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Acharya Nagarjuna University Department of Statistics

Acharya Nagarjuna University Department of Statistics. Workshop on DoE & OR Dec. 8 - 9, 2011. Pharmaceutical Laboratories : Role of Designed Experiments. BIKAS K SINHA Senior Professor of Statistics, ISI & Ex-Member National Statistical Commission Govt. of India.

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Acharya Nagarjuna University Department of Statistics

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  1. Acharya Nagarjuna UniversityDepartment of Statistics Workshop on DoE & ORDec. 8 - 9, 2011

  2. Pharmaceutical Laboratories : Role of Designed Experiments BIKAS K SINHA Senior Professor of Statistics, ISI & Ex-Member National Statistical Commission Govt. of India

  3. Optimal sn factorial designs when observations within-blocks are correlated Computational Statistics & Data AnalysisVolume 50, Issue 10 , 2006, PP 2855-2862 Sethuraman / Raghavarao / Sinha Venkat S. Sethuraman, Biomedical Data Sciences, GlaxoSmithKline R&D, Philadelphia,USA D Raghavarao : Department of Statistics, Temple University, Philadelphia, USA

  4. Laboratory Trials… • In a pharmaceutical industry, the concern was to develop once-daily tablet formulation that would improve patient compliance, compared to a currently available formulation administered twice daily.

  5. Laboratory Expt…. • These laboratory experiments are normally performed on healthy human subjects (called volunteers), who are administered several test formulations separated by drug-washout days. • Approved by Ethical Committee Needed Signed Consent of V’s

  6. Drug Formulations… • Components : Each has at least two or three options to be tried out. • These are technically called levels and the level combinations of all the components constitute the drug formulations to be tested out. • In general, each volunteer is administered all or selected drug formulations of the constituent components under consideration.

  7. Laboratory Results…. • The bio-availability of each formulation is measured by AUC (Area Under time-plasma concentration Curve) which is the response variable of immediate interest. The drug formulations involve several components such as (i) type of polymers to prolong drug release, (ii) amount of film coat on the tablet, (iii) administration under fed or fasted state, etc….. • Use Factorial Expt. Concept

  8. Concept of Blocking…. • The concept of BLOCKS relates to different VOLUNTEERS. • The availability of a volunteer for the laboratory trial determines the block size which is the number of different formulations tested on the same volunteer. • Different experimental conditions may result in unequal block sizes as well. • Some volunteers may opt for a few drug formulations to be tested upon…..

  9. Simplified Models…. • A simple statistical model assumes possible existence of first-order [i.e., linear] effects [called main effects] of the components only, without any interaction effects. • These linear effects are assumed to explain the variation in the drug bioavailability. • Different formulations of the drug are tested on the volunteers in order to examine if there is any differential effect of these formulations.

  10. Designing Problems…. • In order that maximum information on the first order effects may be extracted from the experiment, we need to determine the allocation of the drug formulations to different volunteers forming the blocks. • We assume : “s” levels of each of the “n” components so that altogether sn drug formulations are to be tested on “v” volunteers in an optimal manner.

  11. Basic Assumptions…. • All the sn drug formulations must be tested collectively on these “v” volunteers • Every volunteer must receive at least a minimum number of drug formulations • No two volunteers will receive the same drug formulation • Collectively….the volunteers must exhaust all the sn drug formulations….

  12. Model Descriptions…. • We assume a linear effects model [without any quadratic/higher order terms] for ith volunteer receiving fi drug formulations. E[Yi] = Xiθ, θ= [, 1, 2, …, n] • D[Yi] =iof order fi ; i = 1, 2, …, v • We assume an intraclass correlation structure for i’sof appropriate dimensions.

  13. Unified Notations…. • Levels of the factors [s odd OR even] : • 0, +/- 1, +/- 2, ….OR+/- 1, +/- 3, +/- 5, ….. • E[Y ir] =  + x ir11 + x ir2 2 +…+ x irn n • Volunteer #i; rth drug formulation in the sequence; n = number of components • i ’s…..per unit level effects of the components • i= (1- )I +  J of dim. fi st ifi = sn

  14. Optimal Design Problem… • Most efficient estimation of the beta-parameters…. • D-optimality Criterion involving ^’s for Optimal Choice of ((x irh))’s….. Results available: fi’s multiples of s…. • That means….each volunteer receives at least ‘s’ formulations….. • Recall : x irh refers to ith volunteer, rth drug formulation and hth component

  15. Theorem ….. • D-optimal design : r x irh = 0  i & h • Interpretation : For every volunteer and every component, algebraic sum of formulation levels actually prescribed is 0 ! • Note : fi = Number of drug formulations prescribed for Volunteer # i = multiple of s • Goos (2002:Springer Lecture Notes) :fi = f • Sethuraman et al (2006) : General case BUT all f i’s are still multiples of s ……

  16. Sketch of Proof….. From ith Volunteer : Contribution toI(θ) Ii(θ) = [ X′ii-1Xi] where i= (1- )I +  J is of order fi Summed over all i, this results into I(θ) = A – B – C – D of order (n+1)

  17. Details….. A = (1- )-1 X′X B =(1- )-1 i ci UiUi′ where ci = {1+  (fi -1)}-1 & Ui′ = (xi.1, xi.2,…, xi.n ) C = (1- )-1 ((c iJ)) where c11 = sn ; c12 = 0; c22 = sn-1 (ar2) In ar = ih xirh = sum of x-values for rth factor D = (1- )-1 ((d iJ)) where d11= cifi2; d12= cifi Ui(1)′; d22 = ciUi(1)Ui(1)′; Ui = (1,Ui(1))

  18. Information on -parameters… DecomposeI(θ) and work out I() as I()= I22.11 = I22 – I21I12 / I11 It follows that I22=(1- )-1 [ sn-1( ar2)In - i ciUi(1)Ui(1)′)] For D-optimality, | I22 |  (1- )-n [ sn-1( ar2)]nwith = iff Ui(1) = 0 i i.e., r x irh = 0  i & h

  19. Nature of D-Optimal Designs…. • Define Permutation Matrices P1,P2,…,Ps of order s as follows : P1=Is;Pi = right cyclic rotation of Pi-1; i=2,3,… For given s = 2k+1(odd), define a = (-k, -(k-1), …, -1, 0, 1, …, k-1, k)′ Form sn-1 blocks each of size s by using [a, Pi2a, Pi3a, …, Pina] for i2, i3, …, in = {1, 2, …, s}.

  20. D-Optimal Designs…. • Whenever fi’s are multiples of s, it is enough to decompose the sn-1 blocks so formed into subsets so that s. sn-1= sn = fi. • Illustrative Examples follow for s = 2, 3 and 4.

  21. D-Optimal Designs for s = 2 • Coded levels : -1 & 1 • Recall P1 = Is = I2 = [1 0; 0 1] • P2 = right cyclic rotation of P1= [0 1; 1 0] • a = (-1 1)’ Case of n = 2 components : 22 = 4 formulations • Step I : [a a]; [a P2a]…2 blocks • [(-1 1) (-1 1)]; [(-1 1) (1 -1)] • Volunteer -1 : First 2 col. (-1 -1) (1 1) • Volunteer -2 : Last 2 col. (-1 1) (1 -1)

  22. Case of Three Components…. n = 3: 23 = 8 formulations of 3 components [a a a]; [a a P2a] [a P2a a] [a P2a P2a] : 4 blocks [(-1 1) (-1 1) (-1 1)]; [(-1 1) (-1 1) (1 -1)] [(-1 1) (1 -1) (-1 1)]; [(-1 1) (1 -1) (1 -1)] Two Volunteers ----each with 4 formulations V-1 : (-1 -1 -1) (1 1 1) (-1 -1 1) (1 1 -1) V-2 : (-1 1 -1) (1 -1 1) (-1 1 1) (1 -1 -1) • Likewise....we can handle …. • Four Volunteers – each with 2 Formulations

  23. D-Optimal Designs for s = 3 • Coded levels : -1 0 1 Recall P1 = Is = I3 = [1 0 0; 0 1 0; 0 0 1] • P2 = right cyclic rotation of P1 • = [0 0 1; 1 0 0; 0 1 0] • P3 = right cyclic rotation of P2 • = [0 1 0; 0 0 1; 1 0 0] a = (-1 0 1)’

  24. Case of Two Components 32 = 9 formulationsdivided into 3 blocks Step I : [a a]; [a P2a] [a P3a] …3 blocks • [(-1 0 1) (-1 0 1)]; • [(-1 0 1) (0 1 -1)]; • [(-1 0 1) ( 1 -1 0)] V-1 : First 2 col. [ (-1 -1) (0 0) (1 1)] V-2 : Next set of 2 col. [(-1 0) (0 1) (1 -1)] V-3 : Last set of 2 col. [(-1 1) (0 -1) (1 0)]

  25. Case of Three Components 33 = 27 formulationsdivided into 9 blocks Step I : [a a a]; [a a P2a]; [a a P3a] [a P2a a] [a P2aP2a] [a P2a P3a] [a P3a a] [a P3aP2a] [a P3aP3a] • [(-1 0 1) (-1 0 1) (-1 0 1)]; • [(-1 0 1) (-1 0 1) (0 1 -1)]; • [(-1 0 1) (-1 0 1) ( 1 -1 0)] • etc….till the end • [(-1 0 1) (1 -1 0) (1 -1 0)]

  26. Key Reference

  27. The End…. • Thanks for your attention !!! • BKSinha [ISI, Kolkata] • Guntur, December 8, 2011

  28. INDIAN STATISTICAL INSTITUTEPlatinum Jubilee Celebrations[2006 – 2007] • International Conference on • Environmental and Ecological Statistics with Applications • Venue : Kolkata Campus • Dates : March 21-23, 2007

  29. Proposing Scientist…. • Bikas K Sinha • Senior Professor of Statistics [Stat-Math Division, ISI, Kolkata] AND • Member [2006 – 2009] [National Statistical Commission]

  30. Proposer’s Profile… • Professor [ISI] since 1985 • Mahalanobis Medal Recipient: 1980 • UN Expert on Mission : 1990 • US EPA Consultant [Env. & Ecology]: 1991 • Int’l Stat Inst. Elected Member since 1985 • Statistics Sectional President : Indian Science Congress [2002] • Member : National Stat. Commission [2006]

  31. Organizing CommitteeInt’l Conf. Env. & Eco. Stat. with Appls. • Director, ISI…..Chairman • BKSinha….Vice-Chairman • Professor – in – Charge, Stat-Math Div • Head, Stat-Math Unit, Kolkata • Alok Goswami GMSaha • Ratan Dasgupta Aditya Bagchi • Debapriya Sengupta • Pulakesh Maiti [Convener]

  32. Background Information…. • SURDAC Activities : 1990’s • Collaboration of ISI Scientists with Madhab Gadgil, Anil Gore, Paranjape • 1993 : Int’l Conference in Statistical Ecology • Recent Collaboration with Anil Gore, Paranjape, Abhik Gupta, Dilip Nath & Others in North-East

  33. Conference Thrust Areas 1. Toxic Release Inventory (TRI) • 2. Detection Limits • 3. Combining Environmental Indices • 4. Cancer Growth Models • 5. Pharmacokinetic Models in Environmental Risk Assessment

  34. Thrust Areas …continued • 6. Gene-Environment Interaction Models and Related Data Analysis • 7. fMRI:Statistical Modeling • 8. Health-Related Issues [Arsenic Problem/Ozone Layer /Environmental Health Indices/Occupational Health Hazards & Measures

  35. Thrust Areas …continued 9.Environmental Awareness / Health Issues in Pharmaceutical Industries / Women’s Health / Oceanography & Marine Science / Forestry Health Management] • 10. Ecology : Ecological Imbalances / Flora & Fauna / Biodiversity Measures and Related Issues 

  36. Confirmed Speakers…. • 1. Professor Bimal K Sinha : Univ. Maryland – Baltimore County, USA • 2. Professor Jerzy Filar : Univ. South Australia, Australia • 3. Professor Abhik Gupta : Department of Environmental Science, Assam Univ., Silchar, Assam4. Professor Anil P Gore : Department of Statistics, Univ. Pune • 5. Dr. [Mrs.] S A Paranjape : Department of Statistics, Univ. Pune

  37. Confirmed Speakers….continued • 6. Professor S N Dwivedi: AIIMS, New Delhi • 7. Professor Tapio Nummi: University of Tampere, Finland • 8. Dr S. Asolekar : Centre for Env. Science & Engg., IIT Mumbai • 9. Dr D. N. Guha Mazumdar : Advisor, Pollution Control, Govt. West Bengal • 10. Dr D Chakrabarty, Director, School of Environmental Studies, JU

  38. Confirmed Speakers…continued • 11. Dr.[Mrs.] Gitashree Das : North Eastern Hill University, Assam • 12. Dr. Tapan Chakrabarty : North Eastern Hill University, Assam • 13. Professor Dilip Nath : Gauhati University • 14. Dr. Kishore Das : Gauhati University • 15. Professor Alok Goswami : Stat-Math Unit, ISI, Kolkata • 16. Dr. P Maiti : Economic Research Unit, ISI, Kolkata

  39. Confirmed Speakers…continued • 17. Professor M Ghose : Agri. Science Unit, ISI, Kolkata • 18. Dr. Joydev Chattopdhyaya : Biological Sciences Division, ISI, Kolkata • 19. Professor Debapriya Sengupta : BIRU, ISI, Kolkata • 20. Professor Ratan DasGupta : Stat-Math Unit, ISI, Kolkata • 21. Professor Debasis Sengupta : Applied Statistics Unit, ISI, Kolkata 22. Professor Bikas K Sinha : Stat-Math Unit, ISI, Kolkata

  40. Confirmation yet to be recd. from • Dr. Olaf Berke, University of Guelph, ON, Canada • Prof. Dr. Leonard Held, Munich • Prof. Sudip K Banerjee Chairman, WB Pollution Control Board • Dr. Raman Sukumar, IISc, Centre for Ecological Sciences, Bangalore • Dr. Debashish Chatterjee, Kalyani University

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