Pharmacokinetics Pharmacokinetic techniques attempt to mathematically define the time course for the drug in the body by assaying for drug and metabolites in readilly accesible fluids such as blood and urine. Concentration vs. time relationships are investigated in relation to dosing and route of administration. Specific analytical methods allow to quantify drugs and metabolites in biological samples. The goal is to quantitatively account for the amount of drug which has entered the body (bioavailable dose) and to estimate the rate by which the drug is eliminated from the body.

What is the dose-concentration relationship? What are the concentrations after repeated drug administrations? What is the influence of the route of administration? What is the influence of drug formulation? What is the extent and importance of inter-individual variability in pharmacokinetics and which factors have an influence. Can we predict (account for) this variability? Key pharmacokinetic questions:

Pharmacokinetics techniques The mathematical descriptions use PK variables (PK parameters) for modelling of the time-course of a drug in the plasma or other fluids (concentration versus time curves). PK parameters are than used to calculate appropriate dosing.

Concentration is a function of time after dosing and of PK-parameters V, k which are characteristic of the drug and organism.

Pharmacokinetic parameters apparent volume of distribution V d total clearance CL elimination half-life t 1/2 bioavailability F absorption rate constant k a how PK parameters are obtained experimentally which PK processes describe how PK parameters are used to calculate dosing regimen what are the sources of inter-patient variability of PK parameters

Apparent Volume of Distribution (V d ) Describes the extent of distribution of the drug in the body. V d is calculated assuming a uniform drug distribution in the body at the concentration same as that in the plasma

Apparent Volume of Distribution (V d ) Definition: The apparent volume of distribution indicates into how large a volume the drug distributes if it were at the same concentration as that in plasma (or in other reference fluid which is sampled - blood, serum).

Apparent Volume of Distribution (V d ) This apparent volume of distribution is not a physiological volume. It won't be lower than blood or plasma volume but it can be much larger than body volume for some drugs. It is a mathematical factor relating the amount of drug in the body and the concentration of drug in the measured compartment, usually plasma. V d = AMOUNT OF DRUG IN THE BODY CONCENTRATION IN THE PLASMA Units of V d : volume (L, L/kg of body weight)

L. Dose Loading dose = C P x V D C 0h = Dose / DistributionVolume V d = Dose / C 0h Estimation of V d after i.v. dosing At later times after administration: Amount in the body = Dose - Amount eliminated

Use of V d : 1/ Initiating treatment: V d in conjunction with a target concentration C Therapeutic can be used to compute a loading dose D L : D L = V D. C Therapeutic The loading dose is the first higher dose which rapidly fills the distribution volume with the drug. Therapeutic concentrations are achieved quickly. This is important in acute conditions (status epilepticus, status asthmaticus, bacteraemia) and if the drug has a long t 1/2 (digoxin, amiodarone).

Loading dose 500 mg + maintenance dosing (150 mg every 8 h) 8h16h24h32h40h Conc. maintenance dosing (150 mg every 8 h) Time

Use of V d : EXAMPLE: J.K. (body weight = 90 kg) was admitted to the ICU for pneumonia caused by Gram-negative bacteria. Calculate the loading dose of tobramycin for this patient to achieve the target concentration of 4 mg/l. Tobramycin V D is 0.2 L/kg of body weight. Loading Dose = ? Loading Dose = 0.2 L/kg. Body weight (kg). Concentration Loading Dose = = 72 mg

Use of V d : 2/ Evaluation of the drug distribution pattern (total body water, only plasma plus extracellular water, only plasma, high binding in tissues) V d 0.05 L/kg the drug remains in the blood (heparine) V d L/kgdistribution from blood into extracellular fluid (gentamicin - polar drugs). V d 0.6 L/kgdistribution from blood into intracelular and extracellular fluid (methotrexate) V d >>0.6 L/kgdistribution intracellularly and high binding in tissues (amiodarone L/kg)

Use of V d : 3/ To assess feasibility of using hemoperfusion or dialysis for drug removal from the body: The larger the V d, the smaller fraction of the dose is in the plasma, the less is plasma concentration of the drug and the less efficient is any drug removal through extracorporeal mechanisms (hemodialysis and hemofiltration).

Age related interindividual variability of V d V d per unit of body weight (L/kg) for water- soluble drugs is higher in neonates compared to adults –5 mg/kg gentamicin: C max = mg/L in adults C max = mg/L in neonates V d per unit of body weight (L/kg) for lipid-soluble drugs is higher in the elderly (diazepam)

The effect of obesity on V d Water-soluble drugs have a limited distribution into fat and their V d is better correlated with the ideal body weight than with total body weight (V d is less than it would by predicted using the total body weight). –Aminoglycosides are hydrophilic antibiotics commonly used to treat Gram-negative onfections. The need for dosing adjustment in obesity has been established in gentamicin, tobramycin, and amikacin. –Aminoglycosides distribute primarily into the intravascular space and moderately into the interstitial space. Therefore, aminoglycosides typically display only a slightly larger volume of distribution (in liters) in obese patients than in lean patients.

Drug elimination : Biotransformation + Excretion Drug elimination refers to the irreversible removal of the drug from the body by all routes of elimination. Clearance quantitatively describes the process of drug elimination from the body or an organ regardless of the mechanism.

Clearance (CL) Definition : Clearance of a drug is the ratio of the rate of elimination to the concentration of drug in plasma. CL = Rate of eliminination [mg / h ] C of drug in plasma [mg /L ] Unit: Volume/Time [L/h] or adjusted for body weight [l/h/kg]

Clearance (CL) Rate of eliminination = CL x C (Amount / Unit of time) = (Volume / Unit of time) x C Another possible way of understanding clearance: Clearance is the volume of plasma completly cleared of the drug per unit of time by all routes - by the liver, the kidney…).

Clearance (CL) VdVd High clearanc e Low clearance Concentration Time The higher is CL, the faster is the decrease of the concentration, the shorter is the half-life.

Clearance (CL) VdVd High Vd Low Vd Concentration Time The higher is Vd, the slower is the decrease of the concentration, the longer is the half-life. VdVd

Clearance (CL) Clearance has an additive character: It is the sum of clearences in all eliminating organs. CL = CL RENAL + CL HEPATIC +CL pulmonary...other renal + nonrenal

How is the clearance obtained experimentally? CL= Dose /AUC (i.v. admin.) CL= F× Dose /AUC (perenteral adm.) AUC = Area under the curve The AUC is estimated as a sum of trapezoids

The principle of linear pharmacokinetics Linear (first-order) pharmacokinetics: For most drugs, clearance is constant over the plasma concentration range used in clinical practice. Elimination is not saturable (non-capacity-limited) and the rate of drug elimination is directly proporcionate to its concentration: Rate of elimin. = CL × Concentration

The principle of linear pharmacokinetics Rate of elimin. = CL × Concentration After repeated dosing, at equilibrium (at the steady-state): Rate of dosing = Rate of elimin. = CL × Concentration (2 × Rate of dosing) = CL × (2 × Concentration)

Nonlinear pharmacokinetics Nonlinear pharmacokinetics: (capacity-limited, dose or concentration dependent, saturable) CL varies depending on the concentration of a drug. Rate of elimination = Vmax. C /Michaelis- Menten/ K m + C CL = V max K m + C ethanol, phenytoin, theofylline

Nonlinear pharmacokinetics Linear kinetics Nonlinear kinetics

The importance of Clearance Total clearance determines the average steady-state concentration of a drug during continuous drug administration (multiple intermittent dosing or constant rate i.v. infusion): at the steady-state: Rate of dosing = Rate of elimination = CL. C ss

Continuous i.v. infusion: C ss = Rate of infusion/ CL

Repeated oral dosing (continuous intermittent dosing) C average,ss Oral dosing C average, ss = (F. Dose/  / CL  Time Conc

Use of clearance: 1/ Total clearance, when multiplied by a target steady-state concentration, can be used to calculate the dosing rate required to maintain plasma C SS,Therapeutic i.e. the maintenance dose. Maintenance dose restores the amount of the drug which has been eliminated per unit of time (i.v. infusion) or between doses (intermittent dosing). Rate of dosing = C SS,TARGET. CL Rate of dosing ….. Rate of i.v. infusion (mg/h) ……(F. Oral dose) / Dosing interval …… i.v. dose / Dosing interval

Calculation of the maintenance dose J.K.was admitted to the ICU for pneumonia caused by Gram-negative bacteria. Calculate the maintenance dose (i.v.-infusion in 6 h intervals) of tobramycin for this patient to achive the target average concentration of 4 mg/l. Clearance of tobramycin was estimated to be 70 ml/min. Rate of dosing = Dose / Interval Rate of dosing = Rate of elimination = CL.c T Rate of dosing = / 1000 = 16.8 mg/h Dose = Rate of dosing. Interval = = 101 mg

Use of clearance: 2/ The numerical value of total clearance and its two principal components (hepatic and renal clearances) provide important insights into the elimination processes and into the potential needs for dosage adjustments in case of liver or kidney impairment.

Organ clearance: Hepatic clearance CL h Q… hepatic blood flow per 1 min: 1.5 L/min, Q. C out Amount excreted in bile = Amount extracted= Q.(C in - C out ) LIVER CL h = rate of elimin. / C in = Q. (C in - C out ) / C in CL h = Q × E hepatic extraction ratio Q. C in bile

Hepatic extraction ratio (E) E = (C in - C out ) / C in A/ High extraction: C out  0, E  1 (>0.7) After oral administration, drug is efficiently extracted by the liver and less is available in the systemic circulation - high first-pass effect. The elimination of the drug from the systemic circulation is flow limited: Clearance = Q × E = Q.

Hepatic extraction ratio (E) E = (C in - C out ) / C in B/ Low extraction: C out  C in, E  0 (<0.3) Small first-pass, high systemic availability after oral administration, hepatic clearance is sensitive to change in E (inhibition and induction of metabolism).

Renal clearance = rate of elimination of the drug per unit of its concentration in the plasma entering the kidneys. Renal drug clearance is correlated with kidney function: endogenous creatinine clearance or serum creatinine concentration. Organ clearance: Renal clearance CL R

Renal clearance CL R Amount excreted = V U × C U KIDNEY CL R = rate of elim. / C in plasma = V U × C U / C in plasma GFR, C in plasma V U, C U GFR ml/min URINE V U = volume collected / urine collection period

Factors affecting clearance Clearance and, therefore, maintenance dose requirements can be affected by a range of factors, including age, renal disease, hepatic disease, genetic factors and drug interactions (inhibition or induction of the activity of drug-metabolizing enzymes)

Elimination half-life (t 1/2 ) Definition: Elimination half-life is the time it takes the drug concentration in the blood to decline to one half of its initial value. It is a secondary parameter : The elimination half-life is dependent on the ratio of two primary parameters V d and CL: t 1/2 = 0.7 × V d / CL Unit : time (min, h, day)

How is t 1/2 obtained experimentally? Semi-log graph of Log concentration vs time t 1/2 = 0.7 / k el k el can be estimated as the slope of the Log (concentration) vs. Time relationship by means of the linear- regression analysis

Use of t 1/2 : stopping treatment t 1/2 can be used to predict how long it will take for the drug to be eliminated from the plasma (five half-lives)

Conversely, t 1/2 descibes the kinetics of approaching the steady state t 1/2 can be used to predict how long it will take from the start of dosing to reach steady-state levels during multiple dosing or continuous i.v. infusion. No. of t 1/2 Concentration achieved (% of steady conc.)

Multiple i.v. bolus dose administration : the drug accumulates in the plasma until the steady state is achieved after 5 t 1/2

Important! During continuous (infusion) or continuous intermittent dosing (oral dosing): The steady-concentration depends on the rate of dosing (the dose/dosing interval) and the clearance. Time required to achieve steady-state depends on the half-life and is independent of the rate of dosing and the clearance.

Use of t 1/2 : The relationship between t 1/2 and dosing interval  can be used: 1/ to predict the degree of accumulation of a drug in the blood. The longer t 1/2 and the shorter , the more drug accumulates. 2/ to predict the degree of fluctuation of a drug concentration within a dosing interval. The longer t 1/2 and the shorter , the less the concentration fluctuates (changes) between succesive doses at the stady-state.

The relationship between t 1/2 and dosing interval  determines accumulation of the drug in the blood (diference between the concentration at the steady state vs that after the first dose) and fluctuation (difference between C max and C min at the steady-state). t 1/2 6 h t 1/2 24 h t 1/2 96 h (4 dny)   24 h C ss, max C ss, min

Use of t 1/2 : t 1/2 can be used to predict how long it will take a drug concentration to decline from one specific value to another. t = t 1/2 × ln(C1/C2) / 0.7 It can be usefull in overdoses and dosage adjustments.

Pharmacokinetics after extravascular administration: bioavailability Most of the routes of administration are extravascular; for example i.m., s.c., and most importantly oral. With this type of drug administration the drug isn't placed in the systemic circulation but must be absorbed through at least one membrane. This has a considerable effect on drug pharmacokinetics and may cause a reduction in the actual amount of drug which is absorbed and reaches the systemic circulation.

Pharmacokinetics after extravascular administration: bioavailability F Bioavailability indicates a measurement of the rate and extent (amount) to which the therapeutically active drug reaches the general circulation. Absolute bioavailability (0 < F < 1) descibes the fraction of the dose which reaches systemic circulation: Bioavailable dose = F × Dose, after i.v. administration F = 1.

Absolute bioavailability (F) F determines the AUC and maximum concentration but not the time when the C max is reached (T max )

Bioavailability Absolute bioavailability F is the absolute fraction of dose which is available from a drug formulation in general circulation. It is measured by comparing AUC after i.v. and extravascular administration. CL = Dose / AUC i.v. = (F × Dose) /AUC oral F = AUC oral / AUC i.v.

Bioavailability Conc. Time AUC iv AUC po

Relative bioavailability Relative bioavailability is a relative amount of the dose if two formulations (other than i.v., most frequently oral) are compared.

Bioequivalence Bioequivalence study: new drug formulation (a generic copy) of a known active drug is compared to the reference (original formulation or another marketed formulation) in a study with healthy volunteers. Two drug formulations are bioequivalent if the extent and rate of bioavailability of a drug is comparable (within certain limits). Not only the AUC (the extent of bio- availability) but C max and T max (the rate of bioavailability must be similar).

The effect of changing k a (absorption rate constant) F and Dose are constant If k a decreases, the C max decreases and T max shifts towards later times.

Pharmacokinetic modelling: Compartmental models Dose D X X,V d EXEX keke C(t) = C 0. e - k e. t C 0 = D / V d The organism is replaced by several compartments (boxes with uniform drug concentration which changes with time). The boxes have its volume V d. Transfer of the drug to, between, and out of compartments is described by rate constants. For simple models, concentration of the drug can be expressed as a function of time and parameters of the model. One-compartment model

Pharmacokinetic modelling: Compartmental models Dose D X V1V1 EXEX keke V2V2 k 12 k 21 Two-compartment model

Fitting of parameters of the model to the assayed concentrations of the drug by nonlinear regression (PC) Time (h) Conc. (mg/l)