Est+Opt CIRM 18/8/2009 A. ILIADIS 1 Estimation + optimization in PK modeling Introduction to modeling Estimation criteria Numerical optimization, examples.

Est+Opt CIRM 18/8/2009 A. ILIADIS1

Estimation + optimization in PK modelingEstimation + optimization in PK modeling

Introduction to modeling

Estimation criteria

Numerical optimization, examples

http://pharmapk.pharmacie.univ-mrs.fr/


Real process and mathematical modelReal process and mathematical model

10

100

0 2 4 6 8

t (h)

y (µ

g/m

L)

10

100

0 2 4 6 8

t (h)

y (µ

g/m

L)10

100

0 2 4 6 8

t (h)

y (µ

g/m

L)

Fittedmodel

Fittedmodel

Real processReal process

Math. modelMath. model


Functional schemeFunctional scheme

Measurementnoise

Measurementnoise

PK modelPK model

Equivalence criterion


Nonlinearprogramming


Administrationprotocol


A prioriinformation

A prioriinformation

+ Observation

Prediction

PK processPK process


Mathematical modelingMathematical modeling

Models are defined by : - their structure ( number and connectivity of compartments, etc ) expressed by

mathematical operations involving adjustable parameters :

Ex : 1-cpt,

exponential structure, parameters :

- the numerical value of parameters used :

CHARACTERIZATIONStructure

CHARACTERIZATIONStructure

MODELINGSystem Identification

MODELINGSystem Identification

ESTIMATIONParameters

ESTIMATIONParameters+


Checking identifiabilityChecking identifiability

Structural identifiability : given a hypothetical structure with unknown parameters, and a set of proposed experiments (not measurements !), would the unknown parameters be uniquely determinable from these experiments ?

Parametric identifiability : estimating the parameters from measurements with errors and optimize sampling designs.

: structural non-identifiable Non-consistent estimate

2211

9.10E-2

9.35E-2

9.60E-2

5.70E-2 5.80E-2 5.90E-2

1/V (L-1)

k (h-1)


Structural identifiabilityStructural identifiability

It depends on the observation site !

Solutions :Grouping : But ONLY identifiable

parameters :

Setting :

2211 2211


Functional scheme (dynamic)Functional scheme (dynamic)

Linear / p model : no loop, one stage estimation.

Nonlinear / p model : many loops until convergence.Measurement

noise

Measurementnoise

PK modelPK model







A prioriinformation

A prioriinformation

+PK processPK process

Arbitraryinitial values



Iterations, parameter convergenceIterations, parameter convergence

Ex : Fotemustine neutrophil toxicity :

Nonlinear modeling :

0 5 10 15 20 2510

-2

10-1

100

101

Iteration no

Pa

ram

. va

lue

10-2

10-1

100

101

RM

SE

2 nd

3 rd1 st

Optimizedfinal values



Errors in the functional schemeErrors in the functional scheme

The existing errors :experimental,

structural,

parametric.

Residual error :experimental,

structural (model

misspecification). 0 5 10 15 2010

-2

10-1

100

101

Iteration no

RM

SE

Residual error

Initial parametric error(canceled at the convergence)


Parametric and output spacesParametric and output spaces

Observation

ComparisonComparisonComparisonComparison

Parametric spaceParametric space Output spaceOutput space

Prediction

PK processPK process

PK modelPK model

Real processReal process

Artificial mechanismArtificial mechanism

Random componentRandom component Precision of estimates Measurement error


Optimal estimationOptimal estimation

Estimation is the operation of assigning a numerical values to unknown parameters, based on noise-corrupted observations.

Organization of the variables :The observed drug concentrations over time, ( dimensional vector).

The random parameters to be estimated, ( dimensional vector).

Consider the joint pdf and then :

the marginal is called prior pdf [ the marginal is not of interest ].

the conditional is called posterior pdf :

the conditional leads to the likelihood function :

MAP

MLE


Maximum a posteriori (MAP)Maximum a posteriori (MAP)

Design : A reasonable estimate of would be the mode of the posterior density for the given observation :

Ex :

if

if

The role of the dispersion.

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

f (x/

y)

y1

y2

y3


Maximum likelihood (MLE)Maximum likelihood (MLE)

Design : After the observation has been obtained, a reasonable estimate of would be , the value which gives to the particular observation the highest probability of occurrence :

Ex :

if

if

The role of the precision.

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y

f (y/

x) x1 = 2

x3 = 8

x2 = 3


The influence of x on the conditionalThe influence of x on the conditional

10-4

10-3

0

2000

4000

6000

8000

10000

12000

y (g/L)

f (y/

x)

x = 40 L/h

x = 8 L/h


The influence of y on the likelihoodThe influence of y on the likelihood

100

101

102

0

5000

10000

15000

x (L/h)

L (

y/x)

y = 0.2E-3 g/L

x = 12.59 L/h

x = 47.86 L/h

y = 0.4E-3 g/L


MLE criterion for single - outputMLE criterion for single - output

Initial form :

Hypotheses : H0 : The model is an exact description of the real process.

Error Output

H1 : Additive error :

H2 : Normal error :

H3 : Independence :


Variance heterogeneityVariance heterogeneity

The regression model :

assumes that

Need to relax this assumption (particularly when the model is highly nonlinear).

Transformed modelsFind a transformation function under which the error assumptions hold, i.e. :

where

Box – Cox transformations :

Other transformations : John – Drapper, Carroll, Huber, etc.

2,0~ Nei iiMi extyy ,

, . h

iiMi extyhyh ,,, 2,0~ Nei

0log

01

,

y

yyh

Estimate !


General form of the MLE criterionGeneral form of the MLE criterion

For available observed data and under the H3 hypothesis the estimator becomes :

Where : if

is the criterion function to be minimized.

The 1st term is known as the extended SE term.

The 2nd term is called the weighted SE term. It is the only one involving observed data and it is weighted by the uncertainty of experiment.

Ny ~


Criterion and error variance model Criterion and error variance model

After introducing the error variance model :

is minimum along the direction when :

or with

Then :

Nonlinearly unconstrained optimization : Find :

Assumptions : is computable for all and analytic solution does not exist.


Iterative solutionsIterative solutions

Solution for the nonlinear optimization problem Sequentially approximate starting from an initial value and converging towards

a stationary point .

Design a routine algorithm generating the converging sequence :

Terminology : is the initialization and obtaining from is an iteration.

Assign :


Taylor's expansion for smooth multivariate function: Construct simple approximations of a general function in a neighborhood of the

reference point . With

a vector of unit length supplying the direction of search, and

a scalar specifying the step length:

Associate successive approximations to iterations :

Approximation of functionsApproximation of functions

1

1

kk

kk

xJuhxJxJxJ

xuhxxx


Direction of searchDirection of search

Linear approx. of :

The scalar gives the rate of change of at the point along the direction .

To reduce , move along the direction opposite to : the descent direction.

Quadratic approx. of :

The scalar involves the second derivative of .

characterizes an ellipse. To reduce , move along the direction targeting

the center of ellipse : : the Newton direction


Line searchLine search

Minimization directions : moving along the Newton direction for quadratic surfaces, near . Elsewhere, move along the descent direction.

Line search : to complete the iteration search for in the direction of search :

or

10

15

20

0 1 2 3

x1

x 2

Minimization direction

3.8

3.4

4.5

4.5


Families of algorithmsFamilies of algorithms

Practical : Approximate derivatives by finite-differences instead analytical calculation.

Classify :Twice-continuously differentiable :

Second derivative : quadratic model of , compute and invert (not numerically stable, time consuming processing).

First derivative : quadratic model of , approximate :

without inverting but directly from by finite-differences of .

quasi-Newton methods (appropriate in many circumstances) : BFGS, DFP,...

Non-derivative : linear model of . Approximate by finite-differences of (for smooth functions) : Powell, Brent,...

No assumptions on differentiability : heuristic algorithms : NMS, Hooke-Jeeves,...

J

J

J

J

JJ


The information matrixThe information matrix

The Fisher information matrix : For MLE estimation :

Cramér-Rao inequality : is a lower bound of the covariance matrix , evaluating the precision of .

In practice : With the vector of the sampling times,

Obtain the sensitivity matrix with elements ,

Set and the order diagonal matrices having as elements

and respectively.


Covariance (precision) matrixCovariance (precision) matrix

The order precision matrix is :

Dependence on the sampling protocol:

Graphic interpretation of the precision matrix : is symmetric, and, if , it is also definite positive (by construction).

If , then is an a dimensional ellipsoid.

The volume of the ellipsoid is :

Sums of weighted products of sensitivity functionsover the available sampling times

The lowest , the most precise


Check the structural identifiabilityCheck the structural identifiability

The sensitivity matrix depends :On the experiment (not measurements !) and

On the model parametrization (structural and parametric) .

Ensure definite-positivity of the sensitivity matrix : It must be of full rank for any numeric value of the parameters, e.g., for the arbitrary

initial values (several).

Ex :

Observation in the central cpt free

# central cpt fixed

# peripheral cpt 43

44

54

t

?

?


Simulation in optimizationSimulation in optimization

2-cpt model :

Administration :

IV bolus

Observation :

Horizon

nbr

Heteroscedastic 0 6 12 18 240

0.5

1

1.5

2

2.5

3x 10

-3

t (h)

y 1 (g/

L)

1-21

1-

1-121

h 4.0h 3.0

h 0.1L 20

kk

kV

e

mg 80

h 2411m

%15K

0 6 12 18 2410

-4

10-3

10-2

t (h)

y 1 (g/

L)


Performances of algorithmsPerformances of algorithms

RMSE 1V ek 12k 21k

Reference 0.150 20.000 0.300 1.000 0.400 cpu time

Nelder-Mead 0.105 22.940 0.260 0.977 0.533 12.198 fminsearchHeuristic Genetic 0.938 1.311 2.924 4.455 0.242 129.597 ga

Threshold 0.493 25.512 0.244 1.692 1.006 409.010 threshacceptbndAnnealing 0.413 40.835 0.150 0.201 0.281 289.945 simulannealbnd

Non BFGS 0.137 17.062 0.342 1.503 0.581 11.808 fminuncDFP 0.268 17.042 0.328 1.300 0.641 10.653 fminunc

derivative Steepdesc 0.272 17.018 0.342 0.982 0.365 14.085 fminunc

BFGS 0.105 22.810 0.261 0.990 0.535 2.039 fminuncFirst DFP 0.136 17.621 0.335 1.401 0.532 12.029 fminunc

Steepdesc 0.149 17.065 0.343 1.327 0.502 16.227 fminuncderivative Trust-region 0.105 22.805 0.261 0.990 0.535 8.926 fminunc

BFGS 0.105 22.810 0.261 0.990 0.535 1.000 VA13AD

Est+Opt CIRM 18/8/2009 A. ILIADIS 1 Estimation + optimization in PK modeling Introduction to modeling Estimation criteria Numerical optimization, examples.

Documents

model slide

likelihood slide

convergence slide

conditional slide

map mle slide

nonlinear p model

regression model

unknown parameters