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: 0.02611708641052246
1 4.816669e+05 8.316173e+05
* time: 1.6927011013031006
2 3.598373e+05 6.060150e+05
* time: 2.319962978363037
3 1.682124e+05 2.709072e+05
* time: 2.967258930206299
4 9.168914e+04 1.446832e+05
* time: 3.546808958053589
5 4.791138e+04 6.671463e+04
* time: 4.163985967636108
6 2.909104e+04 3.426663e+04
* time: 4.722419023513794
7 1.843737e+04 1.678953e+04
* time: 5.321857929229736
8 1.282447e+04 1.021463e+04
* time: 5.921519041061401
9 9.723361e+03 9.291650e+03
* time: 6.5806050300598145
10 7.635613e+03 8.325311e+03
* time: 7.246090888977051
11 6.243129e+03 7.337397e+03
* time: 7.847722053527832
12 5.279957e+03 6.304542e+03
* time: 8.492357969284058
13 4.422123e+03 5.001962e+03
* time: 9.07016897201538
14 3.744748e+03 3.550400e+03
* time: 9.696433067321777
15 3.409674e+03 2.461547e+03
* time: 10.2811439037323
16 3.312049e+03 1.856430e+03
* time: 10.913904905319214
17 3.300559e+03 1.648439e+03
* time: 11.500211000442505
18 3.299489e+03 1.599460e+03
* time: 12.117809057235718
19 3.298436e+03 1.567520e+03
* time: 12.696063995361328
20 3.294921e+03 1.501363e+03
* time: 13.305933952331543
21 3.286641e+03 1.407664e+03
* time: 13.897119045257568
22 3.264637e+03 1.259077e+03
* time: 14.513629913330078
23 3.209594e+03 1.043534e+03
* time: 15.116370916366577
24 3.072117e+03 7.454677e+02
* time: 15.736212968826294
25 2.739794e+03 3.818063e+02
* time: 16.308451890945435
26 1.949883e+03 2.246644e+02
* time: 16.900930881500244
27 1.722434e+03 2.224562e+02
* time: 18.660545110702515
28 1.479355e+03 2.042452e+02
* time: 20.67979097366333
29 1.310583e+03 1.531868e+02
* time: 23.08453106880188
30 1.224675e+03 1.756826e+02
* time: 23.659393072128296
31 1.204841e+03 2.738440e+02
* time: 24.22037100791931
32 1.196432e+03 2.403217e+02
* time: 24.732858896255493
33 1.189131e+03 2.301186e+02
* time: 25.231034994125366
34 1.177881e+03 2.083030e+02
* time: 25.765413999557495
35 1.176989e+03 2.201993e+02
* time: 26.321758031845093
36 1.176899e+03 2.230826e+02
* time: 26.801843881607056
37 1.176882e+03 2.233531e+02
* time: 27.337141036987305
38 1.176795e+03 2.238567e+02
* time: 27.815344095230103
39 1.176609e+03 2.236914e+02
* time: 28.375654935836792
40 1.176090e+03 2.210941e+02
* time: 28.927793979644775
41 1.174841e+03 2.112741e+02
* time: 29.48563504219055
42 1.171973e+03 1.825298e+02
* time: 30.002063989639282
43 1.166737e+03 1.210559e+02
* time: 30.5693199634552
44 1.160698e+03 4.450697e+01
* time: 31.069344997406006
45 1.157926e+03 6.411275e+01
* time: 31.616962909698486
46 1.157537e+03 6.121762e+01
* time: 32.091017961502075
47 1.157520e+03 5.867223e+01
* time: 32.60736894607544
48 1.157519e+03 5.814523e+01
* time: 33.07475399971008
49 1.157515e+03 5.686007e+01
* time: 33.587517976760864
50 1.157505e+03 5.497418e+01
* time: 34.05543494224548
51 1.157477e+03 5.129724e+01
* time: 34.581542015075684
52 1.157408e+03 4.463171e+01
* time: 35.04270792007446
53 1.157239e+03 3.279117e+01
* time: 35.54072308540344
54 1.156869e+03 3.681526e+01
* time: 35.998672008514404
55 1.156230e+03 3.198754e+01
* time: 36.51727604866028
56 1.155563e+03 4.576438e+01
* time: 37.00059103965759
57 1.155264e+03 4.832159e+01
* time: 37.57898497581482
58 1.155212e+03 4.052175e+01
* time: 38.06172299385071
59 1.155207e+03 3.797125e+01
* time: 38.58347511291504
60 1.155206e+03 3.778452e+01
* time: 39.05156707763672
61 1.155201e+03 3.723003e+01
* time: 39.559499979019165
62 1.155190e+03 3.642375e+01
* time: 40.0743510723114
63 1.155159e+03 3.487535e+01
* time: 40.61317491531372
64 1.155079e+03 3.209060e+01
* time: 41.096494913101196
65 1.154875e+03 2.670227e+01
* time: 41.61755394935608
66 1.154375e+03 2.366165e+01
* time: 42.11773991584778
67 1.153278e+03 5.499525e+01
* time: 42.618874073028564
68 1.151457e+03 7.903833e+01
* time: 43.09328508377075
69 1.149737e+03 6.771227e+01
* time: 43.587100982666016
70 1.148983e+03 5.387708e+01
* time: 44.05850791931152
71 1.148872e+03 4.683950e+01
* time: 44.543018102645874
72 1.148868e+03 4.512036e+01
* time: 44.99015188217163
73 1.148866e+03 4.489706e+01
* time: 45.45267295837402
74 1.148857e+03 4.426314e+01
* time: 45.896398067474365
75 1.148838e+03 4.321161e+01
* time: 46.36517691612244
76 1.148785e+03 4.095232e+01
* time: 46.809650897979736
77 1.148652e+03 3.780572e+01
* time: 47.271559953689575
78 1.148325e+03 3.474239e+01
* time: 47.72371006011963
79 1.147599e+03 3.678957e+01
* time: 48.236716985702515
80 1.146330e+03 5.134277e+01
* time: 48.7106659412384
81 1.144979e+03 6.444063e+01
* time: 49.19164299964905
82 1.144325e+03 7.574751e+01
* time: 49.65359592437744
83 1.144210e+03 6.968296e+01
* time: 50.11248302459717
84 1.144204e+03 6.648234e+01
* time: 50.5857629776001
85 1.144202e+03 6.560999e+01
* time: 51.005887031555176
86 1.144194e+03 6.305667e+01
* time: 51.495179891586304
87 1.144176e+03 5.956959e+01
* time: 51.92906999588013
88 1.144127e+03 5.307828e+01
* time: 52.41878604888916
89 1.144005e+03 4.218776e+01
* time: 52.89321708679199
90 1.143708e+03 4.120261e+01
* time: 53.39156103134155
91 1.143074e+03 3.675516e+01
* time: 53.848143100738525
92 1.142052e+03 3.855439e+01
* time: 54.36059594154358
93 1.141101e+03 5.193197e+01
* time: 54.79287004470825
94 1.140701e+03 4.606858e+01
* time: 55.29709196090698
95 1.140646e+03 4.857736e+01
* time: 55.73552393913269
96 1.140643e+03 4.817596e+01
* time: 56.23041796684265
97 1.140642e+03 4.793197e+01
* time: 56.64785599708557
98 1.140637e+03 4.723245e+01
* time: 57.13870406150818
99 1.140626e+03 4.619695e+01
* time: 57.57896399497986
100 1.140597e+03 4.418761e+01
* time: 58.08202600479126
101 1.140523e+03 4.050986e+01
* time: 58.54635310173035
102 1.140334e+03 3.335334e+01
* time: 59.06259489059448
103 1.139892e+03 2.982120e+01
* time: 59.5269410610199
104 1.138994e+03 4.360237e+01
* time: 60.038482904434204
105 1.137687e+03 5.246892e+01
* time: 60.49608492851257
106 1.136679e+03 4.825586e+01
* time: 60.9785521030426
107 1.136391e+03 4.467499e+01
* time: 61.425689935684204
108 1.136364e+03 3.961591e+01
* time: 61.912643909454346
109 1.136362e+03 3.951402e+01
* time: 62.365931034088135
110 1.136360e+03 3.943358e+01
* time: 62.84543204307556
111 1.136355e+03 3.923965e+01
* time: 63.28848695755005
112 1.136344e+03 3.894674e+01
* time: 63.76533889770508
113 1.136314e+03 3.841518e+01
* time: 64.2070300579071
114 1.136235e+03 3.748824e+01
* time: 64.70638608932495
115 1.136029e+03 3.577210e+01
* time: 65.1598129272461
116 1.135499e+03 3.252332e+01
* time: 65.64731097221375
117 1.134182e+03 2.630844e+01
* time: 66.09335088729858
118 1.131238e+03 4.869163e+01
* time: 66.5899178981781
119 1.126345e+03 6.537856e+01
* time: 67.06660389900208
120 1.121879e+03 5.123373e+01
* time: 67.57467794418335
121 1.119645e+03 2.457316e+01
* time: 68.06685590744019
122 1.119031e+03 2.124721e+01
* time: 68.57914400100708
123 1.118958e+03 1.857508e+01
* time: 69.04415702819824
124 1.118953e+03 1.896595e+01
* time: 69.51784610748291
125 1.118952e+03 1.849021e+01
* time: 69.97680711746216
126 1.118952e+03 1.852941e+01
* time: 70.46055793762207
127 1.118951e+03 1.856812e+01
* time: 70.91250801086426
128 1.118951e+03 1.862403e+01
* time: 71.39392709732056
129 1.118949e+03 1.871484e+01
* time: 71.8651750087738
130 1.118946e+03 1.886332e+01
* time: 72.36351299285889
131 1.118939e+03 1.910217e+01
* time: 72.84254598617554
132 1.118922e+03 1.948050e+01
* time: 73.34192299842834
133 1.118879e+03 2.006903e+01
* time: 73.8318419456482
134 1.118769e+03 2.097106e+01
* time: 74.33555293083191
135 1.118486e+03 2.231801e+01
* time: 74.82484793663025
136 1.117755e+03 3.049039e+01
* time: 75.31615400314331
137 1.115883e+03 5.345487e+01
* time: 75.8330340385437
138 1.111258e+03 8.489274e+01
* time: 76.29019093513489
139 1.101149e+03 7.643749e+01
* time: 76.80460810661316
140 1.090612e+03 5.802659e+01
* time: 77.30336809158325
141 1.087971e+03 5.776554e+01
* time: 77.8821210861206
142 1.087205e+03 1.148234e+02
* time: 78.39083504676819
143 1.085558e+03 4.724872e+01
* time: 78.94654893875122
144 1.085271e+03 2.018848e+01
* time: 79.53140592575073
145 1.085124e+03 2.004201e+01
* time: 80.11560010910034
146 1.085089e+03 2.025616e+01
* time: 80.66102409362793
147 1.085081e+03 2.049682e+01
* time: 81.2650499343872
148 1.085073e+03 2.086075e+01
* time: 81.83065295219421
149 1.085070e+03 2.095706e+01
* time: 82.41697311401367
150 1.085063e+03 2.094978e+01
* time: 83.0330240726471
151 1.085062e+03 2.093855e+01
* time: 83.62757897377014
152 1.085062e+03 2.093439e+01
* time: 84.33828902244568
153 1.085062e+03 2.093335e+01
* time: 85.02231097221375
154 1.085062e+03 2.093252e+01
* time: 85.75088500976562
155 1.085062e+03 2.093227e+01
* time: 86.53235697746277
156 1.085062e+03 2.093208e+01
* time: 87.28588199615479
157 1.085062e+03 2.093187e+01
* time: 88.07127809524536
158 1.085062e+03 2.093181e+01
* time: 88.8927481174469
159 1.085062e+03 2.093175e+01
* time: 89.70535898208618
160 1.085062e+03 2.093172e+01
* time: 90.62055897712708
161 1.085062e+03 2.046653e+01
* time: 91.21287393569946
162 1.085061e+03 2.045920e+01
* time: 91.91516494750977
163 1.085056e+03 2.021786e+01
* time: 92.48618793487549
164 1.085055e+03 2.018381e+01
* time: 93.17255306243896
165 1.085043e+03 1.997498e+01
* time: 93.72569990158081
166 1.085029e+03 1.961695e+01
* time: 94.30105209350586
167 1.084997e+03 1.929634e+01
* time: 94.83890390396118
168 1.084881e+03 1.855033e+01
* time: 95.40808510780334
169 1.084633e+03 1.799616e+01
* time: 95.96174192428589
170 1.083831e+03 1.856893e+01
* time: 96.5353569984436
171 1.081240e+03 2.069096e+01
* time: 97.09978604316711
172 1.072465e+03 3.262436e+01
* time: 97.74869298934937
173 1.072446e+03 6.686587e+01
* time: 98.41009998321533
174 1.068493e+03 7.417042e+01
* time: 99.02554488182068
175 1.067469e+03 7.109936e+01
* time: 99.79062604904175
176 1.065854e+03 2.774005e+01
* time: 100.43494391441345
177 1.065215e+03 1.328875e+01
* time: 101.10913801193237
178 1.065128e+03 1.390400e+01
* time: 101.76535105705261
179 1.065120e+03 1.383172e+01
* time: 102.42727994918823
180 1.065120e+03 1.384995e+01
* time: 103.04243898391724FittedPumasModel
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: 7.510185241699219e-5
1 4.804005e+05 8.280312e+05
* time: 1.3474841117858887
2 3.595691e+05 6.035843e+05
* time: 2.534959077835083
3 1.691071e+05 2.702705e+05
* time: 3.717705011367798
4 9.193152e+04 1.451040e+05
* time: 4.92848014831543
5 4.768539e+04 6.630202e+04
* time: 6.130375146865845
6 2.904205e+04 3.416919e+04
* time: 7.328541994094849
7 1.840153e+04 1.671742e+04
* time: 8.48523998260498
8 1.280989e+04 1.022268e+04
* time: 9.695870161056519
9 9.713287e+03 9.298485e+03
* time: 10.851838111877441
10 7.628127e+03 8.330656e+03
* time: 12.051741123199463
11 6.236278e+03 7.340367e+03
* time: 13.207720041275024
12 5.271958e+03 6.303773e+03
* time: 14.375622987747192
13 4.411047e+03 4.994130e+03
* time: 15.515275955200195
14 3.732936e+03 3.538184e+03
* time: 16.621971130371094
15 3.399273e+03 2.450045e+03
* time: 17.73664402961731
16 3.302987e+03 1.848476e+03
* time: 18.80033802986145
17 3.291788e+03 1.642864e+03
* time: 19.933995008468628
18 3.290743e+03 1.594765e+03
* time: 20.97487711906433
19 3.289687e+03 1.562641e+03
* time: 22.053393125534058
20 3.286186e+03 1.496632e+03
* time: 23.054267168045044
21 3.277917e+03 1.402865e+03
* time: 24.133937120437622
22 3.255959e+03 1.254297e+03
* time: 25.18813705444336
23 3.200980e+03 1.038621e+03
* time: 26.219139099121094
24 3.063521e+03 7.403605e+02
* time: 27.16952896118164
25 2.730316e+03 3.768701e+02
* time: 28.166651964187622
26 1.934186e+03 2.246121e+02
* time: 29.047192096710205
27 1.704399e+03 2.219826e+02
* time: 32.77497100830078
28 1.471183e+03 2.025826e+02
* time: 39.22922110557556
29 1.316789e+03 1.554714e+02
* time: 44.90329718589783
30 1.226363e+03 1.775666e+02
* time: 45.944130182266235
31 1.206626e+03 2.780980e+02
* time: 46.81198215484619
32 1.197830e+03 2.430550e+02
* time: 47.71193599700928
33 1.190342e+03 2.327905e+02
* time: 48.57404804229736
34 1.178783e+03 2.105809e+02
* time: 49.50160813331604
35 1.177853e+03 2.228585e+02
* time: 50.375293016433716
36 1.177763e+03 2.256116e+02
* time: 51.34533905982971
37 1.177745e+03 2.259359e+02
* time: 52.28489708900452
38 1.177670e+03 2.264633e+02
* time: 53.25929117202759
39 1.177500e+03 2.264574e+02
* time: 54.24158501625061
40 1.177033e+03 2.244136e+02
* time: 55.28711915016174
41 1.175894e+03 2.160226e+02
* time: 56.263370990753174
42 1.173240e+03 1.905912e+02
* time: 57.19015312194824
43 1.168189e+03 1.332783e+02
* time: 58.121915102005005
44 1.161835e+03 5.289952e+01
* time: 59.07906198501587
45 1.158453e+03 6.595387e+01
* time: 60.019392013549805
46 1.157897e+03 6.411419e+01
* time: 61.00095009803772
47 1.157870e+03 6.098509e+01
* time: 61.95716595649719
48 1.157868e+03 6.032325e+01
* time: 62.922281980514526
49 1.157864e+03 5.920812e+01
* time: 63.877477169036865
50 1.157856e+03 5.740225e+01
* time: 64.8551549911499
51 1.157832e+03 5.406795e+01
* time: 65.82625603675842
52 1.157774e+03 4.797879e+01
* time: 66.82449412345886
53 1.157628e+03 3.658732e+01
* time: 67.79346299171448
54 1.157299e+03 3.707874e+01
* time: 68.8206729888916
55 1.156690e+03 3.452947e+01
* time: 69.74993896484375
56 1.155959e+03 4.269004e+01
* time: 70.74152207374573
57 1.155557e+03 5.000322e+01
* time: 71.72419309616089
58 1.155472e+03 4.219349e+01
* time: 72.62027597427368
59 1.155463e+03 3.895049e+01
* time: 73.53994417190552
60 1.155462e+03 3.872099e+01
* time: 74.3825261592865
61 1.155458e+03 3.823491e+01
* time: 75.28940105438232
62 1.155450e+03 3.747370e+01
* time: 76.14975619316101
63 1.155426e+03 3.608648e+01
* time: 77.07763314247131
64 1.155365e+03 3.361118e+01
* time: 77.93202996253967
65 1.155207e+03 2.894678e+01
* time: 78.84740018844604
66 1.154814e+03 2.504284e+01
* time: 79.70571708679199
67 1.153910e+03 4.541126e+01
* time: 80.62178111076355
68 1.152230e+03 7.426044e+01
* time: 81.48539304733276
69 1.150286e+03 7.473044e+01
* time: 82.40299010276794
70 1.149188e+03 5.605967e+01
* time: 83.25646996498108
71 1.148944e+03 4.801123e+01
* time: 84.15591716766357
72 1.148931e+03 4.659478e+01
* time: 85.03090405464172
73 1.148929e+03 4.635180e+01
* time: 85.98858714103699
74 1.148922e+03 4.568579e+01
* time: 86.90504503250122
75 1.148907e+03 4.466128e+01
* time: 87.93957805633545
76 1.148864e+03 4.249742e+01
* time: 88.91295409202576
77 1.148758e+03 3.824840e+01
* time: 89.93128895759583
78 1.148490e+03 3.552459e+01
* time: 90.89270496368408
79 1.147881e+03 3.038181e+01
* time: 91.88046097755432
80 1.146736e+03 4.467281e+01
* time: 92.81267499923706
81 1.145347e+03 5.929158e+01
* time: 93.78445601463318
82 1.144535e+03 7.572784e+01
* time: 94.71028208732605
83 1.144354e+03 7.034986e+01
* time: 95.68195104598999
84 1.144341e+03 6.591584e+01
* time: 96.59441900253296
85 1.144340e+03 6.496353e+01
* time: 97.53505897521973
86 1.144333e+03 6.268267e+01
* time: 98.43770098686218
87 1.144320e+03 5.950121e+01
* time: 99.3805251121521
88 1.144281e+03 5.368613e+01
* time: 100.2833821773529
89 1.144186e+03 4.393956e+01
* time: 101.23476505279541
90 1.143949e+03 4.102324e+01
* time: 102.15888214111328
91 1.143423e+03 3.752005e+01
* time: 103.16964411735535
92 1.142497e+03 3.278884e+01
* time: 104.14113306999207
93 1.141489e+03 5.217814e+01
* time: 105.18705010414124
94 1.140969e+03 4.310032e+01
* time: 106.15299105644226
95 1.140872e+03 4.759545e+01
* time: 107.1604061126709
96 1.140867e+03 4.727034e+01
* time: 108.11766314506531
97 1.140866e+03 4.704365e+01
* time: 109.10540509223938
98 1.140862e+03 4.637923e+01
* time: 110.06973505020142
99 1.140853e+03 4.542582e+01
* time: 111.05847597122192
100 1.140829e+03 4.357378e+01
* time: 112.01939797401428
101 1.140767e+03 4.023060e+01
* time: 113.01356315612793
102 1.140610e+03 3.376323e+01
* time: 113.98938798904419
103 1.140237e+03 2.857028e+01
* time: 114.96671199798584
104 1.139451e+03 3.895699e+01
* time: 115.93339896202087
105 1.138223e+03 5.064036e+01
* time: 116.92458295822144
106 1.137140e+03 4.763314e+01
* time: 117.87785601615906
107 1.136763e+03 4.604968e+01
* time: 118.84796404838562
108 1.136720e+03 3.993479e+01
* time: 119.80368900299072
109 1.136717e+03 3.890565e+01
* time: 120.7070369720459
110 1.136716e+03 3.883645e+01
* time: 121.64087009429932
111 1.136712e+03 3.865517e+01
* time: 122.62205815315247
112 1.136702e+03 3.838833e+01
* time: 123.59617400169373
113 1.136674e+03 3.789981e+01
* time: 124.65812611579895
114 1.136603e+03 3.705558e+01
* time: 125.61389803886414
115 1.136418e+03 3.549940e+01
* time: 126.5742540359497
116 1.135940e+03 3.257136e+01
* time: 127.5256609916687
117 1.134743e+03 2.697248e+01
* time: 128.50862216949463
118 1.132014e+03 4.863416e+01
* time: 129.4931299686432
119 1.127203e+03 6.901202e+01
* time: 130.50301504135132
120 1.122363e+03 5.794311e+01
* time: 131.51960611343384
121 1.119759e+03 2.906766e+01
* time: 132.49630904197693
122 1.118972e+03 2.137843e+01
* time: 133.541846036911
123 1.118870e+03 1.909639e+01
* time: 134.48754811286926
124 1.118863e+03 1.887024e+01
* time: 135.48727416992188
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* time: 136.41160607337952
126 1.118861e+03 1.864783e+01
* time: 137.35923099517822
127 1.118861e+03 1.862870e+01
* time: 138.22820115089417
128 1.118860e+03 1.866000e+01
* time: 139.15695714950562
129 1.118859e+03 1.869625e+01
* time: 140.0890290737152
130 1.118856e+03 1.878717e+01
* time: 141.1207160949707
131 1.118850e+03 1.894283e+01
* time: 142.07358717918396
132 1.118834e+03 1.922180e+01
* time: 143.06053018569946
133 1.118794e+03 1.968060e+01
* time: 143.98213005065918
134 1.118693e+03 2.042353e+01
* time: 144.9784071445465
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* time: 145.90903306007385
136 1.117751e+03 2.596600e+01
* time: 146.9029619693756
137 1.116007e+03 4.716393e+01
* time: 147.8466341495514
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* time: 148.8613531589508
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* time: 149.85454106330872
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* time: 150.9897689819336
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* time: 152.04535698890686
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* time: 153.144198179245
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* time: 154.2367341518402
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* time: 180.94233918190002
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* time: 266.2224850654602
235 1.025404e+03 1.233505e+02
* time: 267.5183789730072
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* time: 282.8454451560974
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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: 291.4479660987854
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* time: 292.6988320350647
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* time: 296.31842517852783
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* time: 297.5577030181885
260 1.012497e+03 4.897808e+00
* time: 298.7741370201111
261 1.012416e+03 2.875911e+00
* time: 300.03369402885437
262 1.012389e+03 9.301211e-01
* time: 301.2374610900879
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* time: 302.47693514823914
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* time: 312.15370512008667
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* time: 313.33257508277893
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* time: 320.78011202812195
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* time: 322.14967012405396
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* time: 323.4102909564972
281 1.011809e+03 8.215973e-01
* time: 324.71212100982666
282 1.011775e+03 9.067203e-01
* time: 325.93768215179443
283 1.011774e+03 9.565073e-01
* time: 327.2270030975342
284 1.011774e+03 9.628263e-01
* time: 328.4484510421753
285 1.011774e+03 9.654767e-01
* time: 329.71515011787415
286 1.011774e+03 9.654769e-01
* time: 331.0440671443939
287 1.011774e+03 9.656366e-01
* time: 332.29603695869446
288 1.011774e+03 9.656622e-01
* time: 333.49823212623596
289 1.011774e+03 9.656624e-01
* time: 334.83915305137634
290 1.011774e+03 9.656627e-01
* time: 336.1911470890045
291 1.011774e+03 9.660395e-01
* time: 337.40075516700745
292 1.011774e+03 9.661257e-01
* time: 338.56860303878784
293 1.011774e+03 9.663115e-01
* time: 339.7686550617218
294 1.011774e+03 9.662851e-01
* time: 340.9534089565277
295 1.011773e+03 9.654773e-01
* time: 342.1841011047363
296 1.011772e+03 9.623124e-01
* time: 343.39579916000366
297 1.011769e+03 9.521889e-01
* time: 344.63091802597046
298 1.011763e+03 9.239616e-01
* time: 345.85337114334106
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* time: 347.06980299949646
300 1.011711e+03 1.257052e+00
* time: 348.2797350883484
301 1.011651e+03 1.361278e+00
* time: 349.5131859779358
302 1.011591e+03 9.076916e-01
* time: 350.72918009757996
303 1.011568e+03 3.084752e-01
* time: 351.9666311740875
304 1.011565e+03 2.946404e-01
* time: 353.18393301963806
305 1.011565e+03 2.821273e-01
* time: 354.4147000312805
306 1.011565e+03 2.766260e-01
* time: 355.6049060821533
307 1.011565e+03 2.778010e-01
* time: 356.8090469837189
308 1.011565e+03 2.830281e-01
* time: 358.00295400619507
309 1.011565e+03 2.842619e-01
* time: 359.26140904426575
310 1.011565e+03 2.861093e-01
* time: 360.4436321258545
311 1.011565e+03 2.861138e-01
* time: 361.67281103134155
312 1.011565e+03 2.867641e-01
* time: 362.8731110095978
313 1.011565e+03 2.867756e-01
* time: 364.15695214271545
314 1.011565e+03 2.884589e-01
* time: 365.3516011238098
315 1.011565e+03 2.897335e-01
* time: 366.55631017684937
316 1.011565e+03 2.927744e-01
* time: 367.7528030872345
317 1.011565e+03 2.967424e-01
* time: 368.9650890827179
318 1.011565e+03 3.027347e-01
* time: 370.16223907470703
319 1.011564e+03 3.099827e-01
* time: 371.3749530315399
320 1.011563e+03 3.155826e-01
* time: 372.5697560310364
321 1.011560e+03 3.082844e-01
* time: 373.79016399383545
322 1.011555e+03 2.625121e-01
* time: 375.00247716903687
323 1.011548e+03 2.136271e-01
* time: 376.2607719898224
324 1.011542e+03 1.165033e-01
* time: 377.5011751651764
325 1.011538e+03 3.914224e-02
* time: 378.72695803642273
326 1.011537e+03 3.083888e-02
* time: 379.9249851703644
327 1.011536e+03 2.766364e-02
* time: 381.10187315940857
328 1.011536e+03 1.873262e-02
* time: 382.3302810192108
329 1.011536e+03 5.453951e-03
* time: 383.66944313049316
330 1.011536e+03 4.882951e-03
* time: 384.9326410293579
331 1.011535e+03 4.331879e-03
* time: 386.2310781478882
332 1.011535e+03 2.711268e-03
* time: 387.5538351535797
333 1.011535e+03 4.474256e-04
* time: 388.8774881362915FittedPumasModel
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: 7.104873657226562e-5
1 4.804005e+05 8.280311e+05
* time: 7.303060054779053
2 3.595691e+05 6.035842e+05
* time: 14.217710018157959
3 1.691071e+05 2.702705e+05
* time: 21.381021976470947
4 9.193152e+04 1.451040e+05
* time: 29.074352025985718
5 4.768538e+04 6.630202e+04
* time: 36.77451300621033
6 2.904205e+04 3.416919e+04
* time: 44.24269914627075
7 1.840152e+04 1.671742e+04
* time: 51.49464511871338
8 1.280989e+04 1.022268e+04
* time: 58.654086112976074
9 9.713285e+03 9.298485e+03
* time: 65.77444505691528
10 7.628127e+03 8.330656e+03
* time: 72.92816305160522
11 6.236278e+03 7.340367e+03
* time: 80.02186298370361
12 5.271958e+03 6.303773e+03
* time: 86.78265404701233
13 4.411047e+03 4.994130e+03
* time: 93.56431102752686
14 3.732935e+03 3.538183e+03
* time: 100.52936005592346
15 3.399272e+03 2.450045e+03
* time: 107.21587204933167
16 3.302987e+03 1.848475e+03
* time: 113.93394017219543
17 3.291788e+03 1.642864e+03
* time: 120.67175102233887
18 3.290743e+03 1.594765e+03
* time: 127.43211913108826
19 3.289686e+03 1.562640e+03
* time: 134.16877102851868
20 3.286186e+03 1.496632e+03
* time: 141.11416101455688
21 3.277917e+03 1.402865e+03
* time: 148.06146097183228
22 3.255958e+03 1.254296e+03
* time: 154.90790915489197
23 3.200979e+03 1.038621e+03
* time: 161.56987810134888
24 3.063520e+03 7.403602e+02
* time: 168.30342602729797
25 2.730315e+03 3.768701e+02
* time: 175.12595105171204
26 1.934186e+03 2.246121e+02
* time: 181.83757519721985
27 1.704399e+03 2.219826e+02
* time: 204.22278904914856
28 1.471184e+03 2.025825e+02
* time: 239.96606302261353
29 1.316789e+03 1.554715e+02
* time: 270.3780381679535
30 1.226363e+03 1.775721e+02
* time: 277.61546897888184
31 1.206626e+03 2.781030e+02
* time: 284.5865089893341
32 1.197830e+03 2.430603e+02
* time: 291.60668301582336
33 1.190342e+03 2.327966e+02
* time: 298.6062650680542
34 1.178784e+03 2.105878e+02
* time: 306.0143361091614
35 1.177853e+03 2.228653e+02
* time: 313.1887249946594
36 1.177764e+03 2.256182e+02
* time: 320.2303321361542
37 1.177746e+03 2.259425e+02
* time: 327.1361041069031
38 1.177670e+03 2.264699e+02
* time: 334.7166361808777
39 1.177500e+03 2.264641e+02
* time: 341.1346380710602
40 1.177034e+03 2.244207e+02
* time: 347.4957320690155
41 1.175895e+03 2.160307e+02
* time: 353.8993980884552
42 1.173241e+03 1.906017e+02
* time: 360.94166111946106
43 1.168190e+03 1.332917e+02
* time: 367.89776396751404
44 1.161836e+03 5.291022e+01
* time: 375.2413260936737
45 1.158453e+03 6.595471e+01
* time: 382.94063115119934
46 1.157897e+03 6.411640e+01
* time: 389.74897599220276
47 1.157870e+03 6.098679e+01
* time: 396.79437017440796
48 1.157868e+03 6.032481e+01
* time: 403.460196018219
49 1.157864e+03 5.920991e+01
* time: 410.1040461063385
50 1.157856e+03 5.740417e+01
* time: 416.6641221046448
51 1.157832e+03 5.407031e+01
* time: 423.2501881122589
52 1.157774e+03 4.798188e+01
* time: 429.78075098991394
53 1.157628e+03 3.659183e+01
* time: 436.37866497039795
54 1.157299e+03 3.707862e+01
* time: 443.1555600166321
55 1.156690e+03 3.453183e+01
* time: 450.6136510372162
56 1.155959e+03 4.268473e+01
* time: 457.9204480648041
57 1.155557e+03 5.000309e+01
* time: 464.72136306762695
58 1.155472e+03 4.219411e+01
* time: 471.3585751056671
59 1.155463e+03 3.894982e+01
* time: 478.0008611679077
60 1.155462e+03 3.872028e+01
* time: 484.5322630405426
61 1.155458e+03 3.823435e+01
* time: 491.1562731266022
62 1.155450e+03 3.747328e+01
* time: 497.70220708847046
63 1.155426e+03 3.608641e+01
* time: 504.95913219451904
64 1.155365e+03 3.361168e+01
* time: 512.3541111946106
65 1.155207e+03 2.894844e+01
* time: 519.2790191173553
66 1.154814e+03 2.504332e+01
* time: 526.0067319869995
67 1.153910e+03 4.539953e+01
* time: 532.7058510780334
68 1.152231e+03 7.425240e+01
* time: 539.4968090057373
69 1.150287e+03 7.473669e+01
* time: 546.1285421848297
70 1.149189e+03 5.606074e+01
* time: 552.926705121994
71 1.148944e+03 4.801310e+01
* time: 559.5189211368561
72 1.148932e+03 4.659470e+01
* time: 566.1089010238647
73 1.148930e+03 4.635167e+01
* time: 573.6650211811066
74 1.148923e+03 4.568576e+01
* time: 580.5513381958008
75 1.148907e+03 4.466140e+01
* time: 587.5642330646515
76 1.148865e+03 4.249800e+01
* time: 594.3189351558685
77 1.148758e+03 3.824645e+01
* time: 600.9238021373749
78 1.148491e+03 3.552369e+01
* time: 607.6346809864044
79 1.147881e+03 3.037355e+01
* time: 614.3372070789337
80 1.146737e+03 4.466673e+01
* time: 621.1490960121155
81 1.145347e+03 5.927932e+01
* time: 627.6799130439758
82 1.144536e+03 7.572710e+01
* time: 634.1699709892273
83 1.144354e+03 7.035223e+01
* time: 640.6072640419006
84 1.144341e+03 6.591571e+01
* time: 647.463709115982
85 1.144340e+03 6.496308e+01
* time: 653.9703011512756
86 1.144333e+03 6.268335e+01
* time: 660.3270909786224
87 1.144320e+03 5.950296e+01
* time: 666.7742261886597
88 1.144281e+03 5.369030e+01
* time: 673.2548730373383
89 1.144186e+03 4.394749e+01
* time: 679.7151501178741
90 1.143949e+03 4.102392e+01
* time: 686.1513340473175
91 1.143423e+03 3.752367e+01
* time: 692.662024974823
92 1.142498e+03 3.276758e+01
* time: 699.1454911231995
93 1.141490e+03 5.217551e+01
* time: 705.8304641246796
94 1.140969e+03 4.309381e+01
* time: 712.3685011863708
95 1.140872e+03 4.759588e+01
* time: 718.7622311115265
96 1.140867e+03 4.727133e+01
* time: 725.1189630031586
97 1.140866e+03 4.704459e+01
* time: 731.4678189754486
98 1.140862e+03 4.638036e+01
* time: 737.9518091678619
99 1.140853e+03 4.542717e+01
* time: 744.2672820091248
100 1.140829e+03 4.357571e+01
* time: 750.5742151737213
101 1.140767e+03 4.023368e+01
* time: 757.0588309764862
102 1.140611e+03 3.376885e+01
* time: 763.4485530853271
103 1.140237e+03 2.856928e+01
* time: 770.2036921977997
104 1.139452e+03 3.894107e+01
* time: 776.9315881729126
105 1.138224e+03 5.063688e+01
* time: 783.5213670730591
106 1.137140e+03 4.762519e+01
* time: 789.8390140533447
107 1.136763e+03 4.605051e+01
* time: 796.2554280757904
108 1.136720e+03 3.993288e+01
* time: 802.7939031124115
109 1.136717e+03 3.890566e+01
* time: 809.0897011756897
110 1.136716e+03 3.883647e+01
* time: 815.5727360248566
111 1.136712e+03 3.865523e+01
* time: 822.0876181125641
112 1.136702e+03 3.838845e+01
* time: 828.6337280273438
113 1.136674e+03 3.790005e+01
* time: 835.3996031284332
114 1.136604e+03 3.705604e+01
* time: 841.7789499759674
115 1.136418e+03 3.550032e+01
* time: 848.1886880397797
116 1.135940e+03 3.257323e+01
* time: 854.5779731273651
117 1.134744e+03 2.697619e+01
* time: 861.1727201938629
118 1.132016e+03 4.861352e+01
* time: 868.3113050460815
119 1.127206e+03 6.900228e+01
* time: 875.400288105011
120 1.122365e+03 5.795238e+01
* time: 882.0887541770935
121 1.119760e+03 2.908163e+01
* time: 888.8622992038727
122 1.118972e+03 2.137897e+01
* time: 895.5849189758301
123 1.118870e+03 1.909827e+01
* time: 902.4367489814758
124 1.118863e+03 1.886936e+01
* time: 909.1608180999756
125 1.118862e+03 1.852746e+01
* time: 916.236958026886
126 1.118861e+03 1.864804e+01
* time: 923.0221049785614
127 1.118861e+03 1.862831e+01
* time: 929.771320104599
128 1.118860e+03 1.865947e+01
* time: 936.0619740486145
129 1.118859e+03 1.869546e+01
* time: 942.5354809761047
130 1.118856e+03 1.878621e+01
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* time: 1691.2276241779327
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* time: 1698.450073003769
235 1.026628e+03 7.187426e+01
* time: 1705.7756099700928
236 1.024510e+03 1.065337e+02
* time: 1712.837678194046
237 1.020621e+03 1.329824e+02
* time: 1720.2242259979248
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* time: 1727.6427659988403
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* time: 1742.156051158905
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* time: 1749.6614170074463
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* time: 1756.8300030231476
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* time: 1764.004240989685
244 1.012698e+03 1.107778e+00
* time: 1771.012379169464
245 1.012697e+03 9.968755e-01
* time: 1777.8426251411438
246 1.012697e+03 9.970599e-01
* time: 1784.708956003189
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* time: 1791.584792137146
248 1.012697e+03 9.972169e-01
* time: 1798.8590910434723
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* time: 1806.9139811992645
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* time: 1814.312350988388
251 1.012697e+03 9.973078e-01
* time: 1822.1184921264648
252 1.012697e+03 9.972048e-01
* time: 1830.1026060581207
253 1.012697e+03 9.970536e-01
* time: 1836.8492109775543
254 1.012697e+03 9.967402e-01
* time: 1843.6247491836548
255 1.012697e+03 9.961565e-01
* time: 1850.383197069168
256 1.012696e+03 9.949478e-01
* time: 1857.0548510551453
257 1.012694e+03 1.312115e+00
* time: 1864.1534781455994
258 1.012690e+03 2.118030e+00
* time: 1871.3820061683655
259 1.012679e+03 3.352490e+00
* time: 1878.3617889881134
260 1.012651e+03 5.076526e+00
* time: 1885.2500751018524
261 1.012594e+03 6.891495e+00
* time: 1892.237053155899
262 1.012501e+03 7.239074e+00
* time: 1899.1164109706879
263 1.012418e+03 4.479611e+00
* time: 1905.7245399951935
264 1.012390e+03 1.267149e+00
* time: 1912.3669211864471
265 1.012386e+03 8.840304e-01
* time: 1918.8868100643158
266 1.012386e+03 8.549097e-01
* time: 1925.4801540374756
267 1.012386e+03 8.516869e-01
* time: 1932.0459461212158
268 1.012386e+03 8.516869e-01
* time: 1939.9808990955353
269 1.012386e+03 8.516869e-01
* time: 1948.0971901416779
270 1.012386e+03 8.516817e-01
* time: 1955.1287310123444
271 1.012386e+03 8.516817e-01
* time: 1963.1566500663757
272 1.012386e+03 8.516817e-01
* time: 1971.608798980713
273 1.012386e+03 8.516817e-01
* time: 1979.764573097229
274 1.012386e+03 8.516817e-01
* time: 1988.4256429672241
275 1.012386e+03 8.516817e-01
* time: 1997.481558084488FittedPumasModel
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: 7.390975952148438e-5
1 4.804005e+05 8.280312e+05
* time: 1.1732280254364014
2 3.595691e+05 6.035843e+05
* time: 2.242790937423706
3 1.691071e+05 2.702705e+05
* time: 3.2214529514312744
4 9.193152e+04 1.451040e+05
* time: 4.302997827529907
5 4.768539e+04 6.630202e+04
* time: 5.33185887336731
6 2.904205e+04 3.416919e+04
* time: 6.380807876586914
7 1.840153e+04 1.671742e+04
* time: 7.409325838088989
8 1.280989e+04 1.022268e+04
* time: 8.47291088104248
9 9.713287e+03 9.298485e+03
* time: 9.523818016052246
10 7.628127e+03 8.330656e+03
* time: 10.59914493560791
11 6.236278e+03 7.340367e+03
* time: 11.592341899871826
12 5.271958e+03 6.303773e+03
* time: 12.615149021148682
13 4.411047e+03 4.994130e+03
* time: 13.591423988342285
14 3.732936e+03 3.538184e+03
* time: 14.60710096359253
15 3.399273e+03 2.450045e+03
* time: 15.556988000869751
16 3.302987e+03 1.848476e+03
* time: 16.566789865493774
17 3.291788e+03 1.642864e+03
* time: 17.50917100906372
18 3.290743e+03 1.594765e+03
* time: 18.467705011367798
19 3.289687e+03 1.562641e+03
* time: 19.45353388786316
20 3.286186e+03 1.496632e+03
* time: 20.416968822479248
21 3.277917e+03 1.402865e+03
* time: 21.426614999771118
22 3.255959e+03 1.254297e+03
* time: 22.38623595237732
23 3.200980e+03 1.038621e+03
* time: 23.38558793067932
24 3.063521e+03 7.403605e+02
* time: 24.32529592514038
25 2.730316e+03 3.768701e+02
* time: 25.304359912872314
26 1.934186e+03 2.246121e+02
* time: 26.17434787750244
27 1.704399e+03 2.219826e+02
* time: 29.862889051437378
28 1.471183e+03 2.025826e+02
* time: 36.26602292060852
29 1.316789e+03 1.554714e+02
* time: 41.7195029258728
30 1.226363e+03 1.775666e+02
* time: 42.72805595397949
31 1.206626e+03 2.780980e+02
* time: 43.64588189125061
32 1.197830e+03 2.430550e+02
* time: 44.60907483100891
33 1.190342e+03 2.327905e+02
* time: 45.48642182350159
34 1.178783e+03 2.105809e+02
* time: 46.38862586021423
35 1.177853e+03 2.228585e+02
* time: 47.291908979415894
36 1.177763e+03 2.256116e+02
* time: 48.21061396598816
37 1.177745e+03 2.259359e+02
* time: 49.10913801193237
38 1.177670e+03 2.264633e+02
* time: 49.96467995643616
39 1.177500e+03 2.264574e+02
* time: 50.85685586929321
40 1.177033e+03 2.244136e+02
* time: 51.71105885505676
41 1.175894e+03 2.160226e+02
* time: 52.63304901123047
42 1.173240e+03 1.905912e+02
* time: 53.53833293914795
43 1.168189e+03 1.332783e+02
* time: 54.453322887420654
44 1.161835e+03 5.289952e+01
* time: 55.37571382522583
45 1.158453e+03 6.595387e+01
* time: 56.261674880981445
46 1.157897e+03 6.411419e+01
* time: 57.15953087806702
47 1.157870e+03 6.098509e+01
* time: 58.00574994087219
48 1.157868e+03 6.032325e+01
* time: 58.88358688354492
49 1.157864e+03 5.920812e+01
* time: 59.71583390235901
50 1.157856e+03 5.740225e+01
* time: 60.59941482543945
51 1.157832e+03 5.406795e+01
* time: 61.42774987220764
52 1.157774e+03 4.797879e+01
* time: 62.27473783493042
53 1.157628e+03 3.658732e+01
* time: 63.1357479095459
54 1.157299e+03 3.707874e+01
* time: 64.0286648273468
55 1.156690e+03 3.452947e+01
* time: 64.96577405929565
56 1.155959e+03 4.269004e+01
* time: 65.83128595352173
57 1.155557e+03 5.000322e+01
* time: 66.71075582504272
58 1.155472e+03 4.219349e+01
* time: 67.55699896812439
59 1.155463e+03 3.895049e+01
* time: 68.45632982254028
60 1.155462e+03 3.872099e+01
* time: 69.33380699157715
61 1.155458e+03 3.823491e+01
* time: 70.20407795906067
62 1.155450e+03 3.747370e+01
* time: 71.04396486282349
63 1.155426e+03 3.608648e+01
* time: 71.85524201393127
64 1.155365e+03 3.361118e+01
* time: 72.70459198951721
65 1.155207e+03 2.894678e+01
* time: 73.51063299179077
66 1.154814e+03 2.504284e+01
* time: 74.36684894561768
67 1.153910e+03 4.541126e+01
* time: 75.16901302337646
68 1.152230e+03 7.426044e+01
* time: 76.01698684692383
69 1.150286e+03 7.473044e+01
* time: 76.80135202407837
70 1.149188e+03 5.605967e+01
* time: 77.60483288764954
71 1.148944e+03 4.801123e+01
* time: 78.43142199516296
72 1.148931e+03 4.659478e+01
* time: 79.23344993591309
73 1.148929e+03 4.635180e+01
* time: 80.04715895652771
74 1.148922e+03 4.568579e+01
* time: 80.85802984237671
75 1.148907e+03 4.466128e+01
* time: 81.74173402786255
76 1.148864e+03 4.249742e+01
* time: 82.57768082618713
77 1.148758e+03 3.824840e+01
* time: 83.47902488708496
78 1.148490e+03 3.552459e+01
* time: 84.30825805664062
79 1.147881e+03 3.038181e+01
* time: 85.15199685096741
80 1.146736e+03 4.467281e+01
* time: 86.00162100791931
81 1.145347e+03 5.929158e+01
* time: 86.99901294708252
82 1.144535e+03 7.572784e+01
* time: 87.90515089035034
83 1.144354e+03 7.034986e+01
* time: 88.71777105331421
84 1.144341e+03 6.591584e+01
* time: 89.54224705696106
85 1.144340e+03 6.496353e+01
* time: 90.31032490730286
86 1.144333e+03 6.268267e+01
* time: 91.14065384864807
87 1.144320e+03 5.950121e+01
* time: 91.8992760181427
88 1.144281e+03 5.368613e+01
* time: 92.69937896728516
89 1.144186e+03 4.393956e+01
* time: 93.55333805084229
90 1.143949e+03 4.102324e+01
* time: 94.36425495147705
91 1.143423e+03 3.752005e+01
* time: 95.21223902702332
92 1.142497e+03 3.278884e+01
* time: 96.01257395744324
93 1.141489e+03 5.217814e+01
* time: 96.90281391143799
94 1.140969e+03 4.310032e+01
* time: 97.7705409526825
95 1.140872e+03 4.759545e+01
* time: 98.61550903320312
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* time: 354.52402782440186FittedPumasModel
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.