Pyry Välitalo Nonlinear mixed effects models in pharmacokinetic modeling Lecture notes

Pyry Välitalo Pharmacokinetics: Traditional, standard two-stage* approach Recruit subjects from a homogenous (healthy?) population. Collect lots of blood samples from each patient. Estimate pharmacokinetic parameters for each patient separately. Calculate means and variances for pharmacokinetic parameters. Use regression to investigate the effects of covariates. ** *Not to be confused with the term ”two-stage estimation procedures” in statistics ** Riviere, J. Comparative Pharmacokinetics: Principles, Techniques and Applications, page 260. Iowa State Press, 1999.

Pyry Välitalo Problems associated with the traditional approach: Parameter variabilities are inflated.* Since recruitment is usually done from a homogenous (healthy?) population, it is harder to extrapolate into target population. Difficult to study special populations who would not handle the blood loss well (neonates, AIDS/cancer patients, critical care patients, etc). *Sheiner LB, Beal SL. J Pharmacokinet Biopharm Dec;8(6):

Pyry Välitalo A solution: population pharmacokinetics ( = Pharmacokinetics using nonlinear mixed effects modeling) Build a pharmacokinetic model with fixed effects, between-subject variabilities, and residual variabilities. Differences to traditional pharmacokinetic modeling: – One model explains all data – Between-subject variability is included in the model as a new kind of parameter: A random effect that varies between patients but stays constant within the patient.

Pyry Välitalo Advantages of a population approach to pharmacokinetics 1/2 Less blood samples per patient are needed – Special patient groups can be studied (children, cancer patients, etc) – Samples taken during routine treatment can be used in studies. Cost-effectiveness increases The results naturally reflect the patient group that is usually receiving the drug.

Pyry Välitalo Advantages of a population approach to pharmacokinetics 2/2 Because data are combined into a single model, more detailed models can be used. E.g. nonlinearity can be detected better than with standard two-stage approach* Easier to design future clinical trials with one single model. *EN Jonsson, JR Wade, MO Karlsson. AAPS PharmSci. 2000;2(3):E32

Pyry Välitalo Advantages of a population approach to pharmacokinetics: an example Docetaxel, a chemotherapeutic agent Main problem: Patients with poor liver function – FDA: do not dose due to unpredictable PK! Aim: Build a clinically relevant model to predict docetaxel clearance in patients with poor liver function. Evaluate CYP3A activity spesifically as a predictor of docetaxel clearance. A population PK model was built with following covariates: – Liver functioning classification based on a few markers – Plasma protein binding – CYP3A liver enzyme activity After these covariates had been accounted for, the unexplained variability in clearance actually became lower for patients with poor liver function than with normal-liver-function patients. Hooker et al. Clin Pharmacol Ther Jul;84(1):111-8.

Pyry Välitalo Using the parameters of NLME models in pop-PK Fixed-effects parameters (Thetas, θ) – Can be for example a typical value for volume of distribution, typical value for clearance or the effect of a covariate (e.g. sex) on a parameter. – A capital theta (Θ) denotes a vector of all fixed-effects parameters in model. – A lowercase theta (θ) denotes an element of Θ, one specific parameter.

Pyry Välitalo Using the parameters of NLME models in pop-PK Between-subject variability (Etas, η) (Empirical Bayes Estimates) – These describe unexplained differences in parameter values between individuals. – E.g. the individual value for clearance could be described as θ CL *e η(CL). – Can also be used to describe inter-occasion variability, between-study variability. – It is expected that etas are distributed N(0, ω 2 )

Pyry Välitalo Using the parameters of NLME models in pop-PK Residual random effects (Epsilons, ε) – The unexplained residual error (different for each observation). – Sources: Misrecording the time of sampling, mistreatment of samples, error induced by analytical methods, model misspesification, etc. – It is expected that epsilons are distributed N(0, σ 2 ) – E.g. observation=prediction+ε

Pyry Välitalo In summary: The population model y ij =f(Θ i )+ε ij Where y ij is the jth observation of ith individual f is a model that describes all observations Θ i is a vector of individual i’s parameter values ε ij is residual error of individual i’s jth observation The elements of Θ i are usually θ i = θ*e η, where θ is the typical value for a parameter ω 2 is the variance of η values

Pyry Välitalo Components of a population pharmacokinetic model Fixed effects Mostly random effects

Pyry Välitalo A hypothetical example: Building a 2-compartment model Let’s say we have pharmacokinetic data from 40 individuals, of which 20 received oral dosing and 20 received intravenous injection. A total of 200 plasma concentrations, 2-8 per individual Known covariates: Weight, sex (SEX=0 to indicate male, SEX=1 to indicate female) This example takes many shortcuts and should not be viewed as a reference of how to build a pop-PK model.

Pyry Välitalo Let’s start by building a single-compartment IV model (and ignore the oral treatment group for now) We use three parameters: Volume of distribution for central compartment (V)= θ V Clearance (CL) = θ CL Residual error (with standard deviation σ) The starting amount (A) in central compartment is Dose. After dosing, amount (A) of drug in central compartment starts to get eliminated. A t0 =Dose dA/dt=-A*CL/V IPRED=A/V The prediction (PRED) is compared to observation (Y) Y=IPRED+ ε ;σ is standard deviation ;of all epsilons

Pyry Välitalo The model failed to converge. Let’s add between-subject variabilities to CL and V. We estimate five parameters: V= θ V *e η(V) ;Volume of distribution and BSV for it CL= θ CL *e η(CL) ;Clearance and BSV for it Residual error (σ) A t0 =Dose dA/dt=-A*CL/V IPRED=A/V Y=IPRED+ ε ; σ is standard deviation ;of all epsilons …Success! The model now converges.

Pyry Välitalo The ”weighted residuals vs time” graph hints that a 2-compartment model might perform better. On the left: WRES vs Time. Notice the red line (locally weighted scatterplot smoothing) starting above zero, falling below and rising back above zero. Below: An illustration of what it usually means if WRES vs time has a shape like that on the left. The black line represents predictions and red dotted line represents observations.

Pyry Välitalo Let us try the two- compartment model. V central = θ V *e η(V) V peripheral = θ V2 ;Peripheral compartment volume Q= θ Q ;Intercompartmental clearance CL= θ CL *e η(CL) A t0, central =Dose dA central /dt=-A central *CL/V central – A central *Q/V central + A periph *Q/V periph dA periph = A central *Q/V central – A periph *Q/V periph IPRED=A central /V central Y=IPRED+ ε

Pyry Välitalo Let us try adding weight as a covariate into the model now. V central = θ V *(WT/70)*e η(V) ; Linear scaling V peripheral = θ V2 *(WT/70) Q= θ Q CL= θ CL *(WT/70) θscale *e η(CL) ; Allometric scaling A t0,central =Dose dA central /dt=-A central *CL/V central – A central *Q/V central + A periph *Q/V periph dA periph = A central *Q/V central – A periph *Q/V periph IPRED=A central /V central Y=IPRED+ε When adding covariates, we should also check if all the BSV’s are still necessary. It might be that covariates can explain most of the between-subject variability.

Pyry Välitalo Is sex a covariate? Let’s find out (remember, SEX=0 for male, SEX=1 for female). V central = θ V *(WT/70)* (1+SEX*θ sex1 )*e η(V) ;affects only ;females V peripheral = θ V2 *(WT/70)* (1+SEX* θ sex2 ) Q= θ Q CL= θ CL *(WT/70) θscale * (1+SEX* θ sex3 )*e η(CL) A t0,central =Dose dA central /dt=-A central *CL/V central – A central *Q/V central + A periph *Q/V periph dA periph = A central *Q/V central – A periph *Q/V periph IPRED=A central /V central Y=IPRED+ε The testing of covariate relationships should be done one at a time. In this example we didn’t find any significant improvement when adding sex as a covariate in any of the parameters.

Pyry Välitalo Once satisfied with the IV model, we add the oral treatment group Ka= θ Ka ;Rate of absorption F= θ F ;Oral bioavailability V central = θ V *(WT/70)*e η(V) V peripheral = θ V2 *(WT/70) Q= θ Q CL= θ CL *(WT/70) θscale *e η(CL) A t0,depot =F*Dose oral A t0,central =Dose IV dA depot /dt= - A depot *Ka dA central /dt=-A central *CL/V central – A central *Q/V central + A periph *Q/V periph +A depot *Ka dA periph = A central *Q/V central – A periph *Q/V periph IPRED=A central /V central Y=IPRED1+ε

Pyry Välitalo Another example: Modeling flurbiprofen pharmacokinetics in children (real case) Data from 64 patients, 1-7 samples per patient Oral dose given to 37, intravenous dose to 27 patients.

Pyry Välitalo Flurbiprofen pharmacokinetics: At the beginning… Observations of flurbiprofen CSF concentrations (60), and observations of both total (304) and unbound (62) flurbiprofen concentrations in plasma Prior knowledge: The doses given for each patient and the volume of CSF compartment.

Pyry Välitalo Flurbiprofen pharmacokinetics: What we ended up with All the parameters were estimated to best describe concentrations in central compartment and CSF. The number of parameters in the final model was: 13 fixed-effect parameters 4 between-subject variability parameters 2 residual error variability parameters

Pyry Välitalo Flurbiprofen pharmacokinetics: Most significant findings Bioavailability of oral flurbiprofen syrup for children was estimated. The model includes children from 3 months to 13 years (previous study: children aged 6-12 years). There was no impairment of clearance seen in infants. Flurbiprofen distributes into cerebrospinal fluid very effectively.

Pyry Välitalo Conclusion With flurbiprofen data, population pharmacokinetic approach yielded several benefits: – Estimating bioavailability of oral flurbiprofen syrup was made possible. – More credibility when describing CSF kinetics with model parameters than with a summary of raw data (e.g. mean ratio of unbound flurbiprofen in plasma versus flurbiprofen in CSF). – More thoroughly investigated covariate model. – Only one model: Possible to use in simulations in future.

Pyry Välitalo Resources NONMEM. Currently the ”golden standard” in population pharmacokinetic modeling. Requires license. – Xpose: An R package that helps in deciphering the output of NONMEM. Free. – R: A program needed by Xpose to operate. Free. – Census. A helpful program for keeping record of NONMEM runs. Free. – PsN (Perl-speaks-Nonmem). A collection of helpful Perl scripts for NONMEM that make life easier in a lot of ways. Free. MONOLIX. Another population pharmacokinetic program. Free. – Advantages: Shorter runtimes than in NONMEM, provides also graphical output by itself – Disadvantages: Currently not as flexible as NONMEM. –

Pyry Välitalo EXTRA: An example of simulatory model diagnostics: Visual Predictive Check Using the model parameters (including random effects), simulate a number of observations, e.g. 200 simulated observations for every true observation. Calculate the prediction intervals for these simulated observations -> see if they agree with real observations. Blue dots: Real observations Black lines: 95th percentile prediction intervals In this text, prediction interval means an interval, inside which a certain percentile of simulated observations fall.

Pyry Välitalo EXTRA: An example of simulatory model diagnostics: Visual Predictive Check Usually a better alternative is to plot confidence intervals for prediction intervals and see if the intervals for real observations fall inside the confidence intervals for prediction intervals. Red lines: Intervals of real observations Blue area: Confidence intervals for prediction intervals

Pyry Välitalo EXTRA: Features of the flurbiprofen model: Intravenous infusion The intravenous dosage had to have an absorption rate constant. The reason for this is that the intravenously given drug is a prodrug and takes some time to hydrolyze into active flurbiprofen (see figure below).

Pyry Välitalo EXTRA: Features of the flurbiprofen model: Implementing unbound observations Central compartment included two kinds of observations: Total flurbiprofen concentrations Unbound flurbiprofen concentrations If the observation was marked as unbound observation, the prediction was multiplied by fraction unbound (FU) before it was compared to the observation. FU= θ FU *e η(FU) … IPRED=A(2)/V2 ;TOTAL IF (FLAG.EQ.3) IPRED=A(2)/V2*FU ;UNBOUND

Pyry Välitalo EXTRA: Features of the flurbiprofen model: CSF kinetics Modeling the distribution of flurbiprofen into CSF was challenging. Only unbound flurbiprofen can enter CSF. However, in CSF the concentrations of flurbiprofen were circa sevenfold compared to unbound flurbiprofen in plasma. This happened probably because of protein binding in CSF (the CSF observations reflect the total amount of flurbiprofen in CSF).

Pyry Välitalo EXTRA: Features of the flurbiprofen model: CSF kinetics An intercompartmental clearance QCSF was estimated to describe the movement between central compartment and cerebrospinal fluid (CSF). The rate from central to CSF was adjusted by fraction unbound and an uptake factor (UPTK). QCSF= θ QCSF UPTK= θ UPTK K25=QCSF*FU/V2*UPTK K52=QCSF/V5 ;K25 and K52 represent rate constants from ;central compartment to CSF and from CSF to ;central compartment