Receptor Occupancy estimation by using Bayesian varying coefficient model

Receptor Occupancy estimation by using Bayesian varying coefficient

model

Young researcher day21 September 2007

Astrid JullionPhilippe LambertFrançois Vandenhende

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

Bayesian models

Bayesian linear regression model

• Y : n-vector of responses• X : n x p design matrix• α : vector of regression coefficients

The model specification is :

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

Bayesian ridge regression model• Using a prespecified value for , the conditional

posterior distributions are :

pp

• 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.

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


• Model specification :

p


• Conditional posterior distributions :

where

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

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).



• 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.

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)

dose

RO

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40 50 60 70 80 90 100

• 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.


• 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



• 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.

• Real study : 6 patients scanned once before treatment and twice after treatment


• Real study : Time-Activity-Curves of one patient in the target (circles) and the reference (stars) regions


• 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.


• The model specification is the following :


<


• Drug concentration-RO curve.

We can select the efficacy dose

• 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.

Conclusion

Receptor Occupancy estimation by using Bayesian varying coefficient model

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