Optimal design in population kinetic experiments by set-valued methods

Peter Gennemark(Mathematical sciences, Göteborg,

Dept. of Mathematics, Uppsala)

in collaboration with

Warwick Tucker, Alexander Danis(Dept. of Mathematics, Uppsala)

Andrew Hooker, Joakim Nyberg(Dept. Of Pharmaceutical Biosciences,

Pharmacometrics, Uppsala)

Optimal experimental design

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Optimal design, definition?

The optimal design problem

Search domain, D, of possible designs.

An objective function, f, that measures the imprecison in the parameter estimates.

Prior knowledge of the model structure and parameters.

f(d)d opt minargDd

PopED (Population Experimental Design)

Nyberg J., Ueckert S., Karlsson M.O., Hooker A.C., Uppsala University

Population optimal design

http://poped.sf.net

Our optimal design approach

We pioneer the use of set-valued methods based on interval analysis and constraint propagationto estimate parameters in NLME models.

We use this for optimal experimental design.

Our optimal design approach

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We pioneer the use of set-valued methods based on interval analysis and constraint propagationto estimate parameters in NLME models.

We use this for optimal experimental design.

Our optimal design approach

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We pioneer the use of set-valued methods based on interval analysis and constraint propagationto estimate parameters in NLME models.

We use this for optimal experimental design.

Simple example of constraint propagation

Consider the model

Data:

Constraints:

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Simple example of constraint propagation

Consider the model

Data:

Constraints:

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Simple example of constraint propagation

Consider the model

Data:

Constraints:

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Simple example of constraint propagation

Consider the model

Data:

Constraints:

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Simple example of constraint propagation

Consider the model

Data:

Constraints:

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slope = k1 = 0

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Simple example of constraint propagation

Solution:Consider the model

Data:

Constraints:

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Simple example of constraint propagation

Consider the model

Data:

Constraints:

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Simple example of constraint propagation

Rearrange the model

Consider the model

Data:

Constraints:

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Simple example of constraint propagation

Rearrange the model

Propagate constraints

Consider the model

Data:

Constraints:

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Simple example of constraint propagation

Rearrange the model

Propagate constraints

Consider the model

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Constraints:

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Constraint propagation

In general, many parameters, data-points and constraints.

One iterates the constraints and also subdivides the search by partitioning the search space.

Implemented by directed acyclic graphs (DAG's).

y= x e− px

Danis A., Hooker A.C., Tucker W. Proc. Int. Symp. on Nonlinear Theory and its Applications 67-70, 2010.

Set-valued parameter estimation

Output consists of boxes that cover the solution.

The optimal design problem

Search domain, D, of possible designs.

An objective function, f, that measures the imprecison in the parameter estimates.

Prior knowledge of the model structure and parameters.

f(d)d opt minargDd

The optimal design problem

Search domain, D, of possible designs.

An objective function, f, that measures the imprecison in the parameter estimates.

Prior knowledge of the model structure and parameters.

Discrete

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Basic search method

Try a design from the search spaceRepeat several times:

Simulate data from the current designCompute the set of consistent parametersEvaluate objective function f

Monitor mean f for the tried design

Basic search method

REPEAT Try a design from the search spaceRepeat several times:

Simulate data from the current designCompute the set of consistent parametersEvaluate objective function f

Monitor mean f for the tried design UNTIL no better design is found

Exhaustive search suggests optimal sampling times

A heuristic search

Global search Entire search domainDecompose the problem into separate groups (same covariates)

D

d

A heuristic search

Global search Entire search domainDecompose the problem into separate groups (same covariates)

best solution

Local search Part of search domainNo decomposition

D

D

dopt

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Example of result

PopED PopED Our method

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A general framework

Covariates like dose and time can be defined as intervals.

Given a dose:

and sampling times with allowed uncertainty

What is the optimal design?

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Conclusions

No prior information in form of point estimates for the parameters is required (as in local optimal design).

Any covariate can be represented by an interval.

Problems with local minima and model linearisation are avoided in the parameter estimation.

Thanks for your attention!

Peter Gennemark(Mathematical sciences, Göteborg,

Dept. of Mathematics, Uppsala)

in collaboration with

Warwick Tucker, Alexander Danis(Dept. of Mathematics, Uppsala)

Andrew Hooker, Joakim Nyberg(Dept. Of Pharmaceutical Biosciences,

Pharmacometrics, Uppsala)