using Pumas
using AlgebraOfGraphics
using CairoMakie
using StableRNGs
Statistical Models Without Differential Equations
1 Introduction
This tutorial represents a slight deviation from most other models presented and performed in Pumas that often includes pharmacokinetic modeling. Here, we will focus on simple Exposure-Response (ER) models. ER models can be models of efficacy, safety, toxicity, or any other response. The response can be continuous in nature, it can be binary, or even multinomial such as ordinal pain score models.
The main point of the tutorial is to introduce the situation where there is data that does not come with an inherent dynamical system component. Instead, we want to study the effect of an exposure measure such as plasma concentration (potentially as a proxy for biophase concentration) or AUC on some kind of response. The response could be almost any response you can think of from the total area of the body affected by eczema, a pain-score, pain relief indicator, body weight, presence of vomiting, or any other efficacy, safety, or toxicity measure. To keep things simple and focus on higher level ideas we go with a sigmoidal Emax model (with a Hill exponent) for some hypothetical response given some level of plasma concentrations.
# Data generating parameters
Emax = 100
C50 = 40
h = 2
# Function to evaluate the Emax model
hill_model(cp, emax, c50, hill) = emax * cp^hill / (c50^hill + cp^hill)hill_model (generic function with 1 method)The Hill function is well-known in pharmacodynamics when we have an exposure-response relationship that has an upper effect limit. The Hill parameter allows us to change where in the exposure domain the change in effect strongest. It includes two additional parameters: C50 and Emax. First, is the half maximal effective concentration C50 (sometimes EC50) parameter that is the exposure (here concentration, hence C) that leads to half of the response in excess of a baseline value. Here, the baseline is zero. Second, is the parameter Emax that is simply the maximum effect or response that can occur. If we sample with a technique that truly has additive error the observed maximal effect could in principle be above Emax.
Let us draw such a function with the data generating parameters.
Show plotting code
hill_plot =
data((exposure = 0:200, effect = hill_model.(0:200, Emax, C50, h))) *
mapping(:exposure, :effect) *
visual(Lines) +
data((exposure = [C50], effect = [0, hill_model(C50, Emax, C50, h)])) *
mapping(:exposure, :effect) *
visual(Lines; linestyle = :dot) +
data((exposure = [0, C50], effect = [hill_model(C50, Emax, C50, h)])) *
mapping(:exposure, :effect) *
visual(Lines; linestyle = :dot) +
data((exposure = [0, 200], effect = [Emax])) *
mapping(:exposure, :effect) *
visual(Lines; linestyle = :dot)
axis_spec = (;
axis = (
limits = (0, 200, 0, Emax + 10),
xticks = ([0, C50, 100, 200], ["0", "C50", "100", "200"]),
yticks = ([0, 25, Emax / 2, 75, Emax], ["0", "25", "Emax/2", "75", "Emax"]),
)
)
draw(hill_plot; axis_spec...)
Since we have no model to explain the exposure but it is rather just a measured quantity we will simply sample exposures for this example. To ensure proper behavior of the estimator, we will make sure to sample well above C50.
2 Data Without Any Events
One basic property of the data for this kind of analysis we need to consider is that there are no events in the provided data. By events we think of the usual dose events associated with infusion rates and durations, bolus dose amounts, and other information about the specific mechanisms at play. We normally parse this information and use it to explicitly build a model for the trajectories of pharmacokinetic variables and how they might influence pharmacodynamics - but not here! We still require a time column, but often it may only be used to separate individual observations within subjects instead of being used for dynamical modeling.
Instead of using a predefined dataset we will construct a simple one as described in the previous section using sampled concentrations, the Hill model, and an additive error model. We need at least id, time, and observations to define an eventless dataset, but to drive the Emax model we need to include cp_i that are the measured or predicted exposures. The observations will be called resp here for response.
# Define the number of concentrations to sample
N = 40
# Define the random number generator
rng = StableRNG(983)
# Sample concentrations from a log-normal distribution
cp_i = rand(rng, LogNormal(log(C50 + 5), 0.6), N)
# Generate response variables given the exposure, cp_i and parameters for the Hill model
resp = @. rand(rng, Normal(hill_model(cp_i, Emax, C50, h), 3.2))
# Combine results into a DataFrame
response_df = DataFrame(id = 1:N, time = 1, cp_i = cp_i, resp = resp)40×4 DataFrame
15 rows omitted
| Row | id | time | cp_i | resp |
|---|---|---|---|---|
| Int64 | Int64 | Float64 | Float64 | |
| 1 | 1 | 1 | 29.9617 | 32.829 |
| 2 | 2 | 1 | 67.714 | 78.3184 |
| 3 | 3 | 1 | 58.7976 | 69.3724 |
| 4 | 4 | 1 | 37.54 | 46.1287 |
| 5 | 5 | 1 | 58.5205 | 67.6474 |
| 6 | 6 | 1 | 51.4522 | 61.8933 |
| 7 | 7 | 1 | 106.938 | 84.1643 |
| 8 | 8 | 1 | 17.5912 | 21.9294 |
| 9 | 9 | 1 | 14.974 | 11.6478 |
| 10 | 10 | 1 | 40.0817 | 47.5374 |
| 11 | 11 | 1 | 67.7002 | 77.037 |
| 12 | 12 | 1 | 38.6564 | 43.8903 |
| 13 | 13 | 1 | 31.8099 | 40.5693 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
| 29 | 29 | 1 | 109.159 | 82.9605 |
| 30 | 30 | 1 | 21.5369 | 22.6964 |
| 31 | 31 | 1 | 79.237 | 82.8221 |
| 32 | 32 | 1 | 94.6278 | 82.5703 |
| 33 | 33 | 1 | 53.9975 | 59.4201 |
| 34 | 34 | 1 | 44.5625 | 52.5194 |
| 35 | 35 | 1 | 130.494 | 92.3875 |
| 36 | 36 | 1 | 66.5144 | 75.9562 |
| 37 | 37 | 1 | 18.7343 | 22.4305 |
| 38 | 38 | 1 | 62.8681 | 71.8646 |
| 39 | 39 | 1 | 63.7703 | 73.3971 |
| 40 | 40 | 1 | 48.1412 | 64.6768 |
Show plotting code
draw(hill_plot + data(response_df) * mapping(:cp_i, :resp) * visual(Scatter); axis_spec...)
2.1 Defining Pumas Population Without Events
To map from tabular data in response_df to a Population in response_pop we use read_pumas just as we did in the case with event data. The important part is to turn off event_data to disable checks that are not relevant to this eventless example. If event_data is not set to false we would get errors about missing event columns for example.
response_pop = read_pumas(
response_df,
id = :id,
time = :time,
covariates = [:cp_i],
observations = [:resp],
event_data = false,
)Population
Subjects: 40
Covariates: cp_i
Observations: resp2.2 A Model Without Dynamics
Above we simply defined the input exposure and output response as simple arrays and function evaluations to defer the PumasModel definition until after seeing how to load eventless datasets. However, to perform a maximum likelihood fit we need to define a proper PumasModel.
response_model = @model begin
@param begin
θemax ∈ RealDomain(lower = 0, init = 90)
θc50 ∈ RealDomain(lower = 0, init = 30)
θhill ∈ RealDomain(lower = 0, init = 3)
σ ∈ RealDomain(lower = 1e-5, init = 0.1)
end
@covariates cp_i
@pre begin
emax_i = hill_model(cp_i, θemax, θc50, θhill)
end
@derived begin
resp ~ @. Normal(emax_i, σ)
end
endPumasModel
Parameters: θemax, θc50, θhill, σ
Random effects:
Covariates: cp_i
Dynamical system variables:
Dynamical system type: No dynamical model
Derived: resp
Observed: resp2.3 Fitting
To fit the model, we simply invoke fit with the model, population, parameters, and likelihood approximation method. Since there are no random effects in this model there is no integral to approximate so we use NaivePooled(). This will perform a maximum likelihood estimation according to the distribution used in @derived. The parameters in this case will match a non-linear least squares fit since we have an additive gaussian error model, but inference may deviate as usual.
emax_fit = fit(response_model, response_pop, init_params(response_model), NaivePooled())[ 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 2.191392e+05 2.289229e+06
* time: 0.022535085678100586
1 6.818291e+04 2.824317e+05
* time: 0.807898998260498
2 4.291959e+04 9.008422e+04
* time: 0.8080430030822754
3 2.526194e+04 7.438611e+04
* time: 0.8081459999084473
4 1.555090e+04 5.381298e+04
* time: 0.8822159767150879
5 9.814559e+03 6.070009e+04
* time: 0.8826761245727539
6 7.614303e+03 5.099953e+04
* time: 0.8828339576721191
7 6.795333e+03 3.666906e+04
* time: 0.8829529285430908
8 6.332366e+03 2.139158e+04
* time: 0.8830690383911133
9 5.860625e+03 1.311493e+04
* time: 0.8831839561462402
10 5.219282e+03 1.507053e+04
* time: 0.8833069801330566
11 4.162225e+03 1.424461e+04
* time: 0.8834319114685059
12 2.938417e+03 1.622318e+04
* time: 0.8835511207580566
13 2.319141e+03 1.430247e+04
* time: 0.8836641311645508
14 1.257223e+03 6.948150e+03
* time: 0.8837769031524658
15 7.239484e+02 3.073878e+03
* time: 0.8838949203491211
16 4.221635e+02 1.591054e+03
* time: 0.8840091228485107
17 2.681901e+02 8.356172e+02
* time: 0.8841230869293213
18 1.904231e+02 4.511418e+02
* time: 0.8842370510101318
19 1.552170e+02 2.590796e+02
* time: 0.8843889236450195
20 1.412507e+02 1.609948e+02
* time: 0.8845109939575195
21 1.368116e+02 1.106316e+02
* time: 0.8846230506896973
22 1.357202e+02 8.485850e+01
* time: 0.884735107421875
23 1.353898e+02 7.013620e+01
* time: 0.8848519325256348
24 1.350937e+02 5.450614e+01
* time: 0.8849639892578125
25 1.347410e+02 3.733061e+01
* time: 0.8850750923156738
26 1.345492e+02 6.266036e+01
* time: 0.8851850032806396
27 1.345033e+02 6.975752e+01
* time: 0.8853249549865723
28 1.344943e+02 6.841408e+01
* time: 0.8854680061340332
29 1.344938e+02 6.745130e+01
* time: 0.8855841159820557
30 1.344931e+02 6.665860e+01
* time: 0.8856940269470215
31 1.344910e+02 6.496068e+01
* time: 0.8858051300048828
32 1.344858e+02 6.244698e+01
* time: 0.885915994644165
33 1.344718e+02 5.817264e+01
* time: 0.8860270977020264
34 1.344356e+02 5.122402e+01
* time: 0.8861410617828369
35 1.343407e+02 4.121832e+01
* time: 0.8862500190734863
36 1.340960e+02 5.595102e+01
* time: 0.8863699436187744
37 1.334812e+02 7.977600e+01
* time: 0.8864800930023193
38 1.320677e+02 1.141588e+02
* time: 0.8865909576416016
39 1.295160e+02 1.330673e+02
* time: 0.8867020606994629
40 1.270177e+02 7.901632e+01
* time: 0.8868129253387451
41 1.264533e+02 2.155415e+01
* time: 0.8869240283966064
42 1.262863e+02 8.225580e+00
* time: 0.8870360851287842
43 1.262719e+02 6.032004e+00
* time: 0.8871490955352783
44 1.262707e+02 4.659021e+00
* time: 0.8872590065002441
45 1.262707e+02 4.722942e+00
* time: 0.8873770236968994
46 1.262706e+02 4.816900e+00
* time: 0.88749098777771
47 1.262705e+02 4.995290e+00
* time: 0.8876171112060547
48 1.262702e+02 5.268485e+00
* time: 0.8877499103546143
49 1.262694e+02 5.722605e+00
* time: 0.8878879547119141
50 1.262674e+02 6.456700e+00
* time: 0.8880059719085693
51 1.262619e+02 7.663981e+00
* time: 0.8881199359893799
52 1.262474e+02 9.659510e+00
* time: 0.8882339000701904
53 1.262080e+02 1.301621e+01
* time: 0.888355016708374
54 1.260905e+02 1.869585e+01
* time: 0.8884730339050293
55 1.254864e+02 1.996336e+01
* time: 0.888617992401123
56 1.242755e+02 2.959843e+01
* time: 0.8887369632720947
57 1.218934e+02 3.470690e+01
* time: 0.8888521194458008
58 1.211416e+02 1.164980e+02
* time: 0.888983964920044
59 1.173281e+02 1.882403e+02
* time: 0.8891229629516602
60 1.148987e+02 6.454397e+01
* time: 0.8892369270324707
61 1.142754e+02 5.925562e+01
* time: 0.8893709182739258
62 1.139379e+02 4.781775e+01
* time: 0.8894879817962646
63 1.134167e+02 4.884944e+01
* time: 0.8896040916442871
64 1.110742e+02 6.710274e+01
* time: 0.8897650241851807
65 1.071014e+02 7.528856e+01
* time: 0.889970064163208
66 1.048398e+02 6.135959e+01
* time: 0.890178918838501
67 1.031549e+02 6.415724e+01
* time: 0.8903930187225342
68 1.023007e+02 1.955391e+01
* time: 0.8906021118164062
69 1.019906e+02 1.355104e+01
* time: 0.8907890319824219
70 1.019243e+02 1.030982e+01
* time: 0.8909769058227539
71 1.019192e+02 1.001753e+00
* time: 0.8911669254302979
72 1.019190e+02 3.009678e-01
* time: 0.8913600444793701
73 1.019189e+02 4.438935e-02
* time: 0.8915491104125977
74 1.019189e+02 1.971922e-03
* time: 0.8917369842529297
75 1.019189e+02 2.678153e-05
* time: 0.8919270038604736FittedPumasModel
Successful minimization: true
Likelihood approximation: NaivePooled
Likelihood Optimizer: BFGS
Dynamical system type: No dynamical model
Log-likelihood value: -101.91895
Number of subjects: 40
Number of parameters: Fixed Optimized
0 4
Observation records: Active Missing
resp: 40 0
Total: 40 0
-------------------
Estimate
-------------------
θemax 104.13
θc50 41.558
θhill 1.8147
σ 3.0927
-------------------and we may use the usual workflow to get estimates of parameter uncertainty
infer(emax_fit)[ Info: Calculating: variance-covariance matrix.
[ Info: Done.Asymptotic inference results using sandwich estimator
Successful minimization: true
Likelihood approximation: NaivePooled
Likelihood Optimizer: BFGS
Dynamical system type: No dynamical model
Log-likelihood value: -101.91895
Number of subjects: 40
Number of parameters: Fixed Optimized
0 4
Observation records: Active Missing
resp: 40 0
Total: 40 0
---------------------------------------------------------
Estimate SE 95.0% C.I.
---------------------------------------------------------
θemax 104.13 4.1798 [95.942 ; 112.33 ]
θc50 41.558 2.1123 [37.418 ; 45.698 ]
θhill 1.8147 0.1155 [ 1.5883; 2.0411]
σ 3.0927 0.27055 [ 2.5624; 3.6229]
---------------------------------------------------------as well as inspect. Diagnostics for these models are typically simple enough that they can be constructed from the DataFrame constructed from inspect output.
2.4 Extensions
Since we used a normal PumasModel we can extend the response analysis with:
- covariate effects including time, dose level, etc
- random effects if there are multiple observations per subject
- more complicated response models such as binary response and ordinal response
3 Concluding Remarks
This tutorial introduced a simple Exposure-Response (ER) modeling approach using the Emax model, focusing on cases where no explicit time component or dynamic system is involved. By working with eventless data, we demonstrated how to structure the dataset, define a model without pharmacokinetics, and fit it using maximum likelihood estimation in Pumas.
The methodology presented provides a foundation for analyzing ER relationships in various contexts, from efficacy assessments to safety evaluations. While we used a basic model, the approach can be extended to incorporate covariate effects, random effects, and more complex response types, making it highly adaptable to different modeling needs. By leveraging these capabilities, users can perform robust ER analyses even in the absence of traditional pharmacokinetic modeling constraints.