est+opt cirm 18/8/2009 a. iliadis 1 estimation + optimization in pk modeling introduction to...
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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/
Est+Opt CIRM 18/8/2009 A. ILIADIS2
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
Est+Opt CIRM 18/8/2009 A. ILIADIS3
Functional schemeFunctional scheme
Measurementnoise
Measurementnoise
PK modelPK model
Equivalence criterion
Equivalence criterion
Nonlinearprogramming
Nonlinearprogramming
Administrationprotocol
Administrationprotocol
A prioriinformation
A prioriinformation
+ Observation
Prediction
PK processPK process
Est+Opt CIRM 18/8/2009 A. ILIADIS4
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+
Est+Opt CIRM 18/8/2009 A. ILIADIS5
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)
Est+Opt CIRM 18/8/2009 A. ILIADIS6
Structural identifiabilityStructural identifiability
It depends on the observation site !
Solutions :Grouping : But ONLY identifiable
parameters :
Setting :
2211 2211
Est+Opt CIRM 18/8/2009 A. ILIADIS7
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
Equivalence criterion
Equivalence criterion
Nonlinearprogramming
Nonlinearprogramming
Administrationprotocol
Administrationprotocol
A prioriinformation
A prioriinformation
+PK processPK process
Arbitraryinitial values
Arbitraryinitial values
Est+Opt CIRM 18/8/2009 A. ILIADIS8
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
Arbitraryinitial values
Est+Opt CIRM 18/8/2009 A. ILIADIS9
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)
Est+Opt CIRM 18/8/2009 A. ILIADIS10
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
Est+Opt CIRM 18/8/2009 A. ILIADIS11
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
Est+Opt CIRM 18/8/2009 A. ILIADIS12
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
Est+Opt CIRM 18/8/2009 A. ILIADIS13
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
Est+Opt CIRM 18/8/2009 A. ILIADIS14
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
Est+Opt CIRM 18/8/2009 A. ILIADIS15
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
Est+Opt CIRM 18/8/2009 A. ILIADIS16
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 :
Est+Opt CIRM 18/8/2009 A. ILIADIS17
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 !
Est+Opt CIRM 18/8/2009 A. ILIADIS18
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 ~
Est+Opt CIRM 18/8/2009 A. ILIADIS19
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.
Est+Opt CIRM 18/8/2009 A. ILIADIS20
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 :
Est+Opt CIRM 18/8/2009 A. ILIADIS21
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
Est+Opt CIRM 18/8/2009 A. ILIADIS22
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
Est+Opt CIRM 18/8/2009 A. ILIADIS23
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
Est+Opt CIRM 18/8/2009 A. ILIADIS24
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
Est+Opt CIRM 18/8/2009 A. ILIADIS25
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.
Est+Opt CIRM 18/8/2009 A. ILIADIS26
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
Est+Opt CIRM 18/8/2009 A. ILIADIS27
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
?
?
Est+Opt CIRM 18/8/2009 A. ILIADIS28
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)
Est+Opt CIRM 18/8/2009 A. ILIADIS29
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
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