Kinetics, Modeling Oct 15, 2006 Casarett and Doull, 6th Edn, Chapter 7, pp. 225-237 7th Edn, Chapter 7, pp. 305-317 Timbrell, Chapter 3, pp 48-61 (3rd Edn)

Exposure External exposure – ambient air, water Dose received by body Dose at target organ Dose at target tissue Dose at target molecule Molecular dose Repair

Exposure – Dose How are they related. Can we measure them Exposure – Dose How are they related ? Can we measure them ? How can we describe the crucial steps so that we can estimate what we can’t measure?

Enzymes: Biological catalysts Proteins May have metals at active site Act on “substrate” May use/require co-factors

Kinetics of Enzyme-catalyzed Reactions Michaelis-Menten Equation: v = Vmax * [S] Km + [S] First-order where Km >> [S] Zero-order where [S] >> Km

First-Order Processes Follow exponential time course Rate is concentration-dependent v = [A]/t = k[A] Units of k are 1/time, e.g. h-1 Unsaturated carrier-mediated processes Unsaturated enzyme-mediated processes

Second-Order Processes Follow exponential time course Rate is dependent on concentration of two reactants v = [A]/t = k[A]*[B]

Steady-state kinetics E + S ES E + P [ES] is constant, i.e. ES/t = 0 k-1

Saturated metabolism Saturated activation Saturated detoxication

Uptake Higher concentration Carrier Pore Diffusion Lipid bilayer Facilitated diffusion Filtration Active transport Lower concentration

Elimination - excretion Absorption - uptake Elimination - excretion Passive diffusion Filtration Carrier-mediated

The single compartment (one compartment) model kin kout

Kinetics of absorption Absorption is generally a first-order process Absorption constant = ka Concentration inside the compartment = C C/t = ka * D where D = external dose

Kinetics of elimination Elimination is also generally a first-order process Removal rate constant k, the sum of all removal processes C/t = -kC where C = concentration inside compartment C = C0e-kt Log10C = Log10C0 - kt/2.303

First-order elimination Half-life, t1/2 Units: time t1/2 = 0.693/k

One compartment system

First-order decay of plasma concentration

Area under the curve (AUC)

Total body burden Integration of internal concentration over time Area under the curve

Volume of Distribution Apparent volume in which a chemical is distributed in the body Calculated from plasma concentration and dose: Vd = Dose/C0 Physiological fluid space: approximately 1L/kg

A more complex time-course

The two-compartment model Tissues Central compartment Peripheral kin kout Plasma

The three-compartment model Deep depot Peripheral compartment kin kout Central Slow equilibrium Rapid equilibrium

The four-compartment model Mamillary model Peripheral compartment kin Central compartment Deep depot Kidney kout

The four-compartment model Catenary model A B C D kout kin

Physiologically-Based Pharmacokinetic Modeling Each relevant organ or tissue is a compartment Material flows into compartment, partitionnns into and distributes around compartment, flows out of compartment – usually in blood If blood flow rates, volume of compartment and partition coefficient are known, can write an equation for each compartment Assuming conservation of mass, solve equations simultaneously – can calculate concentration (mass) in each compartment at any time

Example of equation δkidney/δt = (Cak * Qa) – (Ck * Qvk) IN OUT Rate of change of the amount in the kidney = Concentration in (incoming) arterial blood X arterial blood flow Minus Concentration in (outgoing) venous blood X venous blood flow

Example of a model Air inhaled Lungs Venous blood Arterial blood Rest of body Liver Metabolism Kidneys Urine

Casaret and Doull, 7th Edn, Chapter 7, pp 317-325