Robust specification of the roughness penalty prior distribution in spatially adaptive Bayesian P-splines models

peer reviewedThe potential important role of the prior distribution of the roughness penalty parameter in the resulting smoothness of Bayesian Psplines models is considered. The recommended specification for that distribution yields models that can lack flexibility in specific circumstances. In such instances, these are shown to correspond to a frequentist P-splines model with a predefined and severe roughness penalty parameter, an obviously undesirable feature. It is shown that the specification of a hyperprior distribution for one parameter of that prior distribution provides the desired flexibility. Alternatively, a mixture prior can also be used. An extension of these two models by enabling adaptive penalties is provided. The posterior of all the proposed models can be quickly explored using the convenient Gibbs sampler.IAP research network nr P5/2

Adaptive Bayesian P-splines models for fitting time-activity curves and estimating associated clinical parameters in Positron Emission Tomography and Pharmacokinetic study

In clinical experiments, the evolution of a product concentration in tissue over time is often under study. Different products and tissues may be considered. For instance, one could analyse the evolution of drug concentration in plasma over time, by performing successive blood sampling from the subjects participating to the clinical study. One could also observe the evolution of radioactivity uptakes in different regions of the brain during a PET scan (Positron Emission Tomography). The global objective of this thesis is the modelling of such evolutions, which will be called, generically, pharmacokinetic curves (PK curves). Some clinical measures of interest are derived from PK curves. For instance, when analysing the evolution of drug concentration in plasma, PK parameters such as the area under the curve (AUC), the maximal concentration (Cmax) and the time at which it occurs (tmax) are usually reported. In a PET study, one could measure Receptor Occupancy (RO) in some regions of the brain, i.e. the percentage of specific receptors to which the drug is bound. Such clinical measures may be badly estimated if the PK curves are noisy. Our objective is to provide statistical tools to get better estimations of the clinical measures of interest from appropriately smoothed PK curves. Plenty of literature addresses the problem of PK curves fitting using parametric models. It usually relies on a compartmental approach to describe the kinetic of the product under study. The use of parametric models to fit PK curves can lead to problems in some specific cases. Firstly, the estimation procedures rely on algorithms which convergence can be hard to attain with sparse and/or noisy data. Secondly, it may be difficult to choose the adequate underlying compartmental model, especially when a new drug is under study and its kinetic is not well known. The method that we advocate to fit such PK curves is based on Bayesian Penalized splines (P-splines): it provides good results both in terms of PK curves fitting and clinical measures estimations. It avoids the difficult choice of a compartmental model and is more robust than parametric models to a small sample size or a low signal to noise ratio. Working in a Bayesian context provides several advantages: prior information can be injected, models can easily be generalized and extended to hierarchical settings, and uncertainty for associated clinical parameters are straightforwardly derived from credible intervals obtained by MCMC methods. These are major advantages over traditional frequentist approaches.(STAT 3) -- UCL, 200

Adaptive Bayesian P-splines models for fitting time-activity curves and estimating associated clinical parameters in Positron Emission Tomography and Pharmacokinetic study

In clinical experiments, the evolution of a product concentration in tissue over time is often under study. Different products and tissues may be considered. For instance, one could analyse the evolution of drug concentration in plasma over time, by performing successive blood sampling from the subjects participating to the clinical study. One could also observe the evolution of radioactivity uptakes in different regions of the brain during a PET scan (Positron Emission Tomography). The global objective of this thesis is the modelling of such evolutions, which will be called, generically, pharmacokinetic curves (PK curves). Some clinical measures of interest are derived from PK curves. For instance, when analysing the evolution of drug concentration in plasma, PK parameters such as the area under the curve (AUC), the maximal concentration (Cmax) and the time at which it occurs (tmax) are usually reported. In a PET study, one could measure Receptor Occupancy (RO) in some regions of the brain, i.e. the percentage of specific receptors to which the drug is bound. Such clinical measures may be badly estimated if the PK curves are noisy. Our objective is to provide statistical tools to get better estimations of the clinical measures of interest from appropriately smoothed PK curves. Plenty of literature addresses the problem of PK curves fitting using parametric models. It usually relies on a compartmental approach to describe the kinetic of the product under study. The use of parametric models to fit PK curves can lead to problems in some specific cases. Firstly, the estimation procedures rely on algorithms which convergence can be hard to attain with sparse and/or noisy data. Secondly, it may be difficult to choose the adequate underlying compartmental model, especially when a new drug is under study and its kinetic is not well known. The method that we advocate to fit such PK curves is based on Bayesian Penalized splines (P-splines): it provides good results both in terms of PK curves fitting and clinical measures estimations. It avoids the difficult choice of a compartmental model and is more robust than parametric models to a small sample size or a low signal to noise ratio. Working in a Bayesian context provides several advantages: prior information can be injected, models can easily be generalized and extended to hierarchical settings, and uncertainty for associated clinical parameters are straightforwardly derived from credible intervals obtained by MCMC methods. These are major advantages over traditional frequentist approaches.(STAT 3) -- UCL, 200

Extensions of Bayesian P-splines models for fitting PK curves

In clinical experiments, the evolution of a product concentration in tissue over time is often under study. Different products and tissues may be considered. For instance, one could analyse the evolution of drug concentration in plasma over time. One could also observe the evolution of radioactivity uptakes in different regions of the brain during a PET scan (Positron Emission Tomography). Such evolutions are named, generically, pharmacokinetic curves (PK curves). Some clinical measures of interest are derived from PK curves. For instance, when analysing the evolution of drug concentration in the plasma, PK parameters such as the area under the curve (AUC), the maximal concentration (Cmax) and the time at which it occurs (tmax) are usually calculated. In a PET study, one could measure receptor occupancy (RO) in some regions of the brain. These clinical measures could be badly estimated if the observed PK curves are noisy. The objective of this paper is to provide some extensions of the Bayesian P-splines model for fitting this type of curves. Two extensions are provided. The first one introduces adaptive penalties in the Bayesian P-splines model. Jullion and Lambert (2007) already proposed an adaptive Bayesian P-splines model. Here, two alternative specifications relying on multivariate normal (Nelsen 1999; Sklar 1959) and Archimedean copulas (Genest and MacKay 1986) are proposed. Simulations show that the best performer to fit PK-like curves is the model of Jullion and Lambert (2007). In the second extension, two solutions are proposed to deal with heterogeneous variance. In clinical studies, the variance is often not constant but varies either with the mean response or with time. The first solution relies on the Yeo-Johnson transformation while the second expresses the variance as a parametric function of the mean response. Simulations have been performed on PK-like curves, to compare these two extensions with the Bayesian P-splines model applied on the original and on the log-transformed data, in three different scenarios. In the first one, the conditional variance increases with the conditional mean. In the second one, the variance is homogeneous while in the third one, it increases with the independent variable, x. It turns out that, in the first scenario, the best performer is the model where the variance is a parametric function of the conditional mean followed by the Bayesian P-splines model applied on the logarithm of the data. The simulation performed from scenario 2 shows that, when the variance of the data is constant, the Bayesian P-splines model applied on the original data is the best performer. Finally, in the third scenario, when the variance increases with x, the best choice is the Yeo-Johnson transformation model. These two extensions can be combined to obtain a satisfactory fit of PK curves

Robust specification of the roughness penalty prior distribution in spatially adaptive Bayesian P-splines models

We start the paper by pointing the potential important role of the prior distribution of the roughness penalty parameter in the resulting smoothness of Bayesian P-splines models (Ruppert et al. 2003 ; Lang and Brezger 2004). The recommended specification for that distribution yields models that can lack flexibility in specific circumstances. In such instances, these are shown to correspond to a frequentist P-splines model (Eilers & Marx, 1996) with a predefined and severe roughness penalty parameter, an obviously undesirable feature. We show that the specification of a hyperprior distribution for one parameter of that prior distribution provides the desired flexibility. Alternatively, a mixture prior can also be used. An extension of these two models by enabling adaptive penalties is provided. All the proposed models can be fitted quickly using the convenient Gibbs algorithm

A non-parametric bayesian method to smooth PET time-activity-curves

Positron Emission Tomography (PET) is an imaging technique in which a radionuclide is introduced into a molecule of potential biological relevance (to form what is called a tracer) and administered to a patient. The regional evolution of the uptake of the tracer over time is called a TimeActivity-Curve (TAC) and is used to derive some clinical measures that give information about the process under study. One of these measures is the Distribution Volume (DV ) which can be estimated by several methods, notably the Graphical Analysis Method (GA). It has been shown that using GA method on noisy TAC leads to a systematic underestimation of the Distribution Volume (Hsu et al., 1997; Slifstein and Laruelle, 1999). We propose a method that allows to smooth the Time-Activity-Curves in a non-parametric way by using Bayesian P-splines. This method may be used in all the cases, whatever the compartmental model that might underly the observed data. Simulations have shown that this method gives an unbiased estimation of the true TAC, whatever the level of noise. We show that smoothing TAC with the non-parametric method before computing the Distribution Volume allows to reduce considerably the bias. Logan et al. (2001) proposes to smooth data before computing the Distribution Volume by using the Generalized Linear Least Squares (GLLS) method if a one-tissue compartment model is considered and by applying the GLLS to the data in two parts for a two-tissues compartment model : one set of parameters is estimated from times 0 to T1 and a second set from T1 to the end time where T1 has to be chosen from data. This method provides good results but, if we are not sure about the compartmental model suitable for the data or, if we want to avoid the choice of T1, using the Bayesian nonparameteric model is a good alternative

Estimation of receptor occupancy using varying coefficients models

In many applications of linear regression models, the regression coeffi- cients are not regarded as fixed but as varying with another covariate named the effect modifier. A useful extension of the linear regression models are then varying coefficient models. To link the regression coefficient with the effect modifier, several methods may be considered. Here, we propose to use Bayesian P-splines to relate in a smoothed way the regression coefficient with the effect modifier. We show that this method enables a large level of flexibility: if necessary, adaptive penalties can be introduced in the model (Jullion and Lambert 2007) and linear constraints on the relation between the regression coefficient and the effect modifier may easily be added. We provide an illustration of the proposed method in a PET study where we want to estimate the relation between the Receptor Occupancy and the drug concentration in the plasma. As we work in a Bayesian setting, credibil- ity sets are easily obtained for receptor occupancy, which take into account the uncertainty appearing at all the different estimation steps

Trial predictions vs. trial simulations in Model-based Drug Development: integrating uncertainties to evaluate the predictive probability of success.

In a Model-Based Drug Development strategy, the first objective is to design studies such that the most reliable model estimates are obtained, in order to optimize the design of future studies and to take decisions based on predictions. The objectives of the work is to present from a theoretical and practical point of view how to perform trial predictions, as opposed to trial simulations, by integrating the uncertainty of the parameters. The difference between prediction and simulation is important in early development when limited data or prior information are available. Indeed ignoring the uncertainty of parameter estimates can lead to wrong decisions. First, will be provided methodology, derived from Bayesian statistics, to perform trial predictions from the parameter estimates and their uncertainty, when obtained with conventional frequentist population methods. Second, a practical implementation in R will be shown. This generalized prediction shell can cope with any kind of structural population models: Ordinary Differential Equation, single & multiple doses, infusion, etc... The proposed shell is also flexible to allow the testing of various scenarios and study designs, and therefore evaluate the predictive probability of success of different protocols. When joint models for efficacy and safety are established, the Prediction-based Clinical Utility Index (p-CUI) and its distribution can directly be obtained for more riskless decision making. Examples will be shown to highlight in early phases the differences existing between trial prediction and trial simulation. This approach is required to permit Model-Based Drug Development strategy, and impact successfully decision in early clinical phases

Adaptive Bayesian P-splines to estimate varying regression coefficients: application to receptor occupancy estimation

peer reviewedIn many applications of linear regression models, the regression coefficients are not regarded as fixed but as varying with another covariate named the effect modifier. A useful extension of the linear regression models are then varying coefficient models. To link the regression coefficient with the effect modifier, several methods may be considered. Here, we propose to use Bayesian P-splines to relate in a smoothed way the regression coefficient with the effect modifier. We show that this method enables a large level of flexibility: if necessary, adaptive penalties can be introduced in the model (Jullion and Lambert 2007) and linear constraints on the relation between the regression coefficient and the effect modifier may easily be added. We provide an illustration of the proposed method in a PET study where we want to estimate the relation between the Receptor Occupancy and the drug concentration in the plasma. As we work in a Bayesian setting, credibility sets are easily obtained for receptor occupancy, which take into account the uncertainty appearing at all the different estimation steps