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Pharmacokinetic Conceptsand One Compartment

Model

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Pharmacokinetic Modeling Allows approximation of drug kinetic processes Uses : prediction of plasma, tissue and urine drug levels

correlating drug conc. with therapeutic or toxic effects describe how changes in physiology and disease states affects ADME

Types of Models1. Compartment Models2. Non-Compartment Models3. Physiological Models

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Assumptions Consider kinetics after a rapid IV dosing only Instantaneous distribution – drug in blood in rapid

equilibrium with extravascular tissues immediately after IV injection (no absorption phase)

Drug conc. in tissues is = drug conc. in plasma at any time of sampling.

All drug dosed gets into the blood i.e. amount of drug injected = amount of drug at start (t=0)

Drug elimination follows First Order Kinetics i.e. the rate of change of drug conc. by any process is directly proportional to the drug conc. remaining to undertake that process.

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To illustrate first order kinetics we consider what would happen if we give a drug by IV bolus injection, collect blood samples at various times and measure plasma drug conc.

We might see a steady decrease in conc. as the drug is being eliminated:

Linear plot of conc. Vs time

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Remember Y = mX + C Hence In (Cp) = -kt + In(Cpo)

In(Cp) – In(Cpo) = -ktIn(Cp/Cpo) = -ktexp((In(Cp)/Cpo))= exp -ktCp/Cp0 = exp –kt

The exponential function describing the curve is: Cp = Cp0. e-kt

Calculating Cp0  

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2 methods of extrapolation Calculating ln values for each plasma conc, plot

a ln Cp Vs time graph and extrapolate to y-axis to find ln Cp0, convert to Cp0 OR

Plot plasma conc. values Vs time on a semi-log graph paper and extrapolate to y-axis to find Cp0

Taking logs

ln Cp = ln Cp0 – kt

or

   log Cp = log Cp0 – kt__

2.303

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After an I.V. bolus dose of 500 mg the following data was collected

Concentration versus Time Data

Time hr

1 2 3 4 6 8 10

Cp

g/mL

72 51 33 20 14 9 4

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Semi log graph

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Kel (gradient) = logCp1 – log Cp2

t2 – t1

= log 87.1 – log 4.17 10 - 0

= 0.132 - k = 0.132 2.303 k = 0.304 /hr

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Elimination Rate Constant (k)

Is the rate of change of drug conc in relation to time ie the gradient of the conc vs time slope

Plotting the results of plasma drug conc against time shows that conc first falls rapidly and then progressively more slowly.

k is a 1st order rate constant – expresses the fraction of dose eliminated per unit time e.g. 0.25/hr = ¼ of the drug in the body is eliminated per hour

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Volume of Distribution (Vd) In one compartment model, we assume that the AMOUNT

of drug in the body is related to the plasma drug conc. by a proportionality constant Vd

Is a measure of extent to which a drug is distributed from the plasma to body fluids and tissues.

Vd = Dose / Cp Vd is a direct measure of the extent of distribution. It rarely corresponds to the real vol (plasma vol 3L, ECF

16L, total body water 42L) hence it is called apparent volume of distribution

Units = L

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Significance of Vd

Vd is useful in estimating plasma conc. when a known amount of drug is in the body

Vd is also useful in estimating the dose required to achieve a given plasma concentration – loading dose

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Calculating Vd

– Calculation of Vd requires complete distribution.         Administer dose (t =0)– ·        Measure drug conc at various time intervals– ·        Plot log Conc vs Time– ·        Extrapolate to y-axis to get Cp0 (t=0)– ·        At t = 0 Cp = Cp0 VD = Dose (Db) Cp0

– Eg The C(0) of theophylline is 18mg/L and since 500mg was administered to the patient what is the Vd?

Hence when the plasma conc. is 5mg/L what is the amount of drug in remaining in the body

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Calculating Vd using AUC AUC = Dose = Cp0 Vd . k k

Vd = Dose k. AUC AUC = apply trapeziodal for each segment AUC = Cp1 + Cp2 . (t2 - t1 ) 2 First Segment - Extrapolate C0 Last segment = Cplast / kel

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Linear Plot

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Time (hr) Conc (g/ml)

Delta AUC

AUC (g.hr/ml)

0 100

1 71

2 50

3 35

4 25

6 12

8 6.2

10 3.1

Total

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Clearance Describes drug elimination in volume terms Defined as the volume of fluid cleared of drug from

the body per unit time. Units mls/min, L/hr and maybe corrected for body

weight eg mls/min/kg Total body clearance is a total of all route of

elimination. Clearance = excretion + metabolism Clearance is a constant that relates conc. of plasma

drug to elimination rate.

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Elimination rate = clearance x plasma concPlasma conc (2mg/ml)

clearance (5ml/min)this remains constant in FO ie mls/min

elimination rate (10mg/min) this

remains constant in ZO ie mg/min

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Significance of Clearance Clearance is important because it determines how slowly or

rapidly the conc of a drug declines in the body Therefore, the dose & frequency of drug administration to

maintain certain plasma conc. Eg when Cl or and consequence? Used to determine maintenance dose

At steady state rate of drug admin = rate of drug elimination

Maintenance dose rate = CL x Css (mg/hr) (mls/min) (mg/l)

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How to calculate CL From previous equation

CL = dose rate / Css– Css is achieved after

CL = dose / AUC– AUC after a single dose is like the Css– The amount in the body after admin is determined by

dose and clearance

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Half-life T½ Plasma half-life is the time taken for the

plasma conc. to be decreased by one half of its original value.

Important property of first order kinetics Half life depends on how far a drug

distributes and how rapidly it is cleared. T½ is not dependant on plasma conc, any

changes in clearance or volume of distribution will change half life

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Looking back at equation In (Cp) = -kt + In(Cpo)

At half-life the equation would beIn (Cp) = -k * t½ + In(Cp0) 2In (Cp) – In(Cpo) = -k * t½ 2

In ((Cp) / (Cpo) = k * t½ 2

In (Cp * 1 ) = k * t½ 2 Cpo

0.693 = k * t½

t½ = 0.693 k

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On a semi log graph take any conc and then its half conc., the time taken for it to decrease to half is the same as its t½

The half-life is still the same whether going from 40 to 20 or from 10 to 5mg/L. This is a property of first order process - t½ is constant

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Relationship btw Cl, Vd, half life As discussed earlier Cl and Vd are the primary

parameters that have an effect on t½ and is related as follows:

t½ = 0.693 t½ = 0.693 x V k CL

(k = Cl / V)

e.g. A drug has a Vd of 40L and renal Cl of 650ml/min. Calculate the half-life?

What will be the new half-life in a patient with renal failure where the new clearance is 75ml/min?

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What will happen to t½ if Vd is

a)

b) What will happen to t½ if CL is

a)

b) Drugs A & B have same CL but A has

larger V, which drug will have a longer t½

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Significance of T½ Duration of action

– Conc vs t r/ship is log not linear hence a 10 fold increase in dose is required to produce a 2 fold increase in duration

Time required to reach steady state– Takes 3-5 t½ to reach steady state

Dosing frequency– The dosage interval is determined by t½ – At steady state the fluctuation in plasma conc is 2 fold– This fluctuation can be reduced by dosing more frequently (try

by dosing every 1/3 life)– Fluctuations can also be minimised by SR preps hence t½

affected by absorption rate not CL