Latent variable indirect response modeling of categorical endpoints representing change from baseline

Abstract

Accurate exposure–response modeling is important in drug development. Methods are still evolving in the use of mechanistic, e.g., indirect response (IDR) models to relate discrete endpoints, mostly of the ordered categorical form, to placebo/co-medication effect and drug exposure. When the discrete endpoint is derived using change-from-baseline measurements, a mechanistic exposure–response modeling approach requires adjustment to maintain appropriate interpretation. This manuscript describes a new modeling method that integrates a latent-variable representation of IDR models with standard logistic regression. The new method also extends to general link functions that cover probit regression or continuous clinical endpoint modeling. Compared to an earlier latent variable approach that constrained the baseline probability of response to be 0, placebo effect parameters in the new model formulation are more readily interpretable and can be separately estimated from placebo data, thus allowing convenient and robust model estimation. A general inherent connection of some latent variable representations with baseline-normalized standard IDR models is derived. For describing clinical response endpoints, Type I and Type III IDR models are shown to be equivalent, therefore there are only three identifiable IDR models. This approach was applied to data from two phase III clinical trials of intravenously administered golimumab for the treatment of rheumatoid arthritis, where 20, 50, and 70 % improvement in the American College of Rheumatology disease severity criteria were used as efficacy endpoints. Likelihood profiling and visual predictive checks showed reasonable parameter estimation precision and model performance.

Appendix: Alternative representations of baseline-normalized IDR model forms

We first derive two reduction-from-baseline IDR model forms not given in Woo et al. [2]. From Eq. 1, it follows that the reduction from baseline, R1 = R₀ − R, with R1(0) = 0, is governed by:

$$ {\text{dR1}}/{\text{dt}} = - {\text{dR}}/{\text{dt}} = - \left\{ {{\text{k}}_{\text{in}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] - {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right]{\text{ R}}} \right\} = - {\text{k}}_{\text{in}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] \, + {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{R}}_{0} -{\text{R1}}} \right) $$

(9)

The derivation mathematically holds whether R > R₀ or R ≤ R₀. The importance of this general interpretation will become more apparent in the next section. It is noted that Eq. 10 from Woo et al. [2] was derived by defining baseline subtraction as |R − R₀| and then taking its derivative. While the absolute value function is formally non-differentiable, their derivation is valid when R − R₀ > 0. The case of R − R₀ ≤ 0 is provided here by Eq. 9.

It may appear to follow that, when f_d(t) is interpreted as a Type III IDR model in Eq. 8, the term S_max equals I_max in Eq. 6 and hence may not exceed 1. However this constraint is unnecessary and can be removed by suitable baseline choices. Thus, the equivalence between Type I and Type III IDR model interpretations only holds in the context of an increase/decrease from baseline.

The proportional reduction-from-baseline IDR model form, although not used in the other parts of this manuscript, is next derived for completeness. Let R2 = (R₀ − R)/R₀, then R2(0) = 0, and:

$$ {\text{dR2}}/{\text{dt}} = - {\text{d}}\left( {{\text{R}}/{\text{R}}_{0} } \right)/{\text{dt}} = - \left\{ {{\text{k}}_{\text{in}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] - {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right]{\text{ R}}\left( {\text{t}} \right)} \right\}/{\text{R}}_{0} = - {\text{k}}_{\text{in}} /{\text{R}}_{0} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] + {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( { 1- {\text{R2}}} \right) = - {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] - {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{R2}} - 1} \right) $$

(10)

Note that this model form has one less parameter than Eq. 10 in Woo et al. [2], specifically k_in.

Symmetries between Type I and Type III IDR models

For Type I and III IDR models, H₂(t) = 0, and Eq. 9 becomes:

$$ {\text{dR1}}/{\text{dt}} = - {\text{k}}_{\text{in}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] + {\text{k}}_{\text{out}} \left( {{\text{R}}_{0} -{\text{R1}}} \right) = - {\text{k}}_{\text{in}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] + 2 {\text{ k}}_{\text{out}} {\text{R}}_{0} - {\text{k}}_{\text{out}} \left( {{\text{R1}} + {\text{R}}_{0} } \right) = - {\text{k}}_{\text{in}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] + 2 {\text{ k}}_{\text{in}} - {\text{k}}_{\text{out}} \left( {{\text{R1}} + {\text{R}}_{0} } \right) = {\text{ k}}_{\text{in}} \left[ { 1- {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] - {\text{k}}_{\text{out}} \left( {{\text{R1}} + {\text{R}}_{0} } \right) $$

Therefore, R1 takes the form of a Type III or Type I increase-from-baseline IDR model form as Eq. 10 in Woo et al. [2], depending on whether R is Type I or Type III.

Perhaps surprisingly, such symmetry does not exist between Type II and IV IDR models. This can be seen by taking H₁(t) = 0 in Eq. 9, which then becomes:

$${\text{dR1}}/{\text{dt}} = - {\text{k}}_{\text{in}} + {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{R}}_{0} - {\text{R1}}} \right) = - {\text{k}}_{\text{in}} - {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{R1}} - {\text{R}}_{0} } \right) = - {\text{k}}_{\text{in}} - {\text{k}}_{\text{out}} \left[ { 1- {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{R1}} - {\text{R}}_{0} } \right) - 2 {\text{ k}}_{\text{out}} {\text{H}}_{ 2} \left( {\text{t}} \right) \, \left( {{\text{R1}} - {\text{R}}_{0} } \right) = - {\text{k}}_{\text{in}} - {\text{k}}_{\text{out}} \left[ { 1- {\text{H}}_{ 2} \left( {\text{t}} \right)\left] { \, \left( {{\text{R1}} + {\text{R}}_{0} } \right) + 2 {\text{ k}}_{\text{out}} } \right[ 1- {\text{H}}_{ 2} \left( {\text{t}} \right)} \right]{\text{ R}}_{0} - 2 {\text{ k}}_{\text{out}} {\text{H}}_{ 2} \left( {\text{t}} \right){\text{ R1}} + 2 {\text{ k}}_{\text{in}} {\text{H}}_{ 2} \left( {\text{t}} \right) = - {\text{k}}_{\text{in}} - {\text{k}}_{\text{out}} \left[ { 1- {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{R1}} + {\text{R}}_{0} } \right) + 2 \;{\text{k}}_{\text{in}} - 2 {\text{ H}}_{ 2} \left( {\text{t}} \right) - 2 {\text{ k}}_{\text{out}} {\text{H}}_{ 2} \left( {\text{t}} \right){\text{ R1}} + 2 {\text{ k}}_{\text{in}} {\text{H}}_{ 2} \left( {\text{t}} \right) = - {\text{k}}_{\text{in}} - {\text{k}}_{\text{out}} \left[ { 1- {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{R1}} + {\text{R}}_{0} } \right) + 2 {\text{ k}}_{\text{in}} - 2 {\text{ k}}_{\text{out}} {\text{H}}_{ 2} \left( {\text{t}} \right){\text{ R1}} = {\text{ k}}_{\text{in}} - {\text{k}}_{\text{out}} \left[ { 1- {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{R1}} + {\text{R}}_{0} } \right) - 2 {\text{ k}}_{\text{out}} {\text{H}}_{ 2} \left( {\text{t}} \right){\text{ R1}} $$

This has the extra non-zero term of −2 k_out H₂(t) R1 than Eq. 10 in Woo et al. [2].

Relationship with the latent variable approach

Next, we derive the differential equation for a more general form than the latent variable approach represented by Eqs. 6 and 7. Let R1 = f_d(t), H(t) be the inhibitory term in Eq. 6, and R = R(t), then, noticing k_in = k_out, one has:

$$ {\text{dR}}1/{\text{dt}} = - {\text{DE dR}}/{\text{dt }} = - {\text{DE }}\left\{ {{\text{k}}_{\text{in}} \left[ {1 \, + {\text{ H}}_{1} \left( {\text{t}} \right)} \right] - {\text{k}}_{\text{out}} \left[ {1 + {\text{H}}_{2} \left( {\text{t}} \right)} \right]{\text{ R}}} \right\} = - {\text{DE }}\left\{ {{\text{k}}_{\text{in}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] - {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( { 1- {\text{R1}}/{\text{DE}}} \right)} \right\} = - {\text{DE k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] + {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{DE}} - {\text{R1}}} \right) $$

which takes the form of Eq. 9 with DE = R₀ = k_in/k_out. That is, f_d(t) takes the form of a reduction-from-baseline IDR model.

For the interpretation of Eq. 8, let R2 = −f_d(t) = −R1, one has:

$$ {\text{dR2}}/{\text{dt}} = - {\text{dR1}}/{\text{dt}} = {\text{DE k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] - {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{DE}} - {\text{R1}}} \right) = {\text{ DE k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 1} \left( {\text{t}} \right)} \right] - {\text{k}}_{\text{out}} \left[ { 1+ {\text{H}}_{ 2} \left( {\text{t}} \right)} \right] \, \left( {{\text{R2}} + {\text{DE}}} \right) $$

which takes the form of Eq. 10 from Woo et al. [2], with DE = R₀ = k_in/k_out. That is, g_d(t) = −f_d(t) takes the form of an increase-from-baseline IDR model.