1. Wolverine Pharmacometrics Corporation Within Subject Random Effect Transformations with NONMEM VI B. Frame 9/11/2009

2. Wolverine Pharmacometrics Corporation Dynamic Transform Both Sides (TBS) What is TBS? Why bother with TBS? Brief History, Jacobians, and Likelihoods. Implementation and examples in NONMEM (V or VI)

3. Wolverine Pharmacometrics Corporation What is Transform Both Sides (TBS)? Consider our usual set up: Y ij = PRED( ,  ) ij +  ij Where i indexes subject and j indexes the response or prediction within subject i. The assumption here is that  ij ~ N(0,  2 ) In other words, the within subject variability does not depend on time, the PREDiction, or who the subject is (i).

4. Wolverine Pharmacometrics Corporation What is Transform Both Sides (TBS)? Consider an invertible transformation T, with a domain compatible with what is being transformed (response and prediction)... Then in general TBS is... T( Y ij )= T(PRED( ,  ) ij )+  ij Once again, the assumption here is that  ij ~ N(0,  2 )

5. Wolverine Pharmacometrics Corporation What is Transform Both Sides (TBS)? A simple example (no transformation parameter) ln( Y ij )= ln(PRED( ,  ) ij )+  ij ; Y ij >0, PRED( ,  ) ij >0 A dynamic example (with )

6. Wolverine Pharmacometrics Corporation Why Bother with TBS?

7. Wolverine Pharmacometrics Corporation Useful Resources http://www.stat.uconn.edu/~studentjournal/index_files/pengfi_s05.pdf Carroll Rupert (1988) Transformation and Weighting in Regression Estimating Data Transformations in Nonliner Mixed Effect Models; Oberg and Davidian; Biometrics 56,65-72;March 2000.

8. Wolverine Pharmacometrics Corporation Transformations, Likelihoods and Jacobians. Suppose we have a continuous random variable, X whose logarithm is distributed N( ,  2 ). Letting Y=ln(X) we know that the density for Y is...

9. Wolverine Pharmacometrics Corporation But What is the Distribution for X? To find f X (x) when X=g(Y), and g is monotone, we use the following change of variable formula...

10. Wolverine Pharmacometrics Corporation OK, so we turn the crank! X=g(Y) = exp(Y) Y=g -1 (X) = ln(X) d/dX(g -1 (X)) = 1/X

11. Wolverine Pharmacometrics Corporation Now lets Focus on the Dynamic Box-Cox TBS Our assumption is that... Let, so and

12. Wolverine Pharmacometrics Corporation Example A new ‘patch’ has been developed for bromodrosis. The T/2 is short and we have 7 steady state serum concentrations on each of 100 subjects. This may be the simplest possible PK example!

13. Wolverine Pharmacometrics Corporation Initial Model (CWS7.TXT) $PROBLEM $DATA NMDATA7.CSV $INPUT ID DV ; JUST ID AND SERUM CONCENTRATION! $PRED W=THETA(2) ;ADDITIVE SD CL=THETA(1)*EXP(ETA(1)) ;CL/F WITH BETWEEN SUBJECT VAR PRE=1/CL ;@ SS ASSUMING INPUT RATE = 1 4 ALL RES1=(DV-PRE)/W ;FORM A WITHIN SUBJECT RESIDUAL Y=PRE+EPS(1)*W $THETA (0,.1) ;CL/F (0,1) ;SD ADDITIVE $OMEGA.1 $SIGMA 1 FIX $EST MAXEVALS=9999 METH=1 PRINT=1 ; JUST BECAUSE! $COV PRINT=E $TABLE ID RES1 ONEHEADER NOAPPEND NOPRINT FILE=TWS7.TXT

14. Wolverine Pharmacometrics Corporation Graphics.

15. Wolverine Pharmacometrics Corporation CWS7L.TXT / CWS7L1.TXT $SUB CONTR=CONTR.TXT CCONTR=CCONTRA.TXT $PRED W=THETA(2) ;SD CL=THETA(1)*EXP(ETA(1)) ;CL/F PRE=1/CL ;@ SS ASSUMING INPUT RATE = 1 LAM=THETA(3) ;BOX COX LAMBDA PARAMETER PREL=(PRE**LAM-1)/LAM ;TRANSFORMED PREDICTION Y=PREL+EPS(1)*W ;ADDITIVE WITHIN SUBJECT ERROR IN ;THE TRANSFORMED SPACE RES1=((DV**LAM-1)/LAM-PREL)/W ;RESIDUAL IN THE T SPACE $THETA (0,.1) ;CL/F (0,1) ;SD ADDITIVE (0,1) ;BOX COX LAMBDA PARAMETER $OMEGA.1 ; THIS INIT WORKS FINE WITH NMV NM6?? $SIGMA 1 FIX $EST MAXEVALS=9999 METH=1 PRINT=1 $COV PRINT=E $TABLE ID RES1 ONEHEADER NOAPPEND NOPRINT FILE=TWS7L.TXT

16. Wolverine Pharmacometrics Corporation CCONTRA.TXT subroutine ccontr (icall,c1,c2,c3,ier1,ier2) parameter (lth=40,lvr=30,no=50) common /rocm0/ theta (lth) common /rocm4/ y double precision c1,c2,c3,theta,y,w,one,two dimension c2(*),c3(lvr,*) data one,two/1.,2./ if (icall.le.1) return w=y y=(y**theta(3)-one)/theta(3) call cels (c1,c2,c3,ier1,ier2) y=w c1=c1-two*(theta(3)-one)*log(y) return end

17. Wolverine Pharmacometrics Corporation Regression Engine Bake Off nmvnm6nm6 init2 SD in Transformed Space 1.92.1 0.240.270.26 CL/F0.01 $COVYesNoYes!  0.0041E-50.004

18. Wolverine Pharmacometrics Corporation Last Slide