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: 0.024302959442138672
1 6.818291e+04 2.824317e+05
* time: 0.8377480506896973
2 4.291959e+04 9.008422e+04
* time: 0.8378899097442627
3 2.526194e+04 7.438611e+04
* time: 0.8379960060119629
4 1.555090e+04 5.381298e+04
* time: 0.8381030559539795
5 9.814559e+03 6.070009e+04
* time: 0.838209867477417
6 7.614303e+03 5.099953e+04
* time: 0.8383140563964844
7 6.795333e+03 3.666906e+04
* time: 0.838433027267456
8 6.332366e+03 2.139158e+04
* time: 0.8385329246520996
9 5.860625e+03 1.311493e+04
* time: 0.838629961013794
10 5.219282e+03 1.507053e+04
* time: 0.8387219905853271
11 4.162225e+03 1.424461e+04
* time: 0.838813066482544
12 2.938417e+03 1.622318e+04
* time: 0.8389039039611816
13 2.319141e+03 1.430247e+04
* time: 0.8389959335327148
14 1.257223e+03 6.948150e+03
* time: 0.8390848636627197
15 7.239484e+02 3.073878e+03
* time: 0.8391809463500977
16 4.221635e+02 1.591054e+03
* time: 0.8392829895019531
17 2.681901e+02 8.356172e+02
* time: 0.8393828868865967
18 1.904231e+02 4.511418e+02
* time: 0.8394749164581299
19 1.552170e+02 2.590796e+02
* time: 0.8395669460296631
20 1.412507e+02 1.609948e+02
* time: 0.8396589756011963
21 1.368116e+02 1.106316e+02
* time: 0.8397700786590576
22 1.357202e+02 8.485850e+01
* time: 0.8398690223693848
23 1.353898e+02 7.013620e+01
* time: 0.8399829864501953
24 1.350937e+02 5.450614e+01
* time: 0.8401010036468506
25 1.347410e+02 3.733061e+01
* time: 0.8401939868927002
26 1.345492e+02 6.266036e+01
* time: 0.8402879238128662
27 1.345033e+02 6.975752e+01
* time: 0.8403830528259277
28 1.344943e+02 6.841408e+01
* time: 0.8404750823974609
29 1.344938e+02 6.745130e+01
* time: 0.8405659198760986
30 1.344931e+02 6.665860e+01
* time: 0.8406569957733154
31 1.344910e+02 6.496068e+01
* time: 0.8407459259033203
32 1.344858e+02 6.244698e+01
* time: 0.8408360481262207
33 1.344718e+02 5.817264e+01
* time: 0.8409249782562256
34 1.344356e+02 5.122402e+01
* time: 0.8410129547119141
35 1.343407e+02 4.121832e+01
* time: 0.841101884841919
36 1.340960e+02 5.595102e+01
* time: 0.8411920070648193
37 1.334812e+02 7.977600e+01
* time: 0.8412809371948242
38 1.320677e+02 1.141588e+02
* time: 0.8413960933685303
39 1.295160e+02 1.330673e+02
* time: 0.8415908813476562
40 1.270177e+02 7.901632e+01
* time: 0.8417809009552002
41 1.264533e+02 2.155415e+01
* time: 0.8419690132141113
42 1.262863e+02 8.225580e+00
* time: 0.8421599864959717
43 1.262719e+02 6.032004e+00
* time: 0.8423600196838379
44 1.262707e+02 4.659021e+00
* time: 0.842552900314331
45 1.262707e+02 4.722942e+00
* time: 0.842742919921875
46 1.262706e+02 4.816900e+00
* time: 0.842932939529419
47 1.262705e+02 4.995290e+00
* time: 0.8431239128112793
48 1.262702e+02 5.268485e+00
* time: 0.843311071395874
49 1.262694e+02 5.722605e+00
* time: 0.8435089588165283
50 1.262674e+02 6.456700e+00
* time: 0.8437080383300781
51 1.262619e+02 7.663981e+00
* time: 0.8438959121704102
52 1.262474e+02 9.659510e+00
* time: 0.8440830707550049
53 1.262080e+02 1.301621e+01
* time: 0.8442690372467041
54 1.260905e+02 1.869585e+01
* time: 0.8444619178771973
55 1.254864e+02 1.996336e+01
* time: 0.8446478843688965
56 1.242755e+02 2.959843e+01
* time: 0.8448350429534912
57 1.218934e+02 3.470690e+01
* time: 0.8450210094451904
58 1.211416e+02 1.164980e+02
* time: 0.845221996307373
59 1.173281e+02 1.882403e+02
* time: 0.8454279899597168
60 1.148987e+02 6.454397e+01
* time: 0.8456149101257324
61 1.142754e+02 5.925562e+01
* time: 0.845815896987915
62 1.139379e+02 4.781775e+01
* time: 0.8460009098052979
63 1.134167e+02 4.884944e+01
* time: 0.8461868762969971
64 1.110742e+02 6.710274e+01
* time: 0.8463809490203857
65 1.071014e+02 7.528856e+01
* time: 0.8465690612792969
66 1.048398e+02 6.135959e+01
* time: 0.8467729091644287
67 1.031549e+02 6.415724e+01
* time: 0.8469760417938232
68 1.023007e+02 1.955391e+01
* time: 0.8471739292144775
69 1.019906e+02 1.355104e+01
* time: 0.8473670482635498
70 1.019243e+02 1.030982e+01
* time: 0.8475558757781982
71 1.019192e+02 1.001753e+00
* time: 0.8477439880371094
72 1.019190e+02 3.009678e-01
* time: 0.8479480743408203
73 1.019189e+02 4.438935e-02
* time: 0.8481349945068359
74 1.019189e+02 1.971922e-03
* time: 0.848322868347168
75 1.019189e+02 2.678153e-05
* time: 0.8485159873962402FittedPumasModel
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.