Introduction to Pharmacokinetic Modelling Rationale

Transcription

1 Introduction to Pharmacokinetic Modelling Rationale Michael Weiss Martin Luther University Halle-Wittenberg

2 Philippus von Hohenheim, known as Paracelsus ( ) Everything is a poison... the dose differentiates a poison from a remedy.

3 1937 Archives internationales de pharmacodynamie et de therapie

4 Where the word pharmacokinetics first appeared 1953 This means that the principles governing plasma time-concentrations are not only capable of a mathematical adaptation, but are expected to undergo an advancement through the application of the powerful resources of mathematics.

5 Question 1

6 Question 2 The maximum plasma concentration immediatly (1 min) after a bolus dose of 1 mg digoxin is > 200 ng/ml. Is this dose toxic in view of the therapeutic window (target concentration) of ng/ml (toxicity > 2 ng/ml)? Question 3 When will be more than 90% of this dose be elimitated? CL = 0.2 l/min, V ss = 600 l

7 PK/PD Dose Pharmacokinetics Pharmacodynamics Effect Renal failure Effect Dosis Time Effect Receptor- Upregulation Dosis Time

8 Pharmacokinetic(PK)- Pharmacodynamic(PD) systems analysis Pharmacokinetics Pharmacodynamics Receptor binding & signal transduction PK/PD Pharmacometrics Variability Sparse data Clinical trial simulation Physiologically based (mechanistic) modeling Alterations in diseased states Disease progression Translational PK/PD modeling

9 Pharmacokinetic system Data Model structure Modeling methodology Parameter estimation Modeling purpose Model Cobelli and Carson, 2005

10 Systems Approach Disturbance Input Dose System Output C(t)

11 Pragmatic Validity Behavioural model empirical Heuristic Validity Structural model physiological/mechanistic 2 Black Box 1 Linear system theory Neural network Prediction +Explanation

12 Pharmacodynamics Pharmacokinetics Pharmacy Oral Drug Dose Dissolution sc, im, nasal, pulmonary, transdermal Absorption Distribution Elimination Metabolismn Excretion Effect Site (Receptors) Pharmacological Effect

13 Structure of pharmacokinetic system

14 Pharmacokinetics = Transport with flowing blood Transport across membranes

15 Passive Transport Small Molecules Perfusion or barrier limited Blood Large Molecules Severely permeability limited Blood Tissue Lymph Tissue

16 Active Transport P-Glycoprotein Pump (MDR1) intestine blood-brain barrier kidney liver testis cancer cells (MDR1) etc.

17 Transporter in PK OATPs MDR1 OATs BLOOD BRAIN BARRIER Heart-Lung LIVER BILE MRP2 MDRs MRP3 BSEP OCT1 OATPs OATs MRP3 MRP1 MRP2 MDR1 GUT OATPs OATPs MDR1 MRP2 OATs OCTs OATP: Organic AnionTransporting Polypeptide OAT: Organic Anion Transporter OCT: Organic Cation Transporter MDR: Multidrug Resistance protein MRP: Multidrug Resistance-associated Protein BSEP: Bile Salt Export KIDNEY

18 Metabolism drug metabolite enzymes

19 Diffusion C 1 C 2 da ( t) 1 P S[ C1( t) C2( t )] dt permeability surface da ( t) 1 CL C1( t perm ) dt rate Active Transport V C da1 ( t) max 1 dt K M C1 C 1

20 Plasma Digoxin Concentration (ng/ml) First Principles Based Modeling 3 2 Fit: 3-compartment model 3-exponential function ?? Initial phase < 5 min Time (h) Terminal phase t

21 First Principles Convective transport Convective dispersion Vascular mixing Permeation (Capillary uptake) Diffusion (Extravascular) Not well-mixed! Binding

22 Models of process PK Structural model Compartments (differential equations) Model independent (numerical integration) Subsystems (Laplace transformation) Models of data Statistics Intra- and inter-individual variability Probability distribution of the model parameters in the target population (population approach) Covariates NONMEM ADAPT 5 MONOLIX, etc Poor data perfect model poor result Poor model perfect data poor result

23 Model building Design of PK experiments Feasibility Identifiability

24 Pharmacokinetic System and Experimental Designs Compartmental Isolated perfused organ Destructive sampling PBPK C(t surgery ) In vivo Cardiac output Pulmonary Transit Time Density Systemic Transit Time Density

25 Veins Arteries Lung Structure of the body Brain Routes of drug administration Heart intra venous Kidney Testes Fat Muscle intramuscular oral (enteral) Dose D or, D iv etc. Liver Gut Spleen Organs of drug elimination Pancreas Skin Bone

26 Veins Arteries Lung What can we measure? Brain Heart Kidney Testes Fat Muscle Renal excretion (?) A er (t) Liver Gut Spleen Pancreas Venous blood (plasma) concentration C(t) Skin Bone

27 Modelling of PK transport in terms of mass (amount)! ( Transport or elimination of concentration is nonsense.) Dilemma: we measure concentration. C(t)? A(t)

28 Basic Equation da ( t) e dt CL C( t) Rate of drug elimination = Clearance x Plasma concentration model independent or noncompartmental analysis) (1) 0 dae ( t) A e CL C( t) dt dt 0 D iv CL AUC Well-mixed plasma compartment! Note: A ( ) D (nothing remains in the body) e iv

29 Estimation of Clearance (single dose) CL D iv Intravenous dose AUC Area Under the Curve C(t) Single dose AUC! D iv t AUC C ( t) dt 0

30 Estimation of Clearance (infusion) Steady state after continuous i.v. infusion Elimination rate Dose rate CL C ss DR C(t) Output (elimination rate) = Input (dose rate, infusion rate) C ss CL DR C ss DR t

31 Nonlinear Pharmacokinetics AUC linear nonlinear Dose

32 Saturable Metabolism R H Michaelis-Menten equation R H V K max M C C H, u H, u C(t) C H, u K saturation M C H, u CL int CL Vmax K M C H int V max K M, u linear (dose independent) kinetics exponential high bolus dose

33 Compartmental Models Compartmental Isolated perfused organ Destructive sampling PBPK C(t surgery ) In vivo Cardiac output Pulmonary Transit Time Density Systemic Transit Time Density

34 2-Compartment Model 2 peripheral compartm. V ss V 1 V2 V1 k ( 1 k12 21) k 12 k 21 D iv k 10 1 sampling (central) compartm. V0 V 1 dx1 ( t) dt dx2( t) dt k k x 1 x ( t) k 2 12 ( t) k 12 x 1 x ( t) 1 ( t) k 21 x 2 ( t) x 1 (0) = D iv C( t) x V 1 ( t) 1 reparameterisation k 10 CL V 1 k 12 CL V 1 d k 21 CL V 2 d

35 Distribution Kinetics t ~ 2-10 h Terminal half-life t 1/ 2,z

36 3-Compartment Model Fit excellent for C iv (t) of most drugs-useful as empirical model 2 V 1 : no clear meaning in terms of initial distribution D iv CL 12 V 1 1 CL CL 12, CL 21 : no meaning in terms of underlying distribution processes CL 13 3 Estimation and interpretation of steady-state parameters (CL, V ss ) is straightforward: V ss, CL model independent

37 Physiological Based Pharmacokinetic Modelling Compartmental Isolated perfused organ Destructive sampling PBPK C(t surgery ) In vivo Cardiac output Pulmonary Transit Time Density Systemic Transit Time Density

38 Veins Arteries Lung Brain Physiological Based Pharmacokinetic Modelling (PBPK) Heart Kidney V 2 Testes Fat Flow Q V 1 CL d CL d = f u PS Q Muscle Liver Gut Spleen Organ tissue volume V T Partition coefficient K Vascular volume V 1 V 2 = KV T Pancreas Skin Carcass

39 Veins Arteries Lung PBPK System of Diffeq Brain Heart Kidney Testes e.g., noneliminating organ: dx1 ( t) dt dx2( t) dt Q V 1 CL V x ( t) 2 d 1 x 2 CL V ( t) 1 d CL V x ( t) 1 1 d x ( t) 1 CL V 2 d x 2 ( t) Fat Muscle V 2 Liver Gut Spleen Flow Q V 1 CL d CL d = f u PS Q Pancreas Organ tissue volume V T Skin Partition coefficient K V 2 = KV T Carcass Vascular volume V 1

40 Simulation of Alfentanil Kinetics in Humans Upscaling from rat data Bjorkman, Wada, Stanski Anesthesiology, Human: Tissue volumes (mass) Vascular volumes Blood flows Rat: Partition coefficients Permeabilities (CL d )

41 Upscaling from Animal to Human P a p W CL a CL W b p Since Q L and GFR ~ W Organ function ~ V ss a MDRT V W a MDRT W W b Allometrie 0.25 BSA a BSA W 0.75 Alternatively: CL human CL animal Q Q L, human L, animal

42 Distributed Modelling Normalized outflow concentration Vascular marker Transit Time Distribution Advection-Dispersion Equation (Microvascular Network) C t D 2 C 2 x v C x Dispersion Coefficient (geometrical dispersion) Blood Flow Velocity

43 Solution of Advection-Dispersion Equation: Inverse Gaussian distribution, density f IG (t) (Brownian passage time distribution) Vascular Marker f(t) t t 2 C MTT ( t MTT ) ( t) f ( t) exp IG RD t 2RD 2 MTT t Mean transit time (MTT) Transit time dispersion (RD 2 ) Extent of distribution (V B = MTT Flow) Rate of distribution

44 Subsystems (Transit Time Distribution) Compartmental Isolated perfused organ Destructive sampling PBPK C(t surgery ) In vivo Cardiac output Pulmonary Transit Time Density Systemic Transit Time Density

45 Subsystems Model Formulation in the Laplace Domain Model Structure: Compartments Differential Equations Subsystems Transit Time Density Functions, f i (t) (Impulse Response) Limitation of using compartments as subsystems exponential distributed transit times Advantage of model building in Laplace domain simple rules for connecting subsystems

46 Model building: Laplace Transformation ) ( ˆ2 s f ) ( ˆ1 s f ) ( ˆ1 s f ) ( ˆ2 s f Q Q Q 1 Q 2 ) ˆ( ) ( ) ( ˆ ) ( ˆ ) ˆ( s f L t f s f s f s f ) ˆ( ) ( ) ( ˆ ) (1 ) ( ˆ ) ˆ( s f L t f Q Q q s f q s qf s f Numerical inverse Laplace transformation

47 Numerical Inverse Laplace Transformation f ( t) L 1 { fˆ( s)} Model Equation in Laplace Domain fˆ ( s) L{ f ( t)} Implemented in nonlinear regression software: SCIENTIST 3.0 ADAPT Schalla & Weiss, Eur J Pharm Sci,1999. FORTRAN implementation of Talbot's method

48 Recirculatory Model Compartmental Isolated perfused organ Destructive sampling PBPK C(t surgery ) In vivo Cardiac output Pulmonary Transit Time Density Systemic Transit Time Density

49 Initial Distribution (Front-End Kinetics) Thiopental 3-fold higher V C Determines the anesthetic induction dose! Using Front-end Kinetics to Optimize Target-controlled Drug Infusions Avram & Krejcie, Anesthesiology, 2003

50 Veins Arteries Minimal Circulatory Model Lung Pulmonary Dose C(t) Brain Heart Kidney Testes Fat Muscle Gut Lumping Dose Cardiac output Systemic C(t) Liver Spleen Pancrea s Skin Carcass

51 Hepatic Clearance C out CL Q H H E H Q H = 1500 ml/min F H = 1- E H F H : Hepatic (first pass) availability Fraction escaping elimination by the liver C in A e,b

52 Absorption and Bioavailability F F A F H Systemic circulation Bioavailability F = Fraction of D or that reaches the systemic circulation F H F A D or Gut

53 Subsystems: Isolated Perfused Organs Compartmental Isolated perfused organ Destructive sampling PBPK C(t surgery ) In vivo Cardiac output Pulmonary Transit Time Density Systemic Transit Time Density

54 Hepatic Pharmacokinetics Input Output Output Input

55 Hepatic Clearance: Sinusoidale uptake Hepatocellular metabolism Biliary excretion Cunningham & Van Horn, Alcohol Res & Health, 2003

56 Intravascular Mixing + Cellular Distribution Stochastic model of transit time distribution Advection-Dispersion Equation Microcirculatory network Intravascular mixing (vascular marker) Microscopic volume element V T tissue Cellular distribution Capillary flow V p vascular phase Weiss & Roberts, J Pharmacokin Biopharm, 1996

57 Single Capillary Extravascular Space Intravascular Space

58 k e Well-mixed cellular k in k out Disse space Vascular k off Cell k on V C k e Slow binding Rapid diffusion k in k out L d k e Slow diffusion Rapid binding d L 2 D eff Diffusion time constant k in k out Weiss et al., Br J Pharmacol, 2000

59 Hepatic Transt Time Density of Diclofenac Isolated perfused rat liver outflow curve 10 3 Slow binding 10 2 Well-mixed (cellular) Time (s) Weiss et al., Br J Pharmacol, 2000.

60 Slow Intracellular Carrier Diffusion The free fraction is negligibly small (hydrophobic and amphipatic molecules) soluble cytoplasmic binding proteins act as diffusing transport carriers (D mob ) Albumin (solid line) or [ 14 C]sucrose (dotted line) as the extracellular reference for [ 3 H]palmitate fits of the slow-diffusion model Flow Diffusion, d k in Well-mixed model (dash-dotted line) Slow-binding model (short dashed line) Slow-diffusion/bound model (solid line) Hung et al., Am J Physiol Gastrointest Liver, 2003 Luxon & Weisiger, Am J Physiol, 1993

61 Functional Characterization of Transporters Nonlinear system Flow C(t) k in Multidrug resistance associated protein (MRP) Organic cation transporter (OCT) Organic anion transporter (OAT) Organic anion transporting polypeptide (OATP) sodium-taurocholate cotransporting polypeptide (NTCP) P-glycoprotein (MDR1), Breast cancer resistance protein (BCRP) Bile salt export pump (BSEP) k in K M Vmax C( t)

62 Na pump 4 6 vascular Digoxin Semidistributed liver model Blood flow k in (C) OATP2 Hepatocyte k in ( C) V K M max C k on (C) k out k e Vascular space k off Disse space Weiss et al., Pharm Res, 2010

63 Veins Arteries Lung ICG Kinetics in Dog Brain Heart Kidney Testes Fat Muscle Liver Gut Spleen Lumping Cardiac output Pulmonary Transit Time Density Systemic Transit Time Density Pancrea s Skin Carcass Elimination Uptake LLl Liver Gut Distributed liver model Rest Systemic Weiss et al., Eur J Pharm Sci, 2011

64 Interplay between Hepatic Uptake and Excretion of ICG CL (ml/min) CL uptake (ml/min) k e (min -1 ) Weiss et al., Eur J Pharm Sci, 2011

65 Digoxin concentration (ng/ml) Question 4 Frequency of blood sampling proportional to the rate of change in blood concentration? Time(h)

66 Design of Experiments Picasso The goat Statistics formal Modelling ( Art ) informal intuitive

67 Modelling Art is the lie that helps us see the truth, said Picasso, and the same can be said of modelling. On seeing a Picasso sculpture of a goat, we are amazed that his caricature seems more goatlike than the real animal, and we gain a much stronger feeling for goatness. Similarly, a good mathematical model though distorted and hence wrong like any simplified representation of reality will reveal some essential components of a complex phenomenon. Lee A. Segel, 1984

68 Design of PK Experiments informal intuitive Experience in PK/PD formal mathematical System theory/statistics Identifiability Optimal sampling Models and Modelling

69 Information from Experiments Amount of information ~ log(1/p) p probability (expectation) of the result before the experiment is done Model without model we have no expectations: no basis for choosing what to observe, i.e., design of experiments; an observation generates an infinite amount of information Rescicno and Beck, 1987

70 A priori identifiability Given the structure of the model and experimental design, can the model parameters be estimated if the data are error free? A posteriori identifiability Given model and measured experimental data, can the model parameters be estimated within a reasonable degree of statistical precision?

71 Model misspecification Biased parameter estimates, wrong conclusions Misspecified models can give very precise estimates of the wrong answer. (Halloran et al., 1996) Model identifiability Additional information (data): e.g. on vascular mixing (MID), Fit to low and high dose responses (nonlinear systems) Bayesian estimation, a priori information

72 Issues in Experimental Design Model appropriateness A priori identifiability A posteriori identifiability Input location and duration Output (sampling) location Number, range and spacing of sample times Number of subjects

73 Choice of drug input function? Choice of route of administration Choice of sampling site Choice of sampling scheme Population or individual analysis

74 Model identification complexity reduction Model misspecification Model validity biased estimates modelling objectives

75 The art of asking the right questions in mathematics is more important than the art of solving them. Georg Cantor

76 References Avram, M. J. and T. C. Krejcie (2003). "Using front-end kinetics to optimize target-controlled drug infusions." Anesthesiology 99(5): Bjorkman, S., R. D. Wada, et al. (1998). "Application of physiologic models to predict the influence of changes in body composition and blood flows on the pharmacokinetics of fentanyl and alfentanil in patients." Anesthesiology 88(3): Cobelli, C. and E. Carson (2008). Introduction to modeling in physiology and medicine, Academic Press. Cunningham, C. C. and C. G. Van Horn (2003). "Energy availability and alcohol-related liver pathology." Alcohol Research and Health 27: Hung, D. Y., F. J. Burczynski, et al. (2003). "Fatty acid binding protein is a major determinant of hepatic pharmacokinetics of palmitate and its metabolites." American Journal of Physiology-Gastrointestinal and Liver Physiology 284(3): G423-G433. Rescigno, A., J. S. Beck, et al. (1987). "The use and abuse of models." Journal of Pharmacokinetics and Pharmacodynamics 15(3): Schalla, M. and M. Weiss (1999). "Pharmacokinetic curve fitting using numerical inverse Laplace transformation." European journal of pharmaceutical sciences 7(4): Weiss, M., G. Hübner, et al. (1996). "Effects of cardiac output on disposition kinetics of sorbitol: recirculatory modelling." British journal of clinical pharmacology 41(4): Weiss, M., T. C. Krejcie, et al. (2011). "A physiologically based model of hepatic ICG clearance: Interplay between sinusoidal uptake and biliary excretion." European Journal of Pharmaceutical Sciences 44(3): Weiss, M., P. Li, et al. (2010). "An improved nonlinear model describing the hepatic pharmacokinetics of digoxin: evidence for two functionally different uptake systems and saturable binding." Pharmaceutical research 27(9): Weiss, M. and M. S. Roberts (1996). "Tissue distribution kinetics as determinant of transit time dispersion of drugs in organs: application of a stochastic model to the rat hindlimb." Journal of pharmacokinetics and biopharmaceutics 24(2):