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
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
Bayesian varying coefficient model
• Model specification :
p
Bayesian varying coefficient model
• 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).
Context of RO estimation
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.
Application of the Bayesian varying coefficient model
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.
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
Application of the Bayesian varying coefficient model
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.
• Real study : 6 patients scanned once before treatment and twice after treatment
Application of the Bayesian varying coefficient model
• Real study : Time-Activity-Curves of one patient in the target (circles) and the reference (stars) regions
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.
Application of the Bayesian varying coefficient model
• The model specification is the following :
Application of the Bayesian varying coefficient model
<
Application of the Bayesian varying coefficient model
• 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