using Pumas
using AlgebraOfGraphics
using CairoMakie
using StableRNGs
Statistical Models Without Differential Equations
1 Introduction
This tutorials 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: 0.05872011184692383
1 6.818291e+04 2.824317e+05
* time: 1.0988500118255615
2 4.291959e+04 9.008422e+04
* time: 1.0990428924560547
3 2.526194e+04 7.438611e+04
* time: 1.099174976348877
4 1.555090e+04 5.381298e+04
* time: 1.0993099212646484
5 9.814559e+03 6.070009e+04
* time: 1.0994439125061035
6 7.614303e+03 5.099953e+04
* time: 1.099581003189087
7 6.795333e+03 3.666906e+04
* time: 1.0997118949890137
8 6.332366e+03 2.139158e+04
* time: 1.0998420715332031
9 5.860625e+03 1.311493e+04
* time: 1.0999820232391357
10 5.219282e+03 1.507053e+04
* time: 1.100121021270752
11 4.162225e+03 1.424461e+04
* time: 1.1002519130706787
12 2.938417e+03 1.622318e+04
* time: 1.1003799438476562
13 2.319141e+03 1.430247e+04
* time: 1.1005079746246338
14 1.257223e+03 6.948150e+03
* time: 1.1006360054016113
15 7.239484e+02 3.073878e+03
* time: 1.1007640361785889
16 4.221635e+02 1.591054e+03
* time: 1.1009011268615723
17 2.681901e+02 8.356172e+02
* time: 1.1010310649871826
18 1.904231e+02 4.511418e+02
* time: 1.1011650562286377
19 1.552170e+02 2.590796e+02
* time: 1.101294994354248
20 1.412507e+02 1.609948e+02
* time: 1.101426124572754
21 1.368116e+02 1.106316e+02
* time: 1.1015739440917969
22 1.357202e+02 8.485850e+01
* time: 1.101728916168213
23 1.353898e+02 7.013620e+01
* time: 1.101881980895996
24 1.350937e+02 5.450614e+01
* time: 1.1020519733428955
25 1.347410e+02 3.733061e+01
* time: 1.102186918258667
26 1.345492e+02 6.266036e+01
* time: 1.102320909500122
27 1.345033e+02 6.975752e+01
* time: 1.1024510860443115
28 1.344943e+02 6.841408e+01
* time: 1.1025869846343994
29 1.344938e+02 6.745130e+01
* time: 1.1027159690856934
30 1.344931e+02 6.665860e+01
* time: 1.102863073348999
31 1.344910e+02 6.496068e+01
* time: 1.1030011177062988
32 1.344858e+02 6.244698e+01
* time: 1.1031360626220703
33 1.344718e+02 5.817264e+01
* time: 1.1032700538635254
34 1.344356e+02 5.122402e+01
* time: 1.1034040451049805
35 1.343407e+02 4.121832e+01
* time: 1.1035349369049072
36 1.340960e+02 5.595102e+01
* time: 1.1036689281463623
37 1.334812e+02 7.977600e+01
* time: 1.1038000583648682
38 1.320677e+02 1.141588e+02
* time: 1.104046106338501
39 1.295160e+02 1.330673e+02
* time: 1.1042749881744385
40 1.270177e+02 7.901632e+01
* time: 1.1045000553131104
41 1.264533e+02 2.155415e+01
* time: 1.1047260761260986
42 1.262863e+02 8.225580e+00
* time: 1.1049580574035645
43 1.262719e+02 6.032004e+00
* time: 1.1051840782165527
44 1.262707e+02 4.659021e+00
* time: 1.105407953262329
45 1.262707e+02 4.722942e+00
* time: 1.1056339740753174
46 1.262706e+02 4.816900e+00
* time: 1.1058580875396729
47 1.262705e+02 4.995290e+00
* time: 1.1060900688171387
48 1.262702e+02 5.268485e+00
* time: 1.1063110828399658
49 1.262694e+02 5.722605e+00
* time: 1.1065359115600586
50 1.262674e+02 6.456700e+00
* time: 1.106760025024414
51 1.262619e+02 7.663981e+00
* time: 1.106990098953247
52 1.262474e+02 9.659510e+00
* time: 1.1072180271148682
53 1.262080e+02 1.301621e+01
* time: 1.1074459552764893
54 1.260905e+02 1.869585e+01
* time: 1.107672929763794
55 1.254864e+02 1.996336e+01
* time: 1.1079111099243164
56 1.242755e+02 2.959843e+01
* time: 1.1081459522247314
57 1.218934e+02 3.470690e+01
* time: 1.1083710193634033
58 1.211416e+02 1.164980e+02
* time: 1.1086170673370361
59 1.173281e+02 1.882403e+02
* time: 1.1088590621948242
60 1.148987e+02 6.454397e+01
* time: 1.1090960502624512
61 1.142754e+02 5.925562e+01
* time: 1.1093389987945557
62 1.139379e+02 4.781775e+01
* time: 1.1095659732818604
63 1.134167e+02 4.884944e+01
* time: 1.1097900867462158
64 1.110742e+02 6.710274e+01
* time: 1.110023021697998
65 1.071014e+02 7.528856e+01
* time: 1.1102509498596191
66 1.048398e+02 6.135959e+01
* time: 1.1104960441589355
67 1.031549e+02 6.415724e+01
* time: 1.1107380390167236
68 1.023007e+02 1.955391e+01
* time: 1.1109840869903564
69 1.019906e+02 1.355104e+01
* time: 1.1112089157104492
70 1.019243e+02 1.030982e+01
* time: 1.1114389896392822
71 1.019192e+02 1.001753e+00
* time: 1.1116650104522705
72 1.019190e+02 3.009678e-01
* time: 1.1119070053100586
73 1.019189e+02 4.438935e-02
* time: 1.1121349334716797
74 1.019189e+02 1.971922e-03
* time: 1.1123590469360352
75 1.019189e+02 2.678153e-05
* time: 1.112584114074707FittedPumasModel
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.