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1. Table 2 Pharmacokinetic models with Michaelis Menten elimination included in the library of models Name Input Cpt Elimination Parameterisation Administration Arguments bolus 1cpt VVmkm IV bolus 1 Michaelis Menten V Vm km sd i f A sd doseMM TInf infusion lcpt VVmkm IV infusion 1 Michaelis Menten V Vm km md doseMM TInf tau i sd doseMM orall lcpt kaVVmkm lst order 1 Michaelis Menten ka V Vm km md doseMM tau bolus_2cpt_Vk12k21Vmkm IV bolus 2 Michaelis Menten na KR REDEN bolus 2cpt V10V2Vmkm IV bolus 2 Michaelis Menten an pur er kis sd sd doseMM TInf infusion 2cpt Vk12k21Vmkm IV infusion 2 Michaelis Menten Vr klln Nobeni Km md doseMM TInf tau sd doseMM TInf infusion 2cpt V1QV2Vmkm IV infusion 2 Michaelis Menten Vl Q V2 Vm E m km md doseMM TInf tau A ka V ki2 k21 sd doseMM orall 2cpt kaVk12k21Vmkm lst order 2 Michaelis Menten pedi z Vm km md doseMM tau sd doseMM orall 2cpt kavlov2Vmkm 1st order 2 Michaelis Menten VI Q V2 EE ZAN Vm km md doseMM tau bolus 3cpt Vk12k21k31k13Vmkm IV bolus 3 Michaelis Menten Nye klap KaTy sd SPE k k13 k31 Vm km L 1 1 V2 2 5 bolus 3cpt V101V202V3Vmkm IV bolus 3 Michaelis Menten a B kii Q sd infusi 3cpt Vk12k21k13k31Vmkm infusi 3 Michaelis Ment V k12 k21 sd doseMM TInf infusion 3cpt IV infusion ichaelis Menten klip uie Vme km od doseMM TInf tau a l Mize OU Va 02 sd dos
2. models Linear progression Exponential increase Exponential decrease Name Param Name Param Name Param Linear immed lin lin Alin S0 immed lin exp Alin 50 immed lin dexp Alin S0 kprog kprog kprog Alin Quadratic immed quad lin Alin Aquad immed quad exp Aguad SO immed guad dexp Alin Aquad SO kprog s E x S0 kprog kprog Logarithmic immed log lin Ec immed log exp m immed log dexp ge a Emax immed Emax lin Emak E immed Emax exp pu 250s immed Emax dexp RON CoU SO kprog SO kprog S0 kprog Si id Emax C50 Emax C50 Emax C50 TE immed gammaEmax lin gamma SO immed gammaEmax exp gamma SO immed gammaEmax dexp gamma SO B kprog kprog kprog Imax immed Imax lin DUE C90 immed Imax exp E immed Imax dexp Pu SC SO kprog SO kprog SO kprog Siomoid Imax C50 Imax C50 Imax C50 iis immed gammaImax lin gamma SO immed gammaImax exp gamma SO immed gammaImax dexp gamma SO kprog kprog kprog x In addition to those in Table 3 28 Table 5 Turnover response pharmacodynamic models linked to a pharmacokinetic model included in the library Types of response Emax Sigmoid Emax Imax Sigmoid Imax Full Imax Sigmoid full Imax Full Imax Models with impact on the Input Output Name Parameterisation Name Parameterisati
3. numeric value for the minimum number of sampling times per subject for each response i with i lt A B C 43 9 For Fedorov Wynn algorithm use the initial population design specified in prot must involve elementary designs with number of samples per subject and sampling times in accordance with sampwin nsamp nmaxpts and nminpts for each response 44 6 Output The results are written in the output file called stdout r by default or with the name specified in the input file This file is different when only evaluation is performed or when optimisation is performed It is detailed in next sections respectively for evaluation and for optimisation 6 1 Evaluation output file and objects Figure l represents the output file from the design evaluation built on the Example documentation in the sections 1 2 4 and 1 6 1 The user can read on the Figure 1 D The name of the function used PFIM 4 0 ep The name of the project and the date D A summary of the input response s the evaluated population or individual Bayesian design for each response doses or initial conditions in case of models defined by differential equation system and number of subjects corresponding to those designs between subject variance model and residual error model for each response s rror tolerances for the solver of differential equations system if used name of t
4. source file path directory program LibraryPD PDdesign r For these models the design variables are the concentrations or the doses instead of the sampling times For example if one uses a linear drug action model without baseline immed lin null from the library the model uses one parameters Alin see Example 2 section 4 2 Pharmacodynamic models linked to pharmacokinetic model In this section we consider models with two responses with one response for the PK and the other one for the PD We thus optimise sampling times for both responses using a PK PD model Using the libraries of models we have four cases to compose the PK PD model depending on the form for each submodel either with an analytical form AF or a differential equation system ODE Therefore there are four cases of PK PD models in PFIM library 1 PK model with linear elimination AF and immediate respons PD model AF 2 PK model with linear elimination AF and turnover response PD model ODE W U PK model with Michaelis Menten limination ODE and immediate response PD model AF 4 PK model with Michaelis Menten elimination and turnover response D model ODE U 24 Table 3 Immediat response pharmacodynamic models included in the PD library for PD alone and for PK PD model Drug action models Linear Emax Sigmoid Imax Sigmoid
5. 32 formED lt function t y p Again the user may modify anything within this function but the name and header must remain unchanged The function formED has 3 arguments a vector of time t the current estimate of the variables in the ode system y a vector of parameters p Within the function the user has to define the name of the parameters in vector p and the differential equation system The function returns a list with 2 elements the first element is a vector giving the values of the derivatives for each equation in the differential equation system computed for each time point in t using the parameters p the second element is a vector of predictions computed for each time point in t using the parameters p in PFIM this vector contains the response s we are observing n The initial values of the system have to be specified in the input file stdin r presented in the section 5 uder the name condinit n The implementation of differential equations system requires the use of the lsoda function included in the library deSolve R Thomas Petzoldt and of the fdHess function included in the library nlme developed by Jose Pinheiro and Douglas Bates The lsoda function uses a function of the same name written in Fortran by Linda R Petzold and Alan C Hindmarsh This function solves system of differential equations using the Adams method a predictor corrector method for non
6. X V ka ka k exp k t exp ka t return y Design Sample times for response A times subjects doses 1 c 1 3 8 200 100 Random effect model Trand 2 Variance error model response A 0 5 0 15 f 2 Computation of the Population Fisher information matrix option 1 Previous FIM from file Previous FIM txt FIM saved in FIM txt kkkkkkkkkkkkkkkkkkk FISHER INFORMATION MATRIX kkk kk kd dk kk kk kk k vi v2 v3 v4 v5 v6 1 17523 1601 113 42921 0 0000 0 0000 0 0000 0 0000 2 113 4292 11 91056 0 0000 0 0000 0 0000 0 0000 3 0 0000 0 00000 1499 7312 228 5511 420 7733 594 9001 4 0 0000 0 00000 228 5511 8988 9175 701 5138 2780 3955 5 0 0000 0 00000 420 7733 701 5138 2114 1728 4049 3932 6 0 0000 0 00000 594 9001 2780 3955 4049 3932 12315 5527 9 kkkkkkkkkkkkkkkkkkkkkkkkkk EXPECTED STANDARD ERRORS k ake ae ak he ak dese ak e ake fe ek KK Fe ce kk k Beta StdError RSE k 0 25 0 007798489 3 119395 5 V 15 00 0 299123566 1 994157 5 a Variance of Inter Subject Random Effects omega StdError RSE k 0 25 0 02669459 10 67783 v 0 10 0 01098554 10 98554 Sigma StdError RSE sig interA 0 50 0 03670921 7 341841 sig slopeA 0 15 0 01527153 10 181022 kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk DETERMINANT teke k de de de de ce de skok skok skok de k ve ke de k kkk k kk kk k 2 199791e 19 kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk CRITERION kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkka 1673 903 KK
7. lt if B e A 0 6 11 8 tA z 0 P NSE P if B lt 0 A 7 where 0 is the cumulative distribution function of the standard normal distribution and Z is such that d z 1 a Last we compute the number of subjects needed to be included to obtained a power of P called NNI P using the equation 3 like for comparison test 2 3 1 3 Previous information New feature An option to load previous information through a predicted or an observed Fisher information matrix is now available in PFIM 4 0 Evaluation and optimisation ar then performed combining the previous information matrix with the current Fisher information matrix following the principle of adaptive designs 6 Taking into account previous information the new computation of the Fisher information matrix is then N Meses gt Me E E i l where M Fprev denotes the pr vious Fisher information matrix Note that the previous Fisher information matrix should have the same dimension as the current Fisher information matrix 9 is now possible to save the Fisher information matrix corresponding to an evaluated or optimised design 2 3 2 Bayesian Fisher information matrix New feature The new version 4 0 of PFIM enables design evaluation and optimization for maximum a posteriori estimation of individual parameters based on the Bayesian Fisher information matrix 7 We are interested in the precision estimatio
8. F s the Fisher information matrix and the value of the criterion determinant 1 dim where dim is defined in e n matrix The eigenvalues of the Fisher in formation matrix and the correlation 48 PFIM 4 0 Project Example Optimisation Date Thu Jul 31 09 22 17 2014 kkkkkkkkkkkkkkkkkkkkkkkkkkkk INPUT SUMMARY 2e dete se RK de ce de ke ce se ek Fe de de he He ke Fe ke ke Fe ek k k kk k k Analytical function model function t p X ka lt p 1 k lt p 2 V lt p 3 y lt X V ka ka k exp k t exp ka t return y Initial design Sample times for response A Protocol subjects doses 1 c lt 0 33 1 5 5 12 1 100 Total number of samples 4 Associated criterion value 3 5272 Identical sampling times for each response FALSE Random effect model Trand lt 2 Variance error model response A 0 5 0 15 f 2 Optimization step Sampling windows for the response A Window 1 t 0 33 1 1 5 3 5 8 12 Nb of sampling points to be taken in this window n 1 4 Maximum total number of points in one elementary protocol 4 Minimum total number of points in one elementary protocol 4 BEST ONE GROUP PROTOCOL Sample times for response A times freq Subjects doses 1 c 0 33 1 5 5 8 1 1 100 Associated criterion 3 8066 O O 49 Computation of the Bayesian Fisher information matrix FIM saved in FIM txt kkkkkkkkkkkkkkkkkkk FISHER INFORMATION MATRIX kx
9. A or 8 A while the alternative hypothesis is Hi A SBS4A Ho is composed of two unilateral hypothesis Ho a 8S A and Hosa 2 824A Equivalence between two covariate effects can be concluded if and only if the two hypotheses H and H are rejected The two statistics of the unilateral Wald test under the null hypothesis A A are defined as Sw me and Sy zh with p the covariate effect SE B SE B estimates and its associated standard error Under H when B B with fi Ar we then compute the power of the equivalence Wald test defined as BEALI Pg 1 O z i e A 0 4 ies BEM preja B A Pi Zia i E O A 5 uzela Sen repo where 0 is the cumulative distribution function of the standard normal distribution and z is such that z lt 1 a Vin equivalence test B is usually chosen to be zero Using the covariate effect f fixed by the user the corresponding standard error SE B is predicted since PFIM 3 2 for a given design and the values of population parameters Computation of the number of subjects needed The number of subjects needed to achieve a power P to show eguivalence between two covariate effects using the Wald test is also computed First from eguations 4 and 5 we compute the SE needed on B to obtain a power of P called NSE P using the following relation 8 A z 1 P NSE P
10. Ouadratic Logarithmic Emax Imax Baseline Null baseline Constant baseline Name Parameterisation Name Parameterisation immed lin null immed guad null immed log null immed Emax null immed gammaEmax null immed Imax null immed gammaImax null Alin Alin Aguad Alog Emax C50 Emax C50 gamma Imax C50 Imax C50 gamma immed lin const immed guad const immed log const immed Emax const immed gammaEmax const immed Imax const immed gammaImax const Alin SO Alin Aguad SO Alog SO Emax C50 SO Emax C50 gamma SO Imax C50 SO Imax C50 gamma SO 25 To use PFIM for design evaluation and optimisation for a PK PD model the two models must be in the same format If both models are written in closed form case 1 the user can combine the immediate response pharmacodynamic models in closed form expression from the file LibraryPD PKPDdesign r with the pharmacokinetic models with first order linear elimination Table 1 in closed form expression from the file libraryPK r In the PD functions the expression of the PK model is given as an argument see Example 3 section 4 2 In this case the user must fill in the stdin r using analytical form options and must specify the paths of the library files in model r source file path directory program LibraryPK r source file path directory program LibraryPK PKPDdesign r For the thr other case
11. chosen and error tolerances for the solver of differ C45 The optimised ential equations design and System if used the associated criterion For the simplex algorithm the number of iterations performed and the number of function evaluations the status of the convergence false or achieved are reported For the Fedorov Wynn algorithm for optimal population design the optimal group structure with the proportion of subjects and the equivalence in number are then reported The best one group protocol is also always reported with associated criterion When optimising design correspond to 5 The population or individual or dim dim symmetric ma parameters to be e number of the erro a the bes trix where dim stimated r model paramete param C62 The StdError value e The value of the determinant of rs respectivel Fisher information matrix of and relative si the associated shrinkages values are al The name of is given y 1 each Bayesian or an t one group protocol the number of parameter tandard error individual the resulted design Bayesian Fisher information matrix a is the total number of population individual parametres th or only the number of individual file where is possibly saved the rs th standard error Bayesian design with the expected RSE In case of iso reported
12. concentration in the second central compartment which is defined by the quantity in this compartment y 2 divided by the volume of distribution V Several responses can be given 34 Example 9 PK model after multiple dose administration using a differential eguation system created by the user formED lt function t y p ka lt p 1 V lt p 2 Cl lt p 3 tau lt 12 input orall function ka V dose n tau t if n 0 return dose ka V exp ka t else return dose ka V exp ka t n tau input_orall ka V dose n 1 tau t n lt t tau input lt input orall ka V dose n tau t dy lt C1 V y 1 input list c dy c y 1 Note In this illustration the user creates a function of one response model describing a one compartment oral absorption after multiple dose administration with a between dose interval between two doses equal to twelve hours The number of doses and the between dose interval are defined within the function Example 10 PK model with bolus input linear elimination and turnover response PD model source file path directory program dirsep CreateModel PKPDdesign r create formED bolus lcpt Vk turn input Imax Note In this example the user creates a PK PD model with a one compartment bolus input for the PK and a turnover response model with an inhibition on the input for the PD using the function create formED The dose is specified as initial condition of th
13. depending on administration type additional variables may be reguired They are specified in the arguments N number of doses tau interval between two doses TInf duration of the infusion doseMM dose For models with infusion the user must specify the duration of infusion TInf as an argument The rate of infusion is computed automatically in the function model through the expression dose TInf For PK models with 18 linear elimination the variable dose has to be specified in the input file When a model with multiple dose administration is used for example the first order oral absorption with one compartment model with option md orall lcpt kavCl md from the library the function of th model uses three parameters ka CI and V and two needed variables N tau the number of doses N and the interval between two doses tau see Example 1 section 4 2 Pharmacokinetic models with a linear elimination The library of PK models with linear elimination is composed of one two and three compartment models for the thr types of input bolus infusion and first order oral absorption and the thr types of administration single dose multiple dose steady state The list of these PK models is given in Table 1 19 Table 1 Pharmacokinetic models with first order linear elimination included in the library of models Name Input Cpt Elimination Parameterisation bolus 1cpt Vk IV bolus 1 lst or
14. evaluation OPT for optimisation According to the choice specific objects notified below must be specified 5 1 Option for Fisher information expression You have to choose the option for the Fisher information matrix that is population individual or bayesian and complete the FIM object FIM type of the Fisher information matrix P for population I for individual B for Bayesian In the case of population design i e population Fisher matrix you can complete the previous FIM object if previous information is available If it is not the case leave it as the default previous FIM character string indicating the name of the file containing the previous information To compute Fisher information matrix you have to complete the option object option type of option 1 for block diagonal Fisher information matrix 2 for complete Fisher information matrix A nr Then you have to complete the object nr value indicating the number of responses in the model According to the choice specific objects notified below must be specified 36 5 2 Structural model option modelform character differential either DE or string indicating eguation systems For analytical model specification only dose identical dose boundi a model given under analytical form logical value T if the dose is the same for all elementary designs
15. i 1 N is defined as Y f 0 4 i where the function f defines the nonlinear structural model 0 is the vector of the p individual parameters for individual i is the elementary design of individual i and g is the vector of residual error The vector of individual parameters 0 depends on u the p vector of the fixed effects parameters and on b the p vector of the random effects for individual i The relation between 0 and u b can be additive for a normal distribution of parameters that is 0i H bi or exponential for a lognormal distribution of parameters so that 60 u Xexp b It is assumed that bj N 0 0 with Q defined as a pXp diagonal variance covariance matrix for which each diagonal element oy j lt 1 p represents the inter individual variability of the j component of the vector bi It is also supposed that N 0 2 where X is a njXn diagonal matrix such that i u bi Ointer Ostope amp diag inter Oslope Xf Gi JE 2 The terms Ginter ANA Ostope are the additive and proportional parts of the error model respectively Conditionnally on the value of bj it is assumed that the gj errors are independently distributed In the case of K multiple responses the vector of observations Y can then be composed of K vectors for the different responses Y v5 yb al where Yi k 1 K is the vector of ny observations for the k response Each of the
16. iter print for each respons i with i lt A B C For example if there is a singl respons model representation in 2 intervals 0 12 and 48 60 is specified by upperA c 0 48 and lowerA lt c 12 60 numeric value for the minimum delay between two successive sampling times logical value to print the iterations T or not F simplex parameter percent of change from initial design for the Max iter Rctol initial vertices of the simplex algorithm building default 20 a positiv integer specifying the maximum number of iterations allowed default 5000 a positive numeric value specifying the tolerance level for the relative convergence criterion of the Simplex algorithm default le 6 For Fedorov Wynn nwindi sampwini nsampi fixed timesi nmaxptsi nminptsi algorithm only numeric value for the number of sampling intervals for ach response i with i A B C list of vector of the allowed sampling times for each sampling interval for each response i with i lt A B C list of vector of allowed numbers of points to be taken from each sampling interval for each response i with i A B C m list of times which will be in all evaluated protocols corresponding to fixed constraints for each response i A B C numeric value for the maximum number of sampling times per subject for each response i with i lt A B C
17. o EOPfim4 0opl r To compute the block diagonal Fisher Information matrix option 1 to evaluate a population individual or Bayesian design using a differential equation system to describe the model o EOPfimOPI4 0opl r To compute the block diagonal Fisher Information matrix option 1 to optimise a population individual or Bayesian design using a differential equation system to describe the model o Pfim4 0op2 r To compute the full Fisher Information matrix option 2 to evaluate a population design using an analytical form to describe the model o PfimOPT4 00p2 r To compute the full Fisher Information matrix option 2 to optimise a population design using an analytical form to describe the model 15 o EOPfim4 0op2 r To compute the full Fisher matrix option 2 to evaluate differential equation system to describe the model o EQPfimOPT4 00p2 r To compute the full Fisher matrix option 2 to optimise differential equation system to describe the model o algosimplex4 0 r To use the Simplex algorithm o initfedoR c and classfed h To compile the dll o libFED dll and libFED64 dll The dynamic libraries Information a population design using a Information a population design using a of the Fedorov Wynn algorithm for R 32 bit and 64 bit respectively o algofedorov4 0 r libFED64 dll o LibrayPK r o Li
18. F if not value of the dose if dose identica vector l T if of the q doses for each elementary design If one uses infusion models implemented in not the library of models the dose has to b The rate of infusion is computed model by the expression dose TInf func Specified her in the tion vector of bounds specific to each response i with i A B C If the model is defined over intervals by different expressions give in that vector the values of each time interval for the response i The first expression will be use for the first time interval the second expression for the second time interval logical value T if using numerical derivatives then the model must be written by the user using R function in model r file F if not then the model is specified via the object form which is a vector of expressions For differential equations system specification only time condinit condinit identical condinit RtolEQ AtolEQ initial time at which initial conditions are given logical value T if the initial conditions are gn F if not the same whatever th lementary desi initial values of the system at the initial t given into an expression If condini nter once th ime t identical T xpression of the initial values of initial time l the system at the vectors of the initial el
19. KKKKKKKKKKKKKKKKK EIGENVALUES OF THE FISHER INFORMATION MATRIX dddx x FixedEffects VarianceComponents min 15109 231479 11 17585 max 17523 894830 1509 84067 max min 1 159814 135 09852 kkkkkkkkkkkkkkkkkkk CORRELATION MATRIX Jed kk kkk k kkk k 1 2 3 4 5 6 V1 1 0000000 0 2482863 0 00000000 0 00000000 0 00000000 0 00000000 v2 0 2482863 1 0000000 0 00000000 0 00000000 0 00000000 0 00000000 V3 0 0000000 0 0000000 1 00000000 0 04548621 0 21289290 0 09121384 V4 0 0000000 0 0000000 0 04548621 1 00000000 0 09083326 0 23035289 V5 0 0000000 0 0000000 0 21289290 0 09083326 1 00000000 0 78714187 V6 0 0000000 0 0000000 0 09121384 0 23035289 0 78714187 1 00000000 Figure 1 Example of design evaluation output file 6 2 Optimisation output file and objects Figure 2 represents the output file corresponding to the optimal design described in The user can read on he name of th The name of the Example docum the Figure 2 e function used structural model Bayesian entation in the section 1 3 2 PFIM 4 0 he project and the date between subject and error numbers or proportions of subjects Ba A summary of the input variance model initial design initial and doses total number of allowed samples initial design GD Sampling times specifications criterion associated to the according to the algorithm used within which the optimal samples will be
20. R function formA function t p X ka lt p 1 k p 2 V p 3 y lt X V ka ka k exp k t exp ka t return y form lt formA Note In this illustration the user creates a function of a one response model describing a one compartment oral absorption Example 6 PK model after multiple dose administration using an analytical form with user defined R function form lt function t p X ka lt p 1 V p 2 C1 p 3 N lt 5 tau lt 12 y lt 0 for n in 1 N indic lt t gt n 1 tau yn indic X V ka ka C1 V exp Cl V t n 1 tau exp ka t n 1 tau y lt ytyn return y Note In this illustration the user creates a function of one response model describing a one compartment oral absorption after five administration doses with a between dose interval equal to twelve hours The number of doses and the between dose interval are defined within the function They can also be defined as fixed parameters included in the vector p see Section 5 for more details on fixed parameters 4 2 3 Models defined through a differential equation system Description Model defined as a solution of a differential equation system must be called formED and can be called from the PFIM libraries or defined by the users In the latter case the user need to write an R function in a format suitable for the solver package deSolve and using the following form
21. algorithm Compared to the Simplex algorithm the Fedorov Wynn algorithm better affords high design variables optimisation Moreover it considers only pre specified sampling times avoiding clinically unfeasible sampling times The drawback is the huge number of elementary designs to be created with corresponding huge number of Fisher information matrices to compute when the set of allowed sampling times is very large 13 2 5 1 Simplex algorithm The Simplex algorithm optimises statistical or exact designs in constrained intervals given a total number of samples An initial population design needs to be supplied to start the optimisation The maximum number of elementary designs and the number of sampling times per elementary design are fixed the sampling times and the proportions of subjects in each elementary design are then optimised From this initial design initial vertices for the simplex algorithm are derived reducing successively each component by 20 a default value which can be changed from the original component PFIM uses the Splus function fun amoeba from Daniel Heitjan revised 12 94 which is a translation from the Numerical Recipes for Nelder and Mead Simplex function 11 Note that it is now possible to take into account previous information through a predicted or an observed Fisher information matrix to optimise designs with this algorithm 2 5 2 Fedorov Wynn al
22. brayPD PDdesign r To use the response pharmacodynamic models alone library To use the dynamic library libFED dll or To use the library of pharmacokinetic models of immediate o LibrayPD PKPDdesign r To use the library of pharmacodynamic models linked to pharmacokinetic models both written using analytical form o CreateModel PKPDdesign r To use the libraries of pharmacokinetic and pharmacodynamic models when they ar writing either with different forms or both with differential equation systems 9 The files in the folder Program should not be changed The folder called Examples contains th xamples files The documentation which gives their description is included in PFIM 4 0 with this user guide To install PFIM 4 0 create a directory Documents PFIM 4 0 and download PFIM 4 0 3 3 Working directory Create a working directory for example U NNMy Documents PFIM 4 0 examples NExamplel Copy the files PFIM r In the file PFIM r specify your working directory for example directory U My stdin r and model r in this directory directory lt U My Documents PFIM 4 0_examples Example1 Then specify your program directory i e where is the folder called Program directory program lt U My Documents PFIM 4 0 Program Save the file PFIM r 16 3 4 Run Once the input file and the model file are filled in the user can run PFIM Load the ma
23. corresponding library file in the model file named by default model r New feature In the previous versions of PFIM a user defined model given in analytical form needed to be specified through an R expression An alternative way to write the model is now available through an R function with a specific format see section 4 2 3 4 1 Library of models 4 1 1 Library of pharmacokinetic models m Two types of PK models can be used in PFIM PK models with a first order linear elimination or PK models with a Michaelis Menten elimination The PK models with a linear elimination are written using an analytical form through an R expression whereas the PK models with a Michaelis Menten elimination are written using a differential equation system These PK models are written in the file LibraryPK r available in the Program folder The user has to specify the path of this file in the model file to use this library of models source file path directory program LibraryPK r The following sections show the list of models for each type of PK model in separate tables These tables display all the information in order to use the model function chosen The model is described by a name the type of input the type of elimination the number of compartments the parameters used parameterisation the type of administration sd single dose md multiple dose ss steady state
24. ction 5 for input specification The number of covariates the number of parameters associated to each covariate as well as the number of categories for each covariate are not limited In PFIM the distributions of the covariates are supposed independent 2 3 Fisher information matrix 2 3 1 Population Fisher information matrix 2 3 1 1 Expression The population Fisher information matrix M for multiple response models for an individual with an elementary design with the vector of population parameters Y is given as a weet A E V CCE V JG 2 C E V B E V with E and V the approximated marginal expectation and the variance of the observations of the individual The vector of population parameter VY is defined by VT uT AT with u the p vector of the fixed effects and A the vector of the variance terms Mp is given as a block matrix more details are given in 8 10 with AE V 2 2E pa jet ya ov V with m and l lt l p Ou Ou Ou On BEV etr va PV WA with m and isl dim A OO a on ea jn ae qu dim 2 and m 1 2 ml AN au n p OV If the dependence of V in u is neglected so that a the population u Fisher information matrix is a block diagonal matrix that is to say the block C of the matrix is supposed to be 0 Also the block A is simplified and expressed as T GEN va with m and l p Ca Ou Based on publications showing the bet
25. del for a response is defined over intervals by different expressions ach response should be written as a vector of expressions Each expression can be defined in an object formi where I 1 2 3 For example if the user wants to give thr xpressions for the first response he can write as follows formA c form1 form2 form3 formA can be a model of the PFIM libraries or an user defined model In the latter case the specification of the dose can be anywhere in the analytical expression The name dose should be used unchanged In the computation of the Fisher information matrix th dos given in each elementary design will be used If the user gives a value to the dose directly in the model then all elementary designs will have the same dos Example 1 PK model after multiple dose administration using an analytical form with the library of models source file path directory program LibraryPK r forml lt orall lcpt kaVCl md N 1 tau 12 1 form2 lt orall lcpt kaVCl md N 2 tau 12 1 form3 orall lcpt kaVCl md N 3 tau 12 1 form4 lt orall lcpt kaVCl md N 4 tau 12 1 form5 lt orall lcpt kaVCl md N 5 tau 12 1 formA lt c forml form2 form3 form4 form5 form lt c formA Note In this illustration the user creates a one response model using the model function implemented in the pharmacokinetic library Orall lcpt kaVCl describing a one compartment oral absorption after a m
26. dels is given in Table 5 For the second case where a PK model with linear elimination is associated to a turnover PD respons model defined using differential equation system the PK model must be written with a differential equations system as well Conseguently only some PK models from the Table 1 are implemented in CreateModel PKPDdesign r for bolus input only single dose models for infusion input single dose and multiple dose for first order absorption input single dose and multiple dose Fror models with a bolus input the dose has to be specified in the input file stdin r by default as the initial condition of the differential equation system see Example 10 section 4 2 For models with infusion or first order absorption input dose has to be specified as an argument of the function Create formED and NOT IN THE INITIAL CONDITION OF THE MODEL 26 IN THE INPUT FILE see Example 11 section 4 2 Conseguently it is not possible to specify different doses per group when using models with infusion or first order absorption input from the library All groups of the design are assumed to have the same dose Otherwise the user should use the user defined model option 27 Table 4 Immediat respons pharmacodynamic models linked to a pharmacokinetic model included in the library Drug action Baseline disease models
27. der V k bolus lcpt VCl IV bolus 1 lst order V CI infusion lcpt Vk IV infusion 1 lst order V k infusion lcpt VCl IV infusion 1 lst order V Cl orall lcpt kavk lst order 1 lst order ka V k orall lcpt kavcl lst order 1 lst order ka V Cl bolus 2cpt Vkki2k21 IV bolus 2 lst order V k ki2 k2l bolus 2cpt Clv10V2 IV bolus 2 lst order Gl V1 0 V2 infusion 2cpt Vkki2k21 IV infusion 2 lst order V k kl2 k2l Administration sd md ss sd md ss sd md ss sd md ss sd md ss sd md ss sd md ss sd md ss sd md ss Arguments N tau tau N tau tau TInf TInf N tau TInf tau TInf TInf N tau TInf tau N tau tau N tau tau N tau tau N tau tau TInf TInf N tau TInf tau 20 sd TInf infusion 2cpt Clv10V2 IV infusion 1st order El MIS V2 md TInf N tau ss TInf tau sd orall 2cpt kaVkk12k21 1st order 1st order ka V k k12 k21 md N tau SS tau sd a orall 2cpt kaClv10Vv2 lst order lst order ka Cl V1 0 V2 md N tau ss tau sd bolus 3cpt Vkk12k21k13k31 IV bolus 1st order V k k12 k21 k13 k31 md N tau ss tau sd ME bolus 3cpt C1V101V202V3 IV bolus lst order Cl Vi 01 V2 Q2 V3 md N tau ss tau sd TInf infusion 3cpt Vkki2k21k13k31 IV infusion lst order V k k12 k21 k13 k31 md TInf N tau ss TInf tau sd TInf infusion 3cpt C1V101V202V3 IV infusion lst order CI Vi Ol V2 Q2 V3 md TInf N tau ss TInf tau sd orall 3cpt kaVkk12k21k13k31 1
28. e differential eguation system in the R script file stdin r Example 11 PK model with infusion input Michaelis Menten elimination and immediate response PD model source file path directory program dirsep CreateModel PKPDdesign r create formED infusion lcpt VVmkm immed lin null doseMM 100 TInf 1 Note In this illustration the user creates a one response model using the model function implemented in the pharmacokinetic library infusion lcpt VVmkm describing a one compartment infusion input with Michaelis Menten elimination after a single dose administration sd The dose is specified as an argument of the PK function in the file model r not in the initial condition in the R script file stdin r 35 5 Input This section shows the common objects reguired for both design evaluation and optimisation One input file has to be filled called by default stdin r In the file stdin r the following R objects must be created project character string indicating the name of the project file model character string indicating where to find the structural model output character string indicating in which file the results should be printed outputFIM character string indicating in which file the Fisher information matrix should be saved To specify evaluation or optimisation designs the user has to complete the run object run character string for function selection EVAL for
29. eMM TInf f 3cpt V1Q1V2Q2V3Vmkm IV inf 3 Michaelis Ment infusion 3cpt VIQ Q infusion ichaelis Menten V3 Vm km e d ga tines eau 11 3cpt kakl2k21k13k31Vmkm Ist ord miki Son kaly ER yan orall 3cpt ka st order ichaelis Menten k13 315 vm bom semi eau _ ka V1 01 V2 sd doseMM orall 3cpt kaV1Q1V2Q2V3Vmkm lst order 3 Michaelis Menten Q2 V3 Vm km vid doseMM tau 23 4 1 2 Library of pharmacodynamic models n The library of PD models supports immediate response models either as a function of observed concentrations or linked to a pharmacokinetic model and turnover response models linked to pharmacokinetic models The following tables present these models giving the following elements for each drug model the name of the model function in the library the parameters used parameterisation Examples for the use of the library of pharmacodynamic models are presented in section 4 2 Immediate response pharmacodynamic models alone Linear quadratic logarithmic Emax sigmoid Emax Imax sigmoid Imax models with null or constant baseline are available The list of these models is given in Table 3 These models are written in closed form and can be used in the case of a model with one response PD evaluation or optimisation They are implemented in the file LibraryPD PDdesign r Thus the user has to specify the path of this file in the model file to use this library of models
30. ed standard error Under H when B lt B we then compute the power of the Wald test defined as B P Pag i is E a sa o where is the cumulative distribution function of the standard normal distribution and Zap is such that z 1 a 2 Using the covariate effect f fixed by the user the corresponding standard error SE B is predicted since PFIM 3 2 for a given design and the values of population parameters Computation of the number of subjects needed The number of subjects needed to achieve a power P to detect a covariate effect using the Wald test is also computed First from the equation 1 we compute the SE needed on f to obtain a power of P called NSE P using the following relation B NSE P 2 Zan 1 P 10 Last we compute the number of subjects needed to be included to obtained a power of P called NNI P using SE B NNI P Nx NSE P 3 where N is the initial number of subjects in the given design and SE B the corresponding predicted SE of B for the given design Eguivalence test Computation of the expected power The Wald test can be used to assess the eguivalence of a covariate effect B In PFIM the Wald test is performed on the B of each category for each covariate a global Wald test on the vector B all effect coefficients is not implemented For one covariate and an effect of one category 6 D 2 the null hypothesis is Ho BS
31. ementary design If initial val parameters to be estimated into the expression without any quotation marks relative error tolerance ither a array as long as y function Default value is le 06 absolute error tolerance ither a array as long as y function Default value is le 06 se nter th conditions for each ues depend on enter this parameter scalar or an See details in help for lsoda scalar or an See details in help for lsoda 37 Hmax an optional value for the maximum integration stepsize Default value is Inf From help for lsoda The input parameters RtolEQ and AtolEQ determine the error control performed by the solver The solver will control the vector ef of estimated local errors in y according to an ineguality of the form max norm of e ewt lt 1 where ewt is a vector of positive error weights The values of RtolEQ and AtolEQ should all be non negative The form of ewt is RtolEQ abs y AtolEQ where multiplication of two vectors is element by element If the request for precision exceeds the capabilities of the machine the Fortran subroutine lsoda will return an error code under some circumstances the R function lsoda will attempt a reasonable reduction of precision in order to get an answer It will write a warning if it does SO 5 3 Statistical model option parameters vector of p character strings for the name
32. ent estimation of model parameters especially when the studies are conducted in patients where only a few samples can be taken per subject These approaches rely on the Fisher information matrix FIM for nonlinear mixed effect models NLMEM available in several software tools 2 and are a good alternative to clinical trial simulation They require a priori knowledge of the model and its parameters which can usually be obtained from previous experiments PFIM www pfim biostat fr developed in our group is the first tool for design evaluation and optimisation that has been developed in R It is available since 2001 3 and was extended in version 3 to multi response models inter occasion variability discrete covariates with prediction of power of Wald test 4 5 The current version 4 released in Spring 2014 added several new features In this new version for population designs optimisation can be performed with fixed parameters or fixed sampling times The Fisher information matrix obtained after evaluation or optimisation can be saved in a file Previous information already obtained can be assumed and loaded through a predicted or an observed Fisher information matrix which is important in the perspective of performing adaptive designs 6 Additional features for Bayesian designs are now available The Bayesian Fisher information matrix has been implemented Design for maximum a
33. fy it under the terms of the GNU General Public License as published by the Free Software Foundation either version 3 of the License or at your option any later version You should have received a copy of the GNU General Public License along with this program If not see lt http www gnu org licenses gt THIS SOFTWARE IS PROVIDED AS IS AND ANY EXPRESSED OR IMPLIED WARRANTIES INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED IN NO EVENT SHALL THE UNIVERSITE PARIS DIDEROT OR INSERM OR ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT INDIRECT INCIDENTAL SPECIAL EXEMPLARY OR CONSEOUENTIAL DAMAGES INCLUDING BUT NOT LIMITED TO PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE DATA OR PROFITS OR BUSINESS INTERRUPTION HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY WHETHER IN CONTRACT STRICT LIABILITY OR TORT INCLUDING NEGLIGENCE OR OTHERWISE ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE Redistribution and use in source and binary forms with or without modification are permitted under the terms of the GNU General Public Licence and provided that the following conditions are met 1 Redistributions of source code must retain the above copyright notice this list of conditions and the following disclaimer 2 Redistributions in binary form must reproduce the above copyright no
34. gorithm The Fedorov Wynn algorithm is specifically dedicated to design optimisation problems and has the property to converge towards the D optimal design 12 14 It optimises statistical designs for a given total number of samples The sampling times are chosen among a given finite set of times Minimum and maximum numbers of samples per subject are specified To start the algorithm an initial population design is then required The Fedorov Wynn algorithm is programmed in a C code and is linked to PFIM through a dynamic library called libFED dll and libFED64 dll for R 32 bit and 64 bit respectively Moreover PFIM uses the function combn in the R EU package combinat New feature The best one group protocol which maximises the determinant of the elementary Fisher information matrix of all elementary protocols chosen among the predefined set of samples is given by default when running Fedorov Wynn algorithm before calling the dynamic library This is the optimal protocol for individual design and Bayesian design Moreover in PFIM 4 0 optimisation with Fedorov Wynn algorithm can be performed assuming that some sampling times are fixed 14 3 Use 3 1 Pre requirement The software R is reguired To use PFIM 4 0 additional packages are needed in the R library directory for differential eguation system to describe the model deSolv and nlme packages for the Fedorov Wynn a
35. h options alpha the value of the type on rror for the Wald test Example alpha lt 0 05 It is possible to comput ither th xpected power only or the number of subjects needed for a given power or both of them together compute power logical value if T the expected power for comparison test is computed for each covariate Example compute power lt T compute nni logical value if T the number of subjects needed for a given power for comparison test is computed for each covariate Example compute nni lt T interval eg vector of eguivalence interval Example interval eg lt c log 0 8 log 1 25 compute power e logical value if T the expected power for P P A equivalence test is computed for each covariate Example compute power eg lt T compute nni eg logical value if T the number of subjects needed for a given power for eguivalence test is computed for each covariate Example compute nni eg lt T 41 given power the value of the given power for comparison and or eguivalence test Example given power lt 0 9 5 6 Graph option This list of objects allows to draw a graph with the evaluated design evaluation step ts of sensitivity functions have been added in version 4 0 Options for plo If the user wan graph logical or the optimised design optimisation step ts a graph he has to specify it in the following object logica
36. he previous Fisher information matrix if considered The figure 1 shows a one response model written in user defined model form with a group described by three sampling times for 200 subjects considering a previous Fisher information matrix The parameter k is fixed assuming no variability on k and the dose is egual to 100 a The population or individual or Bayesian Fisher information matrix a dim dim symmetric matrix where dim is the total number of population parameters to be estimated the number of individual parametres th number of error model parameters or only the number of individual parameters respectively The name of the file where is possibly saved the Fisher information matrix is given Eb The value of each parameter with the expected standard error StdError and relative standard error RSE In case of Bayesian design the associated shrinkages values are also reported C62 The value of the determinant of the Fisher information matrix and the value of the criterion determinant 9 where dim is defined in qu D th igenvalues of the Fisher information matrix and the correlation matrix 45 PFIM 4 0 Project Doc example Date Fri Mar 21 13 02 23 2014 kkkkkkkkkkkkkkkkkkkkkkkkkkkk INPUT SUMMARY k k kk kk kk kk kk kk kk kk kk kk kk kk kk Analytical function models function t p X ka lt p 1 k lt p 2 V lt p 3 y lt
37. iability of the j component of b and the inter occasion variability of the jt component of Kin respectively This new development was performed for any number of occasions H It is implemented in PFIM for the case where sam lementary designs are used at each occasion Discrete covariate specification Th present xpression of nonlinear mixed effects models accommodates models with parameters quantifying the influence of discrete covariates Two or more categories can be included In PFIM 3 2 it can be assumed either that covariates are additive on parameters if the random effect model is additive or that covariates are additive on log parameters if the random effect model is exponential For instance the individual parameter 0 is described as the function of a discrete covariate C which takes D values defining D categories with additive effect model D znat Balea h d 2 where here d 1 is defined as the reference group and B lt 0 For each covariate the user has to specify B i e the vector of covariate ffect coefficients and the proportions of subjects associated to the D categories It can be specified if covariates change or not through the different occasions In the latter case additional objects are needed the vector of seguences of values of each covariate at each occasion and the vector of proportions of the elementary designs corresponding to each sequence of covariate values s Se
38. iate category lt list Gender lt c F M list of vectors of proportions Each vector is associated to one covariate and defines the corresponding proportions of subjects involved in each corresponding categories Example covariate proportions list Gender c 0 5 0 5 list of vectors of parameter s associated with each covariate Each vector is associated to one covariate and is defined by the corresponding parameters on which is added the covariate Example parameter associated list Gender c Cl V 9 Name of the parameter s has to be identical to those entered in the object parameters list of the values of parameters for all other categories than th referenc category for which beta 0 Example beta covariate list Gender list c 0 5 0 6 If the user wants to deal with covariates which change with occasion he has to specify the following object Covariate occ model If the user has filled in following objects Covariate occ name Covariate occ category logical value if T covariates changing with Occasion are added to the model by T the previous object he has to specify the list of character indicating the name of the covariate s Example covariate occ name list c Treat list of vectors of categories Each vector is associated to one covariate and defines its corresponding categories They can be written as character or integer Example covaria
39. in function PFIM implemented in the file PFIM r To do that choose the File item from the menu Select Source R code click on the right directories up to the file PFIM r The user can also load the file by typing the command in the Command Window source U My Documents PFIM 4 0 _examples Example1 PFIM r Call the R function in the Commands window PFIM 17 4 Models Models in PFIM can be specified either through their analytical form or as a solution of system of differential eguations PFIM provides libraries of models see Section 4 1 and users may also define their own model analytically or using a system of differential equations see Section 4 2 The PFIM library implements R expressions or differential equation systems for PKPD models The PK model library includes one two and three compartment models with linear elimination and with Michaelis Menten elimination The PD model library supports immediate response models alone or linked to a pharmacokinetic model and the turnover response models linked to pharmacokinetic model These libraries have been derived from the PKPD library developed by Bertrand and Mentr for the MONOLIX software and all analytical expressions are in that document 15 A documentation of PKPD models for PFIM is available when downloading PFIM Presently there is no model with lag time in the library To use the library of models the user has to specify the path of the
40. kxxaxkakkkkakkkka 1 1 2 2 3 0 1 2 3 590507 2 096455 0 2426030 096455 354 843266 4 4964361 242603 4 496436 0 2013882 kkkkkkkkkkkkkkkkkkkkkkkkkk EXPECTED STANDARD ERRORS k RR ake ak de ae de ae ak ae ae fe EK de k KK kk k Beta StdError RSE Shrinkage ka 2 00 0 9638509 48 19255 23 22522 5 k 0 25 0 0688475 27 53900 30 33586 5 V 15 00 3 1862487 21 24166 45 12080 kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk DETERMINANT ake ak he de ke ce de de de de de de ce ve ke de ke ve ke k k ke e skok kk kk k 55 15913 kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk CRITERION kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkka 3 806617 kkkkkkkkkkkkkkkkkkk EIGENVALUES OF THE FISHER INFORMATION MATRIX ixrkkkkkkkkkkikikk min max max min FixedEffects VarianceComponents 9 552493e 02 NA 3 549127e 02 NA 3 715393e 03 NA kkkkkkkkkkkkkkkkkkk CORRELATION MATRIX Joc kk kk kk kk kk 1 1 2 0 3 0 Figure 2 1 2 3 0000000 0 4133690 0 5638373 4133690 1 0000000 0 6330761 5638373 0 6330761 1 0000000 Example of design optimisation output file Moreover the PFIM function returns the following R objects mfisher the population or individual or Bayesian Fisher information matrix corresponding to the optimised protocole determinant the determinant of the Fisher information matrix crit the value of the criterion se the vector of the expected standard er cv the corresponding coefficient of variation exp
41. l representation in the interval 0 60 is specified by graph infA c 0 and graph supA lt c 60 If any lower and upper sampling times for the graph are specified by default for each response the lower sampling time is 0 and the upper sampling time is the maximum sampling time of the initial design vector of lower and upper values for the graphical representations They are identical of each response By default the value is NULL For example representation in 1 interval 0 10 is specified by y range c 0 10 5 7 Objects required only for optimisation The user can optimise design with identical time in each elementary design for each respons To trigger this option the user needs to specify it in the next objects 42 identical times T if identical sampling times for each response within an elementary design else F The choice of the algorithm is specify in the following object algo option character string for algorithm selection SIMP for Simplex algorithm selection FW for Fedorov Wynn algorithm selection For each algorithm the user need to specify particular objects described below For Simplex algorithm specification only subjects opt logical value T is the optimisation of the proportions or the number of subjects is reguired F if not loweri and upperi vector of lower and upper admissible sampling times delta time
42. l value if T draws the graph of the predicted output s and the sampling times graphsensi logical logical value if T draws the graph of the sensitivity function first order derivation of the model with respect to each parameter The user can choose to display only the graphs of models and or sensitivity graph only If the user has following objects functions without computing the Fisher information matrix logical value if T draws the graph of the predicted output s and or the sensitivity functions no design evaluation or optimisation filled in by T the previous object he has to specify the names datax names datay log logical character vector for the names of X axis for each graph that corresponds to each type of measurement the length of this vector must be equal to the number of responses character vector for the names of Y axis for each graph that corresponds to each type of measurement the length of this vector must be equal to the number of responses character string for controls logarithmic axes for the graphical representation of the models Values xy x or y produce log log or log x or log y axes Standard graphic is given by log logical F graph infi and graph supi vector of lower and upper sampling times for y range the graphs for each response i with i A B C For example for a single response mode
43. lgorithm combinat package A Van additional package numDeriv is needed for the computation of the full Fisher information matrix and for numerical derivatives of models written as standard R functions The easiest way to install packages is directly from the web To install the packages deSolve nlme combinat and numDeriv start R and choose the Packages item from the menu Choose Install package s from CRAN to install from the web you will see a list of all available packages pop up choose deSolve nlme combinat and numDeriv To install PFIM 4 0 the user has to download the function named PFIM 4 0 available on the webpage www pfim biostat fr 3 2 Components PFIM 4 0 includes two main folders called PFIM 4 0 Examples The folder PFIM 4 0 is composed of 3 principal files and one folder The 3 principal files are o The main function program file PFIM r o The input file stdin r o The model file model r The folder is called Program and contains the files of functions o Pfim4 0opl r To compute the block diagonal Fisher Information matrix option 1 to evaluate a population individual or Bayesian design using an analytical form to describe the model o PfimOPT4 Qopl r To compute the block diagonal Fisher Information matrix option 1 to optimise a population individual or Bayesian design using an analytical form to describe the model
44. n Fisher information matrix for the evaluation of population pharmacokinetic designs Comput Methods Programs Biomed 65 141 151 Bazzoli C Retout S Mentr F 2010 Design evaluation and optimisation in multiple response nonlinear mixed effect models PFIM 3 0 Comput Methods Programs Biomed 98 55 65 doi 10 1016 j cmpb 2009 09 012 e Nguyen TT Bazzoli C Mentr F 2012 Design evaluation and optimisation in crossover pharmacokinetic studies analysed by nonlinear mixed effects models Stat Med 31 1043 1058 doi 10 1002 sim 4390 Dumont C Chenel M Mentre F 2014 Two stage adaptive designs in nonlinear mixed effects models application to pharmacokinetics in children Commun Stat Simul Comput Epub ahead of print Combes FP Retout S Frey N Mentre F 2013 Prediction of shrinkage of individual parameters using the bayesian information matrix in non linear mixed effect models with evaluation in pharmacokinetics Pharm Res 30 2355 2367 doi 10 1007 s11095 013 1079 3 Mentre F Mallet A Baccar D 1997 Optimal design in random effects regression models Biometrika 84 429 442 doi 10 1093 biomet 84 2 429 Retout S Mentre F 2003 Further developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics J Biopharm Stat 13 209 227 doi 10 1081 BIP 120019267 Bazzoli C Retout S Mentre F 2009 Fisher informa
45. n of individual parameters for a subject i associated to the vector of observation y index i being omitted These individual parameters can be estimated by maximum a posteriori MAP As u is known estimating 0 is similar to estimating n More precisely the MAP estimate of n is given by Ply m pop PW where p is the probability density The Bayesian Fisher information matrix taking into account the a priori distribution of the random effects is expressed as M z S lose AM Lor a 8 toate 1A p 8 leglptn BF n Onon 7 yin Onon n onon E M 9 N 6 Q7 ij argmax p y argmax argmax log ply m log p 12 O lo 0 where M 0 E gley l xpression of the individual Fisher 0000 information matrix in classical nonlinear regression models The expectation E M GG 7 amp can be obtained by first order approximation of the model around the expectation of random effects i e 0 The shrinkage Sh is quantified from the ratio of the estimation variance predicted by May and the a priori variance and can be calculated as the diagonal elements of the matrix I W M Q see 7 for more details V unen a parameter has an a priori variance equal to 0 it will be considered as fixed to the mean value and no predicted shrinkage will be computed 2 4 Design evaluation Population individual and Bayesian design evalua
46. of subjects needed 2 3 1 3 Previous information 3 2 Bayesian Fisher information matrix Design evaluation Design optimisation 5 1 Simplex algorithm 5 2 Fedorov Wynn algorithm USE Pre requirement Components Working directory Run MODELS Library of models 1 1 Library of pharmacokinetic models 1 2 Library of pharmacodynamic models Model writing 2 1 Models defined in analytical form through an R expression 2 Models defined in analytical form through an R function 3 Models defined through a differential eguation system INPUT Option for Fisher information expression Structural model option Statistical model option Design Objects reguired only for computation of power and number of subjects needed 13 13 14 14 15 15 15 16 17 18 18 18 24 30 30 31 32 36 36 37 38 40 41 5 6 5 7 6 1 6 2 6 3 Graph option Objects reguired only for optimisation OUTPUT Evaluation output file and objects Optimisation output file and objects Comments REFERENCES 42 42 45 45 48 51 52 1 Introduction Model based optimal design approaches are increasingly performed in population pharmacokinetic pharmacodynamics PKPD 1 which consist in determining a balance between the number of subjects and the number of samples per subject as well as the allocation of times and doses according to experimental conditions A good choice of design is crucial for an effici
47. on turn input Emax Rin kout Emax C50 turn output Emax Rin kout Emax C50 turn input gammaEmax Rin kout Emax C50 gamma turn output gammakmax Rin ko Emax C50 gamma turn input Imax Rin kout Imax C50 turn output Imax Rin kout Imax C50 turn input gammaImax Rin kout Imax C50 gamma turn output gammalmax Rin kout Imax C50 gamma turn input Imaxfull Rin kout C50 turn output Imaxfull Rin kout C50 turn input gammaImaxfull Rin kout C50 gamma turn output gammaImaxfull Rin kout C50 gamma means Imax is fixed egual to 1 29 4 2 Model writing The structural model should be written in a text file called model r by default but any name can be used It can be specified either through an analytical form as an R expression or an R function or as a solution of systems of differential eguations An analytical expression model or differential eguation model can be called from PFIM libraries see section 4 1 or implemented by the users R functions of models can only be defined by the users and are not available in the pre implemented model libraries 4 2 1 Models defined in analytical form through an R expression Description In case of analytical form the model for each response should be written assigned in an object called formi where i is the letter of the alphabet A B C The formi for all the responses are then grouped in a vector called form form c formA formB formC If the mo
48. posteriori estimation of individual parameters can be evaluated or optimised and the predicted shrinkage is also reported 7 A new way has been added to specify user defined models through an R function It is now possible to visualise the graphs of the model and the sensitivity functions without performing evaluation or optimisation This documentation describes the methodology implemented in PFIM 4 0 in Section 2 Section 3 explains how to install and use PFIM Section 4 describes how to specify models either by using the PKPD library or the user defined model option Lastly sections 5 and 6 present the input and output of PFIM 2 Methodology 2 1 Design The elementary design of individual i i 1 N is defined by the number n of samples and their allocation in time t tini For N individuals the population design is composed of the N elementary designs such as amp y Usually population designs are composed of a limited number Q of groups of individuals with identical design a within each group performed in a number N of individuals The population design can thus be written as Z amp Mi Ko el Individual and Bayesian designs include only one elementary design 2 2 Nonlinear mixed effects models A nonlinear mixed effects model or a population model nis defined as follows The vector of observations Y for the individual i
49. responding elementary design for the respective response The size of the vector for each response has to be the same For example if there are two responses the user must specify protA protB If there ar several responses with several lementary designs the user can specify by NULL if a group do not have samples for one response Example prota lt list c 1 3 6 12 c 18 20 24 protB lt list c 1 3 6 12 c NULL ad In the present version the Fedorov Wynn does not work when there is a NULL design for one response 40 subjects vector of the g numbers of subjects for each elementary design In the case of optimisation this object is a vector of the g initial proportions if subjects input 2 or of the numbers of subjects in each elementary design if subjects input 1 subjects input 1 if the subjects per elementary designs are given as numbers 2 if they are given as proportions Ntot total number of samples Ntot is reguired if the subjects per elementary design are given as proportions i e only if subjects input 2 5 5 Objects required only for computation of power and number of subjects needed To comput th xpected power to detect covariate effects as to compute the number of subjects needed to achieve a given power the previous object covariate model has to be filling in by T Additional R objects are required to be created The following object is needed for bot
50. ressed in percent relative standard error sh the shrinkage values for each parameter in case of design Bayesian EigenValues the eigenvalues of the Fisher information matrix corr matrix the correlation matrix rors for each parameter 50 6 3 Comments It is now possible to visualise the graphs of the model and the sensitivity functions without performing evaluation or optimisation see Example Documentation For an infusion model it is not possible for design evaluation to include the time at the end of the infusion when the end of infusion parameter is a parameter to be estimated If the bound are not correctly specified according to the number of responses in case of analytical form expression by default bounds are initialized to c 0 Inf but that does not appear on the stdin r file In case of analytical form defined by R function or of differential equation system the bound is not used If the between subject variance of a parameter is assumed to be zero enter 0 for this variance in omega PFIM will remove the corresponding row and column in the Fisher information matrix If a parameter is fixed no variability is assumed and no shrinkage is computed for this parameter The number of subjects for each elementary design is the same for all the response s Optimisation with the Fedorov Wynn algorithm can be performed with an initial design composed of several groups with different do
51. s the user has to call a specific function in order to create a system of differential eguations describing the corresponding PK PD model This function named Create formED is implemented in the file CreateModel PKPDdesign r and has to be used in the model file as follows source file path directory program CreateModel PKPDdesign r create formED fun pk fun pd dose NA tau NA TInf NA The arguments to this function are fun pk and fun pd the names of the PK and PD models respectively dose value of the dose only for a PK model with infusion or oral input by default NA tau dosing interval to specify only for multiple dose conditions by default NA TInf time of infusion to specify only for PK model with infusion input by default NA The output of this function is a new file named model created r which is created in the directory currently used This new file contains a function implementing the differential eguation system for the corresponding PK PD model This file can be deleted after running PFIM It will be created overwritten each time the function Create formED is called Because the resulting function is an ODE system the user must fill in the section corresponding to differential equation options in the input file see Section 5 The list of the immediate response PD models in the PFIM library is shown in Tables 3 and 4 The list of the turnover response PD mo
52. s of the fixed effects parameters beta vector of the p fixed effects parameters values beta fixed p vector indicating if the parameter is estimated or not omega vector of the p variances of the random effects should be given n oce integer indicating the number of occasions Example n occ 2 gamma vector of the p variances of the random effects for inter occasion variability sig interi and sig slopei values of the parameters for the residual s 2 variance error model given by var c UO uisa O a f for each response i with i A B C Trand type of between subject variance or random effects model 1 for additive between subject variance model 2 for exponential model If the user wants to deal with covariates which do not change with occasion he has to specify the following object 38 covariate model If the user has filled in following objects covariate name covariate category covariate proportions parameter associated beta covariate logical value if T covariates are added to the model by T the previous object he has to specify the list of character indicating the name of the covariate s Example covariate name list c Gender list of vectors of categories Each vector is associated to one covariate and defines its corresponding categories They can be written as character or integer Example covar
53. se responses is associated with a known function fi which can be grouped in a vector of multiple response model F such as F 0 fi 06 53 fp On 82 fe Oo ik T where is composed of K sub designs such that i i ik The sub design ik is then defined by totis tirn with nj sampling times for the observations of the k response so that n Yang Each response can have its error model and is then the vector composed of the K vectors of residual errors g k 1 K associated with the K responses Inter occasion variability specification The expression of the nonlinear mixed effects model has been extended for model including additional random effects for inter occasion variability or within subject variability 5 The individual parameters of an individual i at occasion h are thus expressed by the following relation which can be additive as O u b Kin or exponential as Oin u Xexp b Kin where is as previously the p vector of fixed effects b the vector of random effects associated to the individual i and Kj the vector of random effects associated to the individual i for the occasion h h 1 H with H b the number of occasions and Kip are supposed independent It is assumed that b N 0 0 and K N 0 with 2 and I defined as diagonal matrices of size pXp Each element w of f and y of l represent the inter individual var
54. ses or initial conditions and with fixed sampling times The dimension of the previous Fisher information matrix is the same as the current Fisher information matrix The previous Fisher information matrix can be taken into account in evaluation and in optimisation with Simplex algorithm or for best one group protocol If a design leads to very poor information with a singular population Fisher information matrix det 0 the expected standard errors and the RSE are returned as NA Standard error of derived parameters can be computed by the delta method available in the R package car using the FIM stored in files or directly obtained in R console after running PFIM see Example documentation Section 1 5 for detailed examples 51 10 7 References Mentr F Chenel M Comets E Grevel J Hooker A et al 2013 Current Use and Developments Needed for Optimal Design in Pharmacometrics A Study Performed Among DDMoRe s European Federation of Pharmaceutical Industries and Associations Members CPT Pharmacomet Syst Pharmacol 2 e46 doi 10 1038 psp 2013 19 Nyberg J Bazzoli C Ogungbenro K Aliev A Leonov S et al 2014 Methods and software tools for design evaluation for population pharmacokinetics pharmacodynamics studies Br J Clin Pharmacol doi 10 1111 bcp 12352 Epub ahead of print Retout S Duffull S Mentr F 2001 Development and implementation of the populatio
55. st order 1st order ka V k k12 k21 k13 k31 md N tau SS tau sd orall 3cpt kaClv101v202V3 lst order lst order ka Cl V1 01 V2 Q2 V3 md N tau SS tau 21 Pharmacokinetic models with a Michaelis Menten elimination One two and three compartment models are implemented for the thr types of input For bolus input only single dose models are implemented For infusion and first order absorption input single dose and multiple dose are implemented There is no steady state form for PK models with Michaelis Menten elimination in this case the user can use a multiple dose model with enough doses to reach SS The list of these PK models is given in Table 2 Wor models with a bolus input the dose has to be specified in the input file stdin r by default as the initial condition of the differential equation system see Example 7 1 section 4 2 For models with infusion or first order absorption input dose has to be specified as an argument of the model function and NOT IN THE INITIAL CONDITION OF THE MODEL IN THE INPUT FILE see Example 7 2 section 4 2 9 As the dose is an argument it is not possible to specify different doses per group for models with infusion or first order absorption input All groups of the design considered have the same dose Otherwise the user should use the user defined model option 22
56. stiff systems it uses the Backward Differentiation Formula BDF for stiff systems The fdHess is used for numerical derivation It evaluates an approximate gradient of a scalar function using finite differences Example 7 1 PK model with bolus input using a differential equation form from the library of models source file path directory program dirsep LibraryPK r formED lt bolus 1cpt VVmkm Note In this illustration the user creates a one response model using the model function implemented in the pharmacokinetic library bolus lcpt VVmkm describing a one compartment bolus input with Michaelis Menten elimination after a single dose administration sd The dose is specified in a part of the R script file stdin r see section 5 for more input details time condinit lt 0 condinit lt expression c 100 dose 100 33 Example 7 2 PK model with infusion input using a differential eguation form from the library of models time condinit lt 0 condinit lt expression c 0 source file path directory program dirsep LibraryPK r formED lt infusion lcpt VVmkm doseMM 100 Tinf 1 Note In this illustration the user creates a one response model using the model function implemented in the pharmacokinetic library infusion lcpt VVmkm describing a one compartment infusion input with Michaelis Menten elimination after a single dose administration sd The dose is specified as an argument of
57. te occ category list Treat c A B 39 covariate occ sequence list of vectors of sequences Each vector is associated to one sequence of values of covariates at each occasion The size of each sequence has to be equal to the number of occasions n occ for each covariate Example covariate occ sequence list Treat list c A B c B A covariate occ proportions list of vectors of proportions Each vector is associated to one covariate and defines the proportions of elementary designs corresponding to each sequence of covariate values The siz of each vector has to be equal to the number of sequences Example covariate occ proportions list Treat list 0 5 0 5 parameter occ associated list of vectors of parameter s associated with beta covariate occ 5 4 Design proti each covariate Each vector is associated to one covariate and is defined by the corresponding parameters on which is added the covariate Example parameter occ associated list Treat c C1 Name of the parameter s has to be identical to those entered in the object parameters list of the values of parameters for all other categories than the reference category for which beta 0 Example beta covariate occ list Treat list c log 1 1 list of vectors of elementary designs for each response i with i A B C Each vector contains the sampling times of the cor
58. ter performance of the block diagonal expression compared to the full one with linearisation 2 the default option in PFIM is the block diagonal information matrix However since PFIM 3 2 the user can choose to compute either a full or a block diagonal matrix for models without covariate and inter occasion variability The size of the block C and the block B of the expression of the Fisher information matrix are thus modified to incorporate the within subject variabilities Prediction of standard errors According to the inequality of Cramer Rao the inverse of Ms is the lower bound of the variance covariance matrix of any unbiased estimate of the parameters From the square roots of the diagonal elements of the inverse of Mp the predicted standard errors SE for estimated parameters can be calculated 2 3 1 2 Computation of power and number of subjects needed Comparison test Computation of the expected power The Wald test can be used to assess the difference of a covariate effect p In PFIM the Wald test is performed on the B of each category for each covariate a global Wald test on the vector P all effect coefficients is not implemented For one covariate and an effect of one category 6 D 2 the null hypothesis is Ho 820 while the alternative hypothesis is Hi 820 The statistic of the Wald test is defined as CEN with p the covariate SE B effect estimates and SE its associat
59. the PK function in the file model r not in the initial condition described in a part of the R script file stdin r Example 8 PK model using a differential eguation system created by the user formED lt function t y p ka lt p 1 km lt p 2 Vm lt p 3 V p 4 ydl lt ka y 1 yd2 lt ka y 1 V Vm y 2 V km y 2 list c ydl yd2 c y 211 V Note This function formED implements a one compartment model with first order absorption and Michaelis Menten elimination The dose is specified as an argument of the PK function in the file model r not in the initial condition described in a part of the R script file stdin r The first four lines in the body of the function assign model parameters from the vector p The next two lines describe the derivatives of the system ydl and yd2 More specifically each derivative represent the drug concentration in the specific compartment at the instant t and its elements can be either positive or negative The notation ydX denotes the derivative of the variable in compartment X while the notation y X denotes the quantity in the same compartment see documentation for the deSolve package for details The last line defines the elements returned by the function the first item is mandatory for the deSolve package and should always consist of a vector with the derivatives of the system here the two elements ydl and ya2 the second item defines the response here the
60. the drug concentration corresponds to the expression of the PK model Example 4 PK model using an analytical form with user defined expression formA lt expression dose v ka ka ke exp ke t exp ka t form lt c formA Note In this illustration the user creates a one response model describing a one compartment oral absorption with expression The dose here needs to be specified in the input file If the dose is defined directely in the model expression as below all elementary designs will have the same dose 100 dose unit formA lt expression 100 v ka ka ke exp ke t exp ka t form lt c formA 4 2 2 Models defined in analytical form through an R function Description The R function for a PFIM model should take the following form formA lt function t p X The function has 3 arguments a vector of times t a vector of parameters p a scalar X which represents the dose Within the function the user can define local variables and use the parameters provided in vector p However the header to the function and 31 its name must remain unchanged The order of the parameters is provided by the user through the parameter vectors in the stdin file The function returns a vector of predictions of each time point in t computed using the dose X and the parameters p Example 5 PK model after single dose administration using an analytical form with user defined
61. tice this list of conditions and the following disclaimer in the documentation and or other materials provided with the distribution 3 The end user documentation included with the redistribution if any must include the following acknowledgment This product includes software developed by Universit Paris Diderot and INSERM http www biostat fr Alternately this acknowledgment may appear in the software itself if and wherever such third party acknowledgments normally appear 4 The names PFIM must not be used to endorse or promote products derived from this software without prior written permission For written permission please contact france mentreGbichat inserm fr 5 Products derived from this software may not be called PFIM nor may PFIM appear in their name without prior written permission of the Universit Paris Diderot and INSERM Copyright PFIM 4 0 Giulia Lestini Thu Thuy Nguyen Cyrielle Dumont Caroline Bazzoli Sylvie Retout Herv Le Nagard Emmanuelle Comets and France Mentr Universit Paris Diderot INSERM www pfim biostat fr 2 1 2 2 2 3 2 4 2 5 3 1 3 2 3 3 3 4 4 1 A 4 2 A 5 1 5 2 5 3 5 4 5 5 2 2 CONTENTS INTRODUCTION METHODOLOGY Design Nonlinear mixed effects models Fisher information matrix 3 1 Population Fisher information matrix VAE N Expression A side LWA Computation of power and number
62. tion is based on the computation of the population individual and Bayesian Fisher information matrix respectively During this process the expected standard errors on the population or individual parameters with the design are evaluated The user can choose to fix one or several parameters in the model that will not be computed in the Fisher information matrix Eigenvalues and conditional number are given by default When considering design for Bayesian estimation of individual parameters the shrinkages are also reported The computed Fisher information matrix can be saved in a file if requested 2 5 Design optimisation Algorithms are required to optimise exact or statistical designs In the case of an exact optimisation the group structure of the design is fixed the number of elementary designs the number of samples per elementary design and the number of subjects per elementary design are given and the design variables to optimise are only the sampling times In the case of statistical optimisation the sampling times number and allocation and the proportions of subjects in each elementary design are optimised PFIM optimises population design using the D optimal criterion i e maximising the determinant of the population Fisher information matrix or similarly minimising its inverse The Fedorov Wynn algorithm has been implemented since PFIM 3 0 in addition to the Simplex
63. tion matrix for nonlinear mixed effects multiple response models evaluation of the appropriateness of the first order linearization using a pharmacokinetic pharmacodynamic model Stat Med 28 1940 1956 doi 10 1002 sim 3573 Nelder JA Mead R 1965 A Simplex Method for Function Minimization Comput J 7 308 313 doi 10 1093 comjnl 7 4 308 Fedorov VV 1972 Theory Of Optimal Experiments Academic Press New York Wynn HP 1972 Results in the theory and construction of D optimum experimental designs JR Stat Soc Series B 34 133 147 Retout S Comets E Samson A Mentr F 2007 Design in nonlinear mixed effects models optimization using the Fedorov Wynn algorithm and power 52 15 of the Wald test for binary covariates Stat Med 26 5162 5179 doi 10 1002 sim 2910 pharmacokinetic and pharmacodynamic models impl Bertrand J Mentr F 2008 Mathematical expressions of the lemented in the MONOLIX software MONOLIX Software Documentation www l lixoft eu 53
64. ultiple dose administration md N and tau are the arguments to be specified by the user in the function model Here there are five oral administration doses with an interval between two doses egual to twelve hours The vector of time intervals of each expression needs to be defined in the input file boundA list c 0 12 c 0 12 12 c 0 12 2 12 c 0 12 3 12 c 0 12 4 12 30 Example 2 PD model using an analytical form with the library of models source file path directory program dirsep LibraryPD PDdesign r formA lt immed lin null 1 form lt c formA Note In this illustration the user creates a one response model using the model function implemented in the library immed lin null describing an immediate response model with a linear drug action and without baseline Example 3 PK model with a linear elimination and immediate response PD model source file path directory program dirsep LibraryPK r source file path directory program dirsep LibraryPD PKPDdesign r formA lt bolus lcpt Vk 1 formB lt immed lin null formA 1 form lt c formA formB Note In this illustration the user creates for the PK model a one compartment model with bolus input and first order elimination for a single dose and for the PD model an immediate response model with a linear drug action and no baseline is used As shown in the example the PK model is given as an argument of the PD model Thus in the PD model
65. universit PARIS DIDEROT IAM E UMR1137 Inserm institut national e la sant at de la recherche m dicale V pFIM 4 0 PFIM Group INSERM and Universit Paris Diderot Paris France August 2014 www pfim biostat fr User guide Written by Cyrielle Dumont and Thu Thuy Nguyen kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkak PFIM 4 0 is free library of functions The Universite Paris Diderot and INSERM are the co owners of this library of functions Contact pfim inserm fr Members of the PFIM Group Ca Cy He Jul Pr France Mentre Bazzoli active member roline ie Ber Emmanuelle Comets Anne Dubo rielle rve Le trand is Dumont act Nagard act Chair active member tive member tive member Giulia Le Sylvie Re tout stini active member Thu Thuy Nguyen ac tive member Disclaimer We inform users that the PFIM 4 0 is a tool developed by the Laboratory Biostatistics Investigation Pharmacometrics UMR 1137 INSERM and University Paris Diderot under R and GCC PFIM 4 0 is a library of functions The functions are published after a scientific validation However it may be that only extracts are published By using this library of functions the user accepts all the conditions of use set forth hereinafter Licence This program is free software you can redistribute it and or modi

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