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: 0.0339818000793457
1 4.816669e+05 8.316173e+05
* time: 1.7958779335021973
2 3.598373e+05 6.060150e+05
* time: 2.507612943649292
3 1.682124e+05 2.709072e+05
* time: 3.1302638053894043
4 9.168914e+04 1.446832e+05
* time: 3.8037188053131104
5 4.791138e+04 6.671463e+04
* time: 4.472915887832642
6 2.909104e+04 3.426663e+04
* time: 5.085271835327148
7 1.843737e+04 1.678953e+04
* time: 5.820868968963623
8 1.282447e+04 1.021463e+04
* time: 6.456052780151367
9 9.723361e+03 9.291650e+03
* time: 7.130998849868774
10 7.635613e+03 8.325311e+03
* time: 7.7556328773498535
11 6.243129e+03 7.337397e+03
* time: 8.4772629737854
12 5.279957e+03 6.304542e+03
* time: 9.089170932769775
13 4.422123e+03 5.001962e+03
* time: 9.71747088432312
14 3.744748e+03 3.550400e+03
* time: 10.330431938171387
15 3.409674e+03 2.461547e+03
* time: 10.950011968612671
16 3.312049e+03 1.856430e+03
* time: 11.597062826156616
17 3.300559e+03 1.648439e+03
* time: 12.17012882232666
18 3.299489e+03 1.599460e+03
* time: 12.790802001953125
19 3.298436e+03 1.567520e+03
* time: 13.358941793441772
20 3.294921e+03 1.501363e+03
* time: 13.995409965515137
21 3.286641e+03 1.407664e+03
* time: 14.590467929840088
22 3.264637e+03 1.259077e+03
* time: 15.220343828201294
23 3.209594e+03 1.043534e+03
* time: 15.79503083229065
24 3.072117e+03 7.454677e+02
* time: 16.42513084411621
25 2.739794e+03 3.818063e+02
* time: 16.97980284690857
26 1.949883e+03 2.246644e+02
* time: 17.57834792137146
27 1.722434e+03 2.224562e+02
* time: 19.189254999160767
28 1.479355e+03 2.042452e+02
* time: 21.31681489944458
29 1.310583e+03 1.531868e+02
* time: 23.79111886024475
30 1.224675e+03 1.756826e+02
* time: 24.416525840759277
31 1.204841e+03 2.738440e+02
* time: 24.94250798225403
32 1.196432e+03 2.403217e+02
* time: 25.503607988357544
33 1.189131e+03 2.301186e+02
* time: 26.02058696746826
34 1.177881e+03 2.083030e+02
* time: 26.570385932922363
35 1.176989e+03 2.201993e+02
* time: 27.106979846954346
36 1.176899e+03 2.230826e+02
* time: 27.701766967773438
37 1.176882e+03 2.233531e+02
* time: 28.23801898956299
38 1.176795e+03 2.238567e+02
* time: 28.7918119430542
39 1.176609e+03 2.236914e+02
* time: 29.3258318901062
40 1.176090e+03 2.210941e+02
* time: 29.870957851409912
41 1.174841e+03 2.112741e+02
* time: 30.37274694442749
42 1.171973e+03 1.825298e+02
* time: 30.902796983718872
43 1.166737e+03 1.210559e+02
* time: 31.444140911102295
44 1.160698e+03 4.450697e+01
* time: 31.998480796813965
45 1.157926e+03 6.411275e+01
* time: 32.57678198814392
46 1.157537e+03 6.121762e+01
* time: 33.10056495666504
47 1.157520e+03 5.867223e+01
* time: 33.621803998947144
48 1.157519e+03 5.814523e+01
* time: 34.09947180747986
49 1.157515e+03 5.686007e+01
* time: 34.64080595970154
50 1.157505e+03 5.497418e+01
* time: 35.138354778289795
51 1.157477e+03 5.129724e+01
* time: 35.726433992385864
52 1.157408e+03 4.463171e+01
* time: 36.210612773895264
53 1.157239e+03 3.279117e+01
* time: 36.76032495498657
54 1.156869e+03 3.681526e+01
* time: 37.2630558013916
55 1.156230e+03 3.198754e+01
* time: 37.82084584236145
56 1.155563e+03 4.576438e+01
* time: 38.33037495613098
57 1.155264e+03 4.832159e+01
* time: 38.89198684692383
58 1.155212e+03 4.052175e+01
* time: 39.3912079334259
59 1.155207e+03 3.797125e+01
* time: 39.91742181777954
60 1.155206e+03 3.778452e+01
* time: 40.38673996925354
61 1.155201e+03 3.723003e+01
* time: 40.910271883010864
62 1.155190e+03 3.642375e+01
* time: 41.397350788116455
63 1.155159e+03 3.487535e+01
* time: 41.91043186187744
64 1.155079e+03 3.209060e+01
* time: 42.38745880126953
65 1.154875e+03 2.670227e+01
* time: 42.90211486816406
66 1.154375e+03 2.366165e+01
* time: 43.48126697540283
67 1.153278e+03 5.499525e+01
* time: 44.012794971466064
68 1.151457e+03 7.903833e+01
* time: 44.506664991378784
69 1.149737e+03 6.771227e+01
* time: 45.01737189292908
70 1.148983e+03 5.387708e+01
* time: 45.48857498168945
71 1.148872e+03 4.683950e+01
* time: 45.988075971603394
72 1.148868e+03 4.512036e+01
* time: 46.451666831970215
73 1.148866e+03 4.489706e+01
* time: 46.940133810043335
74 1.148857e+03 4.426314e+01
* time: 47.46504783630371
75 1.148838e+03 4.321161e+01
* time: 47.96266984939575
76 1.148785e+03 4.095232e+01
* time: 48.45963191986084
77 1.148652e+03 3.780572e+01
* time: 48.975937843322754
78 1.148325e+03 3.474239e+01
* time: 49.467108964920044
79 1.147599e+03 3.678957e+01
* time: 49.98815894126892
80 1.146330e+03 5.134277e+01
* time: 50.480265855789185
81 1.144979e+03 6.444063e+01
* time: 51.01074695587158
82 1.144325e+03 7.574751e+01
* time: 51.511841773986816
83 1.144210e+03 6.968296e+01
* time: 52.025261878967285
84 1.144204e+03 6.648234e+01
* time: 52.524866819381714
85 1.144202e+03 6.560999e+01
* time: 53.03499698638916
86 1.144194e+03 6.305667e+01
* time: 53.522565841674805
87 1.144176e+03 5.956959e+01
* time: 54.02125096321106
88 1.144127e+03 5.307828e+01
* time: 54.512510776519775
89 1.144005e+03 4.218776e+01
* time: 55.00678992271423
90 1.143708e+03 4.120261e+01
* time: 55.544227838516235
91 1.143074e+03 3.675516e+01
* time: 56.00844192504883
92 1.142052e+03 3.855439e+01
* time: 56.54326295852661
93 1.141101e+03 5.193197e+01
* time: 57.021976947784424
94 1.140701e+03 4.606858e+01
* time: 57.55663800239563
95 1.140646e+03 4.857736e+01
* time: 58.076167821884155
96 1.140643e+03 4.817596e+01
* time: 58.64153981208801
97 1.140642e+03 4.793197e+01
* time: 59.11133694648743
98 1.140637e+03 4.723245e+01
* time: 59.64469599723816
99 1.140626e+03 4.619695e+01
* time: 60.12525200843811
100 1.140597e+03 4.418761e+01
* time: 60.66202783584595
101 1.140523e+03 4.050986e+01
* time: 61.164766788482666
102 1.140334e+03 3.335334e+01
* time: 61.73862981796265
103 1.139892e+03 2.982120e+01
* time: 62.23499083518982
104 1.138994e+03 4.360237e+01
* time: 62.77519989013672
105 1.137687e+03 5.246892e+01
* time: 63.30396795272827
106 1.136679e+03 4.825586e+01
* time: 63.896960973739624
107 1.136391e+03 4.467499e+01
* time: 64.4011697769165
108 1.136364e+03 3.961591e+01
* time: 64.94612097740173
109 1.136362e+03 3.951402e+01
* time: 65.45155787467957
110 1.136360e+03 3.943358e+01
* time: 65.9938600063324
111 1.136355e+03 3.923965e+01
* time: 66.49695897102356
112 1.136344e+03 3.894674e+01
* time: 67.04910278320312
113 1.136314e+03 3.841518e+01
* time: 67.51975679397583
114 1.136235e+03 3.748824e+01
* time: 68.10331797599792
115 1.136029e+03 3.577210e+01
* time: 68.59290981292725
116 1.135499e+03 3.252332e+01
* time: 69.10163378715515
117 1.134182e+03 2.630844e+01
* time: 69.5749499797821
118 1.131238e+03 4.869163e+01
* time: 70.07994985580444
119 1.126345e+03 6.537856e+01
* time: 70.55313086509705
120 1.121879e+03 5.123373e+01
* time: 71.04973292350769
121 1.119645e+03 2.457316e+01
* time: 71.53385186195374
122 1.119031e+03 2.124721e+01
* time: 72.0518548488617
123 1.118958e+03 1.857508e+01
* time: 72.56521677970886
124 1.118953e+03 1.896595e+01
* time: 73.07893490791321
125 1.118952e+03 1.849021e+01
* time: 73.56636095046997
126 1.118952e+03 1.852941e+01
* time: 74.05266284942627
127 1.118951e+03 1.856812e+01
* time: 74.52189493179321
128 1.118951e+03 1.862403e+01
* time: 75.00643491744995
129 1.118949e+03 1.871484e+01
* time: 75.4756247997284
130 1.118946e+03 1.886332e+01
* time: 75.9723379611969
131 1.118939e+03 1.910217e+01
* time: 76.44840693473816
132 1.118922e+03 1.948050e+01
* time: 76.9343090057373
133 1.118879e+03 2.006903e+01
* time: 77.41065096855164
134 1.118769e+03 2.097106e+01
* time: 77.93490481376648
135 1.118486e+03 2.231801e+01
* time: 78.47754192352295
136 1.117755e+03 3.049039e+01
* time: 78.94086384773254
137 1.115883e+03 5.345487e+01
* time: 79.46524977684021
138 1.111258e+03 8.489274e+01
* time: 79.93238282203674
139 1.101149e+03 7.643749e+01
* time: 80.45492577552795
140 1.090612e+03 5.802659e+01
* time: 80.96899485588074
141 1.087971e+03 5.776554e+01
* time: 81.53250288963318
142 1.087205e+03 1.148234e+02
* time: 82.08293795585632
143 1.085558e+03 4.724872e+01
* time: 82.67391180992126
144 1.085271e+03 2.018848e+01
* time: 83.2771577835083
145 1.085124e+03 2.004201e+01
* time: 83.8490629196167
146 1.085089e+03 2.025616e+01
* time: 84.40231895446777
147 1.085081e+03 2.049682e+01
* time: 84.9927728176117
148 1.085073e+03 2.086075e+01
* time: 85.59928178787231
149 1.085070e+03 2.095706e+01
* time: 86.14397096633911
150 1.085063e+03 2.094978e+01
* time: 86.73586297035217
151 1.085062e+03 2.093855e+01
* time: 87.32943677902222
152 1.085062e+03 2.093439e+01
* time: 88.04063081741333
153 1.085062e+03 2.093335e+01
* time: 88.75857186317444
154 1.085062e+03 2.093252e+01
* time: 89.52643394470215
155 1.085062e+03 2.093227e+01
* time: 90.3319799900055
156 1.085062e+03 2.093208e+01
* time: 91.15072584152222
157 1.085062e+03 2.093187e+01
* time: 91.90822696685791
158 1.085062e+03 2.093181e+01
* time: 92.74485397338867
159 1.085062e+03 2.093175e+01
* time: 93.61421489715576
160 1.085062e+03 2.093172e+01
* time: 94.50738286972046
161 1.085062e+03 2.046653e+01
* time: 95.04513788223267
162 1.085061e+03 2.045920e+01
* time: 95.69139289855957
163 1.085056e+03 2.021786e+01
* time: 96.24114179611206
164 1.085055e+03 2.018381e+01
* time: 96.87740087509155
165 1.085043e+03 1.997498e+01
* time: 97.42394089698792
166 1.085029e+03 1.961695e+01
* time: 98.00298190116882
167 1.084997e+03 1.929634e+01
* time: 98.58177089691162
168 1.084881e+03 1.855033e+01
* time: 99.16992378234863
169 1.084633e+03 1.799616e+01
* time: 99.81237387657166
170 1.083831e+03 1.856893e+01
* time: 100.33554291725159
171 1.081240e+03 2.069096e+01
* time: 100.92425990104675
172 1.072465e+03 3.262436e+01
* time: 101.53254890441895
173 1.072446e+03 6.686587e+01
* time: 102.17868280410767
174 1.068493e+03 7.417042e+01
* time: 102.81279182434082
175 1.067469e+03 7.109936e+01
* time: 103.55252981185913
176 1.065854e+03 2.774005e+01
* time: 104.19124388694763
177 1.065215e+03 1.328875e+01
* time: 104.8390998840332
178 1.065128e+03 1.390400e+01
* time: 105.47695088386536
179 1.065120e+03 1.383172e+01
* time: 106.13209700584412
180 1.065120e+03 1.384995e+01
* time: 106.79382586479187FittedPumasModel
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: 6.604194641113281e-5
1 4.804005e+05 8.280312e+05
* time: 1.359144926071167
2 3.595691e+05 6.035843e+05
* time: 2.5115089416503906
3 1.691071e+05 2.702705e+05
* time: 3.7016608715057373
4 9.193152e+04 1.451040e+05
* time: 4.9254279136657715
5 4.768539e+04 6.630202e+04
* time: 6.0788819789886475
6 2.904205e+04 3.416919e+04
* time: 7.245415925979614
7 1.840153e+04 1.671742e+04
* time: 8.365715980529785
8 1.280989e+04 1.022268e+04
* time: 9.426537036895752
9 9.713287e+03 9.298485e+03
* time: 10.463114023208618
10 7.628127e+03 8.330656e+03
* time: 11.513552904129028
11 6.236278e+03 7.340367e+03
* time: 12.598955869674683
12 5.271958e+03 6.303773e+03
* time: 13.608495950698853
13 4.411047e+03 4.994130e+03
* time: 14.662185907363892
14 3.732936e+03 3.538184e+03
* time: 15.62900686264038
15 3.399273e+03 2.450045e+03
* time: 16.63200283050537
16 3.302987e+03 1.848476e+03
* time: 17.62273097038269
17 3.291788e+03 1.642864e+03
* time: 18.689509868621826
18 3.290743e+03 1.594765e+03
* time: 19.763722896575928
19 3.289687e+03 1.562641e+03
* time: 20.8901789188385
20 3.286186e+03 1.496632e+03
* time: 21.987637996673584
21 3.277917e+03 1.402865e+03
* time: 23.094738006591797
22 3.255959e+03 1.254297e+03
* time: 24.18255591392517
23 3.200980e+03 1.038621e+03
* time: 25.291775941848755
24 3.063521e+03 7.403605e+02
* time: 26.318245887756348
25 2.730316e+03 3.768701e+02
* time: 27.356482982635498
26 1.934186e+03 2.246121e+02
* time: 28.32728886604309
27 1.704399e+03 2.219826e+02
* time: 32.265588998794556
28 1.471183e+03 2.025826e+02
* time: 39.179713010787964
29 1.316789e+03 1.554714e+02
* time: 45.10893201828003
30 1.226363e+03 1.775666e+02
* time: 46.24491500854492
31 1.206626e+03 2.780980e+02
* time: 47.27215600013733
32 1.197830e+03 2.430550e+02
* time: 48.32266402244568
33 1.190342e+03 2.327905e+02
* time: 49.34406399726868
34 1.178783e+03 2.105809e+02
* time: 50.40811896324158
35 1.177853e+03 2.228585e+02
* time: 51.44653797149658
36 1.177763e+03 2.256116e+02
* time: 52.495179891586304
37 1.177745e+03 2.259359e+02
* time: 53.55114197731018
38 1.177670e+03 2.264633e+02
* time: 54.55349588394165
39 1.177500e+03 2.264574e+02
* time: 55.615402936935425
40 1.177033e+03 2.244136e+02
* time: 56.611177921295166
41 1.175894e+03 2.160226e+02
* time: 57.680712938308716
42 1.173240e+03 1.905912e+02
* time: 58.690356969833374
43 1.168189e+03 1.332783e+02
* time: 59.926997900009155
44 1.161835e+03 5.289952e+01
* time: 60.978893995285034
45 1.158453e+03 6.595387e+01
* time: 62.0408878326416
46 1.157897e+03 6.411419e+01
* time: 63.76893901824951
47 1.157870e+03 6.098509e+01
* time: 65.14499688148499
48 1.157868e+03 6.032325e+01
* time: 66.26513695716858
49 1.157864e+03 5.920812e+01
* time: 67.37696981430054
50 1.157856e+03 5.740225e+01
* time: 68.41786289215088
51 1.157832e+03 5.406795e+01
* time: 69.53313183784485
52 1.157774e+03 4.797879e+01
* time: 70.50396585464478
53 1.157628e+03 3.658732e+01
* time: 71.56993889808655
54 1.157299e+03 3.707874e+01
* time: 72.57355999946594
55 1.156690e+03 3.452947e+01
* time: 73.63233995437622
56 1.155959e+03 4.269004e+01
* time: 74.63366985321045
57 1.155557e+03 5.000322e+01
* time: 75.67968583106995
58 1.155472e+03 4.219349e+01
* time: 76.77702593803406
59 1.155463e+03 3.895049e+01
* time: 77.95550990104675
60 1.155462e+03 3.872099e+01
* time: 78.95889782905579
61 1.155458e+03 3.823491e+01
* time: 79.97477293014526
62 1.155450e+03 3.747370e+01
* time: 80.94057583808899
63 1.155426e+03 3.608648e+01
* time: 81.9850549697876
64 1.155365e+03 3.361118e+01
* time: 82.94562292098999
65 1.155207e+03 2.894678e+01
* time: 83.93896293640137
66 1.154814e+03 2.504284e+01
* time: 84.88165998458862
67 1.153910e+03 4.541126e+01
* time: 85.88738799095154
68 1.152230e+03 7.426044e+01
* time: 86.83668303489685
69 1.150286e+03 7.473044e+01
* time: 87.84930992126465
70 1.149188e+03 5.605967e+01
* time: 88.86928486824036
71 1.148944e+03 4.801123e+01
* time: 89.9883189201355
72 1.148931e+03 4.659478e+01
* time: 90.99528384208679
73 1.148929e+03 4.635180e+01
* time: 92.02197980880737
74 1.148922e+03 4.568579e+01
* time: 93.01829481124878
75 1.148907e+03 4.466128e+01
* time: 94.07059288024902
76 1.148864e+03 4.249742e+01
* time: 95.35555386543274
77 1.148758e+03 3.824840e+01
* time: 96.83344197273254
78 1.148490e+03 3.552459e+01
* time: 97.95430898666382
79 1.147881e+03 3.038181e+01
* time: 99.09204983711243
80 1.146736e+03 4.467281e+01
* time: 100.12004494667053
81 1.145347e+03 5.929158e+01
* time: 101.21826100349426
82 1.144535e+03 7.572784e+01
* time: 102.25839686393738
83 1.144354e+03 7.034986e+01
* time: 103.32884383201599
84 1.144341e+03 6.591584e+01
* time: 104.32950401306152
85 1.144340e+03 6.496353e+01
* time: 105.36305785179138
86 1.144333e+03 6.268267e+01
* time: 106.34987497329712
87 1.144320e+03 5.950121e+01
* time: 107.35847902297974
88 1.144281e+03 5.368613e+01
* time: 108.43915891647339
89 1.144186e+03 4.393956e+01
* time: 109.41955399513245
90 1.143949e+03 4.102324e+01
* time: 110.55263185501099
91 1.143423e+03 3.752005e+01
* time: 111.69638586044312
92 1.142497e+03 3.278884e+01
* time: 112.97472786903381
93 1.141489e+03 5.217814e+01
* time: 114.0677649974823
94 1.140969e+03 4.310032e+01
* time: 115.12609100341797
95 1.140872e+03 4.759545e+01
* time: 116.12950992584229
96 1.140867e+03 4.727034e+01
* time: 117.14846682548523
97 1.140866e+03 4.704365e+01
* time: 118.15179085731506
98 1.140862e+03 4.637923e+01
* time: 119.37341403961182
99 1.140853e+03 4.542582e+01
* time: 120.37834596633911
100 1.140829e+03 4.357378e+01
* time: 121.50409197807312
101 1.140767e+03 4.023060e+01
* time: 122.49957299232483
102 1.140610e+03 3.376323e+01
* time: 123.55615496635437
103 1.140237e+03 2.857028e+01
* time: 124.62670993804932
104 1.139451e+03 3.895699e+01
* time: 125.68165802955627
105 1.138223e+03 5.064036e+01
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111 1.136712e+03 3.865517e+01
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112 1.136702e+03 3.838833e+01
* time: 134.61298894882202
113 1.136674e+03 3.789981e+01
* time: 135.590313911438
114 1.136603e+03 3.705558e+01
* time: 136.60392904281616
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* time: 137.58259892463684
116 1.135940e+03 3.257136e+01
* time: 138.62982892990112
117 1.134743e+03 2.697248e+01
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129 1.118859e+03 1.869625e+01
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130 1.118856e+03 1.878717e+01
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* time: 159.8843059539795
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* time: 161.01157188415527
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* time: 162.0626859664917
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* time: 283.9128420352936
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* time: 285.3278589248657
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* time: 286.708988904953
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* time: 303.01478481292725
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* time: 304.5128300189972
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252 1.012697e+03 9.980635e-01
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253 1.012697e+03 9.984743e-01
* time: 311.15561985969543
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* time: 312.53824186325073
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* time: 317.883083820343
259 1.012590e+03 4.737310e+00
* time: 319.14827394485474
260 1.012497e+03 4.897808e+00
* time: 320.4295380115509
261 1.012416e+03 2.875911e+00
* time: 321.68198800086975
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* time: 324.33451795578003
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* time: 334.4944579601288
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* time: 335.81627798080444
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* time: 343.7205009460449
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* time: 344.99701380729675
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* time: 346.3488748073578
281 1.011809e+03 8.215973e-01
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282 1.011775e+03 9.067203e-01
* time: 349.0686960220337
283 1.011774e+03 9.565073e-01
* time: 350.3848249912262
284 1.011774e+03 9.628263e-01
* time: 351.7163460254669
285 1.011774e+03 9.654767e-01
* time: 353.0205109119415
286 1.011774e+03 9.654769e-01
* time: 354.4735949039459
287 1.011774e+03 9.656366e-01
* time: 355.7786009311676
288 1.011774e+03 9.656622e-01
* time: 357.1124098300934
289 1.011774e+03 9.656624e-01
* time: 358.51899099349976
290 1.011774e+03 9.656627e-01
* time: 360.0175750255585
291 1.011774e+03 9.660395e-01
* time: 361.33866596221924
292 1.011774e+03 9.661257e-01
* time: 362.6686108112335
293 1.011774e+03 9.663115e-01
* time: 363.98257303237915
294 1.011774e+03 9.662851e-01
* time: 365.31491589546204
295 1.011773e+03 9.654773e-01
* time: 366.6513338088989
296 1.011772e+03 9.623124e-01
* time: 368.1098918914795
297 1.011769e+03 9.521889e-01
* time: 369.6161618232727
298 1.011763e+03 9.239616e-01
* time: 371.19315099716187
299 1.011746e+03 9.242954e-01
* time: 373.293741941452
300 1.011711e+03 1.257052e+00
* time: 375.54978489875793
301 1.011651e+03 1.361278e+00
* time: 377.30393385887146
302 1.011591e+03 9.076916e-01
* time: 378.6857178211212
303 1.011568e+03 3.084752e-01
* time: 380.0839250087738
304 1.011565e+03 2.946404e-01
* time: 381.5140948295593
305 1.011565e+03 2.821273e-01
* time: 382.9698758125305
306 1.011565e+03 2.766260e-01
* time: 384.28178787231445
307 1.011565e+03 2.778010e-01
* time: 385.66018080711365
308 1.011565e+03 2.830281e-01
* time: 386.9875729084015
309 1.011565e+03 2.842619e-01
* time: 388.4366409778595
310 1.011565e+03 2.861093e-01
* time: 389.75864601135254
311 1.011565e+03 2.861138e-01
* time: 391.143257856369
312 1.011565e+03 2.867641e-01
* time: 392.45394682884216
313 1.011565e+03 2.867756e-01
* time: 393.91198992729187
314 1.011565e+03 2.884589e-01
* time: 395.2377698421478
315 1.011565e+03 2.897335e-01
* time: 396.62285685539246
316 1.011565e+03 2.927744e-01
* time: 397.94901394844055
317 1.011565e+03 2.967424e-01
* time: 399.4180338382721
318 1.011565e+03 3.027347e-01
* time: 400.74935698509216
319 1.011564e+03 3.099827e-01
* time: 402.12612891197205
320 1.011563e+03 3.155826e-01
* time: 403.43930101394653
321 1.011560e+03 3.082844e-01
* time: 404.84043192863464
322 1.011555e+03 2.625121e-01
* time: 406.2319118976593
323 1.011548e+03 2.136271e-01
* time: 407.6485140323639
324 1.011542e+03 1.165033e-01
* time: 408.9716808795929
325 1.011538e+03 3.914224e-02
* time: 410.3546018600464
326 1.011537e+03 3.083888e-02
* time: 411.66406893730164
327 1.011536e+03 2.766364e-02
* time: 413.0256938934326
328 1.011536e+03 1.873262e-02
* time: 414.33494091033936
329 1.011536e+03 5.453951e-03
* time: 415.70627903938293
330 1.011536e+03 4.882951e-03
* time: 417.0085759162903
331 1.011535e+03 4.331879e-03
* time: 418.3646969795227
332 1.011535e+03 2.711268e-03
* time: 419.6513948440552
333 1.011535e+03 4.474256e-04
* time: 421.00161480903625FittedPumasModel
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: 5.793571472167969e-5
1 4.804005e+05 8.280311e+05
* time: 7.477102994918823
2 3.595691e+05 6.035842e+05
* time: 14.20088005065918
3 1.691071e+05 2.702705e+05
* time: 21.26009202003479
4 9.193152e+04 1.451040e+05
* time: 28.267995834350586
5 4.768538e+04 6.630202e+04
* time: 35.49591302871704
6 2.904205e+04 3.416919e+04
* time: 42.994040966033936
7 1.840152e+04 1.671742e+04
* time: 50.62706398963928
8 1.280989e+04 1.022268e+04
* time: 57.74581003189087
9 9.713285e+03 9.298485e+03
* time: 64.78573393821716
10 7.628127e+03 8.330656e+03
* time: 71.87553095817566
11 6.236278e+03 7.340367e+03
* time: 78.99835085868835
12 5.271958e+03 6.303773e+03
* time: 85.93837285041809
13 4.411047e+03 4.994130e+03
* time: 93.09591698646545
14 3.732935e+03 3.538183e+03
* time: 100.34382605552673
15 3.399272e+03 2.450045e+03
* time: 107.76529693603516
16 3.302987e+03 1.848475e+03
* time: 116.36773204803467
17 3.291788e+03 1.642864e+03
* time: 124.13626599311829
18 3.290743e+03 1.594765e+03
* time: 131.5236358642578
19 3.289686e+03 1.562640e+03
* time: 138.44833898544312
20 3.286186e+03 1.496632e+03
* time: 145.53403496742249
21 3.277917e+03 1.402865e+03
* time: 152.424654006958
22 3.255958e+03 1.254296e+03
* time: 159.40076398849487
23 3.200979e+03 1.038621e+03
* time: 166.37606000900269
24 3.063520e+03 7.403602e+02
* time: 173.53216099739075
25 2.730315e+03 3.768701e+02
* time: 180.51789784431458
26 1.934186e+03 2.246121e+02
* time: 187.11174893379211
27 1.704399e+03 2.219826e+02
* time: 207.3830909729004
28 1.471184e+03 2.025825e+02
* time: 243.10038304328918
29 1.316789e+03 1.554715e+02
* time: 271.69800305366516
30 1.226363e+03 1.775721e+02
* time: 279.2252838611603
31 1.206626e+03 2.781030e+02
* time: 286.22108483314514
32 1.197830e+03 2.430603e+02
* time: 293.89589285850525
33 1.190342e+03 2.327966e+02
* time: 301.04170083999634
34 1.178784e+03 2.105878e+02
* time: 308.3179819583893
35 1.177853e+03 2.228653e+02
* time: 315.1965889930725
36 1.177764e+03 2.256182e+02
* time: 322.1340448856354
37 1.177746e+03 2.259425e+02
* time: 329.45884799957275
38 1.177670e+03 2.264699e+02
* time: 336.3993408679962
39 1.177500e+03 2.264641e+02
* time: 343.32913088798523
40 1.177034e+03 2.244207e+02
* time: 349.971079826355
41 1.175895e+03 2.160307e+02
* time: 356.5888228416443
42 1.173241e+03 1.906017e+02
* time: 363.5140469074249
43 1.168190e+03 1.332917e+02
* time: 370.73134303092957
44 1.161836e+03 5.291022e+01
* time: 377.73172402381897
45 1.158453e+03 6.595471e+01
* time: 384.55436086654663
46 1.157897e+03 6.411640e+01
* time: 391.4858829975128
47 1.157870e+03 6.098679e+01
* time: 398.848669052124
48 1.157868e+03 6.032481e+01
* time: 406.0109009742737
49 1.157864e+03 5.920991e+01
* time: 413.4781210422516
50 1.157856e+03 5.740417e+01
* time: 420.1143798828125
51 1.157832e+03 5.407031e+01
* time: 426.4162678718567
52 1.157774e+03 4.798188e+01
* time: 432.9398410320282
53 1.157628e+03 3.659183e+01
* time: 439.49397683143616
54 1.157299e+03 3.707862e+01
* time: 446.3502309322357
55 1.156690e+03 3.453183e+01
* time: 453.38539600372314
56 1.155959e+03 4.268473e+01
* time: 460.09679102897644
57 1.155557e+03 5.000309e+01
* time: 466.87878799438477
58 1.155472e+03 4.219411e+01
* time: 473.5911400318146
59 1.155463e+03 3.894982e+01
* time: 480.45707082748413
60 1.155462e+03 3.872028e+01
* time: 487.3881139755249
61 1.155458e+03 3.823435e+01
* time: 494.3337490558624
62 1.155450e+03 3.747328e+01
* time: 500.9610879421234
63 1.155426e+03 3.608641e+01
* time: 507.5727210044861
64 1.155365e+03 3.361168e+01
* time: 514.1553828716278
65 1.155207e+03 2.894844e+01
* time: 520.8957798480988
66 1.154814e+03 2.504332e+01
* time: 527.7928838729858
67 1.153910e+03 4.539953e+01
* time: 534.6967718601227
68 1.152231e+03 7.425240e+01
* time: 541.5543138980865
69 1.150287e+03 7.473669e+01
* time: 548.7501409053802
70 1.149189e+03 5.606074e+01
* time: 555.5845458507538
71 1.148944e+03 4.801310e+01
* time: 562.4573459625244
72 1.148932e+03 4.659470e+01
* time: 569.4465439319611
73 1.148930e+03 4.635167e+01
* time: 577.0207569599152
74 1.148923e+03 4.568576e+01
* time: 584.2102520465851
75 1.148907e+03 4.466140e+01
* time: 591.1954958438873
76 1.148865e+03 4.249800e+01
* time: 598.1929728984833
77 1.148758e+03 3.824645e+01
* time: 605.223993062973
78 1.148491e+03 3.552369e+01
* time: 611.9607219696045
79 1.147881e+03 3.037355e+01
* time: 619.3492300510406
80 1.146737e+03 4.466673e+01
* time: 626.246649980545
81 1.145347e+03 5.927932e+01
* time: 633.0484268665314
82 1.144536e+03 7.572710e+01
* time: 639.8638739585876
83 1.144354e+03 7.035223e+01
* time: 646.3561010360718
84 1.144341e+03 6.591571e+01
* time: 652.6499078273773
85 1.144340e+03 6.496308e+01
* time: 658.9436919689178
86 1.144333e+03 6.268335e+01
* time: 665.0039398670197
87 1.144320e+03 5.950296e+01
* time: 671.281919002533
88 1.144281e+03 5.369030e+01
* time: 677.4905068874359
89 1.144186e+03 4.394749e+01
* time: 684.0320069789886
90 1.143949e+03 4.102392e+01
* time: 690.8420238494873
91 1.143423e+03 3.752367e+01
* time: 697.2316830158234
92 1.142498e+03 3.276758e+01
* time: 703.6858758926392
93 1.141490e+03 5.217551e+01
* time: 710.2142848968506
94 1.140969e+03 4.309381e+01
* time: 716.6840188503265
95 1.140872e+03 4.759588e+01
* time: 724.3022840023041
96 1.140867e+03 4.727133e+01
* time: 731.530168056488
97 1.140866e+03 4.704459e+01
* time: 738.2815079689026
98 1.140862e+03 4.638036e+01
* time: 744.6689410209656
99 1.140853e+03 4.542717e+01
* time: 751.1110289096832
100 1.140829e+03 4.357571e+01
* time: 757.662871837616
101 1.140767e+03 4.023368e+01
* time: 764.4223909378052
102 1.140611e+03 3.376885e+01
* time: 771.2942938804626
103 1.140237e+03 2.856928e+01
* time: 777.9078240394592
104 1.139452e+03 3.894107e+01
* time: 784.5200688838959
105 1.138224e+03 5.063688e+01
* time: 791.7831268310547
106 1.137140e+03 4.762519e+01
* time: 798.1829779148102
107 1.136763e+03 4.605051e+01
* time: 804.5067319869995
108 1.136720e+03 3.993288e+01
* time: 810.8247258663177
109 1.136717e+03 3.890566e+01
* time: 817.2954909801483
110 1.136716e+03 3.883647e+01
* time: 823.8782708644867
111 1.136712e+03 3.865523e+01
* time: 830.4435939788818
112 1.136702e+03 3.838845e+01
* time: 836.9344520568848
113 1.136674e+03 3.790005e+01
* time: 843.4014558792114
114 1.136604e+03 3.705604e+01
* time: 849.9094970226288
115 1.136418e+03 3.550032e+01
* time: 856.9287269115448
116 1.135940e+03 3.257323e+01
* time: 863.3192019462585
117 1.134744e+03 2.697619e+01
* time: 869.6146769523621
118 1.132016e+03 4.861352e+01
* time: 876.00865483284
119 1.127206e+03 6.900228e+01
* time: 882.4315519332886
120 1.122365e+03 5.795238e+01
* time: 888.78635597229
121 1.119760e+03 2.908163e+01
* time: 895.0630879402161
122 1.118972e+03 2.137897e+01
* time: 901.3228008747101
123 1.118870e+03 1.909827e+01
* time: 907.5551149845123
124 1.118863e+03 1.886936e+01
* time: 913.7231669425964
125 1.118862e+03 1.852746e+01
* time: 919.9536299705505
126 1.118861e+03 1.864804e+01
* time: 926.0207929611206
127 1.118861e+03 1.862831e+01
* time: 932.1230659484863
128 1.118860e+03 1.865947e+01
* time: 938.2184588909149
129 1.118859e+03 1.869546e+01
* time: 944.3597228527069
130 1.118856e+03 1.878621e+01
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225 1.028175e+03 2.201733e+01
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229 1.028169e+03 2.178132e+01
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230 1.028160e+03 2.162024e+01
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* time: 1629.4710550308228
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235 1.026628e+03 7.187426e+01
* time: 1656.6105518341064
236 1.024510e+03 1.065337e+02
* time: 1663.3808720111847
237 1.020621e+03 1.329824e+02
* time: 1670.1320588588715
238 1.015709e+03 1.045562e+02
* time: 1676.8175840377808
239 1.014365e+03 2.918128e+01
* time: 1683.5658178329468
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* time: 1697.4801189899445
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* time: 1704.1334309577942
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* time: 1710.8311278820038
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* time: 1717.562665939331
245 1.012697e+03 9.968755e-01
* time: 1724.1515300273895
246 1.012697e+03 9.970599e-01
* time: 1730.7131569385529
247 1.012697e+03 9.971842e-01
* time: 1737.2003688812256
248 1.012697e+03 9.972169e-01
* time: 1743.7274239063263
249 1.012697e+03 9.972169e-01
* time: 1750.5629479885101
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* time: 1757.1029658317566
251 1.012697e+03 9.973078e-01
* time: 1763.6030659675598
252 1.012697e+03 9.972048e-01
* time: 1770.164715051651
253 1.012697e+03 9.970536e-01
* time: 1776.710932970047
254 1.012697e+03 9.967402e-01
* time: 1783.2858078479767
255 1.012697e+03 9.961565e-01
* time: 1789.8429749011993
256 1.012696e+03 9.949478e-01
* time: 1796.4373228549957
257 1.012694e+03 1.312115e+00
* time: 1803.016196012497
258 1.012690e+03 2.118030e+00
* time: 1809.6426079273224
259 1.012679e+03 3.352490e+00
* time: 1816.3227899074554
260 1.012651e+03 5.076526e+00
* time: 1822.902127981186
261 1.012594e+03 6.891495e+00
* time: 1829.5635678768158
262 1.012501e+03 7.239074e+00
* time: 1836.202301979065
263 1.012418e+03 4.479611e+00
* time: 1842.8770689964294
264 1.012390e+03 1.267149e+00
* time: 1849.507968902588
265 1.012386e+03 8.840304e-01
* time: 1856.1470608711243
266 1.012386e+03 8.549097e-01
* time: 1862.6924278736115
267 1.012386e+03 8.516869e-01
* time: 1869.1965990066528
268 1.012386e+03 8.516869e-01
* time: 1876.5875198841095
269 1.012386e+03 8.516869e-01
* time: 1884.451367855072
270 1.012386e+03 8.516817e-01
* time: 1891.5559208393097
271 1.012386e+03 8.516817e-01
* time: 1899.4079349040985
272 1.012386e+03 8.516817e-01
* time: 1907.3712079524994
273 1.012386e+03 8.516817e-01
* time: 1915.3160598278046
274 1.012386e+03 8.516817e-01
* time: 1923.8318769931793
275 1.012386e+03 8.516817e-01
* time: 1932.2589979171753FittedPumasModel
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: 6.604194641113281e-5
1 4.804005e+05 8.280312e+05
* time: 1.0638132095336914
2 3.595691e+05 6.035843e+05
* time: 2.0554821491241455
3 1.691071e+05 2.702705e+05
* time: 3.0566630363464355
4 9.193152e+04 1.451040e+05
* time: 3.9885292053222656
5 4.768539e+04 6.630202e+04
* time: 4.974685192108154
6 2.904205e+04 3.416919e+04
* time: 5.890917062759399
7 1.840153e+04 1.671742e+04
* time: 6.903837203979492
8 1.280989e+04 1.022268e+04
* time: 7.835824012756348
9 9.713287e+03 9.298485e+03
* time: 8.825250148773193
10 7.628127e+03 8.330656e+03
* time: 9.739957094192505
11 6.236278e+03 7.340367e+03
* time: 10.675734043121338
12 5.271958e+03 6.303773e+03
* time: 11.618905067443848
13 4.411047e+03 4.994130e+03
* time: 12.53760814666748
14 3.732936e+03 3.538184e+03
* time: 13.472005128860474
15 3.399273e+03 2.450045e+03
* time: 14.369054079055786
16 3.302987e+03 1.848476e+03
* time: 15.303011178970337
17 3.291788e+03 1.642864e+03
* time: 16.19191598892212
18 3.290743e+03 1.594765e+03
* time: 17.121148109436035
19 3.289687e+03 1.562641e+03
* time: 17.9896240234375
20 3.286186e+03 1.496632e+03
* time: 18.884274005889893
21 3.277917e+03 1.402865e+03
* time: 19.794742107391357
22 3.255959e+03 1.254297e+03
* time: 20.692471981048584
23 3.200980e+03 1.038621e+03
* time: 21.607270002365112
24 3.063521e+03 7.403605e+02
* time: 22.485438108444214
25 2.730316e+03 3.768701e+02
* time: 23.38705801963806
26 1.934186e+03 2.246121e+02
* time: 24.203185081481934
27 1.704399e+03 2.219826e+02
* time: 27.571081161499023
28 1.471183e+03 2.025826e+02
* time: 33.41727900505066
29 1.316789e+03 1.554714e+02
* time: 38.25019717216492
30 1.226363e+03 1.775666e+02
* time: 39.17122220993042
31 1.206626e+03 2.780980e+02
* time: 39.96491813659668
32 1.197830e+03 2.430550e+02
* time: 40.77828407287598
33 1.190342e+03 2.327905e+02
* time: 41.60248613357544
34 1.178783e+03 2.105809e+02
* time: 42.42159914970398
35 1.177853e+03 2.228585e+02
* time: 43.257270097732544
36 1.177763e+03 2.256116e+02
* time: 44.06376314163208
37 1.177745e+03 2.259359e+02
* time: 44.89934301376343
38 1.177670e+03 2.264633e+02
* time: 45.693615198135376
39 1.177500e+03 2.264574e+02
* time: 46.53985810279846
40 1.177033e+03 2.244136e+02
* time: 47.32967805862427
41 1.175894e+03 2.160226e+02
* time: 48.143856048583984
42 1.173240e+03 1.905912e+02
* time: 48.97563314437866
43 1.168189e+03 1.332783e+02
* time: 49.80517315864563
44 1.161835e+03 5.289952e+01
* time: 50.66275405883789
45 1.158453e+03 6.595387e+01
* time: 51.46930813789368
46 1.157897e+03 6.411419e+01
* time: 52.313053131103516
47 1.157870e+03 6.098509e+01
* time: 53.09910798072815
48 1.157868e+03 6.032325e+01
* time: 53.93692398071289
49 1.157864e+03 5.920812e+01
* time: 54.713226079940796
50 1.157856e+03 5.740225e+01
* time: 55.515012979507446
51 1.157832e+03 5.406795e+01
* time: 56.34084105491638
52 1.157774e+03 4.797879e+01
* time: 57.1380660533905
53 1.157628e+03 3.658732e+01
* time: 57.96812915802002
54 1.157299e+03 3.707874e+01
* time: 58.75883102416992
55 1.156690e+03 3.452947e+01
* time: 59.618051052093506
56 1.155959e+03 4.269004e+01
* time: 60.40790414810181
57 1.155557e+03 5.000322e+01
* time: 61.21708297729492
58 1.155472e+03 4.219349e+01
* time: 62.03867697715759
59 1.155463e+03 3.895049e+01
* time: 62.838136196136475
60 1.155462e+03 3.872099e+01
* time: 63.65830612182617
61 1.155458e+03 3.823491e+01
* time: 64.4443781375885
62 1.155450e+03 3.747370e+01
* time: 65.28828620910645
63 1.155426e+03 3.608648e+01
* time: 66.07372307777405
64 1.155365e+03 3.361118e+01
* time: 66.87791013717651
65 1.155207e+03 2.894678e+01
* time: 67.69921708106995
66 1.154814e+03 2.504284e+01
* time: 68.5037910938263
67 1.153910e+03 4.541126e+01
* time: 69.33251404762268
68 1.152230e+03 7.426044e+01
* time: 70.13192820549011
69 1.150286e+03 7.473044e+01
* time: 70.96388411521912
70 1.149188e+03 5.605967e+01
* time: 71.74550199508667
71 1.148944e+03 4.801123e+01
* time: 72.57702016830444
72 1.148931e+03 4.659478e+01
* time: 73.34148216247559
73 1.148929e+03 4.635180e+01
* time: 74.12143015861511
74 1.148922e+03 4.568579e+01
* time: 74.92522597312927
75 1.148907e+03 4.466128e+01
* time: 75.70610117912292
76 1.148864e+03 4.249742e+01
* time: 76.51532316207886
77 1.148758e+03 3.824840e+01
* time: 77.29004502296448
78 1.148490e+03 3.552459e+01
* time: 78.11482810974121
79 1.147881e+03 3.038181e+01
* time: 78.883061170578
80 1.146736e+03 4.467281e+01
* time: 79.67513918876648
81 1.145347e+03 5.929158e+01
* time: 80.48186016082764
82 1.144535e+03 7.572784e+01
* time: 81.27076697349548
83 1.144354e+03 7.034986e+01
* time: 82.07738018035889
84 1.144341e+03 6.591584e+01
* time: 82.84351205825806
85 1.144340e+03 6.496353e+01
* time: 83.64573311805725
86 1.144333e+03 6.268267e+01
* time: 84.4027910232544
87 1.144320e+03 5.950121e+01
* time: 85.22377920150757
88 1.144281e+03 5.368613e+01
* time: 85.97701716423035
89 1.144186e+03 4.393956e+01
* time: 86.7586841583252
90 1.143949e+03 4.102324e+01
* time: 87.56198501586914
91 1.143423e+03 3.752005e+01
* time: 88.34125518798828
92 1.142497e+03 3.278884e+01
* time: 89.15589809417725
93 1.141489e+03 5.217814e+01
* time: 89.92629599571228
94 1.140969e+03 4.310032e+01
* time: 90.74452209472656
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* time: 329.331463098526FittedPumasModel
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.