Pharmacokinetics – a practical application of calculus April 6, 2009 Elena Ho

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Oral bio- availability Volume of Distribution Dosing Regimen How much ? How often ? Oral bio- availability Half-life Absorption Clearance Volume of Distribution Permea-bility Efflux Aqueous Solubility Metabolic Stability Renal Excretion Biliary Excretion CNS Penetration Protein Binding Tissue Binding

Typical Study Tools

Barriers between Dose and Target Kerns & Li 2003, DDT 8:316-323

Interesting facts about a human body Absorbing surface area of skin: 1.73 m2 Absorbing surface area of the lung: 70 m2 Absorbing surface area of GI tract: ~200m2 (1/2 basketball court) Small intestine is ~2” around, 22’ long Total length of capillaries is ~ 37,000 miles

Compartment models A compartment is an entity which can be described by a definite volume and a concentration (of drug) Concentration Dose V Dose (mg) = C (ug/ml) x V (ml) V = Dose/Concentration One compartment model: the drug enters the body, distributes instantly between blood and other body fluid or tissues.

__________________________ Model One compartment Two compartment Three compartment Hydrodynamic analogy Drug in Drug in Drug out __________________________ Drug out Drug in central tissue Drug out ____________________________ Drug in Drug in Tissue 1 central Tissue 2 Tissue 1 central Tissue 2 Drug out Drug recycle Drug out

The human body is a multimillion compartment model considering drug concentration in different organelles, cells, or tissues We have access to only two types of body fluid – blood and urine We will begin with the simplest model

Then : dA/dt = - kel A where kel = ke + km Single dose, IV, one compartment : dose of drug introduced rapidly and completely and quickly distributes into its homogenous volume of distribution. Drug is then eliminated by metabolism and excretion. Then : dA/dt = - kel A where kel = ke + km rearrange to : dA/A = - kel dt Integrate: ∫A0 dA/A = - kel ∫ t0 dt Gives: ln A | A0= - kel t | t0 or ln A – ln A0 = - kel . t – t0 A t A t This yields the familiar exponential or logarithmic expressions A = A0 e – Kel t C = C0 e – Kel t log C = log C0 – kel . t /2.3 Kel = 2.3/t . log C/C0 C0 - Kel/2.3 log C time

∫ A dA/A = - kel ∫ t0 dt Biological half-life (T1/2) Consider again the rearranged expression dA/A = - kel dt Integrate between limits A and A/2 ∫ A dA/A = - kel ∫ t0 dt Gives: ln A – ln (A/2) = kel t1/2 ln 2 = kel t1/2 = 0.693 Therefore: t1/2 = 0.693 / kel t/2 A/2

Area Under the Curve (AUC) The integral of drug blood level over time from zero to infinity and a measure of quantity of drug absorbed in the body Area = A o  ∞ Sum of all concentration from t0 to t∞ Linear trapezoidal method: AUC t1t2 = Area of a trapezoid t1t2 = (t2 – t1). (C2+ C1)/2 ii) Log trapezoidal method: AUC t1t2 = (t2 – t1). (C2+ C1)/ln(C2/C1) Lagrange method: cubic polynomial equation Spline method: piecewise polynomials for curve-fitting

Linear and/or Log trapezoidal method T1, T2, T3, T4, T5, T6 T7 Advantages: Easy to use. Reliable for slow declining or ascending curves Disadvantages: error-prone for data points with a wide interval; over or under estimate the true AUC; log 0 is not defined; not good for multiexponential curve

In vivo Pharmacokinetics in Rodents Time Plasma Concentration Distribution Elimination AUC(inf) kel Disposition kinetics: single iv administration repeated blood sampling plasma concentration-time profile Volume of Distribution at steady-state: Plasma Half-life: Plasma Clearance: kel T ln 2 1/2 = T1/2 = 0.693 x Vd/CL AUC Dose CL = kel CL V dss = The clearance of compounds is evaluated in relation to the liver blood flow which is 60 and 90 mL/min/kg in rat and mouse, respectively. The volume of distribution should exceed that of total body water, i.e. 0.6-0.7 L/kg which indicates that the compound distributes freely into tissues.

Absorption GI Tract Stomach: Dissolution Stability at pH 1 Intestines: Permeability Metabolic stability Compound properties controlling absorption: size MW aqueous solubility Sw lipophilicity logP polarity PSA ionization pKa ...

Absorption Deriving Models of the Gastrointestinal Tract

FA% F% Oral bioavailability: Barriers and In vitro Models Fraction of dose absorbed: Gut Lumen FA% Portal Vein Gut Wall Oral bioavailability: Liver F% Oral Absorption limited by: Stability Solubility Permeability ATP-dependent Efflux Drug Metabolism Hepatocellular Uptake, Drug Metabolism and Biliary Excretion Gastric and Intestinal Juice Phys.-Chem. Descr. Caco-2 Intest. Microsomes Liver Microsomes Hepatocytes S9 mix, Cytosol In Vitro Models:

In vivo Pharmacokinetics in Rodents Oral kinetics: Time Plasma Concentration Cmax Tmax Absorption Distribution Elimination AUC(inf) kel single po administration repeated blood sampling plasma concentration-time profile Max. plasma conc. and Time of max. pl. conc. Oral Bioavailability: Tmax Cmax iv po D AUC F / = x 100% There is no possibility to extrapolate the bioavailability in rodents to that in man. The sources of its limitation are often more important than the actual value as this information may allow to study the corresponding mechanism using human in vitro systems and to extrapolate the expected bioavailability .

Example of a pharmacokinetic study: single dose IV in the rat Study design Animal : Sprague-Dawley male rat, approximately 10 weeks old weighing ~250 g each (n=4) Compound : BAY xxxxxx supplied by AABBCC. Dissolve 0.7 mg in 10 ul of DMSO, bring it up to 1 mL with PBS. Dosing : each animal will receive a dose equivalent to 0.7 mg/kg. Time points: pre dose, 5 min, 30 min, 1, 2, 4, 7, 24, 28, 31 hours post dose Blood sample : collect 225 ul of blood in 25 ul of 5% Na Citrate at each time point. Centrifuge blood at 5000 g for 5 minutes. Separate the plasma and keep at -80ºC until analysis

SUMMARY OF RESULTS: Plasma concentration in ng/ml:

Rat plasma concentration was determined using ELISA immunoassay method:

Xxxxxx

Pharmacokinetic parameters: This compound represents a 2-compartment model. Elimination T1/2 = 6.8 hours Total plasma clearance = 135 ml/h/kg Vss = 1.03 L/kg Summary Remark This profile suggests a slow clearance compound with a moderate elimination half life. The volume of distribution at steady state is high, suggesting the compound distribution is beyond the plasma volume compartment

Predicting Human PK Direct Scaling of in vitro rate of metabolism to the CL in vivo in vitro CL in vivo c t physiologically based metabolic CL only  first-pass effect  oral bioavailability Allometric Scaling of human PK based on animal data in vivo body weight CL Vdss empirical total CL and Vss requires mech. to be scalable  t1/2

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