Modeling biological systems - The Pharmaco-Kinetic-Dynamic paradigm

Transcription

1 Modeling biological systems - The Pharmaco-Kinetic-Dynamic paradigm Main features:. Time profiles and complex systems 2. Disturbed and sparse measurements 3. Mixed variabilities One of the most highly developed skills in contemporary Western civilization is dissection: the split-up of problem into their smallest possible components. We are good at it. So good, we often forget to put the pieces back together again. Foreword Science and change Alvin Toffler In: Ilya Prigorine (977 Nobel in Chem), Isabelle Stengers, Order out of chaos. Man s new dialogue with nature, Bantam NY 984

2 CHAPT I : Processes and models Real process and mathematical model The functional scheme. Modeling in modern PKs. Structure and parameters in models. Computational tools: matrix calculus. Linear and nonlinear operators. The parsimony principle Redundancy and misspecification. Indexes for goodness of fit: RMSE and determination coefficient. The fotemustine example: modeling ANC toxicity. Dynamic functional scheme Initialization, iterations, parameter convergence. Residual error. Real-time processing of data Bayesian estimation, the Bayesian attractor. Geometric analysis. Influence of the noise in parameter estimates. Model validation State and parameter spaces. Checking structural and parametric identifiability. Optimize PK experiments. Building and using models Identification, simulation, dosage adjustment. Synthesis of main tasks. Controlled systems, cybernetics. 2

3 Real process and mathematical model y [ng/ml] 0 2 Real process 0 2 Fitted model t [h] y [ng/ml] y Mathematical model () t ( k t) y [ng/ml] 0 2 D = exp V V t [h] k t [h] V = 6 L k = 0. h - 3

4 Standardize observations Ex : Administer the same dose to 2 different subjects and compare kinetics. Sampling times [ h ] Real sampling times [ h ] / Subject n Conc. # - Real sampling times [ h ] / Subject n Conc. # y y2 3 Heterogeneity in real sampling: the comparison is not possible. Compare the kinetics out of the «world» of concentrations. y2 y22 y23 y24 y Transformation Use a mathematical model smoothing observations and computing PK parameters Compare PK parameters. 4

5 Compare PKs for 3 subjects Dataset In general [ ] n ˆ ˆ - V L k h { xˆ j ; j = : n } Subjects are statistically distinguishable : V k thanks to for n 2 and 3, # to for n and 2. k [ h - ] V [L] 3 5

6 Functional scheme Measurement noise noise PK PK process Observation Administration protocol Prior Prior information Equivalence criterion PK PK model Prediction Nonlinear programming 6

7 Topics in in modern PKs PK process Classical PKs : analyze physiology, drug characteristics, biopharmacy, etc. Mathematical models PK : structural and parametric aspects. Error variance models. Equivalence criterion Bayesian, maximum likelihood estimators. Nonlinear programming Iterative methods, heuristic algorithms, quasi-newton methods. Prior information Population approaches, parametric and nonparametric methods. 7

8 Modeling Models Dialectic, mental, physical, mathematical, etc. Elements Structural and parametric knowledge, state variable. Goals Model is proposed to be efficient for a specified action : recognition, control, simulation, etc. Validation Experimental and statistical tools. Applications in : engineering ( fast dynamics ), : biology, sociology, economics, PKs ( slow dynamics ), : physics, astronomy, PDs ( mixed dynamics ). Ex : Tourist map of the city of Marseille. 8

9 Species of models Static / dynamic ( time dimension, differential equations ). Discrete time / continuous time ( computer control ). Linear / nonlinear ( which is the reference? ). Deterministic, statistic ( precision of estimates ), Input-Output form / state-space model : stochastic models ( dispersion of parameters ). presentation ( macro-rates ) / knowledge ( micro-rates ) models. Lumped / distributed parameter models ( space-time differential equations ). Time invariant / time varying ( parameters ). In PKs { Dynamic, continuous time, [ nonlinear / PK parameters, linear / drug amounts ], statistical and stochastic, I-O form and state-space, lumped, time invariant }. 9

10 Mathematical modeling Models are defined by : - their structure ( number and connectivity of compartments, etc ) expressed by mathematical operations involving adjustable parameters Ex : -cpt, D V () t = exp( k t) exponential structure, parameters - the numerical value of parameters used y [ ] [ ] x= V k - V k = 6 L 0. h CHARACTERIZATION Structure + ESTIMATION Parameters MODELING System Identification 0

11 Matrix definitions Definition of terms An array of numbers written as below is called a matrix of order. m= n n = m = m= n= Square matrix if. Column vector if. Row vector if. Scalar if. A square matrix may be : aij = a ji j j aij = 0 i j a ii = i = : n Symmetric if, Diagonal if, etc. m n a a... a a a... a 2 n n = a a... a m m2 mn A diagonal matrix with is the identity matrix. Notation : matrix A ( cap ), matrix element a ( lower ), vector a ( lower, underlined ). A I

12 Matrix algebra Transposition Addition T A m n C A n m c = = ij ji A, B m n C A + B m n = ij ij ij a c = a + b Multiplication A B m n n q C = A B m q c = ij a n k = ik b kj Square matrix Determinant ( characteristic scalar ) Ex. Inversion Rules = a a a for n = 2 A A 22 2 a2 C = A C A = A C = I c = ( ) T T T ( A B) = B A ( B) = B A ij A i+ j = A A ( ) A T A i, j 2

13 Operators : Analytic geometry - Rotation of rectangular coordinate system Rotation angle ω Matrix form or [ X, Y ] [ xy, ] Matrix inversion old and new coordinates x = X cosω + Y sinω y = X sinω+ Y cosω x cosω sinω X = y sinω cosω Y c new = Ω c c old old Ω cosω sin c with = new y Y ω + ω Ω = sinω cosω X M x 3

14 Operators : Analytic geometry - 2 Conversion of rectangular to polar system Inversely r = X + Y ξ = arctan X 2 2 ( Y X) = r cosξ Y r M Impossible to link r ξ and Y X Y = r sinξ by a matrix operator. ξ X The conversion relations are nonlinear. 4

15 Linear operators Let :. 2. e () t ( t) TT e 2 () t ( t) TT s s 2 For the input combination : () t + h e () t g e 2 TT?? The operator T is linear, if and only if?? = g s( t) + h s2( t) 5

16 For the -cpt model y () t s( t) e( t) associate : - to and D - to. The model is a linear operator because y PK example - D V () t = exp( k t) when then D D D () ( ) V V g D + h D2 D + h D2 exp V y t = exp k t and 2 D2 y2() t = exp( k t) ( k t) = g y ( t) + h y ( t) g 2 PK linearity ( proportionality ) 6

17 For the -cpt model associate : - to and - to. PK example - 2 The model is a nonlinear operator because when then k y () t s( t) e( t) k y D V () t = exp( k t) D D V V D g k+ h k2 exp g k+ h k2 t g y t + h y2 t V y() t = exp ( k t) and k2 y2() t = exp( k2 t) [ ( ) ] () () Parametric nonlinearity 7

18 Choose the best model Rule : The model should be a necessary and sufficient description of the real process necessary : good fitting on observed data. sufficient : without redundancy of parameters in the structure. Rule 2 : The model structure should be parsimonious : to adequately represent the real process. by containing the smallest number of parameters. Ex : -cpt model is misspecified. 3-cpt model is redundant. 2-cpt is the best model. y t 8

19 Fotemustine dataset Rectangular array i = : n n = 29 ID ANC ANC BASE Nadir ANC Dose D Clearance CL BSA BSA Weight W i ANC i ANC i D i CL i BSA i W i n ANC n ANC n D n CL n BSA n W n Output / observations Relative ANC count y i = ANC ANC ANC i 9

20 Predictions with using then z the constants / parameters data for individual i Modeling neutrophil toxicity zi e Logit model ymi = + zi e = x + D x + CL x + BSA x + + W i 2 i T 3 i 4 i x p x x x x p, and T i g = [ D CL BSA W ] i z i = 2 = g T i x i i i i p =4 Problem Obtain numeric values for : fit to. Application y M 0 xˆ y Mi Predict toxicity for a new individual with given data y = logit 0 x ( T g ˆ ) M 0 x y i g 0 20

21 Regression line correlation coefficient Coefficient of determination 2 var( yi ymi ) R = var y y Indexes for the goodness of fit y = a y + b + e ( ) is the part of data variability i i R Mi ( with respect to the average y ) explained by the model. n Root Mean Squared Error RMSE = w n p i= RMSE n p y Mi y i R 2 takes into account the sample tail,, the number of model parameters,, and the closeness of predictions to the observations. i y i y Mi 2 i ( y y ) i Mi 2 2

22 Reduce the model structure CV p x x x * * * * * * * 3 x x * * * * * * * * 5 x * * * * * * 6 x * * * * * 7 x * * 8 x x * * * 0 x 7.67 * * * * * * * * * x * * * * x * 3 4 Largest coefficient of variation CV x RMSE (%)

23 Redundancy, misspecification Misspecification High RMSE NC non-satisfied RMSE (%) R 2 Redundancy Increasing RMSE SC not-satisfied p Warning : R 2, the squared correlation coefficient, does not detect redundancy! 23

24 Functional scheme ( dynamic ) Linear / p model : no loop, one stage estimation. Nonlinear / p model : many loops until convergence. PK process Measurement noise noise Administration protocol Prior Prior information Equivalence criterion Arbitrary initial initial values values PK model Nonlinear programming 24

25 Iterations, parameter convergence Ex : Fotemustine neutrophil toxicity Nonlinear modeling n = 29, p = 8 0 Optimized final values 0 2 nd Param. value st 3 rd RMSE Arbitrary initial values Iteration no

26 Errors in in the functional scheme The existing errors : experimental, structural, parametric. Residual error : experimental, structural ( model misspecification ). RMSE Initial parametric error ( canceled at the convergence ) Residual error Iteration no 26

27 The system identification loop Experimental Design Design Prior Knowledge Dataset Choose Model Model Structure Calculate Model Model Choose Criterion of of Fit Fit Validate Model Model Not OK: Revise OK: Use it! 27

28 Real-time processing of data Data processing In batch or off-line { m observations } - data processing Sequentially or in real-time Wait until the last observation for all data processing. Perform modeling while observations are available. { observation - datum processing } { times } Warning : In the first stages, the lack of individual information should be offset by the prior information ( attractor - population studies ). m As the number of individual samples increases, the prior information can be forgotten. Bayesian estimation 28

29 Estimates in in real-time processing Only one observation D y exp t V Two observations D y = exp k t V D y2 = exp k t V ( ) ( t, y ) = k infinite pairs of values for ( V,k ) ( ) More than two observations V,k No exact solution : find minimizing the total discrepancy between observations and predictions i [ ( t, y ) ( t, y ) ] ( ) ( ) D V y exp( k ) 29 t i k V = = t D y t 2 y y ln 2 y y 2 t t t 2 minimize i =: m RMSE

30 Equivalence criterion, real-time Indexing observation times ; individual associated with parameters. Equivalence criterion Intra-kinetic weighting In real-time when, leads to infinite solutions. Bayesian attractor m i j p x j i= Combine criteria to obtain m ( j ) 2 SE = wi obsi pred i x m < p SE Inter-kinetic weighting Select the most appealing solution by using an attractor : located at and associated with force of attraction x A prior = T ( x x ) D ( x x ) j A x j A D j A A 2 ( prior) ln( SE) ln + A 30

31 Concept of simulation Random generator of of measurement noise noise PK PK process y ( t x ) M i, 0 Observation Administration protocol PK PK model No structural error Known parameter values y ( t x ) M i, 0 3 Nonlinear programming Equivalence criterion Prediction

32 Geometric analysis y D V () t = exp( k t) t y Each i i observed pair is represented by a geometric locus in the parameter space Reference values V = 6 L k = Experimental design 0.h D = mg K = 5% - y [ng/ml] k = t D ln V i y i t [h] 32

33 nbr. samples < nbr. parameters 0.4 Infinite solutions in the geometric locus of the st observed pair. 0.3 Geometric locus h ng/ml Bayesian attractor Set solution using the Bayesian attractor [ ] - V L k h Ref k [ h - ] x A D A Est V [L]

34 nbr. samples = nbr. parameters Bias! - V [ L ] k h Ref Calc Remove bias k [ h - ] h ng/ml 0. Intensive sampling h ng/ml V [L]

35 nbr. samples > nbr. parameters Calculated parameters are the coordinates of intersecting loci. 0.4 Intersections are the p combinations of observations : C m p 5 = C 2 = [ ] 0 m - V L k h med iqr MLE k [ h - ] h h 2 h V [L] 6 h 8 h

36 Sensitivity w.r.t. noise 0.4 [ ] - V k L h Ref % med iqr % med iqr k [ h - ] V [L] 36

37 Without measurement error 0.4 Accurate exp devices 0.3 Undisturbed observations m = p k [ h - ] When? V [L] 37

38 Bayesian estimation It analyzes only a few samples per patient It requires Respect of the ethical constraints, Reduction of the cost of the PK analysis. ( ) f x Obtaining the prior information : attractor or. Balancing the population and the individual information balance the prior and likelihood terms Operating with : staffs having a good experience and high specialized health professionals. It may be used in a feedback loop Efficient data analyses. 38

39 State and parameter spaces Parameter space State space Real process PK PK process Observation Comparison Comparison Artificial mechanism PK PK model Prediction Random component Precision of estimates Measurement error 39

40 State vs. parameter space y [µg/ml] y D 0.06 = exp Confidence V 90% () t ( k t) y [µg/ml] 0.04 regions and corridors t [h] t [h] y [µg/ml] D k [h - ] Likelihoodists t [h] V [L] 40

41 Checking 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. u() t -V k 2 k 2 2 k [ h - ] % k p k e k p V [L] structural non-identifiable Non-consistent estimate V 4

42 Overall notes Modeling extensions in PKs Repeated dose ( multi-input ) and simultaneous observation of several kinetics ( multioutput systems ), model the error process. Validation Check whether the model is "good enough", whether it is valid for its purpose : { statistical tests, confidence intervals, sensitivity analysis, analysis of residuals, information concepts, new experiments } Remember The model never explains, but only describes the available observations. The model is not an exact description of the real process. The success of modeling depends on the richness of available knowledge. 42

43 Building and using models Real Process y tt Time schedule + Drug amounts Functional properties Observations, or Predictions, or Clinical Constraints Dosage adjustment L j= ( ) A exp a t ij j Simulation 43

44 Devil's triangle Process : Modeling Simulation Control Input : Adm. Protocol Identification Output : Concentrations 44

45 Dosage adjustment flowchart Individual dosage adjustment Invert the model Identify individual PK parameters by using Bayesian estimation Ethical Ethical constraints Individual observation Prior information Compiling knowledge 45

46 Processes and models Biological real processes Preserve integral functionality of the real process. Avoid experiments leading to : breakdown the feedback mechanisms or excessively stimulate the process to record quantifiable observations. Analysis by modeling Obtain a reliable and efficient ( simple ) tool complying with the functional properties of the real process. Manage complex situations : Optimize experiments : administration and sampling. Individualize pharmaco-therapy : adjust dosage regimens, prediction and control the real process. 46

47 Modeling pros vs. cons Pros : Discovery of fundamental properties of PK process by assaying biological samples. Data reduction : large datasets are summarized with a few PK parameters. Observations standardization : by means of mechanistic models, the overall observed profile is shared to its constitutive [ administration + system + drug ] components. Description of mixed variabilities ( population studies ) : compare, discriminate and classify individuals. Ethical individual recognition : combine population and incomplete individual information in a Bayesian estimator. Cons : Pluridisciplinarity : with emphasis on applied mathematics and engineering techniques. Long training course : [ undergraduate + graduate ] studies followed by Ph.D. thesis. 47

48 Optimize experiments Administration protocol Administration site Dosage adjustment Sampling protocol Schedule + Amounts Sampling site + Schedule Structural identifiability Parametric identifiability 48

49 Controlled systems Control channel Operator Old Transfer and use energy. Human operator. Management of simple situations. Information transmission and data processing. Computer help, communication theory. Control devices New Rapid actions, independent of the nature of support. Complex but optimized situations. Observation devices Remote control. Feedback channel 49

50 N. Wiener, Cybernetics,

51 W. von Brawn, NASA, 960-5

52 R. R. Jelliffe, Clinical PKs, 970- R. Bellman R. Jelliffe 52

53 system identification ( ) ( ) x arg max{ } B = f x y x arg max{ f y x } L = R.A. Fisher 0 2 y (ng/ml) t (h) C.F. Gauss y D V () t = exp( k t) 53