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: 0.0308229923248291
1 6.818291e+04 2.824317e+05
* time: 1.1940038204193115
2 4.291959e+04 9.008422e+04
* time: 1.1942219734191895
3 2.526194e+04 7.438611e+04
* time: 1.194394826889038
4 1.555090e+04 5.381298e+04
* time: 1.1945698261260986
5 9.814559e+03 6.070009e+04
* time: 1.1947410106658936
6 7.614303e+03 5.099953e+04
* time: 1.1949188709259033
7 6.795333e+03 3.666906e+04
* time: 1.1950910091400146
8 6.332366e+03 2.139158e+04
* time: 1.1952710151672363
9 5.860625e+03 1.311493e+04
* time: 1.195436954498291
10 5.219282e+03 1.507053e+04
* time: 1.1956019401550293
11 4.162225e+03 1.424461e+04
* time: 1.1957659721374512
12 2.938417e+03 1.622318e+04
* time: 1.1959328651428223
13 2.319141e+03 1.430247e+04
* time: 1.1960978507995605
14 1.257223e+03 6.948150e+03
* time: 1.1962649822235107
15 7.239484e+02 3.073878e+03
* time: 1.196429967880249
16 4.221635e+02 1.591054e+03
* time: 1.1966049671173096
17 2.681901e+02 8.356172e+02
* time: 1.1967709064483643
18 1.904231e+02 4.511418e+02
* time: 1.1969399452209473
19 1.552170e+02 2.590796e+02
* time: 1.1971030235290527
20 1.412507e+02 1.609948e+02
* time: 1.1972689628601074
21 1.368116e+02 1.106316e+02
* time: 1.1974530220031738
22 1.357202e+02 8.485850e+01
* time: 1.1976368427276611
23 1.353898e+02 7.013620e+01
* time: 1.1978638172149658
24 1.350937e+02 5.450614e+01
* time: 1.1980400085449219
25 1.347410e+02 3.733061e+01
* time: 1.198213815689087
26 1.345492e+02 6.266036e+01
* time: 1.1983859539031982
27 1.345033e+02 6.975752e+01
* time: 1.1985549926757812
28 1.344943e+02 6.841408e+01
* time: 1.1987240314483643
29 1.344938e+02 6.745130e+01
* time: 1.1989119052886963
30 1.344931e+02 6.665860e+01
* time: 1.199084997177124
31 1.344910e+02 6.496068e+01
* time: 1.1992528438568115
32 1.344858e+02 6.244698e+01
* time: 1.1994190216064453
33 1.344718e+02 5.817264e+01
* time: 1.1995868682861328
34 1.344356e+02 5.122402e+01
* time: 1.1997559070587158
35 1.343407e+02 4.121832e+01
* time: 1.1999340057373047
36 1.340960e+02 5.595102e+01
* time: 1.2001240253448486
37 1.334812e+02 7.977600e+01
* time: 1.200286865234375
38 1.320677e+02 1.141588e+02
* time: 1.2005579471588135
39 1.295160e+02 1.330673e+02
* time: 1.2008368968963623
40 1.270177e+02 7.901632e+01
* time: 1.2011189460754395
41 1.264533e+02 2.155415e+01
* time: 1.201387882232666
42 1.262863e+02 8.225580e+00
* time: 1.2016608715057373
43 1.262719e+02 6.032004e+00
* time: 1.201936960220337
44 1.262707e+02 4.659021e+00
* time: 1.2022278308868408
45 1.262707e+02 4.722942e+00
* time: 1.2025399208068848
46 1.262706e+02 4.816900e+00
* time: 1.20281982421875
47 1.262705e+02 4.995290e+00
* time: 1.203110933303833
48 1.262702e+02 5.268485e+00
* time: 1.2033970355987549
49 1.262694e+02 5.722605e+00
* time: 1.203678846359253
50 1.262674e+02 6.456700e+00
* time: 1.2039709091186523
51 1.262619e+02 7.663981e+00
* time: 1.2042629718780518
52 1.262474e+02 9.659510e+00
* time: 1.204542875289917
53 1.262080e+02 1.301621e+01
* time: 1.2048108577728271
54 1.260905e+02 1.869585e+01
* time: 1.2050879001617432
55 1.254864e+02 1.996336e+01
* time: 1.2053589820861816
56 1.242755e+02 2.959843e+01
* time: 1.2056338787078857
57 1.218934e+02 3.470690e+01
* time: 1.2059149742126465
58 1.211416e+02 1.164980e+02
* time: 1.2062358856201172
59 1.173281e+02 1.882403e+02
* time: 1.2065598964691162
60 1.148987e+02 6.454397e+01
* time: 1.2068359851837158
61 1.142754e+02 5.925562e+01
* time: 1.2071619033813477
62 1.139379e+02 4.781775e+01
* time: 1.2074458599090576
63 1.134167e+02 4.884944e+01
* time: 1.2077348232269287
64 1.110742e+02 6.710274e+01
* time: 1.208009958267212
65 1.071014e+02 7.528856e+01
* time: 1.2082769870758057
66 1.048398e+02 6.135959e+01
* time: 1.2085769176483154
67 1.031549e+02 6.415724e+01
* time: 1.2088649272918701
68 1.023007e+02 1.955391e+01
* time: 1.2091679573059082
69 1.019906e+02 1.355104e+01
* time: 1.2094388008117676
70 1.019243e+02 1.030982e+01
* time: 1.2097089290618896
71 1.019192e+02 1.001753e+00
* time: 1.2099838256835938
72 1.019190e+02 3.009678e-01
* time: 1.2102768421173096
73 1.019189e+02 4.438935e-02
* time: 1.2105779647827148
74 1.019189e+02 1.971922e-03
* time: 1.210860013961792
75 1.019189e+02 2.678153e-05
* time: 1.2111499309539795FittedPumasModel
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.