03 2.101190e+01 * time: 109.18417692184448
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: -1140.9777
-----------------------
Estimate
-----------------------
pop_CL 0.13722
pop_V 7.8506
pop_tabs 1.084
pop_lag 0.35905
pop_e0 96.351
pop_emax -1.0615
pop_c50 1.4777
pop_tover 14.461
pk_Ω₁,₁ 0.25384
pk_Ω₂,₂ 0.042739
pk_Ω₃,₃ 0.22544
pd_Ω₁,₁ 0.0027398
pd_Ω₂,₂ 0.0021862
pd_Ω₃,₃ 0.01899
pd_Ω₄,₄ 0.017768
σ_prop 0.013046
σ_add 0.9655
σ_fx 3.4529
-----------------------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: 2.4080276489257812e-5 1 5.174461e+05 8.708698e+05 * time: 1.9264459609985352 2 3.865265e+05 6.344302e+05 * time: 3.6215500831604004 3 1.804274e+05 2.829723e+05 * time: 5.311941146850586 4 9.706640e+04 1.550547e+05 * time: 7.100191116333008 5 4.769637e+04 6.778818e+04 * time: 9.231323003768921 6 2.902319e+04 3.499747e+04 * time: 11.345361948013306 7 1.823472e+04 1.705751e+04 * time: 13.300086975097656 8 1.258819e+04 9.569381e+03 * time: 15.35813307762146 9 9.389984e+03 8.615851e+03 * time: 17.143139123916626 10 7.314702e+03 7.636883e+03 * time: 19.658288955688477 11 5.916029e+03 6.624325e+03 * time: 21.538336992263794 12 4.930519e+03 5.558140e+03 * time: 23.261306047439575 13 4.125060e+03 4.315759e+03 * time: 24.986050128936768 14 3.549280e+03 3.051093e+03 * time: 26.742098093032837 15 3.283489e+03 2.157292e+03 * time: 28.55439805984497 16 3.204886e+03 1.659798e+03 * time: 30.403362035751343 17 3.194875e+03 1.480528e+03 * time: 32.1552300453186 18 3.193944e+03 1.437921e+03 * time: 33.897090911865234 19 3.193070e+03 1.411186e+03 * time: 35.60795497894287 20 3.190129e+03 1.355327e+03 * time: 37.324695110321045 21 3.183228e+03 1.276603e+03 * time: 39.01523494720459 22 3.164897e+03 1.151838e+03 * time: 40.80250692367554 23 3.119250e+03 9.712651e+02 * time: 42.45875406265259 24 3.006297e+03 7.204342e+02 * time: 44.06330108642578 25 2.738913e+03 4.050545e+02 * time: 45.6571409702301 26 2.123834e+03 2.318194e+02 * time: 47.12534809112549 27 1.789138e+03 2.290465e+02 * time: 48.67637300491333 28 1.396455e+03 1.683969e+02 * time: 53.67441201210022 29 1.333545e+03 1.336195e+02 * time: 55.302868127822876 30 1.297771e+03 2.452189e+02 * time: 56.621626138687134 31 1.266002e+03 1.523968e+02 * time: 57.921355962753296 32 1.255506e+03 1.733993e+02 * time: 59.224332094192505 33 1.247789e+03 1.971624e+02 * time: 60.60240292549133 34 1.244490e+03 1.915728e+02 * time: 61.8923180103302 35 1.240568e+03 1.704250e+02 * time: 63.19865608215332 36 1.240503e+03 1.711787e+02 * time: 64.56064796447754 37 1.240492e+03 1.711657e+02 * time: 65.89852809906006 38 1.239992e+03 1.687239e+02 * time: 67.2195451259613 39 1.239199e+03 1.624610e+02 * time: 68.56213092803955 40 1.236971e+03 1.400045e+02 * time: 69.96357011795044 41 1.233203e+03 9.442284e+01 * time: 71.41717910766602 42 1.228682e+03 3.273925e+01 * time: 72.74443507194519 43 1.226466e+03 4.997777e+01 * time: 74.0948920249939 44 1.226104e+03 4.904407e+01 * time: 75.45070314407349 45 1.226088e+03 4.675833e+01 * time: 76.85252714157104 46 1.226088e+03 4.628496e+01 * time: 78.81465411186218 47 1.226085e+03 4.541356e+01 * time: 80.2940320968628 48 1.226080e+03 4.402741e+01 * time: 81.62674403190613 49 1.226064e+03 4.142300e+01 * time: 82.98249101638794 50 1.226026e+03 3.663066e+01 * time: 84.3319480419159 51 1.225931e+03 2.851375e+01 * time: 85.72894096374512 52 1.225713e+03 2.844436e+01 * time: 87.12494897842407 53 1.225303e+03 3.026439e+01 * time: 88.61104798316956 54 1.224791e+03 4.157122e+01 * time: 90.0492091178894 55 1.224489e+03 5.047474e+01 * time: 91.42541694641113 56 1.224420e+03 4.451066e+01 * time: 92.79210114479065 57 1.224413e+03 3.994027e+01 * time: 94.08522295951843 58 1.224412e+03 3.879446e+01 * time: 95.3483989238739 59 1.224408e+03 3.677251e+01 * time: 96.67338299751282 60 1.224400e+03 3.377994e+01 * time: 97.9625039100647 61 1.224379e+03 3.158188e+01 * time: 99.26274108886719 62 1.224324e+03 2.925156e+01 * time: 100.58178091049194 63 1.224180e+03 2.815886e+01 * time: 101.93912601470947 64 1.223813e+03 2.818297e+01 * time: 103.3219940662384 65 1.222906e+03 4.598741e+01 * time: 104.76218891143799 66 1.220910e+03 8.692124e+01 * time: 106.10996508598328 67 1.217609e+03 1.136529e+02 * time: 107.39196300506592 68 1.214313e+03 9.030293e+01 * time: 108.67836713790894 69 1.212533e+03 8.868945e+01 * time: 109.96242499351501 70 1.212272e+03 8.970280e+01 * time: 111.2830970287323 71 1.212264e+03 8.921923e+01 * time: 112.58584213256836 72 1.212259e+03 8.882367e+01 * time: 113.84140491485596 73 1.212242e+03 8.761510e+01 * time: 115.08940410614014 74 1.212205e+03 8.561996e+01 * time: 116.39553594589233 75 1.212103e+03 8.125316e+01 * time: 117.7300591468811 76 1.211850e+03 7.229635e+01 * time: 119.1615560054779 77 1.211238e+03 5.321840e+01 * time: 120.50576114654541 78 1.209945e+03 7.377542e+01 * time: 121.8812530040741 79 1.207890e+03 8.404090e+01 * time: 123.41842699050903 80 1.206058e+03 9.148111e+01 * time: 124.96521091461182 81 1.205389e+03 9.689402e+01 * time: 126.65228700637817 82 1.205303e+03 8.594691e+01 * time: 128.26258897781372 83 1.205297e+03 8.262969e+01 * time: 129.8500349521637 84 1.205293e+03 8.096366e+01 * time: 131.38830304145813 85 1.205277e+03 7.747617e+01 * time: 132.97641110420227 86 1.205241e+03 7.259257e+01 * time: 134.58031010627747 87 1.205143e+03 6.664833e+01 * time: 136.17957496643066 88 1.204895e+03 6.374646e+01 * time: 137.80389499664307 89 1.204268e+03 5.753767e+01 * time: 139.44229412078857 90 1.202818e+03 4.866324e+01 * time: 141.06299901008606 91 1.200116e+03 5.085841e+01 * time: 142.6771800518036 92 1.196895e+03 7.139478e+01 * time: 144.25075101852417 93 1.194610e+03 4.693643e+01 * time: 145.7862319946289 94 1.193880e+03 4.874637e+01 * time: 147.5923149585724 95 1.193833e+03 4.997341e+01 * time: 149.19410705566406 96 1.193831e+03 5.008752e+01 * time: 150.72522807121277 97 1.193829e+03 5.014860e+01 * time: 152.2506000995636 98 1.193824e+03 5.025195e+01 * time: 153.80788111686707 99 1.193812e+03 5.040245e+01 * time: 155.39248609542847 100 1.193778e+03 5.061928e+01 * time: 156.99573802947998 101 1.193693e+03 5.087910e+01 * time: 158.5734519958496 102 1.193472e+03 5.105104e+01 * time: 160.24349403381348 103 1.192926e+03 5.068579e+01 * time: 161.8205690383911 104 1.191681e+03 4.863476e+01 * time: 163.42136001586914 105 1.189351e+03 4.892981e+01 * time: 164.99288511276245 106 1.186420e+03 7.777396e+01 * time: 166.55275011062622 107 1.184527e+03 8.874389e+01 * time: 168.7089409828186 108 1.184034e+03 8.377338e+01 * time: 171.10128712654114 109 1.183978e+03 7.932701e+01 * time: 173.41671514511108 110 1.183969e+03 7.794331e+01 * time: 175.1115961074829 111 1.183966e+03 7.787273e+01 * time: 176.77953791618347 112 1.183962e+03 7.833691e+01 * time: 178.40836191177368 113 1.183955e+03 7.929110e+01 * time: 180.0189290046692 114 1.183940e+03 8.084135e+01 * time: 181.63363313674927 115 1.183904e+03 8.331063e+01 * time: 183.09229397773743 116 1.183812e+03 8.698431e+01 * time: 184.42673993110657 117 1.183576e+03 9.188955e+01 * time: 185.76647400856018 118 1.182997e+03 9.668923e+01 * time: 187.13028192520142 119 1.181702e+03 9.635663e+01 * time: 188.48431611061096 120 1.179436e+03 8.061104e+01 * time: 189.9547119140625 121 1.177026e+03 4.571924e+01 * time: 191.4070589542389 122 1.175794e+03 1.863482e+01 * time: 192.93562698364258 123 1.175474e+03 1.968114e+01 * time: 194.36725401878357 124 1.175429e+03 1.958620e+01 * time: 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488.10483407974243 266 1.068929e+03 8.689733e-01 * time: 490.3046851158142 267 1.068929e+03 8.689308e-01 * time: 492.4863049983978 268 1.068929e+03 8.687927e-01 * time: 494.65591192245483 269 1.068929e+03 1.146689e+00 * time: 496.8524000644684 270 1.068928e+03 1.858717e+00 * time: 499.05850291252136 271 1.068926e+03 3.014163e+00 * time: 501.29847598075867 272 1.068921e+03 4.850801e+00 * time: 503.5252649784088 273 1.068908e+03 7.706317e+00 * time: 505.74298906326294 274 1.068876e+03 1.183609e+01 * time: 507.93011498451233 275 1.068805e+03 1.674967e+01 * time: 510.1117241382599 276 1.068668e+03 1.965804e+01 * time: 512.2641880512238 277 1.068489e+03 1.560006e+01 * time: 514.4509711265564 278 1.068389e+03 6.429456e+00 * time: 516.6820991039276 279 1.068376e+03 1.621982e+00 * time: 518.8158891201019 280 1.068375e+03 1.316816e+00 * time: 520.8768141269684 281 1.068374e+03 1.288387e+00 * time: 522.9203379154205 282 1.068374e+03 1.283839e+00 * time: 524.9372251033783 283 1.068374e+03 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1.068095e+03 7.878246e+00 * time: 564.3307330608368 302 1.068036e+03 2.139748e+00 * time: 566.3571960926056 303 1.068013e+03 8.037342e-01 * time: 568.4251699447632 304 1.068005e+03 1.492898e+00 * time: 570.4507060050964 305 1.067999e+03 1.212543e+00 * time: 572.4567770957947 306 1.067994e+03 4.245521e-01 * time: 574.4850800037384 307 1.067992e+03 8.588518e-02 * time: 576.4877121448517 308 1.067992e+03 2.309018e-01 * time: 578.4897630214691 309 1.067991e+03 1.646081e-01 * time: 580.5213279724121 310 1.067991e+03 9.057167e-02 * time: 582.5455620288849 311 1.067991e+03 9.435652e-02 * time: 584.5624461174011 312 1.067991e+03 9.436421e-02 * time: 586.6975400447845 313 1.067991e+03 9.523930e-02 * time: 588.7049510478973 314 1.067991e+03 9.525578e-02 * time: 590.7153749465942 315 1.067991e+03 9.529733e-02 * time: 592.7162311077118 316 1.067991e+03 9.531681e-02 * time: 594.7506020069122 317 1.067991e+03 9.536338e-02 * time: 597.0153050422668 318 1.067991e+03 9.542813e-02 * time: 599.0687401294708 319 1.067991e+03 9.552402e-02 * time: 601.1418271064758 320 1.067991e+03 9.553892e-02 * time: 603.2980189323425 321 1.067991e+03 9.553930e-02 * time: 605.6175971031189 322 1.067991e+03 9.553937e-02 * time: 608.0138339996338 323 1.067991e+03 9.553945e-02 * time: 610.4070479869843 324 1.067991e+03 9.553947e-02 * time: 612.8970830440521 325 1.067991e+03 9.553951e-02 * time: 615.3992419242859 326 1.067991e+03 9.553952e-02 * time: 617.9610919952393 327 1.067991e+03 1.960276e-01 * time: 620.0245850086212 328 1.067991e+03 2.189747e-01 * time: 622.0962700843811 329 1.067991e+03 3.471907e-01 * time: 624.1733100414276 330 1.067991e+03 4.746893e-01 * time: 626.2408180236816 331 1.067991e+03 6.344881e-01 * time: 628.3335649967194 332 1.067991e+03 6.826402e-01 * time: 630.4052510261536 333 1.067990e+03 4.682357e-01 * time: 632.4804561138153 334 1.067990e+03 1.398439e-01 * time: 634.5638570785522 335 1.067990e+03 4.650616e-03 * time: 636.651242017746 336 1.067990e+03 4.379178e-03 * time: 638.8894591331482 337 1.067990e+03 3.052741e-02 * time: 640.9970660209656 338 1.067990e+03 3.052782e-02 * time: 643.3413770198822 339 1.067990e+03 3.065948e-02 * time: 645.4126040935516 340 1.067990e+03 1.888820e-02 * time: 647.4826550483704 341 1.067990e+03 3.159606e-03 * time: 649.5582120418549 342 1.067990e+03 2.528974e-03 * time: 651.7320921421051 343 1.067990e+03 1.906291e-03 * time: 653.8983700275421 344 1.067990e+03 1.487671e-03 * time: 656.0540759563446 345 1.067990e+03 1.485038e-03 * time: 658.3911209106445 346 1.067990e+03 1.482585e-03 * time: 660.7250039577484 347 1.067990e+03 1.480617e-03 * time: 663.0933740139008 348 1.067990e+03 1.478874e-03 * time: 665.43741106987 349 1.067990e+03 1.477253e-03 * time: 667.7777979373932 350 1.067990e+03 1.477237e-03 * time: 670.2769501209259 351 1.067990e+03 1.477237e-03 * time: 673.5101010799408 352 1.067990e+03 1.477237e-03 * time: 676.4454209804535 353 1.067990e+03 1.477237e-03 * time: 679.8893280029297 354 1.067990e+03 1.477237e-03 * time: 683.0715811252594
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: -1067.99
------------------------
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 0.85627
pd_Ω₁,₁ 0.002937
pd_Ω₂,₂ 6.1888e-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: 1.5974044799804688e-5 1 5.174461e+05 8.708699e+05 * time: 15.898103952407837 2 3.865265e+05 6.344302e+05 * time: 28.17545485496521 3 1.804274e+05 2.829723e+05 * time: 40.11125898361206 4 9.706641e+04 1.550547e+05 * time: 51.854007959365845 5 4.769637e+04 6.778818e+04 * time: 63.555351972579956 6 2.902319e+04 3.499747e+04 * time: 74.90209984779358 7 1.823472e+04 1.705751e+04 * time: 86.68490290641785 8 1.258819e+04 9.569382e+03 * time: 98.32494902610779 9 9.389985e+03 8.615853e+03 * time: 109.63489484786987 10 7.314703e+03 7.636886e+03 * time: 120.94653487205505 11 5.916030e+03 6.624328e+03 * time: 132.2332260608673 12 4.930520e+03 5.558143e+03 * time: 143.41965103149414 13 4.125062e+03 4.315761e+03 * time: 154.58531498908997 14 3.549280e+03 3.051094e+03 * time: 165.70467805862427 15 3.283490e+03 2.157293e+03 * time: 176.89147186279297 16 3.204886e+03 1.659798e+03 * time: 189.29878306388855 17 3.194875e+03 1.480528e+03 * time: 200.855406999588 18 3.193944e+03 1.437922e+03 * time: 212.81818985939026 19 3.193070e+03 1.411186e+03 * time: 224.7903380393982 20 3.190129e+03 1.355327e+03 * time: 236.7638840675354 21 3.183228e+03 1.276603e+03 * time: 248.43321585655212 22 3.164897e+03 1.151838e+03 * time: 259.87145495414734 23 3.119250e+03 9.712652e+02 * time: 271.33470392227173 24 3.006297e+03 7.204344e+02 * time: 283.2904279232025 25 2.738913e+03 4.050546e+02 * time: 294.9082729816437 26 2.123834e+03 2.318194e+02 * time: 306.10301303863525 27 1.789142e+03 2.290466e+02 * time: 318.26736402511597 28 1.396456e+03 1.683975e+02 * time: 352.17130398750305 29 1.333545e+03 1.336198e+02 * time: 365.8561999797821 30 1.297773e+03 2.452181e+02 * time: 376.06141996383667 31 1.266002e+03 1.523972e+02 * time: 386.21088194847107 32 1.255505e+03 1.734000e+02 * time: 396.3862578868866 33 1.247788e+03 1.971637e+02 * time: 407.4413709640503 34 1.244490e+03 1.915744e+02 * time: 418.0139729976654 35 1.240568e+03 1.704280e+02 * time: 428.72329902648926 36 1.240502e+03 1.711813e+02 * time: 438.84351897239685 37 1.240491e+03 1.711683e+02 * time: 449.12860584259033 38 1.239992e+03 1.687296e+02 * time: 459.46736693382263 39 1.239200e+03 1.624744e+02 * time: 469.61480689048767 40 1.236975e+03 1.400443e+02 * time: 479.8007159233093 41 1.233209e+03 9.449763e+01 * time: 490.24963092803955 42 1.228687e+03 3.276923e+01 * time: 500.48719906806946 43 1.226467e+03 4.997214e+01 * time: 510.705286026001 44 1.226104e+03 4.905105e+01 * time: 520.9492609500885 45 1.226088e+03 4.676122e+01 * time: 531.5824739933014 46 1.226087e+03 4.628674e+01 * time: 542.3007709980011 47 1.226085e+03 4.541753e+01 * time: 553.8191368579865 48 1.226080e+03 4.403288e+01 * time: 565.3386740684509 49 1.226064e+03 4.143315e+01 * time: 576.9147930145264 50 1.226026e+03 3.664876e+01 * time: 588.0051710605621 51 1.225931e+03 2.851383e+01 * time: 599.1053659915924 52 1.225714e+03 2.844460e+01 * time: 610.4080190658569 53 1.225304e+03 3.025732e+01 * time: 621.4817650318146 54 1.224793e+03 4.151448e+01 * time: 632.5708010196686 55 1.224489e+03 5.047755e+01 * time: 643.7071468830109 56 1.224420e+03 4.452753e+01 * time: 657.3773748874664 57 1.224413e+03 3.994380e+01 * time: 668.7143979072571 58 1.224412e+03 3.879465e+01 * time: 682.3529999256134 59 1.224408e+03 3.677922e+01 * time: 693.6922309398651 60 1.224400e+03 3.379126e+01 * time: 705.3743889331818 61 1.224379e+03 3.158821e+01 * time: 716.755863904953 62 1.224324e+03 2.926306e+01 * time: 728.3088529109955 63 1.224181e+03 2.815869e+01 * time: 739.8566608428955 64 1.223815e+03 2.818300e+01 * time: 751.2939560413361 65 1.222911e+03 4.583041e+01 * time: 762.9737560749054 66 1.220921e+03 8.675281e+01 * time: 774.2934608459473 67 1.217625e+03 1.136294e+02 * time: 785.4189839363098 68 1.214324e+03 9.050245e+01 * time: 796.6579079627991 69 1.212536e+03 8.867180e+01 * time: 807.8734059333801 70 1.212272e+03 8.970655e+01 * time: 818.9588489532471 71 1.212263e+03 8.922293e+01 * time: 831.8512239456177 72 1.212259e+03 8.882951e+01 * time: 843.568027973175 73 1.212242e+03 8.761908e+01 * time: 854.8586909770966 74 1.212205e+03 8.562573e+01 * time: 866.1403279304504 75 1.212102e+03 8.125881e+01 * time: 877.4882628917694 76 1.211850e+03 7.230599e+01 * time: 888.8495440483093 77 1.211238e+03 5.323492e+01 * time: 900.2711389064789 78 1.209946e+03 7.373618e+01 * time: 911.9688858985901 79 1.207891e+03 8.401772e+01 * time: 923.422544002533 80 1.206059e+03 9.146801e+01 * time: 934.7164340019226 81 1.205389e+03 9.690592e+01 * time: 945.9845368862152 82 1.205302e+03 8.595286e+01 * time: 957.0592968463898 83 1.205296e+03 8.263040e+01 * time: 968.1718828678131 84 1.205292e+03 8.096547e+01 * time: 978.8959140777588 85 1.205276e+03 7.747632e+01 * time: 989.8232369422913 86 1.205240e+03 7.259211e+01 * time: 1000.6120829582214 87 1.205143e+03 6.664635e+01 * time: 1011.551521062851 88 1.204894e+03 6.374389e+01 * time: 1022.4059689044952 89 1.204267e+03 5.753484e+01 * time: 1033.449851989746 90 1.202818e+03 4.869938e+01 * time: 1044.4933080673218 91 1.200117e+03 5.087313e+01 * time: 1055.35191988945 92 1.196896e+03 7.141162e+01 * time: 1066.6613540649414 93 1.194612e+03 4.695316e+01 * time: 1078.9963989257812 94 1.193881e+03 4.874366e+01 * time: 1089.769140958786 95 1.193834e+03 4.997038e+01 * time: 1100.5088410377502 96 1.193832e+03 5.008442e+01 * time: 1111.3647620677948 97 1.193830e+03 5.014549e+01 * time: 1121.9388630390167 98 1.193825e+03 5.024884e+01 * time: 1132.3646049499512 99 1.193813e+03 5.039938e+01 * time: 1143.7149970531464 100 1.193779e+03 5.061632e+01 * time: 1154.6252720355988 101 1.193694e+03 5.087641e+01 * time: 1165.6568858623505 102 1.193473e+03 5.104895e+01 * time: 1176.7027280330658 103 1.192927e+03 5.068500e+01 * time: 1187.7540140151978 104 1.191682e+03 4.863684e+01 * time: 1198.7606010437012 105 1.189353e+03 4.890365e+01 * time: 1209.5893609523773 106 1.186424e+03 7.778082e+01 * time: 1220.4970688819885 107 1.184533e+03 8.873607e+01 * time: 1231.2657580375671 108 1.184042e+03 8.376145e+01 * time: 1241.1561460494995 109 1.183986e+03 7.931813e+01 * time: 1251.4945209026337 110 1.183978e+03 7.793645e+01 * time: 1261.822715997696 111 1.183974e+03 7.786675e+01 * time: 1272.4003469944 112 1.183970e+03 7.833168e+01 * time: 1282.5695269107819 113 1.183963e+03 7.928653e+01 * time: 1292.935930967331 114 1.183949e+03 8.083870e+01 * time: 1303.4991970062256 115 1.183913e+03 8.331100e+01 * time: 1313.9451220035553 116 1.183820e+03 8.698958e+01 * time: 1324.2641868591309 117 1.183584e+03 9.190093e+01 * time: 1334.5928869247437 118 1.183003e+03 9.670514e+01 * time: 1344.9693269729614 119 1.181706e+03 9.636548e+01 * time: 1355.1646559238434 120 1.179435e+03 8.059042e+01 * time: 1365.7321429252625 121 1.177024e+03 4.567337e+01 * time: 1376.4249649047852 122 1.175793e+03 1.863452e+01 * time: 1387.0981938838959 123 1.175474e+03 1.967979e+01 * time: 1398.0291018486023 124 1.175430e+03 1.958939e+01 * time: 1411.7037949562073 125 1.175427e+03 1.993415e+01 * time: 1422.3786568641663 126 1.175427e+03 1.962015e+01 * time: 1433.6993448734283 127 1.175426e+03 1.965822e+01 * time: 1444.659569978714 128 1.175424e+03 1.984190e+01 * time: 1455.6957910060883 129 1.175422e+03 2.000843e+01 * time: 1466.2294499874115 130 1.175413e+03 2.034664e+01 * time: 1476.9448959827423 131 1.175392e+03 2.083524e+01 * time: 1487.3862540721893 132 1.175336e+03 2.163395e+01 * time: 1502.2109038829803 133 1.175190e+03 2.286075e+01 * time: 1513.2677710056305 134 1.174803e+03 2.472708e+01 * time: 1525.3340828418732 135 1.173797e+03 4.124367e+01 * time: 1536.3401539325714 136 1.171232e+03 7.123154e+01 * time: 1547.3174738883972 137 1.165167e+03 1.075848e+02 * time: 1558.9748890399933 138 1.155096e+03 7.751300e+01 * time: 1570.291111946106 139 1.149210e+03 6.815706e+01 * time: 1581.2935810089111 140 1.146904e+03 6.218497e+01 * time: 1592.8316779136658 141 1.144688e+03 5.158719e+01 * time: 1603.9950640201569 142 1.143100e+03 2.555104e+01 * time: 1614.9575819969177 143 1.141816e+03 1.970245e+01 * time: 1625.9977819919586 144 1.141294e+03 2.048648e+01 * time: 1636.9382128715515 145 1.141044e+03 2.076701e+01 * time: 1648.1449539661407 146 1.140979e+03 2.090339e+01 * time: 1659.3159430027008 147 1.140976e+03 2.094515e+01 * time: 1670.3319940567017 148 1.140975e+03 2.099576e+01 * time: 1681.6909809112549 149 1.140974e+03 2.099888e+01 * time: 1692.6850118637085 150 1.140974e+03 2.099931e+01 * time: 1703.6572489738464 151 1.140973e+03 2.099295e+01 * time: 1714.6312458515167 152 1.140972e+03 2.098305e+01 * time: 1725.68922996521 153 1.140970e+03 2.096756e+01 * time: 1736.7590789794922 154 1.140963e+03 2.094619e+01 * time: 1747.8309018611908 155 1.140947e+03 2.092087e+01 * time: 1758.981696844101 156 1.140906e+03 2.090400e+01 * time: 1770.2785658836365 157 1.140797e+03 2.093964e+01 * time: 1781.1919288635254 158 1.140511e+03 2.116231e+01 * time: 1792.4821200370789 159 1.139732e+03 2.193009e+01 * time: 1805.1249928474426 160 1.137531e+03 3.209811e+01 * time: 1817.1483268737793 161 1.132353e+03 5.451891e+01 * time: 1828.9197380542755 162 1.130098e+03 6.354048e+01 * time: 1842.2971720695496 163 1.128599e+03 6.862509e+01 * time: 1855.8877699375153 164 1.127623e+03 8.385755e+01 * time: 1868.4259769916534 165 1.122228e+03 1.204826e+02 * time: 1880.8148839473724 166 1.118774e+03 1.225700e+02 * time: 1892.6564810276031 167 1.115679e+03 4.611370e+01 * time: 1906.646420955658 168 1.113979e+03 3.434856e+01 * time: 1918.9901299476624 169 1.112521e+03 3.523197e+01 * time: 1933.375149011612 170 1.111267e+03 1.228377e+01 * time: 1947.299190044403 171 1.111189e+03 1.078374e+01 * time: 1959.7501039505005 172 1.111187e+03 1.048228e+01 * time: 1972.0397510528564 173 1.111187e+03 1.070660e+01 * time: 1984.285707950592 174 1.111187e+03 1.057817e+01 * time: 1996.8950400352478 175 1.111187e+03 1.055130e+01 * time: 2008.4220058918 176 1.111186e+03 1.049695e+01 * time: 2019.57861495018 177 1.111185e+03 1.042473e+01 * time: 2031.0298550128937 178 1.111181e+03 1.030399e+01 * time: 2042.3906979560852 179 1.111173e+03 1.018024e+01 * time: 2053.839658975601 180 1.111150e+03 1.017076e+01 * time: 2065.3159849643707 181 1.111091e+03 1.547562e+01 * time: 2076.9460849761963 182 1.110938e+03 2.527991e+01 * time: 2088.3663918972015 183 1.110546e+03 4.009030e+01 * time: 2099.723527908325 184 1.109577e+03 6.034329e+01 * time: 2111.0760409832 185 1.107417e+03 8.167302e+01 * time: 2122.2110509872437 186 1.103900e+03 9.585232e+01 * time: 2133.214688062668 187 1.100518e+03 1.081943e+02 * time: 2144.148964881897 188 1.096704e+03 1.051898e+02 * time: 2155.1629779338837 189 1.092613e+03 3.363654e+01 * time: 2166.4654488563538 190 1.092108e+03 2.850803e+01 * time: 2177.6243579387665 191 1.091924e+03 8.252315e+00 * time: 2188.775225877762 192 1.091910e+03 4.387980e+00 * time: 2199.883234977722 193 1.091908e+03 4.381187e+00 * time: 2210.935768842697 194 1.091906e+03 4.351794e+00 * time: 2222.0319118499756 195 1.091905e+03 4.347949e+00 * time: 2233.0940029621124 196 1.091904e+03 4.344937e+00 * time: 2244.131679058075 197 1.091904e+03 4.343565e+00 * time: 2255.047038078308 198 1.091904e+03 4.341813e+00 * time: 2265.963485956192 199 1.091904e+03 4.338152e+00 * time: 2276.894860982895 200 1.091904e+03 4.331276e+00 * time: 2288.087007045746 201 1.091904e+03 4.319882e+00 * time: 2299.9930980205536 202 1.091903e+03 4.299831e+00 * time: 2311.142912864685 203 1.091901e+03 4.267724e+00 * time: 2321.9004850387573 204 1.091897e+03 4.211047e+00 * time: 2333.184275865555 205 1.091886e+03 4.117839e+00 * time: 2343.944930076599 206 1.091856e+03 4.860153e+00 * time: 2354.8643069267273 207 1.091780e+03 7.391080e+00 * time: 2365.4961910247803 208 1.091587e+03 1.165304e+01 * time: 2376.7742319107056 209 1.091126e+03 1.710495e+01 * time: 2388.025913000107 210 1.090065e+03 2.166909e+01 * time: 2399.203495979309 211 1.088566e+03 2.162566e+01 * time: 2410.3907039165497 212 1.087529e+03 2.211146e+01 * time: 2421.583464860916 213 1.087131e+03 6.801922e+00 * time: 2432.6771738529205 214 1.087053e+03 5.487353e+00 * time: 2443.922798871994 215 1.087044e+03 5.347085e+00 * time: 2454.9477128982544 216 1.087043e+03 5.364477e+00 * time: 2465.910474061966 217 1.087043e+03 5.380870e+00 * time: 2477.794676065445 218 1.087043e+03 5.385011e+00 * time: 2491.406637907028 219 1.087043e+03 5.387086e+00 * time: 2502.934545993805 220 1.087043e+03 5.393055e+00 * time: 2515.0431158542633 221 1.087043e+03 5.401766e+00 * time: 2528.381572008133 222 1.087042e+03 5.417259e+00 * time: 2540.3798389434814 223 1.087041e+03 5.443426e+00 * time: 2551.8475239276886 224 1.087039e+03 5.489900e+00 * time: 2563.4301919937134 225 1.087032e+03 5.575645e+00 * time: 2575.184980869293 226 1.087015e+03 5.744246e+00 * time: 2586.303603887558 227 1.086968e+03 8.410978e+00 * time: 2597.777529001236 228 1.086834e+03 1.426362e+01 * time: 2609.0656950473785 229 1.086299e+03 2.760110e+01 * time: 2620.8893399238586 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/mv9zc/src/multivariate/optimize/optimize.jl:136 ┌ Warning: Terminated early due to NaN in gradient. └ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/mv9zc/src/multivariate/optimize/optimize.jl:136 230 1.085441e+03 5.596618e+01 * time: 2639.062248945236 231 1.085250e+03 4.910391e+01 * time: 2650.6933150291443 232 1.084608e+03 5.990859e+01 * time: 2662.36789393425 233 1.084357e+03 6.400304e+01 * time: 2674.7543869018555 234 1.083885e+03 7.439245e+01 * time: 2686.0034489631653 235 1.083196e+03 7.397128e+01 * time: 2697.2808270454407 236 1.081690e+03 6.022479e+01 * time: 2708.937115907669 237 1.080750e+03 5.654666e+01 * time: 2720.6385929584503 238 1.078804e+03 4.051909e+01 * time: 2732.260694026947 239 1.076630e+03 2.980344e+01 * time: 2744.9753119945526 240 1.072899e+03 1.639509e+01 * time: 2756.3455579280853 241 1.070419e+03 2.588930e+01 * time: 2767.8585999011993 242 1.069708e+03 9.901466e+00 * time: 2779.234488964081 243 1.069494e+03 2.272167e+00 * time: 2790.4736919403076 244 1.069489e+03 2.210713e+00 * time: 2801.6291749477386 245 1.069488e+03 2.235087e+00 * time: 2812.7663469314575 246 1.069488e+03 2.236272e+00 * time: 2823.7925839424133 247 1.069488e+03 2.235678e+00 * time: 2834.791656970978 248 1.069488e+03 2.235678e+00 * time: 2847.5247888565063 249 1.069488e+03 2.234082e+00 * time: 2858.687891960144 250 1.069488e+03 2.233537e+00 * time: 2869.867590904236 251 1.069488e+03 2.231859e+00 * time: 2881.267145872116 252 1.069488e+03 2.229400e+00 * time: 2893.1549849510193 253 1.069487e+03 2.224370e+00 * time: 2904.23202586174 254 1.069486e+03 2.214629e+00 * time: 2915.5856239795685 255 1.069483e+03 2.193851e+00 * time: 2926.8602900505066 256 1.069476e+03 3.443280e+00 * time: 2937.6490218639374 257 1.069456e+03 5.476743e+00 * time: 2948.1225068569183 258 1.069409e+03 8.386597e+00 * time: 2958.5637080669403 259 1.069306e+03 1.153064e+01 * time: 2969.038810968399 260 1.069138e+03 1.209396e+01 * time: 2980.281825065613 261 1.068993e+03 7.707092e+00 * time: 2993.647062063217 262 1.068938e+03 2.664854e+00 * time: 3004.880457878113 263 1.068930e+03 8.683686e-01 * time: 3016.0464029312134 264 1.068929e+03 8.687100e-01 * time: 3027.3786170482635 265 1.068929e+03 8.687100e-01 * time: 3041.552710056305 266 1.068929e+03 8.688555e-01 * time: 3053.013979911804 267 1.068929e+03 8.688555e-01 * time: 3066.9610970020294 268 1.068929e+03 8.690015e-01 * time: 3078.3450560569763 269 1.068929e+03 8.690187e-01 * time: 3089.4372470378876 270 1.068929e+03 8.690720e-01 * time: 3100.92586684227 271 1.068929e+03 8.690816e-01 * time: 3112.859762907028 272 1.068929e+03 8.691586e-01 * time: 3124.156702041626 273 1.068929e+03 8.692118e-01 * time: 3135.477593898773 274 1.068929e+03 8.692628e-01 * time: 3146.5786838531494 275 1.068929e+03 8.691948e-01 * time: 3158.363373041153 276 1.068929e+03 8.687698e-01 * time: 3168.7620799541473 277 1.068928e+03 8.677218e-01 * time: 3179.5269708633423 278 1.068928e+03 9.457190e-01 * time: 3190.2318580150604 279 1.068928e+03 1.073922e+00 * time: 3201.1268739700317 280 1.068928e+03 1.217721e+00 * time: 3211.9017560482025 281 1.068928e+03 1.290660e+00 * time: 3222.9746940135956 282 1.068928e+03 1.391629e+00 * time: 3233.5896389484406 283 1.068927e+03 1.539905e+00 * time: 3243.9974200725555 284 1.068926e+03 1.777174e+00 * time: 3254.8306589126587 285 1.068923e+03 2.150672e+00 * time: 3265.4371490478516 286 1.068916e+03 2.731459e+00 * time: 3275.8546919822693 287 1.068897e+03 3.586050e+00 * time: 3286.233785867691 288 1.068853e+03 4.875058e+00 * time: 3296.669252872467 289 1.068759e+03 6.977299e+00 * time: 3307.064785003662 290 1.068602e+03 7.749021e+00 * time: 3317.452831029892 291 1.068438e+03 5.310443e+00 * time: 3329.233829975128 292 1.068380e+03 1.591854e+00 * time: 3339.86913895607 293 1.068375e+03 1.336160e+00 * time: 3350.489224910736 294 1.068375e+03 1.303570e+00 * time: 3361.0248749256134 295 1.068374e+03 1.285208e+00 * time: 3371.4738159179688 296 1.068374e+03 1.284727e+00 * time: 3382.403455018997 297 1.068374e+03 1.284725e+00 * time: 3394.1341259479523 298 1.068374e+03 1.284688e+00 * time: 3405.3920209407806 299 1.068374e+03 1.284643e+00 * time: 3416.93208193779 300 1.068374e+03 1.284639e+00 * time: 3428.9992849826813 301 1.068374e+03 1.284633e+00 * time: 3440.701611995697 302 1.068374e+03 1.284633e+00 * time: 3452.9963839054108 303 1.068374e+03 1.284632e+00 * time: 3465.0321350097656 304 1.068374e+03 1.284631e+00 * time: 3477.975746870041 305 1.068374e+03 1.284631e+00 * time: 3490.9650268554688 306 1.068374e+03 1.284631e+00 * time: 3503.86434006691 307 1.068374e+03 1.284631e+00 * time: 3517.214329957962 308 1.068374e+03 1.284631e+00 * time: 3530.615609884262 309 1.068374e+03 1.284631e+00 * time: 3544.0053720474243 310 1.068374e+03 1.284631e+00 * time: 3557.3843200206757 311 1.068374e+03 1.284631e+00 * time: 3571.674206018448 312 1.068374e+03 1.284631e+00 * time: 3586.224410057068 313 1.068374e+03 1.277430e+00 * time: 3597.3031129837036 314 1.068374e+03 1.278342e+00 * time: 3608.1831319332123 315 1.068374e+03 1.278561e+00 * time: 3620.496243953705 316 1.068374e+03 1.278598e+00 * time: 3632.927733898163 317 1.068374e+03 1.278765e+00 * time: 3645.5450649261475 318 1.068374e+03 1.278767e+00 * time: 3658.835216999054 319 1.068374e+03 1.283325e+00 * time: 3670.0238959789276 320 1.068374e+03 1.284455e+00 * time: 3681.32239985466 321 1.068374e+03 1.297288e+00 * time: 3693.6294519901276 322 1.068373e+03 1.309277e+00 * time: 3706.1950368881226 323 1.068370e+03 1.327095e+00 * time: 3718.5782659053802 324 1.068363e+03 1.340822e+00 * time: 3730.64324593544 325 1.068345e+03 1.329791e+00 * time: 3742.3253350257874 326 1.068304e+03 1.440802e+00 * time: 3754.1032960414886 327 1.068220e+03 1.781108e+00 * time: 3766.8922169208527 328 1.068100e+03 1.555010e+00 * time: 3783.205395936966 329 1.068015e+03 6.383597e-01 * time: 3798.17605304718 330 1.068000e+03 4.313404e-02 * time: 3814.4858660697937 331 1.067999e+03 1.160644e-01 * time: 3846.432205915451 332 1.067998e+03 1.891364e-01 * time: 3870.0866589546204 333 1.067995e+03 2.072805e-01 * time: 3889.680629968643 334 1.067993e+03 1.268387e-01 * time: 3914.831162929535 335 1.067992e+03 7.307859e-02 * time: 3942.960671901703 336 1.067991e+03 8.280285e-02 * time: 3970.462466955185 337 1.067991e+03 8.863498e-02 * time: 3992.5110199451447 338 1.067991e+03 9.334675e-02 * time: 4019.1770708560944 339 1.067991e+03 9.512481e-02 * time: 4041.4833109378815 340 1.067991e+03 9.522332e-02 * time: 4061.150853872299 341 1.067991e+03 9.513094e-02 * time: 4080.4553508758545 342 1.067991e+03 9.508043e-02 * time: 4099.755939006805 343 1.067991e+03 9.508043e-02 * time: 4129.840872049332 344 1.067991e+03 9.508043e-02 * time: 4159.801071882248
FittedPumasModel
Dynamical system type: Nonlinear ODE
Solver(s): OrdinaryDiffEqRosenbrock.Rodas5P
Number of subjects: 32
Observation records: Active Missing
conc: 251 47
pca: 232 66
Total: 483 113
Number of parameters: Constant Optimized
0 18
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: NoObjectiveChange
Log-likelihood value: -1067.9914
-----------------------
Estimate
-----------------------
pop_CL 0.13521
pop_V 8.0133
pop_tabs 0.57109
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 0.86269
pd_Ω₁,₁ 0.0029772
pd_Ω₂,₂ 1.9506e-7
pd_Ω₃,₃ 0.14561
pd_Ω₄,₄ 0.015351
σ_prop 0.088485
σ_add 0.41684
σ_fx 3.5803
-----------------------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: 7.104873657226562e-5 1 5.174461e+05 8.708698e+05 * time: 7.322011947631836 2 3.865265e+05 6.344302e+05 * time: 10.60608696937561 3 1.804274e+05 2.829723e+05 * time: 13.33290696144104 4 9.706640e+04 1.550547e+05 * time: 16.0142719745636 5 4.769637e+04 6.778818e+04 * time: 19.362586975097656 6 2.902319e+04 3.499747e+04 * time: 22.591015100479126 7 1.823472e+04 1.705751e+04 * time: 25.403118133544922 8 1.258819e+04 9.569381e+03 * time: 27.69648814201355 9 9.389984e+03 8.615851e+03 * time: 30.47625994682312 10 7.314702e+03 7.636883e+03 * time: 32.61696910858154 11 5.916029e+03 6.624325e+03 * time: 34.45844316482544 12 4.930519e+03 5.558140e+03 * time: 36.259063959121704 13 4.125060e+03 4.315759e+03 * time: 38.07671809196472 14 3.549280e+03 3.051093e+03 * time: 40.018845081329346 15 3.283489e+03 2.157292e+03 * time: 43.03418207168579 16 3.204886e+03 1.659798e+03 * time: 44.89914608001709 17 3.194875e+03 1.480528e+03 * time: 46.743937969207764 18 3.193944e+03 1.437921e+03 * time: 48.554309129714966 19 3.193070e+03 1.411186e+03 * time: 50.09391403198242 20 3.190129e+03 1.355327e+03 * time: 51.664677143096924 21 3.183228e+03 1.276603e+03 * time: 53.4485399723053 22 3.164897e+03 1.151838e+03 * time: 55.01418709754944 23 3.119250e+03 9.712651e+02 * time: 56.50082802772522 24 3.006297e+03 7.204342e+02 * time: 57.973029136657715 25 2.738913e+03 4.050545e+02 * time: 59.43356800079346 26 2.123834e+03 2.318194e+02 * time: 60.85251808166504 27 1.789139e+03 2.290465e+02 * time: 62.382508993148804 28 1.396455e+03 1.683969e+02 * time: 338.58069801330566 29 1.333545e+03 1.336195e+02 * time: 340.07842898368835 30 1.297771e+03 2.452175e+02 * time: 341.3870310783386 31 1.266002e+03 1.523957e+02 * time: 342.6874141693115 32 1.255506e+03 1.733982e+02 * time: 344.049742937088 33 1.247789e+03 1.971613e+02 * time: 345.4190671443939 34 1.244490e+03 1.915717e+02 * time: 346.79066705703735 35 1.240568e+03 1.704241e+02 * time: 348.1394350528717 36 1.240503e+03 1.711777e+02 * time: 349.95986008644104 37 1.240492e+03 1.711647e+02 * time: 352.6663751602173 38 1.239992e+03 1.687233e+02 * time: 354.3270959854126 39 1.239199e+03 1.624614e+02 * time: 355.6788671016693 40 1.236972e+03 1.400083e+02 * time: 357.0054359436035 41 1.233204e+03 9.443139e+01 * time: 358.3349151611328 42 1.228683e+03 3.274305e+01 * time: 359.63933205604553 43 1.226466e+03 4.997666e+01 * time: 360.93474817276 44 1.226104e+03 4.904455e+01 * time: 362.24123215675354 45 1.226088e+03 4.675831e+01 * time: 363.6303119659424 46 1.226088e+03 4.628481e+01 * time: 365.3394031524658 47 1.226085e+03 4.541370e+01 * time: 366.8299069404602 48 1.226080e+03 4.402775e+01 * time: 368.34204506874084 49 1.226064e+03 4.142396e+01 * time: 369.6363260746002 50 1.226026e+03 3.663267e+01 * time: 370.91848516464233 51 1.225931e+03 2.851376e+01 * time: 372.24897503852844 52 1.225714e+03 2.844439e+01 * time: 373.5399830341339 53 1.225303e+03 3.026353e+01 * time: 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687.9968740940094
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: -1068.9294
------------------------
Estimate
------------------------
pop_CL 0.13521
pop_V 8.0112
pop_tabs 0.56617
pop_lag 0.87614
pop_e0 96.395
pop_emax -1.0613
pop_c50 1.4884
pop_tover 14.053
pk_Ω₁,₁ 0.06929
pk_Ω₂,₂ 0.020319
pk_Ω₃,₃ 0.89959
pd_Ω₁,₁ 0.0028777
pd_Ω₂,₂ 0.00044805
pd_Ω₃,₃ 0.15376
pd_Ω₄,₄ 0.015013
σ_prop 0.088937
σ_add 0.41485
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.