Acharya Nagarjuna University Department of Statistics

Acharya Nagarjuna UniversityDepartment of Statistics

Workshop on DoE & ORDec. 8 - 9, 2011

Pharmaceutical Laboratories : Role of Designed Experiments

BIKAS K SINHASenior Professor of Statistics, ISI

&

Ex-Member

National Statistical Commission

Govt. of India

Optimal sn factorial designs when observations within-

blocks are correlated Computational Statistics & Data Analysis

Volume 50, Issue 10 , 2006, PP 2855-2862 Sethuraman / Raghavarao / Sinha Venkat S. Sethuraman, Biomedical Data Sciences,

GlaxoSmithKline R&D, Philadelphia,USAD Raghavarao : Department of Statistics, Temple

University, Philadelphia, USA

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.

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

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.

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

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…..

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.

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.

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….

Model Descriptions….• We assume a linear effects model [without any

quadratic/higher order terms] for ith

volunteer receiving fi drug formulations.

E [Yi] = Xi θ, θ = [, 1, 2, …, n]

• D[Yi] =i of order fi ; i = 1, 2, …, v

• We assume an intraclass correlation structure

for i’s of appropriate dimensions.

Unified Notations….• Levels of the factors [s odd OR even] : • 0, +/- 1, +/- 2, ….OR +/- 1, +/- 3, +/- 5, …..• E[Y ir] = + x ir11 + 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

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

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 ……

Sketch of Proof…..From ith Volunteer : Contribution to I(θ)

Ii (θ) = [ X′i i-1

Xi ]

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)

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))

Information on -parameters…

Decompose I(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)]n with = iff

Ui(1) = 0 i i.e., r x irh = 0 i & h

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}.

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.

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)

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

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)’

Case of Two Components

32 = 9 formulations divided 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)]

Case of Three Components 33 = 27 formulations divided into 9 blocks

Step I : [a a a]; [a a P2a]; [a a P3a]

[a P2a a] [a P2a P2a] [a P2a P3a]

[a P3a a] [a P3a P2a] [a P3a P3a]

• [(-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)]

Key Reference

The End….

•Thanks for your attention !!!

•BKSinha [ISI, Kolkata] •Guntur, December 8, 2011

INDIAN STATISTICAL INSTITUTE

Platinum Jubilee Celebrations[2006 – 2007]

• International Conference on

• Environmental and Ecological Statistics with Applications

• Venue : Kolkata Campus

• Dates : March 21-23, 2007

Proposing Scientist….

• Bikas K Sinha

• Senior Professor of Statistics

[Stat-Math Division, ISI, Kolkata]

AND

• Member [2006 – 2009]

[National Statistical Commission]

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]

Organizing Committee Int’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]

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

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

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

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 

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

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

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

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

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

TA/DA : Senior Scientists[Rs. 1.50 Lakhs]

• 1. Professor Abhik Gupta :

Assam Univ., Silchar, Assam

2. Professor Anil P Gore : Pune

• 3. Dr. [Mrs.] S A Paranjape : Pune

• 4. Professor Dilip Nath : Assam

• 5.Professor S N Dwivedi:New Delhi

• 6. Dr Shyam R. Asolekar : Mumbai

• 7. Dr. Raman Sukumar : Bangalore

TA/DA: Young Scientists[Rs. 1.20 Lakhs]

• 1. Dr.[Mrs.] Gitashree Das : Assam

• 2.Dr. Tapan Chakrabarty : Assam

• 3. Dr. Kishore Das : Assam

• Additional 10 from North-East

Regions and Mumbai-Pune Region

• Additional 30 from other regions

Proposed Publication : 2007Env. & Ecology Conference

• Edited Volume :World Scientific Pub.

• Editors : BKSinha & AGoswami

• Editorial Board Members :

Bimal Sinha APGore DSengupta

Editorial Assistant : P Maiti

Updated Proposal : Suggested at CC Meeting

• Combine Environment-Ecology, Biodiversity & Sustainable Development Topics together and publish a Single Volume of approx. 500 pages

• Editorial Board : ?• List of Contributors : ?• Responsible Scientists : ?• Deadlines : ?

Budget at a Glance…

• Total Budget : Rs. 5,70 Lakhs• ISI [Sanctioned Budget] :

Rs. 3.50 Lakhs

• DST Funding [Requested] :

Rs. 2,20 Lakhs

DST Funding….

• Specific Items

• TA/DA : Young Scientists Rs. 1.00 Lakh

• TA/DA : Senior Scientists Rs. 0.60 Lakh

• Pre-Conference Printing Rs. 0.10 Lakh

• Publication of Proceedings Rs. 0.50 Lakh

TOTAL Rs. 2.20 Lakhs

•

• Thank You !!!

BKSinha