1 PKPD Workshop Nick Holford Dept Pharmacology & Clinical Pharmacology University of Auckland, New Zealand Holford NHG, Sheiner LB. Kinetics of pharmacologic response. Pharmacol. Ther. 198;16:143-166 Pharmacokinetics Pharmacodynamics PK = Pharmacokinetics: What the body does to the drug PD=Pharmacodynamics: What the drug does to the body Rx=Treatment: What the prescribed and patient need to know. Treatment Rx =Recipe or Jupiter King of Gods What does this mean? 3 Warfarin Data PKPD Studies in Healthy Subjects 1. mg/kg single oral dose Total racemic warfarin plasma concentration Prothrombin complex activity (PCA) 3 subjects, concentrations, 3 PCA A classical clinical pharmacology study from years ago. Prothrombin complex activity is inversely proportional to the International Normalized Ratio (INR) O'Reilly RA, Aggeler PM, Leong LS. Studies of the coumarin anticoagulant drugs: The pharmacodynamics of warfarin in man. Journal of Clinical Investigation 1963;4(1):14-11 O'Reilly RA, Aggeler PM. Studies on coumarin anticoagulant drugs Initiation of warfarin therapy without a loading dose. Circulation 1968;38:169-177
4 mg/l Warfarin Observations 1 18 9 16 14 7 1 1 8 6 3 4 1 4 48 7 96 1 144 168 Hours PCA Two features to notice about the time course of warfarin concentration and effect on PCA: Peak concentration is about 6 h while greatest effect is at 7 h Most marked between subject variability in PCA occurs when PK variability is least CONC PCA $PROB Warfarin PKPD PK Model $INPUT ID time wt age sex amt dvid dv mdv $DATA warfarin_conc_pca.csv IGNORE (ID.EQ.) ; Don t include PCA $EST METHOD=COND INTER MAX=999 NSIG=3 SIGL=9 PRINT= NOABORT $COV $THETA (.1,.134,1) ; _CL L/h/7kg (.1,8.11,) ; _V L/7kg (.1,.3,4) ; _TABS h (.1,.83,4) ; _LAG h $OMEGA.713 ; _CL.181 ; _V.696 ; _TABS.16 ; _LAG $SIGMA.7 ; RUV_CV.661 ; RUV_SD mg/l $SUBR AAN TRAN $PK IF (NEWIND.LE.1) LN=LOG() FSZV=WT/7 FSZCL=FSZV**.7 CL=FSZCL*_CL*EXP(_CL) V=FSZV*_V*EXP(_V) TABS=_TABS*EXP(_TABS) TLAG=_LAG*EXP(_LAG) KA=LN/TABS ALAG1=TLAG S=V $ERROR CP=F One compartment model with first-order absorption and lag time. Note the use of IGNORE on the $DATA record so that only concentration observations are used in the fit (ID= means PCA observation record). 6 Warfarin First-Order Input (KA1L) ID 1 3 Individual post hoc predictions of concentration fit the time course well and should be adequate for driving the pharmacodynamic model. 1 8 1 6 1 4 4 6 7 1 1 1 1 1 Y PRED
7 Warfarin KA1L Predictive Check The visual predictive check reveals that the model overestimates the early variability in concentration. 8 Immediate Effect Model Two Basic Approaches PREDPP AAN AAN6 Differential Equations $PRED Write model for CP An immediate effect model uses central compartment concentrations to predict the drug effect. Concentrations can be predicted using $PRED or one of the PREDPP library AAN subroutines. Using AAN subroutines is usually easier to code but the model is not so clear. 9 Central Compartment Using AAN $SUB AAN TRANS $PK ; Define LN for efficiency IF (NEWIND.EQ.) THEN LN=LOG() ; Define parameters KA=LN/TABS ALAG1=TLAG S=V $ERROR CP=A()/V ; or CP=F IF (ID.LE.1) THEN CE=CP PCA=E + *CE/(C+CE) IF (ID.EQ.) THEN TABS is the half-life of absorption. The absorption rate is parameterised in terms of TABS but because AAN requires a value for KA it must be calculated from TABS in the code. The NEWIND variable is a built in feature of NONMEM. When it is <= 1 it means this is the first record for this individual. The variable LN is calculated from LOG() just once for efficiency. Note how the ID data item is used to distinguish predictions of concentration (ID=1) from predictions of PCA (ID=).
1 11 Data for Joint PKPD Model #ID time wt age sex amt dvid dv mdv 1 66.7 1 1. 1 1 66.7 1.. 1 1. 66.7 1. 1 1 1 66.7 1. 1 1.9 1 66.7 1. 1 3.3 1 3 66.7 1. 1 6.6 1 6 66.7 1. 1 9.1 1 9 66.7 1. 1 1.8 1 1 66.7 1. 1 8.6 1 4 66.7 1. 1.6 1 4 66.7 1. 44 1 36 66.7 1. 1 4 1 36 66.7 1. 7 1 48 66.7 1. 1.7 1 48 66.7 1. 8 1 7 66.7 1. 1.8 1 7 66.7 1. 31 1 96 66.7 1. 1. 1 1 96 66.7 1. 1 1 66.7 1. 1. 1 1 1 66.7 1. 6 1 144 66.7 1. 71 Central Compartment Using AAN6 NONMEM requires that all observations are in a single column. In order to distinguish concentrations from effects (PCA) the ID data item is used to identify the kind of. A ID of is used to indicate a dose record, 1 for a concentration record and for a PCA record. Note that M is used to indicate if there are missing observations e.g. this subject does not have a PCA observation at =. Note the variables defined in $DES cannot have the same name as variables in other blocks e.g. you cannot define CP=A()/V in $DES and also in $ERROR. In this example the variable name DCP is used in $DES and the variable name CP is used in $ERROR. $SUBR AAN6 TOL=9 $MODEL COMP (GUT) COMP (CENTRAL) $PK ; Define LN IF (NEWIND.LE.1) THEN LN=LOG() ; Define parameters KA=LN/TABS ALAG1=TLAG S=V $DES GUT=A(1) DCP=A()/V RATEIN=KA*GUT DADT(1)= -RATEIN DADT()= RATEIN - DCP*CL $ERROR CP=A()/V IF (ID.LE.1) THEN CE=CP PCA=E + *CE/(C+CE) IF (ID.EQ.) THEN 1 Central Compartment Using $PRED Note that DOSE must be defined on every record so it is obtained on the fly from the AMT value. The AMT value is only > at =. $PRED ; Get DOSE from AMT IF (NEWIND.LE.1) THEN LN=LOG() DOSE= IF (AMT.GT.) DOSE=AMT ; Define parameters KA=LN/TABS KE=CL/V ; Adjust time for lagtime IF (.LE.TLAG) THEN TAD= ELSE TAD=-TLAG ; Plasma concentration EXPKA=EXP(-KA*TAD) EXPKE=EXP(-KE*TAD) CP=DOSE*KA/(V*(KA-KE))*(EXPKE-EXPKA) IF (ID.LE.1) THEN CE=CP PCA=E + *CE/(C+CE) IF (ID.EQ.) THEN
13 Warfarin Immediate ID ID 1 1 Individual post hoc predictions from the immediate effect model describe most of the time course of PCA but in general the time of the predicted greatest effect is earlier than the observed effect. 4 1 8 9 14 Warfarin Immediate Individual post hoc predictions from the immediate effect model describe most of the time course of PCA but in general the time of the predicted greatest effect is earlier than the observed effect. PCA 1 1 7 PCA 1 1 7 4 48 7 96 1 144 Hours Median Lo PCA Hi 4 48 7 96 1 144 Hours Hi Median Lo ObsHi ObsMedian ObsLo 1 Delayed Effect Effect Compartment Model Two Basic Approaches PREDPP AAN4 Very small peripheral compartment AAN6 Differential Equations $PRED Write model for CP and CE The visual predictive check points to using a model with a delayed effect. The simplest form of delayed effect model uses an effect compartment. This is usually considered an empirical model but if the delay in effect is only a few minutes then it might be a reasonable way to described the time course of drug distribution from plasma to the site of action in an organ. $PRED and AAN6 can be used to write exact solutions to the effect compartment model. Because AAN6 has to solve differential equations it is slower. AAN4 can be used by making the peripheral compartment have a negligible volume. This is helpful for complex dosing histories that would be hard to code in $PRED.
16 Effect Compartment Using AAN4 $SUB AAN4 TRANS4 $PK ; Define LN IF (NEWIND.EQ.) THEN LN=LOG() ; Define parameters KA=LN/TABS ALAG1=TLAG S=V V=V KEQ=LN/TEQ V3=V*.1 ; negligible volume Q=V3*KEQ $ERROR CP=A()/V IF (ID.LE.1) THEN CE=A(3)/V3 PCA=E + *CE/(C+CE) IF (ID.EQ.) THEN Note how the peripheral compartment volume is scaled to be a very small fraction (.1) of the central compartment volume. This ensures it will have a negligible influence on the model predictions for the central compartment concentrations. 17 Effect Compartment Using AAN6 $SUBR AAN6 TOL=9 $MODEL COMP (GUT) COMP (CENTRAL) COMP (EFFECT) $PK ; Define LN IF (NEWIND.LE.1) THEN LN=LOG() ; Define parameters KA=LN/TABS ALAG1=TLAG S=V KEQ=LN/TEQ $DES GUT=A(1) DCP=A()/V DCE=A(3) RATEIN=KA*GUT DADT(1)= -RATEIN DADT()= RATEIN - DCP*CL DADT(3)= KEQ*(DCP - DCE) $ERROR CP=A()/V IF (ID.LE.1) THEN CE=A(3) PCA=E + *CE/(C+CE) IF (ID.EQ.) THEN The solution to differential equation 3 is scaled in terms of concentration. There is no need to define a volume of this compartment. It has no meaning for this model. 18 Effect Compartment Using $PRED The effect compartment model for a firstorder input one compartment model requires an additional exponential term. $PRED ; Get DOSE from AMT IF (NEWIND.LE.1) THEN LN=LOG() DOSE= IF (AMT.GT.) DOSE=AMT ; Define parameters KA=LN/TABS KE=CL/V KEQ=LN/TEQ ; Adjust time for lagtime IF (.LE.TLAG) THEN TAD= ELSE TAD=-TLAG ; Plasma concentration EXPKA=EXP(-KA*TAD) EXPKE=EXP(-KE*TAD) CP=DOSE*KA/(V*(KA-KE))*(EXPKE-EXPKA) IF (ID.LE.1) THEN ; Effect compartment model concentration EXPKQ=EXP(-KEQ*TAD) CEEXKE=EXPKE/(KA-KE)/(KEQ-KE) CEEXKA=EXPKA/(KE-KA)/(KEQ-KA) CEEXKQ=EXPKQ/(KA-KEQ)/(KE-KEQ) CE=DOSE*KA*KEQ/V*(CEEXKE+CEEXKA+CEEXKQ) PCA=E + *CE/(C+CE) IF (ID.EQ.) THEN
19 ID Warfarin Ce AAN4 Predictive Check 1 1 ID 1 1 The individual post hoc predictions are much better than using the immediate model. This model looks fine. But are we missing something? The visual predictive check confirms that the average prediction matches the observed effect time course but the variability is clearly overestimated. 4 1 PCA 7 8 9 4 48 7 96 1 144 Hours Hi Median Lo ObsHi ObsMedian ObsLo Turnover Using AAN6 NONMEM VI or NONMEM 7 $SUBR AAN6 TOL=9 $MODEL COMP (GUT) COMP (CENTRAL) COMP (TRNOVR) $PK A_(3)=E ;TOVER is turnover half-life KOUT=LN/TOVER RIN=E*KOUT $DES RATEIN=KA*A(1) DCP=A()/V DPCA=A(3) PD=1+*DCP/(C+DCP) DADT(1)=-RATEIN DADT()=RATEIN - CL*DCP DADT(3)=RIN*PD - KOUT*DPCA $ERROR CP=A()/V IF (ID.LE.1) THEN PCA=A(3) IF (ID.EQ.) THEN The turnover family of models describe delayed drug effects where the delay is due to the turnover of a physiological mediator. This is well understood as the mechanism of the delay for warfarin. NONMEM VI allows initialization of the turnover compartment directly using the special variable A_(). 1 ID Warfarin Turnover Predictive Check ID The visual predictive check shows that the turnover model describes the variability of the observations much better than the effect compartment model. 1 1 1 1 4 1 PCA 7 8 9 4 48 7 96 1 144 Hours Hi Median Lo ObsHi ObsMedian ObsLo
Effect Cpt vs Turnover The individual post hoc predictions from the turnover model do not look any better than the predictions from the effect compartment model. ID ID D V ID ID 1 1 1 1 T IM E T IM E 4 1 4 1 T IM E T IM E Y PRED D V Y P R E D 3 Comparison of Models NONMEM 73 ivf11 Obj Min Covariance Eval Sig Ttot (mins) ka1_to_emax1_aan6 1.81 SUCCESSFUL ABORTED 9 4.3 3.66 ka1_to_emax_aan6 13. SUCCESSFUL NONE 638 3. 1.9 ka1_ce_emax_aan4 171.93 SUCCESSFUL NONE 98 3.1.6 ka1_ce_emax_aan6 171.938 TERMINATED NONE 914 NA 1.3 ka1_ce_emax_pred 171.938 SUCCESSFUL NONE 899 3.1.1 ka1_im_emax_aan 1449.94 SUCCESSFUL NONE 9 4.1.19 ka1_im_emax_aan6 1449.94 SUCCESSFUL NONE 6 3..66 ka1_im_emax_pred 1449.94 SUCCESSFUL NONE 7 4.1.13 4 Comparison of Models NONMEM 73 gfortran times are longer than ivf11. Obj Min Covariance Eval Sig Ttot (mins) ka1_to_emax1_aan6 1.81 SUCCESSFUL ABORTED 9 4.3 4.8 ka1_to_emax_aan6 13. TERMINATED NONE 636 NA.4 ka1_ce_emax_aan4 171.93 SUCCESSFUL NONE 9 3.9.3 ka1_ce_emax_aan6 171.938 TERMINATED NONE 1 NA.89 ka1_ce_emax_pred 171.938 SUCCESSFUL NONE 689 3.4.1 ka1_im_emax_aan 1449.94 SUCCESSFUL NONE 76 4.3. ka1_im_emax_aan6 1449.94 TERMINATED NONE 6 NA 4.13 ka1_im_emax_pred 1449.94 SUCCESSFUL NONE 76 4..17
Comparison of Models NONMEM 7 ivf11 Obj Min Covariance Eval Sig Ttot (mins) ka1_to_emax1_aan6 1.81 SUCCESSFUL OK 1 3. 3.38 ka1_to_emax_aan6 13. SUCCESSFUL NONE 616 3.4 1.3 ka1_ce_emax_aan4 171.93 SUCCESSFUL NONE 8 4.14 ka1_ce_emax_aan6 171.938 SUCCESSFUL NONE 73 3.4 1.16 ka1_ce_emax_pred 171.938 SUCCESSFUL NONE 68 4.1 ka1_im_emax_aan 144.873 SUCCESSFUL NONE 83 3.3.1 ka1_im_emax_aan6 144.873 SUCCESSFUL NONE 3 4..48 ka1_im_emax_pred 144.873 SUCCESSFUL NONE 614 3.9.1 6 Comparison of Models NONMEM 7 gfortran Gfortran objective function is higher for immediate models compared with ivf11. times are longer than ivf11. Obj Min Covariance Eval Sig Ttot (mins) ka1_to_emax1_aan6 1.81 SUCCESSFUL OK 9 4.3 3.76 ka1_to_emax_aan6 13. TERMINATED NONE 63 N/A.11 ka1_ce_emax_aan4 171.93 SUCCESSFUL NONE 3.. ka1_ce_emax_aan6 171.938 SUCCESSFUL NONE 94 3.1 1.8 ka1_ce_emax_pred 171.938 SUCCESSFUL NONE 9 3.1.16 ka1_im_emax_aan 1449.94 SUCCESSFUL NONE 89 3.7.18 ka1_im_emax_aan6 1449.94 SUCCESSFUL NONE 67 3.1 3. ka1_im_emax_pred 1449.94 SUCCESSFUL NONE 3.8.1 7 Comparison of Models NONMEM VI Fixing Emax to 1 allowed the turnover model to finish successfully. Whether the covariance step runs is not a helpful criterion of model suitability. Obj Min Covariance Eval Sig Terminated ka1_to_emax_aan6 1.81 Inf Obj Func NONE 67. ka1_to_emax1_aan6 1.816 Successful OK 367 3.1 ka1_ce_emax_aan4 171.931 Successful ABORTED 97 3.3 ka1_ce_emax_aan6 171.936 Successful ABORTED 39 3. ka1_ce_emax_pred 171.939 Successful OK 8 3. ka1_im_emax_aan 144.9 Successful ABORTED 1 3. ka1_im_emax_aan6 144.9 Successful OK 99 3. ka1_im_emax_pred 144.9 Successful ABORTED 1384 3.
8 Comparison of Parameters NONMEM 73 ivf11 The half-life of PCA is reported in the literature to be about 14 hours. This is very close to the TOVER estimate of 13 hours from this data set. Notice also the more physiological meaning of an Emax of -1 i.e. 1% inhibition of PCA formation. S C TEQ/TOVER ka1_to_emax1_aan6 96.6-1 1.18 13 ka1_to_emax_aan6 96.6 -.999 1.18 13 ka1_ce_emax_aan4 96.7-6 11.1 39. ka1_ce_emax_aan6 96.7-6 11.1 39. ka1_ce_emax_pred 96.7-6 11.1 39. ka1_im_emax_aan 96.7-7.7.87. ka1_im_emax_aan6 96.7-7.7.87. ka1_im_emax_pred 96.7-7.7.87. 9 Comparison of Parameters NONMEM 7 ivf11 The half-life of PCA is reported in the literature to be about 14 hours. This is very close to the TOVER estimate of 13 hours from this data set. Notice also the more physiological meaning of an Emax of -1 i.e. 1% inhibition of PCA formation. S C TEQ/TOVER ka1_to_emax1_aan6 96.6-1 1.18 13 ka1_to_emax_aan6 96.6 -.999 1.18 13 ka1_ce_emax_aan4 96.7-6 11.1 39. ka1_ce_emax_aan6 96.7-6 11.1 39. ka1_ce_emax_pred 96.7-6 11.1 39. ka1_im_emax_aan 96. -74.3.1. ka1_im_emax_aan6 96. -74.3.1. ka1_im_emax_pred 96. -74.3.1. 3 Comparison of Parameters NONMEM 73 gfortran Gfortran estimate of Emax is higher (less negative) and EC are lower than with ivf11 S C TEQ/TOVER ka1_to_emax1_aan6 96.6-1 1.18 13 ka1_to_emax_aan6 96.6 -.999 1.18 13 ka1_ce_emax_aan4 96.7-6 11.1 39. ka1_ce_emax_aan6 96.7-6 11.1 39. ka1_ce_emax_pred 96.7-6 11.1 39. ka1_im_emax_aan 96.7-7.7.87. ka1_im_emax_aan6 96.7-7.7.87. ka1_im_emax_pred 96.7-7.7.87.
31 Comparison of Parameters NONMEM 7 gfortran Gfortran estimate of Emax is higher (less negative) and EC are lower than with ivf11 S C TEQ/TOVER ka1_to_emax1_aan6 96.6-1 1.18 13 ka1_to_emax_aan6 96.6 -.999 1.18 13 ka1_ce_emax_aan4 96.7-6 11.1 39. ka1_ce_emax_aan6 96.7-6 11.1 39. ka1_ce_emax_pred 96.7-6 11.1 39. ka1_im_emax_aan 96.7-7.7.87. ka1_im_emax_aan6 96.7-7.7.87. ka1_im_emax_pred 96.7-7.7.87. 3 Comparison of Parameters NONMEM VI S C TEQ/TOVER ka1_to_emax1_aan6 96.6-1 1.18 13 ka1_to_emax_aan6 96.6-1 1.18 13 ka1_ce_emax_aan4 96.7-6 11.1 39. ka1_ce_emax_aan6 96.7-6 11.1 39. ka1_ce_emax_pred 96.7-7 11.1 39. ka1_im_emax_aan 96. -74.3.. ka1_im_emax_aan6 96. -74.3.. ka1_im_emax_pred 96. -74.3.. The half-life of PCA is reported in the literature to be about 14 hours. This is very close to the TOVER estimate of 13 hours from this data set. Notice also the more physiological meaning of an Emax of -1 i.e. 1% inhibition of PCA formation. 33 Comparison of Random Effects NONMEM 73 ivf11 S C TEQ ka1_to_emax1_aan6.3.44.14 3.8 ka1_to_emax_aan6.3.446.14 3.83 ka1_ce_emax_aan4.1..4.69 3.81 ka1_ce_emax_aan6.1..4.69 3.81 ka1_ce_emax_pred.1..4.69 3.81 ka1_im_emax_aan.8. 1.38. 8.37 ka1_im_emax_aan6.8. 1.38. 8.37 ka1_im_emax_pred.8. 1.38. 8.37 RUV FX The residual error is very similar for the effect compartment and turnover models. The turnover model suggests that between subject differences in C are more important than the value predicted using the effect compartment model. Gfortran estimates for immediate models have larger compared with ivf11
34 Comparison of Random Effects NONMEM 7 ivf11 S C TEQ ka1_to_emax1_aan6.3.44.14 3.8 ka1_to_emax_aan6.3.446.14 3.83 ka1_ce_emax_aan4.1..4.69 3.81 ka1_ce_emax_aan6.1..4.69 3.81 ka1_ce_emax_pred.1..4.69 3.81 ka1_im_emax_aan.8.1.8. 8.37 ka1_im_emax_aan6.8.1.8. 8.37 ka1_im_emax_pred.8.1.8. 8.37 RUV FX The residual error is very similar for the effect compartment and turnover models. The turnover model suggests that between subject differences in C are more important than the value predicted using the effect compartment model. Gfortran estimates for immediate models have larger compared with ivf11 3 Comparison of Random Effects NONMEM 73 gfortran The residual error is very similar for the effect compartment and turnover models. The turnover model suggests that between subject differences in C are more important than the value predicted using the effect compartment model. S C TEQ RUV FX ka1_to_emax1_aan6.3.44.14 3.8 ka1_to_emax_aan6.3.446.14 3.83 ka1_ce_emax_aan4.1..4.69 3.81 ka1_ce_emax_aan6.1..4.69 3.81 ka1_ce_emax_pred.1..4.69 3.81 ka1_im_emax_aan.8. 1.38. 8.37 ka1_im_emax_aan6.8. 1.38. 8.37 ka1_im_emax_pred.8. 1.38. 8.37 36 Comparison of Random Effects NONMEM 7 gfortran S C TEQ ka1_to_emax1_aan6.3.44.14 3.8 ka1_to_emax_aan6.3.446.14 3.83 ka1_ce_emax_aan4.1..4.69 3.81 ka1_ce_emax_aan6.1..4.69 3.81 ka1_ce_emax_pred.1..4.69 3.81 ka1_im_emax_aan.8. 1.38. 8.37 ka1_im_emax_aan6.8. 1.38. 8.37 ka1_im_emax_pred.8. 1.38. 8.37 RUV FX The residual error is very similar for the effect compartment and turnover models. The turnover model suggests that between subject differences in C are more important than the value predicted using the effect compartment model.
37 Comparison of Random Effects NONMEM VI The residual error is very similar for the effect compartment and turnover models. The turnover model suggests that between subject differences in EC are more important than the value predicted using the effect compartment model. E C TEQ RUV FX ka1_to_emax_aan6.33 6.4E-7.44.14 3.8 ka1_to_emax1_aan6.33.44.14 3.8 ka1_ce_emax_aan4.1 1.77E-.4.69 3.81 ka1_ce_emax_aan6.1.93e-6.4.69 3.81 ka1_ce_emax_pred.1 1.78E-4.4.69 3.81 ka1_im_emax_aan. 1.73E-.879. 8.37 ka1_im_emax_aan6. 1.7E-.878. 8.37 ka1_im_emax_pred. 1.7E-.878. 8.37 38 Way to go!