Constructing PK Models

Transcription

1 Constructing PK Models Interpretation of biomonitoring data using physiologically based pharmacokinetic modeling Center for Human Health Assessment September 25-29, 29, 2006

2 Pharmacokinetics Studies of the change in chemical/metabolite distribution over time in the body Explores the quantitative relationship between Absorption, Distribution, Metabolism, and Excretion Classical Compartmental models - Data-based, empirical compartments - Describes movement of chemicals with fitted rate constants Physiologically-based models: - Compartments are based on real tissue volumes - Mechanistically based description of chemical movement using tissue blood flow and simulated in vivo transport processes.

3 Approaches to Pharmacokinetic Modeling Non-compartmental - Data summarization Compartmental - Statistical analysis -Interpolation Physiologically Based - Integration of Diverse Data -Extrapolation

4 Example of Simple Kinetic Model: One-compartment model with bolus dose Dose Volume? Purpose: In a 1-compartment 1 model, determine volume of distribution Terminology: Compartment = a theoretical volume for chemical Steady-state = no net change of concentration Bolus dose = instantaneous input into compartment Method: 1. Dose: Add known amount (A) of chemical 2. Experiment: Measure concentration of chemical (C) in compartment 3. Calculate: A compartmental Volume (V)

5 One-compartment model with bolus dose Basic assumption: - Well stirred, instant equal distribution within entire compartment Volume of distribution = A/C - In this model, V is an operational volume - V depends on site of measurement This simple calculation only works IF: - Compound is rapidly and uniformly distributed - The amount of chemical is known - The concentration of the solution is known. What happens if the chemical is able to leave the container?

6 Describing the Rates of Chemical Processes - 1 Chemical in the System Rate equations: - Describe movement of chemical between compartments The previous example had instantaneous dosing Now, we need to describe the rate of loss from the compartment by movement out of compartments or by loss due to metabolism within the compartment Zero-order process: - rate is constant, does not depend on chemical concentration rate = k x C 0 = k First-order process: - rate is proportional to concentration of ONE chemical rate = k x C 1

7 Describing the Rates of Chemical Processes - 2 Chemical Systems Second-order process: - rate is proportional to concentration of both of the chemicals Rate = k x C 1 x C 2 Saturable processes*: - Rate is dependent on interaction of two chemicals M-M kinetics - One reactant, the enzyme, is constant - Described using Michaelis-Menten* equation Rate = (V max x C) / ( C + K m ) 10 * Michaelis-Menten kinetics can describe: -Metabolism - Carrier-mediated transport across membranes -Excretion Rate C

8 1-Comp model with bolus dose and 1 st order elimination Dose Conc? Concentration; Purpose: Examine how concentration changes with time Mass-balance equation (change in C over time): -da/dt=-k e x A, or dividing both sides by Vd -dc/dt=-k e x C where k e = elimination rate constant - Rearrange and integrate above rate equation C = C 0 x e -ke t, or ln C = ln C 0 -k e t Half-life (t 1/2 ): -Time to reduce concentration by 50% -replace C with C 0 /2 and solve for t t 1/2 = (ln 2)/k e = 0.693/k e

9 1-Comp model with bolus dose and 1 st order elimination Dose Conc Clearance: volume cleared per time unit - if k e = fraction of volume cleared per time unit, k e = CL/V (CL=ke ke*v) Calculating Clearance using Area Under the Curve (AUC): AUC = average concentration - integral of the concentration - C dt 10 CL = volume cleared over time (L/min) da/dt = - k e A = -k e V C da/dt = - CL C da = - CL C dt Dose = CL AUC CL = Dose / AUC Conc. 5 0 AUC Time

10 1-Comp model with continuous infusion, 1st order elimination Calculating Clearance at Steady State - At steady state, there is no net change in concentration: dc/dt = k 0 /V k e C = 0 - Rearrange above equation: k 0 /V = k e C ss Steady State -Since CL = k e V, Conc. CL = k 0 /C ss Time

11 2-Comp model with bolus dose and 1 st order elimination k Mass Balance Equations for Compartments k e k 21 Central Compartment (C1): dc1/dt = k 21 C 2 - k 12 C 1 - k e C 1 Peripheral (Deep) Compartment (C2): dc2/dt = k 12 C 1 - k 21 C 2 Conc. Time

12 Linear: Linear and Non-linear Kinetics All elimination and distribution kinetics are 1 st order -Double dose double concentration 100 Conc. 10 AUC 1 Non-linear: Time At least one process is NOT 1 st order -No direct proportionality between dose and compartment concentration Dose Conc. 10 Conc Time Time

13 PBPK Models Building a PBPK Model: 1. Define model compartments Represent tissues 2. Write differential equation for each compartment 3. Assign parameter values to compartments Compartments have defined volumes, blood flows Venous side Chemical in air Lungs Other Fat tissues Liver Arterial side 4. Solve equations for concentration Numerical integration software (e.g. Berkeley Madonna, ACSL) Elimination Simple model for inhalation

14 Structuring PBPK Models What s needed and nothing more - Plausibility vs. Parsimony Considerations: -Uptake routes - Storage/sequestration/binding - Metabolism -Excretion - Target Tissue/Effect Compartment

15 Inhalation PBPK Model for Anesthetics C I Q P C X Q C Q C Lung C A C VL Q L Liver C VF Q F Fat C VR Q R Rapidly Perfused (brain, kidneys, etc.) C VS Q S Slowly Perfused (muscle, bone, etc.)

16 Compartments in a Physiological Model for Methylene Chloride QP CI QC CV Lung Gas Exchange QP CX Lung Tissue QC CA MFO GST QR CVR Richly Perfused Tissues QR CA QS CVS Slowly Perfused Tissues QS CA QF CVF Fat QF CA QL CVL MFO Liver GST QL CA Drink

17 Generic IV PBPK Model (Lutz and Dedrick) RBC Plasma Q pan Stomach Spleen Pancreas Gut Lumen Q f Muscle Fat Bone & Marrow Q int Skin & Fur Liver Q k Intestine Heart Kidneys Sex Organ Bladder Brain Prostate Thyroid Gonads

18 Models in Perspective no model can be said to be correct. The role of any model is to provide a framework for viewing known facts and to suggest experiments. -- Suresh Moolgavkar All models are wrong and some are useful. -- George Box

19 Approach for Developing a PBPK Model Problem Identification Mechanisms of Toxicity Literature Evaluation Biochemical Constants Physiological Constants Model Formulation Simulation Refine Model Compare to Kinetic Data Validate Model Design/Conduct Critical Experiments Extrapolation to Humans

20 Structuring the Model Tissue Grouping: 2 approaches Lumping: Tissues which are pharmacokinetically and toxicologically indistinguishable may be grouped together. Splitting: Tissues which are pharmacokinetically or toxicologically distinct must be separated.

21 Alternative Approaches for Selecting a PBPK Model Structure Lumping Splitting Body Body / Liver Rapid / Slow / Liver Rapid / Slow / Liver / Fat Only a Few Tissues Grouped All Tissues and Organs Separate

22 Splitting Compartments in a PBPK Model Tissue Rapidly Perfused (RP) Slowly Perfused (SP) Liver RP SP Fat Liver Lung RP SP Skin Fat Liver Lung Gut RP SP Skin W.Fat B.Fat

23 Structuring the Model Maintaining mass balance The sum of the tissue blood flows must equal the total cardiac output: ΣQ i = Q C Seems obvious? -- Perhaps, but frequently violated inadvertently in models, particularly when adding new tissue compartments or varying parameters -- e.g., when you split the skin compartment out of the slowly perfused tissue compartment, take its blood flow and volume out too!

24 Structuring the Model Tissue Grouping Criteria: perfusion rate = blood flow / volume R T =Q T / V T rapidly perfused: gut, liver, kidney, etc. slowly perfused: muscle, skin, fat rate constant (/hr) k T = Q T / (P T * V T ) P = C Tissue / C Blood where at equilibrium (e.g., distinguishes fat from the other slowly perfused tissues for a lipophilic compound)

25 Structuring the Model Tissue Grouping Considerations: Storage (e.g., blood cells) Excretion routes (e.g., hair) Flow-limited metabolism (e.g., liver) Uptake routes (e.g., skin) Target Tissues Distributional kinetics Note: the same decisions need to be made for each metabolite, valence, or conjugate formed

26 Building the Model Storage Compartments fat muscle liver kidney blood intestinal lumen

27 Typical Storage Tissue Compartment da T / dt = QT ( CA CVT ) assuming venous equilibration: C = C / VT Note: if V T is constant: T P T so: dat / dt = d( CT VT ) / dt = VT dct / dc / dt = Q ( C C / P ) / V T T A T T T dt

28 Blood compartment: da / dt = Σ( Q C / P ) Q B assuming steady state: therefore: B T T da B T T T / dt = 0 C = Σ( Q C / P ) / T Q C C C B

29 Building the Model Routes of Elimination liver (metabolism) kidney (urinary excretion) bile feces hair exhalation

30 Metabolizing Tissue (e.g., Liver): da L ( C C / P ) da / dt / dt = Q L A L L M where: da / dt = k C V / M and/or F L L P L (linear) max CL / PL / M + ( K C P ) V / L L (saturable)

31 RAL RAM RAO AL' = init CL init AL = 0.0 = CVL = QL = = ( CA CVL) ( VMAX CVL) /( KM CVL) KA MR RAL AL / VL = CL / PL AUCL' = CL PBPK modeling Conventions Ramseyan Code for the Liver AUCL = 0.0 RAM + + RAO + KF CVL VL In Berkeley Madonna, the differential equations have a different standard nomenclature: AL, AM, AO or dal/dt, dam/dt, dao/dt See Berkely Madonna User Guide.

32 Building the Model Distribution Perfusion (flow) limited Transport limited Diffusion limited Partitioning Binding

33 Transport Limited Kinetics VdC T ( CA ( C / P )), / dt = et QT T T where (0 < e T < 1) if e T = 1, kinetics is blood-flow (perfusion) limited

34 Diffusion Limited Kinetics Exchangeable compartment: E Diffusion limited compartment ( Q ( CA C )) ( PA ( C ( C P ))) VdC / dt = / I E ( C ( C P )) VdC / dt = PA / E I I E I I

35 Building the Model Uptake Routes inhalation drinking water oral gavage intravenous intraperitoneal Dermal Others (??)

36 Lung Equations for Inhalation Q P C X LUNG Air Spaces rapid equilibrium Q P C I Q C, C V dlung / dt = Capillary blood Q C, C A ( Q ( C C )) + ( Q ( C C )) C V A ( Q ( C C )) + ( Q ( C ( C P ))) dlung / dt = / A At equilibrium: C V A (( Q C ) + ( Q C ))/( Q + ( Q P )) C = / C V P I P C P C = C / At steady-state: dlung / dt = 0. 0 X A P B I I P A B X B

37 Uptake Routes Drinking water: da Oral gavage: ( Dose )/ k 0 = BW ( Q ( C ( C / P ))) ( k C ( V / P )) 0 / dt = k L L A L L F L L L + A St 0 da da St L = Dose BW / dt = k A A St ( Q ( C ( C / P ))) ( k C ( V P )) + ( k A ) / dt = / L A L L F L L L A St

38 Intravenous: A B 0 Uptake Routes or C = Q C Q C + k / Q V where: k IV = Dose BW (( L VL ) ( F VF ) IV ) C ( Dose BW ) = 0.0, / t IV, t t < > t t IV IV

39 da da Sk SFC One-compartment Dermal Structure / dt = ( K (( ) )) P SA CSk / PSkL CSFC ( K SA ( C ( C / P ))) + ( Q ( C C / P )) / dt = P SFC Sk SkL Sk A Sk SkB IPA Acetone Schematic of the PBPK model for isopropanol and its metabolite acetone Designed for: - oral - inhalation -Dermal exposure routes CV CX QP CI*ff QC Lung QF Fat Surface P QSk Skin QR Rapid QS Slow QBr Brain QL Liver RAO VMaxC, KM CV CX QP Lung Fat Skin Rapid Slow Brain Liver CI QC QF QSk QR QS QBr QL VMax1, KM1 KAD KAS Duodenum KTSD Stomach PDOSE KTD

40 Building the Model Target Tissues -metabolism -binding - pharmacodynamics Metabolite Compartments - compartmental description - physiologically based description Experimental Apparatus -chamber -sampling device

41 Experimental Chamber Compartment C I Qv Chamber Qv C measured Cx N*Qp Cch Animals da Ch / dt = ( N Q ( C C )) + ( Q ( C C )) P Ch where N = the number of animals in the chamber, Q p = the single animal ventilation rate, Q V = the chamber air ventilation rate, C I = the chamber intake air concentration C X = the exhaled air concentration and C ch = the chamber concentration X V I Ch

42 Building the Model Other complications - Experimental problems Loss of material Preening - Total radioactivity data Represents sum of parent and metabolite concentrations May require other metabolites compartment - Tracer data If kinetics are dose-dependent, need to model both unlabeled and labeled material Similar problem for endogenous compounds - Multiple chemical interactions Competition Inhibition/induction

43 Observation: Co-exposure to TCE Decreases the Toxicity of 1,1-DCE 1,1-DCE alone 1,1-DCE plus TCE

44 Hypothesis: Metabolic Interaction

45 For inhibition of metabolism of compound B by compound T: Result: Gas Uptake Kinetic Analysis of 1,1-DCE / TCE Mixtures was Most Consistent with Competitive Inhibition

46 Predicted Inhibition of 1,1-DCE Metabolism by TCE (Assuming Competitive Inhibition)

47 Verification: The Toxicity of 1,1-DCE Is Proportional to the Predicted Amount of 1,1-DCE Metabolized, with or without Co-exposure to TCE

48 Linking Parent Compounds with Metabolites Human model for linear siloxane metabolites produced from metabolism of octamethylcyclotetrasiloxane. An initial metabolite distributes to a central compartment and is then metabolized to multiple downstream products.

49 Summary PBPK Model Development Scientific Method Analysis Observation (Literature) Observation (Literature) Hypothesis (Model Structure) Hypothesis (Model) Experimental Inference (Biology) Test (Experimental Data) Experimental Design (Simulation) Succeed Test (Literature Data) Fail Apply Tweak to Fit