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Short description:

Population approaches to the analysis of dose escalation studies: Emax model

Format:	PharmML 0.8.x (0.8.1)
Related Publication:	Dose escalation studies: a comparison among Bayesian models Russu A., De Nicolao G, Poggesi I, Gomeni R. PAGE. Abstracts of the Annual Meeting of the Population Approach Group in Europe, 6/2009, Volume 10, pages: 1610 Affiliation: (1) Department of Computer Engineering and Systems Science, University of Pavia, Pavia, Italy; (2) GlaxoSmithKline, Clinical Pharmacology/Modelling & Simulation, Verona, Italy Abstract: Objectives: To apply alternative Bayesian models for the population analysis of the dose-exposure relationship in dose escalation Phase I studies. In these studies, subjects receive increasing dose levels and, at each dose escalation, a decision is made on the next dose level to be administered, based on safety/tolerability constraints. In recent years there has been a growing interest in Bayesian methods applied to such experiments [1,2,3]. In the present work, the performance of four alternative Bayesian models was evaluated on real and simulated datasets and a model comparison procedure was developed for the identification of the most appropriate model. Methods: The dose-exposure relationship was explored using a log-linear model (i.e. a linear model in log-log scale), a power model, an Emax model, and a nonparametric model based on population smoothing splines [4]. In all cases, a Bayesian population approach was adopted. The parametric models were estimated using WinBUGS 1.4.3. Sum of squared residuals, predictive root mean square error, AIC and BIC were used as model comparison criteria. Results: Ten phase I dose escalation studies and 60 simulated datasets (generated with the three parametric models) were analyzed. In the experimental datasets, the power and log-linear models provided comparable results in terms of point estimates and credibility intervals, whereas the Emax model proved inadequate when data showed upward curvature. The population spline approach provided good results for both experimental and simulated datasets. In the former case, the goodness of fit was comparable to parametric models. In the simulated benchmark, population splines performed comparably to the true models used to generate the data. The proposed comparison approach correctly identified the true parametric model in most cases. Conclusions: A thorough model comparison procedure was developed, based on model complexity criteria and crossvalidatory techniques. Applying several criteria represents a useful cross-check when one is faced with the problem of finding the most adequate model. It has been shown that the parallel estimation of four models (three parametric, one nonparametric), complemented with model comparison criteria, can robustly handle a variety of dose-exposure relationships overcoming possible misspecification problems. Moreover, population splines may represent an appealing first-try, especially in early escalation stages, when there is not enough information to support a specific parametric model.
Contributors:	Paolo Magni

Context of model development:	Dose & Schedule Selection and Label Recommendation;
Model compliance with original publication:	Yes;
Model implementation requiring submitter’s additional knowledge:	No;
Modelling context description:	Objectives: To apply alternative Bayesian models for the population analysis of the dose-exposure relationship in dose escalation Phase I studies. In these studies, subjects receive increasing dose levels and, at each dose escalation, a decision is made on the next dose level to be administered, based on safety/tolerability constraints. In recent years there has been a growing interest in Bayesian methods applied to such experiments In the present work, the performance of four alternative Bayesian models was evaluated on real and simulated datasets and a model comparison procedure was developed for the identification of the most appropriate model. Methods: The dose-exposure relationship was explored using a log-linear model (i.e. a linear model in log-log scale), a power model, an Emax model, and a nonparametric model based on population smoothing splines . In all cases, a Bayesian population approach was adopted. The parametric models were estimated using WinBUGS 1.4.3. Sum of squared residuals, predictive root mean square error, AIC and BIC were used as model comparison criteria. Results: Ten phase I dose escalation studies and 60 simulated datasets (generated with the three parametric models) were analyzed. In the experimental datasets, the power and log-linear models provided comparable results in terms of point estimates and credibility intervals, whereas the Emax model proved inadequate when data showed upward curvature. The population spline approach provided good results for both experimental and simulated datasets. In the former case, the goodness of fit was comparable to parametric models. In the simulated benchmark, population splines performed comparably to the true models used to generate the data. The proposed comparison approach correctly identified the true parametric model in most cases. Conclusions: A thorough model comparison procedure was developed, based on model complexity criteria and crossvalidatory techniques. Applying several criteria represents a useful cross-check when one is faced with the problem of finding the most adequate model. It has been shown that the parallel estimation of four models (three parametric, one nonparametric), complemented with model comparison criteria, can robustly handle a variety of dose-exposure relationships overcoming possible misspecification problems. Moreover, population splines may represent an appealing first-try, especially in early escalation stages, when there is not enough information to support a specific parametric model.;
Modelling task in scope:	estimation;
Nature of research:	Early clinical development (Phases I and II);

Validation Status:	Annotations are correct.
Certification Comment:	This model is not certified.

Mandatory file
- Executable_Russu_2009_dose_escalation_Emax.xml

Additional Files

Model owner: Paolo Magni
Submitted: Dec 13, 2015 10:44:47 PM
Last Modified: Oct 13, 2016 7:05:32 PM

Revisions

Version: 10
- Submitted on: Oct 13, 2016 7:05:32 PM
- Submitted by: Paolo Magni
- With comment: Edited model metadata online.
Version: 7
- Submitted on: Jul 16, 2016 3:59:51 PM
- Submitted by: Paolo Magni
- With comment: Updated model annotations.
Version: 4
- Submitted on: Dec 13, 2015 10:44:47 PM
- Submitted by: Paolo Magni
- With comment: Edited model metadata online.

Name

Generated from MDL. MOG ID: russu_Emax_mog

Independent Variables

Function Definitions

proportionalError : real (proportional : real, f : real) = proportional \cdot f

Parameter Model: $pm$

Random Variables

{ETA_1}_{vm_mdl.ID} ~ Normal2 (mean = 0, var = pm.OMEGA_1)

{ETA_2}_{vm_mdl.ID} ~ Normal2 (mean = 0, var = pm.OMEGA_2)

{EPS}_{vm_err.DV} ~ Normal2 (mean = 0, var = 1)

Population Parameters

MU_THETA1 = 4.481375

MU_THETA2 = 1.54563

VAR_THETA1 = {2.148283}^{2}

VAR_THETA2 = {2.148283}^{2}

a_1 = 1

b_1 = 0.0262989

a_2 = 1

b_2 = 0.1021269

a_cv = 1

b_cv = 1

{THETA1}_{vm_mdl.MDL__prior} ~ Normal2 (mean = pm.MU_THETA1, var = pm.VAR_THETA1)

{THETA2}_{vm_mdl.MDL__prior} ~ Normal2 (mean = pm.MU_THETA2, var = pm.VAR_THETA2)

{invOMEGA_1}_{vm_mdl.MDL__prior} ~ Gamma2 (shape = pm.a_1, rate = pm.b_1)

{invOMEGA_2}_{vm_mdl.MDL__prior} ~ Gamma2 (shape = pm.a_2, rate = pm.b_2)

{invCV2}_{vm_mdl.MDL__prior} ~ Gamma2 (shape = pm.a_cv, rate = pm.b_cv)

OMEGA_1 = \frac{1}{pm.invOMEGA_1}

OMEGA_2 = \frac{1}{pm.invOMEGA_2}

CV = \sqrt{\frac{1}{pm.invCV2}}

Individual Parameters

ALPHA = pm.THETA1 + pm.ETA_1

BETA = pm.THETA2 + pm.ETA_2

Structural Model: $sm$

Variables

C = \frac{ⅇ^{pm.ALPHA} \cdot T}{(ⅇ^{pm.BETA} + T)}

Observation Model: $om1$

Continuous Observation

Y = sm.C + proportionalError (proportional = pm.CV, f = sm.C) + pm.EPS

External Dataset

OID	$nm_ds$
Tool Format	NONMEM

File Specification

Format	$csv$
Delimiter	comma
File Location	Simulated_russu_2009_emax_data.csv

Column Definitions

Column ID	Position	Column Type	Value Type
$ID$	$1$	$id$	$int$
$DOSE$	$2$	$idv$	$real$
$DV$	$3$	$dv$	$real$

Column Mappings

Column Ref	Modelling Mapping
$ID$	$vm_mdl.ID$
$DOSE$	$T$
$DV$	$om1.Y$

Estimation Step

OID	$estimStep_1$
Dataset Reference	$nm_ds$

Parameters To Estimate

Parameter	Fixed?	Limits
pm.THETA1	false	$()$
pm.THETA2	false	$()$
pm.invOMEGA_1	false	$()$
pm.invOMEGA_2	false	$()$
pm.invCV2	false	$()$

Operations

Operation: $1$

Op Type

generic

Operation Properties

Name	Value
algo	$mcmc$

Operation: $2$

Op Type

BUGS

Operation Properties

Name	Value
nchains	$1$
burnin	$1000$
niter	$5000$
winbugsgui	$false$

Step Dependencies

Step OID	Preceding Steps
$estimStep_1$

DDMODEL00000114: Russu_2009_dose_escalation_Emax

Revisions

Name

Independent Variables

Function Definitions

Parameter Model: pm

Random Variables

Population Parameters

Individual Parameters

Structural Model: sm

Variables

Observation Model: om1

Continuous Observation

External Dataset

File Specification

Column Definitions

Column Mappings

Estimation Step

Parameters To Estimate

Operations

Operation: 1

Operation Properties

Operation: 2

Operation Properties

Step Dependencies

Parameter Model: $pm$

Structural Model: $sm$

Observation Model: $om1$

Operation: $1$

Operation: $2$