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: 0.036778926849365234
1 4.816669e+05 8.316173e+05
* time: 2.5890910625457764
2 3.598373e+05 6.060150e+05
* time: 4.041985988616943
3 1.682124e+05 2.709072e+05
* time: 5.025707960128784
4 9.168914e+04 1.446832e+05
* time: 6.04728102684021
5 4.791138e+04 6.671463e+04
* time: 6.980186939239502
6 2.909104e+04 3.426663e+04
* time: 7.950915098190308
7 1.843737e+04 1.678953e+04
* time: 8.913994073867798
8 1.282447e+04 1.021463e+04
* time: 9.825294971466064
9 9.723361e+03 9.291650e+03
* time: 10.753484010696411
10 7.635613e+03 8.325311e+03
* time: 11.570909023284912
11 6.243129e+03 7.337397e+03
* time: 12.43933892250061
12 5.279957e+03 6.304542e+03
* time: 13.235424041748047
13 4.422123e+03 5.001962e+03
* time: 14.043703079223633
14 3.744748e+03 3.550400e+03
* time: 14.780565023422241
15 3.409674e+03 2.461547e+03
* time: 15.548466920852661
16 3.312049e+03 1.856430e+03
* time: 16.269973039627075
17 3.300559e+03 1.648439e+03
* time: 17.003747940063477
18 3.299489e+03 1.599460e+03
* time: 17.72153091430664
19 3.298436e+03 1.567520e+03
* time: 18.457744121551514
20 3.294921e+03 1.501363e+03
* time: 19.170675039291382
21 3.286641e+03 1.407664e+03
* time: 19.913458108901978
22 3.264637e+03 1.259077e+03
* time: 20.630484104156494
23 3.209594e+03 1.043534e+03
* time: 21.32778000831604
24 3.072117e+03 7.454677e+02
* time: 22.049246072769165
25 2.739794e+03 3.818063e+02
* time: 22.678839921951294
26 1.949883e+03 2.246644e+02
* time: 23.424750089645386
27 1.722434e+03 2.224562e+02
* time: 25.341300010681152
28 1.479355e+03 2.042452e+02
* time: 27.792783975601196
29 1.310583e+03 1.531868e+02
* time: 30.72933006286621
30 1.224675e+03 1.756826e+02
* time: 31.43550205230713
31 1.204841e+03 2.738440e+02
* time: 31.999047994613647
32 1.196432e+03 2.403217e+02
* time: 32.6150860786438
33 1.189131e+03 2.301186e+02
* time: 33.17582893371582
34 1.177881e+03 2.083030e+02
* time: 33.80488991737366
35 1.176989e+03 2.201993e+02
* time: 34.38284206390381
36 1.176899e+03 2.230826e+02
* time: 35.01331090927124
37 1.176882e+03 2.233531e+02
* time: 35.57376194000244
38 1.176795e+03 2.238567e+02
* time: 36.187248945236206
39 1.176609e+03 2.236914e+02
* time: 36.76012706756592
40 1.176090e+03 2.210941e+02
* time: 37.37300491333008
41 1.174841e+03 2.112741e+02
* time: 37.949657917022705
42 1.171973e+03 1.825298e+02
* time: 38.557745933532715
43 1.166737e+03 1.210559e+02
* time: 39.13270711898804
44 1.160698e+03 4.450697e+01
* time: 39.746416091918945
45 1.157926e+03 6.411275e+01
* time: 40.31949710845947
46 1.157537e+03 6.121762e+01
* time: 40.92166495323181
47 1.157520e+03 5.867223e+01
* time: 41.48825192451477
48 1.157519e+03 5.814523e+01
* time: 42.06284809112549
49 1.157515e+03 5.686007e+01
* time: 42.61605906486511
50 1.157505e+03 5.497418e+01
* time: 43.20294690132141
51 1.157477e+03 5.129724e+01
* time: 43.77531695365906
52 1.157408e+03 4.463171e+01
* time: 44.34547996520996
53 1.157239e+03 3.279117e+01
* time: 44.89983606338501
54 1.156869e+03 3.681526e+01
* time: 45.47150707244873
55 1.156230e+03 3.198754e+01
* time: 46.036349058151245
56 1.155563e+03 4.576438e+01
* time: 46.61383891105652
57 1.155264e+03 4.832159e+01
* time: 47.16887903213501
58 1.155212e+03 4.052175e+01
* time: 47.7317750453949
59 1.155207e+03 3.797125e+01
* time: 48.277997970581055
60 1.155206e+03 3.778452e+01
* time: 48.83329105377197
61 1.155201e+03 3.723003e+01
* time: 49.383882999420166
62 1.155190e+03 3.642375e+01
* time: 49.94686007499695
63 1.155159e+03 3.487535e+01
* time: 50.49422812461853
64 1.155079e+03 3.209060e+01
* time: 51.052669048309326
65 1.154875e+03 2.670227e+01
* time: 51.645885944366455
66 1.154375e+03 2.366165e+01
* time: 52.18662095069885
67 1.153278e+03 5.499525e+01
* time: 52.78028607368469
68 1.151457e+03 7.903833e+01
* time: 53.30179810523987
69 1.149737e+03 6.771227e+01
* time: 53.874305963516235
70 1.148983e+03 5.387708e+01
* time: 54.38840293884277
71 1.148872e+03 4.683950e+01
* time: 54.95218896865845
72 1.148868e+03 4.512036e+01
* time: 55.458534955978394
73 1.148866e+03 4.489706e+01
* time: 56.01118612289429
74 1.148857e+03 4.426314e+01
* time: 56.51303195953369
75 1.148838e+03 4.321161e+01
* time: 57.05819892883301
76 1.148785e+03 4.095232e+01
* time: 57.55684995651245
77 1.148652e+03 3.780572e+01
* time: 58.104042053222656
78 1.148325e+03 3.474239e+01
* time: 58.60068106651306
79 1.147599e+03 3.678957e+01
* time: 59.152535915374756
80 1.146330e+03 5.134277e+01
* time: 59.66378998756409
81 1.144979e+03 6.444063e+01
* time: 60.2073540687561
82 1.144325e+03 7.574751e+01
* time: 60.712568044662476
83 1.144210e+03 6.968296e+01
* time: 61.25182795524597
84 1.144204e+03 6.648234e+01
* time: 61.75245213508606
85 1.144202e+03 6.560999e+01
* time: 62.28234505653381
86 1.144194e+03 6.305667e+01
* time: 62.772743940353394
87 1.144176e+03 5.956959e+01
* time: 63.307044982910156
88 1.144127e+03 5.307828e+01
* time: 63.811546087265015
89 1.144005e+03 4.218776e+01
* time: 64.3437271118164
90 1.143708e+03 4.120261e+01
* time: 64.86069798469543
91 1.143074e+03 3.675516e+01
* time: 65.39756393432617
92 1.142052e+03 3.855439e+01
* time: 65.91438698768616
93 1.141101e+03 5.193197e+01
* time: 66.44793796539307
94 1.140701e+03 4.606858e+01
* time: 66.9569981098175
95 1.140646e+03 4.857736e+01
* time: 67.49195313453674
96 1.140643e+03 4.817596e+01
* time: 68.00139594078064
97 1.140642e+03 4.793197e+01
* time: 68.52508807182312
98 1.140637e+03 4.723245e+01
* time: 69.03045296669006
99 1.140626e+03 4.619695e+01
* time: 69.55674290657043
100 1.140597e+03 4.418761e+01
* time: 70.0640001296997
101 1.140523e+03 4.050986e+01
* time: 70.58925604820251
102 1.140334e+03 3.335334e+01
* time: 71.10701894760132
103 1.139892e+03 2.982120e+01
* time: 71.63840007781982
104 1.138994e+03 4.360237e+01
* time: 72.15890002250671
105 1.137687e+03 5.246892e+01
* time: 72.6933479309082
106 1.136679e+03 4.825586e+01
* time: 73.21530795097351
107 1.136391e+03 4.467499e+01
* time: 73.74992108345032
108 1.136364e+03 3.961591e+01
* time: 74.26277899742126
109 1.136362e+03 3.951402e+01
* time: 74.78474307060242
110 1.136360e+03 3.943358e+01
* time: 75.32679295539856
111 1.136355e+03 3.923965e+01
* time: 75.80602192878723
112 1.136344e+03 3.894674e+01
* time: 76.3524649143219
113 1.136314e+03 3.841518e+01
* time: 76.83977103233337
114 1.136235e+03 3.748824e+01
* time: 77.39240407943726
115 1.136029e+03 3.577210e+01
* time: 77.884202003479
116 1.135499e+03 3.252332e+01
* time: 78.42985892295837
117 1.134182e+03 2.630844e+01
* time: 78.94658613204956
118 1.131238e+03 4.869163e+01
* time: 79.58505606651306
119 1.126345e+03 6.537856e+01
* time: 80.15365791320801
120 1.121879e+03 5.123373e+01
* time: 80.78502798080444
121 1.119645e+03 2.457316e+01
* time: 81.33928608894348
122 1.119031e+03 2.124721e+01
* time: 81.93651103973389
123 1.118958e+03 1.857508e+01
* time: 82.49152398109436
124 1.118953e+03 1.896595e+01
* time: 83.08413195610046
125 1.118952e+03 1.849021e+01
* time: 83.62071204185486
126 1.118952e+03 1.852941e+01
* time: 84.21458005905151
127 1.118951e+03 1.856812e+01
* time: 84.78116011619568
128 1.118951e+03 1.862403e+01
* time: 85.3782889842987
129 1.118949e+03 1.871484e+01
* time: 85.92496800422668
130 1.118946e+03 1.886332e+01
* time: 86.5058970451355
131 1.118939e+03 1.910217e+01
* time: 87.04735898971558
132 1.118922e+03 1.948050e+01
* time: 87.61697292327881
133 1.118879e+03 2.006903e+01
* time: 88.15827798843384
134 1.118769e+03 2.097106e+01
* time: 88.73619198799133
135 1.118486e+03 2.231801e+01
* time: 89.28660297393799
136 1.117755e+03 3.049039e+01
* time: 89.877769947052
137 1.115883e+03 5.345487e+01
* time: 90.44304895401001
138 1.111258e+03 8.489274e+01
* time: 91.02758312225342
139 1.101149e+03 7.643749e+01
* time: 91.59094595909119
140 1.090612e+03 5.802659e+01
* time: 92.2185070514679
141 1.087971e+03 5.776554e+01
* time: 92.82703804969788
142 1.087205e+03 1.148234e+02
* time: 93.4706621170044
143 1.085558e+03 4.724872e+01
* time: 94.12859010696411
144 1.085271e+03 2.018848e+01
* time: 94.77615809440613
145 1.085124e+03 2.004201e+01
* time: 95.44430208206177
146 1.085089e+03 2.025616e+01
* time: 96.04406809806824
147 1.085081e+03 2.049682e+01
* time: 96.6955509185791
148 1.085073e+03 2.086075e+01
* time: 97.31198000907898
149 1.085070e+03 2.095706e+01
* time: 97.95375108718872
150 1.085063e+03 2.094978e+01
* time: 98.54541397094727
151 1.085062e+03 2.093855e+01
* time: 99.22301006317139
152 1.085062e+03 2.093439e+01
* time: 99.9742591381073
153 1.085062e+03 2.093335e+01
* time: 100.71732807159424
154 1.085062e+03 2.093252e+01
* time: 101.56392908096313
155 1.085062e+03 2.093227e+01
* time: 102.41550993919373
156 1.085062e+03 2.093208e+01
* time: 103.22258806228638
157 1.085062e+03 2.093187e+01
* time: 104.08704996109009
158 1.085062e+03 2.093181e+01
* time: 104.99790096282959
159 1.085062e+03 2.093175e+01
* time: 105.91559791564941
160 1.085062e+03 2.093172e+01
* time: 106.81795907020569
161 1.085062e+03 2.046653e+01
* time: 107.48654913902283
162 1.085061e+03 2.045920e+01
* time: 108.22480893135071
163 1.085056e+03 2.021786e+01
* time: 108.81096601486206
164 1.085055e+03 2.018381e+01
* time: 109.50923705101013
165 1.085043e+03 1.997498e+01
* time: 110.09617209434509
166 1.085029e+03 1.961695e+01
* time: 110.73259806632996
167 1.084997e+03 1.929634e+01
* time: 111.3284900188446
168 1.084881e+03 1.855033e+01
* time: 111.95556402206421
169 1.084633e+03 1.799616e+01
* time: 112.55737090110779
170 1.083831e+03 1.856893e+01
* time: 113.18822407722473
171 1.081240e+03 2.069096e+01
* time: 113.8402681350708
172 1.072465e+03 3.262436e+01
* time: 114.65537810325623
173 1.072446e+03 6.686587e+01
* time: 115.4879469871521
174 1.068493e+03 7.417042e+01
* time: 116.35348296165466
175 1.067469e+03 7.109936e+01
* time: 117.29630708694458
176 1.065854e+03 2.774005e+01
* time: 118.06751894950867
177 1.065215e+03 1.328875e+01
* time: 118.86322808265686
178 1.065128e+03 1.390400e+01
* time: 119.61015701293945
179 1.065120e+03 1.383172e+01
* time: 120.376699924469
180 1.065120e+03 1.384995e+01
* time: 121.12983798980713FittedPumasModel
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: 0.00011706352233886719
1 4.804005e+05 8.280312e+05
* time: 1.5192348957061768
2 3.595691e+05 6.035843e+05
* time: 2.83803391456604
3 1.691071e+05 2.702705e+05
* time: 4.2914719581604
4 9.193152e+04 1.451040e+05
* time: 5.5921080112457275
5 4.768539e+04 6.630202e+04
* time: 6.934375047683716
6 2.904205e+04 3.416919e+04
* time: 8.278393983840942
7 1.840153e+04 1.671742e+04
* time: 9.582650899887085
8 1.280989e+04 1.022268e+04
* time: 10.996672868728638
9 9.713287e+03 9.298485e+03
* time: 12.30501103401184
10 7.628127e+03 8.330656e+03
* time: 13.64023208618164
11 6.236278e+03 7.340367e+03
* time: 14.880173921585083
12 5.271958e+03 6.303773e+03
* time: 16.160606861114502
13 4.411047e+03 4.994130e+03
* time: 17.385520935058594
14 3.732936e+03 3.538184e+03
* time: 18.640158891677856
15 3.399273e+03 2.450045e+03
* time: 19.945761919021606
16 3.302987e+03 1.848476e+03
* time: 21.230284929275513
17 3.291788e+03 1.642864e+03
* time: 22.455607891082764
18 3.290743e+03 1.594765e+03
* time: 23.669301986694336
19 3.289687e+03 1.562641e+03
* time: 24.95719003677368
20 3.286186e+03 1.496632e+03
* time: 26.265692949295044
21 3.277917e+03 1.402865e+03
* time: 27.66506791114807
22 3.255959e+03 1.254297e+03
* time: 28.98529291152954
23 3.200980e+03 1.038621e+03
* time: 30.30991506576538
24 3.063521e+03 7.403605e+02
* time: 31.462276935577393
25 2.730316e+03 3.768701e+02
* time: 32.64803194999695
26 1.934186e+03 2.246121e+02
* time: 33.711530923843384
27 1.704399e+03 2.219826e+02
* time: 38.09420990943909
28 1.471183e+03 2.025826e+02
* time: 45.78375005722046
29 1.316789e+03 1.554714e+02
* time: 52.24638295173645
30 1.226363e+03 1.775666e+02
* time: 53.75022101402283
31 1.206626e+03 2.780980e+02
* time: 55.33333396911621
32 1.197830e+03 2.430550e+02
* time: 59.57191705703735
33 1.190342e+03 2.327905e+02
* time: 62.08125901222229
34 1.178783e+03 2.105809e+02
* time: 64.50521302223206
35 1.177853e+03 2.228585e+02
* time: 66.45746088027954
36 1.177763e+03 2.256116e+02
* time: 68.1512839794159
37 1.177745e+03 2.259359e+02
* time: 69.66983103752136
38 1.177670e+03 2.264633e+02
* time: 71.21446800231934
39 1.177500e+03 2.264574e+02
* time: 72.701730966568
40 1.177033e+03 2.244136e+02
* time: 74.37466597557068
41 1.175894e+03 2.160226e+02
* time: 75.89445996284485
42 1.173240e+03 1.905912e+02
* time: 77.49400186538696
43 1.168189e+03 1.332783e+02
* time: 78.92913603782654
44 1.161835e+03 5.289952e+01
* time: 80.445631980896
45 1.158453e+03 6.595387e+01
* time: 81.81252598762512
46 1.157897e+03 6.411419e+01
* time: 83.19129300117493
47 1.157870e+03 6.098509e+01
* time: 84.42896509170532
48 1.157868e+03 6.032325e+01
* time: 85.76575899124146
49 1.157864e+03 5.920812e+01
* time: 86.92808103561401
50 1.157856e+03 5.740225e+01
* time: 88.0699999332428
51 1.157832e+03 5.406795e+01
* time: 89.14641809463501
52 1.157774e+03 4.797879e+01
* time: 90.25668597221375
53 1.157628e+03 3.658732e+01
* time: 91.34145092964172
54 1.157299e+03 3.707874e+01
* time: 92.45667600631714
55 1.156690e+03 3.452947e+01
* time: 93.57003903388977
56 1.155959e+03 4.269004e+01
* time: 94.67274188995361
57 1.155557e+03 5.000322e+01
* time: 95.79495692253113
58 1.155472e+03 4.219349e+01
* time: 96.80988001823425
59 1.155463e+03 3.895049e+01
* time: 97.91653108596802
60 1.155462e+03 3.872099e+01
* time: 99.04453897476196
61 1.155458e+03 3.823491e+01
* time: 100.3542640209198
62 1.155450e+03 3.747370e+01
* time: 101.41949796676636
63 1.155426e+03 3.608648e+01
* time: 102.50969004631042
64 1.155365e+03 3.361118e+01
* time: 103.50373101234436
65 1.155207e+03 2.894678e+01
* time: 104.5814778804779
66 1.154814e+03 2.504284e+01
* time: 105.58042001724243
67 1.153910e+03 4.541126e+01
* time: 106.65499997138977
68 1.152230e+03 7.426044e+01
* time: 107.64266586303711
69 1.150286e+03 7.473044e+01
* time: 108.68848896026611
70 1.149188e+03 5.605967e+01
* time: 109.65592694282532
71 1.148944e+03 4.801123e+01
* time: 110.65061688423157
72 1.148931e+03 4.659478e+01
* time: 111.6021020412445
73 1.148929e+03 4.635180e+01
* time: 112.60002589225769
74 1.148922e+03 4.568579e+01
* time: 113.56269907951355
75 1.148907e+03 4.466128e+01
* time: 114.59013986587524
76 1.148864e+03 4.249742e+01
* time: 115.56250286102295
77 1.148758e+03 3.824840e+01
* time: 116.58168697357178
78 1.148490e+03 3.552459e+01
* time: 117.53972887992859
79 1.147881e+03 3.038181e+01
* time: 118.55775094032288
80 1.146736e+03 4.467281e+01
* time: 119.50921106338501
81 1.145347e+03 5.929158e+01
* time: 120.49515104293823
82 1.144535e+03 7.572784e+01
* time: 121.46721291542053
83 1.144354e+03 7.034986e+01
* time: 122.4580090045929
84 1.144341e+03 6.591584e+01
* time: 123.40814089775085
85 1.144340e+03 6.496353e+01
* time: 124.41140294075012
86 1.144333e+03 6.268267e+01
* time: 125.37297487258911
87 1.144320e+03 5.950121e+01
* time: 126.34827303886414
88 1.144281e+03 5.368613e+01
* time: 127.30502200126648
89 1.144186e+03 4.393956e+01
* time: 128.32886385917664
90 1.143949e+03 4.102324e+01
* time: 129.29173707962036
91 1.143423e+03 3.752005e+01
* time: 130.29346108436584
92 1.142497e+03 3.278884e+01
* time: 131.2672619819641
93 1.141489e+03 5.217814e+01
* time: 132.26393795013428
94 1.140969e+03 4.310032e+01
* time: 133.22649693489075
95 1.140872e+03 4.759545e+01
* time: 134.21879506111145
96 1.140867e+03 4.727034e+01
* time: 135.17387199401855
97 1.140866e+03 4.704365e+01
* time: 136.14086389541626
98 1.140862e+03 4.637923e+01
* time: 137.09379386901855
99 1.140853e+03 4.542582e+01
* time: 138.0620620250702
100 1.140829e+03 4.357378e+01
* time: 139.016352891922
101 1.140767e+03 4.023060e+01
* time: 139.99046397209167
102 1.140610e+03 3.376323e+01
* time: 140.95349097251892
103 1.140237e+03 2.857028e+01
* time: 141.9597909450531
104 1.139451e+03 3.895699e+01
* time: 142.94640803337097
105 1.138223e+03 5.064036e+01
* time: 143.93478202819824
106 1.137140e+03 4.763314e+01
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107 1.136763e+03 4.604968e+01
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108 1.136720e+03 3.993479e+01
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111 1.136712e+03 3.865517e+01
* time: 149.86452198028564
112 1.136702e+03 3.838833e+01
* time: 150.878112077713
113 1.136674e+03 3.789981e+01
* time: 151.81833386421204
114 1.136603e+03 3.705558e+01
* time: 152.83747386932373
115 1.136418e+03 3.549940e+01
* time: 153.7962989807129
116 1.135940e+03 3.257136e+01
* time: 154.90574288368225
117 1.134743e+03 2.697248e+01
* time: 155.8784420490265
118 1.132014e+03 4.863416e+01
* time: 156.92252802848816
119 1.127203e+03 6.901202e+01
* time: 157.87802386283875
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128 1.118860e+03 1.866000e+01
* time: 166.7681589126587
129 1.118859e+03 1.869625e+01
* time: 167.68520188331604
130 1.118856e+03 1.878717e+01
* time: 168.68680906295776
131 1.118850e+03 1.894283e+01
* time: 169.65949606895447
132 1.118834e+03 1.922180e+01
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133 1.118794e+03 1.968060e+01
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* time: 173.87150692939758
136 1.117751e+03 2.596600e+01
* time: 174.96889090538025
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* time: 176.03343200683594
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* time: 177.14787602424622
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* time: 178.2622299194336
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* time: 179.45081901550293
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* time: 328.0789530277252
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* time: 334.3984749317169
232 1.027999e+03 3.038649e+01
* time: 335.91544699668884
233 1.027721e+03 5.035972e+01
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235 1.025404e+03 1.233505e+02
* time: 340.1757719516754
236 1.022343e+03 1.646448e+02
* time: 341.80963706970215
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* time: 344.42520093917847
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* time: 346.33879709243774
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* time: 360.2021849155426
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* time: 361.5180449485779
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252 1.012697e+03 9.980635e-01
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253 1.012697e+03 9.984743e-01
* time: 369.49633288383484
254 1.012696e+03 9.988765e-01
* time: 370.99748396873474
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* time: 372.4875409603119
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* time: 373.90662002563477
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* time: 376.7298250198364
259 1.012590e+03 4.737310e+00
* time: 378.16983103752136
260 1.012497e+03 4.897808e+00
* time: 379.5339090824127
261 1.012416e+03 2.875911e+00
* time: 380.86388993263245
262 1.012389e+03 9.301211e-01
* time: 382.17883491516113
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* time: 383.4761519432068
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* time: 384.7260539531708
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* time: 388.10281205177307
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269 1.012386e+03 8.415231e-01
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* time: 391.40973806381226
271 1.012386e+03 8.215489e-01
* time: 392.5239210128784
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* time: 393.6287319660187
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* time: 400.4257559776306
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* time: 401.75014901161194
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* time: 403.2552399635315
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* time: 404.56367588043213
282 1.011775e+03 9.067203e-01
* time: 405.92310404777527
283 1.011774e+03 9.565073e-01
* time: 407.2017879486084
284 1.011774e+03 9.628263e-01
* time: 408.5160689353943
285 1.011774e+03 9.654767e-01
* time: 409.72981905937195
286 1.011774e+03 9.654769e-01
* time: 411.0573809146881
287 1.011774e+03 9.656366e-01
* time: 412.14817690849304
288 1.011774e+03 9.656622e-01
* time: 413.2797749042511
289 1.011774e+03 9.656624e-01
* time: 414.52914690971375
290 1.011774e+03 9.656627e-01
* time: 415.9566168785095
291 1.011774e+03 9.660395e-01
* time: 417.411838054657
292 1.011774e+03 9.661257e-01
* time: 420.7361218929291
293 1.011774e+03 9.663115e-01
* time: 422.9264678955078
294 1.011774e+03 9.662851e-01
* time: 424.89053201675415
295 1.011773e+03 9.654773e-01
* time: 426.63135409355164
296 1.011772e+03 9.623124e-01
* time: 428.21404695510864
297 1.011769e+03 9.521889e-01
* time: 429.60455107688904
298 1.011763e+03 9.239616e-01
* time: 431.01576685905457
299 1.011746e+03 9.242954e-01
* time: 432.3588728904724
300 1.011711e+03 1.257052e+00
* time: 433.70727801322937
301 1.011651e+03 1.361278e+00
* time: 434.99998593330383
302 1.011591e+03 9.076916e-01
* time: 436.30766892433167
303 1.011568e+03 3.084752e-01
* time: 437.6138319969177
304 1.011565e+03 2.946404e-01
* time: 438.9141249656677
305 1.011565e+03 2.821273e-01
* time: 440.64502000808716
306 1.011565e+03 2.766260e-01
* time: 447.7602529525757
307 1.011565e+03 2.778010e-01
* time: 450.07408809661865
308 1.011565e+03 2.830281e-01
* time: 452.05622005462646
309 1.011565e+03 2.842619e-01
* time: 453.7743148803711
310 1.011565e+03 2.861093e-01
* time: 455.3576180934906
311 1.011565e+03 2.861138e-01
* time: 456.8638188838959
312 1.011565e+03 2.867641e-01
* time: 458.39902687072754
313 1.011565e+03 2.867756e-01
* time: 460.0616879463196
314 1.011565e+03 2.884589e-01
* time: 461.60845589637756
315 1.011565e+03 2.897335e-01
* time: 463.0596649646759
316 1.011565e+03 2.927744e-01
* time: 464.4553608894348
317 1.011565e+03 2.967424e-01
* time: 465.7813789844513
318 1.011565e+03 3.027347e-01
* time: 467.1630620956421
319 1.011564e+03 3.099827e-01
* time: 468.45212602615356
320 1.011563e+03 3.155826e-01
* time: 469.80787897109985
321 1.011560e+03 3.082844e-01
* time: 471.32473707199097
322 1.011555e+03 2.625121e-01
* time: 472.89328503608704
323 1.011548e+03 2.136271e-01
* time: 474.3493719100952
324 1.011542e+03 1.165033e-01
* time: 475.70230507850647
325 1.011538e+03 3.914224e-02
* time: 477.2863130569458
326 1.011537e+03 3.083888e-02
* time: 478.6780700683594
327 1.011536e+03 2.766364e-02
* time: 480.2767000198364
328 1.011536e+03 1.873262e-02
* time: 481.8227870464325
329 1.011536e+03 5.453951e-03
* time: 483.27135491371155
330 1.011536e+03 4.882951e-03
* time: 485.43862295150757
331 1.011535e+03 4.331879e-03
* time: 487.00595903396606
332 1.011535e+03 2.711268e-03
* time: 488.7913589477539
333 1.011535e+03 4.474256e-04
* time: 490.4663028717041FittedPumasModel
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: 0.0004229545593261719
1 4.804005e+05 8.280311e+05
* time: 20.43834400177002
2 3.595691e+05 6.035842e+05
* time: 29.471380949020386
3 1.691071e+05 2.702705e+05
* time: 37.48148512840271
4 9.193152e+04 1.451040e+05
* time: 55.84230709075928
5 4.768538e+04 6.630202e+04
* time: 66.06435704231262
6 2.904205e+04 3.416919e+04
* time: 73.87840008735657
7 1.840152e+04 1.671742e+04
* time: 81.54669404029846
8 1.280989e+04 1.022268e+04
* time: 95.6118631362915
9 9.713285e+03 9.298485e+03
* time: 104.83492994308472
10 7.628127e+03 8.330656e+03
* time: 116.6948390007019
11 6.236278e+03 7.340367e+03
* time: 125.74649715423584
12 5.271958e+03 6.303773e+03
* time: 133.56066608428955
13 4.411047e+03 4.994130e+03
* time: 143.04981112480164
14 3.732935e+03 3.538183e+03
* time: 155.91171598434448
15 3.399272e+03 2.450045e+03
* time: 165.12851905822754
16 3.302987e+03 1.848475e+03
* time: 172.9835181236267
17 3.291788e+03 1.642864e+03
* time: 180.44070601463318
18 3.290743e+03 1.594765e+03
* time: 190.4156060218811
19 3.289686e+03 1.562640e+03
* time: 201.36314702033997
20 3.286186e+03 1.496632e+03
* time: 209.5697090625763
21 3.277917e+03 1.402865e+03
* time: 222.28419995307922
22 3.255958e+03 1.254296e+03
* time: 231.74119114875793
23 3.200979e+03 1.038621e+03
* time: 239.39262199401855
24 3.063520e+03 7.403602e+02
* time: 251.87802696228027
25 2.730315e+03 3.768701e+02
* time: 262.83934712409973
26 1.934186e+03 2.246121e+02
* time: 271.126501083374
27 1.704399e+03 2.219826e+02
* time: 297.37584495544434
28 1.471184e+03 2.025825e+02
* time: 352.5652551651001
29 1.316789e+03 1.554715e+02
* time: 390.85428404808044
30 1.226363e+03 1.775721e+02
* time: 399.8102309703827
31 1.206626e+03 2.781030e+02
* time: 408.9569561481476
32 1.197830e+03 2.430603e+02
* time: 416.63732504844666
33 1.190342e+03 2.327966e+02
* time: 423.44104409217834
34 1.178784e+03 2.105878e+02
* time: 435.07282996177673
35 1.177853e+03 2.228653e+02
* time: 443.9634311199188
36 1.177764e+03 2.256182e+02
* time: 451.4774830341339
37 1.177746e+03 2.259425e+02
* time: 459.1620099544525
38 1.177670e+03 2.264699e+02
* time: 476.0657501220703
39 1.177500e+03 2.264641e+02
* time: 485.15774512290955
40 1.177034e+03 2.244207e+02
* time: 493.3200659751892
41 1.175895e+03 2.160307e+02
* time: 501.2916090488434
42 1.173241e+03 1.906017e+02
* time: 514.9100179672241
43 1.168190e+03 1.332917e+02
* time: 523.6566340923309
44 1.161836e+03 5.291022e+01
* time: 531.5969970226288
45 1.158453e+03 6.595471e+01
* time: 540.7448670864105
46 1.157897e+03 6.411640e+01
* time: 553.9287431240082
47 1.157870e+03 6.098679e+01
* time: 561.7589581012726
48 1.157868e+03 6.032481e+01
* time: 568.8653681278229
49 1.157864e+03 5.920991e+01
* time: 575.6833369731903
50 1.157856e+03 5.740417e+01
* time: 582.4387221336365
51 1.157832e+03 5.407031e+01
* time: 591.1384210586548
52 1.157774e+03 4.798188e+01
* time: 598.6969130039215
53 1.157628e+03 3.659183e+01
* time: 606.7312109470367
54 1.157299e+03 3.707862e+01
* time: 619.2901420593262
55 1.156690e+03 3.453183e+01
* time: 628.3015520572662
56 1.155959e+03 4.268473e+01
* time: 636.4888861179352
57 1.155557e+03 5.000309e+01
* time: 644.342481136322
58 1.155472e+03 4.219411e+01
* time: 664.7153830528259
59 1.155463e+03 3.894982e+01
* time: 672.343031167984
60 1.155462e+03 3.872028e+01
* time: 686.8097219467163
61 1.155458e+03 3.823435e+01
* time: 698.6388101577759
62 1.155450e+03 3.747328e+01
* time: 708.0100240707397
63 1.155426e+03 3.608641e+01
* time: 718.1292691230774
64 1.155365e+03 3.361168e+01
* time: 728.1509051322937
65 1.155207e+03 2.894844e+01
* time: 736.8533210754395
66 1.154814e+03 2.504332e+01
* time: 753.8598899841309
67 1.153910e+03 4.539953e+01
* time: 785.2044761180878
68 1.152231e+03 7.425240e+01
* time: 849.2698309421539
69 1.150287e+03 7.473669e+01
* time: 898.6862740516663
70 1.149189e+03 5.606074e+01
* time: 945.3819961547852
71 1.148944e+03 4.801310e+01
* time: 990.8594880104065
72 1.148932e+03 4.659470e+01
* time: 1039.267070055008
73 1.148930e+03 4.635167e+01
* time: 1071.9612309932709
74 1.148923e+03 4.568576e+01
* time: 1108.8116471767426
75 1.148907e+03 4.466140e+01
* time: 1154.0720889568329
76 1.148865e+03 4.249800e+01
* time: 1180.8980050086975
77 1.148758e+03 3.824645e+01
* time: 1208.1219000816345
78 1.148491e+03 3.552369e+01
* time: 1236.714872121811
79 1.147881e+03 3.037355e+01
* time: 1264.948559999466
80 1.146737e+03 4.466673e+01
* time: 1284.4998350143433
81 1.145347e+03 5.927932e+01
* time: 1305.3711149692535
82 1.144536e+03 7.572710e+01
* time: 1330.274708032608
83 1.144354e+03 7.035223e+01
* time: 1374.2716090679169
84 1.144341e+03 6.591571e+01
* time: 1390.9949271678925
85 1.144340e+03 6.496308e+01
* time: 1402.0454921722412
86 1.144333e+03 6.268335e+01
* time: 1411.9175870418549
87 1.144320e+03 5.950296e+01
* time: 1427.5938081741333
88 1.144281e+03 5.369030e+01
* time: 1436.283518075943
89 1.144186e+03 4.394749e+01
* time: 1445.050843000412
90 1.143949e+03 4.102392e+01
* time: 1453.3583381175995
91 1.143423e+03 3.752367e+01
* time: 1462.9932429790497
92 1.142498e+03 3.276758e+01
* time: 1472.3590321540833
93 1.141490e+03 5.217551e+01
* time: 1480.7130789756775
94 1.140969e+03 4.309381e+01
* time: 1488.746691942215
95 1.140872e+03 4.759588e+01
* time: 1497.1185879707336
96 1.140867e+03 4.727133e+01
* time: 1516.8032920360565
97 1.140866e+03 4.704459e+01
* time: 1542.2029631137848
98 1.140862e+03 4.638036e+01
* time: 1581.6669120788574
99 1.140853e+03 4.542717e+01
* time: 1604.6622800827026
100 1.140829e+03 4.357571e+01
* time: 1617.0542480945587
101 1.140767e+03 4.023368e+01
* time: 1629.1456451416016
102 1.140611e+03 3.376885e+01
* time: 1641.0703761577606
103 1.140237e+03 2.856928e+01
* time: 1649.546592950821
104 1.139452e+03 3.894107e+01
* time: 1658.2586181163788
105 1.138224e+03 5.063688e+01
* time: 1668.774108171463
106 1.137140e+03 4.762519e+01
* time: 1678.1966819763184
107 1.136763e+03 4.605051e+01
* time: 1687.0871460437775
108 1.136720e+03 3.993288e+01
* time: 1697.618327140808
109 1.136717e+03 3.890566e+01
* time: 1706.8516991138458
110 1.136716e+03 3.883647e+01
* time: 1715.1650550365448
111 1.136712e+03 3.865523e+01
* time: 1724.0706040859222
112 1.136702e+03 3.838845e+01
* time: 1733.4759459495544
113 1.136674e+03 3.790005e+01
* time: 1746.6718690395355
114 1.136604e+03 3.705604e+01
* time: 1758.4053790569305
115 1.136418e+03 3.550032e+01
* time: 1768.1316890716553
116 1.135940e+03 3.257323e+01
* time: 1776.9608490467072
117 1.134744e+03 2.697619e+01
* time: 1785.4685339927673
118 1.132016e+03 4.861352e+01
* time: 1793.8143169879913
119 1.127206e+03 6.900228e+01
* time: 1801.833261013031
120 1.122365e+03 5.795238e+01
* time: 1810.062390089035
121 1.119760e+03 2.908163e+01
* time: 1818.3642930984497
122 1.118972e+03 2.137897e+01
* time: 1826.815092086792
123 1.118870e+03 1.909827e+01
* time: 1835.1649680137634
124 1.118863e+03 1.886936e+01
* time: 1844.1320791244507
125 1.118862e+03 1.852746e+01
* time: 1856.5950450897217
126 1.118861e+03 1.864804e+01
* time: 1866.5084581375122
127 1.118861e+03 1.862831e+01
* time: 1875.7426221370697
128 1.118860e+03 1.865947e+01
* time: 1884.8379831314087
129 1.118859e+03 1.869546e+01
* time: 1893.4879431724548
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235 1.026628e+03 7.187426e+01
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* time: 3216.6433091163635
247 1.012697e+03 9.971842e-01
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* time: 3233.1766381263733
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* time: 3241.4692890644073
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* time: 3255.4805409908295
251 1.012697e+03 9.973078e-01
* time: 3263.834365129471
252 1.012697e+03 9.972048e-01
* time: 3271.2622430324554
253 1.012697e+03 9.970536e-01
* time: 3278.3843700885773
254 1.012697e+03 9.967402e-01
* time: 3285.1638259887695
255 1.012697e+03 9.961565e-01
* time: 3292.202826023102
256 1.012696e+03 9.949478e-01
* time: 3301.139825105667
257 1.012694e+03 1.312115e+00
* time: 3314.0078580379486
258 1.012690e+03 2.118030e+00
* time: 3340.3247499465942
259 1.012679e+03 3.352490e+00
* time: 3362.176973104477
260 1.012651e+03 5.076526e+00
* time: 3396.4928760528564
261 1.012594e+03 6.891495e+00
* time: 3412.8716809749603
262 1.012501e+03 7.239074e+00
* time: 3443.629009962082
263 1.012418e+03 4.479611e+00
* time: 3459.2118260860443
264 1.012390e+03 1.267149e+00
* time: 3495.185975074768
265 1.012386e+03 8.840304e-01
* time: 3528.58144903183
266 1.012386e+03 8.549097e-01
* time: 3565.8253951072693
267 1.012386e+03 8.516869e-01
* time: 3588.4442241191864
268 1.012386e+03 8.516869e-01
* time: 3617.1038789749146
269 1.012386e+03 8.516869e-01
* time: 3657.302843093872
270 1.012386e+03 8.516817e-01
* time: 3688.437945127487
271 1.012386e+03 8.516817e-01
* time: 3708.840376138687
272 1.012386e+03 8.516817e-01
* time: 3740.1402111053467
273 1.012386e+03 8.516817e-01
* time: 3756.975336074829
274 1.012386e+03 8.516817e-01
* time: 3787.3287370204926
275 1.012386e+03 8.516817e-01
* time: 3801.0230989456177FittedPumasModel
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: 0.00010085105895996094
1 4.804005e+05 8.280312e+05
* time: 1.5635318756103516
2 3.595691e+05 6.035843e+05
* time: 2.973449945449829
3 1.691071e+05 2.702705e+05
* time: 4.2098469734191895
4 9.193152e+04 1.451040e+05
* time: 5.552165985107422
5 4.768539e+04 6.630202e+04
* time: 6.8227269649505615
6 2.904205e+04 3.416919e+04
* time: 8.140636920928955
7 1.840153e+04 1.671742e+04
* time: 9.441663026809692
8 1.280989e+04 1.022268e+04
* time: 10.77936601638794
9 9.713287e+03 9.298485e+03
* time: 12.059754848480225
10 7.628127e+03 8.330656e+03
* time: 13.41556692123413
11 6.236278e+03 7.340367e+03
* time: 14.714256048202515
12 5.271958e+03 6.303773e+03
* time: 16.078125953674316
13 4.411047e+03 4.994130e+03
* time: 17.400704860687256
14 3.732936e+03 3.538184e+03
* time: 18.949026823043823
15 3.399273e+03 2.450045e+03
* time: 20.62395191192627
16 3.302987e+03 1.848476e+03
* time: 25.080665826797485
17 3.291788e+03 1.642864e+03
* time: 28.05945897102356
18 3.290743e+03 1.594765e+03
* time: 30.176079988479614
19 3.289687e+03 1.562641e+03
* time: 32.12293887138367
20 3.286186e+03 1.496632e+03
* time: 33.85771083831787
21 3.277917e+03 1.402865e+03
* time: 35.5943169593811
22 3.255959e+03 1.254297e+03
* time: 37.18429493904114
23 3.200980e+03 1.038621e+03
* time: 38.81414198875427
24 3.063521e+03 7.403605e+02
* time: 40.30588984489441
25 2.730316e+03 3.768701e+02
* time: 41.80408501625061
26 1.934186e+03 2.246121e+02
* time: 43.00061798095703
27 1.704399e+03 2.219826e+02
* time: 47.89810395240784
28 1.471183e+03 2.025826e+02
* time: 56.518481969833374
29 1.316789e+03 1.554714e+02
* time: 66.16471004486084
30 1.226363e+03 1.775666e+02
* time: 68.11983394622803
31 1.206626e+03 2.780980e+02
* time: 69.6958019733429
32 1.197830e+03 2.430550e+02
* time: 71.36149883270264
33 1.190342e+03 2.327905e+02
* time: 72.73438596725464
34 1.178783e+03 2.105809e+02
* time: 74.20251488685608
35 1.177853e+03 2.228585e+02
* time: 76.28807783126831
36 1.177763e+03 2.256116e+02
* time: 78.41123604774475
37 1.177745e+03 2.259359e+02
* time: 80.35065484046936
38 1.177670e+03 2.264633e+02
* time: 82.22239804267883
39 1.177500e+03 2.264574e+02
* time: 84.12630200386047
40 1.177033e+03 2.244136e+02
* time: 85.90185689926147
41 1.175894e+03 2.160226e+02
* time: 87.9045569896698
42 1.173240e+03 1.905912e+02
* time: 89.87055397033691
43 1.168189e+03 1.332783e+02
* time: 91.62784004211426
44 1.161835e+03 5.289952e+01
* time: 95.45163083076477
45 1.158453e+03 6.595387e+01
* time: 101.9491708278656
46 1.157897e+03 6.411419e+01
* time: 103.81520986557007
47 1.157870e+03 6.098509e+01
* time: 105.44981002807617
48 1.157868e+03 6.032325e+01
* time: 107.01936388015747
49 1.157864e+03 5.920812e+01
* time: 108.46431493759155
50 1.157856e+03 5.740225e+01
* time: 109.97215604782104
51 1.157832e+03 5.406795e+01
* time: 111.35535287857056
52 1.157774e+03 4.797879e+01
* time: 113.28410601615906
53 1.157628e+03 3.658732e+01
* time: 116.53428292274475
54 1.157299e+03 3.707874e+01
* time: 119.05746793746948
55 1.156690e+03 3.452947e+01
* time: 121.14047002792358
56 1.155959e+03 4.269004e+01
* time: 123.06517696380615
57 1.155557e+03 5.000322e+01
* time: 124.99146604537964
58 1.155472e+03 4.219349e+01
* time: 126.7311680316925
59 1.155463e+03 3.895049e+01
* time: 128.3339970111847
60 1.155462e+03 3.872099e+01
* time: 129.4409909248352
61 1.155458e+03 3.823491e+01
* time: 130.61320304870605
62 1.155450e+03 3.747370e+01
* time: 131.68385696411133
63 1.155426e+03 3.608648e+01
* time: 132.77817296981812
64 1.155365e+03 3.361118e+01
* time: 133.86697483062744
65 1.155207e+03 2.894678e+01
* time: 135.00072503089905
66 1.154814e+03 2.504284e+01
* time: 136.139466047287
67 1.153910e+03 4.541126e+01
* time: 137.23268485069275
68 1.152230e+03 7.426044e+01
* time: 138.36527681350708
69 1.150286e+03 7.473044e+01
* time: 139.43518090248108
70 1.149188e+03 5.605967e+01
* time: 140.59311890602112
71 1.148944e+03 4.801123e+01
* time: 141.67724895477295
72 1.148931e+03 4.659478e+01
* time: 142.79204893112183
73 1.148929e+03 4.635180e+01
* time: 143.93104100227356
74 1.148922e+03 4.568579e+01
* time: 145.03991103172302
75 1.148907e+03 4.466128e+01
* time: 146.3771059513092
76 1.148864e+03 4.249742e+01
* time: 148.11872696876526
77 1.148758e+03 3.824840e+01
* time: 149.93944883346558
78 1.148490e+03 3.552459e+01
* time: 151.6986379623413
79 1.147881e+03 3.038181e+01
* time: 153.57162594795227
80 1.146736e+03 4.467281e+01
* time: 155.25954794883728
81 1.145347e+03 5.929158e+01
* time: 156.83591294288635
82 1.144535e+03 7.572784e+01
* time: 157.95326590538025
83 1.144354e+03 7.034986e+01
* time: 159.19690585136414
84 1.144341e+03 6.591584e+01
* time: 160.30735087394714
85 1.144340e+03 6.496353e+01
* time: 161.812805891037
86 1.144333e+03 6.268267e+01
* time: 164.6780068874359
87 1.144320e+03 5.950121e+01
* time: 167.5503158569336
88 1.144281e+03 5.368613e+01
* time: 169.54465889930725
89 1.144186e+03 4.393956e+01
* time: 171.03845691680908
90 1.143949e+03 4.102324e+01
* time: 172.5093879699707
91 1.143423e+03 3.752005e+01
* time: 173.9874210357666
92 1.142497e+03 3.278884e+01
* time: 175.46158981323242
93 1.141489e+03 5.217814e+01
* time: 176.93660688400269
94 1.140969e+03 4.310032e+01
* time: 178.29149794578552
95 1.140872e+03 4.759545e+01
* time: 179.76309490203857
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* time: 580.1902258396149FittedPumasModel
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.