Kinetics, Modeling

Oct 15, 2006

Casarett and Doull, 6th Edn, Chapter 7, pp. 225-2377th 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 ?

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

k2k1

k-1

• Saturated metabolism

• Saturated activation

• Saturated detoxication

Uptake

Higher concentration

Lower concentration

Diffusion

Filtration

Pore

Lipid bilayer

Carrier

Facilitated diffusion

Active transpor

t

Absorption - uptake

• Passive diffusion

• Filtration

• Carrier-mediated

Elimination - excretion

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

Centralcompartment

Peripheralcompartment

kin kout

Tissues

Plasma

The three-compartment model

Deepdepot

Peripheralcompartment

kin

kout

Centralcompartment

Slow equilibrium

Rapid equilibrium

The four-compartment model

Mamillary model

Deepdepot

Peripheralcompartment

kin

kout

Centralcompartment

Kidney

The four-compartment model

Catenary model

A B C D

kinkout

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 modelAir inhaled

Lungs

Arterial blood

Venous blood

Urine

Metabolism

Liver

Kidneys

Rest of body

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