1. Introduction to PK/PD Modeling for Statisticians Part 1
Alan Hartford, AbbVie
ASA Biopharm FDA-Industry Statistics Workshop
September 16, 2015
PKPD Modeling Training| Date 9/16/2015

2. Objectives and Outline for Part 1
Provide statisticians with the main concepts of PK/PD modeling (a.k.a., Pharmacometrics) (see outline below)
Encourage statisticians to support pharmacometricians in their modeling efforts
Provide an appreciation for Pharmacometrics
We won’t be reviewing step by step modeling methods using AIC or p-values
Outline: PK Basic Concepts, PK Compartmental Models, Technical Considerations, Software Considerations, Adding covariates to PK models
PKPD Modeling Training| Date 9/16/2015

3. Introduction – PK and PD
Pharmacokinetics is the study of what an organism does with a dose of a drug
kinetics = motion
Absorbs, Distributes, Metabolizes, Excretes
PK Endpoints
AUC, Cmax, Tmax, half-life (terminal), Cmin
The effect of the drug is assumed to be related to some measure of exposure. (AUC, Cmax, Cmin)
Pharmacodynamics is the study of what the drug does to the body (dynamics = change)

4. PK/PD Modeling Procedure: Purpose:
Fit a model to exposure data and estimate exposure at the same time points where we have PD data.
Examine correlation between estimated exposure and PD (or other endpoints, e.g., AE rates).
Might need to fit a mechanistic model with exposure data as explanatory variable and PD as response.
Purpose:
Estimate therapeutic window
Dose selection
Identify mechanism of action
Model probability of AE as function of exposure (and covariates)

5. Concentration of Drug as a Function of Time
Model for Extra-vascular Absorption
Cmax
AUC
Concentration
Tmax
Time

6. Observed or Predicted PK?
Exposure endpoints are not measured – only modeled, i.e., estimated
Concentration in blood or plasma is a biomarker for concentration at site of action
PK parameters are not directly measured
PKPD Modeling Training| Date 9/16/2015

7. Compartmental Modeling
A person’s body is modeled with a system of differential equations, one equation for each “compartment”
If each equation represents a specific organ or set of organs with similar perfusion rates, then called Physiologically Based PK (PBPK) modeling.
The mean function f is a solution of this system of differential equations.
Each equation in the system describes the flow of drug into and out of a specific compartment.

8. Helpful Reminder … 𝑑 𝑑𝑡 𝑒 𝑡 = 𝑒 𝑡 𝑑 𝑑𝑡 𝑒 𝑘𝑡 = 𝑘𝑒 𝑘𝑡 𝑑 𝑑𝑡 𝑒 −𝑡 = −𝑒 −𝑡
𝑑 𝑑𝑡 𝑒 𝑡 = 𝑒 𝑡
𝑑 𝑑𝑡 𝑒 𝑘𝑡 = 𝑘𝑒 𝑘𝑡
𝑑 𝑑𝑡 𝑒 −𝑡 = −𝑒 −𝑡
𝑑 𝑑𝑡 𝑒 −𝑘𝑡 = −𝑘𝑒 −𝑘𝑡

9. First-Order 1-Compartment Model (Intravenous injection)
Input
Central Vc
Elimination
k10
Solution:

10. First-Order 1-Compartment Model (Intravenous injection) Parameterized with Clearance
Input
Central
Another parameterization for the solution uses Clearance = Cl = k10 Vc
Clearance = Volume of drug eliminated per unit time Vc
Elimination
k10
Solution:

11. First-Order 1-Compartment Model (Extravascular Administration)
Input ka
Absorption depot:
Central
Central compartment: Vc
Elimination
k10
Solution:
F = Bioavailability
(i.e., fraction absorbed)

12. First-Order 1-Compartment Model (Extravascular Administration) Parameterized with Clearance
Input ka
Central Vc
Solution:
Elimination
k10
F = Bioavailability
(i.e., amount absorbed)

13. Parameterization ka, k10, V ka, Cl, V
Micro constant
ka, Cl, V
Macro constant
Note that usually F, V, and Cl are not estimable (unless you perform studies with both IV and extravascular administration)
Instead, apparent V (V/F) and apparent Cl (Cl/F) are estimated when only extravascular data are available

14. First-Order 2-Compartment Model (Intravenous Dose)
Input
k12
Peripheral
Central Vc
(Vp)
k21
Elimination
k10
General form of solution:

15. First-Order 2-Compartment Model (Intravenous Dose)
Parameterized in terms of
“Micro constants”
Note that including Vp over-parameterizes the model since
Input
k12
Peripheral
Central Vc
(Vp)
k21
Elimination
Ac = Amount of drug in central compartment
Ap = Amount of drug in peripheral compartment
k10

16. Web Demonstration
(Requires installation of Adobe Shockwave player.)
PKPD Modeling Training| Date 9/16/2015

17. Technical Considerations
Outline
General form of NLME
Parameterization
Error Models
Model fitting
(Approximate) Maximum Likelihood
Fitting Algorithms

18. The Nonlinear Mixed Effects Model
Pharmacokineticists use the term ”population” model when the model involves random effects.

19. For simplification at this stage, assume
and

20. Assay Variability
Assays for measuring PK concentrations are validated for specific concentration ranges.
If the concentration is higher than the upper limit of the validated assay, then the sample is diluted so the resulting diluted sample has PK concentration within the validated limits.
If the concentration is lower than the lower limit of the validated assay, then the concentration is reported to be “below the limit of quantitation” (“BLOQ” or “<LLOQ”).

21. Assay Variability (cont.)
The result of the assay specifications and the needed dilutions is that additional error is added into the measurement system.
These errors can be accounted for in the statistical model.

22. Distribution of Error
In each case, the errors are assumed to be normally distributed with mean 0
In PK literature, the variance is assumed to be constant (s2)
Heteroscedastic variance is modeled using a proportional error term
Another option is to use the additive error model assuming a variance function R(q) where q is an m x 1 vector which can incorporate b, D and other parameters, e.g. R(q)=s2[f(b)]2, q=[s, bT]T

23. Error Models used for PK modeling
Additive error
Proportional error
Additive and Proportional error
Exponential error
PKPD Modeling Training| Date 9/16/2015

24. For the 1-compartment model parameterized with Cl, V, ka
Input ka
Central Vc
Elimination
k10
And cov(logCli, logVi) is assumed to be 0 by definition of the pharmacokinetic parameters.

25. Maximum Likelihood Approach Is Used
We obtain the maximum likelihood estimate by maximizing
Where p(yi) is the probability distribution function (pdf) of y where now we use the notation of yi as a vector of all responses for the ith subject
The problem is that we don’t have this probability density function for y directly.

26. We use the following:
Where p and p are normal probability density functions. Maximization is in 
=[bT, vech(D), vech(R)] T
Notation: the vech function of a matrix is equal to a vector of the unique elements of the matrix.

27. Under Normal Assumptions

28. Maximum Likelihood
Given data yij, we use maximum likelihood to obtain parameters estimates for b, D, and s2. Because the mean function, f, is assumed to be nonlinear in bi in pharmacokinetics, least squares does not result in equivalent parameter estimates.

29. Approximate Methods
Use numerical approaches to approximate the integral and then maximize the approximation
Options:
Approximate the integrand by something we can integrate
First Order method (Taylor series)
Approximate the whole integral
Laplace’s approximation (second order approximation)
Importance Sampling
Gaussian Quadrature
Use Bayesian methodology

30. Algorithms Used
Available in NONMEM
First Order First Order Conditional Estimation Laplace’s Approximation Importance Sampling Gaussian Quadrature Bayesian (Gibb’s Sampler; Not covered in this presentation)
Approximate just the integrand
Or approximate whole integral
(NONMEM is the gold standard software package for PKPD modeling.)

31. First Order Method
Approximate with a first order Taylor series expansion
If the model assumes
And Ri = s2I, then this is pretty straight-forward.
You use a Taylor series expansion about bi.

32. Taylor Series Expansion
With a first order Taylor series approximation expanded about b, the mean of the bi
Let this approximation be
You use this approximation in the integrand.

33. Substituting back in and simplifying …
See slide 26.
And now the exponent term is integrable and now we can maximize the likelihood.

34. Using Laplace’s Approximation
A second order approximation can be constructed by using Laplace’s approximation
In this manner, the whole integral is approximated so no integration is needed.
PKPD Modeling Training| Date 9/16/2015

35. Numerical Integration: Importance Sampling
Consider a function g(b), then
To get a numerical solution to the integral simply use a random number generator to sample many b from the distribution (b) and change the expectation to a sample mean.

36. Where h is the index for the sampling from p(bi).
and

37. Problem!
If each evaluation of the likelihood surface requires a resampling, then you introduce a randomness to your likelihood surface. The likelihood surface would have small perturbations which would affect your determination of a maximum. Solution: sample once and re-use this sample for each evaluation of the likelihood.

38. It turns out that importance sampling is not very efficient
It turns out that importance sampling is not very efficient. To improve on this method, another method takes advantage of the normal assumption of distribution of bi.
This method is called Gaussian Quadrature. Instead of a random sample, specific abscissas have been determined to best evaluate the integral.
In particular, “adaptive Gaussian Quadrature” is a preferred method (not covered here).

39. Review of Approximate Methods
First order: biased, only useful for getting starting values for better methods; converges often even if model is horrible. DON’T RELY ON THIS METHOD! Laplacian: numerically “cheap”, reasonably good fit Importance sampling: Need lots of abscissas, so not useful Gaussian Quadrature: GOLD STANDARD! But when data set large, method is slow and difficult to get convergence.

40. Additional Note
When your model does not converge, often it’s because you have the wrong model. Don’t switch algorithms just because of nonconvergence. First plot data and scrutinize choice of model.

41. Software
R – PKFIT package NONMEM (industry standard, 1979, FORTRAN) Monolix (Bayesian) WinBugs (PKBugs) Phoenix (windows program incorporating methods from NONMEM) SAS and R can be used to fit very simple PK models but, in general, not very useful
Merge this with earlier slide
PKPD Modeling Training| Date 9/16/2015

42. Why is NONMEM the gold standard?
Software needs easy input of PK models.
Not many software packages allow for models written in terms of ODEs instead of closed form solution.
More challenging for multiple dose settings.
Functional form dependent on data.

43. Multiple Dose Model Daily Dose with Fast Elimination

44. Multiple Dose Model Daily Dose with Slower Elimination
Super-position principle

45. Super-position Principle
Assume dosing every 24 hours
Assume concentration for single dose is
Then concentration, C(t) is

46. Multiple Dose Model, Missed Third Dose

47. Dose Delayed by 3 Hours Every Other Day

48. Modeling Covariates
Assumed: PK parameters vary with respect to a patient’s weight or age.
Covariates can be added to the model in a secondary structure (hierarchical model).
“Population Pharmacokinetics” refers specifically to these mixed effects models with covariates included in the secondary, hierarchical structure

49. Nonlinear Mixed Effects Model
With secondary structure for covariates:

50. Part 1 Summary PK Basic Concepts (Cmax, Tmax, AUC …)
PK Compartmental Models (derived mean function from differential equations)
Technical Considerations (approximate maximum likelihood approach for fitting nonlinear mixed effects model)
Software Considerations (many complications!)