Receptor Occupancy estimation by using Bayesian varying coefficient model

1. Receptor Occupancy estimation by using Bayesian varying coefficient model Young researcher day 21 September 2007 Astrid Jullion Philippe Lambert François Vandenhende

2. Table of content • Bayesian linear regression model • Bayesian ridge linear regression model • Bayesian varying coefficient model • Context of Receptor Occupancy estimation • Application of the Bayesian varying coefficient model to RO estimation • Conclusion

3. Bayesian models

4. Bayesian linear regression model • Y : n-vector of responses • X : n x p design matrix • α : vector of regression coefficients The model specification is :

5. Bayesian ridge regression model • Multicollineartiy problem : interrelationships among the independent variables. • One solution to multicollinearity includes the ridge regression (Marquardt and Snee, 1975). • The ridge regression is translated in a Bayesian model by adding a prior on the regression coefficients vector : • Congdon (2006) suggests either to set a prior on or to assess the sensitivity to prespecified fixed values. p

6. Bayesian ridge regression model • Using a prespecified value for , the conditional posterior distributions are : p p

7. Bayesian varying coefficient model • We consider that we have regression coefficients varying as smoothed function of another covariate called “effect modifier” (Hastie and Tibshirani, 1993). • We propose to use robust Bayesian P-splines to link in a smoothed way the regression coefficients with the effect modifier.

8. Bayesian varying coefficient model • Notations : • Y : response vector which depends on two kinds of variables : • X : matrix with all the variables for which the regression coefficients vector α is fixed. • Z: matrix with all the variables for which the regression coefficients vector  varies with an effect modifier E. • We express  as a smoothed function of E by the way of P-splines : : B-splines matrix associated to E : corresponding vector of splines coefficients : roughness penalty parameter

9. Bayesian varying coefficient model • Model specification : p

10. Bayesian varying coefficient model • Conditional posterior distributions : where

11. Bayesian varying coefficient model • Inclusion of a linear constraint : • Suppose that we want to impose a constraint to the relationship between the regression coefficient and the effect modifier. • In our illustration, we shall consider that the relation is known to be monotonically increasing. • This constraint is translated on the splines coefficients vector by imposing the positivity of all the differences between two successive splines coefficients : • To introduce this constraint in the model at the simulation stage, we rely on the technique proposed by Geweke (1991) which allows the construction of samples from an m-variate distribution subject to linear inequality restrictions. : first order difference matrix

12. Context of RO estimation

13. Context of RO estimation • We are interested in drugs that bind to some specific receptors in the brain. • The Receptor Occupancy is the proportion of specific receptors to which the drug is bound. • We consider a blocking experiment : • 1) A tracer (radioactive product) is administered to the subject under baseline conditions. Images of the brain are acquired sequentially to measure the time course of tracer radioactivity. • 2) The same tracer is administered after treatment by a drug which interacts with the receptors of interest. Images of the brain are then acquired. A decrease in regional radioactivity from baseline indicates receptor occupancy by the test drug. The radioactivity evolution with time in a region of the brain during the scan is named a Time-Activity Curve (TAC).

14. Context of RO estimation • To estimate RO, we use the Gjedde-Patlak equations : • The Receptor Occupancy is then computed as : where K1 is the slope obtained for the drug-free condition and K2 after drug administration.

15. Application of the Bayesian varying coefficient model

16. Application of the Bayesian varying coefficient model • Traditional method • Step 1 : Estimate RO • Step 2 : Relation between RO and the dose (or the drug concentration in plasma) RO dose

17. Application of the Bayesian varying coefficient model • Objective : • Application of the Bayesian varying coefficient method in a RO study • We want to use a one-stage method to estimate RO as a function of the drug concentration in plasma, starting from the equations of the Gjedde-Patlak model. • The effect modifier in this context is the drug concentration in plasma.

18. Application of the Bayesian varying coefficient model • Here are the formulas of the Gjedde-Patlak model. Indice 1 (2) refers to the concentrations observed before (after) treatment • The Receptor Occupancy is defined as : • We define : • Then we get

19. Application of the Bayesian varying coefficient model • With simplify the notations with • And ROc(k) is expressed as a smoothed function of the drug concentration in plasma. • As we know that RO has to increase monotonically with the drug concentration, we use the technique of Geweke to include this linear constraint in the model.

20. Application of the Bayesian varying coefficient model • Real study : 6 patients scanned once before treatment and twice after treatment

21. Application of the Bayesian varying coefficient model • Real study : Time-Activity-Curves of one patient in the target (circles) and the reference (stars) regions

22. Application of the Bayesian varying coefficient model • Real study : To take into consideration the correlation between the two observations coming from the same patient, we add in the model the matrix : where T is the time length of the scan.

23. Application of the Bayesian varying coefficient model • The model specification is the following : <

24. Application of the Bayesian varying coefficient model • Drug concentration-RO curve. We can select the efficacy dose

25. Conclusion • In many applications of linear regression models, the regression coefficients are not regarded as fixed but as varying with another covariate called the effect modifier. • To link the regression coefficient with the effect modifier in a smoothed way, Bayesian P-splines offer a flexible tool: • Add some linear constraints • Use adaptive penalties • Credibility sets are obtained for the RO which take into account the uncertainty appearing at all the different estimation steps. In a traditional two-stage method, RO is first estimated for different levels of drug concentration in plasma on the basis of the Gjedde-Patlak method. In a second step, the relation between RO and the drug concentration is estimated conditionally on the first step results. • Same type of results for a reversible tracer.