SUBMITTED BY:

VYJAYANTHI RAO VALLABHANENIREG NO: 256213886030

DEPT. OF PHARMACEUTICS

PHARMACOKINETIC MODELS

Submitted To:Dr. Satyabrata Bhanja

OVERVIEW• Basic considerations in pharmacokinetics• Compartment models• One compartment model

• Assumptions• Intravenous bolus administration• Intravenous infusion• Extravascular administration (zero order and first order absorption

model)

• Multi-compartment model

BASIC CONSIDERATIONS IN PHARMACOKINETICS

• Pharmacokinetic parameters

• Pharmacodynamic parameters

• Zero, first order & mixed order kinetic

• Rates and orders of kinetics

• Plasma drug conc. Time profiles

• Compartmental models – physiological model

• Applications of pharmacokinetics

• Non compartment model

S.no Pharmacokinetic parameter Abbreviation Fundamental units Units example

1. Area under the curve AUC Concentration x time µg x hr/mL

2. Total body clearance ClT Volume x time Litres/time

3. Renal clearance ClR Volume x time Litres/time

4. Hepatic clearance ClH Volume x time Litres/time

5. Apparent volume of distribution VD Volume Litres

6. Vol. of distribution at steady state VSS Volume Litres

7. Peak plasma drug concentration CMAX Concentration mg/L

8. Plasma drug concentration CP Concentration mg/L

9. Steady-state drug concentration Css Concentration mg/L

10. Time for peak drug concentration TMAX Time Hr

11. Dose DO Mass mg

12. Loading dose DL Mass mg

13. Maintenance dose DM Mass mg

14. Amount of drug in the body DB Mass Mg

15. Rate of drug infusion R Mass/time mg/hr

16. First order rate constant for drug absorption Ka 1/time 1/hr

17. Zero order rate constant for drug absorption KO Mass/time mg/hr

18. First order rate constant for drug elimination K 1/time 1/hr

19. Elimination half-life t½ Time hr

20. Fraction of drug absorbed F (no units) Ranges from 0 to 1

Common units in Pharmacokinetics

Mixed Order Kinetics

Kinetics of a pharmacokinetic process changes from First order to Zero order with increasing dose or chronic medication. Deviations from original Linear kinetic profile – Non Linear kinetics. Dose dependent kinetics Seen when P’kinetic process Carriers / Substrates

Capacity Limited – get saturated at

Higher drug Conc.

Michaelis – Menten Kinetics

Describes velocity of Capacity limited, enzyme reactions and non linear pharmacokinetics

MICHAELIS MENTON EQUATION

-DC/DT = VMAX . C / KM + C

KM = Michaelis constant

VMAX = Theoretical maximum

Rate of process

Some examples; Absorption (Vitamin C), Distribution (Naproxen), and Elimination (Riboflavin)

PLASMA DRUG CONCENTRATION – TIME PROFILE

Effectiveness of Dosage Regimen

Concentration of Drug in the Body

Conc. at Site of action

Conc. in whole Blood (Plasma, Serum), Saliva, Urine, CSF

PK Parameters determine drug Conc.

8

A TYPICAL PLASMA DRUG CONC. AND TIME CURVE OBTAINED AFTER A SINGLE ORAL DOSE OF A

DRUG, SHOWING VARIOUS P'KINETIC AND P’DYNAMIC PARAMETERS DEPICTED IN BELOW FIG

PHARMACOKINETIC PARAMETERS

Three important parameters useful in assessing the bioavailability of a drug from its formulation are:

1. Peak plasma concentration ( cmax )

the point at which, maximum concentration of drug in plasma.

Units : µg/ml

• Peak conc. Related to the intensity of pharmacological response, it should be above MEC but less than MSC.

• The peak level depends on administered dose and rate of absorption and elimination.

2. Time of peak concentration (tmax )

the time for the drug to reach peak concentration in plasma (after extra vascular administration).

Units : hrs

• Useful in estimating onset of action and rate of absorption.

• Important in assessing the efficacy of single dose drugs used to treat acute conditions (pain, insomnia ).

3. Area under curve (AUC)

It represents the total integrated area under the plasma level-time profile and expresses the total amount of the drug that comes into systemic circulation after its administration.

Units : µg/ml x hrs

• Represents extent of absorption – evaluating the bioavailability of drug from its dosage form.

• Important for drugs administered repetitively for treatment of chronic conditions (asthma or epilepsy).

PHARMACODYNAMIC PARAMETERS

1. Minimum effective concentration (MEC)

Minimum concentration of drug in plasma/receptor site required to produce therapeutic effect.

• Concentration below MEC – sub therapeutic level

• Antibiotics - MEC

2. Maximum safe concentration (MSC)

Concentration in plasma above which adverse or unwanted effects are precipitated.

• Concentration above MSC – toxic level

3. Onset time

Time required to start producing pharmacological response.

Time for plasma concentration to reach mec after administrating drug

4. Onset of action

The beginning of pharmacologic response.

It occurs when plasma drug concentration just exceeds the required mec.

5. Duration of action

The time period for which the plasma concentration of drug remains above MEC level.

6. Intensity of action

It is the minimum pharmacologic response produced by the peak plasma conc. Of drug.

7. Therapeutic range the drug conc. Between MEC and MSC

CONCEPT OF “HALF LIFE”½ Life = how much time it takes for blood levels of drug to decrease to half

of what it was at equilibrium

There are really two kinds of ½ life…“Distribution” ½ life = when plasma levels fall to half what they were

at equilibrium due to distribution to/storage in body’s tissue reservoirs.“Elimination” ½ life = when plasma levels fall to half what they were

at equilibrium due to drug being metabolized and eliminated.

It is usually the elimination ½ life that is used to determine dosing schedules, to decide when it is safe to put patients on a new drug.

PHARMACOKINETIC MODELS AND COMPARTMENTS

A MODEL IS A MATHEMATIC DESCRIPTION OF A BIOLOGIC SYSTEM AND IS USED TO EXPRESS QUANTITATIVE RELATIONSHIPS.

A COMPARTMENT IS A GROUP OF TISSUES WITH SIMILAR BLOOD FLOW AND DRUG AFFINITY.

A COMPARTMENT IS NOT AREAL PHYSIOLOGIC OR ANATOMIC REGION.

Pharmacokinetic models are hypothetical structures used to describe the fate of a drug in a biological system following its administration.

Pharmacokinetic Modelling

Compartment Models

Non-Compartment

ModelsPhysiologic

Models

Caternary Model

One compt

Mamillary Model

Multi compt Two compt

i v bolus Single oral

Dosei v infusion Intermittent i v

infusionMultiple doses

i v bolus

Oral drug

AUC, MRT, MAT, Cl, VSS

PHARMACOKINETIC MODELS Means of expressing mathematically or quantitatively, time course of drug through out

the body and compute meaningful pharmacokinetic parameters.Useful in :• Characterize the behavior of drug in patient.• Predicting conc. Of drug in various body fluids with dosage regimen.• Calculating optimum dosage regimen for individual patient.• Evaluating bioequivalence between different formulation.• Explaining drug interaction.

Pharmacokinetic models are hypothetical structures that are used to describe the fate of a drug in a biological system following its administration.

Model

• Mathematical representation of the data.

• It is just hypothetical

WHY MODEL THE DATA ?

There are three main reasons due to which the data is subjected to modelling.

1. Descriptive: to describe the drug kinetics in a simple way.

2. Predictive: to predict the time course of the drug after multiple dosing based on single dose data, to predict the absorption profile of the drug from the iv data.

3. Explanatory: to explain unclear observations.

PHARMACOKINETIC MODELING IS USEFUL IN :-

• Prediction of drug concentration in plasma/ tissue/ urine at any point of time.

• Determination of optimum dosage regimen for each patient.

• Estimation of the possible accumulation of drugs/ metabolites.

• Quantitative assessment of the effect of disease on drug’s adme.

• Correlation of drug concentration with pharmacological activity.

• Evaluation of bioequivalence.

• Understanding of d/i.

COMPARTMENTAL MODELS• A compartment is not a real physiological or anatomic region

but an imaginary or hypothetical one consisting of tissue/ group of tissues with similar blood flow & affinity.

• Our body is considered as composed of several compartments connected reversibly with each other.

ADVANTAGES

• Gives visual representation of various rate processes involved in drug disposition.

• Possible to derive equations describing drug concentration changes in each compartment.

• One can estimate the amount of drug in any compartment of the system after drug is introduced into a given compartment.

DISADVANTAGES

• Drug given by IV route may behave according to single compartment model but the same drug given by oral route may show 2 compartment behaviour.

• The type of compartment behaviour i.E. Type of compartment model may change with the route of administration.

1. Central compartment

Blood & highly perfused tissues such as heart, kidney, lungs, liver, etc. 2. Peripheral compartment

Poorly per fused tissues such as fat, bone, etc.

MODELS:

“OPEN” and “CLOSED” models:

• The term “open” itself mean that, the administered drug dose is removed from body by an excretory mechanism ( for most drugs, organs of excretion of drug is kidney)

• If the drug is not removed from the body then model refers as “closed” model.

TYPES OF COMPARTMENT

LOADING DOSE• A drug dose does not show therapeutic activity unless it reaches the desired steady

state.

• It takes about 4-5 half lives to attain it and therefore time taken will be too long if the drug has a long half-life.

• Plateau can be reached immediately by administering a dose that gives the desired steady state instantaneously before the commencement of maintenance dose x0.

• Such an initial or first dose intended to be therapeutic is called as priming dose or loading dose x0,l.

CALCULATION OF LOADING DOSE

• After e.V. Administration, cmax is always smaller than that achieved after i.V. And hence loading dose is proportionally smaller.

• For the drugs having a low therapeutic indices, the loading dose may be divided into smaller doses to be given at a various intervals before the first maintenance dose.

• A simple equation for calculating loading dose is :

xo,l = css,av vd

F

CALCULATION….,

• When vd is not known, loading dose may be calculated by the following equation :

xo,l = 1___________

Xo (1 – e-ket) (1 – e-kat)

• Given equation applies when ka >> ke and drug is distributed rapidly.

• When drug is given i.V. Or when absorption is extremely rapid, the absorption phase is neglected and the above equation reduces to accumulation index:

ASSUMPTIONS

1. One compartment The drug in the blood is in rapid equilibrium with drug in the extra-vascular

tissues. This is not an exact representation however it is useful for a number of drugs to a reasonable approximation.

2. Rapid mixing We also need to assume that the drug is mixed instantaneously in blood or

plasma.

3. Linear model We will assume that drug elimination follows first order kinetics.

LINEAR MODEL - FIRST ORDER KINETICS

• FIRST-ORDER KINETICS

MATHEMATICALLY• This behavior can be expressed mathematically as :

ONE COMPARTMENT MODELOne compartment model can be defined :• One com. Open model – i.V. Bolus.• One com. Open model - cont. Intravenous infusion.• One com. Open model - extra vas. Administration (zero-order absorption)• One com. Open model - extra vas. Administration (First-order absorption ) • INTRAVENOUS (IV) BOLUS ADMINISTRATION

RATE OF DRUG PRESENTATION TO BODY IS:

• Dx =rate in (availability)–rate out( Eli)Dt

• Since rate in or absorption is absent, equation becomes dx = - rate out

dt• If rate out or elimination follows first order kinetic Dx/dt = -kex (eq.1)ELIMINATION PHASE:

Elimination phase has three parameters:

• Elimination rate constant

• Elimination half life

• Clearance

ELIMINATION RATE CONSTANT• Integration of equation (1)

• In x = ln xo – ke t (eq.2)

Xo = amt of drug injected at time t = zero i.E. Initial amount of drug injected

X=xo e-ket ( eq.3)

• Log x= log xo – ke t

2.303 (eq.4)

• Since it is difficult to directly determine amount of drug in body x, we use relationship that exists between drug conc. In plasma C and X; thus

• X = vd C (eq. 5)

• So equation-8 becomes

log c = log co – ke t

2.303 (eq.6)

KE = KE + KM +KB +KL +….. (Eq.7)(KE is overall elimination rate constant)

ELIMINATION HALF LIFE

T1/2 = 0.693 KE (eq.8) • Elimination half life can be readily obtained from the graph of log c

versus t • Half life is a secondary parameter that depends upon the primary

parameters such as clearance and volume of distribution.• T1/2 = 0.693 V d

Cl T (eq.9)

APPARENT VOLUME OF DISTRIBUTION

• Defined as volume of fluid in which drug appears to be distributed. • Vd = amount of drug in the body = x Plasma drug concentration C (eq.10)

Vd = xo/co =I.V.Bolus dose/co (eq.11)

• Example: 30 mg i.V. Bolus, plasma conc.= 0.732 mcg/ml.• Vol. Of dist. = 30mg/0.732mcg/ml =30000mcg/0.732mcg/ml

= 41 liter.

• For drugs given as i.V.Bolus,

Vd (area)=xo/KE.Auc …….12.A

• For drugs admins. Extra. Vas.

Vd (area)=f xo/ke.Auc ……..12.B

CLEARANCE Clearance = rate of elimination

Plasma drug conc.. (Or) cl= dx /dt

C ……., (eq.13)

Thus, renal clearance = rate of elimination by kidney

C

Hepatic clearance = rate of elimination by liver

C

Other organ clearance = rate of elimination by organ

C

Total body clearance:

Clt = clr + clh + clother ……, (eq.14)

• According to earlier definition

cl = dx /dt

C

• Submitting eq.1 dx/dt = KE X , above eq. Becomes ,clt = KE X/ C .., (Eq 15)

• By incorporating equation 1 and equation for vol. Of dist. ( Vd= X/C ) we can get

clt =KE vd (eq.16)

• Parallel equations can be written for renal and hepatic clearance.

Clh =km vd (eq.17)

Clr =ke vd (eq.18)

• But, KE= 0.693/t1/2

• So, clt = 0.693 vd (eq.19)

t1/2

• For non compartmental method which follows one compartmental kinetic is :

• For drug given by i.V. Bolus

clt = xo …..20.A

Auc

• For drug administered by e.V.

Clt = f xo …..20.B

Auc

• For drug given by i.V. Bolus

renal clearance = xu∞ …….(eq. 21) auc

ORGAN CLEARANCE• Rate of elimination by organ= rate of presentation to the organ – rate of exit

from the organ.

• Rate of elimination =q. Cin- Q.Cout

(Rate of extraction) =Q (cin- cout) Clorgan=rate of extraction/cin =q(cin-cout)/cin

=Q.Er …………….(eq 22)• Extraction ratio:

ER= (cin- cout)/ cin

• ER is an index of how efficiently the eliminating organ clear the blood flowing through it of drug.

According to ER, drugs can be classified as• Drugs with high ER (above 0.7)

• Drugs with intermediate ER (between 0.7-0.3)

• Drugs with low ER (below 0.3)

• The fraction of drug that escapes removal by organ is expressed as

F= 1- ER

• Where f=systemic availability when the eliminating organ is liver.

HEPATIC CLEARANCE

Clh = clt – clr

Can also be written down from eq 22 Clh= QH ERH

QH= hepatic blood flow. ERH = hepatic extraction ratio. Hepatic clearance of drug can be divided into two groups :1. Drugs with hepatic blood flow rate-limited clearance 2. Drugs with intrinsic capacity- limited clearance

HEPATIC BLOOD FLOW• F=1-erh

= AUC oral

AUC i.V

INTRINSIC CAPACITY CLEARANCE• Denoted as clint, it is defined as the inherent ability of an organ to

irreversibly remove a drug in the absence of any flow limitation.

ONE COMPARTMENT OPEN MODEL:INTRAVENOUS INFUSION• Model can be represent as : ( i.v infusion)

Drug

Dx/dt =ro-kex …eq 23

X=ro/ke(1-e-ket) …eq 24

Since X =vdc

C= ro/kevd(1-e-ket) …eq 25

= Ro/clt(1-e-ket) …eq 26

Blood & otherBody tissues

R0

Zero order Infusion rate

KE

• At steady state. The rate of change of amount of drug in the body is zero ,eq 23 becomes

Zero=ro-kexss …27

Kexss=ro …28

Css=ro/kevd …29

=Ro/clt i.E infusion rate ....30

Clearance

Substituting eq. 30 in eq. 26

• C=css(1-e-ket) …31

Rearrangement yields:

• [Css-c]=e-ket . ...32

Css

Log CSS-C = -ket …33

Css 2.303

• If n is the no. Of half lives passed since the start of infusion(t/t1/2)

• Eq. Can be written as

• C=CSS [1-(1/2)n] …34

INFUSION PLUS LOADING DOSE

XO,L=CSSVD …35

• SUBSTITUTION OF CSS=RO/KEVD

• XO,L=RO/KE …36

• C=XO,L/VD E-KET+ RO/KEVD(1-E-KET) …37

ONE COMPARTMENT OPEN MODEL EXTRA VASCULAR ADMINISTRATION

• When drug administered by extra vascular route (e.G. Oral, i.M, rectal ), absorption is prerequisite for its therapeutic activity.

ONE COMPARTMENT MODEL: EXTRA VASCULAR ADMIN ( ZERO ORDER ABSORPTION)

• This model is similar to that for constant rate infusion. Drug at site

zero order elimination Absorption

o Rate of drug absorption as in case of CDDS , is constant and continues until the amount of drug at the absorption site (Ex. GIT) is depleted.

o All equations for plasma drug conc. Profile for constant rate i.V. Infusion are also applicable to this model.

Blood & otherBody tissues

R0

ONE COMPARTMENT MODEL: EXTRA VASCULAR ADMIN ( FIRST ORDER

ABSORPTION)

• Drug that enters the body by first order absorption process gets distributed in the body according to one compartment kinetic and is eliminated by first order process.

• The model can be depicted as follows and final equation is as follows

Blood & otherBody tissues

Drug at site

Ka KE

First orderabsorption

elimination

C=Ka F Xo/Vd (Ka-KE) [e -Ket-e-Kat] …41

MULTI- COMPARTMENT MODELS

• Ideally a true pharmacokinetic model should be the one with a rate constant for each tissue undergoing equilibrium.

• Therefore best approach is to pool together tissues on the basis of similarity in their distribution characteristics.

• The drug disposition occurs by first order.

• Multi-compartment characteristics are best described by administration as i.v bolus and observing the manner in which the plasma concentration declines with time.

The no. Of exponentials required to describe such a plasma level-time profile determines the no. Of kinetically homogeneous compartments into which a drug will distribute.

The simplest and commonest is the two compartment model which classifies the body tissues in two categories :

1. Central compartment or compartment 1

2. Peripheral or tissue compartment or compartment 2.

TWO COMPARTMENT OPEN MODEL-IV BOLUS ADMINISTRATION:

Elimination from central compartmentFig:

• After the iv bolus of a drug the decline in the plasma conc. Is bi-exponential. • Two disposition processes- distribution and elimination.• These two processes are only evident when a semi log plot of C vs. T is made.• Initially, the conc. Of drug in the central compartment declines rapidly, due to

the distribution of drug from the central compartment to the peripheral compartment. This is called distributive phase.

1 Central

2

peripheral

Extending the relationship X= vd C

Dcc = K21 xp – K12 xc – KE xc

Dt vp vc vc

X= Amt. Of drug in the body at any time t remaining to be eliminated

C=drug conc in plasma

Vd =proportionality const app. Volume of distribution

Xc and xp=amt of drug in C1 and C2

Vc and vp=apparent volumes of C1 and C2

= K12 xc – K21 xp

Vc vp On integration equation gives conc of drug in central and peripheral compartments at any given time t

Cp = xo [( K21 – a)e-at + (K12 – b)e-bt] Vc b – a a – b

Xo = iv bolus dose

• The relation between hybrid and microconstants is given as : a + b = K12 + K21 + KE A b = K21 KECc = a e-at + be-bt

Cc=distribution exponent + elimination exponent

A and B are hybrid constants for two exponents and can be resolved by graph by method of residuals.

A = X0 [K21 - A] = CO [K21 – A]

VC B – A B – A

B = X0 [K21 - B] = CO [K21 – B]

VC A – B A – B

CO = Plasma drug concentration immediately after i.v. Injection

• Method of residuals : the biexponential disposition curve obtained after i. V. Bolus of a drug that fits two compartment model can be resolved into its individual exponents by the method of residuals.

C = a e-at + b e-bt From graph the initial decline due to distribution is more rapid than the terminal decline

due to elimination i.E. The rate constant a >> b and hence the term e-at approaches zero much faster than e –bt

C = B e-bt

Log C = log B – bt/2.303 C = back extrapolated pl. Conc.• A semilog plot of C vs t yields the terminal linear phase of the curve having slope

–b/2.303 and when back extrapolated to time zero, yields y-intercept log B. The t1/2 for the elimination phase can be obtained from equation

• t1/2 = 0.693/b.

• Residual conc values can be found as-

Cr = C – C = ae-at

Log cr = log A – at

2.303

A semilog plot cr vs t gives a straight line.

Ke = a b c

A b + B a

K12 = a b (b - a)2

C0 (A b + B a)

K21 = A b + B a

C0

• For two compartment model, KE is the rate constant for elimination of drug from the central compartment and b is the rate constant for elimination from the entire body. Overall elimination t1/2 can be calculated from b.

Area under (auc) = a + b

The curve a b

App. Volume of central = X0 = X0

compartment C0 KE (AUC)

App. Volume of = VP = VC K12

Peripheral compartment K21

Apparent volume of distribution at steady state or equilibrium

Vd,ss = VC +VP

Vd,area = X0

B AUC

Total systemic clearence= clt = b vd

Renal clearence= clr = dxu = KE VC

Dt

The rate of excretion of unchanged drug in urine can be represented by :

dxu = KE A e-at + KE B e-bt

Dt

The above equation can be resolved into individual exponents by the method of residuals.

TWO – COMPARTMENT OPEN MODEL- I.V. INFUSION

The plasma or central compartment conc of a drug when administered as constant rate (0 order) i.V. Infusion is given as:

C = R0 [1+(KE - b)e-at +(KE - a)e-bt]

VC KE b – a a - b

At steady state (i.E.At time infinity) the second and the third term in the bracket becomes zero and the equation reduces to:

Css = R0

Vc ke

Now VC KE = vd b

Css = r0 = r0

Vdb clt

The loading dose X0,L = css vc = R0

Ke

1Central

2Peripheral

TWO-COMPARTMENT OPEN MODEL-EXTRAVASCULAR ADMINISTRATION

• First - order absorption :• For a drug that enters the body by a first-order absorption process and distributed

according to two compartment model, the rate of change in drug conc in the central compartment is described by three exponents :

• An absorption exponent, and the two usual exponents that describe drug disposition.

The plasma conc at any time t is C = n e-kat + l e-at + m e-bt

C = absorption + distribution + elimination Exponent exponent exponent • Besides the method of residuals, ka can also be found by loo-riegelman method for

drug that follows two-compartment characteristics.• Despite its complexity, the method can be applied to drugs that distribute in any

number of compartments.

CALCULATING Ka using Wagner-nelson method(Bioavailability

parameters)

WAGNER-NELSONS METHODTHEORY: The working equations can be derived from the mass balance

equation: Gives the following eqaution with time and mass balance

• Above equation Integrating gives

• To the equation amount

absorbed VERSUS TIME

WAGNER-NELSONS METHOD• Taking this to infinity where cp equals 0

• Finally (Amax - A), the amount remaining to be absorbed can also be expressed as the amount remaining in the GI, xg

• We can use this equation to look at the absorption process. If, and only if, absorption is a single first order process

WAGNER-NELSONS METHOD• Example data for the method of wagner-nelson kel (from IV data) = 0.2 hr-

Time(hr)

PlasmaConcentrati

on(mg/L)

Column3

ΔAUC

Column4

AUC

Column 5kel * AUC

A/V[Col2 + Col5]

(Amax - A)/V

0.0 0.0 0.0 0.0 0.0 0.0 4.9

1.0 1.2 0.6 0.6 0.12 1.32 3.58

2.0 1.8 1.5 2.1 0.42 2.22 2.68

3.0 2.1 1.95 4.05 0.81 2.91 1.99

4.0 2.2 2.15 6.2 1.24 3.44 1.46

5.0 2.2 2.2 8.4 1.68 3.88 1.02

6.0 2.0 2.1 10.5 2.1 4.1 0.8

8.0 1.7 3.7 14.2 2.84 4.54 0.36

10.0 1.3 3.0 17.2 3.44 4.74 0.16

12.0 1.0 2.3 19.5 3.9 4.9 -

∞ 0.0 5.0 24.5 4.9 4.9 -

WAGNER-NELSONS METHOD• The data (Amax-A)/V versus time can be plotted on semi-log and linear

graph paper

WAGNER-NELSONS METHOD• Plotting (Amax-A)/V versus time produces a straight line on semi-log graph paper and a curved

line on linear graph paper. This would support the assumption that absorption can be described as a single first process. The first-order absorption rate constant, ka, can be calculated to be 0.306 hr-1 from the slope of the line on the semi-log graph paper.

ADVANTAGES:

• The absorption and elimination processes can be quite similar and accurate determinations of ka can still be made.

• The absorption process doesn't have to be first order. This method can be used to investigate the absorption process.

DISADVANTAGES:

• The major disadvantage of this method is that you need to know the elimination rate constant, from data collected following intravenous administration.

• The required calculations are more complex.

RESIDUAL METHOD OR FEATHERING TECHNIQUE

• Absowhen a drug is administered by extravascular route, absorption is a prerequisite for its therapeutic activity.

• The absorption rate constant can be calculated by the method of residuals.

• The technique is also known as feathering, peeling and stripping.

φ It is commonly used in pharmacokinetics to resolve a multiexponential curve into its individual components.

φ For a drug that follows one-compartment kinetics and administered extravascularly, the concentration of drug in plasma is expressed by a biexponential equation.

C[e-KEt – e-Kat] (1)

If KaFX0/Vd(Ka-KE) = A, a hybrid constant, then:C = A e-KEt – A e-Kat (2)

φ During the elimination phase, when absorption is almost over, Ka<<KE and the value of second exponential e-Kat approaches zero whereas the first exponential e-KEt retains some finite value.

φ At this time, the equation (2) reduces to:

(3)

φ In log form, the above equation is:

Log C− = log A - (4)

70

Where , C− = back extrapolated plasma concentration values

φ A plot of log C versus t yield a biexponential curve with a terminal linear phase having slope –KE/2.303

φ Back extrapolation of this straight line to time zero yields y-intercept equal to log A.

Plasma conc.-Time profile after oral administration of a single dose of a drug

φ Subtraction of true plasma concentration values i.e. equation (2) from the extrapolated plasma concentration values i.e. equation (3) yields a series of residual concentration value Cτ.

(C− - C) = Cτ = A e-Kat(5)

φ In log form , the equation is:

log Cτ= log A - (6)

φ A plot of log Cτ versus t yields a straight line with slope -Ka /2.303 and y-intercept log A.

φ Thus, the method of residual enables resolution of the biexponential plasma level-time curve into its two exponential components.

φ The technique works best when the difference between Ka and KE is large (Ka/KE ≥ 3).

THREE COMPARTMENT MODEL AND APPLICATIONS

OF PHARMACOKINETIC PARAMETERS IN DOSAGE

DEVELOPMENT

THREE COMPARTMENT MODEL

• Gibaldi & feldman described a three compartment open model to explain the influence of route of administration .I.E. Intravenous vs. Oral, on the area under the plasma concentration vs. Time curve.

• Portman utilized a three compartment model which included metabolism & excretion of hydroxy nalidixic acid.

CENTRAL COMPARTMENT

TISSUECOMPARTMENT

DEEPTISSUE

COMPARTMENT

DRUG INPUT

K10

THREE COMPARTMENT CATENARY MODEL

THREE COMPARTMENT MAMMILLARY MODEL

TISSUE COMPARTMENT

CENTRAL COMPARTMENT

DEEPTISSUE

COMPARTMENT

K10

DRUG INPUT

DRUG OUTPUT

K21 K13

K12K31

RAPID IV

Three compartment model consist of the following compartments .Central compartment.Tissue compartment.Deep tissue compartment.

In this compartment model drug distributes most rapidly in to first or central compartment.

Less rapidly in to second or tissue compartment .Very slowly to the third or deep tissue compartment. The third compartment is poor

in tissue such as bone & fat.

• Each compartment independently connected to the central compartment.

• Notari reported the tri exponential equation

c=a e-t+ b e-βt+ c e-γt

• A,B,C are the y-intercept of extrapolated lines.

• Α,β,γ are the rate constants

RAPID I.V BOLUS ADMINISTRATIONS

• When the drug is administered by i.V the drug will rapidly distributed in c.C ,less rapidly in to t.C. Very slowly in to deep tissue compartment.

Plasma profile

• When the drug is administered by i.V the plasma conc. Will increased in c.C this is first order release.

• The conc. Of drug in c.C. Exhibits an initial distribution this is very rapid.

• Drug in central compartment exhibits an initial distribution this is very rapid .

Pharmacokinetic parameters

Bioloigical half-life ::

• It is defined as the time taken for the amount of drug in the body as well as plasma to decline by one half or 50% its initial value.

• Concentration of drug in plasma as a function of time is

c=a e - t+ b e -β t+ c e -γ t

• In this equation α>β>γ some time after the distributive phase (i.e. When time become large) the two right hand side terms values are equal to zero.

• The eq.. Is converted in to c=a e-αt

Taking the natural logarithm on both sides the rate constant of this straight line is ‘α’ and biological half life is t1/2 =0.693/α

VOLUME OF CENTRAL COMPARTMENT• At time=0 C=A e –α t+ B e –β t+ C e –γ t

This equation becomes

CO = A+B+C -----1

CO =conc. Of plasma immediately after the i.V administration• When administered the dose is not distributed in tissue compartment.• Therefore the drug is present in c.C only .• If D is dose administered then CO = D /V C---------2

Vc=volume of drug in c.C

Combining the 1&2 eq.. We get Vc = d/co (c o----- conc. Of drug in plasma)ELIMINATION RATE CONSTANT:

Drug that follows three compartment kinetics and administered by i.V injection the decline in the plasma drug conc. Is due to elimination of drug from the three compartments.

Ke=(a+b+c) α β γ/a β γ +b α γ+ cα β

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PHYSIOLOGICALLY BASEDPHARMACOKINETIC MODELS

• Blood flow rate limited or perfusion rate limited model.

• Drawn on the basis of anatomic and physiologic data.(More realistic)

• Organs or tissues having no perfusion are excluded.

• Drug movement to a particular region is much more rapid than its rate of delivery to that region by blood - perfusion rate limited model.

• Thus, applicable to highly membrane permeable drugs, i.e. Low molecular weight, poorly ionized and highly lipophilic drugs.

• For highly polar, ionized and charged drugs, the model is referred to as membrane permeation rate limited.

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TISSUE DOSIMETRY• Measure of the level of some reactive metabolites reaching the target tissue provide a better dose parameter for risk assessment purpose than administered doses.

• The effects of growth and ageing(since the fat increasing proportional to the body weight, as animal grows), topical adsorption( in inhalation studies), pregnancy and lactation(for example changes in body weight, total body water, plasma proteins, body fat and cardiac output will alter the distribution of many drugs and their metabolites.)And competitive multiple metabolites are illustrated in PBPK modelling.

REFERENCES :BIOPHARMACEUTICS AND PHARMACOKINETICS.

P L MEDAN, 1ST EDN

BIOPHARMACEUTICS AND PHARMACOKINETICS.

D.M BRAHMANKAR AND SUNIL. B .JAISWAL, 1ST EDN

APPLIED BIOPHARMACEUTICS AND PHARMACOKINETICS

LEON SHARGEL AND ANDREW YU,

4TH EDN.

BIOPHARMACEUTICS AND CLINICAL PHARMACOKINETICS BY MILO GIBALDI, 4TH EDN.

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