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Receptor Occupancy estimation by using Bayesian varying coefficient model

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Title: 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
  • a 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 a 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.

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

lt
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.
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