warfarin_pkpd_model = @model begin
...
@dynamics begin
Depot' = -Ka * Depot
Central' = Ka * Depot - CL / Vc * Central
Turnover' =
rin * (1 + emax * (Central / Vc) / (cp50 + Central / Vc)) - kout * Turnover
end
...
end
Differential Equations in Pumas
1 Introduction
Pumas automatically chooses a differential equation solver that is suitable for the simulation or estimation of the dynamical system of the NLME (Nonlinear Mixed Effects) model at hand. This default solver is the preferred choice and optimized for most users and use cases. Nevertheless, in some cases the performance-accuracy trade-off can be improved by adjusting the tolerances, or possibly even the algorithm, of the differential equation solver.
In this tutorial, the Warfarin PK/PD model is used to demonstrate how to configure the differential equation solver.
2 Learning Goals
- Observe the utility of the
@varsblock of a Pumas model with respect to storing dynamic variables associated with differential equations - Understand the main differences between common differential equation solvers for nonlinear dynamical systems
- Learn how to adjust the algorithm and the tolerances of the differential equation solver
3 Warfarin PK/PD Model
We return to the Warfarin PK/PD model. Its dynamical system consists of three states, \(\operatorname{Depot}\), \(\operatorname{Central}\), and \(\operatorname{Turnover}\), whose dynamics are governed by the ordinary differential equations:
\[ \begin{aligned} \operatorname{Depot}'(t) &= - \operatorname{Ka} \operatorname{Depot}(t),\\ \operatorname{Central}'(t) &= \operatorname{Ka} \operatorname{Depot}(t) - \frac{\operatorname{CL}}{\operatorname{Vc}} \operatorname{Central}(t),\\ \operatorname{Turnover}'(t) &= \operatorname{rin} (1 + \operatorname{emax} \frac{\operatorname{Central}(t) / \operatorname{Vc}}{\operatorname{c50} + \operatorname{Central}(t)/\operatorname{Vc}}) - \operatorname{kout} \operatorname{Turnover}(t) \end{aligned} \]
with PK parameters \(\operatorname{Ka}\) (absorption rate), \(\operatorname{CL}\) (clearance), and \(\operatorname{Vc}\) (volume of distribution) and PD parameters \(\operatorname{rin}\), \(\operatorname{emax}\), \(\operatorname{c50}\), and \(\operatorname{kout}\).
The dynamical system can be written more concisely by introducing auxiliary variables for repeated expressions:
\[ \begin{aligned} \operatorname{Depot}'(t) &= -\operatorname{ratein}(t),\\ \operatorname{Central}'(t) &= \operatorname{ratein}(t) - \operatorname{CL} \operatorname{cp}(t),\\ \operatorname{Turnover}'(t) &= \operatorname{rin} \operatorname{pd}(t) - \operatorname{kout} \operatorname{Turnover}(t) \end{aligned} \]
with influx rate \(\operatorname{ratein}(t) := \operatorname{Ka} \operatorname{Depot}(t)\), concentration \(\operatorname{cp}(t) := \operatorname{Central}(t) / \operatorname{Vc}\), and \(\operatorname{pd}(t) := 1 + \operatorname{emax} \frac{\operatorname{cp}(t)}{\operatorname{c50} + \operatorname{cp}(t)}\).
4 Auxiliary Variables in @vars
In Pumas, dynamical systems are defined in the @dynamics block inside of the @model definition. For instance, the dynamical system of the Warfarin PK/PD model can be implemented as follows:
The same concise rewriting can be applied in a Pumas @model by defining auxiliary variables (“aliases”) in the @vars block:
warfarin_pkpd_model = @model begin
...
@vars begin
cp := Central / Vc
ratein := Ka * Depot
pd := 1 + emax * cp / (c50 + cp)
end
@dynamics begin
Depot' = -ratein
Central' = ratein - CL * cp
Turnover' = rin * pd - kout * Turnover
end
...
end
Tip
The walrus operator (:=) ensures that the aliases do not show up in the simulation output of the model. However, if you would like to access an alias in the simulation output, you should define the alias with =. For instance, if you want to obtain concentration cp as part of the simulation output, you can change the @vars block to
@vars begin
cp = Central / Vc
ratein := Ka * Depot
pd := 1 + emax * cp / (c50 + cp)
end5 Differential Equation Solvers
The differential equation in the Warfarin model is non-linear, as detected by Pumas (“Dynamical system type: Nonlinear ODE”):
using Pumas
warfarin_pkpd_model = @model begin
@param begin
# PK parameters
"""
Clearance (L/h/70kg)
"""
pop_CL ∈ RealDomain(lower = 0.0, init = 0.134)
"""
Central Volume L/70kg
"""
pop_V ∈ RealDomain(lower = 0.0, init = 8.11)
"""
Absorption time (h)
"""
pop_tabs ∈ RealDomain(lower = 0.0, init = 0.523)
"""
Lag time (h)
"""
pop_lag ∈ RealDomain(lower = 0.0, init = 0.1)
# PD parameters
"""
Baseline
"""
pop_e0 ∈ RealDomain(lower = 0.0, init = 100.0)
"""
Emax
"""
pop_emax ∈ RealDomain(init = -1.0)
"""
EC50
"""
pop_c50 ∈ RealDomain(lower = 0.0, init = 1.0)
"""
Turnover
"""
pop_tover ∈ RealDomain(lower = 0.0, init = 14.0)
# Inter-individual variability
"""
- ΩCL
- ΩVc
- ΩTabs
"""
pk_Ω ∈ PDiagDomain([0.01, 0.01, 0.01])
"""
- Ωe0
- Ωemax
- Ωec50
- Ωturn
"""
pd_Ω ∈ PDiagDomain([0.01, 0.01, 0.01, 0.01])
# Residual variability
"""
Proportional residual error for drug concentration
"""
σ_prop ∈ RealDomain(lower = 0.0, init = 0.00752)
"""
Additive residual error for drug concentration (mg/L)
"""
σ_add ∈ RealDomain(lower = 0.0, init = 0.0661)
"""
Additive error for PCA
"""
σ_fx ∈ RealDomain(lower = 0.0, init = 0.01)
end
@random begin
# mean = 0, covariance = pk_Ω
pk_η ~ MvNormal(pk_Ω)
# mean = 0, covariance = pd_Ω
pd_η ~ MvNormal(pd_Ω)
end
@covariates FSZV FSZCL
@pre begin
# PK
CL = FSZCL * pop_CL * exp(pk_η[1])
Vc = FSZV * pop_V * exp(pk_η[2])
tabs = pop_tabs * exp(pk_η[3])
Ka = log(2) / tabs
# PD
e0 = pop_e0 * exp(pd_η[1])
emax = pop_emax * exp(pd_η[2])
c50 = pop_c50 * exp(pd_η[3])
tover = pop_tover * exp(pd_η[4])
kout = log(2) / tover
rin = e0 * kout
time = t
end
@dosecontrol begin
lags = (Depot = pop_lag,)
end
@init begin
Turnover = e0
end
# aliases for use in @dynamics and @derived
@vars begin
cp := Central / Vc
ratein := Ka * Depot
pd := 1 + emax * cp / (c50 + cp)
end
@dynamics begin
Depot' = -ratein
Central' = ratein - CL * cp
Turnover' = rin * pd - kout * Turnover
end
@derived begin
"""
Warfarin Concentration (mg/L)
"""
conc ~ @. Normal(cp, sqrt((σ_prop * cp)^2 + σ_add^2))
"""
PCA
"""
pca ~ @. Normal(Turnover, σ_fx)
end
endPumasModel
Parameters: pop_CL, pop_V, pop_tabs, pop_lag, pop_e0, pop_emax, pop_c50, pop_tover, pk_Ω, pd_Ω, σ_prop, σ_add, σ_fx
Random effects: pk_η, pd_η
Covariates: FSZV, FSZCL
Dynamical system variables: Depot, Central, Turnover
Dynamical system type: Nonlinear ODE
Derived: conc, pca
Observed: conc, pcaPumas approximates the solution of the differential equation with a numerical differential equation solver. Generally, one distinguishes between solvers for stiff and non-stiff differential equations.
5.1 Stiff vs. Non-Stiff Systems
A key distinction among numerical solvers is whether they are designed for stiff or non-stiff differential equations:
Non-Stiff Differential Equations: These systems exhibit relatively moderate changes in their variables. Standard non-stiff solvers can efficiently approximate solutions of these systems.
Stiff Differential Equations: These systems contain rapidly changing components alongside more slowly varying dynamics. Non-stiff solvers typically perform poorly on stiff systems, as they may require exceedingly small step sizes to maintain numerical stability. Specialized stiff solvers are therefore employed to handle the sharp gradients and large timescale differences without compromising accuracy.
6 Pumas’s Automatic Solver Selection
By default, Pumas adopts a hybrid approach with automatic stiffness detection to switch between stiff and non-stiff solvers as needed.
6.1 Model Simulation
- Default Solvers: Rosenbrock23 (stiff) and Tsit5 (non-stiff)
- Tolerances: Relative tolerance \(1 \times 10^{-3}\) and absolute tolerance \(1 \times 10^{-6}\)
- Rationale: These higher (less stringent) tolerances permit faster simulations while maintaining sufficient accuracy for exploratory and predictive modeling.
6.2 Model Fitting
- Default Solvers: Rodas5P (stiff) and Vern7 (non-stiff)
- Tolerances: Relative tolerance \(10^{-8}\) and absolute tolerance \(10^{-12}\)
- Rationale: These lower (more stringent) tolerances ensure high precision during parameter estimation, which is critical for matching the model’s predictions to observed data.
The default solvers and tolerances are recommended for most users in most instances. If desired, however, it is possible to adjust these settings with the diffeq_options keyword argument.
Important
Computation time decreases as tolerances are increased. However, higher tolerances come at the cost of a less strict error control, and hence generally a less accurate solution.
6.3 Adjusting the Tolerances
The absolute and relative tolerance of the solver can be specified with abstol and reltol.
7 Absolute and Relative Tolerances
When employing a numerical solver, it is necessary to specify how accurately the solution should be computed. This precision is controlled by two key parameters:
Absolute Tolerance \((\text{abstol})\)
- Interpreted as the maximum allowable error when the solution values are near zero.
- Ensures that numerical approximations stay within a reasonable bound, preventing physically impossible outcomes (e.g., negative concentrations) or excessive drift at small scales.
- For instance, an absolute tolerance of \(10^{-6}\) means the solver attempts to keep the absolute error below \(10^{-6}\) whenever the solution magnitude is close to zero.
Relative Tolerance \((\text{reltol})\)
- Enforces the number of correct digits throughout the simulation, effectively controlling error relative to the current scale of the solution.
- For example, a relative tolerance of \(10^{-3}\) implies the solver aims for three correct decimal places (i.e., the solution is accurate to within 0.1% of its current magnitude).
- As the solution grows or shrinks, the solver adjusts its time-step size and internal computations to maintain this relative accuracy.
Sometimes decreasing tolerances can help to reduce numerical problems, e.g. to keep solutions non-negative that are mathematically guaranteed to be non-negative. Additionally, the choice of tolerances can be motivated by the application of the numerical solution: For plotting a less accurate solution, and hence larger tolerances, might be tolerable, whereas typically for model fitting a more accurate solution, and hence smaller tolerances, are beneficial.
This can be demonstrated when fitting the Warfarin model with an example dataset: Optimization fails with large tolerances of 1e-3 (relative) and 1e-6 (absolute)
fit(
warfarin_pkpd_model,
pop,
init_params(warfarin_pkpd_model),
FOCE();
diffeq_options = (; reltol = 1e-3, abstol = 1e-6),
)[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
Iter Function value Gradient norm
0 3.004020e+06 5.745853e+06
* time: 0.029724836349487305
1 4.816669e+05 8.316173e+05
* time: 2.0062570571899414
2 3.598373e+05 6.060150e+05
* time: 2.7980690002441406
3 1.682124e+05 2.709072e+05
* time: 3.5151119232177734
4 9.168914e+04 1.446832e+05
* time: 4.313179016113281
5 4.791138e+04 6.671463e+04
* time: 5.025882959365845
6 2.909104e+04 3.426663e+04
* time: 5.763444900512695
7 1.843737e+04 1.678953e+04
* time: 6.530328035354614
8 1.282447e+04 1.021463e+04
* time: 7.219933032989502
9 9.723361e+03 9.291650e+03
* time: 7.991725921630859
10 7.635613e+03 8.325311e+03
* time: 8.663815021514893
11 6.243129e+03 7.337397e+03
* time: 9.362900018692017
12 5.279957e+03 6.304542e+03
* time: 10.008405923843384
13 4.422123e+03 5.001962e+03
* time: 10.682711839675903
14 3.744748e+03 3.550400e+03
* time: 11.273859024047852
15 3.409674e+03 2.461547e+03
* time: 11.843358039855957
16 3.312049e+03 1.856430e+03
* time: 12.481034994125366
17 3.300559e+03 1.648439e+03
* time: 13.16940188407898
18 3.299489e+03 1.599460e+03
* time: 13.724724054336548
19 3.298436e+03 1.567520e+03
* time: 14.305112838745117
20 3.294921e+03 1.501363e+03
* time: 14.873414039611816
21 3.286641e+03 1.407664e+03
* time: 15.440361976623535
22 3.264637e+03 1.259077e+03
* time: 15.997187852859497
23 3.209594e+03 1.043534e+03
* time: 16.617003917694092
24 3.072117e+03 7.454677e+02
* time: 17.164456844329834
25 2.739794e+03 3.818063e+02
* time: 17.715576887130737
26 1.949883e+03 2.246644e+02
* time: 18.329725980758667
27 1.722434e+03 2.224562e+02
* time: 19.85228204727173
28 1.479355e+03 2.042452e+02
* time: 21.790685892105103
29 1.310583e+03 1.531868e+02
* time: 24.28032684326172
30 1.224675e+03 1.756826e+02
* time: 24.87396788597107
31 1.204841e+03 2.738440e+02
* time: 25.3345148563385
32 1.196432e+03 2.403217e+02
* time: 25.86268186569214
33 1.189131e+03 2.301186e+02
* time: 26.33276891708374
34 1.177881e+03 2.083030e+02
* time: 26.88728904724121
35 1.176989e+03 2.201993e+02
* time: 27.392173051834106
36 1.176899e+03 2.230826e+02
* time: 27.94138789176941
37 1.176882e+03 2.233531e+02
* time: 28.405926942825317
38 1.176795e+03 2.238567e+02
* time: 28.92625904083252
39 1.176609e+03 2.236914e+02
* time: 29.392478942871094
40 1.176090e+03 2.210941e+02
* time: 29.949235916137695
41 1.174841e+03 2.112741e+02
* time: 30.45778799057007
42 1.171973e+03 1.825298e+02
* time: 30.994550943374634
43 1.166737e+03 1.210559e+02
* time: 31.48364496231079
44 1.160698e+03 4.450697e+01
* time: 32.02490592002869
45 1.157926e+03 6.411275e+01
* time: 32.5980749130249
46 1.157537e+03 6.121762e+01
* time: 33.152344942092896
47 1.157520e+03 5.867223e+01
* time: 33.65998697280884
48 1.157519e+03 5.814523e+01
* time: 34.27387595176697
49 1.157515e+03 5.686007e+01
* time: 34.793344020843506
50 1.157505e+03 5.497418e+01
* time: 35.33447194099426
51 1.157477e+03 5.129724e+01
* time: 35.83544898033142
52 1.157408e+03 4.463171e+01
* time: 36.37207102775574
53 1.157239e+03 3.279117e+01
* time: 36.88265299797058
54 1.156869e+03 3.681526e+01
* time: 37.42654490470886
55 1.156230e+03 3.198754e+01
* time: 37.93988800048828
56 1.155563e+03 4.576438e+01
* time: 38.478617906570435
57 1.155264e+03 4.832159e+01
* time: 38.987661838531494
58 1.155212e+03 4.052175e+01
* time: 39.51592397689819
59 1.155207e+03 3.797125e+01
* time: 40.01277804374695
60 1.155206e+03 3.778452e+01
* time: 40.53997993469238
61 1.155201e+03 3.723003e+01
* time: 41.04446482658386
62 1.155190e+03 3.642375e+01
* time: 41.6595458984375
63 1.155159e+03 3.487535e+01
* time: 42.16261386871338
64 1.155079e+03 3.209060e+01
* time: 42.68754482269287
65 1.154875e+03 2.670227e+01
* time: 43.19315695762634
66 1.154375e+03 2.366165e+01
* time: 43.716959953308105
67 1.153278e+03 5.499525e+01
* time: 44.2250440120697
68 1.151457e+03 7.903833e+01
* time: 44.738083839416504
69 1.149737e+03 6.771227e+01
* time: 45.2470338344574
70 1.148983e+03 5.387708e+01
* time: 45.8035728931427
71 1.148872e+03 4.683950e+01
* time: 46.27865791320801
72 1.148868e+03 4.512036e+01
* time: 46.763136863708496
73 1.148866e+03 4.489706e+01
* time: 47.21384882926941
74 1.148857e+03 4.426314e+01
* time: 47.66077494621277
75 1.148838e+03 4.321161e+01
* time: 48.09964990615845
76 1.148785e+03 4.095232e+01
* time: 48.545389890670776
77 1.148652e+03 3.780572e+01
* time: 48.98205304145813
78 1.148325e+03 3.474239e+01
* time: 49.487318992614746
79 1.147599e+03 3.678957e+01
* time: 50.002771854400635
80 1.146330e+03 5.134277e+01
* time: 50.52331900596619
81 1.144979e+03 6.444063e+01
* time: 51.068039894104004
82 1.144325e+03 7.574751e+01
* time: 51.59901189804077
83 1.144210e+03 6.968296e+01
* time: 52.1621458530426
84 1.144204e+03 6.648234e+01
* time: 52.612109899520874
85 1.144202e+03 6.560999e+01
* time: 53.1344108581543
86 1.144194e+03 6.305667e+01
* time: 53.593031883239746
87 1.144176e+03 5.956959e+01
* time: 54.115488052368164
88 1.144127e+03 5.307828e+01
* time: 54.56674885749817
89 1.144005e+03 4.218776e+01
* time: 55.080068826675415
90 1.143708e+03 4.120261e+01
* time: 55.55855894088745
91 1.143074e+03 3.675516e+01
* time: 56.10743999481201
92 1.142052e+03 3.855439e+01
* time: 56.60622000694275
93 1.141101e+03 5.193197e+01
* time: 57.165180921554565
94 1.140701e+03 4.606858e+01
* time: 57.64548397064209
95 1.140646e+03 4.857736e+01
* time: 58.179245948791504
96 1.140643e+03 4.817596e+01
* time: 58.65185904502869
97 1.140642e+03 4.793197e+01
* time: 59.17725586891174
98 1.140637e+03 4.723245e+01
* time: 59.653767824172974
99 1.140626e+03 4.619695e+01
* time: 60.15275192260742
100 1.140597e+03 4.418761e+01
* time: 60.605921030044556
101 1.140523e+03 4.050986e+01
* time: 61.09307098388672
102 1.140334e+03 3.335334e+01
* time: 61.5455379486084
103 1.139892e+03 2.982120e+01
* time: 62.04730486869812
104 1.138994e+03 4.360237e+01
* time: 62.49710488319397
105 1.137687e+03 5.246892e+01
* time: 62.99178695678711
106 1.136679e+03 4.825586e+01
* time: 63.423651933670044
107 1.136391e+03 4.467499e+01
* time: 63.93027091026306
108 1.136364e+03 3.961591e+01
* time: 64.38046884536743
109 1.136362e+03 3.951402e+01
* time: 64.85402989387512
110 1.136360e+03 3.943358e+01
* time: 65.28286099433899
111 1.136355e+03 3.923965e+01
* time: 65.75208902359009
112 1.136344e+03 3.894674e+01
* time: 66.19487500190735
113 1.136314e+03 3.841518e+01
* time: 66.67323303222656
114 1.136235e+03 3.748824e+01
* time: 67.12131404876709
115 1.136029e+03 3.577210e+01
* time: 67.60785794258118
116 1.135499e+03 3.252332e+01
* time: 68.0646140575409
117 1.134182e+03 2.630844e+01
* time: 68.56378984451294
118 1.131238e+03 4.869163e+01
* time: 69.03646898269653
119 1.126345e+03 6.537856e+01
* time: 69.53587603569031
120 1.121879e+03 5.123373e+01
* time: 70.03179693222046
121 1.119645e+03 2.457316e+01
* time: 70.55405592918396
122 1.119031e+03 2.124721e+01
* time: 71.06296491622925
123 1.118958e+03 1.857508e+01
* time: 71.5822069644928
124 1.118953e+03 1.896595e+01
* time: 72.07880282402039
125 1.118952e+03 1.849021e+01
* time: 72.62868785858154
126 1.118952e+03 1.852941e+01
* time: 73.13885593414307
127 1.118951e+03 1.856812e+01
* time: 73.64048886299133
128 1.118951e+03 1.862403e+01
* time: 74.13744902610779
129 1.118949e+03 1.871484e+01
* time: 74.6356680393219
130 1.118946e+03 1.886332e+01
* time: 75.11932587623596
131 1.118939e+03 1.910217e+01
* time: 75.61092400550842
132 1.118922e+03 1.948050e+01
* time: 76.09845495223999
133 1.118879e+03 2.006903e+01
* time: 76.5951418876648
134 1.118769e+03 2.097106e+01
* time: 77.05900192260742
135 1.118486e+03 2.231801e+01
* time: 77.5318808555603
136 1.117755e+03 3.049039e+01
* time: 77.99631595611572
137 1.115883e+03 5.345487e+01
* time: 78.48412084579468
138 1.111258e+03 8.489274e+01
* time: 78.97799301147461
139 1.101149e+03 7.643749e+01
* time: 79.55168795585632
140 1.090612e+03 5.802659e+01
* time: 80.18081188201904
141 1.087971e+03 5.776554e+01
* time: 80.75567889213562
142 1.087205e+03 1.148234e+02
* time: 81.4542829990387
143 1.085558e+03 4.724872e+01
* time: 82.10414385795593
144 1.085271e+03 2.018848e+01
* time: 82.96370005607605
145 1.085124e+03 2.004201e+01
* time: 83.56954789161682
146 1.085089e+03 2.025616e+01
* time: 84.21434593200684
147 1.085081e+03 2.049682e+01
* time: 84.83074498176575
148 1.085073e+03 2.086075e+01
* time: 85.48445296287537
149 1.085070e+03 2.095706e+01
* time: 86.09431004524231
150 1.085063e+03 2.094978e+01
* time: 86.7024998664856
151 1.085062e+03 2.093855e+01
* time: 87.38792896270752
152 1.085062e+03 2.093439e+01
* time: 88.07346892356873
153 1.085062e+03 2.093335e+01
* time: 88.83808398246765
154 1.085062e+03 2.093252e+01
* time: 89.57580494880676
155 1.085062e+03 2.093227e+01
* time: 90.36978197097778
156 1.085062e+03 2.093208e+01
* time: 91.20789098739624
157 1.085062e+03 2.093187e+01
* time: 92.02035903930664
158 1.085062e+03 2.093181e+01
* time: 92.83537697792053
159 1.085062e+03 2.093175e+01
* time: 93.66535997390747
160 1.085062e+03 2.093172e+01
* time: 94.53683090209961
161 1.085062e+03 2.046653e+01
* time: 95.11549186706543
162 1.085061e+03 2.045920e+01
* time: 95.7680070400238
163 1.085056e+03 2.021786e+01
* time: 96.35349702835083
164 1.085055e+03 2.018381e+01
* time: 96.91465783119202
165 1.085043e+03 1.997498e+01
* time: 97.48475885391235
166 1.085029e+03 1.961695e+01
* time: 98.00829100608826
167 1.084997e+03 1.929634e+01
* time: 98.56966996192932
168 1.084881e+03 1.855033e+01
* time: 99.08399391174316
169 1.084633e+03 1.799616e+01
* time: 99.63483095169067
170 1.083831e+03 1.856893e+01
* time: 100.15179085731506
171 1.081240e+03 2.069096e+01
* time: 100.71300983428955
172 1.072465e+03 3.262436e+01
* time: 101.32411885261536
173 1.072446e+03 6.686587e+01
* time: 101.92538404464722
174 1.068493e+03 7.417042e+01
* time: 102.5943808555603
175 1.067469e+03 7.109936e+01
* time: 103.26672291755676
176 1.065854e+03 2.774005e+01
* time: 103.92106103897095
177 1.065215e+03 1.328875e+01
* time: 104.5223159790039
178 1.065128e+03 1.390400e+01
* time: 105.15694189071655
179 1.065120e+03 1.383172e+01
* time: 105.742928981781
180 1.065120e+03 1.384995e+01
* time: 106.36983585357666FittedPumasModel
Successful minimization: false
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Nonlinear ODE
Solver(s):(OrdinaryDiffEq.Vern7,OrdinaryDiffEq.Rodas5P)
Log-likelihood value: -1065.1201
Number of subjects: 31
Number of parameters: Fixed Optimized
0 18
Observation records: Active Missing
conc: 239 47
pca: 225 61
Total: 464 108
-------------------------
Estimate
-------------------------
pop_CL 0.13142
pop_V 7.9732
pop_tabs 0.57389
pop_lag 0.83565
pop_e0 96.508
pop_emax -1.0982
pop_c50 1.7031
pop_tover 14.804
pk_Ω₁,₁ 0.078665
pk_Ω₂,₂ 0.031456
pk_Ω₃,₃ 0.083202
pd_Ω₁,₁ 0.00316
pd_Ω₂,₂ 0.0023756
pd_Ω₃,₃ 0.025823
pd_Ω₄,₄ 0.013808
σ_prop 0.013696
σ_add 0.84173
σ_fx 3.3937
-------------------------but succeeds 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 2.998927e+06 5.740466e+06
* time: 7.009506225585938e-5
1 4.804005e+05 8.280312e+05
* time: 1.4213109016418457
2 3.595691e+05 6.035843e+05
* time: 2.776289939880371
3 1.691071e+05 2.702705e+05
* time: 3.995556116104126
4 9.193152e+04 1.451040e+05
* time: 5.28485107421875
5 4.768539e+04 6.630202e+04
* time: 6.495600938796997
6 2.904205e+04 3.416919e+04
* time: 7.938838958740234
7 1.840153e+04 1.671742e+04
* time: 9.321347951889038
8 1.280989e+04 1.022268e+04
* time: 10.629220008850098
9 9.713287e+03 9.298485e+03
* time: 12.010786056518555
10 7.628127e+03 8.330656e+03
* time: 13.268932104110718
11 6.236278e+03 7.340367e+03
* time: 14.551939010620117
12 5.271958e+03 6.303773e+03
* time: 15.7865629196167
13 4.411047e+03 4.994130e+03
* time: 17.045713901519775
14 3.732936e+03 3.538184e+03
* time: 18.29279088973999
15 3.399273e+03 2.450045e+03
* time: 19.517412900924683
16 3.302987e+03 1.848476e+03
* time: 20.68213391304016
17 3.291788e+03 1.642864e+03
* time: 21.837488889694214
18 3.290743e+03 1.594765e+03
* time: 22.97154998779297
19 3.289687e+03 1.562641e+03
* time: 24.121416091918945
20 3.286186e+03 1.496632e+03
* time: 25.237525939941406
21 3.277917e+03 1.402865e+03
* time: 26.383261919021606
22 3.255959e+03 1.254297e+03
* time: 27.5228750705719
23 3.200980e+03 1.038621e+03
* time: 28.67260503768921
24 3.063521e+03 7.403605e+02
* time: 29.803741931915283
25 2.730316e+03 3.768701e+02
* time: 30.854473114013672
26 1.934186e+03 2.246121e+02
* time: 31.976321935653687
27 1.704399e+03 2.219826e+02
* time: 36.09633111953735
28 1.471183e+03 2.025826e+02
* time: 43.09261894226074
29 1.316789e+03 1.554714e+02
* time: 48.958149909973145
30 1.226363e+03 1.775666e+02
* time: 50.08729696273804
31 1.206626e+03 2.780980e+02
* time: 51.15478301048279
32 1.197830e+03 2.430550e+02
* time: 52.148098945617676
33 1.190342e+03 2.327905e+02
* time: 53.20940709114075
34 1.178783e+03 2.105809e+02
* time: 54.22853112220764
35 1.177853e+03 2.228585e+02
* time: 55.30527305603027
36 1.177763e+03 2.256116e+02
* time: 56.301328897476196
37 1.177745e+03 2.259359e+02
* time: 57.291074991226196
38 1.177670e+03 2.264633e+02
* time: 58.263449907302856
39 1.177500e+03 2.264574e+02
* time: 59.31709003448486
40 1.177033e+03 2.244136e+02
* time: 60.43332505226135
41 1.175894e+03 2.160226e+02
* time: 61.64549708366394
42 1.173240e+03 1.905912e+02
* time: 63.849257946014404
43 1.168189e+03 1.332783e+02
* time: 65.26796793937683
44 1.161835e+03 5.289952e+01
* time: 66.54442405700684
45 1.158453e+03 6.595387e+01
* time: 67.79223203659058
46 1.157897e+03 6.411419e+01
* time: 68.93668389320374
47 1.157870e+03 6.098509e+01
* time: 70.15670895576477
48 1.157868e+03 6.032325e+01
* time: 71.31102108955383
49 1.157864e+03 5.920812e+01
* time: 72.50489807128906
50 1.157856e+03 5.740225e+01
* time: 73.58168911933899
51 1.157832e+03 5.406795e+01
* time: 74.67741107940674
52 1.157774e+03 4.797879e+01
* time: 75.70085906982422
53 1.157628e+03 3.658732e+01
* time: 76.75829911231995
54 1.157299e+03 3.707874e+01
* time: 77.78539896011353
55 1.156690e+03 3.452947e+01
* time: 78.79574298858643
56 1.155959e+03 4.269004e+01
* time: 79.77993893623352
57 1.155557e+03 5.000322e+01
* time: 80.77716994285583
58 1.155472e+03 4.219349e+01
* time: 81.76612710952759
59 1.155463e+03 3.895049e+01
* time: 82.80450701713562
60 1.155462e+03 3.872099e+01
* time: 83.78659796714783
61 1.155458e+03 3.823491e+01
* time: 84.76249599456787
62 1.155450e+03 3.747370e+01
* time: 85.73295092582703
63 1.155426e+03 3.608648e+01
* time: 86.702064037323
64 1.155365e+03 3.361118e+01
* time: 87.68594408035278
65 1.155207e+03 2.894678e+01
* time: 88.69814491271973
66 1.154814e+03 2.504284e+01
* time: 89.66562390327454
67 1.153910e+03 4.541126e+01
* time: 90.61883306503296
68 1.152230e+03 7.426044e+01
* time: 91.54212093353271
69 1.150286e+03 7.473044e+01
* time: 92.47210192680359
70 1.149188e+03 5.605967e+01
* time: 93.39829397201538
71 1.148944e+03 4.801123e+01
* time: 94.35852193832397
72 1.148931e+03 4.659478e+01
* time: 95.25754404067993
73 1.148929e+03 4.635180e+01
* time: 96.16334390640259
74 1.148922e+03 4.568579e+01
* time: 97.05272197723389
75 1.148907e+03 4.466128e+01
* time: 97.96738290786743
76 1.148864e+03 4.249742e+01
* time: 98.86467599868774
77 1.148758e+03 3.824840e+01
* time: 99.80262207984924
78 1.148490e+03 3.552459e+01
* time: 100.74380207061768
79 1.147881e+03 3.038181e+01
* time: 101.72020411491394
80 1.146736e+03 4.467281e+01
* time: 102.68127107620239
81 1.145347e+03 5.929158e+01
* time: 103.59850001335144
82 1.144535e+03 7.572784e+01
* time: 104.51590609550476
83 1.144354e+03 7.034986e+01
* time: 105.4471960067749
84 1.144341e+03 6.591584e+01
* time: 106.38865089416504
85 1.144340e+03 6.496353e+01
* time: 107.4462399482727
86 1.144333e+03 6.268267e+01
* time: 108.44511008262634
87 1.144320e+03 5.950121e+01
* time: 109.56558108329773
88 1.144281e+03 5.368613e+01
* time: 110.68823194503784
89 1.144186e+03 4.393956e+01
* time: 111.71031594276428
90 1.143949e+03 4.102324e+01
* time: 112.74802303314209
91 1.143423e+03 3.752005e+01
* time: 113.81203007698059
92 1.142497e+03 3.278884e+01
* time: 114.84415102005005
93 1.141489e+03 5.217814e+01
* time: 115.90482306480408
94 1.140969e+03 4.310032e+01
* time: 116.92359709739685
95 1.140872e+03 4.759545e+01
* time: 117.95670008659363
96 1.140867e+03 4.727034e+01
* time: 119.00169801712036
97 1.140866e+03 4.704365e+01
* time: 119.97153997421265
98 1.140862e+03 4.637923e+01
* time: 121.02337098121643
99 1.140853e+03 4.542582e+01
* time: 121.97108006477356
100 1.140829e+03 4.357378e+01
* time: 123.01005291938782
101 1.140767e+03 4.023060e+01
* time: 123.98621702194214
102 1.140610e+03 3.376323e+01
* time: 125.02360200881958
103 1.140237e+03 2.857028e+01
* time: 126.00293207168579
104 1.139451e+03 3.895699e+01
* time: 127.08703994750977
105 1.138223e+03 5.064036e+01
* time: 128.0933859348297
106 1.137140e+03 4.763314e+01
* time: 129.15878796577454
107 1.136763e+03 4.604968e+01
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* time: 132.1346640586853
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* time: 133.10726690292358
111 1.136712e+03 3.865517e+01
* time: 134.02548003196716
112 1.136702e+03 3.838833e+01
* time: 134.99683690071106
113 1.136674e+03 3.789981e+01
* time: 135.96853709220886
114 1.136603e+03 3.705558e+01
* time: 137.00783705711365
115 1.136418e+03 3.549940e+01
* time: 138.00457000732422
116 1.135940e+03 3.257136e+01
* time: 139.12191200256348
117 1.134743e+03 2.697248e+01
* time: 140.13566899299622
118 1.132014e+03 4.863416e+01
* time: 141.18758606910706
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* time: 142.22617602348328
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* time: 143.3027160167694
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122 1.118972e+03 2.137843e+01
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* time: 146.33086705207825
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126 1.118861e+03 1.864783e+01
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127 1.118861e+03 1.862870e+01
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128 1.118860e+03 1.866000e+01
* time: 151.44108605384827
129 1.118859e+03 1.869625e+01
* time: 152.49187302589417
130 1.118856e+03 1.878717e+01
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131 1.118850e+03 1.894283e+01
* time: 154.61494708061218
132 1.118834e+03 1.922180e+01
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133 1.118794e+03 1.968060e+01
* time: 156.62037301063538
134 1.118693e+03 2.042353e+01
* time: 157.6371009349823
135 1.118431e+03 2.156998e+01
* time: 158.6648759841919
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* time: 159.7586851119995
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* time: 160.80476689338684
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* time: 161.82704901695251
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* time: 162.8394649028778
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* time: 164.08347702026367
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* time: 165.37228298187256
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* time: 166.88403296470642
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* time: 286.11368894577026
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* time: 293.06224489212036
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* time: 307.2168290615082
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* time: 308.45721912384033
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252 1.012697e+03 9.980635e-01
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253 1.012697e+03 9.984743e-01
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* time: 316.4778079986572
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* time: 319.1981439590454
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* time: 320.9471130371094
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* time: 324.1880979537964
259 1.012590e+03 4.737310e+00
* time: 325.68547010421753
260 1.012497e+03 4.897808e+00
* time: 327.0984148979187
261 1.012416e+03 2.875911e+00
* time: 328.5509099960327
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* time: 329.95952892303467
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* time: 331.381019115448
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* time: 335.4026880264282
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271 1.012386e+03 8.215489e-01
* time: 342.09105801582336
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* time: 351.2990839481354
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* time: 352.59764099121094
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282 1.011775e+03 9.067203e-01
* time: 356.23260498046875
283 1.011774e+03 9.565073e-01
* time: 357.503525018692
284 1.011774e+03 9.628263e-01
* time: 358.696918964386
285 1.011774e+03 9.654767e-01
* time: 359.97710609436035
286 1.011774e+03 9.654769e-01
* time: 361.3122639656067
287 1.011774e+03 9.656366e-01
* time: 362.5804350376129
288 1.011774e+03 9.656622e-01
* time: 363.8001799583435
289 1.011774e+03 9.656624e-01
* time: 365.1738450527191
290 1.011774e+03 9.656627e-01
* time: 366.58563208580017
291 1.011774e+03 9.660395e-01
* time: 367.88693594932556
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* time: 369.15504002571106
293 1.011774e+03 9.663115e-01
* time: 370.46117091178894
294 1.011774e+03 9.662851e-01
* time: 371.741534948349
295 1.011773e+03 9.654773e-01
* time: 372.99481201171875
296 1.011772e+03 9.623124e-01
* time: 374.2086009979248
297 1.011769e+03 9.521889e-01
* time: 375.45868396759033
298 1.011763e+03 9.239616e-01
* time: 376.604043006897
299 1.011746e+03 9.242954e-01
* time: 377.8690130710602
300 1.011711e+03 1.257052e+00
* time: 379.07086300849915
301 1.011651e+03 1.361278e+00
* time: 380.26551508903503
302 1.011591e+03 9.076916e-01
* time: 381.40745401382446
303 1.011568e+03 3.084752e-01
* time: 382.57904505729675
304 1.011565e+03 2.946404e-01
* time: 383.83995389938354
305 1.011565e+03 2.821273e-01
* time: 385.1048529148102
306 1.011565e+03 2.766260e-01
* time: 386.4107060432434
307 1.011565e+03 2.778010e-01
* time: 387.734482049942
308 1.011565e+03 2.830281e-01
* time: 388.9585530757904
309 1.011565e+03 2.842619e-01
* time: 390.29695296287537
310 1.011565e+03 2.861093e-01
* time: 391.59561491012573
311 1.011565e+03 2.861138e-01
* time: 392.9492540359497
312 1.011565e+03 2.867641e-01
* time: 394.25069403648376
313 1.011565e+03 2.867756e-01
* time: 395.6198670864105
314 1.011565e+03 2.884589e-01
* time: 396.86465191841125
315 1.011565e+03 2.897335e-01
* time: 398.04496002197266
316 1.011565e+03 2.927744e-01
* time: 399.1425449848175
317 1.011565e+03 2.967424e-01
* time: 400.25126600265503
318 1.011565e+03 3.027347e-01
* time: 401.37266993522644
319 1.011564e+03 3.099827e-01
* time: 402.5756850242615
320 1.011563e+03 3.155826e-01
* time: 403.74416303634644
321 1.011560e+03 3.082844e-01
* time: 404.9813470840454
322 1.011555e+03 2.625121e-01
* time: 406.3092210292816
323 1.011548e+03 2.136271e-01
* time: 407.7033441066742
324 1.011542e+03 1.165033e-01
* time: 409.1061511039734
325 1.011538e+03 3.914224e-02
* time: 410.49344205856323
326 1.011537e+03 3.083888e-02
* time: 411.82163405418396
327 1.011536e+03 2.766364e-02
* time: 413.16252303123474
328 1.011536e+03 1.873262e-02
* time: 414.42528796195984
329 1.011536e+03 5.453951e-03
* time: 415.7236270904541
330 1.011536e+03 4.882951e-03
* time: 416.98046708106995
331 1.011535e+03 4.331879e-03
* time: 418.1123208999634
332 1.011535e+03 2.711268e-03
* time: 419.2297189235687
333 1.011535e+03 4.474256e-04
* time: 420.3970100879669FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Nonlinear ODE
Solver(s):(OrdinaryDiffEq.Vern7,OrdinaryDiffEq.Rodas5P)
Log-likelihood value: -1011.5354
Number of subjects: 31
Number of parameters: Fixed Optimized
0 18
Observation records: Active Missing
conc: 239 47
pca: 225 61
Total: 464 108
-------------------------
Estimate
-------------------------
pop_CL 0.13137
pop_V 8.0255
pop_tabs 0.42635
pop_lag 0.92336
pop_e0 96.688
pop_emax -1.0628
pop_c50 1.5016
pop_tover 14.055
pk_Ω₁,₁ 0.050744
pk_Ω₂,₂ 0.021666
pk_Ω₃,₃ 0.97884
pd_Ω₁,₁ 0.0028828
pd_Ω₂,₂ 7.3726e-9
pd_Ω₃,₃ 0.15184
pd_Ω₄,₄ 0.011913
σ_prop 0.088292
σ_add 0.36362
σ_fx 3.5336
-------------------------
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 2.998927e+06 5.740466e+06
* time: 0.00011110305786132812
1 4.804005e+05 8.280311e+05
* time: 7.673443078994751
2 3.595691e+05 6.035842e+05
* time: 14.667448043823242
3 1.691071e+05 2.702705e+05
* time: 21.982383966445923
4 9.193152e+04 1.451040e+05
* time: 29.150563955307007
5 4.768538e+04 6.630202e+04
* time: 36.0916850566864
6 2.904205e+04 3.416919e+04
* time: 43.342525005340576
7 1.840152e+04 1.671742e+04
* time: 51.0553240776062
8 1.280989e+04 1.022268e+04
* time: 57.61330509185791
9 9.713285e+03 9.298485e+03
* time: 65.13860392570496
10 7.628127e+03 8.330656e+03
* time: 72.40481901168823
11 6.236278e+03 7.340367e+03
* time: 79.24578905105591
12 5.271958e+03 6.303773e+03
* time: 85.79942393302917
13 4.411047e+03 4.994130e+03
* time: 91.93942904472351
14 3.732935e+03 3.538183e+03
* time: 98.22914505004883
15 3.399272e+03 2.450045e+03
* time: 107.67582511901855
16 3.302987e+03 1.848475e+03
* time: 116.78793692588806
17 3.291788e+03 1.642864e+03
* time: 124.82811498641968
18 3.290743e+03 1.594765e+03
* time: 131.76890802383423
19 3.289686e+03 1.562640e+03
* time: 138.32961702346802
20 3.286186e+03 1.496632e+03
* time: 144.80122089385986
21 3.277917e+03 1.402865e+03
* time: 150.77416896820068
22 3.255958e+03 1.254296e+03
* time: 156.94855093955994
23 3.200979e+03 1.038621e+03
* time: 163.40392899513245
24 3.063520e+03 7.403602e+02
* time: 170.2171070575714
25 2.730315e+03 3.768701e+02
* time: 177.24845790863037
26 1.934186e+03 2.246121e+02
* time: 184.1445391178131
27 1.704399e+03 2.219826e+02
* time: 202.86816906929016
28 1.471184e+03 2.025825e+02
* time: 234.67003798484802
29 1.316789e+03 1.554715e+02
* time: 262.2527129650116
30 1.226363e+03 1.775721e+02
* time: 269.91614603996277
31 1.206626e+03 2.781030e+02
* time: 276.72540807724
32 1.197830e+03 2.430603e+02
* time: 283.01646304130554
33 1.190342e+03 2.327966e+02
* time: 288.7265040874481
34 1.178784e+03 2.105878e+02
* time: 294.6336200237274
35 1.177853e+03 2.228653e+02
* time: 300.4709289073944
36 1.177764e+03 2.256182e+02
* time: 306.3016629219055
37 1.177746e+03 2.259425e+02
* time: 312.17793107032776
38 1.177670e+03 2.264699e+02
* time: 317.8813691139221
39 1.177500e+03 2.264641e+02
* time: 324.4756679534912
40 1.177034e+03 2.244207e+02
* time: 331.4531469345093
41 1.175895e+03 2.160307e+02
* time: 338.1889669895172
42 1.173241e+03 1.906017e+02
* time: 345.25024604797363
43 1.168190e+03 1.332917e+02
* time: 352.0649869441986
44 1.161836e+03 5.291022e+01
* time: 358.8696930408478
45 1.158453e+03 6.595471e+01
* time: 365.6422998905182
46 1.157897e+03 6.411640e+01
* time: 371.92811393737793
47 1.157870e+03 6.098679e+01
* time: 378.7259681224823
48 1.157868e+03 6.032481e+01
* time: 385.16627192497253
49 1.157864e+03 5.920991e+01
* time: 392.3665211200714
50 1.157856e+03 5.740417e+01
* time: 399.12317299842834
51 1.157832e+03 5.407031e+01
* time: 405.5312600135803
52 1.157774e+03 4.798188e+01
* time: 411.8434569835663
53 1.157628e+03 3.659183e+01
* time: 417.7749660015106
54 1.157299e+03 3.707862e+01
* time: 423.7920289039612
55 1.156690e+03 3.453183e+01
* time: 430.7481029033661
56 1.155959e+03 4.268473e+01
* time: 437.5971009731293
57 1.155557e+03 5.000309e+01
* time: 444.66834592819214
58 1.155472e+03 4.219411e+01
* time: 451.15255403518677
59 1.155463e+03 3.894982e+01
* time: 457.10771799087524
60 1.155462e+03 3.872028e+01
* time: 463.0308039188385
61 1.155458e+03 3.823435e+01
* time: 468.9644401073456
62 1.155450e+03 3.747328e+01
* time: 475.63051199913025
63 1.155426e+03 3.608641e+01
* time: 484.16397309303284
64 1.155365e+03 3.361168e+01
* time: 490.96077609062195
65 1.155207e+03 2.894844e+01
* time: 497.47153401374817
66 1.154814e+03 2.504332e+01
* time: 503.7385051250458
67 1.153910e+03 4.539953e+01
* time: 509.9846749305725
68 1.152231e+03 7.425240e+01
* time: 516.877711057663
69 1.150287e+03 7.473669e+01
* time: 523.3645570278168
70 1.149189e+03 5.606074e+01
* time: 529.445424079895
71 1.148944e+03 4.801310e+01
* time: 535.3944590091705
72 1.148932e+03 4.659470e+01
* time: 541.8447499275208
73 1.148930e+03 4.635167e+01
* time: 549.2147109508514
74 1.148923e+03 4.568576e+01
* time: 555.924201965332
75 1.148907e+03 4.466140e+01
* time: 562.2909870147705
76 1.148865e+03 4.249800e+01
* time: 568.3632690906525
77 1.148758e+03 3.824645e+01
* time: 574.7258539199829
78 1.148491e+03 3.552369e+01
* time: 580.9394030570984
79 1.147881e+03 3.037355e+01
* time: 588.6518459320068
80 1.146737e+03 4.466673e+01
* time: 595.7082009315491
81 1.145347e+03 5.927932e+01
* time: 602.3184039592743
82 1.144536e+03 7.572710e+01
* time: 608.5848269462585
83 1.144354e+03 7.035223e+01
* time: 614.7044529914856
84 1.144341e+03 6.591571e+01
* time: 621.1488010883331
85 1.144340e+03 6.496308e+01
* time: 626.7444078922272
86 1.144333e+03 6.268335e+01
* time: 632.718435049057
87 1.144320e+03 5.950296e+01
* time: 638.7195730209351
88 1.144281e+03 5.369030e+01
* time: 645.1348159313202
89 1.144186e+03 4.394749e+01
* time: 651.7146790027618
90 1.143949e+03 4.102392e+01
* time: 657.6451420783997
91 1.143423e+03 3.752367e+01
* time: 663.5605199337006
92 1.142498e+03 3.276758e+01
* time: 669.622752904892
93 1.141490e+03 5.217551e+01
* time: 675.3638920783997
94 1.140969e+03 4.309381e+01
* time: 681.769905090332
95 1.140872e+03 4.759588e+01
* time: 690.3130490779877
96 1.140867e+03 4.727133e+01
* time: 697.3072800636292
97 1.140866e+03 4.704459e+01
* time: 703.6837470531464
98 1.140862e+03 4.638036e+01
* time: 709.6662900447845
99 1.140853e+03 4.542717e+01
* time: 715.3133509159088
100 1.140829e+03 4.357571e+01
* time: 721.6268210411072
101 1.140767e+03 4.023368e+01
* time: 727.5522429943085
102 1.140611e+03 3.376885e+01
* time: 733.1517210006714
103 1.140237e+03 2.856928e+01
* time: 741.5972790718079
104 1.139452e+03 3.894107e+01
* time: 748.8258669376373
105 1.138224e+03 5.063688e+01
* time: 755.6227099895477
106 1.137140e+03 4.762519e+01
* time: 761.3282198905945
107 1.136763e+03 4.605051e+01
* time: 767.1185319423676
108 1.136720e+03 3.993288e+01
* time: 772.6687409877777
109 1.136717e+03 3.890566e+01
* time: 778.2906858921051
110 1.136716e+03 3.883647e+01
* time: 784.0709800720215
111 1.136712e+03 3.865523e+01
* time: 790.0633509159088
112 1.136702e+03 3.838845e+01
* time: 796.5160501003265
113 1.136674e+03 3.790005e+01
* time: 802.9554579257965
114 1.136604e+03 3.705604e+01
* time: 809.3677079677582
115 1.136418e+03 3.550032e+01
* time: 815.57728099823
116 1.135940e+03 3.257323e+01
* time: 821.7186379432678
117 1.134744e+03 2.697619e+01
* time: 828.0183510780334
118 1.132016e+03 4.861352e+01
* time: 836.7732269763947
119 1.127206e+03 6.900228e+01
* time: 844.3954911231995
120 1.122365e+03 5.795238e+01
* time: 851.1173160076141
121 1.119760e+03 2.908163e+01
* time: 857.3920609951019
122 1.118972e+03 2.137897e+01
* time: 863.4135980606079
123 1.118870e+03 1.909827e+01
* time: 869.0985391139984
124 1.118863e+03 1.886936e+01
* time: 874.7888720035553
125 1.118862e+03 1.852746e+01
* time: 880.3354690074921
126 1.118861e+03 1.864804e+01
* time: 885.8741960525513
127 1.118861e+03 1.862831e+01
* time: 892.4295330047607
128 1.118860e+03 1.865947e+01
* time: 898.5750279426575
129 1.118859e+03 1.869546e+01
* time: 904.3966250419617
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* time: 1715.8391830921173
235 1.026628e+03 7.187426e+01
* time: 1722.5530579090118
236 1.024510e+03 1.065337e+02
* time: 1729.3206329345703
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* time: 1736.0531730651855
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* time: 1742.3341069221497
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* time: 1763.8864710330963
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* time: 1770.166855096817
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* time: 1776.3541219234467
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* time: 1782.540020942688
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* time: 1788.8844759464264
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* time: 1794.840460062027
247 1.012697e+03 9.971842e-01
* time: 1801.0149629116058
248 1.012697e+03 9.972169e-01
* time: 1807.4633610248566
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* time: 1814.7902998924255
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* time: 1822.9457380771637
251 1.012697e+03 9.973078e-01
* time: 1830.299968957901
252 1.012697e+03 9.972048e-01
* time: 1837.031513929367
253 1.012697e+03 9.970536e-01
* time: 1843.599781036377
254 1.012697e+03 9.967402e-01
* time: 1850.5163431167603
255 1.012697e+03 9.961565e-01
* time: 1857.192148923874
256 1.012696e+03 9.949478e-01
* time: 1863.6648120880127
257 1.012694e+03 1.312115e+00
* time: 1870.3512001037598
258 1.012690e+03 2.118030e+00
* time: 1877.5871450901031
259 1.012679e+03 3.352490e+00
* time: 1884.6995720863342
260 1.012651e+03 5.076526e+00
* time: 1891.9568150043488
261 1.012594e+03 6.891495e+00
* time: 1898.9699161052704
262 1.012501e+03 7.239074e+00
* time: 1906.039745092392
263 1.012418e+03 4.479611e+00
* time: 1913.6677079200745
264 1.012390e+03 1.267149e+00
* time: 1921.5598249435425
265 1.012386e+03 8.840304e-01
* time: 1929.2193241119385
266 1.012386e+03 8.549097e-01
* time: 1936.2232670783997
267 1.012386e+03 8.516869e-01
* time: 1942.9492959976196
268 1.012386e+03 8.516869e-01
* time: 1950.5795691013336
269 1.012386e+03 8.516869e-01
* time: 1958.534110069275
270 1.012386e+03 8.516817e-01
* time: 1965.7023429870605
271 1.012386e+03 8.516817e-01
* time: 1974.8013410568237
272 1.012386e+03 8.516817e-01
* time: 1983.5860090255737
273 1.012386e+03 8.516817e-01
* time: 1991.9630680084229
274 1.012386e+03 8.516817e-01
* time: 2000.6564168930054
275 1.012386e+03 8.516817e-01
* time: 2009.2055430412292FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Nonlinear ODE
Solver(s): OrdinaryDiffEq.Rodas5P
Log-likelihood value: -1012.3861
Number of subjects: 31
Number of parameters: Fixed Optimized
0 18
Observation records: Active Missing
conc: 239 47
pca: 225 61
Total: 464 108
--------------------------
Estimate
--------------------------
pop_CL 0.13136
pop_V 8.0217
pop_tabs 0.41997
pop_lag 0.92474
pop_e0 96.692
pop_emax -1.0629
pop_c50 1.5003
pop_tover 14.06
pk_Ω₁,₁ 0.05162
pk_Ω₂,₂ 0.019945
pk_Ω₃,₃ 1.0369
pd_Ω₁,₁ 0.0025468
pd_Ω₂,₂ 0.00037903
pd_Ω₃,₃ 0.15787
pd_Ω₄,₄ 0.012052
σ_prop 0.088654
σ_add 0.36167
σ_fx 3.5399
--------------------------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 2.998927e+06 5.740466e+06
* time: 7.700920104980469e-5
1 4.804005e+05 8.280312e+05
* time: 1.377593994140625
2 3.595691e+05 6.035843e+05
* time: 2.6294469833374023
3 1.691071e+05 2.702705e+05
* time: 3.7249698638916016
4 9.193152e+04 1.451040e+05
* time: 4.860307931900024
5 4.768539e+04 6.630202e+04
* time: 5.905808925628662
6 2.904205e+04 3.416919e+04
* time: 6.976714849472046
7 1.840153e+04 1.671742e+04
* time: 8.043582916259766
8 1.280989e+04 1.022268e+04
* time: 9.118019819259644
9 9.713287e+03 9.298485e+03
* time: 10.142295837402344
10 7.628127e+03 8.330656e+03
* time: 11.188467025756836
11 6.236278e+03 7.340367e+03
* time: 12.173328876495361
12 5.271958e+03 6.303773e+03
* time: 13.190416812896729
13 4.411047e+03 4.994130e+03
* time: 14.178768873214722
14 3.732936e+03 3.538184e+03
* time: 15.164665937423706
15 3.399273e+03 2.450045e+03
* time: 16.155197858810425
16 3.302987e+03 1.848476e+03
* time: 17.17656898498535
17 3.291788e+03 1.642864e+03
* time: 18.17175602912903
18 3.290743e+03 1.594765e+03
* time: 19.131542921066284
19 3.289687e+03 1.562641e+03
* time: 20.08412003517151
20 3.286186e+03 1.496632e+03
* time: 20.993714809417725
21 3.277917e+03 1.402865e+03
* time: 21.921304941177368
22 3.255959e+03 1.254297e+03
* time: 22.820847034454346
23 3.200980e+03 1.038621e+03
* time: 23.769176959991455
24 3.063521e+03 7.403605e+02
* time: 24.65463399887085
25 2.730316e+03 3.768701e+02
* time: 25.59572696685791
26 1.934186e+03 2.246121e+02
* time: 26.460094928741455
27 1.704399e+03 2.219826e+02
* time: 29.80387592315674
28 1.471183e+03 2.025826e+02
* time: 36.17591381072998
29 1.316789e+03 1.554714e+02
* time: 41.47767996788025
30 1.226363e+03 1.775666e+02
* time: 42.369571924209595
31 1.206626e+03 2.780980e+02
* time: 43.180583000183105
32 1.197830e+03 2.430550e+02
* time: 43.98207497596741
33 1.190342e+03 2.327905e+02
* time: 44.794824838638306
34 1.178783e+03 2.105809e+02
* time: 45.633220911026
35 1.177853e+03 2.228585e+02
* time: 46.49257183074951
36 1.177763e+03 2.256116e+02
* time: 47.32323598861694
37 1.177745e+03 2.259359e+02
* time: 48.204272985458374
38 1.177670e+03 2.264633e+02
* time: 49.01487183570862
39 1.177500e+03 2.264574e+02
* time: 49.854599952697754
40 1.177033e+03 2.244136e+02
* time: 50.67004084587097
41 1.175894e+03 2.160226e+02
* time: 51.550333976745605
42 1.173240e+03 1.905912e+02
* time: 52.467926025390625
43 1.168189e+03 1.332783e+02
* time: 53.29774188995361
44 1.161835e+03 5.289952e+01
* time: 54.198277950286865
45 1.158453e+03 6.595387e+01
* time: 55.061887979507446
46 1.157897e+03 6.411419e+01
* time: 56.020084857940674
47 1.157870e+03 6.098509e+01
* time: 56.845794916152954
48 1.157868e+03 6.032325e+01
* time: 57.67557501792908
49 1.157864e+03 5.920812e+01
* time: 58.537895917892456
50 1.157856e+03 5.740225e+01
* time: 59.3876838684082
51 1.157832e+03 5.406795e+01
* time: 60.2262499332428
52 1.157774e+03 4.797879e+01
* time: 61.02031993865967
53 1.157628e+03 3.658732e+01
* time: 61.852914810180664
54 1.157299e+03 3.707874e+01
* time: 62.630345821380615
55 1.156690e+03 3.452947e+01
* time: 63.477810859680176
56 1.155959e+03 4.269004e+01
* time: 64.2647819519043
57 1.155557e+03 5.000322e+01
* time: 65.09320902824402
58 1.155472e+03 4.219349e+01
* time: 65.92519283294678
59 1.155463e+03 3.895049e+01
* time: 66.74335289001465
60 1.155462e+03 3.872099e+01
* time: 67.59260392189026
61 1.155458e+03 3.823491e+01
* time: 68.52840495109558
62 1.155450e+03 3.747370e+01
* time: 69.58695602416992
63 1.155426e+03 3.608648e+01
* time: 70.58643794059753
64 1.155365e+03 3.361118e+01
* time: 71.50067186355591
65 1.155207e+03 2.894678e+01
* time: 72.34362983703613
66 1.154814e+03 2.504284e+01
* time: 73.14786291122437
67 1.153910e+03 4.541126e+01
* time: 73.9666919708252
68 1.152230e+03 7.426044e+01
* time: 74.75626301765442
69 1.150286e+03 7.473044e+01
* time: 75.57277297973633
70 1.149188e+03 5.605967e+01
* time: 76.34451293945312
71 1.148944e+03 4.801123e+01
* time: 77.2857940196991
72 1.148931e+03 4.659478e+01
* time: 78.04050183296204
73 1.148929e+03 4.635180e+01
* time: 78.81697797775269
74 1.148922e+03 4.568579e+01
* time: 79.60090184211731
75 1.148907e+03 4.466128e+01
* time: 80.37755703926086
76 1.148864e+03 4.249742e+01
* time: 81.21344184875488
77 1.148758e+03 3.824840e+01
* time: 82.04326391220093
78 1.148490e+03 3.552459e+01
* time: 82.91700291633606
79 1.147881e+03 3.038181e+01
* time: 83.76479196548462
80 1.146736e+03 4.467281e+01
* time: 84.63676595687866
81 1.145347e+03 5.929158e+01
* time: 85.42790198326111
82 1.144535e+03 7.572784e+01
* time: 86.27797484397888
83 1.144354e+03 7.034986e+01
* time: 87.10642099380493
84 1.144341e+03 6.591584e+01
* time: 87.87741184234619
85 1.144340e+03 6.496353e+01
* time: 88.77870082855225
86 1.144333e+03 6.268267e+01
* time: 89.65780282020569
87 1.144320e+03 5.950121e+01
* time: 90.48373985290527
88 1.144281e+03 5.368613e+01
* time: 91.27355098724365
89 1.144186e+03 4.393956e+01
* time: 92.0815019607544
90 1.143949e+03 4.102324e+01
* time: 92.92830181121826
91 1.143423e+03 3.752005e+01
* time: 93.80473899841309
92 1.142497e+03 3.278884e+01
* time: 94.75245094299316
93 1.141489e+03 5.217814e+01
* time: 95.666818857193
94 1.140969e+03 4.310032e+01
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* time: 369.93529295921326FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Nonlinear ODE
Solver(s): OrdinaryDiffEq.Vern7
Log-likelihood value: -1011.5354
Number of subjects: 31
Number of parameters: Fixed Optimized
0 18
Observation records: Active Missing
conc: 239 47
pca: 225 61
Total: 464 108
-------------------------
Estimate
-------------------------
pop_CL 0.13137
pop_V 8.0255
pop_tabs 0.42635
pop_lag 0.92336
pop_e0 96.688
pop_emax -1.0628
pop_c50 1.5016
pop_tover 14.055
pk_Ω₁,₁ 0.050744
pk_Ω₂,₂ 0.021666
pk_Ω₃,₃ 0.97884
pd_Ω₁,₁ 0.0028828
pd_Ω₂,₂ 7.3726e-9
pd_Ω₃,₃ 0.15184
pd_Ω₄,₄ 0.011913
σ_prop 0.088292
σ_add 0.36362
σ_fx 3.5336
-------------------------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.